mt lectures 11-20

8/9/2019 MT Lectures 11-20

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Modal Testing(Lecture 11)

Dr. Hamid AhmadianSchool of Mechanical Engineering

Iran University of Science and Technology

[email protected]

8/9/2019 MT Lectures 11-20


Dr H Ahmadian,Modal Testing Lab,IUST

Response Function MeasurementTechniques

Response Function

Measurement Techniques

Introduction Test Planning

Basic Measurement System Structure Preparation

Excitation of the structure

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Introduction

The measurements techniques used formodal testing are discussed: Response measurement only

Force and response measurement

The 2nd type of measurement

techniques is of our concern: Single-point excitation( SISO/SIMO)

Multi-point excitation (MIMO)

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Test Planning Objective of the test

Levels according to Dynamic Testing Agency:

UpdatingOut of range

residuesUsabe forvalidation

Mode ShapesDamping

ratioNaturalFreq

Level

0

Only in fewpoints

1

2

3

4

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Test Planning

Extensive test planning is requiredbefore full-scale measurement:

Method of excitation

Signal processing and data analysis

Proper selection of pickup points

Excitation location

Suspension method

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Basic Measurement System

An excitation mechanism A transduction mechanism

An Analyzer

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Basic Measurement System

Source of excitationsignal:

Sinusoidal

Periodic (with specificfreq. content)

Random

Transient

Power Amplifier

Exciter

Transducers Condition Amplifiers

Analyzers

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Structure Preparation

Free Supports Grounded Support

Loaded Support Perturbed Support

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Free Supports

Theoretically the structure will possess 6 rigidbody modes @ 0 Hz.

In practice this is provided by a soft support

Rigid body modes are less then 10% of strainmodes

Suspending from nodal points for minimuminterference

The suspension adds significant damping to thelightly damped structures

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Free Supports

Suspension wires,should be normal tothe primary

vibration direction The mass and

inertia properties

can be determinedfrom the RBMs.

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Free Supports

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Grounded Support The structure is fixed to the ground at selected

points. The base must be sufficiently rigid to provide

necessary grounding.

Usually is employed for large structures Parts of power generation station Civil engineering structures

Another application is simulating the operational

condition Turbine Blade

Static stiffness can be obtained from low frequencymobility measurements.

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Loaded Support

The structure is connected to a simplecomponent with known mobility

A specific mass

The effect of added mass can be removedanalytically

More modes are excited in a certainfrequency range compared to free suspension

The modes of structure are quite different

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Perturbed Support

The data base for the structure can beextended by repetition of modal tests fordifferent boundary conditions

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Perturbed Support

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Excitation of the structure

Various devices are available forexciting the structure:

Contacting

Mechanical (Out-of-balance rotating masses)

Electromagnetic (Moving coil in magnetic field)

Electrohydraulic Non-Contacting

Magnetic excitation

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Electromagnetic Exciters

Supplied input to the shaker is converted toan alternating magnetic field acting on amoving coil.

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Electromagnetic Exciters

There is a small difference between the forcegenerated by the shaker and the appliedforce to the structure The force required to accelerate the shaker

moving

The force required to excite the structuresharply reduces near the resonance point, Much smaller than the generated force in the

shaker and the inertia of the drive rod

Vulnerable to noise or distortion

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Attachment to the structure

Push rod or stingers: Applying force in

only one direction

Flexible driverod/stingerintroduces its ownresonance into themeasurement.

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Support of shakers

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Hammer or Impactor

Excitation

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Other excitation methods

Step Relaxation/sudden release Charge/Explosive impactor

….

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Moving Support

Corresponds togrounded model

Only responses are

measured When the mass

properties are known,

the modal properties canbe calculated frommeasured data

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Moving Support

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[email protected]

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Dr H Ahmadian,Modal Testing Lab,IUSTFRF Measurement Techniques

Digital Signal Processing

Introduction Basics of Discrete Fourier Transform

(DFT)

Aliasing Leakage

Windowing Filtering

Improving Resolution

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Introduction The measured force or

accelerometer signalsare in time domain.

The signals aredigitized by an A/Dconverter

And recorded as a setof N discrete valuesevenly spaced in theperiod T

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Basics of DFT

The spectral properties of the recordedsignal can be obtained using DiscreteFourier Transform/Series (DFT/DFS):

The DFT assumes the signal x(t) is periodic

In the DFT there are just a discrete

number of items of data in either form There are just N values xk

The Fourier Series is described by just N values

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Basics of DFT

*

0

0

0

1

0

)(1

)(

)sin()(2

)cos()(2

,2

)sin()cos(2

)(

)()(

nn

t iT

n

n

t i

n

T

nn

T

nn

n

n

nnnn

X X

dt et xT

X

e X t x

or

dt t t xT

b

dt t t xT

a

T

n

t bt aat x

T t xt x

n

n

=

=

=

=

=

=

++=

+=

−

−

∞

−∞=

∞

=

∫

∑

∫

∫

∑

ω

ω

ω

ω

π

ω

ω ω

∗−

−

=

−

−

=

−

=

−

=

−

=

==

=

=

=

++=

∑

∑

∑

∑

∑

r r N

N

k

N ink

k n

N

n

N ink

nk

N

k

k n

N

k

k n

N

n

nnk

X X e x

N

X

e X x

or

N

nk x

N b

N

nk x

N a

N nk b

N nk aa x

,1

)2

sin(2

)2

cos(2

)2sin()2cos(2

1

0

/2

1

0

/2

1

0

1

0

2

1

1

0

π

π

π

π

π π

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Basics of DFT

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

↑

↑↑↑↑

=⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

=

−−

−

=

−

∑

1

1

0

1

1

0

1

0

/21

N N

N

k

N ink k n

x

x

x

X

X

X

e x N

X

M

NKK

MKKM

OLN

L

M

π

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Basics of DFT

The sampling frequency:

The range of frequency spectrum:

The resolution of frequency spectrum:

T

N

t T

N

t f

s

s

s

s

π π

ω

221==⇒==

T

N f f ss π ω

ω ===⇒=22 maxmax

T T f

π

ω

2,

1=Δ=Δ

Nyquist Frequency

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Basics of DFT

There are a number of features of DFanalysis which if not properly treated,can give rise to erroneous results:

Aliasing Mis-interoperating a high frequency component

as a low frequency one

Leakage

Periodicity of the signal

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Aliasing Digitizing a ‘low’

frequency signalproduces exactly the

same set of discretevalues as result fromthe same process

applied to a higherfrequency signal

2

sω

ω <

ω ω −s

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Aliasing

2

)2sin(

)22sin(

))(2sin()2sin(

:

N p

N pk

N

pk k

N

k p N

N

k p

Compare

<

−

−

−⇔

π

π

π

π π

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Aliasing

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Aliasing The solution to the

problem is to use ananti-aliasing filter

Subjecting the original

signal to low pass withsharp filter

Filters have a finite cut-off rate; it is necessary to

reject the spectral rangenear Nayquist frequency

2)0.108( sω

ω −>

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Leakage A direct consequence of

taking a finite length oftime history coupled with

assumption of periodicity Energy is leaked into a

number of spectral lines

close to the truefrequency.

