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Abstract

A two-stage series-parallel vibration isolation system is already widely used in various industrial fields. However, when the researchers analyze the vibration characteristics of a mechanical system, the system is usually regarded as a single-stage one composed of two substructures. The dynamic modeling of a two-stage series-parallel vibration isolation system using frequency response function–based substructuring method has not been studied. Therefore, this article presents the source-path-receiver model and the substructure property identification model of such a system. These two models make up the transfer path model of the system. And the model is programmed by MATLAB. To verify the proposed transfer path model, a finite element model simulating a vehicle system, which is a typical two-stage series-parallel vibration isolation system, is developed. The substructure frequency response functions and system level frequency response functions can be obtained by MSC Patran/Nastran and LMS Virtual.lab based on the finite element model. Next, the system level frequency response functions are substituted into the transfer path model to predict the substructural frequency response functions and the system response of the coupled structure can then be further calculated. By comparing the predicted results and exact value, the model proves to be correct. Finally, the random noise is introduced into several relevant system level frequency response functions for error sensitivity analysis. The system level frequency response functions that are most sensitive to the random error are found. Since a two-stage series-parallel system has not been well studied, the proposed transfer path model improves the dynamic theory of the multi-stage vibration isolation system. Moreover, the validation process of the model here actually provides an example for acoustic and vibration transfer path analysis based on the proposed model. And it is worth noting that the proposed model can be widely applicable in mechanical systems.

Keywords

Substructuring technique source-path-receiver model substructure property identification model error sensitivity analysis

Introduction

Nowadays, with the development of the modern industry, when all kinds of production machinery and transportation machinery are toward high speed and high efficiency, more and more emphasis is laid on the noise and vibration control of them at the same time. A large amount of effort has been invested into the improvement of the technology of noise and vibration reduction and containment.¹

A two-stage vibration isolation system composed of three substructures is now widely used in various industrial fields. There are two kinds of two-stage isolation systems. One is defined as a two-stage series vibration isolation system (shown in Figure 1) and the other one is defined as a two-stage series-parallel vibration isolation system (shown in Figure 2). In a two-stage series system, substructure C as the vibration source is mounted on the substructure B and through it connected to substructure A (receiver). In a two-stage series-parallel system, substructure C is connected to both substructures A and B. And this article will focus on the series-parallel vibration isolation system.

Figure 1.

Two-stage series isolation system.

Figure 2.

Two-stage series-parallel isolation system.

Actually, at present, when the researchers analyze the vibration transfer characteristics of a system, the lumped mass method is usually applied. However, in a lumped mass model, the stiffness of the springs is a definite value, which is greatly different from an actual complex structure. Therefore, in this article, the dynamic modeling of a two-stage series-parallel vibration isolation system is finished using frequency response function (FRF)-based substructuring method. The FRF-based substructuring method has already been studied by some researchers.

In the end of 1980s, the FRF-based substructure synthesis method began to be studied. Jetmundsen et al.² set up the equations for classic substructure method. Van der Valk³ summed up the current difficulties of substructure method in testing part. And based on a benchmark model, a further study of the difficulties was made. Through an analytical model, Nicgorski and Avitabile⁴ discussed the problems with the system response prediction in implementation with actual test data. The numerical simulation results showed that the low-frequency information of a system is very sensitive to noise on the data. Nicgorski and Avitabile⁵ developed a method of conditioning test data for use with frequency based substurcturing (FBS). The method is referred to as variability improvement of key inaccurate node groups (VIKING). Through experiments, the method was proved to be effective. Lee and Hwang⁶ suggested a designed sensitivity formulation using a multi-domain FRF-based substructuring method for complex systems. The designed sensitivity formulation is applied to an engine mount system of a passenger car. Zhen⁷ and Lei⁸ developed a spectral-based substructuring technique and a FRF-based inverse substructuring theory for analyzing vehicle system NVH (noise, vibration, and harshness) response. A Ford car was regarded as a two-substructure vehicle system and the inverse substructuring technique was applied to analyze its vibration properties. Wang and Chuang⁹ proposed a new method to identify the structural joint parameters directly from the FRFs of the substructures and the whole structure. P Sjövall and T Abrahamsson¹⁰ identified the substructure systems from coupled system test data. The method was based on reconstruction of the interface forces acting between the unknown subsystem and its neighbor. Pradhan and Modak¹¹ posed the issue of identification of damping matrix of a structure as a finite element (FE) damping matrix updating problem. They developed an updating formulation that seek to separate updating of the damping matrix from that of updating of the stiffness and the mass matrix and found that the proposed method was effective in the accurate identification of the damping matrix in cases of complete, incomplete, and noisy data and was not limited by the level of damping in the structure. Mayes et al.¹² presented two new metrics that could be used to determine the cause of indefinite mass or stiffness matrices after substructure uncoupling. ZW Wang et al.¹³ applied the inverse substructuring method in the investigation on dynamic characteristics of a product transport system. ZW Wang and J Wang¹⁴ applied the inverse substructuring approach in a three-substructure system dynamic analysis. Law and Yong¹⁵ presented two substructural identification methods with the structure divided into substructures and with one substructure assessed at one time. The two methods can be used to identify external excitations iteratively in a substructure with time domain information. Both methods were validated with a truss structure. Čelič and Boltežar¹⁶ presented a method for establishing a theoretical model of a joint from the substructures and assembly FRF data. Voormeeren and Rixen¹⁷ found that the decoupling (or identification) of a substructure from an assembled system arisen when substructures could not be measured separately but only when coupled to neighboring substructures. W D’Ambrogio and A Fregolent^18,19 investigated the inverse dynamic substructuring problem, or decoupling problem. The direct decoupling techniques based on a hybrid assembly approach were emphatically considered. And a well-known issue in experimental dynamic substructuring relating to rotational degrees of freedom (DOFs) was settled. It was verified that rotational DOFs are not essential in the decoupling problem. SN Voormeeren et al.²⁰ derived an uncertainty propagation method to investigate the problem of small random errors on substructure measurements in experimental dynamic substructuring using FRFs.

According to the present literatures, when inverse substructuring method is applied to analyze the system vibration, the system is usually regarded as a single-stage vibration isolation system instead of a two-stage one. Although the inverse substructuring method for coupled two-stage product-transport-system has already been studied, but the method is only suitable for the single coordinate coupling system. Moreover, researchers have only validated the method based on a simple lumped mass model. In fact, there are great differences between the model and the actual structure. And the dynamic modeling of a two-stage series-parallel vibration isolation system has not been studied. However, this structure is already widely used in different kinds of mechanical system. Hence, it is necessary to make a research on it in this article.

Here, the source-path-receiver model and substructure property identification model of a two-stage series-parallel isolation system are developed using FRF-based substructuring method. And the model is programmed by MATLAB. To verify these two models, a finite element model (FEM) simulating a vehicle system, which is a typical two-stage series-parallel vibration isolation system, is designed. The structural FRFs and system level FRFs can be obtained by MSC Patran/Nastran and LMS Virtual.lab based on the FEM. The FRFs of substructures and the joint stiffness can be predicted by the system level FRFs through the substructure property identification model; the system response of coupled structure can be predicted by the calculated FRFs of substructures and joint stiffness through the source-path-receiver model. By comparing the predicted results and exact values, these two models will be validated. Furthermore, the random noise is introduced into the relevant system level FRFs for error sensitivity analysis and then its influence on the substructure FRFs is quantified by comparison.

Modeling of coupled series-parallel two-stage isolation system

According to the FRF-based substructuring method, a system can be divided into several independent substructures, which are coupled by springs and damping elements. Each substructure can be represented by its FRF matrix corresponding to specific coordinates.⁸ A two-stage series-parallel system is partitioned into three substructures as shown in Figure 2. The coupling mounts between the three substructures are modeled as scalar springs.

Source-path-receiver model

A substructure M is shown in Figure 3. The relationship between the excitation and the response can be represented by a transfer function matrix $[H_{M}]$ and is expressed as follows

{X_{M}} = [H_{M}] {F_{M}}

(1)

where ${F_{M}}$ is the external excitation vector, ${X_{M}}$ is the response vector, and $[H_{M}]$ is the transfer function matrix between the external excitation vector and the response vector. For a multiple DOFs substructure, the coordinates of interest can be divided into a set of coupling DOFs (denoted by subscript c), a set of excitation coordinates (subscript i), and a set of response coordinates (subscript o).⁸

Figure 3.

General substructure representation.

