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Nonlinear mode coupling in a passively isolated mechanical system

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Abstract

Recent studies in passively isolated systems have shown that mode coupling is desirable for better vibration suppression, thus refuting the long-standing rule of modal decoupling. However, these studies have ignored the nonlinearities in the isolators. In this work, we consider stiffness nonlinearity from pneumatic isolators and study the nonlinear forced damped vibrations of a passively isolated ultra-precision manufacturing (UPM) machine. Experimental analysis is conducted to guide the mathematical formulation. The system comprises linearly and nonlinearly coupled in-plane horizontal and rotational motion of the UPM machine with quadratic nonlinear stiffness from the isolators. We present an analytical study using the method of multiple scales and the method of harmonic balance for different cases of external resonances, viz., the primary and the secondary resonances (superharmonic and combined resonances) with 1 : 2 internal resonance between the modes. We further validate our analytical findings using direct numerical integration and observe an excellent match. On extending our analysis, we observe the existence of subcritical, supercritical, and s-shaped bifurcation depending on the location of the isolators and the case of external resonance. Also, the saturation and quasi-saturation phenomenon are observed for the case of resonances close to the higher natural frequency and combined resonance, respectively. A parametric study is conducted to examine the effect of different parameters on the dynamics of the system, and consecutively the critical parameters of the system are identified for different cases of external resonance.

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Acknowledgements

This work is funded by National Science Foundation (NSF) Award CMMI #2000984: Nonlinear Dynamics of Pneumatic Isolators in Ultra-Precising Manufacturing Machine.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA, 24061, USA

    Sunit K. Gupta, Mohammad A. Bukhari & Oumar R. Barry

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  1. Sunit K. Gupta
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  2. Mohammad A. Bukhari
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  3. Oumar R. Barry
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Correspondence to Oumar R. Barry.

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Appendices

Appendix A: primary resonance near \(\omega _1\)

