ACTIVE VIBRATION CONTROL OF GEAR TRANSMISSION SYSTEM

The 21 st International Congress on Sound and Vibration 13-17 July, 214, Beijing/China ACTIVE VIBRATION CONTROL OF GEAR TRANSMISSION SYSTEM Yinong Li, Feng Zheng, Ziqiang Li, Ling Zheng and Qinzhong Ding State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing, China 444 e-mail: ynli@cqu.edu.cn In this paper, an active control structure used piezoelectric stack actuator is built to suppress vibration due to internal incentive in gear transmission. Because the gear system is timevariant and unknown, an offline modeling technique is unable to tracking the changes in the secondary path, an adaptive filter algorithm used online secondary path modeling method is proposed to the active vibration control. In order to realize the algorithm, a C-MEX S function is used to build an FxLMS (Filtered-x least mean square) algorithm block in Simulink. Besides, reference signal estimation and secondary path modeling are critical components for the proposed control system. Therefore, an adaptive cascaded notch filter technology is used to extract the reference signal from the gear vibration. An adaptive LMS filter is also used to identify the secondary path by online experiment, effective secondary path transfer function is obtained at the same time avoid mutual interference between the secondary path identification and controller. Finally, the custom FxLMS block is downloaded to dspace as controller and a series of experimental studies are carried out. Experiment results show that the performance of the proposed piezoelectric stack actuator controlled by FxLMS to suppress the vibration is obvious, about 13dB-21dB of gearbox vibration is attenuated at different fundamental gear mesh frequency. 1. Introduction Gear system is an essential transmission device in all kinds of mechanical systems. In the working process, Gear system will generate error excitation what will cause vibration, because of the influence caused by the manufacturing and assembly errors, the time-varying mesh stiffness, meshing impact and other factors. The vibration of the gear system not only can generate noise and lead to the instability of the transmission system, but also can accelerate fatigue damage of the transmission system and make it failure, resulting in serious consequences. Reducing the vibration and noise of gear has very important engineering significance for decreasing the risk of gear box fault and improving the working condition. In 1994, Montague et. al 1 applied two piezoelectric chips as the actuators mounted onto one shaft in a gearbox system to control the meshing vibration of gear and reduction in gear mesh vibration up to 75% was reported. In 1999, Rebbechi et.al 2 applied an approach similar to that mentioned above to isolate the vibration transmitting between the gear shaft and housing, different in the usage of a pair of magnetostrictive actuators and an adaptive digital controller. The proposed system was able to simultaneously deal with the responses of the first three gear mesh harmonics. In 2, Chen and Brennan 3 developed an active vibration control scheme that uses three magnetostrictive actuators mounted directly onto one ICSV21, Beijing, China, 13-17 July 214 1

21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 214 of the gears. With this arrangement, the controller can produce circumferential forces that suppress steady-state torsional vibrations. WU 4 applied digital signal processor, combined with analysis theory and filter reference signal least mean square (Filter - x least - mean - square, FxLMS) algorithm, and designed a hybrid controller to reduce the vibration of gear system. In 25, LI 5 proposed a new method of gear vibration active control based on reference signal delay least mean square value (Delayed-x Least- mean - square, DLMS) algorithm, and carried out a large number of experiments. Brennan 6 compared the performances of different actuators used for active vibration control by experiments. The piezoelectric stack actuator devices with broad effective frequency range, low power consumption, large output force, compact structure and other advantages, is more suitable for gear meshing vibration active control which requires small displacement and large output force. FxLMS algorithm as an adaptive filtering algorithmic proposed by WIDROW 7 and BURGESS 8, respectively in the study of the adaptive control and the active noise control, is currently the most widely used in active vibration control. In this paper, piezoelectric stack actuator will be applied to active meshing vibration control of gear and the FxLMS algorithm will be used to control the output of the piezoelectric stack actuator. Through the output displacement of the piezoelectric stack actuator controlling the transverse vibration of the shaft of gear, the purpose of controlling the vibration of gear transmission system will be achieved. In this process, the synthesis reference signal is extracted from the gear meshing vibration signal based on an adaptive cascade notch filter technology, in addition, secondary path contains a piezoelectric stack actuator is online identified using LMS adaptive filter, so that the secondary path model can real-time track the changes of characteristic of the secondary path. 2. Control algorithm FxLMS algorithm with its simple form and strong stability, becomes the most widely used adaptive filtering algorithm 9. In this paper, we use this algorithm to control the output of the piezoelectric stack actuator. Fig. 1 shows the FxLMS algorithm structure for active vibration control of gear mesh. Algorithm consists of three inputs: the reference signal x (, through the secondary path model filter reference signal x (, the residual vibration signal after active control e( and the output control signal y (. Piezoelectric stack actuator as a secondary vibration source produces the reverse vibration d( that directly compensates the gear meshing vibration caused by the excitation force. Reference signal before participation value w( iteration should be filtered by the secondary path model S ˆ( z ), then becomes a filtered reference signal x (. FxLMS algorithm is described by the following formulas: e( d( y ( (1) s T y( w ( x( (2) y s ( y( S( z) (3) T x ( x ( Sˆ( z) (4) w( n 1) w( 2 e( x( (5) where is the step length of weight value w( iteration update, via the formulas Eq. (5) the weight iteration of FxLMS algorithm can be completed. ICSV21, Beijing, China, 13-17 July 214 2

