Equations of Motion for Rigid Bodies

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Rigid-Body Dynamics

Newton's Second Law for Rotations

Body-Fixed and Space-Fixed Frames of Reference

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The moment of inertia tensor is a symmetric positive-definite matrix that multiplies an arbitrary angular-velocity vector to — Click for http://www.kwon3d.com/theory/moi/iten.html

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A torque, or moment is an applied force times the distance of that force from an axis of rotation. The force is applied perpendicular to the moment arm, which is the line from the axis of rotation to the point of force application. For example, pushing the end of a wrench applies a torque to a nut. Mathematically, a torque T is be defined as the cross product of the applied force vector F and the moment-arm vector r, i.e., T = r x F. — Click for http://en.wikipedia.org/wiki/Torque

A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Rigid-Body Dynamics

Newton's Second Law for Rotations

Body-Fixed and Space-Fixed Frames of Reference

Equations of Motion for Rigid Bodies

We are now ready to write down the general equations of motion for rigid bodies in terms of 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

(1): the linear velocity $\underline{v}$ of its center of mass, and
(2): 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 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.

Subsections

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Equations of Motion for Rigid Bodies

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University