Three-Dimensional Kinetics of Rigid Bodies - Theory & Concepts

Three-Dimensional Kinetics of Rigid Bodies

Learning Objectives

Understand the general equations of motion for a rigid body in 3D.
Relate angular momentum and the inertia tensor.
Apply Euler's equations of motion to solve 3D rigid body problems.
Understand Eulerian angles and their application in 3D rotation.
Analyze torque-free motion and gyroscopic motion.
Calculate the kinetic energy of a rigid body in 3D motion.

The kinetics of rigid bodies in three dimensions relates the forces and moments acting on a body to its resulting translational and rotational motion. This is governed by Newton's Second Law and the angular momentum principle. Unlike planar kinetics, the angular momentum vector in 3D is generally not parallel to the angular velocity vector.

Eulerian Angles and 3D Rotation

Describing the orientation of a rigid body in 3D space requires a robust coordinate system. Eulerian angles are one of the standard ways to represent this orientation.

Eulerian Angles

A set of three independent angles used to specify the orientation of a rigid body with respect to a fixed coordinate system. They represent a sequence of three elemental rotations about the axes of a coordinate system.

Understanding Eulerian Angles

The standard Eulerian angle sequence involves:

Precession ( $\phi$ ): A rotation about the vertical fixed $Z$ -axis.
Nutation ( $\theta$ ): A rotation about the moving $x'$ -axis (line of nodes).
Spin ( $\psi$ ): A rotation about the body's $z$ -axis (symmetry axis).

These angles allow us to express the angular velocity vector $\mathbf{\omega}$ in terms of its components along the body-fixed axes, which is crucial for solving Euler's equations of motion.

Equations of Motion

The general equations of motion for a rigid body in 3D are expressed in vector form. The translational motion is governed by the sum of external forces, and the rotational motion is governed by the sum of external moments.

General Equations of Motion

The general equations of motion relate the forces and moments to translational and rotational changes. The translational equation describes the motion of the center of mass, while the rotational equation describes the change in angular momentum.

Translational and Rotational Equations of Motion

Governs the translational and rotational motion of a rigid body.

\sum \mathbf{F} = m \mathbf{a}_G

\sum \mathbf{M}_G = \dot{\mathbf{H}}_G

Variables

Symbol	Description	Unit
$\mathbf{F}$	External force vector	N
$m$	Mass of the rigid body	kg
$\mathbf{a}_G$	Acceleration vector of the mass center	$m/s^2$
$\mathbf{M}_G$	External moment vector about the mass center	$N\cdot m$
$\mathbf{H}_G$	Angular momentum vector about the mass center	$kg\cdot m^2/s$

Angular Momentum and the Inertia Tensor

In 3D, the angular momentum $\mathbf{H}_G$ depends on both the angular velocity $\mathbf{\omega}$ and the mass distribution of the body, which is described by the inertia tensor $\mathbf{I}$ .

Inertia Tensor

A symmetric $3 \times 3$ matrix that describes a rigid body's mass distribution and its resistance to rotational acceleration about various axes. It contains moments of inertia on the diagonal and products of inertia on the off-diagonals.

Principal Axes

A set of mutually orthogonal axes originating from a specific point (usually the center of mass) for which all products of inertia of a given body are zero, leaving only the principal moments of inertia.

Angular Momentum and Inertia

The angular momentum vector $\mathbf{H}_G$ is related to the angular velocity $\mathbf{\omega}$ through the inertia tensor $\mathbf{I}_G$ . When using the principal axes, the products of inertia ( $I_{xy}, I_{xz}, I_{yz}$ ) become zero, significantly simplifying calculations.

Angular Momentum and Inertia Tensor

Relates angular momentum, inertia tensor, and angular velocity in 3D.

\mathbf{H}_G = \mathbf{I}_G \mathbf{\omega}

\mathbf{I}_G = \begin{bmatrix} I_{xx} & -I_{xy} & -I_{xz} \\ -I_{yx} & I_{yy} & -I_{yz} \\ -I_{zx} & -I_{zy} & I_{zz} \end{bmatrix}

Variables

Symbol	Description	Unit
$\mathbf{H}_G$	Angular momentum vector	$kg\cdot m^2/s$
$\mathbf{I}_G$	Inertia tensor matrix	$kg\cdot m^2$
$\mathbf{\omega}$	Angular velocity vector	rad/s
$I_{xx}, I_{yy}, I_{zz}$	Moments of inertia about the x, y, and z axes	$kg\cdot m^2$
$I_{xy}, I_{xz}, I_{yz}$	Products of inertia	$kg\cdot m^2$

Principal Axes Simplification

For any rigid body, there exists a set of mutually orthogonal principal axes for which the products of inertia are zero. When these axes are used, the inertia tensor becomes diagonal, simplifying the angular momentum equation.

Angular Momentum with Principal Axes

Simplified angular momentum equation using principal axes where products of inertia are zero.