Accordingly, equation (1) can be expanded as follows

{\begin{matrix} {X}_{o (m)} \\ {X}_{c (m)} \end{matrix}} = [\begin{matrix} {[H_{M}]}_{o (m) i (m)} & {[H_{M}]}_{o (m) c (m)} \\ {[H_{M}]}_{c (m) i (m)} & {[H_{M}]}_{c (m) c (m)} \end{matrix}] {\begin{matrix} {F}_{i (m)} \\ {F}_{c (m)} \end{matrix}}

(2)

As shown in Figure 4, the two-stage series-parallel isolation system S is composed of substructures A, B, and C. Substructures A and B are connected via a set of flexible springs $[K_{c_{1}}]$ . Substructures B and C are connected via a set of flexible springs $[K_{c_{2}}]$ . Substructures A and C are connected via a set of flexible springs $[K_{c_{3}}]$ . To simplify the model substructures A and B are replaced by a new structure D. Then, the system S composed of subsystem D and substructure C can be regarded as a new coupled one. The coupling coordinates of substructures D and C incorporate c₂ and c₃ and here are defined as c₂₃. According to the formulation of the direct FRF substructuring,⁸ the FRFs of the system S with two substructures can be expressed as follows and the detailed derivation process is shown in Appendix 2

\begin{matrix} [\begin{matrix} {[H_{S}]}_{o (d) i (d)} & {[H_{S}]}_{o (d) c_{23} (d)} & {[H_{S}]}_{o (d) c_{23} (c)} & {[H_{S}]}_{o (d) i (c)} \\ {[H_{S}]}_{c_{23} (d) i (d)} & {[H_{S}]}_{c_{23} (d) c_{23} (d)} & {[H_{S}]}_{c_{23} (d) c_{23} (c)} & {[H_{S}]}_{c_{23} (d) i (c)} \\ {[H_{S}]}_{c_{23} (c) i (d)} & {[H_{S}]}_{c_{23} (c) c_{23} (d)} & {[H_{S}]}_{c_{23} (c) c_{23} (c)} & {[H_{S}]}_{c_{23} (d) i (c)} \\ {[H_{S}]}_{o (c) i (d)} & {[H_{S}]}_{o (c) c_{23} (d)} & {[H_{S}]}_{o (c) c_{23} (c)} & {[H_{S}]}_{o (c) i (c)} \end{matrix}] \\ = [\begin{matrix} {[H_{D}]}_{o (d) i (d)} & {[H_{D}]}_{o (d) c_{23} (d)} & 0 & 0 \\ {[H_{D}]}_{c_{23} (d) i (d)} & {[H_{D}]}_{c_{23} (d) c_{23} (d)} & 0 & 0 \\ 0 & 0 & {[H_{C}]}_{c_{23} (c) c_{23} (c)} & {[H_{C}]}_{c_{23} (d) i (c)} \\ 0 & 0 & {[H_{C}]}_{o (c) c_{23} (c)} & {[H_{C}]}_{o (c) i (c)} \end{matrix}] \\ - [\begin{matrix} [H_{D}]_{o (d) c_{23} (d)} \\ [H_{D}]_{c_{23} (d) c_{23} (d)} \\ - [H_{C}]_{c_{23} (c) c_{23} (c)} \\ - [H_{C}]_{o (c) c_{23} (c)} \end{matrix}] {({[H_{D}]}_{c_{23} (d) c_{23} (d)} + {[H_{C}]}_{c_{23} (c) c_{23} (c)} + {[K_{c_{23}}]}^{- 1})}^{- 1} \\ • [\begin{matrix} {[H_{D}]}_{c_{23} (d) i (d)} & {[H_{D}]}_{c_{23} (d) c_{23} (d)} & - {[H_{C}]}_{c_{23} (c) c_{23} (c)} & - {[H_{C}]}_{c_{23} (c) i (c)} \end{matrix}] \end{matrix}

(3)

Figure 4.

A two-stage series-parallel isolation system.

It can be seen from Figure 4 that excitation at excitation coordinate of subsystem $D (i (d))$ includes excitation at coordinates $i (a)$ , $c_{1} (a)$ , and $c_{1} (b)$ ; response at coordinate $o (d)$ includes response at $o (a)$ , $c_{1} (b)$ , and $c_{1} (a)$ ; excitation and response at coordinate $c_{23} (d)$ consists of coordinates $c_{2} (b)$ and $c_{3} (a)$ ; and excitation and response at coordinate $c_{23} (c)$ is composed of coordinates $c_{2} (c)$ and $c_{3} (c)$ . Based on the statement above, equation (3) can be expanded as follows

\begin{array}{l} [\begin{matrix} {[H_{S}]}_{o (a) i (a)} & {[H_{S}]}_{o (a) c_{1} (a)} & {[H_{S}]}_{o (a) c_{1} (b)} & {[H_{S}]}_{o (a) c_{2} (b)} & {[H_{S}]}_{o (a) c_{3} (a)} & {[H_{S}]}_{o (a) c_{2} (c)} & {[H_{S}]}_{o (a) c_{3} (c)} & {[H_{S}]}_{o (a) i (c)} \\ {[H_{S}]}_{c_{1} (a) i (a)} & {[H_{S}]}_{c_{1} (a) c_{1} (a)} & {[H_{S}]}_{c_{1} (a) c_{1} (b)} & {[H_{S}]}_{c_{1} (a) c_{2} (b)} & {[H_{S}]}_{c_{1} (a) c_{3} (a)} & {[H_{S}]}_{c_{1} (a) c_{2} (c)} & {[H_{S}]}_{c_{1} (a) c_{3} (c)} & {[H_{S}]}_{c_{1} (a) i (c)} \\ {[H_{S}]}_{c_{1} (b) i (a)} & {[H_{S}]}_{c_{1} (b) c_{1} (a)} & {[H_{S}]}_{c_{1} (b) c_{1} (b)} & {[H_{S}]}_{c_{1} (b) c_{2} (b)} & {[H_{S}]}_{c_{1} (b) c_{3} (a)} & {[H_{S}]}_{c_{1} (b) c_{2} (c)} & {[H_{S}]}_{c_{1} (b) c_{3} (c)} & {[H_{S}]}_{c_{1} (b) i (c)} \\ {[H_{S}]}_{c_{2} (b) i (a)} & {[H_{S}]}_{c_{2} (b) c_{1} (a)} & {[H_{S}]}_{c_{2} (b) c_{1} (b)} & {[H_{S}]}_{c_{2} (b) c_{2} (b)} & {[H_{S}]}_{c_{2} (b) c_{3} (a)} & {[H_{S}]}_{c_{2} (b) c_{2} (c)} & {[H_{S}]}_{c_{2} (b) c_{3} (c)} & {[H_{S}]}_{c_{2} (b) i (c)} \\ {[H_{S}]}_{c_{3} (a) i (a)} & {[H_{S}]}_{c_{3} (a) c_{1} (a)} & {[H_{S}]}_{c_{3} (a) c_{1} (b)} & {[H_{S}]}_{c_{3} (a) c_{2} (b)} & {[H_{S}]}_{c_{3} (a) c_{3} (a)} & {[H_{S}]}_{c_{3} (a) c_{2} (c)} & {[H_{S}]}_{c_{3} (a) c_{3} (c)} & {[H_{S}]}_{c_{3} (a) i (c)} \\ {[H_{S}]}_{c_{2} (c) i (a)} & {[H_{S}]}_{c_{2} (c) c_{1} (a)} & {[H_{S}]}_{c_{2} (c) c_{1} (b)} & {[H_{S}]}_{c_{2} (c) c_{2} (b)} & {[H_{S}]}_{c_{2} (c) c_{3} (a)} & {[H_{S}]}_{c_{2} (c) c_{2} (c)} & {[H_{S}]}_{c_{2} (c) c_{3} (c)} & {[H_{S}]}_{c_{2} (c) i (c)} \\ {[H_{S}]}_{c_{3} (c) i (a)} & {[H_{S}]}_{c_{3} (c) c_{1} (a)} & {[H_{S}]}_{c_{3} (c) c_{1} (b)} & {[H_{S}]}_{c_{3} (c) c_{2} (b)} & {[H_{S}]}_{c_{3} (c) c_{3} (a)} & {[H_{S}]}_{c_{3} (c) c_{2} (c)} & {[H_{S}]}_{c_{3} (c) c_{3} (c)} & {[H_{S}]}_{c_{3} (c) i (c)} \\ {[H_{S}]}_{o (c) i (a)} & {[H_{S}]}_{o (c) c_{1} (a)} & {[H_{S}]}_{o (c) c_{1} (b)} & {[H_{S}]}_{o (c) c_{2} (b)} & {[H_{S}]}_{o (c) c_{3} (a)} & {[H_{S}]}_{o (c) c_{2} (c)} & {[H_{S}]}_{o (c) c_{3} (c)} & {[H_{S}]}_{o (c) i (c)} \end{matrix}] \\ = [\begin{matrix} {[H_{D}]}_{o (a) i (a)} & {[H_{D}]}_{o (a) c_{1} (a)} & {[H_{D}]}_{o (a) c_{1} (b)} & {[H_{D}]}_{o (a) c_{2} (b)} & {[H_{D}]}_{o (a) c_{3} (a)} & 0 & 0 & 0 \\ {[H_{D}]}_{c_{1} (a) i (a)} & {[H_{D}]}_{c_{1} (a) c_{1} (a)} & {[H_{D}]}_{c_{1} (a) c_{1} (b)} & {[H_{D}]}_{c_{1} (a) c_{2} (b)} & {[H_{D}]}_{c_{1} (a) c_{3} (a)} & 0 & 0 & 0 \\ {[H_{D}]}_{c_{1} (b) i (a)} & {[H_{D}]}_{c_{1} (b) c_{1} (a)} & {[H_{D}]}_{c_{1} (b) c_{1} (b)} & {[H_{D}]}_{c_{1} (b) c_{2} (b)} & {[H_{D}]}_{c_{1} (b) c_{3} (a)} & 0 & 0 & 0 \\ {[H_{D}]}_{c_{2} (b) i (a)} & {[H_{D}]}_{c_{2} (b) c_{1} (a)} & {[H_{D}]}_{c_{2} (b) c_{1} (b)} & {[H_{D}]}_{c_{2} (b) c_{2} (b)} & {[H_{D}]}_{c_{2} (b) c_{3} (a)} & 0 & 0 & 0 \\ {[H_{D}]}_{c_{3} (a) i (a)} & {[H_{D}]}_{c_{3} (a) c_{1} (a)} & {[H_{D}]}_{c_{3} (a) c_{1} (b)} & {[H_{D}]}_{c_{3} (a) c_{2} (b)} & {[H_{D}]}_{c_{3} (a) c_{3} (a)} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {[H_{C}]}_{c_{2} (c) c_{2} (c)} & {[H_{C}]}_{c_{2} (c) c_{3} (c)} & {[H_{C}]}_{c_{2} (c) i (c)} \\ 0 & 0 & 0 & 0 & 0 & {[H_{C}]}_{c_{3} (c) c_{2} (c)} & {[H_{C}]}_{c_{3} (c) c_{3} (c)} & {[H_{C}]}_{c_{3} (c) i (c)} \\ 0 & 0 & 0 & 0 & 0 & {[H_{C}]}_{o (c) c_{2} (c)} & {[H_{C}]}_{o (c) c_{3} (c)} & {[H_{C}]}_{o (c) i (c)} \end{matrix}] \\ - [\begin{matrix} {[H_{D}]}_{o (a) c_{2} (b)} & {[H_{D}]}_{o (a) c_{3} (a)} \\ {[H_{D}]}_{c_{1} (a) c_{2} (b)} & {[H_{D}]}_{c_{1} (a) c_{3} (a)} \\ {[H_{D}]}_{c_{1} (b) c_{2} (b)} & {[H_{D}]}_{c_{1} (b) c_{3} (a)} \\ {[H_{D}]}_{c_{2} (b) c_{2} (b)} & {[H_{D}]}_{c_{2} (b) c_{3} (a)} \\ {[H_{D}]}_{c_{3} (a) c_{2} (b)} & {[H_{D}]}_{c_{3} (a) c_{3} (a)} \\ - {[H_{C}]}_{c_{2} (c) c_{2} (c)} & - {[H_{C}]}_{c_{2} (c) c_{3} (c)} \\ - {[H_{C}]}_{c_{3} (c) c_{2} (c)} & - {[H_{C}]}_{c_{3} (c) c_{3} (c)} \\ - {[H_{C}]}_{o (c) c_{2} (c)} & - {[H_{C}]}_{o (c) c_{3} (c)} \end{matrix}] {([\begin{matrix} {[H_{D}]}_{c_{2} (b) c_{2} (b)} & {[H_{D}]}_{c_{2} (b) c_{3} (a)} \\ {[H_{D}]}_{c_{3} (a) c_{2} (b)} & {[H_{D}]}_{c_{3} (a) c_{3} (a)} \end{matrix}] + [\begin{matrix} {[H_{D}]}_{c_{2} (c) c_{2} (c)} & {[H_{D}]}_{c_{2} (c) c_{3} (c)} \\ {[H_{D}]}_{c_{3} (c) c_{2} (c)} & {[H_{D}]}_{c_{3} (c) c_{3} (c)} \end{matrix}] + {[\begin{matrix} [K_{c_{2}}] & [K_{c_{2} c_{3}}] \\ [K_{c_{3} c_{2}}] & [K_{c_{3}}] \end{matrix}]}^{- 1})}^{- 1} \\ \cdot [\begin{matrix} {[H_{D}]}_{c_{2} (b) i (a)} & {[H_{D}]}_{c_{2} (b) c_{1} (a)} & {[H_{D}]}_{c_{2} (b) c_{1} (b)} & {[H_{D}]}_{c_{2} (b) c_{2} (b)} & {[H_{D}]}_{c_{2} (b) c_{3} (a)} & - {[H_{C}]}_{c_{2} (c) c_{2} (c)} & - {[H_{C}]}_{c_{2} (c) c_{3} (c)} & - {[H_{C}]}_{c_{2} (c) i (c)} \\ {[H_{D}]}_{c_{3} (a) i (a)} & {[H_{D}]}_{c_{3} (a) c_{1} (a)} & {[H_{D}]}_{c_{3} (a) c_{1} (b)} & {[H_{D}]}_{c_{3} (a) c_{2} (b)} & {[H_{D}]}_{c_{3} (a) c_{3} (a)} & - {[H_{C}]}_{c_{3} (c) c_{2} (c)} & - {[H_{C}]}_{c_{3} (c) c_{3} (c)} & - {[H_{C}]}_{c_{3} (c) i (c)} \end{matrix}] \end{array}