21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 214 Figure 1. Block diagram of the FxLMS algorithm Reference signal x( is the vibration signal expected to be eliminated from gear system, the accuracy of the reference signal directly affects the control output. In this paper, we used the realtime frequency estimator and the waveform generator to acquire the reference signal that is needed by the adaptive controller. Besides, we designed a kind of frequency estimator using an adaptive cascaded notch filter. Compared with the traditional frequency estimation technology, the appliance has a higher estimation accuracy and faster convergence speed 1. Fig. 2 is the diagram of theory of the frequency estimation with adaptive cascaded notch filter. The adaptive cascaded notch filter is consisted of two order notch filters whose number is P, the center frequency of each notch filter is the sinusoidal signal frequency containing in the input signal. dn () en () Nz ( 1 ) e () 1 ( 1 ) 1 Nz k n k e () ( 1 Nz ) p n p the kth order Figure 2. Diagram of cascade adaptive notch filter for frequency estimation 3. Secondary path online modeling The secondary path identification is the necessary step of FxLMS algorithm. ERIKSSON et al 11 proposed an adaptive control strategy based error path online modeling, the principle is shown in Fig. 3. It adds a random noise v( uncorrelated with the reference signal x( at the controller output as an input of the adaptive filter. When the adaptive process is convergence, the filter can uniquely converge to S(z). The reference signal x( is used to update the weight W(z) on the one hand, on the other hand it is used to directly obtain the control output y( in FxLMS algorithm. P(z) represents the primary path from the vibration source to the error signal, d( means the response of the system due to the vibration excitation source without active control and is the vibration signal to be eliminated, e( represents the response when the system is applied active control and is the residual vibration signal. Identification of the secondary path is modeled using adaptive LMS filter. v( is generally selected from a lower power level white noise signal. Vibration source S ˆ( z ) x( W(z) Primary path (P(z)) y( Noise generator d( S(z) y'( Secondary path online modeling v( S ˆ( z ) e( x'( FxLMS algorithm LMS LMS e'( Figure 3. The method of secondary path online modeling ICSV21, Beijing, China, 13-17 July 214 3

21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 214 4. Active vibration control experiment A gear transmission system is designed in this paper consisting of a pair of spur gears based on the transverse vibration suppressing ideas. 4.1Control structure In this research, an active control structure of gear transmission system is designed and manufactured shown in Fig. 4. We built an active vibration control experimental system composed of the active gearbox structure with the piezoelectric stack actuator, controller, sensors and signal acquisition system, as shown in Fig. 5. Figure 4. Diagram of gear active structure based on piezoelectric actuator Detail information of the active gearbox vibration control system: (1) computer (2) torductor (3) dspace (4) charge amplifier (5) piezoelectric stack actuator (6) vibration acceleration sensor (7) LMS Test.Lab (8) current load control device (9) charge amplifier (1) vibration acceleration sensor (11) adjustable speed drive motor (12) gearbox (13) frequency conversion adjustment speed device (14) magnetic brake Figure 5. Active vibration control of gear transmission system test platform After active control, the vibration amplitude of residual error signal at the fundamental frequency may be significantly attenuated, so that the direct using of the estimated frequency of the residual vibration signal will generate a feedback to make the control system instable. Therefore, another acceleration sensor (sign 1 in Fig. 5) is mounted on the shaft and is used for frequency estimation. The specific schematic of the algorithm structure is shown in Fig. 6. Excitation source Signal generator x( W(z) Gear system y( d( S(z) y'( e( Frequency estimator S ˆ( z ) White noise v( S ˆ( z ) Acceleration of axis x'( LMS LMS Figure 6. Test structure with hardware in loop ICSV21, Beijing, China, 13-17 July 214 4

