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Modal Testing(Lecture 1)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and Technology
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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)
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
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,…)
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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 ModeShapes (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 acontrolled environment/Resonance Testing/Mechanical ImpedanceMethod
Testing a component ora structure with theobjective of obtainingmathematical model of
dynamical/vibrationbehavior
Structural Analysis ofULTRA 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 frequencies
Mode shapes
Damping 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 Updating Correlation of experimental/analytical
model
Adjust/correct the analytical model
Optimization 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 ModelIdentification
Substructure process
The component modelis incorporated into thestructural assembly
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
Applications of Modal Testing (continued)
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
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
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
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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 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
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
Summary of Measurement
Methods
Basic measurement system: Single point excitation
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
Summary of Modal Analysis
Processes
Analysis 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 measurement
Curve-Fitting
Construct the required model
Different level of details and accuracy in
above procedure is required dependingon the application.
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
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
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
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.
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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 Technology
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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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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.
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Cubic stiffness nonlinearity
)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 &&&
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Cubic stiffness nonlinearity
⎪⎩
⎪⎨⎧
−=
=++−⇒
−
=⎟ ⎠
⎞⎜⎝
⎛ −
+++−
)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
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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
Coulomb friction nonlinearity
( )
⎟
⎠
⎞⎜
⎝
⎛ ++−
=
=
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&
&
&
&
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Coulomb friction nonlinearity
X increasing
⎟ ⎠ ⎞⎜
⎝ ⎛ ++−
=
X
ccimk F
X
F
πω ω ω 4
1
2
Othe nonlinea ities and othe
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Other nonlinearities and other
descriptions
Backlash Bilinear Stiffness
Microslip friction damping Quadratic (and other power law
damping)
…..
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Modal Testing(Lecture 2)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and Technology
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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.
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Frequency Domain:
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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 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 −=
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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
Stiffness and Mass Lines
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Nyquist Plot For viscous damping the Mobility plot is a
circle.
For structural damping the Receptance andInertance 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 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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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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 Systems
MDOF Systems with ProportionalDamping
MDOF Systems with General StructuralDamping
General Force Vector
Undamped Normal Mode
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
.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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Example (continued):
.64.2118
48
62
05
54
5.0)(
422212ω ω ω ω
ω α +−
=−
−−
=ee
e
ee
MDOF Systems with
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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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
142.0493.0635.0
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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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(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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
General Force Vector In many situations
the system isexcited at severalpoints.
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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 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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
General Force Vector (continued)
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.
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
General Force Vector (continued)
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 =
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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?
Undamped Normal Mode
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
Undamped Normal Mode
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Modal Testing(Lecture 4)
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 General Force Vector
Undamped Normal Mode
MDOF System with General Viscous
Damping
Force Response Solution/ General
Viscous Damping
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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:
[ ] [ ]{ } { } 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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
General Force Vector (continued)
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 =
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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?
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Undamped Normal Mode
512 channel
37 Shakers
Undamped Normal Mode
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
(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)
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
(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)
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
(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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Force Response Solution Single point excitation:
∗
∗∗
= −+−= ∑ r
kr jr N
r r
kr jr
jk si
uu
si
uu
ω ω ω α 1
)(
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Modal Testing(Lecture 5)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
Modal Analysis of Rotating
St t
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Non-symmetric Stator y x k k ≠
FRF of the Rotating Structure
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
FRF of the Rotating Structure
with External Damping
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
with External Damping
Complex ModeShapes
due to significantimaginary part
Out-of-balance excitation
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
Synchronous OOB Excitation
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Synchronous OOB Excitation
Non-Synchronous OOB
Excitation
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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.
