GOVERNORS AND THE GOVERNING OF PRIME MOVERS BY W. TRINKS, M.E. PROFESSOR OF MECHANICAL ENGINEERING, CARNEGIE INSTITUTE OF TECHNOLOGY, PITTSBURGH, PA. 140 Illustrations NEW YORK D. VAN NOSTRAND COMPANY 25 PARK PLACE 1919 COPYRIGHT, 1919 BY D. VAN NOSTRAND COMPANY THE PLIMPTON PRESS NORWOOD MASS U-S-A PREFACE GOVERNORS have been the play toy of inventors for over a century, and have been the hobby of mathematicians for over thirty years. In spite of these facts, the knowledge of gover- nors and of governing among both designing and operating engineers is very incomplete. There are several reasons for this condition. One of them is that governors do not cover a wide enough field to warrant a separate course in any engineer- ing school. Instruction in governors is given in a scattered fashion. In courses in steam engineering, governors for steam engines and turbines are taken up. In courses on hydraulic motors, some time is spent on governors and governing ; and the same is true of governors for internal combustion engines. Textbooks on prime movers reveal the same condition of affairs. Everywhere that which is apparent on the surface is reprinted, but nowhere (with very few exceptions) does the investigation go below the surface. The present book aims to fill this gap. As far as I am aware, there exists to-day no other book of any consequence on gov- ernors and governing in the English language. There are books in French, German, and Swedish, but they are of little use to English-speaking engineers. The present volume is a book of essentials and principles. Practice changes ; there are fashions in engineering almost as changeable as those in women's clothes ; but engineering principles do not change. For that reason, I have tried to dig out that which is essential, and to present it in such a manner that the reader is put in a position to judge existing and future types of governors as well as the properties of prime movers with regard to regulation. In consequence, the book contains not a single catalogue picture. Every drawing was especially prepared to show in a diagrammatic manner that which is important, with the intentional omission of everything else. The book contains more than I give my students in the A 1 vi PREFACE classroom. Students, as a rule, do not know what their life- work will be in later years. While in school, they can be given only the "meat of the essentials." They are, however, anxious to know of a source out of which they can fill the gaps which their necessarily brief instruction at school has left. Never- theless, the book by no means covers the whole subject of governing of prime movers. Many extremely interesting sub- jects were omitted, for two reasons : First, it was necessary to keep the price of the book within such limits as not to make its purchase a burden. Second, it was desirable to restrict the mathematical side of the book to a level which undergraduates can master, so that the reader may be spared the troubles of intellectual indigestion. It is, however, planned to follow this book, later on, with another one on the subjects of " dynamics and design of modern governors for prime movers," which will be primarily intended for those engineers who make governors and governing a life study. In that book many subjects can be taken up which are clearly beyond the limits of the present volume. The foot- notes referring back to the preface indicate some of the subjects which are to be taken up in that volume. In addition, the other book will take up the effect of water inertia in hydraulic turbines and the governing of prime movers driving alternators in parallel. The present book was started in the spring of 1913. Lack of spare time and necessity for original work in the preparation of this book have delayed its completion until now. By the term "original work" I mean that practically all of the theories advanced by me were tried out practically. Some of the tests were made in the Mechanical Engineering Laboratory of the Carnegie Institute of Technology, but most of the trials were made in the field, in the attempt to help those who had trouble with regulation. College graduates will have no trouble with the mathe- matical part. Operating engineers who have not had the good fortune of college training should likewise have no trouble, if they turn to the elementary derivations given in the appendix. By comparison, they will realize that the differential calculus is a wonderful short-cut. PREFACE VII Like other books dealing with a specialty of applied me- chanics, the present book must contain, for some readers, much that is known, and must occasionally pass over the heads of others. It is impossible to pitch the scale right for everybody. The bookmark, containing all symbols, will, I hope, be of assistance. For those who wish to go more deeply into the subject of governing, and who have library facilities, the bibliography will be valuable. It was prepared by the Carnegie Library of Pittsburgh. The librarians worked faithfully on this task. I herewith thank them for their cooperation. W. THINKS PITTSBURGH, MAY, 1919. CONTENTS PAGE PREFACE v LIST OF SYMBOLS xiii INTRODUCTION xv CHAPTER I GENERAL STATEMENTS 1 1. Purposes of Governors 1 2. Forces Used in Governing 2 CHAPTER II THE DIRECT-CONTROL GOVERNOR AS A MOTOR 4 1. Strength of Centrifugal Governors 4 2. Regulating Force of Tangential Inertia 8 3. Work Capacity of Centrifugal Governors 14 4. Detention by Friction 15 5. Overcoming a Passive Resistance 19 CHAPTER III THE CENTRIFUGAL GOVERNOR AS A MEASURING INSTRUMENT (SPEED COUNTER) 23 1. Equilibrium Speed, Static Fluctuation, Stability 23 2. Characteristic Curves 25 3. Constituent Parts of Characteristics 34 4. Influence of Shape of Centrifugal Weights 40 CHAPTER IV PROMPTNESS AND TRAVERSING TIME 45 CHAPTER V ADJUSTMENT OF EQUILIBRIUM SPEED 49 ix x CONTENTS CHAPTER VI PAGE SHAFT GOVERNORS 61 1. Forces Acting in Shaft Governors 61 2. Centrifugal Moment , 63 3. Moment Caused by Inertia of Valve Gear Parts 64 4. Moment Caused by Friction of Valve Gear Parts 66 5. Moment Caused by Friction of Eccentrics 67 6. Spring Moment 69 CHAPTER VII NATURAL PERIOD OF VIBRATION OF GOVERNORS 77 CHAPTER VIII EFFECTS OF OUTSIDE FORCES IMPRESSED UPON GOVERNORS 81 1. Resistibility 81 2. Cyclical Vibrations of Governors 85 CHAPTER IX INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 95 1. Action of Governors Regulating Prime Movers 95 2. Limiting Case 98 3. The Influence of Tangential Inertia upon Stability of Regulation 113 4. Damping of Governor Vibrations by Solid Friction 117 5. Greatest Speed Fluctuation in Direct-control Governing 128 CHAPTER X DISCARDED TYPES OF SPEED GOVERNORS 139 CHAPTER XI RATE-OF-FLOW GOVERNORS 148 CHAPTER XII PRESSURE GOVERNORS 153 CONTENTS xi CHAPTER XIII PAGE RELAY GOVERNING 170 1. Reasons for the Use of Relay Governors, and Forces Acting upon Them 170 2. Mechanisms of Relay Governors 173 3. Work Capacity of Relay Governors 184 4. Speed Fluctuations in Relay Governing 186 CHAPTER XIV GOVERNOR TROUBLES AND THEIR REMEDIES 195 1. Regulation Not Close Enough 195 2. Racing 195 3. Speed Fluctuation too Great when Load Changes 201 4. Vibration of Governor 202 5. Rapid Wear of Joints 203 6. Machine Design Troubles 203 CHAPTER XV SELF-REGULATING FEATURES OP PRIME MOVERS 204 APPENDIX 209 BIBLIOGRAPHY 219 INDEX . 231 LIST OF SYMBOLS To accompany " Governors and the Governing of Prime Movers." By Prof. W. Trinks A .... Governor travel, or displacement. V . . B Distance. C Centrifugal force. W . . D . . . Small difference;; X . . E ....Energy. Z .. F . .. . Force at collar to overcome a . . governor friction. b . . G .... Area. H . . . . Radius of gyration. d . . I . . . . Moment of inertia 1 mass x e . . J ... . Moment of inertia / (radius) 2 K Constant. / . . L . . . . Lever arm. g . . M Moment. h . . P Strength of governor; also i . . Restoring force per unit displace- j . . ment. k . . Q . . . . Weight or force. I . . R Force on governor collar to over- m . . come valve friction only. n . . S .. . . Spring force. p . . T . . . . Special time. Tb . . . . Brake-resistance traversing time. q . . Tf Time required to fill volume of r . . container. (In pressure gov- s . . erning.) t . . T g Traversing time of governor. u . . Z\. . . . Starting time oanjertia mass. v . . T n . . . . Time of one*complete vibration of w . . a governor. (Natural period of vibration.) x . . Tr Relay traversing time. T s Starting time of plrime mover. y . . U. . . . Relative speed deviation. z . . . .Volume; also Volume flowing in unit time. . . Weight. . . Relative governor deviation. . . Relative load change. . . Angular acceleration. . . Radial travel of centrifug; weights. . . Differential. . . Base of Naperian logarithms; also Efficiency. . .Friction coefficient. . . Gravity acceleration. . . Height or space dimensions. . . Angle. . . Angle. . . Angle. . . Length. . . Mass. . . Revolutions per minute. . .Static fluctuation; also Pressure. . . Detention by governor friction. . .Radius. . .Space. . .Time. . . Angular velocity. . . Linear velocity. . . Angular velocity of auxiliary vector. . .Abscissa; also Unknown quantity. . . Ordinate. ..Stability; also Root of an equation. LIST OF SYMBOLS To accompany " Governors and Governing of Prime Movers." By Prof. W. Trinks A .... Governor travel, or displacement. B . . . . Distance. C Centrifugal force. D . . . . Small difference. E Energy. F . . . . Force at collar to overcome governor friction. G ....Area. H . . . . Radius of gyration. 7 Moment of inertia 1 mass x J ... . Moment of inertia j (radius) 2 K . . . . Constant. L . . . . Lever arm. M Moment. P Strength of governor; also Restoring force per unit displace- ment. Q . . . . Weight or force. R Force on governor collar to over- come valve friction only. S . . . . Spring force. T .... Special time. Tb . . . . Brake-resistance traversing time. Tf Time required to fill volume of container. (In pressure gov- erning.) T g Traversing time of governor. T. . . . Starting time of inertia mass. T n . . . . Time of one complete vibration of a governor. (Natural -period of vibration.) Tr Relay traversing time. T s . . . . Starting time of prime mover. U . . . . Relative speed deviation. V . . . .Volume; also Volume flowing in unit time. W Weight. X . . . . Relative governor deviation. Z Relative load change. a Angular acceleration. b Radial travel of centrifugal weights. d Differential. e .... Base of Naperian logarithms; also Efficiency. / . . . . Friction coefficient. g Gravity acceleration. h Height or space dimensions. i . . . . Angle. j Angle. k Angle. I .... Length. m . . . . Mass. n Revolutions per minute. p Static fluctuation; also Pressure. q Detention by governor friction. r . . . . Radius. s . . . .Space. t Time. u .... Angular velocity. v . . . . Linear velocity. w Angular velocity of auxiliary vector. x . . . .Abscissa; also Unknown quantity. y . . . . Ordinate. z . . ..Stability; also Root of an equation. xm GOVERNORS AND THE GOVERNING OF PRIME MOVERS INTRODUCTION THE title of the present book, viz. " Governors, and the Governing of Prime Movers/ 7 clearly indicates that the study of governing comprises two distinct parts, one being a treatment of the governor as a mechanism, and the other being an investigation of the interaction between the governor and the prime mover. That this division is not only logical but is also historical will be realized from a brief reciting of the history of governing for con- stant speed. In the his- torical sketch which now follows, many of the dates are approximate only ; they must necessarily be so, be- cause the development of correct engineering princi- ples and the adoption of FIG. 1 improved apparatus are very slow processes. Besides, the periods of development overlap in different countries. Early mechanical engineers, that is to say the builders of water wheels, wind mills, and steam engines, studied governors XV xvi INTRODUCTION as separate mechanisms and developed their theories with utter disregard of the reciprocal action of governor and prime mover. This condition existed, with very few exceptions, from the time of James Watt (who invented the centrifugal governor in the year 1784) to almost 1880. The Watt governor (Fig. 1) adjusts for higher and higher rotative speeds as the revolving balls move away from the axis of revolution. As might be expected, this property of the governor was con- sidered a drawback, but not until the middle of the nineteenth century were any steps taken to remedy the defect. The search for the isochronous, or "astatic" governor, which would regulate for the same speed at all loads, began. Thus we find the "parabolic governor" which, when operated on a test block, is truly isochronous, introduced about 1850. For the same purpose governors with crossed arms were invented from 1860 to 1870 by different engineers (Kley, Farcot, Head) in different countries. With the same end in view, the high speed governor of Porter, Proell's inverted governor, and the oblong weight "Cosine" governor appeared in the two decades from 1860 to 1880. The same period gave birth to many other forms of governors, for instance the dynamometric (or load-) types (see Chapter X), which attempted to adjust the supply of energy directly by means of the demand for it. All of these types of governor have disappeared for reasons explained in Chapter X. A new period dawned from about 1870 to about 1882, when governors were extensively applied to horizontal shafts (Porter's marine governor, Hartnell's crankshaft governor). Gravity could not, in these adaptations, furnish the controlling centripetal force, and springs had to be employed. The action of spring-loaded governors was so superior to that of weight-loaded governors that inventors placed many new types of spring-loaded governors on the market from 1880 to 1895 (about). The term "inventors" is used purposely instead of "engineers," because the vast majority of the designs of that period reveal inventive ability rather than good engineering judgment. From about 1890 on, the influence of electrical power generation helped to perfect mechanical governors in the same measure in which INTRODUCTION xvii it has helped to perfect all types of mechanical equipment of power plants. While inventors were busy bringing out new types of governors, usually without any knowledge of the mutual relations between prime mover and governor, these very rela- tions were studied by two widely different types of men, namely the operating engineer, and the mathematician. The former found that the wonderful " isochronous" governors, such as the parabolic governor, the cosine governor, and many others, were worthless because they caused perpetual hunting and racing. Oil gag-pots were found to be necessary ; and fre- quently stabilizing springs were concealed in them ; but with all this practical progress, the theoretical foundation or proof for the exasperating behavior remained unknown to engineers, or else was not understood by them. At the other end of the line, mathematicians studied the interaction between governor and prime mover. Starting about 1840 with the regulation of astronomical telescopes, British and French scientists gradually extended their investi- gations to the governing of prime movers. But their researches were of no benefit to the practical engineer, because they were published in the proceedings of astronomical or philosophic societies. More attention was paid to the very ingenious theoretical solution of the governing problem by the Russian engineer Wischnegradsky, which was published in 1876 and 1877 in Russian, French, and German. Although his publica- tions are the foundation upon which modern theories of govern- ing are built, they were scoffed at by engineers, partly because they failed to explain some of the plainly visible phenomena of regulation, and partly because they were clad in very mathe- matical form. Wischnegradsky's work, unfortunately, was given very little or no appreciation in English-speaking countries. From about 1890 to the outbreak of the European war, French and German engineering literature (including Switzer- land and Austria) have been very active on the mutual relations between governor and prime mover, while British and American engineers either took no interest in the theoretical development, or else were too busily engaged with practical xviii INTRODUCTION experimenting to pay attention to theory. Of course there are exceptions, the most prominent of which is the large number of American contributions to the development of the theory of the action of long pipe lines on the governing of hydraulic turbines. Even to-day, the subject of governing, that is to say, the interaction between the governor and prime mover, is very little understood by engineers. In the 1905 edition of the widely used English book on Dynamics by Routh, this state- ment is made: "The defect of a governor is, therefore, that it acts too quickly, and thus produces considerable oscilla- tions of speed in the engine." A similar statement was made by Swinburne, the well-known English engineer, in "Indus- tries," 1890. As a matter of fact, the principal progress in the improvement of regulation has consisted in making gov- ernors act more quickly. It is evident that there is need for clearing up this subject, if even the best of us can go astray on it. In consequence, considerable emphasis is placed on the problem of governing in the present book. To-day the theories of governors as mechanisms and of governing have reached a certain stage of development beyond which it is scarcely necessary to go, for practical purposes. However, we must admit that present theory is by no means complete ; on the contrary, it is far from it. But additional information can only be gained by a combination of experi- ment and mathematics which is at present beyond the reach of most practical engineers. CHAPTER I GENERAL STATEMENTS 1. Purpose of Governors. Governors are used to auto- matically adapt the output of prime movers to the demand. Governors are superfluous, whenever the torque resisting the motion of a prime mover increases considerably with the speed and depends solely upon the speed. This is the case in marine service, where the resistance grows approximately as the square of the speed, and in locomotive service. It is also the case in some classes of pumping machinery. If the resist- ing torque grows with the speed, and is not subject to varia- tions except those caused by the speed of the prime mover, then the supply of energy can be adjusted by hand, because there can be only one speed for a given supply of energy. Hand adjustment is usually practiced in the cases mentioned, which means that under such circumstances the purpose of the governor is narrowed down to that of a safety speed-limit which prevents running away of the engine or turbine, if the resisting torque should be suddenly removed. If, however, the torque is repeatedly removed, as it occurs, when the pro- peller comes near the surface in a high sea, the governor ap- proaches the more normal type mentioned below under (1). Usually some factor in the operation of the prime mover is to be kept constant, or nearly constant, while other factors vary. Thus we find : (1) Constant speed, variable torque. (2) Constant (average per revolution) torque, variable speed. (3) Constant quantity of one factor of the output, with variation of both speed and torque. As examples of heading (1) may be cited engines or turbines operating electric generators, driving lineshafts, etc. In elec- trical power transmission, turning lights on or off, switching i 2 .aOYORtADBE GOVERNING OF PRIME MOVERS motors on and off, etc., varies the resisting torque, while the requirements of constant number of cycles and of constant voltage necessitate keeping the speed of the prime mover constant. Under (2) come pumping machines maintaining a constant pressure, but varying the output by adjustment of rotary speed. An example is found in air compressors supply- ing air to tools in mines. In spite of variable demand for air, the pressure must be kept constant. Under (3) come centrif- ugal pumping machines maintaining a constant flow of fluid against a variable pressure. The relation between pressure, speed, and torque is determined in this case by the character- istics of the prime mover and of the pump. It is evident that purposes differing as widely as those men- tioned under the three headings require different types of governing devices. 2. Forces Used in Governing. Every governor per- forms two separate functions, namely : (a) that of measuring the quantity which it is to keep practically constant, (6) that of varying the supply of energy to the prime mover so that the just mentioned quantity is kept constant. A governor, then, is both a measuring device and a motor. In conformity with the purposes mentioned in the preceding paragraph, governors measure (1) angular velocity (2) the intensity factor generated by the machine which is driven by the prime mover (3) the extensity factor of this energy. Under heading (2) come pressure and electromotive force. Under heading (3) come rate of flow and electric current. In the evolution of the art of governing, many principles have been used, but one by one they were discarded (see Chapter X) , until to-day practically one principle is left, namely this : A force is produced by the quantity to be measured ; it is balanced by an external known force such as is derived from springs or weights. Any change in the quantity to be measured unbalances the system. As soon as the unbalanced force is strong enough to overcome resistance, it constitutes GENERAL STATEMENTS 3 the motive force for adjusting the supply of energy. The principle in question has been pronounced defective and faulty, because, to cause the governor to act, it necessitates a change in the quantity to be kept constant. However, the change can be made exceedingly small ; and the principle has the very great advantage that governing is accomplished regardless of the source of the disturbance. This important theorem will be fully explained later on. If the just mentioned force acts directly on the mechanism which adjusts the supply of energy, the governor is said to be direct acting, or to be a " direct control" governor. If the unbalanced force is too feeble to overcome the re- sistance of the energy controlling mechanism, it becomes necessary to call upon an auxiliary energy for motive power, which energy is released and controlled by the unbalanced governor force. If this is done, the governor is called a relay governor. Motor forces used in relay governing are most varied. The principal ones among them are : (1) Mechanical force derived from the motion of the prime mover. (2) Fluid pressure acting on a piston. (3) Electromagnetic attraction. All governors of the present period have certain general features in common, such as strength, work capacity, detention by friction, behavior in overcoming a passive resistance, prompt- ness, resistibility and others. While, for that reason, it might be logical to discuss these properties in a general way, a different course will be followed for the sake of clearness. More than 90% of the total number of governors in operation are of the centrifugal type, so that it seems advisable to develop the general properties of governors with particular reference to the centrifugal governor. Transfer of the conclusions to other types of governors will not be difficult, once the theory of the centrifugal governor is clearly understood. References to Bibliography at end of book: 35, 36, 63. CHAPTER II THE DIRECT-CONTROL GOVERNOR AS A MOTOR 1. Strength of Centrifugal Governors. When acting as a motor (that is, when shifting the power controlling mechan- ism to a new position), the governor does work. The latter Spring balance ^ Indicates Strength P Suspension points. Centrf mass. Sleeve or collar. FIG. 2 can be expressed as force times distance or as moment times angle. As will be explained later on, the force or moment used in the act of regulating are functions of that force P (or moment 4 THE DIRECT-CONTROL GOVERNOR AS A MOTOR 5 MP) which are required to move the governor, when not rotating, against centripetal force. The place of action of this force (resp. moment) is shown in Figs. 2 and 4 ; the former represents a spindle type of governor which shifts the power controlling mechanism by up and down motion of the sleeve or collar (1) ; Fig. 4 rep- 1. resents a shaft governor which controls the supply of energy by shifting the z c center of the eccentric along the arc (l)-(2). The strength P can be determined experimentally by lifting up the sleeve of the governor and reading the force on a spring balance, as sketched in Fig. 2. The true strength is obtained, if friction is eliminated by vibration, for instance by rapping the governor with a wooden hammer. In German treatises on governors and in translations of such treatises the force P of the not-rotating governor is called the " energy" of the governor. This term is obviously misleading. It has been replaced in the present book by the term " strength." The " strength" of the governor is closely related to that radial or centrifugal force 2C, Fig. 2, which is necessary to balance the frictionless governor, or to keep the governor sleeve floating. Let ds be a small displacement of the sleeve (say 3% of the total working travel of the sleeve), then the rigidity of the mechanism results in a radial displacement dr of the centrifugal weights. The graphical construction of dr for a given value of ds is shown in Fig. 2. From the theory of virtual displacements, we have FIG. 3 P ds = 2 C dr (1) GOVERNORS AND THE GOVERNING OF PRIME MOVERS In equation (1), P is the strength of the governor, and C is that radial (and in Fig. 2 horizontal) force which, if applied at the mass center of each centrifugal weight, just balances P in the frictionless governor. If a spring balance were attached to each centrifugal weight, and if it were pulled radially outward, then it would indicate the force C (provided, of course, that the spring balance correctly indi- cates horizontal forces, and that friction is elimi- nated). In operation of the governor, the force C is furnished by radial inertia and is called cen- trifugal force. From equation (1) follows the important fact that the strength is proportional to the centrifugal force, so long as ds/dr remains constant. 1 In the case of the shaft governor, there is no well-defined, constant direction for the action of the regulating force, so that the regulating force is replaced by a regulating moment. But the " strength moment " coincides with the moment exerted by the spring (spring moment). Hence there exists no reason for the use of the term " strength moment." In shaft governors FIG. 4 1 The relation between strength and centrifugal force can be illustrated by the aid of a simple example. In Fig. 3, let two weights W be suspended by massless rods as shown. Then the weights W can be kept from dropping either by two hori- zontal forces \G or by a vertical force P. From a triangle of forces \C = W; we p may also resolve P into two inclined forces P' = -i= ; take moments about the sus- p pension point (1), then p. I = W I \/2, or P = 2W. Hence C = P in this case. This V2 result can be had directly from the principle of virtual displacements. Move the weights through the small distance db, then radial travel dr of force C and axial travel ds of force P are alike; hence P = C, since P ds = C dr. THE DIRECT-CONTROL GOVERNOR AS A MOTOR 7 the centrifugal moment and the spring moment do not coin- cide, because there are many forces impressed upon the ec- centric. This feature is dealt with explicitly in the chapter on shaft governors. To exert its full strength, a centrifugal governor must slow down from its normal speed to a dead stop. In practice only very small speed fluctuations are allowed, so that only a small part of the strength can be exerted. To find this frac- tion, call u the normal angular velocity of the governor spindle, and du a small change of that angular velocity. From me- chanics it is well known that centrifugal force C = mru 2 , where m = mass of the revolving body, and r = radius to mass center of that body. Corrections to this equation on account of oblong shape of revolving body are given in para- graph 4 of Chapter III. For symmetrical shapes, such as a sphere or a cylinder, no correction is needed. Let the governor be prevented from moving its sleeve, while the speed changes the amount du; then r remains constant ; and since m natu- rally is constant, C and u are the only variables. Hence the change of centrifugal force is by differentiation. 1 dC=2mrudu=2mru 2 du/u = 2 C du/u (2) This equation is true for infinitesimally small changes of u only, but is in practice accurate enough even for du = 10 % of u. The following equation is then approximately correct: DC = 2 C Du/u (3) where Du/u is the relative speed change. Du is not an infinitesi- mal quantity, but a small fraction of u, of the magnitude which is usually allowed in practice. The governor was in internal equilibrium before the speed change ; it is not in internal equilib- rium after the speed change. If the governor is frictionless, the radial force DC produces an axial force DP = DC dr/ds at the sleeve ; this force is known as the regulating force for the relative speed change Du/u. By substitution we obtain DP = 4 C T = 2P Du / u W u as (for 2 weights!) 1 For an elementary derivation see Appendix, p. 209. 8 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Rotation From this equation it follows that the regulating force is proportional to the strength of the governor and to the rela- tive speed change. It equals the product of 4C (where C is the centrifugal force of each of the two weights), Du/u (which is the relative speed change) and dr/ds (which is the ratio of radial travel of centrif- ugal weight and of axial travel of sleeve for a small displace- ment of the governor) . To express this re- lation in figures, imagine a governor with a strength of 250 pounds ; let the gov- ernor be in equilibrium at 300 revolutions per minute, and let the speed drop to 297 revo- lutions per minute. Then the governor will exert a force of 2 X .01 X 250 = 5 pounds on its collar tending toward a new equilibrium position. In this case the regulating force equals five pounds. It should be noted that the available regulating force becomes smaller as the governor approaches its new position. This feature will be explained in detail later on. The corresponding moment equation for shaft governors is never written, because the pulsating moments continually impressed upon a governor of that type upset any elementary theory which might be founded upon a " regulating moment." References to Bibliography at end of book: 1, 24, 35, 36, 37, 52, 53, 73. 2. Regulating Force Due to Tangential Inertia. In Fig. 5 is shown an eccentric with center (1), mounted loose on a shaft the center of which is (2). Let the valve gear be so arranged that clockwise relative rotation of the eccentric FIG. 5 THE DIRECT-CONTROL GOVERNOR AS A MOTOR 9 reduces the supply of power, and let the shaft rotate in the direction indicated by the arrow at the top of the illustration. The details of the valve gear are at present immaterial. A concrete example of variation of power by rotation of an eccentric is furnished by the well known " Buckeye " steam engine. If the demand for power is suddenly increased, an angular retardation, or negative angular acceleration a occurs, the magnitude of which depends upon the dimensions of the prime mover, upon the change of load and upon the moment of inertia "J" of the rotating mass connected to the eccen- tric. If the latter does not rotate relatively to the shaft, a moment (a J) is exerted upon the eccentric, tending to rotate it. Regulating forces or moments caused by tangential inertia differ from those produced by radial inertia (centrifugal force), inasmuch as the latter has to wait for the speed to change before it becomes effective (see equation 3 on page 7), whereas tangential inertia becomes effective at the instant when the change of load occurs. From this fact it might appear at first thought as if tan- gential inertia should be universally used for speed governing purposes. Many such attempts have been made ; but the survival of the fittest has eliminated most of them and has limited the application of tangential inertia almost entirely to shaft governors. The theoretical reasons underlying this decision of practice are partly given in this chapter, and will partly be explained in paragraph 6 of Chapter VI and para- graph 3 of Chapter IX, because they depend upon the dynamics of regulation. The principle of using tangential inertia as a regulating agent differs from the principle enunciated in paragraph 2 of Chapter I and forms the principal exception to that principle. It is discussed in the present place to show the difficulties which even the slightest departure from the general principle introduces. The governor shown in Fig. 5 is worthless, because it does not adjust any particular speed. It simply tries to keep con- stant the accidental speed at which the prime mover happens to be operating. A prime mover provided with such a governor 10 GOVERNORS AND THE GOVERNING OF PRIME MOVERS could not be started, because the resulting acceleration would shut off the power, unless the inertia wheel were kept out of action by some other agency. Furthermore, the inertia wheel is inactive for very gradual changes of speed, because the acceleration "a" is then very small and usually insufficient to overcome friction resistance. With gradual change of load, the speed can thus rise or fall, until either equilibrium is at- tained by means other than the governing equipment, or else the prime mover is wrecked or stalled. These reasons make it necessary to combine tangential inertia (if it is to be used for governing purposes) with centrif- ugal force and, speaking in very general terms, to make the latter so strong that it wm * compensate for the above mentioned shortcom- ings of tangential inertia. Such a combination may be effected either by the use of separate inertia and centrifugal masses, or by the use of a mass which serves both purposes. In illustrations 6 to 9 both of these arrangements FIG. 6 are shown. Figures 6 and 7 show spindle governors; 8 and 9 show shaft governors. The illustrations are purely diagrammatic and are intended to bring out principles only, with omission of all design details. In Figures 6 and 8 each mass is utilized for both radial and tangential inertia action; in Figures 7 and 9 separate masses are provided for the two effects. On mass (1) of Fig. 6 the action of tangential inertia is indicated for a sudden increase of speed (reduction of load); on mass (2) the action of centrifugal force is indicated. Evi- dently the two forces act in the same sense for the direction of rotation marked in the illustration. THE DIRECT-CONTROL GOVERNOR AS A MOTOR 11 From Fig. 8 it is plain that the action of centrifugal force on mass center (1) turns clockwise about suspension point (2), so that an increase of speed likewise produces clockwise rota- tion. A study of the illustration also brings out the fact that for the indicated direction of rotation, tangential inertia acts FIG. 7 in the same sense. Point (3) represents the center of the eccentric which operates the valve gear. In Fig. 7, (1) is an inertia ring, loosely mounted on the revolving spindle. (2) and (2) are the centrifugal masses which are guided radially by bell cranks (3). The two mass systems are connected by a pinion and two racks. In this type 12 GOVERNORS AND THE GOVERNING OF PRIME MOVERS of governor the complete separation of the two force systems is very distinct. In Fig. 9, (1) is the inertia mass, and (2) is the centrifugal mass. They swing about the points (4) and (5) which are fixed in the revolving wheel. Point (3) is again the center of the eccentric. In this type the two force actions are not en- tirely separated, because each mass is partly subjected to the not intended force. From page 9 we remember that the regulating moment is (a/). This expression may FIG. be misconstrued into the belief that the moment could be in- creased to any desired amount by a corresponding increase of J. Such a conclusion would be wrong, because, as J grows, a be- comes less for a given change of load. For, as long as the wheel does not yet shift the eccentric, it acts the same as if it FIG. 9 were keyed to the shaft, or as if it were part of the flywheel. THE DIRECT-CONTROL GOVERNOR AS A MOTOR ' 13 And for a given change of load the rate of change of speed is the smaller, the greater the inertia of the flywheel. The arrangement of Fig. 8 is frequently used; for this reason an expression will now be developed for the regulating moment produced by governors of this type. In the diagram (Fig. 10), (1) is the center of the rotating shaft, (2) is the pivot about which the inertia weight with mass center (3) swings. Let the system start from rest with an angular acceleration a, and let joint (2) be inactive, that is rigid. For conven- ience of calculating, the motion of the mass may be re- solved into a curvi- linear translation of the mass center from (3) to (4), and into a rotation of the mass from (5) to (6) about its center (4). The total inertia moment must be the sum of the inertia moments due to transla- tion and rotation. The moment about point (2) (which latter becomes point (6) at the end of the interval of time under con- sideration) then is M = mr a L + J a = a(mr L + J), where J is the moment of inertia of the pendulum about its own mass center (3) and where L is the lever arm of the trans- latory inertia force mru about the pivot (2). It is evident that the regulating force or moment is made smaller by the regulating motion of the governor; but that does not matter. For, as long as the governor moves toward its new configuration, its purpose is served, even though the available regulating force be lessened thereby. It should be Moss m FIG. 10 14 GOVERNORS AND THE GOVERNING OF PRIME MOVERS noted that the regulating force of a centrifugal governor is likewise reduced as it approaches its correct position. The regulating moment mr a L + J a is the only moment which interests us. Other forces act, but they are not of vital influence. They will, however, be mentioned. Since, in prac- tice, the whole system rotates, while the acceleration occurs, centrifugal forces must be added, and if the inertia mass swings about pivot (#), compound centrifugal forces must be added (the latter are known as Coriolis' forces, see paragraph 4 of Chapter IX). While the centrifugal forces have a moment, the compound centrifugal forces lie in the direction (#)-(#) and have no moment. However, they produce friction. Study of Fig. 8 teaches that in this form of governor the moment due to m r L a is quite small compared to the moment caused by J a. It should be understood that the moment M = a (m r L + J) refers to an ideal f rictionless governor only. The theory of the action of tangential inertia in the inter- action between governor and prime mover will be found in paragraph 3 of Chapter IX, which deals with the dynamics of regulation. References to Bibliography at end of book: 2, 33, 36, 73. 3. Work Capacity of Centrifugal Governors. If the sleeve of a non-rotating centrifugal governor of the spindle type be raised through its working travel s, the work P s is done, where P is the average strength of the governor. This work may also be expressed as I P ds, where P is the variable Jo strength of the governor. Obviously this expression represents the greatest amount of work which can be done by the governor as a motor, unless speeds considerably in . excess of the normal are considered. Hence, the expression I P ds is called the work capacity of the governor. Its numerical value is usually given in governor catalogues. Again it is obvious that for purposes of regulation only a small part of the work capacity can be utilized, because a speed change of 100% cannot be allowed in practice. THE DIRECT-CONTROL GOVERNOR AS A MOTOR 15 Before the available part of the work capacity can be found, the influence of friction in governor and valve gear must be studied. References to Bibliography at end of book: 36, 37, 73. 