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Towards design of a nonlinear vibration stabilizer for suppressing single-mode instability

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Abstract

Many mechanical systems exhibit destabilization of a single mode as some parameters are varied. In such situations, the destabilizing effect is often modeled using a negative damping term, e.g., in flow induced galloping instability of a pipe in a cross-flow. A small, nonlinear, not precisely tuned, lightly damped, secondary system mounted on the primary system can stabilize the primary system over some useful range of parameter values. We study such a system, modeling the primary system as a linear, negatively damped, single-degree-of-freedom oscillator and modeling the small secondary system using small linear damping and significantly nonlinear stiffness. Numerical simulations show useful behavior over a range of parameters even without precise tuning of the secondary system. Analytical treatment using harmonic balance followed by an informal averaging calculation yields amplitude and phase equations that capture the dynamics accurately over a range of parameters. Although intermediate expressions involved are long and unwieldy, numerical and analytical treatment of the equations lead to simplifying insights which in turn allow us to construct useful approximate formulas that can be used by practitioners. In particular, simple approximate expressions are obtained for oscillation amplitudes of the primary and secondary systems and for the negative damping range over which effective vibration stabilization occurs, in terms of system parameters.

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Funding

We have not received any financial support for conducting this research.

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Authors and Affiliations

  1. Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Ashutosh Shirude & Anindya Chatterjee

  2. Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Sangareddy, 502285, India

    C. P. Vyasarayani

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  1. Ashutosh Shirude
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  2. C. P. Vyasarayani
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  3. Anindya Chatterjee
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Correspondence to C. P. Vyasarayani.

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Appendices

Appendix A

The expressions for P and Q in Eq. (12) and Eq. (13) are shown below

$$\begin{aligned}&P=-\frac{3}{4}{\bar{B}}^{2}\bar{C}\cos \left( t+\bar{\phi } \right) {k_d} \\&\quad +\, \frac{3}{4}{\bar{B}}^{2}\bar{C}\cos \left( 3t+3\bar{\phi } \right) {k_d} \frac{3}{4}{\bar{C}}^{3}\cos \left( t+\bar{\phi } \right) {k_d} \\&\quad -\,\frac{1}{4} {\bar{C}}^{3}\cos \left( 3t+3\bar{\phi } \right) {k_d} + \frac{1}{4} {\bar{B}}^{3} {k_d} \sin \left( 3t+3\bar{\phi } \right) \\&\quad -\,\frac{3}{4}{\bar{B}}^{3}{k_d} \sin \left( t+\bar{\phi } \right) -\frac{3}{4}\bar{B}{\bar{C}}^{2}{k_d} \sin \left( 3t+ 3\bar{\phi } \right) \\&\quad -\,\frac{3}{4}\bar{B}\bar{C}^{2}\sin \left( t+\bar{\phi } \right) {k_d} - c\bar{A} \cos \left( t+\bar{\phi } \right) \\&\quad -\,\bar{B}\cos \left( t+\bar{\phi } \right) {c_d}-\bar{C} \cos \left( t+\bar{\phi }\right) {k_d} \\&\quad +\,\dot{\bar{A}}\cos \left( t+\bar{\phi } \right) - \bar{A}\sin \left( t+\bar{\phi } \right) \dot{\bar{\phi }} -\bar{B}\sin \left( t+\bar{\phi } \right) {k_d}\\&\quad +\,\bar{C}\sin \left( t+\bar{\phi } \right) {c_d}, \\&Q=\frac{1}{4m}\,\bigg (3{\bar{B}}^{3}k_d\,\sin \left( t+\bar{\phi }\right) -{\bar{B}}^{3} k_d\,\sin \left( 3\,t+3\,\bar{\phi }\right) \\&\quad +\,3\,{\bar{B}}^{2}\bar{C} \cos \left( t+\bar{\phi }\right) k_d \\&\quad -\,3\,{\bar{B}}^{2}\bar{C}\,\cos \left( 3\,t+3\,\bar{\phi } \right) k_d+3\,\bar{B} {\bar{C}}^{2}\sin \left( t+\bar{\phi } \right) k_d \\&\quad +\,3\,\bar{B}\,{\bar{C}}^{2}k_d \,\sin \left( 3\,t+3\,\bar{\phi } \right) \\&\quad +\,3\,{\bar{C}}^{3}\cos \left( t+\bar{\phi } \right) k_d+{\bar{C}}^{3}\cos \left( 3\,t+3\,\bar{\phi } \right) k_d\\&\quad -\,4\,\bar{A}\,\sin \left( t+\bar{\phi } \right) m \dot{\bar{\phi }}-4\,\bar{B}\,\sin \left( t+\bar{\phi }\right) m\dot{\bar{\phi }}\\&\quad -\,4\,\bar{C}\,\cos \left( t+\bar{\phi } \right) m\dot{\bar{\phi }}-4\,\bar{A}\,\sin \left( t+\bar{\phi } \right) m\\&\quad +\,4\,\dot{\bar{A}}\,\cos \left( t+\bar{\phi }\right) m +4\,\bar{B}\,\cos \left( t+\bar{\phi } \right) c_d \\&\quad +\,4\,\bar{B}\,\sin \left( t+\bar{\phi } \right) k_d-4\, \bar{B}\,\sin \left( t+\bar{\phi } \right) m\\&\quad +\,4\,\dot{\bar{B}}\,\cos \left( t+\bar{\phi } \right) m+4\,\bar{C}\,\cos \left( t+\bar{\phi }\right) k_d \\&\quad -\,4\,\bar{C}\,\cos \left( t+\bar{\phi }\right) m-4\,\bar{C}\,\sin \left( t+\bar{\phi } \right) c_d\\&\quad -\,4\,\dot{\bar{C}}\,\sin \left( t+\bar{\phi } \right) m\bigg ). \end{aligned}$$

