“Theory Guides; Experiment Decides.” - METUae568/16/Lecture1.pdf · AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory

AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring

AE 568 - Experimental Analysis of Vibrating Structures

“Theory Guides; Experiment Decides.”



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.



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.



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.



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.



Structures Lab Capabilities



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 softwareof 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



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.


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



Theoretical Basis and Terminology


Theoretical (Analytical) Experimental

(Modal Testing)

Aim is to develop “Reliable Dynamic Models”!!



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.



Wind induced Vibration

Tacoma Narrows Bridge (Washington State), 1940



Forced Vibration - Resonance

London Millenium Bridge Opening,2000



Ground Resonance

A Chinook Helicopter



Flutter

Piper PA-30 Twin Comanche Aircraft Tail Flutter Test,1966



Flutter



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.



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.



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).



Theoretical Route to Vibration Analysis

Description of

Structure

Vibration

Modes

Response

Level

Structure will vibrate under

given excitation condition

SPATIAL MODEL

• Mass

• Stiffness

• Damping

MODAL MODEL

• Natural Frequencies

• Mode Shapes

• Modal Damping Factors

RESPONSE MODEL

• Set of FRFs

• Impulse Responses

Structure's physical characteristics

Structure's behaviour as a set of vibration modes



Experimental Route to Vibration Analysis

Response

Properties

Vibration

Modes

Structural

Model

Experimental Modal

Analysis

SPATIAL MODEL

• Mass

• Stiffness

• Damping

MODAL MODEL

• Natural Frequencies

• Mode Shapes

• Modal Damping Factors

RESPONSE MODEL

• Set of FRFs

• Impulse Responses



Basic Vibration Theory

SDOF Systems

(Single Degree of Freedom)

MDOF Systems

(Multi Degree of Freedom)

Continuous Systems

(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!)



Basic Vibration Theory

Terms Vibration Type Description

External Excitation

• Free • Forced

• Vibration induced by initial inputs only• Vibration subjected to one or more continuousexternal inputs

Presence of Damping

• Undamped• Damped

• Vibration with no energy loss or dissipation• Vibration with energy loss

Linearity of Vibration

• Linear Vibration • Non-linear Vibration

• Vibration for which superposition principle holds• Vibration that violates superposition principle

Predictability • Deterministic • Random

• The value of vibration is known at any given time• The value of vibration is not known at any giventime but the statistical properties of vibration areknown


The Phasor

Phasor is a vector that rotates in a counterclockwise direction with Angular velocity in the complex plane.

A


ω

{ }

{ } tjjtj

j

tj

eeAeAA

jSinCose

jwhereeAtjSintCosAA

ωφφω

θ

ω

θθ

ωω

⋅⋅=⋅=

±=

−=⋅=+=

+

±

1)(

11

)()(

1)()(


The Phasor


Real

Imaginary


The Phasor


)(2)(2

222

2

)( 2

πωω

ω

ωω

ωω

ω

ωωπ

+

+

=−=

=

==

tjtj

tj

tjtj

AeAe

Aejdt

Ad

AeAejdtAd

where



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)



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;

(Modal Model))or (,)(

0

0 nti

mkxetx

kxxm

ωωω ==

=+0)( =tf





(a) Undamped

Response Model

(In the form of FRF)

x and f are complex to accommodate both the Amplitude and Phase information

Complex if damping is not zero!Real if damping is zero!

)(1)(

)(

)(,)(

2

2

ωαω

ω

ω ωω

ωω

=−

==

=−

==

mkfxH

fexemk

xetxfetf

titi

titi

Force HarmonicResponsent Displaceme Harmonic)( =ωH




(a) Undamped

Circular Frequency (repetitiveness of the oscillation)

]/[11

][2][2

][]/[2

]/[2

]/[

scycleHz

Hzs

cyclecyclerad

sradf

srad

nnnnn

n

=

====πω

πω

πω

πω

ω




(a)Undamped

Simple Harmonic Motion:




(a)Undamped

Recall Phasor!




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.




(b) Viscous Damping (c)

Viscous dashpot c or damper, physical model for dissipating energy

Equation of motion for free vibration case;

In Laplace Domain,

]/[],/[],/[],[0

skgmNscmNkkgmkxxcxm

→→→=++

mk

mc

mcs −

±−=

2

2,1 22




(b) Viscous Damping

Overdamped: Both roots are real.

Underdamped: Two roots are complex conjugate.

Critically damped: Two equal real roots.

