Time-domain Galerkin method for dynamic load

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2015) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4991

Time-domain Galerkin method for dynamic load identification Jie Liu*,† , Xianghua Meng, Chao Jiang, Xu Han and Dequan Zhang State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China

SUMMARY This paper proposes a new method called time-domain Galerkin method (TDGM) for investigating the structural dynamic load identification problems. Firstly, the shape functions are adopted to approximate three parameters, such as the dynamic load, kernel function response, and measured structural response Secondly, defining a residual function could be expressed as the difference of the measured response and the computational response. Thirdly, select an appropriate weighting function to multiply the defined residual function and make integral operation with respect to time to be zero. Finally, when the shape functions are chosen as the weighting function, it establishes the forward model called TDGM. Furthermore, the regularization method could have effectiveness in solving the ill-posed matrix of load reconstruction and obtaining the accurate identified results of the dynamic load. Compared with the traditional Green kernel function method (GKFM), TDGM can effectively overcome the influences of noise and improve the accuracy of the dynamic load identification. Three numerical examples are provided to demonstrate the correctness and advantages of TDGM. Copyright © 2015 John Wiley & Sons, Ltd. Received 22 December 2014; Revised 14 July 2015; Accepted 21 July 2015 KEY WORDS:

dynamic load identification; time-domain Galerkin method; shape function approximation; ill-posedness; regularization

1. INTRODUCTION Nowadays in many practical engineering problems, the dynamic load acting on a structure is always involved [1]. Once the dynamic load is obtained correctly, it is convenient to use a series of advanced methods to ensure the stability and safety of engineering structures [2] and to satisfy the requirement of modern industry. However, for most practical engineering problems, such as the wind load applied to ocean platforms and the interaction between road and tires, it is relatively hard to measure the dynamic load due to the lack of technology or cost restrictions. Thus, the demand of the load identification is presented, which leads the vigorous development of the technique for the dynamic load identification [3]. Dynamic load identification [4–11] belongs to the second category of inverse problems [12–14] in structural dynamic systems. When the measured response of internal point on the structure is obtained and the system properties are known, it is a process to identify the dynamic load acting on the structure. Dynamic load identification is not only one of the keys to the structure dynamic design but also a problem containing important research value and significance for practical engineering. There are two main traditional methods for load identification, namely, frequency-domain method [15, 16] and time-domain method [17–19]. Frequency-domain method calculates the load spectrum by using the frequency response function and the response spectrum of the system in modal coordinate system, and then dynamic load can be transformed from the modal coordinate into the time domain. Moreover, time-domain method based on the kinetic equations is mainly dependent on the

*Correspondence to: Jie Liu, State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China. † E-mail: [email protected] Copyright © 2015 John Wiley & Sons, Ltd.

J. LIU ET AL.

relationships between the dynamic load and the system responses, in order to obtain the time history of load. Compared with frequency-domain method, they can identify various types of loads, for instance, impact load [20, 21]. Above all, the identified results of time-domain method have a clear physical meaning and a relatively higher accuracy. As a result, time-domain method is more practical and convenient for engineering problems. In recent years, some efforts had been achieved for the dynamic load identification, especially for time-domain method with an acceptable accuracy. Based on the modal transformation method for load identification, Desanghere et al. [22] proposed the dynamic load identification of timedomain method. Ory et al. [17] adopted the modal coordinate transformation method to decouple the differential equations. The dynamic load was described as step function during a limited time interval, and the reverse model was established for load identification. Kreitinge et al. [23] proposed the sum of weighted acceleration technique method, which was a strategy to identify the dynamic load by calculating the weighted sum of the measured acceleration responses. However, the results’ accuracy is mainly affected by the position of the measured points and the weighted coefficients. Doyle [24] presented an inverse convolution method based on the wavelet transform or the Fourier transform to achieve the dynamic load reconstruction of plate and beam structure. Law and his colleagues [25–28] investigated the moving load identification of bridge–vehicle system in time domain. Tang [21] proposed a kind of time-domain inverse method of discrete system. On condition that the excitation frequency was lower than cutoff frequency, the simulation calculations were provided and the satisfactory results were achieved. Lindfield et al. [29] proposed the singular value decomposition technique, which is applied for load identification problems. It was adopted to deal with the contradiction equations, and the least-squares solution was obtained if coefficient matrix was rank deficient. Zhang and Zhu [30] have employed the generalized orthogonal polynomial to discrete the dynamic load in time domain and studied the dynamic load identification problems utilizing the idea of finite element method (FEM) theory. Liu and Han [31] applied Green impulse and Heaviside step function to discretize the dynamic response equations of the structural system. Liu et al. [18, 19] proposed the shape function method based on moving least-square fitting method. Their results of the numerical examples showed in good accuracy and efficiency. However, most of the methods mentioned earlier are based on the discretization of the time domain. Nevertheless, there are several existing problems. On the one hand, the size of sampling time interval seems to be one of the main factors limiting the accuracy of the identified results. When the aforementioned methods are used to approximate the dynamic load, the identified results remain accurate only at small time intervals. However, they will result in the large matrix of kernel function response. In this case, the ill-posedness of the kernel function response has been increased. On the contrary, when the large time step is selected, those aforementioned methods not only reduce the accuracy of the identified results but also waste the information of the response points within sampling time step because they only use the information at two endpoints. On the other hand, most methods have a weak anti-noise performance. Noise is the error of measurement caused by all kinds of factors, such as vibration in the measured process. Even with applying the regularization method [32–35] to deal with the ill-posedness [36, 37], the accuracy is still unacceptable with the increase of noisy level. In this paper, a new dynamic load identification method (TDGM) is proposed to solve the shortcomings mentioned earlier. In the first step, the time domain should be discretized into a series of time elements. Note that there may be several sampling time intervals and sampling time points in each time element. Then the dynamic load, kernel function response, and structural response are approximated by a variety of shape functions in each time element. Therefore, TDGM not only has a description of the dynamic load and the responses in a time element but also guarantees enough accuracy with the increase of the time element’s size or the decrease of the size of the kernel function response matrix. The shape function coefficients are calculated by the least-squares fitting method using the data of sampling time points within each time element in order to ensure that all the information of each sample points are used as much as possible. After that, the approximate results are plugged into the original convolution equations of structure dynamic response. A residual function, which is the difference of the measured response and the computational response, can be defined. For establishing the forward model of dynamic load identification, the integral with respect to the time of the residuals multiplied by the weighting functions, which have been determined previously, Copyright © 2015 John Wiley & Sons, Ltd.

Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

TIME-DOMAIN GALERKIN METHOD FOR DYNAMIC LOAD IDENTIFICATION

is equal to zero. When choosing the previous shape functions as the weighting function, it successfully establishes the dynamic forward model based on TDGM. More importantly, the application of regularization technique can efficiently overcome the ill-posedness of load reconstruction and realize the stable identification of dynamic load. Because TDGM contains integral operation, which has an effect of smoothing noise, it can effectively reduce the effect of noise and improve the identified accuracy. Three numerical examples are given for the demonstration. The identified results of the first and the third numerical examples are both compared with ones based on GKFM in order to demonstrate the superiority of the method. The second example gives the results based on different regularization methods. 2. FORMULATION OF THE FORWARD PROBLEM BASED ON GKFM For determining the system, dynamic load can be expressed as a superposition of the unit pulse signal in time domain. Under the linearity and time-invariant suppositions [18] of dynamic load identification problems, the dynamic load is discretized as a series of unit pulse load superposition. After obtaining the response caused by unit impulse load system, the response caused by random dynamic load can be expressed as [38] Z t m p c . /g.t /d (1) y .t / D 0 m

where y .t / is the measured structural response which may be displacement, velocity, acceleration, stress, strain, and so on. g.t / is the corresponding Green’s kernel function response which is calculated by applying the unit pulse load on the structure. p c .t / is the dynamic load acting on the structure which needs to be identified. The whole time domain is separated into Q equal spaced time intervals. Discretizing the aforementioned convolution integral, Equation (1) is transformed into a matrix form as follows [19]: 2 m 3 2 32 c 3 y .t1 / g.t1 / p .t0 / 6 y m .t / 7 6 g.t / g.t / 7 6 p c .t / 7 2 7 1 1 6 6 2 76 7 6 7D6 76 7 t (2) :: :: 6 7 6 :: :: : : 76 7 : 4 5 4: : 54 5 : : y m .tQ / g.tQ / g.tQ1 / : : : g.t1 / p c .tQ1 / where y m .th / is the measured structural response, g.th / and p c .th / are the Green kernel function response and the exciting load at time point th D ht .h D 1; 2; : : : ; Q/, respectively. t is the size of a time element. Consequently, there are some problems. On one hand, in order to describe the load actually, the time sampling interval should be as small as possible. But, this will lead to the large size of the kernel function matrix and the serious ill-posedness [29]. On the other hand, according to Equation (2), because the section of actual load between any two discrete time points is approximated by unit impulse function and this approximating function is no longer continuous and smooth especially at discrete time points. Hence, excessively long discrete time interval would give rise to quite low approximating accuracy. According to the formulas, it can be seen that the method only uses the response at each sampling time interval endpoints, but those within a sampling time interval have not been fully used. In order to deal with the aforementioned problems, TDGM is proposed in this paper. 3. FORMULATION OF THE FORWARD PROBLEM BASED ON TDGM Using the idea of discretizing variable in spatial domain of FEM [17], the time domain is divided into many elements which interconnect with each other, and there are several time points in each element. Then local coordinate systems are defined on each time element. It aims to more easily obtain the varying form of dynamic load, computed kernel function response, and measure structural Copyright © 2015 John Wiley & Sons, Ltd.


