no ?. California. University. Department of Tnechan5cal engirt : iletln. no UNIVERSITY OF CALIFORNIA AT LOS ANGELES NO. qjartment of Jjedtmtkal Jnginwing, UNIVERSITY OK CALIFORNIA. HYDRAl'LIC STEP. BERKELEY: BULLETIN No Iqjartment of jkhanical tnjjtnemng, UNIVERSITY OF CALIFORNIA. HYDRAULIC STEP. BERKELEY: 1887. Analysis of a Hydraulic jStep The machinery which was used in connection with a cer- tain pumping system for irrigation contained a vertical shaft which supported a heavy load and rotated in water at a high velocity. The whole plant was required to be of such con- struction as to dispense with the necessity of supervision. Under these conditions the footstep of the shaft claimed careful consideration. Lubrication, except with water, was out of the question. The use of lignum vitse was also of doubtful utility on account of the grit, of which the water carried considerable quantities. Girard's method of forcing water between the surfaces of contact by means of a force- pump* was too complicated and lacked permanency. In this dilemma it occurred to me that a permanent water pressure might be established through the agency of a forced vortex, created by the rotation of the shaft itself, which, in acting against a disc attached to the shaft, could be made to balance a portion of or the whole load. Fig. 1 is a sectional view of the footstep, through the axis of shaft, S. It represents a cylindrical vessel, composed of a number of identical compartments m, m, etc. (the drawing shows three). The concentric shaft S, passes freely through these and rests on an ordinary footstep, which is attached to the bottom of the lowest compartment. The shaft carries for each division a disc of a diameter just sufficient to clear the surrounding face of the cylinder, and midway between top and bottom. The upper part of each disc is provided with radial ribs or blades r, T, and the bottom of each vessel is pro- vided with stationary ribs r, r. *See Armengaud, " Vignole des Mecaniciens," p. 13, and Note "Sure lea Experiences de," etc., in Comptes Rendues de 1'Academie dea Sciences a Paris, T. 55. 41S65O For our purpose, and, as will be seen also in common prac- tice, it is sufficient to consider only one compartment. If the shaft rotates and the vessel is filled with water, the wings, r, will cause the latter to rotate with the same angular velocity. A forced vortex is created, and the pressure of the water will increase from the center towards the circumference. The water in the space below the rotating disc will be under a Fig. 1 ,' uniform pressure, equal to the maximum pressure above; hence the disc, and with it the shaft, will be under the influ- ence of an upward pressure, equal to the difference of the total pressures acting upon both sides of the disc. Let 67 denote the angular velocity of rotation ; r, the radius of the rotating disc in feet; r , the radius of the hub in feet; n, the number of revolutions per minute; v, velocity of rotation at any given distance; v , the velocity of rotation at r,; y, the acceleration of gravity; y, the density of the fluid; m, the number of rotating discs; PI, the total pressure upon the upper face of the disc; P 2 , the total pressure upon the lower face of the disc; P=P. i PI, the resulting upward pressure. Assuming r to be zero, and neglecting fluid friction, we may obtain the pressure P as follows: The hight to which the fluid would rise at a distance x> from the axis, measured from the plane bVb, is equal to v* &?V 2~~-$ The ordinates representing these heads terminate, therefore, in the surface of a paraboloid of revolution, with its vertex at V, and its volume aVa, is equal to one-half that of the cylinder baab, hence Pi=y n r*T~> and considering that the maximum specific pressure, y r 2 r acts upon the en- u tire lower face of the disc, we have P 2 =y n r* , hence r *g For any other value of r we find Pi=y n (? t2 ?'o) 8 T~ ( see notie below), and since the maximum specific pressure acts upon the entire area of the disc, we have: P,r*(r*^J)r~,aad P=y7f (r 4 r^)~- If we express oo in terms of n and r, r , in terms of d, d a (diameters), we find: P=0.00104 (d*d*) n*, and for m rotating discs P=0.00104 (d*d*} n* m. NOTE. The specific pressure p, at distance x, is equal t y (a? rj) a? 2 rr ca 2 jnce Pi 1 ity | (x 2 r=) x dx=n y (r 2 r=) 2 y^ ' The work in the laboratory has to deal with two subjects of investigation, viz. : Pressure, and work consumed by f ric- tional resistance of the disc rotating in water. The value for pressure, found by the above formula, gives the same result as that found in practice, provided fluid friction does not exist; its presence" will not only decrease the maximum pressure, y T* , but also the total pressure under u the disc, which is now no longer the resultant of a uniform maximum pressure under the disc. The cause of this is to be found in the clearance or space between the movable and stationary ribs and surfaces. The rotating ribs do not impart the same motion to all the water above the disc. There remains a film of fluid next to the upper stationary surface, either wholly stationary, or if slippage takes place, partially so. Within the free space, or clearance, inner friction causes the water to assume gradually, according to some law, the full velocity of rotation. The pressures due to the normal forces being thus modified, radial currents towards the axis of rotation are induced, while a compensating flow from the axis, in conformity to the law of continuity, takes place between the rotating ribs. In other words, potential energy is converted into kinetic energy. The same action, but with currents reversed, takes place in the compartment below the disc. The problem mapped out in the foregoing was given to the students as laboratory work. It calls for mechanical appli- ances, instruments of precision for measurement, etc., of varied character; it appeals to their knowledge acquired in the lecture-room to reach correct conclusions, and the results are not merely of temporary value but find important appli- cations. The students made the experiments and assisted in the de- signing and drawing of the apparatus, all of which was built, with the assistance of the students, by Jos. A. Sladky, Super- intendent of the Machine Shops. His skill and ready com- prehension of the cardinal points aimed at, have largely con- tributed to the successful results of the experiments. Experiments for Pressure. The footstep constructed to suit experimental convenience is shown in Fig . 