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Nonlinear vibration analysis in precision motion stage with PID and time-delayed feedback controls

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Abstract

We investigate the control of friction-induced vibration in a precision motion stage under the effect of the LuGre friction dynamics. We consider a lumped parameter model of the precision motion stage with PID and linear time-delayed state feedback control acting in the direction of the motion of the stage. Linear stability analysis reveals the criticality of integral gain in the stability and, accordingly, the existence of multiple stability lobes and codimension-2 Hopf points for a given choice of system parameters. The nature of the bifurcation is determined by an analytical study using the method of multiple scales and harmonic balance. We observe the existence of both subcritical and supercritical Hopf bifurcations in the system, depending on the choice of control parameters. Hence, the nonlinearity due to dynamic friction model could both be stabilizing or destabilizing in nature, and therefore, stick-slip nonlinearity is essential to capture the global behavior of the system dynamics. Furthermore, numerical bifurcation analysis of the system reveals the existence of period-doubling bifurcation near the Hopf points. We observe complicated solutions such as period-4, quasi-periodic, large-amplitude stick-slip limit cycles along with chaotic attractor in the system.

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Acknowledgements

This work was funded by National Science Foundation (NSF) Award CMMI #1855390: Towards a Fundamental Understanding of a Simple, Effective and Robust Approach for Mitigating Friction in Nanopositioning Stages.

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Authors and Affiliations

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

    Sunit Kumar Gupta, Jiamin Wang & Oumar R. Barry

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  1. Sunit Kumar Gupta
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  2. Jiamin Wang
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  3. Oumar R. Barry
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Correspondence to Oumar R. Barry.