$$\begin{aligned}&D_{0,0}{\eta }_1+\beta ^2(\eta _1-\alpha \vartheta _1)\nonumber \\&\quad =-2i\left( {D_1 A_1}{\varLambda _1}+{A_1}\zeta \beta {\varLambda _1}-{A_1}\zeta \,\beta \,\alpha \right) {\omega _1}e^{i\omega _1T_0}\nonumber \\&\qquad -2i \left( D_1A_2{\varLambda _2}+{A_2}\zeta \beta \,{ \varLambda _2}\nonumber \right. \\&\left. \quad -{A_2} \zeta \beta \alpha \right) {\omega _2}e^{i\omega _2T_0}\nonumber \\&\qquad - {A_1}^{2}{\alpha _1}{ m_r}\left( \alpha -{\varLambda _1} \right) ^{2}e^{2i\omega _1T_0}\nonumber \\&\qquad -{A_2}^{2}{\alpha _1}{ m_r}\left( \alpha -{\varLambda _2} \right) ^{2}e^{2i\omega _2T_0}\nonumber \\&\qquad -2{A_1} {{\bar{A}}_2}{ \alpha _1}{m_r}\left( \alpha - {\varLambda _1} \right) \left( \alpha -{{\bar{\varLambda }}_2}\right) e^{i\left( \omega _1-\omega _2\right) T_0}\nonumber \\&\qquad +2\,{A_1} {A_2}{ \alpha _1}{m_r}\left( \alpha - {\varLambda _1} \right) \left( \alpha -{\varLambda _2} \right) e^{i\left( \omega _1+\omega _2\right) T_0}\nonumber \\&\qquad +2\,{A_1}{{\bar{A}}_1}{ \alpha _1}{m_r} \left( \alpha -{\varLambda }_1 \right) \left( \alpha -{{\bar{\varLambda }}_1} \right) \nonumber \\&\qquad +2{A_2}{{\bar{A}}_4}{ \alpha _1}{m_r}\left( \alpha -{\varLambda _2} \right) \left( \alpha -{{\bar{\varLambda }}_2} \right) \nonumber \\&\qquad +\frac{f_0}{2}m_re^{i\left( \omega _1T_0+\sigma _1T_1\right) }+ C.C.\,, \end{aligned}$$
(56a)
$$\begin{aligned}&D_{0,0}\vartheta _1+{\vartheta }_{{1}}-{k_r}\alpha \left( \eta _1-\alpha \vartheta _1\right) \nonumber \\&\quad =-2i\omega _1\left( D_1{A_1}+{\sqrt{\frac{k_r}{m_r}}} {\zeta {\alpha }^{2}{A_1}}\nonumber \right. \\&\left. \qquad +\kappa {A_1} -{\sqrt{\frac{k_r}{m_r}}} {\zeta {\alpha }{A_1}\varLambda _1}\right) e^{i\omega _1T_0}\nonumber \\&\qquad -2i\omega _2\left( D_1{A_2} +{\sqrt{\frac{k_r}{m_r}}} {\zeta {\alpha }^{2}{A_2}}\nonumber \right. \\&\qquad \left. +\kappa {A_2} -{\sqrt{\frac{k_r}{m_r}}} {\zeta {\alpha }{A_2}\varLambda _2}\right) e^{i\omega _2T_0}\nonumber \\&\qquad +{A_1}^{2} \left( \alpha _1\alpha (\alpha -\varLambda _1)^2-q_{zr} \right) e^{2i\omega _1T_0}\nonumber \\&\qquad +{A_2}^{2} \left( \alpha _1\alpha (\alpha -\varLambda _2)^2-q_{zr} \right) e^{2i\omega _2T_0}\nonumber \\&\qquad +2{A_1}{{\bar{A}}_2}\left( \alpha _1\alpha (\alpha -\varLambda _1)(\alpha -{\bar{\varLambda }}_2) -{q_{zr}} \right) e^{i\left( \omega _1-\omega _2\right) T_0}\nonumber \\&\qquad +2{A_1}{A_2}\left( \alpha _1\alpha (\alpha -\varLambda _1)(\alpha -{\varLambda }_2) -{q_{zr}} \right) e^{i(\omega _1+\omega _2)T_0}\nonumber \\&\qquad +2\,{A_1}{{\bar{A}}_1} \left( \alpha _1\alpha (\alpha -\varLambda _1)(\alpha -{\bar{\varLambda }}_1)-q_{zr} \right) \nonumber \\&\qquad +2\,{A_2}{{\bar{A}}_2}\nonumber \\&\qquad \left( \alpha _1\alpha (\alpha -\varLambda _2)(\alpha -{\bar{\varLambda }}_2)-q_{zr} \right) \nonumber \\&\qquad +\frac{f_0}{2}\alpha e^{i(\omega _1T_0+\sigma _1T_1)}+C.C.\,, \varGamma _1\nonumber \\&\quad =\frac{4\omega _1\zeta \left( l_1\beta -\alpha \sqrt{\frac{k_r}{m_r}} \right) (\alpha -\varLambda _1)-4\kappa \omega _1}{4\omega _1(1+l_1\varLambda _1)}\nonumber \\&\varGamma _2=\frac{\alpha _1\left( \alpha -l_1m_r\right) (\alpha -\varLambda _2)^2-q_{zr}}{4\omega _1(1+l_1\varLambda _1)}\nonumber \\&\varGamma _3=\frac{2(l_1m_r-\alpha )}{4\omega _1(1+l_1\varLambda _1)}\nonumber \\&\varGamma _4=\frac{\alpha _1(l_2m_r-\alpha )(\alpha -{\bar{\varLambda }}_2)(\alpha -\varLambda _1)+q_{zr} }{2\omega _2(1+l_2\varLambda _2)}\nonumber \\&\varGamma _5=\frac{2\zeta \omega _2\left( l_2\beta -\alpha \sqrt{\frac{k_r}{m_r}}\right) (\alpha -\varLambda _2)-2\kappa \omega _2}{4\omega _2(1+l_2\varLambda _2)} \end{aligned}$$
(56b)

Appendix B: primary resonance near \(\omega _2\)

