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Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
Periodic Structures - A Passive Vibration Damper
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Periodic Structures - A Passive Vibration Damper

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What is a periodic structure? …

What is a periodic structure?
How do the reduce vibrations?
How to analyse a periodic beam, bar, or plate?

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https://wikicourses.wikispaces.com/Topic+Periodic+Structures

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  • 1. Passive Vibration AttenuationViscoelastic Damping, Shunt Piezoelectric Patches, and Periodic StructuresMohammad Tawfik
  • 2. Periodic Structures: A Passive Vibration Filter Periodic StructuresContents1. Periodic Structures: A Passive Vibration Filter.................................................................................... 3 1.1. Periodic Structures ....................................................................................................................... 3 1.2. Literature Survey.......................................................................................................................... 3 1.3. Periodic Analysis .......................................................................................................................... 5 1.4. Periodic Bars ................................................................................................................................ 9 1.4.1. Forward approach for a periodic bar .................................................................................... 9 1.4.2. Reverse approach for a periodic bar................................................................................... 12 1.4.3. Experimental Work ............................................................................................................. 13 1.5. Periodic Beams........................................................................................................................... 14 1.5.1. Beams with Periodic Geometry .......................................................................................... 15 1.5.2. Experimental Work ............................................................................................................. 16 1.6. Propagation Surfaces for Periodic Plates ................................................................................... 21 1.6.1. Input-Output Relations ....................................................................................................... 22 1.6.2. Propagation Surfaces .......................................................................................................... 23 1.6.3. Constant angle curves ......................................................................................................... 25 1.6.4. Plates with Periodic Geometry ........................................................................................... 30 1.6.5. Experimental Work ............................................................................................................. 30 1.7. Effect of Shunted Piezoelectric Patches on Propagation Surfaces ............................................ 30 1.7.1. Propagation Surfaces for Coupled System.......................................................................... 30 1.8. Appendices................................................................................................................................. 34 1.8.1. Appendix A .......................................................................................................................... 34 1.8.2. Nomenclature ..................................................................................................................... 37 1.9. References and Bibliography ..................................................................................................... 38Passive Vibration Attenuation 2
  • 3. Periodic Structures: A Passive Vibration Filter Periodic Structures1. Periodic Structures: A Passive Vibration Filter1.1. Periodic StructuresThe first question that anyone may ask is: what is a Periodic Structure? The definition of a periodicstructure, according to Mead [‎ 8], is that it is one that consists fundamentally of a number of 7identical substructure components that are joined together to form a continuous structure. Periodicstructures are seen in many engineering products, examples of periodic structures may includesatellite solar panels, railway tracks, aircraft fuselage, multistory buildings, etc …Following the above definition of periodic structure, there must be a distinction between differentsubstructures that defines the individual unit, that distinction or boundary will introduce a suddenchange in the properties of the structure. Two main types of discontinuities may be identifies,namely: geometric discontinuity and material discontinuity. Figure ‎ .1 shown a sketch of the two 1different types of discontinuities. (a) (b)Figure ‎ .1. Types of discontinuities (a) Material discontinuity (b) Geometric dicontinuity 1 Recall what happens to a wave as it travelsthrough a boundary between two differentmedia; part of the light wave refracts inside thewater and another part reflects back into theair. Mechanical waves behave in a similar way! Now, imagine a rod, as example of 1-Dstructures. As the wave propagates through therod, it faces a discontinuity in the structure. Apart of the wave reflects and another partpropagates into the new part. The reflectedpart of the wave will, definitely, interfere with Figure ‎ .2. Sketch of light wave behaviour when 1the incident wave. incident on water surfaceThe interference between the incident and reflected waves will result, in some frequency band, indestructive interference. In the frequency band where destructive interference occurs, there will bereduced vibration level. This band is what we call Stop-Band. Stop bands are the center of interestfor the periodic analysis of structures (see section ‎ .3) 11.2. Literature SurveyIn his paper, reviewing the research performed in the area of wave propagation in periodicstructures, Mead [‎ 8] defined a periodic structure as a structure that consists fundamentally of a 7number of identical structural components that are joined together to form a continuous structure.Examples of periodic structures can be seen in satellite solar panels, wings and fuselages of aircraft,Passive Vibration Attenuation 3
