Study on a Method of Dynamic Response Function for the

Sensors & Transducers, Vol. 153, Issue 6, June 2013, pp. 111-117

Sensors & Transducers © 2013 by IFSA http://www.sensorsportal.com

Study on a Method of Dynamic Response Function for the Piezoelectric Measurement System Zongjin Ren, Zhenyuan Jia, Jun Zhang, Shengnan Gao, Yongyan Shang Key Laboratory for Precision and Non-traditional Machining Technology of Ministry of Education, Dalian University of Technology, 116024, P. R. China Tel.: +86-0411-84708415 E-mail: [email protected]

Received: 22 May 2013 /Accepted: 14 June 2013 /Published: 25 June 2013 Abstract: Some important areas use more and more piezoelectric force test systems, due to their high sensitivity and natural frequency. To study the dynamic testing performance of the system in the conditions of high natural frequency, one of the effective methods is establishing the mathematical model of dynamic response of test system. By improving specific algorithm of the conventional nonlinear modal fitting method, algorithm of transfer function which is aiming at this test system is deduced, and the transfer function of dynamic response is obtained by using amplitude and phase frequency response curve which is measured by using a piezoelectric test device and the curve is compared with the measured amplitude and phase frequency curve. The rise time, peak time and the overshoot of the step response of the test apparatus is obtained with the inverse Laplace transform. This paper provides a basis of objective evaluation for the dynamic performance of the test device. Copyright © 2013 IFSA. Keywords: Nonlinear modal analysis, Piezoelectric, Measurement, Step response.

1. Introduction The piezoelectric force test system is used more and more in aerospace, high-end CNC machine tools, automobile industry because of its high sensitivity and natural frequency. In some areas, needs of highprecision test promote the in-depth research of dynamic test performance of dynamic testing system. The traditional experimental methods of uninstalling weight quickly have been unable to meet the demand for higher natural frequency of such test systems because of the impact of the unloading time. Schoukens J. analyzed how to reduce leakage errors in frequency response function measurements (FRF) and a new default window is proposed [1]. Pan, Weiqing studied the approximate analytical impulse response function of aperture and misaligned ABCD

Article number P_1227

optical system in phase-space and formulate the input-output relation completely in terms of Wigner distribution functions [2]. Giraudeau Alain proposed an idea to compute the frequency response function of the tool work-workpiece structure, under the machining conditions of interest, for a given material as it is sheared by a single-point cutting tool with the objective of assessing its machinability [3]. Based on the method of conventional nonlinear modal analysis, its calculation method is optimized and dead cycle is avoided in a loop iterative process. The transfer function of the dynamic response of piezoelectric thrust test system is obtained by using this method, and the performance indicators of its step response are calculated. Once the transfer function of the test system is obtained, the vibration isolation and acceleration compensation of the test system could be studied.

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Sensors & Transducers, Vol. 153, Issue 6, June 2013, pp. 111-117

2. The Research on Nonlinear Modal Analysis Method

Cr ( ) 

Nonlinear modal analysis method is an important method for experimental modal analysis, wherein, the application of frequency domain method is common. By obtaining the time-domain signal of stimulus and response and FFT, and the amplitude-frequency and phase frequency characteristic curve of structural response is obtained, and then the algorithm is determined, and the transfer function is fitted and finally to identify the parameters to obtain the modal parameters [4-12].

2.1. Nonlinear Modal Fitting Method Nonlinear modal fitting method is a single point excitation frequency domain modal parameter identification method, that is stimulating a single point of system and obtaining time domain signal of excitation point and response point simultaneously, and the curve of frequency response function is obtained after A/D conversion and FFT transform, and then constituting the error vector together with theoretical transfer function whose modal parameters is as variables, and there is a non-linear relationship between error vector and identified modal parameters of the theoretical transfer function. For any system, the transfer function expression of any excitation and any response is showed as formula (1) below. n

H ( )   ( r 1

Ar A  r *) , s  sr s  sr

(1)

Ar  ur  jvr is established, Ar *  ur  jvr ; sr and sr * sr   r  j  r

is established,

sr *   r  j  r and n is the mode number.

