# Driven Harmonic Oscillator Activities

Install Ejs and then right-click on a launch node (green arrow) to copy the Free Fall model from this package into the Ejs workspace to do the following activities.

### Homogeneous and Inhomogeneous Solutions

Set the initial displacement to A0 ≠ 0 and the velocity and drive amplitude FD to zero in order to observe the homogeneous solutions. Estimate the decay time τ for an oscillatory homogeneous solution. Set the drive amplitude and FD ≠ 0 and observe the steady state behavior when time is greater than the decay time t >> τ.

### Resonance

Measure the steady state amplitude when the drive frequency is equal to the natural frequency of oscillation. Shift the drive frequency away from resonance and note the steady state amplitude decreases. The driving frequency that produces the maximum steady state amplitude is know as the resonant frequency.

What is the driving frequency that is required to decreases the steady state amplitude by 50% from its resonant value. Is the frequency shift for this condition the same above and below resonance?

Set the damping constant to zero and describe the steady state behavior at resonance, above resonance, and below resonance. Does a steady state even exist?

### Phase Shift

Add a graph of Fexternal to the displacement plot. What is the phase shift between the driving frequency and the steady state displacement at resonance? Above resonance when the amplitude is 50% of its resonant value? Below resonance when the amplitude is 50% of its resonant value?

### Nonsinusoidal Driving Force (Intermediate)

Modify the model so that the external driving force is a "square wave" that periodically switches between +A0 and - A0. Explain the shape of the displacement x(t) plot.

Hint: Use a square wave period approximately 10 times longer the natural frequency of oscillation for interesting results.

### Driven Pendulum (Advanced)

Extend the Driven Harmonic Oscillator model so that it models a driven pendulum. (Or extend the Pendulum model to add friction and a sinusoidal driving torque.) Is there a resonance at the natural oscillatory frequency of the pendulum? If so, how is this resonance different from the resonance observed for the mass-spring system?