Vibrational resonance in the Morse oscillator - Indian Academy of ...

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K Abirami, S Rajasekar and M A F Sanjuan • For a fixed value of g when ω is varied, resonance occurs when ω = ω VR where ω VR is given by (obtained from dS/dω = 0) √ ω VR = ωr 2 − d2 2 , ω2 r > d2 2 . (13) • When g is varied the condition for resonance is dS/dg = 4ω r ω rg (ωr 2 − ω 2 ) = 0. Because I 0 is always >0 and increases monotonically with g we have ω r ̸= 0 and ω rg = dω r /dg ̸= 0 and hence the resonance condition is ωr 2 = ω 2 , that is, β I0 2(g/2 )/I 0 (2g/ 2 ) = ω 2 . Resonance occurs whenever the resonant frequency ω r matches with the low-frequency ω of the periodic force. • Figures 3a and b show the variation of the response amplitude Q and ωr 2 respectively with g for three fixed values of β and d = 0.5, f = 0.1, ω= 1 and = 10. The theoretical value of Q is in very good agreement with Q obtained from the numerical solution of eq. (2). In figure 3a, for both β = 2 and 1.5, as g increases from zero the value of Q increases monotonically, it reaches a maximum at g = g VR and then decreases. For β = 2, the theoretical and numerical values of g VR are 176 and 173 respectively. In figure 3b for both β = 2 and 1.5 atg = g VR (indicated by solid circles), ωr 2 = ω2 . • At g = 0, I 0 (μ) = I 0 (2μ) = 1 and ωr 2(g = 0) = β. As g increases, I 0(μ) and I 0 (2μ) increase rapidly with I 0 (2μ) growing faster than I 0 (μ) as shown in figure 2a. Consequently, ωr 2 decreases rapidly from the value of β for a while and then decays to zero slowly. The maximum value of ωr 2 is β and this happens at g = 0. For β< ω 2 , ωr 2 is always less than ω2 , that is, ωr 2 ̸= ω2 for g > 0 implying no resonance. This is shown in figure 3a forβ = 0.5 and ω = 1forwhichQ decreases continuously with g. • Resonance is possible only for β>ω 2 . In this case, as g increases from zero, ωr 2 decreases and becomes ω 2 at only one value. Hence, there is only one resonance. 2 2 1 2 2 1 1 (a) 1 0 3 500 1000 (b) 3 0 0 500 1000 Figure 3. (a) Response amplitude Q vs. the control parameter g for the Morse oscillator for three values of β. The values of the parameters are d = 0.5, f = 0.1, ω= 1 and = 10. The values of β for the curves 1, 2 and 3 are 2, 1.5and0.5 respectively. The continuous curve and solid circles represent theoretically and numerically computed values of Q respectively. The horizontal dashed line denotes the limiting value of Q, Q L (g →∞). (b) Dependence of ωr 2 with g for the three values of β used in figure 3a. The solid circles on the g axis denote the values of g at which resonance occurs. 132 Pramana – J. Phys., Vol. 81, No. 1, July 2013
Vibrational resonance in the Morse oscillator 2 1 2 1 0 0 250 Figure 4. Three-dimensional plot of the theoretically computed response amplitude Q as a function of β and g. 500 This is shown in figure 3a forβ = 1.5 and 2 with ω = 1. Figure 4 presents the dependence of Q on β and g for ω = 1. Resonance is seen only for β>ω 2 (= 1). • At resonance ωr 2 = ω 2 and hence Q max = 1/dω and it depends only on d and ω. The response amplitude at resonance is independent of the parameters β, g and . • An analytical expression for g VR (at which Q becomes maximum) is difficult to obtain because ωr 2 is a complicated function of g. However, g VR can be calculated from the resonance curve. It depends on , ω and β and independent of d and f . Q decreases with increase in the value of d. In figure 5 we plot the theoretically predicted g VR and numerically computed g VR vs. the parameter β for a few fixed values of ω. g VR increases with increase in the value of β. • For very large values of g, ωr 2 → 0 and Q approaches the limiting value Q L given by 1 Q L (g →∞) = ω √ ω 2 + d . (14) 2 That is, Q does not decay to zero but approaches the above limiting value (see figure 3a). This limiting value depends only on the parameters ω and d (note that Q max = 1/dω). The point is that when ωr 2 → 0, eq. (9) becomes the damped free particle driven by the periodic force whose solution is Y (t) = Q L f cos(ωt + ), = tan −1 (d/ω). (15) 250 150 50 0 1 2 Figure 5. Variation of theoretically predicted (continuous curve) and numerically computed (solid circles) g VR with the parameter β for a few fixed values of the parameter ω. The value of is 10ω. Pramana – J. Phys., Vol. 81, No. 1, July 2013 133
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