Modified truncated singular value decomposition method for moving force identification

Abstract

In this study, a modified truncated singular value decomposition (MTSVD) method is proposed for the identification of dynamic moving forces on simply-supported beams. By regularizing the truncated singular value decomposition (TSVD) method, the MTSVD method focuses on overcoming the ill-posed problems that intrinsically exist in moving force identification. Two regularization parameters, namely, regularization matrix and truncating point are the most important regularization parameters affecting the performance of the MTSVD method. The accuracy and efficiency of the MTSVD method is shown by comparing the results with the conventional counterpart SVD and TSVD methods. In addition, the proposed method is also compared with a similar method recently proposed by the author, that is, the piecewise polynomial truncated singular value decomposition (PP-TSVD) method. Numerical simulations demonstrate that the performance of the MTSVD method is significantly improved compared with the PP-TSVD method in high noise level cases.

Keywords

Moving force identification modified truncated singular value decomposition ill-posed problem regularization matrix truncating point piecewise polynomial truncated singular value decomposition

Introduction

Site-specific information of moving vehicles is very desirable for the design of new bridges as well as the evaluation of existing bridges (Ding et al., 2015; He et al., 2019a; Li et al., 2013). However, the calculation or direct measurement of the moving force is usually subjected to bias (Ji et al., 2020; Zhu et al., 2006). Force reconstruction and force identification techniques have been developed to fight this drawback, typically through solving an inverse problem on the basis of the measured dynamic responses of the bridge (Richardson et al., 2014; Weng et al., 2014; Yang and Yang, 2018; Yang et al., 2021).

Force identification techniques can be distinguished into two main classes, namely, the frequency domain methods (Aucejo and De Smet, 2018; Kim and Lee 2017; He et al., 2018) and time domain methods (Chen et al., 2021; Helmi et al., 2015; Law et al., 1997; Liu et al., 2016). Moving force identification (MFI) belongs to the second category of inverse problems in structural dynamic systems. Since not all the initial conditions or state variables are well known, the inverse identification problems usually have no unique solution even the nature of this problem is typical ill-posedness (Chen and Chan 2017; Yu et al., 2016). It is one of the solutions to transform ill-posed problems into well-posed problems by putting forward a new method. The method, which put the ill-posed problems turn into a well-posed problem, is defined as the regularization process (Uhl, 2007).

Regularization process can be formulated into several categories such as the singular value decomposition (SVD) method (Thite and Thompson, 2003), iterative method (Aucejo and De Smet, 2019; Li and Lu, 2018), data filtering approach (Lourens et al., 2012; Mendrok and Uhl, 2010), Tikhonov-like regularization method (Jiang et al., 2020; Kalhori et al., 2018; Obrien et al., 2014; Zhang et al., 2020), and some novel regularization method (He et al., 2019b, 2020; Liu et al., 2015, 2017; Qiao et al., 2017, 2019, 2020). In MFI, the SVD-based method relies on the decomposition of the vehicle-bridge system matrix on singular values and singular vectors. The numerical ill-condition of the system matrix case can be realized by calculating the condition number of the vehicle-bridge system matrix, which is equal to the ratio of the maximum and minimum singular values of the system matrix. If the condition number is small, the solution is unique and then the MFI is well-posed. On the contrary, if the condition number is large, the solution is usually not unique and the inverse identification problem will most likely be ill-posed. On account of the vehicle-bridge system matrix has a large scale, which makes it generally exist some very small singular values. Then, the ratio of the maximum and minimum singular values will be very large, i.e., the condition number of the vehicle-bridge system matrix is very large. Under these circumstances, truncating the small singular values is the most effective and direct approach to reduce the condition number of the system matrix. In this sense the truncated singular value decomposition (TSVD) method can be treated as a useful regularization process.

The SVD technique is commonly used to solve linear discrete ill-posed problems of small-to-moderate scale problems while the TSVD method can be used to solve large-scale linear discrete ill-posed problems (Onunwor and Reichel, 2017). Winkler (Winkler, 1997) indicated that the TSVD method can be used to solve the ill-posed problem by deletion some small singular values and polynomial basis conversion. The improvement of the TSVD method has attracted the attention of many researchers (Bouhamidi et al., 2011; Noschese and Reichel, 2014).

In a previous study, Chen et al. (2019) proposed a piecewise polynomial truncated singular value decomposition (PP-TSVD) method to solve the over-fitting problem existed in the TSVD technique. The PP-TSVD method extracts useful information from the truncated small singular values and superimposing it into the solution of the TSVD method, which is able to address the disadvantage of the TSVD method effectually. However, simulations also show that the identification accuracy is not good enough under high noise level. In order to solve this problem, a modified truncated singular value decomposition (MTSVD) method is presented in this study. Simulations show that the MTSVD method not only has a significant improvement over the SVD method and the TSVD method, but also has a certain improvement over the PP-TSVD method. In this paper, the MFI theory and improvement process of the MTSVD method are firstly demonstrated in section 2. Then, the overall performances of the MTSVD method and the selection of regularization parameters are illustrated in section 3. Finally, section 4 presents the details of the comparative studies of the MTSVD method and the recently proposed PP-TSVD method. From the conclusions of the numerical simulations, some recommendations are provided for properly applying the proposed MFI method.

Basic theory

Time domain method

Force identification in time domain is a classic and effective method, which is based on the analytical model and the relationship between unknown time-varying forces and structural dynamic responses. For the sake of completeness, the time domain method (TDM) is briefly introduced and more details about the TDM method can be seen in Ref (Law et al., 1997).

