Introduction

The car body primarily serves as a load-bearing structure. Due to its design characteristics, vibrations and noise are readily transmitted throughout the vehicle, which not only affects ride comfort but also increases the risk of fatigue damage to the body1,2,3. Dynamic stiffness refers to a structure’s ability to resist deformation under external excitation, and it varies with frequency4,5,6.In body structure design, key points such as engine mounting, damper mounting, trailing arm attachment, and subframe attachment are primary sources of vibration transmission, which should be prioritized to reduce structure-borne noise7. Dynamic stiffness influences the transmission and distribution of structural energy. Therefore, the dynamic stiffness at these key connection points must be sufficiently high to reduce the incoming structural energy8. If the dynamic stiffness is insufficient, it can lead to increased vibration and noise9,10. Recent advancements have introduced novel approaches for analyzing and optimizing the NVH performance of vehicle bodies. For instance, human modal analysis techniques have been applied to assess vibration transmission and its effects on passenger comfort11,12. Both the amplitude and phase of vibration responses at a frequency can be utilized by the ODS to map the deflection pattern of structures13,14.

In the current research, there is a relative paucity of studies focused on the reasons for the failure of dynamic stiffness to meet established standards. Generally, when considering dynamic stiffness, the focus is on the minimum dynamic stiffness15. In the low-frequency range, however, the minimum dynamic stiffness is often associated with common issues related to overall modes. This issue is costly to optimize and, even after optimization, does not change the vibration pattern but only shifts the frequency16.

Identifying whether the issue with minimum dynamic stiffness in the low-frequency band is a general or localized problem can significantly improve the efficiency of NVH simulations. By analyzing the modes at frequencies where minimum dynamic stiffness fails to meet the standard, we can better understand the nature of the problem. However, in engineering practice, the maximum frequency for dynamic stiffness calculations is set at 750 Hz to address high-frequency localized vibrations (e.g., in suspension systems or joints), whereas the Body-in-White (BIW) modal analysis is limited to 200 Hz, aligning with low-frequency structural responses such as road-induced vibrations and powertrain oscillations. Due to modal truncation at specific frequencies, deviations may occur, meaning the relationship between dynamic stiffness and corresponding modal frequencies is not always direct.

To address these issues, this study uses modal contribution analysis to identify the modes at frequencies where minimum dynamic stiffness falls short of the standard. The modal frequencies and mode shapes of the system can then be obtained by solving the characteristic equation17,18. By tracing the noise response peak at the field point back to the structural modes, the modal contribution analysis can pinpoint the mode with the greatest influence on the dynamic stiffness19,20,21. This enables a clear determination of whether the issue is a common problem linked to overall structural modes, based on the vibration pattern. Differentiating between common and unique issues helps to avoid wasting time on recurring problems, thus improving work efficiency.

Basic theory and finite element modelling

Basic theory

According to the dynamical equations of a single-degree-of-freedom viscoelastic system

$$\:\text{m}\ddot{x}+c\dot{x}+kx=f$$
(1)

where m is the mass; c is the viscous damping coefficient; k is the stiffness; x,\(\:\dot{x}\),\(\:\ddot{x}\) are the displacement, velocity, and acceleration responses, respectively; and f is the excitation force.

When the external excitation force \(\:f=F{e}^{j\omega\:t}\) the system stiffness can be obtained as

$$\:{k}_{d}=\frac{f}{x}=\left(k-m{\omega\:}^{2}\right)+j\omega\:c$$
(2)

At this point the system stiffness is a function of the external excitation frequency and its amplitude is

$$\:\left|{k}_{d}\left(\omega\:\right)\right|=\sqrt{{(k-m{\omega\:}^{2})}^{2}+{\left(c\omega\:\right)}^{2}}$$
(3)

Input Point Inertance (IPI) is defined as the ratio of acceleration to force. It reflects the combined effects of mass, stiffness, and damping at a specific location, which in turn influences the dynamic behavior of the structure. One of method to obtain the IPI is to calculate the dynamic stiffness at a connecting point between car body and mount using FEA22,23. This method is a key approach for evaluating the NVH performance of body joints. The specific calculation is shown below

$$\:\text{I}\text{P}\text{I}=\frac{\ddot{x}}{F}=\frac{{\omega\:}^{2}x}{F}=\frac{{\omega\:}^{2}}{k}=\frac{{\left(2\pi\:f\right)}^{2}}{k}$$
(4)

where F is the excitation force on the connection point, ω is the circular frequency of excitation, f is the frequency of excitation and k is the dynamic stiffness of the connection point.

The overall vibration of the vehicle is a superposition of a series of mutually independent vibration modes. The mathematical expression for the linear combination of these modal vibration patterns can be written as

$$\:\varvec{x}={j}_{1}{q}_{1}+\dots\:+{j}_{r}{q}_{r}+\dots\:+{j}_{n}{q}_{n}=\varvec{F}\varvec{q}$$
(5)

where F is the vibration mode matrix, \(\:{j}_{r}\) is the coordinates of the rth-order vibration mode,\(\:{q}_{r}\) is the participation factor of the rth-order vibration mode and q is the participation factor vector for each order of the vibration mode. The ratio of the column occupied by the rth-order mode after normalization is then given by

$$\:{Q}_{r}=\frac{\left|{q}_{r}\right|}{\sum\:_{r=1}^{n}\left|{q}_{r}\right|}$$
(6)