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Appendices

Appendix 1: Expressions used in the linear analysis

$$\begin{aligned} n_1= & {} -2{ a_1}{\omega }^{6}{ k_i}-{\omega }^{10}-2{{ v_{rv}}}^ {2}{{ \sigma _0}}^{2}{{ g_0}}^{2}{\omega }^{6}{ a_2} \\&+\,{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2}{\omega }^{4}{{ a_2}}^{2}-2{\omega }^{6}{ v_{rv}}{ \sigma _0}{ g_0}{ a_1} \\&-\,{\omega }^{6}{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2 }{{ a_1}}^{2}+2{\omega }^{4}{{ v_{rv}}}^{3}{{ \sigma _0}}^{3} {{ g_0}}^{3}{ a_1} \\&+\,2{{ k_i}}^{2}{\omega }^{2}{{ v_{rv}} }^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2}+2{ k_i}{\omega }^{4}{ v_{rv}}{ \sigma _0}{ g_0} \\&+\,2{k_i}{\omega }^{6}{ v_{rv}}{ \sigma _0}{ g_0}+2{ k_i}{{ v_{rv}}}^{3 }{{ \sigma _0}}^{3}{{ g_0}}^{3}{\omega }^{2} \\&+\,2{ k_i}{{ v_{rv}}}^{3}{{ \sigma _0}}^{3}{{ g_0}}^{3}{\omega }^{4}+4{ \omega }^{6}{ a_2}{ v_{rv}}{ \sigma _0}{ g_0}{ a_1} \\&+\,{\omega }^{4}{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2 }-{\omega }^{2}{{ v_{rv}}}^{4}{{ \sigma _0}}^{4}{{ g_0}}^{4} \\&+\,{{k_i}}^{2}{{ v_{rv}}}^{4}{{ \sigma _0}}^{4}{{ g_0}}^{4}+{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2}{\omega }^{8}\\&+\,{{a_1}}^{2}{\omega }^{8}+{{ k_i}}^{2}{\omega }^{4}+2{\omega }^{8}{ a_2}-{\omega }^{6}{{ a_2}}^{2} \\&-\,4{\omega }^{8}{ v_{rv}}{ \sigma _0}{ g_0}{ a_1}-4{\omega }^{4}{ a_2}{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2} \\&+\,4{\omega }^{6}{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2}-2{ a_1}{ \omega }^{4}{ k_i}{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0 }}^{2} \\&-\,2{ k_i}{\omega }^{4}{ v_{rv}}{ \sigma _0}{ g_0}{ a_2}-2{ k_i}{{ v_{rv}}}^{3}{{ \sigma _0}}^{3} {{ g_0}}^{3}{\omega }^{2}{ a_2} \\ d_1= & {} \left( {\omega }^{2}+{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2} \right) \\&\left( {\omega }^{2}{{ v_{rv}}}^{2}{{ \sigma _0}}^{2} {{ g_0}}^{2}+{{ g_0}}^{2}{{ \sigma _0}}^{2}{{ v_{rv}}}^{2} {{ k_i}}^{2} \right. \\&\quad +\,2{ k_i}{\omega }^{2}{ v_{rv}}{ \sigma _0 }{ g_0}+2{ k_i}{\omega }^{4}{ v_{rv}}{ \sigma _0} { g_0} \\&\quad -\,2{ k_i}{\omega }^{2}{ v_{rv}}{ \sigma _0}{ g_0}{ a_2}-2{\omega }^{4}{ v_{rv}}{ \sigma _0}{ g_0}{ a_1} \\&\quad -\,2{\omega }^{6}{ a_2}+{{ k_i}}^{2}{ \omega }^{2}+{\omega }^{6}{{ a_1}}^{2}+{\omega }^{8} \\&\quad \left. +\,{\omega }^{4}{{ a_2}}^{2}-2{ a_1}{\omega }^{4}{ k_i} \right) \\ n_2= & {} -2 \left( { a_1}{\omega }^{4}-{ k_i}{\omega }^{2}-{ k_i}{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2} \right. \\&\qquad - {v_{rv}}{ \sigma _0}{ g_0}{\omega }^{2} - {v_{rv}}{ \sigma _0}{ g_0}{\omega }^{4} \\&\qquad \left. + {v_{rv}}{\sigma _0}{ g_0}{\omega }^{2}{ a_2} \right) \omega \\&\left( -{{ v_{rv}}}^{ 2}{{ \sigma _0}}^{2}{{ g_0}}^{2}+{ v_{rv}}{ \sigma _0}{g_0}{ a_1}{\omega }^{2}+{\omega }^{4}-{\omega }^{2}{ a_2 } \right) \\ d_2= & {} \left( {\omega }^{2}+{{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2} \right) \\&\left( {\omega }^{2}{{ v_{rv}}}^{2}{{ \sigma _0}}^{2} {{ g_0}}^{2}+{{ g_0}}^{2}{{ \sigma _0}}^{2}{{ v_{rv}}}^{2} {{ k_i}}^{2} \right. \\&\quad +\,2{ k_i}{\omega }^{2}{ v_{rv}}{ \sigma _0 }{ g_0}+2{ k_i}{\omega }^{4}{ v_{rv}}{ \sigma _0} { g_0} \\&\quad -\,2{ k_i}{\omega }^{2}{ v_{rv}}{ \sigma _0}{ g_0}{ a_2} -2{\omega }^{4}{ v_{rv}}{ \sigma _0}{ g_0}{ a_1} \\&\quad -\,2{\omega }^{6}{ a_2}+{{ k_i}}^{2}{ \omega }^{2} +{\omega }^{6}{{ a_1}}^{2}+{\omega }^{8} \\&\quad \left. +\,{\omega }^{4}{{ a_2}}^{2}-2{ a_1}{\omega }^{4}{k_i} \right) \\ ct= & {} \cos {\left( \omega T\right) }\quad st=\sin {\left( \omega T\right) }\\ \hbox {Re}_1= & {} -{\frac{{v_{rv}}{g_1}{h_0}{\omega }^{2}}{{{v_{rv}}}^{2} {{\sigma _0}}^{2}{{g_0}}^{2}+{\omega }^{2}}} \\ Im_1= & {} {\frac{-{{v_{rv}}}^{2}{\sigma _0}{g_0}{g_1}{ h_0}\omega }{{{v_{rv}}}^{2}{{\sigma _0}}^{2}{{g_0}}^{2} +{\omega }^{2}}} \end{aligned}$$
$$\begin{aligned} \hbox {Lre}_1= & {} {\frac{{\omega }^{3}{ K_0}st -{ k_i}{\omega }^{2}}{-2{\omega }^{2}{{ K_0}}^{2}+2 \omega { K_0}st{ k_i }-{{ k_i}}^{2}+2{\omega }^{2}{{ K_0}}^{2}ct +2{ K_0}ct {\omega }^{2}-2{ K_0}{\omega }^{2}-{\omega }^{2}}} \\ \hbox {Lim}_1= & {} {\frac{ \left( {\omega }^{3} -{\omega }^{3}{ K_0}ct +{\omega }^{3}{ K_0}\right) }{-2{\omega }^ {2}{{ K_0}}^{2}+2\omega { K_0}st { k_i}-{{ k_i}}^{2}+2{\omega }^{2}{{ K_0}}^ {2}ct+2{ K_0}st{\omega }^{2}-2{ K_0}{\omega }^{2}-{ \omega }^{2}}} \\ \hbox {Lre}_2= & {} {\frac{{ k_i}{\omega }^{2}{ K_0}ct -{ k_i}{\omega }^{2}{ K_0}-{ k_i}{\omega } ^{2}}{-2{\omega }^{2}{{ K_0}}^{2}+2\omega { K_0}st{ k_i}-{{ k_i}}^{2}+2{ \omega }^{2}{{ K_0}}^{2}ct+2 { K_0}ct{\omega }^{2}-2{ K_0}{\omega }^{2}-{\omega }^{2}}} \\ \hbox {Lim}_2= & {} {\frac{{ k_i}{\omega }^{2}{ K_0}st-{{ k_i}}^{2}\omega }{-2{\omega }^{2}{{ K_0}}^{2 }+2\omega { K_0}st{ k_i}-{{ k_i}}^{2}+2{\omega }^{2}{{ K_0}}^{2}ct+2{ K_0}ct{\omega }^{2}-2{ K_0}{\omega }^{2}-{\omega }^{2}}} \\ \hbox {Lre}_3= & {} \frac{-{\omega }^{2}{ h_2} \left( \omega { K_0}st{ v_{rv}}{ \sigma _0}{ g_0}-{ k_i}{ v_{rv}}{ \sigma _0}{ g_0}-{ K_0}ct{\omega }^{2}+{ K_0}{\omega }^{ 2}+{\omega }^{2} \right) }{\left( {{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2}+{\omega } ^{2} \right) \left( -2{\omega }^{2}{{ K_0}}^{2}+2\omega { K_0}st{ k_i}-{{ k_i}}^ {2}+2{\omega }^{2}{{ K_0}}^{2}ct+2{ K_0}ct{ \omega }^{2}-2{ K_0}{\omega }^{2}-{\omega }^{2} \right) } \\ \hbox {Lim}_3= & {} \frac{-{\omega }^{3}{ h_2} \left( { v_{rv}}{ \sigma _0}{ g_0}{ K_0}ct-{ v_{rv}} { \sigma _0}{ g_0}{ K_0}-{ v_{rv}}{ \sigma _0} { g_0}+\omega { K_0}st -{ k_i} \right) }{ \left( {{ v_{rv}}}^{2}{{ \sigma _0}}^{2}{{ g_0}}^{2}+{\omega } ^{2} \right) \left( -2{\omega }^{2}{{ K_0}}^{2}+2\omega { K_0}st{ k_i}-{{ k_i}}^ {2}+2{\omega }^{2}{{ K_0}}^{2}ct+2{ K_0}ct{ \omega }^{2}-2{ K_0}{\omega }^{2}-{\omega }^{2} \right) } \end{aligned}$$

