Mathematical Problems in Engineering

Volume 2019, Article ID 1458149, 14 pages

https://doi.org/10.1155/2019/1458149

## Path Contribution Analysis of Vibration Transfer Path Systems

^{1}School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China^{2}School of Mechanical Engineering, Tianjin University, Tianjin 300350, China

Correspondence should be addressed to Wei Zhao; nc.ude.uen.liam@oahziew

Received 11 August 2018; Revised 20 November 2018; Accepted 18 December 2018; Published 8 January 2019

Academic Editor: Xiao-Qiao He

Copyright © 2019 Wei Zhao et al. 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

To analyze the vibration characteristics of mechanical systems from the aspect of vibration transfer, two types of models for vibration transfer path systems were designed, which comprised three subsystems: excitation sources, transfer paths, and receivers. One type of model represented that the systems only sustained unidirectional force excitation, where the transfer paths remained free from the influence of mass parameters. The other type indicated that the systems were subjected to simultaneous excitations of forces and moments, with transfer paths without mass parameter. Because the transfer characteristics of vibration paths directly determine the output response behaviours of the systems, the analysis of path contributions to the vibration responses of system receivers is important in systemic vibration and noise reduction. In order to evaluate the path contributions quantitatively, the concepts of path transfer ratio (TR) and path insertion loss (IL) were introduced and the convenient formulae were derived based on the path transfer force analysis and the path disconnected method in this work. Thereby the measurement of path contributions to the receiver vibration responses within the frequency domain can be accomplished. Through numerical examples, the ideal calculation results were obtained. These conclusions further indicate that the path TR and path IL can be applied as evaluation indicators of path contributions for the vibration transfer path systems.

#### 1. Introduction

Vibration and noise performance has long remained the main aspect for evaluating the quality of mechanical equipment. Particularly, with the steadily increasing social productivity, the performance of various mechanical equipment exhibits a trend towards high velocity, lightweight, and heavy duty features. Therefore, engineers are required to continuously address vibration control problems. In order to achieve the desired design requirements, the vibration control problem should be largely addressed from the following three aspects: the identification of vibration sources, the optimization of transfer paths, and the vibration elimination of receivers [1, 2]. The technique of vibration source identification is mainly based on the measurement of normal vibration velocity on the source surface and the near-field acoustic pressure measurement. The accuracy can be largely influenced by the environment factors, the sensor location, and the sensor amount [3, 4]. The vibration elimination of receiver is the passive control for the output vibration. The application of dynamic vibration absorber is most common, but it is limited to a certain frequency range. The nonlinear vibration absorber has the characteristics of wide frequency absorption, but because of its complex structure, it is difficult to be applied in practice [5, 6]. By comparison, vibration isolation and structural dynamic optimization remain an effective approach for vibration control. Moreover, the dynamic optimization technique could be guaranteed in the early stage of system design based on the specific analyses of transfer paths in a vibration system [7, 8].

For the problem of path transitivity, the current research includes the transfer path analysis (TPA) of test-based methodologies and statistical energy analysis. The TPA is more used to deal with the low-frequency problems [9, 10]. Moreover, for the purpose of operability and saving time in the experiments, this method usually departs from the traditional source-transfer-receiver model with the assumption of the loads with physical meanings, frequency response function, and the partial response [11, 12]. For example, D. de Klerk and A. Ossipov et al. provided the operational transfer path analysis (OTPA), whose simple and fast modeling is the biggest advantage [13, 14]. Ba-Leum Kim and Yoshida J et al. developed the modified transfer path analysis (MTPA) method to more accurately estimate the operational force of the main vibration source in a complicated system subjected to multiple vibration sources, base excitation, and several disturbances [15]. Therefore, the accuracy of the experimental results must be affected to a certain extent. Hou Lei and Zhang Jiming measured the response of each installation point and evaluation point and the excitation force. According to the frequency response function, the transfer path analysis was completed, and the contributions of equipment and pipeline were obtained [16]. However, since the direct measurement of the excitation force is almost impossible, the excitation force can only be obtained through the admittance characteristics of the installation system. Additionally, the ill-conditioned problem of matrix inversion may affect the accuracy of pipeline contribution to a certain extent. The statistical energy analysis can solve the dynamics problems in high-frequency domains for the complex systems. Due to the large workload, the practical application of this analytical method to the complex systems is very limited. Many researchers usually use the mobility power flow theory to solve the practical problems [17–19]. In fact, the power flow method is only applicable to the modeling and analysis of vibration system in the middle and low-frequency bands. Xiao Bin and Li Biao et al. established vibration energy model of double-layer vibration isolation system based on power flow method. The corresponding vibration transmission was studied combined with vibration isolation system test and modal test [20]. The two research methods mentioned above mainly depend on the support of various experimental methods and the research for the prediction of vibration characteristics is limited.

