pure maths

Binomial series

$$(a+b)^n=a^n+{n \choose 1}a^{n-1}b+{n \choose 2}a^{n-2}b^2 + \cdots + {n \choose r}a^{n-r}b^r + \cdots + b^n$$

polynomial equation

$$y=ax^4+bx^3+cx^2+dx+e$$ $$y=ax^3+bx^2+cx+d$$ $$y=ax^2+bx+c$$

Solutions For Quadratic Equation

$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$

Surface area of revolution

$$ S_x = 2 \pi \int y \sqrt{\bigl(1+ ({dy \over dx})^2 \bigr)} dx$$ $$ S_x = 2 \pi \int y \sqrt{({dx \over dt})^2 + ({dy \over dt})^2 } dt$$ $$ S_x = 2 \pi \int r sin\theta \sqrt{r^2 + ({dr \over d\theta})^2 } d\theta$$

Fourier Series

$$f(x) = a_0 + \sum_{n=1}^\infty \bigl( a_n cos({nx \pi \over L} + b_n sin({nx \pi \over L}) \bigr)$$ $$a_0 = \frac1{2L} \int_{-L}^L f(x) dx$$ $$a_n = \frac1{L} \int_{-L}^L f(x) cos({nx\pi \over L}) dx$$ $$b_n = \frac1{L} \int_{-L}^L f(x) sin({nx\pi \over L}) dx$$

Fourier transform

$$f(x) = \int_{-\infty}^\infty F(k) e^{2\pi ikx}dk$$

Differential equations

$${dy^5 \over dx^5} +{dy^4 \over dx^4} +{dy^3 \over dx^3} +{dy^2 \over dx^2} +{dy \over dx} = f(x)$$

Integration

$$\int sin^2\theta \ d\theta = {\theta \over 2} - \frac14 sin2\theta + C$$ $$\int cos^2\theta \ d\theta = {\theta \over 2} + \frac14 sin2\theta + C$$ $$\int sin\theta \ cos\theta \ d\theta = \frac12 sin^2\theta + C$$

Integrating factors

$${dy \over dx} +P(x)y=Q(x)$$ $${dy \over dx} =-P(x)y$$ $$\int {dy \over y} =\int -P(x) dx$$ $$ln|{y}| =\int -P(x) dx +c$$ $$y = e^{-\int P(x) dx +c}$$ $$y = Ce^{-\int P(x) dx }$$

statistics

Expectation (mean)

$$ E(X) = \mu = \int x f(x) dx$$

Variance

$$ Var(X) = \sigma^2 = \int (x-\mu)^2 f(x) dx = \int x^2 f(x) dx - \mu^2$$

the product moment correlation coefficient

$$r = {S_{xy} \over \sqrt{S_{xx}S_{yy}}} = {\sum(x_i-\overline x)(y_i-\overline y) \over \sqrt{\bigl(\sum(x_i-\overline x)^2\bigr)\bigl(\sum(y_i-\overline y\bigr)^2)}} = {\sum x_iy_i-{(\sum x_i)(\sum y_i) \over n} \over \sqrt{\bigl(\sum x_i^2 - {(\sum x_i)^2 \over n} \bigr)\bigl(\sum y_i^2 - {(\sum y_i)^2 \over n} \bigr)}}$$

The regression coefficient of y on x

$$ b = {S_{xy} \over S_{xx}} = {\sum(x_i-\overline x)(y_i-\overline y) \over \sum(x_i-\overline x)^2} $$

Residual Sum of Squares (RSS)

$$ RSS = S_{yy} - {(S_{xy})^2 \over S_{xx}} = S_{yy}(1-r^2)$$

Spearman’s rank correlation coefficient

$$ r_S = 1 - {6 \sum d^2 \over n(n^2-1)}$$

mass moment of inertia

thick-walled hollow sphere

$$I={2m\over5}({r_2^5-r_1^5\over r_2^3-r_1^3})$$

solid sphere

$$I=\frac25mr^2$$

hollow sphere

$$I=\frac23mr^2$$

solid rectangular plate

$$I_centre={1\over12}m(a^2+b^2)$$

thick-walled hollow cylinder (z is central axis)

