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6 Free Vibration of Multiple Degrees of Freedom System
6.8 Dunkerley’s Equation
A somewhat simpler approach has been suggested by Dunkerley, to obtain an approximation to the fundamental frequency. Good results are expected for the system,
having very small amount of damping and the fundamental frequency differs to a
significant extent from the higher harmonics.
When the determinant given by Eq. (6.14) is expanded for a two-mass system, it
becomes
d 11 −
1
p 2
d 12
d 21
d 22 −
1
p 2
= 0
(6.45)
Expanding this determinant, we obtain
1
p 2
2
− (d 11 + d 22 )
1
p 2
+ (d 11 d 22 − d 12 d 21 ) = 0
(6.46)
If p 1 and p 2 are two natural frequencies of the system, then Eq. (6.46) can be
written in terms of its factors as follows:
1
p 2 −
1
p
2
1
1
p 2 −
1
p
2
2
= 0
(6.47)
or
1
p 2
2
−
1
p
2
1
+
1
p
2
2
1
p 2
+
1
p
2
1 p
2
2
= 0
(6.48)
Equating the coefficients of 1/p
2 terms in Eqs. (6.46) and (6.48), we get
1
p
2
1
+
1
p
2
2
= d 11 + d 22
(6.49)
If, for the system, the values of p 1 and p 2 are such that 1/ p
2
1 ≥≥ 1/ p
2
2 , then
Equation (6.49) becomes
1
p
2
1
= d 11 + d 22
(6.50)
Equation (6.50) is known as Dunkerley’s equation. Equation (6.49) can be
extended to systems with n masses, and it will assume the following form