Viscoelastic Damping: Lecture Notes 140202

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Viscoelastic Damping: Lecture Notes 140202

  1. 1. Viscoelastic Damping Mohammad Tawfik Cairo University Aerospace Engineering Department 2 February, 2014
  2. 2. Introduction Contents Introduction ............................................................................................................................................ 3 Classical Models ...................................................................................................................................... 3 Maxwell Model ................................................................................................................................... 3 Model Characteristics ..................................................................................................................... 4 Kalvin-Voigt Model .............................................................................................................................. 6 Zener Model ........................................................................................................................................ 8 The Area in the curve exist only when   0 ...................................................................................... 11 Golla-Hughes-McTavish (GHM) 1983................................................................................................ 11 Unconstrained Layer Damping.............................................................................................................. 17 Finite Element Model of Bars ........................................................................................................... 17 Composite Bar ................................................................................................................................... 18 Constrained Layer Damping .................................................................................................................. 18 Active Constrained layers damping ...................................................................................................... 27 Bibliography .......................................................................................................................................... 42 Viscoelastic Damping 2
  3. 3. Introduction Introduction Objectives • Recognize the nature of viscoelastic material • Understand the damping models of viscoelastic material • Dynamics of structures with viscoelastic material What is Viscoelastic Material? • Materials that Exhibit, both, viscous and elastic characteristics. • The material may be modeled in many different ways. Classical models include: – Mawxell Model – Kalvin-Voight Model Classical Models Maxwell Model The Maxwell model describes the material as a viscous damper in series with an elastic stiffness (Figure 1). When stress is applied, it is uniform through the element, in turn, we may write the total strain of the viscoeleastic element as:   s  d Figure 1. Schematic for a viscoelastic element using the Maxwell model According to this model, the stress is equal in both elements, which may be expressed by the relation:    Es s  Cd  d According to this relation, we may write: s   Es d    Cd dt According to this, the total strain may be expressed as: Viscoelastic Damping 3
  4. 4. Classical Models   Es   Cd dt Or     Es   Cd Model Characteristics When investigating the model characteristics in our context, we are interested in three aspects; namely: • Creep. When a material is loaded for a prolonged period of time, the strain tends to increase, which leads, in turn, to failure. The phenomenon of the strain increase at constant load is called creep. • Relaxation. When materials are strained for a prolonged periods of time, the internal stresses tend to decrease. The phenomenon of stress decrease at a constant strain value is called relaxation. • Storage and Loss Moduli. When the viscoelastic material is loaded harmonically, the