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AE 568 - Experimental Analysis of Vibrating Structures
“Theory Guides; Experiment Decides.”
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Experimental vibration analysis of engineering structures is a field
of increasing importance and popularity for researchers as
consequence both of
significant
technological
improvement
of
measurement
equipments and theoretical formulations
and of the extreme importance on the structural safety,
serviceability conditions and durability of vibrating structures.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Experimental vibration analysis
is one of the most important tools for analysing dynamic
properties of mechanical structures as the information obtained is
used in the development or modification of structures to obtain a
desired dynamic behaviour.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
This course is designed to use the experimental techniques in
vibration measurements and thus to provide the students
especially for the ones working on structural dynamics,
mechanical vibrations and modal testing areas by providing
unique inside on the general understanding of vibration test
planning, selection and use of exciters, transducers and sensors,
data collection, processing and assessment in particular with
hands on environment for modal analysis and testing.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Therefore, the course mainly focuses on
- investigating structural vibrations by putting particular
emphasize on the real application of experimental techniques in
vibration measurements by maintaining the balance between
theory and practical training.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Structures Lab Capabilities
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Structures Lab Capabilities
 SOFTWARE




MATLAB 2009a
ANSYS 11.0
MSC PATRAN/ NASTRAN 2007r1
NI LabVIEW 8.6
 HARDWARE
 B&K 6 channel Pulse portable data acquisition unit with special software
of FFT Analysis, Time Data Record, Modal Test Consultant,
Operational Modal Analysis, Reflex Modal Analysis Software
 B&K Modal Vibration Exciter (200N)
 B&K Impact Hammer
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
AE 568 - Experimental Analysis of Vibrating Structures
Structures Lab Capabilities
 HARDWARE - Cont’
 Various B&K Single-axis, Triaxial and miniature accelorometers,
Empedance head.
 Keyence Laser Displacement Sensor
 Polytec Scanning Laser Vibrometer
(New! – METUWIND Structural Dynamics LAB)
 Agilent Signal Generator
 Hameg Oscilloscope
Additionally,
 Various Uni-axial Strain Gauges and Installation Kits
 Dedicated equipment for smart structure applications comprising programmable
controller (SS10), high voltage power amplifiers, high voltage power supplies,
preamplifiers and piezoelectric (PZT) patches in various size and shape.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Lecture # 1
1. Modal Analysis Theory
1.1 Theoretical Basis and Terminology
1.2 Modal Analysis of SDoF Dynamic Systems
1.3 Modal Analysis of MDoF Dynamic Systems
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Theoretical Basis and Terminology
Modal Analysis Theory
Theoretical (Analytical)
Experimental
(Modal Testing)
Aim is to develop “Reliable Dynamic Models”!!
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Application of Modal Analysis:
• Identification and evaluation of vibration phenomena
• Validation, correction and updating of analytical dynamic models
• Development of experimentally based dynamic models
• Structural integrity assessment
• Structural modification and damage detection
• Reduction of mathematical models
• Determining, improving and optimising dynamic characteristics of
engineering structures.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Wind induced Vibration
Tacoma Narrows Bridge (Washington State), 1940
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Forced Vibration - Resonance
London Millenium Bridge Opening,2000
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Ground Resonance
A Chinook Helicopter
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Flutter
Piper PA-30 Twin Comanche Aircraft Tail Flutter Test,1966
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Flutter
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Assumptions:
• Structure is linear and time-invariant
• Structure obeys Maxwell’s Reciprocity Theorem.
• About FRFs (Frequency Response Functions)
• The positions of the shaker and accelerometer are reversed
in multiple single-input RECIPROCITY checks!
i.e. Various seperate single-input tests with the shaker
located at different position for each test.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Assumptions:
• Structure obeys Maxwell’s Reciprocity Theorem.
The measured FRF for a force at location j and response at location i should
correspond directly with the measured FRF for a force at location i and response
at location j. The FRF matrix is symmetric and this property can be used as a
check on the quality of the measured data.
