Dynamics of a nonlinear vibration absorption system with time delay

Abstract

This paper reveals the dynamical responses of a time-delayed nonlinear vibration absorption system under harmonic excitation. The slow and fast dynamics of the forced system are analyzed by using complex averaging method. The curves of saddle-node bifurcation and Hopf bifurcation are given. Afterward, the analytical expressions and properties of slow invariant manifold are explored. The existence of strongly modulated response is determined by discussing the geometry of the slow invariant manifold. Furthermore, abundant and interesting behaviors are observed numerically and these phenomena reach a good agreement with theoretical analysis. The results show that time delay control plays important roles in the vibration reduction performance and can regulate the response regimes, such as the generation and transition of periodic orbits and quasi-periodic solutions.

Appendix

$$ s_{1} = (1 + \varepsilon )[\lambda + g\sin (\tau )] $$

$$ \begin{gathered} s_{2} = - \frac{1}{4}\left[ {(3kD + 2g - 4\sigma )\varepsilon^{2} + (6kD + 4g - 4\sigma )\varepsilon + 3kD + 2g - 2} \right]g\cos (\tau ) \hfill \\ \quad \quad + \frac{{\varepsilon^{2} }}{64}\left[ {32g^{2} + g(48kD - 64\sigma ) + 27k^{2} D^{2} - 96kD\sigma + 128\sigma^{2} + 16\lambda^{2} } \right] \hfill \\ \quad \quad + \frac{\varepsilon }{64}\left[ {64g^{2} + g(96kD - 64\sigma ) + 54k^{2} D^{2} - 96kD\sigma + 32\lambda^{2} + 64\sigma } \right] + \frac{g}{4}(3kD - 2) \hfill \\ \quad \quad + \frac{g}{2}\lambda (1 + \varepsilon )^{2} \sin (\tau ) + \frac{{\lambda^{2} }}{4} + \frac{{27k^{2} D^{2} }}{64} - \frac{3kD}{4} + \frac{{g^{2} }}{2} + \frac{1}{4} \hfill \\ \end{gathered} $$

$$ s_{3} = \frac{\varepsilon }{4}[\lambda + g\sin (\tau )][1 + 4\varepsilon^{2} \sigma^{2} + 4\varepsilon \sigma (1 + \sigma )] $$

$$ \begin{gathered} s_{4} = - \frac{{\varepsilon^{2} g}}{64}[8 + 16\left( {1 + \varepsilon } \right)\sigma ]\{ - 4\varepsilon \sigma^{2} + [2g(\varepsilon + 1) + 3\varepsilon kD + 3kD - 2]\sigma + \frac{3kZ}{2} + g]\} \cos (\tau ) \hfill \\ \quad \quad + \frac{{\varepsilon^{2} \lambda g}}{8}[1 + 2(1 + \varepsilon )\sigma ]^{2} \sin (\tau ) + \varepsilon^{4} \sigma^{4} - \frac{{\varepsilon^{3} {\mkern 1mu} \sigma^{3} }}{2}[2g(\varepsilon + 1) + 3\varepsilon kD + 3kD - 2] \hfill \\ \quad \quad + \varepsilon^{2} \sigma^{2} \left\{ {\frac{{(1 + \varepsilon )^{2} g^{2} }}{2} + g[\frac{3}{4}\varepsilon^{2} kD + \varepsilon ( - 1 + \frac{3}{2}kD) + \frac{3}{4}kD - \frac{1}{2}] + (\frac{{\lambda^{2} }}{4} + \frac{27}{{64}}k^{2} D^{2} )\varepsilon^{4} } \right\} \hfill \\ \quad \quad + \frac{27}{{64}}\varepsilon^{2} \sigma^{2} \left[ {\left( {\frac{32}{{27}}\lambda^{2} + 2k^{2} D^{2} - \frac{32}{9}kD} \right)\varepsilon + k^{2} D^{2} - \frac{16kD}{9} + \frac{{16\lambda^{2} }}{27} + \frac{16}{{27}}} \right] \hfill \\ \quad \quad + \varepsilon^{2} \sigma \left[ {\frac{1}{2}(\varepsilon + 1)g^{2} + \frac{1}{4}g( - 1 + 3kD + 3\varepsilon kD) + \left( {\frac{{\lambda^{2} }}{4} + \frac{27}{{64}}k^{2} D^{2} } \right)\varepsilon + \frac{27}{{64}}k^{2} D^{2} - \frac{3}{8}kD + \frac{{\lambda^{2} }}{4}} \right] \hfill \\ \quad \quad + \frac{{\varepsilon^{2} }}{16}\left( {\lambda^{2} + 3gkD + \frac{27}{{16}}k^{2} D^{2} + 2g^{2} } \right) \hfill \\ \end{gathered} $$