  • 4. Periodic Structures: A Passive Vibration Filter Literature Surveypetroleum pipe-lines, and many others. An illustration of a simple periodic bar is presented inFigure ‎ .3. 1Figure ‎ .3. An illustration of a simple periodic bar. 1Studies of the characteristics of one-dimensional periodic structures have been extensively reported[‎ 9-‎ 4]. These structures are easy to analyze because of the simplicity of the geometry as well as the 7 9nature of coupling between neighbouring cells. Ungar [‎ 9] presented a derivation of an expression 7that could describe the steady state vibration of an infinite beam uniformly supported onimpedances. That formulation, easily allowed for the analysis of the structures with fluid loadings.Later, Gupta [‎ 0] presented an analysis for periodically-supported beams that introduced the 8concepts of the cell and the associated transfer matrix. He presented the propagation andattenuation parameters’ plots which form the foundation for further studies of one-dimensionalperiodic structures. Faulkner and Hong [‎ 1] presented a study of mono-coupled periodic systems. 8They analysed the free vibration of spring-mass systems as well as point-supported beams usinganalytical and finite element methods. Mead and Yaman [‎ 2] presented a study for the response of 8one-dimensional periodic structures subject to periodic loading. Their study involved thegeneralization of the support condition to involve rotational and displacement springs as well asimpedances. The effects of the excitation point as well as the elastic support characteristics on thepass and stop characteristics of the beam are presented.Other studies have also shown very promising characteristics of periodic structures for waveattenuation [‎ 6-‎ 4]. Langley [‎ 6] investigated the localization of a wave in a damped one- 8 9 8dimensional periodic structure using an energy approach. Later, Cetinkaya [‎ 0], by introducing 9random variation in the periodicity of one-dimensional bi-periodic structure, showed that thevibration can be localized near to the disturbance source. Using the same concept, Ruzzene and Baz[‎ 2] used shape memory inserts into a one-dimensional rod, and by activating or deactivating the 9inserts they introduced aperiodicity which in turn localized the vibration near to the disturbancesource. Then, they used a similar concept to actively localize the disturbance waves travelling in afluid-loaded shell [‎ 3]. Thorp et al. [‎ 4] applied the same concept to rods provided with shunted 9 9periodic piezoelectric patches which again showed very promising results.The analysis of periodic plates is of a specific importance as it relates to many practical structures[‎ 5-‎ 03]. Mead [‎ 5] presented a general theory for the wave propagation in multiply-coupled and 9 1 9two-dimensional periodic structures by reducing the number of degrees of freedom of the systembased on the propagation relation existing between the two ends of the structure. Mead andParathan [‎ 6] used the energy method [‎ 5] together with characteristic beam modes to describe the 9 9behaviour of plates. In that paper, they introduced the concept of “Propagation Surfaces” thatreflects the change of the dynamical behaviour of the periodic plate with the change in the directionand phase of propagating waves. Finally, Mead et al. [‎ 7] approached the wave propagation 9Passive Vibration Attenuation 4
  • 5. Periodic Structures: A Passive Vibration Filter Periodic Analysisproblem of a periodically stiffened plate using the finite element approach which utilizedhierarchical polynomials. The investigation of the acoustic characteristics of a periodic plate was alsostudied by Mead [‎ 8]. In that study, he used the methods developed in his previous three papers to 9extend the model to predict the structural-acoustic characteristics of a periodically stiffened plate.Mace [‎ 9] presented an analysis of a periodic plate that is supported on periodically-separated point 9supports. The solution procedure involved the use of the Fourier transform of the equation ofmotion and the support conditions. The analysis also extended to the prediction of the acousticloading and radiation from the vibrating surface of the plate.Langley [‎ 00,‎ 01] introduced analytical techniques for predicting the response of two-dimensional 1 1structures under point loading. The response to harmonic point loading [‎ 00] was studied and 1conclusions were drawn that showed the potential of using periodic two-dimensional structures asfilters. Similar results were obtained when analyzing the response of a periodic plate to pointimpulsive loading [‎ 01]. 1The analysis of elastically-supported plates was of great interest to many researchers as it representsmore realistic structures. Warburton and Edney [‎ 02] used the Rayleigh-Ritz method to analyse an 1elastically-supported periodic plate. Later, Mukherjee and Parathan [‎ 03] used the beam functions 1of Mead and Parathan [‎ 6] to analyze the behaviour of periodic plates with rotational stiffeners. 9They concluded that their proposed method is computationally efficient compared to finite elementmethod.1.3. Periodic AnalysisPeriodic structures can be modeled like any ordinary structure, but in a periodic structure, the studyof the behavior of one cell is enough to determine the stop and pass bands of the completestructure independent of the number of cells.Recall the equations of motion for a general body  m11 m12  U1   k11 k12  U1   F1 m            21 m22  U 2  k21 k22  U 2  F2 Where U is a vector presenting the displacements at a certainpoint in the structure, F is a general force vector; m and k aregeneral mass and stiffness terms depending on the modellingmethod. For harmonic excitation, we may write: k11   2 m11 k12   2 m12  U1   F1      k21   m21 k22   m22  U 2  F2  2 2 Figure ‎ .4. 1 General sketch for a structureFrom which, the dynamic stiffness matrix may be written as follows: D11 D12  U1   F1 D    21 D22  U 2  F2  Expanding the two equations, we get:Passive Vibration Attenuation 5
