modal testing
TRANSCRIPT
![Page 1: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/1.jpg)
Modal Testing(Lecture 1)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Overview
Introduction to Modal TestingApplications of Modal TestingPhilosophy of Modal TestingSummary of TheorySummary of Measurement MethodsSummary of Modal Analysis ProcessesReview of Test Procedures and Levels
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Introduction to Modal TestingExperimental Structural Dynamics
To understand and to control the many vibration phenomenon in practice
Structural integrity (Turbine blades- Suspension Bridges)Performance ( malfunction, disturbance, discomfort)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Introduction to Modal Testing (continued)
Necessities for experimental observations
Nature and extend of vibration in operationVerifying theoretical modelsMaterial properties under dynamic loading (damping capacity, friction,…)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Introduction to Modal Testing (continued)
Test types corresponding to objectives:Operational Force/Response measurements
Response measurement of PZL Mielec Skytruck Mode Shapes (3.17 Hz, 1.62 %), (8.39 Hz, 1.93 %)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Introduction to Modal Testing (continued)
Modal Testing in a controlled environment/ Resonance Testing/ Mechanical Impedance Method
Testing a component or a structure with the objective of obtaining mathematical model of dynamical/vibration behaviorStructural Analysis of ULTRA Mirror
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Introduction to Modal Testing (continued)
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)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Applications of Modal Testing
Model Validation/Correlation:Producing major test modes validates the model
Natural frequenciesMode shapesDamping information are not available in FE models
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Applications of Modal Testing (continued)
Model UpdatingCorrelation of experimental/analytical modelAdjust/correct the analytical modelOptimization procedures are used for updating.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Applications of Modal Testing (continued)
Component Model Identification
Substructure processThe component model is incorporated into the structural assembly
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Applications of Modal Testing (continued)
Force DeterminationKnowledge of dynamic force is requiredDirect force measurement is not possibleMeasurement of response + Analytical Model results the external force
[ ] [ ]( ){ } { }fxMK =− 2ω
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Philosophy of Modal TestingIntegration of three components:
Theory of vibrationAccurate vibration measurementRealistic and detailed data analysis
Examples:Quality and suitability of data for processExcitation typeUnderstanding of forms and trends of plotsChoice of curve fittingAveraging
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Summary of Theory (SDOF)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Summary of Theory (MDOF)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Summary of TheoryDefinition of FRF:
[ ] [ ] [ ]( ).
)()(
)(
)(
122
12
∑=
−
−==
+−=N
r r
krjr
k
jjk f
xh
DiMKH
ωωφφ
ωω
ω
ωω
Curve-fitting the measured FRF:
Modal Model is obtainedSpatial Model is obtained
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Summary of Measurement MethodsBasic measurement system:
Single point excitation
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Summary of Modal Analysis ProcessesAnalysis of measured FRF data
Appropriate type of model (SDOF,MDOF,…)Appropriate parameters for chosen model
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Review of Test Procedures and Levels
The procedure consists of:FRF measurementCurve-FittingConstruct the required model
Different level of details and accuracy in above procedure is required depending on the application.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Review of Test Procedures and LevelsLevels according to Dynamic Testing Agency:
UpdatingOut of range residues
Usable for validationMode ShapesDamping
ratioNatural FreqLevel
0
Only in few points
1
2
3
4
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Text BooksEwins, 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 and Experimental Modal Analysis”, Research studies press Ltd.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Evaluation Scheme
Home Works (20%)Mid-term Exam (20%)Course Project (30%)Final Exam (30%)
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Modal Testing(Lecture 10)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Theoretical Basis
Analysis of weakly nonlinear structuresApproximate analysis of nonlinear structuresCubic stiffness nonlinearityCoulomb friction nonlinearityOther nonlinearities and other descriptions
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Analysis of weakly nonlinear structuresThe whole bases of modal testing assumes linearity:
Response linearly related to the excitationResponse to simultaneous application of several forces can be obtained by superposition of responses to individual forces
An introduction to characteristics of weakly nonlinear systems is given to detect if any nonlinearity is involved during modal test.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Cubic stiffness nonlinearity
)sin()3sin(41)sin(
43
)sin()cos()sin()sin(
)(sin)sin()cos()sin()sin()(
)sin(
33
2
333
2
33
φωωω
ωωωωω
φωωωωωωω
ωφω
−=⎟⎠⎞
⎜⎝⎛ −
+++−⇒
−=+++−⇒
=⇒−=+++
tFttXk
tkXtXctXmtF
tXktkXtXctXmtXtx
tFxkkxxcxm &&&
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Cubic stiffness nonlinearity
⎪⎩
⎪⎨⎧
−=
=++−⇒
−
=⎟⎠⎞
⎜⎝⎛ −
+++−
)sin(
)cos(43
)sin()cos()cos()sin(
)3sin(41)sin(
43
)sin()cos()sin(
33
2
33
2
φω
φω
φωφω
ωω
ωωωωω
FXc
FXkkXXm
tFtF
ttXk
tkXtXctXm
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Cubic stiffness nonlinearity
( )
23
22
23
2
43
43
1
Xkkk
cXkkmFX
eq +=
+⎟⎠⎞
⎜⎝⎛ ++−
=
ωω
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Cubic stiffness nonlinearity
Softening effect Hardening effect
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Softening-stiffness effect
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Softening-stiffness effect
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Softening-stiffness effect
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Softening-stiffness effect
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Coulomb friction nonlinearity
( )
⎟⎠⎞
⎜⎝⎛ ++−
=
=Δ
=⇒=Δ
+=
∫
XccimkF
X
Xc
dttx
EcXcE
txtxctxctf
F
FeqF
Fd
πωωω
πωωπ
41
44
)()()()(
2
/2
0
2&
&
&&
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Coulomb friction nonlinearity
X increasing
⎟⎠⎞
⎜⎝⎛ ++−
=
XccimkF
XF
πωωω 4
12
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Other nonlinearities and other descriptions
BacklashBilinear StiffnessMicroslip friction dampingQuadratic (and other power law damping)…..
