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4. SECOND-ORDER LINEAR EQUATIONS
Mechanical systems also experience friction and for vibrational motion is proportional to velocity:
dy
Ff riction = b = by 0,
dt
where b 0 is the damping coefficient. Any other forces on the oscillator are
regarded as external to the system. We lump all such forces together into a
single, known fuction Fext(t). Then the DE for the mass-spring oscillator is
my 00 =
ky
by 0 + Fext
or
my 00 + by 0 + ky = Fext.
If b = Fext(t) = 0, this DE becomes the equation of simple harmonic motion
my 00 + ky = 0,
or
y 00 +
2
= 0,
=
which has a solution of the form
r
k
,
m
y(t) = cos t.
The (idealized) mass-spring motions would be perpetual vibrations like the ones
depicted below.