Appendix 2: Expressions used in the nonlinear analysis

$$\begin{aligned} ct= & {} \cos {\left( \omega T_{cr}\right) }\quad st=\sin {\left( \omega T_{cr}\right) } \\ u_{11}= & {} {K_{0,cr}}{T_{cr}}ct\quad u_{12}=\left( \omega -{K_{0,cr}}{T_{cr}}st \right) \\ u_{21}= & {} -{\frac{{v_{rv}}{g_1}{h_0}{\omega }^{2}}{{{v_{rv}}}^{2}{{\sigma _0}}^{2} {{g_0}}^{2}+{\omega }^{2}}} \\ u_{22}= & {} {\frac{-{{v_{rv}}}^{2}{\sigma _0}{g_0}{g_1}{ h_0}\omega }{{{v_{rv}}}^{2}{{\sigma _0}}^{2}{{g_0}}^{2} +{\omega }^{2}}} \\ b_{11}= & {} -2i{\omega }^{2}\mathrm {e}^{2i\omega T_{cr}} \\&\left( 2i{\omega }^{2}{ h_0 }{\sigma _1}{ h_3}+{ h_2}\omega { h_0}{ h_3} \right. \\&\quad -\,i{ v_{rv}}{\sigma _0}{ g_0}{\sigma _1}{ h_4} {Re_1}+{\sigma _1}{ h_4}{Im_1}{ v_{rv}}{\sigma _0}{ g_0}\\&\quad +\,{ h_0}{\sigma _1}{ h_3}\omega { v_{rv}}{\sigma _0}{ g_0}+2i\omega {\sigma _1}{ h_4}{Im_1} \\&\quad \left. -i{ h_4}{Re_1}{ h_2}+{ h_2}{ h_4}{Im_1}+2{\sigma _1}{ h_4}\omega {Re_1} \right) / \\&\left( \mathrm {e}^{2i\omega T_{cr}}{ k_i}{ v_{rv}}{\sigma _0}{ g_0}+2i\mathrm {e}^{2i\omega T_{cr}}{ k_i}\omega \right. \\&\quad -\,4{\omega }^{2}{ h_1}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0}-8i{ \omega }^{3}{ h_1}\mathrm {e}^{2i\omega T_{cr}} \\&-\,2i\omega { K_0}{ v_{rv}}{\sigma _0}{ g_0}+4{ K_0}{\omega }^{2}+2i \omega \mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0}\\&-\,4{ \omega }^{2}\mathrm {e}^{2i\omega T_{cr}}-8i{\omega }^{3}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{ \sigma _0}{ g_0} \\&+\,16{\omega }^{4}\mathrm {e}^{2i\omega T_{cr}}+4{{\mathrm {e}^{i\omega T_{cr}}}} ^{2}{ h_2}{ v_{rv}}{ g_1}{ h_0}{\omega }^{2} \\&\left. +2i \omega { K_0}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0}-4{\omega }^{2}{ K_0}\mathrm {e}^{2i\omega T_{cr}}\right) \\ b_{12}= & {} 4 {\omega }^{3}\mathrm {e}^{2i\omega T_{cr}} \\&(2i{\omega }^{2}{ h_0} {\sigma _1}{ h_3}+{ h_2}\omega { h_0}{ h_3}\\&\quad -\, i{ v_{rv}}{\sigma _0}{ g_0}{\sigma _1}{ h_4} {Re_1}+{\sigma _1}{ h_4}{Im_1}{ v_{rv}}{\sigma _0}{ g_0} \\&\quad +\,{h_0}{\sigma _1}{ h_3}\omega { v_{rv}}{\sigma _0}{ g_0}+2i\omega {\sigma _1}{ h_4}{Im_1} \\&\quad -\,i{ h_4}{Re_1}{ h_2}+{ h_2}{ h_4}{Im_1}+2{\sigma _1}{ h_4}\omega {Re_1})/ \\&(\mathrm {e}^{2i\omega T_{cr}}{ k_i}{ v_{rv}}{\sigma _0}{ g_0}+2i\mathrm {e}^{2i\omega