A stochastic analysis method of transient responses using harmonic wavelets, part 2: Time-dependent vehicle-bridge systems

doi:10.1016/j.ymssp.2021.107871

Mechanical Systems and Signal Processing

Volume 162, 1 January 2022, 107871

https://doi.org/10.1016/j.ymssp.2021.107871 Get rights and content

Highlights

•
A method is proposed to estimate the response PSDs of vehicle-bridge systems.
•
A semi-analytical solution of the PSDs is presented using harmonic wavelets.
•
The results of two vehicle-bridge systems verify the effectiveness of the method.

Abstract

Conventional stochastic response analysis methods of vehicle-bridge (VB) systems involve a brute-force calculation based on a large number of simulations, which imposes huge computational burdens because the dynamic analysis of a time-dependent VB system requires repeated assembles of system matrices. This paper proposes a novel stochastic analysis method of transient responses for time-dependent VB systems based on periodic generalized harmonic wavelets (GHW). We establish the equations of motion of the VB system and reveal the orthogonal characteristic relationships between the time-dependent system matrices and the wavelet functions. Then, two sets of linear algebraic equations of wavelet coefficients are established for describing the time-dependent VB system, and the time-varying power spectral density functions of system responses are obtained in a semi-analytical form. The stochastic response analysis of the time-dependent VB system is therefore converted into solving a set of linear algebraic equations, which significantly simplifies the stochastic response analysis. The accuracy and efficiency of the proposed method are validated via comparison against the Monte Carlo simulations of nonstationary stochastic analyses of a highway bridge and a railway bridge, respectively. The proposed method provides a high-efficient way to estimate the time-varying power spectral density functions of stochastic responses of time-dependent VB systems.

Introduction

The dynamic analysis of vehicle-bridge (VB) coupled systems is very important for evaluating the dynamic performance of bridges and the running safety of vehicles [1], [2], [3], [4]. The major excitation of a VB system, i.e. track irregularity or road roughness, is a stochastic excitation, therefore, a VB system is essentially a stochastic dynamic system [5], [6], [7]. Compared with other stochastic dynamic systems, a VB system has certain unique features. Firstly, the track irregularity or road roughness has obvious nonstationary stochastic characteristics. Secondly, a VB system is a time-dependent dynamic system, of which the system matrices (mass, damping and stiffness matrices) are all time-dependent [6], [8], [9]. Consequently, the stochastic response analysis of VB systems is more challenging compared with conventional structural dynamic systems.

At present, the stochastic response analysis of VB systems attracts increasing attentions in the field of structural dynamics [8], [9], [10], [11], [12]. Representative analysis methods include the probability density evolution method (PDEM) [5], [6], [7] and the pseudo-excitation method (PEM) [8], [13], [14]. In the PDEM, load samples are generated based on the power spectral density (PSD) of loads, while the corresponding response samples are calculated by deterministic analysis approaches. Then, the probability densities of the responses are obtained by the probability density evolution equations. In the PEM, the pseudo-load samples defined in complex-domain are also generated based on the PSD of loads, and the response samples are accordingly calculated. The PSDs of the responses are then determined by the auto-correlation analysis of response samples. Both the PDEM and the PEM involve the brute-force calculation based on a large number of samples [6], [7], [8], [13], [14]. In addition, in the deterministic analysis of each response sample, the system matrices of the VB system are repeatedly assembled [9] accompanying with an iterative solution procedure [8]. In summary, available stochastic response analysis methods for the VB system cost a great deal of computational resources thus time-consuming and low-efficient.

The newly emerged signal processing technology termed wavelet analysis has significant advantages in joint time–frequency analysis, which is beneficial to the dynamic analysis of structural systems and widely used in civil engineering field [15], [16], [17]. Dos Santos et al. [15] proposed a wavelet-based identification technique for estimating the parameters of structural systems. Notably, Newland developed a special harmonic wavelet [18], [19] for structural dynamic analysis. Because the basis functions of this harmonic wavelet share the same form as that of the analytical solution of bridge vibration response, thus the wavelet-based method has significant advantages in time–frequency analysis considering nonstationary excitations.

