Multiple vibrational resonance and antiresonance in a coupled anharmonic oscillator under monochromatic excitation

Abstract

We examine the existence of multiple vibrational resonance (VR) and antiresonance in two coupled overdamped anharmonic oscillators where each one is individually driven by a monochromatic sinusoidal signal with widely separated frequencies ($\Omega \gg \omega$). In contemporary VR, superposed periodic waves are adopted to infer resonance, but herein we employ non-superposed periodic waves to acquire the elevated response. We study two coupling schemes namely, unidirectional and bidirectional, to substantiate the occurrence of multiple VR and antiresonance. Such occurrences have been shown and the results were ascertained with supportive numerical and experimental outcomes. We also illustrate the effect of coupling strength on the observed phenomenon.

Appendices

Appendix A. Unidirectionally coupled oscillators—Theoretical approach

We have made an attempt to provide an empirical solution to the considered models. While solving eq. (1), the solution for $\dot{x}$ is determined to be

$$\begin{aligned} x=\frac{g\cos (\Omega t-\theta _1)}{\sqrt{(1+\Omega ^2)}}, \end{aligned}$$

(A1)

where $\theta _1=\tan ^{-1}(\Omega )$. Upon substituting the value of x, the standard method of variable decomposition $y(t)=Y(t,\omega t)+\psi (t,\Omega t)$ is applied to eq. (2), where $Y(t,\omega t)$ and $\psi (t,\Omega t)$ correspond to slow and fast moving components of the response, respectively [40]. Employing the decomposition method and averaging the fast moving components, one gets

$$\begin{aligned} \dot{Y}= & {} Y(1-3\bar{\psi } ^2)- Y^3 +f\cos (\omega t), \end{aligned}$$

(A2)

$$\begin{aligned} \dot{\psi }= & {} g_1\cos (\Omega t) \end{aligned}$$

(A3)

with $g_1={\gamma g}/({\sqrt{1+\Omega ^2}})$. Considering the value of $\bar{\psi } ^2$ from eqs (A3), (A2) becomes

$$\begin{aligned} \dot{Y}-Yc_1+Y^3=f\cos (\omega t), \end{aligned}$$

(A4)

where $c_1=1-({3g_1^2}/{2\Omega ^2})$ and it is quite evident from the above equation that the shape of the potential depends on the value of g, $\gamma$ and $\Omega$. When $c_1 >0$, a double-well potential is obtained with its maxima and minima located at $Y_1^*=0$ and $Y_{2,3}^*=\pm \sqrt{c_1}$, respectively. On the other hand, when $c_1 <0$, the shape of the potential becomes a single well with its minima located at $Y_1^*=0$. Also, from eq. (A4), it is evident that the effective parameters of the system get affected due to resonance.

The critical value of g at which the resonance is observed can be deduced from the following relation:

$$\begin{aligned} g_{\max }=\sqrt{\frac{2\Omega ^2(\Omega ^2 +1)}{3\gamma ^2}}. \end{aligned}$$

(A5)

As the oscillators are one-way coupled, f and $\omega$ do not affect $g_{\max }$. As previously mentioned, if $g<\sqrt{({2\Omega ^2(\Omega ^2 +1)})/{3\gamma ^2}}$, there exist two minima in ${\pm }\sqrt{c_1}$ and the deviation from the minima $Y_{2,3}^*$ can be calculated by substituting $Z=Y-Y_{2,3}$ in eq. (A4). Upon substitution of z and using the linearisation, one gets a modified equation as

$$\begin{aligned} \dot{z}=s_1z+f\cos \omega t, \end{aligned}$$

(A6)

where $s_1=2c_1$. From the above equation, the response amplitude and the phase shift are analytically calculated to be

$$\begin{aligned} Q= & {} \frac{1}{\sqrt{\omega ^2+4c_1 ^2}}, \end{aligned}$$

(A7)

$$\begin{aligned} \theta _2= & {} \tan ^{-1} \frac{\omega }{s_1}. \end{aligned}$$

(A8)

Else when $g>\sqrt{({2\Omega ^2(\Omega ^2 +1)})/{3\gamma ^2}}$, there exists only one minimum at $Y_1^*$ and the response amplitude and phase shift now become

$$\begin{aligned} Q= & {} \frac{1}{\sqrt{\omega ^2+c_1 ^2}}, \end{aligned}$$

(A9)

$$\begin{aligned} \theta _2= & {} \tan ^{-1} \frac{\omega }{c_1}. \end{aligned}$$

(A10)

The response amplitude is calculated using eq. (A9) when $\omega =0.1$, $\Omega =0.5$, $f=0.05$ and the outcomes are compared in figure 12.

