PHY301 - Worksheet 2, Fall 2005

Driven, damped, linear oscillations
Initial value problems in ODE's

Due: Oct 7, 2005 (three weeks)

In most oscillatory physical systems, small amplitude oscillations obey simple harmonic motion. Examples which you study in undergraduate physics include a mass+spring system, a pendulum, and an LC circuit. Each of these examples has a ``natural frequency'' (in radians/sec):

$$\omega_0=(k/m)^{1/2},\ \ \omega_0=(g/l)^{1/2},\ \ \ \omega_0=(1/LC)^{1/2}$$ (1)

If we include a small amount of damping or ``dissipation'' in any of these systems, the amplitude of oscillation decreases slowly with time. This is the ``underdamped'' case. If the dissipation is large the system is ``overdamped'' and oscillatory behavior ceases altogether. The crossover point between these two cases is known as ``critically damped''. If we also include a sinusoidal driving force, then the frequency of the driving force competes with the natural frequency of the system. However at the special frequency called the ``resonant frequency'', where the driving frequency matches the natural frequency, the amplitude of oscillations can get very large. Resonance is a very important practical phenomena. For example bridge designers must ensure that the typical winds in an area cannot resonate with a natural frequency of the bridge or disasters can happen. The same is true of skyscraper design.

PROBLEMS

Problem 1. Find the differential equation for driven (with a sinusoidal drive), damped linear oscillations. Give examples of what each term in the equation corresponds to in the cases of: A mass+spring system; A pendulum; A linear(i.e. just resistors, capacitors and inductors) electical circuit. In the mass+spring and pendulum cases, consider dissipation to be due to a ``drag'' force (ie. proportional to velocity).

Problem 2. Mathematica can analytically solve the differential equation for driven damped linear oscillations. Write a Mathematica code to do this and then use your code to plot the behavior as a function of time. For initial conditions, let the position and velocity both be zero at time t=0. Let the frequency of the driving term be equal to the natural frequency of oscillation. Plot data which illustrate the following cases: Underdamped motion; Overdamped motion, and Critically Damped motion.

Problem 3. The final problem in this worksheet will be to write a c++ program to solve the differential equation you found in problem 1. But before undertaking that, it will be helpful to warm up with an easier problem: one that involves a first-order differential equation instead of a second-order equation.

(a) Solve the equation   dx/dt - 2*x + 4*t = 0   for x(t) with the boundary condition x(0)=0, and graph the resulting x(t) in the region 0 < t < 1 using Mathematica's DSOLVE to obtain an explicit solution.

(b) Repeat part (a) using Mathematica's NDSOLVE to obtain a numerical solution.

(c) Repeat part (a) by writing a c++ program that solves the differential equation numerically by dividing the time interval 0 < t < 1 into small equal steps and moving from one time step to the next using Euler's method: x(t + delta) = x(t) + delta*x'(t), with x'(t) taken from the differential equation.

(d) Repeat part (c) but improve it by using the Runge-Kutta method in place of Euler's method to move from each step to the next. You can use either second-order or fourth-order Runge-Kutta. (Discussion of various methods for solving initial value problems like this can be found in Chapter 16 of Numerical Recipes online, particularly sections 16.0-16.1. You may also be able to use GNU Scientific Library -- look under Ordinary Differential Equations).

Problem 4. Now go back to the second order differential equation of problem 1. Rewrite it as a first order equation for a two-dimensional vector object whose components are x and dx/dt. Revise your Euler and Runge-Kutta programs to numerically integrate this first order equation and check that your results agree with those obtained using Mathematica.

Illustrate the case of underdamped driven oscillations using your code (i.e., write a data set to a file and then plot it using xmgrace). Choose the overall time for the plot such that approximately 20 oscillations occur. Choose the damping parameter such that damping is strong enough to be seen. Make plots using the Euler and Runge-Kutta methods, and investigate how small the time step must be in order to get accurate results.

(Note that Runge-Kutta is an old workhorse and works well for many problems. However in some research problems, more elaborate methods such as ``Verlet'', ``Bulirsch-Stoers'' are preferred. Some equations are not well approximated by any explicit method, so that an ``implicit'' method is needed. We will return to this later in the course when we consider partial differential equations, in particular the Schroedinger equation.)