Rectangular Cross-Section

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Physical Audio Signal Processing

Radius of Gyration

Circular Cross-Section

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For a planar distribution of mass rotating about some axis in the plane of the mass, the radius of gyration is the distance from the axis that all mass can be concentrated to obtain the same mass moment of inertia. — Click for http://en.wikipedia.org/wiki/Radius_of_gyration

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For a planar region rotating about some axis in the plane, the area moment of inertia (or second moment of area) is given by the integral over the area of distance-squared from the axis of rotation. Thus, the area moment of inertia can be interpreted as (mass) moment of inertia in which mass is replaced by area. In particular, if an area has uniform mass density d, then the moment of inertia is given by d times the area moment of inertia. — Click for http://en.wikipedia.org/wiki/Second_moment_of_area

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Physical Audio Signal Processing

Radius of Gyration

Circular Cross-Section

Rectangular Cross-Section

For a rectangular cross-section of height and width , area , the area moment of inertia about the horizontal midline is given by

$\displaystyle I_w = w\int_{-h/2}^{h/2} y^2 dy = w\left.\frac{1}{3}y^3\right\vert _{-h/2}^{h/2} = \frac{Sh^2}{12}.$

The radius of gyration about this axis is then

$\displaystyle R_g = \sqrt{\frac{I_w}{S}} = \sqrt{\frac{h^2}{12}} = \frac{h}{2\sqrt{3}}.$

Similarly, the radius of gyration about a vertical axis passing through the center of the cross-section is $R_g=w/(2\sqrt{3})$ .

The radius of gyration can be thought of as the ``effective radius'' of the mass distribution with respect to its inertial response to rotation (``gyration'') about the chosen axis.

Most cross-sectional shapes (e.g., rectangular), have at least two radii of gyration. A circular cross-section has only one, and its radius of gyration is equal to half its radius, as shown in the next section.

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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

Rectangular Cross-Section

``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