Modal Testing Lectures 1-10

8/11/2019 Modal Testing Lectures 1-10

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Modal Testing(Lecture 1)

Dr. Hamid AhmadianSchool of Mechanical Engineering

Iran University of Science and Technology

[email protected]





Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing

Introduction to Modal Testing

Experimental Structural Dynamics To understand and to control the many vibration

phenomenon in practice Structural integrity (Turbine blades- Suspension Bridges)

Performance ( malfunction, disturbance, discomfort)




Introduction to Modal Testing (continued)

Necessities for experimentalobservations

Nature and extend of vibration in operation

Verifying theoretical models

Material properties under dynamic loading

(damping capacity, friction,…)





Test types corresponding to objectives: Operational Force/Response measurements

Response measurement of PZL Mielec Skytruck ModeShapes (3.17 Hz, 1.62 %), (8.39 Hz, 1.93 %)





Modal Testing in acontrolled environment/Resonance Testing/Mechanical ImpedanceMethod

Testing a component ora structure with theobjective of obtainingmathematical model of

dynamical/vibrationbehavior

Structural Analysis ofULTRA Mirror





Milestones in the development: Kennedy and Pancu (1947)

Natural frequencies and damping of aircrafts

Bishop and Gladwell (1962)

Theory of resonance testing

ISMA (bi-annual since 1975) IMAC (annual since 1982)




Applications of Modal Testing

Model Validation/Correlation: Producing major test modes validates the model

Natural frequencies

Mode shapes

Damping information are not available in FE models




Applications of Modal Testing (continued)

Model Updating Correlation of experimental/analytical

model

Adjust/correct the analytical model

Optimization procedures are used for

updating.





Component ModelIdentification

Substructure process

The component modelis incorporated into thestructural assembly





Force Determination Knowledge of dynamic force is required

Direct force measurement is not possible

Measurement of response + Analytical Modelresults the external force

[ ] [ ]{ } { } f x M K =−

2

ω




Philosophy of Modal Testing Integration of three components:

Theory of vibration Accurate vibration measurement Realistic and detailed data analysis

Examples: Quality and suitability of data for process Excitation type Understanding of forms and trends of plots Choice of curve fitting Averaging




Summary of Theory (SDOF)




Summary of Theory (MDOF)




Summary of Theory

Definition of FRF:[ ] [ ] [ ]( )

.)(

)(

)(

)(

122

12

∑=

−

−==

+−=

N

r r

kr jr

k

j

jk f

x

h

Di M K H

ω ω

φ φ

ω

ω

ω

ω ω

Curve-fitting the

measured FRF: Modal Model is obtained

Spatial Model is obtained




Summary of Measurement

Methods

Basic measurement system: Single point excitation




Summary of Modal Analysis

Processes

Analysis of measured FRF data Appropriate type of model (SDOF,MDOF,…)

Appropriate parameters for chosen model




Review of Test Procedures

and Levels

The procedure consists of: FRF measurement

Curve-Fitting

Construct the required model

Different level of details and accuracy in

above procedure is required dependingon the application.




Review of Test Procedures

and Levels

Levels according to Dynamic Testing Agency:Updating

Out of rangeresidues

Usable forvalidation

Mode ShapesDamping

ratioNaturalFreq

Level

0

Only in fewpoints

1

2

3

4




Text Books

Ewins, D.J. , 2000, “Modal Testing; theory,practice and application”, 2nd edition,Research studies press Ltd.

McConnell, K.G., 1995, “ Vibration testing;theory and practice”, John Wiley & Sons.

Maia, et. al. , 1997, “Theoretical andExperimental Modal Analysis”, Researchstudies press Ltd.




Evaluation Scheme

Home Works (20%) Mid-term Exam (20%)

Course Project (30%) Final Exam (30%)






[email protected]



Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis

Theoretical Basis

Analysis of weakly nonlinear structures Approximate analysis of nonlinear

structures

Cubic stiffness nonlinearity

Coulomb friction nonlinearity

Other nonlinearities and otherdescriptions




Analysis of weakly nonlinear

structures

The whole bases of modal testing assumeslinearity:

Response linearly related to the excitation

Response to simultaneous application of severalforces can be obtained by superposition ofresponses to individual forces

An introduction to characteristics of weaklynonlinear systems is given to detect if anynonlinearity is involved during modal test.





)sin()3sin(4

1)sin(

4

3

)sin()cos()sin(

)sin()(sin)sin()cos()sin(

)sin()(

)sin(

3

3

2

33

3

2

33

φ ω ω ω

ω ω ω ω ω

φ ω ω ω ω ω ω ω

ω

φ ω

−=⎟ ⎠

⎞⎜⎝

⎛ −

+++−⇒

−=+++−⇒

=⇒−=+++

t F t t X k

t kX t X ct X m

t F t X k t kX t X ct X m

t X t x

t F xk kx xc xm &&&





⎪⎩

⎪⎨⎧

−=

=++−⇒

−

=⎟ ⎠

⎞⎜⎝

⎛ −

+++−

)sin(

)cos(4

3

)sin()cos()cos()sin(

)3sin(4

1)sin(

4

3

)sin()cos()sin(

3

3

2

3

3

2

φ ω

φ ω

φ ω φ ω

ω ω

ω ω ω ω ω

F X c

F X k kX X m

t F t F

t t X k

t kX t X ct X m







Softening effect Hardening effect






Softening-stiffness effect













( )

⎟

⎠

⎞⎜

⎝

⎛ ++−

=

=

Δ

=⇒=Δ

+=

∫

X

ccimk

F X

X

c

dt t x

E c X c E

t x

t x

ct xct f

F

F eqF

F d

πω

ω ω

πω ω π

41

44

)(

)(

)()(

2

/2

0

2&

&

&

&





X increasing

⎟ ⎠ ⎞⎜

⎝ ⎛ ++−

=

X

ccimk F

X

F

πω ω ω 4

1

2

Othe nonlinea ities and othe




Other nonlinearities and other

descriptions

Backlash Bilinear Stiffness

Microslip friction damping Quadratic (and other power law

damping)

…..






