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Vibration suppression and stability analysis of a beam at large amplitude excitation using a two-degree-of-freedom nonlinear energy sink

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Abstract

A two-degree-of-freedom (TDOF) nonlinear energy sink (NES) is proposed to suppress the vibration of a simply supported beam subjected to a large amplitude excitation corresponding to its fundamental frequency. The Euler–Bernoulli beam theory along with the Euler–Lagrange equation is utilized for the dynamical modeling of the beam-NES system. The numerical solutions are compared with an approximate analytical solution based on the complexification-averaging method. As the beam response significantly depends on the values of the NES parameters, the optimal values are obtained by employing the particle swarm optimization. After comparison with a single-degree-of-freedom (SDOF) NES, it is found that at lower excitation amplitude SDOF NES is more effective than TDOF NES, however, as the excitation amplitude increases, the TDOF NES gets more effective in terms of resonant peak suppression and energy dissipation. The TDOF NES reduces the first peak amplitude of about 95% as well as dissipates the vibration energy of about 90–98.98%. The chaotic response of the beam is examined using the largest Lyapunov exponents with different parameters. The higher values of the NES stiffness, mass and the excitation amplitude make the system chaotic. Moreover, the analysis of steady-state response shows that a higher value of NES damping reduces the unstable band and thus provides stable response.

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Acknowledgements

This work was supported by Science and Engineering Research Board, Govt. of India, core research grant CRG/2018/002539 (SER-1332-MID).

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  1. Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Rajni Kant Kumar & Anil Kumar

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Appendices

Appendix A

For two-mode expansion of beam, the set of real first-order differential equations are as follows:

$$\begin{aligned}\dot{\alpha}_{1} &=-\frac{1}{8{w}^{3}{M}_{11}}\Big\{4{w}^{3}\overline{\gamma }{\alpha }_{1}-4{w}^{4}{M}_{11}{\alpha }_{2}+4{w}^{2}{M}_{12}{\alpha }_{2} -3\overline{{K}_{1}}{\alpha }_{5}^{2}{\alpha }_{6}{\varphi }_{1}\left(\overline{d}\right)-3\overline{{K}_{1}}{\alpha }_{6}^{3}{\varphi }_{1}\left(\overline{d}\right)-4{w}^{3}{\alpha }_{5}\overline{{\lambda }_{1}}{\varphi }_{1}\left(\overline{d}\right)\\ &\quad +3\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{5}^{2}{\varphi }_{1}^{2}\left(\overline{d}\right)+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{5}{\alpha }_{6}{\varphi }_{1}^{2}\left(\overline{d}\right)+9\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{6}^{2}{\varphi }_{1}^{2}\left(\overline{d}\right)+4{w}^{3}{\alpha }_{1}\overline{{\lambda }_{1}}{\varphi }_{1}^{2}\left(\overline{d}\right)-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{5}{\varphi }_{1}^{3}\left(\overline{d}\right)\\ &\quad -3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{6}{\varphi }_{1}^{3}\left(\overline{d}\right)-9\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{6}{\varphi }_{1}^{3}\left(\overline{d}\right)+3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{2}{\varphi }_{1}^{4}\left(\overline{d}\right)+3\overline{{K}_{1}}{\alpha }_{2}^{3}{\varphi }_{1}^{4}\left(\overline{d}\right)+3\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{5}^{2}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)\\&\quad +6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{5}{\alpha }_{6}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)+9\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{6}^{2}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)+4{w}^{3}{\alpha }_{3}\overline{{\lambda }_{1}}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)-6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{5}{\varphi }_{1}^{2}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)\\&\quad -6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}{\alpha }_{5}{\varphi }_{1}^{2}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)- 6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{6}{\varphi }_{1}^{2}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)-18\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}{\alpha }_{6}{\varphi }_{1}^{2}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)\\ &\quad +6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{3}{\varphi }_{1}^{3}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)+3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{4}{\varphi }_{1}^{3}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)+9\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{4}{\varphi }_{1}^{3}\left(\overline{d}\right){\varphi }_{2}\left(\overline{d}\right)\\ &\quad -6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}{\alpha }_{5}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}^{2}\left(\overline{d}\right)-3\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{6}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}^{2}\left(\overline{d}\right)-9\overline{{K}_{1}}{\alpha }_{4}^{2}{\alpha }_{6}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}^{2}\left(\overline{d}\right)+3\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}^{2}{\varphi }_{1}^{2}\left(\overline{d}\right){\varphi }_{2}^{2}\left(\overline{d}\right)\\ &\quad+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{4}{\varphi }_{1}^{2}\left(\overline{d}\right){\varphi }_{2}^{2}\left(\overline{d}\right)+9\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}^{2}{\varphi }_{1}^{2}\left(\overline{d}\right){\varphi }_{2}^{2}\left(\overline{d}\right)+3\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{4}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}^{3}\left(\overline{d}\right)+3\overline{{K}_{1}}{\alpha }_{4}^{3}{\varphi }_{1}\left(\overline{d}\right){\varphi }_{2}^{3}\left(\overline{d}\right)\Big\}\end{aligned}$$

