2023-03-06 22:25:45 +00:00
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---
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author: Akbar Rahman
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date: \today
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2023-03-13 17:11:43 +00:00
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title: MMME2046 // Vibrations // Approximate Methods
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2023-03-06 22:25:45 +00:00
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tags: [ vibrations, approximate_methods, rayleighs_method ]
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uuid: 7cd5b86f-74df-4ec6-b3c6-9204cf949093
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lecture_slides: [ ./lecture_slides/Vibrations - Approximate Methods.pdf ]
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lecture_notes: [ ./lecture_notes/Approximate Methods - No Dunkerley - 6.pdf ]
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exercise_sheets: [ ./exercise_sheets/Vibratioon SHEET 6 - Approximate Methods.pdf, ./exercise_sheets/Vibratioon SHEET 6 - Approximate Methods Solutions.pdf ]
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---
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branson says 3rd year dynamics is way harder
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This module will focus exclusively on Rayleigh's method.
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In general:
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$$\omega_{n,\text{Rayleigh}} \geq \omega_{n,\text{exact}}$$
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This means that it is best to try several mode shapes and the lowest frequency of
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those shapes will be the most accurate.
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The most important frequency is the first natural frequency.
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If the operating frequency never achieves this frequency, then there won't be any issues.
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# Instantaneous Strain Energy
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$$E = 0.5k(\Delta L)^2$$
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where $k$ is the stiffness, $\Delta L$ is the length
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# Rayleigh's Method for Shafts and Beams
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$$T_\text{max} = 0.5\omega^2 \int^L_0 \rho A [Y(x)]^2 \mathrm{d} x$$
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$$U_\text{max} = 0.5\int^L_0 EI \left(\frac{\mathrm{d}^2Y}{\mathrm dx^2}\right)^2 \mathrm{d} x$$
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where $Y(x)$ is the mode shape function, which defines the amplitude of vibration of the shaft or
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beam along its length. This function needs to be guessed.
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## $Y(x) = Cx^2$
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This guess is not that close, but the only one expected to be used in exams.
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It gives:
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$$T_\text{max} = \omega^2\frac{\rho A C^2L^5}{10}$$
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$$U_\text{max} = 2EIC^2L$$
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Equating them gives
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$$\omega_n = \frac{4.47}{L^2}\sqrt{\frac{EI}{\rho A}}$$
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... which is an **error of 27%** (not close).
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## Other choices of $Y(x)$
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Are detailed in lecture slides (p12-14).
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# Dynamically Equivalent Systems (slides p22-37)
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