Generally, there are multiple vibration transfer paths in practical mechanical vibration systems. Each path not only has the effect for vibration transferring, but also affects the dynamic characteristics of the whole system. From these two aspects, this work applied two evaluation indicators to analyze path contributions in the vibration transfer path systems, namely, the path transfer ratio (TR) and insertion loss (IL). Based on the path transfer force analysis and the path disconnected method, the quantitative analysis method could be further employed to investigate the ranking problem with respect to path contributions in the frequency domain. Moreover, this method can be applied to a wider frequency range and could conveniently and effectively identify the critical path. Hence, this could truly solve the problem of low-vibration and low-noise design.

#### 2. Evaluation Indicators of Path Contribution

##### 2.1. Path Transfer Ratio

For the vibration transfer path systems, the transfer capacity of each path is different due to the differences in path structure and path parameters. In order to quantitatively evaluate the contribution of each path to the receiver vibration, we give the concept of path transfer ratio, that is, the ratio between the force to the receiver and the excitation force of the source.

In order to make the analysis process more concise and clear, we assume the vibration transfer path system is excited by the harmonic force and there is a fixed connection between a path and a receiver. According to the traditional theory of vibration mechanics analysis and the definition of path force [21], the force that is transferred to the receiver through the path iswhere* k* is spring stiffness and* c *is damping coefficient.

Based on the calculation method of force transmissibility in the vibration isolation systems [22], the calculation formula of the corresponding path TR is

It can be seen that the path TR is a simple and quick evaluation indicator to analyze the path contribution, which can be used as the theoretical prediction basis for the path transitivity.

##### 2.2. Insertion Loss

The concept of IL was initially applied in electronic systems, where it was defined as the ratio of the power that was transferred to the load with a certain component to the power transferred to the load without the certain component. It was normally quantified using the unit of decibel (dB), expressed as where v1 and v2 denote the power before and after an electronic component is connected to the system, respectively [23, 24].

In this work, this concept was introduced into the vibration transfer path system. When analyzing the path contribution, the path IL could be defined based on the ratio of the receiver responses prior to and after the connection of a certain transfer path.

Evidently, a high path IL indicated the high contribution of the path in the vibration transfer system. According to the varying responses of selected receiver, the ILs could be classified into displacement IL, velocity IL, and acceleration IL.

According to the definition, the calculation formula of velocity IL may be written aswhere denotes the velocity response of the receiver prior to the path connection, while represents that of the receiver after the path connection.

Similarly, (4) may be expressed aswhere is the mass of the receiver, denotes the energy of the receiver prior to path connection, and is the energy of the receiver after the path connection.

Hence, the formula for the calculation of energy IL could be expressed as follows:

When a receiver is characterized by a coupled rectilinear and swaying motion, to avoid the biased consideration of the path transfer in a single direction, energy IL is generally adopted to assess the importance of the paths.

#### 3. Vibration Transfer Path System Model

In engineering practices, many mechanical vibration systems are established with multiple vibration transfer paths. For example, with respect to a vehicle powertrain subsystem, the excitations generated by the operating engine are transferred to the vehicle body via the three-point suspension system; thereby the vibrations of the vehicle body are induced [25]. The function of the suspension subsystem in the powertrain system is to attenuate the vibration energy transmitted to the vehicle body, so it is a vibration isolation system. At the same time, this subsystem is also a dynamic vibration absorption system for the whole vehicle, which can alleviate the impact of road roughness to the car body. These require that the suspension system should have large stiffness and damping to prevent excessive displacement of power plant when subjected to low-frequency impact and low stiffness and small damping to ensure comfort when subjected to high-frequency vibration. Therefore, when designing suspension system, it is necessary to optimize the parameter matching of damping and stiffness in order to obtain the best vibration transmission performance. Path contribution analysis can find out the critical path, determine the direction for optimization design, and then improve work efficiency. These vibration transfer paths have been described without any mass parameter, so we can make use of the system model shown in Figure 1 to analyze the path contributions.