(A1)

$$\begin{aligned}\dot{\alpha}_{2}&=-\frac{1}{8{w}^{3}{M}_{11}}\{4 {\varphi }_{1}\left(\hat{a} \right)\overline{F}{w}^{3}+4{w}^{4}{M}_{11}{\alpha }_{1}-4{w}^{2}{M}_{12}{\alpha }_{1}+4{w}^{3}\overline{\gamma }{\alpha }_{2}+3\overline{{K}_{1}}{\alpha }_{5}^{3}{\varphi }_{1}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{5}{\alpha }_{6}^{2}{\varphi }_{1}(\overline{d})\\ &\quad -4{w}^{3}{\alpha }_{6}\overline{{\lambda }_{1}}{\varphi }_{1}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{5}^{2}{\varphi }_{1}^{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{5}{\alpha }_{6}{\varphi }_{1}^{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{6}^{2}{\varphi }_{1}^{2}(\overline{d})+4{w}^{3}{\alpha }_{2}\overline{{\lambda }_{1}}{\varphi }_{1}^{2}(\overline{d})\\ &\quad +9\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{5}{\varphi }_{1}^{3}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{5}{\varphi }_{1}^{3}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{6}{\varphi }_{1}^{3}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}^{3}{\varphi }_{1}^{4}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}^{2}{\varphi }_{1}^{4}(\overline{d})\\ &\quad -9\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{5}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{5}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{6}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})+4{w}^{3}{\alpha }_{4}\overline{{\lambda }_{1}}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})\\ &\quad +18\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{5}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}{\alpha }_{5}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{6}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})\\ &\quad +6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}{\alpha }_{6}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{3}{\varphi }_{1}^{3}(\overline{d}){\varphi }_{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{3}{\varphi }_{1}^{3}(\overline{d}){\varphi }_{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{4}{\varphi }_{1}^{3}(\overline{d}){\varphi }_{2}(\overline{d})\\ &\quad +9\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{4}^{2}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}^{2}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})\\ &\quad - 6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{4}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}^{2}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{3}^{3}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{3}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{3}(\overline{d})\} \end{aligned}$$
$$\begin{aligned}\dot{\alpha}_{3} &=-\frac{1}{8{w}^{3}{M}_{11}}\{4{w}^{3}\overline{\gamma }{\alpha }_{3}-4{w}^{4}{M}_{11}{\alpha }_{4}+4{w}^{2}{M}_{12}{\alpha }_{4}-3\overline{{K}_{1}}{\alpha }_{5}^{2}{\alpha }_{6}{\varphi }_{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{6}^{3}{\varphi }_{2}(\overline{d})-4{w}^{3}{\alpha }_{5}\overline{{\lambda }_{1}}{\varphi }_{2}(\overline{d})\\ &\quad + 3\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{5}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{5}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{6}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})+4{w}^{3}{\alpha }_{1}\overline{{\lambda }_{1}}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})\\ &\quad -6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{5}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{6}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{6}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{2}{\varphi }_{1}^{3}(\overline{d}){\varphi }_{2}(\overline{d})\\ &\quad +3\overline{{K}_{1}}{\alpha }_{2}^{3}{\varphi }_{1}^{3}(\overline{d}){\varphi }_{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{5}^{2}{\varphi }_{2}^{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{5}{\alpha }_{6}{\varphi }_{2}^{2}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{6}^{2}{\varphi }_{2}^{2}(\overline{d})+4{w}^{3}{\alpha }_{3}\overline{{\lambda }_{1}}{\varphi }_{2}^{2}(\overline{d})\\ &\quad -6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})\\ &\quad -18\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{3}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{4}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{4}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})\\ &\quad -6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}{\alpha }_{5}{\varphi }_{2}^{3}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{6}{\varphi }_{2}^{3}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{4}^{2}{\alpha }_{6}{\varphi }_{2}^{3}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{3}(\overline{d})\\ &\quad +6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{4}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{3}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{3}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{4}{\varphi }_{2}^{4}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{4}^{3}{\varphi }_{2}^{4}(\overline{d})\}\end{aligned}$$