4. Detention by Friction. If a spindle governor be run at such a speed that it is floating, touching neither the upper nor the lower stop, the speed can be varied between the limits u h and u t , while the governor sleeve remains motionless. Any speed higher than u h raises the governor, any speed lower than HI lowers it. The ratio q has received various names, namely " detention by friction," " sluggishness," "time lag," " insensibility," or "degree of in- sensibility." In the present book the name "detention by friction" will be adopted. This quantity depends upon (1) the design of the governor, (2) its lubrication, (3) the vibration to which it is subjected, while the speed changes, (4) the forces transmitted to the governor through the sleeve. Taking up these items in the order given, we find that in a governor of the type shown in Fig. 2 or in Fig. 6, centrifugal and centripetal forces are balanced by forces passing through numerous joints, each of which produces friction, whereas in the type shown in Fig. 7 these forces are balanced directly without the interposition of joints. Consequently the former type has, under otherwise equal conditions, more friction than the latter, and shows more detention due to friction. The friction can be somewhat reduced by the use of knife edges. Item 2, lubrication, certainly affects friction. Governors may be so designed that the joints are always submerged in oil. Such a design furnishes the only case in which the degree of lubrication is definitely known. In all other cases it is uncertain. In many governor catalogues the detention due to friction 16 GOVERNORS AND THE GOVERNING OF PRIME MOVERS is given on the basis of a certain coefficient of friction and of absence of vibration. The calculations made for determining the detention are more or less arbitrary and of doubtful value, unless all are based on the same coefficient of friction. In that case the figures afford a comparison which can be used to advantage. The immediate purpose of any such calculation is to find between what limits the force C in Fig. 2 can be varied without moving the mechanism. The solution of this problem is a matter of applied mechanics. It may be effected by graphic Governor Position statics or by taking moments, or by the principle of virtual displacements. In the appendix two numerical examples are given. Under average conditions the detention varies for different types of governors and for various degrees of lubrication from \% up to 4 %. The detention q usually varies for different positions of one and the same governor. Item 3, vibration, considerably affects detention, as may be judged from the following fact which proves that vibrations always reduce and often eliminate friction. Objects lying quietly on an apparently horizontal table march off promptly, if the table is rapped by a vibrating instrument, such as a pneumatic hammer, riveter, etc. In the same manner a governor subject to vibrations loses the detention by friction in various THE DIRECT-CONTROL GOVERNOR AS A MOTOR 17 degrees. To prove this fact, the author artificially produced vibrations in an ordinary Watt type spindle governor by atttaching an unbalanced revolving weight to its collar. Figs. 11 and 12 show speed and position of the governor plotted against time. Fig. 11 was taken with the weight at rest, and Fig. 12 was taken with the weight revolving at high speed. It will be observed that Fig. 11 shows the peculiar jaggy motion of detention, whereas in Fig. 12 practically every trace of detention is eliminated. Note the practically straight line, which represents motion without detention. Incidentally \ A / 5peed "8! Time FIG. 12 it will be noticed that the speed of the governor varied cyclically during the test, whereas the speed appeared to the observer to be uniform. Circumstances of this sort make even the experimental determination of the detention by friction difficult. Influence 4, namely that of forces transmitted through the governor sleeve, will be considered in Chapter VIII. In spite of the fact that vibrations, variation of lubrication, and other circumstances make the detention by friction an uncertain quantity, we must admit that there are cases where the theory of the detention by friction is well applicable (see 18 GOVERNORS AND THE GOVERNING OF PRIME MOVERS next paragraph). In such cases the following equations are useful : detention q = - u h HI u h 2 uf mr Uh* mr u h \ 2 dr dr C h - Ci ' h ds /l ds P h - PI (u h + ui\ 2C ds or approximately 2(P - P f ) P- P^ F 2P P P In these equations C h , C, and Cz are centrifugal forces for high, average, and low speed, covering the range over which the governor is detained by friction; P is the strength of the governor for the position under consideration. Evidently a governor can have only one value for its strength at a given position ; hence the expressions P h P and P PI must mean additional axial forces at the sleeve, sufficient to overcome the internal friction of the governor. For this reason the said values have been denoted by the letter F. If q has been deter- mined by calculation, the equivalent friction force F can be found from the final equation q = F /P. References to Bibliography at end of book: 36, 73. FIG. 13 THE DIRECT-CONTROL GOVERNOR AS A MOTOR 19 Equal 5. Overcoming a Passive Resistance by a Centrifugal Governor. When moving the throttle valves of steam turbines, or well-balanced throttle valves of engines, governors overcome " passive resistance," that is to say a resistance which exerts no vibratory or shaking force upon the governor. In order to determine the influence of a passive resistance upon a governor, imagine the latter to be replaced by a spring balance at point (1), Fig. 13. It should be understood that the linkage between the governor and the valve gear must be so arranged as to vary the energy supply to the prime -x d =-zo* mover from zero to its maximum value, while the governor sleeve passes through its range of travel. Let the forces which are measured in , , FIG. 14 moving the spring bal- ance up and down be X u and X d respectively. V" J_ V ~T -A d Then = R is the passive resistance of the valve gear, that is to say a force which resists motion either way and affects the motor part of the governor. On the other hand, V V - is a force which does not affect the motor part of the governor, but only the speed counter part. 1 Even if the pas- sive resistance of the valve gear were entirely eliminated by 1 To make these relations clear, imagine the valve resistance replaced by the f rictional resistance of a spring load acting on a heavy block (/), Fig. 14. If the f fictional resistance is 15 pounds, then the forces at the governor collar necessary t$> move the weight are 10 and 20 pounds as indicated. The resistance of 15 pounds concerns the motor part of the governor, and the constant force of 5 pounds reduces the equilibrium speed of the governor. 20 GOVERNORS AND THE GOVERNING OF PRIME MOVERS V V vibrations, - - would still act and would be noticed by the speed counting properties of the governor. See below in paragraph 1 of Chapter III. R is then the force which the governor must overcome in order to shift the valve gear, provided that the governor itself is free from friction; but from the preceding paragraph it is known that most governors have friction and must, for this reason, overcome an additional force F (equivalent friction force). Hence the governor must exert a force R + F to move the valve gear. This fact is mathematically expressed by the following equations : (1) R = D P= 2 P see (4) on page 7. u (2) F = q P = 2 P see page 18. u The 2 in the second equation comes from the definition for q; the value of q covers the whole detention above and below normal speed, so that q = - - is equal to twice the relative u speed change which is necessary to overcome the internal re- sistance F; DIU is that speed change which is necessary to overcome the external resistance R. Addition of equations (1) and (2) furnishes I u + Z> 2 u R + F = 2P u From this follows the total speed change which is necessary to overcome both resistances : R + F ufR Evidently the desirable effect of making the total speed change small can partly be obtained by the increase of the strength P; partly only, because F/P is a constant for a given type of governor and provides a lower limit for the speed change, no matter how strong the governor might be. THE DIRECT-CONTROL GOVERNOR AS A MOTOR 21 According to the theory just presented, it would be impos- sible to obtain close regulation, unless the internal friction of the governor were reduced to almost nothing. This viewpoint is strongly emphasized in catalogues advertising governors with small internal friction. On the other hand, everyday experience proves 'that in many cases close regulation is obtained even with governors having considerable internal friction. As may be surmised, vibratory forces from various sources partially or wholly eliminate friction in these cases. See also chapters on resistibility and on shaft governors. In other cases, where vibratory forces are either absent, or quite small compared to the passive friction resistance, nothing remains but the use of strong governors with small internal friction (see chapter on relay governing). In that case it is advisable to investigate the required speed change for two positions of the governor and valve gear, because the strength of the governor and the resistance of the valve gear usually vary from position to position. In practice the difficulty arises that neither the frictional passive resistance can be exactly predetermined nor the help- ing vibratory force is definitely known, so that judgment is required in the selection of a governor. In the regulation of gas engines working with dirty gas (producer gas, blast furnace gas, coke oven gas) exceptionally great strength of governors is needed, because the regulating valves which are operated by the governor gradually clog with dirt, and stick. Weak governors require too frequent cleaning of the regulating valves. Insufficient strength of governors coupled with absence of vibratory forces can easily be recognized from the speed varia- tions which occur even with the smallest and most gradual changes of load on the prime mover. To make matters even clearer, an example will be given. Let the passive resistance of the governor rig be 10 pounds, measured at the governor collar. Let q, the detention by internal governor friction, be equal to 1| % Let it be stipu- lated that the total speed variation in overcoming the valve resistance with gradual changes of load must not exceed 2%. 22 GOVERNORS AND THE GOVERNING OF PRIME MOVERS What strength P of the governor is necessary to meet these requirements? For a solution we will write equation 3 in this form: _ D U R f u- u r> R q = , from which P = u Z q u By substitution, References to Bibliography at end of book: 36, 73. CHAPTER III THE CENTRIFUGAL GOVERNOR AS A MEASURING INSTRUMENT (SPEED COUNTER) 1. Equilibrium Speed, Static Fluctuation and Stability. If, in a prime mover, the resisting torque is reduced, while the torque which is produced either remains constant or is not reduced as much as the resisting torque, the liberated excess energy is converted into kinetic energy, that is to say, the speed grows. For the sake of stability, the governor must be so arranged that the increase of speed causes it to move the torque-controlling mechanism toward the no-load posi- tion. The inverse series of events has to be gone through, if the load is increased. As mentioned in paragraph 1 of Chapter I, the purpose of speed governing is to keep the speed constant in spite of varia- tions of torque. It will be shown in this paragraph and in paragraph 2 of Chapter IX that centrifugal governors must have a higher equilibrium speed (at least temporarily) at no load than at full load, in order to make use of the principle enunciated at the beginning of the present paragraph. The smaller the speed variation, the better the governing. In consequence, several measures of the closeness of regula- tion have been introduced; most important among them is the term " fluctuation " or "static fluctuation," which means u u Ud . , 1N p = (1) J(t* + u d ) In this equation u u means that speed at which the governor is in equilibrium in its uppermost (or equivalent) position, and u d means that speed at which the governor is in equilibrium when in its lowest position. The term "is in equilibrium" means that the governor is floating between centrifugal and centripetal forces, with friction eliminated. Uppermost and lowermost position refer to no load and to full-load positions. 23 24 GOVERNORS AND THE GOVERNING OF PRIME MOVERS The terms " outermost" and " innermost" are more correct, but are seldom used. If a prime mover could be made (which it cannot) to always run at a speed proportional^to the equilibrium speed of the governor, a small value of p would mean close regulation, and p = would represent the ideal case of constant speed regula- tion. A governor of this description (p = 0) is called isochronous. In speaking of static fluctuation, governor manufacturers usually assume that the whole range of the travel of the governor collar is utilized for varying the torque produced by the prime mover, from zero to its maximum value. If that condition does not exist, that is to say, if a fraction of the travel of the collar varies the torque through its complete range, the static fluctuation of the governor, as given in the governor catalogues, re- mains the same, but the effect upon the prime mover is different and is equivalent to a reduction of static fluctuation, or to an increase of sensitiveness. A governor with a small static fluctuation is spoken of as a " sensitive" governor, so that the sensitiveness of a governor is closely related to the value 1 /p. An example will illustrate these statements. In Fig. 15 the numbers represent the equilibrium speeds for certain posi- tions of the governor collar. If the total available travel of the latter be utilized in governing, the static fluctuation is - = 10 %. If, on the other hand, the position marked luu 102 J rev./niin. i s the no-load position, and the one marked 97^ is the full-load position, then - - = 5% is the static FIG. 15 fluctuation. 100 THE GOVERNOR AS A MEASURING INSTRUMENT 25 The mistake is often made to assume that an engine or turbine will work with close regulation, because it has been equipped with a sensitive governor, which, as before stated, means a governor with small static fluctuation. Such an assumption is wrong. The sensitiveness of a governor gives no guarantee for close regulation, and too great a sensitiveness will often make it impossible to obtain close regulation. For detail discussion of the causes underlying these facts see paragraph 2 of Chapter IX. The static fluctuation of a governor is related to its sta- bility, which means its ability to return to the position of equilibrium after it has been displaced from that position. For a judicious discussion of this point, knowledge of the " characteristic " of a governor is necessary. See the following paragraph. References to Bibliography at end of book: 1, 11, 14, 24, 26, 36, 37, 53, 73. 2. Characteristics of Governors (C-curves). In the fur- ther treatment of the properties of governors as speed coun- ters it is advisable to deal separately with spindle governors and with shaft governors. The latter have numerous moments impressed upon them, and it becomes necessary to express the equation of equilibrium in moments. In the spindle type of governor it is possible and advisable to deal with forces. The treatment of the spindle type is therefore simpler and will be taken up in the following paragraphs. For the shaft governor see Chapter VI. The characteristic properties of a governor as a speed counter can be expressed and plotted in various ways; for instance, we may plot : (1) equilibrium speed against position of collar, (2) equilibrium speed squared against position of collar, (3) equilibrium speed against radial displacement of cen- trifugal weight, (4) equilibrium speed squared against radial displacement of centrifugal weight, (5) centrifugal force against radial displacement of centrif- ugal weight. 26 GOVERNORS AND THE GOVERNING OF PRIME MOVERS At first thought, method 1 appears to be the most con- venient and to offer the final criterion. Yet, closer study reveals the fact that method 1 shows nothing but the final result, whereas method 5 shows most of the hidden mechanics of the governor. This latter method was, as far as the author knows, first used by Mr. W. Hartnell l and was later very much developed by Mr. M. Tolle. 2 The Hartnell-Tolle method will be used in this and the following paragraphs. The mass of the links will at first be neglected. In the present discussion C represents that radial, out- wardly directed force which balances (directly, or by means of a linkage) all other forces acting in the governor, such as W = weight of centrifugal masses Q = weight of counterpoise, see Fig. 16 Si, S 2 , etc. = forces exerted by springs F = equivalent friction forces, see paragraph 4 of Chapter II. The line of action of C passes through the mass center of each centrifugal weight. In order to develop the properties of characteristics logically, we start with a simple case, namely with a modified type of Watt governor (see Fig. 16). In this type the two principal centripetal forces are due to the weights W and Q. It is ad- visable to let W represent the weight of all centrifugal masses, because Q can then represent the weight of the whole counter- poise. Since the horizontal force C balances the vertical forces W and Q, it may be considered as consisting of two components, C w and C q . C w may be found from W by forming moments about fulcrum (13), so that C w = W~. C q can likewise be ^i found by the moment method ; 3 but any method of mechanics, including graphic statics, may be employed. After the forces 1 Proceedings of British Institute of Mechanical Engineers, 1882. 2 Zeitschrift des Vereins Deutscher Ingeriieure, 1895. 3 The lever arm of Q about (13) is (13)-(14), which will be seen from the resolution of Q into a horizontal and an inclined force. Shift the latter in its own direction to point (14), and resolve it into a vertical and a horizontal force. All horizontal forces cancel, because there is an equal and opposite force on the other side of the governor. The vertical component equals Q. THE GOVERNOR AS A MEASURING INSTRUMENT 27 C w and C q have been found, they are laid off directly under the center (12) of weight W from a horizontal base line (1) (15), Fig. 16. Thus (15) (10) represents C w to some scale, (15) (7) represents C q to the same scale, and (15) (4) represents FIG. 16 C = C w +C q . By repetition of this process for several positions of the governor, the curves (8) (10) (9), (5) (7) (6) and (2) (4) (3) are obtained. It should be noted that in the construction of these charac- 28 GOVERNORS AND THE GOVERNING OF PRIME MOVERS teristic curves the governor has been treated solely as a mecha- nism and not as a speed counter. The mechanism in this case consists of a slider crank with fulcrum (13) and slide (11). The location of the governor spindle has not entered the calculation. Consequently, the shape of the characteristic curves of a weighted governor does not depend upon the location of the governor spindle with regard to the mecha- nism (13) (12) (11). Qn the other hand, we certainly must consider the location of the characteristic curve relative to the spindle if we wish to investigate the physical properties which it represents. To that end we must replace the auxiliary horizontal force C by its true function of centrifugal force of the revolving masses. Then C = mru 1 (1) where m = mass of the revolving centrifugal weights ; m = W/g r = radius from axis of rotation to mass center of W. The shape of the centrifugal weights is for the present limited to bodies of circular section. For other shapes see paragraph 4 of present chapter. Ti u =3.14 = that angular velocity at which equilibrium 30 exists between centrifugal and centripetal forces. n = revolutions of governor per minute From (1) follows u 1 = . In order to be able to tell at a r glance the value of u for different positions of the governor, we introduce the following trigonometric relation : Calling "i" the angle (4)(1)(15) of Fig. 16, we have tan i = C/r'= mu 2 , from equation (1); but, since m is constant, tan i is proportional to u 2 , or u is proportional to -\/tan i. Hence u grows, as long as i grows; it falls, whenever i decreases. Observation of the angle i in Fig. 16 teaches that, in the type of governor there represented, the speed grows, as the weights move outward. The question is: Must this be so, or would it be just as good to have the speed drop, as the weights move outward? In order to decide this question refer to Fig. 17, in which (2) (4) (3) is a governor characteristic with THE GOVERNOR AS A MEASURING INSTRUMENT 29 rising speed. Imagine that the governor weights are displaced the small distance dr from position of equilibrium (4), while the speed remains constant. The centrifugal force grows from (4) (7) to (5) (5), but the centripetal force grows at the same time to ()(), so that a centripetal force (6) (5) results which urges the governor back to its position of equilibrium. The governor is stable. If, on the other hand, the characteristic had followed the line (4) (9), the displacement dr (at constant speed) would have produced an excess centrifugal force (5) (9) which would *-dr- FIG. 17 urge the governor still farther away from its position of equi- librium. The governor would be unstable. The rise and fall of the angle "i" of the characteristic in relation to the radius r is thus a certain criterion for the stability of a governor. A quantitative measure of the stability is furnished by the well-known expression of mechanics: " Restoring force at unit displacement." Strictly speaking, this expression is correct only, if the restoring force is proportional to the displacement. In the absence of such proportionality we must use the ratio: restoring force at a given small displacement, divided by that same small displacement. In the case of a centrifugal governor, d(C P -C,) the value of this ratio is z = dr where C p = centrip- GOVERNORS AND THE GOVERNING OF PRIME MOVERS etal force and C/ = centrifugal force. For the interpretation of this equation it should be remembered that the present investigation considers only the governor by itself, running on a test block, so that pushing the governor up and down, away from its position of equilibrium, does not alter the speed at which it runs, and the latter remains constant. For the dis- placement dr, Fig. 17, (0) (5) is the restoring force d(C p - C f ) which, as can be seen from the illustration, equals the differ- ential (with regard to speed) of the centrifugal force for r = con- stant. But dC (for r = constant) equals 2 m r u du. Hence (6) (5) 2mrudu the stability = v V ; = - - = z dr dr du du so that z=2mru*-2- = 2C^- ( 2 ) dr dr The stability of a governor, when running on a test block "by itself " and not governing a prime mover, is proportional du to the total centrifugal force and to the value - By pass- dr ing from differentials to finite differences we obtain du u u - u d u = approx. \ (u u + u d ) p dr TO - Ti 2 Cp and the stability z = . An elementary derivation of To - Ti this equation is given in the appendix. This latter value is an approximation only, but is substantially correct for straight- line characteristics. If a characteristic is curved, the value du - must be used, but since this derivative is not easily con- dr structed in the diagram, it is advisable to fall back on the trigonometric function given on page 28. To this end we form the complete differential of C and obtain THE GOVERNOR AS A MEASURING INSTRUMENT du C dr 31 dC = u or u The left-hand member of this equation, if divided by rfr, equals the stability z, hence z = (3) But dC/dr = tan k, and C/r = tan i (see Fig. 16). The stability of a governor can, therefore, be very simply expressed as the difference of the tangents of two angles. Re- FIG. 18a FIG. 186 ferring again to the same illustration we note that the angle k is formed by the tangent to the characteristic at that point (4), where the stability is to be determined, and that the angle i is formed by the vector drawn from the origin (1) to the point (4) on the characteristic. From the fact that the stability depends upon the difference between the tangents of the angles k and i, it is evident that the governor is stable, whenever the point of intersection (16) lies to the right of point 32 GOVERNORS AND THE GOVERNING OF PRIME MOVERS (1), and that the governor is unstable, if the point (16) lies to the left of (1). " Right" and "left" must, of course, be reversed if the characteristic is drawn to the left of the spindle, instead of to the right as in Fig. 16. From this influence of the intersection point (for instance: point (16), for point (4) of characteristic) follows the important conclusion that any weighted governor may be made more sensitive, or less stable, by moving its mechanism closer to the axis of rotation. The shifting of the relative positions of mechanism and of axis of rotation is illustrated in Figs. 18a and 186. In both cases the governor mechanism and the shape of the characteristic are identical, but the stability in Fig. 18a is considerably greater than that in Fig. 186. It is, therefore, possible to design weight-loaded governors (which are based on a given mechanism) to have any desired stability by simply varying ^" the location of the mecha- nism relative to the spindle. It will be shown later on that spring-loaded governors can, with equal or even greater facility, be designed for any desired stability. However, the question of .. what stability gives the best FIG. 19 . ,. , , , . , , regulation cannot be decided by any study of the governor alone; it requires the study of the interaction of governor and prime mover. See Chapter IX. Knowledge of the properties of the characteristic clears up the widespread confusion between the terms "stability" and "static fluctuation." In Fig. 19 three characteristics, (2) (4) (3), (2) (5) (3) and (2) (6) (3) are shown, all of which have the same static fluctuation, because the angles i u and i d are the same for all three. Tangents drawn to the curves at different points show, however, that the characteristic (2) (5) (3) is unstable near point (3) , whereas the characteristic (2) (6) (3) is unstable near point (2). From this difference follows that knowledge of the static fluctuation alone without knowledge THE GOVERNOR AS A MEASURING INSTRUMENT 33 of the shape of the characteristic is not sufficient to judge the properties of a governor. The difference between the two terms under consideration will be further emphasized by the following statement : As far as the governor alone is concerned apart from its con- nection with a prime mover the static fluctuation may be decreased without decrease of stability, by utilizing only a small part of the radial travel of the weights. Figure 16 shows that confinement of the weight travel between the vertical dash and dot lines reduces the static fluctuation. To see this more clearly, imagine straight lines drawn from point (1) to intersections (17) and (18) of the vertical dash and dot lines and the characteristic. The angles i made by these straight lines with the axis of abscissae vary considerably less than those made by the straight lines drawn from (!) to (2) and (3). It will be remembered that angle i is, to a certain extent, a measure of the equilibrium speed. While limitation of radial travel has been frequently em- ployed for increasing the sensitiveness of governors, and while it does not change the stability of the govern- or proper, it does decrease the stability of regulation of a prime mover (see Chapter IX). It also reduces the work capacity of the govern- or and necessitates the use of a stronger governor, if a passive resistance is to be overcome. Finally, it reduces the ability of the governor to act as a shock absorber (see Chapter VIII). Engineers who wish to go more deeply into the study of characteristics will find the two following geometrical relations of interest: The first of them expresses the static fluctuation by the angles i. To find this relation we transform equation (1) on p. 23 in the following manner (see Fig. 20): FIG. 20 34 GOVERNORS AND THE GOVERNING OF PRIME MOVERS u u - u d , (u u 2 - Ud 2 ) \mr u mr d j tanu- tani d ~%(u u + u d ) f2 (u u + Ud) 2 C a /mr a 2 tan i a In these equations the indices u, a, and d mean "up," "aver- age," and "down." In any regulation aiming at practically constant speed, u a and u u differ only by a few per cent, so that it is permissible to write 4- Q Y\ /i 4- Q Y"| f\ , I ill I v u I ell I. v d p = = 2 tan i u The second relation solves the problem of quickly determin- ing the equilibrium speed for any point of the characteristic, if the equilibrium speed for any other point is known. Referring again to Fig. 20, let u^ point (3) of the characteristic be known, and let it be desired to find u at point (4). Draw ray (1)(4) to intersection (6) with vertical (3) (8), erect semicircle over (3) (8), draw horizontal through (6) to intersection (9) with semicircle, then (8) (9) equals u of point (4) to the same scale to which (SX*) equals u u . Proof: = L = but u u * tani (3) (8) by construction u u = (3) (8), so that it cancels out, and u 2 = u u X (0)(S) remains, which means that u is the mean proportional of (6) (8) and (#)(<). The construction which is given above makes (8) (9) the mean proportional of these two distances. References to Bibliography at end of book: 1, 11, 14, 26, 36, 36, 37, 52, 63, 72, 73. 3. Constituent Parts of the Characteristic Curve. While the analysis of the preceding paragraph teaches that the principal static properties of a governor depend upon its re- sultant characteristic, more detailed study proves that the individual character of the C Q and C w curves is of equally great importance. THE GOVERNOR AS A MEASURING INSTRUMENT 35 In order to obtain sufficient material for this discussion, we shall investigate the characteristics of two governors quite different from the Watt type. (1) Proell type, or inverted suspension type, Fig. 21. In this design of weight-loaded governor the centrifugal weight is w Forces are those of both halves of governor FIG. 21 not carried directly by the pendulum arm (13) (16), but forms part of the connecting rod (11) (16). The forces CV and C Q are found by forming moments about the instantaneous center (14)-, the lever arms of the forces have been marked L c , L w , and LQ. In the illustration the forces have been entered to scale*, and the CV and C Q curves have been drawn. Study of these 36 GOVERNORS AND THE GOVERNING OF PRIME MOVERS curves shows that the C w curve is slightly unstable in its inner part (at least for the selected location of the governor spindle) and that it is stable in its outer part. The C Q curve shows excess stability. By variation of the ratio Q/W the stability of the governor may be varied. In the illustra- tion, Q is so much greater than W that the governor has considerable stability, particularly in the outer position. (2) Hartnell type, Fig. 22. This governor consists of two (sometimes 3 or 4) centrifugal weights TFwith mass centers (12)} III 'Q (17)1/^1 ^vv* C s ^e we ig nts are carried on J I L // kell cranks with fixed fulcrums (18). The principal centripetal force is furnished by an axial spring. Again C Q and C w can be found by moments about the pivots, which are represented by points (13) in this case. Q is variable in this governor so that doubt may arise concerning the meaning FIG. 22 of C Q . For reasons which will appear later it is advisable to plot C Q for a constant force Q. For the latter, we assume an arbitrary constant unit force applied at point (14). Both the C w curve (8) (10) (9) and the C Q curve (5) (7) (6) show a decidedly negative static fluctuation ; note that tan i drops. To render the governor useful, the central force Q must be made variable to such an extent that the resulting C curve furnishes a small positive fluctuation. Disregarding for THE GOVERNOR AS A MEASURING INSTRUMENT 37 the present the limitations which may be imposed by the space available for the spring, we can determine the required spring force as follows; Draw the desired characteristic (15) (17) (16), and the C Q curve for those weights which move up and down with the sleeve (18). These weights include that of the sleeve, of roller (14), of the plate, and one half of the spring. Add the ordinates of this C Q curve and of the Cw curve. Find the differences between this combined curve and the desired final C curve. These differences divided by the ordinates of that CQ curve which belong to Q = 1 furnish the spring forces. The first attempt may result in impossible spring dimensions, or in improper utilization of the available space, but it will show in which direction the assumptions of the resulting C curve must be altered. The description of these governors and of the construction of the characteristic curves is rather sketchy, but illustrations 16, 21 and 22 contain sufficient information for studying the individual influence of the Cw and CQ curves. When a governor governs a prime mover, it is subjected to forces acting upon its sleeve, that is to say forces which either add to or subtract from Q. Such forces are caused by (1) Adjustment of equilibrium speed. (2) Resistance to motion caused by friction of valve gear. (3) Unbalanced weights or fluid pressure. (4) Integrated effect of valve gear reaction. Speed adjustment is usually accomplished by the addition or subtraction of a constant force, see Chapter V, for instance by the shifting of a weight along a lever or by the tightening of a helical spring. The addition of a constant force at the sleeve produces a proportionate increase or reduction of the ordinates of the CQ curve. If the C Q curve and the C w curve are both stable and of the same character as the combined curve, such as, for instance, in Fig. 16, the addition or sub- traction of a constant force does not affect the static character of the governor. But if the C Q curve has excess stability in order to make up for insufficient stability of the Cw curve, see Fig. 21, any addition to Q increases the stability, and a reduc- tion of Q reduces the stability of the governor. If, finally, 38 GOVERNORS AND THE GOVERNING OF PRIME MOVERS the CQ curve has negative stability (which is the case in most spring-loaded governors, such as in 22) an addition to Q makes the governor less stable, and if carried far enough, makes it utterly unfit for regulation. This effect of the addition of a constant force is always the same, no matter whether it conies from item 1, 2, 3 or 4 of the above given classification. The forces entering the governor through the sleeve are a fruitful source of annoyance and of triangular litigation between the builder of the governor, the builder of the prime mover and the user of the latter. Thus, the excessive clogging of regulating valves with tar in gas engines, the excessive and onesided tightening of stuffing box glands of regulating spindles in engines and turbines furnish complaints for which neither the governor builder nor the builder of the prime mover is responsible, except that, perhaps, the latter did not install a large enough governor. If these forces are constant throughout the working range of the governor, they will only alter the equilibrium speed, but will not upset stability of regulation, provided that the CQ and C w curves have like character and stability. As a rule, the forces resulting from items 2, 3 and 4 of the list on page 47 vary with the position of the governor sleeve. In gas engines there may be (and there usually is) more tar deposit near one end of the swing of the regulating valve than there is near the other end. So-called balanced regulating valves are frequently so designed that they appear at first glance to be balanced, but are really not so, because near the closed position forces arise from impact or jet action of the fluid. These forces are transmitted to the governor and, by their action at one end of the travel only, often seriously upset regulation. For the purpose of this book it will suffice to study the action of two types of these so-called balanced valves. Fig. 23 shows a double beat poppet valve, the bottom seat of which is slightly smaller than the upper one, for assembling purposes. Each valve is cup shaped for purposes of strength and rigidity. When the valve is wide open, there is an upward thrust, because there is then maximum flow, and because the concave THE GOVERNOR AS A MEASURING INSTRUMENT 39 Steam Pressure Throttle Pressure FIG. 23 side offers more resistance to flow than the convex side does. With an almost closed valve, that is to say with very low rate of discharge, there is a downward thrust. In this position of the valve, the upstream steam pressure is much greater than the downstream throttle pressure, and a static pressure effect due to the dif- ference of areas of upper and lower valve is felt. So-called bal- anced butterfly valves are usually quite unbalanced, because the " noz- zles" in which pres- sure is converted into velocity have very dissimilar shapes on the two sides of the valve. The lines (Jf)()(3) of Fig. 24 are intended to show respectively equal areas of flow and niveau surfaces of respectively equal pressure. It is evident that the pressure drop takes place with less travel at the bottom of the valve, so that a larger part of the bottom wing is exposed to high pressure than there is at the top. Conse- quently a force results in the direction of arrow (4). When the valve is opened, this force disappears. The injurious influence which these unbalanced-in-certain- positions valves have upon satisfactory regulation has caused many engineers to keep the unbalanced forces away from the governor by interposing a relay, see Chapter XIII. But if a relay is not used, the forces in question should be made as small as possible by the builder of the prime mover; besides, the governor designer should be notified of the extent of these FIG. 24 40 GOVERNORS AND THE GOVERNING OF PRIME MOVERS forces, so that he may provide for them in the design of the governor. In doing the latter he will again make use of the C Q curve. If such a curve be drawn for Q = 1, the ordinate belonging to any given position of the governor sleeve can be multiplied by the force reacting upon the sleeve in the position under consideration. Repetition of this method for several positions of the sleeve furnishes a new C curve for the forces entering the governor from without. Combination of this curve with the other C curves must result in a curve with posi- tive static fluctuation, as previously explained, if correct regu- lation is desired. References to Bibliography at end of book: 11, 36, 53, 72, 73 4. Influence of Shape of Centrifugal Weights on Character- istics. --The subject of the present paragraph is not of vital consequence for any centrifugal governor now on the market; neither is it likely to be of such consequence. It might, there- fore, be passed in silence, if it were not for the human fail- ing of reinventing designs which were relegated to the scrap heap long ago. And " Improving CL governor" by changing the shape of the centrifugal weights is a popular subject for inventors. In paragraphs 2 and 3 of the present chapter the theory of the characteristic curves of governors was developed for centrifugal weights of spherical or other- wise symmetrical shape, and the state- ment was made that for such shapes it was permissible to concentrate the whole mass of the weight in its mass center. The correctness of this statement will now be analyzed. In Fig. 25 the oval shape represents a centrifugal weight with mass center (0), and with suspension point (1). The moment of the centrifugal force of a small mass dm about the suspension point is dM = dm u 2 (r + x) (y + L) . For meaning of r and L see illustration. The moment of the whole mass is the integral FIG. 25 THE GOVERNOR AS A MEASURING INSTRUMENT 41 M = u 2 fdm(r + a;) (y + L), or M = u 2 r fdm y + u 2 r L fdm + u 2 fdm x y + u 2 L fdm x. By definition of the mass center, f dm y = 0, and f dm x=Q. It is also evident that fdm = m, so that the centrifugal moment of the weight is reduced to M = u 2 r L m + u 2 f dm x y. In the preceding paragraphs centrifugal force had been given as m r u 2 and centrifugal moment as u 2 r L m. Evidently, the actual moment differs by the amount u 2 f dm x y, and the equivalent centrifugal force which, when acting at the mass center, produces the correct moment is C = r u 2 (m + fdm xy\ (1) For all shapes which are symmetrical with regard to two axes at right angles to each other, fdm x y = 0. Such shapes are the circle, the square, the octagon, etc. For centrifugal weights of any one of these forms the assumption that the mass may be concentrated in the mass center is, therefore, correct. For all other shapes the theories of paragraphs 2 and 3 must be modified. To carry out this modification, it is desirable to narrow down somewhat the extent of the investigation, because the value of the correction : f dm x y in equation (1) cannot be rL weighed or appreciated in its general form. If the investigation is limited to weights whose thickness h is constant at right angles to the plane in which they swing, the correction can be put into a form which presents a fairly clear view of the influence of the shape. To this end we use the principal axes (3) (4) and (5) (6) of the plane section of the swinging weight as a new system of coordinates. Let m f = mass of unit volume in centrifugal weight; J 8 and Ji = the smallest resp. largest moment of inertia of the plane section of the centrifugal weights; k = the angle between the axis of rotation and the axis