Finite element modelling

The 3D model of the body-in-white is imported into HyperMesh for finite element analysis. The meshing quality control parameters in the software are used as the criteria for grid quality. The body-in-white is primarily modeled using shell elements, which are suitable for thin-walled structures (e.g., BIW sheet metal with thickness much smaller than length/width), to efficiently capture bending and in-plane deformations, with a cell size of 8 mm × 8 mm. For small-sized parts, such as the front bumper beam, energy-absorbing box, and shotgun, a finer mesh with a size of 5 mm × 5 mm is applied. The mesh consists primarily of quadrilateral elements, with triangular elements included to ensure that they constitute no more than 5% of the total number of cells. Welded joints are simulated using RB3-HEXA-RB3 element type, while glued connections are modeled with hexahedral elements. RB3 elements are rigid body elements that are ideal for simulating non-deformable connections (e.g., powertrain-frame joints) or for constraining degrees of freedom in complex assemblies and key connection points (e.g., bolted joints between BIW sheet metal parts) were modeled using RBE2 (Rigid Body Element) rigid elements24, where a central master node was connected to multiple slave nodes to enforce localized rigidity and simplify dynamic interactions, as described in the Altair HyperMesh User Manual (Altair, 2023).The model consists of 1,400,411 shell elements and 769,664 solid elements.

Load and boundary condition setting

Unit loads are applied in the X, Y, and Z directions at the key connection points. The load type, loading frequency (spanning from 20 Hz to 500 Hz), response frequency range (with a maximum frequency of 1.5 times the loading frequency), and structural damping (set to 3%) are subsequently defined25. The boundary conditions for the body-in-white are set to an unconstrained, free state. The excitation point is used as the output response point, with the acceleration response corresponding to the excitation direction being recorded. The maximum frequency considered in the BIW modal analysis is 200 Hz.The key connection points of the body-in-white are detailed in Table 1, with their locations illustrated in Fig. 1.

Table 1 Key attachment points for body-in-white.
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Fig. 1
Fig. 1
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The locations of key attachment points on the BIW. This figure was generated using Altair HyperMesh 2021.1 (Altair, USA. https://altair.com/hypermesh).

Calculation of the front cabin

The dynamic stiffness of each attachment point of the front cabin was analyzed, with the results shown in Fig. 2. The dynamic stiffness curves reveal that each attachment point exhibits a valley in the X, Y, and Z directions at frequency points of 54 Hz, 58 Hz, and 74 Hz. This is a common phenomenon.

Fig. 2
Fig. 2
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Dynamic stiffness of the front cabin attachment points.

When combined with the BIW modal analysis, the results show that the first-order bending mode occurs at 55.2 Hz, the front-end transverse mode at 59.4 Hz, and the front cabin torsion mode at 75.5 Hz, as shown in Fig. 3.

Fig. 3
Fig. 3
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BIW mode. Deformation scale: 5. This figure was solved using Altair Compute Console 2021.1 and post-processed in Altair HyperView 2021.1(Altair, USA. https://altair.com/hyperworks). (A) The first-order bending mode. (B) The front-end transverse mode. (C) The front cabin torsion mode.

Consequently, the valley points caused by these global modes can be disregarded when determining the minimum dynamic stiffness. It is also noted that the frequencies do not align exactly due to differences in the truncation frequencies used in the dynamic stiffness and BIW modal calculations.

The dashed lines in Fig. 2 represent the target dynamic stiffness values. The target values are primarily influenced by the load paths and the principal directions of load transfer at different locations and orientations26. Different attachment points of the subframe experience different load paths. For example, the front attachment points mainly transmit vertical engine vibrations, while the rear points are more subjected to lateral forces from the suspension.

The dynamic stiffness targets in the X, Y, and Z directions are set based on the principal load transfer directions.

As the frequency increases, distinguishing between common and individual issues becomes more difficult. For example, the Y-direction dynamic stiffness of the left front (1002) of the front suspension upper swing arm is 8324 N/mm at 120 Hz, as shown in Fig. 4. Table 2 shows that several BIW modes are present near 120 Hz. Due to the truncation of modal frequencies, the dynamic stiffness may not correspond to the actual frequencies of these modes, leading to potential errors in judgment. Common problems are costly to optimize, and the vibration mode cannot be altered after optimization. Missing individual problems can result in unwanted vibration and noise issues.

Fig. 4
Fig. 4
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Dynamic stiffness curve.

Table 2 Calculation results of BIW modal section.
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The modal contribution is calculated, and the result of 120 Hz modal contribution in the 1002Y direction is shown in Fig. 5. Among the closely spaced modes of the front cabin, identified in the 120–125 Hz frequency range, the 46th-order mode has the largest contribution to the response in the Y direction at measurement point 1002 and can therefore be considered a primary contributor to the dynamic stiffness drop near 120 Hz. Upon reviewing the BIW modal analysis results, it is observed that the 46th order vibration mode corresponds to the breathing mode of the entire front cabin, as shown in Fig. 6. This common issue is mitigated through the implementation of frequency avoidance. The simplified flowchart presented in Fig. 7 helps distinguish between overall modes and local modes.

Fig. 5
Fig. 5
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120Hz modal contributions in the 1002Y direction.

Fig. 6
Fig. 6
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BIW 46th order vibration mode. Deformation scale: 5. This figure was solved using Altair Compute Console 2021.1 and post-processed in Altair HyperView 2021.1(Altair, USA. https://altair.com/hyperworks).

Fig. 7
Fig. 7
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Flowchart of distinguishing between overall modes and local modes.

Conclusion

This paper classifies the cases of substandard minimum dynamic stiffness into common and individual problems, providing a foundation for subsequent optimization and improving efficiency. Differentiating between common and individual problems helps prevent wasting time on recurring issues while ensuring that unique problems are not overlooked, which could lead to vibration and noise issues. Additionally, the challenges arising from frequency discrepancies between dynamic stiffness and BIW modes due to mode truncation can be effectively addressed through contribution analysis.