When using FFT to analyze the signal, s  j , and substituting the above parameters into the formula (1), and the expression form such as formula (2) is obtained. ur  jvr

n

H ( )   ( r 1



ur  jvr

 r  j (   r )  r  j (   r )

),

(2)

The formula (2) can be rewritten as the form with two parts of the real and imaginary components, which is showed as formula (3). n

H ( )   r 1

(Cr ( )ur  Dr ( )vr ) n

 j ( ( Er ( )ur  Fr ( )vr ))

Dr ( ) 

E ( )   r

F ( )  r

  r  r 2  (   r ) 2



  r  r 2  (   r ) 2

  r r  2 2 2   (   )   (   )2 r r r r





r r   2  (   )2  2  (   )2 r r r r

Formula (1), formula (2) and formula (3) all show transfer function expression of any excitation point and any response point, and they are essentially the same, but with different forms. Some modal parameters need to be identified in the transfer function, such as real and imaginary parts of poles and residue of each order modal, and if the mode number is n, the number of parameters need to be identified is 4n, and the form of the vector can be expressed as follows: X  (u , v , ,  , u , v , ,  ,, u , v ,  ,  )T 1 1 1 1 2 2 2 2 n n n n

Due to addition of vector X to the variable of theoretical transfer function H ( ) of the test system, the transfer function H ( ) should be written in the form H ( , X ) . When    j is established, the difference between the theoretical value of the frequency response function H ( j , X ) and the actual value H ( j )

can

be

expressed

by

the

error

function  ( j , X ) , which is showed as formula (4).

In formula (1), Ar and Ar  is the residue of H ( ) at the pole sr and “*” means conjugation, and if are pole, and if

r r  2 2  r  (   r )  r  (   r ) 2 2

,

(3)

 ( j , X )  H ( j , X )  H ( j ) ,

(4)

If  j  1 , 2 ,, m is established, the number of frequency test points is m, thereby, the error vector E ( X ) is constructed, which is showed as the formula (5) below. E ( X )  H ( X )  H  ( ( , X ),  ( , X ), ,  ( , X ))T 1 2 m

,

(5)

Wherein, E ( X ) is nonlinear function of vectors X which is to be identified, and the error vector is founded. But it is inconvenient to study the problem by using the error vector E ( X ) , the degree of deviation between theoretical transfer function and actual curve of frequency response can be expressed by using total variance J ( X ) of the error vector E ( X ) , such as represented by formula (6).

r 1

J ( X )  E( X )H E( X ) ,

wherein,

112

(6)

Sensors & Transducers, Vol. 153, Issue 6, June 2013, pp. 111-117 wherein, “ H ” represents conjugate transpose. Expressed by the formula (6) is a real number, which is the difference of two squares of the degree of deviation of all points of frequency test between theoretical transfer function and the actual curve of frequency response, so formula (6) is the objective function of searching optimization of system model parameter. X 0  (u 0 , v 0 , 0 ,  0 , u 0 , v 0 , 0 , 1 1 1 1 2 2 2

 0 ,, u 0 , v 0 , 0 ,  0 )T 2

n

n

n

n

is initial iteration vector of vector X , and at any point of frequency test points, such as when    j is established, the form of an arbitrary element  ( j , X ) of error vector E ( X ) with Taylor series expansion in the vicinity of X 0 and retaining linear terms can be expressed as equation (7). 4n

 ( j , X )

i 1

xi

 ( j , X )   ( j , X 0 )  

( xi  xi 0 ) ,

(7)

Each of the above is substituted into the error vector E ( X ) , and be written in the matrix form, such as formula (8) below. E ( X )  E ( X 0 )  P 0 X ,

(8)

Wherein, P 0 is the initial iteration matrix, whose element is p ji 0 

 ( j , X ) xi

  j

X X0

The traditional methods of analysis used the form of the matrix norm, but in this paper, direct expanded form is used to avoid dead cycle in solving process, and the calculation of optimization algorithm of the nonlinear modal parameter is improved further, and formula(9) is obtained by the direct reference of formula (8). J ( X )  ( E ( X 0 )  P 0 X ) H ( E ( X 0 )  P 0 X ) ,

(9)

Seeking a derivative of the vector X after expanding formula (9), the form of formula (10) is obtained.

P 0T E ( X 0 )  P 0H E( X 0 )

,

(10)

Solved by the formula (10), iterations length calculating expression can be as shown by the formula (11). X  (( P 0 H P 0 )T  P 0 H P 0 ) 1 ( P E ( X 0 )  P E ( X 0 )) 0H

0T

2.2. The Definition of Initial Value of Iteration The definition of system initial value X 0 directly determines the speed and success of the iterative operation. Because the system amplitude frequency response curves is obtained beforehand, the natural frequency and damping ratio of each mode can be estimated in advance, according to the following formula (12) to (14). r   r 2   r 2 (r  1, 2, n) , r 

r

(12)

(r  1, 2, n) ,

(13)

 r  1   2 r (r  1, 2, n) ,

(14)

 r 2  r 2

The initial value  r 0 and  r 0 (r  1, 2, n) of each mode can be further calculated and estimated, so that the formula (3) become linear for variable ur and vr (r  1, 2, n) , and these variables can be obtained by further using the inner loop mode. ur and vr (r  1, 2, n) form a new vector X R : X R  (u1 , u2 , , u7 , v1 , v2 , vn )T X R is substituted into formula (3), and formula

.