Appendix B

The expressions for \(f_1\), \(f_2\), \(f_3\), and \(f_4\) in Eqs. (14)–(17) are shown below

$$\begin{aligned}&f_1=-\frac{1}{8}{\bar{B}}^{3} k_d\sin \left( 4t+4\bar{\phi } \right) + \frac{1}{4}{\bar{B}}^{3} k_d\sin \left( 2t+2\bar{\phi } \right) \\&\quad -\,\frac{3}{8}{\bar{B}}^{2}\bar{C} k_d\cos \left( 4t+4\bar{\phi } \right) +\frac{3}{8}{\bar{B}}^{2}\bar{C} k_d \\&\quad +\,\frac{3}{8}\bar{B}{\bar{C}}^{2}k_d\sin \left( 4t+4\bar{\phi } \right) + \frac{3}{4}\bar{B}{\bar{C}}^{2}k_d\sin \left( 2t+2\bar{\phi } \right) \\&\quad +\,\frac{1}{2}{\bar{C}}^{3}k_d\cos \left( 2t+2\bar{\phi } \right) \\&\quad +\,\frac{1}{8}{\bar{C}}^{3} k_d\cos \left( 4t+4\bar{\phi } \right) +\frac{3}{8}{\bar{C}}^{3}k_d +\frac{1}{2}\bar{A}c\cos \left( 2t\right. \\&\quad \left. +2\bar{\phi }\right) +\frac{1}{2}\bar{A}c+\frac{1}{2}\bar{B}c_d\cos \left( 2t+2\bar{\phi } \right) \\&\quad +\, \frac{1}{2}\bar{B}c_d +\frac{1}{2}\bar{B} k_d\sin \left( 2t+2\bar{\phi } \right) +\frac{1}{2} \bar{C} k_d \cos \left( 2t\right. \\&\quad \left. +2\bar{\phi } \right) +\frac{1}{2}\bar{C} k_d-\frac{1}{2}\bar{C}c_d\sin \left( 2t+2\bar{\phi } \right) ,\\&f_2=\frac{1}{8 \bar{A}m} \bigg ( 4{\bar{A}}^{2}m\sin \left( 2t+2\bar{\phi } \right) +4{\bar{C}}^{2}c_dm\\&\quad -\,3\bar{A}{\bar{C}}^{3} k_d-4\bar{A}\bar{C} k_d-4\bar{A}\bar{B} c_d+4\bar{A}\bar{C}m \\&\quad -\,4{\bar{A}}^{2}c\,m +3\bar{A}{\bar{B}}^{2}\bar{C} k_d\,m\cos \left( 4t+4\bar{\phi } \right) \\&\quad -\,3\bar{A}\bar{B}{\bar{C}}^{2} k_d\, m \sin \left( 4t+4\bar{\phi }\right) \\&\quad -\,6\bar{A}\bar{B}{\bar{C}}^{2} k_d\,m\,\sin \left( 2t+2\bar{\phi } \right) - \bar{A}{\bar{C}}^{3} k_d\cos \left( 4t+4\bar{\phi } \right) \\&\quad -4\bar{A}{\bar{C}}^{3} k_d \cos \left( 2t+2\bar{\phi } \right) \\&\quad -\,4{\bar{A}}^{2}cm\cos \left( 2t+2\bar{\phi }\right) \\&\quad -\,4\bar{A}\bar{B} k_d\sin \left( 2t+2\bar{\phi } \right) \\&\quad -\, 4\bar{A}\bar{C}k_d\cos \left( 2t+2\bar{\phi } \right) \\&\quad +\,4\bar{A}\bar{C}m\cos \left( 2t+2\bar{\phi } \right) -4{\bar{C}}^{2}c_dm\cos \left( 2t+2\bar{\phi } \right) \\&\quad -4\bar{A}\bar{B} c_d\cos \left( 2t+2\bar{\phi } \right) -3\bar{A}{\bar{B}}^{2}\bar{C} k_d \\&\quad -\,3\bar{B} \bar{C}^{3} k_d m -3\bar{A}{\bar{C}}^{3} k_d m-4\bar{B}\bar{C} k_d m-4\bar{A}\bar{B} c_d m\\&\quad -\,4\bar{A}\bar{C} k_d m-3{\bar{B}}^{3}\bar{C} k_dm \\&\quad +\,4\bar{A}\bar{B}m\sin \left( 2t+2\bar{\phi } \right) +4\bar{A}\bar{C} c_d\sin \left( 2t+2\bar{\phi } \right) \\&\quad -{\bar{C}}^{4} k_dm\sin \left( 4t+4\bar{\phi } \right) \\&\quad -\,2{\bar{C}}^{4}k_d m\sin \left( 2t+2\bar{\phi } \right) +\bar{A}{\bar{B}}^{3} k_d\sin \left( 4t+4\bar{\phi } \right) \\&\quad -\,2\bar{A}{\bar{B}}^{3} k_d\sin \left( 2t+2\bar{\phi } \right) \\&\quad -\,4{\bar{C}}^{2} k_dm\sin \left( 2t+2\bar{\phi } \right) -\bar{A}{\bar{C}}^{3}kdm\cos \left( 4t+4\bar{\phi } \right) \end{aligned}$$
