Rigid-Body Dynamics

Below are selected topics from *rigid-body dynamics*, a subtopic
of *classical mechanics* involving the use of Newton's laws of
motion to solve for the motion of *rigid bodies* moving in 1D,
2D, or 3D space.^{B.11} We may think
of a rigid body as a *distributed mass*, that is, a mass that has
length, area, and/or volume rather than occupying only a single point
in space. Rigid body models have application in stiff strings
(modeling them as disks of mass interconnect by ideal springs), rigid
bridges, resonator braces, and so on.

We have already used Newton's
to formulate mathematical dynamic
models for the ideal *point-mass* (§B.1.1), *spring*
(§B.1.3), and a simple mass-spring system (§B.1.4).
Since many physical systems can be modeled as assemblies of masses and
(normally damped) springs, we are pretty far along already. However,
when the springs interconnecting our point-masses are very stiff, we
may approximate them as rigid to simplify our simulations. Thus,
rigid bodies can be considered mass-spring systems in which the
springs are so stiff that they can be treated as rigid massless rods
(infinite spring-constants
, in the notation of §B.1.3).

So, what is new about *distributed* masses, as opposed to the
point-masses considered previously? As we will see, the main new
ingredient is *rotational dynamics*. The *total momentum*
of a rigid body (distributed mass) moving through space will be
described as the *sum* of the *linear momentum* of its
*center of mass* (§B.4.1 below) plus the *angular
momentum* about its center of mass (§B.4.13 below).

- Center of Mass

- Translational Kinetic Energy
- Rotational Kinetic Energy
- Mass Moment of Inertia

- Perpendicular Axis Theorem
- Parallel Axis Theorem
- Stretch Rule
- Area Moment of Inertia
- Radius of Gyration

- Two Masses Connected by a Rod

- Angular Velocity Vector
- Vector Cross Product
- Cross-Product Magnitude
- Mass Moment of Inertia as a Cross Product
- Tangential Velocity as a Cross Product

- Angular Momentum

- Angular Momentum Vector

- Mass Moment of Inertia Tensor

- Principal Axes of Rotation

- Rotational Kinetic Energy Revisited
- Torque
- Newton's Second Law for Rotations
- Equations of Motion for Rigid Bodies

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University