Hindawi Publishing Corporation Advances in Mechanical Engineering Volume 2014, Article ID 690768, 8 pages http://dx.doi.org/10.1155/2014/690768
Research Article Modeling and Optimization of a Friction Damper for Boring Chatter Control Min Wang, Tao Zan, and Xiangsheng Gao The Beijing Key Laboratory of Advanced Manufacturing Technology, School of Mechanical and Applied Electronics Engineering, Beijing University of Technology, Beijing 100124, China Correspondence should be addressed to Min Wang;
[email protected] Received 11 October 2013; Accepted 24 March 2014; Published 23 April 2014 Academic Editor: David R. Salgado Copyright © 2014 Min Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Machining performance of boring processes is often limited by chatter vibration when the boring bar is with a large slenderness ratio. The dynamic vibration absorber (DVA) is the most widely used damper in practical chatter control of boring bars. The DVA consists of an additional mass-spring-damper subsystem and needs accurate tuning of its natural frequency and damping ratio to match the main structure. Compared with the DVA, the friction damper consisting of a mass attached to the boring bar through a sliding interface near the tool insert is more simple and easy for design and implementation. The main purpose of this paper is to model the friction damper and investigate its chatter suppression effect. A mathematical model is established to depict the friction force at the friction interface between the damper body and the main structure. Based on the modeling of the fundamental principle of the friction, the dynamic behavior and machining stability of the friction damper damped boring system can be predicted and analyzed. Parameter studies are also carried out using the numerical model to evaluate the machining stability improvement by using the friction damper and identify the best configuration to maximize the boring stability.
1. Introduction Boring is a frequently employed finish machining operation in machining shops. However, because of the low rigidity of the boring bar, chatter is difficult to be avoided even if the depth of cut is very small. So, chatter is main one of the obstacles to the improvement of the tool life, surface quality, and material removal rate in boring processes. One extensively investigated chatter control method is to attach various dampers to the machining structure, as demonstrated by Tobias [1]. The dynamic vibration absorber (DVA) is the most widely used damper in practical chatter control and has been proven to be effective for the chatter suppression of the boring bar with a large slenderness ratio, as demonstrated by Rivin and Kang [2] and Migu´elez et al. [3]. The DVA consists of an additional mass-spring-damper subsystem and needs accurate tuning of its natural frequency and damping ratio to match the main structure for chatter control. However, Mei [4] stated that the availability of accurate model parameters of the main structure is very hard to acquire because the global properties are varying with the metal removal process and the
movable components of machine. With regard to the boring system, the dynamic properties of the boring bar are sensitive to the change of the clamping condition and the length of the bar overhang. The friction damping is a very important mechanism of energy dissipation and vibration reduction as demonstrated by Ferri [5]. Wang et al. [6] proposed a new-type nonlinear tuned mass damper (TMD) containing an additional element of elastic support dry friction compared with the common linear TMD, which was proven effective in machining chatter control by theoretical analysis and cutting tests. Edhi and Hoshi [7] proposed a very simple friction damper which has been found successful to prevent high frequency boring chatter occurring at more than 10000 Hz. This friction damper consists of an additional mass attached to the main vibrating structure with small piece of permanent magnet. Compared with the DVA, the proposed friction damper is more simple and easy for design and implementation. Mechanism of chatter suppression of the friction damper was investigated in [7]. The combination of two mathematical models assuming Coulomb and viscous frictions was used
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to explain the fundamental principle of the friction damper. This model evaluates the effect of damper by calculating the damper frictional work as the mass sliding with respect to the main structure under the action of boring bar vibration. However, their proposed model is only a qualitative model with no capability to predict machining stability; it is necessary to establish a new model to predict the performance of the friction damper for machining chatter control. Considering the special nature of the machining stability problem, the stability limit of machining processes can be obtained when the magnitude of the real part of the frequency response function (FRF) of the machining system is minimized by using the damper [8]. However, the analytical treatment is relatively difficult for the dry friction damped system. Here, for obtaining the FRF of damped machining system, a mathematical model is established to depict the friction force at the friction interfaces between the damper mass and the main structure. Based on the proposed friction force model, the FRF of the friction damper damped boring system can be calculated by employing the harmonic balancing method (HBM), which is a semianalytical approximate frequency domain method [9]. After the FRF of damped boring bar is obtained, the boring stability diagram can be predicted based on the regenerative machining chatter theory [1]. Based on the results from numerical simulations, the dynamic behavior and machining stability of the friction damper damped boring system are analyzed in this paper. The effectiveness and characteristics of the friction damper in suppressing chatter vibration are investigated. The design parameters of the friction damper are also evaluated using the numerical model to identify the best configuration to maximize the boring stability.