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Modal Testing(Lecture 2)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
MODAL ANALYSIS THEORYUnderstanding of how the structural parameters of mass, damping, and stiffness relate to
the impulse response function (time domain), the frequency response function (Fourier, or frequency domain), andthe transfer function (Laplace domain)
for single and multiple degree of freedom systems.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Theoretical Basis
SDOF systemTime Domain: Impulse Response FunctionPresentation of FRFProperties of FRF
Undamped MDOF systemMDOF system with proportional damping
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
SDOF SystemThree classes of system:
UndampedViscously-dampedStructurally Damped
Response Models:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
+−
+−
−
==
idmk
icmk
mk
FXH
2
2
2
1
1
1
)()()(
ω
ωω
ω
ωωω
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Time Domain: Impulse Response Function
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Frequency Domain: Frequency Response Function
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Alternative Forms of FRFReceptance
Inverse is “Dynamic Stiffness”
MobilityInverse is “Dynamic Impedance”
InertanceInverse is “Apparent mass”
)()(
ωω
FX
)()(
)()(
ωωω
ωω
FXi
FV
=
)()(
)()( 2
ωωω
ωω
FX
FA
−=
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Graphical Display of FRF
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Graphical Display of FRF
The magnitude of the three mobility functions (accelerance, mobility and compliance)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Stiffness and Mass Lines
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Reciprocal PlotsThe “inverse” or “reciprocal” plots
Real part Imaginary part ⎪
⎪⎩
⎪⎪⎨
⎧
=
−=⇒
ωωω
ωωω
cXF
mkXF
))()(Im(
))()(Re( 2
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Nyquist PlotFor viscous damping the Mobility plot is a circle.
For structural damping the Receptance and Inertance plots are circles.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
3D FRF Plot (SDOF)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Properties of SDOF FRF PlotsNyquist Mobility for viscose damping
( )( )
2
22222
222222
222
2
222
2
2
21
)()(4)()(
)Im(,21)Re(
)()()()Im(
)()()Re(
)(
⎟⎠⎞
⎜⎝⎛=
+−
+−=+
=⎟⎠⎞
⎜⎝⎛ −=
+−−
=+−
=
+−=
ccmkccmkVU
YVc
YU
cmkmkY
cmkcY
icmkiY
ωωωω
ωωωω
ωωωωω
ωω
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Properties of SDOF FRF PlotsNyquist Receptance for structural damping
( )( )
( )( ) ( )
222
222222
2
222
2
2
21
21
,
1)(
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛ ++
+−=
+−
−=
+−
−−=
−+=
ddVU
dmkdV
dmkmkU
dmkidmk
midkH
ωωω
ωω
ωω
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
A DemoBasic Assumptions
The structure is assumed to be linearThe structure is time invariantThe structure obeys Maxwell’s reciprocityThe structure is observable
loose components, or degrees-of-freedom of motion that are not measured, are not completely observable.
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Modal Testing(Lecture 3)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Theoretical Basis
Undamped MDOF SystemsMDOF Systems with Proportional DampingMDOF Systems with General Structural DampingGeneral Force VectorUndamped Normal Mode
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped MDOF SystemsThe equation of motion:
The modal model:The orthogonality:
Forced response solution:
[ ]{ } [ ]{ } { })()()( tftxKtxM =+&&
[ ] ),,,(, 222
21 Ndiag ωωω K=ΓΦ
[ ] [ ][ ] [ ] [ ] [ ][ ] [ ]., Γ=ΦΦ=ΦΦ KIM TT
[ ] [ ]( ){ } { } titi eFeXMK ωωω =− 2
{ } [ ] [ ]( ) { } { } [ ]{ }FXFMKX )(12 ωαω =⇒−=−
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped MDOF Systems (continued)
Response Model
[ ] [ ]( ) [ ][ ] [ ] [ ]( )[ ] [ ] [ ] [ ][ ] [ ]( ) [ ] [ ] [ ][ ] [ ] [ ] [ ]( )[ ][ ] [ ] [ ] [ ]( ) [ ]T
T
T
TT
I
I
I
MK
MK
Φ−ΓΦ=
Φ−ΓΦ=
ΦΦ=−Γ
ΦΦ=Φ−Φ
=−
−
−−−
−
−
−
12
121
12
12
12
)(
)(
)(
)(
)(
ωωα
ωωα
ωαω
ωαω
ωαω
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped MDOF Systems (continued)
The receptance matrix is symmetric.
∑∑== −
=−
=
===
N
r r
jkrN
r r
krjrjk
j
kkj
k
jjk
AFX
FX
122
122)(
,
ωωωωφφ
ωα
αα Modal Constant/Modal Residue
Single Input
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Example:
[ ] [ ]
[ ] [ ]
.64.2118
62.16205
545.0)(
1111
21
6254
/2.18.08.02.1
11
42
2
2211
2
ωωω
ωωωα
ω
+−−
=−
+−
=
⎥⎦
⎤⎢⎣
⎡−
=Φ⎥⎦
⎤⎢⎣
⎡=
⎥⎦
⎤⎢⎣
⎡−
−=⎥
⎦
⎤⎢⎣
⎡=
eee
ee
ee
mMNKkgM
r
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
.64.2118
62.16205
545.0)( 42
2
2211 ωωω
ωωωα
+−−
=−
+−
=ee
eee
Poles
Example (continued):
Zero
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Example (continued):
.64.211848
6205
545.0)( 422212 ωωωω
ωα+−
=−
−−
=eee
ee
![Page 60: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/60.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
MDOF Systems with Proportional DampingA proportionally damped matrix is diagonalized by normal modes of the corresponding undamped system
[ ] [ ][ ] ),,,( 21 NT ddddiagD L=ΦΦ
Special cases:[ ] [ ][ ] [ ][ ] [ ] [ ].
