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# Computational Analysis Of A Thin Plate

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This is my thesis defense presentation from senior year

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• Hi today we will be talking about my research on the computational analysis of a thin plate My faculty sponsors were….
• First I will give an explanation of what resonance in an object is Motivation for this project: previous work and a history of this what led us to study a thin plate. I will then outline the goals of this investigation And present the model that was used to describe a thin plate Finite difference method: numerical method used to create an approximate time-dependent solution to our model of a thin plate Results and Analysis: present a comparison of theoretical results with experimental observations Conclusions: restate our conclusions from this project Future work: and highlight what implications these conclusions hold for future work
• A crotale is a cymbal-like instrument with a central region that is thicker than its outer edge. Usually, many crotales of varying diameters are arranged in rows and held fixed by a bolt through their central holes. They are then played by striking the outer edge of specific crotales to produce different sounds.
• All objects have a natural tendency to vibrate at specific resonant frequencies. When an object is externally driven at any of these frequencies it will form a standing vibrational pattern referred to as a mode Vibrational pattern of a freely vibrating plate is the combination of its resonant modes, with each mode contributing a different amount of power to the vibrations of the plate.
• A group at Rollins previously investigated the use of several different models to predict the resonant frequencies of a crotale Their first model was to treat the crotale as a thin circular plate clamped at its center. They used Rayleigh-Kirchoff thin plate theory, which Defines a thin plate as one in which the thickness of the plate is uniform and much smaller than its diameter and describes the transverse displacement of a thin plate using “this” equation, where: Sigma is the surface density of the plate D is the flexural rigidity of the plate defined by “this” equation that is composed of E, Young’s modulus, which is a measure of the elastic rigidity of the plate H is the thickness, and v is Poisson’s ratio, which is a ratio of the transverse to the longitudinal strain of an object They analytically solved for this differential equation creating a time-independent solution for the resonant frequencies to compare with experiment They did not find agreement with experiment for either the predicted resonant frequencies or ratios between these frequencies They concluded that a thin plate model was not sufficient for the crotale because the thick central region does have a significant effect on the tuning, which was not addressed in their solution for a thin plate Previous work with the crotale was performed by Bradley Deutch, Cherie Ramirez, and Thomas Moore.
• They then applied this thin plate theory to an annular plate model, in which the thick central region of the crotale was treated as a clamped region. They then analytically solved for the resonant frequencies of the plate and compared these values with experiment. Again they did not find agreement with experiment for either the predicted values for the resonant frequencies or the ratios between these frequencies
• So finally, they had to use a commercially developed finite element program called Solidworks to predict the resonant frequencies observed from experiment. They then went back and shaved the thick central region off the crotale and found agreement between thin plate theory and experiment.
• While this previous work prompted our investigation, we sought to develop a time-dependent finite difference solution to describe the evolution of the vibrations in a crotale, whereas they were only interested in predicting the resonant frequencies Since a different group at Rollins was already studying the time-evolution of a thin plate clamped over a central region, we began our investigation with the this plate, which was actually a crotale with the central region shaved off. Since this experimental set up can best be modeled as a thin annular plate, we began our investigation here to avoid the possible complications of the thick central region. If our model was successful, we would then adapt it to describe the crotale.
• Our thin annular plate model applies Rayleigh-Kirchoff thin plate theory to and annular plate clamped at the center and free at the outer edge, as shown in this diagram.
• Since we were studying a circular plate, we wrote equation of motion for a two-dimensional thin plate that was presented earlier in polar coordinates.
• We then added a velocity dependent damping term to describe the minimal dampening effects due to air, where R is the coefficient of the damping term We also added a pressure term used to simulate a strike on the plate at a specific point (r_o,phi_o), where the strike is simulated as a pulse as described in “this” equation This does not provide a mechanism for frequency dependent damping
• They are 2 nd and 3 rd order differential equations
• Since a freely vibrating plate will naturally tend to vibrate at is resonant frequencies, the resonant frequencies will make the largest contribution to the mechanical power and will show up as large maxima on the graph of the Fourier transform
• We have more error on our lower modes, than on our higher modes.
• Vibrational pattern of a freely vibrating plate is the combination of its resonant modes, with each mode contributing a different amount of power to the vibrations of the plate. Therefore, the first goal of our project was to determine if a thin annular plate model can correctly predict the initial relative strengths of the resonant modes of vibration within a freely vibrating plate.
• ### Computational Analysis Of A Thin Plate

1. 1. Computational Analysis of a Thin Plate David J. Parker April 28, 2008 Faculty Sponsors: Donald Griffin & Connor Ballance
2. 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. 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. 4. Resonant Modes <ul><li>Objects naturally vibrate at resonant frequencies </li></ul>1,0 mode 2,0 mode 3,0 mode
5. 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. 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. 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. 8. Goals Crotale Shaved Crotale Use a time-dependent finite difference solution to describe the evolution of the vibrations in a plate
9. 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. 10. Thin Annular Plate Model where Thin Plate in Polar Coordinates:
11. 11. Thin Annular Plate Model where Thin Plate in Polar Coordinates:
12. 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. 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. 14. Finite Differences
15. 15. Finite Differences
16. 16. Finite Differences <ul><li>Problem: How do you solve for 2 points beyond the physical plate? </li></ul>?
17. 17. Finite Differences <ul><li>Virtual Rings: </li></ul>
18. 18. Finite Differences <ul><li>Virtual Rings </li></ul><ul><li>Enforce boundary conditions </li></ul><ul><li>Test accuracy </li></ul>
19. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 30. Transient Behavior Experimental Observation
31. 31. Transient Behavior Experimental Observation Theoretical Prediction
32. 32. Transient Behavior
33. 33. Transient Behavior
34. 34. Transient Behavior
35. 35. Transient Behavior
36. 36. Transient Behavior
37. 37. Transient Behavior
38. 38. Transient Behavior
39. 39. Transient Behavior
40. 40. Transient Behavior
41. 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. 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. 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. 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>