1 Introduction

With the progress of engineering technology and the improvement of living standards, people’s performance expectations for passenger cars have gradually increased. While pursuing power and economy performance, more and more attention is paid to ride comfort in sport-utility vehicles (SUVs). The vibration characteristics of the cab gradually become a key index to evaluate the quality of the vehicle [1].

In SUVs, the frame-to-body mount increases the vibration transfer linkage between the engine and the cab. In the target response analysis, each transmission link has an impact on the vibration characteristics of the cab. Also, there is a coupling effect between each transmission link, and the vibration coupling between transmission links is very complex. Therefore, it is important to study the vibration characteristics of a framed SUV cab.

Research into the vibration characteristics in the cab is mainly focused on improving vehicle vibration performance by means of transfer path analysis. Reference [2] used mechanical-acoustic transfer path analysis, where the contribution of each body panel to the vibration in the cab is analyzed, to diagnose the structural transmission noise. In Ref. [3], transfer path analysis and the experimental modal technique were combined to optimize the noise vibration harshness of the vehicle suspension. Reference [4] proposed a powertrain mounting system based on improved transfer path analysis to optimize the mount transfer characteristics of a car. In Ref. [5], coordinate transformation was used to solve the coupling problem of vibration force between partial transfer paths and analyze the vibration of the vehicle body.

Existing research, combined with an engineering example of a specific type of SUV, was used to explore the significant influence of all transfer paths on the cab vibration in an “engine-power train mount system-frame-body mount-body-driver seat guide.” According to the theory of vibration source coupling transfer path analysis, the coupling effect between each transmission link is analyzed, and the vehicle frame structure and powertrain mounting system are optimized. The combination of simulation, analysis, calculation, and experiment verified the effectiveness of the optimization results and proved the practicability of the method, which is especially useful for assessing vibration of the SUV cab.

2 Multistage Coupling Transfer Path Analysis Method

2.1 Basic Principles of Path Analysis of Vibration Source Transfer

If the SUV system is assumed to be linear and time-invariant, the vibration level of the seat guide in the cab is equal to that of the vibration excitation superposition coupling along different transfer paths [6]. To simplify the model, the junction between the intermediate coupling and the passive side is considered as the path point. The path point is regarded as the starting point of the vibration transfer on the passive side, and each path point corresponds to a transfer path. In other words, if one looks at the vibration of the active side as a whole and simply treats each path point as an independent vibration source, then multiple path points are equivalent to multiple vibration sources. However, the vibration energy of these sources comes from the external active side through intermediate coupling. In this case, the vibration contribution of the path point vibration to the target point actually reflects the fact that the vibration energy of the active side is transferred to the path point through different intermediate coupling paths, and then to the target point through the passive side where the path point is located, as shown in Fig. 1.

Fig. 1
figure 1

Vibration transmission diagram

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Multiple excitation forces, where each excitation force has x, y, and z directional components, were assumed, and each excitation component corresponds to n transfer paths. Therefore, the excitation force component and corresponding transfer path produce a response component of the system. With the target point vibration as the system response, the vibration component can be expressed as:

$$ t_{ti} \left( \omega \right) = H_{t/i} \left( \omega \right) \cdot f_{i} \left( \omega \right)$$
(1)

where \( t_{ti} \left( \omega \right) \) is the vibration component produced by the path of section i on the target point t, \( H_{t/i} \left( \omega \right) \) is the frequency response function of the section i path, and \( f_{i} \left( \omega \right) \) is the working load of the section i path.

