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This simulation uses Easy Java Simulations (Ejs) to model the problem of a damped, driven pendulum. The equation of motion for such a pendulum can be written in terms of the angle φ by considering the net torque on the pendulum bob:
m L2 φ-double dot = − mgL sin (φ) − bL φ-dot + LF cos (ωt),
where m is the mass of the pendulum bob, L is the length of the pendulum, b is the damping coefficient, F is the driving force, and ω is the frequency of the driving force. This equation can be written in terms of φ-double dot as [1]:
φ-double dot = − ω0² sin (φ) − 2β φ-dot + ω0² γ cos (ωt),
where ω0 = (g/L)½ is the natural frequency of the pendulum, β = b/2mL is the damping parameter, and γ = F/mg is the drive strength parameter (the ratio of the driving force to the gravitational force).
The simulation allows the user to change the initial angle, φ, the initial angular velocity, φ-dot, the damping parameter, β, and the drive strength parameter, γ. The pendulum's natural frequency is set at ω0 = 2π*1.5 rad/sec and the driving frequency is set at ω = 2π*1.0 rad/sec.
These parameters are chosen to allow exploration of the period for the damped, driven pendulum. This simulation was created to parallel the discussion in Chapter 12 of J. R. Taylor's Classical Mechanics textbook.
[1] J. R. Taylor, Classical Mechanics, Chapter 12, University Science Books (2004).