periodic structures

Periodic Structures: A PassivePeriodic Structures: A Passive Vibration Filter

Mohammad TawfikAero631 – Vibrations of Structures

What is a Periodic Structure?What is a Periodic Structure?

• A structure that consists fundamentally ofA structure that consists fundamentally of a number of identical substructure components that are joined together tocomponents that are joined together to form a continuous structure


Examples of periodic structuresExamples of periodic structures• Satellite panelsSatellite panels• Railway tracks• Aircraft FuselageAircraft Fuselage• Multistory buildings• Etc• Etc…


Structure Discontinuity!Structure Discontinuity!


Types of Discontinuity / Periodicity

Material Periodicityy

Geometric/SupportPeriodicityPeriodicity


Recall what happens to aRecall what happens to a wave as it travels through awave as it travels through a

boundary between two different media


Wave propagation in different media


Mechanical waves behave inMechanical waves behave in a similar way!a similar way!


Stop BandsStop Bands

• As the wave faces an abrupt change in the geometry a• As the wave faces an abrupt change in the geometry, a part if it is reflected

• The reflected part, interferes with the incident waveAt some frequency bands that interference becomes• At some frequency bands, that interference becomes destructive creating the “Stop Bands”


Stop bands are the center ofStop bands are the center of interest for the periodic panalysis of structures!


Periodic Analysis of Structures


Why Periodic Analysis?Why Periodic Analysis?

• Periodic structures can be modeled likePeriodic structures can be modeled like any ordinary structure, BUT

• In a periodic structure the study of the• In a periodic structure, the study of the behavior of one cell is enough to determine the stop and pass bands of thedetermine the stop and pass bands of the complete structure independent of the number of cellsnumber of cells


How?!How?!


Equations of MotionEquations of Motion

FUkkU

2

1

2

1

2221

1211

2

1

2221

1211

FF

UU

kkkk

UU

mmmm

1122

122

12112

11

FF

UU

kkmkmk

2222

22221

221 FUmkmk

111211 FUDD

222221 FUDD Rearranging the terms



1212111 FUDUD

2222121

1212111

FUDUD

11

121111

122

UDUDFFDUDDU

2221212 UDUDF

11 FDUDDU

11

12221111

1222212

112111122

FDDUDDDDF

FDUDDU



11

12111

122 UDDDU

11

1222111

1222212 FDDDDDDF

12

FU

eFU

12 FF

11 UDDDU

1

11

1222111

122221

121112

1

1

FU

DDDDDDDDD

FU

e



111211

FU

eFU

TTTT

112221 FFTT

TT

2221

1211

TTTT

sEigenvaluee

2221

Propagation factor


Note!Note!

• The transfer matrix is dependent on theThe transfer matrix is dependent on the excitation frequency

• Hence the propagation factor is• Hence, the propagation factor is dependent on the frequencyTh i l f th t f t i ill• The eigenvalues of the transfer matrix will appear in reciprocal pairs (.


Example: Periodic Spring MassExample: Periodic Spring Mass

W it d th ti f ti f th• Write down the equations of motion for the cell given by 2 half masses and one spring

2

1

2

1

2

1

00

ff

uu

kkkk

uu

mm


ExampleExample

• Getting the dynamic stiffness matrixGetting the dynamic stiffness matrix

112

2

ff

uu

mkkkmk

• Rearranging: 22 fumkk

21

222

2 11 uukkm

21

222

1 ffkm

kmkk


ExampleExample

• Getting the transfer matrix:Getting the transfer matrix:

11

2 11 uukkm

1

1

1

1222

1 fu

efu

kmk

kmk

kk

• Using Matlab to calculate the eigenvalues, we will get.g


The EigenvaluesThe Eigenvalues


The Propagation FactorThe Propagation Factor


Frequency Resp of CellFrequency Resp. of Cell


Freq Resp of 6 CellsFreq. Resp. of 6 Cells


HomeworkHomework

• Prepare a MATLAB program to performPrepare a MATLAB program to perform the periodic analysis of a bar.


