Compressed sensing for moving force identification using redundant dictionaries

Mechanical Systems and Signal Processing 138 (2020) 106535

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Compressed sensing for moving force identification using redundant dictionaries Huanlin Liu a, Ling Yu a,⇑, Ziwei Luo a, Chudong Pan a,b a MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, China b School of Civil Engineering, Guangzhou University, Guangzhou 510006, China

a r t i c l e

i n f o

Article history: Received 23 July 2019 Received in revised form 13 October 2019 Accepted 19 November 2019

Keywords: Moving force identification (MFI) Compressed sensing (CS) Redundant dictionary Sparse regularization Structural health monitoring (SHM)

a b s t r a c t Moving force identification (MFI) techniques have been widely studied in recent years. However, the contradiction between response acquisition and energy consumption limits applications of existing MFI methods and has become one of the most prominent issues in the field of structural health monitoring (SHM). In fact, sample length of response data can be shortened by exploiting compressed coefficients of responses based on compressed sensing (CS) theory. In order to mitigate this contradiction and to study if these compressed coefficients can be efficiently exploited for MFI, a novel method is proposed for MFI based on CS with redundant dictionaries in this study. Firstly, a redundant dictionary is designed for creating a sparse expression on each moving force based on prior knowledge of moving forces. Then, by the aid of relationship between moving forces and responses, an indirect way is presented to design dictionaries for different types of structural responses, sparse expression of responses is established simultaneously, and a MFI governing equation is formulated by directly exploiting compressed coefficients of responses via CS. Moreover, sparse regularization is introduced to ensure the accuracy of MFI results. Finally, the proposed method is validated by both numerical simulations and experimental verifications. The illustrated results show that the sample length of each acquired data can be obviously shortened and the compressed coefficients rather than structural responses can be directly used for MFI. The identified moving forces are in good agreement with the true ones, which shows the effectiveness and applicability of the proposed method. In addition, the proposed method can estimate the total weight of the car with a good accuracy and a strong robustness to noise. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction As one of the major loads acting on the bridge deck, moving vehicle forces are important for bridge design [1], so monitoring moving forces is an effective way to ensure the safety of bridge [2,3]. Compared with measuring moving forces directly, indirect methods are more economical and convenient. As an indirect method, moving force identification (MFI) is an attractive issue in the field of structural health monitoring (SHM), and many methods have been developed in recent years [4,5].

⇑ Corresponding author at: School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, China. E-mail address: [email protected] (L. Yu). https://doi.org/10.1016/j.ymssp.2019.106535 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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H. Liu et al. / Mechanical Systems and Signal Processing 138 (2020) 106535

The MFI technique can effectively identify moving forces from structural responses through the relationship between moving forces and responses. Early studies of MFI focus on the establishment of the identified equations. These methods can be classified into two categories [6]: one is based on the analytical model and another is based on the numerical model. These methods include influence line method [7], interpretive method I [8], time domain method [9] and so on [10]. Further studies about MFI focus on how to identify moving forces under the influence of measurement noises, and many techniques have been developed to improve the ill-posedness of the MFI problem. Among these techniques, regularization techniques have been widely applied in the MFI field. Many researchers [11–14] adopted Tikhonov regularization to reduce the effect of noises. Other regularization techniques, such as the updated static component technique [15], Bayesian inference regularization [16], method of moments [17] and preconditioned least square QR-factorization algorithm [18], are also introduced into the MFI problem. More recently, inspired by sparse regularization, which provides a new method for solving the inverse problem on sparse signals in time domain, spatial domain, or some basis transformation [19], several studies introduced the sparse regularization into the MFI problem for more accurate results [19–21]. In these MFI methods, structural responses are always essential because they contain information of both moving forces and structures. That is to say, these methods must identify moving forces from the measured structural responses. However, response data transfer and storage will cause large energy consumption and increase costs. With the development of wireless sensor network, the contradiction between response acquisition and energy consumption is becoming increasingly prominent. This contradiction places restrictions on response acquisition, and also limits applications of the existing MFI methods. Inspiring by the sparse regularization [22–24], compressed sensing (CS) technique [23,24] can be applied to mitigate the contradiction. CS is a new signal processing technique, which can efficiently acquire and reconstruct a signal by finding solutions to underdetermined linear systems. Through optimization, CS exploits the sparsity of a signal to recover it from far fewer samples than required by the Nyquist-Shannon sampling theorem, so CS can effectively reduce the energy consumption and storage costs for data acquisition. CS relies on the sparse regularization technique because the sparse regularization provides guarantees to efficiently reconstruct a signal. Thus, the corresponding signal can be recovered from the compressed coefficients with sparse regularization. Based on the CS theory, structural responses can be effectively compressed, and the applications of the compressed coefficients have been studied in the SHM field [25–27]. Because the structural responses are not sparse in time domain, some dictionaries are established to obtain sparse expressions of responses. The Fourier basis [25], the sinusoid basis [26] and the Daubechies wavelet family [27] are respectively adopted as a dictionary in these studies. These studies involve the applications about modal identification [26], response signal recovery and structural damage detection [25,27]. If the compressed coefficients can be effectively used in MFI, it can not only reduce the energy consumption and storage costs of response data acquisition, but also identify moving forces from the compressed coefficients. It is an effective way to mitigate the above-mentioned contradiction by directly exploiting compressed coefficients in the MFI field. However, to the author’s knowledge, there is no investigation whether compressed coefficients of response data can be effectively exploited for MFI. Therefore, in order to effectively mitigate the contradiction between response acquisition and energy consumption for MFI, this study focuses on studying the effectiveness of directly exploiting compressed coefficients for MFI. A novel method is proposed based on CS with redundant dictionaries. This proposed method takes full advantage of the CS theory to establish a MFI equation by directly using the compressed coefficients rather than structural responses, so both responses and moving forces can be solved from the same equation. Moreover, most dictionaries used for sparse expressions are directly provided [25–27], which is closely related to response types. As a result, it is hard to design a suitable dictionary. As mentioned above, the existing MFI methods have established the identification equations between moving forces and structural responses. Different types of responses are aroused by same moving forces. As for the dictionary in force identification, different basis functions such as cosine discrete function [28,30], cubic B-spline [29] and wavelets [29,30] have been applied to approximate the desired force in the time domain. Therefore, an indirect way is proposed to design dictionaries for structural responses in this study according to the relationships between moving forces and structural responses. To evaluate the proposed MFI method, some numerical simulations and experimental verifications are conducted. This paper is organized as follows: the basic theories and the proposed MFI method are introduced in Section 2, a relationship between the moving forces and the compressed coefficients is established. Then, several numerical validations on the proposed MFI method are carried out in Section 3. Further, experimental studies on a hinge supported hollow steel beam are conducted in Section 4. Finally, some conclusions are made in Section 5.

