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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.

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 [

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.

According to Taylor series expansion theory [

Because the path parameters vector

The eigenvalue problem of the stochastic parameter system can be expressed as

Equation (

Since

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

By formula (

According to the interference theory of the reliability, the state function of the resonance problem for the vibration transmission path system can be described as

Based on the state function determined by the excitation frequency

Given

The quasi failure probability of the resonance for the vibration transmission path system is

The random parameters of the vibration transmission path systems shown in Figures

Vibration transfer path system model with vertical force excitation.

Vibration transfer path system model with vertical force excitation and moment excitation.

In the vibration transmission path system, when any one of the excitation frequencies is close to the natural frequency, the resonance will occur. The system is considered as the series system and the quasi failure probability of the whole system is

The transmission reliability is

In most vibration systems, there are several transfer paths between the vibration source and the receiving structure and the excitations are transmitted in the corresponding ways. For example, the car body vibration comes partly from the force and torque generated by the engine. In this system, the engine mount is transfer path. Based on the typical structure [

In the vibration systems, all the parameters in the random paths are represented by a vector

Considering the lumped mass in the transfer paths as shown in Figure

The mean values of the natural frequencies are

According to formula (

The transmission reliability with the exciting frequency.

Because the system shown in Figure ^{−3} kg^{2} and the receiver rotational inertia ^{−3} kg^{2}; the source suspension positions

The mean values of natural frequencies are

The change trend of transmission reliability with the excitation frequency is shown in Figure

The transmission reliability with exciting frequency.

It can be seen from the calculation results that the system has high reliability when the excitation frequency is far away from the natural frequency of the system. As the excitation frequency approaches the natural frequency, the reliability is gradually reduced until zero. In order to ensure the normal operation of the system, the related frequency area must be avoided in the actual project.

The examples illustrate that the dynamics analysis results of the system are ideal based on the theory of the transmission reliability proposed in this paper. The effective dynamic design and optimization can be carried out after determining the transmission reliability of the transfer path parameters. Particularly when the vibration and noise level do not meet the actual requirements, the engineers can quickly find out the key parameters and improve the design.

The authors declare that they have no conflicts of interest.

This study was supported by the National Natural Science Foundation of China (51305072) and the Basic Scientific Research Foundation of Central University in China (N120303001).