Lab 6 Pre Lab 1

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University Of Connecticut *

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3263

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Mechanical Engineering

Date

Dec 6, 2023

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pdf

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2

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ME-3283 Lab Section 012, Station 02 Andrew Chang 11/8/2023 Lab 6, Pre-Lab 1 1. Using MATLAB or Python, perform a multiple linear regression on the above data to find thickness as a function of the changes in the first three natural frequencies. (4 points) Thickness (t) function: 𝑡 = − 7. 93∆ω 1 − 86. 05∆ω 2 + 29. 25∆ω 3
2. Collected frequency data for an unknown coating thickness show frequency changes of 𝜔 1=-1.34 kHz, 𝜔 2 = -7.71 kHz, and 𝜔 3 = -21.27 kHz. What is the thickness of the gold coating? (2 points) 3. From the lab manual, the material we discussed in class, and your knowledge from previous labs, develop a hypothesis for what you expect to see in Lab 6. What will be the relationship between our controlled variables (the location and size of the added mass) and our measured variables (the beam’s natural frequencies). Make a prediction that we do not know a priori to be true. (3 points). Since we know that mass has a direct function relating to the change of natural frequency eg. , which is derived from Ghatkesar’s findings. From that same 𝑚 = ?∆ω 1 + ?∆ω 2 + ?∆ω 3 article we saw that a uniform deposition, otherwise, a uniformly distributed load across the cantilever beam, showed that increase in mass sensitivity was linear with the square of the mode number. This implies that the equivalent point load at exactly half of the cantilever’s length would also have the same relationship. In Lab 6, I expect to see a linear relationship between mass and change in frequency when regression is applied when the point load is applied at half the cantilever’s length. I believe that variation from this location specifically may interfere with the cantilever’s natural frequencies in an unexpected way. I also believe that the existence of a potential second point load at the end of the beam due to the mounting of the beam may also have an adverse effect on the results, skewing the data or adding unnecessary noise. Ghatkesar et al also discussed the effect the medium the beam is vibrating in having an effect so I expect to see a 0.5% deviation from the theoretical/guessed data. I also believe that applying a point mass to a node may have very little effect on the resonant frequency due to the nature of a node being a natural point of little to no oscillation. However I also believe that this only holds true for masses with small moment of Inertia as we see from Euler-Lagrange Dynamic beam equation that moment of inertia affects the resonant frequency, so if the moment of inertia of the applied mass is too large to no longer be negligible, then due to parallel axis theorem, the effect on the beam will be considerable.
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