[email protected]




MODAL ANALYSIS THEORY Understanding of how the structural

parameters of mass, damping, and stiffnessrelate to

the impulse response function (time domain), the frequency response function (Fourier, or

frequency domain), and

the transfer function (Laplace domain) for single and multiple degree of freedom

systems.







Frequency Domain:




Frequency Domain:

Frequency Response Function




Alternative Forms of FRF Receptance

Inverse is “DynamicStiffness”

Mobility Inverse is “Dynamic

Impedance”

Inertance Inverse is “ Apparent

mass”

)(

)(

ω

ω

F

X

)()(

)()(

ω

ω ω

ω

ω

F

X iF

V =

)()(

)()( 2

ω

ω ω

ω

ω

F

X

F

A −=




Graphical Display of FRF






Stiffness and Mass Lines




Reciprocal Plots The “inverse” or

“reciprocal” plots

Real part

Imaginary part ⎪

⎪

⎩

⎪⎪⎨

⎧

=

−=

⇒ω

ω

ω

ω

ω

ω

c X

F

mk X

F

))(

)(Im(

))(

)(

Re(

2




Nyquist Plot For viscous damping the Mobility plot is a

circle.

For structural damping the Receptance andInertance plots are circles.




3D FRF Plot (SDOF)




Properties of SDOF FRF Plots Nyquist Mobility for viscose damping

( )( )

2

22222

222222

222

2

222

2

2

2

1

)()(4

)()(

)Im(,

2

1)Re(

)()(

)(

)Im()()()Re(

)(

⎟ ⎠

⎞⎜⎝

⎛ =

+−

+−=+

=⎟

⎠

⎞⎜

⎝

⎛ −=

+−

−

=+−=

+−=

ccmk c

cmk V U

Y V c

Y U

cmk

mk

Y cmk

c

Y

icmk

iY

ω ω

ω ω

ω ω

ω ω

ω ω

ω

ω ω

ω

ω




Properties of SDOF FRF Plots Nyquist Receptance for structural damping

( )( )

( )( ) ( )

22

2

222222

2

222

2

2

2

1

2

1

,

1)(

⎟ ⎠

⎞⎜⎝

⎛ =⎟

⎠

⎞⎜⎝

⎛ ++

+−=

+−

−=

+−

−−=

−+=

d d V U

d mk

d V

d mk

mk U

d mk

id mk

mid k H

ω ω

ω

ω

ω

ω

ω







Iran University of Science and [email protected]




Theoretical Basis Undamped MDOF Systems

MDOF Systems with ProportionalDamping

MDOF Systems with General StructuralDamping

General Force Vector

Undamped Normal Mode




Undamped MDOF Systems The equation of motion:

The modal model:

The orthogonality:

Forced response solution:

[ ]{ } [ ]{ } { })()()( t f t xK t x M =+&&

[ ] ),,,(, 22

2

2

1 N diag ω ω ω K=ΓΦ

[ ] [ ][ ] [ ] [ ] [ ][ ] [ ]., Γ=ΦΦ=ΦΦ K I M T T

[ ] [ ]{ } { } t it ieF e X M K

ω ω ω =− 2

{ } [ ] [ ]( ) { } { } [ ]{ }F X F M K X )(

12

ω α ω =⇒−=

−

Undamped MDOF Systems




Undamped MDOF Systems(continued)

Response Model

[ ] [ ]( ) [ ]

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

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

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

T

T

T

T T

I

I

I

M K

M K

Φ−ΓΦ=

Φ−ΓΦ=

ΦΦ=−ΓΦΦ=Φ−Φ

=−

−

−−−

−

−

−

12

121

12

12

12

)(

)(

)(

)(

)(

ω ω α

ω ω α

ω α ω

ω α ω

ω α ω

Undamped MDOF Systems




Undamped MDOF Systems(continued)

The receptance matrix is symmetric.

∑∑ == −=−=

===

N

r r

jk r N

r r

kr jr

jk

j

k

kjk

j

jk

A

F

X

F

X

122

122)(

,

ω ω ω ω

φ φ

ω α

α α Modal Constant/Modal Residue

Single Input




Example:

[ ] [ ]

[ ] [ ]

.64.2118

62.1

62

05

54

5.0)(

11

11

2

1

62

54

/2.18.0

8.02.1

1

1

42

2

2211

2

ω ω

ω

ω ω ω α

ω

+−

−=

−+

−=

⎥⎦

⎤

⎢⎣

⎡

−

=Φ

⎥⎦

⎤

⎢⎣

⎡=

⎥⎦

⎤⎢⎣

⎡

−

−=⎥

⎦

⎤⎢⎣

⎡=

ee

e

ee

e

e

m MN K kg M

r




.64.2118

62.1

62

05

54

5.0)(

42

2

2211ω ω

ω

ω ω ω α

+−

−=

−+

−=

ee

e

ee

Poles

Example (continued):

Zero




Example (continued):

.64.2118

48

62

05

54

5.0)(

422212ω ω ω ω

ω α +−

=−

−−

=ee

e

ee

MDOF Systems with




Proportional Damping A proportionally damped matrix is

diagonalized by normal modes of thecorresponding undamped system

[ ] [ ][ ] ),,,( 21 N

T

d d d diag D L=ΦΦ Special cases:

[ ] [ ][ ] [ ]

[ ] [ ] [ ].