Condition Damped Criticallyfor Constant DampingConstant Damping Ratio Damping

frequency natural Undamped

222tCoefficien Damping Critical

=→

→

===→

ξ

ω

ω

n

nc mmkmkmc

mk

mc

⟩

2

2

mk

mc

⟨

2

2

mk

mc

=

2

2




frequency natural Damped

1s

quantity essDimensionl

21,2

→

−±−=

→=

d

d

nn

ccc

ωω

ξωξω

ξ

(b) Viscous Damping

Overdamped:

Underdamped:

Critically damped:

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.

1⟩ξ

1⟨ξ

1=ξ



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!




(c) Structural (Hysteretic) Damping (d or h)

Describes more closely the energy dissipation mechanism.

By making viscous damping rate vary inversly with the frequency.

Provides much simpler analysis for MDOF systems!

ωdc =

ξηω

ηω

ηξ

ωη

ωω 21222

Factor Loss Damping Structural

at n = →====

==→

=

mk

kmk

kmc

cc

kd

kc

c




A common unit of measurement for vibration amplitudes and RMS values isthe decibel (dB).

The decibel is defined in terms of the base 10 logarithm of the power ratio oftwo electrical signals (or as the ratio of the square of the amplitudes of twosignals)

Voltage ratios in dB are calculated by

dB Scale expands of compresses vibration response information

→

=

→

2

110

2

110

2

2

110

log20dB

log20log10dB

VV

xx

xx




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 foldincrease or decrease in frequency.

Numerically:( )

( )( ) decadedB

decadedB

sradsrad

dBsrad

/40100log20For /01log20For

/1001010 and/11010

2010log20/10

102

101

10

=⇒=⇒

=×=

==

ωω

ω

20 dB decrease!

20 dB increase!




Octave: A doubling or halving, usually applied to frequency.

Example: A 6 dB/octave gain: A gain change of 6 dB for each doubling orhalving of frequency.

Numerically:( )

( )( ) octavedB

octavedB

sradsrad

dBsrad

/2620log20For /145log20For

/20210 and/52

10

2010log20/10

102

101

10

≅⇒≅⇒

=×=

==

ωω

ω

6 dB decrease!

6 dB increase!

Slopes can be defined as either dB/octave or dB/decade. octavedBdecadedB /6/20 ≅∴



Definitions of FRFs;





A most effective way of investigation for modal analysis is using the Frequency Response Function (FRF)

RECEPTANCE:(recall)

)(fx

fexe)(H

fexe)mk(

xe)t(x,fe)t(f

ti

ti

titi

titi

ωαω

ω

ω

ω

ωω

ωω

===

=−

==

2

Force HarmonicResponsent Displaceme Harmonic)( =ωH




Alternative forms of FRF;

MOBILITY:

2 : Phase

)()( :Magnitude

)()(

)(

)(,)(

Y

2

πθθ

ωαωω

ωωαωω

ω

α

ω

ω

ωω

ωω

+=

=

===

=−

==

Y

ifexei

fxY

fexemk

xetxfetf

ti

ti

titi

titi



Summary;

The reciprocals of the three FRFs of an SDOF system.

πθπθθ

ωαωωωω

α +=+=

==

2 : Phase

:Magnitude

A

2

Y

)()(Y)(A

)(A

)(Y

)(

ω

ω

ωα

1 Responseon Accelerati

ForceMassApparent

1 ResponseVelocity

ForceInpedance Mechanical

1 Responsent Displaceme

ForceStiffness Dynamic

==

==

==





(b) Viscous Damping, c

Response Model (In the form of FRF)

( )

( )

( ) )c(imk)(F)(X)(A

)c(imki

)(F)(X)(Y

)c(imk)(F)(X)()(H

ωωω

ωωω

ωωω

ωωω

ωωωωωαω

+−−

==

+−==

+−===

2

2

2

2

:eAcceleranc

:Mobility

1 :Receptance




(c) Structural (Hysteretic) Damping, d or h

Response Model (In the form of FRF)

( )

( )

( ) ihmk)(F)(X)(A

ihmki

)(F)(X)(Y

ihmk)(F)(X)()(H

+−−

==

+−==

+−===

2

2

2

2

:eAcceleranc

:Mobility

1 :Receptance

ωω

ωωω

ωω

ωωω

ωωωωαω



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

Documents

“Theory Guides; Experiment Decides.” - METUae568/16/Lecture1.pdf · AE 568 Experimental Analysis of Vibrating Structures Dr. M. ŞAHİN – 2015/16 Spring Modal Analysis Theory