Vibration fatigue dynamic stress simulation under non-stationary state
Introduction
Nowadays, with the control of metro operation cost, such as prolonging the wheel re-profiling and rail grinding maintenance interval, the wear of wheel and rail increases, resulting in a series of uncontrollable vibration, and the vibration frequency are generally unknown at structural design stage, which leads to some resonance problems, namely, vibration fatigue. In serious cases, it will lead to structural failure [1], [2].
Compared with traditional fatigue analysis, vibration fatigue analysis deals with the material fatigue of flexible structures operating close to natural frequencies [3], [4] and it has two main differences. First, in post-processing of dynamic stress, the traditional statistical method of Rain Flow counting is abandoned and the stress statistics is carried out by spectral method [3], [5], [6], [7], which is used to improve the calculation efficiency. Second, in dynamic stress simulation, vibration fatigue needs to consider the modal parameters of the structure and the influence of structural dynamic characteristics on deformation [3], [8], but traditional fatigue simulation mainly considers structural stiffness, that is, Hooke’s law principle. But, S-N curves and cumulative damage approach are suitable for both traditional fatigue and vibration fatigue.
In order to solve or avoid the problems caused by vibration fatigue, it is necessary to adopt accurate stress simulation method in the early design or optimal design of the structure. In recent years, more and more studies on structural vibration fatigue have been carried out [9], [10], [11], [12]. Frequency domain analysis method is commonly adopted in vibration fatigue, and various spectral methods have been developed, such as Dirlik [13], Tovo-Benasciutti [14] and Zhao-Baker [15] method. And the differences of evaluation methods in frequency domain are also studied [16], [17], [18]. At the same time, there are a lot of research achievements on how to evaluate the fatigue life under multi-axial stress state [19], [20], [21], even in vibration fatigue field [4], [22], [23], [24], [25]. Dealing with stationary random loads, vibration fatigue analysis requires frequency limited steady-state response solutions [3]. Therefore it can take full advantage of the computationally very efficient modal superposition approach, based on linear modal analysis, as opposed to the computationally more demanding direct time integration methods [26]. Li [1] showed that the dynamic stress frequency domain data could be calculated by FRFs and FFT results of external loads, and the dynamic stress time history could be acquired by Inverse FFT (IFFT) of frequency domain data. Zhou [27] investigated the role of Stress Modal Analysis (SMA) in multi-axial random fatigue and proposed an improved approach to exploit Stress Mode Shapes (SMSs) in random fatigue with a multi-axial criterion. Mršnik [3] used the modal model and reduction technique to estimate the fatigue-damage intensity and linked the fatigue damage intensity with the dynamic properties of the system, then the fatigue damage was estimated directly through the properties of the modal model. The study offered a very direct and important relationship between the dynamic characteristics and the total damage. In terms of vibration test excitation research, Zheng [28] presented a new control method for multi-input multi-output stationary non-Gaussian random vibration test using time domain randomization, the advantages of the proposed methods were the high computational efficiency and simultaneous control of the time–frequency characteristics of response signals. For non-stationary vibration excitation, Lamb [29] studied a technique to generate realistic nonstationary dual track road elevation data which focuses on uncovering statistical distributions that describe the nonstationary relationships between the left and right wheel paths. Dai [30] developed a new method for explicitly representing and synthesizing non-Gaussian and non-stationary stochastic processes which was particularly well suited for stochastic finite element analysis of structures as well as for general purpose simulation of realizations of these processes.
Due to the stationarity of the input excitation is one of the fundamental assumptions required for frequency-domain fatigue damage theory [31]. However, for actual situation, the excitation is frequently non-stationary. Capponi [32] studied the fatigue life of structures under non-stationary input excitation by non-stationarity index, and pointed out if the non-stationarity is identified during the excitation, the resulting fatigue life is shown to significantly decrease than that of stationary excitation. Palmieri [31] studied the non-Gaussianity and non-stationarity in vibration fatigue and reach a conclusion that the fatigue life would be significantly impacted by non-stationarity. Therefore, when the traditional frequency domain vibration fatigue assessment method is used, the non-stationary characteristics in vibration fatigue need to be further studied.
However, it is noteworthy that most of the existing studies mainly focus on stationary processes, while the simulation of vibration fatigue for non-stationary processes is relatively rare, so the existing research will encounter problems in practical application. Therefore, the emphasis of this paper is the dynamic stress simulation method of vibration fatigue under non-stationary state. A time history simulation method for calculating the dynamic stress of structures in resonance state under non-stationary condition is presented. The concept of load-stress FRF under traditional stationary state is considered, and the load-stress IRF of structures is obtained by IFFT, and the Duhamel integral (Convolution integral) in dynamics is considered, which is used to calculate the dynamic stress response of the structure under arbitrary excitation. Based on the obtained dynamic stress time history, subsequent fatigue life assessment and structural optimization design can be carried out.
This manuscript is divided into the following sections. In Section 2, the theoretical background of vibration fatigue is shown and the method for determining the window width for calculating the non-stationarity factor is proposed. In Section 3, an improved non-stationary process approach for vibration fatigue dynamic stress calculation is proposed. Section 4 validates the dynamic stress simulation method through line test data. The application of stress simulation method is discussed in Section 5. And in final Section, the conclusions are drawn.
Section snippets
Non-stationarity factor
In this research, run tests are used to identify non-stationarity [32]. The run test is a non-parametric method based on the idea of dividing the signal to be analyzed in time windows, and for every window calculate the variation of one of the statistical properties with respect to the same property of the entire signal [33]. It is based on the definition of a run, as a sequence of identical observations followed and preceded by a different observation or no observation at all [34]. It means
Improved non-stationary process approach
The structural dynamic stress simulation method includes time domain simulation method and frequency domain simulation method. When describing non-stationary state vibration, the time domain simulation method has obvious advantages. For the load-stress FRF, as shown in Eq. (17), When only a certain mode and a single external load are considered, load-stress FRF could be written as the follow form [1]:where o means stress response at position
Experimental design
In order to verify the dynamic stress simulation method of vibration fatigue under the non-stationary process, the line test of a metro vehicle, which has encountered lifeguard failure, with a new lifeguard was carried out. Lifeguard is a sturdy metal bracket fixed in front of each of the leading wheels of a train to deflect small objects away from the wheels to prevent derailment, the installation diagram of a lifeguard was shown in Fig. 1(a). In this research, the lifeguard was fixed at the
Application
In this section, the stress simulation method given above is used for structural optimization. When solving the structural fatigue problem caused by resonance, the structural modal frequency is usually designed far away from the external excitation frequency to avoid resonance, thereby reducing the elastic vibration and reducing the stress level. Therefore, the structural modal design requirements are proposed, so that the structure can be redesigned to meet the design requirements. In this
Conclusion
In order to solve the problem of how to accurately simulate the vibration fatigue dynamic stress of structure under non-stationary state, this paper focuses on the simulation method. For the non-stationary process simulation, the time-domain method is chosen instead of the frequency-domain method commonly used in vibration fatigue simulation. The simulation formula of stress time history was deduced, the concrete calculation steps were provided. The core of simulation is to consider the
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.
CRediT authorship contribution statement
Fansong Li: Conceptualization, Methodology, Writing - review & editing. Hao Wu: Visualization, Investigation, Writing - original draft. Pingbo Wu: Supervision.
Acknowledgements
The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (Grant No. 11790282), Independent Project of State Key Laboratory of Traction Power (Grant No. 2020TPL-T09) and the Fundamental Research Funds for the Central Universities (Grant No. 2682019CX47).
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