Section 1. Linear Oscillators

The harmonic oscillator is the fundamental model for the analysis of oscillating systems. Phase plane trajectories are constructed. DEplot draws direction fields.

1. Obtain the solution to the following initial value problem. Call it soln.

y'' + 4y = 0  , y(0) = 2 , y'(0) = -3

From the form of the solution decide if the system is undamped, underdamped, critically damped, or overdamped. What is the period of the oscillations?

Continuing 1. Plot the solution to the IVP in Exercise 1. Then create a plot showing the cosine term, the sine term and their sum (the solution curve). Make the solution red, the cosine blue, and the sine green.

a. What IVP does the cosine term solve?

b. What IVP does the sine term solve?

c. Determine the amplitude of the oscillations by solving y'(t) = 0 and substituting the time value into the solution. Compare the answer to the amplitude calculated using the standard formula for converting the solution into amplitude/phase angle form.

d. Assuming this is the model of a mass spring system, determine the speed of the mass as it passes through equilibrium.

e. Use unapply to convert the solution into the function g. Use g to plot the phase plane trajectory. What type of curve is this trajectory?

f. Add to the trajectory the points corresponding to t = 0, 0.25, 0.5, 0.75, ... , 2.0.

2.* Consider now the following damped system. Obtain the solution.

y'' + y' + 4y = 0  , y(0) = 2 , y'(0) = -3

From the form of the solution decide if the system is underdamped, critically damped, or overdamped.

a. What is the pseudo-period of the oscillations?

b. What is the time constant?

c. Based upon your answer to part b estimate the time interval required for the oscillations to disappear from view.

d. Plot the solution curve over the interval you named in part c.

e. Add to the curve in part d the curves defined by Aexp(-t/2) and -Aexp(t/2) where

A = sqrt(4+16/15) .

Make them blue. What is the significance of these curves? Where did the formula for A come from?

f. Use unapply to convert the solution into the function g. Use g to plot the phase plane trajectory.

g. Add to the trajectory the points corresponding to t = 0, 0.25, 0.5, 0.75, ... , 2.0.

3. Use DEplot to draw the direction field in phase space for the undamped system. Then add the solution trajectory.

Note. You will have to enter the equivalent system of two first order equations. See the manual, page 74.

4.* Use DEplot to draw the direction field in phase space for the damped system. Then add the solution trajectory.