to which the smallest moment of inertia is referred; then we have fdm x y = \ m' h ( J t - J s ) sin 2 k (2) 42 GOVERNORS AND THE GOVERNING OF PRIME MOVERS For proof of this equation see Appendix, p. 214. From equation (2) it follows that the difference of the two principal moments of inertia and the angle k of the inclination of the axis determine the extent of the correction. For k = 0, and for k = 90, the value of the correction is zero. The same is true for any value of k, if J s = J t , which bears out the state- ments made above concerning symmetrical shapes. The dif- ference between J ' a and Ji is the greater, for a given area of section, the more oblong the shape of the weight. Let C p L = M p be the centripetal moment caused by spring forces, weights, etc. which is to be balanced by the centrifugal moment M c , and let C p be the equivalent centrip- etal force referred to the center of each centrifugal mass ; then C p L = mru 2 L+%m'hu 2 (Ji- J s ) sin 2 k, from which , __ C p __ C P 1 _ 1 = mr^ . m'M/r-J s )sin2A; (3) mr + m f h(Ji-J s )sm2k s It will be remembered that, for symmetrical centrifugal weights, the static properties of a governor were found from C tan i its characteristic by means of the equation v? = = - - mr m Equation (3) teaches that this simple relation does not hold for governors with oblong weights. However, a charac- C" teristic may be so drawn with ordinates C' that u 2 = , pro- mr vided that we make C m'h(J l -J s )sm2k 2Lmr Evidently, the relations between sin 2 k, L and r in conjunc- tion with the value of (J t J s ) determine the difference between C p and C'. In general, the difference is greatest for k = 45. C Whereas for symmetrical shapes of weights tan i = - = con- mr THE GOVERNOR AS A MEASURING INSTRUMENT 43 stant, that is to say, a straight line characteristic passing through the origin, means an isochronous governor, it does not convey the same meaning for a governor with oblong weights. On the contrary, the characteristic for an isochronous governor with oblong weights is curved, the magnitude of the curvature depending principally upon (J l J s ). Equations (3) and (4) express the conditions quite clearly from a mathematical standpoint. A less comprehensive, but ver> much simpler explanation of the effect of the shape of the centrifugal weights upon the speed and upon the shape of the characteristic can be gained by studying the effect of split- ting a concentrated centrifugal mass into two concentrated masses some distance apart and joined by a mass- less rod. In Fig. 26 the centrifugal moment of a mass ra at point (1) is M = ra u^ 4 s 2 . If the mass ra is split into two masses | ra, located at points (2) and (3}j the moment appears in the form M = | ra u 2 z (l 2 + 3 2 ) s 2 = ra u^ 5 s 2 . If, finally, the masses \ ra are located at points (4) and (5), the moment becomes M = J ra M 3 2 (0 2 + 4 2 ) s 2 = ra Uz 8 s 2 . The moment M remains the same in all three cases, because the centripetal (in this case gravity) moment is the same. The speeds are found from the foregoing equations by solving for u. ^~ i/l i/~^~ */l /^" u> = V- V- FIG. 26 m s 4 8 The oblong shape of weight evidently lowers the equilibrium speed, if the weight is arranged as in Fig. 26. The reverse action takes place if the long axis of the mass is placed as in Fig. 27. The moment equation for the mass at point (1) is M = m Ui 2 4 s 2 . For two masses \ ra at points (2) 7TL and (3} it is M = u _ f CO _L Lift of Collar FIG. 42 ADJUSTMENT OF EQUILIBRIUM SPEED 59 type; l the illustration represents a detail of point (4) of Fig. 40. As soon as rod (10), Figs. 40 and 41, reaches a nearly horizontal position corresponding to the highest safe speed of the governor, the rod (13) of catch block (11) strikes collar (14) and releases the rod (15) carrying weights (12). These weights, in dropping, close the power-controlling valve. Before the engine can be started again, the rod (15) must be lifted to again engage -j- catch block (11). ^ The necessity of a discon- necting device can be avoided by a governor whose speed curve has the shape indicated in Fig. 42. Part (1)(2) of this curve is used for speed adjust- ment, as described. Part (2) (3) used to bring the power- is controlling device to no-load position in case of accident. Such a governor was designed by Professor J. Stumpf (Germany). It is shown in Fig. 43. In this governor skillful application is made of variation of leverage of main spring (3), and of lost motion of spring (3) and spring (4) in this manner: In the low positions of collar (5) the spring (4) pushes upward, allowing a low speed to raise the governor. Spring (4) is very short and soon gets out of action with rising speed. On account of slots (6), the main spring (3) is ineffective in the bottom part of the governor travel. With rising collar it takes hold and vastly increases the equilibrium speed. In the higher position of collar (5) the force of spring (3) increases slowly, whereas its lever arm L decreases rapidly, which causes the speed to change very little for the upper part of the travel of collar (5). Another governor with a speed curve similar to that shown in Fig. 42 is described in Chapter XII. 1 This device, as well as the whole method No. 4, was originated and developed by Mr. K. Weiss (Switzerland). The method was first used in the United States by Mr. B. Nordberg, of Milwaukee, Wis. FIG. 43 60 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Governors of this type are useful for pumping machinery only. The modern tendency in such machinery is to adapt the supply to the demand and to regulate either for constant pressure or for constant rate of delivery. Forces other than centrifugal are used for this purpose. Speed-changing gov- ernors of that type are, therefore, not considered in the present chapter, but are discussed in Chapters XI and XII. References to Bibliography at end of book: 30, 36, 51, 73. CHAPTER VI SHAFT GOVERNORS 1. Forces Acting in Shaft Governors. The term " shaft governor" commonly denotes a governor the centrifugal weights of which move in a plane at right angles to the axis of the shaft or spindle, and which adjusts the position of an eccentric or crank relative to other revolving parts. Figures 4, 5, 8, 9, 44 illustrate shaft governors. The arrangement of centrifugal and inertia weights in shaft governors, their method of fastening, the linkage between weights and eccentric, the arrangement of springs, and the transmission of motion to the valve gear belong in treatises on machine design and engine design. They do not form subjects of the present book. Spindle governors could be studied apart from the active forces which the valve gear might impress upon them, because most spindle governors have to overcome mainly passive re- sistance, compared to which the small existing reaction is usually of little or no importance. Shaft governors, on the other hand, have to move valves directly against inertia forces, against varying friction forces and against varying unbalanced steam pressure. These forces react upon the governor and tend to disturb the equilibrium between centrifugal forces and spring forces. Not only does this reaction induce vibration, but it has an " integrated effect, " that is to say, a definite average force or moment which is added to the spring moment and changes the speed from what it would be, if that effect were zero (see also pages 38 and following). The average effect of the reacting force or moment is, in practically every shaft governor, large enough to upset static equilibrium, and must therefore be considered either by theoretical study, or else by experiment. In view of these disturbing influences, a complete static calculation of a shaft governor involves the following steps: 61 62 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Select a static fluctuation p, about the permissible value of which see Chapter IX, and compute the various moments tending to displace the centrifugal weights, viz. centrifugal moments of all masses, moments due to inertia of reciprocating masses, due to valve friction, eccentric strap friction, and that due to unbalanced steam pressure, for three, or better, four positions of the weights. Then locate and dimension a spring so that its moment (negatively) coincides with the sum of all other moments. This latter problem is not difficult, if the FIG. 44 weight arms swing through a small angle and if the springs are protected against disturbing deformations due to centrifugal force. If, however, the weight arms swing through a large angle, say more than 40, or if the springs are allowed to bow out under the action of centrifugal force, no satisfactory co- incidence of moments can be attained with one set of springs, and unusual expedients must be resorted to. On account of the great variety of shaft governors on the market, and on account of the still greater number of valve SHAFT GOVERNORS 63 gears which they control, it is impossible to follow out this method in all its details within the limitations of the present book. It is, therefore, advisable to study the underlying principles with the aid of one particular example. The careful study of one example will show the methods which are to be followed, with slight modifications, in all cases. References to Bibliography at end of book: 1, 13, 24, 30, 36, 40, 50, 51, 58, 59, 68, 73, 74. 2. Centrifugal Moment of Rotating Masses. Figure 44 shows a simple type of governor suitable for study. This governor is used for controlling slide valves, including piston valves. The triangle (1}(2}(3) is fixed relative to the shaft the center of which is (1) - - and rotates with it. (4) is the center of the eccentric which is shifted across the shaft by the regulating forces, maintaining an approxi- mately constant lead of the valve. The centrifugal weight (5) tends to fly out; it is restrained prin- cipally by the spring (6), but to some extent also by the centrifugal force of the eccentric (4) and of its strap. (5) and (4) are joined mechanically by link (7). For the sake of clearness some parts of the governor have been drawn diagrammatically in Fig. 45. With the notations of this latter illustration, the centrifugal moment of the small mass dm about the suspension point (2) is d M c = dm r u 2 L. But the area of the triangle (1) (2) (11} is \ r L = | s b which makes the element of the centrifugal moment d M c = dm s b u 2 . The moment of the whole centrifugal weight is M c = su 2 fdm b = mBsu 2 (1) where B is the perpendicular distance from mass center of weight (5) (including supporting lever) to line (1)(2), and m FIG. 45 64 GOVERNORS AND THE GOVERNING OF PRIME MOVERS is the mass of the centrifugal weight (5), including lever. From this equation follows the fact that in shaft governors the dis- tribution of the mass around the mass center of the centrifugal weight has no influence upon the speed. In this respect shaft governors differ from spindle governors (see paragraph 4 of Chapter III). Equation (1) applies to any revolving mass in a shaft governor and furnishes the moment of the mass about its own suspension point. If applied to the eccentric, it furnishes the moment of the centrifugal force of its mass about point (3), of Fig. 44, which, however, can easily be converted into a moment m FIG. 46 about point (2) by means of the leverages of the connecting linkage. The eccentric strap does not revolve, but each particle describes a path similar to an ellipse and exerts individual inertia forces, the integrated effect of all of which is trans- mitted to the eccentric proper. Generally speaking, the inte- gration can only be accomplished by a tedious point by point process. Fortunately, the distribution of material in the ec- centric strap is symmetrical about the center of the eccentric in the vast majority of cases. The whole mass may then be considered as concentrated in the center of the eccentric and be added to the mass of the latter. 3. Moment due to Inertia of Reciprocating Parts. If the valves moved by the governor are driven by a cam, the SHAFT GOVERNORS 65 Equal integrated effect of the moment cause by their inertia can be ascertained only by a graphical integration, but if the valves are operated directly by the eccentric with practically sine harmonic motion, the integral may easily be found thus: In Fig. 46, (1) is the center of the eccentric and (3) is the suspension point of the eccentric. (3) revolves about (2), which is the center of the shaft. The eccentric reciprocates a mass m v (mass of the valve) by means of long eccentric rod (1)(4). The following derivation is based upon the supposi- tion that mass m v is not large enough to displace point (1) perceptibly from its circular path. When (2)(1) coincides in direction with (#)(4), the inertia force is m v r u 2 . In the position shown in the illustration it is smaller. Usually the eccentric rod (1)(4) is very long com- pared to the eccentric radius (1)(2). The direction of the inertia force is then practically con- stant, and its magnitude varies with the cosine of the angle j. If (1) (5) repre- sents m v r u 2 , then (1) (6) = (/) (5) cos j is the inertia force in the position shown in the illustration, and the circle with diameter (1) (5) is the locus of the terminals of the force vectors, referred to the rotating system. In one revolution of the shaft the just mentioned circle is passed over twice. This will readily be seen by a study of the forces existing when the shaft has turned through 180, see dotted position, because the forces in the two positions have the same magnitude and direction relative to the rotating system. The average effect of a force vector describing a circle with constant angular velocity may be found by the following reasoning (see Fig. 47). Any four vectors (1)(7), (-O(tf), (1)(9), (1)(8)> located symmetrically as shown, have the average horizontal component (1)(10), whereas all the vertical components cancel. Since the same is true for any four sym- FIG. 47 66 GOVERNORS AND THE GOVERNING OF PRIME MOVERS metrically located vectors, (1)(10) is the average of all the force vectors, both with regard to magnitude and direction. Application of this result to Fig. 46 means that one half of 00(5), or (-0(7), or, in symbols, ^ m v r u 2 , represents the average of the inertia forces, both with regard to magnitude and direction. The same effect is produced by a mass J m v , located in the center of the eccentric, and revolving with the latter. The effect of the inertia of the reciprocating valve is, therefore, easily taken care of by adding one half of its mass to the rotating mass concentrated at the center of the eccentric. If mass m v is big compared to the masses swinging in the governor, the path of point (1) is no longer circular. The equa- FIG. 48 tions of motion of the point (1) then lead to an elliptic integral which is quite difficult. References to Bibliography at end of book: 36, 40, 50, 58, 59, 74. 4. Moment Due to Friction of Valve Gear. The average reaction of valves having a sine harmonic motion may be found by a similar analysis. Figure 48 is a repetition of Fig. 46 with the exception that inertia forces have been re- placed by friction forces. If, as before, we neglect the small angularity of the eccentric rod, the direction of the friction forces remains constant; hence this direction revolves uni- formly with regard to the rotating system. In Fig. 48 the fric- tion vectors have been entered. For a constant value of the friction force, the vector describes a semicircle (5) (6) (2), SHAFT GOVERNORS 67 jumps back to (5), again describes the semicircle, etc. If the friction force is not constant, the semicircle is distorted. The integrated effect of the friction force is represented by the average vector. In any case, including that of variable friction force, it can be found by graphical construction. In the case of constant friction force the terminal of the average vector is the mass center of the semicircular line (5) (6) (2). The magnitude of the average vector is therefore 2 /TT times the friction force. It acts at right angle to the diameter (2) (1). The integrated effect of friction and of inertia must be determined for several values of the angle ft, that is to say for several configurations of the governor. This is necessary for the purpose of equating the spring moment to the sum of all other moments over the whole range of the swing of the governor. References to Bibliography at end of book: 36, 40, 50, 58, 59, 74. 5. Moment due to Friction between Eccentric and Strap. - Friction between eccentric and strap is produced by the weight and the centrifugal force of the strap, by the forces transmitted through the eccentric rod (valve inertia, valve friction, unbalanced steam pressure), and by the pressure of the assembling fit between eccentric and strap. The combina- tion of these forces for various angular positions of the whole governor and for a few positions of the parts in the governor can always be made graphically. The friction moment result- ing from these forces is quite uncertain, because the coefficient of friction between eccentric and strap varies within wide limits with quantity and viscosity of oil. It is frequently observed that shaft governor engines have a different speed at starting up (when the oil is cold and thick) from that which finally establishes itself, after the oil has been heated by fric- tion. Another uncertainty is introduced by the deformation of the eccentric straps. The latter are too often designed so weak that they cause an amount of binding and gripping sufficient to frustrate any calculation. Nevertheless, a brief account will be given of the distribution of frictional forces around the eccentric, because the seemingly erratic behavior of many shaft governors cannot be understood without the knowledge of these forces. 08 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Suspension Point FIG. 49 Figure 49 shows a governor eccentric and strap with the principal forces acting upon them, and Fig. 50 illustrates the distribution of forces by means of oJn vectors having their origins at point (1). For a given load, the centrif- ugal force of the eccentric strap has a constant magnitude and direction (1) (2) relative to the revolving eccentric. Valve inertia is, in most cases, a sine function and adds the circle (2) (3) (4) (5) , compare also Fig. 46. The force due to valve friction, if the latter be constant, is also a circular func- tion and acts in the same direction as valve inertia ; but the function is discontinuous and changes its sign abruptly at the ends of the valve travel, see also Fig. 48. Addition of this force to those previously enumerated furnishes the vector terminal curve (6) (7) (8) (9) (2). The weight of the eccentric strap is constant in magnitude and in absolute direction, and, there- fore, rotates contin- uously around the re- volving eccentric. The same is true of un- balanced steam pres- sure, unless the latter be variable. The re- sultant of strap weight and of steam pressure changes the final locus of the terminus of the force vector to the curve (14) (13) (12) (11) (10) (15) (16} (17) (18) (14), etc. Thus (1)(12) is a vector picked out at random. The correct method of finding the average friction moment from these forces would consist in determining the friction moment for each force vector and then averaging the friction FG. 50 SHAFT GOVERNORS 69 moments thus found. This method would be tedious and is entirely out of place in view of the uncertainties with regard to the coefficient of friction. Since the vectors sweep over a small part of the circumference of the eccentric only, the friction moment caused by the average vector coincides nearly enough with the true average friction moment for all practical pur- poses, which leads to the following approximative construction. In Fig. 49 lay off (1)(2) equal to centrifugal force of strap in the direction from center of shaft to center of eccentric. In the same direction lay off (2) (20} equal to maximum inertia force of reciprocating valve parts, reduced to the center of the eccentric. At right angle to this direction, and opposite to the direction in which point (1) rotates about the center of the shaft, lay off (20) (19) equal to 64% of the constant valve friction force (2/?r = .64). Then (1)(19) is the average vector and acts upon the eccentric at point (21). The product of this average force, the friction coefficient and the lever arm 6 furnishes the average friction moment for the position under consideration. References to Bibliography at end of book: 36, 40, 50, 58, 59, 74. 6. Spring Moment. - - The spring in a shaft governor must be of such size, and must be so located, that the equilibrium speed of the governor (when governing the engine) rises gradu- ally and steadily from full-load position to no-load position under any conditions of careful operation, with the exception of the first minute or two after starting. A few additional exceptions due to tangential inertia or to compensating oil pots will be dealt with later on. For the purpose of expressing this rather general statement in a more tangible form, let us equate the spring moment to the rest of the moments. Let M , = spring moment M c = any one of the speed moments (centrifugal, valve inertia) M f = any one of the moments due to friction. For equilibrium, the spring moment must balance all other mo- ments, so that M 8 = S M c + S M f 70 GOVERNORS AND THE GOVERNING OF PRIME MOVERS where S is the mathematical sign for "the sum of." S M c means the sum of all speed moments. The speed moments are all of the form m r L u 2 , see para- graph 2 of present chapter. Those friction moments which are caused by the friction produced by centrifugal forces are of the form fm' r' L' u 2 , where / is a coefficient of friction. All other friction moments are of the form F I, where F is a friction force which is practically independent of the angular velocity u. By substitution of these values we obtain M s = u 2 [2mrL + 2fm'r' L'^ + ZFl ...... (1) In order to see clearly the conditions which the spring moment must meet in order to make u rise steadily, let us solve the equation (1) for u. It is immediately seen that the greatest obstacle to securing a gradual and steady rise of u is the variability of the friction coefficient, which affects both 2 F I and 2 / m! r' L' . In practice, lubrication varies; friction of valves and stuffing boxes varies not only with lubrication, but also with steam pressure, steam temperature, tightening of packing bolts by engineers, and with other, but minor, features. The variation of friction moment affects u the more, the smaller M s and 2 m r L. Vice versa, the greater M s and S m r L for a given engine, the smaller the disturbances caused by changes in friction forces. The latter thought, namely to make the spring moment and the centrif- ugal moment quite great, is carried into practice by all builders of small engines using shaft governors. For small and medium sized engines the magnitude of the spring moment required for faultless operation under all conditions is seldom, if ever, de- termined by calculation, but by practical tests. Both calculation and experience prove that with those shaft governors in which the spring forces are transmitted to the centrifugal weights through pins, there exists for each engine a certain point, above which an increase of spring moment does more harm than good, because the friction in the governor SHAFT GOVERNORS 71 pins becomes so great that the reacting or impressed forces coming from the valve gear cannot keep the governor vibrating or " limbered up" (see also Chapter VIII). The governor becomes sluggish and does not respond quickly to changes of load. This undesirable feature can be avoided by placing the centrifugal mass directly on the spring (see Fig. 54), or by the proper arrangement of inertia masses. The doubly beneficial influence of tangential inertia in shaft governors was recognized at a comparatively early time in the history of such governors. First, tangential inertia steadies the governor against impressed forces reacting from the valve gear (see Chapter VIII) ; and second, it prevents racing and hunting no matter whether the speed drops off as the load comes on, or whether it rises (see paragraph 3 of Chapter IX). The latter effect of tangential inertia was ex- ploited to obtain reversed speed curves (higher speed at full load than at no load), but the use of " we-don't-care-what-the- speed-curve-is " governors on engines driving alternators brought stern realization of the fact that, at least for the purpose of driving alternators in parallel, a gradual and steady rise of the speed is required as the load drops off, and that every other arrangement is a positive failure. One firm after another had this (often quite expensive) experience, between the years 1900 and 1905.. The result is that the statement made above concerning a sufficiently great spring moment to maintain a steady speed rise in spite of reasonable variations of friction holds good for all present-day shaft governors. In view of this explanation, it will pay to make the follow- ing comparatively simple, although rather tedious, calculation, whenever a new size or type of shaft governor is to be built. For all friction forces assume maximum and minimum values within reasonable limits. For guidance in this assumption the coefficient of friction may be varied from the low value of | % to the high value of 12 %. Compute the friction moments with both of these values for three or four positions (configura- tions) of the governor, as indicated in paragraphs 4 and 5 of the present chapter, and substitute in equation (2). In either case u must rise gradually from full load to no load. If it does 72 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Centrifugal .Weight not, either the static fluctuation p, or the spring moment M s , must be increased, or else the location and arrangement of the spring must be changed, as explained below. The rapid growth of the steam turbine in central stations limits the use of shaft governors for close-regulation alternating- current work more and more. For that reason a detailed example of the above outlined calculation would benefit only very few readers. It is, therefore, omitted. The following brief remarks will, however, be of interest. Each speed moment, if plotted against angle of swing of governor weight, is part of a slightly distorted sine curve. The distortion is caused by the fact that the governor is not isochronous. The spring moment can never be made to be a sine curve, because both spring force and its lever arm vary, as the governor moves in or out. Consequently, it is quite difficult to make the spring moment coincide with the sum of the other moments as required by a gradually and steadily rising speed, if either the main centrifugal mass or the eccentric turns FIG. 51 FIG. 52 through a large angle. Large in this case means more than 50. If the main centrifugal mass is large compared to the influence of other moments, and swings through an angle not exceeding 30 , the problem of securing a steadily rising speed is very easily solved. This feature deserves consideration in the design of new shaft governors. As an illustration of the complications introduced by the swinging of weights or eccentrics through large angles, the well SHAFT GOVERNORS 73 Roller known Thompson governor of the Buckeye Engine Co. may be cited. The eccentric is turned through about 110 on the shaft, so that the spring must balance a moment which is roughly of the form K sin i + K' sin (3 i + i f ), where i is the angle through which the lever arm (carrying the centrifugal weight) swings. For many years after its introduction, this governor was not fit for close regulation. It was made fit by the addition of auxiliary springs which oppose the main springs in the inner position of the governor and Fixed go out of action somewhere near the central position of the weights. Even with this ingenious expedient, the governor is sensitive to changes of fric- tion in eccentric and piston valves. In the detail calculation of springs of shaft governors many influences which, at first thought, seem too trivial to have much effect, must be considered. Among them is the in- fluence of the mass of the governor springs. Not only does the spring exert a centrifugal moment which must be taken care of in the balancing of the various moments, but frequently the centrifugal force distorts the spring so that both its force and its line of action are altered. This condi- tion is illustrated in Fig. 51. The system (2) (3) (4) (5) rotates rigidly about point (3). Roller (5) keeps the spring from bowing out. If this roller is removed, the center line of the spring assumes a parabolic shape as indicated in an exaggerated manner by the dotted line (1)(4\ The centrif- W ugal force tending to bow the spring out is s u 2 , where W is the weight of the spring. The meaning of s is clear from Fig. 51. Fixed FIG. 53 74 GOVERNORS AND THE GOVERNING OF PRIME MOVERS If S is the elastic force and I the length of the spring before, and S f and I' the corresponding values after bowing out, then S f V = ; but V is found from the relation that the center line / of the spring becomes an arc of a composite curve which is so shallow that it can be replaced by a parabola. Hence I' = 1(1 + i i 2 ) } where i follows (see Fig. 52) from the relation W su 2 1 = T ;-. From these relations both change of spring force Q * & and change of lever arm can be computed. In modern shaft governors for poppet valve engines with lay shafts it is customary to place the springs just as close to the shaft as possible (see Fig. 53). The centrifugal force of the spring in this case is so small as to be without noticeable effect. Centrifugal force of the spring also has a component in the direction of the spring. However, the influence is small and can be taken care of by adjustments described below. In some of the best shaft governors, leaf springs are used. If the spring is held as indicated in Fig. 54, the nature of the fastening is somewhat uncertain. It is common to figure the spring as if it were a cantilever, built in at point (2) and loaded at point (1). Actually it is held at points (2) and (3) and can deflect between these points. This feature should be taken care of in the calculation. The spring is a beam which is semi- constrained at (3), supported at point (2) and loaded at point (1). While an exact mathematical solution may be diffi- cult, careful judgment allows a very close approximation. The exact predetermination of all forces in a shaft governor places an enormous amount of work on the engineering depart- FIG. 54 SHAFT GOVERNORS 75 ment. For that reason it is often preferable not to be quite so particular in the calculations, but to allow for all kinds of adjustments; so that the man in the field can work out his own salvation. Although the foregoing explanations contain the principles upon which all adjustments should be based, the effects of the more important adjustments will be briefly considered. (1) Change of spring tension. A change of initial spring tension is equivalent to adding or subtracting a constant spring force throughout the range of governor motion (with the exception of the disturbance caused by the centrifugal force of helical springs). An increase of spring tension increases the speed, but decreases stability and static fluctuation; a decrease of spring tension decreases speed but increases static fluctuation and stability. (2) Change of spring leverage. If the movable end of a helical spring is shifted along the weight arm in such a way that its tension in mid-position of governor is not altered, the spring moment changes directly as the spring leverage. The change in spring moment necessitates a corresponding change of centrifugal moment, which means a change of equilibrium speed. A greater spring leverage works the spring through a wider range of deflection, which means a greater variation of spring moment between full-load position and no-load position, or briefly an increase of static fluctuation and of stability. It is possible to locate a slot in the weight arm in such a way that while the spring leverage is increased, its tension is decreased, with the result that the speed of the engine is not varied by the adjustment. In this case only the stability and static fluctuation are varied, the speed remaining constant. In practice a variation of spring tension is a good deal easier than variation of the spring leverage, because it calls simply for the adjustment of the tension screw, whereas it is necessary to slacken the spring almost all the way in order to shift it along the lever. Slight variations in speed are therefore mostly accomplished by variation of spring tension. Engineers know that practice bears out theory, because by increasing spring tension farther and farther they "strike a racing point," which necessitates an increase of spring leverage. 76 GOVERNORS AND THE GOVERNING OF PRIME MOVERS (3) Adjustment of fixed end of spring in a circular slot, concentric with point of attachment of movable end of spring in mid-position of governor (see Fig. 55). This has the effect of varying the lever arm of the spring for a given position of the governor and of varying the ratio of lever arms in the full-load and no-load posi- tions. This adjustment offers very good means for making the spring moment balance all other moments with a steady speed curve throughout the range of governor motion, unless the angle of swing be too great. (4) Adjustment of quantity of centrifugal mass. From equation (2) of this paragraph it follows that addition of centrifugal weight (increase of ra) reduces the speed, and, vice versa, that reduction of centrifugal mass means an increase of speed. From the same equation it follows that the stability and static fluctuation remain unchanged only on condi- tion that each and every centrifugal mass is changed in the same ratio (not only the so-called centrifugal weight). There exist additional adjustments which the man in the field can make. Among them are : shifting the mass center of one or more of the centrifugal weights; changing the number of active leaves or coils in a spring ; bringing into action an auxiliary spring for part of the travel ; and others. However, any one of these adjustments should not be made blindly, without full knowledge of the theoretical effect which they have. They should be tried only after thorough discussion with the designing engineer. Adjustments of oil gag pots and friction brakes are, of course, frequently made. For such ad- justments see the chapter on cyclical vibrations of governors. References to Bibliography at end of book: 36, 40, 50, 61, 58, 59, 74. CHAPTER VII NATURAL PERIOD OF VIBRATION OF A CENTRIFUGAL GOVERNOR IN paragraph 2 of Chapter III that force was discussed which returns a stable governor to its position of equilibrium, after the governor has been displaced from that position. From Fig. 17 it follows that the restoring force is proportional to the displacement, so that the frictionless governor makes sine harmonic vibrations about the position of equilibrium. In reality these vibrations are more or less damped, the damping depending upon the internal friction of the governor. From mechanics the fact is known that solid friction does not change the period of vibration. In general, the natural vibrations of a governor are of no importance. They become of importance if their period coin- cides, or nearly coincides, with that of the forces impressed upon the governor by the valve gear. Knowledge of the natural period is also necessary for determining the stability of regula- tion. In order to prepare for these interactions between governor and prime mover, a method will now be given for computing the period of natural vibrations. From mechanics the time of a complete vibration is known to be T = _ %/ vibrating mass restoring force per unit displacement In this equation both the vibrating mass and the restoring force must be referred to the same representative point. If, for instance, the restoring force is measured in the direction of centrifugal force at the mass center of the revolving weights, the motion of all governor masses must be referred to that direction and to that point. If, on the other hand, the mass motion is referred to the governor sleeve, the restoring force must also be referred to the sleeve. For the method of such reduction see below. 77 78 GOVERNORS AND THE GOVERNING OF PRIME MOVERS From paragraph 2 of Chapter III the restoring force per 2 C v unit displacement is known to be for a straight-line characteristic. If the characteristic is curved, p is taken from the tangent to the characteristic, as previously explained, at the position of equilibrium. The vibration takes place about the latter position. Substitution of the restoring force furnishes 2M *V ri) (i) pC In this equation m e is the equivalent mass. The method of finding the equivalent mass is illustrated by Fig. 56, in which two closely adjacent positions of a governor have been drawn. If for the present only the weights Q l and W are considered, the equivalent mass is W 9 r The square of the travel results from the double effect that first the greater travel requires greater accelerating force and that second this force acts with a longer lever arm. If all forces and mass motions of a governor are referred to radial direction, the equivalent mass m e for any system of Sra(ds) 2 masses is given by the equation m e = 7T\z > where m is any mass in the system and ds is the space through which it travels, while the centrifugal mass travels dr radially. In any governor the value of m e may be expressed by the angles, lengths of arms, etc., but it is usually easier to use the method indicated above, that is to say, take the travels directly from two positions of the governor. If a governor is connected to a valve gear, its period of vibration changes, because the mass of the valve gear parts has to swing with the governor and enters into the sum of all masses times travel squared. As an example, the time of vibration of the loaded Watt governor shown in Fig. 56 will now be computed. i Q = weight of | counterpoise. NATURAL PERIOD OF VIBRATION OF A GOVERNOR 79 10ADED WATT GOVERNOR (Calculation for ^Governor) Weight of one ball- ...... ----- W-127 pounds Weight Of 348 pounds in d,agram r ;. FIG. 56 80 GOVERNORS AND THE GOVERNING OF PRIME MOVERS For this governor, the equivalent mass reduced to radial travel (from p. 78, equation 2) equals 254 X 3.38 2 + 268 X 3.25 2 pounds sec 2 . 