,,

M K D

M DK D

δ β

δ β

+=

==

MDOF Systems with Structurally




y y

Proportional Damping

Response Model

[ ] [ ] [ ]( ) [ ]

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

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

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

∑=

−

−−−

−

−

−

−+=

Φ−+Φ=

Φ−+Φ=

ΦΦ=−+

ΦΦ=Φ−+Φ

=−+

N

r r r

kr jr

jk

T r r

r r

T

T r r

T T

i

I i

I i

I i

M DiK

M DiK

1

222

1222

12221

1222

12

12

)1(

)(

)1()(

)1()(

)()1(

)(

)(

ω η ω

φ φ ω α

ω η ω ω α

ω η ω ω α

ω α ω η ω

ω α ω

ω α ω

Real Residue

Complex Pole

MDOF Systems with Viscously




Proportional Damping Response Model

[ ] [ ] [ ]( ) [ ]

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

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

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

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

∑=

−

−−−

−

−

−

+−

=

Φ−+Φ=

Φ−+Φ=

ΦΦ=−+

ΦΦ=Φ−+Φ

=−+

N

r r r r

kr jr

jk

T r r r

r r r

T

T

r r r

T T

I i

I i

I i

M C iK

M C iK

1

22

122

1221

122

12

12

2

)(

2)(

2)(

)(2

)(

)(

ω ω ζ ω ω

φ φ ω α

ω ω ζ ω ω ω α

ω ω ζ ω ω ω α

ω α ω ω ζ ω ω

ω α ω ω

ω α ω ω






Example:

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−

=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=Γ

==

===

142.0493.0635.0

318.0782.0536.0

318.1218.0464.0

,

6698

3352

950

6,...,1,/31

5.1,1,5.0

1

321

Undamped

jm N ek

kgmkgmkgm

Model

j




Example:[ ] [ ]

[ ]

[ ]

⎥

⎥⎥

⎦

⎤

⎢

⎢⎢

⎣

⎡

−−

−

−

=Φ

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

+

+

+

=Γ

===−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−

−−

=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+=Γ

=

)1.3(142.0)3.1(492.0)0.1(636.0

)7.6(318.0)181(784.0)0.0(537.0

)181(318.1)173(217.0)5.5(463.0

,

)078.01(6690

)042.01(3354

)067.01(957

6,...,2,0.0,3.0Pr

142.0493.0635.0

318.0782.0536.0

318.1218.0464.0

,

6698

3352

950

)05.01(

05.0

Pr

11

ooo

ooo

ooo

i

i

i

jd k d oportional Non

i

K D

oportional

j

Almost real modes




Example:

[ ]

[ ] [ ]

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−

=Φ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+=Γ

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−

=Φ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=Γ

==

===

207.0752.0587.0

827.0215.0567.0

552.0602.0577.0

,

4124

3892

999

)05.01(

,05.0

Pr

207.0752.0587.0

827.0215.0567.0

552.0602.0577.0

,

4124

3892

999

6,...,1,/31

05.1,95.0,1

2

321

i

K D

oportional

Undamped

jm N ek

kgmkgmkgm

Model

j




Example:

[ ]

⎥

⎥⎥

⎦

⎤

⎢

⎢⎢

⎣

⎡

−

−

=Φ

⎥

⎥⎥

⎦

⎤

⎢

⎢⎢

⎣

⎡

+

+

+

=Γ

===

−

)50(560.0)12(848.0)2(588.0

)176(019.1)101(570.0)2(569.0

)40(685.0)162(851.0)4(578.0

,

)019.01(4067

)031.01(3942

)1.01(10066,...,2,0.0,3.0

Pr

11

ooo

ooo

ooo

i

i

i

jd k d

oportional Non

j

Heavily complex modes

MDOF Systems with General




Structural Damping[ ] [ ] [ ]( ) [ ]

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

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

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

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

∑=

−

−−−

−

−

−

−+=

Φ−+Φ=

Φ−+Φ=

ΦΦ=−+

ΦΦ=Φ−+Φ

=−+

N

r r r

kr jr

jk

T

r r

r r

T

T

r r

T T

i

I i

I i

I i

M DiK

M DiK

1222

1222

12221

1222

12

12

)1()(

)1()(

)1()(

)()1(

)(

)(

ω η ω

φ φ ω α

ω η ω ω α

ω η ω ω α

ω α ω η ω

ω α ω

ω α ω

Complex Residues

Complex Poles




General Force Vector In many situations

the system isexcited at severalpoints.




General Force Vector (continued)

The response is governed by:

The solution:

All forces have the same frequency butmay vary in magnitude and phase.

[ ] [ ]( ){ } { } t it ieF e X M iDK

ω ω ω =−+ 2

{ } { } { }{ }∑= −+

= N

r r r

r T r

i

F X

1222 )1( ω η ω

φ φ





The response vector is referred to:

Forced Vibration Mode

or Operating Deflection Shape (ODS)

When the excitation frequency is closeto the natural frequency:

ODS reflects the shape of nearby mode But not identical due to contributions of

other modes.





Damped system normal mode:

By carefully tuning the force vector theresponse can be controlled by a single

mode. The is attained if

Depending upon damping condition theforce vector entries may well be complex(they have different phases)

{ } { } rss

T

r F δ φ =




Undamped Normal Mode Special Case of interest:

Harmonic excitation of mono-phased forces

Same frequency

Same phase Magnitudes may vary

Is it possible to obtain mono-phasedresponse?





p(continued)

The real force response amplitudes:

Real and imaginary parts:

The 2nd equation is an eigen-value problem;its solutions leads to real

{ }

{ } { } )(ˆ)(

ˆ)(

θ ω

ω

−=

=t i

t i

e X t x

eF t f [ ] [ ]( ) t it i eF e X M iDK

ω ω ω ˆˆ2 =−+

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

[ ] [ ]( ) [ ]( ){ } { }0ˆ

cossin

ˆˆsincos

2

2

=+−

=+−

X D M K

F X D M K

θ θ ω

θ θ ω

F ˆ

N solutions





p(continued)

At a frequency that the phase lagbetween all forces and all responses is90 degree then

Results

Undamped normal modes Natural frequencies of undamped system

[ ] [ ]( ) [ ]( ) { }0ˆcossin2 =+− X D M K θ θ ω











Theoretical Basis General Force Vector


MDOF System with General Viscous

Damping

Force Response Solution/ General

Viscous Damping









The response is governed by:

All forces have the same frequency but

may vary in magnitude and phase.