(4)

According to equation (4), the system response $[H_{S}]_{o (a) i (c)}$ representing the response at o(a) coordinate of substructure A due to an excitation at i(c) coordinate of substructure C can be expressed as follows

\begin{matrix} [H_{S}]_{o (a) i (c)} = [[H_{D}]_{o (a) c_{2} (b)} {[H_{D}]}_{o (a) c_{3} (a)}] \\ • {([\begin{matrix} {[H_{D}]}_{c_{2} (b) c_{2} (b)} & {[H_{D}]}_{c_{2} (b) c_{3} (a)} \\ {[H_{D}]}_{c_{3} (a) c_{2} (b)} & {[H_{D}]}_{c_{3} (a) c_{3} (a)} \end{matrix}] + [\begin{matrix} {[H_{C}]}_{c_{2} (c) c_{2} (c)} & {[H_{c}]}_{c_{2} (c) c_{3} (c)} \\ {[H_{c}]}_{c_{3} (c) c_{2} (c)} & {[H_{C}]}_{c_{3} (c) c_{3} (c)} \end{matrix}] + {[\begin{matrix} [K_{c 2}] & [K_{c 2 c 3}] \\ [K_{c 3 c 2}] & [K_{c 3}] \end{matrix}]}^{- 1})}^{- 1} [\begin{matrix} {[H_{C}]}_{c_{2} (c) i (c)} \\ {[H_{C}]}_{c_{3} (c) i (c)} \end{matrix}] \end{matrix}

(5)

Since the structure D consists of substructures A and B, the FRFs of structure D can be expressed by the FRFs of substructures A and B using the same derivation method described in Appendix 2

\begin{matrix} [\begin{matrix} {[H_{D}]}_{c_{3} (a) c_{3} (a)} & {[H_{D}]}_{c_{3} (a) i (a)} & {[H_{D}]}_{c_{3} (a) c_{1} (a)} & {[H_{D}]}_{c_{3} (a) c_{1} (b)} & {[H_{D}]}_{c_{3} (a) c_{2} (b)} \\ {[H_{D}]}_{o (a) c_{3} (a)} & {[H_{D}]}_{o (a) i (a)} & {[H_{D}]}_{o (a) c_{1} (a)} & {[H_{D}]}_{o (a) c_{1} (b)} & {[H_{D}]}_{o (a) c_{2} (b)} \\ {[H_{D}]}_{c_{1} (a) c_{3} (a)} & {[H_{D}]}_{c_{1} (a) i (a)} & {[H_{D}]}_{c_{1} (a) c_{1} (a)} & {[H_{D}]}_{c_{1} (a) c_{1} (b)} & {[H_{D}]}_{c_{1} (a) c_{2} (b)} \\ {[H_{D}]}_{c_{1} (b) c_{3} (a)} & {[H_{D}]}_{c_{1} (b) i (a)} & {[H_{D}]}_{c_{1} (b) c_{1} (a)} & {[H_{D}]}_{c_{1} (b) c_{1} (b)} & {[H_{D}]}_{c_{1} (b) c_{2} (b)} \\ {[H_{D}]}_{c_{2} (b) c_{3} (a)} & {[H_{D}]}_{c_{2} (b) i (a)} & {[H_{D}]}_{c_{2} (b) c_{1} (a)} & {[H_{D}]}_{c_{2} (b) c_{1} (b)} & {[H_{D}]}_{c_{2} (b) c_{2} (b)} \end{matrix}] \\ = [\begin{matrix} {[H_{A}]}_{c_{3} (a) c_{3} (a)} & {[H_{A}]}_{c_{3} (a) i (a)} & {[H_{A}]}_{c_{3} (a) c_{1} (a)} & 0 & 0 \\ {[H_{A}]}_{o (a) c_{3} (a)} & {[H_{A}]}_{o (a) i (a)} & {[H_{A}]}_{o (a) c_{1} (a)} & 0 & 0 \\ {[H_{A}]}_{c_{1} (a) c_{3} (a)} & {[H_{A}]}_{c_{1} (a) i (a)} & {[H_{A}]}_{c_{1} (a) c_{1} (a)} & 0 & 0 \\ 0 & 0 & 0 & {[H_{B}]}_{c_{1} (b) c_{1} (b)} & {[H_{B}]}_{c_{1} (b) c_{2} (b)} \\ 0 & 0 & 0 & {[H_{B}]}_{c_{2} (b) c_{1} (b)} & {[H_{B}]}_{c_{2} (b) c_{2} (b)} \end{matrix}] \\ - [\begin{matrix} {[H_{A}]}_{c_{3} (a) c_{1} (a)} \\ {[H_{A}]}_{o (a) c_{1} (a)} \\ {[H_{A}]}_{c_{2} (a) c_{1} (a)} \\ - {[H_{B}]}_{c_{1} (b) c_{1} (b)} \\ - {[H_{B}]}_{c_{2} (b) c_{1} (b)} \end{matrix}] {({[H_{A}]}_{c_{1} (a) c_{1} (a)} + {[H_{B}]}_{c_{1} (b) c_{1} (b)} + {[K_{c_{1}}]}^{- 1})}^{- 1} \\ • [\begin{matrix} {[H_{A}]}_{c_{1} (a) c_{3} (a)} & {[H_{A}]}_{c_{1} (a) i (a)} & {[H_{A}]}_{c_{1} (a) c_{1} (a)} & - {[H_{B}]}_{c_{1} (b) c_{1} (b)} & - {[H_{B}]}_{c_{1} (b) c_{2} (b)} \end{matrix}] \end{matrix}

(6)

[C_{1}] = ([H_{A}]_{c_{1} (a) c_{1} (a)} + [H_{B}]_{c_{1} (b) c_{1} (b)} + [K_{c_{1}}]^{- 1})^{- 1}

(7)

