A Review of Dynamics Exam 1 Material

As part of my studies in mechanical engineering, I've compiled this cheat sheet on dynamics to help me prepare for my upcoming exam. It covers key concepts and equations from weeks 2 to 6 of the course.

Week 2

Math Review

Dot Product

For $ \vec{A} \cdot \vec{B} = 0 $, $ \vec{A} \perp \vec{B} $.

Cross Product

For $ \vec{A} \times \vec{B} = 0 $, $ \vec{A} \parallel \vec{B} $.

Chain Rule:

\frac{d}{dx} [f(g(x))] = \frac{df(g)}{dg} \frac{dg(x)}{dx}

Product Rule:

\frac{d}{dx} [f(x)g(x)] = f(x) \frac{dg(x)}{dx} + \frac{df(x)}{dx} g(x)

Trigonometry:

Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$
Cosine Rule: $c^2 = a^2 + b^2 - 2ab\cos(C)$

Week 3

Coordinate Systems

Cartesian Coordinates

\vec{r} = x(t) \hat{e}_x + y(t) \hat{e}_y + z(t) \hat{e}_z \] \[ \vec{v} = \dot{\vec{r}}(t) \] \[ \vec{a} = \ddot{\vec{r}}(t)

Cylindrical Coordinates

\vec{r} = r \hat{e}_r + z \hat{e}_z \] \[ \vec{v} = \dot{r} \hat{e}_r + r\dot{\theta} \hat{e}_{\theta} + \dot{z} \hat{e}_z \] \[ \vec{a} = (\ddot{r} - r\dot{\theta}^2) \hat{e}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta}) \hat{e}_{\theta} + \ddot{z} \hat{e}_z

Cylindrical Basis

\hat{e}_r = \cos\theta \hat{e}_x + \sin\theta \hat{e}_y, \quad \hat{e}_{\theta} = -\sin\theta \hat{e}_x + \cos\theta \hat{e}_y \] \[ \dot{\hat{e}}_r = \dot{\theta} \hat{e}_{\theta}, \quad \dot{\hat{e}}_{\theta} = -\dot{\theta} \hat{e}_r \] \[ \hat{e}_z = 0, \quad x = r \cos\theta, \quad y = r \sin\theta

Normal-Tangent Coordinate System

Radius of Curvature:

\kappa = \frac{1}{\rho} = \frac{\left| \frac{d^2 y}{dx^2} \right|}{\left(1 + \left( \frac{dy}{dx} \right)^2 \right)^{3/2}} \approx \frac{d^2 y}{dx^2} \quad

Infinitesimal Path Traveled:

ds = \rho d\beta

Speed:

v = \frac{ds}{dt} = \rho \frac{d\beta}{dt} = \rho \dot{\beta}

Velocity:

\vec{v} = \rho \dot{\beta} \hat{e}_t

Acceleration:

\vec{a} = \dot{v} \hat{e}_t + \frac{v^2}{\rho} \hat{e}_n

Week 4

Free Vibration of Particles: Undamped

Equation of Motion:

m\ddot{x} + kx = 0

Simple Harmonic Motion Equation:

\ddot{x} + \omega_n^2 x = 0 ,\quad \omega_n = \sqrt{\frac{k}{m}}

With solution of the form:

x = A \cos(\omega_n t) + B \sin(\omega_n t) \quad \text{or} \quad x = C \sin(\omega_n t + \psi)

Natural Frequency:

f_n = \frac{1}{\tau} = \frac{\omega_n}{2\pi}

Units: $1\, \text{Hz} = 1\, \text{cycle/second}$

Conservation of Angular Momentum

Angular Momentum:

r^2\dot{\theta} = h = \text{constant}

Acceleration Components:

a_r = \ddot{r} - r\dot{\theta}^2, \quad a_{\theta} = r\ddot{\theta} + 2\dot{r}\dot{\theta} \] \[ a_{\theta} = \frac{1}{r}\frac{d}{dt}(r^2\dot{\theta})

Friction: Static and Kinetic

Static Friction:

|F_{\text{f}}| \leq \mu_s |N|

If $ |F_{\text{trial}}| > \mu_s |N| $, sliding occurs.