$$\begin{aligned}&{\mathcal {O}}(\epsilon ^0)\quad : D_{0,0}{\eta }_{{0}} +{\beta }^{2}\left( {\eta }_{{0}} -\alpha \,{\vartheta }_{{0}}\right) =0\,, \end{aligned}$$
(57a)
$$\begin{aligned}&D_{0,0}{\vartheta }_{{0}} +{\vartheta }_{{0}} -{k_r}\,\alpha \left( {\eta }_{{0}} -{\alpha }{\vartheta }_{{0}} \right) =0 \end{aligned}$$
(57b)
$$\begin{aligned}&{\mathcal {O}}(\epsilon )\quad : D_{0,0}{\eta }_1+\beta ^2(\eta _1-\alpha \vartheta _1)\nonumber \\&\quad =-2\zeta \beta \left( D_0\eta _0-\alpha D_0\vartheta _0\right) \nonumber \\&\qquad -\alpha _1 m_r\left( \eta _0-\alpha \vartheta _0\right) ^2- 2D_{0,1}\eta _0\nonumber \\&\qquad +f_0 m_r \cos (\omega _2T_0+\sigma _1T_1) \end{aligned}$$
(58a)
$$\begin{aligned}&D_{0,0}\vartheta _1+{\vartheta }_{{1}}-{k_r}\alpha \left( \eta _1-\alpha \vartheta _1\right) \nonumber \\&\quad =2\zeta \sqrt{\frac{k_r}{m_r}}\alpha \left( D_0\eta _0-\alpha D_0\vartheta _0\right) -2D_{0,1}\vartheta _0\nonumber \\&\qquad +{\alpha _1}{\alpha }(\eta _0-\alpha \vartheta _0)^{2}-{q_{zr}}{\vartheta }_{{0}}^{2}\nonumber \\&\qquad -2\kappa D_0\vartheta _0-{f_0}\alpha \cos \left( {\omega _2}\,T_{{0}} +\sigma _1T_1\right) \,. \end{aligned}$$
(58b)