In accordance with the equations (6) and (7), the FRFs of the structure D can be expressed as follows

[H_{D}]_{o (a) c_{2} (b)} = [H_{A}]_{o (a) c_{1} (a)} [C_{1}] [H_{B}]_{c_{1} (b) c_{2} (b)}

(8)

[H_{D}]_{o (a) c_{3} (a)} = [H_{A}]_{o (a) c_{3} (a)} - [H_{A}]_{o (a) c_{1} (a)} [C_{1}] [H_{B}]_{c_{1} (a) c_{3} (a)}

(9)

[H_{D}]_{c_{2} (b) c_{3} (b)} = [H_{B}]_{c_{2} (b) c_{2} (b)} - [H_{B}]_{c_{2} (b) c_{1} (b)} [C_{1}] [H_{B}]_{c_{1} (b) c_{2} (b)}

(10)

[H_{D}]_{c_{2} (b) c_{3} (a)} = [H_{B}]_{c_{2} (b) c_{1} (b)} [C_{1}] [H_{A}]_{c_{1} (a) c_{3} (a)}

(11)

[H_{D}]_{c_{3} (a) c_{2} (b)} = [H_{A}]_{c_{3} (a) c_{1} (a)} [C_{1}] [H_{B}]_{c_{1} (b) c_{2} (b)}

(12)

[H_{D}]_{c_{3} (a) c_{3} (a)} = [H_{A}]_{c_{3} (a) c_{3} (a)} - [H_{A}]_{c_{3} (a) c_{1} (a)} [C_{1}] [H_{A}]_{c_{1} (a) c_{3} (a)}

(13)

From the above derivation, the source-path-receiver model of a two-stage series-parallel vibration isolation system is developed. The system level FRFs of coupled structure can be predicted by the calculated FRFs of substructures and coupling stiffness in free state through the source-path-receiver model.

Substructure property identification model

In order to obtain the FRFs of substructures in free state, the coupled structure needs to be decoupled, which usually costs quite a lot of time. Therefore, it is better to identify substructure property in the coupling condition. In this section, the FRFs of substructures and coupling stiffness in free state of the two-stage series-parallel vibration isolation system will be predicted through the coupled system level FRFs.

$[K_{c_{1}}]$ , $[K_{c_{3}}]$ , and substructure A

To calculate the FRFs of the substructure A in free state, the coupling stiffness $[K_{c_{1}}]$ (between substructures A and B) and $[K_{c_{3}}]$ (between substructures A and C), substructures B and C are treated as a new substructure E. The coupling coordinates of substructures A and E incorporate c₁ and c₃ and here are defined as c₁₃. According to the inverse substructuring method of a two-substructure system,⁸ the coupling stiffness $[K_{c_{13}}]$ and FRFs of substructure A can be expressed as follows and the detailed derivation process is shown in Appendix 3

[K_{c_{13}}] = ({[H_{S}]}_{c_{13} (a) c_{13} (a)} [H_{S}]_{c_{13} (e) c_{13} (a)}^{- 1} [H_{S}]_{c_{13} (e) c_{13} (e)} {- {[H_{S}]}_{c_{13} (a) c_{13} (e)})}^{- 1}

(14)

[H_{A}]_{c_{13} (a) c_{13} (a)} = {[H_{S}]}_{c_{13} (a) c_{13} (e)} ([H_{S}]_{c_{13} (e) c_{13} (e)} - [H_{S}]_{c_{13} (a) c_{13} (e)})^{- 1} [K_{c_{13}}]^{- 1}

(15)

[H_{A}]_{o (a) c_{13} (a)} = [H_{S}]_{o (a) c_{13} (a)} [H_{S}]_{c_{13} (a) c_{13} (a)}^{- 1} [H_{A}]_{c_{13} (a) c_{13} (a)}

(16)

Since the coordinate $c_{13} (a)$ includes coordinates $c_{1} (a)$ , $c_{3} (a)$ and coordinate $c_{13} (e)$ is composed of coordinates $c_{1} (b)$ and $c_{3} (c)$ , equations (14)–(16) can be expanded as follows

\begin{array}{l} [\begin{matrix} [K_{c_{1}}] & [K_{c_{1} c_{3}}] \\ [K_{c_{3} c_{1}}] & [K_{c_{3}}] \end{matrix}] = ([\begin{matrix} {[H_{S}]}_{c_{1} (a) c_{1} (a)} & {[H_{S}]}_{c_{1} (a) c_{3} (a)} \\ {[H_{S}]}_{c_{3} (a) c_{1} (a)} & {[H_{S}]}_{c_{3} (a) c_{3} (a)} \end{matrix}] \\ \cdot {[\begin{matrix} {[H_{S}]}_{c_{1} (b) c_{1} (a)} & {[H_{S}]}_{c_{1} (b) c_{3} (a)} \\ {[H_{S}]}_{c_{3} (c) c_{1} (a)} & {[H_{S}]}_{c_{3} (c) c_{3} (a)} \end{matrix}]}^{- 1} [\begin{matrix} {[H_{S}]}_{c_{1} (b) c_{1} (b)} & {[H_{S}]}_{c_{1} (b) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (c) c_{1} (b)} & {[H_{S}]}_{c_{3} (c) c_{3} (c)} \end{matrix}] \\ {- [\begin{matrix} {[H_{S}]}_{c_{1} (a) c_{1} (b)} & {[H_{S}]}_{c_{1} (a) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (a) c_{1} (b)} & {[H_{S}]}_{c_{3} (a) c_{3} (c)} \end{matrix}])}^{- 1} \end{array}

(17)

\begin{matrix} [\begin{matrix} {[H_{A}]}_{c_{1} (a) c_{1} (a)} & {[H_{A}]}_{c_{1} (a) c_{3} (a)} \\ {[H_{A}]}_{c_{3} (a) c_{1} (a)} & {[H_{A}]}_{c_{3} (a) c_{3} (a)} \end{matrix}] = [\begin{matrix} {[H_{S}]}_{c_{1} (a) c_{1} (b)} & {[H_{S}]}_{c_{1} (a) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (a) c_{1} (b)} & {[H_{S}]}_{c_{3} (a) c_{3} (c)} \end{matrix}] \\ • ([\begin{matrix} {[H_{S}]}_{c_{1} (b) c_{1} (b)} & {[H_{S}]}_{c_{1} (b) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (c) c_{1} (b)} & {[H_{S}]}_{c_{3} (c) c_{3} (c)} \end{matrix}] {- [\begin{matrix} {[H_{S}]}_{c_{1} (a) c_{1} (b)} & {[H_{S}]}_{c_{1} (a) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (a) c_{1} (b)} & {[H_{S}]}_{c_{3} (a) c_{3} (c)} \end{matrix}])}^{- 1} {[\begin{matrix} [K_{c_{1}}] & [K_{c_{1} c_{3}}] \\ [K_{c_{3} c_{1}}] & [K_{c_{3}}] \end{matrix}]}^{- 1} \end{matrix}

(18)

\begin{matrix} [\begin{matrix} {[H_{A}]}_{o (a) c_{1} (a)} & {[H_{A}]}_{o (a) c_{3} (a)} \end{matrix}] = [[\begin{matrix} H_{S}]_{o (a) c_{1} (a)} & {[H_{S}]}_{o (a) c_{3} (a)} \end{matrix}] \\ • {[\begin{matrix} {[H_{S}]}_{c_{1} (a) c_{1} (a)} & {[H_{S}]}_{c_{1} (a) c_{3} (a)} \\ {[H_{S}]}_{c_{3} (a) c_{1} (a)} & {[H_{S}]}_{c_{3} (a) c_{3} (a)} \end{matrix}]}^{- 1} [\begin{matrix} {[H_{A}]}_{c_{1} (a) c_{1} (a)} & {[H_{A}]}_{c_{1} (a) c_{3} (a)} \\ {[H_{A}]}_{c_{3} (a) c_{1} (a)} & {[H_{A}]}_{c_{3} (a) c_{3} (a)} \end{matrix}] \end{matrix}

(19)

$[K_{c_{2}}]$ and substructure B

To calculate the FRFs of the substructure B, coupling stiffness $[K_{c_{2}}]$ (between substructures B and C), the substructures A and C are treated as a new substructure. The derivation method is similar to that in section “ $[K_{c_{1}}]$ , $[K_{c_{3}}]$ , and substructure A”

\begin{matrix} [\begin{matrix} [K_{c_{1}}] & [K_{c_{1} c_{2}}] \\ [K_{c_{2} c_{1}}] & [K_{c_{2}}] \end{matrix}] = ([\begin{matrix} {[H_{S}]}_{c_{1} (b) c_{1} (b)} & {[H_{S}]}_{c_{1} (b) c_{2} (b)} \\ {[H_{S}]}_{c_{2} (b) c_{1} (b)} & {[H_{S}]}_{c_{2} (b) c_{2} (b)} \end{matrix}] \\ • {[\begin{matrix} {[H_{S}]}_{c_{1} (a) c_{1} (b)} & {[H_{S}]}_{c_{1} (a) c_{2} (b)} \\ {[H_{S}]}_{c_{2} (c) c_{1} (b)} & {[H_{S}]}_{c_{2} (c) c_{2} (b)} \end{matrix}]}^{- 1} [\begin{matrix} {[H_{S}]}_{c_{1} (a) c_{1} (a)} & {[H_{S}]}_{c_{1} (a) c_{2} (c)} \\ {[H_{S}]}_{c_{2} (c) c_{1} (a)} & {[H_{S}]}_{c_{2} (c) c_{2} (c)} \end{matrix}] \\ {- [\begin{matrix} {[H_{S}]}_{c_{1} (b) c_{1} (a)} & {[H_{S}]}_{c_{1} (b) c_{2} (c)} \\ {[H_{S}]}_{c_{2} (b) c_{1} (a)} & {[H_{S}]}_{c_{2} (b) c_{2} (c)} \end{matrix}])}^{- 1} \end{matrix}

