key: cord-0283909-wrutvzmb
authors: Paliwal, Vineet; Babu, N. Ramesh
title: Prediction of Stability Lobe Diagrams in High-Speed Milling by Operational Modal Analysis
date: 2020-12-31
journal: Procedia Manufacturing
DOI: 10.1016/j.promfg.2020.05.049
sha: 85f56670af207e80767cbc0fd6afbf971c406b81
doc_id: 283909
cord_uid: wrutvzmb

Abstract Self-excited regenerative vibration or chatter limits the primary requirements like productivity, surface finish and dimensional accuracy of high-speed machining. It is the most critical factor that severely affects the tool-life and life of the machine tool. Out of several methods followed for suppressing and avoiding chatter, machining conditions chosen with stability lobe diagram is the most reliable way. Stability lobe diagrams are typically generated by knowing specific cutting force coefficients and tool point frequency response functions (FRFs). Inaccurate use of tool point FRFs significantly affects the stable regions of stability lobe diagrams. As tool-point FRFs are influenced by gyroscopic effect, centrifugal force, thermal change in bearing dynamics, etc. during machining, it is important to consider the effect of these factors on tool point FRFs in order to generate accurate stability lobe diagrams. The present work covers a new approach for estimating tool point FRFs during machining operations, employing cutting tool vibration signals. For measuring the vibration at the tip of end mill at different spindle speeds, a non-contact laser vibrometer is used. In order to remove the tooth pass frequency and its harmonics from the measured signals, a comb filter is employed. The filtered signal is subjected to Operational Modal Analysis (OMA) in order to derive tool point FRFs. With the estimated FRFs at different spindle speeds, the stability lobe diagrams are drawn for high-speed milling machine. Comparative study of stability lobe diagrams, drawn with static modal analysis and Operational Modal Analysis, have shown that the cutting conditions chosen from stability lobe diagram derived with OMA, are found to be more realistic for avoiding chatter during high-speed milling applications.

Milling is one of the most extensively used machining operations in metal cutting industries because of its capability to cut different ranges of materials. High-speed milling is chosen for attaining higher productivity and better surface quality [1] . Undesired vibrations during high-speed milling are the main restrictive factors that limit the productivity and affect the life of the tool and certain components of the machine tool. Among different types of vibrations such as impulsive, forced and self-excited regenerative vibration, self-excited regenerative vibration, also considered as chatter negatively influences the dimensional accuracy and surface quality of the machined part. Various detrimental effects of chatter are given in Fig. 1 . Attempts were made to study the mechanism, behavior, and factors responsible for chatter during machining. Tobias and Tlusty [2, 3] explained the mechanism of chatter in orthogonal and oblique cutting by self-excited regenerative vibration and also proposed the stability lobe for selecting chatter-free machining parameters i.e. depth of cut and spindle speed. Avoidance of chatter is possible by either changing the system behavior such as stiffness enhancement, active or passive damping, etc. [4] [5] [6] [7] [8] or by selecting stable cutting parameters with the application of stability lobe diagrams. Out

Milling is one of the most extensively used machining operations in metal cutting industries because of its capability to cut different ranges of materials. High-speed milling is chosen for attaining higher productivity and better surface quality [1] . Undesired vibrations during high-speed milling are the main restrictive factors that limit the productivity and affect the life of the tool and certain components of the machine tool. Among different types of vibrations such as impulsive, forced and self-excited regenerative vibration, self-excited regenerative vibration, also considered as chatter negatively influences the dimensional accuracy and surface quality of the machined part. Various detrimental effects of chatter are given in Fig. 1 . Attempts were made to study the mechanism, behavior, and factors responsible for chatter during machining. Tobias and Tlusty [2, 3] explained the mechanism of chatter in orthogonal and oblique cutting by self-excited regenerative vibration and also proposed the stability lobe for selecting chatter-free machining parameters i.e. depth of cut and spindle speed. Avoidance of chatter is possible by either changing the system behavior such as stiffness enhancement, active or passive damping, etc. [4] [5] [6] [7] [8] or by selecting stable cutting parameters with the application of stability lobe diagrams. Out