The solution:

[ ] [ ]{ } { } t it ieF e X M iDK

ω ω ω =−+ 2

{ } { } { }{ }∑= −+

= N

r r r

r

T

r

iF X

1222 )1( ω η ω

φ φ







Damped system normal mode:

By carefully tuning the force vector theresponse can be controlled by a single

mode. This is attained if

Depending upon damping condition the

force vector entries may well be complex(they have different phases)

{ } { } rss

T

r F δ φ =




Undamped Normal Mode Special Case of interest:

Harmonic excitation of mono-phased forces

Same frequency

Same phase Magnitudes may vary

Is it possible to obtain mono-phasedresponse?





512 channel

37 Shakers





(continued)

The real force response amplitudes:

Real and imaginary parts:

The 2nd equation is an eigen-value problem;its solutions leads to real

{ }{ } { } )(ˆ)(

ˆ)(

θ ω

ω

−=

=t i

t i

e X t x

eF t f [ ] [ ]( ) t it i

eF e X M iDK ω θ ω ω ˆˆ )(2 =−+ −

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

[ ] [ ]( ) [ ]( ){ } { }0ˆ

cossin

ˆˆsincos

2

2

=+−

=+−

X D M K

F X D M K

θ θ ω

θ θ ω

F ˆ

N solutions

Undamped Normal Mode( i d)




(continued)

At a frequency that the phase lag

between all forces and all responses is90 degree then

Results

Undamped normal modes Natural frequencies of undamped system

[ ] [ ]( ) [ ]( ) { }[ ] [ ]( ){ } { }0ˆ

0ˆcossin2

2

=−⇒

=+−

X M K

X D M K

ω

θ θ ω

Undamped Normal Mode( ti d)




(continued)

The base for multi-

shaker test procedures. Modal Analysis of Large

Structures: MultipleExciter Systems By:M. Phil. K. Zaveri

MDOF System with General

Vi D i




Viscous Damping[ ]{ } [ ]{ } [ ]{ } { }

{ } { } { } { }

{ } [ ] [ ] [ ]( ) { }F C i M K X

e X t xeF t f

f xK xC x M M O E

t it i

12

)()(

...

−

+−=

=⇒=

=++⇒

ω ω

ω ω

&&&

Next the orthogonality properties of thesystem in 2N space is used for force responsesolution.

F R S l ti




Force Response Solution

{ } { } { }

{ } { }

).,,,(0

0,

0

00

0

0

0

0

0

0

.

00

0

0

221 N

T T

r r

sssdiagU M

K U I U

M

M C U

u M

K

M

M C ssolution Eigen

u M

K u

M

M C VibFree

f

x

x

M

K

x

x

M

M C EOM

L

&

&&&

&

=⎥⎦

⎤⎢⎣

⎡

−=⎥

⎦

⎤⎢⎣

⎡⇒

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ⎥⎦

⎤⎢⎣

⎡

−+⎥

⎦

⎤⎢⎣

⎡⇒−

=⎥

⎦

⎤⎢

⎣

⎡

−+⎥

⎦

⎤⎢

⎣

⎡⇒

⎭⎬

⎫

⎩⎨

⎧=

⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡

−+

⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡⇒

F R S l ti




Force Response Solution

*1

2

1

000

r

r

H

r N

r r

r

T

r N

r r

r

T

r

si

u

F

u

si

u

F

u

si

u

F

u

X i

X

−⎭⎬

⎫

⎩⎨

⎧

+−⎭⎬

⎫

⎩⎨

⎧

=−⎭⎬

⎫

⎩⎨

⎧

=⎭⎬⎫

⎩⎨⎧

∗

==∑∑

ω ω ω ω

The above simplification is due to the factthat eigen-values and eigen-vectors occur incomplex conjugate pairs.

F R S l ti




Force Response Solution Single point excitation:

∗

∗∗

= −+−= ∑ r

kr jr N

r r

kr jr

jk si

uu

si

uu

ω ω ω α 1

)(






Modal Analysis of Rotating

St t




Structures Non-symmetry in system

matrices Modes of undamped rotating

system

Symmetric Stator Non-Symmetric Stator

FRF’s of rotating system

Out-of-balance excitation Synchronous excitation

Non-Synchronous excitation

Non-symmetry in System

Matrices




Matrices The rotating structures are subject to

additional forces: Gyroscopic forces

Rotor-stator rub forces

Electrodynamic forces

Unsteady aerodynamic forces

Time varying fluid forces These forces can destroy the symmetry of the

system matrices.

Non-rotating system

properties




properties A rigid disc mounted on

the free end of a rigidshaft of length L,

The other end of iseffectively pin-jointed.

Modes of Undamped Rotating

System




System

.0

0

0

0

0/

/0

/0

0/

0

0

⎭⎬

⎫

⎩⎨

⎧

=⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡

+⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡

Ω−

Ω

+⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡

y

x

Lk

Lk

y

x

L J

L J

y

x

L I

L I

y

x

z

z

&

&

&&

&&

Symmetric stator




Symmetric stator

( ) ( )( ) ( )

.02

,0

0

//

//

,

,

,

2

0

22

2

00

24

2

0

22

220

2

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +⎟⎟

⎠

⎞⎜⎜

⎝

⎛ ⎟⎟ ⎠

⎞⎜⎜⎝

⎛ Ω+−

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡−Ω−

Ω−

=

=

==

I

kL

I

J

I

kL

Y

X

L I k L J i

L J i L I k

Ye y

Xe x

k k k

z

z

z

t i

t i

y x

ω ω

ω ω

ω ω

ω

ω

Support is symmetric

Simple harmonic motion

Natural Frequencies




Natural Frequencies

⎪

⎪

⎩

⎪⎪⎨

⎧

==

Ω+Ω±Ω+=⇒

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛ +

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ Ω+−

00

22

0

222

0

222

0

2

2,1

2

0

22

2

00

24

,

42

.02

I

J

I

kL

I

kL

I

J

I

kL z

γ ω

γ ω γ

γ ω ω

ω ω

Mode Shapes




Mode Shapes( ) ( )

( ) ( )

( ) ( )

( ) ( )

.