Kinetic Friction:

F_{\text{f}} = -\mu_k |N| \frac{\vec{v}_{\text{rel}}}{||\vec{v}_{\text{rel}}||}

Detailed Friction Model:

$ \vec{F}^* $: Sum of all forces except tangential friction $ F_t $.
Tangential component: $F_t^* = \vec{F}^* - (\vec{F}^* \cdot \hat{e}_n)\hat{e}_n$
Trial friction force: $F_{\text{trial}} = m \left( \dot{\vec{v}}_{\text{wall}} - (\hat{e}_n \cdot \dot{\vec{v}}_{\text{wall}}) \hat{e}_n \right) = F_t^*$
Friction force $ F_t $: $F_t = \begin{cases} - F_{\text{trial}}, & v_s = 0, \ |F_{\text{trial}}| \leq \mu_s |N| \\ - \mu_s |N| \dfrac{F_{\text{trial}}}{|F_{\text{trial}}|}, & v_s = 0, \ |F_{\text{trial}}| > \mu_s |N| \\ - \mu_k |N| \dfrac{\vec{v}_s}{||\vec{v}_s||}, & v_s \ne 0 \end{cases}$

Week 5

Relative Motion in Non-Rotating Frames

\vec{r}_A = \vec{r}_B + \vec{r}_{A/B} \] \[ \vec{v}_A = \vec{v}_B + \vec{v}_{A/B} \] \[ \vec{a}_A = \vec{a}_B + \vec{a}_{A/B}

Cartesian Motion:

\vec{a}_{A/B} = (\vec{a}_{A/B})_x \hat{i} + (\vec{a}_{A/B})_y \hat{j}

Normal-Tangential:

\vec{a}_{A/B} = (\vec{a}_{A/B})_t \hat{e}_t + (\vec{a}_{A/B})_n \hat{e}_n

Where:

(\vec{a}_{A/B})_t = \alpha \times \vec{r}, \quad (\vec{a}_{A/B})_n = \omega \times (\omega \times \vec{r})

Relative Motion in Rotating Frames

For velocity:

\vec{v}_A = \vec{v}_B + \omega \times \vec{r}_{\text{rel}} + \vec{v}_{\text{rel}}

For acceleration:

\vec{a}_A = \vec{a}_B + \dot{\omega} \times \vec{r}_{\text{rel}} + \omega \times (\omega \times \vec{r}_{\text{rel}}) + 2\omega \times \vec{v}_{\text{rel}} + \vec{a}_{\text{rel}}

Euler: $ \dot{\omega} \times \vec{r}_{\text{rel}} $

Centrifugal: $ \omega \times (\omega \times \vec{r}_{\text{rel}}) $

Coriolis: $ 2\omega \times \vec{v}_{\text{rel}} $

Work, Energy, and Power

Power:

P = \vec{F} \cdot \vec{v} = F v \cos \theta

Power is the rate at which work is done or energy is transferred.

Work:

W = \int_{s_A}^{s_B} \vec{F} \cdot d\vec{s} = \int_{t_A}^{t_B} \vec{F} \cdot \vec{v} \, dt

Work is the integral of the force along the displacement path.

For constant force and straight-line motion:

W = F d \cos \theta

where $ d $ is the displacement and $ \theta $ is the angle between $ \vec{F} $ and $ d\vec{s} $.

Work-Energy Principle:

W_{\text{total}} = \Delta K = K_B - K_A = \frac{1}{2} m v_B^2 - \frac{1}{2} m v_A^2

The net work done on a particle equals the change in its kinetic energy.

Kinetic Energy:

K = \frac{1}{2} m v^2

Potential Energy:

Gravitational Potential Energy: $\[ U_g = m g h \]$ $ h $ is the height above a reference level.
Elastic Potential Energy: $U_e = \frac{1}{2} k x^2$ $ x $ is the displacement from the spring's equilibrium position.