  • 6. Periodic Structures: A Passive Vibration Filter Periodic AnalysisD11U1  D12U 2  F1D21U1  D22U 2  F2Rearranging terms of the equations gives:  U 2   D121 D11U1  D121F1F2  D21U1  D22U 2Collecting right hand displacements and forces on the right hand side of the equations gives:  U 2   D121 D11U1  D121 F1  1  1F2  D21  D22 D12 D11 U1  D22 D12 F1In matrix form:U 2     D121 D11 D121  U1    1 1    F2   D21  D22 D12 D11 D22 D12   F1 Now, assume the input output relation for the given cell are in the form:U 2    U1  e   F2   F1 Then, we may write: U1     D121 D11 D121  U1  e   1 1     F1   D21  D22 D12 D11  D22 D12   F1 Giving the input output, transfer, relation as:T11 T12  U1   U1 T  F   e  F  21 T22   1   1Where the input output transformation matrix is called the transfer matrix T. From the aboverelation, we can clearly see that: T T e   Eigenvalue s  11 12  T21 T22 Note that the transfer matrix is dependent on the excitation frequency, hence, the propagationfactor is dependent on the frequency. Also, it can be proven that the eigenvalues of the transfermatrix will appear in reciprocal pairs ().Example ‎ .1: Periodic Spring Mass 1Passive Vibration Attenuation 6
  • 7. Periodic Structures: A Passive Vibration Filter Periodic AnalysisFigure ‎ .5. Sketch of the periodic spring mass system. 1Write down the equations of motion for the cell given by 2 half masses and one springm 0  u1   k   k  u1   f1  0 m     k    k  u2   f 2   u2   Then, we may get the dynamic stiffness matrixk   2 m  k  u1   f1       k k   2 m u2   f 2 Rearranging terms  2m 1 1   u   u  k k  1    2  k  k   m 2 2  1  m   f1   f 2  2 k k From which we may write the transfer matrix  2m 1  1   k k   u1   e   u1       k  m  k 2 2   2 m   f1   1  f1  k k Below, is the MATLAB code used to generate the results of this example.m=1; k=1;mc=[m,0;0,m];kc=[k,-k;-k,k];mg=[m 0 0 0 0 0 0 2*m 0 0 0 0 0 0 2*m 0 0 0 0 0 0 2*m 0 0 0 0 0 0 2*m 0 0 0 0 0 0 m];kg=[k -k 0 0 0 0 -k 2*k -k 0 0 0 0 -k 2*k -k 0 0 0 0 -k 2*k -k 0 0 0 0 -k 2*k -k 0 0 0 0 -k k];Passive Vibration Attenuation 7
  • 8. Periodic Structures: A Passive Vibration Filter Periodic Analysisfor ii=1:1001 freq(ii)=(ii-1)*0.002; KD=kc-freq(ii)*freq(ii)*mc; TT=[-KD(1,1)/KD(1,2) 1/KD(1,2) KD(2,2)*KD(1,1)/KD(1,2)-KD(2,1) -KD(2,2)/KD(1,2)]; Lamda(:,ii)=sort(eig(TT)); Mew(ii)=acosh(0.5*(Lamda(1,ii)+Lamda(2,ii))); Resp=inv(KD)*[1;0]; xx(ii)=20*log(abs(Resp(2))); KG=kg-freq(ii)*freq(ii)*mg; Resp=inv(KG)*[1;0;0;0;0;0]; yy(ii)=20*log(abs(Resp(6)));endsubplot(4,1,1); plot(freq,Lamda(1,:),freq,Lamda(2,:)); gridsubplot(4,1,2); plot(freq,real(Mew),freq,imag(Mew)); gridsubplot(4,1,3); plot(freq,xx); gridsubplot(4,1,4); plot(freq,yy); grid Figure ‎ .6. Variation of the eigenvalues with the 1 Figure ‎ .7. Variation of the real and imaginary 1excitation frequency parts of the propagation factor with the excitation frequency Figure ‎ .8. Frequency response of a single cell 1 Figure ‎ .9. Frequency response of the six cells 1From Figure ‎ .6 we may notice that the eigenvalues of the transfer matrix appear as complex 1conjugate for all frequencies below the cut-off frequency of the cell (Only real part is plotted). Frofrequencies above the cut-off frequency, the eigenvalues appear in real reciprocal pairs. Figure ‎ .7 1presents plot for the variation of the real and imaginary parts of the propagation factor μ. Note herethat the real part of the propagation factor is equal to zero for all frequency values below the cut-offPassive Vibration Attenuation 8
  • 9. Periodic Structures: A Passive Vibration Filter Periodic Barsfrequency. Further, we may notice that the imaginary part varies from 0 to π then it stays constantfor the frequency values at which the real part is non-zero. Figure ‎ .8 is a plot of the frequency 1response of the cell. In this plot we may also note that the response of the cell becomes less thanunity (0 dB) for higher frequencies. Finally, Figure ‎ .9 presents the response of the 6-mass spring 1system in which we may notice that the response also becomes less than unity for the higherfrequencies similar to that of a single cell.1.4. Periodic BarsOne-dimensional periodic structures will be our key-way towards better understanding of thephenomena associated with general periodic structures. Consider a unit cell of the periodic structureof Figure ‎ .3 and its free body diagram shown in Figure ‎ .10, we may define a relation between the 1 1force f3 and displacement u3 at the right hand side of the cell and f1 and u1 on the left hand side asfollows, u3    u1   e   (1)  f3   f1 where  is the propagation factor.On the other hand, the force-displacement relations of each of the parts of the cell could be writtenin terms of the dynamic stiffness matrix as follows,  D11 1 D12  u1   f1  1  1 1     , (2)  D12 D22  u2   f 2   D22 2 D23  u2   f 2  2and  2 2    (3)  D23 D33  u3   f 3  rwhere Dij is the dynamic stiffness coefficient relating the i’th force to the j’th displacement of ther’th element that can be determined using any technique such as finite element. Remember that thedynamic stiffness matrix of an element is a function of the excitation frequency.Figure ‎ .10. A free body diagram for a cell of the periodic bar. 1 1.4.1. Forward approach for a periodic barThe approach presented in this section for the analysis of the periodic characteristic of a bar is goingto be named the “forward approach”, in contrast with the “reverse approach” that will be presentedPassive Vibration Attenuation 9
  • 10. Periodic Structures: A Passive Vibration Filter Periodic Barslater. The forward approach starts with a physical input (excitation frequency) and advances todetermine the periodic characteristics of the bar, mainly presented in the propagation factor.For the first element, we may rearrange the equation (2-a) to be in the form,  D1 1    11  1 D12 D12  u1  u2  1      (4)  D1  D11D22 D22   f1   f 2  1 1 1  12 1 D12  1  D12 Similarly, for the second element, equation (3) can take the following form,  2 D11 1    D2 D12  u2  u3  2  12    2  (5)  D 2  D11D22 D22   f 2   f 3  2 2  12 2 D12  2  D12 Combining equations (1), (3) and (4) gives,  D2 1  D1 1    11  11  2 2D12 2 D12   2   1 1D12 1 D12  u1  1 u     e  1  (6)  D11D22  D 2  22   11 1 22  D12 D2 D D  22   1  1 1 D f  f1   D2 12 D12   D12 2 1   12  D12 which can be rewritten as, T11 T12  u1   u1  T  f   e  f  (7)  21 T22   1   1where, [T] is called the transfer matrix of the cell. The above equation is an Eigenvalue problem,similar to that obtained previously for the periodic mass spring system, in [T] which can be solveddirectly yielding the required Eigenvalues. Recall that the transfer matrix was derived from thedynamic stiffness matrix which is a function of the excitation frequency. It may be shown that theeigenvalues (’s) of the transfer matrix [T] appear in pairs such that one is the reciprocal of the other(i.e.  &1 /  ). Suggesting that these eigenvalues are e  and e   , which we can use to write  asfollows,  1   ArcCosh       i (8)  In general, the value obtained for the propagation factor  from equation (8) is a complex valuewhose imaginary part  defines the phase difference between the input and the output vibrationwaves, while the real part  denotes the attenuation in the vibration amplitude between the inputand the output.Passive Vibration Attenuation 10
