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Nonlinear responses and bifurcations of a rotor-bearing system supported by squeeze-film damper with retainer spring subjected to base excitations

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Abstract

A high-speed rotating rotor system mounted on a moving vehicle is inevitably subjected to parametric excitations and exciting forces induced by base motions. Dynamic characteristics of a rotor-bearing system supported by squeeze-film damper with retainer spring subjected to unbalance and support motions are investigated. Using Lagrange’s principle, equations of motion for rotor system relative to a moving support are derived. Under base excitations, steady-state and transient responses are analyzed by frequency–amplitude curve, waveform, orbit, frequency spectrum, and Poincaré map. Changing with rotating speed or base harmonic frequency, journal motions are analyzed by bifurcation diagram. The results indicate that under base axial rotation, increasing base angular velocity, first two critical speeds decrease but resonant amplitudes increase slightly. The journal whirls around the static eccentricity with noncircular orbit. Under base lateral rotation, critical speeds, and resonant amplitudes remain essentially unchanged, but orbit’s deviation is related to base angular velocity. Excited by base harmonic translation, the integral multiples of fundamental frequency \(k{\varOmega }\left( {k = 1,2} \right)\), base harmonic frequency \({\varOmega}^{z}\), and combined frequencies \(k{\varOmega } \pm j{\varOmega }^{z} { }\left( {k,j = 1,2} \right)\) are stimulated, changing the motions from periodic to quasiperiodic. Overall, it provides a flexible approach with good expandability to predict dynamic characteristics of squeeze-film damped rotor system under base motions.

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Acknowledgements

The authors are grateful for the support from Aerospace Propulsion Institute and Advanced Technology Institute at SUSTech. This research is technically supported by Center for Computational Science and Engineering at Southern University of Science and Technology.

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  1. Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, 518055, China

    Xi Chen, Xiaohua Gan & Guangming Ren

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Appendices

Appendix 1

Generalized inertia matrices for disk element:

$$ \varvec{M}_{{{\text{t}},{\text{d}}}} = \left[ {\begin{array}{*{20}c} {m_{{\mathrm{d}}} } \\ 0 \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {m_{{\mathrm{d}}} } \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right], {{\varvec{M}}}_{\mathrm{r},\mathrm{d}}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{d}}\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{d}}\end{array}\right];$$

Generalized gyroscopic matrix and its component for disk element:

$${{\varvec{G}}}_{\mathrm{d}}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{p}}\end{array} \begin{array}{c}0\\ 0\\ {-I}_{\mathrm{d}}^{\mathrm{p}}\\ 0\end{array}\right], \quad {{\varvec{H}}}_{\mathrm{d}}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{p}}\end{array} \begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]$$

where \({{\varvec{G}}}_{\mathrm{d}}={{\varvec{H}}}_{\mathrm{d}}-{{\varvec{H}}}_{\mathrm{d}}^{\mathrm{T}}\);

Parametric damping matrix, i.e., Coriolis effect matrix, induced by base motions for disk element:

$$ ~\varvec{C}_{{{\text{b}}x,{\text{d}}}} = \left[ {\begin{array}{*{20}c} 0 \\ {2m_{{\mathrm{d}}} } \\ 0 \\ 0 \\ \end{array} ~~\begin{array}{*{20}c} { - 2m_{{\mathrm{d}}} } \\ 0 \\ 0 \\ 0 \\ \end{array} ~~\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ { - \left( {I_{{\mathrm{d}}}^{{\text{p}}} - 2I_{{\mathrm{d}}}^{{\text{d}}} } \right)} \\ \end{array} ~~\begin{array}{*{20}c} 0 \\ 0 \\ {\left( {I_{{\mathrm{d}}}^{{\text{p}}} - 2I_{{\mathrm{d}}}^{{\text{d}}} } \right)} \\ 0 \\ \end{array} } \right] $$