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Leakage

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Leakage

To avoid the leakage there are numberof scenarios:

Increasing the record time T

Windowing Multiply the time record by a function that is

zero at the ends of the time record and large inthe middle, the FFT content is concentrated onthe middle of the time record

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Windowing Windowing involves the imposition of a

prescribed profile on the time signal prior toperforming the FT

T

elsewhere

T t t at a

t at at aa

t w

t xt wt x

π ω

ω ω

ω ω ω

2

0

,0)4cos()3cos(

)2cos()cos()cos(

)(

)()()(

0

0403

0201010

=

⎪⎩

⎪⎨

⎧<<

+−

++−

=

×=′

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Windowing

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Windowing

a4a3a2a1a0Function

----1Rectangular

---11Hanning

-0.0030.2441.2981Kaser-Bessel

0.0320.3881.2861.9331Flat top

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Windowing

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Improving Resolution (Zoom) There arises limitations of inadequate

frequency resolution at the lower end of the frequency range

For lightly-damped systems A common solution is to concentrate all

spectral lines into a narrow band Within f min-f max

Instead of 0-f max

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Zoom Method 2:

A controlled aliasingeffect Applying a band pass

filter Because of the

aliasing phenomenon,the frequencycomponent betweenf 1 and f 2 will appearaliased between 0-(f 2-f 1)

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Iran University of Science and [email protected]

Use of Different Excitation

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Signals Introduction

Stepped-Sine Testing

Slow Sine Sweep Testing

Periodic Excitation

Random Excitation

Transient Excitation

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Introduction Periodic:

Stepped sine

Slow sine sweep

Periodic

Pseudo-random

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Introduction Transient:

Burst sine

Burst random

Chirp

Impulse

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Stepped-Sine Testing Classical method of FRF measurement

To encompass a frequency range ofinterest, the command signal frequency is

stepped from one frequency to another The excitation/response(s) are measured

(amplitudes and phase(s)) . It is necessary to ensure that the steady-state

condition is attained before the measurement.

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Stepped-Sine Testing

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Stepped-Sine Testing The extent of unwanted transient

response depends on: Proximity of excitation frequency to a

natural frequency, The abruptness of the changeover from

the previous command signal to the new

one, The lightness of the damping of nearby

modes.

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Stepped-Sine Testing An advantage

of stepped-sinetesting is thefacility of takingmeasurementwhere and as

they arerequired.

Largest Error

dB%

No. pointbetween

HPP’s

3301

1102

0.553

0.225

0.118

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Slow Sine Sweep Testing Involves the use of a sweep oscillator

Provides a sinusoidal signal

Its frequency is varied slowly but

continuously

If an excessive sweep rate is used then

distortions of FRF plot are introduced

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Slow Sine Sweep Testing One way of

checking thesuitability of asweep rate is to

make themeasurementtwice: Once sweeping up And the 2nd time

sweeping down

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Slow Sine Sweep Testing It is possible to

prescribe anoptimum sweep ratefor a given structuretaking into accountits damping levels

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Slow Sine Sweep Testing Recommended sweep rate:

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Slow Sine Sweep Testing ISO prescribes

maximum linear andlog sweep ratethrough a resonanceas:

min/)(310

min/)(216

2

max

2

max

OctavesS

Log

HzS

Linear

r r

r r

ω ζ

ω ζ

×<

×<

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Periodic Excitation A natural extension of the sine wave

test methods: To use a complex periodic input signal

which contains all the frequencies ofinterest,

The DFT of both input and output signals

are computed and the ratio of these givesthe FRF

Both signal have the same frequency

contents

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Periodic Excitation Two types of periodic signals are used:

A deterministic signal (square wave) Some frequency components are inevitably

weak.

Pseudo-Random type of signal The frequency components may be adjusted to

suit a particular requirements-such as equal

energy at each frequency, Its period is exactly equal to the sampling time

resulting zero leakage .

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Random Excitation

)(

)(

)()()(

)(

)()(

)()()(

)()()(

)()()(

2

12

2

12

ω

ω γ

ω ω ω

ω

ω ω

ω ω ω

ω ω ω

ω ω ω

H

H

S S H

S

S H

S H S

S H S

S H S

xf

xx

ff

fx

xf xx

ff fx

ff xx

=

=

=

=

=

=

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Random Excitation There may be noise on one of the two

signals

Near resonance this is likely to influence

the force signal At anti-resonances it is the response signal

which will suffer

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Random Excitation H2 might be a better indication near

resonances while H1 is a better indicationnear anti-resonances:

)(()()(,

)()()()( 21

ω ωω ω

ω ω ω ω

xf

mm xx

nn ff

fx

S

S S H S S

S H +

=

+

=

Auto-spectra of noise on the input signal

Auto-spectra of noise on the output signal

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Random Excitation A closer optimum formula for the FRF is

defined as the geometric mean of the twostandard estimates Phase is identical to that in the two basic

estimates

)()()( 21 ω ω ω H H H v =

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Random Excitation Typical measurement made using random

excitation:

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Random Excitation Details from previous plot around a

resonance:

H1

H2

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Random Excitation Use of zoom spectrum analysis:

Improving the resolution removes the majorsource of low coherence

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Random Excitation Effect of averaging:

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Transient Excitation The excitation

and theresponse arecontainedwithin thesingle

measurement

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Transient Excitation Burst excitation signals:

A short section of a continuous signal (sin,random, …) followed by a period of zero wave.

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Transient Excitation Chirp excitation:

The spectrum can be strictly controlled to be suchwithin frequency range of interest

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Transient Excitation Impulsive excitation

by Hammer: Different impulsive

excitations

Signals and spectrafor double hit case

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Transient Excitation Impulsive excitation by Shaker:

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RESPONSE FUNCTION

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MEASUREMENT TECHNIQUES 3.9 Calibration

3.10 Mass Cancellation

3.11 Rotational FRF Measurement

3.12 Measurement on NonlinearStructures

Effects of Different Excitations Level Control in FRF Measurement

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Calibration The overall system

calibration The scale factor should

be checked against

computed factor usingmanufacturers statedsensitivity

Should be carried outbefore & after each test

ll

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Mass Cancellation Near resonance the actual applied force

becomes very small and is thus very prone toinaccuracy.

Some applied mass is used to moveadditional transducer mass

measured F

X

required

F

X

xm f f

M

M

T

T

M T

→=

→=

−= ∗

&&

&&

&&

α

α

M C ll i

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Mass Cancellation Added mass to be

cancelled and thetypical analoguecircuit

At deriving point arelation between

measured andrequired FRF’s canbe obtained

M C ll ti

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Mass Cancellation

)/1Im()/1Im()/1Re()/1Re(

)Im()Im()Im()Re()Re()Re(

M T

M T

M T

M T

m

or

X mF F X mF F

α α

α α

=−=

−=−=

∗

∗

∗

&&

&&

R t ti l FRF M t

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Rotational FRF Measurement Measurement of rotational FRFs using two or

more transducers:

L

x x x

B Ao

B Ao

θ θ θ +=

+=2

R t ti l FRF M t

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Rotational FRF Measurement Application of moment excitation

M F M

X

F

X θ θ ,,,

Measurement on Nonlinear

St t

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Structures Many structures, especially in vicinity of

resonances, behave in a nonlinear way: Natural frequency varies with position

and strength of excitation Distorted frequency responses (near

resonances)

Unstable or unrepeatable data

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Effects of Different Excitations