  • 11. Periodic Structures: A Passive Vibration Filter Periodic BarsTo demonstrate the previous concepts, a test case was considered in which the modulus of elasticity(E) for both parts of the bar is 71 GPa, density () 2700 Kg/m3, smaller diameter 4 cm, largerdiameter 4 2 cm, and length of each part 1 m.The variation of the eigenvalues of the transfer matrix function of a unit cell with the excitationfrequency is plotted in Figure ‎ .11. For the frequency band in which the eigenvalues are presented 1by one branch, they appear as a complex conjugate pair. While, for the frequency band in whichthey have two distinct branches, the eigenvalues are real. Figure ‎ .11. A plot of the variation of the transfer 1 Figure ‎ .12. The variation of the real and imaginary 1matrix eigenvalues with the excitation frequency. parts of the propagation factor with the excitation frequency.The variation of the propagation parameter can thus be determined through equation (8). The realand imaginary parts of the propagation parameter are plotted in Figure ‎ .12. It should be noted at 1this point that the real and imaginary parts of the propagation parameter are varying withfrequency. The frequency band in which the real part is zero, the imaginary part varies from 0 to and from  to 0. While, through the frequency bands in which the real part is positive, the imaginarypart is constant at the values of  or 0. This note is going to help us understanding the behaviour ofthe propagation surfaces of two-dimensional plates later.Another way for obtaining the propagation factor is through dynamic condensation of the dynamicstiffness matrix after assembling the cell global matrix. The condensation is obtain through thefollowing procedure; assemble the dynamic stiffness matrix to obtain D11 D12 0  u1   f1 D     12 D22 D23  u2    0   0 D23 D33  u3   f 3     Then evaluate the internal degrees of freedom in terms of the boundary degrees of freedom usingthe second equation  D12u1  D23u3 u2  D22Then substitute into the other equations to obtainPassive Vibration Attenuation 11
  • 12. Periodic Structures: A Passive Vibration Filter Periodic Bars D11  D 212 / D22  D12 D23 / D22  u1   f1        D12 D23 / D22 D33  D 2 23 / D22  u3   f 3 Which may be written as D11 D12  u1   f1       D12 D22  u3   f 3 If the reduced stiffness matrix is then handled in the same manner as explained in the previoussection, the same results presented in Figure ‎ .11 and Figure ‎ .12 will be obtained. 1 1 1.4.2. Reverse approach for a periodic barIn this section, the reverse approach will be introduced in order to illustrate the concept ofpropagation lines which will be extended to the propagation surfaces for plates. Using the finiteelement model presented earlier, we may assemble the global dynamic stiffness matrix of the cell asfollows,  D11 D12 0  u1   f1  D      12 D22 D23  u2    f 2   (9)  0 D23  D33  u3   f 3     Substituting equation (1) into (9) gives,  D11 D12 0  u1   f1  D      12 D22 D23e   u2    f 2   (10)  0  D23e  D33  u1   f1     Since the resultant force f2 at point two is zero, we may add the first and last equations of the abovesystem and simplify the result to get,  D11  D33 D12  D23e    u1  0        (11)  D12  D23e D22  u 2  0Separating the mass and stiffness terms in the above equation, we get   K11  K 33 K12  K 23e    2  M 11  M 33 M 12  M 23e    u1  0            K  K e     12 23 K 22  M 12  M 23e M 22  u 2  0Or K     2 u  0 M    1     (12) u2  0Equation (12) presents an eigenvalue problem of the vibration as a function of the propagationparameter . We will call this approach the “reverse approach” as the independent variable of thePassive Vibration Attenuation 12
  • 13. Periodic Structures: A Passive Vibration Filter Periodic Barsproblem,  , is a quantity that we have no direct access to, in contrast with the “forward approach”in which the independent variable is the excitation frequency which is a quantity we can physicallycontrol and measure.To demonstrate the relationship between both approaches, the values of the propagation factor isconstrained to be imaginary values varying from 0 to . The resulting values of the naturalfrequencies of oscillation are shown in Figure ‎ .13. Few important notes have to be emphasized at 1this point. The curves presenting the variation of the excitation frequency are identical to thosepresenting the variation of the imaginary part of the propagation factor (Figure ‎ .12) with the 1independent and dependent variable reversed. Also, the gap existing between both curves ofFigure ‎ .13 corresponds to the frequency band in which the value of the propagation factor has a 1real part (Figure ‎ .12). The characteristic graphs shown in Figure ‎ .13 are called the propagation 1 1curves. Figure ‎ .13. The variation the natural frequency of 1 Figure ‎ .14. the variation of the natural frequency 1oscillation with the propagation factor. of oscillation with the real part of the propagation factor (imaginary part =)Now, varying the values of the real part of the propagation factor, for a constant value of theimaginary part, results in the characteristics shown in Figure ‎ .14. Similar notes can be taken when 1comparing the results of Figure ‎ .14 with those of Figure ‎ .12. But it has to be noted that increasing 1 1the value of the real part above the maximum obtained by the “forward approach” results inobtaining complex pairs for the excitation frequencies indicating going beyond the physicalboundaries. Nevertheless, the graphs of Figure ‎ .14 fill the gap that exists in Figure ‎ .13. Thus, we 1 1may call that gap “the attenuation band”, or “stop band”, and the curves, “the attenuation curves”. 1.4.3. Experimental WorkIn an extended research of the characteristics of periodic bars, Asiri conducted different experimentson bars with periodic configurations. His results were assessed by numerical results for the pass andstop bands obtained from a spectral finite element model. His results emphasized the effectivenessof the periodic configurations in attenuating the vibration response in the stop bands indicated bythe numerical model for the different configurations. Figure ‎ .15 presents the geometry of one of 1the experiments conducted by Asiri, and Figure ‎ .16 presents the experimentally obtained frequency 1response of the bar together with the numerically obtained attenuation curves. The results shown inFigure ‎ .16 show the degree of accuracy by which the attenuation bands may be predicted by the 1attenuation curves for the bar.Passive Vibration Attenuation 13