32.2 X 2.5 2 ~1T~ This expression gives a mean value, because finite paths Dr, Ds and Dh have been used instead of the corresponding dif- ferentials dr, ds and dh. From p. 31 the restoring force per unit displacement is /7 C* C* known to be z = - - , or z = tan k tan i (Fig. 16), where a r r k is the angle which the tangent to the characteristic curve makes with the base line, and i the angle which the straight line drawn from the same point of tangency to the origin makes with the base line. The tangents are obtained by forming the ratios of the drawing lengths of the two short sides in the right angle triangles and by multiplying each length by its scale. Thus we obtain: z = -^- -r-^r = 340 pounds per ft. of radial displacement. .OO 1..4O By substitution we obtain the period of vibration T n =2w\/= 6.28 i/H = 1.82 seconds. z 340 References to Bibliography at end of book: 14, 36, 37, 83. CHAPTER VIII EFFECTS OF OUTSIDE FORCES IMPRESSED UPON GOVERNORS 1. Resistibility. The power-controlling mechanisms which are adjusted by governors may be broadly divided into two classes, namely (1) Devices which offer passive or friction resistance only to the governor, (2) Devices which react upon the governor and tend to cyclically displace it from its position of equilibrium. The features of the governor which are valuable in dealing with the first class of mechanisms are strength, work capacity, and, for certain cases, tangential inertia, see Chapter II. The feature which is valuable in dealing with the second class of mechanisms has not received much attention from writers and has, therefore, no recognized name in any language. Since the feature in question is the ability to resist, I have adopted the term "resistibility." Dr. Proell uses the term " moment of resistance" for shaft governors. The necessity for this property in governors subjected to great reaction will easily be realized. Heavy intermittent forces from the valve gear throw a light governor back and forth, causing unequal and irregular power distribution, and, consequently, great speed fluctuations. And yet the light, but, in this case, useless governor may be very strong, because the strength, in the sense of paragraph 1 of Chapter II depends upon m r u 1 ; and r u 2 may be very great, offsetting the small- ness of m. It will presently be shown that resistibility is indeed closely related to the mass of a governor. Returning to the statement that light governors are thrown back and forth under the influence of vibratory forces, we are forced to admit that the actual motion of such a governor is not amenable to mathematical treatment, except through repeated semigraphical point-by-point integration, which is, 81 82 GOVERNORS AND THE GOVERNING OF PRIME MOVERS of course, too slow for any practical purpose. The difficulty is partly caused by the interaction of governor and prime mover. Any motion of the governor affects the speed of the prime mover, and variation of the speed moves the governor. This motion is superposed over the cyclical vibration impressed upon the governor by the valve gear. The latter motion is modified by the natural vibration of the governor (see Chapter VII) and by the resistance of solid and liquid friction. From these facts it is evident that an analytical solution of the problem of finding the motion of the governor is out of question. Fortunately, a complete solution of this problem is not needed for practical purposes. Since it is not permissible to have governor motions of such a magnitude that they disturb power distribution and vary the speed of the prime mover, governors must be made resistant enough to prevent excessive vibrations, and the influence of such speed variations may, in consequence, be omitted from the study of cyclical governor vibrations. The problem then is to make governors resistant enough to limit vibrations (caused by reacting forces) to a small fraction of the swing or travel of the governor parts. For the solution of this problem the knowledge of certain facts con- cerning vibrations is required. For that reason they will be briefly reviewed. If a mass is subjected to a cyclically fluctuating force, and is not affected by any other force, it performs a vibration which can easily be computed in every detail. It is called the im- pressed vibration. Mass furnishes the only opposition limiting the amplitude of the impressed vibration. The impressed vibration is modified by friction, both liquid and solid, and by the natural vibration. The latter may magnify or diminish the amplitude of the impressed vibration, depending upon the ratio of their frequencies. If that ratio is near one, and if there is very little friction, very large amplitudes result. As a matter of fact, governors with very small friction, when used in connection with reacting valve gears, have occasionally struck that coincidence of frequencies which in mechanics is called resonance. Under such conditions a frictionless governor is useless and friction must be added to make it useful, usually EFFECTS OF OUTSIDE FORCES 83 much to the dismay of the designer who tried to produce cor- rect regulation by the use of a frictionless governor. Fortu- nately, a relatively small amount of friction suffices to prevent undue magnification of the impressed amplitude. In paragraph 2 of Chapter IX proof is furnished that a certain amount of friction is needed in any governor to insure stability of regulation. And such an amount of friction is needed that resonance (magnification of impressed amplitude) cannot occur, if regulation is to be stable. For this reason resonance may also be dismissed from the discussion. There remains then only the amplitude of the impressed vibration, damped by solid and liquid friction. Friction reduces the amplitude of any vibration and thus increases resistibility. This brings up the question to what extent friction should be depended upon to furnish resistibility. No generally applicable answer can be given, for the following reasons: The amount of friction which is necessary in a governor to produce stability of regulation varies with the promptness of the governor and with the kinetic energy stored up in the rotating masses of the prime mover per horse power of capacity (see paragraph 2 of Chapter IX). A considerable increase of friction over the amount which is necessary for stability of regulation does harm, because it delays the governor, when the latter is adjusting the power- controlling mechanism after a change of load, and thereby causes undue and excessive speed fluctuation. This uncertainty of the permissible amount of frictional damping in governors makes it advisable to depend mainly upon mass for keeping down the amplitude of cyclical vibrations and to depend upon friction to a limited extent only. If a rule is wanted, the fol- lowing will serve as a rough guide. Use a governor with suf- ficient mass to reduce the amplitude of the impressed vibration to 1J times the permissible amplitude, and obtain further reduction of amplitude by solid or liquid friction. Just what amplitude is permissible will be considered in the next paragraph. Solid friction is very effective as a damping agent; however, several precautionary measures must be observed to make its use successful. In governors, such as shown in Figs. 2 and 4, FIG. 57 84 GOVERNORS AND THE GOVERNING OF PRIME MOVERS great centrifugal forces are transmitted through the pin joints. The diameters and lengths of these pins are usually made very small for the purpose of reducing friction, and the joints wear considerably, if the governor vibrates. It is, therefore, much better to make the pressure between the rubbing surfaces so small that wear is very slow. Furthermore, it is advisable to make the friction adjustable and to arrange the parts so that slight wear does not noticeably vary the fric- tion. Finally, it is desirable to keep oil away from the rubbing surfaces, so that a varying amount of oil may not disturb the proper fric- tional damping. Figure 57 shows an arrange- ment which embodies these features and which has given satisfaction. While resistibility is easily defined as ability to resist im- pressed forces, a mathematical definition is impossible, because both mass and friction share in the resistance. If friction is omitted from the discussion, resistibility becomes (in spindle governors) simply equivalent mass of governor at the sleeve or collar of the latter, and, in shaft governors, equivalent mo- ment of inertia of governor parts about the suspension point of the eccentric. For this reason, j"y & 9 makers of direct-acting governors J~~ should give the equivalent mass of each size of governor in their cata- logues. While it is evident that valve gears which impose great alter- nating or vibratory forces upon the governors require the latter to be quite heavy and massy, it is less evident how that mass should be distributed in the governors for best advantage. For the purpose of reducing amplitude ' of impressed vibration, the governor mass need not be so arranged that all of it pro- duces centrifugal force. For reduction of cyclical vibration, the mass is very effective, if arranged as shown in Fig. 58, where both m : and m 2 contribute to the resistibility, whereas FIG. 58 EFFECTS OF OUTSIDE FORCES 85 mi only produces centrifugal force. However, the expedient of placing a large part of the governor mass outside of the governor proper considerably reduces the promptness of the latter and causes great fluctuations of the speed, whenever the load changes suddenly (see Chapter IX). It must not be used where close regulation is essential. In the latter case it is much better either to use a larger-size governor, or else to arrange the additional mass in the governor in such a manner that it furnishes tangential inertia acting in the right direction (see paragraph 2 of Chapter II). In that paragraph proof was fur- nished that the regulating force due to tangential inertia is proportional to the change of load of the engine. Paragraph 3 of Chapter IX contains proof that tangential inertia increases the stability of regulation. As before stated, it increases resistibility without increasing centrifugal force. Since in a vibrating system a comparatively small centrifugal force suffices to adjust the position of equilibrium, tangential inertia furnishes the ideal vibration-resisting force. Of all types of governors, shaft governors are subjected to greatest fluctuating forces. The use of tangential inertia in shaft governors, which was introduced by American engineers shortly after the year 1890, is, in consequence, well justified and is correct engineering. References to Bibliography at end of book: 23, 28, 50. 2. Cyclical Vibrations of Governors. As explained in the preceding paragraph, the necessary resistibility of a governor is determined by the impressed force (resp'y moment) and by the permissible amplitude of the vibration of the govern- or. The latter is determined chiefly by the consideration that, for a constant load, the supply of energy must be the same, stroke after stroke of the engine. A governor vibrat- ing too much will strike the stops, if the load is either very light or very heavy. The result is irregular motion of governor and of engine. Besides, vibrations of large amplitude wear out the governor in short time, unless it was specially designed for such service. Both requirements, namely that of clearing the stops, and 86 GOVERNORS AND THE GOVERNING OF PRIME MOVERS that of avoiding wear, refer particularly to spindle governors. The latter are commonly so adjusted that their total available travel just varies the energy supply between the limits of no load and full load, which means that a vibrating governor strikes the stops near the extreme ends of its travel: In spindle governors a third feature enters, namely the visibility of the vibration. Engineers seem to have an instinctive feeling that a vibrating governor is wrong, and take steps to quiet it. Matters are very different with shaft governors. The vibrations are not visible and, for that reason, do not interfere with any preconceived notion of operating engineers. Experi- ence with vibrating shaft governors has taught designers to make the supporting pins of liberal proportions so that wear is slight. Furthermore, the valve gear is frequently of such nature that the governor stops can be placed considerably outside the limits of no power and of maximum power. In that case the governor can (and usually does) vibrate with a considerable amplitude without producing any harmful effects upon regulation. Evidently, relatively greater vibrations can be permitted in shaft governors than in spindle governors, unless the latter are designed for cyclical vibrations. It is also evident that a large number of circumstances influence the greatest permissible amplitude so that no universal figure can be given for that quantity. The following values may, however, be used as a guide for average conditions: For spindle governors, maximum permissible amplitude equals ^V of governor travel, so that maximum displacement equals yV of governor travel. For shaft governors, maximum permissible angular ampli- tude equals -jV of total angle, so that maximum angular dis- placement equals J of total angle. These are limiting values, and it will pay in practice to keep the amplitude of cyclical vibrations somewhat below these values. It should be noted that the tendency of some designers to keep any and all impressed forces away from governors for the purpose of entirely eliminating cyclical vibrations is funda- mentally wrong. In paragraph 4 of Chapter II, mention was EFFECTS OF OUTSIDE FORCES 87 made of the fact that vibrations reduce or eliminate friction and the time lag (detention) caused by it. Consequently, moderate cyclical vibrations of governors are beneficial. Their action in eliminating friction may be analyzed as follows: Let Q be the regulating force which tends to move the governor collar at a given instant. Let F be the resisting frictional force, and let F be greater than Q, so that the governor would not move, if it were not for the effect of the impressed cyclical vibration. Let f s be the amplitude of the latter. While the governor collar moves in the direction of the force Q, the latter acts into the space s and puts into the governor the work Qs in the shape of kinetic energy, so that the governor moves its center of vibration the distance Qs/F in the direction of Q. During the return travel against the force Q, the latter reduces the kinetic energy of the governor the amount Qs so that the center of vibration is again shifted the distance Qs/F in the direction of the force Q. With each complete vibration, the collar moves the distance 2 Qs/F, which means that any force, no matter how small, will affect a vibrating governor and will move it. Skillful and experienced designers make use of this fact. The problem of so selecting the resistibility of the governor that the amplitude is kept within the desired moderate limits consists of two separate problems, namely (1) Ascertaining the impressed force or moment as a func- tion of time, (2) Double integration of the resulting linear or angular acceleration. The first part, namely the determination of the impressed force or moment, is really a problem of engine or valve gear design. The great diversity of valve gears prohibits a detailed discussion, so that the latter must be limited to a few remarks concerning general guiding principles. Forces reacting upon the governor are caused by friction of valves and valve gear parts, by inertia of valves and valve gear parts, by unbalanced steam pressure and by unbalanced weights (in shaft governors). In the case of releasing gears the principal force is the friction between the catchblocks which, in turn, is determined by the force required to drive the valves. GOVERNORS AND THE GOVERNING OF PRIME MOVERS , the This latter force is found as below given for automatic valve gears. In the case of non-releasing valve gears all inertia forces are computed on the basis of the valves being moved by a non-vibrating governor. To this end the displacements of all moving valves and valve gear parts from their respective midpositions are plotted against time. The second derivatives of these curves furnish the accelerations which later, upon multiplication by the respective moving masses, furnish the inertia forces. Forces caused by friction of valves and by un- balanced steam pressure on valve Stems are COm- Curve of impressed , ., Moment (also of puted from Angular Acceleration.) well known of mechanics. In Fig. 59 the upper curve is an example of an im- pressed moment plotted against time (one revolu- tion of the engine). The curve should not be taken as repre- senting all pos- sible cases. On the contrary, curves of impressed moments are most varied in shape and depend very much upon the type of valve gear. From the moment curve the amplitude is found by a double integration. If the impressed moments are divided by the moment of inertia of the governor parts, 1 the moment curve 1 For the governor, Fig. 8, the moment of inertia is with sufficient accuracy mass (4) times If plus mass (5) times l plus moment of inertia of connecting link. For a governor of the type of Fig. 9 both impressed moment and moment of inertia must be referred to either point (4) or else to point (5). If (4) is made the point of reference, the moment of inertia is roughly J = m\ If \ Curve of Angular Velocity of Vibration. Curve of Angular Displacement. FIG. 59 EFFECTS OF OUTSIDE FORCES 89 becomes a curve of angular accelerations which, upon being integrated once furnishes angular velocity of vibration. The second integration furnishes angular displacement. Each integration can be carried out either by the addition of mean ordinates, or by the use of a planimeter. Or else the double integration may be accomplished in one operation by the drawing of an equilibrium polygon. It is evident that in a cyclical vibration the mean value of the angular velocity of vibration must be zero and that likewise the mean value of the impressed moment must vanish. This fact must be observed in the integration. In the foregoing the influence of friction has been neglected. Reasons for the advisability of neglecting it were given in paragraph 1 of the present chapter. For a fairly complete solution the whole calculation should be made for at least three positions or configurations of the governor. While the work is not difficult, it is certainly long and tedious so that a healthy guess is usually con- sidered the lesser evil. A great deal of trouble has been caused by cyclical vibrations due to unbalanced weights in shaft governors. Although this type of govern- or has not the importance to-day which it had at the end of the nineteenth cen- tury, a brief discussion of the underlying principles will even at this date be helpful to a large number of engineers. Figure 60 diagrammatically represents a shaft governor with an unbalanced weight such as is used for instance in the Rites governor. (1) is the center of the shaft. (2) is the sus- pension point of weight W whose mass center is (3). The sys- tem rotates about (1) with uniform angular velocity u. Spring (4) is so dimensioned that it balances the centrifugal force of W in any position. The motion of weight W can be found under the following simplifying assumptions : FIG. 60 90 GOVERNORS AND THE GOVERNING OF PRIME MOVERS (1) The angular displacement of the weight is so small that it can be neglected compared to the average value of u t which, of course, is 45. (2) Effects of friction and of resonance are neglected. Jus- tification for these simplifications is given in paragraph 1. The angular acceleration of the weight W (the moment of inertia 'of which about point (2) is J) is d 2 i du W L sin u t = a = dP dt By integrating l twice we obtain first Du = cos ut+K uJ and then W L Di = - 7- sin u t + K t + K', u- J where K and K' are constants the values of which depend upon the location of the zero line as shown in Fig. 59. Du is the change in angular velocity, or the difference of angular velocity between weight W and shaft (1), or, in other words, the rela- tive angular velocity of weight W. For a cyclical vibration the average value of Du must vanish, and to have that occur, K must equal zero. The same reasoning holds true for Di and K f , so that the angular displacement of vibration becomes W L W L Di = -y~r sin u t with angular amplitude of 2 , (1) u J U" J Equation (1) teaches that Di reaches a maximum (negatively), when sin u t becomes a maximum which occurs in point (5) of Fig. 60. In points (6) and (7) sin u t = so that there is no displacement. The effect of gravity in this simple case then consists in lifting the center of the (slightly distorted) circle, W L 2 which the weight describes, the distance 2 . If H is the u J radius of gyration of the weight W, the distance which the 1 For the benefit of those who are not versed in integration, it has been done graphically in the Appendix, Fig. 139. EFFECTS OF OUTSIDE FORCES 91 m g L 2 (j L 2 weight is raised becomes ; = ' The center of the u 2 m H 2 u 2 H 2 circle described by the eccentric which is linked to the weight W is likewise shifted. The direction of the displacement de- pends upon the kinematic connection. Early experimenters were surprised to find the raising of the path of the vibrating weight; but this fact appears quite simple and natural, if comparison is made with the ordinary pendulum, in which the FIG. 61 average location of the mass center is higher during vibration than it is when the pendulum is at rest. Other experimenters doubted the raising of the center of the path, because they had found a shifting to one side. Again the explanation is not difficult. From the theory of vibrations it is known that friction causes a phase lag which, in the case of resonance, reaches the value of 90. Evidently, the actual direction of the displacement varies with the friction so that 92 GOVERNORS AND THE GOVERNING OF PRIME MOVERS it is advisable not to count upon any given direction, but rather to keep the amplitude of vibration within safe limits as pre- viously given in this paragraph. For a given angular velocity u of the governor shaft the displacement may be kept small by making radius of gyra- tion H large compared to mass radius L . For small values of u, say for less than 150 revolutions per minute, the requirement of small amplitude means (with a single centrifugal weight, as in Fig. 60) so small a value of L that the centrifugal moment becomes insufficient for speed counting. The use of a single weight is, therefore, inadvisable for slow speeds, and double weights, partly or wholly balancing each other, must be used. FIG. 62 FIG. 63 An actual case from the author's practice will serve to illus- trate these theories. The Rites shaft governor shown in Fig. 61 was originally built with a single annular weight W\. The suspension point of the latter is at (2) ; its mass center is at (3) . The center of the eccentric swinging rigidly with weight W\ is at (8). The governor was noisy, making a hammering sound. EFFECTS OF OUTSIDE FORCES 93 Indicator cards taken from the engine had the shape given in Fig. 62. A pencil point fastened to the eccentric rod close to FIG. 64 the governor described the egg-shaped curves reproduced in Fig. 63. The outer curve is described with full load on the engine. The spring holds the governor against the stop most of the time so that the motion of the eccentric is almost symmetrical. The inner curve is described with a partial load on the engine. It is shifted both upward and sideways. To remedy matters, the weight T7 2 with suspension point (6) and mass center (7) was added, and was connected to Wi by a link. The two weights now almost balance each other against gravity. The results were very satisfactory. Figure 64 shows the curves described by the same point on the eccentric rod after the change, and Fig. 65 shows the correspond- ing indicator cards. The hammering of the weights in the 94 GOVERNORS AND THE GOVERNING OF PRIME MOVERS governor had entirely disappeared. Of course, a new and stronger spring (4) was installed, because the addition of the second weight Wi increased the centrifugal moment for the desired speed. Cyclical vibrations of governors are also caused by the interaction of governor and prime mover. Their period differs from that of the vibrations studied in the present paragraph. They are dealt with in Chapter IX. References to Bibliography at end of book: 28, 40, 60, 59, 63, 70. CHAPTER IX INTERACTION BETWEEN DIRECT-CONTROL GOVERNOR AND PRIME MOVER 1. Action of Governors Regulating Prime Movers. In paragraph 1 of Chapter I the statement was made that govern- ors are used to keep some one quantity practically con- stant while the output of a prime mover varies. The purpose of the following investigations is to find out to what extent the aim of keeping angular velocity practically constant has been realized. In Fig. 66 load or output has been plotted against time, and two changes of load have been indicated, one, (1)(2), sudden, and the other, (3) (4), gradual. In an ideal regulation the gover- nor would follow the change of load with- out time lag. In that ideal case the broken line (/)(*) (3) (4) would also represent governor position. In reality, governors have mass which requires time for its acceleration, so that a time lag exists between load and governor position. During the interval of lack of coincidence between load (power consumed) and power generated, the difference of work causes a rise or fall of speed which, in turn, affects the motion of the governor. In the language of mechanics, the governor and the prime mover form a system with two degrees of freedom. Translated into everyday language it means that the position of the governor determines the rate at which the speed of the prime mover varies, and that the speed of the prime mover determines 95 LOAD (Ideal Governor Position") - NO Load ( P moment for removal of - of the total load is 2 C h ~ ~ = p" C l s 2i u (compare equation (3) of paragraph 1, Chapter II), where lz is the lever arm of the centrifugal mass about its suspension point (see Fig. 78). Turning to the inertia mass, we find that removal of one half of the load produces an angular accelera- 1 M , tion of ^ (see paragraph 2 of Chapter II) , and a regulating 2i L moment equal to - ^ about the suspension point of the 2i L inertia weight. As before, I is the moment of inertia of the rotating parts of the prime mover, while J is the total moment of inertia of the inertia mass about its suspension point. To refer the regulating moment of the inertia mass to the pivot of the centrifugal mass, multiply it by (see Fig. 78) . We then have P "C1 3 = -J Zi 1 LI k-nn " IM JI * from which follows p = - p T r- 2i L is LI Finally, we obtain the equivalent static fluctuation \M Jl, If the rotative speed of the governor spindle differs from that of the engine or turbine shaft, M JI must be multiplied by a factor involving the speed ratio. Since the stability of regulation depends solely upon the inclination of line (20) (21), it is quite immaterial how p f is constituted, and we obtain the most interesting result that p 116 GOVERNORS AND THE GOVERNING OF PRIME MOVERS may be zero, or even negative, provided that p" is large enough to make p f positive. In other words, if a governor is built with a sufficient amount of tangential inertia, the static fluctua- tion may be zero (isochronous governor), or even negative (reversed speed curve), and yet the stability of regulation will be preserved. Many inertia governors can and this circumstance is well known to operating engineers be so adjusted that the engine runs at a higher speed with full load than it does at no load. Between the years 1892 and 1905 this fact was widely advertised, so much so that some engines appeared to consist principally of a wonderful governor and a few other, relatively unimportant parts. The total disappearance of these advertise- ments suggests the inference that regulation with a reversed speed curve is of no lasting commercial value. Reasons for the latter statement are apparent. For many purposes, particularly for parallel operation of alternating current generators, a uniform, positive static fluc- tuation is necessary. A positive speed drop furnishes the only means of distributing the load between the different generating units in the power plant. For this reason, prime movers for alternating current generators are seldom, if ever, equipped with inertia governors. The rapid growth of central station power plants generating alternating currents has deprived the inertia governor of its importance. The passing of the inertia governor from the central station should not be misconstrued into the belief that inertia governors are not suited to alternating current work; they are so suited, but they offer no advantages whatsoever except as shaft governors. For mathematical proof of this statement refer to equation (4) of paragraph 2 of present chapter. This equation, it will be remembered, gives the speed fluctuation of the limiting case. /yy\ Under the radical stands the expression - ~ which, in words, p (_/ equals the equivalent governor mass divided by static regulating force. To make the speed fluctuation small (that is to say: to W) make regulation good), we must make - very small. But 2i j) C INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 117 in the case of the inertia governor, equivalent governor mass includes not only centrifugal masses, but also the mass of the inertia weights, so that m e is increased, which is undesirable. If tangential inertia of centrifugal masses is utilized, instead of using separate centrifugal and tangential masses, the equiva- lent mass is increased just the same, because, for a given product of centrifugal force times radial path of this force, the product of total mass times total travel of mass must be increased over what it would be if the weights traveled radially, in which latter case they are subjected to centrifugal force only, but not to tangential inertia. On the other hand, the regulating force has also been increased by the addition of inertia force. Thus both numerator and denominator of what may be termed the promptness factor have been increased. Evidently, addition of tangential inertia is useful, where reduction of p to a very small value is demanded, because it keeps the denominator of the promptness factor from becoming almost zero. But where p must have a definite, positive value, say from .02 to .04, total equivalent mass the fraction - . ,. cannot be reduced by the ad- total regulating force dition of tangential inertia. Attention is again called to the fact that this fraction represents the "quality factor" of the governor in equations (4) and (4a) of paragraph 2. Nothing can be gained by application of tangential inertia to the spring- loaded, high-speed governors of our modern steam turbines, hydraulic turbines, and gas engines. The logical conclusion to be drawn from this reasoning is that tangential inertia should be limited to shaft governors which have to handle heavy valve gears. See also paragraph 1 of Chapter VIII. References to Bibliography at end of book: 2, 7, 32, 33, 36, 43, 65. 4. Solid Friction as a Damping Agent. Wischnegrad- sky's theorem that stable governing is impossible without the use of an oil gag pot was not taken seriously by engineers at the time of its publication, because of the great number of governors working satisfactorily and giving excellent regu- lation without the use of an oil brake. At the same time, 118 GOVERNORS AND THE GOVERNING OF PRIME MOVERS governors in those days (1877) were not free from friction. The thought, therefore, suggests itself that solid friction may, under certain conditions, satisfactorily take the place of liquid friction. Unfortunately, the exact calculations, as soon as solid fric- tion is substituted for liquid friction, become so complicated, even in the comparatively simple limiting case, that we must be satisfied with approximations. Three cases will be con- sidered, first, that of friction caused by compound centrifugal force; second, the case of constant friction; and third, the case of friction which is greater at rest than it is dur- ing motion. For the purpose of study- ing the action of compound centrifugal force (sometimes called Coriolis' force), sup- pose that in the governor of which Fig. 79 is a plan view, two weights, each of mass m, are rotating with an angular velocity u. The springs are so dimensioned that they balance the centrifugal forces of the weights in any position. Let the weights move out- ward with a linear velocity v. Then a force is exerted between each weight and its guiding wall by reason of two sepa- rate actions. First, the weight moves into a larger orbit so that its absolute velocity must be increased; and second, the absolute velocity of each weight changes its direction. In time dt the radius of the orbit of the weight is changed from r to r + v dt which means that the peripheral velocity of the weight is increased from u r to u (r + v dt) so that change of velocity dv u v dt. But force X time = mass X change of velocity, or in symbols Qidt = muvdt, and Qi = m u v for each weight. In the same time dt the velocity v has been turned through the angle u dt. Figure 80 shows that the change of velocity is u v dt from which follows the accelera- FIG. 79 INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 119 tion u v dt/dt = u v, and the force Q 2 = m u v for each weight. The total force exerted by each weight against its guiding sur- face equals the sum Qi + Qz = 2 m u v. The force which each moving weight exerts against the constraining side wall is directly proportional to the velocity v of the vibrating motion. If we now assume that the coefficient of friction is constant, then we find that the damping friction is Ancjle^ U-dt. also directly proportional to the F IG< go velocity of vibration of the govern- or. If, therefore, all other friction be eliminated and only that friction which is caused by compound centrifugal force be allowed to act, then and in that case the damping action is identical with that furnished by liquid friction. If, in addition, the coefficient of friction is of such magnitude that 2fmuv just equals the dynamic regulating force, all conditions which are necessary for the limiting case are fulfilled, and all the calculations of paragraph 2 of the present chapter apply without change. Friction caused by compound centrifugal force is present in every governor with practically no exception. If, for in- stance, the governor illustrated in Fig. 2 is considered, it will be seen that compound centrifugal force tends to bind the joints at the top of the spindle across the corners to such an extent that a very effective damping action is secured often, however, at the expense of rapid wear in case of repeated sudden changes of load. In practice, friction due to compound centrifugal force can never entirely take the place of the oil gag pot, because it is not adjustable. Since the value of the friction force isZfmuv, adjustment could be secured only by variation of the friction coefficient /; and such variation at the will of the operator is difficult, if not impossible. Nevertheless, the force under dis- cussion is helpful, and any complications in governor design for the purpose of eliminating this friction are not only super- fluous, but entirely out of place. Friction by compound cen- trifugal force is probably responsible for the correct action of many governors without oil pots and, at least partly so, for 120 GOVERNORS AND THE GOVERNING OF PRIME MOVERS the lack of confidence placed in Wischnegradsky's theories at the time of their publication. In the majority of governors and governing devices, many forces in addition to compound centrifugal force produce solid friction. For the purpose of studying the effect of the latter, let us assume for a while that the resistance caused by friction is constant over the whole travel of the governor. Referring back to Fig. 71, we remember that the limiting case consists of continued vibrations of constant amplitude about the position of equilibrium. We also remember that this condition is brought about by dissipating (by means of some damping force) the work done by the dynamic regula- ting force. Figure 81, which illust- rates the limiting case with constant (path of representative point) FIG. 81 solid friction as a The principal addi- damping agent, is very similar to Fig. 71. tion is the line (9) (15), which has been drawn in such a way that the work areas shaded horizontally and vertically are equal, which makes the rectangle (11) (9) (15) (13) equal to the half ellipse (11)(18)(13). At this point the approximation begins. With solid friction, curve (11) (18) (13) cannot be an exact ellipse, because the latter is based upon the equality (at any time) of dynamic regulating force and of damping force. However, the difference between the ellipse and the curve re- placing it will, except under special conditions explained below, be so small that we may well use the ellipse as an approxima- tion; the true curve most certainly must have a horizontal tangent at point (18). With the work of the dynamic regulating forces dissipated INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 121 by friction, the governor moves under the influence of the static regulating force only. The vibrations are very nearly sine harmonic, with constant amplitude. It is, therefore, apparent that solid friction, on general principles, can serve for damping governor vibrations. This statement must be qualified by the restricting explanationHhat there are two vital differences between liquid friction (including the equivalent action of friction by compound centrifugal force) and con- stant frictipnal resistance. The first difference is that, while with an oil pot stability can be enforced, even if the static fluctuation is infinitely small, there exists with solid friction a minimum, below which the static fluctuation cannot be re- duced, if stability is to be preserved. The second difference is that, with solid friction, there occur continuous speed fluc- tuations no matter whether the load varies or remains constant. For proof of the assertion that, with solid friction, stability requires a minimum value of the static fluctuation, again study Fig. 81. As before, (11) (18) (13) is the near-ellipse of dynamic regulating forces, (11) (9) = (13} (15} is the constant damping force of friction so that area (11} (9} (15} (13} equals area (11} (18} (13} which (from the well-known formula for the area of an ellipse) means that (11} (9} = (10} (18}. Let (11} (13} equal the whole travel of the . representative point of the governor. Then line (14) (15} indicates the regulating forces caused by the smallest allowable static fluctuation p. The motion of the governor could not be continuous, if p were smaller than here shown; for, if (11) (14) were smaller than (11) (9), there would not be enough regulating force at either end of the governor travel to start it. The two halves (11) (18) (13) and (18)(19)(ll) e oithe near-ellipse would be pulled apart up and down in the illustration. Area (11) (9} (15) (13} of the damping forces then would no longer equal the area of the dynamic regulating forces, and the speed would fluctuate between wider and wider limits. The mathematical determination of the smallest static fluctuation which is allowable with solid friction as a damping force must evidently start from the relation that the static 122 GOVERNORS AND THE GOVERNING OF PRIME MOVERS regulating force (11) (14) equals damping force (11) (9), which latter equals -- times regulating force (10) (18). But (10) (18) is caused by u e of equation (5) of paragraph 2. The cited equation is - - ^ y - For the conditions indi- u 2i A. o 1 s p A 1 cated in Fig. 81 (governor swinging from stop to stop) - - = - A. o Since the damping force (11) (9) only serves the purpose of relating speed difference (14) (11) to speed difference (10) (18), we conclude that and or finally u 2 4 2 2 T s 8rrt * s 16 (1) This, as before stated, is the smallest permissible static fluctua- tion, if solid friction of constant magnitude (referred to the ^ representative point) 1 is employed, and if the governor has con- stant stability over its whole travel. This latter condition is seldom fulfilled so that actually the smallest permissible value of p must exceed that found _ from equation (1). -Total t ravel of govern orJ ft i s interesting to (path of representative point; FIG. 82 note that p can be the smaller, the prompter the governor (T g small), and the greater the moment of inertia of the rotating masses of the prime mover compared to the capacity of the latter (T 8 long). INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 123 For proof of the second assertion, namely that constant friction as a damping agent results in never-ending vibrations, even at constant load, refer to Fig. 82, which is similar to the preceding illustrations, but is more complete. The outer curve (11) (18) (13) (19) is, as before, the near ellipse of the forces caused by the speed changes resulting from a sudden change of 50% of the total load. (9) (15) is the line of con- stant friction resistance. Now let the engine or turbine operate at a little above half of maximum load (governor position (21)), and let the load be suddenly reduced to exactly one half of maximum load. Then the governor will, in the limiting case, vibrate in the range (21) (22). However, the static regulating force now is small compared to the resisting friction, and, before motion of the governor can begin, the speed must rise, until a dynamic regulating force (21) (23) has been produced which is determined by the equation (23) (2Jf) = (21) (2 5). The relation between governor motion and speed of prime mover is given by the near ellipse (23) (26) (27). At this last point the governor is detained by friction; the speed must drop through a range corresponding to the distance (27) (28) before the return stroke of the governor can take place. The motion is intermittent, discontinuous. Now let the change of load become smaller and smaller. Then the curve of motion will finally shrink into the line (29) (30), which means that even at constant load there will be a continued speed fluctuation of sufficient magnitude to overcome frictional resistance. Whenever the just described conditions are realized in practice, the speed fluctuation is even greater than is indicated by the distance (29) (30), because the latter equals twice the minimum friction necessary for stability with endless vibra- tions for 50 % change of load, and because in practice a some- what greater friction will be provided for safety. From the foregoing explanations it is evident that solid constant friction as a damping agent for governors is rather undesirable under the assumed conditions. However, it will do, after a fashion. On the other hand, friction which is greater between surfaces-at-rest than it is between surf aces- in-relative- motion will not do at all, as may be seen from Fig. 83. If the 124 GOVERNORS AND THE GOVERNING OF PRIME MOVERS governor is suddenly displaced the distance (2) (3) from posi- tion of equilibrium and is then released, friction of rest holds it. As before, the speed must rise to a value indicated by point (4)) determined by the relation (4) () = C0(#)> before motion can begin. As soon as motion is under way, the dynamic regulating force grows see curve (4) (6) while the damp- ing force of friction drops rapidly. The work of the dynamic regulating force cannot be dissipated by the damping force of friction, and the vibrations increase in amplitude, until one of the limiting