2. Background A schematic view of a boring operation is illustrated in Figure 1. For general problem of vibration control, it is normally desirable to minimize the amplitude of the FRF of the main structure with a damper attached, and this can be achieved by tuning the parameters of the damper. However, the machining stability limit is determined by the magnitude of the real part of the FRF of machining systems rather than that of the amplitude. Tobias [1] has formulated the following chatter stability law that has been widely used since 1950s: 𝑏lim =
−1 , 2𝐾𝑓 Re [𝐺𝑠 (𝑗𝜔𝑐 )]
(1)
where 𝑏lim is the value of the cut depth at the stability limit, 𝜔𝑐 is the chatter frequency, 𝐾𝑓 is the cutting force coefficient, and 𝐺𝑠 (𝑖𝜔𝑐 ) is the FRF of the machining system. The relation among the phase shift 𝜀, the integer number 𝐾 of the oscillations between subsequent cutter tip passes and the chatter frequency 𝜔𝑐 can be expressed as 60𝜔𝑐 (2) = 2𝜋𝐾 + 𝜀, 𝑛 where 𝑛 is the spindle speed of the machine tool. Equation (2) can be used to determine the relation between spindle speed
Phase shift 𝜀 between subsequent tool tip passes
Workpiece
Previous surface
New surface
Friction damper Boring bar Tool post
Figure 1: Schematic boring processes and cutting process model. F0 sin(𝜔t)
𝜇s
m
md N
x
k
c
Figure 2: Dynamic model of the friction damper damped boring system.
𝑛 and chatter frequency 𝜔𝑐 . Based on (1) and (2), the stability diagram can be drawn to show the relation between the cut depth at the stability limit and the spindle speed for different values of 𝐾. As shown in Figure 1, the friction damper consists of an additional mass attached to the boring bar with small piece of permanent magnet. The damped boring system can be depicted as shown in Figure 2. The boring bar is simplified as a single-degree-of-freedom main structure to which a mass is attached. In Figure 2, 𝑚, 𝑘, and 𝑐 are the dynamic parameters of the boring bar. 𝑚𝑑 is the mass of the friction damper. 𝜇𝑠 is the kinetic friction coefficient. 𝑁 is the normal force applied to the friction interface by using the magnet. 𝐹0 sin(𝜔𝑡) is the external force applied to the vibration system. It can be used to depict the dynamic cutting force exerted on the bar because the dynamic cutting force is an approximately harmonic force when chatter occurs during cutting processes.
3. Representation of the Friction Force For the damper mass, there are three vibration modes: stick mode, stick-slip mode, and pure-slip mode. There exist two threshold amplitudes of the bar vibration that decide the shifts between three vibration modes. When the boring process is stable and the vibration amplitude of the bar is small, the damper body sticks to the bar and vibrates with the bar together under the action of the static friction force. With the development of chatter, the vibration amplitude of the boring bar gradually increases. Once the vibration amplitude
3
𝑥 = 𝑥𝑚 sin 𝜔𝑡,
(3)
where 𝑥𝑚 is the amplitude of 𝑥 and 𝜔 and 𝑡, respectively, are the frequency and time variable. As shown in Figure 3, the stick and slip motions occur alternately in one circle of vibration during the stick-slip vibration mode. When the damper body sticks to the bar, the static friction force equals the inertia force. If the sliding occurs at the friction interface, the damper body is accelerated and decelerated by an alternating kinetic friction force having a rectangular waveform as illustrated in the bottom of Figure 3. During one circle of the vibration of the bar, the piecewise-linear nonlinear friction force 𝐹𝑑 shown in Figure 4 can be expressed as follows (refer to Appendix A): 𝐹𝑑 (𝜔𝑡) = −𝑚𝑑 𝜔2 𝑥𝑚 sin 𝜔𝑡
Velocity
3.1. Stick-Slip Vibration Mode. The stick-slip vibration mode of the damper body is illustrated in Figure 3. Here, the displacement of the bar x is assumed to be sinusoidal displacement. It can be expressed as
𝜔t
x
Acceleration
reaches the first threshold amplitude, the friction force is not enough to keep the damper body vibrating with the bar in whole stroke; the damper body will start sliding. With the vibration amplitude of the bar increasing, the portion taken by the slip motion in one circle of vibration is gradually enlarged. After the vibration amplitude reaches the second threshold amplitude, the damper body starts to vibrate in the pure-slip mode.
