2.4 Free Damped Vibration of Sdf System
33
Substituting y 1 from Eq. (a) into Eq. (b), we get
m ¨
y + c ˙
y + k
b
a
2
y = 0
( c )
Equation (c) can be written as follows
¨
y + 2n ˙
y + p
2 y = 0
( d )
where
n =
c
2m
and p
2
=
k
m
b
a
2
Equation (c) is of the same form as Eq. (2.21), except that k is replaced by k
b
a
2
The natural angular frequency of damped oscillation is
p d =
p 2 − n 2 =
k
m
b
a
2
−
c
2m
2
At critical damping
n = p
or
c
2m
=
b
a
k
m
or
c =
2b
a
√
km
Example 2.9 A piston of mass 5 kg is travelling inside a cylinder with a velocity
of 15 m/s and engages a spring and damper as shown in Fig. 2.17. Determine the
maximum displacement of the piston after engaging the spring–damper.
How many seconds does it take?
The mass has an initial velocity. Referring to Eq. (2.32)
n =
c
2m
=
175
2 × 5
= 17.5
p =
k
m
=
3000
5
= 24.295 rad/s
33
Substituting y 1 from Eq. (a) into Eq. (b), we get
m ¨
y + c ˙
y + k
b
a
2
y = 0
( c )
Equation (c) can be written as follows
¨
y + 2n ˙
y + p
2 y = 0
( d )
where
n =
c
2m
and p
2
=
k
m
b
a
2
Equation (c) is of the same form as Eq. (2.21), except that k is replaced by k
b
a
2
The natural angular frequency of damped oscillation is
p d =
p 2 − n 2 =
k
m
b
a
2
−
c
2m
2
At critical damping
n = p
or
c
2m
=
b
a
k
m
or
c =
2b
a
√
km
Example 2.9 A piston of mass 5 kg is travelling inside a cylinder with a velocity
of 15 m/s and engages a spring and damper as shown in Fig. 2.17. Determine the
maximum displacement of the piston after engaging the spring–damper.
How many seconds does it take?
The mass has an initial velocity. Referring to Eq. (2.32)
n =
c
2m
=
175
2 × 5
= 17.5
p =
k
m
=
3000
5
= 24.295 rad/s