We know that simple oscillatory motion is described by a general quadratic potential of the form $$V=\frac{1}{2}kx^2,$$ where $k$ is a “force constant”. In a spring, this is a measure of the stiffness of the spring. In a pendulum, it is given by $$k=\frac{mg}{L}$$ where $m$ is the mass of the particle and $L$ its length. From this potential we can extract the force $$F=-\frac{dV}{dx}=-kx$$ and the equation of motion can be written as $$\frac{d^2x}{dt^2}=-\omega _0 ^2x, $$ where the quantity $\omega _0$ is defined by $$\omega _0^2=\frac{k}{m}$$ The dynamical motion described by these equations is called “simple harmonic motion” and can be solved analytically: $$x(t)=A\cos{(\omega _0t+\delta)}, $$ where $a$ is the amplitude of th oscillations and $\delta$ is a phase that depends on the initial conditions.
We can now define a period $T$ such as $$x(t+T)=x(t).$$ Since $\omega _0T$ corresponds to one cycle, we obtain $$T=\frac{2\pi}{\omega _0}=\frac{2\pi}{\sqrt{k/m}}.$$ The frequency $\nu$ is the number of cycles per unit of time and is given by $\nu=1/T$. Note that the period of the motion depends on the ratio $k/m$ and is completely independent of the initial conditions.
In Exercise 1.6 we wrote program to simulate simple harmonic motion. Now use the program to calculate the relative change in the total energy during one cycle $\Delta _n=(E_n-E_0)/E_0$. Is the function $\Delta$ uniformly small during the cycle? Choose $x_0=1$, $v_0=0$, and $\omega_0^2=9$ Compare using Euler’s method, 2nd. order Runge-Kutta, and Verlet's algorithm.
Plot position and velocity as a function of the time $t$.
Compute the amplitude $A$ for the initial conditions $x_0=4$, $v_0=0$, and $x_0=0$, $v_0=4$; choose $\omega _0^2=4$ in both cases. What quantity determines the value of $A$?.
Compute the average value of the kinetic energy and the potential energy during a complete cycle. Is there a relation between the two averages?
Plot the path of the oscillator in phase space $(x,v)$. Set $\omega _0^2=9$ and use different initial conditions $(x_0,v_0)=(1,0);(0,1);(4,0)$. Do you find different paths for each of them? What physical quantity distinguishes or characterizes each path? What is the shape of the phase paths? Is the motion of a representative point $(x,v)$ always in the clockwise or counterclockwise direction?
Modify the code for Exercise 1.6 to solve Exercise 3.1
The pendulum responds to the equation of motion for simple harmonic motion only in the limit of small angles. In the case of large oscillations, the equation has to be modified becoming $$\frac{d^2\theta}{dt^2}=-\frac{g}{L}\sin{\theta}.$$
The energy of the pendulum is then given by $$E=\frac{1}{2}mL^2(\frac{d\theta}{dt})^2+mgL(1-\cos{\theta})$$
Modify your program to simulate large amplitude oscillations in a pendulum. Set $g/L=9$ and choose $\Delta t$ so that the numerical solution is stable, *i.e.* it does no diverge with time from the “true” solution. Check the stability by calculating the total energy and ensuring that it does not drift from its initial value.
Set $d\theta /dt|_{t=0}=0$ and make plots of $\theta (t)$ and $d\theta /dt(t)$ for the initial conditions $\theta(t=0)=0.1$, 0.2, 0.4, 0.8, 1.0. Describe the qualitative behavior of $\theta$ and $d\theta /dt$. What is the period $T$ and the maximum amplitude $\theta _{max}$ in each case? Plot $T$ versus $\theta _{max}$ and discuss the qualitative dependence of the period on the amplitude. How do the results compare in the linear and non-linear cases, *e.g.* which period is larger? Explain the relative values of $T$ in physical terms.
Modify the code and solve Exercise 3.2
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.
Incorporate the effects of damping in your program and plot the time dependece of position and velocity. Make runs for $\omega _0^2=9$, $x_0=1$, $v_0=0$, and $\gamma=0.5$.
Compare the period and angular frequency 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?
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 $\gamma$ used in the previous item and discuss the qualitative dependence of $\tau$ with $\gamma$.
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, explain the cause of the time-dependence.
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?
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.
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.
Modify the code and solve Exercise 3.3
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, applying a small displacement and observe the resulting motion. We will finr 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 $F(t)$ $$\frac{d^2x}{dt^2}=-\omega _0^2x-\gamma\frac{dx}{dt}+\frac{1}{m}F(t).$$ It 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: $$\frac{1}{m}F(t)=A_0 \cos{\omega t}, $$ where $\omega$ is the angular frequency of the driving force.
Modify your program so that an external force of the form ([driving force]) is included. Set $\omega _0^2=9$, $\gamma=0.5$, $A_0=1$ and $\omega = 2$ (we’ll use these values for 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 frequency 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.
Compute $x(t)$ for $\omega = 1$ and $\omega = 4$. What is the period and angular frequency of the steady state in each case?
Compute $x(t)$ for $\omega _0=4$. What is the angular frequency of the steady state motion? On the basis of these results, explain which parameters determine the frequency of the steady state behavior.
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.
The long term behavior of the driven harmonic oscillator depends on the frequency of the driving force. One measure of this behavior is the maximum of the steady state displacement $A(\omega )$.
Adopt the initial condition $(x_0=0,v_0=0)$. Compute $A(\omega )$ for $\omega =0$, $1.0$, $2.0$, $2.2$, $2.4$, $2.6$, $2.8$, $3.0$, $3.2$, $3.4$ with $\omega _0=3$ and $\gamma =0.5$. Plot $A(\omega )$ versus $\omega$ and describe its qualitative behavior. If $A(\omega )$ has a maximum, determine the “resonance angular frequency” $\omega _{max}$, which is the frequency at the maximum of $A$. Is the value of $\omega _{max}$ close to the natural angular frequency $\omega _0$?
Compute $A_{max}$, the value of the amplitude at $\omega _{max}$, and the ratio $\Delta \omega /\omega _{max}$, where $\Delta \omega$ is the “width” of the resonance. Define $\Delta \omega$ as the frequency interval between points on the amplitude curve which are $1/\sqrt{2}A_{max}$. Set $\omega _0=3$ and consider $\gamma=0.1$, 0.5, 1.0, 2.0. Describe the qualitative dependence of $A_{max}$ and $\Delta \omega/\omega _{max}$ on $\gamma$. The quantity $\Delta \omega /\omega_{max}$ is proportional to $1/Q$, where $Q$ is the “quality factor” of the oscillator.
Decribe the qualitative behavior of the steady state amplitude $A(\omega )$ near $\omega=0$ and $\omega\gg \omega _0$. Why is $A(\omega =0) > A(\omega )$ for small $\omega$? Why does $A(\omega ) \rightarrow 0$ for $\omega \gg \omega _0$?
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