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Simultaneous Identification of Vertical and Horizontal Complex Stiffness of Preloaded Rubber Mounts: Transformation of Frequency Response Functions and Decoupling of Degrees of Freedom

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Abstract

Background

Rubber mounts are widely used to isolate vibrating components. Their complex stiffness characteristics, including dynamic stiffness and loss factors, are highly concerning in terms of vibration analysis and optimization. Rubber mounts show non-linear behavior with preload, leading to difficulty to predict their complex stiffness. Dynamic testing is generally necessary.

Objective

An approach to identify the complex stiffness of preloaded rubber mounts in both vertical and horizontal directions simultaneously is developed.

Methods

Tested Frequency Response Functions (FRF) of a mass suspended by rubber mounts are transformed to an FRF matrix of the mass center to decouple the Z Degrees of Freedom (DOF) and RZ DOF from other DOFs, which allows complex stiffness to be identified from the two decoupled DOFs. A software tool to implement automatically the FRF transformation and parameter identification is developed. An EPDM rubber mount is tested using the device and its complex stiffness is identified using the software to validate the proposed approach.

Results

The driving-point FRFs of the mass center calculated from the identified complex stiffness are very close to the corresponding FRFs determined from the test data. The comparison between the Finite-Element Analysis (FEA) results of the surficial FRFs and the test results shows good consistency as well. Therefore, the proposed approach and its supporting algorithm are validated.

Conclusion

the proposed approach allows for swift identification of high-accuracy complex stiffness of preloaded rubber mounts in both vertical and horizontal directions simultaneously.

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Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

DMA:

Dynamic mechanical analyzer

DOF:

Degree of freedom

FEA:

Finite element analysis

FRF:

Frequency response function

EPDM:

Ethylene Propylene Diene Monomer

NVH:

Noise, Vibration, Harshness

RMS:

Root mean square

SNR:

Signal-to-noise ratio

\(\overline{{\varvec{a}} }\) :

\(\{{\overline{a} }_{x}, {\overline{a} }_{y}, {\overline{a} }_{z}, {\overline{\gamma }}_{x}, {\overline{\gamma }}_{y}, {\overline{\gamma }}_{z}{\}}^{\mathrm{T}}\), Six-DOF acceleration of the mass center

\({{\varvec{a}}}_{ik}\) :

Acceleration of channel i in the impact test of the k-th impact point

\({{\varvec{a}}}_{k}\) :

\({\left\{{a}_{1k}, {a}_{2k}, \cdots , {a}_{Nk}\right\}}^{\mathrm{T}}\), Acceleration of all channels excited at the k-th impact point

\({\overline{{\varvec{a}}} }_{k}\) :

\(\{{\overline{a} }_{xk}, {\overline{a} }_{yk}, {\overline{a} }_{zk}, {\overline{\gamma }}_{xk}, {\overline{\gamma }}_{yk}, {\overline{\gamma }}_{zk}{\}}^{\mathrm{T}}\), Acceleration of mass center excited at the k-th impact point

\({{\varvec{D}}}_{i}\) :

\({\left\{{d}_{xi},{d}_{yi},{d}_{zi}\right\}}^{\mathrm{T}}\), Normalized vector along the acceleration of channel i

\({{\varvec{D}}}_{k}{\prime}\) :

\({\left\{{d}_{xk}{\prime},{d}_{yk}{\prime},{d}_{zk}{\prime}\right\}}^{\mathrm{T}}\), Normalized vector along the impact force of the k-th impact point

\(\overline{{\varvec{f}} }\) :

\(\{{\overline{f} }_{x},{\overline{f} }_{y},{\overline{f} }_{z},{\overline{m} }_{x},\boldsymbol{ }{\overline{m} }_{y},\boldsymbol{ }{\overline{m} }_{z}{\}}^{\mathrm{T}}\), Six-DOF excitation at the mass center

\({f}_{k}{\prime}\) :

Impact force of the k-th impact point

\({\overline{{\varvec{f}}} }_{k}\) :

\({\left\{{\overline{f} }_{xk},{\overline{f} }_{yk},{\overline{f} }_{zk},{\overline{m} }_{xk},\boldsymbol{ }{\overline{m} }_{yk},\boldsymbol{ }{\overline{m} }_{zk} \right\}}^{\mathrm{T}}\), Equivalent impact load acting on the mass center in the impact test of the k-th impact point

\(g\) :

Loss factor

\({g}_{x}\), \({g}_{z}\) :

Loss factors in horizontal and vertical directions, respectively

\({\varvec{H}}\) :

\(\left[{{\varvec{H}}}_{1},{{\varvec{H}}}_{2},\cdots ,{{\varvec{H}}}_{M}\right]\), FRF matrix of all acceleration channels excited at the M impact points

\(\overline{{\varvec{H}} }\) :

Driving-point FRF matrix of mass center from transformation

\({\overline{{\varvec{H}}} }^{\boldsymbol{^{\prime}}}\) :

Driving-point FRF matrix of mass center calculated from the complex stiffness

\({{\varvec{h}}}_{k}\) :

\({\left\{{h}_{1k},{h}_{2k},\cdots ,{h}_{Nk}\right\}}^{\mathrm{T}}\), FRF matrix of all acceleration channels excited at the k-th impact point

