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# Periodic Structures

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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.

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https://wikicourses.wikispaces.com/Topic+Periodic+Structures
https://eau-esa.wikispaces.com/Vibration+of+structures

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### Periodic Structures

1. 1. Periodic Structures: A Passive Vibration Filter Mohammad Tawfik Aero631 – Vibrations of Structures
2. 2. What is a Periodic Structure? • A structure that consists fundamentally of a number of identical substructure components that are joined together to form a continuous structure Mohammad Tawfik Aero631 – Vibrations of Structures
3. 3. Examples of periodic structures • • • • • Satellite panels Railway tracks Aircraft Fuselage Multistory buildings Etc… Etc Mohammad Tawfik Aero631 – Vibrations of Structures
4. 4. Structure Discontinuity! Mohammad Tawfik Aero631 – Vibrations of Structures
5. 5. Types of Discontinuity / Periodicity Material Periodicity y Geometric/Support Periodicity Mohammad Tawfik Aero631 – Vibrations of Structures
6. 6. Recall what happens to a wave as it travels through a boundary between two different media Mohammad Tawfik Aero631 – Vibrations of Structures
7. 7. Wave propagation in different media Mohammad Tawfik Aero631 – Vibrations of Structures
8. 8. Mechanical waves behave in a similar way! Mohammad Tawfik Aero631 – Vibrations of Structures
9. 9. Stop Bands • As the wave faces an abrupt change in the geometry a geometry, part if it is reflected • The reflected part, interferes with the incident wave • At some frequency bands that interference becomes bands, destructive creating the “Stop Bands” Mohammad Tawfik Aero631 – Vibrations of Structures
10. 10. Stop bands are the center of interest for the periodic p analysis of structures! Mohammad Tawfik Aero631 – Vibrations of Structures
11. 11. Periodic Analysis of Structures Mohammad Tawfik Aero631 – Vibrations of Structures
12. 12. Why Periodic Analysis? • Periodic structures can be modeled like any ordinary structure, BUT • In a periodic structure, the study of the structure behavior of one cell is enough to determine the stop and pass bands of the complete structure independent of the number of cells Mohammad Tawfik Aero631 – Vibrations of Structures
13. 13. How?! Mohammad Tawfik Aero631 – Vibrations of Structures
14. 14. Equations of Motion   m11 m12  U1   k 11 k 12  U1   F1       m m       k 22  U 2  F2  22  U 2  k 21  21  k 11   2 m11  k 21   2 m21   D11 D  21 k 12   2 m12  U1   F1       2 k 22   m22  U 2   F2  D12  U1   F1   U    F  D22   2   2  Rearranging the terms Mohammad Tawfik Aero631 – Vibrations of Structures
15. 15. Equations of Motion D11U1  D12U 2  F1 D21U1  D22U 2  F2   U 2   D121 D11U1  D121 F1 F2  D21U1  D22U 2 1 12 1 12 1 U 2   D D11U1  D F     F2  D21  D22 D121 D11 U1  D22 D121 F1 Mohammad Tawfik Aero631 – Vibrations of Structures
16. 16. Equations of Motion  U 2    D121 D11     F2   D21  D22 D121 D11   U1   1   D22 D12   F1  1 12 D U 2    U1   e    F2   F1  1 U1    D12 D11  e    F1   D21  D22 D121 D11   U1   1    D22 D12   F1  1 12 D Mohammad Tawfik Aero631 – Vibrations of Structures
17. 17. Equations of Motion T11 T12  U1   U1  T T   F   e  F   21 22   1   1 T11 T12  e  Eigenvalues   T21 T22   Propagation factor Mohammad Tawfik Aero631 – Vibrations of Structures
