Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6814547, 6 pages

https://doi.org/10.1155/2017/6814547

## Reliability Analysis of Random Vibration Transmission Path Systems

School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China

Correspondence should be addressed to Wei Zhao

Received 1 March 2017; Accepted 1 June 2017; Published 12 July 2017

Academic Editor: Yuri Vladimirovich Mikhlin

Copyright © 2017 Wei Zhao and Yi-Min Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The vibration transmission path systems are generally composed of the vibration source, the vibration transfer path, and the vibration receiving structure. The transfer path is the medium of the vibration transmission. Moreover, the randomness of transfer path influences the transfer reliability greatly. In this paper, based on the matrix calculus, the generalized second moment technique, and the stochastic finite element theory, the effective approach for the transfer reliability of vibration transfer path systems was provided. The transfer reliability of vibration transfer path system with uncertain path parameters including path mass and path stiffness was analyzed theoretically and computed numerically, and the correlated mathematical expressions were derived. Thus, it provides the theoretical foundation for the dynamic design of vibration systems in practical project, so that most random path parameters can be considered to solve the random problems for vibration transfer path systems, which can avoid the system resonance failure.

#### 1. Introduction

In the modern industry, the problem of vibration and noise has always been focused on widely by technical personnel. The suitable control method for the system vibration and noise is of great theoretical value and economic benefit. Generally speaking, the vibration system is composed of three parts, the vibration source, the vibration transfer path, and the vibration receiving structure. The transfer path is a specific medium, through which the vibration sources are transferred to the system receiving structure. In order to control the vibration and noise effectively, it is necessary to recognize and analyze the vibration transfer path accurately. Consequently, the prediction and dynamic design of the vibration transmission path system become especially important. At present, the analysis of the transfer path of the vibration system mainly concentrated in the experimental method and the energy transfer method [1–5]. Overall, the research on the prediction and the randomness of the parameters is limited.

It is generally known that the uncertain factors will appear inevitably in the vibration system. The random nature of the path parameters will change the system’s vibration transmission characteristics and ultimately affect the output of the system, which can bring out the system reliability problem. According to the actual working condition, the research on the reliability problems of vibration transmission path system can be made in two ways. On the one hand, it can be studied when the transfinite is considered as the measure indicator about the path transfer force or transfer rate. This kind of research is based on the system responses and depends on the random response analysis for the random systems. On the other hand, the study can be finished according to the structural failure coming from the resonance or potential resonance, which exists widely in the various structures with uncertainty factors. When the system is in a state of resonance, the dynamic stress is very large and the service life can be shortened greatly, which will influence the performance of mechanical equipment. In this paper, our works focus on the latter.

#### 2. Stochastic Finite Element Method for System Eigensolutions Analysis

According to Taylor series expansion theory [6–9], if the vector is the function of the vector , then the two-order Taylor expression of at iswhere is the second-order power. The symbol represents Kronecker power, which is defined as and then the corresponding matrix derivatives are defined as .

Because the path parameters vector of the vibration transmission path system is random, the eigenvalue and characteristic vector of the system are also random and they also can be expanded into Taylor series.

The eigenvalue problem of the stochastic parameter system can be expressed asBoth sides of (4) are expanded into two-order Taylor expression at the . We can obtain the following equations through comparing the same order of :

Equation (5) describes the problem with determined eigenvalue when the mean value of the structure parameters is taken. Thus, according to the conventional way, we can get the determined eigenvalue and characteristic vector , written as and . The first-order sensitivities and can be obtained by (4), written as and .

Since and both are real symmetric matrixes, (4) is multiplied by on the left; the following equation can be obtained:

By the regularization condition of the characteristic vector, the first-order sensitivity of the system eigenvalue to the mean of the random parameters can be obtained:

According to the Taylor series expansion and the second moment method, the mean value and variance of the system eigenvalue can be written as where is the variance matrix of the random parameters, including all of the variances and covariances.

By formula (8), the first-order sensitivity of the eigenvalue variances to the random parameter variances is

#### 3. System Transmission Reliability Analysis

According to the interference theory of the reliability, the state function of the resonance problem for the vibration transmission path system can be described aswhere represents the excitation frequency and represents the natural frequency.

Based on the state function determined by the excitation frequency and the natural frequency , the two states of the vibration transmission path system can be defined aswhere is a specific interval and is generally 5%–15% times of the natural frequency average.

Given , the mean value and variance of the function are, respectively,

The quasi failure probability of the resonance for the vibration transmission path system iswhere is the joint probability density of the basic random parameter vector .

The random parameters of the vibration transmission path systems shown in Figures 1 and 2 are subject to the normal distribution, which brings convenience for the system reliability analysis. When the excitation frequency and natural frequency are also subject to normal distribution and independent, the system failure index isThe system quasi failure probability iswhere is standard normal distribution function.