Appendix C: secondary resonance

$$\begin{aligned}&D_{0,0}{\eta }_1+\beta ^2(\eta _1-\alpha \vartheta _1)\nonumber \\&\quad =-2i\left( {D_1 A_1}{\varLambda _1}+{A_1}\zeta \beta {\varLambda _1}-{A_1}\zeta \,\beta \,\alpha \right) {\omega _1}e^{i\omega _1T_0}\nonumber \\&\qquad -2i \left( D_1A_2{\varLambda _2}+{A_2}\zeta \beta \,{ \varLambda _2}-{A_2} \zeta \beta \alpha \right) {\omega _2}e^{i\omega _2T_0}\nonumber \\&\qquad - {A_1}^{2}{\alpha _1}{ m_r}\left( \alpha -{\varLambda _1} \right) ^{2}e^{2i\omega _1T_0}-{A_2}^{2}{\alpha _1}{ m_r}\nonumber \\&\qquad \left( \alpha -{\varLambda _2} \right) ^{2}e^{2i\omega _2T_0}\nonumber \\&\qquad -2{A_1} {{\bar{A}}_2}{ \alpha _1}{m_r}\left( \alpha - {\varLambda _1} \right) \left( \alpha -{{\bar{\varLambda }}_2}\right) e^{i\left( \omega _1-\omega _2\right) T_0}\nonumber \\&\qquad +2\,{A_1} {A_2}{ \alpha _1}{m_r}\left( \alpha - {\varLambda _1} \right) \left( \alpha -{\varLambda _2} \right) e^{i\left( \omega _1+\omega _2\right) T_0}\nonumber \\&\qquad +{A_1}{\alpha _1}{m_r}{f_0} \left( \alpha -{\varLambda _1}\right) \left( {\varPhi _1}-\alpha {\varPhi _2} \right) e^{i\left( \omega _1-\omega _r\right) T_0}\nonumber \\&\qquad +{A_1} {\alpha _1}{m_r}{f_0} \left( \alpha -{\varLambda _1} \right) \left( {\varPhi _1}-\alpha {\varPhi _2} \right) e^{i\left( \omega _1+\omega _r\right) T_0}\nonumber \\&\qquad +{A_2}{\alpha _1}{m_r}{f_0} \left( \alpha -{\varLambda _2} \right) \left( {\varPhi _1}-\alpha {\varPhi _2} \right) e^{i\left( \omega _2-\omega _r\right) T_0}\nonumber \\&\qquad +{A_2} {\alpha _1}{m_r}{f_0} \left( \alpha -{\varLambda _2} \right) \left( {\varPhi _1}-\alpha {\varPhi _2} \right) e^{i\left( \omega _2+\omega _r\right) T_0}\nonumber \\&\qquad -\frac{1}{4}f^2_0\alpha _1m_r(\varPhi _1-\alpha \varPhi _2)^2e^{2i\omega _r T_0}\nonumber \\&\qquad -i \left( \varPhi _1-{\varPhi _2}\,\alpha \right) \beta {f_0}{ \omega _r}\,\zeta e^{i\omega _r T_0}\nonumber \\&\qquad +C.C.+CVF_1 \end{aligned}$$
(59)
$$\begin{aligned}&D_{0,0}\vartheta _1+{\vartheta }_{{1}}-{k_r}\alpha \left( \eta _1-\alpha \vartheta _1\right) \nonumber \\&\quad =-2i\omega _1\left( D_1{A_1}+{\sqrt{\frac{k_r}{m_r}}} {\zeta {\alpha }^{2}{A_1}}\nonumber \right. \\&\qquad \left. +\kappa {A_1} -{\sqrt{\frac{k_r}{m_r}}} {\zeta {\alpha }{A_1}\varLambda _1}\right) e^{i\omega _1T_0}\nonumber \\&\qquad -2i\omega _2\left( D_1{A_2} +{\sqrt{\frac{k_r}{m_r}}} {\zeta {\alpha }^{2}{A_2}}\nonumber \right. \\&\qquad \left. +\kappa {A_2} -{\sqrt{\frac{k_r}{m_r}}} {\zeta {\alpha }{A_2}\varLambda _2}\right) e^{i\omega _2T_0}\nonumber \\&\qquad +{A_1}^{2} \left( \alpha _1\alpha (\alpha -\varLambda _1)^2-q_{zr} \right) e^{2i\omega _1T_0}\nonumber \\&\qquad +{A_2}^{2} \left( \alpha _1\alpha (\alpha -\varLambda _2)^2-q_{zr} \right) \nonumber \\&\qquad e^{2i\omega _2T_0}\nonumber \\&\qquad +2{A_1}{{\bar{A}}_2}\left( \alpha _1\alpha (\alpha -\varLambda _1)(\alpha -{\bar{\varLambda }}_2) -{q_{zr}} \right) \nonumber \\&\qquad e^{i\left( \omega _1-\omega _2\right) T_0}\nonumber \\&\qquad +2{A_1}{A_2}\left( \alpha _1\alpha (\alpha -\varLambda _1)(\alpha -{\varLambda }_2) -{q_{zr}} \right) \nonumber \\&\qquad e^{i(\omega _1+\omega _2)T_0}\nonumber \\&\qquad -A_1f_0\left( {\alpha _1}\alpha \left( \alpha -{\varLambda _1} \right) \left( \varPhi _1-{ \varPhi _2}\,\alpha \right) +\varPhi _2q_{zr}\right) \nonumber \\&\qquad e^{i(\omega _1-\omega _r)T_0}\nonumber \\&\qquad -A_1f_0\left( {\alpha _1}\alpha \left( \alpha -{\varLambda _1} \right) \left( \varPhi _1-{ \varPhi _2}\,\alpha \right) +\varPhi _2q_{zr}\right) \nonumber \\&\qquad e^{i(\omega _1+\omega _r)T_0}\nonumber \\&\qquad -A_2f_0\left( {\alpha _1}\alpha \left( \alpha -{\varLambda _2} \right) \left( \varPhi _1-{ \varPhi _2}\,\alpha \right) +\varPhi _2q_{zr}\right) \nonumber \\&\qquad e^{i(\omega _2-\omega _r)T_0}\nonumber \\&\qquad -A_2f_0\left( {\alpha _1}\alpha \left( \alpha -{\varLambda _2} \right) \left( \varPhi _1-{ \varPhi _2}\,\alpha \right) +\varPhi _2q_{zr}\right) \nonumber \\&\qquad e^{i(\omega _2+\omega _r)T_0}\nonumber \\&\qquad +\frac{1}{4}f^2_0\left( \alpha _1\alpha (\varPhi _1-\alpha \varPhi _2)^2-q_{zr}\varPhi ^2_2\right) e^{2i\omega _rT_0} \nonumber \\&\qquad +i \left( -{\varPhi _2}\kappa -{\varPhi _2}{\alpha }^{2}{\zeta }{\sqrt{\frac{k_r}{m_r}}}\nonumber \right. \\&\left. \qquad + \alpha {\zeta }{\sqrt{\frac{k_r}{m_r}}}{\varPhi _1} \right) {f_0}\,{\omega _r}e^{i\omega _rT_0}+C.C.+CVF_2 \end{aligned}$$
(60)