T_{cr}}{ k_i}\omega \\&\quad -\,4{\omega }^{2}{h_1}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0}-8i{ \omega }^{3}{ h_1}\mathrm {e}^{2i\omega T_{cr}} \\&-\,2i\omega { K_0}{ v_{rv}}{\sigma _0}{ g_0}+4{ K_0}{\omega }^{2}\\&\quad +\,2i \omega \mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0}-4{ \omega }^{2}\mathrm {e}^{2i\omega T_{cr}} \\&\quad -\,8i{\omega }^{3}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{ \sigma _0}{ g_0}+16{\omega }^{4}\mathrm {e}^{2i\omega T_{cr}} \\&\quad +\,4{{\mathrm {e}^{i\omega T_{cr}}}} ^{2}{ h_2}{ v_{rv}}{ g_1}{ h_0}{\omega }^{2} \\&\quad +\,2i \omega { K_0}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0}-4{\omega }^{2}{ K_0}\mathrm {e}^{2i\omega T_{cr}}) \\ b_{13}= & {} -\,\omega \mathrm {e}^{2i\omega T_{cr}} \\&\left( 2i{\omega }^{2}{ h_0}{\sigma _1}{ h_3}+{ h_2}\omega { h_0}{ h_3} \right. \\&\quad -\,i{ v_{rv}}{\sigma _0}{ g_0}{\sigma _1}{ h_4}{Re_1}+{\sigma _1}{ h_4}{Im_1}{ v_{rv}}{\sigma _0 }{ g_0} \\&\quad +\, {h_0}{\sigma _1}{ h_3}\omega { v_{rv}}{\sigma _0}{ g_0}+2i\omega {\sigma _1}{ h_4} {Im_1} \\&\quad \left. -i{h_4}{Re_1}{ h_2}+{ h_2}{ h_4}{ Im_1}+2{\sigma _1}{ h_4}\omega {Re_1} \right) / \\&\left( {{ \mathrm {e}^{2i\omega T_{cr}}}}{ k_i}{ v_{rv}}{\sigma _0}{ g_0}+2i{{ \mathrm {e}^{2i\omega T_{cr}}}}{ k_i}\omega \right. \\&\quad -\,4{\omega }^{2}{ h_1}{{\mathrm {e}^{2i\omega T_{cr}}}}{ v_{rv}}{\sigma _0}{ g_0}-8i{\omega }^{3}{ h_1} \mathrm {e}^{2i\omega T_{cr}} \\&\quad -\,2i\omega { K_0}{ v_{rv}}{\sigma _0}{ g_0}+4{ K_0}{\omega }^{2} \\&\quad +\,2i\omega \mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0}-4{\omega }^{2}\mathrm {e}^{2i\omega T_{cr}} \\&\quad -\,8i{ \omega }^{3}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0}+16{ \omega }^{4}\mathrm {e}^{2i\omega T_{cr}} \\&\quad +\,4\mathrm {e}^{2i\omega T_{cr}}{ h_2}{ v_{rv}}{ g_1}{ h_0}{\omega }^{2} \\&\quad +\,2i\omega { K_0}{{\mathrm {e}^{2i\omega T_{cr}}}}{ v_{rv}}{\sigma _0}{ g_0} \\&\quad \left. -4{\omega }^{2}{ K_0}{ {\mathrm {e}^{i\omega T_{cr}}}}\right) \end{aligned}$$
