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A reduced-order model of the three-parameter fluid viscous damper with consideration of fluid compressibility and bellows volume deformation

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Abstract

When the control moment gyroscope, momentum wheel, solar array drive mechanism, antenna drive mechanism, and refrigerator on spacecraft work, they produce micro-vibration, which seriously affects the accuracy of observation equipment. It is an effective way to suppress micro-vibration to install a vibration isolation device at the payload. In this paper, a three-parameter fluid viscous damper (TPFVD) for spacecraft micro-vibration suppression is researched. Considering the compressibility of the fluid and the volume deformation of the bellows, according to the effective area principle of bellows, the bellows is equivalent to a single tube, and a reduced-order model of the vibration isolator is established by using the approximate analytical modeling method. The nonlinearity caused by the inlet and outlet effects of the damping orifice is also considered in the model. The model error is further modified by introducing the correction coefficient of bellows volume deformation. Compared with the FEM model, the validity of the model and the modified method is verified. The results show that the models in the existing literature are applicable only when the frequency is less than 100 Hz. The calculation error of the model proposed in this paper is less than 8.20% in the frequency range of 1–300 Hz, and the maximum error of the unmodified model in the literature can reach 34.28%. In addition, the influences of damping orifice parameters, viscosity, payload mass, and nonlinear characteristics on the force transmissibility and payload displacement of the isolator are analyzed.

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Derived data supporting the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

This work was supported by the Provincial Joint Fund of Guangdong Province (No. 2020A1515110582), Natural Science Foundation of Guangdong Province (No. 2021A1515012051), and National Natural Science Funds of China (No. 62022013).

Funding

This work was funded by the Provincial Joint Fund of Guangdong Province (Grant No. 2020A1515110582), Natural Science Foundation of Guangdong Province (Grant No. 2021A1515012051), and National Natural Science Funds for Excellent Yong Scholars of China (Grant No. 62022013).

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Authors and Affiliations

  1. School of Aeronautics and Astronautics, Sun Yat-Sen University, Shenzhen, 518107, China

    Xiaolei Jiao & Jinxiu Zhang

  2. Beijing Institute of Control Engineering, Beijing, 100190, China

    Wenbo Li

  3. China Aerospace Science and Technology Corporation, Beijing, 100048, China

    Jingjing Mu

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  1. Xiaolei Jiao
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  2. Jinxiu Zhang
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  3. Wenbo Li
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Correspondence to Wenbo Li.

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Appendix

Appendix

The expressions of the coefficient \(\lambda_{1}\) and \(\lambda_{2}\) in Eq. (18) are as follows

$$ \begin{aligned} \lambda_{1} & = \frac{{2GF_{0} }}{{x_{0} }} \\ & \quad \cdot \frac{{\left\{ {4\omega^{2} \left[ {m_{2} \left( {x_{3}^{2} - \omega^{2} + y_{3}^{2} } \right) + x_{3} \left( {m_{2}^{2} - \omega^{2} + n_{2}^{2} } \right)} \right]} \right\} \cdot \left( { - \omega } \right)}}{\begin{gathered} + \left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \end{gathered} } \\ \end{aligned} $$
$$ \lambda_{2} = \frac{{\left\{ {2\omega \left[ {\left( {x_{3}^{2} - \omega^{2} + y_{3}^{2} } \right)\left( {m_{2}^{2} - \omega^{2} + n_{2}^{2} } \right) - 4m_{2} x_{3} \omega^{2} } \right]} \right\} \cdot \left( {\frac{{A_{g} }}{{\rho L_{g} }}} \right)}}{\begin{gathered} \left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ + \left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \end{gathered} } $$

The expressions of the coefficient \(\lambda_{3}\) and \(\lambda_{4}\) in Eq. (19) are as follows:

$$ \begin{aligned} \lambda_{3} & = \frac{{ - 2GF_{0} \omega }}{{x_{0} }} \\ & \quad \cdot \frac{{\left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\} \cdot \left( {\frac{{A_{g} }}{{\rho L_{g} }}} \right)}}{\begin{gathered} \left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ + \left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \end{gathered} } \\ \end{aligned} $$
$$ \begin{aligned} \lambda_{4} & = \frac{{ - 2GF_{0} \omega }}{{x_{0} }} \\ & \quad \cdot \frac{{\left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\} \cdot \left( { - \omega } \right)}}{\begin{gathered} \left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ + \left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \end{gathered} } \\ \end{aligned} $$

The expressions of the coefficient \(\lambda_{5}\) and \(\lambda_{6}\) in Eq. (20) are as follows:

$$ \begin{aligned} \lambda_{5} & = \frac{{2GF_{0} }}{{x_{0} }} \\ & \quad \cdot \frac{{\left\{ {4\omega^{2} \left[ {m_{2} \left( {x_{3}^{2} - \omega^{2} + y_{3}^{2} } \right) + x_{3} \left( {m_{2}^{2} - \omega^{2} + n_{2}^{2} } \right)} \right]} \right\} \cdot \left( { - \omega } \right)}}{\begin{gathered} \left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ + \left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \end{gathered} } \\ \end{aligned} $$
$$ \begin{aligned} \lambda_{6} & = \frac{{2GF_{0} }}{{x_{0} }} \\ & \quad \cdot \frac{{\left\{ {2\omega \left[ {\left( {x_{3}^{2} - \omega^{2} + y_{3}^{2} } \right)\left( {m_{2}^{2} - \omega^{2} + n_{2}^{2} } \right) - 4m_{2} x_{3} \omega^{2} } \right]} \right\} \cdot \left( {\frac{{A_{g} }}{{\rho L_{g} }}} \right)}}{\begin{gathered} \left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ + \left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \end{gathered} } \\ \end{aligned} $$

The expressions of the coefficient \(\lambda_{7}\) and \(\lambda_{8}\) in Eq. (21) are as follows:

$$ \begin{aligned} \lambda_{7} & = \frac{{ - 2GF_{0} \omega }}{{x_{0} }} \\ & \quad \cdot \frac{{\left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\} \cdot \left( {\frac{{A_{g} }}{{\rho L_{g} }}} \right)}}{\begin{gathered} \left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ + \left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \end{gathered} } \\ \end{aligned} $$
$$ \begin{aligned} \lambda_{8} & = \frac{{ - 2GF_{0} \omega }}{{x_{0} }} \\ & \quad \cdot \frac{{\left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\} \cdot \left( { - \omega } \right)}}{\begin{gathered} \left\{ {2\left[ {\omega \left( {x_{1} + x_{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right) + 2m_{1} \omega \left( {x_{1} x_{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ + \left\{ {2\left[ { - 2m_{1} \omega^{2} \left( {x_{1} + x_{2} } \right) + \left( {x_{1} x_{2} - \omega^{2} } \right)\left( {m_{1}^{2} + n_{1}^{2} - \omega^{2} } \right)} \right]} \right\}^{2} \hfill \\ \end{gathered} } \\ \end{aligned} $$

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Jiao, X., Zhang, J., Li, W. et al. A reduced-order model of the three-parameter fluid viscous damper with consideration of fluid compressibility and bellows volume deformation. Nonlinear Dyn 111, 7027–7053 (2023). https://doi.org/10.1007/s11071-022-08211-x

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