Mechanics Cheatsheet - Hibbele

Cheatsheet Content

### Vectors & Components - **Cartesian Vector:** $\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$ - **Magnitude:** $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$ - **Unit Vector:** $\vec{u}_A = \frac{\vec{A}}{|\vec{A}|}$ (direction only) - **Dot Product:** $\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos\theta = A_x B_x + A_y B_y + A_z B_z$ - Used to find angle between vectors or projection of one vector on another. - **Cross Product:** $\vec{C} = \vec{A} \times \vec{B} = (A_y B_z - A_z B_y)\hat{i} + (A_z B_x - A_x B_z)\hat{j} + (A_x B_y - A_y B_x)\hat{k}$ - Magnitude: $|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin\theta$ - Direction: Right-hand rule, perpendicular to plane containing $\vec{A}$ and $\vec{B}$. ### Equilibrium of a Particle - **Condition for Equilibrium:** $\sum \vec{F} = \vec{0}$ - **2D Analysis:** - $\sum F_x = 0$ - $\sum F_y = 0$ - **3D Analysis:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum F_z = 0$ - **Free-Body Diagram (FBD):** Essential for visualizing all external forces acting on a particle. ### Force System Resultants - **Resultant Force:** $\vec{F}_R = \sum \vec{F}$ - **Resultant Couple Moment:** $\vec{M}_R = \sum \vec{M} + \sum (\vec{r} \times \vec{F})$ - For a system of forces, a resultant force and resultant couple moment can be found at any point. - In 2D, a couple moment is often simplified to $M = Fd$. ### Equilibrium of a Rigid Body - **Conditions for Equilibrium:** - $\sum \vec{F} = \vec{0}$ (Sum of all external forces is zero) - $\sum \vec{M}_O = \vec{0}$ (Sum of all external moments about any point O is zero) - **2D Analysis:** - $\sum F_x = 0$ - $\sum F_y = 0$ - $\sum M_O = 0$ (scalar sum of moments about point O) - **3D Analysis:** - $\sum F_x = 0, \sum F_y = 0, \sum F_z = 0$ - $\sum M_x = 0, \sum M_y = 0, \sum M_z = 0$ (sum of moments about x, y, z axes) - **Support Reactions:** Common supports include rollers (normal force), pins (normal forces in x & y), fixed supports (normal forces in x, y, and moment). ### Trusses & Frames - **Trusses:** Pinned at joints, members assume to be two-force members (tension or compression). - **Method of Joints:** Apply particle equilibrium ($\sum F_x=0, \sum F_y=0$) at each joint. - **Method of Sections:** Cut through members, apply rigid body equilibrium ($\sum F_x=0, \sum F_y=0, \sum M=0$) to a section. - **Frames & Machines:** At least one multi-force member. - Disassemble into individual members. - Apply rigid body equilibrium to each member. - Internal forces between connected members appear as equal and opposite pairs on their respective FBDs. ### Centroids & Center of Gravity - **Center of Gravity (CG):** Point where the entire weight of a body acts. - $\bar{x} = \frac{\sum W_i x_i}{\sum W_i}$, $\bar{y} = \frac{\sum W_i y_i}{\sum W_i}$, $\bar{z} = \frac{\sum W_i z_i}{\sum W_i}$ - **Centroid:** Geometric center of an area or volume (if uniform density, CG = Centroid). - **Area:** $\bar{x} = \frac{\int x dA}{\int dA}$, $\bar{y} = \frac{\int y dA}{\int dA}$ - For composite areas: $\bar{x} = \frac{\sum \bar{x}_i A_i}{\sum A_i}$, $\bar{y} = \frac{\sum \bar{y}_i A_i}{\sum A_i}$ - **Volume:** $\bar{x} = \frac{\int x dV}{\int dV}$, etc. - **Pappus-Guldinus Theorems:** - **Area of surface of revolution:** $A = 2\pi \bar{r} L$ (L = arc length, $\bar{r}$ = centroid distance from axis) - **Volume of body of revolution:** $V = 2\pi \bar{r} A$ (A = area, $\bar{r}$ = centroid distance from axis) ### Moments of Inertia - **Area