2. Theoretical background 2.1. Governing equation in time domain For the MFI problem, a relationship between moving forces and structural response should be established firstly. As shown in Fig. 1, a simply-supported beam is taken as an example to form this relationship. The beam is subjected to n moving forces, and m measurement points are used to acquire the structural responses caused by the moving forces. To obtain this relationship, two hypotheses are assumed similarly as the previous studies at first [8–14,16–21]. The first one is that

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Fig. 1. Simply-supported beam subjected to n moving forces.

each moving force has the same constant speed c. As the moving force acting on structures is a function related to both time and space, and MFI is more concerned about the time history of the moving force, the complexity of the problem can be effectively simplified based on this assumption. The second one is that structures in service can be reasonably approximated as a linear system, which has been applied for dealing with the problem in the real engineering [12]. Thus, for a linear structure, the response ri ðtÞ ði ¼ 1; 2; . . . ; mÞ can be calculated by the principle of linear superposition [8–14,16–21]:

r i ðt Þ ¼

n Z X 0

j¼1

t

  hi g j ðsÞ; t  s f j ðsÞds

ð1Þ

  where, g j ðtÞ is the time-varying position of the moving force f j ðtÞ, hi g j ðtÞ; t is the unit impulse response function corresponding to the ith measurement point when a unit impact force is acting at x ¼ g j ðt Þ. Assuming that M þ 1 sample points are collected in a time interval ½0; T , the sampling time interval is Dt ¼ T=M. Let N j þ 1 represent the sample points of the jth moving force when f j ðt Þ is acting on the beam in a time interval h i   fin fin 0 6 T ini T ini j ; Tj j < T j 6 T . With discretization, a matrix expression between moving forces and the ith structural response can be obtained from Eq. (2):

r i ¼ ½ H i1

n    H in  f T1

T

 fn

oT

ð2Þ

where, r i 2 RðMþ1Þ1 (i ¼ 1; 2; . . . ; m) is the structural response vector acquired by the ith measurement point, H ij 2 RðMþ1ÞðNj þ1Þ is the transfer matrix, and f j 2 RðNj þ1Þ1 (j ¼ 1; 2; . . . ; n) represents the jth moving force vector. Eq. (2) gives a relationship between moving forces and structural responses in the time domain, so it can be used for MFI if accurate structural response is measured. However, the moving forces directly identified by Eq. (2) often deviate from the true values. This is because structural responses are always polluted by measurement noise, and the MFI problem is of ill-posed. As a result, the features used for expressing moving forces in the time domain are greatly influenced by noise. In other words, the features of moving forces cannot be highlighted in the time domain by directly solving Eq. (2). 2.2. Moving force expression by redundant dictionaries The salient features of moving forces are concerned because they can reflect the characteristics of moving forces [14]. Proper dictionaries have abilities to express possible features of moving forces and reduce the influence of noise. In this study, a corresponding dictionary Uj is given to express the possible features of each moving force vector f j (j ¼ 1; 2; . . . ; n). The dictionary Uj is made of cj column vectors, and each column vector is called as an atom, i.e. h i ujk (k ¼ 1; 2; . . . ; cj ). As a result, by the aid of the dictionary Uj ¼ uj1 uj2    ujcj , the moving force vector f j can be represented as a linear combination of selected atoms ujk [28–30]:

fj ¼

cj X

ajk ujk ¼ Uj aj

ð3Þ

k¼1

where, ajk is the coefficient of ujk . Substituting Eq. (3) into Eq. (2), a relationship between coefficient vectors of the given dictionaries and structural response is obtained as:

r i ¼ ½ H i1 U1

    H in Un  aT1

   aTn

T

ð4Þ

Some studies [20,21] indicated that a proper redundant dictionary is more suitable for feature extraction than a complete dictionary. As a result, sparse coefficient vector can be obtained from the given dictionaries and the features of moving force are highlighted. As mentioned above, proper selection of the dictionary is important for expressing the features of moving force and reducing the influence of noise. In this study, the dictionary Uj is designed based on the prior knowledge and the signal char-

