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
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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
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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
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.
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