18. 18. Note! • The transfer matrix is dependent on the excitation frequency • Hence the propagation factor is Hence, dependent on the frequency • Th eigenvalues of th t The i l f the transfer matrix will f ti ill appear in reciprocal pairs (. Mohammad Tawfik Aero631 – Vibrations of Structures
19. 19. Example: Periodic Spring Mass • W it down th equations of motion f the Write d the ti f ti for th cell given by 2 half masses and one spring  m 0  u1   k  0 m      k   u2    k  u1   f1       k  u2   f 2  Mohammad Tawfik Aero631 – Vibrations of Structures
20. 20. Example • Getting the dynamic stiffness matrix k   2 m  k  u1   f1     2  k   m u2   f 2   k • Rearranging:   2m  1 k  k   2m k   k    2  u  u   1    2   2 m   f1   f 2  1 k   1  k Mohammad Tawfik Aero631 – Vibrations of Structures
21. 21. Example • Getting the transfer matrix:   2m  1 k  2 2  k  m  k  k    1    k   u1   e   u1      2  m   f1   f1   1 k  • Using Matlab to calculate the eigenvalues, g we will get. Mohammad Tawfik Aero631 – Vibrations of Structures
22. 22. The Eigenvalues Mohammad Tawfik Aero631 – Vibrations of Structures
23. 23. The Propagation Factor Mohammad Tawfik Aero631 – Vibrations of Structures
24. 24. Frequency Resp of Cell Resp. Mohammad Tawfik Aero631 – Vibrations of Structures
25. 25. Freq. Resp Freq Resp. of 6 Cells Mohammad Tawfik Aero631 – Vibrations of Structures
26. 26. Homework • Prepare a MATLAB program to perform the periodic analysis of a bar. Mohammad Tawfik Aero631 – Vibrations of Structures
27. 27. Modeling K  M  & Eigen nvalues(  Rearrangement Eigenvalue problem T   (Hz) Mohammad Tawfik Aero631 – Vibrations of Structures
28. 28. Modeling K  M  & Rea al( Rearrangement Ima aginary( Eigenvalue problem  e (Hz) T      Propagation Factor Mohammad Tawfik Aero631 – Vibrations of Structures
29. 29.  k11 k  21 k12  u1   f1   u    f  k 22   2   2  k   11  u2  k12   k k  f 2   11 22  k12  k12  u3    u1   e   f3   f1    1  k12  u1   k 22   f1    k12   EigenvalueT12  T23     Forward Approach pp  k11 k  21  k31  k11   k 21 k31e    k13  u1   f1      k 23  u2    f 2   k33  u3   f 3      k12 k 22 k32 k13e   u1   f1      k 23e   u 2    0  k33  u1   f1      k12 k 22 k32 k11  k13e   k31e    k33  k 21  k 23e    k12  k32 e    u1  0      k 22  u2  0  Eigenvalue M 1  K     Reverse Approach Mohammad Tawfik Aero631 – Vibrations of Structures
30. 30. Modeling K  & M  ( (Hz) Rearrangement K     2 M    (Hz) Eigenvalue problem Imaginary( Mohammad Tawfik Aero631 – Vibrations of Structures
31. 31. Propagation Curves Forward Approach Reverse Approach Imag ginary( Attenuation Band Propagation Curves p g  (Hz) Propagation Bands (Hz) Imaginary( Mohammad Tawfik Aero631 – Vibrations of Structures
32. 32. Note! All the above mentioned analysis is independent of the y p structure type (beams, bars, or plates) Mohammad Tawfik Aero631 – Vibrations of Structures
33. 33. So … What really happens? Mohammad Tawfik Aero631 – Vibrations of Structures
34. 34. Experimental Investigation • Bars with periodic geometry and material changes. • Beams with periodic geometry geometry. • Plates with periodic geometry. Mohammad Tawfik Aero631 – Vibrations of Structures
35. 35. Periodic Bar Mohammad Tawfik Aero631 – Vibrations of Structures
36. 36. Results Mohammad Tawfik Aero631 – Vibrations of Structures
