Three-Dimensional Kinetics of Rigid Bodies

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.

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

Translational Equation of Motion:
∑F=maG\sum \mathbf{F} = m \mathbf{a}_G
Rotational Equation of Motion:
∑MG=H˙G\sum \mathbf{M}_G = \dot{\mathbf{H}}_G
Where:
  • F\mathbf{F} is the external force vector.
  • mm is the mass of the rigid body.
  • aG\mathbf{a}_G is the acceleration vector of the mass center GG.
  • MG\mathbf{M}_G is the external moment vector about GG.
  • HG\mathbf{H}_G is the angular momentum vector about GG.

Angular Momentum and the Inertia Tensor

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

Angular Momentum and Inertia

Angular Momentum Vector:
HG=IGω\mathbf{H}_G = \mathbf{I}_G \mathbf{\omega}
The inertia tensor is a symmetric 3x3 matrix containing the moments and products of inertia:
Inertia Tensor:
IG=[Ixx−Ixy−Ixz−IyxIyy−Iyz−Izx−IzyIzz]\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}

Principal Axes

For any rigid body, there exists a set of mutually orthogonal axes (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 to:
HG=Ixωxi+Iyωyj+Izωzk\mathbf{H}_G = I_x \omega_x \mathbf{i} + I_y \omega_y \mathbf{j} + I_z \omega_z \mathbf{k}

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

∑Mx=Ixω˙x−(Iy−Iz)ωyωz∑My=Iyω˙y−(Iz−Ix)ωzωx∑Mz=Izω˙z−(Ix−Iy)ωxωy\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}

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 (∑MG=0\sum \mathbf{M}_G = 0). This is called torque-free motion. Examples include spacecraft, satellites, and tossed objects (like a football in flight).

Torque-Free Motion Principles

When ∑MG=0\sum \mathbf{M}_G = 0, Euler's equations simplify to:
0=Ixω˙x−(Iy−Iz)ωyωz0 = I_x \dot{\omega}_x - (I_y - I_z) \omega_y \omega_z0=Iyω˙y−(Iz−Ix)ωzωx0 = I_y \dot{\omega}_y - (I_z - I_x) \omega_z \omega_x0=Izω˙z−(Ix−Iy)ωxωy0 = I_z \dot{\omega}_z - (I_x - I_y) \omega_x \omega_y
Because the net moment is zero, two fundamental conservation laws apply to the body during torque-free motion:
  • Conservation of Angular Momentum: The angular momentum vector HG\mathbf{H}_G is constant in both magnitude and direction in space. HG=constant\mathbf{H}_G = \text{constant}
  • Conservation of Kinetic Energy: Since there are no external moments to do work on the body, the rotational kinetic energy TT is also constant. T=12ω⋅HG=constantT = \frac{1}{2} \mathbf{\omega} \cdot \mathbf{H}_G = \text{constant}
Axisymmetric Bodies: For bodies with an axis of symmetry (e.g., a cylinder where Ix=Iy=II_x = I_y = I), the motion simplifies significantly. The body will rotate about its axis of symmetry at a constant rate ωz\omega_z, while simultaneously undergoing regular precession about the fixed angular momentum vector HG\mathbf{H}_G.

Kinetic Energy in 3D

The kinetic energy (TT) 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 12Izω2\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

T=12mvG⋅vG+12ω⋅HGT = \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=12mvG2+12(Ixωx2+Iyωy2+Izωz2)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)

Gyroscopic Motion

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:
M=Ω×H\mathbf{M} = \mathbf{\Omega} \times \mathbf{H}
Where:
  • Ω\mathbf{\Omega} is the precession angular velocity.
  • H\mathbf{H} is the spin angular momentum.

Gyroscopic Motion Simulation

Interact with the simulation below to explore gyroscopic motion.

Gyroscopic Precession

M = Ω × H

Faster spin = slower precession

Angle between vertical (Ω) and spin axis (L)

Calculated Precession Rate (Ω):0.67 rad/s
Key Takeaways
  • Equations of Motion: ∑MG=H˙G\sum \mathbf{M}_G = \dot{\mathbf{H}}_G
  • Euler's Equations: Simplified rotational equations aligned with principal axes.
  • Torque-Free Motion: Occurs when net external moment is zero, resulting in constant angular momentum and constant kinetic energy. Axisymmetric bodies undergo steady precession in this state.
  • 3D kinetics requires the use of the inertia tensor to relate angular velocity and angular momentum.
  • Euler's equations describe the rotational motion in a body-fixed principal axis frame.
  • Gyroscopic motion is characterized by precession, where an applied torque causes rotation about a perpendicular axis.
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