(20)

\begin{matrix} [\begin{matrix} {[H_{B}]}_{c_{1} (b) c_{1} (b)} & {[H_{B}]}_{c_{1} (b) c_{2} (b)} \\ {[H_{B}]}_{c_{2} (b) c_{1} (b)} & {[H_{B}]}_{c_{2} (b) c_{2} (b)} \end{matrix}] = [\begin{matrix} {[H_{S}]}_{c_{1} (b) c_{1} (a)} & {[H_{S}]}_{c_{1} (b) c_{2} (c)} \\ {[H_{S}]}_{c_{2} (b) c_{1} (a)} & {[H_{S}]}_{c_{2} (b) c_{2} (c)} \end{matrix}] \\ • ([\begin{matrix} {[H_{S}]}_{c_{1} (a) c_{1} (a)} & {[H_{S}]}_{c_{1} (a) c_{2} (c)} \\ {[H_{S}]}_{c_{2} (c) c_{1} (a)} & {[H_{S}]}_{c_{2} (c) c_{2} (c)} \end{matrix}] \\ {- [\begin{matrix} {[H_{S}]}_{c_{1} (b) c_{1} (a)} & {[H_{S}]}_{c_{1} (b) c_{2} (c)} \\ {[H_{S}]}_{c_{2} (b) c_{1} (a)} & {[H_{S}]}_{c_{2} (b) c_{2} (c)} \end{matrix}])}^{- 1} {[\begin{matrix} K_{c_{1}} & K_{c_{1} c_{2}} \\ K_{c_{2} c_{1}} & K_{c_{2}} \end{matrix}]}^{- 1} \end{matrix}

(21)

Substructure C

To calculate the FRFs of the substructure C, the substructures A and B are treated as a new substructure G. Using the derivation method mentioned in Appendix 3, equations (22)–(24) can be obtained

\begin{matrix} [\begin{matrix} [K_{c_{2}}] & [K_{c_{2} c_{3}}] \\ [K_{c_{3} c_{2}}] & [K_{c_{3}}] \end{matrix}] = ([\begin{matrix} {[H_{S}]}_{c_{2} (b) c_{2} (b)} & {[H_{S}]}_{c_{2} (b) c_{3} (a)} \\ {[H_{S}]}_{c_{3} (a) c_{2} (b)} & {[H_{S}]}_{c_{3} (b) c_{3} (a)} \end{matrix}] \\ • {[\begin{matrix} {[H_{S}]}_{c_{2} (c) c_{2} (b)} & {[H_{S}]}_{c_{2} (c) c_{3} (a)} \\ {[H_{S}]}_{c_{3} (c) c_{2} (b)} & {[H_{S}]}_{c_{3} (c) c_{3} (a)} \end{matrix}]}^{- 1} [\begin{matrix} {[H_{S}]}_{c_{2} (c) c_{2} (c)} & {[H_{S}]}_{c_{2} (c) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (c) c_{2} (c)} & {[H_{S}]}_{c_{3} (c) c_{3} (c)} \end{matrix}] \\ {- [\begin{matrix} {[H_{S}]}_{c_{2} (b) c_{2} (c)} & {[H_{S}]}_{c_{2} (b) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (a) c_{2} (c)} & {[H_{S}]}_{c_{3} (a) c_{3} (c)} \end{matrix}])}^{- 1} \end{matrix}

(22)

\begin{matrix} [\begin{matrix} {[H_{C}]}_{c_{2} (c) c_{2} (c)} & {[H_{C}]}_{c_{2} (c) c_{3} (c)} \\ {[H_{C}]}_{c_{3} (c) c_{2} (c)} & {[H_{C}]}_{c_{3} (c) c_{3} (c)} \end{matrix}] = {[\begin{matrix} [K_{c_{2}}] & [K_{c_{2} c_{3}}] \\ [K_{c_{3} c_{2}}] & [K_{c_{3}}] \end{matrix}]}^{- 1} \\ • {([\begin{matrix} {[H_{S}]}_{c_{2} (b) c_{2} (b)} & {[H_{S}]}_{c_{2} (b) c_{3} (a)} \\ {[H_{S}]}_{c_{3} (a) c_{2} (b)} & {[H_{S}]}_{c_{3} (b) c_{3} (a)} \end{matrix}] - [\begin{matrix} {[H_{S}]}_{c_{2} (b) c_{2} (c)} & {[H_{S}]}_{c_{2} (b) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (a) c_{2} (c)} & {[H_{S}]}_{c_{3} (a) c_{3} (c)} \end{matrix}])}^{- 1} \\ • [\begin{matrix} {[H_{S}]}_{c_{2} (b) c_{2} (c)} & {[H_{S}]}_{c_{2} (b) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (a) c_{2} (c)} & {[H_{S}]}_{c_{3} (a) c_{3} (c)} \end{matrix}] \end{matrix}

(23)

\begin{matrix} [\begin{matrix} {[H_{C}]}_{c_{2} (c) i (c)} \\ {[H_{C}]}_{c_{3} (c) i (c)} \end{matrix}] = [\begin{matrix} {[H_{C}]}_{c_{2} (c) c_{2} (c)} & {[H_{C}]}_{c_{2} (c) c_{3} (c)} \\ {[H_{C}]}_{c_{3} (c) c_{2} (c)} & {[H_{C}]}_{c_{3} (c) c_{3} (c)} \end{matrix}] \\ • {[\begin{matrix} {[H_{S}]}_{c_{2} (c) c_{2} (c)} & {[H_{S}]}_{c_{2} (c) c_{3} (c)} \\ {[H_{S}]}_{c_{3} (c) c_{2} (c)} & {[H_{S}]}_{c_{3} (c) c_{3} (c)} \end{matrix}]}^{- 1} [\begin{matrix} {[H_{S}]}_{c_{2} (c) i (c)} \\ {[H_{S}]}_{c_{3} (c) i (c)} \end{matrix}] \end{matrix}

(24)

Up to this point, the substructure property identification model of the two-stage series-parallel system is developed (equations (17)–(24)). The FRFs of substructure and coupling stiffness in free state can be predicted by the system level FRFs of the coupled structure.

To sum up, the source-path-receiver model and the substructure property identification model make up the transfer path model of a two-stage series-parallel vibration isolation system. The transfer path model relates the substructure level dynamic characteristics to the overall system dynamic responses. The source-path-receiver model shows the relationship between the system level FRFs and the substructure level FRFs in free state. And through the substructure property identification model, the FRFs of substructures and coupling stiffness in free state can be calculated by system level FRFs of coupled structure.

FEM-based validation

In previous section, the transfer path model of a two-stage series-parallel isolation system is developed. In this section, a FEM is designed to validate the transfer path model.

Validation process

The transfer path model of a two-stage series-parallel isolation system model can be validated by the following steps. As shown in Figure 5, first, a simplified model of a two-stage series-parallel isolation system is built by CATIA and then the FE model of both the substructures and coupled system can be generated in Hypermesh. The FRFs of each substructure in free state and system level FRFs of coupled structure can be obtained by MSC Patran/Nastran and LMS Virtual.lab based on the FEM. Then, the system level FRFs are substituted into the substructure property identification model to predict the substructure FRFs matrix and coupling stiffness using MATLAB program. By comparing the predicted substructure FRFs and the exact values, the substructure property identification model is verified. Then, we substitute the calculated FRFs of substructures and coupling stiffness into the source-path-receiver model to predict the system response of coupled structure using MATLAB program. By comparing the predicted system response and the direct response calculations, the source-path-receiver model is verified.

Figure 5.

The process of validation based on FEM.

Frequency response analysis based on FEM

Finite element model

As mentioned above, the two-stage series-parallel vibration isolation system is a common structure in various mechanical systems and a vehicle system is one of them. Here, we will take it as an example to build the FEM to carry out the validation. As shown in Figure 6, a three-dimensional (3D) model of the system is set up. Substructure A represents the vehicle body. Substructure B is regarded as the subframe. The substructure C simulates the powertrain. Substructures A and B are connected via four bushings simulating the subframe mounts; substructures B and C are connected via a bushing simulating the powertrain mount; substructures A and C are connected via two bushings simulating the powertrain mounts.

Figure 6.

The two-stage series-parallel system model simulating a vehicle system.

Then, the entity models of substructures A, B, and C from CATIA are imported into Hypermesh to generate the FE models. The FEM of the coupled system can be got by assembling FE models of three substructures through seven bushings as shown in Figure 7.

Figure 7.

FEM of coupled system.

In the FE model, each bushing is represented as a set of three orthogonal scalar springs and damping elements. They locate at the coupling coordinates between the substructures. The dynamic stiffness of the springs and damping elements can be obtained from the test: (a) axial stiffness, 709 N/mm; axial damping, 0.48 N s/mm and (b) radial stiffness, 7650 N/mm; radial damping, 3.8 N s/mm.