Milling is one of the most extensively used machining operations in metal cutting industries because of its capability to cut different ranges of materials. High-speed milling is chosen for attaining higher productivity and better surface quality [1] . Undesired vibrations during high-speed milling are the main restrictive factors that limit the productivity and affect the life of the tool and certain components of the machine tool. Among different types of vibrations such as impulsive, forced and self-excited regenerative vibration, self-excited regenerative vibration, also considered as chatter negatively influences the dimensional accuracy and surface quality of the machined part. Various detrimental effects of chatter are given in Fig. 1 . Attempts were made to study the mechanism, behavior, and factors responsible for chatter during machining. Tobias and Tlusty [2, 3] explained the mechanism of chatter in orthogonal and oblique cutting by self-excited regenerative vibration and also proposed the stability lobe for selecting chatter-free machining parameters i.e. depth of cut and spindle speed. Avoidance of chatter is possible by either changing the system behavior such as stiffness enhancement, active or passive damping, etc. [4] [5] [6] [7] [8] Altintas and Budak [9] proposed an analytical approach in the frequency domain, known as Zero Order Approximation (ZOA). Though this approach is much simpler and popular, it does not provide accurate results for highly intermittent milling operations. To overcome this limitation, Altintas and Budak [10] proposed a Multi-frequency Solution (MFS) for the prediction of stability lobe diagrams. Insperger and Stepan [11] proposed a time-domain solution to find out the stability limits based on the semi-discretization method. Various other timedomain approaches such as full-discretization [12] , Chebyshev collocation method [13] and integration method [14] were also proposed for generating stability lobe diagrams. All these approaches mainly consider time-invariant specific cutting force coefficients and system dynamics to derive stability lobe diagrams.

System dynamics is mainly defined by tool-point FRFs and are usually obtained by performing an impact test at the tip of the milling cutter. The major limitation of this test is that it can only be performed during the static condition of the machine tool. However, during machining, several factors such as machining parameters [15] , centrifugal force, gyroscopic moment [16, 17] , thermal loads and deformations, change in bearing dynamics [18, 19] , etc. influence tool point FRFs which in turn affect the stability lobe diagrams. Therefore, the toolpoint FRFs estimated by performing impact tests during static condition do not represent the exact dynamic behavior of spindle tool assembly. To consider the effect of the abovementioned factors on the dynamic behavior of cutting tool, tool-holder and, spindle assembly, attempts were made to estimate tool-point FRFs during machining operation. Burney et al. [20] evaluated the dynamics of machine tool under actual operating conditions and found that dynamic parameters are different during static and operating conditions. They also studied the effect of spindle speed and feed rate on the dynamic stability of milling machine. Ismail and Soliman [21] proposed an approach that uses a chatter indicator factor to determine the stability lobe diagrams. Albertelli et al. [22] proposed a chatter indicator algorithm based on cyclostationary theory for detecting chatter in variable speed milling. Cao and Altintas [23] proposed a general Finite Element (FE) model to predict the static and dynamic behavior of spindle assembly. They also studied the effect of centrifugal and gyroscopic effects on the natural frequency of the spindle assembly. Movahhedy et al. [24] investigated the influence of gyroscopic effect on the stability lobe diagrams with the help of FE based model and concluded that this effect is dominant at high spindle speed. Zaghbani et al. [25] applied the Operational Modal Analysis (OMA) technique for estimating tool point FRFs during machining operation and generated stability lobe diagrams at various spindle speeds. Ozsahin et al. [26] used the actual cutting forces to excite the machine tool structure and estimated different FRFs during static and operating conditions. Li et al. [27] proposed the output-only modal identification method to predict the dynamics of the machine tool at different feeds and speeds. Mao et al. [28] proposed a random decrement method to find the FRFs in the operating condition by using output data only. Grossi et al. [29] proposed a method to find the speed varying FRFs under operational condition by performing speed ramp tests.