,,
MKDMDKD
δβδβ
+===
![Page 61: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/61.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
MDOF Systems with Structurally Proportional Damping
Response Model[ ] [ ] [ ]( ) [ ][ ] [ ] [ ] [ ]( )[ ] [ ] [ ] [ ][ ] [ ]( ) [ ] [ ] [ ][ ] [ ] [ ] [ ]( )[ ][ ] [ ] [ ] [ ]( ) [ ]
∑=
−
−−−
−
−
−
−+=
Φ−+Φ=
Φ−+Φ=
ΦΦ=−+
ΦΦ=Φ−+Φ
=−+
N
r rr
krjrjk
Trr
rrT
Trr
TT
i
Ii
Ii
Ii
MDiK
MDiK
1222
1222
12221
1222
12
12
)1()(
)1()(
)1()(
)()1(
)(
)(
ωηωφφ
ωα
ωηωωα
ωηωωα
ωαωηω
ωαω
ωαω
Real Residue
Complex Pole
![Page 62: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/62.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
MDOF Systems with Viscously Proportional DampingResponse Model[ ] [ ] [ ]( ) [ ][ ] [ ] [ ] [ ]( )[ ] [ ] [ ] [ ][ ] [ ] [ ]( ) [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]( )[ ][ ] [ ] [ ] [ ] [ ]( ) [ ]
∑=
−
−−−
−
−
−
+−=
Φ−+Φ=
Φ−+Φ=
ΦΦ=−+
ΦΦ=Φ−+Φ
=−+
N
r rrr
krjrjk
Trrr
rrrT
Trrr
TT
Ii
Ii
Ii
MCiK
MCiK
122
122
1221
122
12
12
2)(
2)(
2)(
)(2
)(
)(
ωωζωωφφ
ωα
ωωζωωωα
ωωζωωωα
ωαωωζωω
ωαωω
ωαωω
![Page 63: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/63.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
MDOF Systems with General Structural DampingThe equation of motion:
The orthogonality:
Forced response solution:
[ ]{ } [ ] [ ]( ){ } { })()()( tftxDiKtxM =++&&
[ ] [ ][ ] [ ] [ ] [ ][ ] [ ]., Γ=Φ+Φ=ΦΦ iDKIM TT
[ ] [ ] [ ]( ){ } { } titi eFeXMDiK ωωω =−+ 2
{ } [ ] [ ] [ ]( ) { } { } [ ]{ }FXFMDiKX )(12 ωαω =⇒−+=−
Complex Eigen-valuesComplex Mode Shapes
![Page 64: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/64.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Example:
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−−=Φ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=Γ
=====
142.0493.0635.0318.0782.0536.0318.1218.0464.0
,6698
3352950
6,...,1,/315.1,1,5.0
1
321
UndampedjmNek
kgmkgmkgmModel
j
![Page 65: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/65.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
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
142.0493.0635.0318.0782.0536.0318.1218.0464.0
,6698
3352950
)05.01(
05.0Pr
11
ooo
ooo
ooo
ii
i
jdkdoportionalNon
i
KDoportional
j
Almost real modes
![Page 66: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/66.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Example:
[ ]
[ ] [ ]
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
−=Φ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+=Γ
=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
−=Φ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=Γ
=====
207.0752.0587.0827.0215.0567.0
552.0602.0577.0,
41243892
999)05.01(
,05.0Pr
207.0752.0587.0827.0215.0567.0
552.0602.0577.0,
41243892
999
6,...,1,/3105.1,95.0,1
2
321
i
KDoportional
UndampedjmNek
kgmkgmkgmModel
j
![Page 67: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/67.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
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(1006
6,...,2,0.0,3.0Pr
11
ooo
ooo
ooo
ii
i
jdkdoportionalNon
j
Heavily complex modes
![Page 68: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/68.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
MDOF Systems with General Structural Damping[ ] [ ] [ ]( ) [ ][ ] [ ] [ ] [ ]( )[ ] [ ] [ ] [ ][ ] [ ]( ) [ ] [ ] [ ][ ] [ ] [ ] [ ]( )[ ][ ] [ ] [ ] [ ]( ) [ ]
∑=
−
−−−
−
−
−
−+=
Φ−+Φ=
Φ−+Φ=
ΦΦ=−+
ΦΦ=Φ−+Φ
=−+
N
r rr
krjrjk
Trr
rrT
Trr
TT
i
Ii
Ii
Ii
MDiK
MDiK
1222
1222
12221
1222
12
12
)1()(
)1()(
)1()(
)()1(
)(
)(
ωηωφφ
ωα
ωηωωα
ωηωωα
ωαωηω
ωαω
ωαω
Complex Residues
Complex Poles
![Page 69: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/69.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force VectorIn many situations the system is excited at several points.
![Page 70: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/70.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force Vector (continued)
The response is governed by:
The solution:
All forces have the same frequency but may vary in magnitude and phase.
[ ] [ ]( ){ } { } titi eFeXMiDK ωωω =−+ 2
{ } { } { }{ }∑= −+
=N
r rr
rTr
iFX
1222 )1( ωηω
φφ
![Page 71: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/71.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force Vector (continued)
The response vector is referred to:Forced Vibration Modeor Operating Deflection Shape (ODS)
When the excitation frequency is close to the natural frequency:
ODS reflects the shape of nearby modeBut not identical due to contributions of other modes.
![Page 72: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/72.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force Vector (continued)
Damped system normal mode:By carefully tuning the force vector the response can be controlled by a single mode.The is attained ifDepending upon damping condition the force vector entries may well be complex (they have different phases)
{ } { } rssTr F δφ =
![Page 73: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/73.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode
Special Case of interest:Harmonic excitation of mono-phased forces
Same frequencySame phaseMagnitudes may vary
Is it possible to obtain mono-phased response?
