determination of natural frequencies-analytical methods

8/17/2019 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



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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 + … .



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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



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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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determination of natural frequencies-analytical methods

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