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- 1. Computational Analysis of a Thin Plate David J. Parker April 28, 2008 Faculty Sponsors: Donald Griffin & Connor Ballance
- 2. Overview <ul><li>Goals and motivation for this project </li></ul><ul><li>Present the model used for a thin plate </li></ul><ul><li>Finite difference method </li></ul><ul><li>Procedures and Results </li></ul><ul><li>Conclusions and Future Work </li></ul>
- 3. Motivation <ul><li>This project began as an investigation of the evolution of the vibrational patterns of a crotale after it had been struck. </li></ul>
- 4. Resonant Modes <ul><li>Objects naturally vibrate at resonant frequencies </li></ul>1,0 mode 2,0 mode 3,0 mode
- 5. Previous Work with the Crotale 1 <ul><li>Predict the resonant frequencies of the crotale </li></ul><ul><li>Models: </li></ul><ul><ul><li>Thin circular plate clamped at its center </li></ul></ul>where Thin Plate: one in which the thickness of the plate uniform and much smaller than its diameter
- 6. Previous Work with the Crotale 1 <ul><li>Predict the resonant frequencies of the crotale </li></ul><ul><li>Models: </li></ul><ul><ul><li>Thin circular plate clamped at its center </li></ul></ul><ul><ul><li>Annular plate clamped over the thick central region </li></ul></ul>
- 7. Previous Work with the Crotale 1 <ul><li>Predict the resonant frequencies of the crotale </li></ul><ul><li>Models: </li></ul><ul><ul><li>Thin circular plate clamped at its center </li></ul></ul><ul><ul><li>Annular plate clamped over the thick central region </li></ul></ul><ul><ul><li>Finite element using Solidworks TM </li></ul></ul>
- 8. Goals Crotale Shaved Crotale Use a time-dependent finite difference solution to describe the evolution of the vibrations in a plate
- 9. Thin Annular Plate Model <ul><li>Apply Rayleigh-Kirchoff thin plate theory to an annular plate clamped at the center and free at the outer edge </li></ul>
- 10. Thin Annular Plate Model where Thin Plate in Polar Coordinates:
- 11. Thin Annular Plate Model where Thin Plate in Polar Coordinates:
- 12. Decreased Radial Spacing Constant differences in radial and angular direction Decreasing radial spacing <ul><li>Decreasing radial spacing results in constant areas between mesh </li></ul><ul><li>points </li></ul><ul><li>New equation of motion in ρ -space is derived </li></ul>
- 13. Boundary Conditions <ul><li>Inner clamped region: </li></ul><ul><ul><li>Slope and displacement equal 0 </li></ul></ul><ul><li>Outside edge of plate: </li></ul><ul><ul><li>Radial bending moment vanishes </li></ul></ul><ul><ul><li>Transverse shearing and twisting moment must also vanish </li></ul></ul>
- 14. Finite Differences
- 15. Finite Differences
- 16. Finite Differences <ul><li>Problem: How do you solve for 2 points beyond the physical plate? </li></ul>?
- 17. Finite Differences <ul><li>Virtual Rings: </li></ul>
- 18. Finite Differences <ul><li>Virtual Rings </li></ul><ul><li>Enforce boundary conditions </li></ul><ul><li>Test accuracy </li></ul>
- 19. Numerical Test for Accuracy <ul><li>Test our equations, mesh, and boundary conditions </li></ul><ul><li>Is a plate thin enough to be called thin? </li></ul><ul><li> thickness = 2 mm </li></ul><ul><li> radius = 86.1 mm </li></ul><ul><li>radius of clamped region = 23.8 mm </li></ul>
- 20. Numerical Test for Accuracy <ul><li>Record the amplitude of driving point at 2 µs intervals for 0.25 s after a simulated strike. </li></ul>
- 21. Numerical Test for Accuracy <ul><li>Perform Fourier transform </li></ul><ul><ul><li>Square the amplitude so that it is proportional to power </li></ul></ul><ul><li>Maxima occur at resonant frequencies </li></ul>
- 22. Numerical Test for Accuracy 1.1% 792 783 3,0 0.2% 475 474 2,0 -2% 398 406 0,0 -5.3% 378 398 1,0 % Different from analytical Resonant Frequency (Hz) Numerical Analytical Mode
- 23. Long-term Behavior <ul><li>Compare thin annular plate model with experiment for the shaved crotale: </li></ul><ul><li>To determine if it can correctly predict the initial relative strengths of the resonant modes </li></ul><ul><li>Confirm that the relative strengths of the resonant modes do not damp as a function of frequency </li></ul>