If there are n transfer paths, the total response at the target point is a linear superposition of each component on each path, that is:

$$ \begin{aligned} T_{t} (\omega ) & = \sum\limits_{i = 1}^{n} {\left( {H_{t/i} (\omega ) \cdot f_{i} (\omega )} \right)} \\ & = H_{t} (\omega ) \cdot F(\omega )\\ \end{aligned} $$
(2)

where \( F(\omega ) = \left[ {F_{1} (\omega )F_{2} (\omega ) \cdots F_{n - 1} (\omega )F_{n} (\omega )} \right]^{\text{T}} \) is the working load column vectors for each transfer path, \( H_{t} (\omega ) = \left[ {H_{t/1} (\omega )H_{t/2} (\omega ) \cdots H_{t/n - 1} (\omega )H_{t/n} (\omega )} \right] \) is the frequency response function matrix, and \( T_{t} (\omega ) \) is the response of the target point under the action of \( F(\omega ) \).

2.2 Analysis of Substructure Multistage Coupling Model

In a transfer path analysis model established by substructure theory, the whole system is generally divided into several substructures, each of which is characterized by a frequency response function. Information is transmitted between substructures by means of various elastic and damping elements. Figure 2a is a framed SUV substructure model divided into three substructures: body, frame, and powertrain (including engine and transmission). Figure 2b is a load-supporting structure of the SUV substructure model, which is divided into two substructures: (1) body and (2) power transmission system and suspension. The connections between the two substructures, such as the powertrain mounts and suspension bushings, are expressed as a spring model.

Fig. 2
figure 2

Simplified vehicle substructure model

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For a substructure (denoted as M), the relation between the input excitation and the output response can be characterized by a transfer function matrix \( \left[ {H_{\text{M}} } \right]: \)

$$ \left\{ {X_{\text{M}} } \right\} = \left[ {H_{\text{M}} } \right]\left\{ {F_{\text{M}} } \right\}$$
(3)

where \( \left\{ {F_{\text{M}} } \right\} \) is an external input excitation vector, \( \left\{ {X_{\text{M}} } \right\} \) is the output response vector of the substructure, and \( \left[ {H_{\text{M}} } \right] \) is a transfer function matrix that characterizes the relation between the excitation and response of the substructure.

For a substructure with multiple degrees of freedom (DOF), the coordinate system can be divided into three parts: (1) the coupling coordinate (hereinafter denoted c), (2) the self-excited coordinate (hereinafter denoted i), and (3) the self-response coordinate (hereinafter denoted o). The degrees of freedom of the substructure are shown in Fig. 3.

Fig. 3
figure 3

Relation between input and output of substructure M

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Therefore, formula (3) can be expanded to

$$ \left\{ {\begin{array}{*{20}c} {\left\{ X \right\}_{o(m)} } \\ {\left\{ X \right\}_{c(m)} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\left[ {H_{m} } \right]_{o(m)i(m)} } & {\left[ {H_{m} } \right]_{o(m)c(m)} } \\ {\left[ {H_{m} } \right]_{c(m)i(m)} } & {\left[ {H_{m} } \right]_{c(m)c(m)} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\left\{ F \right\}_{i(m)} } \\ {\left\{ F \right\}_{c(m)} } \\ \end{array} } \right\}$$
(4)

where \( \left[ {H_{\text{M}} } \right]_{c(m)c(m)} \) is the origin admittance of the coupled coordinates of substructure M; \( \left[ {H_{\text{M}} } \right]_{c(m)i(m)} \) is the frequency response function from the excitation coordinate to the coupling coordinate; \( \left[ {H_{\text{M}} } \right]_{o(m)c(m)} \) is the frequency response function from the coupling coordinate to the response coordinate; and \( \left[ {H_{\text{M}} } \right]_{o(m)i(m)} \) is the frequency response function of the excitation coordinate to the response coordinate.

Several substructures are connected to each other to form system S, as shown in Fig. 4, where A, B, and C represent its substructures. In the system, two substructures are connected by flexible couplings. The excitation of each substructure is composed of two parts, external excitation and reaction to other substructures transferred through the coupling. Linear system theory holds that the response of the system S is produced by the superposition of both excitations.