Appendix D: superharmonic resonance near \(\omega _1\)

$$\begin{aligned} \varGamma _1&=\frac{4\zeta l_1\beta \omega _1(\alpha -\varLambda _1)-4\alpha \sqrt{\frac{k_r}{m_r}}\zeta \omega _1 (\alpha -\varLambda _1)-4\kappa \omega _1}{4\omega _1(1+l_1\varLambda _1)}\,,\nonumber \\ \varGamma _2&=\frac{-\alpha _1(\alpha -\varLambda _2)^2(l_1m_r-\alpha )-q_{zr}}{4\omega _1(1+l_1\varLambda _1)}\,,\nonumber \\ \varGamma _3&=\frac{-\alpha _1(\varPhi _1-\alpha \varPhi _2)^2(l_1m_r-\alpha )-q_{zr}\varPhi ^2_2}{4\omega _1(1+l_1\varLambda _1)}\,,\nonumber \\ \varGamma _4&=\frac{\alpha _1(\alpha -{\bar{\varLambda }}_2)(\alpha -\varLambda _1)(l_2m_r-\alpha )+q_{zr}}{2\omega _2(1+l_2\varLambda _2)}\,,\nonumber \\ \varGamma _5&=\frac{2\zeta l_2\beta \omega _2(\alpha -\varLambda _2)-2\alpha \sqrt{\frac{k_r}{m_r}}\zeta \omega _2 (\alpha -\varLambda _2)-2\kappa \omega _2}{2\omega _1(1+l_2\varLambda _2)}\,. \end{aligned}$$
(61)

Appendix E: combined resonance

$$\begin{aligned} \varGamma _1&={\frac{-4\,\kappa \,{\omega _1}-4\,\alpha \,{\zeta }\sqrt{\frac{k_r}{m_r}}{\omega _1 }\, \left( \alpha -\varLambda _1\right) +4 \zeta { l_1}\,\beta \,{\omega _1}\, \left( \alpha -\varLambda _1 \right) }{4\,{\omega _1}\,{l_1}\,{\varLambda _1}+4\,{\omega _1}}}\\ \varGamma _2&=\frac{-\alpha _1(\alpha -\varLambda _2)^2(l_1m_r-\alpha )-q_{zr}}{4\omega _1(1+l_1\varLambda _1)}\\ \varGamma _3&=\frac{-2\,{\alpha _1}\, \left( \alpha -{{\bar{\varLambda }}_2}\right) \left( \alpha -{ l_1}\,{m_r} \right) \left( {\varPhi _1} -{\varPhi _2}\,\alpha \right) -2\,{q_{zr}}\,{\varPhi _2} }{4\omega _1(1+l_1\varLambda _1)}\\ \varGamma _4&=\frac{\alpha _1(\alpha -{\bar{\varLambda }}_2)(\alpha -\varLambda _1)(l_2m_r-\alpha )+q_{zr}}{2\omega _2(1+l_2\varLambda _2)}\\ \varGamma _5&=\frac{-{\alpha _1}\, \left( \alpha -{{\bar{\varLambda }}_1}\right) \left( \alpha -{ l_2}\,{m_r} \right) \left( {\varPhi _1} -{\varPhi _2}\,\alpha \right) -{q_{zr}}\,{\varPhi _2} }{2\omega _1(1+l_2\varLambda _2)}\\ \varGamma _6&={\frac{-2\,\alpha \,{\zeta }{\sqrt{\frac{k_r}{m_r}}}\,{\omega _2}\, \left( \alpha -{\varLambda _2} \right) +2\, \zeta \,{\varLambda _2}\,\beta \,{\omega _2}\, \left( \alpha -{\varLambda _2} \right) -2\,\kappa \,{\omega _2}}{2 \,{\omega _2}+2\,{\omega _2}\,{l_2}\,{\varLambda _2}}} \end{aligned}$$

