18 Chapter 0 Ordinary Differential Equations
Figure 2 Mass–spring–damper system with an external force.from rest, with an external sinusoidal force (see Fig. 2) is described by this
initial value problem:
d^2 u
dt^2+bdu
dt+ω^2 u=f 0 cos(μt),u( 0 )= 0 ,du
dt(^0 )=^0.
See the Section 1 example on the mass–spring–damper system. The coeffi-
cientf 0 is proportional to the magnitude of the force. There are three impor-
tant cases.
b= 0 ,μ=ω:undamped, no resonance. The form of the trial solution is
up(t)=Acos(μt)+Bsin(μt).Substitution and simple algebra lead to the particular solution
u 0 (t)=ω (^2) −f^0 μ 2 cos(μt)
(that is,B=0).Thegeneralsolutionofthedifferentialequationis
u(t)= f^0
ω^2 −μ^2
cos(μt)+c 1 cos(ωt)+c 2 sin(ωt).
Applying the initial conditions determinesc 1 andc 2. Finally, the solution of
the initial value problem is
u(t)= f^0
ω^2 −μ^2
(
cos(μt)−cos(ωt)