Most of the researchers obtained FRFs from vibration signals collected either at the workpiece or at the stationary part of the spindle unit or tool-holder assembly. According to the mechanism responsible for the chatter phenomenon, it is important to acquire vibration signals at the tip of the milling cutter. In the present work, a new approach is used to estimate tool-point FRFs during milling operation. For measuring the vibration at the tip of the end mill at different spindle speeds, a non-contact laser vibrometer is used. In order to remove the tooth pass frequency and its harmonics from the measured signals, a comb filter is employed. The filtered signal is subjected to Operational Modal Analysis (OMA) in order to derive Impulse Response Function (IRF) and tool point FRFs. With the application of estimated FRFs at different spindle speeds, the stability lobe diagrams are drawn for high-speed milling machine.

The paper is organized into five sections. In section 2, the theoretical background and steps followed for plotting stability lobe diagrams are presented. In the next section, OMA technique and the methodologies followed for estimating specific cutting force coefficients and tool-point FRFs are given. In the subsequent section, experimental work performed for estimating tool-point FRFs and specific cutting force coefficients are presented. Finally, the results and conclusion drawn from the work are given in sections 4 and 5 respectively.

Stability lobe diagram is a plot of limiting axial depth of cut versus spindle speed. Various lobes in the diagram define the boundaries between the stable and unstable cutting conditions for a range of spindle speeds (Fig. 2) . For estimating the dynamics of tool, tool-holder and spindle assembly, end mill cutter is considered as a two degree of freedom system with flexibility in the feed as well as in the normal direction, as shown in Fig. 3 . The first step is to model the flexibility in form of FRFs. For this purpose, it is required to perform modal analysis at the tip of the end mill cutter. Next step is to model the cutting forces in feed and normal directions by calculating the specific cutting force coefficients experimentally. Finally, the stability criteria are evaluated analytically by the application of cutting force and FRFs. In the present work Zero Order Approximation [9] method is used for generating the stability lobe diagrams. The steps followed for generating stability lobe diagram are given in Fig. 4 . The first step is to calculate specific cutting force coefficients in the feed (Kt) and normal (Kr) directions. The methodology followed for estimating the specific cutting force coefficients is given in Fig. 5 . The next step is to estimate the tool-point FRFs (ϕxx, ϕyy) in the feed and normal directions. By the application of cutting force coefficients and FRFs, a 

Here, 0 and 1 is calculated by equation (2) and (3) 

Here, αxx, αyy, αxy and αyx are directional dynamic milling force coefficients. Eigenvalue (A) in equation (1) is complex with real (AR) and imaginary (AI) part. After calculating the eigenvalue from equation (1), axial limit depth of cut is estimated from equation (4) by considering the phase shift (k) of the eigenvalue which depends on the chatter frequency (fc) selected around the dominant mode of the FRF.

Here, 'AR' is the real part of the complex eigenvalue. 'N' is the number of flutes on the cutter. 'k' is the phase shift that can be calculated by taking the ratio of imaginary part to the real part of the eigenvalue.

The corresponding spindle speed in revolution per minute (n) is calculated by using equation (5) 

Here, 'T' is the tooth passing period which is calculated by equation (6) 

In the above equation, 'ε' is the phase shift between the inner and outer modulations and is calculated based on the phase shift of the eigenvalue, 'fc' is the chatter frequency in Hz and 'l' is the lobe number. Chatter frequency is selected near to the natural frequency of the dominant mode of the tool holding system.

Equation (6) is derived by equating the phase distance per tooth period (2πfcT) for the chatter frequency (fc) and the phase shift of the present and previous vibration path. Equation (6) will provide the value of the tooth passing period in such a way that the two paths will be in the phase with each other.

