Lecture Notes in Differential Equations - Bruce E. Shapiro

214 LESSON 25. HARMONIC OSCILLATIONS
Figure 25.1: Harmonic oscillations with ω = 1 and initial conditions y(0) =
1, y ′ (0) = 0. (a) simple harmonic motion; (b) under-damped system with
b = 0.25; (c) critically damped system with b = 2.0; (d) under-damped
system with b = 0.025.
Forced Oscillations
Finally, it is possible to imaging adding a motor to the spring that produces
a force f(t). The resulting is system, including damping, called the forced
harmonic oscillator:
my ′′ = −Cy ′′ − ky + f(t) (25.17)
If we define a force per unit mass F (t) = f(t)/m and ω and the parameter
b as before, this becomes, in standard form,
y ′′ + by ′ + ω 2 y = F (t) (25.18)
This is the general second-order linear equation with constant coefficients
that are positive. Thus any second order linear differential equation with
positive constant coefficients describes a harmonic oscillator of some sort.
The particular solution is
∫ ∫
y P = e r2t e (r1−r2)u e −r1s F (s)ds (25.19)
t
where r 1 and r 2 are the roots of the characteristic equation. For example,
suppose the system driven by a force function
Then
u
F (t) = F 0 sin αt (25.20)
∫ ∫
y P =F 0 e r2t e (r1−r2)u e −r1s sin αs dsdu (25.21)
t
u
= F ∫
0
α 2 + r1
2 e r2t e −r2u (α cos αu + r 1 sin αu)du (25.22)
t