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

uni/mmme/2047_thermodynamics_and_fluid_dynamics/turbomachinery.md

@@ -0,0 +1,124 @@
+---
+author: Akbar Rahman
+date: \today
+title: MMME2047 // Turbomachinery
+tags: [ turbomachinery ]
+uuid: 11f0f745-2364-4594-8e47-127a4af39417
+lecture_slides: [ ./lecture_slides/T5 - Turbomachinery - with solutions.pdf ]
+lecture_notes: [ ./lecture_notes/turbomachinery lecture notes(H Power).pdf ]
+exercise_sheets: [ ./exercise_sheets/Turbomachinery-problems.pdf]
+---
+
+Turbomachinery are rotating devices that add (pump for liquids; fan, blower, or compressor for gases at <0.02, <1 bar, and > 1 bar respectively) or extract (turbine) energy from a fluid.
+
+# Positive Displacement (PD) Pumps
+
+- PD pumps force fluid along using volume changes (e.g. bike pumps, the heart)
+- All PD pumps deliver a periodic flow
+- They deliver any fluid regardless of viscosity (dynamic pumps struggle with viscous fluids)
+- They are self priming (will be filled automatically)
+- They can operate under high pressures (300 atm) but low flow rates (25 m$^3$h$^{-1}$)
+- flow rate can be only be changed by vary speed or displacement
+
+# Dynamics Pumps
+
+- add momentum to fluid by fast moving blades or vanes
+- classified based on direction of flow at exit:
+
+    - centrifugal
+    - axial
+    - mixed flow
+
+- fluid increases momentum while moving through open passages and extra velocity is converted to
+  pressure through exiting it into a diffuser section
+- provide high flow rates (up to 70000 m$^3$h$^{-1}$) but usually at moderate pressure rises (a few atm)
+- require priming
+
+# Centrifugal Pumps
+
+- fluid enters through eye of casing and gets caught in impeller blades
+- fluid is whirled outwards until it leaves via the expanding area section, known as the diffuser or volute
+
+## Blades
+
+- backward inclined blades - most common and efficient, intermediate pressure rise, less robust
+- straight blades - simplest geometry, high pressure rise, less robust
+- forward inclined blades - more blades but smaller, lowest pressure rise, lowest efficiency, more robust
+
+
+## Integral Analysis of Centrifugal Pumps
+
+Derivation in slides (p. 23-25).
+
+\begin{align}
+\frac{w_s}{g} - \left(\frac{u_2-_1-q}{g}\right) = \left(\frac{p_2}{\rho g} + z_2 + \frac{v_2^2}{2g}\right) -\left(\frac{p_1}{\rho g} + z_1 + \frac{v_1^2}{2g}\right) \\
+H_s - H_f &= H = H_{T,2} - H_{T,1} \nonumber
+\end{align}
+
+where $H_s$ is supplied head to pump, $H_f$ friction loss head, $H$ is head supplied to fluid, $H_{T,1}$ is total head at inlet, and $H_{T,2}$ is total head at outlet.
+
+Assuming that $z_1 \approx z_2$, $v_1 \approx v_2$ (from inlet and outlet diameters are equal) then:
+
+\begin{equation}
+H \approx \frac{p_2-p_1}{\rho g}
+\end{equation}
+
+and the power to the fluid (water horsepower) is:
+
+\begin{equation}
+P_w = \rho QgH
+\end{equation}
+
+where $Q$ is volumetric flow rate.
+
+Power supplied to the pump (brake horsepower), $P = \omega T$, lets us find the overall pump efficiency:
+
+\begin{equation}
+\eta = \frac{P_w}{P} = \frac{\rho QgH}{\omega T} = \eta_h \eta_m \eta_v
+\end{equation}
+
+where:
+
+- $\eta_h = 1 - \frac{H_f}{H_s}$ is hydraulic efficiency
+- $\eta_m = 1- \frac{P_f}{P}$ is mechanical efficiency
+- $\eta_v = \frac{Q}{Q+Q_L}$ (where $Q_L$ is loss due to leakage flow) is the volumetric efficiency
+
+## Performance
+
+![](./images/vimscrot-2023-03-13T10:07:58,750255090+00:00.png)
+
+# Cavitation
+
+Cavitation is when bubbles form in liquid by sudden pressure drop, followed by their implosion when
+original pressure is restored.
+The implosion generates a high pressure wave that can damage nearby solid surfaces.
+
+In a centrifugal pump, the fluid pressure drops at the impeller's eye, where it has the minimum
+value.
+If pressure falls below saturation pressure, bubbles appear.
+Pressure grows as the fluid flows between the blades as the ducts are diverging.
+Pressure is maximum at the trailing edge of the blades, on their front side.
+This is where cavitation occurs and causes wear on the blade.
+
+<https://www.michael-smith-engineers.co.uk/resources/useful-info/pump-cavitation>  
+<https://www.youtube.com/watch?v=g1o5Z9o7b0>  
+<https://www.youtube.com/watch?v=eMDAw0TXvUo>  
+<https://www.youtube.com/watch?v=1Lbxtjfdat4>
+
+![](./images/cavitation.png)
+
+# Net Positive Suction Head (NPSH)
+
+The following conditions must be satisfied to prevent cavitation:
+
+\begin{equation}
+H_i - \frac{p_v}{\rho g} > \text{NSPSH}
+\end{equation}
+
+where $H_i = \frac{p_i}{\rho g} + \frac{v_i^2}{2g}$ is total head at inlet, $p_v$ is saturation
+pressure at $T_i$.
+
+It is important that the inlet pressure is as high as possible.
+To do this, one can reduce frictional losses (e.g. shorter smoother pipes) or install the pump lower down
+(even below the reservoir) (slides p. 36).
+