As shown in Figure 1, the vehicle-bridge system is simulated as a Bernoulli-Euler simply-supported beam and subjected to unknown time-varying forces. The dynamic responses of the beam can be measured when the vehicle moves across the bridge deck. Assuming that the force $f (t)$ moves at a constant velocity $c$ and the system equation can be written as (Law et al., 1997)

{\ddot{q}}_{n} (t) + 2 ξ_{n} ω_{n} {\dot{q}}_{n} (t) + ω_{n}^{2} q_{n} (t) = \frac{2}{ρ L} p_{n} (t) (n = 1,2, \dots, \infty)

(1)

where

ξ_{n} = \frac{C}{2 ρ ω_{n}}

is the nth modal damping ratio;

C

is the viscous damping of the simply-supported beam and

ρ

is the constant mass per unit length of the beam;

ω_{n} = \frac{n^{2} π^{2}}{L^{2}} \sqrt{\frac{E I}{ρ}}

is the nth modal frequency. The length of the beam is

L

and the second moment of area of the beam cross-section is

I

E

is Young’s modulus and

p_{n} (t) = f (t) \sin \frac{n π c t}{L}

is the modal force.

Figure 1.

Moving force identification with a simply-supported beam.

Combining the modal superposition and convolution integral, the dynamic deflection $v (x, t)$ of the beam can be solved by equation (1) in the time domain

v (x, t) = \sum_{n = 1}^{\infty} \frac{2}{ρ L ω_{n}^{'}} sin \frac{n π x}{L} \int_{0}^{t} e^{- ξ_{n} ω_{n} (t - τ)} sin ω_{n}^{'} (t - τ) sin \frac{n π c τ}{L} f (τ) d τ

(2)

where

ω_{n}^{'} = ω_{n} \sqrt{1 - ξ_{n}^{2}}

Both the time-varying force $f (t)$ and dynamic deflection $v (x, t)$ are step functions, and then the corresponding bending moment and acceleration responses at point $x$ and time $t$ can be derived from equation (2) by integral and differential calculation, respectively.

m (x, t) = - E I \frac{\partial^{2} ν (x, t)}{\partial x^{2}} = \sum_{n = 1}^{\infty} \frac{2 E I π^{2}}{ρ L^{3}} \frac{n^{2}}{ω_{n}^{'}} sin \frac{n π x}{L} \int_{0}^{t} e^{- ξ_{n} ω_{n} (t - τ)} sin ω_{n}^{'} (t - τ) sin \frac{n π c τ}{L} f (τ) d τ

(3)

\ddot{v} (x, t) = \sum_{n = 1}^{\infty} \frac{2}{ρ L} \sin \frac{n π x}{L} [f (t) sin \frac{n π x}{L} + \int_{0}^{t} {\ddot{h}}_{t} (t - τ) f (τ) sin \frac{n π c τ}{L} d τ]

(4)

where

{\ddot{h}}_{n} (t) = \frac{1}{ω_{n}^{'}} e^{- ξ_{n} ω_{n} t} \times {[{(ξ_{n} ω_{n})}^{2} - ω_{n}^{' 2}] \sin ω_{n}^{'} t + (- 2 ξ_{n} ω_{n} ω_{n}^{'}) \cos ω_{n}^{'} t}

and

\ddot{v}

is the second derivative of

v

For the real bridge, the strain gauge which can be used to measure bending moment responses indirectly. The relationship between bending moment responses and voltage signals of strain gauge can be calibrated by static step-by-step loading test of bridge or derived from the mechanical analysis. Assuming that the force $f (t)$ is a step function in a small time interval $Δ t$ , equation (3) can be rewritten in discrete terms as

m (i) = \frac{2 E I π^{2}}{ρ L^{3}} \sum_{n = 1}^{\infty} \frac{n^{2}}{ω_{n}^{'}} sin \frac{n π x}{L} \sum_{z = 0}^{i} e^{- ξ_{n} ω_{n} Δ t (i - z)} sin ω_{n}^{'} Δ t (i - z) sin \frac{n π c Δ t z}{L} f (z) Δ t, i = 0,1,2, \dots, N

(5)

Definition the time interval is $Δ t$ , the number of sample points is $N + 1$ . Assuming that $m (0) = 0$ and $m (N) = 0$ at the entry and exit of a vehicle, equation (5) can be rewritten in matrix form as

{\begin{matrix} m (1) \\ ⋮ \\ m (N - 1) \end{matrix}} = \frac{2 E I π^{2}}{ρ L^{3}} \sum_{n = 1}^{\infty} \frac{n^{2}}{ω_{n}^{'}} sin \frac{n π x}{L} Δ t [\begin{matrix} e^{- ξ_{n} ω_{n} Δ t} sin ω_{n}^{'} Δ t sin \frac{n π c Δ t}{L} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ e^{- (N - 1) ξ_{n} ω_{n} Δ t} sin (N - 1) ω_{n}^{'} Δ t sin \frac{n π c Δ t}{L} & \dots & e^{- (N - N_{B} + 1) ξ_{n} ω_{n} Δ t} sin (N - N_{B} + 1) ω_{n}^{'} Δ t sin \frac{(N_{B} - 1) n π c Δ t}{L} \end{matrix}] \times {\begin{matrix} f (1) \\ ⋮ \\ f (N_{B} - 1) \end{matrix}}

(6)

where

N_{B} = \frac{L}{c Δ t}

MFI from bending moment responses by the TDM method can be simply rewritten as

\underset{(N - 1) \times 1}{M} = \underset{(N - 1) \times (N_{B} - 1)}{B} \cdot \underset{(N_{B} - 1) \times 1}{f}

(7)

where

B

is a

(N - 1) \times (N_{B} - 1)

matrix and its elements can be obtained by equation (6).

f

and

M

should be a

(N_{B} - 1) \times 1

and

(N - 1) \times 1

vector, respectively.