FRF analysis

The structural and acoustic FRFs can be obtained by MSC Patran/Nastran and LMS Virtual.lab. The FRFs of substructures A, B, and C and coupled system S are shown in Table 1. The system target response includes the vibration acceleration in X-, Y-, and Z-direction at coordinate o(a) and the sound pressure at coordinate o_s(a). It can be seen from Figure 8 that the excitation points of the coupled system include point A₁, A₂, A₃, A₄, A₅, A₆, B₁, B₂, B₃, B₄, B₇, C₅, C₆, C₇, i(c) and the response points include the vibration acceleration point o(a), A₁, A₂, A₃, A₄, A₅, A₆, B₁, B₂, B₃, B₄, B₇, C₅, C₆, C₇, i(c) and the sound pressure point o_s(a); the excitation points of substructure A include A₁, A₂, A₃, A₄, A₅, A₆ and the response points include the vibration acceleration point o(a), A₁, A₂, A₃, A₄, A₅, A₆ and the sound pressure point o_s(a); Figure 9(b) shows that the excitation points of substructure B include point B₁, B₂, B₃, B₄, B₇ and the response points also include B₁, B₂, B₃, B₄, B₇; Figure 9(c) shows that both the excitation points and the response points of substructure C incorporates C₅, C₆, C₇, i(c). When analyzing the frequency response of the system and the substructures, the excitation in X-, Y- and Z-direction will be exerted on every excitation point. Then, response of the vibration acceleration points and the sound pressure points can be obtained.

Table 1.

The list of FRFs.

Coupled system S	Dimension	Coupled system S	Dimension
$[H_{S}]_{o (a) c_{1} (a)}$	4 × 12	$[H_{S}]_{c_{3} (c) i (c)}$	6 × 3
$[H_{S}]_{o (a) c_{3} (a)}$	4 × 6	$[H_{S}]_{c_{1} (b) c_{3} (c)}$	12 × 6
$[H_{S}]_{c_{1} (a) c_{3} (a)}$	12 × 6	$[H_{S}]_{c_{1} (a) c_{2} (c)}$	12 × 3
$[H_{S}]_{c_{1} (a) c_{3} (c)}$	12 × 6	$[H_{S}]_{c_{1} (b) c_{2} (c)}$	12 × 3
$[H_{S}]_{c_{1} (b) c_{3} (a)}$	12 × 6
Coupled system	Dimension	Substructure A	Dimension
$[H_{S}]_{c_{3} (a) c_{3} (a)}$	6 × 6	$[H_{A}]_{o (a) c_{1} (a)}$	4 × 12
$[H_{S}]_{c_{3} (a) c_{3} (c)}$	6 × 6	$[H_{A}]_{o (a) c_{3} (a)}$	4 × 6
$[H_{S}]_{c_{3} (c) c_{3} (c)}$	6 × 6	$[H_{A}]_{c_{1} (a) c_{1} (a)}$	12 × 12
$[H_{S}]_{c_{1} (a) c_{1} (a)}$	12 × 12	$[H_{A}]_{c_{1} (a) c_{3} (a)}$	12 × 6
$[H_{S}]_{c_{1} (b) c_{1} (b)}$	12 × 12	$[H_{A}]_{c_{3} (a) c_{3} (a)}$	12 × 12
$[H_{S}]_{c_{1} (a) c_{1} (b)}$	12 × 12
Coupled system	Dimension	Substructure B	Dimension
$[H_{S}]_{c_{1} (a) c_{2} (b)}$	12 × 3	$[H_{B}]_{c_{1} (b) c_{1} (b)}$	12 × 12
$[H_{S}]_{c_{1} (b) c_{2} (b)}$	12 × 3	$[H_{B}]_{c_{2} (b) c_{2} (b)}$	3 × 3
$[H_{S}]_{c_{2} (b) c_{2} (b)}$	3 × 3	$[H_{B}]_{c_{1} (b) c_{2} (b)}$	12 × 3
$[H_{S}]_{c_{2} (c) c_{2} (c)}$	3 × 3
Coupled system	Dimension	Substructure C	Dimension
$[H_{S}]_{c_{2} (b) c_{2} (c)}$	3 × 3	$[H_{C}]_{c_{2} (c) c_{2} (c)}$	3 × 3
$[H_{S}]_{c_{2} (b) c_{3} (a)}$	3 × 6	$[H_{C}]_{c_{2} (c) c_{3} (c)}$	3 × 6
$[H_{S}]_{c_{2} (b) c_{3} (c)}$	3 × 6	$[H_{C}]_{c_{3} (c) c_{3} (c)}$	6 × 6
$[H_{S}]_{c_{2} (c) c_{3} (a)}$	3 × 9	$[H_{C}]_{c_{2} (c) i (c)}$	3 × 3
$[H_{S}]_{c_{2} (c) c_{3} (c)}$	3 × 9	$[H_{C}]_{c_{3} (c) i (c)}$	6 × 3
$[H_{S}]_{c_{2} (c) i (c)}$	3 × 3

FRFs: frequency response functions.

Figure 8.

The FRFs of the coupled structure.

Figure 9.

The FRFs of substructures: (a) substructure A, (b) substructure B, and (c) substructure C.

Comparison of predicted results and exact values

Validation of the substructure property identification model

In previous sections, the FEM is already developed and the frequency response analysis is finished. Based on the validation process mentioned above, the validation of the substructure property identification model of the two-stage series-parallel isolation system is carried out here. The predictions of following FRFs are verified.

Free substructure FRFs of substructure A: $[H_{A}]_{o (a) c_{1} (a)}$ , $[H_{A}]_{o (a) c_{3} (a)}$ , $[H_{A}]_{c_{1} (a) c_{1} (a)}$ , $[H_{A}]_{c_{1} (a) c_{3} (a)}$ , and $[H_{A}]_{c_{3} (a) c_{3} (a)}$

Free substructure FRFs of substructure B: $[H_{B}]_{c_{1} (b) c_{1} (b)}$ , $[H_{B}]_{c_{2} (b) c_{2} (b)}$ , and $[H_{B}]_{c_{1} (b) c_{2} (b)}$

Free substructure FRFs of substructure C: $[H_{C}]_{c_{2} (c) c_{2} (c)}$ , $[H_{C}]_{c_{2} (c) c_{3} (c)}$ , $[H_{C}]_{c_{3} (c) c_{3} (c)}$ , $[H_{C}]_{c_{2} (c) i (c)}$ , and $[H_{C}]_{c_{3} (c) i (c)}$

Coupling stiffness: $[K_{c_{1}}]$ , $[K_{c_{2}}]$ , and $[K_{c_{3}}]$ .

Due to the length limitation, only a part of the figures are shown here. Figures 10 –15 show the comparison between the calculated substructure level FRFs and the exact values. It can be seen that the predicted results match very well with the direct frequency response. Thereby, we can draw the conclusion that the FRFs of substructures and coupling stiffness in free state can be predicted by the system level FRFs of the coupled structure through the substructure property identification model of the two-stage series-parallel isolation system.

Figure 10.

$[K_{c_{1}}]$ .

Figure 11.

$[K_{c_{3}}]$ .

Figure 12.

$[K_{c_{2}}]$ .

Figure 13.

$[H_{A}]_{c_{3} (a) c_{3} (a)}$ .

Figure 14.

$[H_{B}]_{c_{2} (b) c_{2} (b)}$ .

Figure 15.

$[H_{C}]_{c_{2} (c) c_{2} (c)}$ .

Validation of the source-path-receiver model

As mentioned in the section “Validation process,” the calculated FRFs of substructures and coupling stiffness from previous section are substituted into the source-path-receiver model to predict the system response $[H_{S}]_{o (a) i (c)}$ . Figures 16 and 17 show the comparison of the calculated and exact values. It is obvious that the predicted results match very well with the direct frequency response.

Figure 16.

$[H_{S}]_{o (a) i (c)}$ .

Figure 17.

$[H_{S}]_{o (a) i (c)}$ .

Up to this point, the transfer path model of the two-stage series-parallel isolation system is already verified. It is worth noting that the proposed transfer path model is suitable for any linear mechanical systems, which can be simplified as a two-stage series-parallel structure although the validation is finished based on a FEM simulating the vehicle system.

Application

The verified transfer path model is of practical utility. It can be applied at several steps of a product development, such as trouble-shooting process and pre-prototype design.

According to the transfer path model, the system response can be first cascaded to the FRFs of substructures C, D and coupling stiffness between substructures D and C. The FRFs of substructure D can be further partitioned into the FRFs of substructures A, B and coupling stiffness between substructures A and B in free state (shown in Figure 18).

Figure 18.

Cascading relationship of the system response.

It is obvious that with the application of the transfer path model, every possible transfer path that contributes to the system response can be identified. The transfer path model can ease the trouble-shooting process of a product development. As mentioned above, the system response can be cascaded to the substructure level. Therefore, based on the model the problem area, namely the most effective location for palliative treatment, can be easily addressed.

The cascading of the system response based on the transfer path model also helps in a pre-prototype design. The designer can substitute different components into the various paths and use a sound quality replay system to listen to the effects subjectively during the early design stages.²¹

In general, the transfer path model of a series-parallel two-stage isolation system is a powerful tool for the trouble-shooting process and pre-prototype design of a product development.

Error sensitivity analysis

In a FEM-based simulation, noise is not considered. However, random noise is unavoidable in the actual measurement of FRFs. Random noise may reduce the accuracy of the predicted FRFs. Regardless of how precise the FRFs are measured, the estimated FRFs will not correspond perfectly to the true transfer function between the input and output.⁸ Hence, in this section, the random noise is introduced into the relevant system level FRFs for error sensitivity analysis and then its influence on the substructure FRFs is quantified by comparison. Additional care should be taken, when acquiring the system level FRFs which are most sensitive to the random errors.