\({\overline{h} }_{33}\) :

Third diagonal element of the matrix \(\overline{{\varvec{H}} }\)

\({\overline{h} }_{66}\) :

Sixth diagonal element of the matrix \(\overline{{\varvec{H}} }\)

\({\left|{\overline{h} }_{33}\right|}_{\mathrm{p}}\) :

Peak value of \(\left|{\overline{h} }_{33}\left(\omega \right)\right|\)

\({\left|{\overline{h} }_{66}\right|}_{\mathrm{p}}\) :

Peak value of \(\left|{\overline{h} }_{66}\left(\omega \right)\right|\)

\(k\) :

Dynamic stiffness

\(K\) :

Complex stiffness

\({\varvec{K}}\) :

Complex stiffness matrix

\({K}_{x}\), \({K}_{y}\), \({K}_{z}\) :

Translational complex stiffness in XY, and Z directions, respectively

\({K}_{\theta }\) :

Angular complex stiffness about the axis of the rubber mount

\({k}_{x}\), \({k}_{z}\) :

Dynamic stiffness in horizontal and vertical directions, respectively

\(l\) :

Height of the mass center from the supporting surface of the rubber mounts

M :

Number of impact points

\({\varvec{M}}\) :

\(\mathrm{d}iag (m, m, m, {I}_{x}, {I}_{y}, {I}_{z})\), mass matrix

N :

Acceleration channel quantity

\(r\) :

Distance between the axis of the mass assembly and the axis of any rubber mount

\(R\) :

Distance between the compressor axis and the mount axis

\({{\varvec{T}}}_{\mathrm{a}}\) :

Transformation matrix for acceleration

\({{\varvec{T}}}_{\mathrm{f}}\) :

\(\left[{{\varvec{T}}}_{\mathrm{f}1},{{\varvec{T}}}_{\mathrm{f}2},\cdots ,{{\varvec{T}}}_{\mathrm{f}M}\right]\), transformation matrix for impact forces of all impact points

\({{\varvec{T}}}_{\mathrm{f}k}\) :

Transformation matrix for impact force of the k-th impact point

\(\overline{{\varvec{u}} }\) :

\(\{{\overline{u} }_{x},\boldsymbol{ }{\overline{u} }_{y},{\overline{u} }_{z},\boldsymbol{ }{\overline{\theta }}_{x},\boldsymbol{ }\boldsymbol{ }{\overline{\theta }}_{y},\boldsymbol{ }{\overline{\theta }}_{z}{\}}^{\mathrm{T}}\), six-DOF displacement of the mass center

\(\overline{x },\overline{y },\overline{z }\) :

Coordinates of mass center

\({x}_{i},{y}_{i},{z}_{i}\) :

Accelerometer coordinates of channel i

\({x}_{k}{\prime},{y}_{k}{\prime},{z}_{k}{\prime}\) :

Coordinates of the k-th impact point

\({\lambda }_{33}\) :

Frequency ratio of Z DOF

\({\lambda }_{33,\mathrm{p}}\) :

Frequency ratio at the peak of \(\left|{\overline{h} }_{33}\left(\omega \right)\right|\) curve

\(\Delta \theta\) :

Angular displacement of a compressor about its axis

\(\omega\) :

Angular frequency in rad/s

\({\omega }_{33,\mathrm{p}}\) :

Angular frequency at the peak of \(\left|{\overline{h} }_{33}\left(\omega \right)\right|\)

\({\omega }_{66,\mathrm{p}}\) :

Angular frequency at the peak of \(\left|{\overline{h} }_{66}\left(\omega \right)\right|\)

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Acknowledgements

The study received financial support from Scientific and Technological Innovation Foundation of Foshan, USTB (BK22BE023), Major Fundamental Research of Equipment (514010208-301), International Science and Technology Cooperation Project of Huangpu (2021GH13), and GDAS' Project of Science and Technology Development (2022GDASZH- 2022010108).

Funding

Scientific and Technological Innovation Foundation of Foshan, USTB, BK22BE023, Dong Xiang, Major Fundamental Research of Equipment, 514010208-301, Danfeng Long, International Science and Technology Cooperation Project of Huangpu, 2021GH13, Danfeng Long, GDAS' Project of Science and Technology Development, 2022 GDASZH- 2022010108, Danfeng Long.

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Authors and Affiliations

  1. Institute of Intelligent Manufacturing, Guangdong Academy of Sciences, Guangzhou, China

    D. Long, Q. Chen & H. Zhang

  2. Guangdong Key Laboratory of Modern Control Technology, Guangzhou, China

    D. Long, Q. Chen & H. Zhang

  3. University of Science and Technology Beijing, Beijing, China

    D. Xiang

  4. Shunde Innovation School, University of Science and Technology Beijing, Foshan, China

    D. Xiang

  5. GD Midea Heating & Ventilating Equipment Co., Ltd., Foshan, China

    M. Zhong

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  1. D. Long
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Long, D., Chen, Q., Xiang, D. et al. Simultaneous Identification of Vertical and Horizontal Complex Stiffness of Preloaded Rubber Mounts: Transformation of Frequency Response Functions and Decoupling of Degrees of Freedom. Exp Mech 63, 1479–1492 (2023). https://doi.org/10.1007/s11340-023-01002-4

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