Amplitude/(g/V) Phase/ Weights Weight 21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 214 4.2 The Order of The Secondary Path According to the filter design theory, if the order of FIR filter is higher, the secondary path can approximate the actual transfer function better. But limited by the computing ability of control chip, the order of the FIR filter cannot be too high. Therefore, when the secondary path is online modeling, the order of the secondary path should be determined reasonably, that is the length M of the filter S ˆ( z ). The reference signal of the control system is set to, the superimposed noise variance is set to.5, the identification step is set to.5, then the offline identification is conducted in the shutdown state and the results of the different orders of filter are studied. Fig. 7 shows the distribution of each order weight of the secondary path FIR filter with convergence, that is the impulse response of the secondary path. Obviously, the weights after the 6th order are very little. Fig. 8 shows the weights of secondary path update process without control in offline identification case, after about 8 seconds, the weights converge to a very stable value. Fig. 9 shows the frequency characteristics of S ˆ( z ) with the different values of M, comparing M = 64 and M = 1, there is the small difference in the frequency characteristic. The FIR filter with M = 64 is enough to simulate transfer characteristics of the secondary path..15.1.5 M=1 M=64 M=32 M=16 -.5 -.1 2 4 6 8 1 The number of weight Figure 7. The weight of different filter length.1.5 -.5 -.1 1 2 3 4 5 Time/s Figure 8.The weights of secondary path update with time.8.6.4.2 M=16 M=32 M=64 M=1 3 2 1-1 M=16 M=32 M=64 M=1 2 4 6 8 1-2 2 4 6 8 1 (a) (b) Figure 9.Frequency characteristics of secondary path:(a) amplitude, (b) phase ICSV21, Beijing, China, 13-17 July 214 5

Weight Weights Vibration acceleration/db 21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 214 4.3 Control Results and Analysis During the experiment, the length of S ˆ( z ) is set to 64, the step size is set to.5, the superimposed noise variance is set to.5, the test load is set to 1 Nm and the gear speed is adjusted by the drive motor, so that the effect of active control at different gear mesh frequency can be tested. Fig. 1 shows the time domain effect of control when gear speed is 486 r/min (mesh frequency 3 Hz). The control is started at 8s, in the meanwhile the peak of the vibration acceleration on the housing immediately reduces from original 1.4g (1g = 9.8m/s2) to.9g with the control effect attaining a steady state within.2s and the control effect is stable for a long time. Fig. 11 is the acceleration spectrum at 3 Hz of vibration without control and with control, from the spectrum, it can be seen that the fundamental frequency is the main frequency component before control and the fundamental frequency signal is significantly attenuated after control, which reaches 19 db. The residual signal is almost only some broadband noise signal. Fig. 12 is the weights after convergence of secondary path at a mesh frequency of 3 Hz. Fig. 13 shows the weights of secondary path in time domain and comparing to the offline identification, the weights fluctuate greatly and converge slowly. Convergence speed of the secondary path is much smaller than that of the controller. Fig. 14 shows the frequency characteristics of the secondary path with the online identification. The difference compared with the offline identification results is not obvious, but at 3 Hz the amplitude-frequency, the amplitude frequency characteristic has larger distinction. Figure 1. the time domain control effect at fundamental frequency 3 Hz -1-2 -3 without control with control -4-5 -6.1.5-7 1 2 3 4 5 Figure 11.the frequency domain control effect at fundamental frequency 3 Hz before and after control.1.5 -.5 -.1 1 2 3 4 5 6 The number of weight Figure 12. the weight of secondary path model -.5 -.1 1 2 3 4 5 Time/s Figure 13. the process of weights update ICSV21, Beijing, China, 13-17 July 214 6

