Statistical distribution modeling and two-step parameter identification of vehicle bridge displacement spectrum

Abstract: The study of the statistical distribution is the basis for further loading spectrum and fatigue reliability platform test. Normal distribution and weibull distribution are 2 kinds of probability statistical distribution models widely used in reliability engineering. The idea of weighted superposition is used to approximate the actual distribution by so-called mixed model, and it has a strong practical application value, so it has been paid increasing attention by many scholars. The introduction of mixed distribution model brings many challenges in model parameter identification. Finding a simple, efficient and accurate mixed distribution model parameter estimation method has become a focus in the field of reliability research. The traditional reliability model parameter identification methods include graphic method, nonlinear least square method, maximum likelihood estimation and bias estimation, and so on. The main disadvantages of these algorithms are as follows: (1) The calculation efficiency needs to be improved, and the traditional algorithms mostly rely on iterative solution. Requirement to improve the accuracy of parameter estimation distinct increases the time cost. (2) The selection of parameter identification and optimization targets is improper. Most of the existing studies have defined the objective function of parameter identification as the square sum of the model and the measured data, which inevitably ignores the simulation error of the transverse coordinates between the sample points and the simulation points, that only considers the simulation error of the ordinate. (3) The empirical dependence of parameter identification is high, and the initial value of parameter identification has a great influence on the results. However, the intelligent algorithm shows great potential in the problem of parameter identification of the model with multidimensional nonlinearity and uneasy initial value. In view of this, the measured vehicle bridge displacement signal was taken as the research object in this paper, the time domain analysis and frequency domain power spectrum analysis were carried out respectively. In order to further study the statistical law of the displacement signals, the signal was grouped and the frequency was counted, the statistical histogram and the cumulative probability distribution curve were obtained. The normal distribution, mixed normal distribution, weibull distribution and mixed weibull distribution were employed respectively. A novel two-step parameter identification method was proposed, and the grey correlation degree objective function was introduced. The grey correlation coefficient objective function could ensure the maximum geometric similarity between the fitting curve and the original curve. By doing this, the inherent malpractice of the optimization process with the square sum of error as the fitness was overcome to some extent. The proposed parameter estimation method's tep was as following: Firstly, the parameters obtained by the artificial fish swarm algorithm were applied as the initial values of the model parameters. Secondly, the iterative nonlinear least square method, namely, levenberg-marquardt (LM) algorithm was used to identify the parameters accurately. Thirdly, the goodness of fit for each model were calculated by using the kolmogorov-smirnov test index and correlation coefficient. The result showed that the mixed weibull model could be used to describe the tested displacement signal best. The correlation coefficient between the mixed Weibull distribution and the statistical histogram was (0.9800, 0.9908,0.9867,0.9665), whereas, the mixed normal distribution was (0.9793,0.9904,0.9783,0.9661), the weibull model was (0.8613,0.9113,0.8618,0.8854), and the normal model was (0.8611,0.9127,0.8624,0.8869). The proposed two-step parameter identification method combined the advantages of the artificial fish swarm optimization algorithm and the traditional iterative algorithm, and used the artificial fish swarm optimization result as the initial value of the LM algorithm. It solved the problem of the difficulty in selecting the initial value of the nonlinear least square method and improved the efficiency of the parameter identification. This study can provide reference for the fatigue load spectrum and the bench test of off-road vehicles.