2.3 Free Undamped Vibration of the Sdf System
19
or
x = A 1 (cos pt + i sin pt) + A 2 (cos pt − i sin pt)
Rearranging the terms of Eq. (2.5) yields
x = C 1 cos pt + C 2 sin pt
(2.6)
where C 1 and C 2 are constants.
The factors cos pt and sin pt are periodic functions. As such the motion is
periodic, and it repeats itself after a certain interval of time.
Since pT = 2π , then
T =
2π
p
= 2π
m
k
(2.7)
This time interval T is called the period of undamped free vibration. The number
of times that the motion repeats itself in one second is called natural frequency of
vibration f .
f =
1
T
=
1
2π
k
m
(2.8)
The parameter p can now be given a physical significance, since
p =
2π
T
= 2π f
(2.9)
p is called the circular or angular frequency of vibration.
The vibratory motion represented by Eq. (2.6) is a harmonic motion. The constants
C 1 and C 2 can be determined from the initial conditions of the motion. If at t =
0, x = x 0 and ˙
x = ˙
x 0 , then applying the initial conditions
C 1 = x 0 and C 2 =
˙
x 0
p
(2.10)
Therefore, the equation of motion becomes
x = x 0 cos pt +
˙
x 0
p
sin pt
(2.11)
Assuming, x 0 = A cos ∈ and
˙
x 0
p
= A sin ∈, Eq. (2.11) becomes
x = A cos ∈ cos pt + A sin ∈ sin pt