stressstrain relation may be presented by complex modulus of elasticity. The real part of the complex modulus is called storage modulus while the imaginary part is called the loss modulus. To study the creep characteristics of the Maxwell model, we need to set the rate of change of stress to zero in the stress-strain differential relation. Thus:     Es    Cd zero Solving the differential equation, we get:   Cd t The resulting strain time function indicates that the strain will grow to an unbound value as time increases! To investigate the relaxation characteristics, the strain rate is set to be zero in the differential relation, the resulting relation becomes: 0   Es   Cd When solved, the above relation gives the stress time relation as: Viscoelastic Damping 4
  5. 5. Classical Models    0e tE s Cd Where,  0 indicates the initial stress value. The above relation indicates that the stress will decrease exponentially with time with an asymptotic value of zero. When studying the response of the model under harmonic excitation, the excitation stress is presented as:    0e jt Thus, the strain response is presented as:    0 e jt Substituting in the differential equation, we get: o  Es Cd j o Es  jCd Giving: o C d E s  2  E s C d j  o 2 2 Es   2Cd 2 2 Separating the real and imaginary parts, we get: 2  C d 2 E s 2 Es Cd    2  o o   j 2 2 2 E s   2Cd E s   2Cd    Where, the storage modulus is: Cd Es 2 2 2 Es   2Cd 2 E'  And the loss modulus becomes: Es Cd  2 2 E s   2Cd 2 E"  The loss modulus, defines as the ratio between the storage and loss moduli, may be given as:  Es Cd  Now, the stress strain relation may be expressed as:  o  E 1  j  o Viscoelastic Damping 5
  6. 6. Classical Models Where the complex modulus is given by: E *  E 1  j  1 0.9 0.8 Modulus 0.7 0.6 E 0.5 u 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 Frequency Figure 2. The variation of the storage modulus and the loss factor with frequency according to Maxwell’s model Figure 2 presents the variation of the storage modulus and the loss factor with frequency. Note that according to Maxwell’s Model: • Under static loading, the stiffness, storage modulus, is zero and the loss factor is infinity! • For very high frequencies, the loss factor becomes zero! Kalvin-Voigt Model The Kalvin-Voigt model describes the material as a viscous damper in parallel with an elastic stiffness (Figure 3). When stress is applied, it is distributed through the element, while the strain in both elements is equal. Figure 3. Schematic for a viscoelastic element using the Kalvin-Voigt model The stress strain relation may be written as:   s d Viscoelastic Damping 6
  7. 7. Classical Models    Es s  Cd  d No we come to the studying the Kalvin-Voigt Model characteristics. To study the creep we solve the above equation for constant stress to get:   1  e Es  E s t Cd  Which indicates that the strain will grow to a constant value as time increases! When studying the relaxation, we set the strain rate to zero, giving:   Es 0 Which means that the stress will stay constant as time grows for the same strain! Now, we come to investigating the Storage modulus and Loss Factor. For harmonic stress and strain we get:    0 e jt    0e jt Resulting in the relation:   Es  jCd  o 14 12 Modulus 10 8 6 E 4 u 2 0 0 2 4 6 8 10 Frequency Figure 4. The variation of the storage modulus and the loss factor with frequency according to the Kalvin-Voigt model Figure 4 presents the variation of the storage modulus and the loss factor with frequency. Note that according to the Kalvin-Voigt Model: Viscoelastic Damping 7
  8. 8. Classical Models • Under all loading, storage modulus is equal to the stiffness of the spring, and the loss factor is zero. • For very high frequencies, the loss factor becomes unbound! Zener Model The Zener model describes the material as a viscous damper in parallel with an elastic stiffness and both are in series with stiffness (Figure 5). The strain may be written as:    s  1 Figure 5. Schematic for a viscoelastic element using the Zener model Stress-Strain relation, according to the zener model, may be written as:    Es  s  E p  1  Cd  1 From which we may write in Laplace domain: s   Es , 1   E p  sCd Or:   Es   E  sCd  Es    p  E E  sC   E p  sCd d   s p  Back to time domain, we get: Es E p  sCd   E p  sCd  Es  From which we get the differential equation:   Es E p  Es Cd   E p  Es   Cd  Or:   E  E     Viscoelastic Damping 8