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Effects of Different Excitations Most types of nonlinearity are amplitude

dependent: A linearized behaviour is observed when

the response level is kept constant The obtained linear model is valid for that

particular vibration level

Level Control in FRF

Measurement

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Measurement Response level

control, Best linear representation

(nonlinearities are

displacement dependent) Force level control

Or no level control


Measurement

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Measurement Inverse FRF plots for a SDOF

Real part is expected to be liner wrt frequency squared Imaginary part should be linear/constant

Any deviation from the expected behaviour can be detectedas nonlinearity in the system


Measurement

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Measurement Use of Hilbert transform to detect non-linearity

The Hilbert transform express the relations betweereal and imaginary parts of the Fourier Transform

Notes: Hilbert Transform

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Notes: Hilbert Transform The Hilbert transform express the

relations between real and imaginaryparts of the Fourier Transform

Fourier Transform is considered to mapfunctions of time to functions of frequencyand vice versa

Hilbert transform map functions of time orfrequency to the same domain


8/9/2019 MT Lectures 11-20



Notes: Hilbert Transform For causal functions:

⎩⎨

⎧

<−

>

=

⎩⎨

⎧

<

>

=

+=

0,2/)(

0,2/)()(

0,2/)(

0,2/)(

)(

),()()(

t t g

t t gt g

t t g

t t g

t g

t gt gt g

odd

even

odd even


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Notes: Hilbert Transform{ } { }

{ } { }

{ }

.)(Re)(Im

,)(Im)(Re

)(

,)()()()(Im

,)()()()(Re

πω ω ω

πω

ω ω

πω

ω

ω

iGG

iGiG

theormnconvolutioonbased

i

t signSince

t signt gt gG

t signt gt gG

evenodd

odd even

−∗=

−∗=

−=ℑ

×ℑ=ℑ=

×ℑ=ℑ=

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Modal Parameter Extraction

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Dr H Ahmadian,Modal Testing Lab,IUSTModal Parameter Extraction Methods

Modal Parameter Extraction Introduction

Preliminary checks of FRF data Visual checks

Assessment of multiple-FRF data set using SVD

Mode indicator functions

SDOF modal analysis methods Peak amplitude method

Circle fit method

Inverse or line fit method

Introduction

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Introduction Some of the many available procedures for

fitting a model to the measured data arediscussed:

Their various advantages and limitations are

explained, No single method is best for all cases.

This phase of the modal test procedure isoften called modal parameter extraction or modal analysis

Introduction

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Introduction Types of modal analysis:

Frequency domain (of FRFs)

Time domain (of Impulse Response

Function) The analysis will be performed using

SDOF methods, and

MODF methods.

Introduction

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Introduction Another classification of methods

relates to the number of FRFs used inthe analysis:

Single-FRF methods, and Multi-FRF methods:

Global methods which deals with SIMO data

sets and Polyreference which deals with MIMO data

Introduction

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Introduction Difficulty due to damping:

In practice we are obliged to make certainassumption about the damping model,

Significant errors can be incurred in the

modal parameter estimates as a result ofconflict between assumed and actualdamping effects.

Decision on the issue of real and complexmodes.

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Preliminary checks of FRF

data

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data Mode Indicator Functions:

The Peak-Picking Method Sum of amplitudes of all measured FRFs to

locate the resonance points

The frequency-domain decompositionmethod

Defined by the SVD of the FRF matrix

Case Study: MODES OF A

RAILWAY VEHICLE

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RAILWAY VEHICLE

Case Study: Test set-up

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Case Study: Test set up

Case Study: Sensor Locations

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Case Study: Excitation

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Case Study: Excitation

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The Peak-Picking Method

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g Sum of amplitudes of all measured FRFs to

locate the resonance points

987654321Mode#

24.6716.0014.0013.3312.338.335.334.672.67Frequency)Hz(

The frequency-domain

decomposition method

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p A more advanced method consists of

computing the Singular ValueDecomposition of the spectrum matrix.

The method is based on the fact thatthe transfer function or spectrum matrixevaluated at a certain frequency is onlydetermined by neighboring modes.

The frequency-domain

decomposition method[ ] { } { } )()()()( HHHH

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p[ ] { } { }

[ ] [ ][ ][ ][ ] [ ] [ ])()()(

)()()()(

)()()()( 2111

ω ω ω ω ω ω ω

ω ω ω ω

ΣΣ=Σ=

=

T

T

np

MIF V U H

H H H H K

SDOF modal analysis

methods

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The SDOF assumption

jk r

r r r

jk r

jk

N

r ss sss

jk s

r r r

jk r

jk

N

s sss

jk s

jk

Bi

A

i

A

i

A

i

A

++−

=

+−+

+−=

+−=

∑

∑

≠=

=

22

1 2222

122

)(

)(

)(

ω η ω ω ω α

ω η ω ω ω η ω ω ω α

ω η ω ω ω α

SDOF modal analysis

methods

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SDOF modal analysis methods

Peak amplitude method

Circle fit method

Inverse or line fit method

SDOF modal analysis

methods: Peak Amplitude

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p Individual resonance peaks are

detected from the FRF The frequency of the maximum responses

is takes as the natural frequency of thatmode,

The peak amplitude and the half power

points are determined,

SDOF modal analysis


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p

⎪

⎪

⎩

⎪⎪⎨

⎧

==

=−=−=

⇒

⎪⎩

⎪⎨⎧

⇒

H A A

H

then

H knowns

r r r

r r

r

r r

r

ba

r

bar

ba

r

ˆ,ˆ

2,2

,

,

ˆ

2

2

2

22

ω η

ω η

η ζ ω

ω ω

ω

ω ω η

ω ω

ω

SDOF modal analysis


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p Another estimate for

modal residue:

( )( )min(Re)max(Re)

min(Re)max(Re)ˆ

2 +=

+=

r r r A H

ω η

8/9/2019 MT Lectures 11-20


Modal Parameter Extractionf

8/9/2019 MT Lectures 11-20



Circle-fit method

Properties of the modal circle

Circle-fit analysis procedure

Interpretation of damping plots

Properties of the modal circleA i t ith t t l d i

8/9/2019 MT Lectures 11-20



Assuming a system with structural damping

the basic function to deal with is:

Since the effect of modal constant is to scalethe size and rotate the circle, we consider:

))/(1()(

222

r r r

jk r

i

A

η ω ω ω ω α

+−=

))/(1(

1)(222

r r r iη ω ω ω ω α

+−=

Properties of the modal circleFi di th t l f

8/9/2019 MT Lectures 11-20



Finding the natural frequency:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ −+−=⇒−=⇒

−==−

−=

2222

22

2

2

)/(11

2))

2tan(1(

)/(1)

2

tan()90tan(,

)/(1

tan

r

r r r r r

r

r

r

r

d

d

η

ω ω ω η

θ

ω θ η ω ω

η

ω ω θ γ

ω ω

η γ

Properties of the modal circled ⎟

⎞⎜⎛ ⎞⎛

2222 1)/(1

8/9/2019 MT Lectures 11-20



Dampingd d

frequency Naturald

d

d

d

ratesweepd

d

r r

r

r

r r r

r

⇒−=⎟ ⎠ ⎞⎜

⎝ ⎛

⇒==⎟⎟ ⎠

⎞⎜⎜⎝

⎛

⇒⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ −+−=

=22

2

222

2

@.0

)(

1)/(11

2

ω η ω θ

ω ω θ

ω

ω

η

ω ω ω η

θ

ω

ω ω

Properties of the modal circleθ⎧ )/(1 2

8/9/2019 MT Lectures 11-20



r

bar

ba

bar

bar

r

bar

bar

r

r aa

r

r bb

when

for

ω

ω ω η

θ θ

θ θ ω

ω ω η

η

θ θ ω

ω ω η

η

ω ω θ

η

ω ω θ

−=⇒

==

+

−=⇒

≤

+

−=⇒

⎪⎪⎩

⎪⎪

⎨

⎧

−=

−=

o

K

90

))2

tan()2

(tan(

)(2

%3%2

))2

tan()2

(tan(1)/(

)2

tan(

)/(1)

