282
6 Oscillations
Substituting m = 1.0, r = 14, k = 49, (1) becomes
d 2 x
dt 2 + 14
dx
dt
+ 49x = 0
( 2 )
ω 0 =
k
m
=
49
1
= 7 rad/s
b =
r
2m
=
14
2 × 1
= 7
(a) Therefore the motion is critically damped.
(b) For critically damped motion, the equation is
x = x 0 e
−bt
(1 + bt)
(3)
With b = 7 and x 0 = 1.5, (3) becomes
x = 1.5 e
−7t
(1 + 7t)
6.56 ω 0 =
k
m
=
150
60
= 5
Damping force f r = r · v
or r =
f r
v
=
80
2
= 40
b =
r
2m
=
40
2 × 6
= 3.33 rad/s
ω(res) =
ω 2
0 − 2b 2 =
5 2 − 2 × (3.33) 2 = 1.66 rad/s
f (res) =
ω(res)
2π
= 0.265 vib/s
6.57 Equation of motion is
2d 2 x
dt 2 + 1.5
dx
dt
+ 40x = 12 cos 4t
Dividing throughout by 2
d 2 x
dt 2 + 0.75
dx
dt
+ 20x = 6 cos 4t