Similarly, MFI from acceleration responses by the TDM method can be expressed as

\underset{N \times 1}{\ddot{V}} = \underset{N \times (N_{B} - 1)}{D} \cdot \underset{(N_{B} - 1) \times 1}{f}

(8)

where

D

is a

N \times (N_{B} - 1)

matrix.

f

and

\ddot{V}

should be a

(N_{B} - 1) \times 1

and

N \times 1

vector, respectively.

Bending moments and acceleration responses can be united to identify the moving force. The vectors $M$ in equation (7) and $\ddot{V}$ in equation (8) are combined as

[\begin{matrix} B / ‖ M ‖ \\ D / ‖ \ddot{V} ‖ \end{matrix}] \times f = {\begin{matrix} M / ‖ M ‖ \\ \ddot{V} / ‖ \ddot{V} ‖ \end{matrix}}

(9)

where

‖ \cdot ‖

is the norm of the vector.

Based on the existing TDM and equation (9), the relationship between the unknown moving force and the dynamic response can be transformed to solve the linear algebraic equation

\underset{m \times n}{A} \cdot \underset{n \times 1}{x} = \underset{m \times 1}{b}

(10)

where

A

is the transfer matrix;

x

is the unknown moving force vector and

b

is the structural dynamic response vector. By solving equation (10), the unknown moving force can be identified.

Truncated singular value decomposition method

By applying the SVD algorithm to the vehicle-bridge system matrix $A$ , we have: $A = UΣ V^{T} = \sum_{i = 1}^{n} u_{i} σ_{i} v_{i}^{T}$ , where $U = (u_{1}, u_{2}, \dots u_{m})$ and $V = (v_{1}, v_{2}, \dots v_{n})$ are orthonormal columns matrices with $U^{T} U = I_{m}$ and $V^{T} V = I_{n}$ , $u_{i}$ and $v_{i}$ are the $i$ th column vectors of the matrices $U$ and $V$ , respectively. $Σ = diag (σ_{1}, σ_{2}, \dots, σ_{n})$ is a diagonal matrix of singular values of the matrix $A$ , which has non-negative diagonal elements $σ_{1}, σ_{2}, \dots, σ_{n}$ and suffers from without loss of generality. The non-negative diagonal elements are assumed to be positive and arranged in decreasing order as $σ_{1} \geq σ_{2} \geq \dots \geq σ_{n} \geq 0$ . $σ_{n}$ is the $n$ th singular value of $A$ and the condition number of $A$ is equal to the ratio $\frac{σ_{1}}{σ_{n}}$ . Then, the SVD solution of the vehicle-bridge system equation $Ax = b$ can be written as

x = \sum_{i = 1}^{n} \frac{u_{i}^{T} b}{σ_{i}} v_{i}

(11)

Due to the large scale of the vehicle-bridge system matrix, the matrix $A$ is characterized by having some small singular values, which directly leads to the instability results and the ill-posedness nature of MFI. On this account, the TSVD method is introduced to solve the problem by truncating small singular values of the vehicle-bridge system matrix. The TSVD method combats the ill-posed problems by replacing vehicle-bridge system matrix $A$ with truncated matrix $A_{k}$ ( $k \leq n$ ). Then, the matrix $A_{k}$ can be decomposed as $A_{k} = UΣ V^{T} = \sum_{i = 1}^{k} u_{i} σ_{i} v_{i}^{T}$ . In accordance with it, the $min Ax - b_{2}$ can be transformed into $min A_{k} x - b_{2}$ and the TSVD solution of the equation $A_{k} x = b$ can be written as

x_{k} = \sum_{i = 1}^{k} \frac{u_{i}^{T} b}{σ_{i}} v_{i}

(12)

That optimal value of the regularization is that minimizes the predicated errors. That is, the selected truncating point k controls the stabilization of the TSVD solution and the ill-posedness immunity of the TSVD method (Xu, 1998). To overcome the ill-posed problems, the most important of the TSVD solution is selection the optimal truncating point

k

to minimize

A_{k} x - b_{2}

Piecewise polynomial truncated singular value decomposition method

With the TSVD method, the truncated system matrix $A_{k}$ is used to replace the original system matrix $A$ . By truncating some small singular values, the solution of the TSVD method has a significant improvement over the SVD method in inverse identification problems. Although the TSVD method works well in many applications, it still has some typical disadvantages such as the responses over-fitting and weak noise immunity. In other words, some useful information in the original system matrix $A$ is discarded. What the most straightforward and effective approach to solve the data over-fitting problem is to extract useful information from the truncated singular values and superimpose it into the initial solution of the TSVD method.

One of the most effective methods for inverse identification is the Tikhonov-like method. By drawing into the Tikhonov regularization process to the TSVD method, a hybrid regularization method named the PP-TSVD method is proposed with regularization matrix $L$ . Then, the regularization of $x_{2}$ is appropriate by minimizing the seminorm $L x_{2}$ , which can be shown as

\min ‖ L x_{2} ‖ subject to \min ‖ A_{k} x - b_{2} ‖

(13)

The PP-TSVD algorithm replaces the 2-norm of $Lx$ with the 1-norm, which can be expressed as

\min ‖ L x_{1} ‖ subject to \min ‖ A_{k} x - b_{2} ‖

(14)

By means of superimposing the TSVD solution $x_{k}$ and a modification solution $- V_{k} w_{k}$ , the final solution of the PP-TSVD method can be expressed as (Hansen and Mosegaard, 1996)

x_{P P - T S V D} = x_{k} - V_{k} ω_{k}

(15)

where the vector

ω_{k} = {(L V_{k})}^{+} L x_{k}

can be obtained by the following solution

min ‖ (L V_{k}) ω - L x_{k 1} ‖

(16)

Based on the simplex algorithm, the PP-TSVD method is transformed into a constrained linear problem. In addition, two regularization parameters, named as the truncating point k and the regularization matrix $L$ are the important factors that affect the identification accuracy of the PP-TSVD method.