2tan(

2

22

2

2

8/9/2019 MT Lectures 11-20


Circle-fit analysis procedureSelect points to

8/9/2019 MT Lectures 11-20



Select points to

be used Fit circle,

calculate quality

of fit

Locate natural

frequency, …

Circle-fit analysis procedure Obtain damping

8/9/2019 MT Lectures 11-20



Obtain damping

estimates Calculate

multipledamping

estimate andscatter

Determine modal

constant moduleand argument.

Interpretation of damping

plots Noise may contribute

8/9/2019 MT Lectures 11-20



Noise may contribute

to the roughness ofthe surface.

Systematic distortions

due to: Leakage

Erroneous estimates fornatural frequency

Nonlinearity

Circle-fit Minimizing the algebraic distance:

8/9/2019 MT Lectures 11-20



Minimizing the algebraic distance:

.0

1

1

1

1:

.0

)()(

22

22

2

2

2

2

11

2

1

2

1

22

222

=

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎥⎥⎥

⎥

⎦

⎤

⎢⎢⎢

⎢

⎣

⎡

+

+

+

=++++=+++

C

B

A

y x y x

y x y x

y x y xSolutionSquares Least

C By Ax y x

Rb ya x

nnnn

MMMM

Circle-fit Minimizing the geometric distance:

8/9/2019 MT Lectures 11-20



Minimizing the geometric distance:

2

1

3

2

12

3

2

12

222

21

)(min

,

)()(

U d

u

u

u

U Let uu

u X d

Rb xa x

n

i

i

ii

∑=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=⎟⎟ ⎠

⎞

⎜⎜⎝

⎛ −

⎭⎬⎫

⎩⎨⎧

−=

=+++

Circle-fit+Δ

∂+= 0 pd

dd L

8/9/2019 MT Lectures 11-20



⎦

⎤

⎢⎢⎢

⎢⎢⎢

⎣

⎡

−−+−

−−+−

−

−−+−

−−+−

−

=

∂

∂

Δ∂

∂+≈

+Δ∂

+=

1)()()()(

1)()()()(

minmin

2

22

2

11

22

2

22

2

11

11

2

122

2

111

122

2

122

2

111

111

0

00

0

mm

m

mm

m

xu xu

xu

xu xu

xu

xu xu

xu

xu xu

xu

p

d

p p

d d d

p p

d d

MMM

L

Circle-fitLeast Squares Fitting of

8/9/2019 MT Lectures 11-20



Least-Squares Fitting ofCircles and Ellipses By: Walter

Gander, Gene H. Golub, and Rolf

Strebel

You may find it in ftpmech.iust.ac.ir

Home work 1 Determine the modal properties of the

8/9/2019 MT Lectures 11-20



Determine the modal properties of the

beam tested in the lab Frequency range of 0-400Hz

Natural frequencies

Damping (carpet plots)

Mode Shapes

Due time 87/2/22

8/9/2019 MT Lectures 11-20


Importing the ASCII files

8/9/2019 MT Lectures 11-20



Modal Testing

8/9/2019 MT Lectures 11-20





[email protected]

MDOF Modal Analysis in the

Frequency Domain (SISO) In some cases the SDOF approach to

8/9/2019 MT Lectures 11-20



In some cases the SDOF approach to

modal analysis is simply inadequate orinappropriate:

closely-coupled modes,

the natural frequencies are very closely spaced,or

which have relatively heavy damping, those with extremely light damping

MDOF Modal Analysis in the

Frequency Domain (SISO) One step MDOF curve fitting methods:

8/9/2019 MT Lectures 11-20



p g

Non-linear Least Squares Method Rational Fraction Polynomial Method

A method particularly suited to very lightly

damped structures

Global Modal Analysis in Frequency

Domain Global Rational Fraction Polynomial Method

Global SVD Method

Non-linear Least Squares

MethodAm 112

8/9/2019 MT Lectures 11-20



etc A A AqdqdE w E

H H

M K i

A

H H

jk jk jk

p

l

ll

l

m

ll

r lr mr r r lr

jk r

ll jk

,,,,,.,0,

11

)(

1321

1

2

2222

2

1

ω ε

ε

ω ω η ω ω ω

L===

−=

+++−==

∑

∑

=

=

The difference betweenmeasurement and analyticalmodel

Non-linear Least Squares

Method The set of obtained equations are

8/9/2019 MT Lectures 11-20



q

nonlinear No direct solution (iterative procedures)

Non-uniqueness of solution Huge computational load

Rational Fraction Polynomial

Method

8/9/2019 MT Lectures 11-20



N

N

N

N

N

r r r r

jk r

iaiaiaa

ibibibb H

i

A H

2

2

2

210

12

12

2

210

122

)()()(

)()()()(

2)(

ω ω ω

ω ω ω ω

ζ ωω ω ω ω

++++

++++=

+−=

−−

=∑

L

L


MethodOrder of model is selected

8/9/2019 MT Lectures 11-20



( )( )m

k mk k k

m

k mk k k

k m

k mk k

m

k mk k k

iaiaiaa H

ibibibbe

or

H iaiaiaaibibibbe

2

2

2

210

12

12

2

210

2

2

2

210

12

12

2

210

)()()(

)()()(

)()()()()()(

ω ω ω

ω ω ω

ω ω ω ω ω ω

++++−

++++=′

−++++

++++=

−−

−

−

L

L

LL


Methodb

b

1

0

⎪⎪⎫

⎪⎪⎧

8/9/2019 MT Lectures 11-20



{ }

{ } m

k k m

m

m

k k k k

m

mk k k k

i H a

a

a

a

a

iii H

b

b

b

iiie

2

2

12

2

1

0

122

12

2

1

122

)()()()(1

)()()(1

ω ω ω ω

ω ω ω

−

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−

⎪⎪⎪

⎭

⎪

⎪

⎬

⎪⎪⎪

⎩

⎪

⎪

⎨=′

−

−

−

−

ML

M

L


Method A set of linear equations using each

8/9/2019 MT Lectures 11-20



individual measured FRF is formed. The unknowns ai and bi are obtained

using a least square solution.

The modal properties are extractedfrom obtained coefficients ai and bi.

The analysis may repeat for a differentmodel order.

Lightly Damped Structures In these structures it is easy to locate

8/9/2019 MT Lectures 11-20



the natural frequencies, Its accuracy is equal to the frequency

resolution of the analyzer

The damping ratio is assumed to bezero.

The modal constants are obtained usingcurve fittings.

