182
Appendix G
where the constant of integration ξ 0 is the value of ξ at the instant t = 0. This is the
required general solution; the function x(t) is given by the imaginary part of G.4,
divided by ω 0
1 .
G.2 Damped Oscillations
Consider the equation
¨
x = −ω
2
0 x − γ ˙
x
(G.5)
The solution is x = exp(rt) and obtain r for the characteristic equation r 2 + γ r +
ω 2
0 = 0. whence r 1,2 = −
γ
2 ±
γ
2
2 − ω 2
0
. The general solution of Eq. G.5 is
x = c 1 exp(r 1 t + c 2 exp(r 2 t)
(G.6)
Two cases must be distinguished. if
γ
2 < ω 0 , we have two complex conjugate values
of r. The general solution solution of the equation of motion can then be written as
x = re
A exp
−
γ
2
t + i
ω 2
0 −
γ
2
2
t
(G.7)
where A is an arbitrary complex constant, or as
x = a exp
−
γ
2
t
cos (ωt + θ )
(G.8)
with ω =
ω 2
0 −
γ
2
2
and a and θ real constants. If
γ
2 ω 0 , the amplitude of
the damped oscillation is almost unchanged during the period
2π
ω .
Next, let
γ
2 > ω 0 . Then the values of r are both real and negative. The general
form of the solution is
x=c 1 exp
−
γ
2
−
γ
2
2 −ω 2
0
t
+c 2 exp
−
γ
2
+
γ
2
2 −ω 2
0
t
(G.9)
In this case when the friction is sufficiently strong, the motion consists of a decrease
in |x|, i.e. an asymptotic appoach as (t → ∞) to the equilibrium position.This type
of motion is called aperiodic damping.
Finally, in the special case
γ
2 = ω 0 , the characteristic equation has the double
root r = −
γ
2 . The general solution of the differential equation is then
x = (c 1 + c 2 t) exp
−
γ
2
t
(G.10)
This is a special case of aperiodic damping.
1 The force F (t) must, of course, be written in real form.
Appendix G
where the constant of integration ξ 0 is the value of ξ at the instant t = 0. This is the
required general solution; the function x(t) is given by the imaginary part of G.4,
divided by ω 0
1 .
G.2 Damped Oscillations
Consider the equation
¨
x = −ω
2
0 x − γ ˙
x
(G.5)
The solution is x = exp(rt) and obtain r for the characteristic equation r 2 + γ r +
ω 2
0 = 0. whence r 1,2 = −
γ
2 ±
γ
2
2 − ω 2
0
. The general solution of Eq. G.5 is
x = c 1 exp(r 1 t + c 2 exp(r 2 t)
(G.6)
Two cases must be distinguished. if
γ
2 < ω 0 , we have two complex conjugate values
of r. The general solution solution of the equation of motion can then be written as
x = re
A exp
−
γ
2
t + i
ω 2
0 −
γ
2
2
t
(G.7)
where A is an arbitrary complex constant, or as
x = a exp
−
γ
2
t
cos (ωt + θ )
(G.8)
with ω =
ω 2
0 −
γ
2
2
and a and θ real constants. If
γ
2 ω 0 , the amplitude of
the damped oscillation is almost unchanged during the period
2π
ω .
Next, let
γ
2 > ω 0 . Then the values of r are both real and negative. The general
form of the solution is
x=c 1 exp
−
γ
2
−
γ
2
2 −ω 2
0
t
+c 2 exp
−
γ
2
+
γ
2
2 −ω 2
0
t
(G.9)
In this case when the friction is sufficiently strong, the motion consists of a decrease
in |x|, i.e. an asymptotic appoach as (t → ∞) to the equilibrium position.This type
of motion is called aperiodic damping.
Finally, in the special case
γ
2 = ω 0 , the characteristic equation has the double
root r = −
γ
2 . The general solution of the differential equation is then
x = (c 1 + c 2 t) exp
−
γ
2
t
(G.10)
This is a special case of aperiodic damping.
1 The force F (t) must, of course, be written in real form.