Modified truncated singular value decomposition method

By introducing the Tikhonov regularization process into the TSVD method, a similar hybrid regularization approach named the MTSVD method is proposed in this study. In this case, the regularization matrix $L$ is the most important regularization parameter. Regularization matrix $L$ is a banded matrix with a full row rank. Assuming that the matrix size of matrix $L$ is $(n - (j - 1)) \times n$ , the value of ( $j - 1$ ) corresponds to the $(j - 1)$ th derivative operator. Moreover, if $j = 1$ , $L_{1}$ becomes the unity matrix $I_{n}$ , the MTSVD method then degrades into the TSVD method in this special case. By introducing different regularization matrices in the TSVD method, the solution of the MTSVD method is obviously disparate. The $L_{1}$ , $L_{2}$ , $L_{3}$ and $L_{4}$ correspond to the TSVD, MTSVD ( $L_{2}$ ), MTSVD ( $L_{3}$ ), and MTSVD ( $L_{4}$ ) methods with different regularization matrices, respectively, which can be expressed as

\begin{array}{l} L_{1} = {(\begin{matrix} 1 \\ 1 \\ ⋱ \\ 1 \end{matrix})}_{n \times n} L_{2} = {(\begin{matrix} - 1 & 1 \\ - 1 & 1 \\ ⋱ & ⋱ \\ - 1 & 1 \end{matrix})}_{(n - 1) \times n} \\ L_{3} = {(\begin{matrix} 1 & - 2 & 1 \\ 1 & - 2 & 1 \\ ⋱ & ⋱ & ⋱ \\ 1 & - 2 & 1 \end{matrix})}_{(n - 2) \times n} L_{4} = {(\begin{matrix} - 1 & 3 & - 3 & 1 \\ - 1 & 3 & - 3 & 1 \\ ⋱ & ⋱ & ⋱ & ⋱ \\ - 1 & 3 & - 3 & 1 \end{matrix})}_{(n - 3) \times n} \end{array}

(17)

Then, the regularization of ${‖ x ‖}_{2}$ is appropriate by minimizing the seminorm ${‖ Lx ‖}_{2}$ . In order to derive an analytical expression for the MTSVD solution, two matrices and a $(n - k) \times (n - k)$ diagonal matrix $\sum$ need to be defined firstly as follows

V_{k} \equiv (V_{k + 1}, \dots, V_{n}), U_{k} \equiv (U_{k + 1}, \dots, U_{n}), \sum_{k} \equiv (σ_{k + 1}, \dots, σ_{n})

(18)

where the diagonal matrix

\sum

consists of the

(n - k)

smallest singular values of the original system matrix

A

, which are truncated by the TSVD method. The MTSVD solution

x_{L, k}

can be expressed as (Hansen et al., 1992)

x_{L, k} = (I_{n} - V_{k} {(L V_{k})}^{+} L) A_{k}^{+} b = x_{k} - V_{k} {(L V_{k})}^{+} L x_{k}

(19)

where

x_{k}

is the ordinary TSVD solution and

{(L V_{k})}^{+}

represents the pseudo-inverse of the matrix

L V_{k}

. The

- V_{k} {(L V_{k})}^{+} L x_{k}

is the useful information extracted from the

(n - k)

smallest singular values.

Following Eldén (1982), the MTSVD solution also can be rewritten formally as

x_{L, k} = (I_{n} - {(L (I_{n} - A_{k}^{+} A_{k}))}^{+} L) A_{k}^{+} b

(20)

With the relations $I_{n} - A_{k}^{+} A_{k} = V_{k} V_{k}^{T}$ and ${(L V_{k} V_{k}^{T})}^{+} = V_{k} {(L V_{k})}^{+}$ , the MTSVD solution can then be shown as

x_{L, k} = x_{k} - V_{k} {(L V_{k})}^{+} L x_{k}

(21)

In this solution, $x_{L, k}$ consists of the TSVD solution $x_{k}$ minus a “modification” vector $ω_{k} = {(L V_{k})}^{+} L x_{k}$ . As such, the MTSVD solution $x_{L, k}$ can finally be expressed as

x_{M T S V D} = x_{k} - V_{k} ω_{k}

(22)

In equation (22), the vector $ω_{k} = {(L V_{k})}^{+} L x_{k}$ can be solved as following

\min (L V_{k}) ω - L x_{k 2}

(23)

Introducing a $(n - k) \times (n - k)$ matrix $E_{k} \equiv antidiag (1, \dots, 1)$ that reverts the columns of $L V_{k}$ , the matrix $L V_{k} E_{k}$ can be factorized by QR-factorization as

L V_{k} E_{k} = Q_{k} R_{k}

(24)

Then, the vector $ω_{k}$ is solved as

ω_{k} = E_{k} {(R_{k})}^{- 1} {(Q_{k})}^{T} x_{k}

(25)

where

{(R_{k})}^{- 1}

is the inverse of the matrix

R_{k}

and

{(Q_{k})}^{T}

represents the transpose of the matrix

Q_{k}

The MTSVD algorithm picks out the useful response from the truncated small singular values of the original system matrix $A$ and then superimposes it to the TSVD solution, which overcome the limitation of the TSVD method. The MTSVD solution includes the determination of the optimal regularization matrix $L$ and truncating point k. The basic procedure of the PP-TSVD method and MTSVD method is expressed in Figure 2.

