292 Ordinary Differential Equations6.1.2 Forced Motion
Suppose some mechanical or electrical system whose state is represented by
the parameter y(t) is mathematically modelled by the differential equation
(24) ay" +by'+ dy = f(t)
where a > 0, b 2 0, and d > 0 are constants and f(t) =/:-0. Since f(t) =/:-0,
the system is said to execute forced motion. We will assume the forcing
function f(t) is periodic. More specifically we will assume f(t) = Esinw*t,
where E and w are constants. Other periodic functions such as E cos wt
or E sin ( w* t + e) will serve as the forcing function as well as the function
which we have chosen. For a pendulum system the forcing function might
represent a weight or main spring in a clock which is connected to and driving
the pendulum. For a spring-mass system the forcing function might represent
a motor which vibrates the support or a magnetic field which acts upon a
suspended iron mass. For an electrical circuit or network the forcing function
represents the electromotive force appli ed to the system by a battery or a
generator.
6.1.2.1 Undamped Forced Motion
When there is no damping force (b = 0) and the forcing function f(t)
Esinw*t, equation (24) becomes
(25) ay" + dy = Esinw*t.
The general solution of the associated homogeneous equation ay" + dy = 0,
Ye ( t) = A sin wt + B cos wt = C sin (wt + </>)
where w = .Jd70, and A , B , C, and </>are arbitrary constants, is the comple-
mentary solution of equation (25).
Case 1. If w* =/:-w, then a particular solution of (25) will have the form
yp(t) = F sinwt + G coswt.
Differentiating twice, we find
y~(t) = w* F cosw*t - w*G sinw*tand
y~(t) = -(w)^2 Fsinwt-(w)^2 Gcoswt.
Substituting Yp and y~ into equation (25) and rearranging, we see that the
constants F and G must satisfy
F[d-a(w)^2 ] sinwt + G[d-a(w)^2 ] coswt = Esinw*t.