  9. 9. Classical Models Studying Zener Model characteristics, we get for the creep:  E  E    0 Giving:  0 E  e t  Es And for the relaxation, we get:  E     Giving:    0  E 0 1  e t   While for the storage modulus and loss factor we get: E o  jE o   o  jo Rearranging, we get: 1  j 1   2  j     o  E o  E o 1  j 1   2 2 Or:  1   2 j       o  E  1   2 2  1   2 2  o    Or simply:  o  E 1  j  o Viscoelastic Damping 9
  10. 10. Classical Models 2 1.8 1.6 Modulus 1.4 1.2 1 0.8 0.6 E 0.4 u 0.2 0 0 1 2 3 4 Frequency Figure 6. The variation of the storage modulus and the loss factor with frequency according to the Zener model This is more realistic for the presentation of the material characteristics, however, is does not satisfy the detailed studies needed for analysis of complex structures. Let’s recall the harmonic relations:    o e i t    o e it And the differential equation   E  E     Which give:  oeit   oieit  E oeit  E oieit Expanding the complex exponentials, we get:  o cost  i sin t    oi cost  i sin t   E o cost  i sin t   E oi cost  i sin t  Equating the real and imaginary parts:  o cos t   o sin t  E o cos t  E o sin t  o sin t   o cos t  E o sin t  E o cos t Viscoelastic Damping 10
  11. 11. Classical Models  o sin t  E ' o sin t  E '' o cos t  Total   e   d   elastic dissipativ e  dissipativ e  E  o cos t ''  E '' ' E  o 1  sin 2 t ' E   ( E ' o ) 2  ( E ' o sin t ) 2   ( E ' o ) 2   e d    2 2  2   ( E ' o ) 2   e   2   2   d    e  ( E ' o ) 2     Divide by ( E ' o ) 2  2 2  d   e   '    '  1  E    E   o  o    This equation represent ellipse with major diameter  2 * E ' o & minor diameter  2 *E ' o Figure 7. The Area in the curve exist only when   0 Golla-Hughes-McTavish (GHM) 1983 Simple mass+visco elastic material Viscoelastic Damping 11
  12. 12. Classical Models  S 2  2n S   S    1   2 0 2 S  2n S  n   2 n 2 let  Z  2  2 S  2n S  n Z 00  2n Z 0  n Z  n  2 2 Z 00  2n Z 0  n   Z  Z  int ernal vaiable substitute( Z )  in DE.of .system   S 2  2n S   n 2    S    1    2   S 2  2 S   2    0 n  n n      2  S 2  2n S Z  0 2 n  2  00    2 n   Z   0 n  S 2       00   1     Z  0 & . 00  n   2n  0  n   0 2 2 00 -     0 0    0       0     0             0 2   1  00  0 2   0  - n 2   0 n        0  00 0  0  0    -     0                   00   2   0      0     0  0    -   2   n     n         Complex stiffeners is given as Viscoelastic Damping 12
  13. 13. Classical Models  S 2  2n S     1   2 2 S  2n S  n   *  S 2  2n S  G  G 1   2 2 S  2n S  n   *  S 2  2n S     1   2 2 S  2n S  n    ,  , n are unknown prameters * mass supported on a visco elastic spring :  S 2  2n S   S    1   2 0 2 S  2n S  n   assume the internal variable Z 1 2 n 2 Z  internal DOF  2  2 S  2n S  n S2 Z  2n SZ  n     2 2 3 From equation 1&2 Viscoelastic Damping 13
  14. 14. Classical Models   S 2  2 n S n 2  S    1   0 2 2  2 n S  2 n S   n     S 2  2 n S 2  S      Z 0 2 2 n  S 2           0  S 2    1       0 in time domain  00   1       0 eqn.  3  00  2 n  0   n    n   0 2  0    0   2 0  00  0 0   0     1 -         0  00     0    2   1    0 2 n    -  n 2  n      0  00 0 0  0               1 -      0    00   2   0            0    -   2   n   n       Viscoelastic Damping 14
  15. 15. Classical Models  Visco elastic material has its own internal DOF =Z  In general X&Z are vectors   S 2  2 n n S    1   n   n 2 2   S  2 n n S   n       Summary        The original system has (X ) DOF The system+vesco elastic material has (X+Z) DOF Entire system order has increased X=primary DOF Z=secondary DOF Use static condensation method (Guyan reduction method ) I.e condensation =eliminate the secondary DOF&only maintain the primary DOF. Static Condensation *consider only the stiffnes matrix  1    -       -       F         0        1       F F    redused stiffnes matrix     1               1 consider     1  1  1    1 -     1    0  Viscoelastic Damping -   1    1  15