Figure 2.

Basic procedure of the modified truncated singular value decomposition method and piecewise polynomial truncated singular value decomposition method.

Selection of regularization parameters

Simulation parameters of vehicle and bridge

In Figure 1, the length of the simply-supported beam is 40 m, $E I = 1.274916 \times 10^{11} N \cdot m^{2}$ , $ρ A = 12 000 kg \cdot m^{- 1}$ . The constant speed of the forces is $40 m \cdot s^{- 1}$ and the forces pass on the bridge in 1 s. The first natural frequency of the beam is 3.2 Hz, the second frequency of the beam is 12.8 Hz and the third frequency of the beam is 28.8 Hz. The sampling frequency of the vehicle-bridge system is 200 Hz and the analysis frequency of the dynamic system is from 0 Hz to 40 Hz. The biaxial moving forces are simulated as follow, which are extracted from Ref (Law et al., 1997). The axle spacing of the biaxial moving forces is 8 m.

f_{1} (t) = 20 [1 + 0.1 \sin (10 π t) + 0.05 sin (40 π t)] kN

f_{2} (t) = 20 [1 - 0.1 \sin (10 π t) + 0.05 sin (50 π t)] kN

(26)

Random noise is used to simulate the polluted responses as

R_{measured} = R_{calculated} \cdot (1 + E_{p} \cdot N_{noise})

(27)

where

E_{p}

represents error level selecting as 1%, 5%, 10%, and 20%, respectively;

N_{noise}

is random measurement noise, which is a standard normal distribution vector.

The relative percentage error (RPE) values between the true force and the identified force are calculated as

R P E = \frac{‖ f_{identified} - f_{true} ‖}{‖ f_{true} ‖} \times 100 %

(28)

In this study, the bending moment responses and acceleration responses are adopted. The locations of the measuring points are arranged at 1/4, 1/2 and 3/4 span of the simply-supported beam, respectively.

Selection of the optimal regularization matrix $L$ for the modified truncated singular value decomposition method

In this subsection, the regularization matrix

L

of the MTSVD method will be selected by numerical simulation prudently. Actually, matrix

L_{1} = I_{n}

is not an appropriate selection in some regularization processes. For the sake of selecting the optimal regularization matrix

L_{j}

for the MTSVD method, the influence of regularization matrix on MFI with time-varying forces is studied with three dynamic responses including two bending moment responses at the 1/4 span and 1/2 span positions and an acceleration response at the 1/2 span position, respectively. In this subsection, the selected case is case 3 listed in the first column in Table 1. The influence of regularization matrix

L_{j}

on the MTSVD method with case 3 is shown in Figure 3, where the RPE values of identified results are used as the ordinate and the subscript

j

of regularization matrix

L_{j}

is used as the abscissa. The abscissa values correspond to the TSVD, MTSVD (

L_{2}

), MTSVD (

L_{3}

), MTSVD (

L_{4}

), MTSVD (

L_{5}

), and MTSVD (

L_{6}

) methods, respectively.

Table 1.

Relative percentage error (%) values of the time domain method (singular value decomposition) method, truncated singular value decomposition method, and modified truncated singular value decomposition method with two proper regularization matrices.

Case	Response	Regularization matrix	5% noise		10% noise		20% noise
Case	Response	Regularization matrix	Front axle	Rear axle	Front axle	Rear axle	Front axle	Rear axle
1	1/4m&1/2m&3/4m	TDM (SVD)	*	*	*	*	*	*
		TSVD ( $L_{1}$ )	(26.5)	(27.2)	(31.5)	(32.6)	(*)	(*)
		MTSVD ( $L_{2}$ )	6.8	6.6	9.4	8.3	10.9	10.6
		MTSVD ( $L_{3}$ )	14.2	15.8	30.1	23.5	48.8	38.8
2	1/4a&1/2a	TDM (SVD)	96.1	*	*	*	*	*
		TSVD ( $L_{1}$ )	(12.8)	(5.4)	(19.6)	(10.0)	(34.6)	(31.1)
		MTSVD ( $L_{2}$ )	6.0	4.1	9.6	6.6	13.6	14.4
		MTSVD ( $L_{3}$ )	9.3	5.0	18.4	10.0	32.7	20.3
3	1/4m&1/2m&1/2a	TDM (SVD)	*	*	*	*	*	*
		TSVD ( $L_{1}$ )	(18.6)	(20.9)	(24.5)	(24.6)	(77.3)	(75.9)
		MTSVD ( $L_{2}$ )	7.0	7.0	7.6	7.6	8.8	9.4
		MTSVD ( $L_{3}$ )	12.4	9.6	16.1	13.0	28.6	26.1
4	1/4m&1/2m&1/4a&1/2a	TDM (SVD)	58.6	74.9	*	*	*	*
		TSVD ( $L_{1}$ )	(13.1)	(8.0)	(19.0)	(19.4)	(34.3)	(30.1)
		MTSVD ( $L_{2}$ )	4.5	3.5	6.5	6.9	6.9	7.7
		MTSVD ( $L_{3}$ )	6.1	5.2	11.8	9.9	23.5	19.4
5	1/4m&1/4a&1/2a	TDM (SVD)	91.9	*	*	*	*	*
		TSVD ( $L_{1}$ )	(12.4)	(8.9)	(20.5)	(14.8)	(33.2)	(31.3)
		MTSVD ( $L_{2}$ )	4.6	4.2	7.1	6.3	7.8	11.7
		MTSVD ( $L_{3}$ )	7.5	6.7	15.4	11.8	31.2	22.1
6	1/2m&1/4a&1/2a	TDM (SVD)	59.5	74.7	*	*	*	*
		TSVD ( $L_{1}$ )	(13.2)	(6.4)	(18.8)	(18.7)	(33.8)	(29.8)
		MTSVD ( $L_{2}$ )	4.7	3.8	7.6	6.4	7.9	9.7
		MTSVD ( $L_{3}$ )	9.3	6.0	20.5	11.3	30.9	22.4

TDM: time domain method; TSVD: truncated singular value decomposition; MTSVD: modified truncated singular value decomposition.

