This paper performs a theoretical and experimental investigation of the natural frequency and stability of rocking semicircular, parabolic, and semi-elliptical disks. Horace Lamb's method for deriving the natural frequency of an arbitrary rocking disk is applied to three shapes with semicircular, parabolic, and semi-elliptical cross sections, respectively. For the case of the semicircular disk, the system's equation of motion is derived to verify Lamb's method. Additionally, the rocking semicircular disk is found to always have one stable equilibrium position. For the cases of the parabolic and semi-elliptical disks, this investigation reveals a supercritical pitchfork bifurcation for changes in a single geometric parameter which indicates that the systems can exhibit bistable behavior. Comparisons between experimental validation and theory show good agreement.
Dynamics of Rocking Semicircular, Parabolic, and Semi-Elliptical Disks: Equilibria, Stability, and Natural Frequencies
Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 18, 2014; final manuscript received March 10, 2015; published online April 24, 2015. Assoc. Editor: Guilhem Michon.
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Mazzoleni, M. J., Krone, M. B., and Mann, B. P. (August 1, 2015). "Dynamics of Rocking Semicircular, Parabolic, and Semi-Elliptical Disks: Equilibria, Stability, and Natural Frequencies." ASME. J. Vib. Acoust. August 2015; 137(4): 041017. https://doi.org/10.1115/1.4030169
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