J. LIU ET AL.

response in time element. In the ith time element under the local coordinate system, the dynamic load, computed kernel function response, and the structure dynamic response can be approximately described by using the linear combination of basic functions, respectively: k1 X

pic .t / D

qj .t /aij D qT .t /ai ; 0 6 t 6 .n 1/t

(3)

wj .t /bij D wT .t /bi ; 0 6 t 6 .n 1/t

(4)

j D1

gi .t / D

k2 X j D1

k3 X

yim .t / D

vj .t /cij D vT .t /ci ; 0 6 t 6 .n 1/t

(5)

j D1

where q.t / D q1 .t / q2 .t / : : : qk1 .t / , w.t / D w1 .t / w2 .t / : : : wk2 .t / ; and v.t / D v1 .t / v2 .t / : : : vk3 .t / are the vector of the linearly independent basic functions. k1 , k2 , and k3 are the numbers of the basic functions of dynamic load, computed kernel function response, and measured structural response, respectively. n is the number of sampling time points in each a a : : : a is the undetermined coefficient vector of dynamic load. D time element. a i1 i 2 i i k 1 bi D bi1 bi 2 : : : bi k2 and ci D ci1 ci 2 : : : ci k3 are the coefficient vectors of the computed kernel function response and measured structural response. Note that gi .t / is the computed kernel function response, which is calculated by applying shape function loads on the structure. In order to ensure that the coefficients bi and ci can be determined via least-squares method, k1 , k2 , and k3 should not be bigger than n. Because gi .t / and yim .t / are obtained directly, coefficients bi and ci can be determined by the least-squares method, respectively. The kernel function response g.t / and structural response y m .t / in the global coordinate system can be approximately expressed as follows: g.t / D

Q X

gi .t / D

i D1

m

y .t / D

Q X

Q X

wT .t /bi ; t > 0

(6)

i D1

yim .t /

D

i D1

Q X

vT .t /ci ; t > 0

(7)

i D1

Substituting Equation (6) into Equation (1), the computed response y c .t / can be obtained: Z t Q X c p c . / wT .t /bi d y .t / D 0

(8)

i D1

Defining the residual function R.t / in the time domain R.t / D y m .t / y c .t / D

Q X

Z

T

v .t /ci

c

p . / 0

i D1

t

Q X

wT .t /bi d

(9)

i D1

Using the weighted residual method, make the weighted integral of the residual function R.t / to be zero in the whole time domain: Z t ˛.s/R.s/ ds D 0 (10) 0

where ˛.t / is the weighting function defined in time domain, and the weighting function can be any independent function. Copyright © 2015 John Wiley & Sons, Ltd.



Therefore, following equation can be‘ obtained: " Q # Z s Z t Q X X T c T ˛.s/ v .s/ci p . / w .s /bi d ds D 0 0

0

i D1

(11)

i D1

Choosing different weighting functions leads to a different forward model of dynamic load identification, and there are many choosing methods such as collocation method, the subdomain method, the least-square method and Galerkin method [39]. For the convenience of calculation and simple statement, it might as well make v.t / equal to w.t /. Therefore, when choosing the basic function wj .t /.j D 1; 2 : : : ; k2 / as the weighting function ˛.t /, it establishes the forward model of TDGM, and Equation (11) is written as follows: " Q # Z t Z s Q X X T c T wj .s/ v .s/ci p . / w .s /bi d ds D 0; j D 1; 2 : : : ; k2 (12) 0