Moment of Inertia:** Measures resistance to bending/deflection. - $I_x = \int y^2 dA$, $I_y = \int x^2 dA$ - **Polar Moment of Inertia:** $J_O = \int r^2 dA = I_x + I_y$ (resistance to torsion) - **Parallel-Axis Theorem:** $I = \bar{I} + Ad^2$ - $\bar{I}$ = moment of inertia about centroidal axis - $A$ = area - $d$ = perpendicular distance between parallel axes - **Standard Shapes (Examples):** - **Rectangle (centroidal x-axis):** $\bar{I}_x = \frac{1}{12}bh^3$ - **Circle (centroidal):** $\bar{I}_x = \bar{I}_y = \frac{1}{4}\pi r^4$, $\bar{J}_O = \frac{1}{2}\pi r^4$ ### Kinematics of a Particle - **Rectilinear Motion:** - **Constant Acceleration:** - $v = v_0 + a_c t$ - $s = s_0 + v_0 t + \frac{1}{2} a_c t^2$ - $v^2 = v_0^2 + 2 a_c (s - s_0)$ - **Variable Acceleration:** - $a = \frac{dv}{dt}$ or $dv = a dt$ - $v = \frac{ds}{dt}$ or $ds = v dt$ - $a ds = v dv$ - **Curvilinear Motion:** - **Rectangular Components:** $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ - $\vec{v} = \dot{x}\hat{i} + \dot{y}\hat{j} + \dot{z}\hat{k}$ - $\vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} + \ddot{z}\hat{k}$ - **Normal & Tangential Components (Path-dependent):** - $\vec{v} = v \vec{u}_t$ - $\vec{a} = \dot{v} \vec{u}_t + \frac{v^2}{\rho} \vec{u}_n$ - $\rho$ = radius of curvature ($\rho = \frac{[1+(dy/dx)^2]^{3/2}}{|d^2y/dx^2|}$) - **Cylindrical Components:** $\vec{r} = r\vec{u}_r + z\vec{u}_z$ - $\vec{v} = \dot{r}\vec{u}_r + r\dot{\theta}\vec{u}_{\theta} + \dot{z}\vec{u}_z$ - $\vec{a} = (\ddot{r} - r\dot{\theta}^2)\vec{u}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\vec{u}_{\theta} + \ddot{z}\vec{u}_z$ ### Kinetics of a Particle - **Newton's Second Law:** $\sum \vec{F} = m\vec{a}$ - **Rectangular:** $\sum F_x = m a_x$, $\sum F_y = m a_y$, $\sum F_z = m a_z$ - **Normal-Tangential:** $\sum F_t = m a_t$, $\sum F_n = m a_n = m \frac{v^2}{\rho}$ - **Cylindrical:** $\sum F_r = m a_r$, $\sum F_{\theta} = m a_{\theta}$, $\sum F_z = m a_z$ - **Work & Energy:** - **Work of a Force:** $U_{1-2} = \int_{s_1}^{s_2} \vec{F} \cdot d\vec{r}$ - Constant Force: $U_{1-2} = (F \cos\theta) \Delta s$ - Weight: $U_{1-2} = -W \Delta y$ - Spring: $U_{1-2} = \frac{1}{2} k (s_1^2 - s_2^2)$ - **Principle of Work & Energy:** $T_1 + U_{1-2} = T_2$ - $T = \frac{1}{2} m v^2$ (Kinetic Energy) - **Conservation of Energy (Conservative Forces Only):** $T_1 + V_1 = T_2 + V_2$ - $V_g = W y$ (Gravitational Potential Energy) - $V_e = \frac{1}{2} k s^2$ (Elastic Potential Energy) - **Impulse & Momentum:** - **Linear Momentum:** $\vec{L} = m\vec{v}$ - **Linear Impulse:** $\vec{I} = \int \vec{F} dt$ - **Principle of Linear Impulse & Momentum:** $m\vec{v}_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m\vec{v}_2$ - For a system of particles: $\sum m_i (\vec{v}_i)_1 + \sum \int \vec{F}_i dt = \sum m_i (\vec{v}_i)_2$ - **Conservation of Linear Momentum:** $\sum m_i (\vec{v}_i)_1 = \sum m_i (\vec{v}_i)_2$ (if net external impulse is zero) - **Coefficient of Restitution (e):** For impact, $e = \frac{(v_B')_n - (v_A')_n}{(v_A)_n - (v_B)_n}$ - $e=1$ (elastic impact), $e=0$ (plastic impact) - **Angular Momentum:** $\vec{H}_O = \vec{r} \times m\vec{v}$ - **Angular Impulse & Momentum:** $(\vec{H}_O)_1 + \sum \int_{t_1}^{t_2} \vec{M}_O dt = (\vec{H}_O)_2$ - **Conservation of Angular Momentum:** $(\vec{H}_O)_1 = (\vec{H}_O)_2$ (if net external moment about O is zero) ### Kinematics of a Rigid Body - **Types of Motion:** - **Translation:** - Rectilinear: All points move in parallel straight lines. $\vec{v}_B = \vec{v}_A$, $\vec{a}_B = \vec{a}_A$. - Curvilinear: All points move in parallel curved paths. $\vec{v}_B = \vec{v}_A$, $\vec{a}_B = \vec{a}_A$. - **Rotation about a Fixed Axis:** - Angular velocity: $\omega = \frac{d\theta}{dt}$ - Angular acceleration: $\alpha = \frac{d\omega}{dt}$ - For constant $\alpha$: $\omega = \omega_0 + \alpha_c t$, $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha_c t^2$, $\omega^2 = \omega_0^2 + 2\alpha_c(\theta - \theta_0)$ - Velocity: $\vec{v} = \vec{\omega} \times \vec{r}$ (where $\vec{r}$ is from fixed axis to point) - Acceleration: $\vec{a} = \vec{\alpha} \times \vec{r} + \vec{\omega} \times (\vec{\omega} \times \vec{r}) = a_t \vec{u}_t + a_n \vec{u}_n$ - $a_t = \alpha r$ (tangential) - $a_n = \omega^2 r$ (normal, towards center) - **General Plane Motion:** (Translation + Rotation) - $\vec{v}_B = \vec{v}_A + \vec{\omega} \times \vec{r}_{B/A}$ - $\vec{a}_B = \vec{a}_A + \vec{\alpha} \times \vec{r}_{B/A} + \vec{\omega} \times (\vec{\omega} \times \vec{r}_{B/A})$ - **Instantaneous Center of Zero Velocity (IC):** Point where velocity is instantaneously zero. Locating IC simplifies velocity analysis. - **Relative-Motion Analysis:** (Dependent Motion) - Position: $\vec{r}_P = \vec{r}_A + \vec{r}_{P/A}$ - Velocity: $\vec{v}_P = \vec{v}_A + \vec{v}_{P/A}$ - Acceleration: $\vec{a}_P = \vec{a}_A + \vec{a}_{P/A}$ ### Kinetics of a Rigid Body - **Equations of Motion:** - $\sum \vec{F} = m\vec{a}_G$ (Sum of forces equals mass times acceleration of center of mass G) - $\sum M_G = I_G \alpha$ (Sum of moments about G equals moment of inertia about G times angular acceleration) - Alternatively, if moments are taken about an instantaneously accelerating point P: $\sum M_P = I_P \alpha + m(\vec{r}_{G/P} \times \vec{a}_P)$ - **Plane Motion Equations:** - $\sum F_x = m(a_G)_x$ - $\sum F_y = m(a_G)_y$ - $\sum M_G = I_G \alpha$ (for the body rotating about its center of mass G) - **Work & Energy:** - **Kinetic Energy:** $T = \frac{1}{2} m v_G^2 + \frac{1}{2} I_G \omega^2$ (for general plane motion) - **Work done by a Moment:** $U_M = \int_{\theta_1}^{\theta_2} M d\theta$ - **Principle of Work & Energy:** $T_1 + \sum U_{1-2} = T_2$ - **Impulse & Momentum:** - **Linear Impulse & Momentum:** $m(\vec{v}_G)_1 + \sum \int_{t_1}^{t_2} \vec{F} dt = m(\vec{v}_G)_2$ - **Angular Impulse & Momentum (about G):** $(I_G \omega)_1 + \sum \int_{t_1}^{t_2} M_G dt = (I_G \omega)_2$ - **Angular Impulse & Momentum (about a fixed point O):** $(\vec{H}_O)_1 + \sum \int_{t_1}^{t_2} \vec{M}_O dt = (\vec{H}_O)_2$ - $\vec{H}_O = I_O \vec{\omega}$ for rotation about fixed axis O. - **Impact (Rigid Bodies):** - **Coefficient of Restitution (e):** $e = \frac{(\omega_B')_n - (\omega_A')_n}{(v_A)_n - (v_B)_n}$ for rotational impacts, or for a general point on a rigid body. - Apply impulse-momentum for the system and individual bodies. ### Vibrations - **Undamped Free Vibration:** - Equation of Motion: $m\ddot{x} + kx = 0$ - Natural Frequency: $\omega_n = \sqrt{\frac{k}{m}}$ (rad/s) or $f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$ (Hz) - Period: $\tau = \frac{2\pi}{\omega_n}$ - Solution: $x(t) = A \sin(\omega_n t) + B \cos(\omega_n t)$ or $x(t) = C \sin(\omega_n t + \phi)$ - **Damped Free Vibration:** - Equation of Motion: $m\ddot{x} + c\dot{x} + kx = 0$ - Damping Ratio: $\zeta = \frac{c}{c_c} = \frac{c}{2m\omega_n}$ - Critical Damping: $c_c = 2m\omega_n = 2\sqrt{km}$ - Damped Natural Frequency: $\omega_d = \omega_n \sqrt{1 - \zeta^2}$ - Overdamped $(\zeta > 1)$, Critically Damped $(\zeta = 1)$, Underdamped $(\zeta