0

0

1//

//,

0

01

//

//2

0

2

2

2

2

2

2

2

0

2

2

2

0

2

1

2

1

2

1

2

0

2

1

⎭

⎬⎫

⎩

⎨⎧

=

⎭

⎬⎫

⎩

⎨⎧⎥

⎦

⎤⎢

⎣

⎡

−Ω−

Ω−

⎭

⎬⎫

⎩

⎨⎧

=

⎭

⎬⎫

⎩

⎨⎧⎥

⎦

⎤⎢

⎣

⎡

−Ω−

Ω− i

L I k L J i

L J i L I k

i L I k L J i

L J i L I k

z

z

z

z

ω ω

ω ω

ω ω

ω ω

Non symmetric Stator




Non-symmetric Stator y x k k ≠

FRF of the Rotating Structure





,0

0

/

/

/0

0/2

2

2

0

2

0

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−

Ω+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

y

x

z

z

f

f

y

x

k

k

y

x

c L J

L J c

y

x

L I

L I

&

&

&&

&&

[ ] ( ) ( )

( ) ( )

⎩⎨⎧

−=

=⇒

⎥⎦

⎤⎢⎣

⎡

+−Ω−

Ω+−=

−

)()(

)()(

//

//)(

1

2

0

22

22

0

2

ω α ω α

ω α ω α

ω ω ω

ω ω ω ω α

yx xy

yy xx

z

z

ic L I k L J i

L J iic L I k

Loss of Reciprocity

External Damping

Coupling Effect


with External Damping




with External Damping

Complex ModeShapes

due to significantimaginary part

Out-of-balance excitation




Out-of-balance excitation Response analysis for the particular

case of excitation provided by out-of-balance forces is investigated: When the force results from an out-of-

balance mass on the rotor, it is of asynchronous nature

When the force results from an out-of-balance mass on a co/counter rotatingshaft, it is of a non-synchronous nature

Synchronous OOB Excitation





{ }

( ))1(

:

1

)sin(

)cos(

22

00

2

2

γ ω −Ω−=

⎭⎬⎫

⎩⎨⎧

−=

⎭⎬⎫

⎩⎨⎧

⇒

−

⎭⎬

⎫

⎩⎨

⎧

−=

⎭⎬

⎫

⎩⎨

⎧

Ω

ΩΩ=

ΩΩ

Ω

I

L A

eiA

AF e

Y

X

Stator Symmetric

eiF t

t

mr F

t i

OOB

t i

t i

OOB






Non-Synchronous OOB

Excitation




Excitation Force is generated by another rotor at

different speed

( ))(2200

2

γ β β ω

β

β β

−Ω−=

⎭⎬⎫

⎩⎨⎧

−=⎭⎬⎫

⎩⎨⎧

Ω⇒

ΩΩ

I

L

A

eiA

AF e

Y

X

Excitation

t iOOB

t i

The essential results are the same as for

synchronous case.






Theoretical Basis




Theoretical Basis Analysis using rotating frame

Damping in rotating and stationaryframes

Dynamic analysis of general rotor-statorsystems Linear Time Invariant Rotor-Stator

Systems LTI Rotor-Stator Viscous Damp System

LTI Systems Eigen-Properties

Analysis using rotating frame





⎭⎬⎫

⎩⎨⎧⎥⎦⎤⎢

⎣⎡ ΩΩ

Ω−Ω=

⎭⎬⎫

⎩⎨⎧

r

r

y x

t t t t

y x

)cos()sin()sin()cos(

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

ΩΩ−

ΩΩ=

⎭⎬⎫

⎩⎨⎧

y

x

t t

t t

y

x

r

r

)cos()sin(

)sin()cos(

Y

X

Yr

Xr

t Ω






[ ]

[ ] [ ]

