Kinematics of Rigid Bodies

Kinematics of Rigid Bodies

Learning Objectives

Define and categorize different types of rigid body motion.
Apply angular kinematic equations for rotation about a fixed axis.
Determine velocity and acceleration of a point on a rotating body.
Analyze relative motion using translating and rotating axes.
Calculate velocities using the Instantaneous Center of Zero Velocity (IC).

While particle kinematics ignores rotational effects, rigid body kinematics accounts for both the translation and rotation of bodies. A rigid body is idealized as a system of particles wherein the distance between any two specific particles remains perfectly constant under applied loads.

Types of Rigid Body Motion

Rigid Body Motion Categories

Translation: Any straight line segment drawn on the body remains parallel to its original orientation throughout the motion. All particles in the body have the exact same velocity and acceleration.
- Rectilinear Translation: Paths of particles are straight lines.
- Curvilinear Translation: Paths of particles are congruent curves.
Rotation About a Fixed Axis: All particles move in circular paths centered on the axis of rotation, except for those lying on the axis (which have zero velocity).
General Plane Motion: A combination of translation and rotation. The body undergoes a displacement consisting of a translation within a reference plane and a rotation about an axis perpendicular to that plane.

Interactive Simulation

Interact with the simulation below to explore rigid body translation.

Rigid Body Translation & Rotation

Motion Control

Motion Type

Animation Speed1.0x

Kinematics Equation Solver

Governing Principle:

In translation, all lines in the body remain parallel. Thus:

\mathbf{v}_A = \mathbf{v}_B \quad \text{and} \quad \mathbf{a}_A = \mathbf{a}_B

Live Values:

Point A

v: 96.0 m/s

a: 0.0 m/s²

Point B

v: 96.0 m/s

a: 0.0 m/s²

✓ Verified: Point A and Point B have identical motion vectors.

Velocity at A (v_A)

Velocity at B (v_B)

Acceleration (a_A)

Acceleration (a_B)

Rotation About a Fixed Axis

When a rigid body rotates about a fixed axis, its motion is described by angular parameters.

Angular Position ( $\theta$ )

The angle defining the position of a line on the body, usually measured in radians.

Angular Velocity Equation

Defines angular velocity as the time rate of change of angular position.

\omega = \frac{d\theta}{dt}

Variables

Symbol	Description	Unit
$\omega$	Angular velocity	rad/s
$\theta$	Angular position	rad
$t$	Time	s

Angular Acceleration Equation

Defines angular acceleration as the time rate of change of angular velocity.

\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}

Variables

Symbol	Description	Unit
$\alpha$	Angular acceleration	rad/s²
$\omega$	Angular velocity	rad/s
$\theta$	Angular position	rad
$t$	Time	s

Differential Angular Kinematics

Similar to linear motion, eliminating $dt$ yields a purely spatial relation for angular kinematics.

Differential Angular Kinematic Equation

Relates angular acceleration, position, and velocity independent of time.

\alpha \, d\theta = \omega \, d\omega

Variables

Symbol	Description	Unit
$\alpha$	Angular acceleration	rad/s²
$d\theta$	Differential change in angular position	rad
$\omega$	Angular velocity	rad/s
$d\omega$	Differential change in angular velocity	rad/s

If angular acceleration $\alpha_c$ is constant, we obtain angular equations entirely analogous to linear constant acceleration formulas.

Constant Angular Acceleration Equations

Equations of motion for a rigid body rotating with a constant angular acceleration.

\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)

Variables

Symbol	Description	Unit
$\omega$	Final angular velocity	rad/s
$\omega_0$	Initial angular velocity	rad/s
$\alpha_c$	Constant angular acceleration	rad/s²
$\theta$	Final angular position	rad
$\theta_0$	Initial angular position	rad
$t$	Time	s

Velocity and Acceleration of a Point

Point Kinematics in Rotation

For a point $P$ located at a distance $r$ from the axis of rotation, its kinematics can be determined directly from the angular parameters.

Point Kinematics Equations

Relates linear velocity and acceleration components to angular parameters.

v = \omega r

a_t = \alpha r

a_n = \omega^2 r = \frac{v^2}{r}

Variables

Symbol	Description	Unit
$v$	Velocity magnitude, tangent to the circular path	m/s
$\omega$	Angular velocity	rad/s
$r$	Distance from the axis of rotation	m
$a_t$	Tangential acceleration magnitude	m/s²
$\alpha$	Angular acceleration	rad/s²
$a_n$	Normal acceleration magnitude, directed towards the axis	m/s²

Relative Motion Analysis Using Translating Axes

A powerful method for analyzing general plane motion is to break the motion down into two parts: a translation of a chosen base point, and a pure rotation about that base point. This relies on non-rotating (translating) axes attached to the base point.