$$\begin{aligned}&\quad -\,4\bar{A}{\bar{C}}^{3} k_dm\cos \left( 2t+2\bar{\phi } \right) \\&\quad -\,4\bar{B}\bar{C} c_dm\sin \left( 2t+2\bar{\phi }\right) +3{\bar{B}}^{2}{\bar{C}}^{2}k_dm\sin \left( 4t+4\bar{\phi }\right) \\&\quad - 6{\bar{B}}^{2}{\bar{C}}^{2} k_dm\sin \left( 2t+2\bar{\phi } \right) \\&\quad +\,3\bar{B}{\bar{C}}^{3}k_dm\cos \left( 4t+4\bar{\phi } \right) +3\bar{A}{\bar{B}}^{2}\bar{C} k_d\cos \left( 4t+4\bar{\phi } \right) \\&\quad -\,3\bar{A}\bar{B}{\bar{C}}^{2}k_d\sin \left( 4t+4\bar{\phi } \right) -6\bar{A}\bar{B}{\bar{C}}^{2}k_d\sin \left( 2t\right. \\&\quad \left. +2\bar{\phi } \right) -4\bar{A}\bar{B} k_dm\sin \left( 2t+2\bar{\phi } \right) \\&\quad -\,4\bar{A}\bar{C}cm\sin \left( 2t+2\bar{\phi } \right) +4\bar{A}\bar{C}c_dm\sin \left( 2t+2\bar{\phi } \right) \\&\quad +\bar{A}{\bar{B}}^{3}k_dm\sin \left( 4t+ 4\bar{\phi } \right) \\&\quad -\,2\bar{A}{\bar{B}}^{3} k_dm\sin \left( 2t+2\bar{\phi } \right) +4\bar{B}\bar{C} k_dm\cos \left( 2t+2\bar{\phi } \right) \\&\quad -\,4\bar{A}\bar{B}c_dm\cos \left( 2t+2\bar{\phi } \right) \\&\quad -\,4\bar{A}\bar{C}k_dm\cos \left( 2t+2\bar{\phi } \right) -{\bar{B}}^{3}\bar{C} k_dm\cos \left( 4t+4\bar{\phi } \right) \\&\quad +\,4{\bar{B}}^{3}\bar{C} k_dm\cos \left( 2t+2\bar{\phi } \right) -3\bar{A}{\bar{B}}^{2}\bar{C} k_dm \bigg ),\\&f_3=\frac{1}{8\bar{A}m}\bigg (-4\bar{A}\bar{B}m+4\bar{A}\bar{B} k_dm-4\bar{A}\bar{C} c_dm\\&\quad -\,4\bar{B}\bar{C} c_dm+\bar{A}{\bar{B}}^{3} k_d\cos \left( 4t+4\bar{\phi } \right) \\&\quad +\,\bar{A}{\bar{C}}^{3} k_d \sin \left( 4t+4\bar{\phi } \right) +2\bar{A}{\bar{C}}^{3}k_d\sin \left( 2t+2\bar{\phi } \right) \\&\quad +\,4{\bar{A}}^{2}cm\sin \left( 2t+2\bar{\phi } \right) \\&\quad +\,4{\bar{B}}^{2} c_dm\sin \left( 2t+2\bar{\phi } \right) +4\bar{A}\bar{B} c_d\sin \left( 2t+2\bar{\phi } \right) \\&\quad +\,4\bar{A}\bar{C}k_d\sin \left( 2t+2\bar{\phi } \right) -4\bar{A}\bar{C}c_d+4\bar{A}\bar{B}k_d \\&\quad +\,4{\bar{B}}^{2} k_dm+3{\bar{B}}^{4}k_dm+4{\bar{A}}^{2}m\cos \left( 2t+2\bar{\phi } \right) \\&\quad +\,3\bar{A}{\bar{B}}^{3} k_d+3\bar{A}{\bar{B}}^{3}k_dm+3{\bar{B}}^{2}{\bar{C}}^{2} k_dm \\&\quad -\,3\bar{A}\bar{B}{\bar{C}}^{2}k_d\cos \left( 4t+4\bar{\phi } \right) -4\bar{A}\bar{B} k_dm\cos \left( 2t\right. \\&\quad \left. +2\bar{\phi } \right) +4\bar{A}\bar{C}c_dm\cos \left( 2t+2\bar{\phi }\right) \\&\quad +\,4\bar{B}\bar{C}c_dm\cos \left( 2t+2\bar{\phi } \right) +\bar{A}{\bar{C}}^{3}k_dm\sin \left( 4t+4\bar{\phi } \right) \\&\quad +\,2\bar{A}{\bar{C}}^{3}k_dm\sin \left( 2t+2\bar{\phi } \right) \\&\quad -\,3{\bar{B}}^{3}\bar{C} k_dm\sin \left( 4t+4\bar{\phi } \right) +6{\bar{B}}^{3}\bar{C} k_dm\sin \left( 2t\right. \\&\quad \left. +2\bar{\phi }\right) -3{\bar{B}}^{2}{\bar{C}}^{2}k_dm\cos \left( 4t+4\bar{\phi } \right) \\&\quad +\,\bar{B}{\bar{C}}^{3} k_dm\sin \left( 4t+4\bar{\phi } \right) +2\bar{B}{\bar{C}}^{3}k_dm\sin \left( 2t\right. \\&\quad \left. +2\bar{\phi } \right) -3\bar{A}{\bar{B}}^{2}\bar{C} k_d\sin \left( 4t+4\bar{\phi } \right) \\&\quad +\,6\bar{A}{\bar{B}}^{2}\bar{C} k_d\sin \left( 2t+2\bar{\phi } \right) +4\bar{A}\bar{B}c_dm\sin \left( 2t\right. \\&\quad \left. +2\bar{\phi } \right) +4\bar{A}\bar{C}k_dm\sin \left( 2t+2\bar{\phi } \right) \\&\quad +\,4\bar{B}\bar{C}k_dm\sin \left( 2t+2\bar{\phi } \right) -4\bar{A}{\bar{B}}^{3} k_dm\cos \left( 2t\right. \\&\quad \left. +2\bar{\phi } \right) +\bar{A}{\bar{B}}^{3} k_dm\cos \left( 4t+4\bar{\phi } \right) \\&\quad +\,4\bar{A}\bar{B}cm\sin \left( 2t+2\bar{\phi } \right) -4\bar{C}m\bar{A}\sin \left( 2t+2\bar{\phi } \right) \\&\quad -\,4{\bar{B}}^{2} k_dm\cos \left( 2t+2\bar{\phi } \right) \\&\quad -\,4\bar{A}\bar{B} k_d\cos \left( 2t+2\bar{\phi } \right) +4\bar{B}m\bar{A}\cos \left( 2t+2\bar{\phi } \right) \\&\quad +\,4\bar{A}\bar{C} c_d\cos \left( 2t+2\bar{\phi } \right) \\&\quad -\,4{\bar{B}}^{4} k_dm\cos \left( 2t+2\bar{\phi } \right) +{\bar{B}}^{4} k_dm\cos \left( 4t+4\bar{\phi } \right) \end{aligned}$$
$$\begin{aligned}&\quad -\,4\bar{A}{\bar{B}}^{3}k_d\cos \left( 2t+2\bar{\phi } \right) \\&\quad +\,6\bar{A}{\bar{B}}^{2}\bar{C} k_dm\sin \left( 2t+2\bar{\phi } \right) -3\bar{A}\bar{B}{\bar{C}}^{2}k_dm\cos \left( 4t\right. \\&\quad \left. +4\bar{\phi } \right) -3\bar{A}{\bar{B}}^{2}\bar{C}k_dm\sin \left( 4t+4\bar{\phi } \right) \\&\quad +\,3\bar{A}\bar{B}{\bar{C}}^{2}k_dm+3\bar{A}\bar{B}{\bar{C}}^{2} k_d-4{\bar{A}}^{2}m\bigg ),\\&f_4=\frac{1}{8\bar{A}}\bigg (-{\bar{B}}^{3} k_d\cos \left( 4t+4\bar{\phi } \right) +4{\bar{B}}^{3}k_d\cos \left( 2t\right. \\&\quad \left. +2\bar{\phi } \right) +3{\bar{B}}^{2}\bar{C} k_d \sin \left( 4t+4\bar{\phi } \right) \\&\quad -\,6{\bar{B}}^{2}\bar{C} k_d\sin \left( 2t+2\bar{\phi } \right) + 3\bar{B}{\bar{C}}^{2} k_d\cos \left( 4t+4\bar{\phi }\right) \\&\quad -\,{\bar{C}}^{3}k_d\sin \left( 4t+4\bar{\phi } \right) \\&\quad -\,2{\bar{C}}^{3}k_d\sin \left( 2t+2\bar{\phi } \right) -3{\bar{B}}^{3} k_d-3\bar{B}{\bar{C}}^{2} k_d\\&\quad -\,4\bar{A}c\sin \left( 2t+2\bar{\phi } \right) -4\bar{B} c_d \sin \left( 2t+2\bar{\phi } \right) \\&\quad +\,4\bar{B} k_d\cos \left( 2t+2\bar{\phi } \right) -4\bar{C} k_d\sin \left( 2t+2\bar{\phi } \right) \\&\quad -\,4\bar{C} c_d\cos \left( 2t+2\bar{\phi } \right) -4\bar{B}k_d+4\bar{C} c_d \bigg ). \end{aligned}$$