  • 14. Periodic Structures: A Passive Vibration Filter Periodic BeamsFigure ‎ .15. Geometry of one of the experiments conducted by Asiri. 1Figure ‎ .16. Experimental frequency response of the bar with the above mentioned geometry and the corresponding 1attenuation curves.1.5. Periodic BeamsPeriodic beams have been of special interest to researchers in the past decades due to their relationto railroad structures. The fact that the railway is supported at equal distances presents an almost-Passive Vibration Attenuation 14
  • 15. Periodic Structures: A Passive Vibration Filter Periodic Beamsideal case for the study of infinite simply supported beams, further, the effect of the foundationelasticity, presenting the ground elasticity, was widely introduced to the studies. 1.5.1. Beams with Periodic GeometryNumerical ModelThe spectral finite element model presented earlier for the plate case was simplified to be suitablefor the beam case. The degrees of freedom and generalized forces of the beam cell at the threenodes are shown in Figure ‎ .17. 1Figure ‎ .17. A sketch of the forces and displacements of a beam cell. 1The equations of motion of the beam elements could be written as follows,  D11 1 D12  W1   f1  1  1 1    (13)  D21 D22  W2   f 2   D11 2 D12  W2   f 2  2  2 2    (14)  D21 D22  W3   f 3  w  F where Wi   i  , f i   i  , and k ij is the dynamic stiffness matrix term relating the ith r wi  M i displacement vector with the jth generalized force vector. The dynamic stiffness matrix can beassembled for the whole cell  D11 D12 0  W1   f1  D      12 D22 D23  W2    f 2    0  D23 D33  W3   f 3     Condensing the above system to remove the internal displacement vector (W2) and assuming nointernal forces on the cell, i.e. f2 is zero, we get,  D11 D12  W1   f1         D21 D22  W3   f 3  1where D11  D11  D12 D22 D12 ,Passive Vibration Attenuation 15
  • 16. Periodic Structures: A Passive Vibration Filter Periodic Beams 1 D12   D12 D22 D32 , 1 D21   D32 D22 D12 , 1and D22  D33  D32 D22 D32 .Rearranging the equations to put them in an input output relation, we get, T11 T12  W1  W3  T       21 T22   f1   f 3  1where T11   D12 D11 , 1 T12   D12 , 1 T21  D12  D22 D12 D11 , 1and T22  D22 D12 .We may assume that, W3    W1   e    f3   f1 where  is the propagation factor of the cell. T  W1   W1   e    f1   f1 The above equation can be solved as an eigenvalue problem for the eigenvalues e. It can be proventhat the eigenvalues of this problem will appear in pairs each if which is the reciprocal of the other. 1.5.2. Experimental WorkDue to the lack of experimental studies that emphasize the periodic characteristics of structures, itwas decided to study the characteristics of the periodic beam to give a broader and more in-depthunderstanding of the behaviour of the periodic structures. In the forthcoming sections, theunderstanding of the periodic beam and plate structures will be emphasized through theexperimental and numerical results obtained.At this point, differentiations between two techniques of analysis have to be outlined; the periodicanalysis and the finite element analysis. When periodic analysis is mentioned, it is to point towardsthe process of investigating the pass and stop bands through the study of the propagation curvesand surfaces and related characteristics. On the other hand, the “finite element analysis” term willbe used to point towards the use of ordinary finite element techniques that would apply to anyPassive Vibration Attenuation 16
  • 17. Periodic Structures: A Passive Vibration Filter Periodic Beamsstructure’s geometry rather than to periodic structures in specific. This distinction had to be made asmost of the periodic analysis will be derived from a finite element model.Experimental SetupIn order to develop more understanding of the of the behaviour of the periodic beams as well asdeveloping a numerical model to study its characteristics, an experiment was set for a periodic beamwith free-free boundary conditions (Figure ‎ .18). 1Figure ‎ .18. The setup of the periodic beam experiment. 1The beam is aluminium beam which is 40 cm long and 5 cm wide with 1 mm thickness. Theperiodicity was introduced onto the beam by bonding 5 cm by 5 cm pieces of the same material onboth surfaces separated by 5 cm (Figure ‎ .19). The beam is then suspended by a thin wire from one 1of its end to simulate free-free boundary conditions. Thus, the beam is set up with four identical cellseach of which has free-free boundary conditions.Figure ‎ .19. A sketch for one cell of the periodic beam. 1The beam is then excited by a piezostack (model AE0505D16 NEC Tokin, Union City, CA, 94587) atone end and the measurement was taken by an accelerometer from the other end (Figure ‎ .20). 1Passive Vibration Attenuation 17
  • 18. Periodic Structures: A Passive Vibration Filter Periodic BeamsFigure ‎ .20. The excitation piezostack and the output accelerometer. 1Comparison of ResultsThe experiment described above was set up and measurements were taken from the two ends ofthe beam. Figure ‎ .21 shows the transfer function frequency response of the beam for the plain and 1periodic beams. The attenuation factor of the beam, as calculated by the real part of thepropagation factor of the periodic model, is plotted below the frequency response for the sake ofcomparison. The results shown emphasize the accuracy of the periodic model used to predict thebehaviour of the beam. Figure ‎ .22 presents the frequency response obtained by the finite element 1model of the described beam. Comparing the results of both figures, we can note clearly theconsistency of results obtained by the three models, experimental, periodic and finite element. 40 2.5 20 2 0 Respence Ampl. (dB) Attenuation Factor 1.5 -20 1 -40 Plain Beam Periodic Beam Attenuation Factor 0.5 -60 -80 0 0 1000 2000 3000 4000 5000 6000 Frequency (Hz)Figure ‎ .21. The frequency response together with the numerical results of the stop bands for the proposed beam. 1Passive Vibration Attenuation 18