![Page 74: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/74.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode (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)(θω
ω
−=
=ti
ti
eXtx
eFtf[ ] [ ]( ){ } { } titi eFeXMiDK ωωω ˆˆ2 =−+
[ ] [ ]( ) [ ]( ){ } { }[ ] [ ]( ) [ ]( ){ } { }0ˆcossin
ˆˆsincos2
2
=+−
=+−
XDMK
FXDMK
θθω
θθω
{ }F̂
N solutions
![Page 75: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/75.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode (continued)
At a frequency that the phase lag between all forces and all responses is 90 degree then
Results Undamped normal modesNatural frequencies of undamped system
[ ] [ ]( ) [ ]( ){ } { }0ˆcossin2 =+− XDMK θθω
![Page 76: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/76.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode (continued)
The base for multi-shaker test procedures.Modal Analysis of Large Structures: Multiple Exciter Systems By: M. Phil. K. Zaveri
![Page 77: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/77.jpg)
Modal Testing(Lecture 4)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
![Page 78: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/78.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Theoretical Basis
General Force VectorUndamped Normal ModeMDOF System with General Viscous DampingForce Response Solution/ General Viscous Damping
![Page 79: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/79.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force VectorIn many situations the system is excited at several points.
![Page 80: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/80.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force VectorOtherwise you end up damaging the structure!!!!
![Page 81: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/81.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force Vector (continued)
The response is governed by:
All forces have the same frequency but may vary in magnitude and phase.The solution:
[ ] [ ]( ){ } { } titi eFeXMiDK ωωω =−+ 2
{ } { } { }{ }∑= −+
=N
r rr
rTr
iFX
1222 )1( ωηω
φφ
![Page 82: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/82.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force Vector (continued)
The response vector is referred to:Forced Vibration Modeor Operating Deflection Shape (ODS)
When the excitation frequency is close to the natural frequency:
ODS reflects the shape of nearby modeBut not identical due to contributions of other modes.
![Page 83: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/83.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
General Force Vector (continued)
Damped system normal mode:By carefully tuning the force vector the response can be controlled by a single mode.This is attained ifDepending upon damping condition the force vector entries may well be complex (they have different phases)
{ } { } rssTr F δφ =
![Page 84: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/84.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode
Special Case of interest:Harmonic excitation of mono-phased forces
Same frequencySame phaseMagnitudes may vary
Is it possible to obtain mono-phased response?
![Page 85: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/85.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode
512 channel37 Shakers
![Page 86: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/86.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode (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)(θω
ω
−=
=ti
ti
eXtx
eFtf[ ] [ ]( ){ } { } titi eFeXMiDK ωθωω ˆˆ )(2 =−+ −
[ ] [ ]( ) [ ]( ){ } { }[ ] [ ]( ) [ ]( ){ } { }0ˆcossin
ˆˆsincos2
2
=+−
=+−
XDMK
FXDMK
θθω
θθω
{ }F̂
N solutions
![Page 87: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/87.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode (continued)
At a frequency that the phase lag between all forces and all responses is 90 degree then
Results Undamped normal modesNatural frequencies of undamped system
[ ] [ ]( ) [ ]( ){ } { }[ ] [ ]( ){ } { }0ˆ
0ˆcossin2
2
=−⇒
=+−
XMK
XDMK
ω
θθω
![Page 88: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/88.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Undamped Normal Mode (continued)
The base for multi-shaker test procedures.Modal Analysis of Large Structures: Multiple Exciter Systems By: M. Phil. K. Zaveri
![Page 89: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/89.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
MDOF System with General Viscous Damping
[ ]{ } [ ]{ } [ ]{ } { }{ } { } { } { }{ } [ ] [ ] [ ]( ) { }FCiMKX
eXtxeFtffxKxCxMMOE
titi
12
)()(...
−+−=
=⇒=
=++⇒
ωω
ωω
&&&
Next the orthogonality properties of the system in 2N space is used for force response solution.
![Page 90: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/90.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Force Response Solution
{ } { } { }
{ } { }
).,,,(0
0,
0
00
00
00
00
.
000
0
221 NTT
rr
sssdiagUM
KUIU
MMC
U
uM
KM
MCssolutionEigen
uM
Ku
MMC
VibFree
fxx
MK
xx
MMC
EOM
L
&
&&&
&
=⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡⇒
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦
⎤⎢⎣
⎡−
+⎥⎦
⎤⎢⎣
⎡⇒−
=⎥⎦
⎤⎢⎣
⎡−
+⎥⎦
⎤⎢⎣
⎡⇒
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡−
+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡⇒
![Page 91: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/91.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Force Response Solution
*1
2
1
000
r
rHrN
r r
rTrN
r r
rTr
si
uF
u
si
uF
u
si
uF
u
XiX
−⎭⎬⎫
⎩⎨⎧
+−⎭⎬⎫
⎩⎨⎧
=−⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧
∗
==∑∑ ωωωω
The above simplification is due to the fact that eigen-values and eigen-vectors occur in complex conjugate pairs.
![Page 92: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/92.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Force Response SolutionSingle point excitation:
∗
∗∗
= −+
−= ∑
r
krjrN
r r
krjrjk si
uusi
uuωω
ωα1
)(
![Page 93: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/93.jpg)
Modal Testing(Lecture 5)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
![Page 94: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/94.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Modal Analysis of Rotating StructuresNon-symmetry in system matricesModes of undamped rotating system
Symmetric StatorNon-Symmetric Stator
FRF’s of rotating systemOut-of-balance excitation
Synchronous excitationNon-Synchronous excitation
![Page 95: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/95.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Non-symmetry in System Matrices The rotating structures are subject to additional forces:
Gyroscopic forcesRotor-stator rub forcesElectrodynamic forcesUnsteady aerodynamic forcesTime varying fluid forces
These forces can destroy the symmetry of the system matrices.
![Page 96: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/96.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Non-rotating system properties
A rigid disc mounted on the free end of a rigid shaft of length L, The other end of is effectively pin-jointed.