Fig. 4
figure 4

Multistage coupling system with tertiary substructures

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The vibration transmission link from the engine to the cab of a framed SUV is increased by linkages between the substructures, that is, powertrain (A) → frame (B) → body (C). According to the theory of substructure coupling, the coupled vibrations of A, B, and C superpose; the transfer paths increase; and the coupling between the transfer links becomes more complicated. In contrast, the vibration characteristics between the SUV load-supporting structure and secondary substructure are much simpler. Therefore, to improve the vibration characteristics of the whole vehicle, it is necessary to fully consider the multistage vibration coupling relations between powertrain, frame, and body, so as to effectively improve the vibration characteristics in multiple coupling links.

3 An Example of Analysis and Optimization

3.1 Optimum Design of Body Structure Size

The excitation frequency of the SUV engine covers a range of 26.7–116.7 Hz. The natural frequency of the frame and the body-in-white (BIW) cannot completely avoid this excitation range, and the distribution of the natural frequency of the body is dense, as shown in Table 1. There is a coupling relationship between the structural modal frequency of the body before optimization and the excitation frequency of the engine idling speed. The first three natural frequencies of the BIW are coupled with the others to varying degrees, and the vibration excitation at these three frequencies has a strong impact on cab vibration.

To solve the problem of frequency coupling, the adjustment of the natural frequency of the body is realized by changing the size of the plate thickness of some parts. There are more than 300 parts in the SUV body. To save material and time costs, considering the feasibility of optimization measures, it is necessary to select several easily modified plates as optimization variables and select a sensitivity analysis method based on plate thickness [7]. The design variable is the maximum direct sensitivity of 20 component thickness, as shown in Fig. 5.

Fig. 5
figure 5

Main plates of SUV BIW size optimization

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The objective of optimization is to minimize the difference between the first natural frequency and 25.0 Hz. The constraint range of the second- and third-order natural frequency of the BIW and frame are 27.5–28.5 Hz and 30.5–32.0 Hz, respectively. Therefore, the optimization objectives and constraints are

$$ \begin{aligned} f(x) & = \left| {f_{1} - 25.0} \right| \to \hbox{min} \\ 27.5 & \le f_{2} \le 28.5,30.5 \le f_{3} \le 32.0\\ \end{aligned} $$
(5)

where f(x) is the objective function and f1, f2, and f3 are the natural values of the first-, second-, and third-order frequencies of the BIW, respectively.

The thicknesses of the optimized plates (after rounding) are shown in Table 2. The thickness of the front plate increased the most, and the thickness of the strengthened plate on both sides of A-pillar and the right inner plate of the front wind window also increased. Regarding the larger plates, the roof, the outer panels of the two sides of the door frame, and the thickness of these three parts was reduced. When the natural frequency of the body was allocated optimally, the weight was reduced by 22.4 kg.

Table 1 Optimization of body structure modal frequency and engine idling excitation frequency before and after optimization
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As shown in Table 1, the first-order natural frequency of the BIW is 25.2 Hz, which is more than 1.4 Hz from the first natural frequency of the frame and the idling frequency of the engine. Additionally, the optimization avoids the BIW’s second- and third-order natural frequencies and the frame’s second- and third-order natural frequencies. It can be seen that the coupling of the first three frequency orders of the body with the excitation frequency of the frame and engine idling speed has achieved better effects.

Table 2 Optimization results of plate thickness
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3.2 Analysis and Optimization of Powertrain Mounting System

The influence of the powertrain mounting system on the vibration characteristics of the cab is mainly controlled by two aspects: the design defect of the mounting support and the insufficient decoupling rate of the mounting system. In the suspension bracket design, the stiffness and first modal frequency of the front and rear suspension brackets are in accordance with the requirements of the engineering for the mounting bracket, and vibration isolation is achieved to a certain extent.

