Exercise 3.4: Linear response to external forces

How can we determine the period of a pendulum that is not already in motion? The obvious way is to disturb the system, for instance, apply a small displacement and observe the resulting motion. We will find that the ``response'' of a system is actually an intrinsic property of the system and can tell us about its nature in the absence of perturbations.

Consider a damped linear oscillator with an external force $F(t)$

$$\frac{d^2x}{dt^2}=-\omega _0^2x-\gamma\frac{dx}{dt}+\frac{1}{m}F(t).$$

Is is customary to interpret the reponse of the system in terms of the displacement $x$. The time dependence in $F(t)$ is arbitrary. A particular case is when the force is harmonic:

$$\frac{1}{m}F(t)=A_0 \cos{\omega t},$$ (42)

where $\omega$ is the angular frequency of the driving force.

When a weak driving force has frequency $\omega$ equal to the natural frequency $\omega _0$, it is possible to have a periodic steady motion with frequency $\omega _0$. In this case it is possible to pick the magnitude of the external force such that after the initial transient dies off, the average energy put into the system during one period exactly balances the energy dissipated by friction. This leads to a ``limiting cycle'' in which the motion is stable even in the presence of damping.

However, when the magnitude of the force $f$ is large, the driving overpowers the natural motion of the system and the steady state motion is at the frequency of the driving force. This is a case of ``mode locking''. The force can lock the system in an excited ``overtone'', a vibrational mode with higher frequency. In this case the driving and the natural frquencies are rationally related:

$$\frac{\omega}{\omega _0}=\frac{n}{m},$$

where $n$ and $m$ are integers.

1. Modify your program so that an external force of the form (42) is included. Set $\omega_0^2=9$, $\gamma=0.5$, $A_0=1$ and $\omega = 2$ (we'll use these values fro the rest of the exercise). These values correspond to a lightly damped oscillator. Plot $x(t)$ versus $t$ for the initial conditions $(x_0=1,v_0=0)$. How does the qualitative behavior differ from the unperturbed case? What is the period and angular frequency of $x(t)$ after several oscillations? Obtain a similar plot with $(x_0=0,v_0=1)$. What is the period and angular frequancy after several oscillations? Does $x(t)$ approach a limiting behavior that is independent of the initial conditions? Identify a ``transient'' part of $x(t)$ which depends on the initial conditions and decays in time, and a ``steady state'' part which dominates at longer times and which is independent of the initial conditions.

2. Compute $x(t)$ for $\omega = 1$ and $\omega = 4$. What is the period and angular frequancy of the steady state in each case?

3. Compute $x(t)$ for $\omega _0=4$. What is the angular frequancy of the steady state motion? Onthe basis of these results, explain which parameters determine the frequency of the steady state behavior/

4. Verify that the steady state behavior is given by
   
   $$x(t)=A(\omega )\cos{\omega t+\delta},$$
   
   
   where $\delta$ is the phase difference between the applied force and the steady state motion. Compute $\delta$ for $\omega_0^2=9,\gamma=0.5$,$\omega=0$, $1.0$, $2.0$, $2.2$, $2.4$, $2.6$, $2.8$, $3.0$, $3.2$, $3.4$. repeat the computation for $\gamma=1.5$ and plot $\delta$ versus $\omega$ for the two values of $\gamma$. Discuss the qualitative dependence of $\delta(\omega )$ in the two cases.