periodic structures
DESCRIPTION
What are periodic structures? Why are they important? How to analyze them? Simple examples and procedure to get you to understand periodic structures and their applications.TRANSCRIPT
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
Mohammad TawfikAero631 – Vibrations of Structures
Examples of periodic structuresExamples of periodic structures• Satellite panelsSatellite panels• Railway tracks• Aircraft FuselageAircraft Fuselage• Multistory buildings• Etc• Etc…
Mohammad TawfikAero631 – Vibrations of Structures
Structure Discontinuity!Structure Discontinuity!
Mohammad TawfikAero631 – Vibrations of Structures
Types of Discontinuity / Periodicity
Material Periodicityy
Geometric/SupportPeriodicityPeriodicity
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
Wave propagation in different media
Mohammad TawfikAero631 – Vibrations of Structures
Mechanical waves behave inMechanical waves behave in a similar way!a similar way!
Mohammad TawfikAero631 – Vibrations of Structures
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”
Mohammad TawfikAero631 – Vibrations of Structures
Stop bands are the center ofStop bands are the center of interest for the periodic panalysis of structures!
Mohammad TawfikAero631 – Vibrations of Structures
Periodic Analysis of Structures
Mohammad TawfikAero631 – Vibrations 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
Mohammad TawfikAero631 – Vibrations of Structures
How?!How?!
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
Equations of MotionEquations of Motion
1212111 FUDUD
2222121
1212111
FUDUD
11
121111
122
UDUDFFDUDDU
2221212 UDUDF
11 FDUDDU
11
12221111
1222212
112111122
FDDUDDDDF
FDUDDU
Mohammad TawfikAero631 – Vibrations of Structures
Equations of MotionEquations of Motion
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
Mohammad TawfikAero631 – Vibrations of Structures
Equations of MotionEquations of Motion
111211
FU
eFU
TTTT
112221 FFTT
TT
2221
1211
TTTT
sEigenvaluee
2221
Propagation factor
Mohammad TawfikAero631 – Vibrations of Structures
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 (.
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
The EigenvaluesThe Eigenvalues
Mohammad TawfikAero631 – Vibrations of Structures
The Propagation FactorThe Propagation Factor
Mohammad TawfikAero631 – Vibrations of Structures
Frequency Resp of CellFrequency Resp. of Cell
Mohammad TawfikAero631 – Vibrations of Structures
Freq Resp of 6 CellsFreq. Resp. of 6 Cells
Mohammad TawfikAero631 – Vibrations of Structures
HomeworkHomework
• Prepare a MATLAB program to performPrepare a MATLAB program to perform the periodic analysis of a bar.
Mohammad TawfikAero631 – Vibrations of Structures
MK &Modeling
MK &
Rearrangement
TEigenvalue problem
nval
ues(
Eige
n
(Hz)
Mohammad TawfikAero631 – Vibrations of Structures
MK &Modeling
MK &
Rearrangement
al(
TEigenvalue problem
Rea
eagin
ary(
(Hz)
Ima
FactornPropagatio
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
MK &Modeling
MK &
Rearrangement
(Hz)
MK 2Eigenvalueproblem
(
(H
z)
Imaginary(
Mohammad TawfikAero631 – Vibrations of Structures
Propagation CurvesPropagation CurvesForward Approach Reverse Approach
Attenuation Band
Propagation CurvesPropagation
Bandsgina
ry(
p gBands
Imag
(Hz)
(Hz) Imaginary(
Mohammad TawfikAero631 – Vibrations of Structures
Note!Note!
All the above mentionedAll the above mentioned analysis is independent of the y p
structure type(beams, bars, or plates)
Mohammad TawfikAero631 – Vibrations of Structures
So What really happens?So … What really happens?
Mohammad TawfikAero631 – Vibrations of Structures
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.
Mohammad TawfikAero631 – Vibrations of Structures
Periodic BarPeriodic Bar
Mohammad TawfikAero631 – Vibrations of Structures
ResultsResults
Mohammad TawfikAero631 – Vibrations of Structures
Experimental Setup for the Periodic Beam
Mohammad TawfikAero631 – Vibrations of Structures
Overview PictureOverview Picture
Mohammad TawfikAero631 – Vibrations of Structures
Beam CellBeam Cell
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
Periodic PlatePeriodic Plate
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
Wave propagates in 2-DWave propagates in 2 D
Wave is splitinto its componentsinto its componentsin X and Y-directions
Mohammad TawfikAero631 – Vibrations of Structures
No forward approach Reverse approach
(Hz)
MK 2Eigenvalueproblem
(
(H
z)
Imaginary(
Mohammad TawfikAero631 – Vibrations of Structures
Propagation Surfaces -AnalyticalAnalytical
Mead and Parathan 1979
xy
Parathan 1979
Mohammad TawfikAero631 – Vibrations of Structures
Requires higher order elements!Requires higher order elements!
64 DOF element used
Mohammad TawfikAero631 – Vibrations of Structures
Propagation Surfaces -Numerical
xy
Mohammad TawfikAero631 – Vibrations of Structures
ExperimentsExperiments
Mohammad TawfikAero631 – Vibrations of Structures
Periodic PlatePeriodic Plate
Mohammad TawfikAero631 – Vibrations of Structures
Periodic PlatePeriodic Plate
Mohammad TawfikAero631 – Vibrations of Structures
Propagation SurfacesPropagation Surfaces
Mohammad TawfikAero631 – Vibrations of Structures
ComparisonComparison
Mohammad TawfikAero631 – Vibrations of Structures
Effect of shunted inductanceEffect of shunted inductance
Mohammad TawfikAero631 – Vibrations of Structures
Vibration AbsorberVibration Absorber
0
bb WM
0
0
bbDb
DD
WW
KKKK
WM
DDDb WKK
Mohammad TawfikAero631 – Vibrations of Structures
Adding the InductanceAdding the Inductance
Inductancex
y
Mohammad TawfikAero631 – Vibrations of Structures
Further developmentsFurther developments
Mohammad TawfikAero631 – Vibrations of Structures
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
Mohammad TawfikAero631 – Vibrations of Structures
Effect of Shunt Circuit on Propagation Surfaces
Not Shunted Shunted
Mohammad TawfikAero631 – Vibrations of Structures