Conservative Forces:

W_{\text{cons}} = -\Delta U

Non-Conservative Forces:

W_{\text{nc}} = \Delta E_{\text{mechanical}} = (K_B + U_B) - (K_A + U_A)

Mechanical Energy Conservation:

In the absence of non-conservative forces:

E_{\text{mechanical}} = K + U = \text{constant} \] \[ K_A + U_A = K_B + U_B

Total Mechanical Energy:

E_{\text{mech}} = K + U_g + U_e

Work Done by Variable Forces:

For forces that vary with position:

W = \int_{x_A}^{x_B} F(x) \, dx

Relation Between Work and Potential Energy:

For a conservative force $ \vec{F} = -\nabla U $:

W = -\Delta U

Week 6

Angular Momentum

Linear Momentum:

\vec{G} = m\vec{v}

Linear momentum is the product of mass and velocity.

Angular Momentum:

\vec{H}_O = \vec{r} \times m\vec{v}

In plane polar coordinates, angular momentum is expressed as:

\vec{H}_O = m r \vec{e}_r \times (\dot{r}\vec{e}_r + r\dot{\theta}\vec{e}_{\theta}) = m r^2 \dot{\theta} \vec{e}_z = H_O \vec{e}_z

Time Derivative of Angular Momentum:

\dot{\vec{H}}_O = m\vec{v} \times \vec{v} + \vec{r} \times m\vec{a} = \vec{r} \times m\vec{a}

Conservation of Angular Momentum:

If $ \vec{r} \times m\vec{a} = 0 $ in the $ \hat{e}_z $ direction, angular momentum is conserved and:

\frac{d\vec{H}_O}{dt} = 0

Orbital Motion

Acceleration in Polar Coordinates:

\vec{a} = (\ddot{r} - r\dot{\theta}^2)\hat{e}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{e}_{\theta}

Equations of Motion:

m(\ddot{r} - r\dot{\theta}^2) = -\frac{G M m}{r^2} \] \[ m(r\ddot{\theta} + 2\dot{r}\dot{\theta}) = 0

Conservation of Angular Momentum:

h = r^2 \dot{\theta} = \text{constant}

Energy Equation:

E = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) - \frac{G M m}{r} = \text{constant}

Orbital Parameters

Velocity Components:

v_r = \dot{r}, \quad v_{\theta} = r \dot{\theta} \] \[ v = \sqrt{v_r^2 + v_{\theta}^2}

Specific Mechanical Energy:

\epsilon = \frac{v^2}{2} - \frac{G M}{r}

Orbital Equation (Conic Sections):

r(\theta) = \frac{h^2 / (G M m^2)}{1 + e \cos(\theta - \theta_0)}

where $ e $ is the eccentricity.

Eccentricity:

e = \sqrt{1 + \frac{2 E h^2}{G^2 M^2 m^3}}

Periapsis and Apoapsis Distances:

r_{\text{min}} = \frac{h^2}{G M m^2 (1 + e)} \] \[ r_{\text{max}} = \frac{h^2}{G M m^2 (1 - e)}

Types of Orbits:

Ellipse ($ e < 1 $)
Parabola ($ e = 1 $)
Hyperbola ($ e > 1 $)

Additional Key Formulas

Relationship Between Energy and Semi-Major Axis:

E = -\frac{G M m}{2a} \] \[ v^2 = G M \left( \frac{2}{r} - \frac{1}{a} \right)

Angular Momentum per Unit Mass:

h = r v_{\theta} = \sqrt{G M a (1 - e^2)}

Velocity at Periapsis and Apoapsis:

v_{\text{min}} = \sqrt{\frac{G M (1 - e)}{a (1 + e)}} \] \[ v_{\text{max}} = \sqrt{\frac{G M (1 + e)}{a (1 - e)}}

Orbital Period (Elliptical Orbits):

T = 2\pi \sqrt{\frac{a^3}{G M}}

Velocity Components at Any Point:

v_r = \frac{G M e \sin \theta}{h} \] \[ v_{\theta} = \frac{G M (1 + e \cos \theta)}{h}

Power-Law Force

Normal Form (Force):

\vec{F}(r) = -k (r - r_0)^{n} \hat{e}_r

Potential Energy Form:

When $ n \ne -1 $: $U(r) = \frac{k}{n + 1} (r - r_0)^{n + 1} + C$
When $ n = -1 $: $U(r) = k \ln (r - r_0) + C$