  • 19. Periodic Structures: A Passive Vibration Filter Periodic BeamsFigure ‎ .22. Frequency response of the beam using finite element model. 1Another experiment was set up for a set of beams with cantilever boundary conditions. Theexperiments was set up with two accelerometers and excited by a piezoelectric actutator as shownin Figure ‎ .23 and Figure ‎ .24. Different cases with varying the lengths L1 and L2 were constructed 1 1to examine the effect of the geometry on the attenuation characteristics (Figure ‎ .25). 1Figure ‎ .23. Sketch of the experimental setup for the cantilever beam. 1Passive Vibration Attenuation 19
  • 20. Periodic Structures: A Passive Vibration Filter Periodic BeamsFigure ‎ .24. A picture of the experimental setup. 1Figure ‎ .25. a sketch for the cell geometry of the experiment for the cantilever beam. 1 30 10 9 20 8 10 Transfer Function Amplitude (dB) 7 Attenuatin Factor (rad) 0 6 0 500 1000 1500 2000 2500 3000 3500 4000 -10 5 4 -20 Plain Beam 3 -30 Periodic Beam Attenuation Factor 2 -40 1 -50 0 Frequency (Hz)Figure ‎ .26. Experimental results obtained for case #1 compared to plain beam and attenuation curves obtained by 1numerical model.Passive Vibration Attenuation 20
  • 21. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates 20 10 9 10 8 Transfer Function Amplitude (dB) 0 7 0 500 1000 1500 2000 2500 3000 3500 4000 Attenuatin Factor (rad) 6 -10 5 -20 4 Plain Beam -30 3 Periodic Beam Attenuation Factor 2 -40 1 -50 0 Frequency (Hz)Figure ‎ .27. Experimental results obtained for case #2 compared to plain beam and attenuation curves obtained by 1numerical model. 20 10 9 10 8 Transfer Function Amplitude (dB) 0 7 0 500 1000 1500 2000 2500 3000 3500 4000 Attenuatin Factor (rad) 6 -10 5 -20 4 -30 3 Plain Beam Periodic Beam 2 Attenuation Factor -40 1 -50 0 Frequency (Hz)Figure ‎ .28. Experimental results obtained for case #3 compared to plain beam and attenuation curves obtained by 1numerical model.1.6. Propagation Surfaces for Periodic PlatesIt is naturally understood that the beam is a special case of the plate structure. The thin beam andplate structures have similar approximate theories that describe their behaviour. From dynamicsPassive Vibration Attenuation 21
  • 22. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Platespoint of view, the beam would be characterized by having the bending waves travelling in onedimension, along the direction of the beam axis. Due to the characteristic of the beam being of shortwidth relative to the length, its modes of vibration in the shorter direction are associated with veryhigh frequencies.On the other hand, the plate is the general case in which the length and width dimensions are of thesame order giving way for bending waves to travel in both directions with similar characteristics.That specific nature of the plate introduces a lot of complexities to the study. A basic problem thatarises from the 2-dimensional effect is the fact that the source of vibration at a certain point on theplate can not be pointed out due to the fact that reflections from the tips of the structure areinterfering together with the fact that in a periodic structure we are introducing more reflectionsthat would travel in all directions increasing the degree of complexity. 1.6.1. Input-Output RelationsTo establish a system of equations that can be used for the “reverse approach” study of the periodicbehaviour of the plate, relations between the displacements of the different nodes are developedand implemented similar to those introduced in equation(1). Mead [‎ 5] and Mead et al. [‎ 7] 9 9introduced relations that could be developed for use with higher order elements.The input-output relations summarized in Figure ‎ .29 are presented in the following two sets of 1equations, w2  e  x w1 , f 2   e  x f1 , x  y x  y w3  e w1 , f3  e f1 ,   w4  e y w1 , f 4   e y f1 ,   w10  e y w5 , f10  e y f 5 ,   w9  e y w6 , f 9  e y f 6 , w7  e  x w12 , f 7  e  x f12 ,and w8  e  x w11 and f 8  e  x f11where  x  ix and  y  i y with  x and  y denoting the phase factor in the x and y-directionrespectively.Figure ‎ .29. A sketch representing the relations between the input and output displacements. 1Passive Vibration Attenuation 22
  • 23. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic PlatesNote that in the above relations, wi stands for the vector of degrees of freedom if the ith node; i.e.{w,wx,wy,wxy,D}. Implementing those relations in the element equations of motion, and assumingharmonic vibration, we may obtain the following relation,  w1  w   5  w6       m11  m19   k11  k19   w11  2                   w12   0 (15)    m91  m99  k91     k99  w13      w14    w15  w16    1.6.2. Propagation SurfacesThe concept of propagation surfaces was introduced by Mead and Parathan [‎ 6] as a graphical 9presentation of the change in the dynamic characteristics of the periodic plate with the change inthe wave direction. For a planar wave travelling in a periodically supported plate at an inclinationangle  from the x-axis, the phase difference between two adjacent periods in the x and y-directionsare x ,  y respectively, and the average wave numbers in the x and y-directions could be given by x y kx  & ky  (16) a bwhere a and b are the plate-period length in the x and y-directions respectively.Mead and Parathan [‎ 6] used displacement functions to describe the vibration of beams then 9extended the model to two dimensions by multiplying two polynomials (the x-polynomial and the y-polynomial). Then, the stiffness and mass matrices were constructed and the natural frequencieswere calculated. A plot of the non-dimensional frequency  with the phase difference (i.e. thepropagation surfaces) for a simply-supported periodic plate was then presented. The non-dimensional frequency  is defined as, t 2 a 4  ( D p 417)where t and Dp are the plate thickness and flexural rigidity respectively.Using the developed 16-node element, the set of 16 matrix equations can be reduced to a set of 9matrix equations, and can be solved as an eigenvalue problem for the non-dimensional frequency Such that:Passive Vibration Attenuation 23
