mmme2044 notes on shaft design

uni/mmme/2xxx/2044_design_manufacture_and_project/shaft_design.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2044 // Shaft Design
+tags: [ mmme2044, shafts, uni ]
+uuid: 8e0928a6-c20c-4f80-9691-beb2defa4022
+---
+
+# Shaft Design Considerations
+
+- Function and loading
+- Size and connection to components
+- Material selection and treatments
+- Deflection and rigidity
+- Stress and strength
+- Critical speed
+- Manufacturing constraints
+
+# Shaft-Hub Connections
+
+![](./images/vimscrot-2022-10-31T15:09:11,091932569+00:00.png)
+
+![](./images/vimscrot-2022-10-31T15:10:02,106108248+00:00.png)
+
+# Shaft-Shaft Connections
+
+Shaft-shaft connections can use either rigid or flexible couplings.
+
+![](./images/vimscrot-2022-10-31T15:11:05,058803741+00:00.png)
+
+# Location of Bearing on Shaft
+
+![](./images/vimscrot-2022-10-31T15:18:36,571125403+00:00.png)
+
+# Shaft Loading
+
+- Axial stresses
+
+    - cause by self-weight in vertical shafts
+    - due to axial restraint at bearings and associated axial load
+
+- bending stress
+
+    - due to bending moment in belt drivers, gear forces, mounted component weights
+    - dynamic forces which can load to fatigue and resonance
+
+- shear stresses
+
+    - due to torque load/direct shear
+
+# Shaft Diameter (ASME Design Code)
+
+$$d = \left[
+        \frac{32n_s}{\pi}
+        \sqrt{
+                \left(\frac{M}{\sigma_e}\right)^2 
+                + \frac34 \left(\frac{T}{\sigma_y}\right)^2
+        }
+      \right]^{\frac13}
+$$
+
+where:
+
+- $n_s$ is [safety factor](#safety-factor)
+- $M$ is max bending moment (Nm)
+- $T$ is max torque (Nm)
+- $\sigma_e$ is endurance limit stress (Pa)
+- $\sigma_y$ is yield strength of shaft (Pa)
+
+
+<details>
+
+<summary>
+
+#### ASME Design Code Derivation
+
+</summary>
+
+- Bending moment creates alternating tensile/compressive stresses ($\sigma_a$):
+
+  \begin{align}
+  \sigma &= \frac{My}{I} \\
+  I &= \frac{\pi d^4}{64}
+  \end{align}
+
+  Combine to get:
+
+  $$\sigma_a = \frac{32M}{\pi d^3}$$
+
+- Torque normally generates constant shear stress ($\tau_m$):
+
+  \begin{align}
+  \tau &= \frac{TR}{J} \\
+  J &= \frac{\pi d^4}{32}
+  \end{align}
+
+  Combine to get:
+
+  $$\rightarrow \tau_m = \frac{16T}{\pi d^3}$$
+
+- von Mises stress in plane stress condition:
+
+  $$\sigma_\text{von Mises} = (\sigma^2 + 3\tau^2)^{\frac12}$$
+
+- ASME Fatigue Failure Criterion
+
+  $$\left(\frac{n_s\sigma_a}{\sigma_e}\right)^2 + \left(\frac{n_s\sigma_m}{\sigma_Y}\right)^2 = 1$$
+
+  Finally substitute $\sigma_a$ and $\tau_m$ to make $d$ the subject.