2 b , which represents a section through the axis of the shaft. The inner diameter of the cylinder is 12 inches in order to admit a 12-inch disc; but discs of less diam- eter can be inserted together witli corresponding rings V, and stationary ribs R', R. The shaft >S", passes through another shaft, S, which has its bearings in the hub F, of the frame, and on which are mounted the driving pulleys P, P. Rotation is communi- cated to S' by means of a couple G. In this way the tension of the belt is entirely taken up by the shaft S, and its bear- ings F, and since the resisting forces upon the disc occur only in couples, the shaft S', is left free to slide in its bearings without friction. The upper end of the shaft S', engages one end of the balance lever L, through the medium of the socket or step D, which is prevented from rotating, as seen in the figure, in order to protect the pointed pivot which acts upon L. The other end of lever L, is supported by the platform of a scale C, Fig. 2, which measures the pressure upon the disc. The experimental data are given in the following table, in which d denotes the diameter of the disc in inches, n " ' number of revolutions per minute, R " " reading of the scale, 4| 21.2 20.1 0.95 7" 1155 160.5 148.6 0.93 12" 462| 222 8 2208 0.98 932 105.7 96.1 0.92 443 204.1 197.8 0.97 764 70.2 668 0.95 300; 94.7 i 95.1 1.00 424 21 6 22.2 1.03 252 680 65.6 i 0.96 284 9.7 10.2 1.05 186; 21 3 20.1 j 0.94 1 i 1 Combining these results, we find the ratio for the several discs: For 5" disc 926 adopted value, 0.95. I H U*J ^ 12 " 0973J Multiplying .95 into the coefficient of P (see page 5), we find: P=0.001 (d* d*} n* m. Resistance. The arrangement as represented in Fig. 2, which was de- signed to measure pressure and resistance at the same time, would have necessitated the elimination of the friction mo- ment of the upper footstep of shaft S', which is great, con- sidering the pressure to which the step is exposed. For this reason the experiments for determining the resistance were made independently of those for pressure, which made it pos- sible to eliminate the latter by arranging the ribs in the same way both under and above the disc. The resulting resistance of the footstep would therefore be equal to one-half of the sum of the two resistances found for movable and stationary ribs on each side of the disc. Fig. 3 represents a vertical section of the dynamometer, which was designed and built for these tests. Fig. 4 is a front view and Fig. 4 a a top view of the index plate. 11 The two journal bearings J J, support the shaft S', which carries the pulley P, the graduated disc D, and the cylin- drical casing C. This shaft S', is bored to receive the shaft S, shown in dotted lines. Connection between the two is established through the medium of a spiral spring t, the ends of which are attached to the box C, and shaft S, respectively. Fig. 3 The lower end of S is provided with a disc u, which car- ries two coupling pins for the purpose of conveying rotation to the driving pulley for the footstep A, Fig. 2. The relative position of the two shafts, that is, the tension of the spring t, measures the moment required to communicate motion to the footstep. Since precision of measurement was aimed at in the construction of the dynamometer, the recording apparatus was 12 not designed for reading while in motion, because such an ar- rangement would have impaired its sensitiveness. The bar x, which carries the index hand (see Fig. 4 a ), rests on the plate D, and is pivoted on the shaft S'. n, n, are two springs attached to x ; their outer ends are bent over as seen in the drawings. If these springs are forced down so as to bear upon x, their outer ends enter the forked ends of the arms A, A, which are connected with the disc u (see Fig. 4). If released, they become disengaged from A, A, but bind x to the disc D (see Fig. 3). L, L are two levers, pivoted to the bar x. In the position in Fig. 4, their short ends C, C, abut against the edge of a slot cut in the springs n, n, and keep them down. By a slight depression of L, L, however, the arms C, C, will release their hold, pass through the slots and the springs will fly up (see Fig. 3). 13 At any time, during the action of the dynamometer, the position of the index hand may be secured by pressing upon a knob y of a rod, which, by means of lever z, sleeve S and levers L, disengages C, C. During the use of the dynam- ometer the arms A, of shaft S, are connected with the spring, plate x, and index hand, Fig. 4, which relative position is secured as explained. Resistance Determined by Dynamometer. Let a= index reading of dynamometer. o- 1 =index reading of dynamometer without water in the footstep, to eliminate frictional resistance and the correction for zero point of the graduated disc. M =corrected moment of resistance of footstep. f _0. 02298 value of one division of index plate. r= radius of pulley on footstep shaft. r!=radius of pulley on dynamometer. r =0.494 r, Then we have M=e r (a I )=0.01135 (,). 