147 lines
3.8 KiB
TeX
147 lines
3.8 KiB
TeX
\documentclass{IEEEtran}
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\usepackage{amsmath}
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\title{ME for ECEs Equation Sheet}
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\begin{document}
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\maketitle
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\section{Vectors}
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$$\vec{v}_R = \sum v_x \hat{x} + \sum v_y \hat{y} + \sum v_z \hat{z}$$
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$$\|\vec{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}$$
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$$\begin{align}
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\vec{u} \cdot \vec{v} &= \|\vec{u}\| \|\vec{v}\| \cos(\theta) \\
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&= (u_x v_x) + (u_y v_y) + (u_z v_z)
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\end{align}$$
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$$\begin{align}
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\vec{u} \times \vec{v} &=
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\begin{vmatrix}
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\hat{x} & \hat{y} & \hat{z} \\
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u_x & u_y & u_z \\
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v_x & v_y & v_z
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\end{vmatrix} \\
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&= (u_y v_z - u_z v_y)\hat{x} - (u_x v_z - u_z v_x)\hat{y} + (u_x v_y - u_y v_x)\hat{z}
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\end{align}$$
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\section{Statics}
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$$\vec{F} = m \vec{a}$$
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$$\vec{M} = \vec{r} \times \vec{F}$$
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$$\vec{r} = \vec{s}_f - \vec{s}_0$$
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$$\sum F = 0$$
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$$\sum M = 0$$
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\section{Dynamics}
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\subsection{Kinematic Equations}
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$$\vec{s}_f - \vec{s}_0 = \vec{v}_0 t + {1\over2} \vec{a} t^2$$
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$$\vec{v}_f^2 = \vec{v}_0^2 + 2 \vec{a} \cdot (\vec{s}_f - \vec{s}_0)$$
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$$\vec{v}_f = \vec{v}_0 + \vec{a}t$$
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$$\vec{s}_f = \vec{s}_0 + \vec{v}t$$
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\subsection{Projectile Motion}
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$$\vec{g} = -9.81\hat{y}\left[{\text{m} \over \text{s}^2}\right] = -32.2\hat{y} \left[{ \text{ft} \over \text{s}^2}\right]$$
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Without air resistance:
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$$v_{x_0} = v_{x_f}$$
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\section{Heat Transfer}
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\subsection{Conduction}
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$$q_x'' = k {dT \over dx}$$
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$${dT \over dx} = {T_2 - T_1 \over L}$$
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$$q_x = q_x'' A$$
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Where:
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\begin{itemize}
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\item[$q_x''$] is heat flux
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\item[$q_x$] is heat rate
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\item[$T$] is temperature
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\item[$L$] is length
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\item[$A$] is the contact area
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\end{itemize}
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\subsection{Convection}
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$$q'' = hA(T_s - T_\infty)$$
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Where:
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\begin{itemize}
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\item[$h$] is the heat transfer coefficient
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\item[$A$] is the contact area between the surface and the fluid
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\item[$T_s$] is the surface temperature
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\item[$T_\infty$] is the fluid temperature very far away from the surface
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\end{itemize}
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\subsection{Radiation}
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$$q_\text{ideal}'' = \sigma T_s^4$$
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$$q_\text{real}'' = \varepsilon \sigma T_s^4$$
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Where:
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\begin{itemize}
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\item[$T_s$] is the absolute temperature
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\item[$\sigma$] is the Stefan-Boltzmann constant
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\item[$\varepsilon$] is the emissivity
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\end{itemize}
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\section{Fluids}
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\noindent
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Pascal's Law:
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$${F_1 \over A_1} = {F_2 \over A_2}$$
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Density:
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$$\rho = {m \over v}$$
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Specific Weight:
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$$\gamma = {mg \over v} = \rho g$$
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Pressure in a fluid:
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$$P_2 = P_1 + \rho g z$$
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Bernoulli Equation:
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$$p_1 + {1\over2}\rho v_1^2 + \rho g h_1 = p_2 + {1\over2}\rho v_2^2 + \rho g h_2$$
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Fluid Velocity Between Plates:
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$${U \over b} = {u \over y}$$
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Where:
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\begin{itemize}
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\item[$U$] is the velocity of the moving plate
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\item[$b$] is the distance between the plates
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\item[$u$] is the velocity of the fluid at between the plates at some distance above the stationary plate
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\item[$y$] is the distance above the stationary plate.
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\end{itemize}
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Continuity Equation
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$$A_1 v_1 \Delta t = A_2 v_2 \Delta t$$
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$$\dot{m} = \rho_1 A_1 v_1 = \rho_2 A_2 v_2$$
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\section{Gears}
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$$N = P d$$
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$$N = {d \over m}$$
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$$c = {d_1 + d_2 \over 2} = {N_1 + N_2 \over 2 P} = {(N_1 + N_2) m \over 2}$$
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Where:
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\begin{itemize}
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\item[$N$] is the number of teeth
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\item[$d$] is the pitch diameter
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\item[$c$] is the center distance
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\item[$P$] is the diametral pitch (Customary)
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\item[$m$] is the module (SI)
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\end{itemize}
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\noindent
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Gear Ratio:
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$$R = {T_2 \over T_1} = {N_2 \over N_1} = {d_2 \over d_1} = {\omega_1 \over \omega_2}$$
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$$\omega = {\pi \over 30}\text{RPM}$$
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\noindent
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Big Gear to Small Gear:
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\begin{itemize}
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\item Speed increases
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\item Torque decreases
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\end{itemize}
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\noindent
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Small Gear to Big Gear:
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\begin{itemize}
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\item Speed decreases
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\item Torque increases
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\end{itemize}
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\noindent
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Power:
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$$P = \omega T$$
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If no power losses:
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$$P_\text{in} = P_\text{out}$$
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$$\omega_1 T_1 = \omega_2 T_2$$
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\end{document}
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