diff --git a/uni/mmme/2046_dynamics_and_control/approximate_methods.md b/uni/mmme/2046_dynamics_and_control/approximate_methods.md index f4143b8..09de555 100755 --- a/uni/mmme/2046_dynamics_and_control/approximate_methods.md +++ b/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 ] diff --git a/uni/mmme/2046_dynamics_and_control/exercise_sheets/Vibratioon SHEET 7 - Isolation Part I - Solutions.pdf b/uni/mmme/2046_dynamics_and_control/exercise_sheets/Vibratioon SHEET 7 - Isolation Part I - Solutions.pdf new file mode 100644 index 0000000..9dd0e86 Binary files /dev/null and b/uni/mmme/2046_dynamics_and_control/exercise_sheets/Vibratioon SHEET 7 - Isolation Part I - 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FOR PRINT.pdf b/uni/mmme/2046_dynamics_and_control/lecture_slides/Vibration Isolation - FOR PRINT.pdf new file mode 100644 index 0000000..32d7b4f Binary files /dev/null and b/uni/mmme/2046_dynamics_and_control/lecture_slides/Vibration Isolation - FOR PRINT.pdf differ diff --git a/uni/mmme/2046_dynamics_and_control/vibration_isolation.md b/uni/mmme/2046_dynamics_and_control/vibration_isolation.md new file mode 100755 index 0000000..b58a219 --- /dev/null +++ b/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