Rigid-Body Dynamics: Motion, Orientation, and Generalized Equations

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Rigid body

An idealized body in which distances between all pairs of points remain constant; its motion is described by translation and rotation.

Point acceleration on a rigid body

It combines the reference-point acceleration with tangential acceleration from angular acceleration and centripetal acceleration from angular velocity.

Center-of-mass equations for planar motion

With the center of mass as reference, the resultant force is $F=MA$ and the torque is $T=I_C\alpha k$.

Euler's equation of rotational motion

Torque equals the inertial-frame rate of change of angular momentum: $\tau=\frac{DL}{Dt}=\frac{dL}{dt}+\omega\times L$.

Lagrange's equations for conservative forces

With $L=T-V$, the equations are $\frac{d}{dt}\frac{\partial L}{\partial\dot q_j}-\frac{\partial L}{\partial q_j}=0$.

Quiz questions

1. Why must a three-dimensional rotation description specify the order of its rotations?Answer: Because rotations about different axes generally do not commute.
2. Why do internal forces cancel when particle equations are summed for a system?Answer: Internal forces occur in equal-and-opposite pairs.
3. What does a rigid body's inertia matrix represent in three dimensions?Answer: How the body's mass distribution affects rotational response, including axis coupling.
4. When torque is perpendicular to angular momentum, what change does it produce?Answer: It changes the direction of angular momentum and can produce precession.
5. In generalized coordinates, what are generalized forces?Answer: The coefficients of the generalized virtual displacements in virtual work.