notes on vibration isolation

uni/mmme/2046_dynamics_and_control/approximate_methods.md

@@ -1,7 +1,7 @@
 ---
 author: Akbar Rahman
 date: \today
-title: MMME2046 // Approximate Methods
+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 ]

uni/mmme/2046_dynamics_and_control/vibration_isolation.md

@@ -0,0 +1,77 @@
+---
+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