Parametric stiffness matrix induced by base motions for disk element:

$$\begin{aligned}& \varvec{K}_{{{\text{a}},{\text{d}}}} = \left[ {\begin{array}{*{20}c} 0 \\ {m_{{\mathrm{d}}} } \\ 0 \\ 0 \\ \end{array} ~~\begin{array}{*{20}c} { - m_{{\mathrm{d}}} } \\ 0 \\ 0 \\ 0 \\ \end{array} ~~\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ {I_{{\mathrm{d}}}^{{\text{d}}} } \\ \end{array} ~~\begin{array}{*{20}c} 0 \\ 0 \\ { - I_{{\mathrm{d}}}^{{\text{d}}} } \\ 0 \\ \end{array} } \right] , \quad {{\varvec{K}}}_{\mathrm{r},\mathrm{d}}=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{p}}\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{p}}\end{array}\right] \\ &{{\varvec{K}}}_{\mathrm{b}x,\mathrm{d}}=\left[\begin{array}{c}{m}_{\mathrm{d}}\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ {m}_{\mathrm{d}}\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{d}}-{I}_{\mathrm{d}}^{\mathrm{p}}\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{d}}-{I}_{\mathrm{d}}^{\mathrm{p}}\end{array}\right],\end{aligned}$$
$$\begin{aligned} & \varvec{K}_{{{\text{b}}y,{\text{d}}}} = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} ~~\begin{array}{*{20}c} 0 \\ {m_{{\mathrm{d}}} } \\ 0 \\ 0 \\ \end{array} ~~\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} ~~\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ {I_{{\mathrm{d}}}^{{\text{p}}} - I_{{\mathrm{d}}}^{{\text{d}}} } \\ \end{array} } \right] , \quad {{\varvec{K}}}_{\mathrm{b}z,\mathrm{d}}=\left[\begin{array}{c}{m}_{\mathrm{d}}\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{p}}-{I}_{\mathrm{d}}^{\mathrm{d}}\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ 0\end{array}\right]\\ &{{\varvec{K}}}_{\mathrm{b}yz,\mathrm{d}}=\left[\begin{array}{c}0\\ {m}_{\mathrm{d}}\\ 0\\ 0\end{array} \begin{array}{c}{m}_{\mathrm{d}}\\ 0\\ 0\\ 0\end{array} \begin{array}{c}0\\ 0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{p}}-{I}_{\mathrm{d}}^{\mathrm{d}}\end{array} \begin{array}{c}0\\ 0\\ {I}_{\mathrm{d}}^{\mathrm{p}}-{I}_{\mathrm{d}}^{\mathrm{d}}\\ 0\end{array}\right]\end{aligned};$$

External force vectors caused by base motions for disk element:

$$ \varvec{S}_{{V,{\text{d}}}} = \left\{ {\begin{array}{*{20}c} {m_{{\mathrm{d}}} } & \quad 0 & \quad 0 & \quad 0 \\ \end{array} } \right\}^{{\text{T}}} , \quad {{\varvec{S}}}_{W,\mathrm{d}}={\left\{\begin{array}{cccc}0& \quad {m}_{\mathrm{d}}& \quad 0& \quad 0\end{array}\right\}}^{\mathrm{T}}$$
$$ \varvec{S}_{{{\rm B},{\text{d}}}} = \left\{ {\begin{array}{*{20}c} 0 & \quad 0 & \quad {I_{{\mathrm{d}}}^{{\text{p}}} } & \quad 0 \\ \end{array} } \right\}^{{\text{T}}} , \quad {{\varvec{S}}}_{\Gamma ,\mathrm{d}}={\left\{\begin{array}{cccc}0& \quad 0& \quad 0& \quad {I}_{\mathrm{d}}^{\mathrm{p}}\end{array}\right\}}^{\mathrm{T}}$$
$$ \varvec{S}_{{V\Gamma ,{\text{d}}}} = \left\{ {\begin{array}{*{20}c} {m_{{\mathrm{d}}} x_{{\mathrm{d}}} } & \quad 0 & \quad 0 & \quad {I_{{\mathrm{d}}}^{{\text{d}}} } \\ \end{array} } \right\}^{{\text{T}}} , \quad {{\varvec{S}}}_{W{\rm B},\mathrm{d}}={\left\{\begin{array}{cccc}0& \quad {m}_{\mathrm{d}}{x}_{\mathrm{d}}& \quad -{I}_{\mathrm{d}}^{\mathrm{d}}& \quad 0\end{array}\right\}}^{\mathrm{T}}$$

