---
author: Akbar Rahman
date: \today
title: MMME2046 // Vibrations // 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)