i D1

0

i D1

Because each time element contains k2 basic functions and the total time domain is divided into Q time elements, it eventually forms an overdetermined equation of system with the order of .k2 Q/ .Q C 1/ in time domain. Particularly, when choosing the interpolation function as the basic function of dynamic load, the coefficient vector ai is equal to the corresponding value of the load. When k1 D 2, which means q.t / D q1 .t / q2 .t / , the final forward model can be established as follows: 2 q1 q2 3 2 m 3 g11 g11 y11 6 q1 q2 7 7 g12 g12 6 m7 6 6 7 y 6 12 7 6 7 6 7 6:: :: 7 6 :: 7 6: : 7 6 : 7 6 7 6 7 6g q1 g q2 7 6y m 7 6 1k2 1k2 7 6 1k27 6 7 6 m 7 6g q1 g q2 Cg q1 g q2 7 6 y21 7 6 21 21 11 11 7 6 7 6 7 6 y m 7 6g q1 g q2 C g q1 g q2 7 6 22 7 6 21 22 12 12 7 6 7 6 72 : : : : 6 : 7 6: 7 pc 3 : : 6 : 7 6: : : 7 6 7 6 q 7 6 1c 7 q2 6y m 7 6g 1 g q2 C g q1 7 6 p2 7 g 6 2k27 6 2k2 2k2 1k2 1k2 76 6 m7 6 q q 7 6 pc 7 q q q q 2 1 2 6 y31 7 6g 1 g 2 C g 1 76 3 7 g21 C g11 g11 6 m 7 6 31 31 21 76 : 7 6 y 7D6 q q 76 : 7 q2 q1 q2 6 32 7 6g 1 g 2 C g q1 76 : 7 g22 C g12 g12 6 : 7 6 32 32 7 22 76 6 : 7 6: 76 : 7 :: :: :: 6 : 7 6: 7 4 :: 7 6 m 7 6: : : : 5 7 6y 7 6 q 7 c q2 q1 q2 6 3k27 6g 1 g q2 C g q1 p 7 g C g g 6 : 7 6 3k2 3k2 2k2 2k2 1k2 1k2 7 QC1 6 : 7 6: 7 6 : 7 6: 7 6 7 : 7 6 :: 7 6 6 7 6 : 7 6: 7 6 7 6:: 7 6ym 7 6 7 6 Q1 7 6 q1 q2 q1 q2 q1 q2 q1 q2 q1 q2 7 6 m 7 6gQ1 gQ1 C g.Q1/1 g.Q1/1 C g.Q2/1 g.Q2/1 C g.Q3/1 g21 C g11 g11 7 6 yQ2 7 6 7 6 7 q1 q2 q1 q2 q1 q2 q1 q2 q1 q2 7 gQ2 gQ2 C g.Q1/2 g.Q1/2 C g.Q2/2 g.Q2/2 C g.Q3/2 g22 C g12 g12 7 6 : 7 6 6 6 :: 7 6 7 4 5 6:: 7 : m 4 5 yQk2 q1 q2 q1 q2 q1 q2 q1 q2 q1 q2 gQk2 gQk2Cg.Q1/k2 g.Q1/k2Cg.Q2/k2 g.Q2/k2Cg.Q3/k2 g2k2Cg1k2 g1k2 (13) gijq1 D Copyright © 2015 John Wiley & Sons, Ltd.

Z

.n1/t 0

wj . / wT . /bqi 1 d

(14) Int. J. Numer. Meth. Engng (2015) DOI: 10.1002/nme

J. LIU ET AL.

gijq2 D

yijm

Z

.n1/t

0

Z D

wj . / wT . /bqi 2 d

(15)

wj . / wT . /ci d

(16)

.n1/t

0

where gijq1 and gijq2 are the computed kernel function responses caused by shape function load q1 .t / and q2 .t / respectively. yijm is the measured dynamic response caused by actual load. bqi 1 , bqi 2 , and ci are the corresponding coefficient vectors. Equation (13) can be expressed in matrix form as follows: Ym D GPc

(17)

In this paper, TDGM not only revises the kernel matrix G but also revises Ym in the sense of the least-squares method. Because the least-square method has a filtering effect, it can smooth the noise in the measured response. In addition, integral has the function of the smoothing noise, which is conducive to the stable reverse of the dynamic load. What’s more, the equation established by TDGM is an overdetermined system, which makes full use of the information of computed kernel function responses and measured structural response and improves the identified accuracy. 4. REGULARIZATION METHOD FOR ILL-POSED PROBLEM Because the kernel function matrix established by TDGM or GKFM is generally ill-posed and the measured responses contain inevitable noise, the identified load may be inaccurate via direct inverse operation. Therefore, it cannot be deconvoluted through conventional matrix operations. In order to obtain an efficient and stable solution, regularization methods are adopted. In this paper, we choose the least-square QR (LSQR) [40] iterative regularization method. LSQR method is used to solve the linear system equations and the least-squares problem based on Lanczos iterative process, which is a relatively simple and practical process to realize tri-diagonal matrix. This method has the similar numerical characteristics of conjugate gradient method. Its convergence speed is faster and computational accuracy is higher. Especially when the matrix is illposed, it can get relative accuracy in less iteration steps, and the numerical stability is good, thus it is suitable for solving ill-posed problems. The least-squares problem .min kYm GPc k2 / of dynamic load identification can realize the matrix bidiagonalization [37] of kernel function matrix G through the following algorithm in Table I. Assuming that the matrix bidiagonalization process has been underway for ksteps, two standard orthogonal matrixes UkC1 ; Vk and a bidiagonal matrix Bk can be obtained: UkC1 D u1 u2 : : : ukC1 2 Rm.kC1/ Vk D v1 v2 : : : vk 2 Rmk Table I. Lanczos algorithm of matrix bidiagonalization process. Chose the starting vector Ym , and let 1 D kYm k2 ; u1 D Ym =1 ; v0 D 0 for i D 1; 2; : : : ; k ri D GT ui i vi1 i D kri k2 vi D ri =i si D Gvi i ui iC1 D ksi k2 uiC1 D si =iC1 end Copyright © 2015 John Wiley & Sons, Ltd.



