Exercise 3.3: Damped oscillations

If a drag force is included in the problem, the equation of motion becomes:

$$\frac{d^2x}{dt^2}=-\omega _0^2x-\gamma \frac{dx}{dt},$$

where the ``damping coefficient'' gamma is a measure of the friction. Note that the drag force opposes the motion.

1. Incorporate the effects of damping in yout program and plot the time dependece of position and velocity. Make runs for $\omega_0^2=9$, $x_0=1$, $v_0=0$, and $\gamma=0.5$.

2. Compare the period and angular frquency to the undamped case. is the period longer or shorter? Make additional runs for $\gamma = 1,2,3$. Does the frequency increase or decrease with greater damping?

3. Define the amplitude as the maximum value of $x$ in one cycle. Compute the ``relaxation time'' $\tau$, the time it takes for the amplitude to change from its maximum to $1/e\approx 0.37$ of its maximum value. To do this, plot the maximum amplitud of each cycle, and fit it with an exponential of the form $A_0 \exp{-t/\tau}$. Compute $\tau$ for each of the values of $\gamma$ use in th previous item and discuss the qualitative dependence of $\tau$ with $\gamma$.

4. Plot the total energy as a function of time for the values of $\gamma$ considered previously. If the decrease of the energy is not monotonic, explainthe cause of the time-dependence.

5. Compute the average value of the kinetic energy, potential energy, and total energy over a complete cycle. Plot these averages as a function of the number of cycles. Due to the presence of damping, these averages decrease with time. Is the decrease uniform?

6. Compute the time-dependence of $x(t)$ and $v(t)$ for $\gamma=4,5,6,7,8$. Is the motion oscillatory for all $\gamma$? Consider a condition for equilibrium $x<0.0001$; how quickly does $x(t)$ decay to equilibrium? For fixed $\omega _0$, the oscillator is said to be ``critically damped'' at the smallest values of $\gamma$ for which the decay to equilibrium is monotonic. For what value of $\gamma$ does critical damping occur for $\omega_0^2=9$ and $\omega^2 _0=4$? For each value of $\omega _0$ compute the value of $\gamma$ for which the system approaches equilibrium more quickly.

7. Construct the phase space diagram for cases $\omega_0^2=9$ and $\gamma=0.5$, 2, 4, 6, 8. Area the qualitative features of the paths independent of $\gamma$? If not, discuss the qualitative differences.