For generating stability lobe diagrams, the two primary requirements: specific cutting force coefficients and FRFs at the tip of milling cutter are needed. The methodology followed for estimating specific cutting force coefficients is given in Fig.  5 and the methodology followed for estimating the tool-point FRFs is explained in the next section. 

In the present work, the tool point FRFs at the tip of the milling cutter are estimated by means of Operational Modal Analysis (OMA). OMA is a modal testing procedure that allows the experimental estimation of modal parameters of the structure from the measurement of vibration response only. The foundation of this technique is based on the fact that all the vibration modes of the structure can be excited simultaneously when it is excited by a stochastic white noise input. The autoand cross-correlation functions between the response signals behave like Impulse Response Functions (IRF) of the structure [31, 32] . Dynamic parameters can be estimated by these IRFs by applying standard parameter extraction techniques like Least Square Complex Exponential (LSCE), Eigensystem realization algorithms, etc. In Fig. 6 , step by step approach for conducting OMA is shown by taking an example of a single degree of freedom (SDOF) system. First, the output vibrations corresponding to the input white noise excitations are recorded then the auto-correlation function of output vibration is calculated. It shows the properties of the IRF of the system. The FRF of the SDOF system is estimated by taking the Fast Fourier Transformation (FFT) of IRF. In the present work, a similar approach is followed for estimating the FRFs from output vibration signals. In the present work, tool point FRFs are estimated during static condition (at 0 rpm) and at different spindle speeds (4,000, 6,000, 10,000 and 12,000 rpm). Fig. 7 . For the measurement of vibrations at the tip of end mill cutter at different spindle speeds, a noncontact laser vibrometer was used. A comb filter is then applied to remove the tooth pass frequency and its harmonics from the measured vibration signals in order to make the signals suitable for OMA. In the next step, auto-correlation functions of the filtered signals are calculated and these auto-correlation functions are scaled by taking reference from the IRF estimated during static condition. Subsequently, FRFs are estimated from these properly scaled and filtered signals with operational modal analysis (OMA). Comb filter was designed for removing the harmonics from the signal. LSCE algorithm is applied for estimating the dynamic parameters at the tip of the milling cutter. 

Two different sets of experiments were performed on a threeaxis CNC vertical machining center, having a maximum spindle power of 15 kW and a maximum spindle speed of 15,000 revolutions per minute (rpm). First set of experiments was performed to determine specific cutting force coefficients in both feed and normal directions and the second set of experiments were conducted to estimate the dynamic parameters at the tip of milling cutter in static as well as in the operating conditions. Fig. 8 shows the experimental setup used for collecting vibration signals and force signals during the end milling operation. For collecting the vibration signals at the tip of the end mill cutter, a mould of nominal weight was prepared and fitted 5 millimeter above of the tooltip. A stripe of reflective material was pasted on that mould for better acquisition of the response signals (Fig. 8) .

For estimating specific cutting force coefficients in both feed and normal directions, the slotting operation was performed on hardened mild steel workpiece employing carbide end mill cutter of 12 mm diameter. The technical specification of the end mill cutter is given in Table 1 . Cutting parameters selected for performing slotting operation were:

• Spindle speed: 10,000 rpm.

• The depth of cut was varied, in steps of 0.25 mm, from 0.5 to 1.5 mm. • Ten different feed rates, in steps of 50 mm/min, from 50 mm/min. to 500 mm/min. According to the methodology explained in Fig. 5 , the first step is to calculate the average cutting force for one revolution from the measured cutting force. Fig. 9 cutting forces in both feed and normal directions. The corresponding figures below the measured forces show the cutting force for one revolution. Next step is to calculate the average cutting force per revolution per tooth by dividing the number of teeth to the average cutting force. Finally, with the application of equation (7) and equation (8), specific cutting force coefficients were estimated [8] . The linear regression model for carbide end milling cutter, for calculating specific cutting force coefficients in the feed direction (Kr) is shown in Fig. 10 . The dotted points represent the measured cutting forces per tooth per revolution for a particular feed rate. For this case, cutting parameters were: spindle speed of 10,000 rpm; axial depth of cut of 1 mm and the feed rate varied from 50 mm/min to 500 mm/min in steps of 50 mm/min. The estimated Kr and Kt values for carbide end milling cutter were 2985 N/mm 2 and 4629 N/mm 2 respectively.