$$I_z=\frac12m(r_1^2+r_2^2)$$ $$I_x=I_y={1\over12}m[3(r_1^2+r_2^2)+h^2]$$

solid cylinder

$$I_z=\frac12mr^2$$

thin-walled hollow cylinder

$$I=mr^2$$

Rod

$$I_centre={1\over12}mL^2$$ $$I_end={1\over3}mL^2$$

solid circular plate (z is central axis)

$$I_z=\frac12mr^2$$ $$I_x=l_y=\frac14mr^2$$

hollow circular plate (ring)

$$I_z=mr^2$$ $$I_x=l_y=\frac12mr^2$$

second moment of area

Parallel axis theorem

$$I_{x'}=Ix+Ad^2$$

filled rectangle with centroid at the origin (base b, height h)

$$I_x=\underset R\iint y^2dA=\int_{-{b\over 2}}^{{b\over 2}}\int_{-{h\over 2}}^{{h\over 2}}y^2dydx=\int_{-{b\over 2}}^{{b\over 2}}\frac13{h^3\over4}dx={bh^3\over12}$$ $$I_y=\underset R\iint x^2dA=\int_{-{b\over 2}}^{{b\over 2}}\int_{-{h\over 2}}^{{h\over 2}}x^2dydx=\int_{-{b\over 2}}^{{b\over 2}}hx^2dx={b^3h\over12}$$ polar coordinates $$J_z=I_x+I_y={bh^3\over12}+{hb^3\over12}={bh\over12}(b^2+h^2)$$

filled rectangle with vertex at the origin (base b, height h)

$$I_x={bh^3\over3}$$ $$I_y={b^3h\over3}$$

hollow rectangle with centroid at the origin (base b, height h, inner width b1, inner height h2)

$$I_x={bh^3-b_1h_1^3\over12}$$ $$I_y={b^3h-b_1^3h_1\over12}$$

annulus centered at origin (z is central axis)

$$I_{x,circle}=\underset R\iint y^2dA=\underset R\iint (rsin\theta)^2dA=\int_0^{2\pi}\int_0^r(rsin\theta)^2(rdrd\theta)=\iint (rsin\theta)^2dA=\int_0^{2\pi}\int_0^rr^3(sin^2\theta) rdrd\theta=\int_0^{2\pi}{r^4(sin^2\theta)\over4}d\theta={\pi\over4}r^4$$ polar coordinates $$I_{y,circle}=\underset R\iint r^2dA=\int_0^{2\pi}\int_0^rr^2(rdrd\theta)=\int_0^{2\pi}\int_0^rr^3drd\theta=\int_0^{2\pi}{r^4\over4}d\theta={\pi\over2}r^4$$ if it has inner radius r1 and outer radius r2 (change limits on intergral) $$I_{y,circle}=\underset R\iint r^2dA=\int_0^{2\pi}\int_{r_1}^{r_2}r^2(rdrd\theta)=\int_0^{2\pi}\int_{r_1}^{r_2}r^3drd\theta=\int_0^{2\pi}[{r_2^4\over4}-{r_1^4\over4}]d\theta={\pi\over2}(r_2^4-r_1^4)$$

filled circle

$$I_x=I_y=I_z={\pi\over4}r^4$$

filled ellipse (a along x, b along y)

$$I_x={\pi\over4}ab^3$$ $$I_y={\pi\over4}a^3b$$

filled regular hexagon with side length a

$$I_x=I_y={5\sqrt3\over16}a^4$$

dynamics

SUVAT

$$x=\bar v t$$ $$v=v_0+at$$ $$x=v_0t+\frac12at^2$$ $$v^2=v_0^2 +2ax$$ $$w=w_0+\alpha t$$ $$\theta =w_0t+\frac12at^2$$ $$w^2 =w_0^2+2\alpha\theta$$