  16. 16. Classical Models compare energy terms :  1  1   reduced        total  2 2   1      total   2     strain energy of entire system  strain energy of primary DOF. " " redused can be obtained also as follow : -       F   1       -       0    0     F       1 1   0            0  0  0   00     0 2   n   0   1    2   0    -  n    -   F      0         redused  00  C redused  00   redused  Visco Elastic Material Damping *Golla-Hughes-McTavish (GHM) model Stiffners complex modulas (longitudinal or sheer)  S 2  2 n S     1   2 2  S  2 n S   n     * Viscoelastic Damping 16
  17. 17. Unconstrained Layer Damping For structure& V E M-------------system dynamics   0   0  00 0         00       0 n 2        0        1 2   0        -  n      0 -       F         0   **GHM model when augmented with structural model can be written as:1-frequency domain 2-time domain Other Models • Some, more accurate, models were developed to represent the behavior of viscoelastic material • The greatest concern was paid for the modeling in the time domain. • The most famous models are: – Golla-Hughes-McTavish – Augmented Temperature Field Fractional Derivative Unconstrained Layer Damping • The most common way of using viscoelastic material in damping is by bonding it to the surface of the structure! • The viscoelastic material will be strained with the structure resulting in energy losses in the surface layer Finite Element Model of Bars • Recall the stiffness and mass matrices of a bar: • It is possible, in the above model, to superimpose more than one element! Viscoelastic Damping 17
  18. 18. Constrained Layer Damping K EA  1  1 AL 2 1   1 1  & M  6 1 2  L     Composite Bar • The effect of each part of the bar may be added to the other part linearly incorporating the effect of both materials 1  1   A  V AV 2 MC  B B L 6 1 KC  E B AB  EV AV L  1 1  1 2  Homework #9 • Use the datasheet of the DYAD606 viscoelastic material to calculate the bar response with modulus of elasticity varying with frequency Constrained Layer Damping • When the viscoelastic layer is covered, constrained, from the top side, sheer stresses are generated between the different surfaces. • Viscoelastic materials are characterized by having much higher losses in the case of sheer than in the case of axial strain. Constrained Layer Damping Viscoelastic Damping 18
  19. 19. Constrained Layer Damping Sheer Stresses Viscoelastic Damping 19
  20. 20. Constrained Layer Damping    x h2   E2 u x  2u   2 x E2 h2  2u G *  u  u0     x 2 E2 h2  h1    Axial Displacement • The axial displacement relation becomes:  2u G *  u  u0     x 2 E2 h2  h1    E2 h2 h1  2u  u  u0 G * x 2 B*  2u  u  u0 x 2 • The axial displacement relation becomes: • Solving: B*u xx  u   0 x  x   x  u  a1Sh *   a1Ch *    0 x B  B    x  B * Sh *     B  u  0 x    l  Ch *     2B    Sheer Strain Viscoelastic Damping 20
  21. 21. Constrained Layer Damping  u  u0 u   0 x  h1 h1  x  *  B     l  Ch *   2B   0 B * Sh Lost Energy G o2 B* l/2 W  h1G   2 dx  l / 2 h1Ch  * 2 2 l 2B   Sh x l/2 2 * B* dx l / 2 Note that l/2    B* l * / 2Sh x B dx  2 Sh l B  2 l 2  ass  A W (1 / 2) 2  0 h1 h2 l 2  4 l cos( / 2) 0  0  sh( A) sin( / 2)  sin( ) cos( / 2)   l  ch( A)  cos( )    l sin( / 2) 0 G *  G (cos  i sin  )  G cos(1  tan  )  G cos(1  i ) 0  h1 h2 E 2 G  v  tan Example .01 * .01 *107 0   1' ' 103 Loptimum  3.28 * 0  3.28' ' Viscoelastic Damping 21
  22. 22. Constrained Layer Damping For unconstrained layer damping d (W )  hh11  0 '' 2 W  total energy dissipated  hh11  0 L '' 1 2 In the constrained case l/2 Wconstrained  l/2  d (W ) hh G   '' 1 l / 2 2 dx l / 2 hh1G ' ' l/2  2 dx Wconstrained l / 2  Wunconstrained hh11 '' L o 2 G*  G 'iG' '  G ' 1  i  *  'i' '  ' 1  i  G ' '  G ' Viscoelastic Damping ' '  ' 22