Dynamic parameters of the tool, tool-holder, and spindle assembly were estimated in static and operating conditions. During static condition, impact hammer tests were performed at the tip of milling cutter in both feed and normal directions. Vibration signals were collected at the tip of milling cutter by using an uni-axial piezoelectric accelerometer For the estimation of dynamic parameters of the tool, tool holder and spindle assembly during operating condition, vibration signals were collected at the tip of milling cutter by using a laser vibrometer.

The experiments were performed with the following equipment:

1. Impact hammer (DYTRAN 5800B4) with a sensitivity of 10mv/lbf. The vibration signals were collected by rotating the spindle at 4,000, 6000, 10,000 and 12,000 revolutions per minute. Vibrations signals were collected by using DEWEsoft software. The vibrations signals at 4,000 and 6,000 rpm were collected at 20 kHz sampling frequency, whereas vibration signals at 10,000 and 12,000 rpm were collected at a sampling frequency of 30 kHz. Tool-point FRFs estimated at different spindle speeds were compared to study the variations in the natural frequencies and damping ratios.

Dynamic parameters were estimated by experimental modal analysis (EMA) and operational modal analysis (OMA) techniques. The schematic diagram and the methodology followed for acquiring the vibration signals by using a laser vibrometer are explained in section 2. The estimated FRFs during the static condition and at different spindle speeds are given in Fig. 11 . By primary observation, it can be realized that the dynamic behavior of cutting tool, tool-holder, and spindle assembly is different at different spindle speeds. Table 2 and Table 3 provide the corresponding natural frequencies and damping ratios respectively during static condition and at various spindle speeds. During operating condition, because of the several inter-related factors like centrifugal force, gyroscopic effect, change in the bearing dynamics and friction, etc., dynamic parameters are different than those are in the static condition. As can be seen from Table 2 , the natural frequencies of the tool, tool-holder, and spindle assembly decrease during spindle operation implying that the system becomes less stiff in the operating condition. These results are in accordance with the results obtained by Zaghbani et al. [25] . However, in case of damping ratio, no definite pattern was observed, for some modes damping ratio decreases while for others it increases (Table 3) . Fig.  11 (b) (FRF at 4,000 rpm) and Fig. 11 (c) (FRF at 6,000 rpm) that mode number '5' and '7' are the two dominant modes that define the position of the lobes in the stability lobe diagrams. Therefore, the relative variations in the natural frequencies of these dominant modes are calculated. For FRFs at 0 rpm and 4,000 rpm, the relative variation in natural frequencies for mode number '5' and '7' is 2.46 % and 0.5 % respectively, while for FRFs at 0 rpm and 6,000 rpm it is 3.6 % and 1.56 %. Similarly, for FRFs at 4,000 rpm and 6,000 rpm, it is 1.19 % and 1.06 % respectively. As the variation between the FRFs at 0 rpm and 4,000 is below 2.5 %, it can be concluded that the dynamic behavior of tool, tool-holder and spindle assembly up to 4,000 rpm can be expressed by the dynamic parameters estimated at 0 rpm. Similarly, the FRF obtained at 4,000 rpm will be valid up to 6,000 rpm for generating the stability lobe diagrams. 

