Unit 5 - Notes

MEC107 7 min read

Unit 5: Introduction to Dynamics

1. Introduction and Basic Terms

Dynamics is the branch of engineering mechanics that deals with the study of bodies in motion and the forces that cause or change that motion. Unlike statics, which assumes zero acceleration, dynamics involves objects experiencing varying degrees of acceleration.

Dynamics is subdivided into two main branches:

Kinematics: The study of the geometry of motion without reference to the forces causing the motion. It relates displacement, velocity, acceleration, and time.
Kinetics: The study of the relationship between the forces acting on a body, the mass of the body, and the motion produced by these forces.

Basic Terminology

Particle: A body whose mass is concentrated at a single point. Its dimensions are considered negligible in the analysis of its motion.
Rigid Body: A collection of particles in which the distance between any two particles remains constant under the action of forces.
Position: The exact location of a particle in space at a given time, usually defined relative to a fixed reference frame or origin.
Displacement ( $\Delta s$ ): A vector quantity representing the change in position of a particle. It depends only on the initial and final positions, not the path taken.
Distance ( $s$ ): A scalar quantity representing the total length of the path traveled by a particle.
Velocity ( $v$ ): The rate of change of position with respect to time. It is a vector quantity.
- Average velocity: $v_{avg} = \Delta s / \Delta t$
- Instantaneous velocity: $v = ds / dt$
Acceleration ( $a$ ): The rate of change of velocity with respect to time. It is a vector quantity.
- Average acceleration: $a_{avg} = \Delta v / \Delta t$
- Instantaneous acceleration: $a = dv / dt = d^2s / dt^2$

2. General Principles in Dynamics

The study of dynamics is built upon several foundational laws and principles.

Newton's Laws of Motion

First Law (Law of Inertia): A particle originally at rest, or moving in a straight line with a constant velocity, will remain in this state provided the particle is not subjected to an unbalanced external force.
Second Law (Law of Acceleration): The time rate of change of momentum of a particle is directly proportional to the applied force. For a particle of constant mass, this is expressed as $\Sigma \mathbf{F} = m\mathbf{a}$ (Force equals mass times acceleration).
Third Law (Action and Reaction): For every action, there is an equal, opposite, and collinear reaction.

D'Alembert's Principle

D'Alembert's principle provides a way to reduce a dynamics problem into a statics problem by introducing a "fictitious" or "inertia" force.

Equation: $\Sigma \mathbf{F} - m\mathbf{a} = 0$
The term $-m\mathbf{a}$ is called the inertia force. By adding this force to the free body diagram, the system is in a state of "dynamic equilibrium."

Work-Energy Principle

The work done by all the forces acting on a particle equals the change in kinetic energy of the particle.

$U_{1 \to 2} = T_2 - T_1$
Where $U_{1 \to 2}$ is the work done, and $T = \frac{1}{2}mv^2$ is the kinetic energy.

Impulse-Momentum Principle

The linear impulse applied to a particle over a given time interval is equal to the change in the particle's linear momentum.

$\int_{t_1}^{t_2} \mathbf{F} dt = m\mathbf{v}_2 - m\mathbf{v}_1$

3. Types of Motion

The motion of a rigid body can be classified into the following fundamental categories:

Translation: Any straight line drawn on the body remains parallel to its original position throughout the motion.
- Rectilinear Translation: All particles move in parallel straight lines.
- Curvilinear Translation: All particles move on congruent curved paths.
Rotation about a Fixed Axis: All particles of the body move in circular paths centered on a single fixed axis.
General Plane Motion: A combination of translation and rotation. The body undergoes simultaneous translation and rotation within a plane.

4. Rectilinear Motion

Rectilinear motion is the continuous motion of a particle along a straight line.

Kinematic Relationships (Calculus Approach)

For variable acceleration, motion is solved using basic differential equations:

Velocity: $v = \frac{ds}{dt}$
Acceleration: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$
Differential relation: $a \, ds = v \, dv$ (derived by eliminating $dt$ )

Uniformly Accelerated Rectilinear Motion

If the acceleration $a$ is constant ( $a = a_c$ ), the calculus equations can be integrated to yield the standard equations of kinematics (where $s_0$ and $v_0$ are initial position and velocity at $t=0$ ):

$v = v_0 + a_c t$
$s = s_0 + v_0 t + \frac{1}{2} a_c t^2$
$v^2 = v_0^2 + 2a_c(s - s_0)$

Uniform Rectilinear Motion

If the acceleration is zero ( $a = 0$ ), the velocity is constant:

$s = s_0 + vt$

5. Plane Curvilinear Motion

Plane curvilinear motion occurs when a particle moves along a curved path that lies entirely within a single plane. Because the velocity vector changes direction (and possibly magnitude), there is always an acceleration, even if the speed is constant.

