Lecture Notes in Differential Equations - Bruce E. Shapiro

215
As with the unforced case, we can define the amplitude and phase angle by
A sin θ = −α(r 1 + r 2 ) (25.23)
A cos θ = α 2 − r 1 r 2 (25.24)
Then
where
y P = F 0A sin(αt + θ)
(α 2 + r 2 1 )(α2 + r 2 2 ) (25.25)
A 2 = [−α(r 1 + r 2 )] 2 + [α 2 − r 1 r 2 ] 2 (25.26)
= α 2 b 2 + (α 2 − ω 2 ) 2 (25.27)
because r 1 + r 2 = −b and r 1 r 2 = ω 2 . Furthermore,
(α 2 + r 2 1)(α 2 + r 2 2) = α 4 + (r 2 1 + r 2 2)α 2 + (r 1 r 2 ) 2 (25.28)
and therefore
y P =
F 0 sin(αt + θ)
√
α2 b 2 + (α 2 − ω 2 ) 2 (25.29)
Forcing the oscillator pumps energy into the system; it has a maximum
at α = ω, which is unbounded (infinite) in the absence of damping. This
phenomenon – that the magnitude of the oscillations is maximized when
the system is driven at its natural frequency – is known as resonance. If
there is any damping at all the homogeneous solutions decay to zero and
all that remains is the particular solution – so the resulting system will
eventually be strongly dominated by (25.29), oscillating in synch with the
driver. If there is no damping (b = 0) then
y = C sin(ωt + φ) + F 0 sin(αt + θ)
|α 2 − ω 2 |
(25.30)
where C is the natural magnitude of the system, determined by its initial
conditions.