[ ] [ ] [ ]⎭⎬⎫

⎩⎨⎧

Ω+⎭⎬⎫

⎩⎨⎧

Ω+⎭⎬⎫

⎩⎨⎧

=⎭

⎬⎫

⎩

⎨⎧⎥

⎦

⎤⎢

⎣

⎡

ΩΩ−

ΩΩΩ−

⎭

⎬⎫

⎩

⎨⎧⎥

⎦

⎤⎢

⎣

⎡

Ω−Ω−

ΩΩ−Ω+

⎭

⎬⎫

⎩

⎨⎧⎥

⎦

⎤⎢

⎣

⎡

ΩΩ−

ΩΩ=

⎭

⎬⎫

⎩

⎨⎧

⎭⎬⎫

⎩⎨⎧

Ω+

⎭⎬⎫

⎩⎨⎧

=

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤

⎢⎣

⎡

Ω−Ω−

ΩΩ−Ω+

⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤

⎢⎣

⎡

ΩΩ−

ΩΩ=

⎭⎬⎫

⎩⎨⎧

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

ΩΩ−

ΩΩ

=⎭⎬⎫

⎩⎨⎧

x

xT

y

xT

y

xT

y

x

t t

t t

y

x

t t

t t

y

x

t t

t t

y

x

y

xT

y

xT

y

x

t t

t t

y

x

t t

t t

y

x

y

xT

y

x

t t

t t

y

x

r

r

r

r

r

r

1

2

21

2

21

1

2

)cos()sin(

)sin()cos(

)sin()cos(

)cos()sin(2

)cos()sin(

)sin()cos(

,)sin()cos(

)cos()sin(

)cos()sin(

)sin()cos(

,)cos()sin(

)sin()cos(

&

&

&&

&&

&

&

&&

&&

&&

&&

&

&

&

&

&

&

Transformation Matrices






z z z z z z Ω−Ω+Ω+=Ω+Ω−Ω+= 2/)2/(2/)2/( 22

02

22

01 γ γ ω ω γ γ ω ω

.00

//)()(//

0//2

//20

/0

0/

22222

0

2

22222

0

2

22

0

22

0

2

0

2

0

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧⎥⎦⎤⎢

⎣⎡

++Ω+Ω−−−++Ω+Ω−+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−Ω

Ω+Ω−+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

r

r

y x z z x y

x y y x z z

r

r

z z

z z

r

r

y x

sk ck L J L I k k csk k cssk ck L J L I

y

x

L J L I

L J L I

y

x

L I

L I

&

&

&&

&&

Equation of Motion in Stationary Coordinates

Equation of Motion in Rotating Coordinates

2/)2/(2/)2/(

,0

0

0

0

0/

/0

/0

0/

22

02

22

01

2

2

2

0

2

0

z z z z

y

x

z

z

y

x

k

k

y

x

L J

L J

y

x

L I

L I

Ω+Ω+=Ω−Ω+=

⎭⎬

⎫

⎩⎨

⎧

=⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡

+⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡

Ω−

Ω

+⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡

γ γ ω ω γ γ ω ω

&

&

&&

&&

Note: Eigenvectors remain unchanged






⎭⎬⎫

⎩⎨⎧

Ω++Ω−

Ω++Ω−=

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

ΩΩ−

ΩΩ=

⎭⎬⎫

⎩⎨⎧

⎭⎬

⎫

⎩⎨

⎧

⎥⎦

⎤

⎢⎣

⎡

ΩΩ−

ΩΩ=

⎭⎬

⎫

⎩⎨

⎧

t t

t t F

t F

t t

t t

F

F

F

F

t t

t t

F

F

yr

xr

y

x

yr

xr

)sin()sin(

)cos()cos(

2

)cos(0)cos()sin(

)sin()cos(

.)cos()sin(

)sin()cos(

0

0

ω ω

ω ω

ω

For Example: Response harmonies notpresent in the excitation

Internal Damping in rotating

and stationary frames




and stationary frames

.00

//00//

//2

//2

/0

0/

222

0

2

222

0

2

22

0

22

0

2

0

2

0

⎭⎬⎫

⎩⎨⎧=

⎭⎬⎫

⎩⎨⎧⎥⎦⎤⎢

⎣⎡

+Ω+Ω−+Ω+Ω−+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−Ω

Ω+Ω−+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

r

r

z z

z z

r

r

I z z

z z I

r

r

y x

k L J L I k L J L I

y

x

c L J L I

L J L I c

y

x

L I

L I

&

&

&&

&&

,0

0

/

/

/0

0/2

2

2

0

2

0

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−

Ω+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

Ω−

Ω+

⎭⎬⎫

⎩⎨⎧⎥⎦

⎤⎢⎣

⎡

y

x

k c

ck

y

x

c L J

L J c

y

x

L I

L I

y I z

I z x

I z

z I

&

&

&&

&&

Equation of Motion in Rotating Coordinates

Equation of Motion in Stationary Coordinates

Internal/External Damping in

2DOF System




2DOF System

,

0

0

//

/00/

2

2

2

0

2

0

⎭

⎬⎫

⎩

⎨⎧

=

⎭

⎬⎫

⎩

⎨⎧⎥

⎦

⎤⎢

⎣

⎡

Ω−

Ω+

⎭⎬⎫

⎩⎨⎧⎥⎦⎤⎢

⎣⎡

+Ω−Ω++

⎭⎬⎫

⎩⎨⎧⎥⎦⎤⎢

⎣⎡

y

x

k c

ck

y x

cc L J L J cc

y x

L I L I

y I z

I z x

I E z

z I E

&&

&&&&

At super critical speeds the real parts ofeigen-values may become positive,i.e. unstable system

Dynamic Analysis of General

Rotor-Stator Systems




Rotor Stator Systems The rotating machines and their modal

testing is much more complex Non-symmetric bearing support

Fixed/Rotating observation frame

Non-axisymmetric rotors Internal/External damping

These lead to:

Time-varying modal properties

Response harmonies not present in the excitation

Instabilites (negative modal damping)






y Equation of motion of rotating systems

are prone: to lose the symmetry

to generate complex eigen-values/vectorsfrom velocity/displacement related non-symmetry

to include time varying coefficients asappose to conventional Linear TimeInvariant (LTI) systems






yRotating Coord.Stationary Coord.System Type

LTILTIR-symm;S-symm

L(t)LTIR-symm;S-nonsymm

LTIL(t)R-nonsymm;S-symm

L(t)L(t)R-nonsymm;S-nonsymm

LTI: Linear Time InvariantL(t): Linear Time Dependent

Linear Time Invariant Rotor-

Stator Systems[ ]{ } [ ] [ ]( ){ }




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

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

[ ] [ ] [ ] [ ][ ] [ ] .)(,)(

.,,,

)()(

)(

symmSkew E G

Symm DK C M

t f x E DiK

xGC x M

−⇒ΩΩ⇒

=Ω++

+Ω++ &&&

Solution of equations will follow differentrouts depending upon the specific features.