+
+</details>
+
+## Endurance Limit Stress, $\sigma_e$
+
+$\sigma_e$ is the level of stress with which fatigue failure wouldn't occur in cycling or
+alternating load conditions:
+
+![](./images/vimscrot-2022-11-04T10:13:44,527741950+00:00.png)
+
+- $\sigma$ at failure decreases with the number of cycles up until a certain point ($\sigma_e$ after
+  around $10^6$ cycles)
+- if you never exceed this point then the material will last for "infinite" cycles
+- most steels have this fatigue behaviour so they are often used for shafts
+
+Time for a silly equation:
+
+$$\sigma_e = k_ak_bk_ck_dk_ek_fk_g\sigma_e'$$
+
+where:
+
+- $k_a$ --- surface factor
+- $k_b$ --- size factor
+- $k_c$ --- reliability factor
+- $k_d$ --- temperature factor
+- $k_e$ --- duty cycle factor
+- $k_f$ --- fatigue stress concentration factor
+- $k_g$ --- miscellaneous effects factor
+- $\sigma_e'$ --- endurance limit of test specimen
+
+this is a joke of an equation
+
+## Reducing Stress Concentrations
+
+![](./images/vimscrot-2022-11-04T10:31:41,364654063+00:00.png)
+
+![](./images/vimscrot-2022-11-04T10:31:50,672814681+00:00.png)
+
+## Critical Speed of Shaft (Natural Frequency)
+
+- operational speed of shat should be half the critical speed
+- Centre of Mass should be on the Centre of Rotation
+- in practice this is not the case
+- this imbalance causes a deflection (due to centrifugal force, $mr\omega^2$
+- the critical speed (or natural frequency) is the speed at which the shaft is unstable
+
+  when this is the case it may cause damage to the shaft, bearings, and other destructive
+  vibrations
+
+## Critical Speed Equation
+
+$$\omega_c = \sqrt\frac{g}{\delta_\text{st}}$$
+
+where $g$ is acceleration due to gravity and $\delta_\text{st}$ is 
+[static deflection](#shaft-deflection) of the shaft.
+
+## Rayleigh Ritz Equation
+
+When you have multiple masses the Rayleigh-Ritz equation may be more suitable:
+
+$$\omega_c = \sqrt{g\frac{\Sigma w_i\delta_i}{\Sigma w_i\delta_i^2}} $$
+
+where $w_i$ is the weight of node $i$ and $\delta_i$ is the static deflection at node $i$.
+
+# Shaft Deflection
+
+- Shaft deflection is required to determine the critical speed.
+- Macaulay's method for the deflection in beam bending:
+
+  $$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = \frac{M}{EI}$$
+  $$y = \int^x_0\int^x_0\left(\frac{M}{EI}\right) \mathrm{d}x + C_1x + C_2$$
+
+## Shaft Deflection Equations
+
+![](./images/shaft-design-064.jpg)
+
+- Maximum deflection:
+
+  $$\delta_\text{max} = \frac{PL^3}{3EI}$$
+
+- Deflection at any point $x$
+
+  $$\delta = \frac{Px^3}{6EI}(3L-x)$$
+
+![](./images/shaft-design-065.jpg)
+
+- Maximum deflection:
+
+  $$\delta_\text{max} = \frac{PL^3}{48EI}$$
+
+- Deflection at any point:
+
+  $$\delta = \frac{Px}{12EI} \left(\frac{3L^2}{4} - x^2 \right)$$
+
+![](./images/shaft-design-066.jpg)
+
+
+- Maximum deflection
+
+  $$\delta_\text{max} = \frac{Pb^2L}{3EI}$$
+
+- Deflection at any point
+
+  - For $0 \le x \le a$:
+
+    $$\delta = \frac{Pbx}{6aEI}(x^2-a^2)$$
+
+  - For $0 \le z \le b$:
+
+    $$\delta = \frac{Pbx}{6aEI}\left(z^3 - b(2L+b) + 2b^2L\right)$$
+
+# Safety Factor
+
+Safety Factor (also known as reserve factor) is a simple way to accommodate for uncertainties in
+design.
+
+Simply multiply the maximum stresses and loads you expect by the safety factor, $n_s$, and assume
+that as your maximum stress and load.
+
+$n_s$ | Operational conditions and use of materials
+----- | -------------------------------------------
+1.25-1.50 | Reliable materials under controlled conditions, known stresses with certainty
+1.50-2.00 | Well-known materials under reasonably constant environmental condition, known stresses
+2.00-2.50 | Average materials subjected to known loads and stresses and environment (LSE)
+2.50-3.00 | Lesser well-known materials under average conditions LSE
+3.00-3.40 | Untried materials under average conditions of stresses and environment, or well known materials under uncertain LSE