![Page 97: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/97.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Modes of Undamped Rotating System
.00
00
0//0
/00/
0
0
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡Ω−
Ω+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡yx
LkLk
yx
LJLJ
yx
LILI
y
x
z
z
&
&
&&
&&
![Page 98: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/98.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Symmetric stator
( ) ( )( ) ( )
.02
,00
////
,,
,
2
0
22
2
00
24
20
22
220
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ Ω+−
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
−Ω−Ω−
=
=
==
IkL
IJ
IkL
YX
LIkLJiLJiLIk
YeyXex
kkk
z
z
z
ti
ti
yx
ωω
ωωωω
ω
ω
Support is symmetric
Simple harmonic motion
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Natural Frequencies
⎪⎪⎩
⎪⎪⎨
⎧
==
Ω+Ω±
Ω+=
⇒
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ Ω+−
00
220
2220
2220
22,1
2
0
22
2
00
24
,
42
.02
IJ
IkL
IkL
IJ
IkL z
γω
γωγγωω
ωω
![Page 100: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/100.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Mode Shapes( ) ( )( ) ( )
( ) ( )( ) ( ) .
00
1////
,001
////
20
22
22
22
20
22
20
21
21
21
20
21
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
−Ω−Ω−
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
−Ω−Ω− i
LIkLJiLJiLIk
iLIkLJiLJiLIk
z
z
z
z
ωωωω
ωωωω
![Page 101: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/101.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Non-symmetric Stator
yx kk ≠
![Page 102: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/102.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
FRF of the Rotating Structure
,0
0/
//00/
2
2
20
20
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
Ω−Ω
+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
y
x
z
z
ff
yx
kk
yx
cLJLJc
yx
LILI
&
&
&&
&&
[ ] ( ) ( )( ) ( )
⎩⎨⎧
−==
⇒
⎥⎦
⎤⎢⎣
⎡
+−Ω−Ω+−
=−
)()()()(
////
)(1
20
22
220
2
ωαωαωαωα
ωωωωωω
ωα
yxxy
yyxx
z
z
icLIkLJiLJiicLIk
Loss of Reciprocity
External Damping
Coupling Effect
![Page 103: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/103.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
FRF of the Rotating Structurewith External Damping
Complex Mode Shapes
due to significant imaginary part
![Page 104: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/104.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Out-of-balance excitationResponse 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 a synchronous natureWhen the force results from an out-of-balance mass on a co/counter rotating shaft, it is of a non-synchronous nature
![Page 105: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/105.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Synchronous OOB Excitation
{ }
( ))1(
:
1)sin()cos(
2200
2
2
γω −Ω−=
⎭⎬⎫
⎩⎨⎧−
=⎭⎬⎫
⎩⎨⎧
⇒
−⎭⎬⎫
⎩⎨⎧−
=⎭⎬⎫
⎩⎨⎧
ΩΩ
Ω=
ΩΩ
Ω
ILA
eiAA
FeYX
StatorSymmetric
ei
Ftt
mrF
tiOOB
ti
tiOOB
![Page 106: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/106.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Synchronous OOB Excitation
![Page 107: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/107.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Non-Synchronous OOB ExcitationForce is generated by another rotor at different speed
( ))(2200
2
γββω
β
ββ
−Ω−=
⎭⎬⎫
⎩⎨⎧−
=⎭⎬⎫
⎩⎨⎧
Ω⇒
ΩΩ
ILA
eiAA
FeYX
Excitation
tiOOB
ti
The essential results are the same as for synchronous case.
![Page 108: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/108.jpg)
Modal Testing(Lecture 6)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
![Page 109: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/109.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Theoretical Basis
Analysis using rotating frameDamping in rotating and stationary framesDynamic analysis of general rotor-stator systems
Linear Time Invariant Rotor-Stator SystemsLTI Rotor-Stator Viscous Damp SystemLTI Systems Eigen-Properties
![Page 110: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/110.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Analysis using rotating frame
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡ΩΩΩ−Ω
=⎭⎬⎫
⎩⎨⎧
r
r
yx
tttt
yx
)cos()sin()sin()cos(
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡ΩΩ−ΩΩ
=⎭⎬⎫
⎩⎨⎧
yx
tttt
yx
r
r
)cos()sin()sin()cos(
Y
X
Yr
Xr
tΩ
![Page 111: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/111.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Analysis using rotating frame
[ ]
[ ] [ ]
[ ] [ ] [ ]⎭⎬⎫
⎩⎨⎧
Ω+⎭⎬⎫
⎩⎨⎧
Ω+⎭⎬⎫
⎩⎨⎧
=
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡ΩΩ−ΩΩ
Ω−⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡Ω−Ω−ΩΩ−
Ω+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡ΩΩ−ΩΩ
=⎭⎬⎫
⎩⎨⎧
⎭⎬⎫
⎩⎨⎧
Ω+⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡Ω−Ω−ΩΩ−
Ω+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡ΩΩ−ΩΩ
=⎭⎬⎫
⎩⎨⎧
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡ΩΩ−ΩΩ
=⎭⎬⎫
⎩⎨⎧
xx
Tyx
Tyx
T
yx
tttt
yx
tttt
yx
tttt
yx
yx
Tyx
Tyx
tttt
yx
tttt
yx
yx
Tyx
tttt
yx
r
r
r
r
r
r
12
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
![Page 112: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/112.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Analysis using rotating frame
zzzzzz Ω−Ω+Ω+=Ω+Ω−Ω+= 2/)2/(2/)2/( 2202
2201 γγωωγγωω
.00
//)()(//
0//2//20
/00/
222220
2
222220
2
220
220
20
20
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
++Ω+Ω−−−++Ω+Ω−
+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
Ω−ΩΩ+Ω−
+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
r
r
yxzzxy
xyyxzz
r
r
zz
zz
r
r
yx
skckLJLIkkcskkcsskckLJLI
yx
LJLILJLI
yx
LILI
&
&
&&
&&
Equation of Motion in Stationary Coordinates
Equation of Motion in Rotating Coordinates
2/)2/(2/)2/(
,00
00
0//0
/00/
2202
2201
2
2
20
20
zzzz
y
x
z
z
yx
kk
yx
LJLJ
yx
LILI
Ω+Ω+=Ω−Ω+=
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
Ω−Ω
+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
γγωωγγωω
&
&
&&
&&
Note: Eigenvectors remain unchanged
![Page 113: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/113.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Analysis using rotating frame
⎭⎬⎫
⎩⎨⎧
Ω++Ω−Ω++Ω−
=
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡ΩΩ−ΩΩ
=⎭⎬⎫
⎩⎨⎧
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡ΩΩ−ΩΩ
=⎭⎬⎫
⎩⎨⎧
ttttF
tF
tttt
FF
FF
tttt
FF
yr
xr
y
x
yr
xr
)sin()sin()cos()cos(
2
)cos(0)cos()sin(
)sin()cos(
.)cos()sin()sin()cos(
0
0
ωωωω
ω
For Example: Response harmonies not present in the excitation
![Page 114: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/114.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Internal Damping in rotating and stationary frames
.00
//00//
//2//2
/00/
2220
2
2220
2
220
220
20
20
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
+Ω+Ω−+Ω+Ω−
+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
Ω−ΩΩ+Ω−
+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
r
r
zz
zz
r
r
Izz
zzI
r
r
yx
kLJLIkLJLI
yx
cLJLILJLIc
yx
LILI
&
&
&&
&&
,00
//
/00/
2
2
20
20
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡Ω−
Ω+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
Ω−Ω
+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡yx
kcck
yx
cLJLJc
yx
LILI
yIz
Izx
Iz
zI
&
&
&&
&&
Equation of Motion in Rotating Coordinates
Equation of Motion in Stationary Coordinates
![Page 115: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/115.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Internal/External Damping in 2DOF System
,00
//
/00/
2
2
20
20
⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡Ω−
Ω+
⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
+Ω−Ω+
+⎭⎬⎫
⎩⎨⎧⎥⎦
⎤⎢⎣
⎡
yx
kcck
yx
ccLJLJcc
yx
LILI
yIz
Izx
IEz
zIE
&
&
&&
&&
At super critical speeds the real parts of eigen-values may become positive, i.e. unstable system
![Page 116: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/116.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Dynamic Analysis of General Rotor-Stator SystemsThe rotating machines and their modal testing is much more complex
Non-symmetric bearing supportFixed/Rotating observation frameNon-axisymmetric rotorsInternal/External damping
These lead to:Time-varying modal propertiesResponse harmonies not present in the excitationInstabilites (negative modal damping)
![Page 117: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/117.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Dynamic Analysis of General Rotor-Stator Systems
Equation of motion of rotating systems are prone:
to lose the symmetry to generate complex eigen-values/vectors from velocity/displacement related non-symmetryto include time varying coefficients as appose to conventional Linear Time Invariant (LTI) systems
![Page 118: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/118.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Dynamic Analysis of General Rotor-Stator Systems
Rotating 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
![Page 119: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/119.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Linear Time Invariant Rotor-Stator Systems
[ ]{ } [ ] [ ]( ){ }[ ] [ ] [ ]( ){ } { }[ ] [ ] [ ] [ ][ ] [ ] .)(,)(
.,,,)()(
)(
symmSkewEGSymmDKCM
tfxEDiKxGCxM
−⇒ΩΩ⇒
=Ω+++Ω++ &&&
Solution of equations will follow different routs depending upon the specific features.
![Page 120: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/120.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
LTI Rotor-Stator Systems (Viscous Damping Only)[ ]{ } [ ]{ } { }
[ ]
[ ]
{ }⎭⎬⎫
⎩⎨⎧
=
⎥⎦
⎤⎢⎣
⎡ Ω+=
⎥⎦
⎤⎢⎣
⎡−
Ω+=
=+
xx
u
MEK
B
MMGC
A
uBuA
&
&
00)(
0)(
,0
The system matrices are non-symmetricComplex eigenvalsTwo eigenvect sets:
RH; mode shapesLH; normal excitation shapes
![Page 121: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/121.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
LTI Rotor-Stator Systems (Viscous Damping Only)Symmetric Rotor/ Non-symmetric Support
Backwards Whirl
Forwards Whirl
![Page 122: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/122.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
FRF of LTI Rotor-Stator Systems
[ ] [ ] ( )[ ] [ ]HLHrRH ViV 1)( −−= ωλωα
![Page 123: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/123.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
LTI Systems Eigen-PropertiesSkew-symmetry in damping Matrix
[ ] [ ]
[ ] ⎥⎦
⎤⎢⎣
⎡−
−Δ−+⎥
⎦
⎤⎢⎣
⎡−
Δ=
⎥⎦
⎤⎢⎣
⎡−
−=⎥
⎦
⎤⎢⎣
⎡=
15.05.01
)1(05.05.00
,3113
,1
1
CCC
KM
![Page 124: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/124.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
LTI Systems Eigen-Properties
-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
CΔ 1X 2X2λ
![Page 125: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/125.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
LTI Systems Eigen-PropertiesSkew-symmetry in stiffness Matrix
[ ] [ ]
[ ] ⎥⎦
⎤⎢⎣
⎡−
−Δ−+⎥
⎦
⎤⎢⎣
⎡−
Δ=
=⎥⎦
⎤⎢⎣
⎡=
3113
)1(0110
,0,1
1
KKK
CM
![Page 126: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/126.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
LTI Systems Eigen-Properties
-1.0012.00i0.0-1.1211.90i0.1-1.5811.65i0.3
Infinity11.23i0.51.58i10.32+1.00i0.71.12i10.57+0.79i0.9
i10.70+0.70i1.0
KΔ 1X 2X2λ
![Page 127: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/127.jpg)
Modal Testing(Lecture 7)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
![Page 128: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/128.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Complex Measured Modes
![Page 129: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/129.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Complex Measured Modes
![Page 130: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/130.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Display of Mode Complexity
![Page 131: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/131.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Analytical Real Modes
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
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 modes fromcomplex measured modes
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Extracting real modes
Normalizing the complex measured mode shape:
The problem is rewritten as:
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Extracting real modes
Skew-symmetricSymmetric
Rank 2 matrices
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Extracting real modes