Vibration acceleration/db Vibration acceleration/db Amplitude/(g/V) Phase/ 21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 214.7.6.5.4.3.2.1 2 4 6 8 1 (a) 2 1-1 -2 2 4 6 8 1 Figure 14.the amplitude- frequency characteristic of secondary path:(a)amplitude; (b)phase Fig. 15 shows the spectrograms of the vibration acceleration of gearbox under different rotational speed before and after controlling. It can be seen from the spectrograms that at different rotational speeds, gear vibration has a varying levels of decrease after controlling. When the rotational speed is 324r/min (mesh frequency 2Hz), the vibration has a decrease of 13dB at fundamental frequency. Besides, when the rotational speed rises to 648r/min(mesh frequency 4Hz), the vibration at fundamental frequency attenuates 12dB. However, when the mesh frequency is 4Hz, a relatively apparent increase of side-band frequencies can be seen between 3Hz and 35Hz, which is almost identical with the resonance region of the secondary path identification, after the controlling action is applied. According to literature 13, this is over-control of out-of-band owing to large step length. (b) -1-2 -3-4 -5-6 -7 without control with control 1 2 3 4 5 5. Conclusions -1-2 -3-4 -5-6 -7-8 without control with control 1 2 3 4 5 (a) 2Hz (b) 4Hz Figure 15.the frequency domain control effect at different fundamental frequency before and after control To suppress the periodical vibration and noise generated by the meshing error of the gear transmission system, a method of feedback FxLMS control algorithm with secondary path online identification and a structure of active vibration control were adopted in this paper. Conclusions are listed as below: The order of FIR filter plays a critical role in the accuracy of the secondary path model directly. Compared to offline identification, the weight of secondary path modeling with online identification has a relatively greater fluctuation and a slower convergence speed. However, because the error of the frequency characteristic is very small, online identification still has a good control effect. After controlling the weight, the controller converges quickly, while the weight of the secondary path model converges relatively slowly, which implies that the emphasis of the ICSV21, Beijing, China, 13-17 July 214 7

21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 214 subsequent work should be put on how to improve the algorithm to quicken the convergence speed of the secondary path. (3) The vibration acceleration evoked by the superposing noise of online identification cannot be offset, which leads to increase of vibration acceleration of some frequency bands but does not have a big influence on the effect of control. (4) When adopting the FxLMS algorithm with online secondary path identification at different rotational speeds, the vibration accelerations at fundamental frequency are almost suppressed, which accords with the theoretical results perfectly. The attenuation can respectively be 13dB, 19dB and 21dB when the mesh frequencies are 2Hz, 3Hz and 4Hz. (5) When the fundamental frequency is 4Hz, some sideband frequencies of vibration signals have some increase due to the over control of out-of-band caused by oversize convergence step length. Acknowledgements This research is supported by the National Natural Science Foundation of China (Grant No. 587527), Foundation of state key laboratory of Mechanical Transmission (Grant No. SKLMT-ZZKT- 32). REFERENCES 1 Montague G T, Kascak A F, Palazzolo A, et al, Feed-forward control of gear mesh vibration using piezoelectric actuators, NASA Technique Memorandum 16366, (1994). 2 B. Rebbechi, C. Howard, C. Hansen, Active control of gearbox vibration, Proceedings of the Active Control of Sound and Vibration Conference, Fort Lauderdale, 295-34, (1999). 3 Chen M H, Brennan M J, Active control of gear vibration using specially configured sensors and actuators, Smart Materials and Structures, 9, 342-35, (2). 4 Jian-Da Wu, Jia-Hong Lin, Implementation of an active vibration controller for gear-set shaft using -analysis, Journal of Sound and Vibration, 281 (3-5), 137-155, (25) 5 Li M F, Lim T C, Shepard W S, Experimental active vibration control of gear mesh harmonics in a power recirculation gearbox system using a piezoelectric stack actuator, Smart Materials and Structures, 14(5), 917 927, (25) 6 M J Brennan, J Garcia-Bonito, S J Elliott, Experimental investigation of different actuator technologies for active vibration control, Smart Materials and Structures, 8(1), 145 153, (1999) 7 Widrow B, Shur D, Shaffer S, On adaptive inverse control, Proc.15th Asilomar Conf, Sata Clara CA, 185-189, (1981). 8 Burgess J C, Active Adaptive Sound Control in a Duct: A Computer Simulation, The Journal of the Acoustical Society of America, 7(3), 715 726, (1981) 9 Bernard Widrow, Samuel D. Stearns, Adaptive signal processing, China Machine Press, Beijing, (28) 1 Chu Zhao-Bi, Zhang Chong-Wei, Feng Xiao-Ying, Adaptive notch filter-based frequency and amplitude estimation, Acta Automatica Sinica, 1(36), 6-66, (21)(in Chinese) 11 Eriksson L J, Alile M A, Use of random noise for online transducer estimate in an adaptive active attenuation systems, J. Acoust. Soc. Amer., 85(2), 797-82, (1989). ICSV21, Beijing, China, 13-17 July 214 8