$$\begin{aligned}\dot{\alpha}_{4} &=-\frac{1}{8{w}^{3}{M}_{11}}\{4{\varphi }_{2}\left(\hat{a} \right)\overline{F}{w}^{3}+4{w}^{4}{M}_{11}{\alpha }_{3}-4{w}^{2}{M}_{12}{\alpha }_{3}+4{w}^{3}\overline{\gamma }{\alpha }_{4}+3\overline{{K}_{1}}{\alpha }_{5}^{3}{\varphi }_{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{5}{\alpha }_{6}^{2}{\varphi }_{2}(\overline{d}) \\ &\quad -4{w}^{3}{\alpha }_{6}\overline{{\lambda }_{1}}{\varphi }_{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{5}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{5}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{6}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})\\ &\quad +4{w}^{3}{\alpha }_{2}\overline{{\lambda }_{1}}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{5}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{5}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{6}{\varphi }_{1}^{2}{\varphi }_{2}(\overline{d})\\ &\quad -3\overline{{K}_{1}}{\alpha }_{1}^{3}{\varphi }_{1}^{3}(\overline{d}){\varphi }_{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}^{2}{\varphi }_{1}^{3}(\overline{d}){\varphi }_{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{5}^{2}{\varphi }_{2}^{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{5}{\alpha }_{6}{\varphi }_{2}^{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{6}^{2}{\varphi }_{2}^{2}(\overline{d})\\ &\quad +4{w}^{3}{\alpha }_{4}\overline{{\lambda }_{1}}{\varphi }_{2}^{2}(\overline{d})+18\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})\\ &\quad +6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{3}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{3}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{4}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}^{2}(\overline{d})\\ &\quad +9\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{5}{\varphi }_{2}^{3}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{4}^{2}{\alpha }_{5}{\varphi }_{2}^{3}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}{\alpha }_{6}{\varphi }_{2}^{3}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{3}(\overline{d})\\ &\quad -6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{4}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{3}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{3}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{3}^{3}{\varphi }_{2}^{4}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}^{2}{\varphi }_{2}^{4}(\overline{d})\}\end{aligned}$$
$$\begin{aligned}\dot{\alpha}_{5}&=-\frac{1}{8{w}^{3}{\varepsilon }_{1}}\{3\overline{{K}_{1}}{\alpha }_{5}^{2}{\alpha }_{6}+3\overline{{K}_{2}}{\alpha }_{5}^{2}{\alpha }_{6}+3\overline{{K}_{1}}{\alpha }_{6}^{3}+3\overline{{K}_{2}}{\alpha }_{6}^{3}-6\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{6}{\alpha }_{7}+3\overline{{K}_{2}}{\alpha }_{6}{\alpha }_{7}^{2}-3\overline{{K}_{2}}{\alpha }_{5}^{2}{\alpha }_{8}-9\overline{{K}_{2}}{\alpha }_{6}^{2}{\alpha }_{8}\\ &\quad +6\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{7}{\alpha }_{8}-3\overline{{K}_{2}}{\alpha }_{7}^{2}{\alpha }_{8}+9\overline{{K}_{2}}{\alpha }_{6}{\alpha }_{8}^{2}-3\overline{{K}_{2}}{\alpha }_{8}^{3}-4{w}^{4}{\alpha }_{6}{\varepsilon }_{1}+4{w}^{3}{\alpha }_{5}\overline{{\lambda }_{1}}+4{w}^{3}{\alpha }_{5}\overline{{\lambda }_{2}}\\ &\quad -4{w}^{3}{\alpha }_{7}\overline{{\lambda }_{2}}-3\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{5}^{2}{\varphi }_{1}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{5}{\alpha }_{6}{\varphi }_{1}(\overline{d})\\ &\quad -9\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{6}^{2}{\varphi }_{1}(\overline{d})-4{w}^{3}{\alpha }_{1}\overline{{\lambda }_{1}}{\varphi }_{1}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{5}{\varphi }_{1}^{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{6}{\varphi }_{1}^{2}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{6}{\varphi }_{1}^{2}(\overline{d})\\ &\quad -3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{2}{\varphi }_{1}^{3}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{2}^{3}{\varphi }_{1}^{3}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{5}^{2}{\varphi }_{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{5}{\alpha }_{6}{\varphi }_{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{6}^{2}{\varphi }_{2}(\overline{d})-4{w}^{3}{\alpha }_{3}\overline{{\lambda }_{1}}{\varphi }_{2}(\overline{d})\\ &\quad +6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})\\ &\quad +18\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{3}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{4}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{4}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})\\ &\quad +6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}{\alpha }_{5}{\varphi }_{2}^{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{6}{\varphi }_{2}^{2}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{4}^{2}{\alpha }_{6}{\varphi }_{2}^{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})\\ &\quad -6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{4}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{4}{\varphi }_{2}^{3}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{4}^{3}{\varphi }_{2}^{3}(\overline{d})\}\end{aligned}$$
$$\begin{aligned}\dot{\alpha}_{6}&=-\frac{1}{8{w}^{3}{\varepsilon }_{1}}\{-3\overline{{K}_{1}}{\alpha }_{5}^{3}-3\overline{{K}_{2}}{\alpha }_{5}^{3}-3\overline{{K}_{1}}{\alpha }_{5}{\alpha }_{6}^{2}-3\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{6}^{2}+9\overline{{K}_{2}}{\alpha }_{5}^{2}{\alpha }_{7}+3\overline{{K}_{2}}{\alpha }_{6}^{2}{\alpha }_{7}-9\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{7}^{2}+3\overline{{K}_{2}}{\alpha }_{7}^{3}\\&\quad+6\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{6}{\alpha }_{8}-6\overline{{K}_{2}}{\alpha }_{6}{\alpha }_{7}{\alpha }_{8}-3\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{8}^{2}\\&\quad+3\overline{{K}_{2}}{\alpha }_{7}{\alpha }_{8}^{2}+4{w}^{4}{\alpha }_{5}{\varepsilon }_{1}+4{w}^{3}{\alpha }_{6}\overline{{\lambda }_{1}}+4{w}^{3}{\alpha }_{6}\overline{{\lambda }_{2}}-4{w}^{3}{\alpha }_{8}\overline{{\lambda }_{2}}+9\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{5}^{2}{\varphi }_{1}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{5}{\alpha }_{6}{\varphi }_{1}(\overline{d})\\&\quad+3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{6}^{2}{\varphi }_{1}(\overline{d})-4{w}^{3}{\alpha }_{2}\overline{{\lambda }_{1}}{\varphi }_{1}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{5}{\varphi }_{1}^{2}(\overline{d})-3\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{5}{\varphi }_{1}^{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{6}{\varphi }_{1}^{2}(\overline{d})\\&\quad+3\overline{{K}_{1}}{\alpha }_{1}^{3}{\varphi }_{1}^{3}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}^{2}{\varphi }_{1}^{3}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{5}^{2}{\varphi }_{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{4}{\alpha }_{5}{\alpha }_{6}{\varphi }_{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{6}^{2}{\varphi }_{2}(\overline{d})\\ &\quad -4{w}^{3}{\alpha }_{4}\overline{{\lambda }_{1}}{\varphi }_{2}(\overline{d})-18\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{4}{\alpha }_{5}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})\\&\quad-6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}{\alpha }_{6}{\varphi }_{1}(\overline{d}){\varphi }_{2}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{1}^{2}{\alpha }_{3}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{2}^{2}{\alpha }_{3}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})\\&\quad+6\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{2}{\alpha }_{4}{\varphi }_{1}^{2}(\overline{d}){\varphi }_{2}(\overline{d})-9\overline{{K}_{1}}{\alpha }_{3}^{2}{\alpha }_{5}{\varphi }_{2}^{2}(\overline{d})\\&\quad-3\overline{{K}_{1}}{\alpha }_{4}^{2}{\alpha }_{5}{\varphi }_{2}^{2}(\overline{d})-6\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}{\alpha }_{6}{\varphi }_{2}^{2}(\overline{d})+9\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{3}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+6\overline{{K}_{1}}{\alpha }_{2}{\alpha }_{3}{\alpha }_{4}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})\\&\quad+3\overline{{K}_{1}}{\alpha }_{1}{\alpha }_{4}^{2}{\varphi }_{1}(\overline{d}){\varphi }_{2}^{2}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{3}^{3}{\varphi }_{2}^{3}(\overline{d})+3\overline{{K}_{1}}{\alpha }_{3}{\alpha }_{4}^{2}{\varphi }_{2}^{3}(\overline{d})\}\end{aligned}$$
$$\begin{aligned}\dot{\alpha}_{7}&=-\frac{1}{8{w}^{3}{\varepsilon }_{2}}\{-3\overline{{K}_{2}}{\alpha }_{5}^{2}{\alpha }_{6}-3\overline{{K}_{2}}{\alpha }_{6}^{3}+6\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{6}{\alpha }_{7}-3\overline{{K}_{2}}{\alpha }_{6}{\alpha }_{7}^{2}\\&\quad+3\overline{{K}_{2}}{\alpha }_{5}^{2}{\alpha }_{8}+9\overline{{K}_{2}}{\alpha }_{6}^{2}{\alpha }_{8}-6\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{7}{\alpha }_{8}+3\overline{{K}_{2}}{\alpha }_{7}^{2}{\alpha }_{8}-9\overline{{K}_{2}}{\alpha }_{6}{\alpha }_{8}^{2}\\&\quad+3\overline{{K}_{2}}{\alpha }_{8}^{3}-4{w}^{4}{\alpha }_{8}{\varepsilon }_{2}-4{w}^{3}{\alpha }_{5}\overline{{\lambda }_{2}}+4{w}^{3}{\alpha }_{7}\overline{{\lambda }_{2}}\}\end{aligned}$$