3

2

1 6 2 2 6 6 : 6 3 : : Bk D 6 6 6 :: 4 :

k kC1

7 7 7 7 7 2 R.kC1/k 7 7 5

The three matrixes meet the relationship UT GV D B

(18)

The iterative process can be denoted as 9 > =

UkC1 .1 e1 / D Ym GVk D UkC1 Bk T

G UkC1 D

Vk BTk

C

kC1 vkC1 eTkC1

(19)

> ;

where ei is the ith column of an identity matrix. Assuming that 9 Pck D Vk qk > = rk D ym GPck > tkC1 D 1 e1 Bk qk ;

(20)

we can obtain the following equations: rk D Ym Gpck D UkC1 .1 e1 / GVk qk D UkC1 .1 e1 / UkC1 Bk qk D UkC1 tkC1 min kYm GPc k2 D min ktkC1 k2 D min k1 e1 Bk qk k2

(21)

(22)

It will transform a complex least-squares problem into a simple least-squares problem on subspace. At last, determine a suitable tolerance to obtain a fairly accurate result. The application of standard QR factors decomposition technique is the main idea of LSQR method. In fact, when identifying dynamic load based on iterative regularization method, the lowfrequency part of the solution converges faster than the high-frequency part, and the structural response caused by the dynamic load is always belong to the low-frequency signal compared with noise. Therefore, this process based on the LSQR iterative method has the effect of implicit regularization, and the iterative step plays the role of regularization parameters. Reasonably choosing tolerance and iteration step will realize the stable reverse of the dynamic load.

Figure 1. The finite element model of plane truss. Copyright © 2015 John Wiley & Sons, Ltd.


J. LIU ET AL.

5. NUMERICAL EXAMPLES AND DISCUSSIONS In order to verify the proposed approach, three numerical examples are prepared. A plane truss problem is firstly investigated to illustrate how to identify dynamic load by TDGM, and the identified results are compared with GKFM. Then, the proposed approach is applied to a truss beam structure, which has been studied for load identification based on a presented frequency method with a quite accuracy in reference [41]. However, this example repeats the load identification process based on TDGM. What’s more, it gives the results based on conjugate gradient (CG) iterative regularization method [42] and Monte-Carlo generalized cross-validation (GCV) criterion [43] for comparison The third example is a vehicle door problem and compares the results based on TDGM and GKFM as the first example. In those three numerical examples, noise is directly added to the computer-generated response to simulate the noise-contaminated measurement according to the following equation: 1

The shape function load

q (t )/N

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1 -3

Time t /s

x 10

Figure 2. The shape function loads. -3

The displacement response y m / m

8

x 10

6 4 2 0 -2 -4 -6 -8

0

0.1

0.2 Time t /s

0.3

0.4

Figure 3. The displacement of Point 7 in y-direction with 25% noisy level. Copyright © 2015 John Wiley & Sons, Ltd.



ym D ycom C lno std.ycom / randn

(23)

where ycom is the computer-generated response, std .ycom / is the standard deviation of ycom ; randn denotes a random vector from a standard normal distribution and its size is the same as ycom , and lno is a parameter to control the noisy level. In addition, in order to assess the approximation of the identified result to the true load, the relative error (RE) and the correlation coefficient (CC) are defined as follows: RE D Q1 P

CC D

kPc Pk kPk

(24)

ŒP c .th / E.P c / ŒP .th / E.P/

hD0

(25)

kP c E.P c /k kP.th / E.P/k

where E.: / denotes the mean value. 6000 Actual load Identified load 4000

Load p / N

2000

0

-2000

-4000

-6000

0

0.05

0.1

0.15

0.2 0.25 Time t /s

0.3

0.35

0.4

6000 Actual load Identified load 4000

Load p / N

2000

0

-2000

-4000

-6000

0

0.05

0.1

0.15

0.2 Time

0.25

0.3

0.35

0.4

t /s

Figure 4. The identified results with 5% noisy level based on (a) time-domain Galerkin method (TDGM) and (b) Green kernel function method (GKFM). Copyright © 2015 John Wiley & Sons, Ltd.


