32
2 Free Vibration of Single Degree of Freedom System
Fig. 2.16 Example 2.8
The stiffness of the spring is k = 0.7 kN/mm = 700 N/mm. The natural angular
frequency is
p =
k
m
=
700
1
= 26.46 rad/s
At critical damping
n = p
or
c
2m
= 26.46
or
c = 26.46 × 2 × 1 = 52.92 Ns/m
Example 2.8 A mass, spring and dashpot attached to a rigid bar are shown in
Fig. 2.16. Write the equation of motion. Determine the natural frequency of damped
free oscillation and critical damping coefficient.
The bar AC being rigid, the point B where the mass and the dashpot are attached
and the end C where the spring is attached will undergo different displacements and
they are related. From similar triangles, it can be shown that
y 1 =
b
a
y
(a)
During oscillation, the forces acting on the system are the inertia force m ¨
y, the
damping force c ˙
y and the spring force ky 1 . Taking moment about A [Fig. 2.16c]
yields
m ¨
ya + c ˙
ya + ky 1 b = 0
( b )
2 Free Vibration of Single Degree of Freedom System
Fig. 2.16 Example 2.8
The stiffness of the spring is k = 0.7 kN/mm = 700 N/mm. The natural angular
frequency is
p =
k
m
=
700
1
= 26.46 rad/s
At critical damping
n = p
or
c
2m
= 26.46
or
c = 26.46 × 2 × 1 = 52.92 Ns/m
Example 2.8 A mass, spring and dashpot attached to a rigid bar are shown in
Fig. 2.16. Write the equation of motion. Determine the natural frequency of damped
free oscillation and critical damping coefficient.
The bar AC being rigid, the point B where the mass and the dashpot are attached
and the end C where the spring is attached will undergo different displacements and
they are related. From similar triangles, it can be shown that
y 1 =
b
a
y
(a)
During oscillation, the forces acting on the system are the inertia force m ¨
y, the
damping force c ˙
y and the spring force ky 1 . Taking moment about A [Fig. 2.16c]
yields
m ¨
ya + c ˙
ya + ky 1 b = 0
( b )