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acteristics of moving force described in the existed studies [14]: in a bridge-vehicle vibration system, a moving force is always related to its axle weight. Therefore, it is assumed that the moving force consists of two components. One component, which will be referred to as constant force component in this paper, is unchanged with time. Another component is referred to as time-varying component and it is changed with time. The dictionary Uj is designed as follows: (1) The first atom uj1 of the dictionary Uj is used to express the constant force component, in which all the elements are pffiffiffiffiffiffiffiffiffiffiffiffiffiffi identical, i.e., uj1s ¼ 1= N j þ 1(s ¼ 1; 2; . . . ; N j þ 1). (2) In Ref. [20], the trigonometric and rectangular functions showed good ability to express the time-varying component of the moving force. Thus, these functions are employed here to establish the dictionary Uj and to express the timevarying component. 2.3. Compressed sensing (CS) As a data compression technique, compressed sensing (CS) acquires and reconstructs a signal by exploiting its sparsity. CS can effectively reduce energy consumption for data acquisition, storage, and transmission. The advantages and effectiveness by means of CS in the SHM field have been investigated in some studies [25–27]. The basic theories of CS have been developed by some excellent tutorials [23,24]. To summarize, for a signal x 2 RMx 1 , it can be expressed as follows:

x¼

Nx X

si Wi ¼ Ws

ð5Þ

i¼1

where, W 2 RMx Nx is a dictionary, and s 2 RNx 1 is the representation of the signal x in the W domain, i.e. the coefficient vector of the dictionary W. As mentioned above, CS acquires the signal x by exploiting its sparsity. Thus, it is assumed that the coefficient vector s is sparse in the given dictionary W, and the number of non-zero elements in the vector s is equal to K x (K x << N x ). Then, a measurement matrix K 2 RPx Mx (Px << M x ) is used to obtain the compressed coefficients y below:

y ¼ KWs ¼ Hs

ð6Þ

To ensure the signal x can be reconstructed from the compressed coefficients y, the sensing matrix H should obey the restricted isometry property (RIP) for a complete dictionary [31], or the restricted isometry property adapted to D (D-RIP) for a redundant dictionary [32]. For a given dictionary, the measurement matrix can be selected as a random matrix. In this study, the random matrix is generated by collecting the independent and identically distributed (i.i.d.) entries from a normal distribution with zero mean and variance 1=Px [27]. Moreover, as mentioned in Ref. [33], for any choice of the dictionary W, this random matrix satisfies the D-RIP with a high probability, and the algorithm proposed in Ref. [33] can be used for CS under these conditions. Because the coefficient vector s is sparse, it is possible to recover the signal x by the following convex optimization: 



mink s k1 ; s:t:;k H s y k2 6 e 

ð7Þ



x ¼ Ws

ð8Þ 

where, k  k2 is the standard Euclidean norm, k  k1 denotes the l1 norm, e is a likely upper bound on the noise power, and x is the reconstructed signal. Moreover, the value of Px is determined by the following relationship [23]:

Px P a0 K x log



Nx Kx

ð9Þ

where, a0 is a constant value which is approximately equal to 4.0 [24,25]. 2.4. Compressed sensing for moving force identification using redundant dictionaries For the MFI problem, because CS can effectively shorten data transmission and storage time compared with the traditional sampling technique, it can mitigate the contradiction between response acquisition and energy consumption, and it is important to study how to effectively exploit the compressed coefficients for identifying moving forces. In this study, a novel MFI method is proposed directly using the compressed coefficients. According to Eq. (4), a matrix Bi is obtained from the redundant dictionary U ¼ ½ U1    Un  and linear map ½ H i1    H in , i.e. Bi ¼ ½ H i1 U1    H in Un . Then, Eq. (4) can be rewritten as:

r i ¼ ½ Bi1

    Bin  aT1

   aTn

T

¼ Bi a

ð10Þ

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In this study, the dictionary U is normalized to make each atom of the dictionary Bi have the same norm, i.e., the 2-norm of each atom is equal to one [20,33]. From Eq. (10), it can be seen that the structural response vector r i (i ¼ 1; 2; . . . ; m) can be represented by a redundant dictionary Bi and its corresponding coefficient vector a. It should be noted that, Eq. (10) gives an indirect way to design the dictionary Bi based on the dictionary U and the linear map. For different types of structural responses, the dictionary U is invariant and the linear map is determined by the transfer matrices H ij (j ¼ 1; 2; . . . ; n). Therefore, rather than directly designing dictionaries for different responses, this indirect way is more convenient. Surprisingly, if the coefficient vector a is sparse, Eq. (10) will have the same expression with Eq. (5). That is to say, the redundant dictionary Bi is designed by the aid of Eqs. (2) and (3), and a sparse representation of structural response vector r i in the redundant dictionary Bi is provided. Therefore, based on the CS theory, a measurement matrix Ri is generated by collecting the independent and identically distributed (i.i.d.) as mentioned in Section 2.3 [27], and the following equation is obtained:

bi ¼ Ri r i ¼ Ri Bi a

ð11Þ

where, bi is the compressed coefficient vector of the structural response vector r i . With the help of the following sparse regularization, the coefficient vector can be obtained from the compressed coefficient vector bi : 



mink a k1 ; s:t:;k Ri Bi a bi k2 6 e

ð12Þ 

It is notable that the coefficient vector a is a sparse representation not only in the Bi domain but also in the U domain. In other words, the sparse representations of both structural response and moving forces can be obtained from Eq. (12) at the same time. For multiple structural responses, the coefficient vectors of responses are identical because they are determined by the same moving forces. As a result, a combination form of structural responses is expressed as follows:

8 9 2 b1 =k b1 k2 > R > > > < = 6 1 .. 6 ¼4 . > > > > : ; bm =k bm k2

8 9 9 38 r1 =k b1 k2 > r1 =k b1 k2 > > > > > > > < < = = 7 .. .. .. 7 ¼R . . . 5> > > > > > > > : : ; ; Rm r m =k bm k2 rm =k bm k2

ð13Þ

where, R is a block diagonal matrix. A block diagonal matrix has been employed in CS, and detail introductions can be seen in Refs. [34–36]. To alleviate storage limitations, computational considerations in practical applications, each measurement matrix Ri is identical in Eq. (13), and the block diagonal matrix R is called as Repeated Block Diagonal (RBD) matrix [36]. Then, by the aid of Eq. (10), the following equation is obtained:

n

T

b1 =k b1 k2

T

   bm =k bm k2

oT

 ¼ R BT1 =k b1 k2

   BTm =k bm k2

T

a

ð14Þ

which can be simplified as:

b ¼ RBa where, b ¼

n

T

b1 =k b1 k2

ð15Þ 

T

bm =k bm k2

oT

is a vector related to the compressed coefficients of responses.

