vibrations lecture --- approximate methods

uni/mmme/2046_dynamics_and_control/approximate_methods.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2046 // Approximate Methods
+tags: [ vibrations, approximate_methods, rayleighs_method ]
+uuid: 7cd5b86f-74df-4ec6-b3c6-9204cf949093
+lecture_slides: [ ./lecture_slides/Vibrations - Approximate Methods.pdf ]
+lecture_notes: [ ./lecture_notes/Approximate Methods - No Dunkerley - 6.pdf ]
+exercise_sheets: [ ./exercise_sheets/Vibratioon SHEET 6 - Approximate Methods.pdf, ./exercise_sheets/Vibratioon SHEET 6 - Approximate Methods Solutions.pdf ]
+---
+
+branson says 3rd year dynamics is way harder
+
+This module will focus exclusively on Rayleigh's method.
+In general:
+
+$$\omega_{n,\text{Rayleigh}} \geq \omega_{n,\text{exact}}$$
+
+This means that it is best to try several mode shapes and the lowest frequency of
+those shapes will be the most accurate.
+
+The most important frequency is the first natural frequency.
+If the operating frequency never achieves this frequency, then there won't be any issues.
+
+# Instantaneous Strain Energy
+
+$$E = 0.5k(\Delta L)^2$$
+
+where $k$ is the stiffness, $\Delta L$ is the length
+
+# Rayleigh's Method for Shafts and Beams
+
+$$T_\text{max} = 0.5\omega^2 \int^L_0 \rho A [Y(x)]^2 \mathrm{d} x$$
+
+$$U_\text{max} = 0.5\int^L_0 EI \left(\frac{\mathrm{d}^2Y}{\mathrm dx^2}\right)^2 \mathrm{d} x$$
+
+where $Y(x)$ is the mode shape function, which defines the amplitude of vibration of the shaft or
+beam along its length. This function needs to be guessed.
+
+## $Y(x) = Cx^2$
+
+![](./images/vimscrot-2023-03-06T16:38:11,596006361+00:00.png)
+
+This guess is not that close, but the only one expected to be used in exams.
+
+It gives:
+
+$$T_\text{max} = \omega^2\frac{\rho A C^2L^5}{10}$$
+
+$$U_\text{max} = 2EIC^2L$$
+
+Equating them gives
+
+$$\omega_n = \frac{4.47}{L^2}\sqrt{\frac{EI}{\rho A}}$$
+
+... which is an **error of 27%** (not close).
+
+## Other choices of $Y(x)$
+
+Are detailed in lecture slides (p12-14).
+
+# Dynamically Equivalent Systems (slides p22-37)
+
+![](./images/vimscrot-2023-03-06T17:07:02,646825881+00:00.png)
+
+
+
+