Finite Element Modelling of Traveling Waves

1. FINITE ELEMENT MODELLING OF TRAVELING WAVES By Ashwin Ajith Chandran H00114624 Year 4, B.Eng (Hons.) Mechanical Engineering Supervisor : Prof. Andrew. J. Moore 1

2. PROJECT BREAKDOWN Natural frequencies of an edge clamped circular plate Natural frequencies of a centre clamped circular plate Dual frequency excitation traveling wave (model) FE Animations 2

3. INTRODUCTION Natural Frequency Natural frequency is the frequency at which a system tends to oscillate in the absence of any driving or damping force. 3

4. THEORY ( ) = 2 f (Hz) = 2 Where, D = 3 12(12) And, Flexural rigidity, D = 6.5462 Radius, R =0.07m Density, = 2700 kg/m3 Thickness, h = 0.001m Youngs Modulus, E = 70 GPa Poissons ratio, = 0.33 Base Mesh 4

5. OBJECT 5 Aluminium Disc

6. STANDING WAVES 6

7. STANDING WAVE 3D DISC 7 Fundamental mode 2 Diametral nodes and 1 circumferential node 1 circumferential node and 1 diametral node

8. TRAVELING WAVE 8 + = Standing Standing Traveling

9. TRAVELING WAVE EQUATION w(, t) = A(K ) cos( t) + B(K ) sin( t) (1) Where, K = Spatial Wave number A(K ) and B(K ) are functions that are both position dependent A(K ) = A1 cos(K ) + A2 sin(K ) (2) B(K ) = B1 cos(K ) + B2 sin(K ) (3) 9

10. TRAVELING WAVE EQUATION Substituting (2) and (3) in (1) w(, t) = 1 2 (( A1 + B2 ) cos( t K ) + ( B1 A2 ) sin( t K ) + ( A1 B2 ) cos( t + K ) + ( B1 + A2 ) sin( t + K ) ) 10

11. TRAVELING WAVE EQUATION The terms ( t K ) and ( t + K ) determine the positive or negative direction of the traveling wave, w+ = 1 2 (( A1 + B2 ) cos( t K ) + ( B1 A2 ) sin( t K )) w = 1 2 (( A1 B2 ) cos( t + K ) + ( B1 + A2 ) sin( t + K ) ) 11

12. SIMULATIONS Case 1 : Edge Clamped Disc Case 2 : Centre and Unclamped Case 3 : Traveling Wave 12

13. EDGE CLAMPED CIRCULAR DISC Natural frequency comparison Theory: 516.665 1075.228 1764.062 2581.356 Abaqus: 513.36 1082.6 1780.7 2600.9 Theory: 2011.42 3075.98 4277.64 5614.35 Abaqus: 2002.9 3051.8 4275.4 5669.3 Theory: 4506.45 6073.06 7778.96 9624.45 Abaqus: 4550.3 6083.5 7790.3 9564.1 13

14. ERROR, ANIMATION, FREQUENCY GRAPH 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 0 50 100 150 EigenFrequencies Mode Number Frequency f (hz) n=0 n=1 n=2 n=3 s=0 0.639836 -0.68561 -0.94317 -0.7571 s=1 0.423832 0.786149 0.052548 -0.97875 s=2 -0.97305 -0.1718 -0.14569 0.627097 14

15. UNCLAMPED AND CENTRE CLAMPED DISCS According to theoretical calculations, Unclamped natural frequency = Centre clamped natural frequency 15

16. UNCLAMPED AND CENTRE CLAMPED DISCS Natural frequency comparison Theory: 0 0 265.67 618.53 Unclamped: 0 0 263.21 615.28 C.Clamped: 0 0 267.34 621.32 Theory: 459.42 1037.8 1782.7 2675.93 Unclamped: 455.61 1030.8 1765.6 2692.1 C.Clamped: 455.8 1030.9 1771.6 2695.8 Theory: 1949.67 3027.43 4243.25 5629.01 Unclamped: 1939.3 3039.3 4274.2 5616.3 C.Clamped: 1932.7 3050.3 4281.3 5601.1 16

17. UNCLAMPED AND CENTRE CLAMPED DISCS FE time history animation comparison and error % Unclamped Centre Clamped f (hz) n=0 n=1 n=2 n=3 s=0 0 0 0.926469 0.526134 s=1 0.830351 0.67474 0.96337 -0.60419 s=2 0.532026 -0.39209 -0.72923 0.225907 f (hz) n=0 n=1 n=2 n=3 s=0 0 0 -0.62808 -0.45037 s=1 0.788995 0.665105 0.626816 -0.74246 s=2 0.870545 -0.75544 -0.89656 0.495936 17

18. UNCLAMPED AND CENTRE CLAMPED DISCS Eigen frequency chart comparison Unclamped Disc Centre Clamped Disc 18

19. EIGEN FREQUENCY CONVERGENCE GRAPHS 0 3000 6000 9000 0 10000 20000 30000 40000 50000 60000 70000 EigenFrequency Number of elements Edge Clamped Disc(m=2,n=0) Frequency(m=2,n=0) 0 2000 4000 6000 0 10000 20000 30000 40000 50000 60000 70000 EigenFrequency Number of elements Unclamped/Centre Clamped Disc (m=2,n=0) Frequency (m=2,n=0) 19

20. STEADY STATE MODAL ANALYSIS Modal Dynamics Applying Constant (1N) Steady State Dynamics, Modal Frequency range 0 Hz 10000 Hz New load, concentrated force, continuous 20

21. STEADY STATE MODAL ANALYSIS FE Animations 21

22. TRAVELING WAVE ANIMATION 22

23. CHALLENGES Hourglassing A spurious deformation mode of a Finite Element Mesh, resulting from the excitation of zero-energy degrees of freedom. Caused due to inability to resist deformation where no stiffness in the mode Resolved by mesh refinement 23

24. FURTHER SCOPE Structural Intensity Injected power Vibrational energy flow Power flow 24

25. THANK YOU 25

Finite Element Modelling of Traveling Waves

Engineering

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