The vector b can be deemed as the compressed coefficient vector of a signal. Thus, similar to Eq. (12), the convex optimization is used to obtain the coefficient vector: 



mink a k1 ; s:t:; k RB a b k2 6 e

ð16Þ

In this study, the Signal Space Compressive Sampling Matching Pursuit (SSCoSaMP) algorithm [33], which provides a provable recovery guarantees for signal, is adopted to solve Eq. (12) or Eq. (16). As mentioned in Ref. [33], near-optimal supports for the SSCoSaMP algorithm should be found. Therefore, two variants of the SSCoSaMP algorithm are used in this study: one in which Orthogonal Matching Pursuit (OMP) is used for computing the near-optimal supports (referred to as ‘‘SSCoSaMP (OMP)”), and the other in which Compressive Sampling Matching Pursuit (CoSaMP) is used for computing the near-optimal supports (referred to as ‘‘SSCoSaMP (CoSaMP)”). Because these algorithms may struggle in some scenarios [33], it should be pointed out that when the CoSaMP performs well, the SSCoSaMP (CoSaMP) may perform even better, and when the OMP performs poorly, the SSCoSaMP (OMP) may still perform poorly. A suitable algorithm should be selected to find the near-optimal supports at first. Therefore, the OMP and the CoSaMP algorithms are adopted for selecting the variants of the SSCoSaMP algorithm. In the following examples, as the coefficient vector is quite sparse, the CoSaMP algorithm is more time-consuming than the OMP algorithm in these situations [37]. From this point of view, the SSCoSaMP (OMP) algorithm is firstly taken as the solution algorithm if it can obtain reasonable results. Only the identified results obtained by the SSCoSaMP (CoSaMP) algorithm are better than that obtained by the SSCoSaMP (OMP) algorithm, the SSCoSaMP (CoSaMP) algorithm will be taken into account.

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H. Liu et al. / Mechanical Systems and Signal Processing 138 (2020) 106535

As a result, a possible sparse representation of moving forces is directly obtained from the compressed coefficients of structural response(s) by means of Eq. (12) or Eq. (16). Thus, the features of moving forces are extracted, and each moving force vector is reconstructed from the following expression: 



f j ¼ Uj aj

ð17Þ



where, f j ðj ¼ 1; 2; . . . ; nÞ is the reconstructed moving force vector. To summarize, the basic flowchart for the proposed MFI method based on CS and redundant dictionaries is shown in Fig. 2. 3. Numerical simulations In order to assess the effectiveness of the proposed MFI method, some numerical simulations on identification of vehicle forces moving across a simply-supported beam are conducted. Two kinds of simulated moving forces, which are both travelling at a speed of 20 m/s (72 km/h), are considered: (a) single moving force simulated,

( f 1 ðt Þ ¼

0 s 6 t < 1:2 s 40½1 þ 0:3sinð25pt Þ þ 0:2sinð60pt Þ kN  40 1 þ 0:3sinð25pt Þ þ 0:2sinð60pt Þ þ 3e35ðt1:2Þ sinð125ðt  1:2ÞÞ kN 1:2 s 6 t < 2 s

ð18Þ

(b) two moving forces with an axle distance of 4 m,

f 1 ðtÞ ¼ 5½1 þ 0:15sinð10ptÞ þ 0:05sinð35pt Þ kN 0 s6t<2 s f 2 ðt Þ ¼ 20½1  0:1sinð10ptÞ þ 0:15sinð40pt Þ kN 0:2 s 6 t < 2:2 s

ð19Þ

The span of the beam is 40 m, and the beam is divided into 20 finite elements. The flexural rigidity and the density of unit length are EI = 1.274916  1011 Nm2 and qA = 12000 kgm1, respectively. The first three modes of the beam are used here for MFI [9]. The first three natural frequencies are 3.2 Hz, 12.8 Hz and 28.8 Hz, respectively. The first three damping ratios,

Fig. 2. Basic flowchart of MFI method based on CS and redundant dictionaries.

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calculated from Rayleigh damping, are 0.02, 0.02 and 0.0378, respectively. The sampling frequency is 200 Hz for both moving forces and responses, and the sampling time interval is 0.05 s. Moreover, the Newmark-b method is used to calculate the transfer matrices, and the corresponding time step is set as 0.005 s for sufficient calculated accuracy. To simulate the polluted measurement responses, the white noise, taking as the measurement noise, is added into each calculated response in the following form [20]:

rni ¼ ri þ lev 

N 1X

r ij  randn N j¼1

ð20Þ

where, r i and r ni (i ¼ 1; 2; . . . ; m) are the ith noise-free and noisy responses, respectively, lev is the noise level, N is the element number in vector r i , and randn is a standard normal distribution vector. The number of non-zeros in a is set to be 20, so the measurement matrix Ri is generated for the corresponding response r ni , and the compressed coefficients of the response r ni are obtained by means of Eq. (11). The compressed coefficients obtained from both the bending moment and acceleration responses are employed for MFI. The Bayesian information criterion (BIC) is adopted to determine the sparsity of the coefficient vector a [38]:



