Lecture Notes in Differential Equations - Bruce E. Shapiro

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It is standard to define a new constant b = C/m and a frequency ω = √ k/m
as before so that we can rearrange our equation into standard form as
y ′′ + by ′ + ω 2 y = 0 (25.10)
Equation (25.10) is the standard equation of damped harmonic motion.
As before, it is a linear second order equation with constant coefficients, so
we can solve it exactly by finding the roots of the characteristic equation
r 2 + br + ω 2 = 0 (25.11)
The roots of the characteristic equation are given by the quadratic equation
as
r = −b ± √ b 2 − 4ω 2
(25.12)
2
The resulting system is said to be
• underdamped when b < 2ω;
• critically damped when b = 2ω; and
• overdamped when b > 2ω.
In the underdamped system (b < 2ω) the roots are a complex conjugate
pair with negative real part, r = µ ± iϖ where
µ = b/2 > 0, ϖ = ω √ 1 − (b/2ω) 2 (25.13)
and hence the resulting oscillations are described by decaying oscillations
y = Ae −µt sin(ϖt + φ) (25.14)
The critically damped system has a single positive real repeated root r =
µ = b/2 , so that
y = (C 1 + C 2 t)e −µt (25.15)
The critically damped systems decays directly to zero without crossing the
y-axis.
The overdamped system has two negative real roots −µ±|ϖ| (where |ϖ| <
µ) and hence
y = e −µt ( C 1 e |ϖ|t + C 2 e −|ϖ|t) (25.16)
The system damps to zero without oscillations, but may cross the y-axis
once. The first term in parenthesis in (25.16) does not give an exponential
increase because |ϖ| < µ.