Note. 1/4, 1/2 and 3/4 represent the measurement location at 1/4, 1/2 and 3/4 span, respectively. The letters ‘m’ and ‘a’ represent the bending moment and acceleration responses respectively. The symbol ‘*’ represents that the RPE (%) value is larger than 100%. RPE (%) values in parentheses are for the TSVD method with unity matrix $L_{1}$ . Underlined RPE (%) values are for the MTSVD method with bidiagonal matrix $L_{2}$ . Italics RPE (%) values are for the MTSVD method with tri-diagonal matrix $L_{3}$ and other values are for SVD embedded in TDM method.

Figure 3.

Influence of regularization matrix $L_{j}$ on the accuracy of the modified truncated singular value decomposition method (Case 3 in Table 1).

As shown in Figure 3, the RPE values firstly decrease and then increase with the rise of the derivative operator from the unity matrix to the sixth derivative operator. When using the unit matrix $L_{1}$ , in which case the regularization process with regularization matrix $L_{j}$ is ignored, the RPE values of the TSVD method with time-varying forces are quite large, especially when the random noise is 10%, the RPE values of the MTSVD method exceed 60%. The illustration results indicate that the regularization process is very important to the TSVD method. It is quite difficult to achieve high precision identified results unless the regularization matrix $L_{j}$ is selected properly. The results indicate that the regularization matrices $L_{2}$ and $L_{3}$ have much better noise immunity compared with the other derivative operators. In order to avoid the error caused by the single case, six typical cases are chosen for selecting the optimal regularization matrix in the following study.

As shown in Table 1, for the existence of extremely small singular values in the vehicle-bridge system matrix, the identification accuracy of the TDM method is very poor in most cases. By truncating the ( $n - k$ ) smallest singular values to reduce the condition number of the system matrix, the identification accuracy of the TSVD method has obviously been improved compared with the TDM method. Moreover, by extracting the useful responses from the truncated ( $n - k$ ) smallest singular values and superimposing them to the TSVD method solution, the identification accuracy of the MTSVD method has also been remarkable improved compared with the TSVD method. In other words, compared with the conventional TDM method embedding with the SVD algorithm, both the TSVD method and the MTSVD method have better ill-posed immunity. In addition, as shown in Table 1 and Figures 4–6, the regularization process is very efficient to improve the calculation accuracy of the TSVD method. Figures 4(b) and 5(b) show the identified results of the rear axle force. At 0.7–0.8 s, the moment corresponds the time that the front axle force leaves the simply-supported beam. At the special moment, the identified results of the TDM method may have large errors under circumstance of some interference such as the presence or absence of other loads.

Figure 4.

Moving force identification with 5% noise level by the time domain method, truncated singular value decomposition, and modified truncated singular value decomposition methods embedding with two proper regularization matrices (Case 5 in Table 1). (a) Front axle; (b) Rear axle.

Figure 5.

Moving force identification with 10% noise level by the time domain method, truncated singular value decomposition, and modified truncated singular value decomposition methods embedding with two proper regularization matrices (Case 2 in Table 1). (a) Front axle; (b) Rear axle.

Figure 6.

Moving force identification with 20% noise level by the time domain method, truncated singular value decomposition, and modified truncated singular value decomposition methods embedding with two proper regularization matrices (Case 3 in Table 1). (a) Front axle; (b) Rear axle.

By comparing the identification results of the front axle and rear axle in Table 1, the identification accuracy of the MTSVD ( $L_{2}$ ) method is relatively better than the MTSVD ( $L_{3}$ ) method in all 6 cases especially under high noise level. In view of the fact that the MTSVD ( $L_{2}$ ) method has better adaptability with sensors and stronger robustness comparing with the MTSVD ( $L_{3}$ ) method, the bidiagonal matrix $L_{2}$ is selected as the optimal regularization matrix for the MTSVD method. In order to ensure that the identification results are not disturbed by other regularization parameters, the optimal truncating points of the TSVD method and the MTSVD method are adopted in all cases. For the integrity of the study, the selection of the truncating point $k$ for the TSVD method and the MTSVD ( $L_{2}$ ) method will be scrutinized in the next subsection.

Selection of the optimal truncating point k for the modified truncated singular value decomposition method

With this vehicle-bridge system matrix, the total number of samples $n$ is 396, i.e., the truncating point k satisfies $n = 396 \geq k \geq 1$ . It should be noted that, when $k = 396$ is adopted, there is no small singular value truncated. In this special condition, the TDM, TSVD, and MTSVD methods show the same identification results.

As shown in Table 2, under high noise levels, smaller truncating point should be chosen. In addition, the optimal truncating points of the TSVD method are larger than those of the MTSVD method in each case. The truncated (

n - k

) smallest singular values is discarded directly in the TSVD method. To avoid discarding too much meaningful information contained in small singular values, the optimal truncating points should be relatively large in order to improve the identification accuracy of the TSVD method. In contrast, the MTSVD method does not need to take this into account because it can extract the discarded useful responses and superimposes them into the TSVD solution. As shown in Table 2, the identification accuracy of the MTSVD method is much improved from the TSVD method.

Table 2.

The optimal truncating point k of the truncated singular value decomposition and modified truncated singular value decomposition methods.

