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Equations of Motion for Rigid Bodies

We are now ready to write down the general equations of motion for rigid bodies in terms of $ f=ma$ for the center of mass and $ \tau=I\alpha$ for the rotation of the body about its center of mass.

As discussed above, it is useful to decompose the motion of a rigid body into

the linear velocity $ \underline{v}$ of its center of mass, and
its angular velocity $ \underline {\omega }$ about its center of mass.

The linear motion is governed by Newton's second law $ \underline{f}=M\dot{\underline{v}}$ , where $ M$ is the total mass, $ \underline{v}$ is the velocity of the center-of-mass, and $ \underline {f}$ is the sum of all external forces on the rigid body. (Equivalently, $ \underline {f}$ is the sum of the radial force components pointing toward or away from the center of mass.) Since this is so straightforward, essentially no harder than dealing with a point mass, we will not consider it further.

The angular motion is governed the rotational version of Newton's second law introduced in §B.4.19:

$\displaystyle \underline{\tau}\eqsp \dot{\underline{L}} \eqsp \mathbf{I}\,\dot{\underline{\omega}} \eqsp \mathbf{I}\,\dot{\underline{\omega}} \protect$ (B.29)

where $ \tau $ is the vector torque defined in Eq.(B.27), $ \underline{L}$ is the angular momentum, $ \mathbf{I}$ is the mass moment of inertia tensor, and $ \underline {\omega }$ is the angular velocity of the rigid body about its center of mass. Note that if the center of mass is moving, we are in a moving coordinate system moving with the center of mass (see next section). We may call $ \underline{L}$ the intrinsic momentum of the rigid body, i.e., that in a coordinate system moving with the center of the mass. We will translate this to the non-moving coordinate system in §B.4.20 below.

The driving torque $ \underline{\tau}$ is given by the resultant moment of the external forces, using Eq.(B.27) for each external force to obtain its contribution to the total moment. In other words, the external moments (tangential forces times moment arms) sum up for the net torque just like the radial force components summed to produce the net driving force on the center of mass.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University