  • 24. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates     2 m  k  0The propagation surfaces resulting from the solution of the above eigenvalue problem are shown inFigure ‎ .30. 1In Figure ‎ .31, which is the same as Figure ‎ .30 but from a different viewing point, we can clearly see 1 1the bands over which the propagation surfaces reside. These frequency bands are the bands inwhich the vibration would propagate from the input to the output nodes in an analogous manner tothe propagation bands identified earlier for the periodic bar. Gaps that exist between the surfacesover bands of frequencies can also be identified as “attenuation or stop bands”.  yFigure ‎ .30. Propagation surfaces 1 resulting from the solution of the eigenvalue problem of the finite element model. xPassive Vibration Attenuation 24
  • 25. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates  x yFigure ‎ .31. The plot of the propagation surfaces from a planar point of view. 1 1.6.3. Constant angle curvesTo simplify the graphical representation of the “reverse approach”, we are going to examine thepropagation surfaces at constant angle. A wave propagation angle of 45o is considered. By varyingthe imaginary part of the propagation factor from 0 to , setting the real part to 0 and taking y tobe equal to x, we can obtain the propagation curves for the different bands. Figure ‎ .32 shows the 1resulting curves drawn with the independent variable (x) on the vertical axis.Passive Vibration Attenuation 25
  • 26. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates oFigure ‎ .32. The curves of the propagation surfaces at angle 45 . 1Approaching the problem from the perspective of the attenuation factor (the real part of thepropagation factor), we can draw the “Attenuation Surfaces” or the “Attenuation Curves”. Settingthe imaginary part of the propagation factor to zero, we can obtain the attenuation curves (or thestop bands). Figure ‎ .33 presents the attenuation curves for a wave propagating at 45o and with the 1imaginary part of the propagation factor set equal to zero. While Figure ‎ .34 presents the 1 oattenuation curves with a wave propagating at 45 and with the imaginary part of the propagationfactor set to .An interesting feature appears in these graphs, namely, the overlapping of the propagation andattenuation bands. This property of the bands comes from the fact that the wave is now propagatingin a square plate in contrast with the one-dimensional structures considered earlier. In a simply-supported square plate, the 2nd and the 3rd vibration modes coincide (namely the (1,2) and (2,1)modes). Nevertheless, the energy flow in both directions is distinct and occurs between twodifferent set of nodes.Getting back to the three dimensional surfaces, we can now obtain the “Attenuation Surfaces” forthe plate by setting the values of the imaginary part of the propagation factor to 0 or . The resultingsurfaces present the attenuation bands associated with the periodic plate of interest. It has to benoted, again at this point, that the overlapping of the surfaces does not contradict the fact that thebands are distinct. In other words, within the stop bands, the vibration of certain propagation modewhile the other modes that undergo propagation phases are still propagating vibration.Passive Vibration Attenuation 26
  • 27. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates oFigure ‎ .33. The “attenuation surface” at angel 45 with the imaginary part set to zero. 1 oFigure ‎ .34. The “attenuation surface” at angel 45 with the imaginary part set to . 1Passive Vibration Attenuation 27
  • 28. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates  y xFigure ‎ .35. The attenuation surfaces for the plate with the imaginary part set to zero. 1Passive Vibration Attenuation 28
  • 29. Periodic Structures: A Passive Vibration Filter Propagation Surfaces for Periodic Plates  y xFigure ‎ .36. The attenuation surfaces for the first two attenuation bands of the plate with the imaginary part set to . 1Passive Vibration Attenuation 29
  • 30. Periodic Structures: A Passive Vibration Filter Effect of Shunted Piezoelectric Patches on Propagation Surfaces 1.6.4. Plates with Periodic Geometry 1.6.5. Experimental Work1.7. Effect of Shunted Piezoelectric Patches on Propagation Surfaces 1.7.1. Propagation Surfaces for Coupled SystemIt is interesting to visualize the result of adding an inductor to the shunt circuit. It simply splits themode that is targeted into two modes with one surface above and another below the originalpropagation surface; just like the case of adding a secondary mass-spring system to a primary systemas takes place in the classical vibration absorber problem (Figure ‎ .37). 1Figure ‎ .37. Frequency response for a vibration absorber. 1When an inductance is added to the system, a similar result is obtained for the propagation surfacesas shown in Figure ‎ .38. It is obvious that the shape of the propagation surfaces exhibits shape 1similar to that of the frequency response of a two-degree of freedom spring-mass system.Passive Vibration Attenuation 30
  • 31. Periodic Structures: A Passive Vibration Filter Effect of Shunted Piezoelectric Patches on Propagation Surfaces y xFigure ‎ .38. The two surfaces resulting from adding the inductance compared to the original surface. 1Drawing the curves with a wave propagating at 45o (Figure ‎ .39), we can visualize the gap introduced 1in the band that was originally covered by the first propagation surface. That gap now presents astop band.Passive Vibration Attenuation 31
  • 32. Periodic Structures: A Passive Vibration Filter Effect of Shunted Piezoelectric Patches on Propagation SurfacesFigure ‎ .39. The propagation curves resulting from introducing the shunted inductance. 1Plotting the attenuation curves for this case, we can notice the introduction of an attenuation factorin that band indicating that the propagating wave is expected to decay. (Figure ‎ .40 and Figure ‎ .41). 1 1Figure ‎ .40. The attenuation curves resulting from introducing the shunted inductance at phase angle . 1Passive Vibration Attenuation 32
  • 33. Periodic Structures: A Passive Vibration Filter Effect of Shunted Piezoelectric Patches on Propagation SurfacesFigure ‎ .41. The attenuation curves resulting from the introduction of the shunted inductance at phase angle zero. 1Passive Vibration Attenuation 33
