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. #WikiCourses https://wikicourses.wikispaces.com/Topic+Periodic+Structures https://eau-esa.wikispaces.com/Vibration+of+structures
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Periodic Structures
- 1. Periodic Structures: A Passive Vibration Filter Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. Examples of periodic structures • • • • • Satellite panels Railway tracks Aircraft Fuselage Multistory buildings Etc… Etc Mohammad Tawfik Aero631 – Vibrations of Structures
- 4. Structure Discontinuity! Mohammad Tawfik Aero631 – Vibrations of Structures
- 5. Types of Discontinuity / Periodicity Material Periodicity y Geometric/Support Periodicity Mohammad Tawfik Aero631 – Vibrations of Structures
- 6. Recall what happens to a wave as it travels through a boundary between two different media Mohammad Tawfik Aero631 – Vibrations of Structures
- 7. Wave propagation in different media Mohammad Tawfik Aero631 – Vibrations of Structures
- 8. Mechanical waves behave in a similar way! Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. Stop bands are the center of interest for the periodic p analysis of structures! Mohammad Tawfik Aero631 – Vibrations of Structures
- 11. Periodic Analysis of Structures Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. How?! Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. 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. 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. 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. 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. 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. 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. 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. The Eigenvalues Mohammad Tawfik Aero631 – Vibrations of Structures
- 23. The Propagation Factor Mohammad Tawfik Aero631 – Vibrations of Structures
- 24. Frequency Resp of Cell Resp. Mohammad Tawfik Aero631 – Vibrations of Structures
- 25. Freq. Resp Freq Resp. of 6 Cells Mohammad Tawfik Aero631 – Vibrations of Structures
- 26. Homework • Prepare a MATLAB program to perform the periodic analysis of a bar. Mohammad Tawfik Aero631 – Vibrations of Structures
- 27. Modeling K M & Eigen nvalues( Rearrangement Eigenvalue problem T (Hz) Mohammad Tawfik Aero631 – Vibrations of Structures
- 28. Modeling K M & Rea al( Rearrangement Ima aginary( Eigenvalue problem e (Hz) T Propagation Factor Mohammad Tawfik Aero631 – Vibrations of Structures
- 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 EigenvalueT12 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. Modeling K & M ( (Hz) Rearrangement K 2 M (Hz) Eigenvalue problem Imaginary( Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. 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. So … What really happens? Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. Periodic Bar Mohammad Tawfik Aero631 – Vibrations of Structures
- 36. Results Mohammad Tawfik Aero631 – Vibrations of Structures
- 37. Experimental Setup for the Periodic Beam Mohammad Tawfik Aero631 – Vibrations of Structures
- 38. Overview Picture Mohammad Tawfik Aero631 – Vibrations of Structures
- 39. Beam Cell Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. 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. 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. Periodic Plate Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. Wave propagates in 2-D 2D Wave is split into its components in X and Y-directions Mohammad Tawfik Aero631 – Vibrations of Structures
- 46. ( (Hz) No forward approach Reverse approach K 2 M (Hz) Eigenvalue problem Imaginary( Mohammad Tawfik Aero631 – Vibrations of Structures
- 47. Propagation Surfaces Analytical x y Mead and Parathan 1979 Mohammad Tawfik Aero631 – Vibrations of Structures
- 48. Requires higher order elements! 64 DOF element used Mohammad Tawfik Aero631 – Vibrations of Structures
- 49. Propagation Surfaces Numerical x y Mohammad Tawfik Aero631 – Vibrations of Structures
- 50. Experiments Mohammad Tawfik Aero631 – Vibrations of Structures
- 51. Periodic Plate Mohammad Tawfik Aero631 – Vibrations of Structures
- 52. Periodic Plate Mohammad Tawfik Aero631 – Vibrations of Structures
- 53. Propagation Surfaces Mohammad Tawfik Aero631 – Vibrations of Structures
- 54. Comparison Mohammad Tawfik Aero631 – Vibrations of Structures
- 55. Effect of shunted inductance Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. Adding the Inductance Inductance x y Mohammad Tawfik Aero631 – Vibrations of Structures
- 58. Further developments Mohammad Tawfik Aero631 – Vibrations of Structures
- 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. Effect of Shunt Circuit on Propagation Surfaces Not Shunted Shunted Mohammad Tawfik Aero631 – Vibrations of Structures