Stability lobe diagrams were obtained by plotting axial limiting depth of cut versus spindle speed by the application of dynamic parameters and specific cutting force coefficients (Kr and Kt). Fig. 12 shows the stability lobe diagrams generated by applying EMA (Fig. 12 (a) at 0 rpm) and OMA (at rest of the spindle speeds) techniques. Because of discrepancies in the natural frequencies and damping ratios during static and operating conditions as presented in Table 2 and Table 3 , different stability boundaries are obtained in the stability lobe diagrams. As the horizontal and vertical positions of the stability lobes depend on the natural frequency and damping ratio of the dominant modes respectively, variation in these parameters results in different stability lobe diagrams. The dominant mode also changes by varying the spindle speed, for example, at 10,000 rpm and 12,000 rpm a significant variation in the critical depth of cut can be observed. It is also observed that some depth of cuts that were stable during the static condition becomes unstable during the operating condition. The variation of critical depth of cut with spindle speed is plotted in Fig. 13 . By comparing stability lobe diagrams at 0 rpm and 4,000 rpm, it was observed that the stability limit decreases. However, at 6,000 rpm, a small variation in the critical depth of cut was observed with respect to the 4,000 rpm. At 10,000 rpm, because of the thermal effects, a slightly higher critical depth of cut is observed. At 12,000 rpm, a significant change in the stability limits was observed. By performing a number of slotting operations at different cutting conditions, a valid speed range of each stability lobe diagram was obtained. For example, the machining conditions selected based on Fig. 12(a) , below 4,000 rpm provides results according to the stability lobe diagram. However, after 4,000 rpm, the results provided by Fig. 12(a) are not accurate. For example, the machining conditions that correspond to the 'star' points should be stable. However, chatter was observed at these points. From these observations, it is concluded that the stability limits plotted based on impact test (at 0 rpm) were accurate up to a spindle speed of 4,000 rpm, this also agrees with the dynamic parameters estimated at 0 rpm. Identical experiments were performed based on the other stability lobe diagrams also (Fig. 12 (b) to Fig. 12 (e) ). And it was observed that each stability lobe diagram is valid only up to a range of spindle speed, for example, the stability lobe diagram at 4,000 rpm is valid up to a spindle speed of 6,000 rpm. Identical results were obtained at 6,000 rpm and 10,000 rpm.

To overcome these discrepancies, a new stability lobe diagram was generated by the application of obtained stability lobe diagrams. In the final stability lobe diagram (Fig. 14 ) different lobes were selected from different stability lobe diagrams. For example, from 0 to 4,000 rpm, the lobes are plotted based on the stability lobe diagram at 0 rpm i.e. Fig. 12 (a) and similarly the lobes from 4,000 to 6,000 rpm are plotted based on Fig. 12(b) and so on. To confirm the accuracy of the final stability lobe diagram, experiments were performed at three different spindle speeds, 6,000, 8,000, and 10,000 rpm. At each spindle speed, the slotting operation was performed for three depths of cut that are stable, conditionally stable, and unstable according to the final stability lobe diagram (Fig. 14) . For example, at 6,000 rpm three depths of cut i.e. 1.0, 1.2 and 1.5 mm (point 'a', 'b' and 'c' in Fig. 14) were selected. Similarly, at 8,000 rpm 0.75, 0.85 and 0.95 mm depth of cuts and at 10,000 rpm three different depths of cut (0.9, 1.00 and 1.2 mm) were selected. In order to examine the machined surfaces, the samples were investigated under a stereomicroscope. Fig. 15 shows the stereomicroscopic images of the machined surfaces of the workpiece. Chatter marks were observed for 1.5 mm @ 6,000 rpm, 0.95 mm @ 8000 rpm and 1.2 mm @ 10,000 rpm. Fig.16 shows the vibration signals recorded in the feed direction by accelerometer corresponding to points 'f' and 'i', according to these signals amplitude of the vibration increasing with the time that confirms the self-excited regenerative vibration at these points. In accordance with the stability lobe diagram (Fig. 14) , these are the unstable points. The other machined surfaces do not show any chatter marks, it also agrees with the final stability lobe diagrams. 