LTI Rotor-Stator Systems

(Viscous Damping Only)[ ]{ } [ ]{ } { }




( p g y)[ ]{ } [ ]{ } { }

[ ]

[ ]

{ }⎭⎬⎫

⎩⎨⎧

=

⎥⎦

⎤⎢⎣

⎡ Ω+=

⎥⎦

⎤⎢⎣

⎡−

Ω+=

=+

x

xu

M

E K B

M

M GC A

u Bu A

&

&

0

0)(

0

)(

,0

The system matricesare non-symmetric

Complex eigenvals

Two eigenvect sets:

RH; mode shapes LH; normal excitation

shapes

LTI Rotor-Stator Systems

(Viscous Damping Only)




( p g y) Symmetric Rotor/ Non-symmetric Support

Backwards Whirl

Forwards Whirl

FRF of LTI Rotor-Stator

Systems[ ] [ ] ( )[ ] [ ]H1




y[ ] [ ] ( )[ ] [ ] H

LH r RH V iV 1

)( −

−= ω λ ω α



LTI Systems Eigen-Propertiesλ




y g p

-1.001-0.75+1.85i0.0-1.05+0.08i1-0.68+1.88i0.1

-1.08+0.28i1

-0.52+1.94i0.3

-1.03+0.49i1-0.37+1.99i0.5

-0.90+0.63i1-0.23+2.04i0.7

-0.76+0.71i1-0.07+2.08i0.9

-0.69+0.73i12.11i1.0

C Δ 1 X 2 X 2λ

LTI Systems Eigen-Properties




Skew-symmetry in stiffness Matrix

[ ] [ ]

[ ] ⎥

⎦

⎤⎢

⎣

⎡

−

−Δ−+⎥

⎦

⎤⎢

⎣

⎡

−Δ=

=⎥

⎦

⎤⎢

⎣

⎡=

31

13)1(

01

10

,0,

1

1

K K K

C M

LTI Systems Eigen-PropertiesXλ




-1.0012.00i0.0-1.1211.90i0.1

-1.581

1.65i0.3

Infinity11.23i0.5

1.58i10.32+1.00i0.7

1.12i10.57+0.79i0.9

i10.70+0.70i1.0

K Δ 1 X 2 X 2λ

Modal Testing






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Complex Measured Modes




Complex Measured Modes




Display of Mode Complexity




Analytical Real Modes




H Ahmadian GML Gladwell Proceedings of

Extracting real modes from

complex measured modes




H Ahmadian, GML Gladwell - Proceedings of

the 13th International Modal Analysis (1995): The optimum real mode is the one with maximum

correlation with the complex measured one:

Extracting real modesNormalizing the complex measured




Normalizing the complex measured

mode shape:

The problem is rewritten as:

Extracting real modes




Skew-symmetricSymmetric

Rank 2 matrices

Extracting real modes Since V is skew symmetric




Since V is skew symmetric,

Therefore the problem is equivalent to:

















Follow-ups: E. Foltete, J. Piranda, “Transforming Complex




o tete, J a da, a s o g Co p e

Eigenmodes into Real Ones Based on an Appropriation Technique”, Journal of Vibrationand Acoustics, JANUARY 2001, Vol. 123

S.D. GARVEY, J.E.T. PENNY, “THERELATIONSHIP BETWEEN THE REAL ANDIMAGINARY PARTS OF COMPLEX MODES”,

Journal of Sound and Vibration

1998,212(1),75-83

Modal Testing






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Theoretical Basis Non-sinusoidal Vibration and FRF




Non sinusoidal Vibration and FRF

Properties: Periodic Vibration

Transient Vibration Random Vibration

Violation of Dirichlet’s conditions

Autocorrelation and PSD functions H1 and H2

Incomplete Response Models



Periodic Vibration Excitation is not simply sinusoidal but retain




p y

periodicity. The easiest way of computing the response

is by means of Fourier Series,

t i

n

nk n jk j

n

t i

n

nk k

n

n

eF t x

T eF t f

ω

ω

ω α

π ω

∑

∑∞

=

∞

=

=

==

1

1

)()(

2)(

Periodic Vibration To derive FRF from periodic vibration




p

signals: Determine the Fourier Series components

of the input force and the relevant

response Both series contain components at the

same set of discrete frequencies

The FRF can be defined at the same set offrequency points by computing the ratio ofresponse to input components.

Transient Vibration Analysis via Fourier Transform




∫

∫

∞+

∞−

+∞

∞−

−

=

=

=

ω ω ω

ω ω ω

π ω

ω

ω

d eF H t x

F H X

dt et f F

t i

t i

)()()(

)()()(

)(2

1)(

Transient Vibration Response via time domain (superposition)




)()(2

1)(

2

1)()0()(

)()()(

t hd e H t xThen

F t f Let

d f t ht x

t i ==→

=⇒=→

−=

∫

∫

∞+

∞−

+∞

∞−

ω ω π

π ω δ

τ τ τ

ω

Transient Vibration To derive FRF from transient vibration




signals: Calculation of the Fourier Transforms of

both excitation and response signals

Computing the ratio of both signals at thesame frequency

In practice it is common to compute aDFT of the signals.

Random Vibration Neither excitation nor response signal




can be subject to a valid FourierTransform:

Violation of Dirichlet Conditions

Finite number of isolated min and max

Finite number of points of finite discontinuity

Here we assume the random signals tobe ergodic

Random Vibrationt f )( Time Signal




∫

∫∞+

∞−

−

∞+

∞−

=

+=

τ τ π ω

τ τ

ωτ d e RS

dt t f t f R

f

i ff ff

ff

)(2

1)(

)()()(

)( Autocorrelation Function

Power Spectral Density

Random VibrationTi Si l




Time Signal

Autocorrelation


Random Vibration




Sinusoidal Signal

Autocorrelation




Random Vibration




Autocorrelation

Noisy Signal

Random Vibration

The autocorrelation function is real and even:




τ

τ τ

τ τ

+=

−=−=

+=

∫

∫∞+

∞−

+∞

∞−

t u

Rduu f u f

dt t f t f R

ff

ff

)()()(

)()()(

The Auto/Power Spectral Density function isreal and even.