motion equation

$$m {\ddot x} + c{\dot x} +kx =0$$ $$x(t)=ae^{\lambda t}$$ $$(m \lambda^2 + c \lambda^2 +k)ae^{\lambda t} =0$$ $$\lambda = {-c \pm \sqrt{c^2-4mk} \over 2m}$$ $$\lambda_{1,2} = - {c \over 2m} \pm \sqrt{{c \over 2m}^2 -{k \over 2m}}$$ $$\lambda_{1,2} = - \zeta \omega_n \pm \omega_n \sqrt{\zeta^2 -1}$$
Critical damping $$c_c = 2m \sqrt{\frac km} = 2m \omega_n = 2\sqrt{km}$$ $${c \over 2m} = {c \over c_c} \cdot {c_c \over 2m} = \zeta \omega_n$$ $$ \lambda_{1,2} = - \omega_n$$ $$x(t) = a_1 e^{-\omega _n t} + a_2 e^{-\omega _n t}$$ $$a_1 = x_o$$ $$a_2 = \dot x_0 + \omega_n x_0$$
Overdamped $$x(t) = a_1 e^{(- \zeta \omega_n + \omega_n \sqrt{\zeta^2 -1})} + a_2e^{(- \zeta \omega_n - \omega_n \sqrt{\zeta^2 -1})}$$ $$a_1 = {x_0 \omega_n (\zeta + \sqrt{\zeta^2 -1}) +\dot x_0 \over 2 \omega_n \sqrt{\zeta^2 -1}}$$ $$a_2 = {x_0 \omega_n (\zeta + \sqrt{\zeta^2 -1}) - \dot x_0 \over 2 \omega_n \sqrt{\zeta^2 -1}}$$
Underdamped $$x(t) = e^{- \zeta \omega_nt}(a_1 e^{i \omega_d t} + a_2e^{- i \omega_d t})$$ $$x(t) = Ae^{- \zeta \omega_nt} sin(\omega_d t + \phi)$$ $$\omega_d = \omega_n \sqrt{1 - \zeta^2}$$ $$A = {1 \over \omega_d } \sqrt{(\dot x_0 + \zeta \omega_n x_0)^2 + (x_0 \omega_d)^2}$$ $$\phi = tan^{-1 } ({x_0 \omega_d \over \dot x_0 + \zeta \omega_n x_0})$$
The Lagrange’s Equations $${d \over dt} ({\partial T \over \partial \dot q_j} ) - {\partial T \over \partial \dot q_j} + {\partial V \over \partial \dot q_j} = Q_j^{(n)}$$

structural (solid) mechanics

Radial and hoop stresses for thick cylinders

$$\sigma_r = A- {B \over r^2}$$ $$\sigma_\theta = A + {B \over r^2}$$

Radial and circumferential stresses for rotating discs

$$\sigma_r = A - {B \over r^2} + {3+\nu \over 8} \rho \omega^2 r^2$$ $$\sigma_r = A + {B \over r^2} - {1+3\nu \over 8} \rho \omega^2 r^2$$

Circular Plates of uniform thickness

deflection for a uniformly distributed normal load p, per unit area $${pr \over 2D} = {d \over dr}[\frac1r {d \over dr}(r {dz \over dr})]$$ $${d \over dr}[\frac1r {d \over dr}(r {dz \over dr})] = {F \over 2\pi D}$$ $$D = {Et^3 \over 12(1- \nu^2)}$$

Bending moments per unit length

$$M_r = -D [{d^2z \over dr^2} + {\nu \over r} {dz \over dr}]$$ $$M_\theta = -D [\nu {d^2z \over dr^2} + {1 \over r} {dz \over dr}]$$

Stress distribution

$$\sigma_r = {12M_r h \over t^3}$$ $$\sigma_\theta = {12M_\theta h \over t^3}$$

Force equilibrium relationships

$${\partial \sigma_{xx} \over \partial x} + {\partial \tau_{yx} \over \partial y} + {\partial \tau_{zz} \over \partial z} + f_x =0$$ $${\partial \sigma_{xy} \over \partial x} + {\partial \tau_{yy} \over \partial y} + {\partial \tau_{zy} \over \partial z} + f_y =0$$ $${\partial \sigma_{xz} \over \partial x} + {\partial \tau_{yz} \over \partial y} + {\partial \tau_{zz} \over \partial z} + f_z =0$$