1 2 BIC ðkÞ ¼ Pln k RBak  b k2 þ klnðPÞ P

ð21Þ

where, P is the element number of vector b, ak is the solution with a given sparsity k, and k is the number of non-zeros in ak . The value of k is an integer in the interval ½1; 20, and the value corresponding to a minimum BIC value is chosen as the optimal sparsity kopt . Therefore, when kopt is given, the reconstructed moving forces are obtained from Eq. (17). To estimate the identification accuracy, the relative percentage error (RPE) between the true and identified moving forces is quantitatively defined as Eq. (22). Similarly, the RPE value between the true and reconstructed responses is defined as Eq. (23): true

RPEf ¼

RPEr ¼ true

k fi

k

iden

 fi

true fi

k1

k1

 100%

k rtrue  rrecon k1 i i  100% k r true k i 1

ð22Þ

ð23Þ

iden

where, f i and f i are the ith true and identified moving force vectors, respectively, r true and r recon are the ith true and i i reconstructed response vectors, respectively. 3.1. Single force identification The moving force in Eq. (18) is used for single force identification. The sampling duration is 2 s when the moving force is crossing through the beam. Here, the measured point of bending moment is placed on 1/4 span of the beam, and the measured point of acceleration is on 1/2 span of the beam. The sampling frequency is 200 Hz, so the sample length of each response is 400. By means of the CS theory, the length of each compressed coefficient vector is 286. In other words, the sample length of each acquired data is compressed from 400 to 286, which show that CS can substantially reduce the sample length of data. It is no doubt that the transmission time of the data can be shortened. Three noise levels, i.e., 5%, 10% and 15%, are respectively considered here. The noise-free and noisy bending moment responses are labeled as 1/4m. and 1/4m*., respectively. Similarly, the noise-free and noisy acceleration responses are labeled as 1/2a. and 1/2a*., respectively. The basic arguments for the SSCoSaMP (OMP) algorithm are given as follows [33]: 1) The maximum numbers of iteration are 20 for the SSCoSaMP algorithm. 2) The norm of residual is set to be 10-6 for stopping criteria of iterations. As given in Table 1, the optimal sparsity kopt is determined by BIC under each noise level. The RPEs between the true and the identified moving forces are listed. Meanwhile, the RPEs between the calculated responses 1/4m., 1/2a. and the corresponding reconstructed responses are given. The RPEs between the measured responses 1/4m*., 1/2a*. and corresponding reconstructed responses are also listed in Table 1. From Table 1, it can be seen that the values of kopt are within the interval ð1; 20Þ, and the RPEs are less than 13% under different noise levels. It indicates that Eq. (21) gives a suitable sparsity for MFI in the simulated examples. Fig. 3 gives the MFI results obtained from the given compressed coefficients when different noise levels are added into the corresponding responses. It should be pointed out that the optimal sparsity selected by BIC is less than the sparsity used to express the moving force, as given in Table 1. Since the trigonometric and rectangular functions are used to express the timevarying components of the moving force in this study. For the component 120e35ðt1:2Þ sinð125ðt  1:2ÞÞ kN, rather than one

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Table 1 Relative percentage errors on MFI results under different noise levels. Noise level

5% 10% 15%

Optimal sparsity

16 17 16

Relative percentage error (RPE)/% moving force

1/4m.

1/2a.

1/4m*.

1/2a*.

1.71 4.08 5.70

1.09 1.43 3.47

1.60 4.92 6.30

3.93 7.80 12.25

4.09 8.75 12.96

Fig. 3. Comparison on MFI results obtained from responses at both 1/4 m*. and 1/2a*. under different noises.

atom, several atoms of the given dictionary should be used to approximately express this component. As a result, the identified and true moving forces are inconsistent, as shown in Fig. 3. As an inverse problem, it is important to distinguish true responses and noises in MFI. For example, under the 10% noise, the CS-reconstructed data for both responses and noises are shown in Figs. 4 and 5. It can be seen that the proposed method can effectively reconstruct the true responses and the adding noises. With the increase in noise levels, the RPEs of moving force and responses are increasing. This is mainly due to two reasons: 1) responses caused by some components of moving force are very weak, and they are more easily polluted by the white noises. With the increase of noise levels, these components are deemed as noises, so the coefficients of these atoms are set to be zero. 2) some components of white noises are identified as weak parts of responses, so the deviations between the true noises and reconstructed noises can be observed in both Figs. 4(b) and 5(b), and the RPEs between the calculated responses 1/4m., 1/2a. and corresponding reconstructed responses are not equal to zero. Furthermore, it should be pointed out that the matrices R and B are needed to be independently considered for the SSCoSaMP algorithm, so the computational efficiencies are traded off and just improved slightly by using the compressed data [33]. However, if the matrix RB is directly used in an algorithm, the computational efficiencies can be improved evidently when the compressed data is used. This response combination under the 10% noise is taken as an example here. Because the matrix RB can be directly used in the OMP algorithm, this algorithm is applied for the original response data and for the compressed data, and the block diagonal matrix R should be replaced by the identity matrix for the original response data. Although this algorithm cannot provide a recovery guarantee for signal, the identified result using this response

Fig. 4. Reconstructed bending moment at 1/4 span. (a) Reconstructed response. (b) Reconstructed noise.