Lightly Damped Structures=∑ N

jk r A H

22)(ω

8/9/2019 MT Lectures 11-20



( ) ( )( ) ( )

⎪⎪⎭

⎪⎪

⎬

⎫

⎪⎪⎩

⎪⎪

⎨

⎧

⎥⎥

⎥⎥⎥

⎦

⎤

⎢⎢

⎢⎢⎢

⎣

⎡

Ω−Ω−Ω−Ω−

=

⎪⎪⎭

⎪⎪

⎬

⎫

⎪⎪⎩

⎪⎪

⎨

⎧ Ω

Ω

−

−−

−−

=

∑

M

M

MMM

MMM

LL

M

M

jk

jk

r r

A A

H H

2

112

2

2

2

12

2

2

1

12

1

2

2

12

1

2

1

2

1

1

22

)()(

)(

ω ω ω ω

ω ω The natural frequencies are know

Global Modal Analysis in

Frequency Domain So far each measured FRF is curve

8/9/2019 MT Lectures 11-20



fitted individually, Multi-estimates for global parameters

(natural frequencies and damping)

Another way is to use measured FRFcurves collectively.

Frequency and damping characteristicsappear explicitly.

Global Rational Fraction

Polynomial Method If we take several FRF’s from the same

8/9/2019 MT Lectures 11-20



structure then the denominatorpolynomial will be the same in every case.

A natural extension of RFP method is tofit all n FRFs simultaneously

2m-1 values of ai and,

and n(2m-1) values of bi

Global SVD Method( )⎫⎧H ω

8/9/2019 MT Lectures 11-20



( ){ }

( )( )

( )

[ ] ( )[ ] { } { }11

11

2

1

)(××

−

××

×

+−Φ=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

N k N k N N r N n

nnk

k

k

k

Rsi

H

H

H

H

ω φ ω

ω

ω

ω

ω M

8/9/2019 MT Lectures 11-20


Global SVD Method

−=Δ

8/9/2019 MT Lectures 11-20



( ){ } ( ){ } ( ){ }k cik ik i H H H +−=Δ

ω ω ω

( ){ } [ ] ( ){ }( ){ } [ ] [ ] ( ){ }ik r N nk i

ik N nk i

gs H g H

ω ω ω ω

ΔΦ=ΔΔΦ=Δ

×

×

&

Global SVD Method

Consider data from several different frequencies

8/9/2019 MT Lectures 11-20



to obtain frequencies and damping:

( )[ ] [ ] ( )[ ]

( )[ ] [ ] [ ] ( )[ ]

[ ] [ ]( ){ } [ ] [ ] Tr T k r

T k

L N k r N n Lnk

L N k N n Lnk

z z H s H

gs H

g H

+

×××

×××

Φ==Δ−Δ

ΔΦ=Δ

ΔΦ=Δ

,0&

& ω ω

ω ω

Global SVD Method

The eigen-problem is solved using the SVD.

8/9/2019 MT Lectures 11-20



The rank of the FRF matrices and eigenvalues areobtained.

Then the modal constants can be recovered from:

Modal Testing

8/9/2019 MT Lectures 11-20


(Lecture 18-1)



[email protected]

MDOF Modal Analysis in theTime Domain

The basic concept: Any Impulse R esponse

8/9/2019 MT Lectures 11-20



Function can be expressed by a series ofComplex Exponentials

The Complex Exponential Series contain the

eigenvalues and eigenvectors information. The IRF is obtained by taking inverse Fourier

transform of the measured FRF.

( )2

2

1

1;)(r r r r

N

r

t s

jk r jk

ise At h r ζ ζ ω −+−==

∑=

Complex Exponential Method∗

+=⇒ jk r N

jk r A A

8/9/2019 MT Lectures 11-20



∑

∑

∑

=

=

∗=

=⇒

−=

−+

−⇒

N

r

t s

jk r jk

N

r r

jk r jk

r r r jk

r e At h IRF

si

Aor

sisiFRF

2

1

2

1

1

)(

)(:

)(

ω ω α

ω ω ω α

Complex Exponential Method(Single FRF)

==⇒= ∑∑∑ Δ N

l

r r

N t ls

r l

N t s

r V Ae Ahe At h r r

222

)(

8/9/2019 MT Lectures 11-20



⎪⎪⎪⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪⎩

⎪⎪⎪

⎨

⎧

∑∑∑===

N

q

N

qq

N

N

q

r r r

A

A

A

V V V

V V V

V V V

h

hh

h

2

2

1

221

2

2

2

2

2

1

221

2

1

0

111

111

M

M

LL

MLLMM

KK

LL

LL

M


⎪⎫

⎪⎧

⎥⎤

⎢⎡

⎪⎫

⎪⎧

⎪⎫

⎪⎧

⎪⎫

⎪⎧

T T

Ah1000 111

LL

LL β β

8/9/2019 MT Lectures 11-20



⎪⎪⎪

⎭

⎪⎪⎬

⎪⎪⎪

⎩

⎪⎪⎨

⎥

⎥⎥⎥⎥

⎦⎢

⎢⎢⎢⎢

⎣⎪⎪⎪

⎭

⎪⎪⎬

⎪⎪⎪

⎩

⎪⎪⎨=

⎪⎪⎪

⎭

⎪⎪⎬

⎪⎪⎪

⎩

⎪⎪⎨

⎪⎪⎪

⎭

⎪⎪⎬

⎪⎪⎪

⎩

⎪⎪⎨

N q N

qq

N

N

qqq A

A

V V V

V V V V V V

h

hh

2

2

221

2

2

2

2

2

1

221

2

1

2

1

2

1

M

M

LL

MLLMM

KK

LL

MMM

β

β β

β

β β

∑ ∑∑ = ==⎟⎟ ⎠

⎞⎜⎜⎝

⎛ =

N

j

q

i

i ji j

q

i

ii V Ah

2

1 00 β β


The are selected to be coefficients of thei β

8/9/2019 MT Lectures 11-20



polynomial:

N

N

i

ii

N

i

ii

N

i

i

ji N

j

q

i

i

ji j

q

i

ii

q

q

hh

h

V V Ah N qSet

V V V

2

12

0

2

0

2

02

1 00

2

210

.0

02:

.0

−=

⎪⎪⎩

⎪⎪⎨

⎧

=

=⇒⎟⎟ ⎠

⎞⎜⎜⎝

⎛ =⇒=

=++++

∑

∑∑∑ ∑∑

−

=

=

=

= ==

β

β

β β β

β β β β L


−=−12 N

8/9/2019 MT Lectures 11-20



⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−

+

−−+−

−

=∑

14

12

2

12

1

0

2412212

2321

12210

20

N

N

N

N N N N N

N

N

N i

ii

h

h

h

hhhh

hhhh

hhhh

hh

M

M

M

M

L

MLMMM

MLMMM

L

L

β

β

β

β


The values and A i are obtained

t s

ir eV

Δ=

8/9/2019 MT Lectures 11-20



from: .02

2

2

210 =++++ N

N V V V β β β β L

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥

⎥⎥⎥⎥

⎦

⎤

⎢⎢

⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−−−− N

N

N

N N

N

N

N A

A

A

V V V

V V V

V V V

h

h

h

h

2

2

1

12

2

12

2

12

1

2

2

2

2

2

1

221

12

2

1

0 111

M

M

LLMLLMM

KK

LL

LL

M


Implementation Procedure:

8/9/2019 MT Lectures 11-20



Order of modal model is selected, Modal model is identified using the defined

steps in previous slides,

FRF is regenerated from modal informationand compared with the measured FRF

The procedure repeated using another

order for the modal model until stableresults are obtained.

