Resonance oscillation of a damped driven simple pendulum

Abstract

The resonance characteristics of a driven damped harmonic oscillator are well known. Unlike harmonic oscillators which are guided by parabolic potentials, a simple pendulum oscillates under sinusoidal potentials. The problem of an undamped pendulum has been investigated to a great extent. However, the resonance characteristics of a driven damped pendulum have not been re- ported so far due to the difficulty in solving the problem analytically. In the present work we report the resonance characteristics of a driven damped pendulum calculated numerically. The results are compared with the resonance characteristics of a damped driven harmonic oscillator. The work can be of pedagogic interest too as it reveals the richness of driven damped motion of a simple pendulum in comparison to and how strikingly it differs from the motion of a driven damped harmonic oscillator. We confine our work only to the nonchaotic regime of pendulum motion.

Extracted equations

• d^2x/dt^2 + γ dx/dt - cos(x) = F0 cos(ωt)

Paper claims vs. our run

• Resonance frequency of damped driven pendulum depends on amplitude, unlike harmonic oscillator
  not-testable
  no fidelity score recorded
• Three distinct regimes of damping: small γ (γ < 0.165), intermediate γ (0.165 ≤ γ ≤ 0.38), and large γ (γ > 0.38)
  not-testable
  no fidelity score recorded
• For intermediate and large damping, resonance frequency increases as amplitude decreases
  not-testable
  no fidelity score recorded
• For small damping, resonance frequency curve shows discontinuity and crosses free oscillation frequency curve
  not-testable
  no fidelity score recorded
• Coexistence of large-amplitude (LA) and small-amplitude (SA) states occurs in certain parameter ranges
  not-testable
  no fidelity score recorded

Parameters

Run notes

equations tried: ['d^2x/dt^2 + γ dx/dt - cos(x) = F0 cos(ωt)']