Case	Response	Method	5% noise			10% noise			20% noise
			k	RPE (%)		k	RPE (%)		k	RPE (%)
			k	Front	Rear	k	Front	Rear	k	Front	Rear
1	1/4m&1/2m&3/4m	TSVD	(137)	(26.5)	(27.2)	(121)	(31.5)	(32.6)	(112)	(*)	(*)
1	1/4m&1/2m&3/4m	MTSVD	89	6.8	6.6	64	9.4	8.3	50	10.9	10.6
2	1/4a&1/2a	TSVD	(386)	(12.8)	(5.4)	(386)	(19.6)	(10.0)	(373)	(34.6)	(31.1)
2	1/4a&1/2a	MTSVD	366	6.0	4.1	366	9.6	6.6	339	13.6	14.4
3	1/4m&1/2m&1/2a	TSVD	(278)	(18.6)	(20.9)	(272)	(24.5)	(24.6)	(245)	(77.3)	(75.9)
3	1/4m&1/2m&1/2a	MTSVD	240	7.0	7.0	231	7.6	7.6	231	8.8	9.4
4	1/4m&1/2m&1/4a&1/2a	TSVD	(388)	(13.1)	(8.0)	(382)	(19.0)	(19.4)	(373)	(34.3)	(30.1)
4	1/4m&1/2m&1/4a&1/2a	MTSVD	369	4.5	3.5	311	6.5	6.9	311	6.9	7.7
5	1/4m&1/4a&1/2a	TSVD	(388)	(12.4)	(8.9)	(386)	(20.5)	(14.8)	(378)	(33.2)	(31.3)
5	1/4m&1/4a&1/2a	MTSVD	363	4.6	4.2	338	7.1	6.3	317	7.8	11.7
6	1/2m&1/4a&1/2a	TSVD	(388)	(13.2)	(6.4)	(382)	(18.8)	(18.7)	(373)	(33.8)	(29.8)
6	1/2m&1/4a&1/2a	MTSVD	368	4.7	3.8	368	7.6	6.4	313	7.9	9.7

RPE: relative percentage error; TSVD: truncated singular value decomposition; MTSVD: modified truncated singular value decomposition.

Note. Values in parentheses are for the TSVD method and underlined values are for the MTSVD method. Bold font values are the optimal truncating points and the others are the RPE values corresponding to respective cases.

As shown in Figure 7, the truncating point has significant influence on the identification accuracy of the two methods. If the noise level is 5%, the RPE values will be greater than 100% for both methods when the selected truncating point is greater than 350. If the noise level is 10%, the RPE values will be greater than 100% for both methods when the selected truncating point is greater than 250. The illustration results clearly indicate that it is necessary to truncate the small singular values to avoid noise pollution on the dynamic responses of bridge. However, if too many small values are truncated, the effective responses that can be used to identify the force are too slightly, which greatly reduces the validity of the study. It should be noted that the rule of truncating point is directly related to the case. Therefore, the unique optimal truncating point cannot be selected for the identification method.

Figure 7.

Influence of truncating point k on the accuracy of the truncated singular value decomposition and modified truncated singular value decomposition methods (Case 1 in Table 2). (a) Front axle; (b) Rear axle.

As shown in Figure 8, the identification results vary greatly with different truncating points. If the truncating point is too small, the responses that can be used to identify the moving forces are very limited and therefore not able to accurately identify the moving forces. In this special case, the identification results are similar to the average forces, which cannot truly reflect the forces fluctuation. Provided that the truncating point is too large, the effect of very small singular values cannot be eliminated and the ill-posed problems existed in MFI will appear, which will lead to drastic fluctuation of the identified forces and thus the force identification with the MTSVD method cannot be figured out.

Figure 8.

Moving force identification with 5% noise level by the modified truncated singular value decomposition method with three different truncating points (Case 1 in Table 2). (a) Front axle; (b) Rear axle.

Comparative studies

In this study, the MTSVD method is presented in MFI by solving the over-fitting problem existed in the TSVD technique. There is another similar approach named the PP-TSVD method proposed by the first author in a previous study. To compare the advantages, disadvantages and differences of the two methods, a detailed comparative study is carried out in this section. Both methods have two regularization parameters, i.e., the regularization matrix and the truncating point. In order to ensure that the identification results are not disturbed by these regularization parameters, the optimal regularization matrix and the optimal truncating points of two methods are adopted by default in this section.

Table 3 tabulates the RPE values of the TSVD, PP-TSVD, and MTSVD methods with three random noise levels in all six cases. By extracting the helpful information from the discarded small singular values and superimposes it to the TSVD method solution, the PP-TSVD and MTSVD methods address the over-fitting problem existed in the TSVD technique. As shown in Table 3 and Figures 9–11, the identification accuracy of the PP-TSVD and MTSVD methods is much improved compared with the TSVD method. Moreover, the identification accuracy of the MTSVD method has a certain improvement over the PP-TSVD method. The MTSVD method is more suitable for identifying moving forces under high noise contamination. In other words, it has better robustness than the PP-TSVD method under noisy conditions.

Table 3.

Comparison of relative percentage error (%) values identified by the truncated singular value decomposition, piecewise polynomial truncated singular value decomposition, and modified truncated singular value decomposition methods.