Relative Velocity Concept

Let points A and B lie on the same rigid body undergoing general plane motion. A translating reference frame is attached to base point A. The velocity of B equals the velocity of A plus the velocity of B relative to A due to rotation.

Relative Velocity Equation

Calculates the velocity of point B using a translating reference frame at point A.

\mathbf{v}_B = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}_{B/A}

Variables

Symbol	Description	Unit
$\mathbf{v}_B$	Velocity of point B	m/s
$\mathbf{v}_A$	Velocity of base point A	m/s
$\mathbf{\omega}$	Angular velocity vector of the rigid body	rad/s
$\mathbf{r}_{B/A}$	Position vector of B with respect to A	m

Relative Acceleration Concept

The acceleration of B equals the acceleration of A plus the relative acceleration of B with respect to A. The relative acceleration consists of tangential and normal components.

Relative Acceleration Equation

Calculates the acceleration of point B using a translating reference frame at point A.

\mathbf{a}_B = \mathbf{a}_A + \mathbf{\alpha} \times \mathbf{r}_{B/A} - \omega^2 \mathbf{r}_{B/A}

Variables

Symbol	Description	Unit
$\mathbf{a}_B$	Acceleration of point B	m/s²
$\mathbf{a}_A$	Acceleration of base point A	m/s²
$\mathbf{\alpha}$	Angular acceleration vector of the rigid body	rad/s²
$\mathbf{r}_{B/A}$	Position vector of B with respect to A	m
$\omega$	Magnitude of angular velocity	rad/s

Relative Motion Analysis Using Rotating Axes

While translating axes are sufficient for simple problems where points A and B lie on the same rigid body, many mechanisms involve sliding contacts on rotating links. In these cases, it's necessary to express relative motion using a coordinate system $(x,y)$ that is attached to a rotating body.

Coriolis Acceleration Concept

When a particle moves with a velocity relative to a frame rotating with an angular velocity, an extra acceleration component arises, known as the Coriolis acceleration ( $\mathbf{a}_c$ ).

Relative Velocity with Rotating Axes

Calculates velocity when the reference frame is itself rotating.

\mathbf{v}_B = \mathbf{v}_A + \mathbf{\Omega} \times \mathbf{r}_{B/A} + (\mathbf{v}_{B/A})_{xyz}

Variables

Symbol	Description	Unit
$\mathbf{v}_B$	Absolute velocity of point B	m/s
$\mathbf{v}_A$	Absolute velocity of origin A of moving frame	m/s
$\mathbf{\Omega}$	Angular velocity of the rotating frame	rad/s
$\mathbf{r}_{B/A}$	Position of B relative to A	m
$(\mathbf{v}_{B/A})_{xyz}$	Velocity of B relative to the moving frame	m/s

Relative Acceleration with Rotating Axes (Coriolis)

Calculates acceleration when the reference frame is rotating, including the Coriolis component.

\mathbf{a}_B = \mathbf{a}_A + \dot{\mathbf{\Omega}} \times \mathbf{r}_{B/A} + \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{r}_{B/A}) + 2\mathbf{\Omega} \times (\mathbf{v}_{B/A})_{xyz} + (\mathbf{a}_{B/A})_{xyz}

Variables

Symbol	Description	Unit
$\mathbf{a}_B$	Absolute acceleration of point B	m/s²
$\mathbf{a}_A$	Absolute acceleration of origin A	m/s²
$\dot{\mathbf{\Omega}}$	Angular acceleration of the rotating frame	rad/s²
$\mathbf{\Omega}$	Angular velocity of the rotating frame	rad/s
$\mathbf{r}_{B/A}$	Position of B relative to A	m
$(\mathbf{v}_{B/A})_{xyz}$	Velocity of B relative to moving frame	m/s
$(\mathbf{a}_{B/A})_{xyz}$	Acceleration of B relative to moving frame	m/s²

Coriolis Acceleration Direction

A common mistake is applying the wrong sign or direction to the Coriolis acceleration component: $2\mathbf{\Omega} \times (\mathbf{v}_{B/A})_{xyz}$ . Because it is a cross product, the direction is strictly determined by the right-hand rule. Rotate the vector $\mathbf{\Omega}$ into the vector $(\mathbf{v}_{B/A})_{xyz}$ ; your thumb points in the direction of the Coriolis acceleration.