Appendix C

The expressions for \(m_0, m_1\), and \(m_2\) in Eq. (34) are shown below

$$\begin{aligned}&m_0 =\frac{c_{{d}}}{8\left( -m+\sqrt{r}m-1 \right) ^{3}{m} ^{3}\left( \sqrt{r}-1 \right) ^{2}}\\&\quad \bigg (-2{c_{{d}}}^{2}{m}^{3}r+{c_{{d}}}^{2}{m}^{6}{r}^{5} -{c_{{d}}}^{2}mr+ 28{c_{{d}}}^{2}{m}^{6}{r}^{2} \\&\quad -\,3{c_{{d}}}^{2}{m}^{2}r+4{r}^{3/2}{c_{{d}}}^{2}m+10{c_{{d}}}^{2}{m}^{4}{r}^{2}+57{c_{{d}}}^{2}{m}^{5}{r}^{2}\\&\quad -\,7{r}^{3}{c_{{d}}}^{2}{m}^{2}+8{r}^{7/2}{c_{{d}}}^{2}{m}^{4}-44{c_{{d}}}^{2}{m}^{3}{r}^{2} \\&\quad -\,34{c_{ {d}}}^{2}{m}^{3}{r}^{3}+16{r}^{3/2}{c_{{d}}}^{2}{m}^{3} +56{r}^{5/2 }{c_{{d}}}^{2}{m}^{3}\\&\quad +\,{c_{{d}}}^{2}{m}^{6}r- 8{r}^{3/2} {c_{{d}}}^{2}{m}^{6}+24{r}^{5/2}{c_{{d}}}^{2}{m}^{2} \\&\quad -\,5{r}^{2}{c_{{d}}}^{2}m+2{r}^{5/2}{c_{{d}}}^{2}m-30{r}^{2}{c_{{d}}}^{2}{m}^{2}-2{c_{{d}}}^{2}{m}^{4}{r}^{4}\\&\quad +\,70{c_{{d}}}^{2}{m}^{6}{r}^{3}-8{r}^{9/2}{c_{{d}}}^{2}{m}^{6}-20{r}^{3/2}{c_{{d}}}^{2}{m}^{5} \\&\quad +\,28{c_{{d}}}^{2}{m}^{6}{r}^{4}+3{c_{{d}}}^{2}{m}^{5}r+85{c_{{d}}}^{2} {m}^{5}{r}^{3}-90{r}^{5/2}{c_{{d}}}^{2}{m}^{5}\\&\quad +\,8{r}^{7/2}{c_{{d}}}^{2}{m}^{3}-2{r}^{9/2}{c_{{d}}}^{2}{m}^{5} \\&\quad -\,48{r}^{7/2}{c_{{d}}}^{2}{m}^{5}-10{c_{{d}}}^{2}{m}^{4}{r}^{3}+15{c_{{d}}}^{2}{m}^{5}{r}^{4}-8{r}^{3/2}{c_{{d}}}^{2}{m}^{4}\\&\quad +\,16{r}^{3/2}{c_{{d}}}^{2}{m}^{2}-56{r}^{5/2}{c_{{d}} }^{2}{m}^{6} \\&\quad -\,56{r}^{7/2}{c_{{d}}}^{2}{m}^{6} +2{c_{{d}}}^{2}{m}^{4}r-32\sqrt{r}{m}^{4}-4{r}^{5/2}{m}^{5}\\&\quad -\,40{r}^{3/2}{m}^{5}-32{r}^{3/2}{m}^{4}+20{m}^{5}{r}^{2}+40{m}^{5}r \\&\quad +\,8{m}^{4}{r}^{2}+48{m}^{4}r-4{m}^{3}{r}^{3/2}+12{m}^{3}r -12{m}^{3}\sqrt{r}\\&\quad -\,20\sqrt{r}{m}^{5}+4{r}^{3/2}k_{{d}}{m}^{2} -48k_{{d}}{m}^{4}{r}^{2} \\&\quad -\,40k_{{d}}{m}^{5}{r}^{2}-36k_{{d}}{m}^{3}r-12k_{{d}}{m}^{3}{r}^{2}+72{r}^{3/2}k_{{d}}{m}^{4}\\&\quad -\,20k_{{d}}{m}^{5}r+12\sqrt{r}k_{{d}}{m}^{4}+40{r}^{3/2}k_{{d}}{m}^{5} \\&\quad +\,20{r}^{5/2}k_{{d}}{m}^{5}-48k_{{d}}{m}^{4}r+4\sqrt{r}k_{{d}}{m}^{2}-8k_{{d}}{m}^{2}r\\&\quad +\,12k_{{d}}{m}^{3}\sqrt{r}+4{m}^{3}+4{m}^{5}+8{m}^{4} \end{aligned}$$