MODAL ANALYSIS is the process of determining the
inherent dynamic characteristics of a system in the forms of
Natural Frequencies, Damping Factors and Mode Shapes.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
The three main phases of modal testing
• The theoretical basis of vibration,
• Accurate measurement of vibration (Controlled testing conditions),
• Realistic and detailed data analysis (Signal processing, Range of
curve fitting procedures in an attempt to find the mathematical model which
provides the closest description of the actually observed behaviour).
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Theoretical Route to Vibration Analysis
Structure will vibrate under
Structure's physical characteristics
given excitation condition
Description of
Vibration
Response
Structure
Modes
Level
SPATIAL MODEL
MODAL MODEL
RESPONSE MODEL
• Mass
• Natural Frequencies
• Set of FRFs
• Stiffness
• Mode Shapes
• Impulse Responses
• Damping
• Modal Damping Factors
Structure's behaviour as a set of vibration modes
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Experimental Route to Vibration Analysis
Experimental Modal
Analysis
Response
Vibration
Structural
Properties
Modes
Model
RESPONSE MODEL
MODAL MODEL
SPATIAL MODEL
• Set of FRFs
• Natural Frequencies
• Mass
• Impulse Responses
• Mode Shapes
• Stiffness
• Modal Damping Factors
• Damping
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Basic Vibration Theory
SDOF Systems
MDOF Systems
Continuous Systems
(Single Degree of Freedom)
(Multi Degree of Freedom)
(Infinitely many
number of DOF)
DOF: The minimum number of independent coordinates required to determine
completely the motion of all parts of the system at any instant of time.
A different selection of coordinates will lead to different equations of motion
but end up with same natural frequencies regardless of the choice of coordinates
(i.e. same system!)
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Basic Vibration Theory
Terms
Vibration Type
Description
External
Excitation
• Free
• Forced
• Vibration induced by initial inputs only
• Vibration subjected to one or more continuous
external inputs
Presence of
Damping
• Undamped
• Damped
• Vibration with no energy loss or dissipation
• Vibration with energy loss
Linearity of
Vibration
• Vibration for which superposition principle holds
• Linear Vibration
• Non-linear Vibration • Vibration that violates superposition principle
Predictability
• Deterministic
• Random
AE 568 Experimental Analysis of Vibrating Structures
• The value of vibration is known at any given time
• The value of vibration is not known at any given
time but the statistical properties of vibration are
known
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
The Phasor
Phasor A is a vector that rotates in a counterclockwise direction with
Angular velocity ω in the complex plane.
A = A{Cos (ωt ) + jSin(ωt )} = A ⋅ e jωt
where j = − 1
e ± jθ = Cos (θ ) ± jSin(θ )
{
}
A1 = A1 ⋅ e j (ωt +φ ) = A1 ⋅ e jφ ⋅ e jωt
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
The Phasor
Imaginary
Real
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
The Phasor
dA
j (ωt + π2 )
j ωt
= jωAe = ωAe
dt
d2A
j ωt
2 2
j
Ae
=
ω
dt 2
= −ω 2 Ae j (ωt ) = ω 2 Ae j (ωt +π )
where
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Although very few practical structures could realistically be modeled by
SDOF System, a mode complex multi-degree-of-freedom (MDOF)
system can always be represented as the linear superposition of a
number of SDOF systems.
(a) Undamped
(b) Viscously Damped (c)
(c) Hysterically (or structurally) Damped (d or h)
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Dynamic Properties of Mechanical Systems:
•
•
•
Mass (Responsible for Inertia),
Stiffness (Responsible for Elastic Forces),
Damping (Responsible for Dissipative Forces)
(a) Undamped
Spatial Model: m, k (Simple Harmonic Oscillator = Spring&Mass System)
If no forcing;
f (t ) = 0
mx + kx = 0
x(t ) = xeiωt , ω0 =
AE 568 Experimental Analysis of Vibrating Structures
k
m
(or ωn )
(Modal Model)
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(a) Undamped
Response Model
f (t ) = feiωt , x(t ) = xeiωt
(k − ω 2 m) xeiωt = feiωt
H (ω ) =
x
1
=
= α (ω )
2
f k −ω m
(In the form of FRF)
x and f are complex to accommodate both the Amplitude and Phase information
Harmonic Displacement Response
H (ω ) =
Harmonic Force
AE 568 Experimental Analysis of Vibrating Structures
Complex if damping is not zero!