Force coefficients caused by base motions for disk element:

$$ f_{V} = \ddot{y}_{{\mathrm{b}}} + \omega _{{{\text{b}}z}} \dot{x}_{{\mathrm{b}}} - \omega _{{{\text{b}}x}} \dot{z}_{{\mathrm{b}}}, \quad {f}_{W}={\ddot{z}}_{\mathrm{b}}+{\omega }_{\mathrm{b}x}{\dot{y}}_{\mathrm{b}}-{\omega }_{\mathrm{b}y}{\dot{x}}_{\mathrm{b}}$$

Appendix 2

Translational shape function matrix:

$${{\varvec{N}}}_{\mathrm{t}}=\left[\begin{array}{c}{\theta }_{1}\\ 0\end{array} \begin{array}{l}0\\ {\theta }_{1}\end{array} \begin{array}{l}0\\ -{\theta }_{2}\end{array} \begin{array}{l}{\theta }_{2}\\ 0\end{array} \begin{array}{l}{\theta }_{3}\\ 0\end{array} \begin{array}{l}0\\ {\theta }_{3}\end{array} \begin{array}{l}0\\ -{\theta }_{4}\end{array} \begin{array}{l}{\theta }_{4}\\ 0\end{array}\right], \quad \left\{\begin{array}{l}{\theta }_{1}=\left[1-3{\upsilon }^{2}+2{\upsilon }^{3}+\varPhi \left(1-\upsilon \right)\right]/\left(1+\varPhi \right)\\ {\theta }_{2}=\left[\left(\upsilon -2{\upsilon }^{2}+{\upsilon }^{3}\right)l+\varPhi l\left(\upsilon -{\upsilon }^{2}\right)/2\right]/\left(1+\varPhi \right)\\ {\theta }_{3}=\left[3{\upsilon }^{2}-2{\upsilon }^{3}+\varPhi \upsilon \right]/\left(1+\varPhi \right)\\ {\theta }_{4}=\left[\left(-{\upsilon }^{2}+{\upsilon }^{3}\right)l-\varPhi l\left(\upsilon -{\upsilon }^{2}\right)/2\right]/\left(1+\varPhi \right)\end{array}\right.;$$

Rotational shape function matrix:

$$ {\varvec{N}}_{{\mathrm{r}}} = \left[ {\begin{array}{*{20}c} 0 \\ {\varphi_{1} } \\ \end{array} \begin{array}{*{20}c} { - \varphi_{1} } \\ 0 \\ \end{array} \begin{array}{*{20}c} {\varphi_{2} } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {\varphi_{2} } \\ \end{array} \begin{array}{*{20}c} 0 \\ {\varphi_{3} } \\ \end{array} \begin{array}{*{20}c} { - \varphi_{3} } \\ 0 \\ \end{array} \begin{array}{*{20}c} {\varphi_{4} } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {\varphi_{4} } \\ \end{array} } \right],\quad \left\{ {\begin{array}{*{20}c} { \varphi_{1} = \left( {6\upsilon^{2} - 6\upsilon } \right)/\left[ {l\left( {1 + \varPhi } \right)} \right]} \\ {\varphi_{2} = \left[ {\left( {1 - 4\upsilon + 3\upsilon^{2} } \right) + {\varPhi }\left( {1 - \upsilon } \right)} \right]/\left[ {\left( {1 + \varPhi } \right)} \right]} \\ { \varphi_{3} = \left( { - 6\upsilon^{2} + 6\upsilon } \right)/\left[ {l\left( {1 + \varPhi } \right)} \right]} \\ { \varphi_{4} = \left( {3\upsilon^{2} - 2\upsilon + {\varPhi }\upsilon } \right)/\left[ {\left( {1 + \varPhi } \right)} \right]} \\ \end{array} } \right.; $$