$$\begin{aligned} b_{14}= & {} \left( -4{{ h_0}}^{2}{\sigma _1}{ h_3}{\omega }^{3}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{ g_1} \right. \\&\quad -\,4{\sigma _1}{ h_4 }{\omega }^{2}{Im_1}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{ g_1}{ h_0} \\&\quad +\,2i{\omega }^{2}{ K_0}\mathrm {e}^{2i\omega T_{cr}}{ h_0}{ h_3}-4 {\omega }^{3}{ h_1}\mathrm {e}^{2i\omega T_{cr}}{ h_0}{ h_3} \\&\quad -\,4{\omega }^{2}{ h_1}\mathrm {e}^{2i\omega T_{cr}}{ h_4}{Im_1}-8i{\omega }^{3}{{ \mathrm {e}^{2i\omega T_{cr}}}}{ h_4}{Im_1} \\&\quad -\,8i{\omega }^{4}\mathrm {e}^{2i\omega T_{cr}}{ h_0}{ h_3}+\mathrm {e}^{2i\omega T_{cr}}{ k_i}{ h_0}{ h_3}\omega \\&\quad +\,\mathrm {e}^{2i\omega T_{cr}}{ k_i}{ h_4}{Im_1}+2\omega {{\mathrm {e}^{i\omega T_{cr}}}}{ h_4}{Re_1} \\&\quad -\,i\mathrm {e}^{2i\omega T_{cr}}{ k_i}{ h_4}{Re_1} +4i{\omega }^{2}{ h_1}\mathrm {e}^{2i\omega T_{cr}}{ h_4}{Re_1} \\&\quad -\,2 \omega { K_0}{ h_4}{Re_1}+2i\omega { K_0}{{ ed}}^{2}{ h_4}{Im_1}\\&\quad +\,2i\omega \mathrm {e}^{2i\omega T_{cr}}{ h_4} {Im_1} +2\omega { K_0}\mathrm {e}^{2i\omega T_{cr}}{ h_4}{Re_1} \\&\quad -\,2i{\omega }^{2}{ K_0}{ h_0}{ h_3}-2i\omega { K_0}{ h_4}{Im_1} \\&\quad -\,8{\omega }^{3}\mathrm {e}^{2i\omega T_{cr}}{ h_4} {Re_1} \\&\quad +\,4i{\sigma _1}{ h_4}{\omega }^{2}{Re_1}{{ \mathrm {e}^{2i\omega T_{cr}}}}{ v_{rv}}{ g_1}{ h_0} \\&\quad \left. +2i{\omega }^{2}{{\mathrm {e}^{i\omega T_{cr}}}}{ h_0}{ h_3} \right) \omega / \\&\left( \mathrm {e}^{2i\omega T_{cr}}{ k_i}{ v_{rv}}{\sigma _0}{ g_0}+2i\mathrm {e}^{2i\omega T_{cr}}{ k_i} \omega \right. \\&\quad -\,4{\omega }^{2}{ h_1}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0} \\&\quad -\,8i{\omega }^{3}{ h_1}\mathrm {e}^{2i\omega T_{cr}} -2i \omega { K_0}{ v_{rv}}{\sigma _0}{ g_0}+4{ K_0}{\omega }^{2} \\&\quad +\,2i\omega \mathrm {e}^{2i\omega T_{cr}}{ v_{rv}}{\sigma _0}{ g_0} -4{\omega }^{2}\mathrm {e}^{2i\omega T_{cr}}\\&\quad -\,8i{\omega }^{3}{{\mathrm {e}^{i\omega T_{cr}}}}{ v_{rv}}{\sigma _0}{ g_0} +16{\omega }^{4}{{\mathrm {e}^{i\omega T_{cr}}}} \\&\quad +\,4\mathrm {e}^{2i\omega T_{cr}}{ h_2}{ v_{rv}}{ g_1}{ h_0}{\omega }^{2} \\&\quad \left. +2i\omega { K_0}\mathrm {e}^{2i\omega T_{cr}}{ v_{rv} }{\sigma _0}{ g_0}-4{\omega }^{2}{ K_0}{{\mathrm {e}^{2i\omega T_{cr}}}}\right) \\ b_{31}= & {} b_{32}=0, \\ \end{aligned}$$