Inputs and outputs of error sensitivity analysis

Before the error sensitivity analysis, the inputs and outputs are confirmed. As shown in Figure 19, based on the transfer path model, the FRFs of substructures and coupling stiffness, which are used to predict the system response $[H_{S}]_{o (a) i (c)}$ , are found out. Then the system level FRFs, which are used to calculate these relevant substructure level FRFs are identified and considered as the inputs. Because there are several pairs of reciprocal transpose matrix among them, there are actually only 25 independent inputs. The inputs are as follows: $[H_{S}]_{o (a) c_{1} (a)}$ , $[H_{S}]_{o (a) c_{3} (a)}$ , $[H_{S}]_{c_{3} (a) c_{3} (a)}$ , $[H_{S}]_{c_{3} (c) c_{3} (c)}$ , $[H_{S}]_{c_{3} (a) c_{3} (c)}$ , $[H_{S}]_{c_{1} (a) c_{3} (a)}$ , $[H_{S}]_{c_{1} (a) c_{3} (c)}$ , $[H_{S}]_{c_{1} (b) c_{3} (a)}$ , $[H_{S}]_{c_{1} (b) c_{3} (c)}$ , $[H_{S}]_{c_{1} (a) c_{1} (a)}$ , $[H_{S}]_{c_{1} (b) c_{1} (b)}$ , $[H_{S}]_{c_{1} (a) c_{1} (b)}$ , $[H_{S}]_{c_{1} (a) c_{2} (b)}$ , $[H_{S}]_{c_{1} (a) c_{2} (c)}$ , $[H_{S}]_{c_{1} (b) c_{2} (b)}$ , $[H_{S}]_{c_{1} (b) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{2} (b)}$ , $[H_{S}]_{c_{2} (c) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{3} (a)}$ , $[H_{S}]_{c_{2} (b) c_{3} (c)}$ , $[H_{S}]_{c_{2} (c) c_{3} (a)}$ , $[H_{S}]_{c_{2} (c) c_{3} (c)}$ , $[H_{S}]_{c_{2} (c) i (c)}$ , $[H_{S}]_{c_{3} (c) i (c)}$ .

Figure 19.

The inputs and outputs of the error sensitive analysis.

The substructure level FRFs mentioned above are considered as the outputs of the analysis. The 16 independent outputs are as follows: $[K_{c_{1}}]$ , $[K_{c_{3}}]$ , $[H_{A}]_{c_{1} (a) c_{1} (a)}$ , $[H_{A}]_{c_{3} (a) c_{3} (a)}$ , $[H_{A}]_{c_{1} (a) c_{3} (a)}$ , $[H_{A}]_{o (a) c_{1} (a)}$ , $[H_{A}]_{o (a) c_{3} (a)}$ , $[K_{c_{2}}]$ , $[H_{B}]_{c_{1} (b) c_{1} (b)}$ , $[H_{B}]_{c_{2} (b) c_{2} (b)}$ , $[H_{B}]_{c_{1} (b) c_{2} (b)}$ , $[H_{C}]_{c_{2} (b) c_{2} (c)}$ , $[H_{C}]_{c_{3} (c) c_{3} (c)}$ , $[H_{C}]_{c_{2} (c) c_{3} (c)}$ , $[H_{C}]_{c_{2} (c) i (c)}$ , $[H_{C}]_{c_{3} (c) i (c)}$ .

Sensitivity analysis of different input FRFs

As we can see from Figure 19, each output is affected by several inputs. The error sensitivity analysis aims to find out the system level FRFs which are most sensitive to random noise and it may have the greatest influence on the outputs. For the analysis, some levels of random errors are introduced into the input FRFs (direct frequency response) individually. Here, we take 5% random error as an example. The exact system level FRFs with random errors are substituted into the substructure property identification model (equations (17)–(24)) to calculate the substructure FRFs. The system response can be predicted by calculated substructure level FRFs through the source-path-receiver model. The prediction error of the outputs is shown in Figures 20 –26 and represented in the form of relative Frobenius norm error defined as follows

r e (f_{i}) = \frac{{‖ [H] (f_{i}) - [H_{r e a l}] (f_{i}) ‖}_{F}}{{‖ [H_{r e a l}] (f_{i}) ‖}_{F}} \times 100 %

(25)

where $[H]$ is the predicted FRF; $[H_{real}]$ is the exact value. To present the analysis results more clearly, the histograms are used to show the overall error level within the frequency range.

Figure 20.

$[K_{c_{1}}]$ .

Figure 21.

$[K_{c_{3}}]$ .

Figure 22.

$[H_{A}]_{c_{1} (a) c_{1} (a)}$ .

Figure 23.

$[K_{c_{2}}]$ .

Figure 24.

$[H_{B}]_{c_{1} (b) c_{2} (b)}$ .

Figure 25.

$[H_{C}]_{c_{3} (c) i (c)}$ .

Figure 26.

$[H_{S}]_{o (a) i (c)}$ .

Due to the length limitation, only a part of the figures are shown here. Table 2 summarizes the results of the error sensitivity analysis.

Figures 20, 21, and 23 show the effect of 5% input error on the prediction error of $[K_{c_{1}}]$ , $[K_{c_{3}}]$ , and $[K_{c_{2}}]$ . Random error is added to relevant system level FRFs (as shown in Table 2) individually. It can be seen that when estimating the stiffness at coupling coordinate c₁, more attention should be paid to the system level FRF $[H_{S}]_{c_{1} (a) c_{1} (b)}$ since it is most sensitive to random noise. For the coupling stiffness at coordinate c₃, more emphasis should be put on the system level FRFs $[H_{S}]_{c_{1} (a) c_{3} (c)}$ and for the coupling stiffness $[K_{c_{2}}]$ , the most sensitive system level FRFs are $[H_{S}]_{c_{2} (c) c_{2} (c)}$ and $[H_{S}]_{c_{2} (b) c_{2} (b)}$ .

Figure 22 shows the effect of 5% input error on the prediction error of substructure A. Random errors are added to relevant system level FRFs (as shown in Table 2) individually. It is obvious that additional care should be taken, when acquiring the system level FRFs $[H_{S}]_{c_{1} (a) c_{1} (b)}$ and $[H_{S}]_{c_{1} (b) c_{3} (a)}$ .

Figure 24 shows the effect of 5% input error on the prediction error of substructure B. Random errors are added to relevant system level FRFs (as shown in Table 2) individually. Obviously, the prediction error of substructure B is mainly determined by system level FRF $[H_{S}]_{c_{1} (b) c_{2} (b)}$ .

Figure 25 shows the effect of 5% input error on the prediction error of substructure C. Random errors are added to relevant system level FRFs (as shown in Table 2) individually. It can be seen that the prediction error is more sensitive to the random error added to $[H_{S}]_{c_{2} (c) c_{2} (c)}$ and $[H_{S}]_{c_{2} (c) c_{3} (c)}$ . So, additional care should be taken, when acquiring these two system level FRFs.

Figure 26 shows the effect of 5% input error on the prediction error of system response $[H_{S}]_{o (a) i (c)}$ . Random errors are added to relevant system level FRFs (as shown in Table 2) individually. Obviously, more attention should be paid to the system level FRFs $[H_{S}]_{c_{1} (a) c_{2} (c)}$ , $[H_{S}]_{c_{1} (a) c_{2} (b)}$ , $[H_{S}]_{c_{2} (b) c_{3} (c)}$ , and $[H_{S}]_{c_{1} (b) c_{3} (a)}$ , when estimating the system response.

Table 2.

Error sensitivity analysis of different system level FRFs.