Stabilization Diagram

8/9/2019 MT Lectures 11-20



Global Analysis in Time Domain(Ibrahim Time Domain Method)

The basic concept is to obtain a unique

8/9/2019 MT Lectures 11-20



set of modal parameters from a set ofvibration measurements:

Scaled (mass normalized) mode shapeswhen the force is known,

Un-scaled mode shapes when the force is

not measured.

Ibrahim Time Domain Method

=m

t s

ii

r et x2

)( ψ

8/9/2019 MT Lectures 11-20



∑=r ir i1

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

×

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

qmm

q

q

t st s

t st s

t st s

mnnn

m

m

qnnn

q

q

ee

ee

ee

t xt xt x

t xt xt x

t xt xt x

212

212

111

2,21

2,22221

2,11211

21

22212

12111

)()()(

)()()(

)()()(

LL

MLLM

LL

LL

L

MLMM

L

L

L

MLMML

L

ψ ψ ψ

ψ ψ ψ

ψ ψ ψ

[ ] [ ] [ ]Λ×Ψ= X


A 2nd set of eqns:m

l

2

)()(

8/9/2019 MT Lectures 11-20



( ) ∑∑

∑

==

Δ

=

Δ+

==

=Δ+

m

r

t sir

m

r

t st sir

r

t t sir li

lr lr r

lr

eee

et t x

2

1

2

1

1

)(

ˆ

)(

ψ ψ

ψ

[ ]Λ×Ψ= ˆˆ X


[ ] [ ]Ψ=Ψ× A ˆ

8/9/2019 MT Lectures 11-20



[ ] [ ][ ] [ ] [ ]

[ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ]+×=

=×⇒⎭⎬

⎫

Λ×Ψ=

Λ×Ψ=

X X A

X X A X

X

ˆ

ˆˆˆ


[ ]{ } { }t sr

A Δ

8/9/2019 MT Lectures 11-20



[ ]{ } { }r r e A ψ ψ = Eigenvectors of matrix [A] are the mode

shapes,

The natural frequencies and damping ratiosare obtained from eigenvalues of [A].

Modal Testin

(MDOF Modal Analysis in the Time Domain)

8/9/2019 MT Lectures 11-20


(MDOF Modal Analysis in the Time Domain)

School of Mechanical Engineering


[email protected]

Time Domain The basic concept: Any Impulse R esponse

Function can be expressed by a series of

8/9/2019 MT Lectures 11-20


Function can be expressed by a series ofComplex Exponentials

22

1;)( r r r r

N t s

jk r jk ise At h r

The Complex Exponential Series contain the.

The IRF is obtained by taking inverse Fourier

Dr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods

.

Complex Exponential

M t

8/9/2019 MT Lectures 11-20


Met o


Modal Parameter ExtractionMethods

Complex Exponential Method

k r N

k r A A

jk sisi

8/9/2019 MT Lectures 11-20


r r r jk sisi1

N jk r A

or

2

r r si1

t s jk r jk

r e At h IRF )(


r 1

8/9/2019 MT Lectures 11-20


(Single FRF)T T

AVVVh

1

1

0

1

0

1

0

8/9/2019 MT Lectures 11-20


N AV V V h 2221111

N 221222

N

q

N

qqqqq V V V h

2221

i

ji jii V Ah


8/9/2019 MT Lectures 11-20


8/9/2019 MT Lectures 11-20


(Single FRF) The values and A i are obtained

t s

ir eV

from: 022 N VVV

8/9/2019 MT Lectures 11-20


from: .02210 N

N V V V

Ah 10 111

N

N

V V V h

2

2

2

2

2

2

1

221

2

1

N

N

N

N N

N AV V V h 2

12

2

12

2

12

112


8/9/2019 MT Lectures 11-20


he Least S uaresComplex Exponential

Met o

8/9/2019 MT Lectures 11-20


Met o



Exponential Method (LSCE) The LSCE is the extension of CE to a

global procedure

8/9/2019 MT Lectures 11-20


global procedure.’

using SIMO method.

The coefficients β that provide thesolution of characteristic ol nomial are

global quantities.


Exponential Method (LSCE) N N hhhhh 2012210

One Typical IRF

N N

hhor

hhhhh

1212321

8/9/2019 MT Lectures 11-20


qq hhor ,

N N N N N N hhhhh

14122412212

hh 11

Extending to all measured IRFs

G

T

GG

T

GGG hhhhhhor

hh

122

,


q p

he Pol R eferenceComplex Exponential

Met o PR E

8/9/2019 MT Lectures 11-20


Met o PR E



Exponential Method (PRCE) Constitutes the extension of LSCE to

MIMO.

8/9/2019 MT Lectures 11-20


MIMO.

analyzing dynamics of a structure.

MIMO test method overcomes theroblem of not excitin some modes as

usually happens in SIMO.


Exponential Method (PRCE)Considering q input reference points:

N

t s jr jr e At h

2

11 )(

N

t s jr jr e At h

2

11 )(

8/9/2019 MT Lectures 11-20


r 1

)( r 1

)(

N

t s

jr jr e At h

2

22 )( jlr klr jk r

r jr r jr

AW A N

t s

jr r jr e AW t h

2

1212 )(

N t sr

2

lr

kr klr W

N t sr

2

r

jqr jq

1 r

jr qr jq

1

11


Modal Participation factor

Exponential Method (PRCE)

2

11 )( N

t s

jr j e At h r

1

2

1212 )()(j

t

j

N

t s jr r j AeW t he AW t h r

8/9/2019 MT Lectures 11-20


11

)( j jr

2

11)( N

t s

jr qr jq e AW t h r

111

1

1 00111)( jt s

j

r

Aet h

1221221221122

j N j e


12121211200 j N

sq N qq jq et N

Exponential Method (PRCE)

AV W t h )( 1

8/9/2019 MT Lectures 11-20


VWt )( 1e,

j j AV W t Lh )( 1

N q LV W V W V W L L 2,02

210


Exponential Method (PRCE))0( AW h

111 )( j j AV W t h

8/9/2019 MT Lectures 11-20


1

2

22 )( j j AV W t h

1)( j

L

L j L AV W t Lh

L L

jk

k jk AV W t k h 1)(


k k 0 0

Exponential Method (PRCE) ,0)( L

L

jk I t k h

1

)()( L

j jk t Lht k h

8/9/2019 MT Lectures 11-20


t j j j t N ht hh

))1(()()0(

t j j j

L

110

t j j j

t j j j

t N Lht Lht Lh

))1(())1(()(


j jT

Exponential Method (PRCE) j jT

Considerin for each res onse location =1 :

8/9/2019 MT Lectures 11-20


Considerin for each res onse location =1 … :

2121 p pT hhhhhh B

1

T T

T T T B

T T T T T

Knowing the coefficient matrix [B], we must now determine [V]


8/9/2019 MT Lectures 11-20


Exponential Method (PRCE)0

Lk

W V 0 k

L L

12

8/9/2019 MT Lectures 11-20


r r

0 r z W

1 0

2

2 1

r r r

r r r

z z

z V W V z

0 0 1 1 z z

11 2

L L r r r L z V W V z

1 1 1 L L r L z z


1

L

L r r r L z V W V z

8/9/2019 MT Lectures 11-20


Exponential Method (PRCE) Lk AV W t k h j

k

j ,,1,0,1

t k h j1

t k h

j

j

2

10 j

V W W

t hh

8/9/2019 MT Lectures 11-20


VW

t k h jq11 jV j j

L

AW H or A

j

jV j1

The residue calculation is repeated for


a measure po n s, = , ,…, .