0  (( P 0 H P 0 )T  P 0 H P 0 )X 

Step of the iteration of the loop is obtained by the formula (11), and the loop iteration is conducted until the total variance J ( X ) is less than a pre-determined value  .

,

(11)

(3) is written in the form of a matrix which is showed as formula (15). Rt  X R  A ,

 E11 E  12    E wherein， Rt   1m C  11  C12    C1m

(15)

 En1  En 2 

F11 F12 

 Enm  Cn1

F1m  D11 

 Cn 2 

D12  

 Cnm

D1m 

Fn1  Fn 2     Fnm  Dn1   Dn 2     Dnm 

 

A  [H1I (1), H2I (2 ),HmI (m ), H1R (1), H2R (2 ),, HmR (m )]T

In the formula (15), the elements in the matrix A are the calculation value of the transfer function H ( ) which is unknown with specific frequency, and H ( ) can be replaced by the measured amplitudefrequency curve, such as formula (16). H ( j )  H ( j )  H R ( j )  jH I ( j ) ( j  1, 2, m)

,

(16)

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Sensors & Transducers, Vol. 153, Issue 6, June 2013, pp. 111-117 Formula (17) can be obtained by using formula (15) and formula (16). 1

X R  ( Rt Rt ) Rt A , T

T

(17)

X R can be calculated according to formula (17),

which together with  r 0 and  r 0 (r  1, 2, n) contribute the initial iteration vector X 0 .

3. Test System Dynamics Modeling 3.1. Frequency Response Experiments Piezoelectric measuring equipment for measuring the performance of the rocket engine thrust is showed in Fig. 1. Dynamic response experiments of the test apparatus is conducted in the manner which is showed in Fig. 2.

From the viewpoint of research on dynamic characteristics of testing system, and it is equivalent to measuring excitation at the flange and testing vibration at the three-way force sensor in the force measuring device. This exciting point and vibration measurement point is the excitation point and response point of dynamic characteristics which need to be studied. Two-way output data of the hammer and the test device are collected at the same time. The input and output data are transformed by FFT, and frequency domain signal of input and output are conducted by self-spectral and cross-spectral analysis, and ultimately, amplitude and phase frequency characteristic curve is obtained from the flange excitation and the output of test device, which is showed in Fig. 3. The numbers of peak point is 7, that is, the mode of this direction can be set at 7 orders. The information of frequency, phase and amplitude of each modal is listed in Table 1. As can be seen from Fig. 3, the modal of main thrust direction is dense, and it is suitable for using nonlinear modal parameter identification method to identify its modal parameters.

Fig. 1. Piezoelectric measuring equipment for measuring thrust. Fig. 3. Amplitude-frequency and phase-frequency.

3.2. Modal Parameter Identification

Fig. 2. Dynamic response experiments of the test apparatus.

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Firstly, frequency point which need modal identification is determined, according to the frequency response curve showed in Fig. 3, and each point is chose at appropriate position on the trough of both sides of natural frequency points of each order, respectively, and together with the modal natural frequency point, composing frequency test points of non-linear modal parameter identification. The values of trough are showed in Table 2. Then,  r 0 and  r 0 (r  1, 2, 7) of each mode is estimated, according to the damping ratio of the general mechanical structure which is between 0.01 and 0.09, and presuppose that the damping ratio of test device is 0.01, and according to equation (12) to (14), the results are listed in Table 3.

Sensors & Transducers, Vol. 153, Issue 6, June 2013, pp. 111-117 Table 1. Modal information of order. Modal order