Since V is skew symmetric,
Therefore the problem is equivalent to:
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Extracting real modes
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Extracting real modes
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Extracting real modes
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Extracting real modes
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Extracting real modes
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Dr H Ahmadian ,Modal Testing Lab ,IUSTOverview of Modal Testing
Follow-ups:E. Foltete, J. Piranda, “Transforming Complex Eigenmodes into Real Ones Based on an Appropriation Technique”, Journal of Vibration and Acoustics, JANUARY 2001, Vol. 123S.D. GARVEY, J.E.T. PENNY, “THE RELATIONSHIP BETWEEN THE REAL AND IMAGINARY PARTS OF COMPLEX MODES”, Journal of Sound and Vibration 1998,212(1),75-83
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Modal Testing(Lecture 8)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Theoretical Basis
Non-sinusoidal Vibration and FRF Properties:
Periodic VibrationTransient VibrationRandom Vibration
Violation of Dirichlet’s conditionsAutocorrelation and PSD functionsH1 and H2
Incomplete Response Models
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Non-sinusoidal Vibration and FRF PropertiesWith the FRF data, response of a MDOF system to a set of harmonic loads:
{ } [ ]{ } titi eFeX ωω ωα )(=Different amplitudes and phases
The same frequency
We shall now turn our attention to a range of other excitation/response situvations.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Periodic VibrationExcitation is not simply sinusoidal but retain periodicity.The easiest way of computing the response is by means of Fourier Series,
ti
nnknjkj
nti
nnkk
n
n
eFtx
TeFtf
ω
ω
ωα
πω
∑
∑∞
=
∞
=
=
==
1
1
)()(
2)(
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Periodic VibrationTo derive FRF from periodic vibration signals:
Determine the Fourier Series components of the input force and the relevant responseBoth series contain components at the same set of discrete frequenciesThe FRF can be defined at the same set of frequency points by computing the ratio of response to input components.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Transient VibrationAnalysis via Fourier Transform
∫
∫
∞+
∞−
+∞
∞−
−
=
=
=
ωωω
ωωωπ
ω
ω
ω
deFHtx
FHX
dtetfF
ti
ti
)()()(
)()()(
)(21)(
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Transient VibrationResponse via time domain (superposition)
)()(21)(
21)()0()(
)()()(
thdeHtxThen
FtfLet
dfthtx
ti ==→
=⇒=→
−=
∫
∫
∞+
∞−
+∞
∞−
ωωπ
πωδ
τττ
ω
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Transient Vibration
To derive FRF from transient vibration signals:
Calculation of the Fourier Transforms of both excitation and response signalsComputing the ratio of both signals at the same frequency
In practice it is common to compute a DFT of the signals.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random Vibration
Neither excitation nor response signal can be subject to a valid Fourier Transform:
Violation of Dirichlet ConditionsFinite number of isolated min and maxFinite number of points of finite discontinuity
Here we assume the random signals to be ergodic
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random Vibration
∫
∫∞+
∞−
−
∞+
∞−
=
+=
ττπ
ω
ττ
ωτdeRS
dttftfR
tf
iffff
ff
)(21)(
)()()(
)( Time Signal
Autocorrelation Function
Power Spectral Density
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random Vibration
Time Signal
Autocorrelation
Power Spectral Density
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random Vibration
Sinusoidal Signal
Autocorrelation
Power Spectral Density
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random Vibration
Random Signal
Autocorrelation
Power Spectral Density
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random Vibration
Autocorrelation
Noisy Signal
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random VibrationThe autocorrelation function is real and even:
τ
ττ
ττ
+=
−=−=
+=
∫
∫∞+
∞−
+∞
∞−
tu
Rduufuf
dttftfR
ff
ff
)()()(
)()()(
The Auto/Power Spectral Density function is real and even.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random Vibration
Cross Correlation / Spectral Densities
Cross Correlation functions are real but not always even.Cross Spectral Densities are complex functions.
∫∫+∞
∞−
−+∞
∞−
=+= ττπ
ωττ ωτdeRSdttftxR ixfxfxf )(
21)()()()(
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random Vibration
)()()()()()(
)()()()()()(
)()()()()()(
*
*
*
ωωωττ
ωωωττ
ωωωττ
XXSdttxtxR
FXSdttftxR
FFSdttftfR
xxxx
xfxf
ffff
=⇒+=
=⇒+=
=⇒+=
∫
∫
∫
∞+
∞−
∞+
∞−
+∞
∞−
Time Domain Frequency Domain
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Random VibrationTo derive FRF from random vibration signals:
)()(
)()()()()(
)()(
)()()()()(
)()()(
*
*
2
*
*
1
ωω
ωωωωω
ωω
ωωωωω
ωωω
ff
fx
xf
xx
SS
FFXFH
SS
FXXXH
FXH
==
==
=
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Complete/ Incomplete Models
It is not possible to measure the response at all DOF or all modes of structure (N by N)Different incomplete models:
Reduced size (from N to n) by deleting some DOFsNumber of modes are a reduced as well (from N to m, usually m<n)
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Incomplete Response Models
[ ][ ]⎩
⎨⎧
Φ⇒
+−=
×
×
<
=∑
mn
mmr
Nm
r rrr
jkrjk i
A
2
1222)(
ω
ωηωωωα
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Incomplete Response Models
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Modal Testing(Lecture 9)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Theoretical Basis
Sensitivity of ModelsModal Sensitivity
SDOF eigen sensitivityMDOF system natural frequency sensitivityMDOF system mode shape sensitivity
FRF SensitivitySDOF FRF sensitivityMDOF FRF sensitivity
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Sensitivity of Models
The sensitivity analysis are required:to help locate errors in models in updatingto guide design optimization proceduresthey are used in the course of curve fitting
A short summery on deducing sensitivities from experimental and analytical models is given.