Displacement
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𝜔t
𝜋 + 𝜃1
𝜋 + 𝜃2 𝜔t
𝜃1
𝜃2
Figure 3: Acceleration, velocity, and displacement of the damper body and the bar: — bar; - - - damper body.
0 ≤ 𝜔𝑡 < 𝜃1 ,
𝐹𝑑 (𝜔𝑡) = −𝑓 𝜃1 ≤ 𝜔𝑡 < 𝜃2 , 𝐹𝑑 (𝜔𝑡) = −𝑚𝑑 𝜔2 𝑥𝑚 sin 𝜔𝑡
𝜃2 ≤ 𝜔𝑡 < 𝜋 + 𝜃1 ,
(4)
𝐹𝑑 (𝜔𝑡) = 𝑓 𝜋 + 𝜃1 ≤ 𝜔𝑡 < 𝜋 + 𝜃2 , 𝐹𝑑 (𝜔𝑡) = −𝑚𝑑 𝜔2 𝑥𝑚 sin 𝜔𝑡 𝜃1 = sin−1 (
𝜋 + 𝜃2 ≤ 𝜔𝑡 < 2𝜋,
𝑓 ), 𝑚𝑑 𝜔2 𝑥𝑚
cos 𝜃2 − cos 𝜃1 = − (𝜃2 − 𝜃1 ) sin 𝜃1 ,
(5) (6)
where𝑓 = 𝜇𝑠 𝑁 is the kinetic friction force between the friction interface and 𝑚𝑑 is the mass of the damper body. 𝜃1 and 𝜃2 are two critical phase angles of the bar vibration. The damper body starts sliding at 𝜃1 and ends sliding at 𝜃2 as shown in Figure 3. This model assumes the Coulomb friction law with equal static and kinetic coefficients of friction. 3.2. Stick Vibration Mode. When the vibration amplitude of the bar is less than the first threshold amplitude, the static friction force is enough to keep the damper body vibrating with the bar in the whole stroke. The first threshold amplitude 𝑥1 can be expressed as follows: 𝑥1 =
𝑓 . 𝑚𝑑 𝜔2
Figure 4: Nonlinear friction force.
(7)
For the stick vibration mode, 𝜃1 and 𝜃2 can be assumed to be zero in (4).
3.3. Pure-Slip Vibration Mode. When the vibration amplitude of the bar is more than the second threshold amplitude, the damper body is accelerated and decelerated by an alternating kinetic friction force in the whole stroke. The second threshold amplitude 𝑥2 can be expressed as follows (refer to Appendix B): 𝑥2 = √ 1 +
𝑓 𝜋2 𝑓 ≅ 1.8621 . 2 4 𝑚𝑑 𝜔 𝑚𝑑 𝜔2
(8)
For the pure-slip vibration, 𝜃1 and 𝜃2 in (4) can be expressed as follows (refer to Appendix A): 𝜃1 = cos−1 ( 𝜃2 = 𝜋 + 𝜃1 .