A new approach is used to estimate tool-point Frequency Response Functions (FRFs) at different spindle speeds (4,000, 6,000, 10,000 and 12,000 rpm) during machining operation by the application of OMA technique. To apply OMA technique, vibration signals were acquired at the tip of the milling cutter during machining operation by using a non-contact laser vibrometer. Stability lobe diagrams were drawn by the application of these estimated FRFs for carbide end mill cutter of diameter 12mm. As the proposed approach considers the actual boundary conditions for estimating the dynamic behavior at the tip of the milling cutter, the stability lobe diagrams generated by the application of these parameters are more accurate and realistic.

Following observations were made based on the study: • Dynamic parameters were compared during the static condition and at different spindle speeds. The dynamic parameters were found to be different during static and operating conditions. Natural frequencies for all the modes decrease with increasing the spindle speed and for damping ratio, there was no specific pattern of variation was observed. • Dominant modes responsible for determining the stability limits also changes by varying the spindle speeds.

Because of the discrepancies in dynamic parameters during static and operating conditions, different stability lobe diagrams were obtained.

Based on the slotting operation performed according to different stability lobe diagrams, it was observed that the stability lobes estimated based on the dynamic parameters calculated during static condition (0 rpm) and at different spindle speeds are valid only up to a particular range of spindle speed.

Based on the validity of the stability lobe diagrams at different spindle speeds, a final stability lobe diagram was plotted by selecting appropriate lobes from various stability lobe diagrams. The accuracy of the proposed approach was evaluated at three different spindle speeds by performing slotting operations, and by examining the machined surfaces under a stereomicroscope, it was observed that the proposed stability lobe diagram provides accurate results. The proposed approach can easily be used to automate the generation of stability lobe diagrams accurately during actual machining condition.

High speed machining

Machine-tool vibration

Self-excited vibrations on machine tools

Parameter optimization of time-varying stiffness method for chatter suppression based on magnetorheological fluid-controlled boring bar

Optimization of multiple tuned mass dampers to suppress machine tool chatter

Milling vibration attenuation by eddy current damping

Chatter suppression techniques in metal cutting

Chatter in machining processes: A review. International Journal of Machine Tools and Manufacture

Analytical prediction of stability lobes in milling

Analytical prediction of chatter stability in milling¬-Part I: General formulation

Semi-discretization method for delayed systems

A full-discretization method for prediction of milling stability

Analysis of milling stability by the Chebyshev collocation method: algorithm and optimal stable immersion levels

Milling stability analysis with simultaneously considering the structural mode coupling effect and regenerative effect

Stability Analysis in Face Milling Operations, Part 2: Experimental Validation & Influencing Factors

Study of the Gyroscopic Effect of the Spindle on the Stability Characteristics of the Milling System

Centrifugal Force Induced Dynamics of a Motorized High-Speed Spindle

Chatter Stability of Milling with Speed-Varying Dynamics of Spindles

Modeling & Comparison of High Speed Spindle Systems with Different Bearing Preload Mechanisms

A stochastic approach to characterization of machine tool system dynamics under actual working conditions

A new method for the identification of stability lobes in machining

Development of a generalized chatter detection methodology for variable speed machining. Mechanical Systems and Signal Processing

A general method for the modeling of spindle-bearing systems

Prediction of chatter in high speed milling including gyroscopic effects. International Journal of Machine Tools and Manufacture

Estimation of machine-tool dynamic parameters during machining operation through operational modal analysis

In-process tool point FRF identification under operational conditions using inverse stability solution

A new approach to identifying the dynamic behavior of CNC machine tools with respect to different worktable feed speeds

An approach for measuring the FRF of machine tool structure without knowing any input force

Improved experimentalanalytical approach to compute speed-varying tool-tip FRF. Precision Engineering

A contribution to the mathematical analysis of variable spindle speed machining

Introduction to operational modal analysis

The Natural Excitation Technique (NExT) for Modal Parameter Extraction from Operating Structures

Chatter stability prediction for high-speed milling through a novel experimentalanalytical approach. The International Journal of Advanced Manufacturing Technology

Manufacturing automation: Metal cutting mechanics, machine tool vibrations, and CNC design