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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Elementary Physics, Mechanics, and Acoustics
       Rigid-Body Dynamics
          Radius of Gyration
             Circular Cross-Section

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Circular Cross-Section

For a circular cross-section of radius $ a$, Eq.$ \,$(B.11) tells us that the squared radius of gyration about any line passing through the center of the cross-section is given by

\begin{eqnarray*} R_g^2 &=& \frac{1}{\pi a^2} \int_{-a}^a dx \int_{-\sqrt{a^2-x^... ...frac{4a^2}{3\pi} \int_{0}^{\frac{\pi}{2}} \cos^4(\theta)d\theta. \end{eqnarray*}

Using the elementrary trig identity $ \cos(2\theta)=2\cos^2(\theta)-1$, we readily derive

$\displaystyle \cos^4(\theta) = \frac{1}{8}\cos(4\theta) + \frac{1}{2}\cos(2\theta) + \frac{3}{8}. $

The first two terms of this expression contribute zero to the integral from 0 to $ \pi /2$, while the last term contributes $ 3\pi/16$, yielding

$\displaystyle R_g^2 = \frac{4a^2}{3\pi} \frac{3\pi}{16} = \frac{a^2}{4}. $

Thus, the radius of gyration about any midline of a circular cross-section of radius $ a$ is

$\displaystyle R_g = \frac{a}{2}. $

For a circular tube in which the mass of the cross-section lies within a circular annulus having inner radius $ b$ and outer radius $ a$, the radius of gyration is given by

$\displaystyle R_g = \frac{\sqrt{a^2-b^2}}{2}. \protect$ (B.12)

Previous: Rectangular Cross-Section
Next: Two Masses Connected by a Rod

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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