Classical Dynamics of Particles and Systems · Problem 3-41

Nonlinear Oscillations and Chaos — Marion & Thornton

Physics — Mechanics · damped-oscillator-initial-conditions-computer

Problem

(a) Use the general solutions

x(t)

to the differential equation

d^2x/dt^2 + 2\beta\,dx/dt + \omega_0^2 x = 0

for underdamped, critically damped, and overdamped motion and choose the constants of integration to satisfy the initial conditions

x = x_0

and

v = v_0 = 0

t = 0

. (b) Use a computer to plot the results for

x(t)/x_0

as a function of

\omega_0 t

in the three cases

\beta = (1/2)\omega_0

\beta = \omega_0

, and

\beta = 2\omega_0

. Show all three curves on a single plot.