stops is struck. If solid friction really acted as indicated by Figs. 82 and 83, Wischnegradsky's theorem of the necessity of an oil gag pot would not have been doubted at the time of its publication. But as a matter of fact it is extremely difficult to make solid friction behave in the above assumed man- ner. It can be done in a governor regulat- ing a very well-bal- anced steam or hydraulic turbine, said governor being driven (path of representative poll FIG. 83 by well-hobbed gears running in oil, and said governor being entirely free from impressed vibrations. But these ideal condi- tions did not exist and could not be produced in 1877. Steam and gas engines with considerable cyclical speed fluctuation were the principal prime movers. Governors were driven by belts or by inaccurately cut gears; and most governors were sub- jected to impressed vibratory forces. We shall presently see that under such conditions solid friction is indeed quite fre- quently equivalent to an oil gag pot. For the sake of simplicity consider instead of a governor a particle of mass m which is moved back and forth by a force Q reversing its direction every t seconds. Let the motion be INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 125 resisted by a frictional force F which is quite small compared to Q. Then total change of velocity between opposite direc- Qt tions of motion equals ; maximum velocity in each direc- 7YI tion equals - - - and average velocity in each direction equals 2i Tfl . Hence, the total displacement of the vibrating particle 1 Qt 2 equals - - . Now let a small, constant force Qi which is smaller than the friction force F, act upon the particle ra in the direction of its vibratory motion. Then the center of IQ? QiX~ 4 m vibration is shifted the distance - every half vibration ; r for proof, see paragraph 2 of Chapter VIII. The particle, therefore, moves forward with an average velocity v = which may be written v = K Qi, where K is a proportionality factor. The " average" forward velocity of the jerky motion is proportional to the moving force, from which we conclude that the equivalent resistance is also proportional to the velocity, for, if it were not, the velocity could not remain con- stant for a given Qi, but would change. Under the described conditions solid friction is absolutely equivalent to an oil gag pot. While the conditions are some- what idealized, the action of the forces impressed upon the governor by many valve gears is sufficiently close to the ideal case to make the latter dependable for practical conclusions. The diagram of one complete vibration will, in the language of the other illustrations of this paragraph, look somewhat like Fig. 84, the number of zigzags depending upon the relative frequencies of impressed vibration and natural vibration. Since it is somewhat difficult to visualize the diagram 84, a tachometer record taken from a steam engine, and showing the same action, is reprinted in Fig. 85. The upper curve represents the motion of the governor, and the lower curve illustrates the speed flue- 126 GOVERNORS AND THE GOVERNING OF PRIME MOVERS tuation of prime mover and governor. The governor was on purpose subjected to greater impressed forces than will ever occur in practice. These forces were not the same at all loads, which ac- counts for the dif- ference in the gov- ernor motion before and after the change of load. Governor travel *- The mathema- tically treated ideal case does not hold, if the regulating force Qi is greater than the resisting friction F, because the average velocity then grows continually, and the equivalence of solid friction and liquid friction is lost. To CDO ee < FIG. 84 Speed of Engine y^^s^ FIG. 85 again make F exceed Qi in value, the former may be increased, but there are limits, because the impressed force Q must stay great compared to the friction force F, to preserve proportion- INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 127 ality. Fortunately, friction due to compound centrifugal force steps in helpfully just about at the place where solid friction of the vibrating system leaves off. In addition to the case of a vibrating governor, there is another set of circumstances under which solid friction acts very much like liquid friction, viz. the case of a governor in which solid friction of considerable magnitude is alternately applied and removed, for instance by jar or vibration. To treat this case mathematically assume that the governor is absolutely free from friction during t seconds and is held by quite great friction during the next t seconds, and so on. During each free period the governor which we will replace by a particle of mass m reaches a final velocity , where Qi, as before, is the regulating force. This velocity is then almost instantly dissipated, and the governor remains at rest during the next t seconds. The average velocity is, therefore, one quarter of the above value, or average v = - . Again, the average ve- 4 in locity is proportional to the regulating force so that the action of heavy friction applied at intervals and relieved at in- tervals closely re- sembles the action of an oil brake. The regulation dia- gram of the limiting case will look some- what like Fig. 86, the number of steps depending upon the ratio of the frequencies of friction releases and of natural vibra- tion of governor. While the just explained method of damping is effective, it is not as efficient as an oil brake, because it produces greater speed fluctuations. It very much increases the time of natural vibration of the governor, because the latter has to start from Ol coo > 1 Governor iravel - FIG. 86 128 GOVERNORS AND THE GOVERNING OF PRIME MOVERS rest after each stoppage. However, satisfactory regulation T 'g can be obtained in spite of this drawback, if the ratio -- is J- s made quite small. It is now evident that the practical engineers who objected to Wischnegradsky's theorem on the ground that so many governors worked well without gag pots were right in their practical observations. But they did not realize, first, that the governors to which they pointed as examples of correct regulation without damping by liquid friction were damped by other agencies, particularly by solid friction and, second, that solid friction works well only on condition that its action is made equivalent to that of an oil gag pot. We may add that, the more uniform the angular velocity of the prime mover, the freer the latter from vibration, the smoother the drive of the governor, and the freer the latter from impressed vibrations on the one hand, the smaller is, on the other hand, the value of solid friction as a damping agent, and the more must we approach a frictionless governor with liquid fric- tion for damping. And such is indeed the trend of evolution in the governing of modern steam turbines and hydraulic turbines. References to Bibliography at end of book: 28, 36, 65. 5. Greatest Speed Fluctuation with Direct-Control Gov- erning. In the sales specifications of engines and turbines the following clause is frequently used with regard to regula- tion, " If A per cent of the rated load are suddenly removed or put on, the speed will not vary more than B per cent either way from mean speed." No standard values for A and B have ever been adopted by engine and turbine builders, but the common use of this clause is proof that there exists a definite practical demand for computing the speed fluctuation of at least the first wave after a disturbance. Unfortunately the analytical calculation is extremely unsatisfactory from a prac- tical standpoint. To get an analytical expression, many as- sumptions must be made, and even then the calculation is far from simple; so far from it, in fact, that it deters all engi- INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 129 neers with the possible exception of a few governor specialists. In spite of this condition the elements of the analytical calcu- lation will be given in the present chapter, because they afford a very good insight into the mechanism of regulation. The assumptions are the same as those made in the pre- ceding paragraphs of this chapter, namely that the action of the governor is continuous, that there is no time lag due to stored-up fluid energy beyond control of the governor, and that all resisting friction is equivalent in its action to an oil gag pot. Figure 87 shows the relations between the factors entering Ao= maximum governor travel / /U Q old ZZ~; /Maximum \ V torque / FIG. 87 into the equations. Let both prime mover and governor work in equilibrium with initial or "old" torque M , and let the resisting or counter torque suddenly be reduced to a "new" value M n . Then the torque (M M n ) is unbalanced, and the speed rises as indicated by curve (1)(2). Under the influence of unbalanced centrifugal force and of tangential inertia the governor moves, as indicated by curve (1)(4)> The lag of the governor motion (1)(4) behind the speed change (!)() is, of course, due to its mass. When the governor has reached position (4), the unbalanced torque is M = - - M m . The ordinate x is measured from the 130 GOVERNORS AND THE GOVERNING OF PRIME MOVERS new position as a basis, because, as will be seen later, the gov- ernor has a tendency to perform vibrations about the new position. The minus sign is used, because in the chosen posi- tion M is positive, and x is negative, so that the negative sign is required to make M positive. From the fundamental rela- tion that rate of change of angular velocity equals unbalanced du x M m torque divided by moment of inertia, we have = p- dt A. o JL To make this equation applicable to all cases, we introduce the x following notations: '- = X = relative change of governor position, and = U = relative speed change; then du = d(u u n ) = u n - = u n dU, from which we derive Un u n = X ^- ; but as formerly (in paragraph 2) we have the at 1 u I starting time T s = -rr-. Throughout this calculation we neglect M. m the (comparatively) very small difference between u and u n , and finally obtain the equation for the " rate of change of speed deviation " in the following simplified form: m dt 2V On the governor act (1) unbalanced centrifugal force, (2) tangential inertia, (3) damping resistances. Turning first to the centrifugal forces, we note that in the position under consideration the speed (2) (4) is unbalanced. In the illustra- tion, the scale for velocities has been so related to the scale of governor positions that the curve showing the latter also repre- sents the curve of equilibrium speeds of the governor. At point (5) the governor has reached its correct position, but the actual speed having reached point (6>), there is still a force urging the governor on. Going back to position (4), we recognize that the distance (2) (3) represents the speed dif- ference producing the dynamic regulating force (due to "wrong INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 131 m ( speed ") and that (3) (4) represents the speed difference pro- ducing the static regulating force (due to " wrong position"). x The speed difference driving the governor is thus (uu n )pu n - The second term is positive, because x itself is negative. The force driving the governor then is 2C ( - p \ U n A > if we refer the motion to radial travel of the governor. The motion may be referred to the sleeve, in which case C is replaced by the strength P, or it may be referred to angular swing of weight arm, in which case C is replaced by spring moment of governor. It is evident that the speed relation be- tween prime mover shaft and gov- ernor spindle drops out of the equa- tion, because we deal only with relative speed changes. For the sake of completeness let us consider a combined centrifugal and inertia governor of any one of the types illustrated in paragraph 2 of Chapter II. The diagrammatic drawing of such a governor (Fig. 88) will assist in deriving the equations. From paragraph 2 of Chapter II it is known that the du inertia regulating moment is (ra r L+J) . The governor shown in Fig. 88 is of a simple type, in which mrL = 0, so that the du du regulating moment is reduced to J = - m Li 2 . Note that the at at du moment for positive produces a positive change of #, that at is to say it moves the centrifugal weight away from the axis of rotation. If all equations are referred to radial motion, the du Li 2 force exerted by tangential inertia is ~rr'nii~r~ at point (1), and dt Jui u n du LI Z/2 mi ~at the mass center of the centrifugal weight. U n dt L Z FIG. 88 132 GOVERNORS AND THE GOVERNING OF PRIME MOVERS This expression for the regulating force due to tangential inertia dU LiL 2 may be written u n m* at L/3 The force exerted by an oil gag pot or by equivalent solid friction appears in the form K or K A . It has the negative sign because it opposes increase of x. Adding up we obtain the sum of all unbalanced forces acting upon the governor, namely This sum of forces equals equivalent mass (m e ) 1 times accelera- tion so that we obtain d*x LiL 2 dU dX m e -:r^ = 2C(U-Xp) + u n mi -- -K A - di L/3 at at Further developments will show that the character of the regulation depends upon the relation between certain time elements. With two of these we are familiar, namely the starting time T S) and the governor traversing time T g . To introduce the latter, divide the equation by 2 C ; then the left- m e A ~ 2m ( Chapter IV) so that the left-hand member can be written *d?X ', UnmtLnL* Similar time values can be introduced for hand member becomes - f^ "T^T ', but A = ---T? (see m at 2 2 C and for * ; studying the latter expression, we realize that C is that velocity which an unbalanced force C would finally K KA produce when opposed by the oil-brake. And is the time which is required to traverse the whole travel A of the governor 1 m e in this case equals m c + n INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 133 c with said velocity This time may be called the brake- A resistance traversing time. It will be denoted by T b , so that 7 = - T b . Similar reasoning can be applied to the expres- 2 C 2 UnWliLiLz sion ^-777 -- u n LI is the linear velocity of the inertia mass m\. 2 CL 3 A centrifugal force C would exert the force ~ on this mass, so L/2 CT Ij that - is a linear acceleration. C~ T t = u n LI where 7\ might be called the starting time of the inertia mass. Hence W n 77l;Z/iZ/2 Ti = With these notations the equation of motion 2 CL/3 2 of the governor masses is IT A 2 &X T j and , dU [/ _ _ The equations for 7 and X are identical. At first thought this may appear strange, because we know from previous ex- . planation that the curve of govern- or motion and the curve of speed of prime mover do not coincide. However, this seeming discrep- ancy will disappear shortly. Equations of the type of (5) and (6) are known in mathe- matics as linear differential equa- tions of the third order with right- hand member equal to zero. Their integral equations are well known. Let equation (5) be written d*X .d z X dX -+#! + JK 2 - + K 3 X = ; dts dt 2 dt then the character of the motion represented by the equation depends entirely upon the relation between the three coef- ficients KI, K 2 , and K^ all of which are built up of time ele- ments, with the exception of p. Correct governing is a game of getting there first with the lowest terminal velocity. The theory of vibrations teaches that, if the product KI K 2 is greater than K 3 , the governor comes to rest after a change of load. The motion is then represented either by Fig. 89 or Fig. 90. If KI K 2 = K z then the motion is kept up indefinitely with con- stant amplitude (limiting case, see Fig. 91). If KI K 2 is less than K 3 , the amplitude increases continually. INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 135 The case of Fig. 90 is the one most usually occurring in practice. It is represented by the equation X (or U) = K* e^ + (j 6 sin z 3 1 +K 6 cos z 3 1) e* (7) where e is the basis of natural logarithms, (2.71828), and where 21, 2 + 23 V - 1, 2 3 V - 1 are the three roots of the equa- U- Relafive Speed Change X'Governor Displacement Time FIG. 90 tion z 3 + K l z 1 +K 2 z + K 3 = 0. The coefficients K 4 , K 6 and K 6 are to be found from the state of motion at the instant when the change of load occurred. They differ in the solution for X, from those obtained in the solution for C7, as will easily be under- U- Relative Speed Change -Governor Displacement Time FIG. 91 stood from a study of Fig. 91. At the beginning of a disturb- ance the velocity curve starts with a finite angle, whereas the governor motion curve must start tangent to the horizontal axis. This explains how the two curves of U (velocity) and X 136 GOVERNORS AND THE GOVERNING OF PRIME MOVERS (governor displacement) can be quite different in spite of having the same differential equations. The coincidence of the latter simply expresses that both vibrations are made stable or un- stable by the same combination of time elements, that their period of vibration is identical, that they have the same rate of damping, etc. Although the proof cannot be given here, it may be mentioned that (1) the period (or length of each wave) is always the same, independent of the load change, (2) that both the speed fluctuation and the governor displacement (referred for instance to the first wave) are directly propor- tional to the load change. To find this speed variation of the first wave, the derivative of equation (7) for U must be equated to zero, that is to say, we must form = ; we must then solve the resulting equa- dt tion for t, which can be done by trial only, because t appears both as an exponent and as a factor of an angle. The resulting value of t is substituted in equation (7), and U maximum is found. Creative practice pays no attention to these equations, because they are too complicated. It takes much longer to master and apply them than it takes to build a governor and try it. Besides, the practically important features can be ascer- tained much more easily from the limiting case. In spite of this condition, no apology is offered for the introduction of this brief sketch of the theory, because no one can ever appre- ciate the great usefulness of the simple equations of the limit- ing case, unless he has wrestled for days, or even for weeks, with the solution of the complete equation. In the advanced volume a few examples of the complete calculation will be given for the benefit of those who wish to make the governing of prime movers a life study. In estimating the probable maximum fluctuation from the limiting case, we must take care of the assumptions upon which it is based. Referring to Fig. 91, note that the vibra- tion is not sine harmonic at the outset, but soon becomes so. Consequently, the speed changes (1)(2), (3) (4), (5) (6), etc., are not equal to one another. Definitely known from theory INTERACTION BETWEEN GOVERNOR AND PRIME MOVER 137 ^| are the speed change (!)() = -~p, and the amplitude (7) (8) A- o of the sine harmonic vibration which finally establishes itself, i A T /T~ and which equals - -~ \ But what we are desirous of L A. o 1 3 p knowing is the fluctuation (1) (3) of Fig. 90. As in Fig. 91, (1) (2) A equals p in Fig. 90 also; but (#)(#) is the trouble maker. A o A i A T rr~ We may replace (1) (3) by - - p + - K - %/ where K is a ^LO 2 ^LO i s v p stop-gap coefficient, the lowest value of which is approximately 1.2, and the average value of which is 1.6. K may reach 2 or 3, if much energy is stored up beyond control of the governor, as for instance in compound or triple-expansion engines, or in four-cycle gas engines. In the latter case it is better to use the average coefficient 1.6 and to add separately the speed change caused by the stored-up energy. Then we have du A A T g 7l E - = (approximately) U = -j-p + .8 y +77^ (8) U ->- o ilo-LapJ-U where E is the available stored-up energy, I the moment of inertia of the rotating parts, and u the average angular velocity. In a tandem double-acting four-cycle gas engine, E, for a sudden change from full load to no load equals twice the maxi- mum work done by each cylinder per power stroke, because one cylinder carries a compressed rich charge and another cylinder has taken in a full charge of rich mixture. Both of these charges will do maximum work, no matter what the governor does. If there is a large amount of explosive mixture stored between the governor valve and the cylinder, matters are worse. Since the steam turbine is to-day the principal prime mover, a brief sketch of a method of computing the speed rise due to energy beyond control of the governor will be appropriate. In a steam turbine both the steam in the steam chest, W 8 (pounds), and the steam in the turbine, W t , will do work after the closing of the governor valve. This work is passed on to the turbine with an efficiency e. Let EI be the Rankine cycle work of a pound 138 GOVERNORS AND THE GOVERNING OF PRIME MOVERS of steam between inital and exhaust conditions (which work can be found from a total heat-entropy chart), then the work done by the steam not under governor control equals E = (W 8 + i W t )e E,. Not all of W a does work, because a small amount remains at the end of the expansion; however, it can be neglected. Only one half of W t is put into the equation, because the steam in the turbine has only about half as much available energy per pound as that in the chest, e is A for very small turbines, .65 for the largest turbines. , Du The relative speed rise is - - = U = - ^ u I u The numerical addition of the three items in speed rise, namely static speed rise, dynamic speed rise under control of the governor, and dynamic speed rise beyond control of the governor is an approximation only. The true calculation is hopeless, as before mentioned. If, for any reason, the exact knowledge of the greatest speed fluctuation should be desired, a point by point method may be used. (See 83, Ruelf, Der Reguliervorgang bei Dampfmaschinen, and 34, Koob, Das Regulierproblem in vorwiegend graphischer Behandlung.) References to Bibliography at end of book: 15, 18, 32, 34, 36, 57, 65, 73, 83. CHAPTER X DISCARDED TYPES OF SPEED GOVERNORS THE speed governors which were discussed and the theory of which was developed up to this point are based on " Watt's principle," which means that the centrifugal force of revolving masses is opposed by a centripetal force, and that any differ- ence between the two constitutes the motive force of the governor. The inertia governors treated in paragraphs 2 of Chapter II and 3 of Chapter IX are seemingly an exception to this statement; in reality they are not, because they are useful only on condition that tangential inertia is coupled with centrifugal force. At first thought, Watt's principle seems to be very unsatis- factory, because a change of speed (which the governor is intended to prevent) must occur before the governor can act. It is, therefore, quite natural that inventors of many countries should have worked hard to design governors on the basis of apparently more promising principles. However, the governors based on Watt's principle have survived, while governors based on other principles have disappeared. Nevertheless, knowledge of the discarded principles is valuable, particularly if it is coupled with knowledge of the reasons why they failed to meet the requirements of practice. For only by such knowledge can people with inventive minds be prevented from reinventing, with the courage of ignorance, mechanisms which were relegated to the scrap heap long ago. Tangential Inertia The principle of using tangential inertia for governing was investigated in paragraph 2 of Chapter II. It suffices here to refer to that paragraph. The general idea of using tangential inertia was first published by Werner and William Siemens in 139 140 GOVERNORS AND THE GOVERNING OF PRIME MOVERS 1845. Hence it is known as " Siemens' principle." It is c.learly illustrated by Fig. 92. The inertia mass (1) tends to maintain a constant speed, no matter what the latter may be. To keep Power Control Mechanism pinion rotates continually FIG. 92 wheels (1) and (2) from losing speed and running down, shaft (4) must drive shaft (6) by means of sufficient friction. Any acceleration or retardation of shaft (4) causes the supporting load Power conio! nism .-4 4J f W *-4 Spring Prime Mover FIG. 93 axle of planetary gear (5) to rotate about point (3), which motion is utilized to adjust the energy control mechanism. The reasons why Siemens' principle, by itself, cannot be used for governing were explained in paragraph 2 of Chapter II. DISCARDED TYPES OF SPEED GOVERNORS 141 Dynamometric (or Load-) Governors Poncelet, in 1829, proposed a governing principle which is illustrated by Fig. 93. The torque from prime mover to load passes through a flexible spring coupling. The twist causes gears (1) and (5) to be displaced with regard to each other. Hence gears (2) and (4) (in mesh with (1) and (5)) are also displaced, which causes gear (2) to travel along screw (3). Axial displacement of (2) shifts power control mechanism. Power Control Mechanism Seeming advantage: The action is practically instantaneous as soon as load changes. Drawback : The feature which makes this governor impossible is its failure to adjust the power-control mechanism, if intensity of available energy varies (for instance steam pressure, back pressure, head of water in hydraulic tur- bines, etc.). It has other drawbacks, namely vibra- tions due to flexible couplings, and never-ending vibrations in case the load changes suddenly (no damping). It does not maintain a constant speed. FIG. 94 Chronometric Governors If the flywheel of Fig. 92 is replaced by a time-piece, which maintains a uniform speed of bevel gear (#), then the governor is called a " chronometric governor." This type was invented by Sir C. W. Siemens in 1843. One of the many possible forms is shown in Fig. 94. Bevel gear and spindle (3) are driven by the prime mover. Planetary gear (2) drives the time-piece (6) through bevel gear (1), and operates the power-control mecha- nism through segment (5) and gear (4). The latter pair does 142 GOVERNORS AND THE GOVERNING OF PRIME MOVERS not revolve continually, but moves only to adjust the position of the power-control mechanism. In the mechanism shown in the illustration, the uniform speed of (6') and (1) is obtained by the friction brake (7), which, if desired, may be replaced by an oil brake. If the weights (6) go faster than desired, the friction force is greater, so that their speed is reduced. If they go slower than desired, the friction is reduced, so that the driving mecha- nism can speed them up. Compared to the purely centrifugal gov- ernor, the chronometric governor possesses several disadvantages. First, the friction brake consumes power, or, if an independent source of energy is employed for fime- the time-piece, its energy is wasted, in addition to FIG. 95 the disadvantage of requiring attention. Second, a "jack in the box" gear is an unwelcome com- plication. Third, sudden changes of load produce never-ending speed fluctuations. Let part of the load suddenly be removed at points (1) and (11), Fig. 95; then gear wheel (3) of Fig. 94 will go ahead of wheel (1). At points (2) and (12) of Fig. 95, du torque equilibrium exists, and = 0; but wheel (3), Fig. 94, at still runs ahead (see excess speed (2) (3), Fig. 95), so that the governor travels too far, namely to point (13) instead of point (12). The governor oscillates about its new position, and the speed of the prime mover oscillates about the " synchronous speed." The three disadvantages taken together have made it impossible for the chronometric governor to compete with the high-grade centrifugal governor. DISCARDED TYPES OF SPEED GOVERNORS 143 Cataract Governors (Also known as pump governors, hydraulic governors, atmospheric governors or pneumatic governors) The principle of this method of regulation is clear from Fig. 96. A pump (1) driven by the prime mover delivers fluid to a receptacle (3), from which it returns to the pump through an orifice (#), which is usually made adjustable. The depth of the liquid in the upper receptacle (respectively the pressure, if the fluid be a gas), regulates the supply of energy to the Power Control Mechanism FIG. 96 prime mover. In the illustration, the float (4) operates the power-control mechanism. The cataract principle antedates the centrifugal governor of James Watt. It was used on the Cornish pumping engines. It can be used to-day to govern the number of strokes per minute of direct-acting pumps and compressors (which, as is well known, have neither crank nor flywheel). For turbines, and for engines with flywheels, the cataract principle has been very little used. The principal drawback is variation of viscosity (and of density) of the working fluid in the governor. If the type in question is to be used for engines or turbines, centrifugal pumps or rotary pumps are preferable to reciprocating pumps. 144 GOVERNORS AND THE GOVERNING OF PRIME MOVERS With proper design, the principle is capable of development, for special purposes of regulation, particularly in connection with direct-acting pumps. Vane Governors The resistance which a fluid offers to revolving vanes has been used as a principle upon which to design governors for prime movers. Figure 97 illustrates this type. Vanes (1) in box (2) are driven by the prime mover. The liquid which partly fills the box is taken along by the vanes with a fraction of their angular velocity. The moving liquid exerts a drag on the Power Control Mechanism Driven prime mover FIG. 97 Fixed casing (which is provided with many projections for that pur- pose). The drag is resisted either by a spring or by a weight- Governors designed on this principle bear different names (derived from the names of their inventors) in different coun- tries. They always involve a loss of energy. Besides, the dissi- pated energy is converted into heat, which changes the density and viscosity of the oil (or other fluid) and with them the equilibrium speed of the governor. The principle has been abandoned and is not likely to be introduced again. Centrifugal Compensator Recognition of the reasons why an isochronous governor fails to produce isochronous governing led Knowles in 1884 to DISCARDED TYPES OF SPEED GOVERNORS 145 the invention of a practical solution, the principle of which is illustrated in Fig. 98. Knowles' principle has been discarded, not because it is impracticable, but because the same end can be attained by much simpler means (see paragraph 2 of Chapter IX). In the illustration, (1) is a centrifugal governor with sufficient static fluctuation to insure stability of regulation ; (6) is a similar governor, the regulating travel of which is kept within narrow limits. (1) is the main governor, (6) is the auxiliary governor. From previous explanations it is known that different speeds belong to the posi- tions (2) (3) (4) of gover- nor (1 ) , and would likewise belong to positions (10) (9) (8) of the power- control mechanism, if the length of the link (3) (9) remained constant. If, however, after a change of load, governor (1), and with it the upper end of link (3) (9), is returned to one and the same position, then one and the same speed corresponds to all positions of the power-control mechanism. This desirable object is gained by making link (3) (9) adjustable in length and by automatically returning governor (1) to the same position. The auxiliary governor (6), by means of friction wheel (5), pulleys (7), and turnbuckle (11), slowly adjusts the length of link (3) (9). Equili- brium obtains if the speed of the prime mover keeps the auxiliary governor free from the friction wheel. The equilibrium speed of governor (6*) should coincide with the mid-position speed of governor (1). Governor (6) takes the place of the compensating oil pots explained in Chapters IX and XIII ; so that it may be called a centrifugal compensator. While Knowles' principle has been discarded in direct-con- Power Control Mechanism 146 GOVERNORS AND THE GOVERNING OF PRIME MOVERS trol governing, a modified form, with one governor only, is extensively used in relay governing (see Fig. 123). Electrical Governors At the present time power is transmitted long distances almost exclusively by electrical means, so that prime movers for power generation and transmission (as distinguished from prime movers for pumping, blowing, etc.) are used more and more for driving electric generators to the gradual exclusion of everything else. Electrical energy is quick acting and is free from mechanical friction. It lends itself beautifully to the design of regulators and governors. In consequence, many attempts have been made to use electricity directly for governing, and to circumvent the Ffxed comparatively slow mechanical governor. The principle of an electrical type of governor which was in use for more than a decade will be under- stood from Fig. 99. Core (4) and coil (5) form a solenoid, the attrac- tion of which is counteracted by a spring (1). If the current in (5) is increased, core (4) is pulled down, and the throttle valve (2) moves toward closed position. In practice, the travel of the core must be kept very small, to avoid hunting. The use of an oil gag pot (8) reduces hunting, but does not eliminate it. To keep the travel of (4) small, a relay was interposed in practice (see Chapter XIII). The governor in question was used mainly on constant (direct) current arc lighting machines. Maintaining constant current meant great variation in speed, depending upon the number of lamps in the circuit, and for that purpose the governor was well suited. Direct current arc lighting for streets is to-day a matter of past history. When the direct current arc lighting machine went out of usage, the governor, Fig. 99, went with it. JE 1 FIG. 99 DISCARDED TYPES OF SPEED GOVERNORS 147 In alternating current work, it is desirable to maintain a given number of cycles, and to govern voltage by electrical means. But a constant number of cycles means a constant speed of the prime mover, and for that purpose the revolution counter, or, in other words, the centrifugal governor, is the best solution. If maintaining a constant voltage, without any regard to the number of cycles, were the prime consideration, then and in that case it is highly probable that governing would be accom- plished to-day by electrical means. References to Bibliography at end of book: 4, 5, 6, 22, 31, 36, 39, 42, 61, 62, 80. CHAPTER XI GOVERNING FOR CONSTANT RATE OF FLOW IN connection with prime movers driving pumps, cases are encountered in which it is necessary to keep the rate of delivery of the pump constant. The type of governor employed for this purpose varies with the type of pump. In positive displacement pumps (including reciprocating and rotary blowers), the delivery is, within reasonable limits, proportional to the angular velocity, and independent of the resistance to flow on the discharge side. Consequently, con- stant delivery is, within the same limits, identical with constant speed, and prime movers driving constant displacement pumps or blowers are, for the purpose in question, equipped with constant speed governors. If the resistance to flow varies widely, constant speed does not mean constant delivery, because the volumetric or delivery efficiency drops in all posi- tive displacement machines, as the resistance is increased. The reduction of volumetric efficiency is due to several causes such as leakage and slip, reexpansion in clearance volume, heating of incoming gas, reevaporation (if vapor is pumped) etc., which need not be discussed here. Since increased re- sistance means increased pressure on the delivery side, the reduction of volumetric efficiency can automatically be com- pensated for by a slowly applied increase of speed under the influence of increased pressure. (In the case of vacuum pumps, substitute " suction side" and " decrease of pressure" for " dis- charge side " and " increase of pressure.") There should be no difficulty in accomplishing this result, although no applica- tions of this thought appear in practice. Matters are quite different with velocity pumps, of which type the centrifugal pump and the turbo-blower are repre- sentatives. The rate of delivery is a function of both angular 148 RATE-OF-FLOW GOVERNORS 149 velocity and resistance to flow, so that, for a given speed, the flow may vary within wide limits. Speed governors, therefore, will not do for maintaining constant rate of flow, but a force must be derived from the flow itself. Here a wide field is open to the ingenuity of inventors, because there exist so many methods for deriving a force from the rate of flow. If we glance over the variety of centrifugal governors and remember that they are all based upon one single force action, namely that of centrifugal force, then we begin to realize what a number of " constant volume" governors is possible using either the im- pact disk, or the Venturimeter, or the Pitotmeter or nozzles -and orifices, or constant pressure difference with variable orifice area, or temperature rise methods, etc. With all this variety of possibilities there is a certain similarity between volume governing and speed governing, as will be apparent from the following : In centrifugal pumps and low pressure turbo-blowers the relation between angular velocity u, pressure p and rate of delivery V is (near enough for all practical purposes) given by the equation p = s- D 1/5 l/> OJ i_ d. o: where Ki and K 2 are design constants peculiar to each machine, 1 see also Fig. 100. If now the pump or blower operates solely against a friction resistance (centrifugal pumps in closed systems, blowers for fur- naces, etc.), the pressure generated by the pump is used for overcoming friction resistance only, which may be expressed by the equation p = K,V\ (In high lift centrifugal pumps this rela- Resistance Characteristic Pump Characteristic V- Rate of Del ivery FIG. 100 tion does not hold. For such pumps, however, volume governing is practically never used.) 1 This equation does not apply to turbo-compressors, on account of the change of density with pressure. But turbo-compressors are governed for constant pres- sure, and not for constant volume. Therefore, the discrepancy does not matter. 150 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Combining the two equations furnishes (K z + K 2 )V 2 = K : u 2 or finally _ ^~ ............... (1) Evidently the rate of delivery is directly proportional to the speed, as long as the external resistance K z remains constant. But the latter is usually the case. If, for instance, a turbo- blower delivers air to a blast furnace, the changes of resistance in the furnace, although of considerable magnitude, occur so slowly that during the period of one process of regulation (caused, for instance, by a sudden change of steam pressure), the resistance may be considered constant. But if relation (1) holds, all the theories applying to speed governing must also apply to volume governing. There must be a static fluc- tuation (difference of pressure between great and small resist- ance), there exists a limiting case of never-ending vibrations, a gag pot is needed to enforce stability, etc. For this reason it is unnecessary to go further into the theory of volume govern- ing, because the theory of speed governing can be applied with small changes. The two types of " volume" governors which are at the present time in use in the United States are diagrammatically illustrated in Figs. 101 and 102. In both illustrations the force derived from the rate of flow is shown as operating the energy-controlling device directly, whereas, in practice, a relay is used. The latter part of the governors will be found discussed in Chapter XIII. In Fig. 101 (General Electric) an impact disk is used. The disk produces a pressure difference due to throttling and, at the same time, acts as piston for converting the pressure differ- ence into a regulating force. In Fig. 102 (Rateau, Ingersoll- Rand) a partial vacuum is produced by a Rateau multiplier (two or three Venturi tubes in a set). The suction is transmitted to a separate piston (4). Many of the definitions from the statics of centrifugal governors can be applied here, if properly modified. In both types the " strength" of the governor is found by the force RATE-OF-FLOW GOVERNORS 151 which is required to move the governor when it is not in opera- tion, which in the present case means without flow. The whole strength cannot be utilized in regulation, because that would require cessation of flow. The governor has a work capacity depending upon the governor piston displacement and upon the pressure difference due to the allowable difference in the rate of flow. Both governors have stability and a static fluc- tuation. In Fig. 101 stability is obtained by cone angle i and the spring (1). In Fig. 102 it is obtained by the spring (1). If the steam pressure varies, or the resistance of the furnace varies, the control valve (2) must change its position, which Fulcrum Partial Vacuum Steam Line CD FIG. 101 FIG. 102 (on account of the just mentioned stability) means that the rate of flow must change. If the stability is too small, that is to say if it is attempted to regulate too closely, never-ending vibrations will result ; the governor will hunt, unless special gag pots, etc., are used (see paragraph 2 of Chapter IX). Just as in speed governors adjustment of speed is obtained by variation of strength, so in volume governors adjustment of rate of flow is obtained by the same means. In Fig. 101 the strength is varied by the shifting of the weight (3) along its supporting lever, and in Fig. 102 the same is accomplished by turning hand wheel (3), which varies the initial compression of spring (1). 