𝑓 𝜋 ), 2 𝑚𝑑 𝜔2 𝑥𝑚
(9) (10)
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Advances in Mechanical Engineering 𝜔0 = √𝑘/𝑚: main structure natural frequency,
3.4. Approximation of the Nonlinear Friction Force. Using (4)–(6), the nonlinear force 𝐹𝑑 can be approximated by the first harmonic term of their corresponding Fourier series expansion solution: 𝐹𝑑 (𝜔𝑡) ≅ 𝑎1 cos 𝜔𝑡 + 𝑏1 sin 𝜔𝑡,
𝜆 = 𝜔/𝜔0 : nondimensional excitation frequency, 𝛾 = 𝑓/𝐹0 : force ratio, 𝑎𝑔 = 𝑎1 /𝐹0 : nondimensional coefficient of 𝑎1 ,
(11)
𝑥𝑔 = 𝑥𝑚 /𝛿st : nondimensional amplitude of 𝑥,
where
𝛿st = 𝐹0 /𝑘: static deflection,
1 2𝜋 𝑎1 = ∫ 𝐹𝑑 (𝜔𝑡) cos 𝜔𝑡 𝑑𝜔𝑡 𝜋 0 =
𝜇 = 𝑚𝑑 /𝑚: mass ratio, 𝜉 = 𝑐𝜔0 /(2𝑘): main structure damping ratio,
2
2𝑓 𝑚𝑑 𝜔 𝑥𝑚 (cos 2𝜃1 − cos 2𝜃2 ) + (sin 𝜃1 − sin 𝜃2 ) , 2𝜋 𝜋
1 2𝜋 𝑏1 = ∫ 𝐹𝑑 (𝜔𝑡) cos 𝜔𝑡 𝑑𝜔𝑡 𝜋 0
𝑏𝑔 = 𝑏1 /𝐹0 : nondimensional coefficient of 𝑏1 . After substituting (16) into (15), based on the harmonic balancing method, we can get the equations as follows: 𝑏1 = 𝛿𝑠𝑡 cos 𝜑, 𝑘 𝑎 2𝜉𝜆𝑥𝑚 + 1 = 𝛿𝑠𝑡 sin 𝜑. 𝑘
− 𝜆2 𝑥𝑚 + 𝑥𝑚 +
𝑚 𝜔2 𝑥𝑚 (sin 2𝜃1 − sin 2𝜃2 − 2𝜋 − 2𝜃1 + 2𝜃2 ) = 𝑑 2𝜋 +
2𝑓 (cos 𝜃2 − cos 𝜃1 ) . 𝜋
(12)
When 𝑥𝑚 is less than 𝑥1 , the friction interfaces are stuck and 𝐹𝑑 is the inertia force: 𝐹𝑑 (𝜔𝑡) ≅ 𝑎1 cos 𝜔𝑡 + 𝑏1 sin 𝜔𝑡 = −𝑚𝑑 𝜔2 𝑥𝑚 sin 𝜔𝑡.
Then substituting nondimensional terms, abovementioned, into (17), nondimensional equations can be derived as follows: − 𝜆2 𝑥𝑔 + 𝑥𝑔 + 𝑏𝑔 = cos 𝜑, 𝜉𝜆𝑥𝑔 + 𝑎𝑔 = sin 𝜑,
(13)
𝑎𝑔 =
𝐹𝑑 (𝜔𝑡) ≅ 𝑎1 cos 𝜔𝑡 + 𝑏1 sin 𝜔𝑡 (14)
4. Modeling of the Friction Damper Damped Boring Bar Due to the nonlinear nature of the related motion, it is difficult to get an analytical solution of the FRF for the system containing friction damping. For solving this problem, the harmonic balance method (HBM) is applied to get a numerical solution of the real part of the FRF of the main structure. The motion equation of the system shown in Figure 2 is (15)
where 𝐹0 is the amplitude of excitation force. For using HBM technique to solve the equation above, let 𝑥 = 𝑥𝑚 sin(𝜔𝑡 − 𝜑). 𝜑 is the phase angle between excitation and response. The nonlinear friction force 𝐹𝑑 in (15) will be written as follows: 𝐹𝑑 = 𝑎1 cos (𝜔𝑡 − 𝜑) + 𝑏1 sin (𝜔𝑡 − 𝜑) .
𝑏𝑔 =
𝜇𝜆2 𝑥𝑔 2𝜋 𝜇𝜆2 𝑥𝑔 2𝜋 +
where 𝜃1 can be obtained by using (9).
𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 + 𝐹𝑑 = 𝐹0 sin 𝜔𝑡,
(18)
where
When 𝑥𝑚 is more than 𝑥2 , the friction force is kinetic friction force: 4𝑓 =− sin (𝜃 − 𝜃1 ) , 𝜋
(17)
(16)
For getting nondimensional equations, some nondimensional terms will be introduced into the motion equations above. The definitions of the terms are as follows:
(cos 2𝜃1 − cos 2𝜃2 ) +
2𝛾 (sin 𝜃1 − sin 𝜃2 ) , 𝜋
(sin 2𝜃1 − sin 2𝜃2 − 2𝜋 − 2𝜃1 + 2𝜃2 )
(19)
2𝛾 (cos 𝜃2 − cos 𝜃1 ) . 𝜋
By solving (18), the nondimensional response amplitude, 𝑥𝑔 (𝜆), and phase angle, 𝜑(𝜆), of the main structure can be derived. Then, the nondimensional real part of FRF, 𝑅𝑔 (𝜆), can be obtained by using the equation as follows: 𝑅𝑔 (𝜆) = 𝑥𝑔 (𝜆) cos [−𝜑 (𝜆)] = 𝑥𝑔 (𝜆) cos [𝜑 (𝜆)] .
