Lecture Notes in Differential Equations - Bruce E. Shapiro

212 LESSON 25. HARMONIC OSCILLATIONS
The characteristic equation is r 2 + ω 2 = 0, and since the roots are purely
imaginary (r = ±iω) the motion is oscillatory,
y = C 1 cos ωt + C 2 sin ωt (25.3)
It is sometimes easier to work with a single trig function then with two. To
do this we start by defining the parameter
A 2 = C 2 1 + C 2 2 (25.4)
where we chose the positive square root for A, and define the angle φ such
that
cos φ = C 1
A
sin φ = − C 2
A
(25.5)
(25.6)
We know such an angle exists because 0 ≤ |C 1 | ≤ A and 0 ≤ |C 2 | ≤ A and
by (25.4) φ satisfies the identity cos φ + sin 2 φ = 1 as required. Thus
y = A cos φ cos ωt − A sin φ sin ωt (25.7)
= A cos(φ + ωt) (25.8)
Then φ = tan −1 (C 1 /C 2 ) is known as the phase of the oscillations and A
is called the amplitude. As we see, the oscillation is described by a single
sine wave of magnitude (height) A and phase shift φ. With a suitable redefinition
of C 1 and C 2 , we could have made the cos into sin rather than a
cosine, e.g., C 1 /A = sin φ and C 2 /phi = cos φ.
Damped Harmonic Model
In fact, equation (25.2) is not such a good model because it predicts the
system will oscillate indefinitely, and not slowly damp out to zero. A good
approximation to the damping is a force that acts linearly against the motion:
the faster the mass moves, the stronger the damping force. Its direction
is negative, since it acts against the velocity y ′ of the mass. Thus we
modify equation (25.2) to the following
my ′′ = −Cy ′ − ky (25.9)
where C ≥ 0 is a damping constant that takes into account a force that
is proportional to the velocity but acts in the opposite direction to the
velocity.