37. 37. Experimental Setup for the Periodic Beam Mohammad Tawfik Aero631 – Vibrations of Structures
38. 38. Overview Picture Mohammad Tawfik Aero631 – Vibrations of Structures
39. 39. Beam Cell Mohammad Tawfik Aero631 – Vibrations of Structures
40. 40. Case#1 30 10 9 20 10 7 0 0 500 1000 1500 2000 2500 3000 3500 6 4000 5 -10 4 20 -20 Plain Beam Periodic Beam Attenuation Factor -30 -40 Atte enuatin Factor (ra ad) Transfer Function Amplitu (dB) ude 8 3 2 1 -50 0 Frequency (Hz) Mohammad Tawfik Aero631 – Vibrations of Structures
41. 41. Case#2 20 10 9 10 0 0 500 1000 1500 2000 2500 3000 3500 7 4000 6 -1 0 5 -2 0 4 P lain B eam P e riod ic B eam A ttenu atio n F ac tor -3 0 Att tenuatin Factor (ra ad) Transfer Function Amplitu (dB) r ude 8 3 2 -4 0 4 1 0 -5 0 F re q u e n c y (H z) Mohammad Tawfik Aero631 – Vibrations of Structures
42. 42. Case#3 20 10 9 10 0 0 500 1000 1500 2000 2500 3000 3500 7 4000 6 -10 5 -20 4 -30 Atte enuatin Factor (ra ad) Transfer Function Amplitu (dB) ude 8 3 Plain Beam Periodic Beam Attenuation Factor 2 -40 40 1 -50 0 Frequency (Hz) Mohammad Tawfik Aero631 – Vibrations of Structures
43. 43. Periodic Plate Mohammad Tawfik Aero631 – Vibrations of Structures
44. 44. Problems Associated with 2-D Structures • Wave propagates in 2-dimensions 2 dimensions. • Input-Output relations are not readily available (no forward approach) • Requires higher order elements for numerical analysis i l l i Mohammad Tawfik Aero631 – Vibrations of Structures
45. 45. Wave propagates in 2-D 2D Wave is split into its components in X and Y-directions Mohammad Tawfik Aero631 – Vibrations of Structures
46. 46. ( (Hz) No forward approach  Reverse approach K     2 M    (Hz) Eigenvalue problem Imaginary( Mohammad Tawfik Aero631 – Vibrations of Structures
47. 47. Propagation Surfaces Analytical  x y Mead and Parathan 1979 Mohammad Tawfik Aero631 – Vibrations of Structures
48. 48. Requires higher order elements! 64 DOF element used Mohammad Tawfik Aero631 – Vibrations of Structures
49. 49. Propagation Surfaces Numerical  x y Mohammad Tawfik Aero631 – Vibrations of Structures
50. 50. Experiments Mohammad Tawfik Aero631 – Vibrations of Structures
51. 51. Periodic Plate Mohammad Tawfik Aero631 – Vibrations of Structures
52. 52. Periodic Plate Mohammad Tawfik Aero631 – Vibrations of Structures
53. 53. Propagation Surfaces Mohammad Tawfik Aero631 – Vibrations of Structures
54. 54. Comparison Mohammad Tawfik Aero631 – Vibrations of Structures
55. 55. Effect of shunted inductance Mohammad Tawfik Aero631 – Vibrations of Structures
56. 56. Vibration Absorber  0   Wb   M b   0    M D  WD      K b   K bD   Wb  0    K Db   K D   WD     Mohammad Tawfik Aero631 – Vibrations of Structures
57. 57. Adding the Inductance  Inductance x y Mohammad Tawfik Aero631 – Vibrations of Structures
58. 58. Further developments Mohammad Tawfik Aero631 – Vibrations of Structures
59. 59. Further Development • More analytical numerical and analytical, numerical, experimental studies need to further investigate the periodic plate • Periodic Shells –L Longitudinal periodicity i cylindrical shell it di l i di it in li d i l h ll – Circumferential periodicity in axisymmetric shells Mohammad Tawfik Aero631 – Vibrations of Structures
60. 60. Effect of Shunt Circuit on Propagation Surfaces Not Shunted Shunted Mohammad Tawfik Aero631 – Vibrations of Structures