## Network Modeling | | Mechanical | Electrical | Fluidic | Thermal | | --------- | ------------------ | ------------------------- | ------------------------------------------ | --------------------------- | | Potential | $F$, [N] | $V$, [V] | $\Delta P$ [Pa] | $\Delta T$ [K] | | Flux | $v$, [m/s] | ${} i$, [A] | ${} q$, [$\text{m}^{3}/\text{s}$] | ${} Q$, [W] | | Inertia | ${} F=m\dot{v} {}$ | ${} V=L \frac{di}{dt} {}$ | ${} \Delta P = \frac{\rho L}{A}\dot{q} {}$ | N/A | | Losses | ${} F=bv {}$ | ${} V=iR {}$ | ${} \Delta P=qR {}$ | ${} \Delta T = R\cdot Q {}$ | | Capacity | $F=kx$ | ${} i=C \frac{dV}{dt} {}$ | ${} q = \frac{A}{\rho g}\Delta P {}$ | $Q=mc\Delta T$ | ## Conservation Laws ### Energy $ \frac{dE_{\text{cv}}}{dt}=\dot{Q}-\dot{W}+\sum_{\text{in}}\dot{m}_{\text{in}}\left( h+\frac{v^{2}}{2}+gz \right)_{\text{in}} -\sum_{\text{out}}\dot{m}_{\text{out}}\left( h+\frac{v^{2}}{2}+gz \right)_{\text{out}} $ $ h = u + \frac{P}{\rho} $ ### Linear Momentum $ \sum \vec{F} = m\vec{a} $ For an inertial control volume: $ \sum \vec{F} = \frac{d}{dt} \int_{CV}\vec{v} \rho dV + \sum_{out}\dot{m}_{i}\vec{v}_{i} - \sum_{in} \dot{m}_{i} \vec{v}_{i} $ ##### Navier Stokes $ \rho \frac{D\vec{v}}{Dt} = -\vec{\nabla}P + \rho \vec{g} + \mu \nabla^{2} \vec{v} $ ### Angular Momentum $\sum \vec{T}=\frac{d}{dt}(r\times m\nu)$ ### Entropy $ \frac{dS_{\text{cv}}}{dt}=\frac{\dot{Q}}{T_{\text{s}}}+\dot{S}_{\text{gen}}+\sum_{\text{in}} \dot{m}_{\text{in}}S_{\text{in}} - \sum_{\text{out}} \dot{m}_{\text{out}} S_{\text{out}} $ ## Statics ### Von Mises Stress $ \bar{\sigma}=\sqrt{ \frac{1}{2}((\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-\sigma_{33})^{2}+(\sigma_{11}-\sigma_{33})^{2})+3(\sigma_{12}^{2}+\sigma_{23}^{2}+\sigma_{13}^{2}) } $ ## Math ### Taylor Series $ f(x + h) = f(x) + f'(x)h + f''(x) \frac{h^{2}}{2!} + \dots $ ### Material Derivative $ \frac{D}{Dt} = \left( \frac{\delta}{\delta t} + v_{x} \frac{\delta}{\delta x} + v_{y} \frac{\delta}{\delta y} + v_{z} \frac{\delta}{\delta z} \right) = \frac{\delta}{\delta t} + (\vec{v} \cdot \vec{\nabla}) $ ## Material Properties ### Solids | | **$E$** | **$\nu$** | **$\rho$** | **$F_{ty}$** | **$F_{tu}$** | **$\alpha$** | **$c$** | **$k$** | | ------------- | ---------------: | --------: | -----------------------: | ---------------: | ---------------: | -----------: | -----------------------: | ----------------------: | | | *$\mathrm{GPa}$* | | *$\mathrm{ kg / m^{3}}$* | *$\mathrm{MPa}$* | *$\mathrm{MPa}$* | | *$\mathrm{J / (kg ~K)}$* | *$\mathrm{W / (m ~K)}$* | | Al 6061 | 68 | 0.33 | 2700 | 240 | 290 | 23.6 | 896 | 167 | | Al 7075 | 71 | 0.33 | 2800 | 345 | 415 | 23.5 | 960 | 130 | | Steel 4130 | 200 | 0.32 | 7850 | 517 | 655 | 12.2 | 477 | 42.7 | | Stainless 304 | 193 | 0.30 | 8000 | 215 | 505 | 17.2 | 500 | 16 | | Ti‑6Al‑4V | 114 | 0.34 | 4430 | 880 | 950 | 8.6 | 526 | 6.7 | | ABS | 2.1 | 0.35 | 1040 | 46 | 65 | 80 | 1350 | 0.17 | | Nylon 6 | 2.8 | 0.39 | 1140 | 70 | 95 | 80 | 1700 | 0.25 | | Polycarbonate | 2.3 | 0.37 | 1200 | 65 | 70 | 65 | 1200 | 0.20 | ### Fluids | | **$\rho$** | **$c_{p}$** | **$c_{v}$** | **$\mu$** | **$k$** | **$\mathrm{Pr}$** | | ------------ | -----------------------: | -----------------------: | -----------------------: | ----------------------: | ----------------------------: | ----------------: | | | *$\mathrm{ kg / m^{3}}$* | *$\mathrm{J / (kg ~K)}$* | *$\mathrm{J / (kg ~K)}$* | *$\mathrm{Pa \cdot s}$* | *${} \mathrm{W / (m ~K)} {}$* | | | Air (20°C) | 1.2 | 1005 | 718 | 0.0000181 | 0.026 | 0.71 | | Water (20°C) | 1000 | 4182 | 4175 | 0.001 | 0.60 | 7.0 | | Oil | 900 | 2000 | 1800 | 0.1 | 0.15 | 100 | ## Units | Name | Symbol | Equation | SI Units | Dimension | | ---------------------------------------------- | -------------- | ------------------------------------------------------- | --------------------------------- | --------------------------------------- | | Velocity | $v$ | $v =\dot{x}$ | ${} \mathrm{m / s} {}$ | ${} L / T {}$ | | Acceleration | $g$ | ${} a = \dot{v} {}$ | ${} \mathrm{m / s^{2}} {}$ | ${} \mathrm{L / T^{2}} {}$ | | Force | $F$ | ${} F = ma {}$ | ${} \mathrm{N} {}$ | ${} \mathrm{ML / T^{2}} {}$ | | Energy / Work | $W$ | ${} W = \int Fdx {}$ | ${} \mathrm{J} {}$ | $\mathrm{ML^{2} / T^{2}}$ | | Power | $P$ | ${} P = \dot{W} {}$ | ${} \mathrm{W} {}$ | ${} \mathrm{ML^{2} / T^{3}} {}$ | | Density | ${} \rho {}$ | ${} \rho = m /V {}$ | ${} \text{kg} / \text{m}^{3} {}$ | ${} \mathrm{M / L^{3}} {}$ | | Viscosity | $\mu$ | ${} \tau_{x}=\mu \frac{ \partial u }{ \partial x } {}$ | ${} \mathrm{Pa \cdot s} {}$ | ${} \mathrm{\frac{M}{LT}} {}$ | | Kinematic Viscosity / <br>Momentum diffusivity | $\nu$ | ${} \nu = \mu / \rho {}$ | ${} \mathrm{m^{2} / s} {}$ | $\mathrm{L^{2} / T}$ | | Thermal Diffusivity | ${} \alpha {}$ | ${} \alpha = k / \rho {}$ | ${} \mathrm{m^{2} / s} {}$ | $\mathrm{L^{2} / T}$ | | Heat Transfer Coefficient | $h$ | ${} R = \frac{1}{hA} {}$ | ${} \mathrm{\frac{W}{m^{2}K}} {}$ | ${} \mathrm{\frac{M}{T^{3}\theta}} {}$ | | Thermal Conductivity | $k$ | ${} R = \frac{L}{kA} {}$ | ${} \mathrm{\frac{W}{mK}} {}$ | ${} \mathrm{\frac{ML}{T^{3}\theta}} {}$ | | Specific Heat Capacity | $c$ | ${} \Delta U = mc\Delta T {}$ | ${} \mathrm{\frac{J}{kg~K}} {}$ | $\mathrm{\frac{L^{2}}{T^{2}\theta}}$ | ### Conversions | Metric | Conversion | Imperial | | ------ | ---------: | -------- | | mm | 25.4 | in | | m/s | 0.45 | mph | | kg | 0.45 | lbm | | N | 4.45 | lb | | GPa | 6.9 | ksi | | Nm | 0.11 | in-lb | | Nm | 1.36 | ft-lb | | °C | -32, ×0.56 | °F | | deg | 57.3 | rad | ## GD&T https://www.drafterinc.com/post/why-your-dowel-pin-doesnt-fit--and-how-to-fix-it