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Modal Sensitivities (SDOF)
.21
21
,21
21
00
03
0
0
kmkk
mmk
m
mk
ωω
ωω
ω
==∂∂
−=−=∂∂
=
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Modal Sensitivities (MDOF)
[ ] [ ]( ){ } { }
[ ] [ ]( ){ } { }
[ ] [ ]( ) { } [ ] [ ] [ ] { } { },0
,0
,0
22
2
2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
−∂∂
+∂∂
−
=−∂∂
=−
rrrr
r
rr
rr
pMM
ppK
pMK
MKp
MK
φωωφω
φω
φω
![Page 168: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/168.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Eigenvalue Sensitivity (MDOF){ }
{ } [ ] [ ]( ) { }
{ } [ ] [ ] [ ] { } { }
{ } [ ] [ ] { }
{ } [ ]{ }rTr
rrT
rr
rrrT
r
rr
Tr
Tr
MpM
pK
presults
pMM
ppK
pMK
byMultiply
φφ
φωφω
φωωφ
φωφ
φ
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
=∂∂
⇒
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
−∂∂
+∂∂
−
22
22
2
,0
![Page 169: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/169.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Eigenvector Sensitivity (MDOF)
[ ] [ ]( ) { } [ ] [ ] [ ] { } { }
{ } { }
[ ] [ ]( ) { } [ ] [ ] [ ] { } { }0
,0
:
22
1
2
1
22
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
−∂∂
+−⇒
=∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
−∂∂
+∂∂
−
∑
∑
≠=
≠=
rrr
N
rjj
jrjr
N
rjj
jjr
rrrr
r
pMM
ppKMK
ptakingand
pMM
ppK
pMK
fromStarting
φωωφγω
φγφ
φωωφω
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Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Eigenvector Sensitivity (MDOF)
[ ] [ ]( ) { } [ ] [ ] [ ] { } { }
{ } [ ] [ ]( ) { } { } [ ] [ ] [ ] { } { }
( ) { } [ ] [ ] { } { }0
0
0
222
22
1
2
22
1
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
+−⇒
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
−∂∂
+−⇒
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
−∂∂
+−
∑
∑
≠=
≠=
rrT
srsrs
rrrT
s
N
rjj
jrjrT
s
rrr
N
rjj
jrjr
pM
pK
pMM
ppKMK
pMM
ppKMK
φωφγωω
φωωφφγωφ
φωωφγω
![Page 171: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/171.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Eigenvector Sensitivity (MDOF)
( ) { } [ ] [ ] { }
{ } [ ] [ ] { }
( )
{ } { } [ ] [ ] { }
( ) { }∑≠= −
⎟⎠
⎞⎜⎝
⎛∂∂−
∂∂
=∂∂
⇒
−
⎟⎠
⎞⎜⎝
⎛∂∂
−∂∂
=⇒
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
+−⇒
N
rss
ssr
rrT
sr
sr
rrT
s
rs
rrT
srsrs
pM
pK
p
pM
pK
pM
pK
122
2
22
2
222
φωω
φωφφ
ωω
φωφγ
φωφγωω
![Page 172: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/172.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Updating, Redesign, Reanalysis
{ }{ }
{ } { } { }
{ } { } { }⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
ΔΔΔ
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
∂Δ∂
∂Δ∂
∂Δ∂
∂Δ∂
∂Δ∂
∂Δ∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
=
⎪⎪⎪⎪
⎭
⎪⎪⎪⎪
⎬
⎫
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
ΔΔ
ΔΔ
M
OMMM
K
K
KMMM
K
K
M
M
3
2
1
3
2
2
2
1
2
3
1
2
1
1
1
3
22
2
22
1
22
3
21
2
21
1
21
2
1
22
21
ppp
ppp
ppp
ppp
ppp
φφφ
φφφ
ωωω
ωωω
φφ
ωω
![Page 173: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/173.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
Updating, Redesign, Reanalysis
The change in parameters must be very small for accurate analysisWhen the change in parameters is not small:
Higher order sensitivity analysis Iterative linear sensitivity analysis
![Page 174: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/174.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
FRF Sensitivities (SDOF)
( )
( )
( )22
2
22
22
2
)(
)(
1)(
1)(
mcikm
mciki
c
mcikk
mcik
ωωωωα
ωωωωα
ωωωα
ωωωα
−+=
∂∂
−+
−=
∂∂
−+
−=
∂∂
−+=
![Page 175: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/175.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
FRF Sensitivities (MDOF)( )[ ] [ ] [ ] [ ]
[ ] [ ]( ) [ ] [ ] [ ]( ) [ ][ ]
[ ] ( )[ ] [ ] ( )[ ]
( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ]( ) ( )[ ]
( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ]AxAx
AAxxAx
xA
Z
ZZZZZZthen
ZBAZAtake
ABBAABA
MCiKZ
ωαωωαωαωα
ωωωωωω
ωω
ωωω
Δ−=−
−−=⇒
⇒+⇒
+−=+⇒
−+=
−−−−−
−−−−
11111
1111
2
,
,
![Page 176: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/176.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
FRF Sensitivities (MDOF)
( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ]
( ) ( ){ } ( ){ } ( )[ ] ( )[ ]ATjx
TjAx
AxAx
Z
Z
ωαωωαωαωα
ωαωωαωαωα
Δ=−
Δ−=− ,
![Page 177: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/177.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
FRF Sensitivities (MDOF)
![Page 178: Modal Testing](https://reader034.vdocuments.us/reader034/viewer/2022042504/5868e0411a28ab12578be47e/html5/thumbnails/178.jpg)
Dr H Ahmadian ,Modal Testing Lab ,IUSTTheoretical Basis
FRF Sensitivities (MDOF)( )[ ] ( )[ ]( ) ( )[ ] ( )[ ] ( )[ ]
( )[ ] ( )[ ] ( )[ ] ( )[ ]
( )[ ] ( )[ ] [ ] [ ] [ ] ( )[ ]ωαωωωαωα
ωαωωαωα
ωωωωωα
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−∂∂
+∂∂
−=∂
∂
∂∂
−=∂
∂
∂∂
−=∂
∂=
∂∂ −−
−
pM
pCi
pK
p
pZ
p
Zp
ZZp
Zp
2
111