Target	Predicted FRFs	Relevant inputs	Most sensitive FRFs
Coupling stiffness	$[K_{c_{1}}]$	$[H_{S}]_{o (a) c_{1} (a)}$ , $[H_{S}]_{o (a) c_{3} (a)}$ , $[H_{S}]_{c_{3} (a) c_{3} (a)}$ , $[H_{S}]_{c_{3} (c) c_{3} (c)}$ , ${[H_{S}]}_{c_{3} (a) c_{3} (c)}$ , ${[H_{S}]}_{c_{1} (a) c_{3} (a)}$ , ${[H_{S}]}_{c_{1} (a) c_{3} (c)}$ , ${[H_{S}]}_{c_{1} (b) c_{3} (a)}$ , $[H_{S}]_{c_{1} (b) c_{3} (c)}$ , $[H_{S}]_{c_{1} (a) c_{1} (a)}$ , $[H_{S}]_{c_{1} (b) c_{1} (b)}$ , $[H_{S}]_{c_{1} (a) c_{1} (b)}$	$[H_{S}]_{c_{1} (a) c_{1} (b)}$
Coupling stiffness	$[K_{c_{3}}]$		$[H_{S}]_{c_{1} (a) c_{3} (c)}$
Substructure A	$[H_{A}]_{c_{1} (a) c_{1} (a)}$		$[H_{S}]_{c_{1} (a) c_{1} (b)}$ , $[H_{S}]_{c_{1} (b) c_{3} (a)}$
	$[H_{A}]_{c_{1} (a) c_{3} (a)}$
	$[H_{A}]_{c_{3} (a) c_{3} (a)}$
	$[H_{A}]_{o (a) c_{1} (a)}$
	$[H_{A}]_{o (a) c_{3} (a)}$
Coupling stiffness	$[K_{c_{2}}]$	$[H_{S}]_{c_{1} (a) c_{1} (a)}$ , $[H_{S}]_{c_{1} (b) c_{1} (b)}$ , $[H_{S}]_{c_{1} (a) c_{1} (b)}$ , $[H_{S}]_{c_{1} (a) c_{2} (b)}$ , $[H_{S}]_{c_{1} (a) c_{2} (c)}$ , $[H_{S}]_{c_{1} (b) c_{2} (b)}$ , $[H_{S}]_{c_{1} (b) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{2} (b)}$ , $[H_{S}]_{c_{2} (c) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{2} (c)}$	$[H_{S}]_{c_{2} (c) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{2} (b)}$
Substructure B	$[H_{B}]_{c_{1} (b) c_{1} (b)}$		$[H_{S}]_{c_{1} (b) c_{2} (b)}$
	$[H_{B}]_{c_{2} (b) c_{2} (b)}$
	$[H_{B}]_{c_{1} (b) c_{2} (b)}$
Substructure C	$[H_{C}]_{c_{2} (c) c_{2} (c)}$	$[H_{S}]_{c_{2} (b) c_{2} (b)}$ , $[H_{S}]_{c_{2} (c) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{3} (a)}$ , $[H_{S}]_{c_{2} (b) c_{3} (c)}$ , $[H_{S}]_{c_{2} (c) c_{3} (a)}$ , $[H_{S}]_{c_{2} (c) c_{3} (c)}$ , $[H_{S}]_{c_{3} (a) c_{3} (a)}$ , $[H_{S}]_{c_{3} (c) c_{3} (c)}$ , $[H_{S}]_{c_{3} (a) c_{3} (c)}$ , $[H_{S}]_{c_{2} (c) i (c)}$ , $[H_{S}]_{c_{3} (c) i (c)}$	$[H_{S}]_{c_{2} (c) c_{2} (c)}$ , $[H_{S}]_{c_{2} (c) c_{3} (c)}$
	$[H_{C}]_{c_{3} (c) c_{3} (c)}$
	$[H_{C}]_{c_{2} (c) c_{3} (c)}$
	$[H_{C}]_{c_{2} (c) i (c)}$
	$[H_{C}]_{c_{3} (c) i (c)}$
System response	$[H_{S}]_{o (a) i (c)}$	$[H_{S}]_{o (a) c_{1} (a)}$ , $[H_{S}]_{o (a) c_{3} (a)}$ , $[H_{S}]_{c_{3} (a) c_{3} (a)}$ , $[H_{S}]_{c_{3} (c) c_{3} (c)}$ , $[H_{S}]_{c_{3} (a) c_{3} (c)}$ , $[H_{S}]_{c_{1} (a) c_{3} (a)}$ , $[H_{S}]_{c_{1} (a) c_{3} (c)}$ , $[H_{S}]_{c_{1} (b) c_{3} (a)}$ , $[H_{S}]_{c_{1} (b) c_{3} (c)}$ , $[H_{S}]_{c_{1} (a) c_{1} (a)}$ , $[H_{S}]_{c_{1} (b) c_{1} (b)}$ , $[H_{S}]_{c_{1} (a) c_{1} (b)}$ , $[H_{S}]_{c_{1} (a) c_{2} (b)}$ , $[H_{S}]_{c_{1} (a) c_{2} (c)}$ , $[H_{S}]_{c_{1} (b) c_{2} (b)}$ , $[H_{S}]_{c_{1} (b) c_{2} (c)}$ , ${[H_{S}]}_{c_{2} (b) c_{2} (b)}$ , $[H_{S}]_{c_{2} (c) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{2} (c)}$ , $[H_{S}]_{c_{2} (b) c_{3} (a)}$ , $[H_{S}]_{c_{2} (b) c_{3} (c)}$ , $[H_{S}]_{c_{2} (c) c_{3} (a)}$ , $[H_{S}]_{c_{2} (c) c_{3} (c)}$ , $[H_{S}]_{c_{2} (c) i (c)}$ , $[H_{S}]_{c_{3} (c) i (c)}$	$[H_{S}]_{c_{1} (a) c_{2} (c)}$ , $[H_{S}]_{c_{1} (a) c_{2} (b)}$ , $[H_{S}]_{c_{2} (b) c_{3} (c)}$ , $[H_{S}]_{c_{1} (b) c_{3} (a)}$

FRFs: frequency response functions.

Conclusion

A single-stage vibration isolation system model is already well developed by some researchers. However, a two-stage series-parallel isolation system has not been studied although it is already a widely used structure in mechanical field. Hence, in this article, the source-path-receiver model and the substructure property identification model of a two-stage series-parallel vibration isolation system are developed using FRF-based substructuring method. These two models make up the transfer path model of the system. And the model is programmed by MATLAB. Next, the transfer path model is validated through a FE model simulating a vehicle system. The FRFs of substructures in free state and system level FRFs of coupled structure are obtained by MSC Patran/Nastran and LMS Virtual.lab based on the FEM. Then, the FRFs of substructures and the joint stiffness are predicted by the system level FRFs through the substructure property identification model; the system response of coupled structure is predicted by the calculated FRFs of substructures and joint stiffness through the source-path-receiver model. The results of the simulation reveal that the predicted system response and substructural FRFs match very well with the exact values. Therefore, the proposed transfer path model is verified. Furthermore, considering that noise is unavoidable in the actual measurement, random noise is introduced into relevant system level FRFs for error sensitivity analysis. The analysis results show that the transfer path model is very sensitive to the random noise and the system level FRFs, which are most sensitive to the random errors, are found. Additional care should be taken, when acquiring these FRFs.

In general, the transfer path model proposed here further develops the dynamic theory of the multi-stage vibration isolation system. Such a widely used two-stage series-parallel system has not been deeply studied previously. Moreover, based on the model, a detailed acoustic and vibration transfer path analysis process is also presented here. Above all, the model can be applied in any mechanical systems.

Footnotes

Appendix 1

Appendix 2

Appendix 3

Academic Editor: Tian Han

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 51205290) and the Fundamental Research Funds for the Central Universities (grant number 1700219118).

References

Al-Zubi

Investigation and improvement of noise, vibration and harshness (NVH) properties of automotive panels. PhD Dissertation, Wayne State University, Detroit, MI, 2012.

Jetmundsen

Bielawa

Flannelly

WG.

Generalized frequency domain substructure synthesis. J Am Helicopter Soc 1988; 33: 55–64.

Van der Valk

PLC

Wuijckhuijse

Klerk

A benchmark test structure for experimental dynamic substructuring. Struct Dyn 2011; 3: 1113–1122.

Nicgorski

Avitabile

Experimental issues related to frequency response function measurements for frequency based substructuring. Mech Syst Signal Pr 2010; 24: 1324–1337.

Nicgorski

Avitabile

Conditioning of FRF measurements for use with frequency based substructuring. Mech Syst Signal Pr 2010; 24: 340–351.

Lee

Hwang

WS.

Parametric optimization of complex systems using a multi-domain FRF-based substructuring method. Comput Struct 2003; 81: 2249–2257.

Zhen

Development of a spectral-based substructuring technique for modeling vibratory response of complex vehicle structures. PhD Dissertation, The University of Alabama, Tuscaloosa, AL, 2000.

Lei

A frequency response function-based inverse substructuring approach for analyzing vehicle system NVH response. PhD Dissertation, The University of Alabama, Tuscaloosa, AL, 2002.

Wang

Chuang

SC.

Reducing errors in the identification of structural joint parameters using error functions. J Sound Vib 2004; 273: 295–316.

10.

Sjövall

Abrahamsson

Substructure system identification from coupled system test data. Mech Syst Signal Pr 2008; 22: 15–33.

11.

Pradhan

Modak

SV.

A method for damping matrix identification using frequency response data. Mech Syst Signal Pr 2012; 33: 69–82.

12.

Mayes

Allen

Kammer

DC.

Correcting indefinite mass matrices due to substructure uncoupling. J Sound Vib 2013; 332: 5856–5866.

13.

Wang

Zhang

. Application of the inverse substructure method in the investigation of dynamic characteristics of product transport system. Packag Technol Sci 2012; 25: 351–362.

14.

Wang

Inverse substructure method of three-substructures coupled system and its application in product-transport-system. J Vib Control 2010; 17: 943–952.

15.

Law

Yong

Substructure methods for structural condition assessment. J Sound Vib 2011; 330: 3606–3619.

16.

Čelič

Boltežar

Identification of the dynamic properties of joints using frequency-response functions. J Sound Vib 2008; 317: 158–174.

17.

Voormeeren

Rixen

DJ.

A family of substructure decoupling techniques based on a dual assembly approach. Mech Syst Signal Pr 2012; 27: 379–396.

18.

D’Ambrogio

Fregolent

Inverse dynamic substructuring using the direct hybrid assembly in the frequency domain. Mech Syst Signal Pr 2014; 45: 360–377.

19.

D’Ambrogio

Fregolent

The role of interface DOFs in decoupling of substructures based on the dual domain decomposition. Mech Syst Signal Pr 2010; 24: 2035–2048.

20.

Voormeeren

de Klerk

Rixen

DJ.

Uncertainty quantification in experimental frequency based substructuring. Mech Syst Signal Pr 2010; 24: 106–118.

21.

LMS International. Transfer path analysis: the qualification and quantification of vibro-acoustic transfer paths, 1995, http://www.plm.automation.siemens.com/en_us/products/lms/testing/transfer-path-analysis.shtml

Dynamic modeling and simulation of a two-stage series-parallel vibration isolation system

Abstract

Keywords

Introduction

Modeling of coupled series-parallel two-stage isolation system

Source-path-receiver model

Substructure property identification model

[ K c 1 ] , [ K c 3 ] , and substructure A

[ K c 2 ] and substructure B

Substructure C

FEM-based validation

Validation process

Frequency response analysis based on FEM

Finite element model

FRF analysis

Comparison of predicted results and exact values

Validation of the substructure property identification model

Validation of the source-path-receiver model

Application

Error sensitivity analysis

Inputs and outputs of error sensitivity analysis

Sensitivity analysis of different input FRFs

Conclusion

Footnotes

Appendix 1

Appendix 2

Appendix 3

Declaration of conflicting interests

Funding

References

$[K_{c_{1}}]$ , $[K_{c_{3}}]$ , and substructure A

$[K_{c_{2}}]$ and substructure B