$$\begin{aligned}&\quad +\,24{r}^{2}{c_{{d}}}^{2}k_{{d}}m+280{r}^{7/2} {c_{{d}}}^{2}k_{{d}}{m}^{5}+150{r}^{7/2}{c_{{d}}}^{2}k_{{d}}{m}^{4} \\&\quad -\,10{r}^{7/2}{c_{{d}}}^{2}k_{{d}}{m}^{2}-280{c_{{d}}}^{2}k_{{d}}{m}^{5}{r}^{3} \\&\quad -\,56{r}^{2}{c_{{d}}}^{2}k_{{d}}{m}^{5}-60{r}^{2}{c_{{d}}}^{2}k_{{d}} {m}^{4}+8{c_{{d}}}^{2}k_{{d}}m{r}^{3}\\&\quad +\,56{r}^{9/2}{c_{{d}}}^{2} k_{{d}}{m}^{5}+10{r}^{9/2}{c_{{d }}}^{2}k_{{d}}{m}^{4} \\&\quad -\,60{c_{{d}}}^{2}k_{{d}}{m}^{4}{r}^{4}-200{c_{{d}}}^{2}k_{{d}} {m}^{4} {r}^{3} +40{c_{{d}}}^{2}k_{{d}}{m}^{2}{r}^{3}\\&\quad -\, 8{c_{{d}}}^{2} k_{{d}}{m}^{5}{r}^{5}- 168{c_{{d}}}^{2}k_{{d}}{m}^{5}{r}^{4} \\&\quad -\,112{c_{{d}}}^{2}k_{{d}}{m}^{6}{r}^{3}-16{c_{{d}}}^{2}k_{{d}}{m}^{6}{r}^{5}-16{c_{{d}}}^{2}k_{{d}}{m}^{6}{r}^{2}\\&\quad -\,8{r}^{3/2}{c_{{d}}}^{2}k_{{d}}m+8{r}^{3/2}{c_{{d}}}^{2}k_{{d}}{m}^{5} \\&\quad +\,10{r}^{3 /2}{c_{{d}}}^{2}k_{{d}}{m}^{4} -10{r}^{3/2}{c_{{d}}}^{2} k_{{d}}{m}^{2}+168{r}^{5/2}{c_{{d}}}^{2}k_{{d}}{m}^{5} \\&\quad +\,150{r}^{5/2}{c_{{d}}}^{2 }k_{{d}}{m}^{4}-60{r}^{5/2}{c_{{d}}}^{2} k_{{d}}{m}^{2} \\&\quad -\,24{r}^{5/2}{c_{{d}}}^{2}k_{{d}}m-112{c_{{d}}}^{2}k_{{d}}{m}^{6} {r}^{4}+2{r}^{3/2}{c_{{d}}}^{2}k_{{d}}{m}^{6}\\&\quad +\,2{r}^{11/2} {c_{{d}}}^{2} k_{{d}}{m}^{6}+56{r}^{9/2}{c_{{d}}}^{2}k_{{d}}{m}^{6} \\&\quad +\,140{r}^{7/2}{c_{{d}}}^{2 }k_{{d}}{m}^{6}+56{r}^{5/2}{c_{{d}}}^{2} k_{{d}} {m}^{6}\\&\quad +\,40{r}^{2}{c_{{d}}}^{2}k_{{d}}{m}^{2}-2{r}^{5/2} {c_{{d}}}^{2}k_{{d}}+4{r}^{2}{c_{{d}}}^{2}k_{{d}}\\&\quad +\,12{r}^{5/2}k_{{d}}{m}^{4}-2{r}^{3/2}{c_{{d}}}^{2} k_{{d}} \\&\quad +\,4\sqrt{r}k_{{d}}{m}^{5}+36k_{{d}}{m}^{3}{r}^{3/2}-4k_{{d}}{m}^{5}{r}^{3} \bigg ),\\&m_1 =\frac{c_d}{4{r \left( -m+\sqrt{r}m-1 \right) ^{2}{m}^{3} \left( \sqrt{r}-1 \right) }} \\&\quad \bigg (6{c_{{d}}}^{2}{m}^{3}r +{c_{{d}}}^{2}mr +4{c_{{d}}}^{2}{m}^{2}r-3{r}^{3/2}{c_{{d}}}^{2}m \\&\quad +\,70{c_{{d}}}^{2} {m}^{4}{r}^{2}+ 26{c_{{d}}}^{2}{m}^{5}{r}^{2} -2{r}^{3}{c_{{d}}}^{2} {m}^{2} \\&\quad -\,{r}^{7/2}{c_{{d}}}^{2}{m}^{4}+64{c_{{d}}}^{2}{m}^{3}{r}^{2}+6{c_{{d}}}^{2}{m}^{3}{r}^{3}-33{r}^{3/2}{c_{{d}}}^{2}{m}^{3} \\&\quad -\,48{r}^{5/2}{c_{{d}}}^{2}{m}^{3}-7{r}^{5/2}{c_{{d}}}^{2}{m}^{2}+2{r}^{2}{c_{{d}}}^{2}m\\&\quad +\,22{r}^{2}{c_{{d}}}^{2}{m}^{2}-4{c_{{d}}}^{2}{m}^{4}{r}^{4}-8{r}^{3/2}{c_{{d}}}^{2}{m}^{5}+{c_{{d}}}^{2}{m}^{5}r \\&\quad +\,37{c_{{d}}}^{2}{m}^{5}{r}^{3}-43{r}^{5/2}{c_{{d}}}^{2}{m}^{5}+5{r}^{7/2}{c_{{d}}}^{2}{m}^{3}\\&\quad +\,{r}^{9/2}{c_{{d}}}^{2}{m}^{5}-14{r}^{7/2}{c_{{d}}}^{2}{m}^{5}+42{c_{{d}}}^{2}{m}^{4}{r}^{3} \\&\quad -\,27{r}^{3/2}{c_{{d}}}^{2}{m}^{4}-17{r}^{3/2}{c_{{d}}}^{2}{m}^{2}+4{c_{{d}}}^{2}{m}^{4}r\\&\quad +\,18\sqrt{r}{m}^{4}+{r}^{5/2}{m}^{5}+18{r}^{3/2}{m}^{5}+6{r}^{3/2}{m}^{4} \\&\quad -\,7{m}^{5}{r}^{2}-22{m}^{5}r-18{m}^{4}r-2{m}^{3}r+5{m}^{3}\sqrt{r}\\&\quad -\,2\sqrt{r}k_d{m}^{5}-6k_d{m}^{3}{r}^{3/2}+6k_d{m}^{4}{r}^{2} \\&\quad +\,8k_d{m}^{5}{r}^{2}+12k_d{m}^{3}r-18{r}^{3/2}k_d{m}^{4}+8k_d{m}^{5}r\\&\quad -\,6\sqrt{r}k_d{m}^{4} -12{r}^{3/2}k_d{m}^{5}-2{r}^{5/2}k_d{m}^{5} \\&\quad +\,18k_d{m}^{4}r-2 \sqrt{r}k_d{m}^{2}+2k_d{m}^{2}r-6k_d{m}^{3}\sqrt{r}\\&\quad -\,12{r}^{2}{c_{{d}}}^{2}k_d{m}^{5}-50{r}^{2}{c_{{d}}}^{2}k_d{m}^{4}+2{c_{{d}}}^{2}k_dm{r}^{3} \\&\quad +\,4{r}^{9/2}{c_{{d}}}^{2}k_d{m}^{5}-2{r}^{9/2}{c_{{d}}}^{2}k_d{m}^{4}-8{c_{{d}}}^{2}k_d{m}^{4}{r}^{4}\\&\quad -\,110{c_{{d}}}^{2}k_d{m}^{4}{r}^{3}-10{c_{{d}}}^{2}k_d{m}^{2}{r}^{3} \\&\quad -\,20{c_{{d}}}^{2}k_d{m}^{5}{r}^{4}+10{r}^{3/2}{c_{{d}}}^{2}k_dm+2{r}^{3/2}{c_{{d}}}^{2}k_d{m}^{5}\\&\quad +\,10{r}^{3/2}{c_{{d}}}^{2}k_d{m}^{4}+20{r}^{3/2}{c_{{d}}}^{2}k_d{m}^{2} \\&\quad +\,32{r}^{5/2}{c_{{d}}}^{2}k_d{m}^{5}+104{r}^{5/2}{c_{{d}}}^{2}k_d{m}^{4}\\&\quad +\,56{r}^{5/2}{c_{{d}}}^{2}k_d{m}^{2}+8{r}^{5/2}{c_{{d}}}^{2}k_dm-60{r}^{2}{c_{{d}}}^{2}k_d{m}^{2} \\&\quad -\,20{r}^{2}{c_{{d}}}^{2}k_dm+42{r}^{7/2}{c_{{d}}}^{2}k_d{m}^{5}+56{r}^{7/2}{c_{{d}}}^{2}k_d{m}^{4}\\&\quad -\,6{r}^{7/2}{c_{{d}}}^{2}k_d{m}^{2}-48{c_{{d}}}^{2}k_d{m}^{5}{r}^{3} \end{aligned}$$
$$\begin{aligned}&\quad +\,6{c_{{d}}}^{2}k_d{m}^{3}{r}^{4}+120{r}^{5/2}{c_{{d}}}^{2}k_d{m}^{3}+20{r}^{3/2}{c_{{d}}}^{2}k_d{m}^{3}\\&\quad +\,8{r}^{7/2}{c_{{d}}}^{2}k_d{m}^{3}-74{c_{{d}}}^{2}k_d{m}^{3}{r}^{3} \\&\quad -\,80{r}^{2}{c_{{d}}}^{2}k_d{m}^{3}+13\sqrt{r}{m}^{5}-3{m}^{3}-3{m}^{5}-6{m}^{4}\\&\quad -\,2{r}^{2}{c_{{d}}}^{2}k_d+2{r}^{3/2}{c_{{d}}}^{2}k_d- 84{r}^{5/2}{c_{{d}}}^{2}{m}^{4}\bigg ),\\&m_2=-\frac{c_{{d}}}{ \left( -m+\sqrt{r}m-1 \right) \left( \sqrt{r}-1 \right) {m}^{2}} \\&\quad \bigg ( k_d{m}^{3}{r}^{5/2}+6k_d{m}^{3}{r}^{3/2}+9{r}^{3/2}k_d{m}^{2} -6{m}^{3}{r}^{3/2} \\&\quad -\,4k_d{m}^{3}{r}^{2}+3{r}^{3/2}k_dm-3 {r}^{3/2}{m}^{2}+k_d{m}^{3} \sqrt{r}\\&\quad -4k_d{m}^{3}r-3{r}^{2}k_d{m}^{2}+2{m}^{3}{r}^{2}+3\sqrt{r}k_d{m}^{2} \\&\quad -\,2{m}^{3}\sqrt{r}-9k_d{m}^{2}r+6{m}^{3}r+3\sqrt{r}k_dm-3\sqrt{r}{m}^{2}\\&\quad -\,6rk_dm+6r{m}^{2}+\sqrt{r}k_d-\sqrt{r}m -rk_d+mr \bigg ), \end{aligned}$$

where \(r=c/c_d\).

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Shirude, A., Vyasarayani, C.P. & Chatterjee, A. Towards design of a nonlinear vibration stabilizer for suppressing single-mode instability. Nonlinear Dyn 103, 1563–1583 (2021). https://doi.org/10.1007/s11071-021-06207-7

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