Case	Response	Method	5% noise		10% noise		20% noise
Case	Response	Method	Front axle	Rear axle	Front axle	Rear axle	Front axle	Rear axle
1	1/4m&1/2m&3/4m	TSVD	(26.5)	(27.2)	(31.5)	(32.6)	(*)	(*)
		PP-TSVD ( $L_{2}$ )	7.9	8.9	14.0	13.3	20.6	19.0
		MTSVD ( $L_{2}$ )	6.8	6.6	9.4	8.3	10.9	10.6
2	1/4a&1/2a	TSVD	(12.8)	(5.4)	(19.6)	(10.0)	(34.6)	(31.1)
		PP-TSVD ( $L_{2}$ )	6.0	4.9	9.8	7.2	14.8	14.9
		MTSVD ( $L_{2}$ )	6.0	4.1	9.6	6.6	13.6	14.4
3	1/4m&1/2m&1/2a	TSVD	(18.6)	(20.9)	(24.5)	(24.6)	(77.3)	(75.9)
		PP-TSVD ( $L_{2}$ )	7.2	7.3	8.7	8.5	9.4	9.5
		MTSVD ( $L_{2}$ )	7.0	7.0	7.6	7.6	8.8	9.4
4	1/4m&1/2m&1/4a&1/2a	TSVD	(13.1)	(8.0)	(19.0)	(19.4)	(34.3)	(30.1)
		PP-TSVD ( $L_{2}$ )	4.6	4.2	9.6	7.7	9.2	9.1
		MTSVD ( $L_{2}$ )	4.5	3.5	6.5	6.9	6.9	7.7
5	1/4m&1/4a&1/2a	TSVD	(12.4)	(8.9)	(20.5)	(14.8)	(33.2)	(31.3)
		PP-TSVD ( $L_{2}$ )	4.8	4.2	7.3	6.3	10.0	13.7
		MTSVD ( $L_{2}$ )	4.6	4.2	7.1	6.3	7.8	11.7
6	1/2m&1/4a&1/2a	TSVD	(13.2)	(6.4)	(18.8)	(18.7)	(33.8)	(29.8)
		PP-TSVD ( $L_{2}$ )	5.4	3.9	8.1	6.7	11.3	9.9
		MTSVD ( $L_{2}$ )	4.7	3.8	7.6	6.4	7.9	9.7

TSVD: truncated singular value decomposition; PP-TSVD: piecewise polynomial truncated singular value decomposition: MTSVD: modified truncated singular value decomposition.

Figure 9.

Moving force identification with 5% noise level by the truncated singular value decomposition, piecewise polynomial truncated singular value decomposition, and modified truncated singular value decomposition methods (Case 4 in Table 3). (a) Front axle; (b) Rear axle.

Figure 10.

Moving force identification with 10% noise level by the truncated singular value decomposition, piecewise polynomial truncated singular value decomposition, and modified truncated singular value decomposition methods (Case 2 in Table 3). (a) Front axle; (b) Rear axle.

Figure 11.

Moving force identification with 20% noise level by the truncated singular value decomposition, piecewise polynomial truncated singular value decomposition, and modified truncated singular value decomposition methods (Case 6 in Table 3). (a) Front axle; (b) Rear axle.

As shown in Figure 12, the distinction of identification accuracy between the two methods increases with the rise of noise level. If the noise level is 1%, the RPE values curves of two methods are almost the same. If the noise level increases to 5%, the RPE values curves of the PP-TSVD method are slightly above the curves of the MTSVD method. If the noise level equals to 10%, the difference of two RPE curves is very obvious. With the increases of the truncating point, the RPE values of the PP-TSVD method increase more rapidly than the MTSVD method. In other words, the noise immunity of the MTSVD method is better than the PP-TSVD method. To summarize, the performance of the MTSVD method is not only better than conventional counterpart the SVD and the TSVD methods, but also better than the latest proposed PP-TSVD method.

Figure 12.

Influence of truncating point k on the accuracy of the piecewise polynomial truncated singular value decomposition and modified truncated singular value decomposition methods (Case 1 in Table 3). (a) Front axle; (b) Rear axle.

Conclusions

In this study, a new algorithm named modified truncated singular value decomposition method for MFI is presented and a comparative study was conducted. Based on the simulation results, some conclusions can be drawn as follows:

1. In view of the influence of some small singular values, the identification results of the conventional TDM method embedding with the SVD algorithm are obviously ill-posed problems, which can be mitigated by truncating some singular values with the TSVD method. Unfortunately, the TSVD method still has some drawbacks such as the responses over-fitting and weak noise immunity.

2. The MTSVD method extracts useful responses from the discarded small singular values of the original system matrix $A$ and superimposes them to the TSVD method solution, which has theoretical completeness and offset the disadvantages of the TSVD method.

3. Both the regularization matrix $L$ and truncating point k are important regularization parameters of the MTSVD method. The bidiagonal matrix $L_{2}$ is selected as the optimal matrix for the MTSVD method, which is case independent and can be adopted by default. However, the optimal truncating point k cannot be selected by default, which is case dependent.

Finally, the performance of the MTSVD method proved by numerical simulations is not only more accurate than the conventional counterpart SVD and TSVD methods, but also more comprehensive than the latest PP-TSVD method. The MTSVD method is suitable for identifying moving forces under high noise contamination thanks to its strong robustness.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is jointly supported by National Natural Science Foundation of China, China (Grant Numbers U2004184, 51778222 and 52008160), and Training Plan for Young Key Teachers in Colleges and Universities in Henan Province (Grant Number 2021GGJS078).

ORCID iD

Zhen Chen

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Modified truncated singular value decomposition method for moving force identification

Abstract

Keywords

Introduction

Basic theory

Time domain method

Truncated singular value decomposition method

Piecewise polynomial truncated singular value decomposition method

Modified truncated singular value decomposition method

Selection of regularization parameters

Simulation parameters of vehicle and bridge

Selection of the optimal regularization matrix L for the modified truncated singular value decomposition method

Selection of the optimal truncating point k for the modified truncated singular value decomposition method

Comparative studies

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Selection of the optimal regularization matrix $L$ for the modified truncated singular value decomposition method