For the design of the powertrain mounting system, it is possible to avoid the resonance of the mounting system and other parts of the vehicle (such as the body and engine). The vibration of the suspension system in six directions is decoupled as far as possible [8, 9]. The dynamic model of the powertrain 6-DOF Mount system is established in the Adams/View software to analyze the decoupling rate of the mounting system, as shown in Fig. 6. Analysis shows that there is a great degree of coupling between the RXX, Y, and Z direction modes of the mounting system. The main methods for improving the vibration characteristics of the cab and the decoupling rate of the mounting system are to optimize the mounting stiffness and adjust the mounting angle of the mount [10]. The relevant parameters for powertrain and mount modeling are shown in Tables 3 and 4.

Fig. 6
figure 6

Powertrain mounting system dynamics model

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Table 3 Powertrain basic parameters
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Table 4 Suspension point position and angle
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Due to the problem of insufficient decoupling rate of the powertrain mounting system, it is necessary to decouple and optimize it. When establishing the optimization model, the objective function, design variables, and constraint conditions should be set up as described below.

  1. (1)

    Objective function

The maximum sum of modal kinetic energy percentages in the six directions of the powertrain mounting system is chosen as the objective function, that is,

$$ f\left( x \right) = \hbox{min} \left[ {6 - \sum\limits_{i = 1}^{6} {\hbox{max} \left( {T_{{P_{i} }} } \right)} } \right]$$
(6)

where \( i = 1, \ldots ,6\;{\text{and}}\;{ \hbox{max} }(T_{{P_{i} }} ) \) is the main vibration energy distribution corresponding to six degrees of freedom.

  1. (2)

    Design variable

Each direction stiffness, inclination angle, and Y-direction coordinate of the front and rear mounts are selected as design variables. There are 11 design variables, including front-mount three-direction coordinates (ku1, Kv1, Kw1), rear-mount three-direction coordinates (Ku2, Kv2, Kw2), front-mount tilt angle (θ1, θ2), front left and right suspension Y coordinates (Y1, Y2), and rear suspension Y coordinate (Y3).

  1. (3)

    Constraint conditions

To improve the vibration isolation, the frequency ratio of the suspension system is generally greater than \( \sqrt 3 \) and the rigid body modal frequency of the vehicle body is generally < 3.3 Hz. Thus, the highest-order natural frequency value range of the powertrain suspension system is f:

$$ 3.3\sqrt 2 \approx 4.67 < f < {{26.7} \mathord{\left/ {\vphantom {{26.7} {\sqrt 2 }}} \right. \kern-0pt} {\sqrt 2 }} \approx 18.66$$
(7)

In addition, to ensure that the natural frequencies of each order do not couple with each other and avoid the situation in which the second-order mode shape is excited at the same time, the difference between adjacent second-order frequencies must be more than 1 Hz:

$$ 4.67 \le f \le 18.86,|f_{i} - f_{j} | \ge 1$$
(8)

where fi and fj are the ith- and jth-order natural frequencies of the powertrain mounting system and \( i \ne j \).

After the optimization model is established, the genetic algorithm in the MATLAB toolbox is used to optimize the parameters. The optimization results are shown in Table 5. Compared with the mounting system before optimization, the optimized mounting stiffness is reduced and the optimized mounting tilt angle decreased by about 20°.

Table 5 Comparison the value of each suspension stiffness and installation angle before and after optimization
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3.3 Simulation of Optimization Results

To ensure the effectiveness of the optimized calculation results, the optimized system should be simulated before reproducing the suspension component sample. The optimized mounting dynamics model was re-establish in Adams and the frequency domain-free mode was calculated. The simulation results are shown in Table 6. It was found that the decoupling rate corresponding to second-order natural frequencies of the more important Z and RXX directions is above 90%, which were increased by 47.63% and 23.65%, respectively, before optimization. The corresponding decoupling rates of all natural frequencies were over 89%, which shows that the optimization results meet both the set optimization goals and the requirements of operation.