152 GOVERNORS AND THE GOVERNING OF PRIME MOVERS The complete theories of these two governors, including the measuring apparatus for the rate of flow, are not difficult. But they will not be given here, because the governors are patented, which means that the theory would benefit only a very limited circle of engineers. In connection with volume governors it should be remarked that frequently more is expected of them than they can accom- plish. This refers particularly to the application of such gov- ernors to blast furnace blowing equipment. In blast furnace practice the furnace, the boilers and the blowers are interde- pendent. If the furnace hangs (resistance becomes very great), more steam is used, while the gas supply from the furnace to the boilers remains constant. The steam pressure drops, so that still more steam is used. Soon a limit is reached, and the rate of flow of air to the furnace cannot be maintained, unless the heat input into the boilers is increased by other means, such as coal fire. If the quantity of blast falls off, the supply of gas also falls off, the steam pressure is lowered still more, etc. Most obviously neither a constant volume governor on a turbo-blower nor a constant speed governor on a reciprocating blower can obviate or cure such a condition, and should never be expected to do so. Reference to Bibliography at end of book: 81. CHAPTER XII GOVERNING FOR CONSTANT PRESSURE IN the governing of pumping machinery, the most widespread requirement of regulation is to keep constant the pressure (or suction) produced by the pump, no matter how much the demand or rate of flow of fluid pumped may vary. Pressure governing differs from speed governing in many respects. Although the reasons for the differences will not appear until later in the chapter, the principal differences will be mentioned here for the sake of greater clearness. First, the influence of pressure governing upon the prime mover is felt more slowly than that of speed governing, because change of pressure is the accumulated effect of speed deviation. Second, different pumps have widely varying characteristics (relations between pressure, rate of flow and speed), so that a governing device which is successful with one type of pump may be a failure with another type of pump. Third, while in speed governing the two variables, namely torque and angular ve- locity, are, within reasonable limits, independent of each other, the variables in pressure governing are not independent of each other ; the resisting torque (average per revolution) of a pump is a function both of the pressure and of the rate of flow. Finally, the word "regulation," when applied to govern- ing for constant pressure, is very elastic. While in air com- pressor plants for mines and factories variations in pressure of 15% to 20% or even more are allowed, and are considered " close enough," pressure fluctuations on the suction side of gas exhausters for by-product coke ovens must be kept down to 1/10% of the prevailing pressure, and even that small varia- tion is called "not close enough." The understanding of the phenomena of pressure regulation presupposes familiarity with the characteristics of different 153 154 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Atmospheric Pressure FIG. 103 types of pumps. For that reason, a brief discussion of the most important features of pump characteristics will now be given. The two types of pumps on which pressure regulation is practiced are the displacement type and the velocity type. Figures 103 and 104 are indicator cards of a water pump and of an air compressor, both of which are dis- placement pumps. The section lined areas indicate losses due to friction through valves and ports. These losses increase with the speed, so that displacement pumps have an automatic stability. If the prime mover delivers a given amount of work per revolu- tion, and if the demand upon the pump remains constant, the speed will not increase, for two reasons : First (if the demand remains constant), the pressure produced by the pump would rise with speed, and second, the friction losses would increase with speed. But both of these mean more work per revolution, which excess work the prime mover, under the assumed conditions, is unable to give. In some air and gas compressors the just mentioned stability is very small or even zero. By study of the dotted line in Fig. 104, it is easily seen that, while the friction work below the atmosphere increases, the work of compression above the atmosphere falls off, on account of the reduced delivery, which latter is occasioned by the pres- sure drop through the inlet valve and ports. The stability is nega- tive in certain compression ranges i of vacuum pumps and compressors with fixed discharge pressure, but variable intake pressure. It is well known that such pumps require maximum work per stroke, if the intake pressure equals one third of the discharge pressure (see Fig. 105). Below that intake pressure, more speed means reduced intake pressure, less work per stroke ; this in turn means still more speed, etc. The system is naturally unstable. Fortunately, close governing for FIG. 104 PRESSURE GOVERNORS 155 FIG. 105 constant discharge pressure is never required with this wide range of intake pressure. The prime mover also contributes to stability or lack of stability. Although study of this problem rightfully belongs in the chapter on the self-regulating properties of prime movers, it will be taken up here, as far as it concerns pumping machinery. The prime movers driving displacement pumps are steam engines and internal com- bustion engines. Only very occa- sionally steam or hydraulic turbines are used. In steam engines, control is effected either by throttling or by variation of cut-off. In throttle control, there belongs a certain posi- tion of the throttle to a given speed and pressure drop. If the engine speeds up with a fixed position of the throttle, the pressure drop through the valve increases, the work done per revolution is decreased, the deficiency of work is taken out of the flywheel, and the speed is reduced again. Hence throttle control results in inherent stability. It is quite different with cut-off control, because one and the same position answers for all speeds and rates of delivery as long as steam- and back-pressure remain constant. In practice, this holds true within certain limits only, because there is always some throttling in the ports, which makes itself felt more and more at higher and higher speeds. But the effect of speed is small, so small as to give very little inherent stability. The latter must be enforced by the proper regulating mechanisms which are described in the second paragraph of this chapter. The characteristic properties of pumps are expressed in characteristic curves. For displacement pumps such " charac- teristics " are not in common use, because the ideal character- istic of a displacement machine is a vertical straight line, see Fig. 106, which means that the rate of delivery does not depend upon the pressure. (Fig. 106 is drawn for variable discharge pressure and for constant suction pressure.) Actually, several Discharge Pressure * V C ^ \ Actual charac' _J Ideal charcict Volume or weight taken ir> FIG. 106 156 GOVERNORS AND THE GOVERNING OF PRIME MOVERS circumstances, depending upon the type of pump and upon the fluid pumped, change the shape of the characteristic. Such circumstances are slip, reexpansion in clearance space, heating of incoming air or gas, etc. But even with consideration of these features, the characteristics of displacement pumps are almost vertical within the practical range. Three character- istics are shown, namely for unit or normal speed, for \ speed and for double speed. For a given discharge pressure, the delivery is almost proportional to the speed. As a rule, the delivery does not grow as rapidly as the speed, partic- ularly in air and gas compressors (see Fig. 104), unless an intake pipe is provided of sufficient length to use the inertia of the fluid in it for shoving an extra quantity into the pump at the end of the retardation period (end of the stroke in reciprocating machines). Turbo-pumps and turbo-compressors have no " indicator cards," so that the features affecting regulation must be studied directly from the pump characteristics. While the character- istics of centrifugal (or turbo-) pumps for liquids differ from those of turbo-compressors, they will not be studied separately in this sketch, because governing for constant pressure is not common in connection with centrifugal pumps, whereas it is quite common with turbo-compressors. In this latter type of turbo-machine speed, torque, pressure, and rate of delivery cannot be related by any rational algebraic formula, because the characteristics vary greatly with the design of the blades, of the diffusor, with the rate of cooling between the stages, with the quantity of air slipping back per stage, and with other details. But in spite of this variety there is sufficient similarity between the characteristics to allow the drawing of conclusions for pressure regulation. In Fig. 107 the charac- teristics of a turbo-compressor are shown drawn to scale. Arbitrarily, a certain rate of delivery has been called unity, and just as arbitrarily a certain speed has been called unity. PRESSURE GOVERNORS 157 Let PC be the discharge pressure which is to be maintained constant, on the assumption that intake pressure is naturally constant. (If both intake and discharge pressures are subject to change, matters are more complicated.) The characteristics show that 2 1 % of change of speed changes the rate of delivery from .81 to 1.06, or roughly 26%. This behavior of turbo- compressors has been cited as a great advantage, because, with constant speed governing, the pressure changes so little over quite a wide range of delivery. This property is indeed an advantage in cases where the demand (for air, for instance) stays within narrow limits, because then pressure governing is reduced to the common and well-worked-out problem of speed governing. It may be noted in passing that this part of a turbo- compressor characteristic differs radically from the displace- ment pump characteristic (Fig. 106). In one case the curve is almost horizontal ; in the other case it is almost vertical. Re- turning to the case in question, it must be stated that cases of limited range of delivery almost invariably turn into cases of wide range of delivery in the course of time, so that the ad- vantage is more apparent than real. The relation between speed variation and delivery variation for a given constant dis- charge pressure is ap- parent from the nest of characteristics. If the demand drops below a certain critical value V, which equals about .75 in the diagram Fig. 107, the pressure can no longer be kept constant, but drops rapidly to p , rises again to p c under the influence of the governor, drops back to p , and so forth. The turbo-machine surges or pumps, unless special governing devices are provided which either throttle x the inlet, or bleed the discharge. From a comparison of displacement and velocity charac- teristics it is evident that maintaining constant pressure against E U 5 S- =1 - ~. ^i.^,^^ V C ET ^^ s S N O. ajnssa F c s l 5 ,. X 15 a Volume or weight taken in In unit time. FIG. 107 158 GOVERNORS AND THE GOVERNING OF PRIME MOVERS a variable demand requires much greater change of speed in displacement machinery than it does in turbo-machinery. Turning now to the regulating devices used for maintaining constant pressure against a variable demand, we see immediately that we have two possibilities for regulation, namely (1) to keep the speed of the pump and prime mover constant and to vary the output of the pump proper, and (2) to vary the speed of the prime mover. Method No. 1 is obviously independent of the type of prime mover, and may be applied just as well if the pump is driven by an electric motor or by belt from a line shaft. Devices for varying the output of the pump in spite of constant angular velocity were indeed originally developed for belt-driven and for motor-driven pumps. Although they are now, for the sake of uniformity of pump design, frequently used on engine- or turbine-driven pumps, they do not belong in a book on the governing of prime movers, and will not be described here. 1 In the application of the second method the most natural step to be taken consists in deriving a force from the pressure produced by the pump and in opposing it by a known force, as indicated in paragraph 1 of Chapter I. Any difference between the two forces is used to move the energy- controlling mechanism. This method leads to devices which are diagrammatically illustrated in Figs. 108, 109 and 110. In these illustrations no attempt has been made to impart features of machine design. In these devices the pressure, acting upon some form of piston, tends to lift a weight (1) or to deform a spring (2), or to do both. If the weights (1) only are used, the tendency is to maintain a constant pressure irre- spective of the position of the float, bellows, or piston. If the force of a spring such as indicated by (#), Fig. 108, is added to the weight, the tendency is to adjust a different pressure for each position of the float, bellows, or piston. In this description, the phrase "the tendency is to" was used advisedly, because these devices will, under certain con- ditions, govern satisfactorily, and will result in abominably 1 The principal means used are by-passes, throttling of inlet, keeping inlet or outlet valves open, varying clearance space, starting and stopping; many of them are known as unloading devices. PRESSURE GOVERNORS 159 poor regulation under different conditions. The safest way to determine which of the two will happen would be to derive and discuss the complete differential equations of motion. But pressure regulation, as before stated, has one more variable than speed regulation, which makes the differential equations of pressure regulation more complicated than those of speed regulation. For that reason an attempt will be made to con- struct a limiting case between stability and instability in a manner similar to that followed in speed governing. This FIG. 108 FIG. 109 limiting case will be derived with the aid of Fig. Ill, which represents a diagram of an air or gas pump (2) delivering into a container of volume F 2 from which the fluid is discharged in a steady stream through a nozzle (1). Let the pump deliver V f u cubic feet of fluid per second, where u is the average angular velocity of the pump. Valve (4) controls the flow of steam to the prime mover (not shown) operating the pump. Now let the governor be set in motion up and down. Under what con- ditions will governor, speed and pressure perform sine harmonic vibrations about the values of equilibrium? 160 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Let x be the displacement of the governor from equilibrium position, then du = _^M^ dt Ao I which is true upon the assumption that x is small, which means that the pressure variations are so small as not appreciably to affect the unbalanced moment. A , as before, is the whole governor travel, M m is the maximum torque, and I is the moment of inertia of all rotating parts. Let Du be the angular velocity in excess of the average ; then we find the * Position of Equilibrium FIG. 110 FIG. Ill pressure rise in time dt from (DuV'dt + V 2 )p = Vz(p + dp) from which we have V dp = Du pdt (2) V 2 Turning to Fig. 112, assume that the governor performs sine harmonic vibrations of the amplitude \A. Then the speed will also fluctuate up and down with a phase lag of 90 behind the governor motion. But with a container as shown in Fig. PRESSURE GOVERNORS 161 111, the pressure lags 90 behind the speed, the pressure being a maximum when the excess speed disappears. This means that the pressure lags 180 behind the governor. Hence, the pressure change is (negatively) proportional to the governor displacement, and the pressure curve in Fig. 112 is a straight line (1)(4)> Comparing Fig. 112 with Fig. Ill, we note that excess pressure, such as that at point (4), urges the governor still farther away from mid-position. But to get sine harmonic motion, we must subject the governor to a force which pulls FIG. 112 it back to equilibrium position and which is proportional to the displacement x. A stability spring must therefore be added to return the governor. If line (2) (3) in Fig. 112 represents (negatively) the spring forces, then only the difference between spring force and excess pressure force is available for returning the governor, as indicated by vertical section-lining. The angular velocity w of the auxiliary vector of the sine harmonic vibration is determined by the section-lined ordinates, because w = \J , where PI is the restoring force at unit m displacement. At the end of the governor swing the restor- 162 GOVERNORS AND THE GOVERNING OF PRIME MOVERS ^ ing force is (3) (4) = (3) (6) - (4) (6) or Si - G D p, where Si is the spring force for unit deformation, G = cross-sectional area of governor float, D p = excess pressure at the end of governor swing. Now suppose the stiffness Si of the spring to be adjustable. Make the spring more flexible. Then w will become smaller, the excess speed will grow, and the excess pressure will grow. The section-lined strip becomes smaller, first, because point (3) comes down, and second, because point (4) goes up. A limit is soon reached. There must be a minimum scale Si min to the spring to make return of the governor possible. If, on the other hand, the spring is made stiffer, the motion of the governor is quickened, and both excess speed and excess pressure tend to become smaller. It should, however, be understood that during the process of changing from the steady vibration with large amplitude to another steady vibration with smaller amplitude, the three deviations of governor displacement, speed and pressure will fall out of step for a while and will cause quite heavy fluctuations. The minimum spring scale necessary for stability is easily computed. For sine harmonic vibration with angular velocity w of auxiliary vector, equation (1) becomes du 1 A . M m 1 A M From which Du = - sin (wf) > 2 A Iw Substitute this in equation (2) then V 1 A M m p here represents the average value of the pressure in F 2 . By inf.fkcrrot.irm \\rn rVKfain integration we obtain V 1 A M PRESSURE GOVERNORS 163 with a maximum value at end of swing of 1 A V'p M m P " 2A V* I w 2 ' flSi D p G But we also have w = \l - m A m 2~ The two expressions for w must be equal, hence Si Pop G _ 1 A V'p M m 1 m \A m 2A V* I D p This equation may be solved for Si or for D p. It is more jj /Y\ A /yy) instructive to do the latter. By multiplication with - ^-^ > Z(JT we get an equation of the form (D p) 2 K\.D p = K 2) from which D p= %Ki V (%Ki) 2 - K 2 , or, by substitution of the values of the constants, _ 1 S,A /lSiAy I A V'p M,, Am "4"e" ^Vi^V'aZITT / 20' From this equation it follows that there may be a great or a small pressure rise for a given spring force, as long as the value of the radical is real. If the latter becomes zero, then we work with the minimum spring scale. Hence we have 1 A V'p M m A m 16 G 2 2A 7 2 I 2 G By the introduction of time elements, this equation may be simplified. We know that ~ = , also that - - = . > L JL 8 fYlA. \J.g) 164 GOVERNORS AND THE GOVERNING OF PRIME MOVERS where T g is governor traversing time under the influence of pressure ; V'uT f = F 2 , where T f is the. time required to fill the volume of the container. Then we obtain (5) for the smallest permissible scale of the spring. From this equation the static pressure fluctuation can be found, because pressure difference (Between extreme positions) X governor area = spring scale X governor travel, hence when static pressure difference between extreme positions p = > average pressure The conclusion from this reasoning is that, for stability of regulation, the governor must have a minimum static pressure fluctuation, just as in speed governing the governor had to be designed for a minimum static speed fluctuation. For many classes of service such a fluctuation is admissible, but occa- sionally very close pressure regulation is desired ; a static fluctuation is not permissible in such a case, and temporary stability must be secured by other means. Among such means are compensating dashpots, similar to those used in speed governing (Chorlton-Whitehead governor, Bee governor, etc., see paragraph 2, Chapter IX), or throttling of the inlet to the governor such as at point (3), Fig. Ill, or inclosure of the governor in an almost airtight box. The question of stability of regulation being settled, the problem arises, what must be done to damp the vibrations out of existence? The device used in speed governing, namely damping by an oil gag pot, does not work in this case, as will readily be understood from Fig. 112. The period of a damped vibration is longer than that of an undamped vibration, hence the speed rise is greater, and the excess pressure becomes PRESSURE GOVERNORS 165 greater (instead of smaller, as desired) ; the governor does not return promptly. Another method of arriving at the same conclusion consists in arresting the governor at point (5), Fig. 112, until the speed has dropped to normal. During that time the pressure rises considerably. As soon as the governor is released there is no force to return it, and it crawls farther away from the equilibrium position under the influence of accumulated pressure. While governor displacement damping fails, " speed damp- ing" does the trick. If in Fig. 113, which is a reproduction of part of Fig. 112, the speed should not rise quite as high on the travel to the right, and should not fall quite as low on the travel to the left, then the pressure fluctuation would be re- duced, more restoring force would be effective, the governor would quicken in its movements, the pressure fluctuation would become still less, etc., and the vibration would cease. This desirable effect is produced by throttling both in the ports and valves of the prime mover, and in those of the pump. For a given position of the power-controlling mechanism, throttling makes more of the engine or turbine torque available at low speed, and reduces it at high speed. The speed is reduced during the excess speed period, and is increased during the speed deficiency period ; it will follow a line some- what like the dotted spiral of Fig. 113, and the vibrations will rapidly die out. Summing up, the devices shown in Figs. 108, 109 and 110 may work satisfactorily, and may not. Satisfactory regula- tion is favored by long starting time T s (large flywheels), long filling time T f (large container), and small governor traversing time. Stability of regulation requires static pressure fluctuation or, where such fluctuation is not permissible, tem- porary stability. Throttling in valves and ports of prime mover and pump damps vibrations out of existence. Pure pressure governing of the character here described FIG. 113 166 GOVERNORS AND THE GOVERNING OF PRIME MOVERS occurs seldom. As a rule, the container is a piping system of small cross-sectional area compared to its length, see dotted lines in Fig. 111. In that case any increase of speed immedi- ately causes an increase of pressure at the mouth of the pump, on account of the friction in the pipe. The pressure does not lag 90 behind the speed. It has a component which is in phase with the speed and a component which is 90 behind the speed. In a vibration diagram with abscissae representing governor displacement, the component in phase with the speed is represented by an ellipse with the governor displace- ment as an axis, while the component lagging 90 behind the speed is represented by an inclined straight line, as before. The resultant phase depends upon the relative magnitude of the two components (see Fig. 114, where the inclined ellipse resulting from a com- bination of the two phases has been shown.) We then face the peculiar situa- tion that the speed component requires speed regulation, and the pressure com- ponent requires pure pressure regulation. For the sake of the speed component FIG. 114 an adjustable oil gag pot is usually necessary, unless displacement damping can be secured by other means. Friction in pressure governors is just as harmful as friction in speed governors, if close regulation is desired. It produces a " detention by friction" and endless fluctuations of pressure, unless it is eliminated. Just as before, impressed cyclical vibrations of pressure (caused, for instance, by the pulsating delivery of displacement machines), jarring and vibration transmitted mechanically to the governor, and impressed cyclical forces eliminate friction. In the absence of these agents, a frictionless governor must be used ; this means knife edges, roller or ball bearings, and floats or bellows (Figs. 108 and 109) in preference to pistons (Fig. 110). If the latter design must be used on account of high pressure, good lubrica- tion and plenty of oil grooves for distributing the pressure uni- formly around the piston are a great help ; because absence PRESSURE GOVERNORS 167 of such grooves, coupled with local deformation due to high pressure, forces the plunger to one side and causes it to stick. All prime movers regulating for constant pressure must also be equipped with a speed limit centrifugal governor so that excessive demands upon pump or compressor will not wreck the outfit. Such runaway governors may be (and usually are) of very simple design, because they enter into action only occasionally, and because all pretense to close regulation is abandoned when they do enter into action. Of course they must be strong enough to handle the valve gearing. While pressure governing by the devices shown in Figs. 108, 109, and 110 is successful with proper design when a single prime mover and pump work against the demand for air, water, or gas, the assistance of a centrifugal governor cannot be spared as soon as two or more units work in parallel (deliver to the same container). For proof of this statement study the action of two steam-driven compressors jointly furnishing compressed air to a mine. Let these compressors be of a high class type, with steam cut-off governed by pressure governors, such as shown in Fig. 110. Let one governor have a little more friction than the other. Now let the steam pressure drop gradually. Then the governor with less friction will adjust its engine and will maintain the pressure. The governor with more friction will not move, because the pressure is main- tained constant by the other governor ; its engine will not receive enough steam, and the compressor will slow down considerably, causing the other machine to speed up an equal amount. As a makeshift, one of the machines is sometimes operated at constant speed by a centrifugal governor, all the regulating being done by the second machine. This will do in the case of two machines, but when three or more run in parallel, the fluctuations are too severe to be taken care of by one pressure governor and compressor only. In order to make all machines share evenly (or nearly so) in the delivery, a combination centrifugal and pressure governor is frequently used. The principle of such a combination gov- ernor is shown in Fig. 115. A centrifugal governor (1) of great 168 GOVERNORS AND THE GOVERNING OF PRIME MOVERS static fluctuation, such as described in Chapter V, controls one end of a floating lever (2). The other end of the latter is controlled by a pressure governor (4) of the type shown in Fig. 110. Point (3) of the floating lever controls the supply of energy to the prime mover. Variable demand for air (or other fluid pumped) at practically constant pressure means speed of prime mover fluctuating with the demand. Point (3) will, in high class reciprocating units, maintain practically tho same position for any speed, except that it will move temporarily for the purpose of accelerating or retarding the pump. The result is that, as a first approximation, the floating lever always passes through the same point (5), while the two ends move. It is to obtain great speed varia- tion over this range of move- ment that the centrifugal gover- nor is arranged with excessive speed fluctuation. In principle this governor is identical with the governor described in Chap- ter V. Naturally a safety device must be used to prevent over- speeding in case of the pressure main or the pipe to the pressure governor breaking. Any of the devices mentioned in Chapter V may be used. For the sake of completeness an additional and different arrangement is illustrated in Fig. 115. The necessary, great static fluctuation is obtained by a combination of static properties of the governor proper (1) and of a central spring (7) ; the maximum tension of the latter is determined by a weight (5) , which normally rests on a fixed stop (6) and which, in the just mentioned case of overspeeding, is carried by the spring (7) acting upon a lever with fulcrum (8). The combined speed and pressure governor distributes the work evenly, provided that the pressure governor is set with a sufficient static pressure fluctuation. To see that the governor Less? "Steam More Steam FIG. 115 PRESSURE GOVERNORS 169 fills the bill, imagine that one compressor has lagged behind the others ; the speed governor end of lever (2) will drop and point (3) will move down and give more steam, unless point (4) on the pressure side goes up. But the latter cannot occur, because it would unbalance the pressure governor. Hence more steam is given, and the slow machine is speeded up. References to Bibliography at end of book: 54, 75, 81. CHAPTER XIII RELAY GOVERNING 1. Reasons for the Use of Relay Governors and Forces Acting upon Them. In the preceding chapters the governor was studied in its two capacities of measuring instrument and of motor for adjusting the valve gear. In the present chapter, we will investigate the properties of governors the motive power of which need only be large enough to release and control another source of energy which, in turn, adjusts the power- controlling mechanism of the prime mover. The use of relay governors, as this type is called, dates back to about 1870, but their development remained slow until the electric generator demanded precise regulation of large hydraulic turbines. In that type of prime mover, frictional resistance to motion of power-controlling mechanism is espe- cially great, so that direct-control governors were out of the question. In the development of this type of governor, American engineers played a leading part. Relay governing for hydraulic turbines was so successful that the same method of governing was transferred to steam turbines and to large gas engines. While in steam turbines the resistance to adjustment of steam- controlling mechanisms can be kept comparatively small, the dynamic action of steam flow causes widely varying forces upon the governor in different parts of its travel (compare paragraph 3 of Chapter III), unless the controlling mechanism is designed with great care and knowledge. In the latter case direct-control governors are quite feasible and are indeed used with steam turbines passing up to 30,000 pounds of steam per hour. In larger machines direct-control governors, even with careful design, become so large and expensive that it pays to use relay governors instead. Frequently, relay governors are adopted for smaller sizes also in order to obtain uniformity of design. 170 RELAY GOVERNING 171 Many large gas engines are equipped with relay governors, because the resistance of the gas-controlling valves is uncertain on account of dust and tar in the gas. Tarry matter in the gas will easily increase the resistance of the valves to 50 (or more) times the value of the resistance of the valves with clean gas. Last, not least, a relay frees the governor (measuring instru- ment) from the troublesome integrated effect of valve-gear reactions, which effect is so difficult to predetermine correctly. The very nature of relay governing involves certain differ- ences between it and direct-control governing, and these dif- ferences should be clearly understood. The purpose of relay governing is to keep all valve gear forces away from the gov- ernor. In consequence, the friction-eliminating influence of vibratory impressed forces is absent. In relay governors there are no cyclical vibrations due to vibratory reactions, and detention by friction (see paragraph 4 of Chapter II) becomes of great importance. The absence of vibration forces has be- come more complete with the perfection in the production and lubrication of toothed gears used for driving the governors. The old-time jarring of the governor due to rough gears is not found in steam turbines. And finally, turbines have no cyclical variation of angular velocity such as steam and gas engines have ; this does away with another friction-reducing agent. These three facts have necessarily resulted in the use of relay governors with smallest possible internal friction. Knife- edge joints have become universal in the type of governors under discussion. Even small friction in centrifugal governors of this type produces never-ending vibrations. The latter are especially noticeable when the load is constant. Mr. C. A. Parsons, one of the foremost pioneers in steam turbine design, recognized this situation at a very early date, and overcame governor friction by means of impressing a vibration upon the governor. This was accomplished by an eccentric in connection with an oil gag pot. His system of governing became known as "puff governing/' because steam was admitted in puffs. Several turbine builders make use of this principle at the present time ; they have, however, reduced the governor vibration to a very small amount, just sufficient to eliminate friction. 172 GOVERNORS AND THE GOVERNING OF PRIME MOVERS A second difference, and an important one, results from the addition of a variable. The governor releases a force, which action takes time. Then the force adjusts the energy-controlling mechanism, which, again, takes time. Now it may be that the sum of these two time elements is less than the time which a direct-control governor would require for the same work, but a new variable has been added nevertheless. Instead of dealing only with the promptness or traversing time of the governor, we now deal with the latter plus the promptness or traversing time of the relay. In future calculations this time will be used. It is denned as that time which is required by the relay mechanism (hydraulic cylinder, electric motor, gear- ing from prime mover) to move the power-controlling mechanism (valve gear, throttle valve, hydraulic gate) from full-load position to no-load position. The symbol T r will be used for the relay traversing time, while T a will be retained for governor traversing time. What has just been stated with regard to introducing an additional time element applies with equal force to additional detention by friction. The regulating or relay mechanism may have considerable friction which, in some forms of relay governors, is just as harmful as, or even more harmful than friction in the governor proper. A third, and extremely vital, difference between direct- control governing and relay governing is that the latter enables the designer practically to get rid of the troublesome influence of mass of governor. This is accomplished by methods which use only a very short governor travel and which will be described in detail later on. A fourth difference lies in the fact that relay governors are easily adapted to obtaining reversed speed curves. This latter property of relay governors was brought out in the governing of hydraulic turbines with long supply lines. In such governors, gate movements must be slow (unless stand- pipes or surge tanks are provided), to protect the line from water hammer. This requirement necessitates a governor of great static fluctuation for the sake of stability of regulation. But great static fluctuation means a great speed difference RELAY GOVERNING 173 between no load and full load, which is not permissible in elec- tric lighting. This predicament led to the invention of auto- matic speed-restoring devices which gradually bring the speed back to normal after every change of load. These devices will be found described later on. The sources of energy which are released by the governor are either fluid pressure, or else electrical energy, or finally mechanical energy derived from the prime mover. Fluid pres- sure may be derived from steam, oil, or water under pressure, or from compressed air. Steam and compressed air are elastic, or expansive, so that relay pistons operated by these fluids tend to cause overtravel and, in consequence of that, racing and hunting. Frictional damping will stop this, but introduces speed fluctuations from another source. Water and oil are superior in this respect. Water often carries impurities which wear out the pilot valve. Oil under pressure is doubtless the best fluid for hydraulic relay governors. Electrical energy has occasionally been used for relay gov- erning of prime movers, but has been given up on account of the greater convenience of steam and pressure oil. It is used to a much greater extent for regulation purposes outside of power plants where the transmission of steam and pressure oil is incon- venient and for regulating prime movers which keep constant some quantity at a considerable distance from the power house. The use of the mechanical energy of the prime mover is very inviting at first thought, but in practice many difficulties must be overcome. Heavy clutches must be thrown in, held in place, and pulled out. Wear occurs in these clutches, and must be taken care of automatically. So difficult is a satisfac- tory solution that in modern mechanical relay governors the clutch is operated by a hydraulic cylinder, which latter is con- trolled by a relay valve. Such a governor may be called a combination hydraulic and mechanical relay governor. References to Bibliography at end of book: 8, 9, 10, 12, 14, 16, 19, 20, 21, 24, 27, 29, 36, 38, 41, 44, 45, 46, 47, 48, 53, 55, 56, 60, 66, 67, 69, 71, 73, 75, 76, 77, 78, 81, 82, 83. 