$$\begin{aligned} b_{33}= & {} 2{\frac{\omega \left( { h_2}{ h_4}{Im_1}+{ h_2 }\omega { h_0}{ h_3}+{\sigma _1}{ h_4}{Im_1} { v_{rv}}{\sigma _0}{ g_0}+{ h_0}{\sigma _1}{ h_3}\omega { v_{rv}}{\sigma _0}{ g_0} \right) }{{k_i}{ v_{rv}}{\sigma _0}{ g_0}}}, \\ b_{34}= & {} -2{\frac{\omega \left( { h_4}{Im_1}+{ h_0}{ h_3} \omega \right) }{{ v_{rv}}{\sigma _0}{ g_0}}} \\ v_{11}= & {} i\omega +{{\mathrm{e}}^{-i\omega { T_{cr}}}}{K_0}{T_{cr}}\quad v_{21}=Re_1+i Im_1 \\ v_{12}= & {} i{\sigma _1}{ h_4} { A_1} \left( T_{{2}} \right) ^{2}{ b_{12}}{ A_2} \left( T_{{2}} \right) { Im_1} -3i{\sigma _1}{ h_5}{\omega }^{3} { A_1 } \left( T_{{2}} \right) ^{2}{ A_2} \left( T_{{2}} \right) \\&+\,i{\sigma _1}{ h_4}\omega { A_2} \left( T_{{2 }} \right) { A_1} \left( T_{{2}} \right) ^{2}{ b_{14}} +2i{ h_0}{\sigma _1}{ h_3}\omega { A_2 } \left( T_{{2}} \right) { A_1} \left( T_{{2}} \right) ^{2}{ b_{12}} \\&-\,{\sigma _1}{\sigma _0}{ h_3}{ \omega }^{2} { A_1} \left( T_{{2}} \right) ^{2}{ A_2} \left( T_{{2}} \right) {Re_1} -{\sigma _1}{ h_4} { A_1} \left( T_{{2}} \right) ^{2}{ b_{12}} { A_2} \left( T_{{2}} \right) {Re_1} \\&-\,i{\sigma _1}{ h_4 }\omega { A_1} \left( T_{{2}} \right) ^{2}{ A_2} \left( T_{{2}} \right) { b_{34}} -3i{\sigma _1}{\sigma _0}{ h_3}{\omega }^{2} { A_1} \left( T_{{2}} \right) ^{2}{ A_2} \left( T_{{2}} \right) {Im_1} \\&+\,i{ {\mathrm{e}}^{-i\omega { T1c}}}{ K_0}{ k_1}{ A_1} \left( T_{{2}} \right) \omega \\ v_{22}= & {} -2i{ h_0}{ h_3}\omega { A_2} \left( T_{{2}} \right) { A_1} \left( T_{{2}} \right) ^{2}{ b_{12}} -i{ h_4}\omega { A_2} \left( T_{{2}} \right) { A_1 } \left( T_{{2}} \right) ^{2}{ b_{14}} \\&+\,i{ h_4}\omega { A_1} \left( T_{{2}} \right) ^{2}{ A_2} \left( T_{{2}} \right) { b_{34}} +{\sigma _0}{ h_3}{ \omega }^{2} { A_1} \left( T_{{2}} \right) ^{2}{ A_2} \left( T_{{2}} \right) {Re_1} \\&+\,{ h_4}{ A_1} \left( T_{{2}} \right) ^{2}{ b_{12}}{ A_2} \left( T_{{2}} \right) {Re_1} -i{ h_4}{ A_1} \left( T_{{2}} \right) ^{2}{ b_{12}}{ A_2} \left( T_ {{2}} \right) {Im_1} \\&+\,3i{ h_5}{\omega }^{3} { A_1} \left( T_{{2}} \right) ^{2}{ A_2} \left( T_{{2}} \right) +3i{\sigma _0}{ h_3}{\omega }^{2} { A_1} \left( T_{{2}} \right) ^{2}{ A_2} \left( T_{{2}} \right) {Im_1} \end{aligned}$$

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Gupta, S.K., Wang, J. & Barry, O.R. Nonlinear vibration analysis in precision motion stage with PID and time-delayed feedback controls. Nonlinear Dyn 101, 439–464 (2020). https://doi.org/10.1007/s11071-020-05779-0

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