\(\begin{aligned}\dot{\alpha}_{8}&=-\frac{1}{8{w}^{3}{\varepsilon }_{2}}\{3\overline{{K}_{2}}{\alpha }_{5}^{3}+3\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{6}^{2}-9\overline{{K}_{2}}{\alpha }_{5}^{2}{\alpha }_{7}-3\overline{{K}_{2}}{\alpha }_{6}^{2}{\alpha }_{7}+9\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{7}^{2}-3\overline{{K}_{2}}{\alpha }_{7}^{3}-6\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{6}{\alpha }_{8}+6\overline{{K}_{2}}{\alpha }_{6}{\alpha }_{7}{\alpha }_{8}+3\overline{{K}_{2}}{\alpha }_{5}{\alpha }_{8}^{2}-3\overline{{K}_{2}}{\alpha }_{7}{\alpha }_{8}^{2}+4{w}^{4}{\alpha }_{7}{\varepsilon}_{2}-4{w}^{3}{\alpha }_{6}\overline{{\lambda }_{2}}+4{w}^{3}{\alpha }_{8}\overline{{\lambda}_{2}}\}.\end{aligned}\)

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Kumar, R.K., Kumar, A. Vibration suppression and stability analysis of a beam at large amplitude excitation using a two-degree-of-freedom nonlinear energy sink. Acta Mech 235, 971–990 (2024). https://doi.org/10.1007/s00707-023-03765-2

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