Coordinate Systems for Curvilinear Motion

1. Rectangular Coordinates (x, y)
Useful when the x and y components of motion are known or easily expressed as functions of time.

Position: $\mathbf{r} = x\mathbf{i} + y\mathbf{j}$
Velocity: $\mathbf{v} = \dot{x}\mathbf{i} + \dot{y}\mathbf{j} = v_x\mathbf{i} + v_y\mathbf{j}$
Magnitude of Velocity: $v = \sqrt{v_x^2 + v_y^2}$
Acceleration: $\mathbf{a} = \ddot{x}\mathbf{i} + \ddot{y}\mathbf{j} = a_x\mathbf{i} + a_y\mathbf{j}$

2. Normal and Tangential Coordinates (n, t)
Useful when the path is known. The origin is placed at the particle. The tangential axis ( $t$ ) is tangent to the path pointing in the direction of motion, and the normal axis ( $n$ ) points toward the center of curvature.

Velocity: $\mathbf{v} = v\mathbf{e}_t$ (Velocity is always tangent to the path).
Acceleration:
- Tangential acceleration ( $a_t$ ): Represents the change in the magnitude of velocity (speed). $a_t = \dot{v} = \frac{dv}{dt}$
- Normal acceleration ( $a_n$ ): Represents the change in the direction of velocity. $a_n = \frac{v^2}{\rho}$ (where $\rho$ is the radius of curvature).

3. Polar Coordinates (r, $\theta$ )
Useful when motion is constrained by a central pivot or described by radial distance and angle.

Position: $\mathbf{r} = r\mathbf{e}_r$
Velocity:
- $v_r = \dot{r}$ (radial velocity)
- $v_\theta = r\dot{\theta}$ (transverse velocity)
Acceleration: $\mathbf{a} = (\ddot{r} - r\dot{\theta}^2)\mathbf{e}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\mathbf{e}_\theta$

6. General Plane Motion

General plane motion of a rigid body is the superposition of translation and rotation. A classic example is a wheel rolling without slipping on a flat surface.

Relative Motion Analysis

General plane motion can be analyzed by picking a base point $A$ on the rigid body and a second point $B$ . The motion of $B$ is the vector sum of the translation of $A$ and the rotation of $B$ about $A$ .

Relative Velocity:
$\mathbf{v}_B = \mathbf{v}_A + \mathbf{v}_{B/A}$
Since the body is rigid, the relative motion of $B$ with respect to $A$ is pure rotation.
$\mathbf{v}_{B/A} = \mathbf{\omega} \times \mathbf{r}_{B/A}$
(where $\omega$ is the angular velocity of the body and $\mathbf{r}_{B/A}$ is the position vector of $B$ from $A$ ).
Relative Acceleration:
$\mathbf{a}_B = \mathbf{a}_A + \mathbf{a}_{B/A}$
The relative acceleration has two components (tangential and normal):
$\mathbf{a}_{B/A} = (\mathbf{a}_{B/A})_t + (\mathbf{a}_{B/A})_n = \mathbf{\alpha} \times \mathbf{r}_{B/A} - \omega^2 \mathbf{r}_{B/A}$
(where $\alpha$ is the angular acceleration of the body).

Instantaneous Center of Zero Velocity (ICR)

Any body undergoing general plane motion will have a point, either on or off the body, that has zero instantaneous velocity. This point is called the Instantaneous Center of Rotation (ICR).

At any given instant, the body appears to be in pure rotation about the ICR.
For any point $P$ on the body: $v_P = \omega \times r_{ICR \to P}$
Location of ICR: It is located at the intersection of the perpendiculars drawn to the velocity vectors of any two points on the rigid body.
Note: The ICR has zero velocity, but it generally does not have zero acceleration.

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