Random Vibration

Time Domain Frequency Domain




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

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

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

*

*

*

ω ω ω τ τ

ω ω ω τ τ

ω ω ω τ τ

X X S dt t xt x R

F X S dt t f t x R

F F S dt t f t f R

xx xx

xf xf

ff ff

=⇒+=

=⇒+=

=⇒+=

∫

∫

∫

∞+

∞−

∞+

∞−

+∞

∞−

Random Vibration

To derive FRF from random vibration signals:




)()(

)()()()()(

)()(

)()()()()(

)()()(

*

*

2

*

*

1

ω ω

ω ω ω ω ω

ω ω

ω ω ω ω ω

ω ω ω

ff

fx

xf

xx

S S

F F X F H

S S

F X X X H

F X H

==

==

=

Complete/ Incomplete Models

It is not possible to measure the




response at all DOF or all modes ofstructure (N by N)

Different incomplete models: Reduced size (from N to n) by deleting

some DOFs

Number of modes are a reduced as well(from N to m, usually m<n)



Incomplete Response Models






Theoretical Basis

Sensitivity of Models




Modal Sensitivity SDOF eigen sensitivity

MDOF system natural frequency sensitivity

MDOF system mode shape sensitivity

FRF Sensitivity

SDOF FRF sensitivity MDOF FRF sensitivity

Sensitivity of Models

The sensitivity analysis are required:




to help locate errors in models in updating to guide design optimization procedures

they are used in the course of curve fitting

A short summery on deducingsensitivities from experimental and

analytical models is given.

Modal Sensitivities (SDOF)

k ω =




.21

21

,2

1

2

1

00

030

0

k mk k

mm

k

m

m

ω ω

ω ω

==∂∂

−=−=∂

∂

Modal Sensitivities (MDOF)

[ ] [ ]{ } { },02 =− r r M K φ ω




[ ] [ ]{ } { }

[ ] [ ]( ){ } { }

[ ] [ ]( ) { } [ ] [ ] [ ] { } { },0

,0

222

2

=⎟⎟ ⎠ ⎞

⎜⎜⎝ ⎛

∂∂−

∂∂−

∂∂+

∂∂−

=−∂

∂

r r r r

r

r r

p

M M p p

K

p M K

M K p

φ ω ω φ ω

φ ω



Eigenvector Sensitivity(MDOF)

: fromStarting




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

[ ]{ } { }

{ } { }

[ ] [ ]( ) { } [ ] [ ] [ ] { } { }0

,0

2

2

1

2

1

22

2

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛

∂

∂−∂

∂−∂

∂+−⇒

=∂

∂

=⎟⎟ ⎠

⎞⎜⎜⎝

⎛

∂

∂−

∂

∂−

∂

∂+

∂

∂−

∑

∑

≠=

≠=

r r r

N

r j j

jrjr

N

r j j

j jr

r r r r

r

p M M

p pK M K

ptakingand

p

M M

p p

K

p M K

φ ω ω φ γ ω

φ γ φ

φ ω ω φ

ω


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

[ ]{ } { }02

22 =⎟⎟

⎞⎜⎜⎛

∂

∂−

∂

∂−

∂

∂+− ∑

=r r

r N

jrjr p

M M

p p

K M K φ ω

ω φ γ ω




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

[ ]

[ ]

[ ]

{ } { }

( ) { } [ ] [ ] { } { }0

0

222

22

1

2

1

=⎟⎟ ⎠ ⎞⎜⎜

⎝ ⎛

∂∂−

∂∂+−⇒

=⎟⎟ ⎠

⎞

⎜⎜⎝

⎛

∂

∂−

∂

∂−

∂

∂+−⇒

⎠⎝

∑≠=

≠

r r

T

srsr s

r r

r T

s

N

r j j

jrjr

T

s

r j j

p M

pK

p

M

M p p

K

M K

φ ω φ γ ω ω

φ ω

ω

φ φ γ ω φ


( ) { } [ ] [ ]

{ }⎟⎟ ⎠

⎞⎜⎜⎝

⎛

∂

∂−

∂

∂+−⇒ r r

T

srsr s p

M

p

K 222 φ ω φ γ ω ω




{ } [ ] [ ]

{ }

( )

{ } { }

[ ] [ ]{ }

( ) { }∑

≠= −

⎟

⎠

⎞⎜

⎝

⎛

∂

∂−

∂

∂

=∂

∂⇒

−

⎟ ⎠

⎞⎜⎝

⎛

∂

∂−

∂

∂

=⇒

N

r ss

s

sr

r r

T

s

r

sr

r r

T

s

rs

p

M

p

K

p

p

M

p

K

122

2

22

2

φ ω ω

φ ω φ φ

ω ω

φ ω φ

γ



Updating, Redesign,Reanalysis

The change in parameters must be very

ll f t l i




small for accurate analysis When the change in parameters is not

small:

Higher order sensitivity analysis

Iterative linear sensitivity analysis

FRF Sensitivities (SDOF)

2

1)(

mcik ω ω ω α

−+=




( )

( )

( )22

2

22

22

)(

)(

1)(

mcik m

mcik

i

c

mcik k

ω ω

ω ω α

ω ω

ω ω α

ω ω

ω α

−+=

∂∂

−+

−=∂

∂

−+

−=

∂

∂

FRF Sensitivities (MDOF)

( )[ ] [ ] [ ] [ ] M C iK Z ω ω ω −+= 2 ,




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

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

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

( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ] A x A x

A A x x A x

x A

Z

Z Z Z Z Z Z then

Z B A Z Atake

A B B A A B A

ω α ω ω α ω α ω α

ω ω ω ω ω ω

ω ω

Δ−=−

−−=⇒

⇒+⇒

+−=+⇒

−−−−−

−−−−

11111

1111

,





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

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

T

j x

T

j A x

A x A x

Z

Z



Δ=−

Δ−=− ,






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

( )[ ]( )[ ]ω

ω ω

ω ω α

∂

∂−=

∂

∂=

∂

∂ −−−

Z

p

Z Z

p

Z

p

111




( )[ ]( )[ ]

( )[ ]( )[ ]

( )[ ] ( )[ ] [ ] [ ] [ ] ( )[ ]ω α ω ω ω α ω α

ω α

ω

ω α

ω α

⎟⎟ ⎠

⎞⎜⎜⎝

⎛ ∂

∂−∂∂+

∂∂−=

∂∂

∂

∂

−=∂

∂

p

M

p

C i

p

K

p

p

Z

p

p p p

2

Documents

Modal Testing Lectures 1-10