Limit cycles in nonlinear vibrating systems excited by a nonideal energy source with a large slope characteristic

Abstract

This study is the natural continuation of a previous paper of the authors González-Carbajal and Domínguez in J. Sound Vib. (Under revision), where the possibility of finding Hopf bifurcations in vibrating systems excited by a nonideal power source was addressed. Herein, some analytical tools are used to characterize these Hopf bifurcations, deriving a simple rule to classify them as supercritical or subcritical. Moreover, we find conditions under which the averaged system can be proved to be always attracted by a limit cycle, irrespective of the initial conditions. These limit cycle oscillations in the averaged system correspond to quasiperiodic motions of the original system. To the authors’ knowledge, limit cycle oscillations have not been addressed before in the literature about nonideal excitations. Through supporting numerical simulations, we also investigate the global bifurcations destroying the limit cycles. The analytical results are verified numerically.

Appendix 1: coefficients \(f_{ij} \) and \(g_{ij} \)

The coefficients of functions $f\left( {z_1 ,z_2 } \right)$ and $g\left( {z_1 ,z_2 } \right)$ in (33) have the following expressions.

$$\begin{aligned} f_{20}= & {} -\frac{3\rho aR}{4c_2 }\left( {3c_1^2 +c_2^2 } \right) -2c_1 \left( {\frac{2\xi }{aR}+\frac{3}{4}\rho a^{2}} \right) \nonumber \\ \end{aligned}$$

(55)

$$\begin{aligned} f_{02}= & {} -\frac{9\rho aR\omega _0^2 }{4c_2 } \end{aligned}$$

(56)

$$\begin{aligned} f_{11}= & {} -\frac{3}{4}\rho \omega _0 a\left( {a+3R\frac{c_1 }{c_2 }} \right) -\frac{2\xi \omega _0 }{aR} \end{aligned}$$

(57)

$$\begin{aligned} f_{30}= & {} \frac{9\rho c_1 }{4c_2 }\left( {c_1^2 +c_2^2 } \right) \end{aligned}$$

(58)

$$\begin{aligned} f_{03}= & {} \frac{9\rho \omega _0^3 }{4c_2 } \end{aligned}$$

(59)

$$\begin{aligned} f_{21}= & {} \frac{3\rho \omega _0 }{4c_2 }\left( {3c_1^2 +c_2^2 } \right) \end{aligned}$$

(60)

$$\begin{aligned} f_{12}= & {} \frac{9c_1 \rho \omega _0^2 }{4c_2 } \end{aligned}$$

(61)

$$\begin{aligned} g_{20}= & {} \frac{\left( {c_1^2 +c_2^2 } \right) }{\omega _0 }\left\{ {\frac{9\rho a}{4}\left( {R\frac{c_1 }{c_2 }+a} \right) +\frac{4\xi }{aR}} \right\} \end{aligned}$$

(62)

$$\begin{aligned} g_{02}= & {} \frac{3\rho \omega _0 a}{4}\left( {a+3R\frac{c_1 }{c_2 }} \right) \end{aligned}$$

(63)

$$\begin{aligned} g_{11}= & {} \frac{3}{2}\rho a^{2}c_1 +\frac{3\rho aR}{4c_2 }\left( {3c_1^2 +c_2^2 } \right) +\frac{2\xi c_1 }{aR} \end{aligned}$$

(64)

$$\begin{aligned} g_{30}= & {} -\frac{9\rho }{4\omega _0 c_2 }\left( {c_1^2 +c_2^2 } \right) ^{2} \end{aligned}$$

(65)

$$\begin{aligned} g_{03}= & {} -\frac{9c_1 \rho \omega _0^2 }{4c_2 } \end{aligned}$$

(66)

$$\begin{aligned} g_{21}= & {} -\frac{9c_1 \rho }{4c_2 }\left( {c_1^2 +c_2^2 } \right) \end{aligned}$$

(67)

$$\begin{aligned} g_{12}= & {} -\frac{3\rho \omega _0 }{4c_2 }\left( {3c_1^2 +c_2^2 } \right) \end{aligned}$$

(68)

where $a_\mathrm{eq}$ and $R_\mathrm{eq}$ have been shortly written as a and R, respectively.