Rigid Body · aiwiki.page

A rigid body is an idealization in classical mechanics in which the distance between every pair of material points remains constant throughout motion. Its shape and size therefore do not change, although its position and orientation may change. Unlike a point-particle model, a rigid-body model retains the object's spatial extent and mass distribution, making it possible to describe rotation as well as translation. Real objects deform, but the approximation is useful when those deformations are negligible for the motion being studied. (cs.cmu.edu)

Definition and configuration

For material points with positions $\mathbf r_i(t)$ and $\mathbf r_j(t)$ , rigidity requires

\left|\mathbf r_i(t)-\mathbf r_j(t)\right| =\text{constant}

for every pair $i,j$ . The condition concerns distances within the body, not distances from an external observer. A rigid body can therefore move or tumble without violating rigidity. (mitp-content-server.mit.edu)

In three-dimensional Euclidean space, the configuration of a general rigid body has six degrees of freedom: three specifying position and three specifying orientation. Constraints can reduce this number. Fixing one material point leaves three rotational degrees of freedom; restricting the body to rotation about a fixed axis leaves one. Planar rigid-body motion has two translational degrees of freedom and one rotational degree of freedom. (mitp-content-server.mit.edu)

A convenient representation of a material point is

\mathbf r(t)=\mathbf R(t)+Q(t)\boldsymbol{\rho},

where $\mathbf R$ locates a reference point, $\boldsymbol{\rho}$ is the point's constant coordinate vector in a body-fixed frame, and $Q$ is a rotation matrix. The conditions

Q^{\mathsf T}Q=\mathbf 1,\qquad \det Q=1

make $Q$ a proper orthogonal matrix, preserving distances without introducing a reflection. (ocw.mit.edu)

Kinematics: translation and rotation

Rigid-body kinematics describes motion without specifying its causes. All points share the same instantaneous angular velocity vector $\boldsymbol{\omega}$ , but generally have different linear velocities. For any two material points $A$ and $B$ ,

\mathbf v_B =\mathbf v_A+ \boldsymbol{\omega}\times(\mathbf r_B-\mathbf r_A).

Their accelerations satisfy

\mathbf a_B =\mathbf a_A+ \boldsymbol{\alpha}\times(\mathbf r_B-\mathbf r_A) +\boldsymbol{\omega}\times \left[\boldsymbol{\omega}\times(\mathbf r_B-\mathbf r_A)\right],

where $\boldsymbol{\alpha}=d\boldsymbol{\omega}/dt$ . The additional terms describe tangential and centripetal acceleration relative to $A$ . (mitp-content-server.mit.edu)

In pure translation, orientation remains fixed and every point has the same velocity. In rotation about a fixed axis, points move in circles centered on that axis. General motion combines translation and rotation; choosing the center of mass as the reference point is especially useful for dynamics. Orientation can be expressed using rotation matrices or Euler angles. Angular-velocity components are not generally identical to the time derivatives of Euler angles. (mitp-content-server.mit.edu)

Mass distribution and rotational inertia

The total mass determines translational inertia, whereas rotational response depends on how mass is distributed. About an axis, the moment of inertia is

I=\int r_\perp^2\,dm,

where $r_\perp$ is the perpendicular distance from the axis. Mass farther from an axis contributes more strongly because the distance is squared. (ocw.mit.edu)

General three-dimensional rotation requires an inertia tensor. Taking coordinates $\boldsymbol{\rho}$ relative to the center of mass, its components are

I_{ij}=\int \left(\rho^2\delta_{ij}-\rho_i\rho_j\right)\,dm,

where $\delta_{ij}$ is the Kronecker delta. The body's angular momentum about its center of mass is

\mathbf L_C=\mathbf I_C\boldsymbol{\omega}.

Consequently, angular momentum and angular velocity need not be parallel. (ocw.mit.edu)

The inertia tensor is symmetric and can be diagonalized. Its eigenvalues and eigenvectors determine the principal moments of inertia and principal axes. In a principal-axis frame,

\mathbf I_C=\operatorname{diag}(I_1,I_2,I_3).

These values remain constant in a body-fixed frame, although the tensor's components in a space-fixed frame generally change as the body rotates. (ocw.mit.edu)

The total kinetic energy separates into translational and rotational parts:

T=\frac12 M|\mathbf V_C|^2 +\frac12\boldsymbol{\omega}\cdot \mathbf I_C\boldsymbol{\omega}.