J. LIU ET AL.

5.1. A plane truss structure problem A 15-bar truss problem as show in Figure 1 is investigated. The length of all the horizontal and vertical bars L D 20 m, and all the bars have the same Young’s Modulus E D 2 1011 Pa and the same Poisson’s ratio D 0:3. Assume that the structural damping is the proportional damping and that the parameters related to the mass and stiffness matrices are 0 and 3 103 , respectively. Point 4 is fixed and the vertical displacement of Point 8 is limited to be zero. Point 1 is subjected to an external vertical load p.t /. The cross sectional area of bars (1)–(2) is A1 D 0:0004 m2 , bars (3)–(5) is A2 D 0:0005 m2 , bars (7)–(9) is A3 D 0:0006 m2 , and bars (10)–(15) is A4 D 0:0008 m2 . The density of all the bars is D 2800 kg=m3 . The truss element is adopted to create FEM model. Each bar is an element, and there are totally 15 elements and 8 nodes. The dynamic displacement of Point 7 in y-direction is measured via finite element computing model. In this numerical example, the actual load history is defined as follows:


Load

p/N

2000

0

-2000

-4000

-6000

0

0.05

0.1

0.15

0.2 Time

0.25

0.3

0.35

0.4

t /s


Load p / N

2000

0

-2000

-4000

-6000

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time t /s




² p.t / D

0 6 t 6 3td t > 3td

5000 sin 2 t =td 0

(26)

where td is the cycle of the actual load. Divide the dynamic load into 400 time elements in a total time 0.4 s, which means the magnitude of time element is t D 0:001 s. Choose linear interpolation function q.t / which is shown in Figure 2, as the weighting function: q.t / D q1 .t / q2 .t / ² t 0 6 t 6 t 1 t q1 .t / D 0 t > t (27) ² t 0 6 t 6 t q2 .t / D t 0 t > t 6000 Actual load Identified load 4000

Load

p/N

2000

0

-2000

-4000

-6000

0

0.05

0.1

0.15

0.2 Time

0.25

0.3

0.35

0.4

t /s

6000 Actual load Identified load

Load p / N

4000

2000

0

-2000

-4000

-6000

0

0.05

0.1

0.15

0.2 Time

0.25

0.3

0.35

0.4

t /s



J. LIU ET AL.

In order to obtain the computed kernel function response and structure dynamic response, let the sampling time interval be 0.0002 s, which means each time element contains five sampling time intervals and six sampling points. So for the sake of making it simple, w.t / is equal to the previous q.t /. Then apply the shape function load q1 .t /; q2 .t / at the exciting point of the finite element model respectively to obtain the computed kernel function. Next, each six sampling points in each time element are linear fitting by least-square method to obtain gijq1 gijq2 . Then, apply the actual load p.t / at the same point to obtain the computer-generated response ycom . Add 5%, 10%, and 25% noisy Table II. The identified results and the relative errors of plane truss structure. GKFM Time point t (s)

Actual load p (N)

0.035

4221.6

0.083

4524.1

0.114

3644.8

0.133

4524.1

0.174

4960.6

0.229

4911.4

0.268

4381.5

TDGM

Identified load (N)

RE (%)

Identified load (N)

RE (%)

lno D 5% lno D 10% lno D 25%




0.58 1.87 0.32 0.80 2.76 0.43 5.78 2.18 5.12 0.07 2.73 1.03 5.38 2.37 4.88 2.62 2.03 17.15 1.86 8.14 12.48

4223.3 4162.8 4408.9 4526.5 4738.9 4678.4 3639.9 3564.2 3846.7 4521.6 4385.3 4489.9 4953.7 5022.6 5047.5 4905.4 4801.8 4617.0 4388.2 4578.5 4225.0

0.04 1.39 4.44 0.05 2.75 3.41 0.14 2.21 5.54 0.06 3.07 0.76 0.14 1.25 1.76 0.12 2.23 6.00 0.15 4.50 3.57

4197.1 4300.6 4208.2 4487.7 4399.3 4504.5 3855.6 3724.2 3831.3 4521.1 4400.5 4477.5 5227.6 5078.3 5202.5 4782.7 4811.9 4069.0 4299.8 4738.2 3834.8 5.93 7.04 14.84 99.85 99.79 98.90 9 6 5

RE CC Iteration step

3.64 4.96 6.55 99.93 99.88 99.79 29 21 14

TDGM, time-domain Galerkin method; GKFM, Green kernel function method; RE, relative error; CC, correlation coefficient.

Figure 7. The pin-jointed truss structure. Copyright © 2015 John Wiley & Sons, Ltd.



levels into ycom to simulate the measured response yijm , respectively . The noisy response with 25% noisy level is shown in Figure 3. After that, substitute gijq1 ; gijq2 ; yijm into Equation (13) to obtain the forward model based on TDGM successfully. Via the LSQR iterative regulation method, the identified results with different noisy levels based on different methods are shown in Figure 4-6. Besides, the identified results at seven discrete time points are listed in Table II. It can be seen from table and figures of the identified results, when

Figure 8. The pin-jointed truss structure: first mode shape.

Figure 9. The identified results by using LSQRCL-Curve and CGCMCGCV with 5% noisy level. Copyright © 2015 John Wiley & Sons, Ltd.