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9

Fig. 5. Reconstructed acceleration at 1/2 span. (a) Reconstructed response. (b) Reconstructed noise.

combination is successfully obtained, as shown in Fig. 6. As a result, the computational time for MFI is shortened from 4.50 s to 3.39 s with the help of CS. It can be predicted that the computation efficiencies will be greatly improved with the development of the CS theory and the solution algorithms when the matrix RB is directly used with a recovery guarantee for signal. 3.2. Two forces identification Only trigonometric functions are used to simulate the two moving forces, as given in Eq. (19). The dictionary U mentioned in Section 2.2 is employed here. Therefore, the given sparsity of moving forces is 6 because each component can be expressed by one atom in the given dictionary. Because the distance between two moving forces is 4 m, the sampling duration is 2.2 s. The basic arguments for the SSCoSaMP (CoSaMP) algorithm are given as follows [33]: 1) The maximum numbers of iteration for the SSCoSaMP and the CoSaMP algorithms are 50 and 100, respectively. 2) The norm of residual is set to be 10-6 for stopping criteria of iterations. Several combinations of the bending moment and acceleration responses are adopted. The measured bending moment and acceleration responses at both 1/4 and 1/2 span of the beam are selected for MFI. The noise level is set to be 10%. As a result, the optimal sparsity and RPEs of moving forces are given in Table 2. Meanwhile, the RPEs between the calculated responses at 1/4m., 1/2m., 1/4a. and 1/2a. and their corresponding reconstructed responses are listed in Table 3. The RPEs between the measured and reconstructed responses are also listed in Table 3. Using the response combination of 1/4m*. & 1/2 a*., the identified moving forces are shown in Fig. 7. From Table 2, it can be seen that, the RPEs on the first force are always larger than that on the second force under different response combinations. The main reason is that the magnitude of the first moving force is much smaller than that of the second one as in Eq. (19). Thus, the response caused by the first moving force is smaller, which is more easily polluted by the measurement noises. Under the influence of noise, the optimal sparsity selected by BIC is less than the given sparsity. As shown in Fig. 7, it is found that the 250sin(35pt) N, the higher frequency component of the first force in Eq. (19), cannot be effectively identified from the responses due to the influence of noise. The reason is the same as stated in Section 3.1, and it can be seen that the RPEs between the calculated responses and corresponding reconstructed responses are not equal to zero as listed in Table 3.

Fig. 6. Comparison on MFI results obtained from responses at both 1/4 m*. and 1/2a*. under 10% noise.

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Table 2 Relative percentage errors on MFI results under different response combinations. Response combination

Optimal sparsity

1/4m*. & 1/2a*. 1/2m*. & 1/4a*. 1/2m*. & 1/2a*. 1/4m*., 1/2m*. & 1/4a*. 1/4m*., 1/2m*. & 1/2a*.

Relative percentage error (RPE)/%

5 4 5 5 5

First force

Second force

Total force

6.12 11.18 3.54 5.52 5.83

0.86 2.47 0.62 1.04 1.02

1.70 3.62 1.05 1.67 1.72

Table 3 Relative percentage errors on responses under different response combinations. Response combination

1/4m*. & 1/2a*. 1/2m*. & 1/4a*. 1/2m*. & 1/2a*. 1/4m*., 1/2m*. & 1/4a*. 1/4m*., 1/2m*. & 1/2a*.

Relative percentage error (RPE)/% 1/4m.

1/2m.

1/4a.

1/2a.

1/4m*.

1/2m*.

1/4a*.

1/2a*.

0.54 – – 0.44 0.47

– 0.97 0.45 0.32 0.36

– 10.78 – 9.42 –

6.17 – 6.15 – 6.34

8.58 – – 8.47 8.59

– 8.24 8.22 8.27 8.31

– 13.58 – 12.35 –

9.56 – 10.20 – 10.04

Fig. 7. Comparison on MFI results obtained from responses at 1/4 m*. and 1/2a*.

Moreover, the RPEs on accelerations are always higher than that on bending moments due to ineffective identification of higher frequency of the first moving force. Generally speaking, it is effective to directly use the compressed coefficients of structural responses for MFI. This method can effectively identify the given moving forces by means of different response combinations, and it has a great robustness to measurement noises, which shows that the proposed indirect way to design dictionaries for different structural responses is an effective way, and it is suitable for both bending moment and acceleration responses. Moreover, these MFI results show that after compressing structural responses based on the CS theory, the compressed coefficients still contain enough information of both moving forces and responses.

4. Experimental verifications 4.1. Experimental setup Experiments on a hinge supported hollow steel beam have been conducted in the laboratory [21], as shown in Fig. 8. The sectional form of the main beam is an approximate rectangular tube. Its span is 3 m, and the width, height and wall thickness are 140 mm, 60 mm and 3 mm, respectively. The measured flexural stiffness is 1.4055  105 Nm2, and the mass per unit length is 7.689 kgm1. A two-axle model car, as shown in Fig. 8(b), is used, and the distance between two axles is 0.42 m. Seven photoelectric sensors are fixed on a truss for measuring car speed. These sensors are used to capture the moments when the axles of the model car are passing the particular positions of the beam. The distance of two adjacent photoelectric sensors is 0.5 m. An experimental modal analysis (EMA) on the main beam is conducted at first. The first three modal parameters are analyzed, and these parameters are used to calibrate the finite element model (FEM) of the main beam. According to Fig. 8(c), approximate support in the FEM is used to simulate the support in the main beam, and the FEM is shown in Fig. 9. Sufficient accuracy of spring coefficients is essential so that approximate modal parameters of the FEM can be calculated. As a result, the vertical spring coefficients are set as kv1 = 8.4264813193741  106 Nm1 and kv2 = 8.2665819756769  106 Nm1,

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H. Liu et al. / Mechanical Systems and Signal Processing 138 (2020) 106535

Fig. 8. Experimental setup. (a) Experimental setup. (b) Model car. (c) Support. (d) Experimental rig.