2. Relay Mechanisms. While relay mechanisms necessar- ily vary with the forces which are brought into action by the 174 GOVERNORS AND THE GOVERNING OF PRIME MOVERS governor (measuring instrument), there are certain principles which must be observed in any type of relay governor ; and since the hydraulic governor is the one most commonly used, the principles in question will be demonstrated with reference to that type of governor. Application of the same principles to mechanical or electrical relays will not be difficult. In the following illustrations the governor (measuring instrument) is represented by the symbol G'r. The type shown is a centrifugal governor which measures angular velocity. But the operation of the different mechanisms would remain exactly the same if the centrifugal governor were replaced by a constant- volume governor as shown in Chapter XI, or by a constant- pressure governor as shown in Chapter XII. Figure 116 illustrates a very simple arrangement, which, unfortunately, is even now used by those who have not taken time to study governors. The arrangement is faulty, as will appear from the following reasoning : Consider a governor and prime mover in which the governor and the relay have the right position for equilibrium ; then imagine the governor sud- denly depressed from equilibrium position, point (2), Fig. 117, to point (1), and then released. Under the influence of its own stability (due to static fluctua- tion) and of an increase of speed, see curve (9) (12,) it will return toward mid-position (3). While it does so, the power piston has traveled downward, from point (6) to point (8) . It reaches its greatest displacement when the governor passes through mid- position at point (3). During all this time the speed grows. The rate of increase of speed is greatest at point (12). Beyond the line (3) (8) (12) the governor travels upward under the influence of the continually growing speed. The latter reaches a maxi- mum in point (10) t when the power piston has returned to mid-position, point (7). The governor is still driven upward FIG. 116 RELAY GOVERNING 175 by a regulating force due to the excess speed. The latter does not come back to normal at point (11) until the power piston has traveled upward a considerable distance. But at that time the governor is not only away a much greater distance - (4) (^) from the position of equilibrium than it was at the beginning of the disturbance (1)(2) but it also has acquired considerable kinetic energy, which carries it still farther away from correct position. The amplitude of the vibration grows, until the governor strikes a stop. . , The mechanism is useless. Governor \V It is true that the arrange- position omd\ ment under discussion can P'lot position be made to regulate after a fashion by giving the gover- nor great stability and much damping, by making the pilot valve very small and by using a prime mover and torque absorber with deci- dedly self-regulating proper- ties. Even then the device will work only on condition that the changes of load are very, very gradual. The prime mover works in that case with long, slow vibrations at constant load and with speed changes which are severe, even if the load varies only a small fraction of the total. Relay position Speed of prime mover Time FIG. 117 Rigid Return of Pilot Valve The shortcomings of the arrangement shown in Fig. 116 were recognized at an early date. The movement of the pilot valve was " compensated " for by the addition of rod (5) with fulcrum (4) for lever (3), Fig. 118. The latter is now a float- ing lever. If the governor rises, (4) is temporarily a fixed point. The pilot valve is raised, the power piston moves down, turn- ing lever (3) about fulcrum (1) and returning the freely movable 176 GOVERNORS AND THE GOVERNING OF PRIME MOVERS pilot valve to mid- or closed position. The device is known as an "anti-racing" device, or as a " compensating return/' or simply as a " return," meaning, of course, a return of the pilot valve to mid-position. It will presently be shown that other quantities are compensated for in relay governors so .that the name of "compensator" for the return should be abandoned. Since the pilot valve must return to mid-position for power equilibrium, there belongs a definite position of the governor to each position of the power piston (see dotted positions of lever (5)). And since a given angular velocity corresponds to each position of the governor, it follows that a definite angular velocity of the governor (and of the prime mover) belongs to each position of the power piston. The arrangement is fully equivalent to the direct-control governor, except that the resistance to governor motion is minimized and that the tra- versing time of the governor is replaced by it plus relay tra- versing time. Evidently, it is desirable to make the latter as short as possible, but there are limits, because a short travers- ing time can be obtained only by means of a large pilot valve, and that, in turn, means greater resistance to governor motion. The arrangement of Fig. 118 is not the only method of re- turning the pilot valve. The stem (2) of the pilot valve may be split and may be provided with right and left hand threads and a turnbuckle for changing the length. Many arrangements for returning pilot valves have been made by inventors, and the patent records are full of return mechanisms in disguise. The arrangement which has just been described should be called a "relay with rigid return" in contradistinction to those with slowly yielding return which will be described below. The simple rigid return has been almost universally adopted for steam turbine regulation. In order to get along with a FIG. 118 RELAY GOVERNING 177 Pilot Valve very small governor travel, designers, as a rule, place the pilot valve between the governor and the power cylinder. Stabilizing Relay with Limited Governor Travel Figure 119 illustrates a relay mechanism which is commonly ascribed to Dr. Proell (Zeitschrift des Vereines Deutscher In- genieure, 1884), and which forms a bridge between the mechan- isms used for steam turbines and those used in hydraulic turbines. In this mechanism the lever (3) turns about a fixed point (1). The travel of the lever is limited by two lugs (4), and is just sufficient to open wide the ports in the pilot valve. The power piston is joined to the lever (3) by a weak spring (#), the force of which restores the governor to mid- position after a disturb- ance. When the gover- nor has again reached mid-position, the power piston has assumed a Fia. 119 different position and the length of the spring has been changed. By studying either a decrease of load or else an increase of load one sees readily that for a decrease of load the governor is loaded, which means an in- crease of speed, and that for an increase of load the governor is unloaded, which means a reduction in speed. The increase or decrease is proportional to the spring force and the latter, in turn, is proportional to the change of load. The regulating mechanism has, in consequence, a "static speed fluctuation," which is not, however, produced by the properties of the governor, but by those of the relay mechanism. If the travel of the governor is made small enough, the stability or instability of the Jatter is of no consequence. While, with this exception, the action of this relay mechanism is just the 178 GOVERNORS AND THE GOVERNING OF PRIME MOVERS same as the one of the mechanism shown in Fig. 118, it should be noted that the travel of the governor can be made very small, so that the troublesome effect of governor mass can be prac- tically eliminated. Relay Mechanism with Slowly Yielding Return As before mentioned, the governing of hydraulic turbines with long penstocks requires, on one hand, a governor of great static fluctuation (in order to save the water lines), and on the other hand a governor of small static fluctuation in order to maintain a constant voltage or frequency. For the purpose of attaining both of the apparently conflicting requirements, the same principle is used which was described in paragraph 2, Chapter IX, in con- nection with the Chorlton-Whitehead governor. This principle, applied to a relay mechanism, is illustrated in Fig. 120. 1 It will be re- membered that the principle consists in first imparting tem- porary stability to the governor mecha- Va I ve nism, and then gradu- ally, by means of an oil pot, removing the L - L ' stability. Figure 120 is very similar to Fig. 119. The difference consists in the connection between the power piston and the spring (2). In Fig. 119 the connection was rigid, whereas in Fig. 120 an oil gag pot (5) is interposed. If the by-pass on the gag pot is closed (and if we disregard the clearance between the gag pot piston and its cylinder), the action of the mechanism of Fig. 120 is identical with that 1 For the present, parts (6) and (7) should be disregarded. Power Cylinder No load ^" Full Load d ^ L_e__j . tie ss EH 1 | ' ! f ore J nr*^ 1 \ *- } RELAY GOVERNING 179 of Fig. 119. As a matter of fact, we can best understand the action of the mechanism shown in Fig. 120 if we assume the by-pass closed during a change of load and suppose it to be opened after the regulating process with rigid connection has been completed. After the by-pass has been opened, spring (2) can assume its free length, the loading or stability-producing influence of the spring on the governor disappears, and the latter assumes the original speed. The action is just the same as if the tension of a loading spring in any of the devices in Chapter V were changed while the governor is in operation. Centrifugal force and centripetal force are no longer equal, and a regulating process is started. In the case in question the change of speed (or the change of load) becomes more and more gradual as the spring tension disappears. The result is a perfectly isochronous governor, with stability during the process of regulation. Oil pot (5) is commonly called a " compensator," because it compensates for the change of speed which would otherwise be caused by the different position of the power piston. It might be called a " speed restorer," but the term compensator is so generally used that it should be retained. It should be thoroughly understood that illustration 120 is purely diagrammatic. In practice the compensator is usually located some distance away from the power cylinder and is operated by the latter through a system of levers. Slowly-yielding Return with Adjustable Static Fluctuation Isochronous regulation, while apparently ideal, is not always desirable. For parallel operation of synchronous (alternating current) generators, a small, positive static fluctuation is necessary for proper distribution of work among the several generators operating in parallel. On the other hand, asyn- chronous generators require a negative static fluctuation (re- versed speed curve). The mechanism of Fig. 120 can be adapted to either one of these requirements by the addition of a second spring (6) stretched between governor lever (3) and a second lever with fulcrum (7). The bottom lever is attached to the gag pot (not to its piston). Then spring (2), in conjunction 180 GOVERNORS AND THE GOVERNING OF PRIME MOVERS with the gag pot, determines the temporary stability of the governor, whereas the spring (6) determines the static fluc- tuation ; for, while spring (2) loses its tension after every process of regulation, spring (6) does not. Observe that spring (6), in the location shown in Fig. 120, is compressed at no load, and is extended at full load. This means that the speed is reduced at no load and is boosted up at full load. The static fluctuation is negative. If, on the other hand, the spring (6) were fastened to the bottom lever between fulcrum (7) and the gag pot, then the static fluctuation would be positive. Obviously, this arrangement lends itself to an adjustment of the static fluctua- tion while the prime mover is in operation. To that end the auxiliary spring must be made movable right and left. Adjustment of static fluctuation may be obtained with other mechanisms, for instance with the one shown in Fig. 121, which illustrates a design with only one spring, (1). The lat- ter is fastened to a lever (4) with fulcrum (2). The extreme end of the lever is operated by the gag pot. Just as before, we split the process of regulation, assuming gag pot and gag piston to be solid during the first part of theprocess. If the load drops, the gover- nor rises, the pilot valve is lifted, the power piston rises, returns the pilot valve and compresses the spring (1). Up to this point the mechanism acts exactly like a rigid return, such as was shown in Fig. 118, because the force of the deformed spring is taken up by the power cylinder and has no v action upon the governor (measuring instrument). The speed has risen on account of the static fluctuation of the cen- trifugal governor. If now the by-pass in the gag pot is opened, FIG. 121 RELAY GOVERNING 181 the spring (1) is free to act slowly on point (5) of the floating lever. The pilot valve is lifted, the power piston rises, returns the pilot valve, and reduces the supply of energy ; the speed drops, the governor drops, raising the pilot valve ; the power piston drops, bringing the supply of energy back to the right amount. If point (3) were fixed, the spring would be free from stress when point (5) had returned to the initial position. But, since point (3) has been moved up a small distance on account of lever (4) being linked to the gag pot, point (5) will not quite return to its original position ; it will assume a slightly higher position than before. The pilot valve, of course, is finally in mid-position just as before the disturbance. With the pilot valve again in mid-position and point (5) slightly raised, the governor G'r must assume a slightly higher position, so that a small positive static fluctuation is left. In this design, the static fluctuation of the governor is used to furnish stability of regulation, whereas relative displacements of points (3) and (5) determine speed difference which finally establishes itself after a disturbance. For purposes of adjustment, lever (4) and the bracket supporting it are provided with numerous holes. The connecting pin can be put into any one of these holes so that the fulcrum can be shifted. With the fulcrum to the left of (3), such as at (2), the speed curve rises from full load to no load. With the fulcrum to the right of (3), the speed curve becomes reversed. With the fulcrum at (3), the governor is isochronous. Brief mention should be made of the fact that isochronism or, if desired, a reversed speed curve, can also be obtained by letting the slowly-yielding gag pot vary the ratio of speeds between the prime mover and the governor. The Double Relay For close regulation, the governor traversing time and the relay traversing time must be small. To secure a short travers- ing time for a powerful relay, the pilot valve (or equivalent releasing mechanism) must be of considerable size. And the governor proper must be of sufficient size to handle the pilot 182 GOVERNORS AND THE GOVERNING OF PRIME MOVERS valve quickly. The result is that the commonplace statements about relay governing permitting the use of very small gov- ernors are untrue for powerful relays, as many an engineer has found to his sorrow. This difficulty was overcome by the introduction of a relay for the relay. Of course, each relay must have its return, so that the double relay becomes very complicated and "grass- hoppery," if both the relays are returned by levers. External complication is avoided by the floating relay valve shown in Fig. 122. If the small central stem moves, ports in the floating relay valve are uncovered, oil or water pressure is admitted to one end, and the other end is open to exhaust. The ports are so ar- ranged that the floating (outer) valve promptly follows any move- ment of the central stem. The floating relay valve indeed re- moves practically all resistance from the governor, if everything is in working order. New com- plications are introduced, because there are more parts that can stick or bind. The ports of the central stem are so small that scale or lint interferes with the action. Water should never be used with the floating relay valve ; nothing but pure oil is satisfactory. In the theory of the double relay, an additional time element enters, namely the traversing time of the floating valve. The floating relay is commonly used on very large steam turbines and hydraulic turbines. In very large water turbines, a double floating relay valve has recently been introduced. A very good diagrammatic illustration of all the parts neces- sary for a complete relay governor was given by Thoma in Zeitschrift des Vereines Deutscher Ingenieure, 1912. This illustration is here reproduced in Fig. 123. The diagram is so clear that very little explanation is needed. The following notes may be helpful. The centrifugal governor (2) has no Pressure Power Cyl. Waste FIG. 122 RELAY GOVERNING 183 collar or sleeve, but a spherical end bearing stem. The floating lever is held in place by a spring (1), so that lost motion is eliminated even if wear takes place. This arrangement is also useful in avoiding strain of the mechanism in case the power- control mechanism reaches a stop before the governor mechan- ism does. Instead of oil-pot compensation a pair of friction wheels (3) is used which, after a change of 'load has occurred, gradually returns the centrifugal governor to the same position Connected to. turbine gates FIG. 123 and restores the original speed. The friction wheel compensator is an American invention and was introduced by the Wood- ward Governor Co. of Rockford, 111., long ago. By means of hand wheel (6) that position of the governor which maintains equilibrium can be adjusted, which means that the speed is adjusted by making use of the fact that the governor has a static fluctuation. Link (4) serves the purpose of adjusting the speed and effectiveness of the return of the pilot valve (7), 184 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Oil for the power cylinder (5) is furnished by the gear pump (11), which delivers oil through the overflow valve (10) and the check valve (9) to air chamber (8). The overflow valve (10) is so designed that the pump is relieved of load as soon as the oil in the air chamber has risen to a predetermined highest level. References to Bibliography at end of book: 8, 9, 12, 21, 24, 36, 38, 41, 44, 45, 46, 47, 48, 53, 55, 56, 66, 67, 69, 71, 73, 75, 77, 81. 3. Work Capacity of Relay Governors. The multi- plicity of designs of relay governors precludes a simple set of rules for the work capacity of such governors. Only a few guide posts can be given here. The largest direct-control governor built has a work capacity of about 30 ft. pounds (for 2% speed variation). Relay governors have been built with 20,000 ft. pounds (or even greater) work capacity. Right here an important difference between the two types of governors should be mentioned ; namely that in direct-control governing more work may be done by the governor, if we allow a greater speed variation, whereas in a properly designed relay governor the speed change has no influence on the work capacity. Properly considered, the term "work capacity" is incom- plete, because it omits the element of time. This fact becomes manifest if we consider mechanical relay governors in which the governor (measuring instrument) throws in a clutch and makes the whole energy of the prime mover available for moving the energy-controlling device. In such a governor the torque which can be transmitted to the control is limited by the mechanical strength of the mechanism. Shafts, gears, keys, etc., must not break under the strain But work transmitted in unit time equals torque times angular velocity, and total work transmitted equals torque times angle passed over. Hence, the same mechanism, running at a given rotary speed, will transmit more work, if it is given more time. But in the govern- ing of hydraulic turbines a comparatively long time is required whenever the water is supplied through a long pipe. This condition is favorable to the use of a mechanical governor, which explains the fact that this type of governor was developed RELAY GOVERNING 185 in connection with hydraulic turbines and is to-day limited to such use. In hydraulic governors the work capacity is the product of displacement of power cylinder and pressure of fluid. If water or steam is used the pressure is usually limited to that of the available supply, but if oil is used, the pressure is de- termined solely by questions of leakage and strength ; it often equals 200 pounds per square inch. In this definition of work capacity, three features are not included, namely, (1) detention by friction, (2) capacity of oil pump, and (3) size of pilot valve and ports. (1) Packing against high pressure may produce great friction, particularly if cup-leathers are used in the piston. By improper design, 25 % to 30 % of the total force may be absorbed by friction. (2) The work capacity of an oil-pressure governor in a given period of time is limited by the capacity of the oil pump. If frequent and heavy changes of load occur, a given oil pump can serve a small governor only, und, vice versa, a given governor requires a large pump. But in the design of standard governors, or even of a given governor for a given purpose, the frequency and amount of load changes r.re unknown. Hence it has become customary to make the displacement per minute of the oil pump equal to 5 to 10 times the displacement of the power cylinder per stroke. The pump cperates all the time. The excess passes off through a spring- loaded relief valve and is returned to the pump inlet. In a Swiss design, the pilot valve has negative lap, which means that the valve is open in mid-position. The oil can flow through the open valve. In centrifugal governors, the oil pump is usually a gear pump and is arranged at the bottom of the governor spindle (see Fig. 124). (3) The size of the pilot valves and of the ports in a hydraulic governor has no influence on the total work capacity, but has a very marked influence on the work capacity in unit time and on the relay traversing time. The larger the pilot valves and ports, the shorter the relay traversing time, T r . It will be proved that, for closest regula- tion, T r must be small, which means large pilot valves. But large pilot valves offer considerable resistance to the governor. The operating fluid exerts a dynamic thrust on the edges of 186 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Governor Spindle Drivinq Shaft the pilot valve, and, besides, the danger from sticking due to one-sided pressure and unequal expansion grows with the size of the pilot valve. For this reason the diameter of the latter never exceeds one-fourth of the diameter of the power piston, and occasionally goes down to one-tenth of that diame- ter. The diameter of a floating relay valve may go up to one-third of that of the power cylinder, or even more. It should be remem- bered that the resistance to motion offered by the pilot valve (or clutch, or other force-releasing me- Oil Punnp chanism) is a load on the governor (measuring in- strument), which requires a change of speed, before motion of the pilot valve can begin (see paragraph 5 of Chapter II, on overcoming passive resistance). In relay governing, the pilot valve is part of the governor, and its frictional resistance becomes part of the internal friction of the governor. References to Bibliography at end of book: 8, 9, 12, 36, 38, 41, 66, 60, 66, 67, 69, 73. 4. Speed Fluctuations. A complete theory of relay gov- erning is very complex. The most ambitious attempt at a solution was made by Professor Stodola and was published in the Schweizerische Bauzeitung, 1893 and 1894. Professor Stodola considers the action of the water in long-pipe lines, and even introduces the action of surge tanks with compressed air. He gets a linear differential equation of the 7th order, and, by means of a determinant, derives stability relations. While his work deserves to be better known, it exceeds the limits of this volume. FIG. 124 RELAY GOVERNING 187 However, a fair knowledge of the influence of the funda- mental quantities on the process of regulation may be gained from the following simplified treatment. The statement was made in the description of types of relay governors that, with proper design, the utilized travel of the governor could be made very small, so small that the mass of the governor becomes almost negligible. The simplified theory in question neglects the mass of the governor entirely, or, if the governor has mass, assumes that the natural period of No Load New Torque Old Torque Full Load -f No Load.CciseZ ..__^-_- Mew Torque Cose? FullLoadl.CaseZ FIG. 125 vibration of the governor is very small compared to the relay traversing time. The second assumption is that the speed of the relay mechanism or power mechanism is constant. This is very nearly so in mechanical relay governors, and is true in hydraulic governors whenever the pilot valve is wide open. The usual assumptions about linear relations between speed and governor position, and between governor position and torque are likewise made here. In Fig. 125, the abscissae represent time. The ordinates represent two different quantities, namely (1) relay position 188 GOVERNORS AND THE GOVERNING OF PRIME MOVERS (which also indicates position of power control), and (2) speed of governor or of prime mover. The speed curve can be used for both governor and prime mover if we let it represent the ratio of actual speed to average speed ; for this ratio the letter U has been used. At time (3) let the torque suddenly be changed from "old torque" to " new torque." Then the torque (2) (3) is unbalanced and causes a change of speed, the rate of which is given by the equation du_ M dM_ M_ dU J_ ~dt =: T r ~udt = I u dt = ~Ts where U is the relative speed change, and where T s is the starting time of the prime mover (compare paragraph 2 of Chapter IX). In the illustration, the straight line (8) (9) represents this rate of change of speed. The latter sets the governor and relay in motion. According to the assumptions of the present elementary theory, the rate of motion of the relay is constant. It traverses the whole range from full load to no load in the "relay traversing time" T r . Hence the straight line (1) (12) indicates the rate of change of position of the power-controlling mechanism. Evidently the unbalanced torque decreases on the way from point (2) to point (4), and vanishes at the latter place. But unbalanced torque causes change of speed and is proportional to the rate of change. Hence, the curve of speed must have a horizontal tangent vertically above point (4). From mathematics it is known that a curve the tangent of which changes proportionally to the change of ab- scissae is a parabola ; but that is the case with the speed curve (3) (8) (10), in question, which may, therefore, be called the parabola of speed deviation. To the right of (10), the speed drops, because the unbalanced torque is negative. For the sake of a lucid graphical representation, the scale for the speed curve and the scale for the change of torque should be so coor- dinated that the distance (6) (12) represents the static fluc- tuation if measured with the speed scale. If the scales are chosen in this manner, the straight line (1) (12) represents not only position of relay, but also equilibrium speed of governor. The governor passes through an equilibrium position at the RELAY GOVERNING . 189 intersection of the straight line (1)(12), with the parabola of speed deviation (see point (5)). The governor and relay reverse themselves, and a new branch of the parabola is started. No matter how many branches of the parabola there are, they are all parts of the same parabola, because the latter is de- termined solely by T s and T r , and both of these quantities are constant. Straight line (5) (7) has negatively the same slope as line (1)(12). If it be extended to the left to point (8) (intersection with parabola), the two parabola sections (8) (11) (5) and (5) (7) are identical. The same is true for the next branch of the parabola, so that the total number of waves can easily be found by drawing the zigzag line (3) (5) (8) (11), etc., in the first parabola. The speed change (13) (10) can be expressed by a simpb formula. Let Z be the relative load change, or the ratio load change (0) (3) . ,, which means Z= //>W ^ N , then time (2) (4) = ZT r , maximum load (6) (12) and unbalanced torque at beginning of disturbance equals ZM m ; if this latter torque acted during the whole time T r , the speed change would be Du = j-^ZT r and the relative speed change M m T r would be U = Z 2 T r -^ = Z 2 -^. But the unbalanced torque J. u j. a drops off, and the speed curve follows parabola (3) (8) (10) instead of straight line (3) (9) . From a well-known property of the parabola, (13) (10) equals one half of (13) (9), so that the actual relative speed change is found from If, for instance, the relay time is two seconds and the starting time equals 10 seconds, then for 50% load change we have It is interesting to note that the static fluctuation of the governor and relay mechanism does not appear in this equa- 190 GOVERNORS AND THE GOVERNING OF PRIME MOVERS tion (1). In this respect, relay governing with constant speed of relay mechanism differs radically from direct-control governing. Yet it stands to reason that the static fluctuation which, 'to a certain extent, is a measure of the stability of the system, will have some influence on the process of regulation. In the diagram (Fig. 125) this influence appears as follows : The scales for torque and for speed were so selected that (6) (12) represents both full load torque, and p, where p is the static fluctuation. Therefore, if p is changed, one of the two scales must be changed, and since a straight line is more easily changed than a parabola, the scale for speed is left unchanged, which means that we must change the scale for torque. In conse- quence, inclination of line (1) (12} will vary with the variation of static fluctuation p. If p is very small, then a very small change of speed will move the governor through the whole range of adjustment. In Fig. 125 a regulation diagram for a greater static fluctuation has been sketched in dotted lines. Evidently, the first wave remains just as it was before, but the regulation is completed in less time. For governors working with a con- stant speed of relay (mechanical governors), a great static fluctuation is advantageous. The regulation diagram, Fig. 12,5, is particularly useful for impressing upon engineers the very harmful influence of friction in the governor and of lost motion in the relay mechan- ism. The effects of governor friction are illustrated in the dia- gram Fig. 126, which is of the same type as Fig. 125. Since it is rather difficult to see before the mind's eye the connection between the diagram and the governor mechanism, a constant speed relay has been shown in Fig. 127 for the purpose of sup- plementing Fig. 126. If in Fig. 127 the speed grows, the governor rises, making electric contact with the upper one of two contacts. Wires and control switches are provided for starting an electric motor in the proper direction to reduce the supply of energy. By the movement of the motor the electric contact is broken, unless the governor keeps on going with a speed as great as (or greater than) that of the relay. The moving of the contacts by the motor constitutes the return. RELAY GOVERNING 191 During the process of regulation, the governor will go ahead of the relay, and we will assume here that it can do so by virtue of the contacts being slip contacts or mercury dip contacts. More Power Electric Motor Less Power FIG. 126 By making this assumption we make the present study applic- able to the hydraulic relay in which the pilot valve can likewise overtravel and can go ahead of the relay. Attention is again called to the assumption that the governor is practically mass- less, which means that its natural period of vibration is very small compared to the relay traversing time. If there is friction in the governor, the angular velocity must grow by Du = \ qu (see paragraph 4 of Chapter II) be- fore the governor begins to move. This phase of the regulating pro- cess is represented by the straight line (1) (2), because during the time (1) (3) the un- balanced torque is constant and equal to the difference (old torque minus new torque). From point (2) on, the speed curve is parabolic, just as before. The governor displacement lags behind the motion of a frictionless governor, as indicated by the difference between FIG. 127 192 GOVERNORS AND THE GOVERNING OF PRIME MOVERS the speed curve (which also represents position of frictionless governor) and the dotted parabola (5) (4), which represents the actual motion of the governor. At point (4) the governor stops ; it remains motionless until the speed has dropped \qu below the equilibrium speed belonging to the position of point (4), which occurs at point (5). The governor changes contacts as soon as the upward motion of the contacts and the down- ward motion of the governor intersect, which occurs at point (6). The relay (in this case the electric motor) reverses and continues in its new direction until, in the diagram, the straight line representing its motion drops %q below the speed parab- ola, see points (7) and (8). Again, the governor is reversed, and so forth. It is evident that the governor never comes to rest. The tangent points of two successive parabola branches move closer and closer to the line (12) (11} representing the new torque. As soon as the tangent points fall on this line, the reverse of 'the relay occurs simultaneously with the inflexion points of the parabola. All following parabola branches are alike. From the diagram, Fig. 126, it is clear that the remaining speed fluctuation is greater than q, the detention by friction. Distance (1) (2) = (9) (10) represents not only %q but also that fraction of the total torque which is unbalanced when the parabola intersects line (12) (11), but since the whole torque corresponds to a speed deviation p (static fluctuation), the -a unbalanced torque ratio is Z .= . From equation (1) we l/i q\2 y find the corresponding speed deviation to be U = -(--] ^ to 2\ p_J 1 s either side, which means a total speed fluctuation of U due to friction of Let T 8 = 8 sec., T r = 1.5 sec., p = 2%, q = 1 %, then U f 1\ 2 1.5 T = L17%; for q = 2% > U ' would be almost 4f%. The influence of friction grows with the square in this type of regulation. RELAY GOVERNING 193 The harmful effect of friction is very much reduced, if the governor is prevented from going ahead of the relay. This favorable condition can.be obtained in Fig. 127 by letting con- tact tongue (1) fit between the relay contacts so that it abuts and presses against one or the other. The same harmful effect which is caused by friction in the governor is also produced by several other resistances. Evi- dently, any force which must be exerted by the governor to release the action of the relay mechanism requires a speed difference of the sort indicated in Fig. 126. Such a force must be exerted, if the pilot valve in hydraulic relay governors moves hard (for instance by being forced to one side through imperfections in workmanship). Such a force must also be FIG. 128 exerted in mechanical relays to keep the clutch engaged, etc. A very similar, although not strictly identical, effect is produced in hydraulic relays by lap of the pilot valve and by excessive clearance of the pilot valve in its bushing. Lost motion in the joints of the relay mechanism has an effect which is similar in one respect and different in another. Inasmuch as the governor has to travel a certain distance to take up lost motion, the problem resembles that covered by Fig. 126 ; but inasmuch as time is lost in the return by the power part of the relay, the problem differs. The effect of time lag in the mechanism is commonly explained by the method shown in Fig. 128. The diagram is so similar to Fig. 126 that no explanation is needed except this: At point (1) the governor does not reverse the relay ; the reversal occurs after the lost time TI, that is 194 GOVERNORS AND THE GOVERNING OF PRIME MOVERS to say at point (2). The same action occurs at points (3) and (4), (5) and (6), (7) and (8), etc. The process is drawn out longer than it would be without time loss, but it finally ends without lasting vibration. For practical application it should be remembered, however, that lost motion involves the features of both Fig. 126 and Fig. 128, so that lost motion, as a rule, results in never-ending speed fluctuations of small amplitude. In many relay governors, particularly in those of the hydraulic type, the velocity of " opening" and " closing" is not constant, but grows with the relative displacement of gov- ernor and piston in power cylinder. The regulation diagram then looks somewhat like Fig. 129. The slope of the power- piston velocity curve varies with the lengths of the intercepts be- tween the two curves. For those who are mathematically inclined, it may be remarked that the assumption of proportionality between slope and intercept leads to a very interesting theory of relay governing. It will be ex- plained in the author's book on advanced theory and practice of prime-mover governing. For the purposes of the present rather elementary theory we replace the curve (1) (3) (5) of Fig. 129 by the broken line (!)(&) (8) (4) (6). The speed curve then is resolved into a succession of straight lines, such as (1)(6) and (7) (5), and parabolas, such as (6) (7). The diagram is identical with Fig. 128, and the greatest speed fluctuation is easily found by graphical construction. References to Bibliography at end of book: 8, 9, 10, 12, 16, 20, 21, 36, 38, 41, 47, 56, 66, 67, 69, 73, 78, 82, 83. Position of Power Mechcrnism CHAPTER XIV GOVERNOR TROUBLES AND THEIR REMEDIES WHILE the principles which underlie all cures for governor troubles are clearly indicated in the foregoing chapters, the principal troubles, their causes and remedies, are dealt with here for the benefit of the operating engineer. 1. Regulation is not Close Enough. There is too much speed difference (pressure difference, rate of flow difference) between full load and no load. Several methods are open to make regulation closer. A smaller portion of the travel of the governor may be utilized for the whole range from full load to no load, or the governor proper may be altered. If it is of the spring-loaded type, regu- lation is made closer by using a softer (more flexible) spring. In a helical spring use more coils, or thinner wire, or grind off the outer layers of the spring (reduce its diameter). In leaf springs grind some off the width of the spring. If the spring acts on a lever, reduce the length of the latter and tighten spring. All of these changes reduce the static fluctuation of the governor. The latter (for a given prime mover and governor) must not be reduced beyond a certain limit which is set partly by the design of the governor and partly by that of the prime mover (see paragraph 2 of Chapter IX). If further reduction of the static fluctuation is desired, the flywheel effect must be increased, or a prompter governor must be used, or else tan- gential inertia must be used, or finally a compensating oil pot may be applied. For all of these, see paragraph 2 of Chapter IX. 2. Racing. Two broad classes must be distinguished : (a) Racing which appears from the very beginning of the operation, that is, with a new machine ; (b) Racing which appears after the machine has been in operation several months or years. 195 196 GOVERNORS AND THE GOVERNING OF PRIME MOVERS Racing which appears after the machine has been in operation several months or years Discussing this latter case first, we find that racing may gradually develop from several causes : First, the oil may leak out of the gag pot, or may be pumped out of it by quick motion of the governor. Refill oil pot and make such changes in its design that emptying during regular operation becomes impossible (see paragraph 2 of Chapter IX). The adjustable passage for oil may be too large, or the oil may be too light. In governors with compensating oil pot (paragraph 2 of Chapter IX, paragraph 2 of Chapter XIII) too free an opening of the oil pot will likewise cause racing. The same result is produced by air entering the gag pot with the oil. A second reason for racing (or running away akin to racing) results from such changes in the transmitting linkage between governor and power-control mechanism l as do not sufficiently shut off the supply of energy near or at the no-load position. Racing due to this cause occurs only if the load is very light. Obviously, proper change in the linkage will cure this condition. A third reason for racing (and a very prolific one) is friction, either in the governor or in the valve gear (paragraph 4 of Chapter II, 5 of Chapter II, 3 of Chapter XIII). To ascertain whether or not friction is responsible for the trouble, jiggle or pump the governor up and down by hand, taking great care that the upward and downward thrust exerted on the governor are alike. This eliminates friction (paragraph 4 of Chapter II, 2 of Chapter VIII, 4 of Chapter IX) and stops racing, if the latter be due to friction in the governor or power-control mechanism (in the case of direct-control governing) or due to friction in the governor proper, in the case of relay governing. Pumping the governor will not always eliminate frictional speed fluctuation (or pressure fluctuation, or delivery fluctua- tion in other forms of governors), in the case of relay governors, because there may be friction in the relay or in the power-con- 1 Misadjustments of this sort frequently occur in the wake of valve adjustments and repairs. GOVERNOR TROUBLES AND THEIR REMEDIES 197 trol mechanism. To discover such friction, try to move the latter mechanism by hand or by a lever in the direction of least resistance (care being taken not to overstrain the mechan- ism). If doing so stops the speed fluctuation, the frictional 'resistance is too great for the available relay force, or the available force has become less (see also under leakage). In this case the frictional resisting force should be measured. The remedy consists, of course, in removing the undue friction. The causes for a gradual or for a sudden rise of friction are so manifold that they cannot all be enumerated here. Among them are : gumming of valves or valve stems due to improper oil ; sticking of valve stems due to solid matter in steam ; lack of lubrication ; deposit of tar (in regulating valves of gas engines) ; wearing of knife edges ; binding due to lack of align- ment ; and others. It occasionally requires very patient study to hunt down and locate the spot where the undue friction occurs. A fourth reason is leakage in the pilot valve in hydraulic relay governors. Placing of pressure gauges at both ends of the power cylinder is a good means for judging whether or not leakage is responsible for the trouble. If it is, a new pilot valve and bushing are needed. In oil-pressure governors the oil pump may be worn out or deranged. A fifth reason is lost motion in the relay mechanism (para- graph 4 of Chapter XIII) . A weak spring or a weight keeping the parts always in contact in the same direction will eliminate the influence of lost motion (see (1), Fig. 123), and will, inci- dentally, indicate whether or not lost motion is responsible for the racing. A sixth reason is the dulling of knife edges in the releasing gear of steam engines. At the right cut-off the dulled edges slip so that alternately too much and not enough steam is admitted. The remedy is plain. A seventh reason is found likewise in releasing gears and more particularly in those which depend upon a partial vacuum in a dashpot for closing the valve. If the dashpots are worn or improperly adjusted, anything from scarcely perceptible racing up to disastrous running away of the engine may occur. 