---
author: Akbar Rahman
date: \today
title: MMME2046 // Vibrations // Isolation
tags: [ vibration, vibration_isolation ]
uuid: fcdf1af0-9d54-4a6b-82fe-ef2c9f30ecb7
lecture_slides: [ ./lecture_slides/Vibration Isolation - FOR PRINT.pdf ]
lecture_notes: [ ./lecture_notes/Isolation 7.pdf ]
exercise_sheets: 
  - ./exercise_sheets/Vibratioon SHEET 7 - Isolation Part I.pdf
  - ./exercise_sheets/Vibratioon SHEET 7 - Isolation Part I - Solutions.pdf
  - ./exercise_sheets/Vibratioon SHEET 7 - Isolation Part II.pdf
  - ./exercise_sheets/Vibratioon SHEET 7 - Isolation Part II - Solutions.pdf
---

Vibration isolators are used to reduce the vibration transmitted from a source.
They work by introducing flexibility between a device and its support.

There are a two potential aims for vibration isolation:

1. Reduce force transmitted to the support (e.g. a passing train that vibrates the ground)
1. Minimise displacement transmitted to the device (e.g. a satellite mounted in its launch vehicle)

# Types of Isolators

- Elastomeric --- most common type of isolater
- Pneumatic
- Coil spring

# Transmissibility Analysis

Isolators tend to be much more flexible than the devices they support.
A good first approximation is to use a single degree of freedom model:

- the device to be isolated is treated as a rigid body
- the isolators are represented by a spring-damper combination
- steady-state harmonic response is used to characterise the isolation performance at different frequencies

Derivations for force and displacement transmissibility equations are in lecture slides (p. 6-11).
It is always best to derive $T_D$ and $T_F$ for each system.

![Transmissibility curves show how excitation frequency affects the transmitted force or displacement. It has significant effect near resoonance, but little effect at high frequencies. Infinite damping is a special case and corresponds to a rigid connection between the device and its support.](./images/vimscrot-2023-03-13T16:33:44,739577370+00:00.png)

The aim when selecting isolators is to ensure that the system operates in the isolation region:
![](./images/vimscrot-2023-03-13T16:37:12,862474811+00:00.png)

# Isolation Efficiency

$$\eta_\text{isolation} = 1-T$$

![](./images/vimscrot-2023-03-13T16:37:58,091991533+00:00.png)

# Isolator Selection

- to reduce vibrations, $\omega_n << \omega_\text{min}$
- $m$ and $k$ determine $\omega_n$
- $k$ is given by the isolator 
- the mass supported by the isolator can be increased by mounting it on an inertia base.
- for most commercial isolators, $\gamma < 0.$ (it is normal to assume zero damping)
- it is also normal to treat each isolator independently of the others

## Maximum Static Deflection

Manufacturers often specify a maximum static deflection, where the spring will not behave linearly:

$$X_0 = \frac{g}{\omega_\text{min}^2}\left(1+\frac{1}{T_\text{max}}\right)$$

## Design Procedure

1. Find centre of mass of the machine
1. Select number and position of attachment points for isolators
1. Estimate load supported by each isolator
1. For each isolator position

   1. Calculate maximum stiffness
   1. Select isolator with lower stiffness
   1. Check that this does not exceed static deflection limit