Fig. 9. Finite element model of main beam.

respectively. Meanwhile, the rotational spring coefficients are set as kr1 = 4.1613804484088  103 Nm/rad and kr2 = 4.14 09539066102  103 Nm/rad, respectively. As a result, the comparisons between each measured frequency f m and the corresponding calculated one f c are listed in Table 4. The comparisons on modal assurance criterion (MAC) values between each measured mode shape um and the corresponding calculated one uc are also given in Table 4. Moreover, the comparisons on the first three mode shapes are plotted in Fig. 10. According to Table 4 and Fig. 10, the values of frequencies and mode shapes in the FEM are close to that in measured results. Therefore, dynamic similarity between the main beam and the FEM is achieved to a certain extent.

Table 4 Comparison on first three modal parameters of steel beam. Modes

1st 2nd 3rd

Frequencies

MAC

Measured damping ratio 2 

f m (Hz)

f c (Hz)

jf m  f c j=f m



23.3838 90.2466 188.6639

23.3817 90.2211 188.7011

0.0090% 0.0283% 0.0197%

0.9997 0.9970 0.9638

uTm uc = k um k22 k uc k22



Fig. 10. Comparison on first three mode shapes. (a) First mode. (b) Second mode. (c) Third mode.

0.0041 0.0031 0.0073

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After calibrating the FEM, a calibration of strain gauges is performed. Three-level masses are placed at 0.5 m, 1.0 m, 1.5 m, 2.0 m and 2.5 m of the main beam successively, and each level mass is 5.1 kg. Meanwhile, the corresponding bending moment responses are also calculated from the FEM. Consequently, the calibration coefficients between calculated bending moments and measured strains are shown in Fig. 11, which shows a great linear relationship in each measured point. Strain and acceleration responses are acquired when the model car is moving cross the main beam under different experimental cases in the laboratory. Herein, four strain gauges and four accelerometers at 1/4, 2/5, 1/2 and 4/5 span of the main beam are used to acquire structural responses by LMS Test.Lab. The sampling rate is 2048 Hz. The first three modal parameters are used for MFI, so all the measured responses are firstly filtered by a low-pass Butterworth filter, in which with no more than 3 dB of ripple in a passband from 0 to 190 Hz, and at least 40 dB of attenuation in the stopband. Then, the sequential data at a re-sampling frequency of 512 Hz are obtained from the measured responses. By means of calibration coefficients, the strain responses are converted into the bending moment responses. Then, the CS theory is introduced to obtain the compressed coefficients of responses. The number of non-zeros in the vector a is set to be 250. Similar to the numerical simulations, the compressed coefficients of both bending moments and accelerations are employed for MFI. The OMP algorithm can effectively find the near-optimal supports [33]. Thus, the SSCoSaMP (OMP) algorithm is used here, and same basic arguments are set as same in Section 3.1. The true moving forces of the model car are unknown, but the true axle weights of the model car are known, and the constant force components of moving forces can be identified by the proposed method. Therefore, the ith identified axle 

weight can be calculated from the constant force component of the ith moving force, i.e. widen ¼ ui11 ai1 =g(i ¼ 1; 2), where i g is the gravitational acceleration. The sum of the first and second identified axle weights is taken as the identified total weight of the model car. To assess the effectiveness of MFI results, the relative percentage error (RPE) about weight is defined as:

RPE ¼

true

w  widen  100% wtrue

ð24Þ

where, wtrue and widen are the true and identified axle or total weights of the model car, respectively. 4.2. Verification of proposed method First, the case with a car speed 1.4902 m/s is taken as an example to illustrate the MFI results calculated by the proposed method. The compressed coefficients of bending moment at 4/5 span and acceleration at 1/4 span are used. The first and second true axle weights are 4.5955 kg and 6.0290 kg, respectively.

Fig. 11. Calibration coefficients between calculated bending moments and corresponding measured strains. (a) 1/4 span (b) 2/5 span (c) 1/2 span (d) 4/5 span.

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H. Liu et al. / Mechanical Systems and Signal Processing 138 (2020) 106535 Table 5 Identified axle weights from responses at 4/5m*. & 1/4a*. Sample numbers of data

Identified weights/kg (RPE)

Measured

Compressed

First axle

Second axle

Total weight

2350

1714

4.40 (4.18%)

5.82 (3.50%)

10.22 (3.79%)

As a result, the MFI results are shown in Table 5. Here, effective data compression is achieved by means of the CS, and the sample length of each acquired data can be compressed from 2350 to 1714. From Table 5, it can be seen that the identified axle weights of the car are close to the true ones with RPE less than 4.2%, which shows that the proposed MFI method can identify the total weight with a good accuracy. Meanwhile, the identified moving forces are shown in Fig. 12. From Fig. 12, it can be found that when two axles of the car are acting on the main beam, the identified time-varying components fluctuate around the corresponding constant force components. This observation is in agreement with the prior knowledge given in Section 2.2, and it shows that the proposed MFI method is feasible and reasonable. Moreover, the identified total moving force of the car and the corresponding bending moment responses, including the measured and reconstructed ones from the identified moving forces, are illustrated in time domain as shown in Fig. 13. From Fig. 13(a), it can be seen that there are three time intervals for the identified moving forces. The first time interval [0 s, 0.28 s] is corresponding to the case when only the first axle of the car is acting on the main beam, the second interval [1.28 s, 2.01 s] is for the case when two axles of the car are both acting on the beam, and the third interval [2.01 s, 2.30 s] is for the case when only the second axle of the car is still on the main beam. The above observations show that the identified total weights of the car are all reasonable in three time intervals. In addition, the measured bending moment at 4/5 span is compared with the reconstructed one from the identified moving forces, as shown in Fig. 13(b), which shows that they are in good agreement with each other in time domain. The correlation coefficient between them is 0.9493, which further illustrates the effectiveness of the proposed MFI method. To further evaluate the effectiveness of the proposed MFI method, four combinations of responses are considered with car speed 1.4902 m/s, the MFI results are given in Table 6. On the other hand, different car weights given in Table 7 with different

Fig. 12. Moving force identification results. (a) For first axle. (b) For second axle.