  23. 23. Constrained Layer Damping '   poisson's ratio 21    for VEM ,  0.5 G'  G'  ' 3  G' '  G'  ' ' '  3 3 G' '  1/ 3 ' ' Wcon 1 Ratio    2 dx 2  Wuncon 3l o  l / 2 l/2  2 h2  2 o  sh Asin / 2   sin cos / 2    3 h1 G l sin  cosh   cos    o  sh Asin / 2   sin cos / 2     0.124 l sin  cosh   cos    v  1    45o h2 1 h1 R 2  104 G 2 * 1 * 104 * 0.124  1000 3 Summary *constraining the VEM makes it deforms in sheer & results in significantly high energy dissipation characteristics Notes The plunkett & Lee analysis assumes:1-quasi-static analysis (satisfied by the force that the constraining layer thickness is small (its inertia can be neglected) Viscoelastic Damping 23
  24. 24. Constrained Layer Damping 2-A general base structure 3-longitudinal vibration beam 2w  max .ofVEM  d 2  2w M   2   max  cons tan t   0 For the beam:Energy dissipated Viscoelastic Damping 24
  25. 25. Constrained Layer Damping l/2 l/2  d (W ) hh G   Wconstrained  '' 2 1 l / 2 dx l / 2  *      l  h1ch *   2   0*sh W  hh1G ' '  * 2 l/2  2 d Wxx l / 2 2 sh2 (  / *) dx 2 ch (l / *) Exercise Show that the above composite has t t r rh * 3  1  re rh  3(1  rr ) 2 e * 1 1 1  re rh *   re  2  2 (1  i )  re (1  i ) 1 1 * where :  rh  h2 h1 Viscoelastic Damping * re  2 1 25
  26. 26. Constrained Layer Damping And show that:-   re rh 3  6rh  4r  2re rh  re rh 2  3 2 4  (1  re rh ) 1  4re rh  6re rh  4re rh  re rh 2 3 2 4 2  Take;- rh  h2 1 h1 re  3.585 * 10 4   0.00502 2   0.00519 2 Kinematics of CLD h2 Wx 2 h U   U 3  3 Wx 2 U   U1  U   U A  (U 3  U1 )  ( Wx    h1  h3 )Wx 2 U U A h2 Viscoelastic Damping 26
  27. 27. Active Constrained layers damping    U   U   h 2W  h2 (U 1  U 3 )  ( h1 h3   h)W  2 2 h2 U1 U 3 h  W h2 h2 Active Constrained layers damping Viscoelastic Damping 27
  28. 28. Active Constrained layers damping * U   U   0  2 2 *  h1h2 2 G*    L/2  U 0 X Solution procedure *solve for U *determine γ W  G ' 'h2 2 Compute * W   Wdx Compute * *put in dimensionless form η Viscoelastic Damping 28
  29. 29. Active Constrained layers damping For ACLD    passive  active If controller fails---------------system still “fail-safe” because of passive damping   p   0  ( p   d ) 0 t Notes (viscoelastic) if    ' (1   i ) F   ' (1   i )    '    ' i  if    0 e iwt  o  iw o e iwt  iw ' 0 F     Felastic  Fdamping w ' 2 /  Energy dissipated per cycle   0 dx Fd dt  dt 2 /   0  '   o 2 dt for   0 sin t 2 /  Energy  W    ' 0 potential Energy  W  2  We   2  0 2 cos 2 (t )dt  2 ' 2 2   o    ' o 2 1 ' 2   0  We 2 W  2We W  specific damping capacity We Viscoelastic Damping 29
  30. 30. Active Constrained layers damping Viscous Damping Fd  C o 2 /   C W  dissipated energy  o  o dt 0  C  o 2 2 /  2  cos 2 t dt 0   2 2 C 2  o  C o  Equivalent viscous damping to viscoelastic material  '  C C  '  C  ' 1 damping ratio     Co  2  '  2 '  2 at resonance     2 2-Transeverse Vibration Kinematics equation:   U1  U 3 h  Wx h2 h2 h h1  h3  h2 2 U=longitudinal deflection of base structure Viscoelastic Damping 30
  31. 31. Active Constrained layers damping  =shear angle Wx=slope of deflection line h2   U 1  hW  U 1  h2   hW  U 1  h2   hW   1 =F 1h1U 1 x Force on top layer per unit width = dF  1h1U1  G *  shearstress  2 d 1h1 (h2   hW )  G *   G* h    W 1h1h2 h2 let  1  1h1  longitudinal Rigidity    G* h   W 1 h2 NOTE Bending in beam: ( ) *  Dt  Dt (1   i) * Equation of motion; Dt W   m 2W  0 * W   m 2 Dt * W 0 W    B W  0 *4 Viscoelastic Damping 31
  32. 32. Active Constrained layers damping where  B  bending wave number W  W0 e i ( wt  B  ) * one propagation solution W ( D * t / m) 1 / 4 let  *B   W 1/ 2     B (1  i ) 4 ( Dt / m)1 / 4 (1  i  )1 / 4 * B W  W0 e i ( wt  B  ) e (  B  / 4 ) W  W0 e (  B / 4 )  Energy  CW 2  CW0 e (  B / 2)  2 d energy 2   B  CW0 ( )e dx 2   B  2 d Energy/dx   B  Energy 2  2 d Energy/dx C   constrained layers assembly B Energy solution for  C ; * put W  W0 e i ( wt   * solve   * B ) G* h    W   1 h2 h2 for  * calculate d energy/dx   G'  v h2  2 Using the solution given in ''Damping of flexural waves by constrained layers '' Journal of acoustic society of America, Vol 31, 7 pp952-962, 1959 Viscoelastic Damping 32