Fig. 12

Comparison of numerical and analytical values of Q for fixed values of $f=0.05$, $\omega =0.1$ and $\Omega =0.5$ when the coupling strength $\gamma =1.0.$

Appendix B. Mutually coupled oscillators—Theoretical approach

Further, as the process of obtaining analytical solution for mutually coupled oscillators is tedious, a highly approximated method of solution is proposed here. It starts with the addition of eqs (8) and (9) as

$$\begin{aligned} \dot{x}+\dot{y}=(x+y)-(x^3+y^3)+f\cos \omega t+g\cos \Omega t. \end{aligned}$$

(B1)

Replacing $x=X(t,\omega t)+\psi (t, \Omega t)$ and $y=Y(t,\omega t)+\psi (t, \Omega t)$ and on resolving slow and fast components, we obtain

$$\begin{aligned}&\dot{X}+\dot{Y}=(X+Y)[1-3\bar{\psi }^2]-(X^3+Y^3)+f\cos \omega t, \end{aligned}$$

(B2)

$$\begin{aligned}&\dot{\psi }=\frac{g\cos \Omega t}{2}. \end{aligned}$$

(B3)

After substituting the value of $\psi$ using eq. (B3) and by considering $X+Y=A$, the linearised form of equation will be obtained as $\dot{A}=c_1A+f\cos \omega t$. Then the solution A is determined to be

$$\begin{aligned} A=\frac{f\cos (\omega t-\phi )}{\sqrt{c_1 ^2+\omega ^2}}, \end{aligned}$$

(B4)

where $c_1=[({3g^2}/{4\Omega ^2}) -1]$ and $\phi =\tan ^{-1}({\omega }/{c_1})$.

Fig. 13

Variation of Q with g for $\omega =0.1$, $\Omega =1.0$, $\gamma =1.0$ and $f=0.001$. Theoretical output has been appropriately scaled down for better fit.

Similarly, on subtracting eqs (8) and (9), we get

$$\begin{aligned} \dot{x}-\dot{y}= & {} (x-y)-(x^3-y^3)+2\gamma (x-y)\nonumber \\&+\,f\cos \omega t-g\cos \Omega t. \end{aligned}$$

(B5)

Again by replacing $x=X+\psi$ and $y=Y-\psi$ and on resolving slow and fast components, we obtain

$$\begin{aligned}&\dot{X}-\dot{Y}=(X-Y)[1-3\bar{\psi } ^2+2\gamma ]\nonumber \\&\qquad \qquad \quad \ -\,(X^3-Y^3)+f\cos \omega t, \end{aligned}$$

(B6)

$$\begin{aligned}&\dot{\psi }=-\,\frac{g\cos \Omega t}{2}. \end{aligned}$$

(B7)

After substituting the value of $\psi$ using eq. (B7) and by having $X-Y=B$, the solution becomes

$$\begin{aligned} B=\frac{f\cos (\omega t-\phi _1)}{\sqrt{(c_2 ^2+\omega ^2)}}, \end{aligned}$$

(B8)

where $c_2=({3g^2}/{4\Omega ^2})-1-2\gamma$ and $\phi _1=\tan ^{-1}({\omega }/{c_2})$. On solving A and B, the response amplitude of the coupled system is found to be

$$\begin{aligned} Q= \frac{\kappa \cos (\omega t-\theta )}{2}, \end{aligned}$$

(B9)

where

$$\begin{aligned} \kappa= & {} \sqrt{\alpha ^2+\beta ^2+2\alpha \beta \cos (\phi -\phi _1)},\\ \theta= & {} \tan ^{-1}\frac{\alpha \sin \phi + \beta \sin \phi _1}{\alpha \cos \phi +\beta \cos \phi _1}. \end{aligned}$$

The values of $\alpha$ and $\beta$ are ${1}/({\sqrt{c_1^2+\omega ^2}})$ and ${1}/({\sqrt{c_2^2+\omega ^2}})$, respectively. The fitness of the theoretical model to numerical outcomes is shown in figure 13.