Stress function

$$\nabla^2 \phi = {\partial^2 \phi \over \partial y^2} + {\partial^2 \phi \over \partial x^2}$$

Stress compatibility in 2D

$$({\partial^2 \over \partial x^2} + {\partial^2 \over \partial y^2})(\sigma_z + \sigma_y) = -\beta ({\partial f_x \over \partial x} + {\partial f_y \over \partial y})$$ $$\beta = 1+ \nu $$ $$ \beta = {1\over 1 - \nu}$$

Biharmonic operator in 2D

$$\nabla^4= {\partial^4 \over \partial x^4} + 2{\partial^4 \over \partial x^2 \partial y^2} + {\partial^4 \over \partial y^4}$$

Biharmonic operator in polar coordinates

$$({\partial^2 \over \partial r^2 } + \frac1r {\partial \over \partial r} + {1 \over r^2} {\partial^2 \over \partial \theta^2}) ({\partial^2 \phi \over \partial r^2 } + \frac1r {\partial \phi \over \partial r} + {1 \over r^2} {\partial^2 \phi \over \partial \theta^2}) = 0$$

fluid mechanics

Absolute pressure

$$P_{abs}=P_g+P_{atm}$$

Bernoulli's equation

$$P+\frac12 \rho(u^2+v^2+w^2)+\rho gz=constant$$

Hagen–Poiseuille equation

$$Q={(p_2-p_1)\pi r^4\over8L\mu}$$ $$\Delta P={8QL\mu\over\pi r^4}$$

Reynolds number

$$Re={\rho u D\over \mu}={uD\over\nu}$$

Continuity equation

$$\rho_1A_1v_1=\rho_2A_2v_2$$ $${\partial \rho \over \partial t}+\Delta\cdot(\rho\vec V)={\partial \rho \over \partial t}+{\partial \rho u\over \partial x}+{\partial \rho v\over \partial y}+{\partial \rho w\over \partial z}=0$$ $${\partial \rho \over \partial t}+\Delta\cdot(\rho\vec V)={\partial \rho \over \partial t}+{\partial (\rho u_r)\over \partial r}+{\rho u_r\over r}+\cfrac1r{\partial u_\theta \over \partial \theta} + {\theta \rho u_z\over \partial z}=0$$

3D Momentum Equation (Navier-Stokes Equations)

$$\rho({\partial u \over \partial t}+u{\partial u \over \partial x}+v{\partial u\over \partial y}+w{\partial u\over\partial z})=-{\partial \rho \over \partial x}+\mu({\partial^2 u \over \partial x^2}+{\partial^2 u \over \partial y^2}+{\partial^2 u \over \partial z^2})$$ $$\rho({\partial v \over \partial t}+v{\partial v \over \partial x}+v{\partial v\over \partial y}+w{\partial v\over\partial z})=-{\partial \rho \over \partial x}+\mu({\partial^2 v \over \partial x^2}+{\partial^2 v \over \partial y^2}+{\partial^2 v \over \partial z^2})$$ $$\rho({\partial w \over \partial t}+v{\partial w \over \partial x}+v{\partial v\over \partial y}+w{\partial v\over\partial z})=-{\partial \rho \over \partial x}+\mu({\partial^2 w \over \partial x^2}+{\partial^2 w \over \partial y^2}+{\partial^2 w \over \partial z^2})-\rho g$$ $$\rho({\partial w \over \partial t}+v{\partial w \over \partial x}+v{\partial w\over \partial y}+w{\partial w\over\partial z})=-{\partial \rho \over \partial x}+\mu({\partial^2 w \over \partial x^2}+{\partial^2 w \over \partial y^2}+{\partial^2 w \over \partial z^2})-\rho g$$

2D Momentum Equation (f_x(g) and f_y(g) are functions related to gravity)

$$\rho({\partial u \over \partial t}+u{\partial u \over \partial x}+v{\partial u\over \partial y})=-{\partial \rho \over \partial x}+\mu({\partial^2 u \over \partial x^2}+{\partial^2 u \over \partial y^2})+f_x(g)$$ $$\rho({\partial v \over \partial t}+u{\partial v \over \partial x}+v{\partial v\over \partial y})=-{\partial \rho \over \partial x}+\mu({\partial^2 v \over \partial x^2}+{\partial^2 v \over \partial y^2})+f_y(g)$$

