06_Experimental Modal Analysis

Experimental Modal Analysis

Slides to accompany lectures in ME 599/699:

Vibro-Acoustic Design in Mechanical Systems© 2002 by A. F. Seybert

Department of Mechanical EngineeringUniversity of Kentucky

Lexington, KY 40506-0503Tel: 859-257-6336 x 80645

Fax: [email protected]

Dept. of Mech. EngineeringUniversity of Kentucky

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ME 599/699 Vibro-Acoustic Design

Goals of the Lecture

• Understand how experimental modal analysis complements other methods of vibration analysis

• Learn to perform experimental modal analysis on a simple system

• Understand how modal information is extracted from measured data

• Correlate measured natural frequencies and modes with those obtained analytically and numerically


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Principles

• Every mechanical system (structure) has a large number (theoretically infinite) natural frequencies.

• Associated with each natural frequency is a displacement pattern (mode) in which the structure prefers to vibrate.

First vibration mode Second vibration mode


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• A natural frequency may be detected experimentally by exciting the structure with a harmonic force and varying the frequency until “resonance” is achieved. At resonance, the mode associated with the natural frequency may be observed.

• By roving an accelerometer over the surface, the mode may be recorded and plotted.

Principles (cont.)

F tsinω 1 F tsinω 2

First vibration mode Second vibration mode


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• Excitation by an impulsive load (impact force) will produce a transient response consisting of a superposition of all the modes of vibration and their corresponding natural frequencies.

• Accelerometer is moved and impact test is repeated at as many points where a mode is desired.

Principles (cont.)

impulse response

Acc

eler

ation

(m/s

2 )


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Hammer Delivers Measurable Impact Force

Piezoelectric load cell


7



Freq (Hz)

Hij

FFT

Principles (cont.)

Impulse response and frequency response function (FRF) at point i due to impact at point j. Hij is invariant because it is a ratio of the response to the input.

Repeat for all points i = 1 to N while holding j fixed (requiresmoving the accelerometer of having a large number of them.)

HAFij

i

j

=( )( )ωω

Acc

eler

ation

(m/s

2 )


8



Point-Picking to get Mode Shape

Freq (Hz)

Hij

These are the relative values of each mode at point i: Hij(? 1), Hij(? 2), etc.

We can visualize each mode k by plotting Hij(? k), i = 1 to N, over the surface of the structure.

Hij(? k):+

-i = 1


9



Point-Picking Method


10



Reciprocity

Static beam deflection by reciprocity

y

x

P(xo)

y(x)y

x

P(x)

y(xo)

)|()|( xxyxxy oo =

For dynamic (FRF) measurement: jiij HH =

Thus, we can fix the accelerometer (i) and move the impact point (j).


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Modal Analysis Example

Foam Pad

Steel beam (1/2”x1/2”x18-1/8”)

&& ( )x tj F ti ( )

HAFji

j

i

( )( )( )

ωωω

=

PC

• Determine first four modes and natural frequencies experimentally• Compare with theory (see vibration book) for free-free beam• Compare with ANSYS model


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Modal Parameter Extraction

For each mode of thestructure we want:

• Natural frequency• Damping• Mode shape

Methods:

• Peak picking (SDOF)• Circle fit (SDOF)• MDOF curve fitting


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Peak Picking Method

H ji k( )ω

H ji k( )ω 2

ωk

ω2ω1ω ω

ςω ω

ω

n k

k

≈

≈−2 1

2The mode shapes are found by plotting the peak value at ωk for all the FRF’s measured (from the excitation point j to each point i).


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Peak Picking Method - Features

• assumes a SDOF FRF• accuracy depends on the reliability of the peak value• modes must be well-separated (little modal overlap)• damping must be small (so that ωn ≈ ωk) but not too small

so that peak value has high measurement uncertainty• a quick approach for preliminary evaluation and trouble shooting


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Circle Fit Method

• Also assumes SDOF FRF (same shortcomings of PPM)• Plot of mobility (reciprocal of impedance) is a circle

in the complex (Nyquist) plane• Process fits SDOF mobility function to measured

data using least squares (better than PPM)• Natural frequency, damping, and mode shapes are

found from fitted functions

Re (1/Zm)

Im (1/Zm)

FRF data

SDOF model


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MDOF Curve Fitting

Every FRF has the form:

H sb b s b sa a s a s

m pmm

pp( ) ( )=

+ + ⋅ ⋅ ⋅ ++ + ⋅ ⋅ ⋅ +

≤0 1

0 1

The roots of the characteristic equation are the eigenvalues:∆ ( ) ( )( ) ( )

,

s a a s a s s s s

j

pp

p

i i i i i i

= + + ⋅ ⋅ ⋅ + = + + ⋅ ⋅ ⋅ + =

= ± −∗

0 1 1 2

2

0

1

λ λ λ

λ λ ς ω ω ς Natural frequency and damping ratio for each mode

Partial fraction expansion to obtain residues:

( )H sA

s sA H s s si

i i ii

n

i i i i s i

( ) ( )=+ +

→ = ⋅ + += =∑ 2 2

1

2 2

22

ς ω ως ω ω

λ

a’s and b’s from curve fit

Ai for all FRF’s are used to plot mode shape for each mode i


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Example – MDOF Curve Fit


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Example – MDOF Curve Fit

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06_Experimental Modal Analysis

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