200
6 Free Vibration of Multiple Degrees of Freedom System
Then, the natural frequencies of the shaft having only the first disc or a second
disc are given by
1
p
2
11
= f ii I =
I L
G J
1
p
2
22
= f 22 10I =
100I L
G J
From Dunkerley’s equation, we get
1
p
2
1
∼ =
1
p
2
11
+
1
p
2
22
=
I L
G J
+
100I L
G J
= 101
I L
G J
or, fundamental frequency is given by
p
2
1 =
1
101
G J
I L
= 0.0099
G J
I L
6.9 Holzer Method
Holzer method falls under the determinant search technique. The method can be
applied to rectilinear or angular motions for damped or undamped systems. The
method is best suited for systems where the components are arranged along a basic
axis.
The method starts by assuming a trial frequency. Displacement of unit amplitude
is assumed at one end, and the forces and the displacements of different masses are
then calculated in a progressive manner. If the force or the displacement at the other
end is compatible with the conditions prevailing there, the assumed frequency is one
of the natural frequencies of the system. The calculation steps can be presented in a
systematic tabular form, and the method can be easily programmed in a computer.
The method is explained for the spring–mass system of Fig. 6.5.
Assume a frequency p and a displacement x 1 = 1.0 at the top. The inertia force
of mass m 1 is m 1 p
2
. 1 (m ¨
x = − p
2 m 1 A 1 e
i pt
= − p
2 m 1 x 1, if x 1 = A 1 e
i pt is
assumed). Deformation of the spring k 1 due to this force is
m 1 p
2
k 1
= 1 − x 2
or