(20)
5. Parametric Studies Mass ratio 𝜇 and force ratio 𝛾 are two parameters which influence the effectiveness of the friction damper in the machining chatter control. Higher mass ratios 𝜇 increase the effectiveness of the friction damper in the vibration suppression. Force ratio 𝛾 is the only nondimensional parameter taken into consideration for the optimization. Calculations were made for the nondimensional frequency range between 0.6 and 1.2 at the frequency interval of 0.001 for five levels of the normal force exerted on the friction interfaces, 𝛾 = 0.0, 0.1, 0.5, 1.0, 5.0, 10.0. Figure 5 gives several curves of 𝑥𝑔 (𝜆) and 𝑅𝑔 (𝜆) with damping ratio 𝜉 of 0.01 and mass ratio 𝜇 of 0.5.
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5 decreases as the mass ratio increases. The percentage of the magnitude reduction of the negative real part decreases from 95.4% to 50.5% as the mass ratio decreases from 1.0 to 0.05, compared to the undamped boring system.
Nondimensional real part, Rg
Nondimensional amplitude, xg
50
40 30 20 10 0 30 20 10 0 −10 −20 −30 0.6
6. Machining Stability Simulations
0.7
0.8
0.9
1.0
1.1
1.2
Nondimensional frequency, 𝜆 𝛾 = 0.1 𝛾 = 0.5 𝛾 = 1.0
𝛾 = 5.0 𝛾 = 10
Figure 5: FRF of main structure with 𝜇 of 0.5, 𝜉 of 0.01, and five levers of force ratio 𝛾.
As shown in Figure 5, the peak amplitude depends on the magnitude of the force ratio. Initially, the peak decreases as the force ratio increases, and the resonant frequencies of the system are invariable and the same as those of the system when 𝛾 equals 0.0. The peak starts to increase after 𝛾 reaches a value between 0.5 and 1.0. At the same time, the resonant frequencies decrease to a relatively small value. This indicates that the vibration mode of the friction damper gradually shifts from sliding mode to sticking mode. As for the maximum of the negative real part of FRF, there is the same varying process when 𝛾 increases from 0.1 to 10.0. It can be inferred that there is an optimum value of the force ratio to ensure the minimization of the maximum of the negative real part. For obtaining the optimal value of the force ratio, a standard optimization routine of MATLAB Software was employed, which is named Minimax [10] and can find a minimum or maximum solution of a function of several variables. The optimal value of the force ratio is 0.714 when the damping ratio 𝜉 and mass ratio 𝜇 are, respectively, 1% and 0.5. Figure 6 indicates the optimization result of the FRF of the main structure. As shown in Figure 6, there is apparent effect of the friction damper on the dynamics of the main structure. For different mass ratio, the force ratio can be optimized to ensure the best performance of the friction damper. The higher mass ratios improve the performance of the friction damper for the chatter control. However, the damper mass cannot be increased arbitrarily due to the constraints in practice. So, the mass ratio is the main parameter determining the performance of the friction damper in practice. Figure 7 graphically shows the results of optimization calculation for six levels of the mass ratio, 𝜇 = 0.05, 0.2, 0.4, 0.6, 0.8, 1.0 when the damping ratio 𝜉 is 1%. Figure 7(a) shows that the optimal force ratio increases as the mass ratio increases. Figure 7(b) shows that the maximum of the negative real part of FRF
To evaluate how well the friction damper improves the machining stability, the chatter stability was predicted for boring process as shown in Figure 1. The dominant mode of the boring bar is the bending mode in the radial direction. So, chatter is most likely to be caused by the bending vibration of the boring bar in the radial direction. For this bending mode, the dynamics of the boring bar 𝐺𝑠 (𝑖𝜔𝑐 ) can be represented by the natural frequency 𝜔𝑛 of 515 Hz, the stiffness 𝑘 of 2.86 N/𝜇m, and the damping ratio 𝜉 of 1.31%. The mass ratio of the damper mass to the dominant mode of the boring bar is set to three levels, 𝜇 = 0.02, 0.1, 0.5. For three levels of the mass ratio, the optimal value of the force ratio 𝛾 is, respectively, 0.187, 0.479, 0.695. Figure 8 gives the predicted FRFs of the boring system for properly designed friction damper damped boring system and the undamped boring system. After obtaining the FRFs, the boring chatter stability can be predicted using (1) and (2). Here, boring process is a boring of aluminum workpiece by a carbide insert. The cutting force coefficient 𝐾𝑓 is 1200 MPa. The chatter stability diagrams of the boring system damped by using the friction damper are plotted in Figure 9 along with the undamped boring system. At the cutting condition above the stability borderline, chatter should occur with the progress of the boring process. It can be seen that the friction damper with mass ratio of 0.5 provides a significant improvement in the critical limiting cutting depth, which increases from 0.06 mm to 0.56 mm. However, the performance of the friction damper is degraded sharply along with the mass ratio decreases.