MK &Modeling

MK &

Rearrangement

TEigenvalue problem

nval

ues(

Eige

n

(Hz)


MK &Modeling

MK &

Rearrangement

al(

TEigenvalue problem

Rea

eagin

ary(

(Hz)

Ima

FactornPropagatio


13 uu

111211 fukk

11131211 fukkk

11 1k

1

1

3

3

fe

f

fuekkk

2

1

2

1

2221

1211

ff

ukk

3

2

3

2

333231

232221

ff

uu

kkkkkk

1

1

12

2212

12

2211

1212

11

2

2

1

fu

kkk

kkk

kkk

fu

1

1

1

2

1

333231

232221

131211

0f

f

uuu

kkekekkkekkk

00

2

1

222321

321233311311

uu

kekkekkkekekk

2312 TTEigenvalueForward Approach KMEigenvalue 1pp

Reverse Approach KMEigenvalue


MK &Modeling

MK &

Rearrangement

(Hz)

MK 2Eigenvalueproblem

(

(H

z)

Imaginary(


Propagation CurvesPropagation CurvesForward Approach Reverse Approach

Attenuation Band

Propagation CurvesPropagation

Bandsgina

ry(

p gBands

Imag

(Hz)

(Hz) Imaginary(


Note!Note!

All the above mentionedAll the above mentioned analysis is independent of the y p

structure type(beams, bars, or plates)


So What really happens?So … What really happens?


Experimental InvestigationExperimental Investigation

• Bars with periodic geometry and materialBars with periodic geometry and material changes.

• Beams with periodic geometry• Beams with periodic geometry.• Plates with periodic geometry.


Periodic BarPeriodic Bar


ResultsResults


Experimental Setup for the Periodic Beam


Overview PictureOverview Picture


Beam CellBeam Cell


Case#1

20

30

9

10

0

10

20

ude

(dB

)

7

8

ad)

20

-10

00 500 1000 1500 2000 2500 3000 3500 4000

Func

tion

Am

plitu

4

5

6

enua

tin F

acto

r (ra

-30

-20

Tran

sfer

2

3

Atte

Plain BeamPeriodic BeamAttenuation Factor

-50

-40

Frequency (Hz)0

1


Case#2

10

20

9

10

0

10

0 500 1000 1500 2000 2500 3000 3500 4000

ude

(dB

)

7

8

ad)

-20

-10

r Fun

ctio

n A

mpl

itu

4

5

6

tenu

atin

Fac

tor (

ra

-40

-30Tran

sfer

2

3

Att

Plain B eamPeriod ic BeamAttenuation Factor

-50

40

Frequency (H z)0

1


Case#320

9

10

0

10

0 500 1000 1500 2000 2500 3000 3500 4000

ude

(dB

)

7

8

ad)

-20

-10

Func

tion

Am

plitu

4

5

6

enua

tin F

acto

r (ra

40

-30Tran

sfer

2

3

Atte

Plain BeamPeriodic BeamAttenuation Factor

-50

-40

Frequency (Hz)0

1


Periodic PlatePeriodic Plate


Problems Associated with 2-D Structures

• Wave propagates in 2-dimensionsWave propagates in 2 dimensions.• Input-Output relations are not readily

available (no forward approach)available (no forward approach)• Requires higher order elements for

i l l inumerical analysis


Wave propagates in 2-DWave propagates in 2 D

Wave is splitinto its componentsinto its componentsin X and Y-directions


No forward approach Reverse approach

(Hz)

MK 2Eigenvalueproblem

(

(H

z)

Imaginary(


Propagation Surfaces -AnalyticalAnalytical

Mead and Parathan 1979

xy

Parathan 1979


Requires higher order elements!Requires higher order elements!

64 DOF element used


Propagation Surfaces -Numerical

xy


ExperimentsExperiments


Periodic PlatePeriodic Plate


Propagation SurfacesPropagation Surfaces


ComparisonComparison


Effect of shunted inductanceEffect of shunted inductance


Vibration AbsorberVibration Absorber

0

bb WM

0

0

bbDb

DD

WW

KKKK

WM

DDDb WKK


Adding the InductanceAdding the Inductance

Inductancex

y


Further developmentsFurther developments


Further DevelopmentFurther Development

• More analytical numerical andMore analytical, numerical, and experimental studies need to further investigate the periodic plateinvestigate the periodic plate

• Periodic ShellsL it di l i di it i li d i l h ll– Longitudinal periodicity in cylindrical shell

– Circumferential periodicity in axisymmetric shellsshells


Effect of Shunt Circuit on Propagation Surfaces

Not Shunted Shunted


periodic structures

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