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Modal Testing(Lecture 6)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and [email protected]
Theoretical Basis
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical 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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
Analysis using rotating frame
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Analysis using rotating frame
[ ]
[ ] [ ]
[ ] [ ] [ ]⎭⎬⎫
⎩⎨⎧
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
Analysis using rotating frame
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Analysis using rotating frame
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
Analysis using rotating frame
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Analysis using rotating frame
⎭⎬⎫
⎩⎨⎧
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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)
Dynamic Analysis of General
Rotor-Stator Systems
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
Dynamic Analysis of General
Rotor-Stator Systems
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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[ ]{ } [ ] [ ]( ){ }
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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)[ ]{ } [ ]{ } { }
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
( 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)
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
( p g y) Symmetric Rotor/ Non-symmetric Support
Backwards Whirl
Forwards Whirl
FRF of LTI Rotor-Stator
Systems[ ] [ ] ( )[ ] [ ]H1
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
y[ ] [ ] ( )[ ] [ ] H
LH r RH V iV 1
)( −
−= ω λ ω α
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LTI Systems Eigen-Propertiesλ
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Skew-symmetry in stiffness Matrix
[ ] [ ]
[ ] ⎥
⎦
⎤⎢
⎣
⎡
−
−Δ−+⎥
⎦
⎤⎢
⎣
⎡
−Δ=
=⎥
⎦
⎤⎢
⎣
⎡=
31
13)1(
01
10
,0,
1
1
K K K
C M
LTI Systems Eigen-PropertiesXλ
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
-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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Modal Testing(Lecture 7)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and Technology
Complex Measured Modes
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
Complex Measured Modes
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
Display of Mode Complexity
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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
Extracting real modes from
complex measured 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 modesNormalizing the complex measured
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
Normalizing the complex measured
mode shape:
The problem is rewritten as:
Extracting real modes
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
Skew-symmetricSymmetric
Rank 2 matrices
Extracting real modes Since V is skew symmetric
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
Since V is skew symmetric,
Therefore the problem is equivalent to:
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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
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Follow-ups: E. Foltete, J. Piranda, “Transforming Complex
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Dr H Ahmadian,Modal Testing Lab,IUSTOverview of Modal Testing
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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Modal Testing(Lecture 8)
Dr. Hamid AhmadianSchool of Mechanical Engineering
Iran University of Science and Technology
Theoretical Basis Non-sinusoidal Vibration and FRF
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Periodic Vibration Excitation is not simply sinusoidal but retain
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
∫
∫
∞+
∞−
+∞
∞−
−
=
=
=
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)
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
)()(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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
∫
∫∞+
∞−
−
∞+
∞−
=
+=
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Time Signal
Autocorrelation
Power Spectral Density
Random Vibration
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Sinusoidal Signal
Autocorrelation
Power Spectral Density
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Random Vibration
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
Autocorrelation
Noisy Signal
Random Vibration
The autocorrelation function is real and even:
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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.
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Random Vibration
Time Domain Frequency Domain
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
)()()()()()(
)()()()()()(
)()()()()()(
*
*
*
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:
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
)()(
)()()()()(
)()(
)()()()()(
)()()(
*
*
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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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)
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Incomplete Response Models
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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Theoretical Basis
Sensitivity of Models
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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:
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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 =
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
.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
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
[ ] [ ]{ } { }
[ ] [ ]( ){ } { }
[ ] [ ]( ) { } [ ] [ ] [ ] { } { },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
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Eigenvector Sensitivity(MDOF)
: fromStarting
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
[ ] [ ]( ) { } [ ][ ]
[ ]{ } { }
{ } { }
[ ] [ ]( ) { } [ ] [ ] [ ] { } { }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
Eigenvector Sensitivity(MDOF)
[ ] [ ]( ) { } [ ][ ]
[ ]{ } { }02
22 =⎟⎟
⎞⎜⎜⎛
∂
∂−
∂
∂−
∂
∂+− ∑
=r r
r N
jrjr p
M M
p p
K M K φ ω
ω φ γ ω
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
{ } [ ] [ ]( ) { } { }
[ ]
[ ]
[ ]
{ } { }
( ) { } [ ] [ ] { } { }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
Eigenvector Sensitivity(MDOF)
( ) { } [ ] [ ]
{ }⎟⎟ ⎠
⎞⎜⎜⎝
⎛
∂
∂−
∂
∂+−⇒ r r
T
srsr s p
M
p
K 222 φ ω φ γ ω ω
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
{ } [ ] [ ]
{ }
( )
{ } { }
[ ] [ ]{ }
( ) { }∑
≠= −
⎟
⎠
⎞⎜
⎝
⎛
∂
∂−
∂
∂
=∂
∂⇒
−
⎟ ⎠
⎞⎜⎝
⎛
∂
∂−
∂
∂
=⇒
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
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Updating, Redesign,Reanalysis
The change in parameters must be very
ll f t l i
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
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
−+=
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
( )
( )
( )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 ,
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
[ ] [ ]( ) [ ] [ ] [ ]( ) [ ][ ]
[ ] ( )[ ] [ ] ( )[ ]
( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ]( ) ( )[ ]
( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ] 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
,
FRF Sensitivities (MDOF)
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
( )[ ] ( )[ ] ( )[ ] ( )[ ] ( )[ ]
( ) ( ){ } ( ){ } ( )[ ] ( )[ ] A
T
j x
T
j A x
A x A x
Z
Z
ω α ω ω α ω α ω α
ω α ω ω α ω α ω α
Δ=−
Δ−=− ,
FRF Sensitivities (MDOF)
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
FRF Sensitivities (MDOF)
( )[ ] ( )[ ]( )[ ]
( )[ ]( )[ ]ω
ω ω
ω ω α
∂
∂−=
∂
∂=
∂
∂ −−−
Z
p
Z Z
p
Z
p
111
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Dr H Ahmadian,Modal Testing Lab,IUSTTheoretical Basis
( )[ ]( )[ ]
( )[ ]( )[ ]
( )[ ] ( )[ ] [ ] [ ] [ ] ( )[ ]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