Motion of a parametrically driven damped coplanar double pendulum

Abstract

We present the results of linear stability of a damped coplanar double pendulum and its non-linear motion, when the point of suspension is vibrated sinusoidally in the vertical direction with amplitude $a$ and frequency $ω$. A double pendulum has two pairs of Floquet multipliers, which have been calculated for various driving parameters. We have considered the stability of a double pendulum when it is in any of its possible stationary states: (i) both pendulums are either vertically downward or upward and (ii) one pendulum is downward, and the other is upward. The damping is considered to be velocity-dependent, and the driving frequency is taken in a wide range. A double pendulum excited from its stable state shows both periodic and chaotic motion. The periodic motion about its pivot may be either oscillatory or rotational. The periodic swings of a driven double pendulum may be either harmonic or subharmonic for lower values of $a$. The limit cycles corresponding to the normal mode oscillations of a double pendulum of two equal masses are squeezed into a line in its configuration space. For unequal masses, the pendulum shows multi-period swings for smaller values of $a$ and damping, while chaotic swings or rotational motion at relatively higher values of $a$. The parametric driving may lead to stabilization of a partially or fully inverted double pendulum.

Extracted equations

• (1 + µ sin²(θ₁ - θ₂)) θ̈₁ + (1 - A sin Ωτ) [(1 + µ) sin θ₁ - µ cos(θ₁ - θ₂) sin θ₂] + (µ/λ)[θ̇₂² + λ θ̇₁² cos(θ₁ - θ₂)] sin(θ₁ - θ₂) + 2β(1 + sin²(θ₁ - θ₂)) θ̇₁ + 2βΩA [2 sin θ₁ - cos(θ₁ - θ₂) sin θ₂] cos Ωτ = 0
• (1 + µ sin²(θ₁ - θ₂)) θ̈₂ + λ(1 + µ)(1 - A sin Ωτ) [sin θ₂ - cos(θ₁ - θ₂) sin θ₁] - [λ(1 + µ) θ̇₁² + µ θ̇₂² cos(θ₁ - θ₂)] sin(θ₁ - θ₂) + 2β(µ/λ)[1 + µ sin²(θ₁ - θ₂)] θ̇₂ + 2β(1 - µ) θ̇₁ cos(θ₁ - θ₂) + 2βΩA/λ [µ(1 + µ) sin θ₂ - 2µ² cos(θ₁ - θ₂) sin θ₁] cos Ωτ = 0

Paper claims vs. our run

• For equal masses, limit cycles corresponding to normal mode oscillations are squeezed into a line in configuration space
  not-testable
  no fidelity score recorded
• For unequal masses, two marginal stability curves merge to form a double-well-shaped instability zone
  not-testable
  no fidelity score recorded
• Parametric driving can stabilize partially or fully inverted double pendulum configurations
  not-testable
  no fidelity score recorded
• Periodic motion can be harmonic or subharmonic for lower driving amplitudes
  not-testable
  no fidelity score recorded
• Chaotic swings or rotational motion occur at higher driving amplitudes
  not-testable
  no fidelity score recorded
• Multi-period swings appear for unequal masses at smaller driving amplitudes and damping
  not-testable
  no fidelity score recorded

Parameters

Run notes

equations tried: ['(1 + µ sin²(θ₁ - θ₂)) θ̈₁ + (1 - A sin Ωτ) [(1 + µ) sin θ₁ - µ cos(θ₁ - θ₂) sin θ₂] + (µ/λ)[θ̇₂² + λ θ̇₁² cos(θ₁ - θ₂)] sin(θ₁ - θ₂) + 2β(1 + sin²(θ₁ - θ₂)) θ̇₁ + 2βΩA [2 sin θ₁ - cos(θ₁ - θ₂) sin θ₂] cos Ωτ = 0', '(1 + µ sin²(θ₁ - θ₂)) θ̈₂ + λ(1 + µ)(1 - A sin Ωτ) [sin θ₂ - cos(θ₁ - θ₂) sin θ₁] - [λ(1 + µ) θ̇₁² + µ θ̇₂² cos(θ₁ - θ₂)] sin(θ₁ - θ₂) + 2β(µ/λ)[1 + µ sin²(θ₁ - θ₂)] θ̇₂ + 2β(1 - µ) θ̇₁ cos(θ₁ - θ₂) + 2βΩA/λ [µ(1 + µ) sin θ₂ - 2µ² cos(θ₁ - θ₂) sin θ₁] cos Ωτ = 0']