\mathbf{H}_G = I_x \omega_x \mathbf{i} + I_y \omega_y \mathbf{j} + I_z \omega_z \mathbf{k}

Variables

Symbol	Description	Unit
$\mathbf{H}_G$	Angular momentum vector about the mass center	$kg\cdot m^2/s$
$I_x, I_y, I_z$	Principal moments of inertia	$kg\cdot m^2$
$\omega_x, \omega_y, \omega_z$	Components of angular velocity along principal axes	rad/s

Euler's Equations of Motion

When the reference frame is attached to the rigid body and aligned with its principal axes, the rotational equations of motion simplify to Euler's Equations.

Euler's Equations Concepts

Euler's equations expand the vector equation $\sum \mathbf{M}_G = \dot{\mathbf{H}}_G$ into three scalar equations along the principal axes. These nonlinear differential equations relate the applied moments to the resulting angular acceleration and the gyroscopic effects caused by rotation.

Euler's Equations

Scalar equations of rotational motion aligned with the principal axes.

\begin{aligned} \sum M_x &= I_x \dot{\omega}_x - (I_y - I_z) \omega_y \omega_z \\ \sum M_y &= I_y \dot{\omega}_y - (I_z - I_x) \omega_z \omega_x \\ \sum M_z &= I_z \dot{\omega}_z - (I_x - I_y) \omega_x \omega_y \end{aligned}

Variables

Symbol	Description	Unit
$\sum M_x, \sum M_y, \sum M_z$	Sum of moments about the principal axes	$N\cdot m$
$I_x, I_y, I_z$	Principal moments of inertia	$kg\cdot m^2$
$\omega_x, \omega_y, \omega_z$	Components of angular velocity along principal axes	rad/s
$\dot{\omega}_x, \dot{\omega}_y, \dot{\omega}_z$	Components of angular acceleration along principal axes	$rad/s^2$

Solving 3D Kinetics Problems using Euler's Equations

STEP-BY-STEP

Define a Coordinate System: Establish a coordinate system attached to the rigid body, with the origin at the center of mass or a fixed point.
Determine Principal Axes and Moments of Inertia: Align the coordinate axes with the principal axes of the body to eliminate products of inertia, and calculate the principal moments of inertia ( $I_x, I_y, I_z$ ).
Express Angular Velocity and Acceleration: Write the angular velocity vector $\mathbf{\omega}$ and the angular acceleration vector $\dot{\mathbf{\omega}}$ in terms of components along the chosen body-fixed axes.
Identify External Moments: Determine the components of all external moments acting on the body about the coordinate origin ( $\sum M_x, \sum M_y, \sum M_z$ ).
Apply Euler's Equations: Substitute the moments, moments of inertia, angular velocities, and angular accelerations into Euler's equations.
Solve the Equations: Solve the resulting system of differential equations for the unknown angular accelerations, velocities, or applied moments.

Torque-Free Motion

An important class of problems in 3D rigid body dynamics involves bodies that undergo rotation while the net external moment applied to their center of mass is zero ( $\sum \mathbf{M}_G = 0$ ). Examples include spacecraft, satellites, and tossed objects (like a football in flight).

Torque-Free Motion

The rotational motion of a rigid body when the resultant external moment acting about its center of mass is zero. Under these conditions, the angular momentum of the body is conserved in both magnitude and direction.

Torque-Free Motion Principles

Because the net moment is zero ( $\sum \mathbf{M}_G = 0$ ), two fundamental conservation laws apply to the body during torque-free motion: the conservation of angular momentum ( $\mathbf{H}_G$ ) and the conservation of kinetic energy ( $T$ ).

Axisymmetric Bodies: For bodies with an axis of symmetry (e.g., a cylinder where $I_x = I_y = I$ ), the motion simplifies significantly. The body will rotate about its axis of symmetry at a constant rate $\omega_z$ , while simultaneously undergoing regular precession about the fixed angular momentum vector $\mathbf{H}_G$ .

Torque-Free Euler's Equations

Euler's equations when the net external moment is zero.

0 = I_x \dot{\omega}_x - (I_y - I_z) \omega_y \omega_z

0 = I_y \dot{\omega}_y - (I_z - I_x) \omega_z \omega_x

0 = I_z \dot{\omega}_z - (I_x - I_y) \omega_x \omega_y

Variables

Symbol	Description	Unit
$I_x, I_y, I_z$	Principal moments of inertia	$kg\cdot m^2$
$\omega_x, \omega_y, \omega_z$	Components of angular velocity along principal axes	rad/s
$\dot{\omega}_x, \dot{\omega}_y, \dot{\omega}_z$	Components of angular acceleration along principal axes	$rad/s^2$

Torque-Free Conservation Laws

Angular momentum and kinetic energy remain constant in torque-free motion.

\mathbf{H}_G = \text{constant}

T = \frac{1}{2} \mathbf{\omega} \cdot \mathbf{H}_G = \text{constant}

Variables

Symbol	Description	Unit
$\mathbf{H}_G$	Constant angular momentum vector	$kg\cdot m^2/s$
$T$	Constant rotational kinetic energy	J
$\mathbf{\omega}$	Angular velocity vector	rad/s

Kinetic Energy in 3D

The kinetic energy ( $T$ ) of a rigid body undergoing general 3D motion is the sum of its translational and rotational kinetic energy. Unlike 2D where rotational energy is simply $\frac{1}{2}I_z\omega^2$ , in 3D, we must use the dot product of angular velocity and angular momentum vectors.