7. Conclusions For investigating the effectiveness of the friction damper in the machining chatter control, this paper gives a mathematical model to depict the friction force at the friction interfaces between the damper body and the main structure. Based on this friction model, the FRF of the damped machining system can be calculated. And the machining stability can be predicted by calculating the stability borderline according to the regenerative chatter theory. The mass ratio and force ratio are two parameters which influence the effectiveness of the friction damper in the machining chatter control. For different mass ratio, the force ratio can be optimized to ensure the best performance of the friction damper by using the mathematical model presented in this paper. However, the performance of the friction damper is degraded sharply along with the mass ratio decreases. The mass ratio is the main parameter determining the performance of the friction damper because the damper cannot be increased arbitrarily due to the constraints in practice.
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30
20 Nondimensional real part, Rg
Nondimensional amplitude, xg
40
30
20
10
10
0
−10
−20
0 0.6
0.7 0.8 0.9 1.0 1.1 Nondimensional frequency, 𝛾
1.2
−30 0.6
𝛾 = 0.714 Without damper
0.7 0.8 0.9 1.0 1.1 Nondimensional frequency, 𝛾
1.2
𝛾 = 0.714 Without damper
(a)
(b)
Figure 6: Optimization results of the FRF for mass ratio = 0.5. 0.80
14
0.75
12
0.70 10 Max [−Rg (𝜆)]
Force ratio, 𝛾
0.65 0.60 0.55 0.50
8 6 4
0.45 2
0.40 0.35 0.0
0.2
0.4 0.6 Mass ratio, 𝜇
0.8
1.0
0 0.0
0.2
0.4 0.6 Mass ratio, 𝜇
(a)
0.8
1.0
(b)
Figure 7: Optimization results with seven levers of mass ratio.
Appendices
Consider
A. Derivation of the Critical Phase Angles 𝜃1 and 𝜃2
cos 𝜔𝑡2 − cos 𝜔𝑡1 = −
A.1. Stick-Slip Vibration Mode. As shown in Figure 2, the velocities of the bar and the damper body are equal at the phase angle of 𝜃1 and 𝜃2 . The following equation can be given: 𝑡2
∫ − 𝜔2 𝑥𝑚 sin 𝜔𝑡 𝑑𝑡 = − 𝑡1
𝑓 (𝑡 − 𝑡 ) , 𝑚𝑑 2 1
(A.1)
where 𝑡2 and 𝑡1 are the time corresponding to the phase angles 𝜃2 and 𝜃1 .
𝑓 (𝜔𝑡2 − 𝜔𝑡1 ) . 𝑚𝑑 𝜔2 𝑥𝑚
(A.2)
Because 𝑡1 corresponds to the time when the damper body starts sliding, the friction force is equal to the inertia force: −𝑓 = − 𝑚𝑑 𝜔2 𝑥𝑚 sin (𝜔𝑡1 ) , 𝜃1 = 𝜔𝑡1 = arcsin (
𝑓 ). 𝑚𝑑 𝜔2 𝑥𝑚
(A.3) (A.4)
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7 8
15
6
Real part of FRF (𝜇m/N)
Amplitude of FRF (𝜇m/N)
4 10
5
2 0 −2 −4 −6
0 300
400
600 500 Frequency (Hz)
700
400
500
600
700
Frequency (Hz)
𝜇 = 0.1 𝜇 = 0.02
Without damper 𝜇 = 0.5
−8 300
Without damper 𝜇 = 0.5
(a)
𝜇 = 0.1 𝜇 = 0.02
(b)
Figure 8: Simulated FRF of boring system.
