G.2 Damped Oscillations
183
G.2.1 Clasification
(a) overdamped
γ
2
ω 0
(b) underdamped
γ
2
< ω 0
(c) critical − damping
γ
2
= ω 0
(G.11)
G.2.2 Summary
G.2.2.1 (a) Overdamped Oscillations,
γ
2 ω 0
The equations for the coordenates: position, x, and velocity, v, are given (Fig. G.1)
x = a exp
−
γ
2
t
cos
γ
2
t + θ
(G.12)
v = −a
γ
2
exp
−
γ
2
t
cos
γ
2
t + θ
+ sin
γ
2
t + θ
(G.13)
G.2.2.2 (b) Underdamped Oscillations,
γ
2 < ω 0
x = a exp
−
γ
2
t
cos (ωt + θ )
(G.14)
v = −a
γ
2
exp
−
γ
2
t
[cos (ωt + θ ) + sin (ωt + θ )]
(G.15)
with ω =
ω 2
0 −
γ
2
2
G.2.2.3 (c) Critical Damping Oscillations,
γ
2 = ω 0
x = exp
−
γ
2
t
x 0 +
v 0 +
γ
2
x 0
t
(G.16)
v = exp
−
γ
2
t
v 0 −
γ
2
v 0 +
γ
2
x 0
t
(G.17)