  • 34. Periodic Structures: A Passive Vibration Filter Appendices1.8. Appendices 1.8.1. Appendix AWhen the input output relations of the different nodes are implemented into the equations ofmotion of the plate elements, then terms get collected, the 16 equations reduce to 9 equationsgiven by,  w1  w   5  w6     m11  m19   k11  k19   wb                     wc   0  m91  m99  k 91     k 99   wd      we  w   f  wg   Where the terms of the above equation are 4x4 matrix each given by the following set of relations(not that the letters a to g are used instead of the number 10 to 16 for the sake of clarity), x  y y y x  yk11  k11  k12e  x  k13e  k14e  k 21e   x  k 22  k 23e  k 24e  x  y  y  y  x  y k 31e  k 32e  k 33  k 34e   x  k 41e  k 42e  k 43e  x  k 44 y  x   y  x  y  yk12  k15  k1a e  k 25e   x  k 2 a e  k 35e  k 3a e   x  k 45e  k 4a y  x  y  x  y  yk13  k16  k19e  k 26e   x  k 29e  k 36e  k 39e   x  k 46e  k 49  y  x  y  y  yk14  k18e  x  k1b  k 28  k 2b e   x  k 38e  k 3b e  k 48e  k 4b e  y  x  y  x  y  yk15  k17e  x  k1c  k 27  k 2c e   x  k 37e  k 3c e  k 47e  k 4c e  x  y  yk16  k1d  k 2 d e   x  k 3d e  k4d e  x  y  yk17  k1e  k 2e e   x  k 3e e  k 4e e  x  y  yk18  k1 f  k 2 f e   x  k 3 f e  k4 f e  x  y  yk19  k1g  k 2 g e   x  k 3 g e  k4g ePassive Vibration Attenuation 34
  • 35. Periodic Structures: A Passive Vibration Filter Appendices x  y y y  x  yk 21  k 51  k 52e  x  k 53e  k 54e  k a1e  ka2e  k a 3e  x  k a 4 y  yk 22  k 55  k 5 a e  ka5e  k aa y  yk 23  k 56  k 59e  ka6e  k a9 x  x  y  yk 24  k 58e  k 5b  k a 8 e  k abe  x  y  yk 25  k 57e  x  k 5c  k a 7 e  k ace  yk 26  k 5 d  k ad e  yk 27  k 5e  k aee  yk 28  k 5 f  k af e  yk 29  k 5 g  k ag e x  y y y  x  yk 31  k 61  k 62e  x  k 63e  k 64e  k 91e  k 92e  k 93e  x  k 94 y  yk 32  k 65  k 6 a e  k 95e  k9a y  yk 33  k 66  k 69e  k 96e  k 99  x  y  yk 34  k 68e  x  k 6b  k 98e  k 9b e  x  y  yk 35  k 67e  x  k 6c  k 97e  k9c e  yk 36  k 6 d  k 9 d e  yk 37  k 6e  k 9 e e  yk 38  k 6 f  k 9 f e  yk 39  k 6 g  k 9 g e y x  y x  y yk 41  k 81e   x  k 82  k 83e  k 84e  k b1  k b 2 e  x  k b 3 e  kb4 e  x  y yk 42  k 85e   x  k 8 a e  k b 5  k ba e  x  y yk 43  k 86e   x  k 89e  k b 6  k b9 ek 44  k 88  k 8b e   x  k b8 e  x  k bbk 45  k 87  k 8c e   x  k b 7 e  x  k bck 46  k 8 d e   x  k bdk 47  k 8e e   x  k bek 48  k 8 f e   x  k bfk 49  k 8 g e   x  k bgPassive Vibration Attenuation 35
  • 36. Periodic Structures: A Passive Vibration Filter Appendices y x  y x  y yk 51  k 71e   x  k 72  k 73e  k 74e  k c1  k c 2 e  x  k c 3e  kc 4e  x   y yk 52  k 75e   x  k 7 a e  k c 5  k ca e  x  y yk 53  k 76e   x  k 79e  kc6  kc9ek 54  k 78  k 7 b e   x  k c8 e  x  k cbk 55  k 77  k 7 c e   x  k c 7 e  x  k cck 56  k 7 d e   x  k cdk 57  k 7 e e   x  k cek 58  k 7 f e   x  k cfk 59  k 7 g e   x  k cg x  y yk 61  k d 1  k d 2 e  x  k d 3 e  kd 4e yk 62  k d 5  k dae yk 63  k d 6  k d 9 ek 64  k d 8 e  x  k dbk 65  k d 7 e  x  k dck 66  k ddk 67  k dek 68  k dfk 69  k dg x  y yk 71  k e1  k e 2 e  x  k e3 e  ke4 e yk 72  k e5  k ea e yk 73  k e 6  k e9 ek 74  k e8 e  x  k ebk 75  k e 7 e  x  k eck 76  k edk 77  k eek 78  k efk 79  k egPassive Vibration Attenuation 36
  • 37. Periodic Structures: A Passive Vibration Filter Appendices x  y yk81  k f 1  k f 2 e  x  k f 3 e  k f 4e yk82  k f 5  k fa e yk83  k f 6  k f 9 ek84  k f 8 e  x  k fbk85  k f 7 e  x  k fck86  k fdk87  k fek88  k ffk89  k fg x  y y k91  k g1  k g 2 e  x  k g 3e  k g 4e y k92  k g 5  k gae y k93  k g 6  k g 9 e k94  k g 8e  x  k gb k95  k g 7 e  x  k gc k96  k gd k97  k ge k98  k gf k99  k gg 1.8.2. NomenclatureA Areaai Undetermined coefficients of the transverse displacement shape functionbi Undetermined coefficients of the electric displacement shape functionD Electric displacementDP Plate flexural rigidityd Piezoelectric coefficientdi Nodal electric displacementE Young’s modulus of elasticity E Electric fielde Piezoelectric material constant relating stress to electric fieldHw,HD Transverse displacement and electric displacement interpolation functions respectivelyk,kx,ky Wave number, component of wave number in x and y-directions respectivelykb,kD,kbD Element bending, electric, and displacement-electric coupling stiffness matrices respectivelymb,mD Element bending and electric mass matrices respectivelyNw,ND Lateral displacement and electric displacement shape functions respectivelyQ Plane stress plane strain constitutive relationT Kinetic energyPassive Vibration Attenuation 37
  • 38. Periodic Structures: A Passive Vibration Filter References and BibliographyU Potential energyV VolumeW External workw Transverse displacementwb,wD Nodal transverse and electric displacements respectively 1 (.) First variation Strainxy Shear strain Curvature The propagation factor Mass density Stress Wave propagation angle Poisson’s ratio Frequency Dielectric constant Phase angleSubscriptsD Related to electric degrees of freedomw Related to transverse deflectionb Related to bending degrees of freedomx In the x-direction,x Derivative in the x-directiony In the y-direction,y Derivative in the y-directionSuperscriptD At constant electric displacementE At constant electric fieldT Matrix transpose1.9. References and Bibliography1. Wada, B. K., Fanson, J. L., and Crawley, E. F., “Adaptive Structures,” Journal of Intelligent Material Systems and Structures, Vol. 1, No. 2, 1990, pp. 157-174.2. Crawley, E. F., “Intelligent Structures for Aerospace: A Technology Overview and Assessment,” AIAA Journal, Vol. 32, No. 8, 1994, pp. 1689-1699.3. Rao, S. S., and Sunar, M., “Piezoelectricity and Its Use in Disturbance Sensing and Control of Flexible Structures: A Survey,” Applied Mechanics Review, Vol. 47, No. 4, 1994, pp. 113-123.4. Park, C. H., and Baz, A., “Vibration Damping and Control Using Active Constrained Layer Damping: A Survey,” The Shock and Vibration Digest, Vol. 31, No. 5, 1999, pp. 355-364.5. Benjeddou, A., “Recent Advances in Hybrid Active-Passive Vibration Control,” Journal of Vibration and Control, Accepted for Publishing.6. Chee, C. Y. K., Tong, L., and Steven, G. P., “A Review on The Modeling of Piezoelectric Sensors and Actuators Incorporated in Intelligent Structures,” Journal of Intelligent Material Systems and Structures, Vol. 9, No. 1, 1998, pp. 3-19.Passive Vibration Attenuation 38
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