General Plane Motion: Instantaneous Center of Zero Velocity (IC)

General plane motion can be viewed at any specific instant as a pure rotation about a single, unique axis. The point where this axis intersects the plane of motion is called the Instantaneous Center of Zero Velocity (IC).

Limitations of the IC Method

The IC is a point in the plane of motion that momentarily has zero velocity ( $v_{IC} = 0$ ). By locating the IC, the velocity of any other point $A$ on the body can be found simply as $v_A = \omega \cdot r_{A/IC}$ , with the direction of $v_A$ perpendicular to the line connecting $A$ and the IC.

Note: The IC only has zero velocity at that instant. Its acceleration is generally not zero. Therefore, the IC method is strictly for velocity analysis, not acceleration analysis.

Locating the Instantaneous Center of Zero Velocity (IC)

STEP-BY-STEP

Given Velocity Directions: If the directions of the velocities of two points $A$ and $B$ are known, and they are not parallel, draw lines perpendicular to $\mathbf{v}_A$ and $\mathbf{v}_B$ . The intersection of these lines is the IC.
Given Parallel Velocities (Perpendicular to AB): If $\mathbf{v}_A$ and $\mathbf{v}_B$ are parallel and perpendicular to the line segment connecting $A$ and $B$ , draw a line connecting the heads of the velocity vectors and another connecting their tails (the line $AB$ ). The intersection of these two lines is the IC.
Given Velocity and Angular Velocity: If the velocity $\mathbf{v}_A$ of a point and the angular velocity $\omega$ of the body are known, the IC lies on the line perpendicular to $\mathbf{v}_A$ at a distance $r = v_A / \omega$ .

Interactive Simulation

Interact with the simulation below to explore the Instantaneous Center of Zero Velocity.

Instantaneous Center of Zero Velocity (IC) Visualizer

Mechanism Setup

System Mode

Angular Velocity (

\omega

)2.0 rad/s

Wheel Radius (R)2.0 m

Slip Ratio (s)0.00

s < 0 (Braking)s = 0 (Pure)s > 0 (Spin/Skid)

Show Velocity VectorsShow IC Distance Rays

Mathematical Equations

Wheel IC Location & Slip Mechanics:

IC offset from center O is given by slip ratio:

y_{IC} - y_O = R(1 + s) = 2.0(1 + 0.00) = 2.00\text{ m}

Pure rolling: IC is exactly at the point of contact.

Dragged Point P Velocity:

v_P = \omega \times r_{P/IC}

d from IC: 3.27 m|v_P: 6.55 m/s

💡 Drag point P to analyze local velocity.

IC (Instant Center, v=0)

Velocity vector v_P

Distance ray r_P/IC

Key Takeaways

Rotation About a Fixed Axis: Governed by equations $\omega = d\theta/dt$ and $\alpha = d\omega/dt$ .
Point Acceleration: Has normal ( $a_n = \omega^2 r$ ) and tangential ( $a_t = \alpha r$ ) components.
General Plane Motion is the superposition of translation of a reference point and rotation about that point.
Relative Velocity and Acceleration use vector equations ( $\mathbf{v}_B = \mathbf{v}_A + \mathbf{\omega} \times \mathbf{r}$ ) to relate points on the same rigid body.
Instantaneous Center (IC): A powerful technique to find velocities by treating general plane motion instantaneously as pure rotation about the IC.

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Learning Objectives

Types of Rigid Body Motion

Rigid Body Motion Categories

Interactive Simulation

Motion Control

Kinematics Equation Solver

Rotation About a Fixed Axis

Angular Position (θ\thetaθ)

Angular Velocity Equation

Angular Acceleration Equation

Differential Angular Kinematics

Differential Angular Kinematic Equation

Constant Angular Acceleration Equations

Velocity and Acceleration of a Point

Point Kinematics in Rotation

Point Kinematics Equations

Relative Motion Analysis Using Translating Axes

Relative Velocity Concept

Relative Velocity Equation

Relative Acceleration Concept

Relative Acceleration Equation

Relative Motion Analysis Using Rotating Axes

Coriolis Acceleration Concept

Relative Velocity with Rotating Axes

Relative Acceleration with Rotating Axes (Coriolis)

Coriolis Acceleration Direction

General Plane Motion: Instantaneous Center of Zero Velocity (IC)

Limitations of the IC Method

Locating the Instantaneous Center of Zero Velocity (IC)

Interactive Simulation

Mechanism Setup

Mathematical Equations

Angular Position ( $\theta$ )