Potential flow and stream function relationships

$$u={\partial \phi \over \partial x}={\partial \psi\over \partial y},\ v={\partial \phi \over \partial y}=-{\partial \psi\over \partial x}$$ $$u_r={\partial \phi \over \partial r}=\cfrac1r{\partial \psi\over \partial \theta},\ u_\theta=\cfrac1r {\partial \phi \over \partial \theta}=-{\partial \psi\over \partial r}$$

flow in pipes (circular)

$$u(r)=\cfrac1{4\mu}{\Delta P\over L}\cfrac{D^2}4(1-{4r^2\over D^2})$$ $$\tau_w={\Delta P \over L}\cfrac D4$$

Energy balance equation for an open system with multiple inlets and exits

$${dE_{CV} \over dt}=\dot Q+\dot W+\sum_i \dot m_i(h_i+\cfrac {V_i^2}2 +gz_i ) - \sum_e \dot m_e (h_e + \cfrac {V_e^2}2 +gz_e )$$

Thermodynamic relations for isentropic processes of ideal gases

$${T_2 \over T_1} = ({p_2 \over p_1})^{\gamma-1 \over \gamma}=({v_2 \over v_1})^{-(\gamma-1)}=({\rho_2 \over \rho_1})^{(\gamma-1)}$$

combination of first law of thermodynamics and perfect gas law

$${\gamma \over \gamma -1}\cfrac p \rho + \frac 12 q^2 = constant$$

speed of sound

$$c^2=\gamma RT$$

shock analysis (reference state, stagnation and sonic)

$$\cfrac {T_0}T = 1+\frac12 (\gamma -1)M^2$$ $$\cfrac p{p_0} = (1+\frac12 (\gamma -1)M^2)^\cfrac \gamma {\gamma - 1}$$ $$\cfrac \rho{\rho_0} = (1+\frac12 (\gamma -1)M^2)^\cfrac \gamma {\gamma - 1}$$

Rayleigh line

$${p_2 \over p_1} = {1+\gamma M_1^2 \over1+\gamma M_2^2}$$ $${T_2 \over T_1} = {1+\frac12(\gamma-1)M_1^2 \over 1+\frac12(\gamma-1)M_2^2}$$

Fanno line

$${p_2 \over p_1} ={M_2 \over M_1} = \Bigl({1+\frac12(\gamma-1)M_1^2 \over 1+\frac12(\gamma-1)M_2^2}\Bigr)^\frac12$$ $${M_2^2} = {1+\frac12(\gamma-1)M_1^2 \over \gamma M_1^2-\frac12(\gamma-1)}$$ $${\rho_2 \over \rho_1} ={\frac12(\gamma+1)M_1^2 \over 1+\frac12(\gamma-1)M_1^2}$$

Fanno and Rayleigh Flows

Duct flow

$${dM_2 \over M_2} =-{2(1+\frac12(\gamma-1)M^2) \over 1-M^2}{dA\over A}$$ $$T_c=T\Bigl(1+\frac12(\gamma-1)M^2\Bigr)$$ $${p_c\over p}=\Bigl(1+\frac12(\gamma-1)M^2\Bigr)^{\gamma\over \gamma-1}$$

Nozzles and exit flows

$${4fL_*\over D}= {1-M^2\over \gamma M^2}+{\gamma+1\over 2\gamma}log\Bigl({\frac12(\gamma+1)M^2 \over 1+\frac12(\gamma-1)M^2}\Bigr)$$ $${T_0 \over T_0^*} = {(\gamma+1)M^2\bigl(2+\frac12(\gamma-1)M^2\bigr) \over (1+\gamma M^2)^2}$$ $${A_0 \over A_0^*} = \frac1M \Bigl( {1+\frac12(\gamma-1)M^2 \over \frac12(\gamma+1)}\Bigr)^{\gamma+1\over2(\gamma-1)}$$