1

2

3

4

5

6

7

Damped natural frequency /Hz

1074

1702

1934

2050

2087

2142

2606

Phase /º

-70.8

-65.1

-87.0

-110.6

-121.9

-118.5

-95.5

real part

1.24

2.31

0.89

-1.36

-1.81

-2.41

-0.86

Imaginary part

-3.56

-4.97

-17.03

-3.63

-2.91

-4.44

-8.87

Amplitude

3.770

5.481

17.053

3.876

3.427

5.052

8.912

Table 2. Frequency response information of troughs. The point of frequency

1

2

3

4

5

6

7

8

9

10

Frequency /Hz

1000

1100

1570

1780

1990

2060

2100

2200

2580

2700

Phase /º

-7.3

-108.2

-6.4

-30.0

-145.9

-120.1

-126.0

-165.7

-48.9

-168.8

real part

2.08

-0.45

2.62

1.63

-4.14

-1.92

-1.83

-2.81

3.50

-2.80

Imaginary part

-0.27

-1.37

-0.29

-0.94

-2.80

-3.31

-2.52

-0.72

-4.01

-0.55

Amplitude

2.097

1.442

2.636

1.882

4.998

3.827

3.114

2.901

5.323

2.854

Table 3. Iterative initial value. Modal order

1

2

3

4

5

6

7

Modal frequencies /Hz Damping ratio  r

1074

1702

1934

2050

2087

2142

2606

0.01

0.01

0.01

0.01

0.01

0.01

0.01

r

0

67.5

106.9

121.5

128.8

131.1

134.6

163.7

r

0

6748.1

10694.0

12151.7

12880.5

13113.0

13458.6

16374.0

substituting the data  r 0 and 0  r (r  1, 2, 7) of Table 3, the imaginary part and the real part of the data of frequency test points of Table 1 and Table 2 into equation (17),it can be obtained as follow: When

X R  [22.24,13.62, 24.48, 71.37, 68.62, 38.44, 0.15  256.12, 526.21, 2048.77, 280.15, 224.16, 575.90, 1420.82]

Finally, the final similar result of total variance J ( X 6 ) is 6.9  1019 after six iteration process. [3.94

452.77

128.24

7

H ( s)   ( r 1

ur  jvr ur  jvr ),  s  ( r  j  r ) s  ( r  j  r )

(18)

wherein u  [ 3.94 58.10 15.20 35.50 32.00 19.80 18.10] v  [452.77 957.35 2267.30 287.61 192.52 524.51 1324.14]

  [128.24 203.22 133.68 128.81 131.14 134.59 147.37]   [6748.14 10693.98 12151.68 12880.53 13113.01 13458.58 16373.98]

Academic and actual amplitude and frequency curve of the test system is superimposed onto a map, as shown in Fig. 4.

6748.14

58.10 957.35 203.22 10693.98 15.20 2267.30 133.68 12151.68 X 6  35.50 287.61 128.81 12880.53 32.00 192.52 131.14 13113.01 19.80 524.51 134.59 13458.58 18.10 1324.14 147.37 16373.98]

Therefore, the theoretical transfer function of the test system is showed as formula (18).

Fig. 4. Comparison of the theoretical and the practical.

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Sensors & Transducers, Vol. 153, Issue 6, June 2013, pp. 111-117 Theoretical amplitude and phase frequency characteristic curve with nonlinear fitting is consistent with the curve of actual test. The transfer function with nonlinear modal fitting can be used to study the dynamic performance of the test apparatus.

1 F ( s)  H ( s) , s

According to the nature of definite integral of the Laplace transform, the expression of time domain can be expressed as formula (22).

4. Step Response Performance of the Test System

t

f (t )   h(t )dt , 0

According to inverse transform formula of the rational function, the form of the expression of timedomain of transfer function is showed as formula (19) [13, 14].

(21)

(22)

The parameters with modal fitting of the test device are substituted into equation (22), and the matlab programming is employed, and the image of step response is obtained as shown in Fig. 5.

n

h(t )   (e i t e j i t (ui  jvi )  i 1

e

 i t  j  i t

e

,

(19)

(ui  jvi ))

After finishing, the form as formula (20) is obtained. n

h(t )   2e i t (ui cos( it )  vi sin( i t )) ,

(20)

Fig. 5. Step-response of the system.

i 1

Wherein, n is modal order. Under the step response, time-domain response of image function of the test apparatus is showed as formula (21)

It can be seen from the image of step response which is obtained based on the calculation, the key performance indicators of test system are showed in Table 4.

Table 4. Step-response performance of the system. Index

Rising Time /s 8.60×10-5

Peak time /s 2.46×10-4

All the test system is manufactured with stainless steel whose damping coefficient of energyconsuming is small, and pure rigid connection is used between the various components, thus resulting in the damping of each order is relatively small, and overshoot and adjustment time whose dynamic characteristic indicators is step response is relatively large.

5 Conclusions Based on the traditional method, algorithm of nonlinear modal analysis method is improved, and it avoid infinite loop in the loop iterative process. With this method, theoretical transfer function of the piezoelectric thrust measuring apparatus is calculated, and the mathematical model of the dynamic response of the apparatus is founded, and the step response performance of the test apparatus is obtained, so it provides the basis for an objective

116

Adjustment time /s 2.66×10-2

Overshoot 76.2%

evaluation of the dynamic test performance of the test apparatus.

Acknowledgement This work is supported by the national natural science founds of china under grants no. 51205044 and key laboratory research projects for Liaoning province department of education under grants no. L2012015.

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