Exponential Method (PRCE) The method provide more accurate modal

representation of the structure.

It can determine multiple roots or closely

8/9/2019 MT Lectures 11-20


spaced modes.

h r min : Sensitive to nonlinearities and any lack of

,

Some difficulties in analyzing structures.


Global Analysis in Time

Doma n

8/9/2019 MT Lectures 11-20


Method)



(Ibrahim Time Domain Method) The basic concept is to obtain a unique

set of modal parameters from a set of

8/9/2019 MT Lectures 11-20


Scaled (mass normalized) mode shapes

, Un-scaled mode shapes when the force is

.


8/9/2019 MT Lectures 11-20


8/9/2019 MT Lectures 11-20


Ibrahim Time Domain Methodˆ

8/9/2019 MT Lectures 11-20


X X ˆ

ˆˆ X

X X A ˆDr H Ahmadian, Modal Testing Lab, IUSTModal Parameter Extraction Methods


r r r

e

8/9/2019 MT Lectures 11-20


e Eigenvectors o matrix A are t e mo e

shapes,

he natural frequencies and damping ratiosare obtained from eigenvalues of [A].



8/9/2019 MT Lectures 11-20




[email protected]

8/9/2019 MT Lectures 11-20


Derivation of Mathematical

Models Modal Models

Requirements to construct Modal Models

Refinement of Modal Model Conversion to real modes

8/9/2019 MT Lectures 11-20


Dr H Ahmadian,Modal Testing Lab,IUSTDerivation of Mathematical Models

Compatibility of DOFs Reduction

Expansion

Response Models FRF

Transmissibility Base Excitation

Requirement to construct

Modal Models Minimum requirements

One column in case of fixed excitation or

One row when response is measured at a fixed point.

8/9/2019 MT Lectures 11-20



8/9/2019 MT Lectures 11-20


Requirement to construct

Modal Models Several additional elements of FRF or even columns

are measured to:

Replace poor data, To provide checks

M d h t b i d

8/9/2019 MT Lectures 11-20



Modes have not been missed

Refinement of Modal Models Complex to real conversion:

Taking the modulus of each element andassigning a phase of 0 or 180.

Finding a real mode with maximum projection to

8/9/2019 MT Lectures 11-20



g p jthe measured one:

Multi point excitation (Asher’s method)

C R

C T

R

φ φ

φ φ max

Compatibility of DOFs Employment of the measured modes in

updating/modification of analyticalmodels requires the compatibility ofDOFs

8/9/2019 MT Lectures 11-20



DOFs.

There are two approaches incompatibility excursive:

Analytical model reduction

Expansion of measured modes

Reduction of Analytical Model

(Guyan Reduction)

112

1

222

1

2

1

2212

1211

0 xK K x

f

x

x

K K

K K

T

T

−=⎭

⎬⎫

⎩

⎨⎧

=

⎭

⎬⎫

⎩

⎨⎧⎥

⎦

⎤⎢

⎣

⎡

−

8/9/2019 MT Lectures 11-20



{ } [ ]{ }

{ } { }112212

1211

1

2

1

1

12

1

222

1

,

f xT K K

K K

T

xT x

x xK K

I

x

x

T

T

T

=⎥⎦

⎤

⎢⎣

⎡

=⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡

−=⎭⎬⎫

⎩⎨⎧

−

Dynamic Model Reduction

( ) ( )

.0

112

2

12

1

22

2

222

2

1

2212

12112

2212

1211

−−−=

=

⎭

⎬⎫

⎩

⎨⎧

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡

−φωωφ

φ

φ ω

MKMK

M M

M M

K K

K K

T T

T T

8/9/2019 MT Lectures 11-20



( ) ( )

( ) ( ) { } [ ]{ }

{ } .0

,

1

2212

12112

2212

1211

1

2

11

12

2

12

1

22

2

222

1

1121222222

=⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

⎥⎦

⎤

⎢⎣

⎡

−⎥⎦

⎤

⎢⎣

⎡

=⎭⎬⎫

⎩⎨⎧⎥

⎦⎤⎢

⎣⎡

−−−=

⎭⎬⎫

⎩⎨⎧ −

φ ω

φ φ φ φ

ω ω φ φ

φ ω ω φ

T M M

M M

K K

K K

T

T M K M K

I

M K M K

T T

T

T T

Expansion of Models In order to compare analytical model

with the measured modal data on mayexpand the measured data by:

Geometric interpolation using spline

8/9/2019 MT Lectures 11-20



Geometric interpolation using spline

functions Using analytical model spatial model

Using analytical model modal model

Expansion in Spatial Domain

.02

1

2212

12112

2212

1211

=⎭⎬

⎫

⎩⎨

⎧

⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

⎥⎦

⎤

⎢⎣

⎡

−⎥⎦

⎤

⎢⎣

⎡

φ

φ

ω M M

M M

K K

K K T T

8/9/2019 MT Lectures 11-20



( ) ( ) 112

2

12

1

22

2

222 φ ω ω φ T T M K M K −−−=

−

Expansion in Modal Domain

:

0

T

Sol

RΦ=Φ Assuming MassMatrix is correct

8/9/2019 MT Lectures 11-20



,

~~.:~~min

0

0

T T T

T R

VU RV U

I R Rst R

=⇒Σ=ΦΦ

=Φ−Φ

Response Model Frequency response functions

Transmissibilities

[ ] [ ]( ) [ ]T r H Φ−Φ= −122 ω λ

8/9/2019 MT Lectures 11-20



Transmissibilities

( ) ( ) ( )

( )ω

ω ω ω

ω

ω

ki

ji

jk it i

k

t i

j

jk H

H T

e X

e X T == ,

Transmissibility Plots

8/9/2019 MT Lectures 11-20



Response Model The amplitude’s peaks

of the transmissibilities

do not correspond withthe resonantfrequencies.

T i ibiliti ∑∑ −==

irkr

r r

ir jr

ki

ji

jk iH

H T

22

)(

)()(

φ φ

ω ω

φ φ

ω

ω ω

8/9/2019 MT Lectures 11-20



Transmissibilities cross

each other at theresonant frequencies(becomes independentof the location of theinput)

∑−

r r

ir kr ki H 22

)(

ω ω

φφω

kr

jr

jk i r T φ

φ

ω ω ω ≈

≈)(

Base Excitation An application area

of transmissibility.

Input is measuredas response at the

8/9/2019 MT Lectures 11-20



drive point.

{ } { }

⎪

⎪

⎭

⎪⎪

⎬

⎫

⎪

⎪

⎩

⎪⎪

⎨

⎧

+=

1

11

Mref rel x x x

Base Excitation

[ ]{ } [ ]{ } [ ]{ } { } .

1

1

1

, gg M x xK x M ref relrel

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

=−=+ M&&&&

8/9/2019 MT Lectures 11-20



( )[ ] { } { }( ) [ ]{ }

{ } { }( )

( )[ ][ ]{ }.