J. LIU ET AL.

Figure 10. The identified results by using LSQRCL-Curve and CGCMCGCV with 15% noisy level. Table III. The identified results and the relative errors of truss beam structure by using LSQRCL-Curve and CGCMC-GCV. LSQRCL-Curve Time point t (s)

Actual load p (N)

0.030

69.1031

0.090

40.6446

0.150

22.3130

0.205

12.8735

0.340

3.3373

CGCMC-GCV

Identified load (N)

Relative error (%)

Identified load (N)

Relative error (%)

lno D 5% lno D 15%

lno D 5% lno D 15%

lno D 5% lno D 15%

lno D 5% lno D 15%

69.0042 68.2293 41.7035 41.7481 21.5292 21.6133 13.4675 15.0702 3.5628 4.3246

0.14 1.26 2.61 2.71 3.51 3.14 4.61 17.06 6.75 29.58

65.9463 64.9419 40.3003 40.5710 22.3115 22.0211 13.0642 13.7953 3.4976 3.7268

4.57 6.02 0.85 0.18 0.01 1.31 1.34 7.16 4.80 11.67

RE (%) CC Iteration step

8.14 10.87 99.53 99.16 13 9

10.44 10.95 99.22 99.15 6 5

LSQR, least-square QR; CG, conjugate gradient; MC-GCV, Monte-Carlo generalized crossvalidation; RE, relative error; CC, correlation coefficient. Copyright © 2015 John Wiley & Sons, Ltd.



the noisy level is low (5% and 10%). The GKFM and the TDGM can accurately realize stable identification of the dynamic load, and the relative errors of the seven points are lower than 5%, indicating that the two methods are feasible and effective. With the increase of noisy level (25%), the identification accuracy of GKFM declines endlessly, especially the identified load in the initial and the ending part gravely deviates from the actual load, with the relative error being as high as 17.15%. But TDGM is different. Under the high noisy level, it still accurately realizes the dynamic load identification and almost all the relative errors of time points are within 5%, which supports that TDGM has a strong anti-noise ability. Even though noisy level is very high, it still can maintain excellent identified accuracy. In addition, from the perspective of the reverse results, the best LSQR iterative steps under different noisy levels based on TDGM are higher than those based on GKFM, so the TDGM can guarantee the stability of the load identification at the same time keep high frequency components, and the identified accuracy is high naturally. 5.2. A truss beam problem In this example, pin-jointed truss structures reported in Figure 7 is chosen, for their importance in many fields of engineering. The exciting loads are directed as the first mode shape, reported in Figure 8, according to the following evolution law [41]:

Figure 11. The finite element model of vehicle door. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1 x 10-3

Figure 12. The curves of the shape function loads. Copyright © 2015 John Wiley & Sons, Ltd.


J. LIU ET AL.

p.t / D 100 e 10t e 100t

(28)

reported graphically in Figure 9. The total time duration is 0.75 s, time of each element 0.005 s, and t D 0:001 s. Add 5% and 15% noisy levels to simulate the measured response, respectively. Using the same q.t / and w.t / as the first example, the forward model based on the TDGM is set up. The identified results based on LSQR iterative method and L-Curve criterion [44] (LSQRCL-Curve) and the conjugate gradient method and the Monte-Carlo generalized crossvalidation criterion (CGCMC-GCV) are reported in Figure 9 (5% noisy level), Figure 10 (15% noisy level), and Table III. According to the data of Figure 9, Figure 10, and Table III, both the two approaches have a good identified accuracy within 10% in most instances. LSQRCL-Curve method has better identified results at the peak than ones at the end. Oppositely, CGCMC-GCV method has a larger error at the peak than LSQRCL-Curve method. The reason why the relative errors at the end time points are much larger than the beginning time points is that the actual load at the end




time points approach zero. Note that LSQRCL-Curve method and CGCMC-GCV method only use different strategies to choose the iterative step. By the way, the identified accurate is still high with the noisy level increases to 15% from 5%, and it shows that TDGM has a strong anti-noisy ability. Overall, both LSQRCL-Curve method and CGCMC-GCV method are effective approaches for ill-posed problems. 5.3. A vehicle door problem The design of a vehicle door is an important part during vehicle design, and its dynamic characteristics are definitely associated with the safety of occupants. Dynamic load identification is a helpful technique to assess the exciting loads on the door and to make structural modification to satisfy design requirement. The finite element model of a simplified door of some passenger vehicle is shown in Figure 11.



