Classical Dynamics of Particles and Systems · Problem 3-42

Nonlinear Oscillations and Chaos — Marion & Thornton

Physics — Mechanics · undamped-driven-oscillator

Problem

An undamped driven harmonic oscillator satisfies the equation of motion

m(d^2x/dt^2 + \omega_0^2 x) = F(t)

. The driving force

F(t) = F_0\sin(\omega t)

is switched on at

t = 0

. (a) Find

x(t)

for

t > 0

for the initial conditions

x = 0

and

v = 0

t = 0

. (b) Find

x(t)

for

\omega = \omega_0

by taking the limit

\omega \to \omega_0

in your result for part (a). Sketch your result for

x(t)

. Hint: In part (a) look for a particular solution of the differential equation of the form

x = A\sin(\omega t)

and determine

A

. Add the solution of the homogeneous equation to this to obtain the general solution of the inhomogeneous equation.