ACTIVE VIBRATION CONTROL OF GEAR …. 1 shows the FxLMS algorithm structure for active vibration control of gear mesh. ... Active vibration control of gear transmission system test

ICSV21, Beijing, China, 13-17 July 2014 1

The 21st International Congress on Sound and Vibration

13-17 July, 2014, 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 400044

e-mail: [email protected]

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 time-

variant 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.

al1

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.al2

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 2000, Chen and Brennan3

developed an

active vibration control scheme that uses three magnetostrictive actuators mounted directly onto one

21st International Congress on Sound and Vibration (ICSV21), Beijing, China, 13-17 July 2014


of the gears. With this arrangement, the controller can produce circumferential forces that suppress

steady-state torsional vibrations. WU4 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 2005, LI5

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.

Brennan6 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 WIDROW7 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 algorithm9. 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 )(nx , through the

secondary path model filter reference signal )(nx , the residual vibration signal after active

control )(ne and the output control signal )(ny . Piezoelectric stack actuator as a secondary vibration

source produces the reverse vibration )(nd that directly compensates the gear meshing vibration

caused by the excitation force. Reference signal before participation value )(nw iteration should be

filtered by the secondary path model )(ˆ zS , then becomes a filtered reference signal )(nx . FxLMS

algorithm is described by the following formulas:

)()()( nyndne s (1)

)()()( nxnwny T (2)

)()()( zSnynys (3)

)(ˆ)()( zSnxnx T (4)

)()(2)()1( nxnenwnw (5)

where is the step length of weight value )(nw iteration update, via the formulas Eq. (5) the weight

iteration of FxLMS algorithm can be completed.



Figure 1. Block diagram of the FxLMS algorithm

Reference signal x(n) 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 real-

time 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 speed10

. 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.

1

1( )N z1( )e n( )d n

1( )kN z( )ke n

1( )pN z( )pe n

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

al11

proposed an adaptive control strategy based error path online modeling, the principle is shown

in Fig. 3. It adds a random noise v(n) uncorrelated with the reference signal x(n) 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(n) 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(n) in FxLMS algorithm. P(z)

represents the primary path from the vibration source to the error signal, d(n) means the response of

the system due to the vibration excitation source without active control and is the vibration signal to

be eliminated, e(n) 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(n) is generally selected from a lower power level white noise signal.

S(z)

LMS

W(z)

Primary path (P(z))

e(n)

y(n)x(n)

Noise generator

LMS

)(ˆ zS

d(n)

'( )x n

)(ˆ zS

FxLMS algorithm

Secondary path

online modeling

Vibration source

'( )e n

v(n)

'( )y n

Figure 3. The method of secondary path online modeling



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 (10) 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 10 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.

S(z)

LMS

W(z)

Gear system

y'(n)y(n)x(n)

White

noise

LMS

)(ˆ zS

d(n)

'( )x n

Frequency

estimator

Signal

generator

)(ˆ zS

v(n)

Acceleration

of axis

e(n)

Excitation

source

Figure 6. Test structure with hardware in loop



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 )(ˆ zS . The reference signal of the control system is set to 0, the superimposed noise

variance is set to 0.0005, the identification step is set to 0.05, 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 60th 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 )(ˆ zS with the different values of M, comparing M = 64 and M = 100,

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.

0 20 40 60 80 100

-0.1

-0.05

0

0.05

0.1

0.15

The number of weight

Weig

ht

M=100 M=64 M=32 M=16

Figure 7. The weight of different filter length

0 10 20 30 40 50

-0.1

-0.05

0

0.05

0.1

Time/s

Weig

hts

Figure 8.The weights of secondary path update with time

0 200 400 600 800 1000

0.2

0.4

0.6

0.8

Frequency/Hz

Am

plit

ude/(

g/V

)

M=16

M=32

M=64

M=100

0 200 400 600 800 1000-200

-100

0

100

200

300

Frequency/Hz

Phase/°

M=16 M=32 M=64 M=100

(a) (b)

Figure 9.Frequency characteristics of secondary path:(a) amplitude, (b) phase



4.3 Control Results and Analysis

During the experiment, the length of )(ˆ zS is set to 64, the step size is set to 0.05, the

superimposed noise variance is set to 0.0005, 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. 10 shows the time domain effect of control when gear speed is 486 r/min (mesh

frequency 300 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 0.9g with the

control effect attaining a steady state within 0.2s and the control effect is stable for a long time. Fig.

11 is the acceleration spectrum at 300 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 300 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

300 Hz the amplitude-frequency, the amplitude frequency characteristic has larger distinction.

Figure 10. the time domain control effect at fundamental frequency 300 Hz

0 100 200 300 400 500-70

-60

-50

-40

-30

-20

-10

Frequency/Hz

Vib

ration a

ccele

ration/d

B

without control

with control

Figure 11.the frequency domain control effect at fundamental frequency 300 Hz

before and after control

0 10 20 30 40 50 60-0.1

-0.05

0

0.05

0.1

The number of weight

Weig

ht

0 10 20 30 40 50

-0.1

-0.05

0

0.05

0.1

Time/s

Weig

hts

Figure 12. the weight of secondary path model Figure 13. the process of weights update



0 200 400 600 800 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency/Hz

Am

plit

ude/(

g/V

)

0 200 400 600 800 1000-200

-100

0

100

200

Frequency/Hz

Phase/°

(a) (b)

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 200Hz), the vibration has a decrease of 13dB at

fundamental frequency. Besides, when the rotational speed rises to 648r/min(mesh frequency

400Hz), the vibration at fundamental frequency attenuates 12dB. However, when the mesh

frequency is 400Hz, a relatively apparent increase of side-band frequencies can be seen between

300Hz and 350Hz, 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.

0 100 200 300 400 500

-70

-60

-50

-40

-30

-20

-10

Frequency/Hz

Vib

ration a

ccele

ration/d

B

without control

with control

0 100 200 300 400 500

-80

-70

-60

-50

-40

-30

-20

-10

Frequency/Hz

Vib

ration a

ccele

ration/d

B

without control

with control

(a) 200Hz (b) 400Hz

Figure 15.the frequency domain control effect at different fundamental frequency

before and after control

5. Conclusions

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



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 200Hz, 300Hz and 400Hz.

(5) When the fundamental frequency is 400Hz, 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. 50875270),

Foundation of state key laboratory of Mechanical Transmission (Grant No. SKLMT-ZZKT- 032).

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Documents

ACTIVE VIBRATION CONTROL OF GEAR …. 1 shows the FxLMS algorithm structure for active vibration control of gear mesh. ... Active vibration control of gear transmission system test