J. LIU ET AL.

Table IV. The identified results and the relative errors of vehicle door structure. GKFM Time point t (s)

Actual load p (N)

0.03

195.29

0.04

337.23

0.06

633.38

0.07

699.65

0.09

393.46

0.13

578.96

0.15

494.97

0.21

672.21

0.26

479.18

0.29

672.21

TDGM

Identified load (N)

Relative error (%)

Identified load (N)

Relative error (%)

lno D 10% lno D 25%

lno D 10% lno D 25%

lno D 10% lno D 25%

lno D 10% lno D 25%

220.98 217.47 359.33 323.56 598.85 569.05 708.49 638.70 416.35 395.20 617.07 688.03 482.89 500.15 659.24 667.00 418.63 458.06 693.13 673.49

RE (%) CC (%) Iteration step

13.16 11.34 6.55 4.05 5.45 10.16 1.26 8.71 5.82 0.44 6.58 18.84 2.44 1.05 1.93 0.77 12.64 4.41 3.11 0.19 8.26 11.74 99.65 99.34 13 9

199.95 200.59 349.24 350.54 641.20 640.80 698.24 692.44 400.59 387.32 582.97 603.48 491.95 488.98 669.23 649.13 469.93 476.18 663.56 647.94

2.38 2.71 3.56 3.95 1.24 1.17 0.20 1.03 1.81 1.56 0.69 4.24 0.61 1.21 0.44 3.43 1.93 0.63 1.29 3.61 2.40 4.20 99.97 99.92 42 27

TDGM, time-domain Galerkin method; GKFM, Green kernel function method; RE, relative error; CC, correlation coefficient.

The model mainly adopts the triangle and quadrilateral element and uses RIGID element to simulate the solder joint connection. Bolt connection parts using RBE2 elements to build the rigid connection. The model contains 29530 quadrilateral elements, 1442 triangular elements, and 572 welding elements. Six degrees are fixed between the door hinge and body joints The material of the door has the density D 8023 kg=m3 the Young’s Modulus E D 2:07 1011 Pa, and the Poisson’s ratio D 0:29. The damping parameters of all the structures are 0.06. In the Figure 11, Point A is the measured point of the ´-direction response, and Point B is the exciting point subjected to a variable frequency sinusoidal load p.t / in ´-direction: ² 0 6 t 6 0:3 700 sin 100 t 2 p.t / D (29) 0 t < 0 or t > 0:3 Divide the dynamic load into 400 time elements in the whole time 0.4 s, and each element has two sampling time points, that is t D 0:001 s. Choose the shape function q.t / to interpolate the unknown load inside the element. q.t / D q1 .t / q2 .t / 8 1 < 0 6 t 6 0:001 12000.t 0:0005/ q1 .t / D 1 C e : 0 t > 0:001 (30) 8 1 < 0 6 t 6 0:001 1 q2 .t / D 1 C e 12000.t 0:0005/ : 0 t > 0:001 Copyright © 2015 John Wiley & Sons, Ltd.



and the curve of the shape loads are plotted in the Figure 12. In this numerical example, respectively add 10% and 25% levels of noise into to simulate the response. Assuming each sampling time interval is 0.0001 s, namely each time element contains ten sampling intervals. We choose the same weighting function w.t / as the previous numerical example. When lno D 10% and lno D 25%, the identified results with different noisy levels and different methods are shown in Figures 13 and 14, respectively. Moreover, the identified results and the relative errors of the dynamic load at ten discrete time points and the values of relative errors and correlation coefficients are all listed in Table IV. According to the figures of identified results, although both of them can realize the dynamic load identification, TDGM is obviously better than GKFM. The identify load based on GKFM is not very smooth at the beginning, and peak load significantly deviates from the actual value. But on the opposite, the identified results based on TDGM are very smooth in the whole recognition process, almost overlapping the true value. In addition, according to the data of Table IV, TDGM keep its high accuracy and strong anti-noise ability. Even though the number of the sampling time interval in a time element increases from five to ten, the accuracy of the identified results remains pretty high, such as 2.04% and 4.20%. On the other hand, under the 10% noisy level, the relative error of TDGM is only about a third of GKFM, the accuracy increases more than three times. With the noisy level increases to 25%, the relative error jumps to 11.74% based on GKFM. However, the relative error is only 4.2% based on TDGM, increasing quite less. 6. CONCLUSIONS The challenges of dynamic load identification methods could be concluded as: on the one hand, enlarging the time element might loss the information of many sampling time points and decreasing the identified accuracy. On the other hand, the noise will affect the stability and accuracy of identified results. In particular when the noisy level is high, the regularization method is not enough to guarantee the stability of the whole identification process and high accuracy of the identified results. TDGM is an advanced method that can take the two aforementioned factors into consideration. TDGM is an improvement can be summarized as follows: first, three factors (the dynamic load, the kernel function response, and measured response) are approximated by the shape function; and second, regarding residual error as integrand to construct an integral type, which can smooth noise and have the filter function. Compared with GKFM from the numerical examples, TDGM has obvious advantages in terms of the identified accuracy and the anti-noisy ability. Moreover, different iterative methods combined with various criterions show the correctness and advantages of TDGM in the second example. Therefore, TDGM can be an effective identification method in engineering applications. ACKNOWLEDGEMENTS

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