Real if damping is zero!
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(a) Undamped
Circular Frequency ωn
fn =
[rad / s ] (repetitiveness of the oscillation)
ωn
ωn [rad / s ]
ω [cycle] ωn
=
= n
=
[ Hz ]
2π
2π 2π [rad / cycle]
2π [ s ]
1 Hz = 1 [cycle / s ]
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(a) Undamped
Simple Harmonic Motion:
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(a) Undamped
Recall Phasor!
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Useful quantities describing the vibration;
Average Value:
Mean Square Value:
Square of displacement is associated with a system’s potential energy.
Average of the displecement squared is also a useful vibration property.
Root Mean Square (RMS) Value:
Square root of the Mean Square value is commonly used in specifying
vibration.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(b) Viscous Damping (c)
Viscous dashpot c or damper, physical model for dissipating energy
Equation of motion for free vibration case;
mx + cx + kx = 0
m → [kg ], k → [ N / m], c → [ Ns / m], [kg / s ]
In Laplace Domain,
2
s1, 2
c
k
 c 
−
=−
± 

2m
 2m  m
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(b) Viscous Damping
2
 c  k
Overdamped: Both roots are real.  2m  〉 m
2
 c  k
〈
Underdamped: Two roots are complex conjugate.  2m 
m
2
Critically damped: Two equal real roots.  c  = k
m
 2m 
cc → Critical Damping Coefficient = 2 km = 2m
k
= 2mωn
m
ωn → Undamped natural frequency
ξ → Damping Ratio =
Damping Constant
Damping Constant for Critically Damped Condition
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(b) Viscous Damping
Overdamped:
ξ=
ξ 〉1
Underdamped: ξ 〈1
Critically damped:
ξ =1
c
→ Dimensionless quantity
cc
s1,2 = −ωnξ ± ωn ξ 2 − 1

ωd
ωd → Damped natural frequency
Damping ratio for critically damped systems seperates oscillatory motion
from nonoscillatory motion and critical damping is the value of damping that
provides the fastest return to zero without oscillation.
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Critical Damping
Army gun firing – Explosion and recoil take a few miliseconds
With recovery of less than 1 second.
In SDOF systems, it all about using all the spring’s potential energy!
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(c) Structural (Hysteretic) Damping (d or h)
Describes more closely the energy dissipation mechanism.
By making viscous damping rate vary inversly with the frequency.
c=
d
ω
Provides much simpler analysis for MDOF systems!
η → Structural Damping Loss Factor =
ξ=
ηk
η 1
c
c
=
=
=
cc 2 km 2 kmω 2 ω
AE 568 Experimental Analysis of Vibrating Structures
c
d
ω=
k
k
k at ω = ω n
  
→ η = 2ξ
m
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
A common unit of measurement for vibration amplitudes and RMS values is
the decibel (dB).
The decibel is defined in terms of the base 10 logarithm of the power ratio of
two electrical signals (or as the ratio of the square of the amplitudes of two
signals)
2
x 
x 
dB → 10 log10  1  = 20 log10  1 
 x2 
 x2 
Voltage ratios in dB are calculated by
V 
dB → 20 log10  1 
 V2 
dB Scale expands of compresses vibration response information
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Decade: A 1:10 increase or decrease of a variable, usually in frequency.
Example: A 20 dB/decade gain: A gain change of 20 dB for each 10 fold
increase or decrease in frequency.