A.2. Pure-Slip Vibration Mode. When 𝑥𝑚 > 𝑥2 , the damper body is accelerated and decelerated by an alternating kinetic friction force in the whole stroke, and 𝜃2 = 𝜃1 + 𝜋. Equation (A.2) can be reduced to
Cut depth at the stability limit (mm)
1.0
0.8
0.6
𝑓𝜋 , 𝑚𝑑 𝜔2 𝑥𝑚
(A.7)
𝑓 𝜋 ). 2 𝑚𝑑 𝜔2 𝑥𝑚
(A.8)
−2 cos 𝜔𝑡1 = − 0.4
𝜃1 = arccos (
0.2
0.0 1000
1200
1100
1300
1400
1500
Spindle speed (rpm) Without damper 𝜇 = 0.5
𝜇 = 0.1 𝜇 = 0.02
𝜃1 = 𝜔𝑡1 = arccos (
Figure 9: Stability predictions.
cos 𝜃2 − cos 𝜃1 = − sin 𝜃1 (𝜃2 − 𝜃1 ) .
(A.5)
When substituting nondimensional terms into (A.4), nondimensional form of 𝜃1 can be derived as follows: 𝜃1 = 𝜔𝑡1 = arcsin (
𝛾 ). 𝜇𝜆2 𝑥𝑔
𝜋 𝛾 ). 2 𝜇𝜆2 𝑥𝑔
(A.9)
B. Derivation of the Second Threshold Amplitude 𝑥2
After substituting (A.4) into (A.2), it is reduced to cos 𝜔𝑡2 − cos 𝜔𝑡1 = − sin 𝜔𝑡1 (𝜔𝑡2 − 𝜔𝑡1 ) ,
When substituting nondimensional terms into (A.8), the obtained is
(A.6)
The second threshold amplitude 𝑥2 is the critical amplitude between pure-slip mode and stick-slip mode. So, 𝑥2 can be derived by substituting (A.4) into (A.8). Thus, obtained is
𝑥2 = √ 1 +
𝑓 𝜋2 𝑓 ≅ 1.8621 . 4 𝑚𝑑 𝜔2 𝑚𝑑 𝜔2
(B.1)
8
Nomenclature 𝑏lim : Cutting depth at the stability limit, m 𝑎1 , 𝑏1 : Coefficients of first harmonic term of Fourier series expansion of nonlinear friction force, N 𝑎𝑔 , 𝑏𝑔 : Nondimensional coefficient of 𝑎1 and 𝑏1 𝑐: Main structure damping coefficient, N/ms−1 𝑓: Sliding/static friction force, N 𝑁: Normal force, N Nonlinear friction force, N 𝐹𝑑 : Amplitude of dynamic cutting force, N 𝐹0 : 𝐺: Frequency response function, m/N 𝑅𝑔 : Nondimensional real part of main structure response 𝐾𝑓 : Cutting force coefficient, MPa 𝑘: Main structure stiffness, N/m 𝜉: Main structure damping ratio, dimensionless 𝑚: Main structure effective mass, kg Static deflection of main structure, m 𝛿st : 𝑥: Displacement of main structure, m Nondimensional amplitude of main 𝑥𝑔 : structure response First threshold vibration amplitude of the 𝑥1 : main structure, m Second threshold vibration amplitude of 𝑥2 : the main structure, m Kinetic friction coefficient, dimensionless 𝜇𝑠 : 𝛾: Force ratio, dimensionless 𝜇: Mass ratio, dimensionless 𝜃1 , 𝜃2 : Critical phase angles of the main structure vibration, rad 𝜆: Nondimensional excitation frequency 𝜑: Phase angle of 𝑥, rad 𝜔: Excitation force frequency, rad s−1 𝜔0 : Main structure natural frequency, rad s−1 𝜔𝑐 : Chatter frequency, rad s−1 𝑚𝑑 : Mass of the damper body, kg 𝑥𝑚 : Amplitude of 𝑥, m.
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments This research was supported by the Major Special Technological Program (Grant reference: 2012ZX04010-021-04) and the Beijing Natural Science Foundation (Grant reference: 3122005).
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