- 24. Shaved Crotale Parameters 6.35 mm Radius of clamped region 61.9 mm Radius of plate 1 ms Strike time 0.33 Poisson’s ratio 10 11 Young’s modulus 4.7 mm Thickness of plate 8861 kg/(m 2 s) Density
- 25. Long-term Behavior <ul><li>Theoretically </li></ul><ul><ul><li>Simulate a strike and record the displacements at the driving point </li></ul></ul><ul><ul><li>Perform Fourier transform </li></ul></ul><ul><li>Experimentally </li></ul><ul><ul><li>Strike the plate and record the radiated power above the strike point </li></ul></ul><ul><ul><li>Calculate the power spectrum </li></ul></ul>
- 26. Long-term Behavior Theoretical 0 - 0.1 s after the strike Theoretical 0.9 - 1 s after the strike Experimental 0 - 0.1 s after the strike Experimental 0.9 – 1 s after the strike
- 27. Long-term Behavior Numerical 0 – 0.1 s Experimental 0 – 0.1 s 0.7% 4270 4300 4,0 0.4% 2440 2450 3,0 -6.7% 1050 1120 2,0 - (731) * 870 0,0 - (700) * 730 1,0 % Different from Experiment Resonant Frequency (Hz) Numerical Experimental ( ± 10 Hz) Mode
- 28. Long-term Behavior Ratios of the power of each mode with respect to the 2,0 from 0-0.1 s after the strike * The initial relative strengths of the resonant modes were discovered experimentally to vary greatly with the strength, shape, and area of the strike 1.62 3.36e-2 4,0 1.00 1.00 3,0 8.96e-2 4.63e+1 2,0 Experimental Study 0 – 0.1 s Numerical Study 0 – 0.1 s Mode
- 29. Transient Behavior <ul><li>Experimentally </li></ul><ul><ul><li>Strike the plate </li></ul></ul><ul><ul><li>Use high-speed electronic speckle pattern interferometry to view the vibrational pattern of the plate at 30 µs intervals </li></ul></ul><ul><li>Theoretically </li></ul><ul><ul><li>Simulate a strike on the plate </li></ul></ul><ul><ul><li>Record the displacements at every point in the mesh </li></ul></ul><ul><ul><ul><li>Generate graphical representation to simulate interferometric view </li></ul></ul></ul>
- 30. Transient Behavior Experimental Observation
- 31. Transient Behavior Experimental Observation Theoretical Prediction
- 32. Transient Behavior
- 33. Transient Behavior
- 34. Transient Behavior
- 35. Transient Behavior
- 36. Transient Behavior
- 37. Transient Behavior
- 38. Transient Behavior
- 39. Transient Behavior
- 40. Transient Behavior
- 41. Conclusions <ul><li>Develop time-dependent finite difference program that used decreasing radial spacing and virtual rings </li></ul><ul><li>Confirmed that our program was a valid approximation for a thin annular plate </li></ul><ul><li>Confirmed that a thin annular plate model is not sufficient to account for damping of resonant modes </li></ul>
- 42. Conclusions <ul><li>Results from study of the initial relative strengths of the resonant modes were inconclusive </li></ul><ul><ul><li>We discovered that the initial strengths of the resonant modes is very dependent on how the plate is struck </li></ul></ul><ul><li>Transient vibrational patterns can be simulated using a thin annular plate model </li></ul><ul><ul><li>However, predicted amplitudes did not agree with experiment </li></ul></ul>
- 43. Future Work <ul><li>Further investigation of the effects and modeling of a strike on a plate needs to be conducted </li></ul><ul><ul><li>To compare the effects on the initial strengths of the resonant modes from experiment with theory </li></ul></ul><ul><ul><li>To determine the effects on the transient behavior of thin annular plate theory </li></ul></ul>
- 44. References <ul><li>[1] B. Deutsch, C Ramirez, T. Moore, “The dynamics and tuning of orchestral crotales,” J. Acoust. Soc. Am. 116, 2427-2433 (2004). </li></ul><ul><li>[2] B. Deutsch, A. Robinson, R. Felce, and T. Moore, “Nondegenerate normal mode doublets in vibrating flat circular plates,” Am. J. Phys. 72, 220-225 (2004). </li></ul><ul><li>[3] A. Leissa, Vibration of Plates (Acoustical Society of America, Melville, NY, 1993). </li></ul>

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