Appendix F: linear stability of steady states

$$\begin{aligned}&\mathbf{J}_{P1}= \left[ \begin{array}{cccc} {\varGamma _1}&{}2\,{\varGamma _2}\,{\sin (\gamma ^{*}_1)}\, {a^{*}_2}&{}{\varGamma _2}\,{\cos (\gamma ^{*}_1)}\,{{a^{*}_2}}^{2}&{}{f_0}\,{\varGamma _3}\,{\cos (\gamma ^{*}_2)}\\ {\varGamma _4}\,{\sin (\gamma ^{*}_1)}\,{a^{*}_2}&{}{\varGamma _4}\,{\sin (\gamma ^{*}_1)}\,{a^{*}_1}+{\varGamma _5}&{}{\varGamma _4}\,{ \cos (\gamma ^{*}_1)}\,{a^{*}_2}\,{a^{*}_1}&{}0\\ {\frac{4\,{\varGamma _4}\,{\cos (\gamma ^{*}_1)}\,{a^{*}_1}+\sigma _2}{{a^{*}_1}}}&{}{\frac{2{\varGamma _2}\,{\cos (\gamma ^{*}_1)}\,{a^{*}_2}}{{a^{*}_1}}}&{}{\frac{-{\varGamma _2}\,{ \sin (\gamma ^{*}_1)}\,{{a^{*}_2}}^{2}-2\,{\varGamma _4}\,{\sin (\gamma ^{*}_1)}\,{{a^{*}_1}}^{2}}{{a^{*}_1}}}&{}-{\frac{{f_0}\,{\varGamma _3}\,{\sin (\gamma ^{*}_2}}{{a^{*}_1}}} \\ {\frac{{\sigma _1}}{{a^{*}_1}}}&{}\,{\frac{2{ \varGamma _2}\,{\cos (\gamma ^{*}_1)}\,{a^{*}_2}}{{a^{*}_1}}}&{}-{\frac{{\varGamma _2}\, {\sin (\gamma ^{*}_1}\,{{a^{*}_2}}^{2}}{{a^{*}_1}}}&{}-{\frac{{f_0}\,{\varGamma _3}\,{\sin (\gamma ^{*}_1)}}{{a^{*}_1}}}\end{array} \right] \,, \end{aligned}$$
(62)
$$\begin{aligned}&\mathbf{J}_{S1}= \left[ \begin{array}{cccc} {\varGamma _1}&{}2\,{\varGamma _2}\,{\sin (\gamma ^{*}_1)}\, {a^{*}_2}&{}{\varGamma _2}\,{\cos (\gamma ^{*}_1)}\,{{a^{*}_2}}^{2}&{}{f^2_0}\,{\varGamma _3}\,{\cos (\gamma ^{*}_2)}\\ {\varGamma _4}\,{\sin (\gamma ^{*}_1)}\,{a^{*}_2}&{}{\varGamma _4}\,{\sin (\gamma ^{*}_1)}\,{a^{*}_1}+{\varGamma _5}&{}{\varGamma _4}\,{ \cos (\gamma ^{*}_1)}\,{a^{*}_2}\,{a^{*}_1}&{}0\\ {\frac{4\,{\varGamma _4}\,{\cos (\gamma ^{*}_1)}\,{a^{*}_1}+\sigma _2}{{a^{*}_1}}}&{}{\frac{2{\varGamma _2}\,{\cos (\gamma ^{*}_1)}\,{a^{*}_2}}{{a^{*}_1}}}&{}{\frac{-{\varGamma _2}\,{ \sin (\gamma ^{*}_1)}\,{{a^{*}_2}}^{2}-2\,{\varGamma _4}\,{\sin (\gamma ^{*}_1)}\,{{a^{*}_1}}^{2}}{{a^{*}_1}}}&{}-{\frac{{f^2_0}\,{\varGamma _3}\,{\sin (\gamma ^{*}_2}}{{a^{*}_1}}} \\ {\frac{{\sigma _1}}{{a^{*}_1}}}&{}\,{\frac{2{ \varGamma _2}\,{\cos (\gamma ^{*}_1)}\,{a^{*}_2}}{{a^{*}_1}}}&{}-{\frac{{\varGamma _2}\, {\sin (\gamma ^{*}_1}\,{{a^{*}_2}}^{2}}{{a^{*}_1}}}&{}-{\frac{{f^2_0}\,{\varGamma _3}\,{\sin (\gamma ^{*}_1)}}{{a^{*}_1}}}\end{array} \right] \,. \end{aligned}$$
(63)
$$\begin{aligned}&\mathbf{J}_{P2}= \end{aligned}$$
(64)