198 GOVERNORS AND THE GOVERNING OF PRIME MOVERS To tell whether racing or overspeeding is caused by dashpot trouble, tie a very long, flexible spring to the dashpot arms so that the spring assists the dashpot in closing the valve. Make the force of the spring in pounds approximately equal to 5 to 7 times the area of the vacuum pot in square inches. If the irregular speed was due to vacuum pot trouble, the spring will cure it. Of course, the spring looks like an afterthought, so that the vacuum pot should be renewed. An eighth reason is frequently found when the governor is belt driven. It consists in a belt which slips at intervals. Application of a belt tightener will soon tell whether or not the belt is responsible for the trouble. A ninth reason appears whenever springs are tightened or weights are added for the purpose of increasing the speed of the prime mover, unless the governor is so designed that the speed can be increased without variation of stability (see Chapter V). Increase the stiffness of the adjusting spring (by using fewer coils) or increase its lever arm, or (in the case of shifting a weight) put a spring into the oil pot. Every one of these means restores the stability (and static fluctuation) which was lost by the previous adjustment. Racing which appears immediately after the installation of the prime mover Any of the reasons which produce racing in an old prime mover will, of course, have the same action in a new prime mover. But there exist other causes which will produce fluc- tuations at once, and which, once removed, will not return. They will now be enumerated. In order to tell the source of the trouble, observe whether the racing occurs at all loads or whether it occurs only at certain loads (for instance, near the no-load position). If the racing occurs at all loads, and if none of the causes before given exist, then the static fluctuation is too small (paragraph 2 of Chapter IX), or there is not enough damping (by friction or oil gag pot) . If permissible, the static fluctuation should be increased, but this measure, of course, increases the GOVERNOR TROUBLES AND THEIR REMEDIES 199 speed (respectively pressure, or delivery) difference between full load and no load. If permissible, the oil pot should be made more effective, either by adjustment, or by a larger pot, if close adjustment overheats the oil. Again, more effective damping by oil pot increases speed fluctuations whenever the load changes. In that case racing can only be cured by more inertia in the prime mover (heavier flywheel) or by a prompter governor (smaller traversing time, Chapter IV, paragraph 2 of Chapter IX). If temporary speed fluctuation after a change of load is permissible, racing can be cured by a compensating dashpot, or similar arrangement (see paragraph 2 of Chapter IX Bee governor, Chorlton-Whitehead governor, Armstrong governor, also figures 120, 121, 123). This type of governor and oil pot deserves to be more widely used than it now is. Use of tangential inertia in governors also cures racing, but can be recommended for shaft governors only (see paragraph 3 of Chapter IX) ; and even there it is sometimes of doubtful value. In relay governors racing is caused by absence of a return in the governor mechanism (see paragraph 2 of Chapter XIII). Study of the governor will immediately tell whether a return (compensator, anti-racing device) has been provided. The pilot valve, or other relay, must be returned to make the governor satisfactory. If racing or speed fluctuations occur in one certain position of the governor and valve gear, one of two causes is present (in the case of speed governing) . Either the power-control mechan- ism does not vary the supply of energy uniformly with the travel of the governor, reducing stability to an unpermissibly small amount in one spot (paragraph 2 of Chapter IX) or else the power-control mechanism reacts in certain positions upon the governor in such a way as to reduce the stability or to remove it altogether (see paragraph 3 of Chapter III). The first case is very common in governors operating a throttle, either for steam in steam engines or turbines, or for air and gas in internal combustion engines. When a throttle is just cracked, a small increase of area changes the flow through the valve very much more than when it is almost wide open (see Fig. 130). The curve in this illustration shows the general 200 GOVERNORS AND THE GOVERNING OF PRIME MOVERS ) - J( y >)} In the text Ji ong and J short were used in place of J x > and J y >. This is correct for the illustration, but must be replaced by J 8 Ji, if the X' direction is longer than the F' axis. FIG. 140 DETAIL CONSTRUCTION OF CHARACTERISTIC FOR FIGURE 31 In the illustration (Fig. 138), (1), (2), (3), (4), (5) are different positions of the mass center of the centrifugal weight; (11), APPENDIX 215 CqforincreasedQ. FIG. 138 (13), (14), (15) are the corresponding positions of the inner end of the bell crank; (21) (22), (23), (24), (25) are the corresponding positions of the upper end of the connecting link; point (16) is the intersection of the horizontal through (1) and the straight line (11) (21)', points (17), 216 GOVERNORS AND THE GOVERNING OF PRIME MOVERS (18), (19), (20) are found by intersection of corresponding lines for the other positions. The values of W = 15 pounds, Q = 45 pounds and increased Q = 45 + 85 pounds have been laid off near point (10), (right-hand bottom Fig. 138) to the scale of one inch representing 100 pounds. The triangle of forces which is shown in Fig. 32 has been reproduced in convenient form for construction on Fig. 138, as shown in this sketch. The same construction has been carried out with increased Q, and with the various spring forces, each spring force being computed from the " initial tension plus scale times deflection." The spring force for position (2) has been marked. The centrifugal forces necessary to balance the weight of the cen- trifugal masses are found by a series of triangles of forces below point (10). The various centrifugal forces have been laid off and combined above the respective positions of the centrifugal mass. For position (3), Angular velocity = J 32 ' 2 x 12 y^ = 2.09 Vc x 15 X 5.9 APPENDIX 217 Ordinate represents Cross-sectioned area. Or d incite represents Cross/Sectioned area Method of Finding Angular Displacementfrom Impressed Moment. FIG. 139 The illustration, Fig. 139, shows a method of finding angular dis- placement from a curve of impressed moments, as exemplified by the moment caused by gravity. The illustration is so clear that it needs no comment, with the exception that the units and scales must be carefully watched. BIBLIOGRAPHY The following is offered not as a complete and comprehensive bibliography, but as a list of the most important and most readily accessible contributions to governor science. While some of the articles referred to contain theories and statements of doubtful value, they have been included for the purpose of illustrating the historical development of the theory of governing. 1. ANGUS, R. W. Theory of Machines, part I, Principles of Mechan- ism; part II, Elementary Mechanics of Machines. Ed. 2. 340 pages. 1917. McGraw. Chapter XII, pp. 201-239, treats of " Governors"; Chapter XIII, pp. 240-260, treats of " Speed Fluctuation in Machinery." Considerable Mathematics involved. 2. ARMSTRONG, E. J. Use for Inertia in Shaft Governors. 1890. (In Transactions of the American Society of Mechanical En- gineers, vol. 11, pp. 1068-1070. Discussion, pp. 1070-1080.) Non-mathematical treatment of the function of inertia in shaft- governor design. 3. ARMSTRONG, E. J. New Shaft Governor. 1895. (In Transactions of the American Society of Mechanical Engineers, vol. 16, pp. 729-733. Discussion, pp. 733-738.) Description of construction and operation of a governor em- ploying a shifting weight, the operation of which is said to be practically isochronous. 4. Automatic Electric Regulator for Steam Engines. 1903. (In Western Electrician, vol. 33, pp. 480-481.) Devoted mainly to an illustrated description of the method of operation. 6. BALL, F. H. Improved Form of Shaft Governor. 1888. (In Transactions of the American Society of Mechanical Engineers, vol. 9, pp. 300-309. Discussion, pp. 309-323.) Illustrated description of several different shaft governors, with special attention to a dashpot scheme devised by the author. 219 220 GOVERNORS AND THE GOVERNING OF PRIME MOVERS 6. BALL, F. H. Steam Engine Governors. 1897. (In Transactions of the American Society of Mechanical Engineers, vol. 18, pp. 290-308. Discussion, pp. 308-313.) Goes into the history of the development of governors in ""general, and draws conclusions as to the relative importance of the several governing forces. ~7J7' 7. BEGTRUP, J. Theory of Steam Engine Governors. 1893-1894. (In American Machinist, vol. 16, Oct. 19, 1893, pp. 2-3; vol. 16, Dec. 14, 1893, pp. 2-3; vol. 17, Jan. 18, 1894, pp. 1-3; Mar. 1, 1894, pp. 1-3; May 3, 1894, pp. 2-3.) / J Not excessively mathematical in treatment. 8. BUDAU, A. Die Geschwindigkeitsregulierung der Turbinen vom Ende der achtziger Jahre des vorigen Jahrhunderts bis auf den heutigen Tag. 1905. (In Zeitschrift des Oesterreichischen Ingenieur- und Architekten-Vereines, vol. 57, pt. 2, pp. 621-631.) Construction and theory of operation of governors. 9. BUDAU, A. Ueber die Americanischen Turbinenregulatoren mit besonderer Beriicksichtigung des Lombard- und Sturgess-regu- lators. 1908. (In Elektrotechnik und Maschinenbau, vol. 26, pp. 8-12, 28-33.) Theoretical treatment. 10. BUVINGER, G. A. Turbine Design as Modified for Close Regula- tion. 1906. (In Transactions of the American Society of Me- chanical Engineers, vol. 27, pp. 698-710.) Concerned principally with design of gates, but includes con- sideration of flywheel effect of generator and turbine, effect of draft tube and character of load. CHARNOCK, G. F. Governors and the Speed Regulation of Steam Engines. 1908. (In Mechanical Engineer, vol. 22, pp. 674-677, 687-690, 738-740, 754-756.) Chiefly theoretical in treatment. U 12. CHURCH, I. P. Governing of Impulse Wheels. 1905. (In En- gineering Record, vol. 51, p. 214-215.) Concerned with hydraulic conditions produced in a water wheel supply pipe when the governor is in action. 13. COLLINS, H. E. Shaft Governors, Centrifugal and Inertia; Simple Methods for the Adjustment of all Classes of Shaft Governors. 127 pages. 1908. Hill. (Power Handbooks.) Compiled from " Power." 14. DALBY, W. E. Steam Power. 1915. Arnold. Theory of governors treated, pp. 368-404. BIBLIOGRAPHY 221 15. DELAPORTE. Determination de certains elements des moteurs a vapeur en vue de la regulation. 1900. (In Revue de Me- canique, vol. 7, pp. 691-697.) Mathematical considerations of conditions of stability of regulation. 16. EHRLICH, P. Der Einfluss des Tachometers auf den Reguliervorgang indirekt wirkender Regulatoren. 1907. (In Elektrotechnik und Maschinenbau, vol. 25, pp. 25-30, 53-56, 76-82.) Theoretical treatment. 17. EHRLICH, P. Die elastische Verbindung der rotierenden Massen und ihr Einfluss auf den Reguliervorgang des Motors. 1906. (In Zeitschrift des Oesterreichischen Ingenieur- und Architekten- Vereines, vol. 58, pp. 152-157.) Theoretical treatment. 18. Encyklopadie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. 6 vols. 1904-1915. Teubner, Leipzig. "Regulierung des Maschinenganges " treated in theoretical manner by R. von Mises, vol. 4, pt. 2, pp. 254-296. 19. GARRATT, A. V. Elements of Design Favorable to Speed Regu- lation in Plants Driven by Water Power. 1899. (In Transac- tions of the American Institute of Electrical Engineers, vol. 16, pp. 361-394. Discussion, pp. 394-405.) Describes those peculiarities of design of plant which have special bearing on speed regulation. Does not discuss theory or construction of the governor itself. 20. GARRATT, A. V. Speed Regulation of Water Turbines. 1914. (In General Electric Review, vol. 17, pp. 557-566.) Discusses speed regulation formulas and the various factors en- tering into them. Treatment theoretical to only a small degree. 21. GENSECKE, W. Untersuchung einer mittelbaren Dampfmaschinen- regler. 1907. (In Zeitschrift des Vereines Deutscher In- genieure, vol. 51, pt. 2, pp. 1819-1824, 1895-1901.) Theoretical treatment. 22. Gray Electro-mechanical Governor. 1903. (In Western Elec- trician, vol. 32, p. 129.) Devoted mainly to an illustrated description of the method of operation. 23. GUTERMUTH, M. F. Ueber Kraftmaschinenregelung. 1914. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 58, pt. 1, pp. 408-414, 441-447, 497-501.) Theoretical treatment and tests on resistibility. 222 GOVERNORS AND THE GOVERNING OF PRIME MOVERS 24. HALL, H. R. Governors and Governing Mechanism. 119 pages. 1903. Technical Pub. Co., Manchester. Treats of governors and valve gears, and of their underlying principles. Illustrates many commercial forms. 25. HARTNELL, WILSON. On Governing Engines by Regulating the Expansion. 1882. (In the Proceedings of the Institution of Mechanical Engineers, vol. 33, pp. 408-430. Abstract of Dis- cussion, pp. 431-439.) Describes two methods of controlling the expansion gear by means of a governor. Involves considerable theory. 26. HECK, R. C. H. Steam-engine and Other Steam-motors; a Text- book for Engineering Colleges and a Treatise for Engineers. 2 vols. 1907. Van Nostrand. Vol. 2, Chapter 10, pp. 367-437, on " Governors or Regulators" contains considerable theoretical data. 27. HENRY, G. J. Regulation of High-pressure Water WTieels for Power-transmission Plants. 1906. (In Transactions of the American Society of Mechanical Engineers, vol. 27, pp. 662-680. Discussion, pp. 680-681.) Discusses the feasibility of various governor schemes. 28. ISAACHSEN, J. Das Regulieren von Kraftmaschinen. 1899. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 43, pt. 2, pp. 913-918.) Theoretical treatment. 29. JOHNSON, R. D. Surge Tank in Water Power Plants; a Device for Aid in Speed Regulation and Pressure Relief in Water Powers with Long Pressure Pipes and High Velocities. 1908. (In Transactions of the American Society of Mechanical Engineers, vol. 30, pp. 443-474. Discussion, pp. 474-501.) Includes theoretical material on the effort of turbine governors and on the differential regulator. 30. KAISER, K. Achsenregler mit wahrend des Betriebes zu bedienender Verstellung der Umlaufzahl. 1911. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 55, pt. 1, pp. 254-259, 341-345, 507-514.) Theoretical treatment. 31. KITSON, F. W. On the Allen Governor and Throttle Valve for Steam Engines. 1873. (In Proceedings of the Institution of Mechanical Engineers, vol. 24, pp. 47-55. Discussion, pp. 55-62.) Illustrated description of the governor and of its operation. BIBLIOGRAPHY 223 22. KOERNER, C. Dynamik direkt und continuirlicher wirkender Regulatoren. 1899. (In Zeitschrift des Oesterreichischen In- genieur- und Architekten-Vereines, vol. 51, pp. 413-417, 428- 432, 443-447.) 33. KOERNER, C. Untersuchung der Beharrungsregler an Dampf- maschinen. 1901. (In Zeitschrift des Vereines Deutscher In- genieure, vol. 45, pt. 2, pp. 1842-1849.) Theoretical treatment. 34. KOOB, A. Das Regulierproblem in vorwiegend graphischer Be- handlung. 1904. (In Zeitschrift des Vereines Deutscher In- genieure, vol. 48, pt. 1, pp. 296-303, 373-379, 409-416.) Theoretical treatment. 35. LANZA, G. Dynamics of Machinery. 246 pages. 1911. Wiley. ''Governors/' Chapter IV, pp. 135-199; "Deduction of the Formulae for Governor Oscillations," Appendix B, pp. 232-237. Mainly theoretical, with considerable mathematics. 36. LECORNU, L. Les Regulateurs des Machines a Vapeur. 1899- 1903. (In Revue de Mecanique, vol. 4, pp. 333-361, 466-485; vol. 5, pp. 263-285, 611-632; vol. 6, pp. 285-320, 661-712; vol. 7, pp. 557-570; vol. 8, pp. 265-279; vol. 10, pp. 220-237, 309-314; vol. 11, pp. 243-257, 468-484; vol. 12, pp. 309-328; vol. 13, pp. 109-121.) Very comprehensive. Includes illustrated descriptions of con- struction and operation of many types of governors, and the theory of governing in general. (See also Sparre, Comte de.) 37. LOEWY, R. La Theorie des Regulateurs a Ressorts Lamellaires. 1910. (In Revue de Mecanique, vol. 26, pp. 409-428.) Mathematical in treatment. 38. LOEWY, R. Der Reguliervorgang bei modernen indirekt wirkenden hydraulischen Turbinenregulatoren. 1908. (In Elektrotechnik und Maschinenbau, vol. 26, pp. 195-201, 220-226.) Theoretical treatment. 39. LONGRIDGE, M. On the Moscrop Engine Recorder, and the Knowles Supplementary Governor. 1884. (In Proceedings of the Insti- tution of Mechanical Engineers, vol. 35, pp. 150-159. Discus- sion, pp. 160-166.) Part II, "Knowles Supplementary Governor," pp. 157-159, gives brief description of the method of operation of that instru- ment. 224 GOVERNORS AND THE GOVERNING OF PRIME MOVERS 40. MANSFIELD, A. K. Notes on the Theory of Shaft Governors. 1894. (In Transactions of the American Society of Mechanical Engineers, vol. 15, pp. 929-949. Discussion, pp. 949-960.) ^^ Theoretical treatment. 41. MEAD, D. W. Water Power Engineering; the Theory, Investiga- tion and Development of Water Powers. Edition 2, 1915. McGraw. " Speed Regulation of Turbine Water Wheels," Chapter 14, pp. 408-459. " Literature/' pp. 456-459. "The Water Wheel Governor," Chapter 15, pp. 460-501. Mainly theoretical in treatment. 42. MERKL, F. R. v. Die Tourenregulierung von Kraftmaschinen mit Hilfe einer Leitgeschwindigkeit mit moglichster Vermeidung der periodischen Schwankungen. 1908. (In Elektrotechnik und Maschinenbau, vol. 26, pp. 763-769.) Comparatively nontechnical treatment of chronometric or interference governors. ^3. MISES, R. v. Zur Theorie der Regulatoren. 1908.. (In Elektro- technik und Maschinenbau, vol. 26, pp. 783-789.) Theoretical treatment. 44. MOOG, 0. Neue Turbinenpendel der Regulatorenbaugesellschaft deTemple in Leipzig. 1913. (In Zeitschrift fur das gesamte Turbinenwesen, vol. 10, pp. 161-163, 183-186, 200-202.) Describes pendulum governor of the deTemple Governor Construction Co., Leipzig, and devotes considerable space to a consideration of the theory involved. 45. New Governor for Water Turbines. 1914. (In Engineer, vol. 117, pp. 407^08.) Illustrated description of the Pitman water turbine governor. 46. New Type of Pelton-wheel Governor. 1913. (In Engineering Record, vol. 68, pp. 186.) Brief illustrated description of governor, the main purpose of which is to prevent water-hammer in the penstock, but at the same time to cut off the water from the wheel immediately. 47. PFARR, A. Der Reguliervorgang bei Turbinen mit indirekt wir- kendem Regulator. 1899. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 43, pt. 2, pp. 1553-1558, 1594-1599.) Theoretical treatment. 48. PITMAN, P. H. Improved Governor for Water Turbines. 1913. (In Proceedings of the Institution of Mechanical Engineers, vol. 77, parts 1-2, pp. 565-577.) BIBLIOGRAPHY 225 (The same, abstract, 1914. In Zeitschrift fur das gesamte Turbinenwesen, vol. 11, pp. 519-520.) Describes author's hydraulic relay governor, which does away with floating levers. Nonmathematical. 49. PORTER, C. T. Description of an Improved Centrifugal Governor and Valve. 1895. (In Transactions of the American Society of Mechanical Engineers, vol. 16, pp. 134-136.) Brief, illustrated description of an improved frictionless governor so constructed as to permit an adjustment which is close to isochronous. The valve used in connection with it has a balanced stem. 50. PROELL, R. Fortschritte im Bau von Flachreglerventilsteuerungen nebst einem Beitrage zur Theorie der Fliehkraftregler. 1913. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 57, pt. 2, pp. 1287-1295, 1339-1343.) Many illustrations of apparatus, but treatment mainly theoretical. 51. PROELL, R. Neuere Flachregler mit regelbarer Umlaufzahl. 1909. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 53, pt. 2, pp. 568-572.) Presents practical application and theory. 52. RANSOM, H. B. Method of Testing Engine Governors. 1893. (In Minutes of Proceedings of the Institution of Civil Engineers, vol. 113, pp. 194-216.) Describes test apparatus and its operation, and presents data obtained in tests of several different types of governors. 53. RATEAU, A. Les Turbo-machines. 1900. (In Revue de Mecanique, vol. 6, pp. 393-441, 539-561.) Being "Ch. V. Regularisation automatique de la Vitesse des Turbines" of a longer serial on turbines in general. Contains material on the construction of governors, but is principally concerned with the theory of governing. 54. RAUSER. Ueber Regler fur Dampfmaschinen bei Gassaugeran- lagen, Bauart Pintsch. 1902. (In Journal fur Gasbeleuchtung und Wasserversorgung,.vol. 45, pp. 89-90.) The same, abstract translation. 1902. (In Journal of Gas Lighting, Water Supply, etc., vol. 79, p. 964.) Description of construction and operation of Pintsch's ex- hauster engine governor. 55. REICHEL, E. Die Weltausstellung in Paris, 1900; Turbinenbau. 1900-1901. (In Zeitschrift des Vereines Deutscher Ingenieure, 226 GOVERNORS AND THE GOVERNING OF PRIME MOVERS vol. 44, pt. 1, pp. 657-659; vol. 44, pt. 2, pp. 1113-1118, 1348- 1359; vol. 45, pt. 2, pp. 1386-1393, 1562-1567, 1631-1636, 1837-1842.) Contains material giving descriptions and illustrations of several types of turbine governors. 56. REPLOGLE, M. A. Some Stepping Stones in the Development of a Modern Water-wheel Governor. 1906. (In Transactions of the American Society of Mechanical Engineers, vol. 27, pp. 642-660. Discussion, pp. 660-661.) Treats of the fundamental requisites which must be met by the governor, and tells how these requisites may be attained. 57. RILEY, J. C. Apparatus for Obtaining a Continuous Record of the Position of an Engine Governor and the Speed of the Engine Which it is Governing. 1903. (In Transactions of the American Society of Mechanical Engineers, vol. 24, pp. 78-97.) Describes construction and operation of instrument. "Bibliography of Experimental Work on Engine Governors," pp. 96-97. r. RITES, F. M. Analysis of the Shaft Governor. 1893. (In Transac- tions of the American Society of Mechanical Engineers, vol. 14, pp. 92-113. Discussion, pp. 114-117.) Theoretical treatment. 59. SCHNEIDER, O. Theorie der Flachregler. 1895. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 39, pt. 2, pp. 1256-1261, 1288-1292.) Theoretical treatment. 60. SEARING, E. D. Analysis of Water-wheel Governor Effort. 1915. (In Proceedings of National Electric Light Association, 38th convention, San Francisco, June, 1915, pt. 2, pp. 468-493.) Gives methods for determining governor effort in plants under operation. 61. SIEMENS, C. W. Description of an Improved Chronometric Governor for Steam Engines, etc. 1866. (In Proceedings of the Institution of Mechanical Engineers, vol. 17, pp. 19-31. Discussion, pp. 32-42.) 62. SIEMENS, C. W. On an Improved Governor for Steam Engines. 1853. (In Proceedings of the Institution of Mechanical Engineers, vol. 4, pp. 75-83. Discussion, pp. 83-87.) Illustrated description of several of the earlier types of governors, with special attention to the author's " Chronometric Governor." BIBLIOGRAPHY 227 63. SMITH, J. M. Governor for Steam Engines. 1890. (In Transac- tions of the American Society of Mechanical Engineers, vol. 11, pp. 1081-1087. Discussion, pp. 1087-1101.) Treats of the design and theory of a new type of shaft governor. 64. SPARRE, COMTE DE. Note au Sujet de la Theorie des Re*gulateurs. 1903. (In Revue de Mecanique, vol. 12, pp. 558-560.) Comment on paper by L. Lecornu, Mathematical treatment. See also reply by L. Lecornu, pp. 560-561. 65. STODOLA, A. Das Siemenssche Regulierprinzip und die Amerika- nischen " Inertie-regulatoren." 1899. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 43, pt. 1, pp. 506-516, 573-579.) Theoretical treatment. 66. STODOLA, A. Die Dampfturbinen und die Aussichten der Warme- kraftmaschinen; Versuche und Studien. Ed. 4, enl. 1910. Julius Springer, Berlin. Governors considered, pp. 335-356. Technical in treatment. 67. STODOLA, A. Die neue hydraulische Regelung der Sulzerdampf- turbine und Versuche an der 2000 K.W.-turbine des Easier Elektrizitatswerkes. 1911. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 55, pt. 2, pp. 1709-1716, 1794-1800, 1846-1852.) Profusely illustrated, and practically nonmathematical in treatment. 68. STRNAD,F. Neuere Geschwindigkeitsregler. 1907. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 51, pt. 1, pp. 23-28, 62-67.) Concerned principally with illustrated descriptions of apparatus. 69. STURGESS, J. Speed Regulation of Water-power Plants. 1906. (In Transactions of the American Society of Mechanical En- gineers, vol. 27, pp. 682-697.) Treats of the fundamental requisites of successful speed regu- lation. Gives data showing how these requisites are met in various plants under operation. 70. SWEET, J. E. Effect of an Unbalanced Eccentric or Governor Ball on the Valve Motion of Shaft-governed Engines. 1890. (In Transactions of the American Society of Mechanical En- gineers, vol. 11, pp. 1053-1055. Discussion, pp. 1055-1067.) Principally theoretical in treatment. 71. THOMA. Die neuen Turbinenregler von Briegleb, Hansen & Co., in Gotha. 1912. (In Zeitschrift des Vereines Deutscher In- genieure, vol. 56, pt. 1, pp. 121-127, 169-175.) Nonmathematical. 228 GOVERNORS AND THE GOVERNING OF PRIME MOVERS 72. TOLLE, M. Beitrage zur Beurteilung der Zentrifugalpendelregu- latoren. 1895-1896. (In Zeitschrift des Vereines Deutscher Inge- nieure, vol. 39, pt. 1, pp. 735-741, 776-779; vol. 39, pt. 2, pp. 1492-1498, 1543-1551; vol. 40, pt. 2, pp. 1424-1428, 1451-1455.) Theoretical treatment. 73. TOLLE, M. Die Regelung der Kraftsmaschinen; Berechnung und Konstruktion der Schwungrader, des Massenausgleichs und der Kraftmaschinenregler in elementarer Behandlung. Ed. 2, re- vised and enl., 699 pages. 1909. Springer, Berlin. Governors considered, pp. 275-699. Comprehensive treat- ment of design and operation. Considerable mathematical theory included. 74. THINKS, W. and HOUSUM, C. Shaft Governors. 97 pages. 1905. Van Nostrand. (Van Nostrand Science Series.) Mathematical discussion of the forces to be considered in their design. Considers only the statics of shaft governing. 75. TURBO-BLOWER GOVERNING DEVICES. 1915. (In Journal of the American Society of Mechanical Engineers, vol. 37, pp. 181-182.) Illustrated abstract translation of part of a serial article on "Die Dampfturbinen und die Turbogeblase an der Schweiz," by A. Stodola, in Schweizerische Bauzeitung, vol. 65, p. 24, Jan. 16, 1915. 76. UHL, W. F. Speed Regulation in Hydro-electric Plants. 1912. (In Transactions of the American Society of Mechanical En- gineers, vol. 34, pp. 379-432. Discussion, pp. 433-434.) Includes much theoretical material. Presents an appendix containing tables of useful data. 77. WAGENBACH, W. Fortschritte im Bau der Wasserturbinen, zugleich Bericht iiber die Schweizerische Landesausstellung in Bern, 1914- 1915. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 59, pt. 2, pp. 937-941, 955-962, 997-1002, 1018-1024, 1054-1057.) Contains considerable on governors for hydraulic turbines. Well illustrated and nonmathematical. 78. WARREN, H. E. Speed Regulation of High Head Water-wheels. 1907. (In Technology Quarterly, vol. 20, pp. 187-208.) Discusses fundamentals of speed regulation, giving some theory and outlining test methods for ascertaining the regulation of wheels under operation. 79. WHITE, W. M. and MOODY, L. F. Glocker- White Turbine Governor. 1908. (In Power and the Engineer, vol. 29, pp. 201-205.) Ex- planation of the principle of this water-wheel governor. BIBLIOGRAPHY 229 80. WOODS, J. Chronometric Governor. 1846. (In Minutes of Pro- ceedings of the Institution of Civil Engineers, vol. 5, pp. 255-261. Discussion, pp. 261-265.) Describes the method of operation of the chronometric gov- ernor invented by E. W. and C. W. Siemens. 81. WUNDERLICH, H. Ncuere Fortschritte im Bau von Turbogeblasen und Turbokompressoren. 1915. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 59, pt. 1, pp. 129-135, 174-181.) Contains considerable on governors for turbo-blowers. Non- mathematical in treatment. 82. BAUERSFELD, W. Ueber die automatische Regulierung der Tur- binen. 1905. Berlin. Relay governors theoretical treatment. 83. RULF, B. Der Reguliervorgang bei Dampfmaschinen. 1902. (In Zeitschrift des Vereines Deutscher Ingenieure, vol. 46, pt. 2, pp. 1307-1314, 1399-1403.) Graphical method of determining speed change. INDEX Actual motion of governor, 81 Addition of constant force at sleeve, 37 Additional power-control mechanism, 201 Adjustment of equilibrium speed, 49 of rate of flow, 151 of static fluctuation, 180 Adjustments in shaft governors, 76 Amplitude, maximum permissible, 86 of first speed wave, 96 of vibration due to impressed force, 87, 217 Angular displacement from curve of impressed moments, 217 Anti-racing device, 176 Armstrong governor with compensat- ing oil pot, 111 Assembly drawing, relay governor, 183 Automatic disconnecting device for capacity governor, 58 Auxiliary governor, 145 Auxiliary spring, 200 B Balanced regulating valve, 38 Bee governor, 109 Belt slip a cause of racing, 198 Binding due to lack of alignment, 197 Brake resistance traversing time, 133 Capacity of oil pump, relay governing, 185 Cataract governors, 143 Centrifugal compensator, 144 force, 7 force, equivalent, for oblong weights, 41 mass, size of, in shaft governors, 72 moment of eccentric and strap, 64 moment of oblong weights, 214 moment of rotating masses, shaft governors, 63 Change gears, 50 of angle of characteristic, 29 of pressure, accumulated effect of speed change, 153 of spring tension and leverage, shaft governors, 75 Changing lever arm and initial tension, 54 Character of regulation, factors deter- mining, 132 Characteristic, characteristics curves of governors, 26, 52, 214 detail construction of, 214 of displacement pumps, 156 of Hartnell type governor, 36, 52 of isochronous governor with ob- long weights, 43 of Proell type governor, 35 of turbo-compressor, 156 Chorlton-Whitehead governor, 109 Chronometric governor, 141 Close regulation, 21 Closeness of regulation, pressure gov- erning, 153 Combination centrifugal and pressure governor, 167, 168 hydraulic and mechanical relay governor, 173 Compensating oil pot, shaft governors, 111 return, 176 Compensator, 179, 195, 199 friction wheel, 183 Compound centrifugal forces, 14, 118 Constant delivery compared with con- stant speed, 148 force at sleeve, addition of, 37, 38 rate of flow governing, 148 solid friction, cause of vibrations, 123 231 232 INDEX Coriolis' forces, 14, 118 Cosine governor, xvi Criterion of stability, 25 Cut-off control, lack of stability, 155 Cyclical vibrations of governors, 85 eliminate friction, 87 Detail construction of characteristic, 214 Detention by friction, 15, 209 due to forces at collar, 55 eliminated by vibration, 16 importance in relay governors, 171 Diagram of vibration, 125 Direct-control governors, 3 limits of, 170 Discarded types of speed governors, 139 Distribution of forces between eccen- tric and strap, 68 of mass in governors, 84 Displacement pumps, automatic sta- bility, 154 characteristics of, 156 Double floating relay valve, 182 relay, 181 Dust and tar in gas, 171 Dynamic action of steam flow, 170 regulating force, 102 Dynamometric governors, 141 E Eccentriq and strap, force distribu- tion, 68 Effect of adding constant force at sleeve, 38 of outside forces impressed on governor, 81 of valve inertia, 66 Electrical governor, 146 Energy released by governor, 173 beyond control of governor, 201 Equation of motion of governor mass, 133 Equilibrium speed, 23 adjustment of, 49 Equivalent centrifugal force with ob- long weights, 41 friction, 18 mass of governor parts, 46, 48, 78, 84, 117, 132 static fluctuation, 114, 115 Erratic behavior of shaft governors, 67 Experimental determination of gov- ernor's strength, 5 Floating relay valve, 182 Flywheel effect not great enough, 201 weight per horsepower, 111, 199 Force, Forces acting on governor sleeve, 37 on collar, variation of, 50, 51 outside, impressed on governor, 81 required for damping vibrations, 103 Fraction of load suddenly applied, 106 Friction a cause of racing, 196 between eccentric and strap, 67 detention by, 15, 55, 209 drive of governors, 51 eliminated by cyclical vibrations, 87, 203 eliminated by impressing vibra- tion, 171 harmful in pressure governing, 166 increases resistibility, 83 influence in relay governors, 192 internal, 18 moment, shaft governors, 68 of compound centrifugal force, 118; not adjustable, 119 of governor collar, 55 of joints, 210 of pilot valve, 186 of valve gear, shaft governors, 66 of working fluid of prime movers, 204 wheel compensation, 183 Frictionless governor, regulation im- possible with, 101 INDEX 233 Gas engines, regulation of, 21 General Electric Co.'s volume gov- ernor, 150 Governing, for constant rate of flow, 148 for constant pressure, 153 of hydraulic turbines with long supply lines, 172, 178 principles of, 2 Governor, governors as a mechanism, 28 as speed counters, 25 displacement proportional to load change, 136 too light for vibratory forces, 202 too small, 202 troubles and remedies, 195 vibrates, jerks, 202 with considerable internal friction, 21 with small internal friction, 21 Greatest speed variation, 96, 128 limiting case, 103, 104 estimated from limiting case, 137 H Harmonic vibrations of limiting case, 103 Hartnell type governor, characteristic, 36 Helical spring stiffness, changing, 195 Hydraulic (cataract) governor, 143 turbines with long supply lines, 173, 178 I Impressed forces keep governor vi- brating, 71 Inertia governors, 9, 11 moment, 13 moment of reciprocating parts, shaft governors, 64 (tangential) in shaft governors, 71 Influence of friction in relay governing, 192 of throttling in non-condensing engines, 205 Ingersoll-Rand volume governors, 150 Insufficient size of governor, 202 Interaction between governor and prime mover, 95 Internal combustion engines, self -regu- lating properties, 206 governor friction, 18, 20 Inverted suspension type governor, characteristic, 35 Isochronous governor, xvi, 24 characteristic, 43 impracticability of, 108 with tangential inertia, 116 Knife edge joints, 171 edges, in releasing gears, 197 edges wear too fast, 203 Knowles compensator, 144 Lag of governor motion behind speed change, 129 Lap of pilot valve, 193 Large pilot valves, effect of, 176 Leakage in pilot valve, 197 Limitations of volume governing in blast-furnace practice, 152 Limiting case, 97 greatest speed fluctuation, 104 harmonic vibrations, 103 pressure regulation, 159 relative speed fluctuation, 108 with solid friction, 120 Limits of direct-control governors, 170 Load governors, 141 Lost motion, eliminated, 183 in joints, relay governing, 193, 197 M Mass, equivalent, effect of, 84 distribution in governors, 84 of governor parts, 46 of governor springs, influence of, 73 Maximum permissible amplitude, 86 Minimum spring scale for stability, 162 Moment due to friction between ec- centric and strap, 67 due to friction of valve gear, 66 due to inertia of reciprocating parts, 64 234 INDEX Natural period of vibration of governor, 77 in relay governing, 191 Nordberg governor, 59 Oblong swinging weights reduce promptness, 44 weights, centrifugal moment, 214 Oil as fluid for relay governors, 173 Oil gag pot, 103, 105, 196 compensating, in shaft governors, 111 in pressure governing, 164 size of, 113 with floating springs, 109, 195, 199, 200 Outside forces impressed on governor, 81 Parabola of speed deviation, 188 Parallel operation of A. C. generators, 116, 179 of compressors and blowers, 167 Parsons system of puff governing, 171 Passive resistance, 18 Pilot valve, 174, 185, 186 can overtravel, 191 lap, 193 leakage, 197 Positive displacement pumps, 148 Pressure governing, 153 effect of friction, 166 limiting case, 159 minimum spring scale, 162 oil gag pot in, 164 speed damping, 165 traversing time, 163 without receivers for intake and discharge, 165 Principles of relay governing, 174 Process of regulation, 180 Proell type governor, characteristic, 21 Promptness, 44, 45, 195, 199 Puff governing, 171 Pump governors, 143 R Racing, 195, 196, 198 caused by absence of a return, re- lay governors, 199 caused by friction in governor or relay, 196, 197 caused by incorrect throttle con- trol, 200 caused by releasing gears, 197 caused by slipping belt, 198 caused by valve adjustments, 196 of centrifugal pumps and blow- ers, 201 point, shaft governors, 76 Rateau multiplier, 150 Rateau volume governor, 150 Reciprocating valves, inertia of, 66 Regulating force, 7 dynamic, 102 equation, derivation, 209 static, 102 Regulating moment of tangential in- ertia, 9 valve, balanced, 38 Regulation by frictionless governors, 101 Regulation not close enough, 195 of gas engines with dirty gas, 21 Relative change of governor position, 130 change of load, 100 change of speed, 130 speed fluctuation, limiting case, 108 Relay, double, 181 governing, 170 governing, effects of governor mass, 172 governing, principles of, 174 governor, 3 governor, assembly drawing, 183 governor, effect of friction in, 171 governor, influence of static fluc- tuation, 190 governor, lost motion in joints, 193, 197 governor, relative speed change, 189 governor, theory, 186 INDEX 235 governor, variable relay velocity, 194 governor, with rigid return, 175, 176 governor, without return, 175 governor, work capacity, 184, 185 mechanisms, 173 traversing time, 172 valve, floating, 182 with slowly yielding return, 178 Representative point, 98 Resistibility, 81 increased by friction, 83 Reversed speed curve, 116, 180, 181 Safety device to prevent overspeeding, 168 Self -regulating features of prime mov- ers, 204 of the load, 204, 207 Sensitiveness, 24 no guarantee for close regulation, 25 Shaft governors, 5, 10, 61 adjustment of, 75 calculation of, 62 compensating oil pot in, 111 erratic behavior of, 67 friction moment, 68 inertia moment of reciprocating parts, 64 influence of tangential inertia in, 71, 85 spring moment, 69 unbalanced weights, 89 Shape of centrifugal weights, 40 Siemens' principle, 140 Sluggishness, 15 Solid friction, as damping agent, 117, 118, 123, 124, 127 limiting case, 120 minimum static fluctuation, 121 for damping vibrations, 83 Speed counter, 23, 25 Speed damping, 165 Speed fluctuation, excessive, 201 greatest, 128 limiting case, 103, 104 Speed limit, centrifugal governors, 167 Speed of relay mechanism, 187 Speed restorer, 179 Speed rise due to energy beyond con- trol, 137 Speeder, 54 Spindle governors, 10 Spring in oil gag pot, 200 Spring leverage, change of, 75 Spring moment, 69 Spring stiffness, changing, 195 Spring tension, change of, 75 Springs, influence of mass of, 73 Stability, 23, 29, 31, 32 found from characteristic, 31 of regulation, 33, 96, 151, 154 of regulation, enforced by oil pot, 105 of regulation, influence of tan- gential inertia on, 113 temporary, by compensating dash- pots, 164, 178 Starting time, 107 of inertia mass, 133 Static fluctuation, 23, 32 equivalent, 114, 115 minimum with solid friction, 121, 122 negative, 116, 180 pressure governing, 164 reduced by tightening spring, 52 Static regulating force, 102 Static speed fluctuation of regulating mechanism, 177 Steam relay pistons, 173 Stodola's theory of relay governing, 186 Strength, experimental determination of, 5 of centrifugal governors, 4 of volume governors, 150 Tangential inertia, 8, 195 in shaft governors, 71, 85 influence upon stability of regula- tion, 113, 116 Temporary stability, compensating oil pot, 164, 178, 180 236 INDEX Thompson governor, 73 Throttle control, inherent stability of, 155, 205 Total inertia moment, 13 Traversing time, 45 brake resistance, 133 calculation of, 47 of relay, 172 pressure governing, 163 Turbo-blower, characteristics, 157 surging of, 157 U Unbalanced forces impressed upon governors, 39 Unbalanced weights in shaft governors, 89 Unloading devices, 158 Vane governors, 144 Valve adjustments, racing caused by, 196 Variable speed governors, 49 Velocity pumps, 148 Vibrating weight, path of, 91 Vibration diagram, 125 effect upon governor friction, 16, 203 natural period of, 77 Vibrations, cyclical, 86, 86 harmonic, 103 limited by governor resistibility, 82 Volume governors, 150 W Watt governor, xvi, 78 Watt type governor, detention by friction, 209 Watt's principle, 139 Weight of flywheel required per horse- power, 111, 199 Weiss governor, 59 Wischnegradsky's theorem, 101 Woodward governor, 183 Work capacity, centrifugal governors, 14,33 relay governors, 184 Work done by steam beyond governor control, 138 RETURN MAIN CIRCULATION ALL BOOKS ARE SUBJECT TO RECALL RENEW BOOKS BY CALLING 642-3405 DUE AS STAMPED BELOW UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 GENERAL LIBRARY - U.C. BERKELEY BOOOIOIMSM UNIVERSITY OF CALIFORNIA LIBRARY