Numerically:
ω = 10 rad / s
20 log10 (10 ) = 20 dB
10
= 1 rad / s and 10 ×10 = 100 rad / s
10
For ω1 ⇒ 20 log10 (1) = 0 dB / decade
For ω2 ⇒ 20 log10 (100 ) = 40 dB / decade
AE 568 Experimental Analysis of Vibrating Structures
20 dB decrease!
20 dB increase!
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Octave: A doubling or halving, usually applied to frequency.
Example: A 6 dB/octave gain: A gain change of 6 dB for each doubling or
halving of frequency.
Numerically:
ω = 10 rad / s
20 log10 (10) = 20 dB
10
= 5 rad / s and 10 × 2 = 20 rad / s
2
For ω1 ⇒ 20 log10 (5) ≅ 14 dB / octave
For ω2 ⇒ 20 log10 (20 ) ≅ 26 dB / octave
6 dB decrease!
6 dB increase!
Slopes can be defined as either dB/octave or dB/decade. ∴ 20 dB / decade
AE 568 Experimental Analysis of Vibrating Structures
≅ 6 dB / octave
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Definitions of FRFs;
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
A most effective way of investigation for modal analysis is using the Frequency
Response Function (FRF)
RECEPTANCE:
(recall)
f ( t ) = fe iωt , x( t ) = xeiωt
( k − ω 2 m )xeiωt = fe iωt
xeiωt x
H ( ω ) = i ωt = = α ( ω )
fe
f
Harmonic Displacement Response
H (ω ) =
Harmonic Force
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Alternative forms of FRF;
f (t ) = fe iωt , x(t ) = xeiωt
MOBILITY:
(k − ω 2 m) xeiωt = fe iωt
x iωxeiωt
Y (ω ) = =
= iωα (ω )
i ωt
f
fe
Magnitude : Y (ω ) = ω α (ω )
Phase
AE 568 Experimental Analysis of Vibrating Structures
: θ Y = θα +
π
2
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
Summary;
Magnitude : A( ω ) = ω Y ( ω ) = ω 2 α ( ω )
Phase
: θ A = θY +
π
2
= θα + π
The reciprocals of the three FRFs of an SDOF system.
Dynamic Stiffness =
Force
1
=
Displacement Response α ( ω )
Mechanical Inpedance =
Apparent Mass =
AE 568 Experimental Analysis of Vibrating Structures
Force
1
=
Velocity Response Y ( ω )
Force
1
=
Acceleration Response
A( ω )
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(b) Viscous Damping, c
Response Model (In the form of FRF)
Receptance :
Mobility :
H( ω ) = α( ω ) =
1
X(ω )
=
F ( ω ) k − ω 2 m + i( ωc )
(
)
X ( ω )
iω
=
Y(ω ) =
F ( ω ) k − ω 2 m + i( ωc )
Accelerance :
(
)
−ω2
X ( ω )
=
A( ω ) =
F ( ω ) k − ω 2 m + i( ωc )
AE 568 Experimental Analysis of Vibrating Structures
(
)
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
Modal Analysis of SDOF Dynamic Systems
(c) Structural (Hysteretic) Damping, d or h
Response Model (In the form of FRF)
Receptance :
Mobility :
H( ω ) = α( ω ) =
1
X(ω )
=
F ( ω ) k − ω 2 m + ih
(
)
X ( ω )
iω
=
Y(ω ) =
F ( ω ) k − ω 2 m + ih
Accelerance :
(
)
−ω2
X ( ω )
=
A( ω ) =
F ( ω ) k − ω 2 m + ih
AE 568 Experimental Analysis of Vibrating Structures
(
)
Dr. M. ŞAHİN – 2015/16 Spring
Modal Analysis Theory
1. Modal Analysis Theory
1.1 Theoretical Basis and Terminology
1.2 Modal Analysis of SDoF Dynamic Systems (cont’)
1.3 Modal Analysis of MDoF Dynamic Systems
AE 568 Experimental Analysis of Vibrating Structures
Dr. M. ŞAHİN – 2015/16 Spring
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