$$\begin{aligned}&\quad \left[ \begin{array}{cccc} {\varGamma _1}&{}2{\varGamma _2}{\sin (\gamma ^{*}_1)} {a^{*}_2}&{}{\varGamma _2}{\cos (\gamma ^{*}_1)}{{a^{*}_2}}^{2}&{}0 \\ {\varGamma _4}{\sin (\gamma ^{*}_1)}{a^{*}_2}&{}{\varGamma _5}+{ \varGamma _4}{a^{*}_1}{\sin (\gamma ^{*}_1)}&{}{\varGamma _4}{a^{*}_1}{\cos (\gamma ^{*}_1)} {a^{*}_2}+{f_0}{\varGamma _3}{\cos (\gamma ^{*}_1)}&{}0\\ { \frac{-2{f_0}{\varGamma _3}{\cos (\gamma ^{*}_2)}+4{\varGamma _4}{a^{*}_1}{\cos (\gamma ^{*}_1)}{a^{*}_2}+\sigma _2{a^{*}_2}}{{a^{*}_2}{a^{*}_1}}}&{}{ \frac{3{\varGamma _2}{\cos (\gamma ^{*}_1)}{{a^{*}_2}}^{2}+\sigma _2{a^{*}_1}+2 {\varGamma _4}{{a^{*}_1}}^{2}{\cos (\gamma ^{*}_1)}}{{a^{*}_2}{a^{*}_1}}}&{}{ \frac{-2{\varGamma _4}{{a^{*}_1}}^{2}{\sin (\gamma ^{*}_1)}{a^{*}_2}-{\varGamma _2}{\sin (\gamma ^{*}_1)}{{a^{*}_2}}^{3}}{{a^{*}_2}{a^{*}_1}}}&{}2{\frac{{f_0}{\varGamma _3}{\sin (\gamma ^{*}_2)}}{{a^{*}_2}}}\\ -{\varGamma _4}{\cos (\gamma ^{*}_1)}&{}-{\frac{-{\sigma _1}+{\varGamma _4}{a^{*}_1}{\cos (\gamma ^{*}_1)}}{{a^{*}_2}}}&{}{\varGamma _4}{a^{*}_1}{\sin (\gamma ^{*}_1)}&{}-{\frac{ {f_0}{\varGamma _3}{\sin (\gamma ^{*}_2)}}{{a^{*}_2}}}\end{array} \right] \end{aligned}$$
(65)
$$\begin{aligned}&\mathbf{J}_{S2}= \end{aligned}$$
(66)
$$\begin{aligned}&\quad \left[ \begin{array}{cccc} {\varGamma _1}&{}2{\varGamma _2}{\sin (\gamma ^{*}_1)} {a^{*}_2}&{}{\varGamma _2}{\cos (\gamma ^{*}_1)}{{a^{*}_2}}^{2}&{}0 \\ {\varGamma _4}{\sin (\gamma ^{*}_1)}{a^{*}_2}&{}{\varGamma _5}+{ \varGamma _4}{a^{*}_1}{\sin (\gamma ^{*}_1)}&{}{\varGamma _4}{a^{*}_1}{\cos (\gamma ^{*}_1)} {a^{*}_2}+{{f_0}}^{2}{\varGamma _3}{\cos (\gamma ^{*}_1)}&{}0 \\ {\frac{\sigma _2{a^{*}_2}-2{{f_0}}^{2}{ \varGamma _3}{\cos (\gamma ^{*}_2)}+4{\varGamma _4}{a^{*}_1}{\cos (\gamma ^{*}_1)}{a^{*}_2 }}{{a^{*}_2}{a^{*}_1}}}&{}{\frac{2{\varGamma _4}{{a^{*}_1}}^{2}{ \cos (\gamma ^{*}_1)}+3{\varGamma _2}{\cos (\gamma ^{*}_1)}{{a^{*}_2}}^{2}+\sigma _2{a^{*}_1} }{{a^{*}_2}{a^{*}_1}}}&{}{\frac{-{\varGamma _2}{\sin (\gamma ^{*}_1)}{{a^{*}_2}} ^{3}-2{\varGamma _4}{{a^{*}_1}}^{2}{\sin (\gamma ^{*}_1)}{a^{*}_2}}{{a^{*}_2} {a^{*}_1}}}&{}2{\frac{{{f_0}}^{2}{\varGamma _3}{\sin (\gamma ^{*}_2)}}{{ a12}}}\\ -{\varGamma _4}{\cos (\gamma ^{*}_1)}&{}-{\frac{-{\sigma _1}+{\varGamma _4}{a^{*}_1}{\cos (\gamma ^{*}_1)}}{{a^{*}_2}}}&{}{\varGamma _4} {a^{*}_1}{\sin (\gamma ^{*}_1)}&{}-{\frac{{{f_0}}^{2}{\varGamma _3}{\sin (\gamma ^{*}_2)}}{{a^{*}_2}}}\end{array} \right] \end{aligned}$$
(67)
$$\begin{aligned}&\mathbf{J}_{C}= \left[ \begin{array}{cccc}C_{11}&{}C_{12}&{}C_{13}&{}C_{14}\\ C_{21}&{}C_{22}&{}C_{23}&{}C_{24}\\ C_{31}&{}C_{32}&{}C_{33}&{}C_{34}\\ C_{41}&{}C_{42}&{}C_{43}&{}C_{44} \end{array} \right] \, \end{aligned}$$
(68)

