Internal autoresonance in coupled oscillators with slowly decaying frequency

In this work, we study resonance energy transfer from an impulsively loaded strongly nonlinear oscillator to a weakly coupled linear attachment with a slowly time-decaying stiffness. It is shown that even in the absence of external periodic forcing both oscillators may exhibit the resonance phenomenon, with the permanent response enhancement of the linear oscillator and the corresponding response reduction of the nonlinear actuator. This effect is said to be internal autoresonance. The influence of the system parameters on the emergence and stability of autoresonance is investigated both analytically and numerically.

Figure 1

Responses $u_1$ and $u_2$ in system ((2.3)) with parameters $M=1,\varepsilon =0.1,\beta =0.05$, and different coefficients of nonlinearity: (a) displacements $u_2$ of the linear oscillators in system ((2.3)) with different coefficients of nonlinearity; (b) displacements of both oscillators at nonlinearity $\alpha =2$.

Figure 2

Responses of the oscillators in the basic system with parameters $M=1,\alpha =0.5,\varepsilon =0.1,\beta =0.05$: (a) the amplitudes squared $a^2$ and $b^2$, (b) the function $z\left(\tau \right)=b^2\left(\tau \right)$, the backbone curve $\overline{z}\left(\tau \right)$ (the solid line), and its linear approximation ${\overline{z}}_l\left(\tau \right)$ (the dashed line) for the linear oscillator.

Figure 3

Backbone curves of the linear oscillator ((4.1)) with parameters: (a) $\varepsilon =0.1,M=1,\beta =0.05$ (basic); (b) $\varepsilon =0.1,M=1,\beta =0.01$; (c) $\varepsilon =0.05,M=1,\beta =0.05$; (d) $\varepsilon =0.1,M=0.25,\beta =0.05$.

Figure 4

Responses $u_1$ and $u_2$ in the system with parameters $\alpha =0.5,\varepsilon =0.1,\beta =0.05$, and different values of the relative mass $M$: (a) responses of both oscillators at $M=m_2/m_1=0.25$; (b) responses of the linear oscillators at $M=0.25$ and $M=1$ (equal masses).

Figure 5

Tunneling generated by the unit initial impulse imposed on the linear oscillator: the solid line corresponds to the excited linear oscillator; the dotted line depicts the response of the nonlinear oscillator.

Figure 6

Responses $u_1$ and $u_2$ in the time-invariant coupled oscillators with parameters $M=1,\varepsilon =0.1,\beta =0$, and different coefficients of nonlinearity. $u_1$: dashed line; $u_2$: solid line.