Recently, the harmonic wavelet including its subseries (the generalized harmonic wavelet (GHW) and periodic GHW) has been successfully applied to the nonstationary PSD estimation [20], the time–frequency analysis of excitations and responses [21] and so on. More recently, Pasparakis et al. [17] proposed a new approximate analytical technique for determining the response PSD of stochastically excited structural systems. Kong [22] and Spanos [23] proposed a novel approach for the stochastic response analysis of structural systems subject to nonstationary excitations. Their approach has also been applied to assess the stochastic responses of civil structures under wind loads [24]. Considering the significance of transient responses, we further developed a novel wavelet-based approach for the stochastic transient analysis of structural systems in the companion paper [25].

To date, however, the stochastic transient analysis of time-dependent dynamic systems, such as VB systems, has not been well addressed yet. The VB system possesses both time-dependent and transient characteristics, which are significantly different from conventional time-independent dynamic systems. The time-dependent characteristic of the VB system poses a great challenge for its stochastic transient analysis.

This paper proposes a stochastic analysis method of transient responses for VB systems based on the periodic GHW for the first time. Section 2 systematically analyzes the characteristic relationships between the time-dependent system matrices and the periodic GHW functions. Then two sets of linear algebraic equations of wavelet coefficients are established for describing the VB system. In Section 3, a stochastic response analysis method for the VB system is established by using the linear algebraic computations based on the periodic GHW. In Section 4, numerical studies of two typical VB systems namely a highway VB system and a railway VB system are conducted to verify the accuracy and efficiency of the proposed method.

Section snippets

Wavelet analysis of VB systems

In this section, the equations of motion of a time-dependent VB system are established. The characteristic relationships between the time-dependent system matrices and the periodic GHW functions are systematically analyzed. Then, two sets of linear algebraic equations of wavelet coefficients are established for describing the time-dependent VB system.

Deterministic analysis and stochastic analysis

In this section, an analysis procedure is proposed for the time-dependent VB system. Firstly, a deterministic analysis procedure is introduced, and a stochastic analysis procedure follows.

Examples

In this section, a highway VB system and a railway VB system are analyzed to verify the accuracy and efficiency of the proposed method, respectively. Both the highway VB system and the railway VB system consist of a single-span bridge and a single-axle road vehicle, which are widely used to verify the effectiveness of VB models [9]. In this study, the deterministic analyses of the two representative systems are firstly conducted, and then the stochastic characteristics of the two VB systems are

Conclusions

This paper proposes a stochastic analysis method of transient responses for time-dependent VB systems based on the periodic GHW for the first time. The major advantage of the proposed method is that it can transform the complex stochastic analysis of the time-dependent VB system into solving a set of linear algebraic equations, resulting in very high efficiency in terms of computational cost. The proposed method is compared against the MCM in the numerical examples of a highway VB system and a

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.

Acknowledgements

The authors are grateful for the financial support from the National Natural Science Foundation of China (Grant Nos.: 51838006), and the Fundamental Research Funds for the Central Universities (Grant No.: HUST_2018KFYYX JJ007). Findings and opinions expressed here, however, are those of the authors alone, not necessarily the views of the sponsor.

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Representation of nonstationary stochastic excitations is crucial for stochastic response analyses of (time-varying) linear and nonlinear structural systems. This paper proposes a new representation method of non-stationary stochastic excitations based on the generalized harmonic wavelet (GHW) that takes the phase angles and frequencies as basic random variables. The orthogonal properties of the discrete-form spectral process increments describing non-stationary stochastic processes are formulated. Then the GHW-based representation is derived by using the orthogonal properties. This method can be used to accurately reproduce non-stationary stochastic excitations with the target asymptotic Gaussianity and evolutionary power spectrum density. The effectiveness and accuracy of the proposed method have been validated via numerical examples. This study provides a novel way for the representation of non-stationary processes and deserves to be applied in the stochastic response analyses of structures.
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A stochastic analysis method of transient responses using harmonic wavelets, part 2: Time-dependent vehicle-bridge systems

Highlights

Abstract

Introduction

Section snippets

Wavelet analysis of VB systems

Deterministic analysis and stochastic analysis

Examples

Conclusions

Declaration of Competing Interest

Acknowledgements

Mech. Syst. Signal Process.

Eng. Struct.

Mech. Syst. Signal Process.

J. Sound Vib.

Eng. Struct.

Eng. Struct.

Int. J. Struct. Stab. Dyn.

Comput. Struct.

Struct. Saf.

Mech. Syst. Signal Process.

Mech. Syst. Signal Process.

J. Wind Eng. Ind. Aerod

Mech. Syst. Signal Process.

J. Sound Vib.

Probabilist Eng. Mech.