The influence of two kinds of time delays on the vibrational resonance of a fractional Mathieu–Duffing oscillator

Abstract

Vibrational resonance is studied in a fractional Mathieu–Duffing oscillator with two types of time delays: fixed and distributed delays. The theoretical expression of the response amplitude is obtained by utilising the method of direct partition of slow and fast motions. Relative errors between the theoretical prediction and the numerical simulation are introduced to verify the validity of analytical approaches. The relative error of the displacement and the relative error of the response amplitude are calculated. Small relative errors show that the theoretical analysis is statistically correct. Therefore, the effects of fractional order, linear stiffness coefficient, low-frequency signal, time delay intensity and damping coefficient on the Mathieu–Duffing oscillator with distributed delay are studied successively. In order to better illustrate the impact of distributed time delay on the model, the case of fixed time delay is analysed and compared, and it can be found that the distributed delay has more significant influence than fixed delay on the system. In addition, the influence of distributed delay on the system is more significant than that of the fixed delay.

Appendix A. An appendix section

In light of ${\mathrm{eq}.}$ (5) and $x=X+\Psi$,

$$\begin{aligned} Z\left( {t,x\left( t \right) } \right)= & {} \int _{ - \infty }^t {\sigma {\mathrm {e}^{ - \sigma \left( {t - \tau } \right) }} \left( {x\left( \tau \right) } \right) \,\mathrm {d}\tau } \nonumber \\= & {} \int _{ - \infty }^t {\sigma {\mathrm {e}^{ - \sigma \left( {t - \tau } \right) }}\left( {X\left( \tau \right) + \Psi \left( \tau \right) } \right) \,\mathrm {d}\tau } \nonumber \\= & {} Z\left( {t,X\left( t \right) } \right) + Z\left( {t,\Psi \left( t \right) } \right) . \end{aligned}$$

(A.1)

For $\Psi (t) = {A_H}\cos \left( {\Omega t + {\Phi _H}} \right)$, then

$$\begin{aligned} Z\left( {t,\Psi \left( t \right) } \right)= & {} \int _{ - \infty }^t {\sigma {\mathrm {e}^{ - \sigma \left( {t - \tau } \right) }}\Psi \left( \tau \right) \,\mathrm {d}\tau }\nonumber \\= & {} \sigma {A_H}\int _{ - \infty }^t {{\mathrm {e}^{ - \sigma \left( {t - \tau } \right) }}\cos ( {\Omega \tau + {\Phi _H}} ) \,\mathrm {d}\tau } \nonumber \\= & {} \sigma {A_H}{\mathrm {e}^{ - \sigma t}} \int _{ - \infty }^t {{\mathrm {e}^{\sigma \tau }} \cos ( {\Omega \tau + {\Phi _H}} )\,\mathrm {d}\tau }.\nonumber \\ \end{aligned}$$

(A.2)

According to the divisional integration method, the integral value in (A.2) is

$$\begin{aligned}&\int _{ - \infty }^t {{\mathrm {e}^{\sigma \tau }} \cos ( {\Omega \tau + {\Phi _H}} ) \,\mathrm {d}\tau } \nonumber \\&\quad = \frac{{{\mathrm {e}^{\sigma t}}}}{{\sqrt{{\Omega ^2} +{\sigma ^2}} }}\sin \left( {\Omega t + {\Phi _H} +\arctan \frac{\sigma }{\Omega }} \right) . \end{aligned}$$

We have

$$\begin{aligned} \mathrm{Z}\left( {t,\Psi \left( t \right) } \right) =\frac{{\sigma {A_H}}}{{\sqrt{{\Omega ^2} +{\sigma ^2}} }}\sin ( {\Omega t + {\Phi _H} + \theta }) \end{aligned}$$

(A.3)

where

$$\begin{aligned} \theta = \arctan \frac{\sigma }{\Omega }. \end{aligned}$$