with

$$\begin{aligned}&C_{11}={\varGamma _1},\,C_{12}={\varGamma _3}{f_0}{ \sin (\gamma ^{*}_2)}+2{\varGamma _2}{\sin (\gamma ^{*}_1)}{a^{*}_2},\,\nonumber \\&C_{13}={\varGamma _2}{\cos (\gamma ^{*}_1)} {{a^{*}_2}}^{2},\,C_{14}={\varGamma _3}{f_0}{\cos (\gamma ^{*}_2)}{a^{*}_2},\nonumber \\&C_{21}={\varGamma _4}{\sin (\gamma ^{*}_1)}{a^{*}_2}+{f_0}{ \varGamma _5}{\sin (\gamma ^{*}_2)},\,\nonumber \\&C_{22}={\varGamma _6}+{\varGamma _4}{\sin (\gamma ^{*}_1)}{a^{*}_1},\,\nonumber \\&C_{23}={\varGamma _4}{\cos (\gamma ^{*}_1)}{a^{*}_2}{a^{*}_1},\,C_{24}={f_0}{\varGamma _5 }{\cos (\gamma ^{*}_2)}{a^{*}_1},\nonumber \\&C_{31}={\frac{4{\varGamma _4}{ \cos (\gamma ^{*}_1)}{a^{*}_2}{a^{*}_1}-4{f_0}{\varGamma _5}{\cos (\gamma ^{*}_2)} {a^{*}_1}+\sigma _2{a^{*}_2}}{{a^{*}_2}{a^{*}_1}}},\nonumber \\&C_{32}={\frac{3{\varGamma _2}{\cos (\gamma ^{*}_1)}{{a^{*}_2}}^{2}+\sigma _2{a^{*}_1}+2{\varGamma _4} {\cos (\gamma ^{*}_1)}{{a^{*}_1}}^{2}+2{\varGamma _3}{f_0}{\cos (\gamma ^{*}_2)}{ a^{*}_2}}{{a^{*}_2}{a^{*}_1}}},\nonumber \\&C_{33}={\frac{-2{\varGamma _4}{\sin (\gamma ^{*}_1)} {{a^{*}_1}}^{2}{a^{*}_2}-{\varGamma _2}{\sin (\gamma ^{*}_1)}{{a^{*}_2}}^{3}}{{ a^{*}_2}{a^{*}_1}}},\,\nonumber \\&C_{34}={\frac{2{f_0}{\varGamma _5}{\sin (\gamma ^{*}_2)} {{a^{*}_1}}^{2}-{\varGamma _3}{f_0}{\sin (\gamma ^{*}_2)}{{a^{*}_2}}^{2}}{ {a^{*}_2}{a^{*}_1}}}\nonumber \\&C_{41}=-{\frac{-2{f_0}{ \varGamma _5}{\cos (\gamma ^{*}_2)}{a^{*}_1}-{\sigma _1}{a^{*}_2}+2{\varGamma _4}{\cos (\gamma ^{*}_1)}{a^{*}_2}{a^{*}_1}}{{a^{*}_2}{a^{*}_1}}},\nonumber \\&C_{42}=-{ \frac{-3{\varGamma _2}{\cos (\gamma ^{*}_1)}{{a^{*}_2}}^{2}+{\varGamma _4}{ \cos (\gamma ^{*}_1)}{{a^{*}_1}}^{2}-{\sigma _1}{a^{*}_1}-2{\varGamma _3}{ f_0}{\cos (\gamma ^{*}_2)}{a^{*}_2}}{{a^{*}_2}{a^{*}_1}}},\nonumber \\&C_{43}=-{\frac{-{ \varGamma _4}{\sin (\gamma ^{*}_1)}{{a^{*}_1}}^{2}{a^{*}_2}+{\varGamma _2}{\sin (\gamma ^{*}_1)}{{a^{*}_2}}^{3}}{{a^{*}_2}{a^{*}_1}}},\nonumber \\&C_{44}=-{\frac{{f_0}{ \varGamma _5}{\sin (\gamma ^{*}_2)}{{a^{*}_1}}^{2}+{\varGamma _3}{f_0}{ \sin (\gamma ^{*}_2)}{{a^{*}_2}}^{2}}{{a^{*}_2}{a^{*}_1}}}\,. \end{aligned}$$
(69)

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Gupta, S.K., Bukhari, M.A. & Barry, O.R. Nonlinear mode coupling in a passively isolated mechanical system. Nonlinear Dyn 101, 2055–2086 (2020). https://doi.org/10.1007/s11071-020-05908-9

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