14 One hundred and thirty-four tests for moment of resistance of movable and stationary wings, were made. The motive power was derived from a small Pelton wheel, but the water supply proved insufficient to obtain the desired velocities. The very laborious computations by the method of least squares also established the fact that the results could not be trusted on account of the unavoidable slipping of the belts. For these reasons I concluded to apply a different method, applicable not only for the step, which was the original object, but in general for the determination of fluid resistance to ro- tating discs, cylinders, etc. . IS Resistance Measured by Read ion. The apparatus built for this purpose is shown in Fig. 5 in vertical section. M, M, represents a closed cylinder, 15" by 7", in two sec- tions bolted together, accurately turned and balanced and resting on a conical pivot U which is adjustable by a set screw. The shaft 8 rests upon an inverted conical pivot attached to cylinder M, and has its upper bearing in a sleeve, Pig. 6 which is inserted in the hub of the frame M. This shaft is provided with a shoulder and set screws for attaching objects to be experimented with. D, D, represent plates or discs capable of being adjusted to any position required for the test. The upper journal bearing of the shaft passes through a central opening in the cylinder, just large]enough not to touch. The position of the cylinder is maintained by four friction rollers R, R. 16 The frame F is provided with bearings for a shaft S', which carries the driving pulley P. Motion is transmitted to shaft S by means of a coupling which is represented in Fig. 6 on a larger scale. The disc or crosspiece a of the driving shaft S' carries two pins p, p. The disc d of the driven shaft S has pivoted to it at c, c, two levers, which engage each other at the center of the shaft r, while the outer ends receive the thrust of the pins p, p, at i, i. The points i c r c i, and also the centers of the pins p, p, are contained in diameters. It is evident that both~pins bear against the levers with equal pressure* and that in consequence the resultant pressure transmitted to the shaft journals is zero. The moment of resistance is measured by the reaction upon the vessel M. The rotation of this vessel is not influenced by journal friction in the stationary bearings, but the point to be aimed at was precision of motion that is, the avoidance of vibrations caused by bearings enlarged by wear. Speed Indicator. The recording apparatus for the velocity of rotation is ap- plied at the upper end of shaft S'. The counter in common use which is applied at the end of the shaft, etc., could not be trusted, as the possibility of slipping and the source of error in time-interval, caused by a lack of prompt application and withdrawal of the instrument, excludes observations for small time-intervals, which become tedious, and are, under certain conditions, even inadmissible. The following is a description of the recording apparatus designed and built to obviate these defects: A , N, Figs. 8 and 9, represent two identical speed indicat- ors, which are pivoted at 0, 0. In the position shown they *In an ordinary pin coupling skill may produce practically an equal contact, but an almost imperceptible change of the centers of the shafts will produce an unequal distribution of the pressures. In practice these centers cannot remain in line, hence the above result. In the coupling described the equality of pressure is not interfered with by such derangement. 17 are in gear with a pinion P, Fig. 8, which is attached to the end of shaft S'. These positions are maintained by springs 8, S, and checks. The position which corresponds to contacts of the armatures A with their electromagnets M, M, leaves the spur wheels and pinion P out of gear. The contact is maintained by a projection on each indicator, which Fig. 7 locks behind a tooth on the spring bar H (see Fig. 7). The action of the indicator is as follows: In Fig. 9 both indicators are in gear with pinion P. While the pointer of the pendulum is between the two contact springs, and just preceding that reading of the dial-plate of the clock, from which it is intended to count, the push button Rl must be depressed. 18 As soon as the pendulum has completed its stroke and pro- duced contact, a current from battery B passes in the direc- tion of the arrows, the electromagnet attracts the armature, and the indicator N\ is thrown out of gear against the spring and retained as described. Just preceding the specified in- terval of time as before, the push button R2 must be depressed, which causes the indicator N 2 to be thrown out of gear and held in that position as explained. Fig. 9 The difference of the two readings gives the number of revo- lutions made during the interval. The result is liable to be influenced by an error in the time intervening between the closing of the current and the throw- ing out of gear of the index. If the time-interval were the same for both indicators, this error would be eliminated; but such can hardly be expected on account of difference in fric- tion, distribution of masses, intensity of electromagnets, etc. This error will influence the result the more, the smaller the clock interval is. We can, However, from observed data, 19 compute a correction which will make it possible to obtain ex- cellent results for very small intervals. Let the time intervening between the closing of the currents and the release of the spur wheel from the pinion be repre- sented respectively by ^ and ,, for counters 1 and 2. Let i\ denote the first reading of clock ; r 2 denote the second reading of clock; T denote the difference of these readings ; then n + ^^rtimeat instant of release of counter 1; and r. 2 + 2 =time at instant of release of counter 2. Hence, r. 2 + t. 2 (r 1 +* 1 )=2 T +^ ^=true time interval, which corresponds to the difference of readings of the dial. If we repeat the observations for the same T, but in order reversed regarding the push buttons, and if we denote by d and d iy respectively, the difference of dial readings, we have: T+ (t z ,) corresponding to d ; T (t. t