determination of natural frequencies-analytical methods
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Determination of naturalFrequencies-Analytical Method
• Dunkerley’s Formula
• Rayleigh’s Method
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• Expansion of characteristic determinant and the solution of the res
degree polynomial to get natural frequencies is tedious for large vaDOF (n).
• These methods will give approximate value (s) of natural frequenc
mode shapes
Preamble
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• Gives approximate value of fundamental frequency of a composite
terms of natural frequencies of its component parts.
• Derived from the fact that the higher natural frequencies of most vi
systems are large compared to their fundamental frequencies.
• Frequency equation
I - [k] +2 I = 0 (
I -
[I] + I = 0 where [a] is flexibility matrix (
Dunkerley’s Formula
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For a lumped mass system with a diagonal mass matrix, (1) becomes
I -
1 ⋯ 0⋮ ⋱ ⋮0 ⋯ 1
+
⋯ ⋮ ⋱ ⋮
⋯
⋯ 0⋮ ⋱ ⋮0 ⋯
I = 0
I
−
+ ⋯
⋮ ⋱ ⋮
⋯ −
+
I = 0 (3
Dunkerley’s Formula
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The expansion of equation (3) leads to
n
- + 222 + … .
n-1
....= 0
Polynomial of nth degree in
. Let the roots of (4) be denoted by
Thus
−
−
….
−
=
n -
+
+ …
n-1
Equating coefficients of
n-1 in (4) and (5)
+
+ …
= + 222 + … .
Dunkerley’s Formula
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In most cases, the higher frequencies 2 , 3 … are considerably
than fundamental frequency
,
∴
≪
, I = 2,3….n
∴ Equation (6) can be written approximately as
≅ + 222 + … . (
• Fundamental frequency given by (7) is always smaller than the ex
Dunkerley’s Formula
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Estimate the fundamental frequency of a simply supported beam car
identical equally spaced masses.
Problem 1
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Rayleigh’s Principal:
• The frequency of vibration of a conservative system vibrating about
equilibrium position has a stationary value in the neighborhood of amode. This stationary value, in fact, is a minimum value in the neighof the fundamental natural mode.
• The PE and KE of the n-DOF system is
V =
2 {} [k] {} (8)
T =
2 { ሶ}
[m] { ሶ} (9)• Assume harmonic solution
x = X cos (10)
X- mode shape vector / modal vector,
- natural frequency of vibration
Rayleigh’s Method
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If the system is conservative, the maximum KE = maximum PE
Tmax = Vmax (12)
Substituting (10) in (8 and 9) and simplifying
Tmax =
2 [m] 2
Vmax =
2 [k]
Equating
2=
[k] [m] (13) Rayleigh
Rayleigh’s Method
Rayleigh’s quotient provides an upper bound for2
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Procedure:
1. Select a trial vector X to represent the first natural mode X1 and sRHS of (13).
2. Calculate 2
3. If the trial vector is more closer to first natural mode X1 , value of
more accurate
Rayleigh’s Method
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Estimate the fundamental frequency of vibration of
the system shown in Figure. Assume that for
k1=k2=k3=k, m1=m2=m3=m , and the modeshape is X = { 1 2 3}T
Problem 2
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