Fig. 13. Moving force identification results and responses. (a) Total identified result. (b) Comparison of measured and reconstructed bending moments at 4/ 5 span.

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Table 6 Identified axle weights under different combinations of responses. Combinations

Identified weights/kg (RPE)

4/5m*. & 1/4a*. 1/4m*., 4/5m*. & 1/2a*. 2/5m*., 1/2m*. & 4/5a*. 2/5m*., 1/2m*., 1/4a*. & 4/5a*.

First axle

Second axle

Total weight

4.40 4.13 4.04 4.14

5.82 5.95 6.13 5.98

10.22 10.07 10.17 10.11

(4.18%) (10.22%) (12.09%) (9.97%)

(3.50%) (1.33%) (1.76%) (0.86%)

(3.79%) (5.17%) (4.23%) (4.80%)

Table 7 Axle weights of model car. Weight situations

First axle/kg

Second axle/kg

Total weight/kg

Situation 1 Situation 2 Situation 3

4.5955 4.0105 3.4255

6.0290 5.4540 5.4540

10.6245 9.4645 8.8795

Table 8 Identified axle weights under different car speeds and axle weights. Weight situations

Speed/ms1

Situation Situation Situation Situation Situation

0.8099 1.4992 1.9349 1.5030 1.4992

1 1 1 2 3

Identified weights/kg (RPE) First axle

Second axle

Total weight

4.47 4.04 3.89 3.46 2.90

5.75 6.13 6.27 5.57 5.59

10.22 (3.79%) 10.17 (4.23%) 10.16 (4.42%) 9.04 (4.52%) 8.49 (4.35%)

(2.71%) (12.09%) (15.34%) (13.64%) (15.29%)

(4.62%) (1.76%) (3.93%) (2.19%) (2.52%)

moving speeds are also considered, a combination of responses, i.e. 2/5m*., 1/2m*. & 4/5a*. is adopted, as a result, the corresponding MFI results are given in Table 8. From Table 6, it can be seen that the combinations of responses affect the MFI results because limited information is acquired in given measured points. However, sufficient accuracy of the identified weights has been achieved in all given combinations. Meanwhile, from Table 8, it can be found that acceptable identified accuracy is achieved when different moving speeds and car weights are considered, which indicates that the proposed method can be widely used for MFI. Moreover, the identified weights deviate from the true values to a certain extent [14,20,21], as shown in Tables 5, 6 and 8. Three reasons can explain these results. 1) The measured responses are inevitably influenced by noise. 2) As mentioned in Ref. [14], the acceleration responses can effectively obtain the high-frequency components of moving forces. Nevertheless, the model of accelerometers in this experiment is PCB-333B30, the frequency range of these accelerometers is from 0.5 Hz to 3000 Hz. In other words, these accelerometers cannot detect the DC component of acceleration response, so it cannot obtain the axle weights of model car if acceleration responses are used only and it affects the MFI results. 3) The influence of the model car on the beam is simplified as moving forces, the difference between the beam and the car-beam coupled system is neglected. Thus, the mass of the model car is a factor to affect the MFI results [39]. 5. Conclusions Aiming at mitigating the contradiction between response acquisition and energy consumption in the process of moving force identification (MFI), and at studying whether the compressed coefficients of responses can be effectively exploited for MFI simultaneously, a novel method is proposed based on both compressed sensing (CS) and redundant dictionaries in this study. According to the prior knowledge and signal characteristics, dictionaries are designed to represent moving forces firstly. Then, the relationship between moving forces and responses is employed, and an indirect way to design dictionaries for different structural responses is proposed, so the compressed coefficients of responses are acquired based on the CS theory. Meanwhile, a MFI governing equation directly using the compressed coefficients is formulated. Moreover, the l1-norm regularization is used to ensure the accuracy of MFI results. Finally, the identified moving forces are obtained from the identified coefficient vector. In order to assess the effectiveness and feasibility of the proposed MFI method, both numerical simulations and experimental verifications are conducted. Some conclusions can be made as follows: (1) The proposed MFI method can effectively shorten the sample length of each acquired data, in which the compressed coefficients rather than structural responses can be directly used for MFI and they can provide enough information of both moving forces and responses.

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(2) The indirect way to design dictionaries for different structural responses is an effective way, and it is suitable for different types of responses in the field of structural health monitoring (SHM). (3) The proposed MFI method can accurately obtain the total weight of the model car. The time-varying components of the identified moving forces fluctuate around the constant force components, and structural responses can be successfully reconstructed from the identified moving forces. These show that the proposed MFI method is a feasible and reliable technique, which provides a potential tool of MFI in the SHM field.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work was jointly supported by the National Natural Science Foundation of China under grant numbers of 51678278 and 51278226 and the Open Projects Foundation (No. 2017-01-KF) of State Key Laboratory for Health and Safety of Bridge Structures, China. References [1] H.J. Ouyang, Moving-load dynamic problems: a tutorial (with a brief overview), Mech. Syst. Signal Process. 25 (2011) 2039–2060. [2] Z. Chen, Z. Xie, J. Zhang, Measurement of vehicle-bridge-interaction force using dynamic tire pressure monitoring, Mech. Syst. Signal Process. 104 (2018) 370–383. [3] S.Z. Chen, G. Wu, D.C. Feng, L. Zhang, Development of a bridge weigh-in-motion system based on long-gauge fiber bragg grating sensors, J. Bridg. Eng. 23 (2018) 04018063. [4] X.Q. Zhu, S.S. 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