This decomposition uses the center of mass; using an arbitrary moving reference point generally introduces a cross term. For parallel axes separated by perpendicular distance $d$ , the parallel-axis theorem gives $I=I_C+Md^2$ , where the first axis passes through the center of mass. (ocw.mit.edu)

Equations of motion

Rigid-body dynamics combines translational and rotational balance. In an inertial reference frame, Newton's laws of motion give

M\mathbf a_C=\sum\mathbf F_{\mathrm{ext}}, \qquad \left(\frac{d\mathbf L_C}{dt}\right)_{\mathrm{inertial}} =\boldsymbol{\tau}_{C,\mathrm{ext}},

where $\mathbf F_{\mathrm{ext}}$ denotes external force and $\boldsymbol{\tau}_{C,\mathrm{ext}}$ is the total external torque about the center of mass. Together these are commonly called the Newton–Euler equations. (ocw.mit.edu)

In a body-fixed principal-axis frame, the rotational equations become

\begin{aligned} I_1\dot\omega_1+(I_3-I_2)\omega_2\omega_3&=\tau_1,\\ I_2\dot\omega_2+(I_1-I_3)\omega_3\omega_1&=\tau_2,\\ I_3\dot\omega_3+(I_2-I_1)\omega_1\omega_2&=\tau_3. \end{aligned}

These Euler equations include the coupling between rotation about different axes. They determine angular-velocity evolution; additional kinematic equations determine orientation. (ocw.mit.edu)

For rotation about a fixed axis, the equation projected along that axis reduces to $\tau_\parallel=I_\parallel\alpha$ . This scalar relation is not a general replacement for the three-dimensional equations. A body in static equilibrium must have both zero net external force and zero net external torque. (ocw.mit.edu)

Characteristic motions

Torque-free rotation. With zero external torque, angular momentum is constant in an inertial frame, but angular velocity need not be. For three distinct principal moments, steady rotation about the axes of greatest and least inertia is stable to small perturbations; rotation about the intermediate axis is unstable. This is the intermediate-axis theorem, also known as the tennis-racket theorem. (ocw.mit.edu)

Rolling. A wheel rolling without slipping on a stationary surface combines translation and rotation. For straight rolling with radius $a$ , the center's speed satisfies $V_C=a|\omega|$ , and the material point instantaneously touching the surface has zero velocity. This does not mean that point has zero acceleration. Friction may provide the force and torque needed to maintain the no-slip condition. (ocw.mit.edu)

Gyroscopic motion. A spinning body subject to torque can exhibit precession, a change in the direction of its rotation axis, and nutation, an oscillation of its inclination. These motions are described through the evolution of angular momentum, rather than by assuming that the axis moves directly toward the applied force. (ocw.mit.edu)

Development and applications

Rigid-body mechanics developed from Newtonian mechanics and the eighteenth-century study of rotation. Leonhard Euler established central results concerning rotational motion, including the equations that bear his name. The subject also became a major application of analytical mechanics, in which orientation and constraints are handled through generalized coordinates. (damtp.cam.ac.uk)

Rigid-body models are used in mechanical dynamics, robotics, and computer animation. Assemblies can be modeled as multiple rigid bodies connected by joints; contact constraints prevent bodies from interpenetrating. Numerical simulation evolves position, orientation, linear momentum, and angular momentum while computing the forces or impulses required by interactions. (cs.cmu.edu)

Rigidity alone does not specify collision behavior. An impact model must supply additional assumptions about restitution, friction, or other contact properties. Thus, a rigid-body simulation can represent either elastic or inelastic impacts without explicitly resolving deformation during contact. (cs.cmu.edu)

Limits of the idealization

A rigid-body model excludes changes of shape, internal vibration, bending, and compression. Its suitability therefore depends on which effects matter for the problem: it can describe overall motion while omitting deformation that would be important for stresses or contact behavior. Flexible-body models are required when those omitted effects must be resolved. (cs.cmu.edu)

Classical rigidity also cannot be transferred unchanged to special relativity. Distances measured simultaneously are frame-dependent, and instantaneous transmission of a mechanical disturbance would conflict with the finite speed of light. Relativistic notions such as Born rigidity instead preserve local distances measured in instantaneous rest frames. They impose restrictions on permitted motion and do not imply an infinitely stiff material. (arxiv.org)

References

Structure and Interpretation of Classical Mechanics: Chapter 2mitp-content-server.mit.edu
09(F14) Chapter 2: Rigid Body Dynamicsocw.mit.edu
09(F14) Advanced Classical Mechanicsocw.mit.edu
Rigid Body Kinematics Instructor Guideocw.mit.edu
Lecture L28 – 3D Rigid Body Dynamics: Equations of Motion; Euler’s Equationsocw.mit.edu
Week 12: Rotations and Translation - Rollingocw.mit.edu
Physically Based Modeling: Principles and Practicecs.cmu.edu
Dynamic simulation of non-penetrating flexible bodiespublications.ri.cmu.edu
Rigid body motion in special relativityarxiv.org