  33. 33. Active Constrained layers damping W   i *3 BW0 e i B  e iWt * W   i B W *3   G* h     B 3 iW  1 h2 h2   ih B W   G* h2 1   *2 ( B  1 2 )   check that it satisfies equation Summary Loss factor for constrained layer damping during transverse vibration WD dissipated Energy  W Elastic Energy WD loss factor    2 W specifi damping  2-for beam in bending Equation of motion Viscoelastic Damping 33
  34. 34. Active Constrained layers damping ( ) *  Dt  Dt (1   i) * Equation of motion; Dt W   m 2W  0 * W   m 2 Dt * W 0 W    B W  0 *4 1/ 4  mW 2  where  *B  bending wave number    D*   t   * B W 1/ 2     B (1  i ) 1/ 4 1/ 4 4 ( Dt / m) (1  i  ) W  W0 e i ( wt  B  ) e (  B  / 4 ) W  W0 e (  B / 4 )   B  wave number of constrained layer sassembly without losses   Energy  CW 2  C W0ei ( wt   B  ) e B / 2  Ce B / 2 2 denergy / dx Ce B / 2 ( B / 2)   ( B / 2) energy Ce B / 2 3-calculate loss factor of CLD C   2 d Energy/dx B Energy 4- For 3 layers CLD Viscoelastic Damping 34
  35. 35. Active Constrained layers damping   U1  U 3 h  Wx h2 h2 h h1  h3  h2 2 If U3=0 a-   U1 h  W h2 h2 U1  h2  hW U   h2   hW Quasi-static Equilibrium Longitudinal load on layer2=shear load  1 d   (db)   U    G *   h1 1  h1  1  h1  1U    from geomtry; (h1 * b) U 1  h2    hW  Viscoelastic Damping 35
  36. 36. Active Constrained layers damping 1h1 h2   hW   G *  1h1h2  hh    1 1 W   G* G* G * h     W 1h1h2 h2 let 1h1   1    G* h   W   1 h2 h2 W  W0 e i ( wt   * B ) W   i *3 BW0 e i one propagation solution * B e iWt W   i B W *3    G* h    B 3 iW  1 h2 h2 It has a solution;   ih B W   G* h2 1   *2  ( B  1 2 )  check that it satisfies equation Viscoelastic Damping 36
  37. 37. Active Constrained layers damping   G'  v h2 2 Energy dissipated per unit length Energy in bending waves W  W0 e i ( wt   * B ) W  W0 sin( wt   B * ) W 0  W0  cos(t   B * )  linear velocity W 0   W0  B sin(t   B * )  Anguler velocity W    B *W0 cos(t    * ) W     W0 sin(t    * ) *2 moment    Dt *W   Dt *  *2 W0 sin(t    )    Dt *  *3W0 cos(t    )  W 0 power  FW 0    shear  F    W 2 0   *3 Dt * cos2  Dt *   W 2 o sin 2  W0   *3 Dt *3 2 Energy  power * 2 /   -2  *3 DtWo2     h2G ' 'V  2  2 2  3 DtWo2  const .layer (h 2 / Dt ) g  V 1  g 2 Viscoelastic Damping 37
  38. 38. Active Constrained layers damping g G*  shear parameter  *2 1 2  constr is Max. when constr 0 g g optimum  1 NOTE  1  1h1 what is the physical meaning of g , if W  0  CLD in long.vibration   U1  U 3 hW  h2 h2   U1 h2 Also G*  0  1h2    o e  G*/ 1h2     Viscoelastic Damping 38
  39. 39. Active Constrained layers damping U  U oe Uo e U  G*   h 1 2    G * /  1 h2   e  1    1h2 G* 1 g  *2 2  B e e  *  B let g 2 g  2 2 2  e2 bendin gwave length shear wave length b- if U3=0   U1  U 3 h  W h2 h2 where U1 & U 3 are dependent in order to have 1h1U1  3h3U 3   0 F1  F3 0  1U1   3U 3   0 where U3     1  1h1  3  3h3  1 U1 3 1  1 /  3 h U1  W h2 h2 Viscoelastic Damping 39
  40. 40. Active Constrained layers damping  1   3   U1  h2  hW    3    2   3h2  3h  U1   W     1   3 1   3     1   h1    eqm.of top layer     1U1  h1  G *    1U1   G *  But  1 3h2 h    1 3 W  G* 1   3 1   3    G * ( 1   3 ) h   W  1h3h2 h2 Follow same procedure as case of U3=0 to get  constr  V ( 1 h 2 / Dt )( g / 1  g ) 2  1 g  1     3 1 g  2 Summary 1- Longitudinal vibration -to find optimum length of constraining layers Viscoelastic Damping 40
  41. 41. Active Constrained layers damping (Following plunket &lec. paper) Loptimum  3.28 B* G* 1 B*   h1h2 E2 characterstic length 2-comparing between CLD &un CLD Energy dissipated in un CLD<<< CLD Tension shear 3-transiverse vibration A -definition -specific damping -loss factor -loss factor &damping ratio selection B-CLD with U3=0 *shear parameter g=1 for optimum *g=ratio of bending to shear wave length ---optimum is ensured if there is balance between shear and bending C-U3=0 Viscoelastic Damping 41
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