,

1

2

21

g M H x

g x X

or

g M xg x X H

ref

ref

ref ref

ω ω

ω ω

=−

=−

⎪⎭⎪⎩−

Base Excitation

{ } { }( )[ ]( ) [ ]{ }( )

{ } { }( )f

ref

ref

gxX

g M H

x

g x X

−

=−2

,ω

ω

8/9/2019 MT Lectures 11-20



{ }

{ } { }( )

{ } [ ]{ }

( ) j j r r

jr ir

i

ref

ref

uQ

g M u x

g x X Q

∑∑ −=

==

22

2,,

ω ω

φ φ

ω

ω

Spatial Models

[ ] [ ] [ ] 1−−

ΦΦ=T M

8/9/2019 MT Lectures 11-20



[ ] [ ] [ ][ ] [ ] [ ][ ] 1−− ΦΦ= r

T K λ


8/9/2019 MT Lectures 11-20




[email protected]

Derivation of MathematicalModels

Introduction

Equation Error Method (Sec. 6.3.6 page 456)

Identification of Rod FE Model

Parameter Identification

8/9/2019 MT Lectures 11-20


Dr H Ahmadian, Modal Testing Lab, IUSTDerivation of Mathematical Models

Solution of Over-determined set of Equations

Solution of Under-determined set ofEquations

Error Analysis

Introduction

Construction of Spatial Model from modaldata:

111 ,, −−−−−− ΓΦΦ=ΦΦ=ΛΦΦ= T T T C M K

Modal model must be complete:

8/9/2019 MT Lectures 11-20



p

All modes must be present Mode shapes are measured in all DOF’s

Measurement of complete Modal Model is

impractical.

Introduction

Alternative methods are required to constructthe spatial model from

incomplete and

noisy measured modes.

Th diffi lt ith i l t i d

8/9/2019 MT Lectures 11-20



The difficulty with incompleteness is removed

by reducing the number of unknowns inspatial model.

The noise effects are removed by averaging.

Equation Error Method

We have some information regardingthe spatial model format: Symmetry

Pattern of zeros

8/9/2019 MT Lectures 11-20



…

We may incorporate these informationinto the identification procedure andreconstruct the spatial model.

8/9/2019 MT Lectures 11-20


Rearrangement Example:

:

,00

0

2

1

2

12

22

221

⎫⎧

=

⎭

⎬⎫

⎩

⎨⎧

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ ⎥

⎦

⎤⎢

⎣

⎡−⎥

⎦

⎤⎢

⎣

⎡

−

−+

k

or

m

m

k k

k k k r

φ

φ ω

8/9/2019 MT Lectures 11-20



.000

0

2

1

2

1

2

2

12

1

2

211 =

⎪

⎪

⎭

⎪⎪⎬

⎫

⎪

⎪

⎩

⎪⎪⎨

⎧

⎥⎦

⎤⎢⎣

⎡

−−

−−

m

m

k

k

r

r

φ ω φ φ

φ ω φ φ φ

Identification of A Rod FEModel

Consider a fixed-free rod with n elements.

The mass and stiffness matrices are:

3322

221

k k k k

k k k

⎥⎥⎤

⎢⎢⎡

−+−

−+

8/9/2019 MT Lectures 11-20



),,,,( 121

11

3322

nn

nn

nnnn

mmmmdiag M

k k

k k k k

K

−

−−

=

⎥⎥⎥⎥

⎥

⎦⎢⎢⎢⎢

⎢

⎣ −

−+−=

L

LLL

8/9/2019 MT Lectures 11-20



( )⎧ −

=

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡

−−

−−

−

−

−

1

,1,,

,1,,.0

nsnsnr

n

n

nssnsns

nr r nr nr

m

k

φ φ φ

φ λ φ φ

φ λ φ φ

8/9/2019 MT Lectures 11-20



( )

( )

⎪⎪⎩

⎪⎪⎨

⎧

−=

−=⇒≠≠

−

−

1,,

,

1,,,

1,,,

0,0

nr nr

nr r

n

n

nr nr ns

nsnsnr

r s

nn

m

k mk

φ φ

φ λ

φ φ φ

φφφ

λ λ


From other rows one obtains:

n

n

nnn

n

nn m

m

m

m

m

m

m

k

m

k

m

k 121121 ,,,,,,, −− KK

Using total mass information mm is obtained:

8/9/2019 MT Lectures 11-20



Using total mass information m m is obtained:

⎟⎟

⎠

⎞⎜⎜

⎝

⎛ += ∑

−

=

1

1

1n

l n

lntotal

m

mmm


Only two modes and one naturalfrequency are required to construct themass and stiffness matrices.

More details can be found in:

8/9/2019 MT Lectures 11-20



GML Gladwell, YM Ram, ”ConstructingFinite Element Model of a Vibrating Rod”,Journal of Sound and Vibration, 169,229-

237,1994.


In a general case the mass and stiffnessmatrices are parameterized and areobtained by rearranging:

Equation of motion in modal domain

8/9/2019 MT Lectures 11-20



Orthogonality requirements, etc.

8/9/2019 MT Lectures 11-20



=

Λ=ΛΦΦ=

=

−

mmmk k k x x

I mbb A A

b Ax

t arrangemen

nn

xxtotal R

),,,,,,,(

),,,(),,,(

:Re

2121 KK

K

8/9/2019 MT Lectures 11-20



⎪

⎩

⎪⎨⎧

→<

→>=→=

⇒×⇒

−

ed er Over nm

ed er Under nm

b A xnm

mnrank full A

mindet

mindet)(

1

8/9/2019 MT Lectures 11-20


Solution of Over-determinedset of Equations

[ ] [ ]Solution

E E Assumeb Ax

nmb Ax

T

ij ji

minmin

.,0

,

=⇒

==⇒

=−

<=

εεε

σ δ εε ε ε

8/9/2019 MT Lectures 11-20



( ) ( ) b A A A xb A Ax A x

bbb A x Ax A x

Solution

T T T T T

T T T T T T

1

22

2

minmin

−

=⇒−=∂

∂

+−=

=⇒

ε ε

ε ε

ε ε ε

8/9/2019 MT Lectures 11-20


Example:

The parameters to beupdated are the 10

stiffness and 6 masses The measured data

consists of the 1stthree natural

8/9/2019 MT Lectures 11-20



three naturalfrequencies and modeshapes (added withuniformly distributed

random noise)

Example:

Eigenvalue equations arearrangment:

31 equations ( 3*6 equations for each eigenvector term, 2*6symmetric orthogonality equations, and 1 total mass equation)

16 parameters and

The terms in A and b contain noisy data.

{ },,,,,,,,: mmmkkkxParameters = KK

8/9/2019 MT Lectures 11-20



{ }

.:

.,,0:

,,,,,,,,:6211021

m M Extras

K I M M K EOM

mmmk k k xParameters

R

T

R

T T

=

Λ=ΦΦ=ΦΦ=ΦΛ−Φ

φ φ

KK

Example

-0.010.20.990.10.21-632035-41499141008100610008151812661041S/N=100

0.10.110.10.2110001000700050001000100010000150012501000Exact

m6

m5

m4

m3

m2

m1

k 10

k 9

k 8

k 7

k 6

k 5

k 4

k 3

k 2

k 1

8/9/2019 MT Lectures 11-20



0.28-0.020.960.10.210.972323-8321352-102410111160984415021243954S/N=20

8/9/2019 MT Lectures 11-20


mt lectures 11-20

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