Shear shape function matrix:

$$ {\varvec{B}}_{{\mathrm{s}}} = \left[ {\begin{array}{*{20}c} {\zeta_{1} } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {\zeta_{1} } \\ \end{array} \begin{array}{*{20}c} 0 \\ { - \zeta_{2} } \\ \end{array} \begin{array}{*{20}c} {\zeta_{2} } \\ 0 \\ \end{array} \begin{array}{*{20}c} {\zeta_{3} } \\ 0 \\ \end{array} \begin{array}{*{20}c} 0 \\ {\zeta_{3} } \\ \end{array} \begin{array}{*{20}c} 0 \\ { - \zeta_{4} } \\ \end{array} \begin{array}{*{20}c} {\zeta_{4} } \\ 0 \\ \end{array} } \right],\quad \left\{ {\begin{array}{*{20}c} { \zeta_{1} = - \varPhi /\left[ {l\left( {1 + \varPhi } \right)} \right]} \\ { \zeta_{2} = - \varPhi /\left[ {2\left( {1 + \varPhi } \right)} \right]} \\ { \zeta_{3} = \varPhi /\left[ {l\left( {1 + \varPhi } \right)} \right]} \\ { \zeta_{4} = - \varPhi /\left[ {2\left( {1 + \varPhi } \right)} \right]} \\ \end{array} } \right.; $$

Generalized inertia matrices for shaft element:

$$ {\varvec{M}}_{{{\text{t}},{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s){\varvec{N}}_{{\mathrm{t}}} (s){\text{d}}s,\quad {\varvec{M}}_{{{\text{r}},{\text{sh}}}}^{{\text{e}}} = \rho I_{{{\text{sh}}}}^{{\text{d}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s){\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s; $$

Generalized gyroscopic matrix and its component for shaft element:

$$ {\varvec{G}}_{{{\text{sh}}}}^{{\text{e}}} = \rho I_{{{\text{sh}}}}^{{\text{p}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad { - 1} \\ 1 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s,\quad {\varvec{H}}_{{{\text{sh}}}}^{{\text{e}}} = \rho I_{{{\text{sh}}}}^{{\text{p}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad 0 \\ 1 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s; $$

Parametric damping matrix, i.e., Coriolis effect matrix, induced by base motions for shaft element:

\({\varvec{C}}_{{{\text{b}}x,{\text{sh}}}}^{{\text{e}}} = 2\rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad { - 1} \\ 1 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{t}}} (s){\text{d}}s - \rho \left( {I_{{{\text{sh}}}}^{{\text{p}}} - 2I_{{{\text{sh}}}}^{{\text{d}}} } \right) \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad { - 1} \\ 1 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s\);

Parametric stiffness matrix induced by base motions for shaft element:

$$ \begin{aligned} & {\varvec{K}}_{{{\text{a}},{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad { - 1} \\ 1 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{t}}} (s){\text{d}}s + \rho I_{{{\text{sh}}}}^{{\text{d}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad { - 1} \\ 1 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s, \\ & {\varvec{K}}_{{{\text{r}},{\text{sh}}}}^{{\text{e}}} = \rho I_{{{\text{sh}}}}^{{\text{p}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left[ {{\varvec{N}}_{{\mathrm{r}}} (s)} \right]{\text{d}}s, \\ & {\varvec{K}}_{{{\text{b}}x,{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s){\varvec{N}}_{{\mathrm{t}}} (s){\text{d}}s - \rho \left( {I_{{{\text{sh}}}}^{{\text{p}}} - I_{{{\text{sh}}}}^{{\text{d}}} } \right) \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s){\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s, \\ & {\varvec{K}}_{{{\text{b}}y,{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad 0 \\ 0 & \quad 1 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{t}}} (s){\text{d}}s + \rho \left( {I_{{{\text{sh}}}}^{{\text{p}}} - I_{{{\text{sh}}}}^{{\text{d}}} } \right) \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad 0 \\ 0 & \quad 1 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s, \\ & \quad {\varvec{K}}_{{{\text{b}}z,{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 1 & \quad 0 \\ 0 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{t}}} (s){\text{d}}s + \rho \left( {I_{{{\text{sh}}}}^{{\text{p}}} - I_{{{\text{sh}}}}^{{\text{d}}} } \right) \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 1 & \quad 0 \\ 0 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s, \\ & {\varvec{K}}_{{{\text{b}}yz,{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad 1 \\ 1 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{t}}} (s){\text{d}}s + \rho \left( {I_{{{\text{sh}}}}^{{\text{p}}} - I_{{{\text{sh}}}}^{{\text{d}}} } \right) \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left[ {\begin{array}{*{20}c} 0 & \quad 1 \\ 1 & \quad 0 \\ \end{array} } \right]{\varvec{N}}_{{\mathrm{r}}} (s){\text{d}}s; \\ \end{aligned} $$

Bending and shear stiffness matrix for shaft element:

$$ \left[ {{\varvec{K}}_{{\mathrm{B}}} } \right]_{{{\text{sh}}}}^{{\text{e}}} = EI_{{{\text{sh}}}}^{{\text{d}}} \int \limits_{0}^{l} \left[ {{\varvec{N}}_{{\mathrm{r}}}^{\prime} (s)} \right]^{{\text{T}}} {\varvec{N}}_{{\mathrm{r}}}^{\prime} (s){\text{d}}s + \kappa GA \int \limits_{0}^{l} {\varvec{B}}_{{\mathrm{s}}}^{{\text{T}}} (s){\varvec{B}}_{{\mathrm{s}}} (s){\text{d}}s; $$

External force vectors caused by base motions for shaft element:

$$ \begin{aligned} & {\varvec{S}}_{{V,{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left\{ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right\}{\text{d}}s,\quad {\varvec{S}}_{{W,{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left\{ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right\}{\text{ds}}, \\ & {\varvec{S}}_{{{\rm B},{\text{sh}}}}^{{\text{e}}} = \rho I_{{{\text{sh}}}}^{{\text{p}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left\{ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right\}{\text{d}}s,\quad {\varvec{S}}_{{\Gamma ,{\text{sh}}}}^{{\text{e}}} = \rho I_{{{\text{sh}}}}^{{\text{p}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left\{ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right\}{\text{d}}s, \\ & {\varvec{S}}_{{V\Gamma ,{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} x{\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left\{ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right\}{\text{d}}s + \rho I_{{{\text{sh}}}}^{{\text{d}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left\{ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right\}{\text{d}}s, \\ & {\varvec{S}}_{{W{\rm B},{\text{sh}}}}^{{\text{e}}} = \rho A \int \limits_{0}^{l} x{\varvec{N}}_{{\mathrm{t}}}^{{\text{T}}} (s)\left\{ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right\}{\text{d}}s - \rho I_{{{\text{sh}}}}^{{\text{d}}} \int \limits_{0}^{l} {\varvec{N}}_{{\mathrm{r}}}^{{\text{T}}} (s)\left\{ {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} } \right\}{\text{d}}s \\ \end{aligned} $$

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Chen, X., Gan, X. & Ren, G. Nonlinear responses and bifurcations of a rotor-bearing system supported by squeeze-film damper with retainer spring subjected to base excitations. Nonlinear Dyn 102, 2143–2177 (2020). https://doi.org/10.1007/s11071-020-06052-0

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