Table 6 Comparison of natural frequency and decoupling rate of the suspension system before and after optimization
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To test the rationality of the natural frequency distribution after optimization, the forced vibration analysis of the powertrain mount system was carried out by Adams/Vibration software. The analysis results are shown in Fig. 7. The 4.67–18.86 Hz frequency range is intercepted, and the higher frequency of the second order is close to 18.86 Hz before optimization, which affects the vibration isolation effect of the idling excitation of the generator. After optimization, the natural frequency of the order is far away from the frequency corresponding to the engine excitation; there are third-order natural frequencies in the range of 8–10 Hz before optimization, the corresponding modes are very easily coupled, and the difference between the natural frequencies of each order is above 1 Hz and the distribution is more uniform and reasonable.

Fig. 7
figure 7

Comparison of frequency response curves at centroids before and after optimization

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4 Test Scheme and Validation

To verify the correctness of the optimization results, vibration testing of a sample car before and after optimization is needed. According to the result of optimization, the mounting components were manufactured and applied to the original vehicle. The component was also tested in the same position in the original vehicle using a dynamic signal tester. The test conditions and test data processing rules were the same as those used for the unmodified vehicle. The improvement of cab vibration characteristics was analyzed by comparing the data. The sensors were arranged as shown by the red circles in Fig. 8.

Fig. 8
figure 8

Sensor arrangement in the test SUV

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The total vibration level and the ratio of vibration isolation were used to evaluate the vibration characteristics. The total vibration level in the cab was measured in three directions before and after optimization, and the total value of the synthetic vibration of each measuring point under each working condition is as follows:

$$ v_{ij} = \sqrt {(v^{X}_{ij} )^{2} + (v^{Y}_{ij} )^{2} + (v^{Z}_{ij} )^{2} }$$
(9)

where \( v_{ij} \) is the total value of the synthetic vibration of the jth measuring point of the ith working condition and \( v_{ij}^{X} ,\;v_{ij}^{Y} , \) and \( v_{ij}^{Z} \) are the total vibrational value in the X, Y, and Z directions, respectively.

Another important indicator for the evaluation of the mounting system–that is, the formula for calculating the vibration isolation rate—is as follows

$$ T = 20\lg \left| {\frac{{a_{1} }}{{a_{2} }}} \right|$$
(10)

where a1 and a2 represent the total vibration values of the active and passive sides, respectively.

As shown in Table 7, the vibration level of most of the measured points improved by varying degrees after optimization under the measured working conditions, and the total vibration value of most of the measured points decreased. This shows that the vibration energy transferred to the frame of the powertrain mounting system decreases after the decoupling rate is increased, which verifies the effectiveness of decoupling optimization. The total vibration value at 1000 and 2500 rpm was reduced to 0.225 m/s2 and 0.273 m/s2, respectively, which is 42.2% and 32.9% lower than that before optimization, indicating that the vibration characteristics of the cab are improved.

Table 7 Comparison of the total vibration of each measuring point before and after optimization
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As can be seen from Table 8, the vibration isolation rate of each side mount was improved. The main reason is that the decoupling optimization reduces the stiffness of each side mount and is basically maintained around 20 dB. In general, a good improvement has been achieved.

Table 8 Comparison of vibration isolation ratios of each suspension before and after optimization
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5 Conclusions

This paper proposed a multistage coupling transfer path analysis approach to improve the vibration characteristics of an SUV cab. The related engineering example that combined simulation, analysis, and experiment proved that the multistage coupling transfer path analysis approach is effective and valid. From the optimization results, the following conclusions can be stated:

  1. 1.

    On the basis of the dynamic characteristic parameters of the body structure, the coupling between the modal frequency of the body structure and the excitation frequency of the engine idling speed was analyzed. Thereafter, the partial plate thickness parameters were optimized, the weight of the body was reduced, and the influence of low-order frequency coupling on cab vibration characteristics was essentially eliminated.

  2. 2.

    Decoupling of the power assembly suspension system was optimized. The finite element model simulation proved that the decoupling rate of each degree of freedom was increased to more than 89%.