3D Kinetic Energy Formula

Calculates the total kinetic energy for a rigid body in 3D motion.

T = \frac{1}{2}m\mathbf{v}_G \cdot \mathbf{v}_G + \frac{1}{2}\mathbf{\omega} \cdot \mathbf{H}_G

If the angular velocity vector is expressed in terms of components along the principal axes, this expands to:

T = \frac{1}{2}mv_G^2 + \frac{1}{2}(I_x \omega_x^2 + I_y \omega_y^2 + I_z \omega_z^2)

Variables

Symbol	Description	Unit
$T$	Total kinetic energy	J
$m$	Mass of the body	kg
$\mathbf{v}_G$	Velocity vector of the center of mass	m/s
$v_G$	Magnitude of the center of mass velocity	m/s
$\mathbf{\omega}$	Angular velocity vector	rad/s
$\mathbf{H}_G$	Angular momentum vector	$kg\cdot m^2/s$
$I_x, I_y, I_z$	Principal moments of inertia	$kg\cdot m^2$
$\omega_x, \omega_y, \omega_z$	Components of angular velocity along principal axes	rad/s

Gyroscopic Motion

Gyroscopic motion involves the rotation of a symmetric body at high speed about its axis of symmetry, while being subjected to a torque that causes it to rotate about a mutually perpendicular axis.

Precession

The change in the orientation of the rotational axis of a spinning body. It occurs when a torque is applied perpendicular to the axis of spin.

Gyroscopes

A gyroscope is a device containing a rapidly spinning rotor that tends to maintain its orientation in space due to conservation of angular momentum. When a torque is applied perpendicular to the spin axis, the gyroscope exhibits precession—a rotation about an axis mutually perpendicular to the spin and torque axes.

Steady Precession of a Gyroscope

Relates the applied moment to the precession angular velocity and spin angular momentum.

\mathbf{M} = \mathbf{\Omega} \times \mathbf{H}

Variables

Symbol	Description	Unit
$\mathbf{M}$	Applied moment vector	$N\cdot m$
$\mathbf{\Omega}$	Precession angular velocity vector	rad/s
$\mathbf{H}$	Spin angular momentum vector	$kg\cdot m^2/s$

Interactive Simulation

Interact with the simulation below to explore gyroscopic motion, observing how changes in spin speed and applied torque affect the precession rate.

Gyroscopic Precession & Torque Dynamics

Controls

Rotor Spin Rate (

\omega_s

)20.0 rad/s

Tilt Angle (

\theta

)60°

Pivot Length (

d

)0.35 m

Rotor Mass (

m

)1.5 kg

Show Vector Diagrams (H, τ, Ω)

Equations

Rotor Moment of Inertia:

I_s = \frac{1}{2} m R^2 = 0.5 \times 1.5 \times 0.4^2 = 0.120\,\text{kg}\cdot\text{m}^2

Gravity Torque Magnitude:

\tau = m g d \sin\theta = 1.5 \times 9.81 \times 0.35 \times \sin(60^\circ) = 4.46\,\text{N}\cdot\text{m}

Spin Angular Momentum:

H_s = I_s \omega_s = 0.120 \times 20.0 = 2.40\,\text{kg}\cdot\text{m}^2/\text{s}

Precession Rate (steady):

\Omega = \frac{\tau}{H_s \sin\theta} = \frac{m g d}{I_s \omega_s} = \frac{5.150}{2.40} = 2.15\,\text{rad/s}

Precession Vector (Ω)

Angular Momentum (H)

Gravity Torque Vector (τ)

Concept: The gravitational force creates a torque $\tau$ perpendicular to the spin axis.

Instead of falling down, the gyroscope precesses horizontally because the torque changes the direction of the spin angular momentum vector $\mathbf{H}$ .

Drag to rotate the view camera.

Key Takeaways

Equations of Motion: In 3D, the equations are $\sum \mathbf{F} = m \mathbf{a}_G$ and $\sum \mathbf{M}_G = \dot{\mathbf{H}}_G$ .
Inertia Tensor: Describes the mass distribution in 3D and is used to relate angular velocity to angular momentum.
Principal Axes: A specialized coordinate frame where products of inertia are zero, significantly simplifying the angular momentum and kinetic energy equations.
Euler's Equations: Provide the rotational equations of motion aligned with the principal axes.
Eulerian Angles: Precession, nutation, and spin define the 3D orientation of a rigid body.
Torque-Free Motion: Occurs when the net external moment is zero, resulting in constant angular momentum and constant kinetic energy. Axisymmetric bodies undergo steady precession in this state.
Gyroscopic Motion: Characterized by precession, where an applied torque causes rotation about a perpendicular axis.

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