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Research Papers

Dynamics of Rocking Semicircular, Parabolic, and Semi-Elliptical Disks: Equilibria, Stability, and Natural Frequencies

[+] Author and Article Information
Michael J. Mazzoleni

Dynamical Systems Laboratory,
Department of Mechanical Engineering
and Materials Science,
Duke University,
Durham, NC 27708
e-mail: michael.mazzoleni@duke.edu

Michael B. Krone

Dynamical Systems Laboratory,
Department of Mechanical Engineering
and Materials Science,
Duke University,
Durham, NC 27708
e-mail: michael.krone@duke.edu

Brian P. Mann

Dynamical Systems Laboratory,
Department of Mechanical Engineering
and Materials Science,
Duke University,
Durham, NC 27708
e-mail: brian.mann@duke.edu

1Corresponding author.

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.

J. Vib. Acoust 137(4), 041017 (Aug 01, 2015) (7 pages) Paper No: VIB-14-1180; doi: 10.1115/1.4030169 History: Received May 18, 2014; Revised March 10, 2015; Online April 24, 2015

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.

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References

Karnopp, D. C., Margolis, D. L., and Rosenberg, R. C., 2012, System Dynamics, Wiley, Hoboken, NJ.
Lamb, H., 1914, Dynamics, Cambridge University Press, Cambridge.
Satterly, J., 1950, “Some Experiments in Dynamics, Chiefly on Vibrations,” Am. J. Phys., 18(7), pp. 405–416. [CrossRef]
Balachandran, B., and Magrab, E. B., 2009, Vibrations, Cengage Learning, Toronto.
Stanton, S. C., McGehee, C. C., and Mann, B. P., 2010, “Nonlinear Dynamics for Broadband Energy Harvesting: Investigation of a Bistable Piezoelectric Inertial Generator,” Phys. D: Nonlinear Phenom., 239(10), pp. 640–653. [CrossRef]
Norton, R. L., 2012, Design of Machinery, McGraw-Hill, New York.
Taylan, M., 2007, “On the Parametric Resonance of Container Ships,” Ocean Eng., 34(7), pp. 1021–1027. [CrossRef]
Francescutto, A., and Contento, G., 1999, “Bifurcations in Ship Rolling: Experimental Results and Parameter Identification Technique,” Ocean Eng., 26(11), pp. 1095–1123. [CrossRef]
Mitra, S., Hai, L. V., Jing, L., and Khoo, B., 2012, “A Fully Coupled Ship Motion and Sloshing Analysis in Various Container Geometries,” J. Mar. Sci. Technol., 17(2), pp. 139–153. [CrossRef]
Mazzoleni, M. J., Krone, M. B., and Mann, B. P., 2014, “Investigation of Rocking Semicircular and Parabolic Disk Equilibria, Stability, and Natural Frequencies,” ASME Paper No. DETC2014–34396. [CrossRef]
Sah, S. M., and Mann, B. P., 2012, “Potential Well Metamorphosis of a Pivoting Fluid-Filled Container,” Physica D, 241(19), pp. 1660–1669. [CrossRef]
Nayfeh, A. H., and Mook, D. T., 1995, Nonlinear Oscillations, Wiley, New York.
Myers, J. A., 1962, “Handbook of Equations for Mass and Area Properties of Various Geometric Shapes,” U.S. Naval Ordinance Test Station, China Lake, CA, Technical Report No. 7827.
Strogatz, S. H., 2001, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview, Cambridge, MA.
Nayfeh, A. H., and Balachandran, B., 1995, Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, Wiley, New York.

Figures

Grahic Jump Location
Fig. 1

Schematic of a rocking semicircular disk showing both the upright and displaced position of the object. The position vector to the mass center can be written as rG=rA+rG/A.

Grahic Jump Location
Fig. 2

Plot showing the theoretical natural frequency of a rocking semicircular disk as a function of the radius

Grahic Jump Location
Fig. 3

Schematic of a rocking disk with uniform density and an arbitrary symmetric cross section defined by the surface r=r(φ)

Grahic Jump Location
Fig. 4

Schematic of a rocking parabolic disk showing both the upright and displaced position of the object. The instantaneous contact point on the surface of the parabola is identified by (x0, y0).

Grahic Jump Location
Fig. 5

Plot showing the theoretical natural frequency trends for rocking semi-elliptical and parabolic disks defined by the parameters a and b. The natural frequencies of the semi-elliptical and parabolic disks are represented by solid lines and dashed lines, respectively.

Grahic Jump Location
Fig. 6

Bifurcation diagrams showing the stable (solid lines) and unstable (dashed lines) equilibria as a function of the dimensionless ratio a/b for the (a) rocking parabolic and (b) rocking semi-elliptical disks. Both systems are found to exhibit a supercritical pitchfork bifurcation.

Grahic Jump Location
Fig. 7

Schematic of a rocking semi-elliptical disk showing both the upright and displaced position of the object. The instantaneous contact point on the surface of the semi-ellipse is identified by ψ0.

Grahic Jump Location
Fig. 8

(a) Photograph showing the semicircular disks used for experimental validation. The disks were tagged with a black/white marker on their center. (b) Schematic of the experimental setup. The laser tachometer was able to measure the period of the oscillations as it detected the change from black to white while the disks oscillated. (c) Plot providing an example of the raw data collected by the laser tachometer. When the laser tachometer detected the black portion of the disk, it provided a voltage output, and when it detected the white portion of the disk, it provided no voltage output.

Grahic Jump Location
Fig. 9

Plots comparing the experimentally measured and theoretically predicted natural frequencies for the (a) rocking semicircular disk and (b) rocking semi-elliptical and parabolic disks. The theoretical natural frequencies of the semi-elliptical and parabolic disks are represented by solid lines and dashed lines, respectively. For each of the subplots, experimentally measured data points are represented by dots.

Grahic Jump Location
Fig. 10

Bifurcation diagrams showing the stable (solid lines) and unstable (dashed lines) equilibria as a function of the dimensionless ratio a/b for the (a) rocking parabolic and (b) rocking semi-elliptical disks. Both systems are found to exhibit a supercritical pitchfork bifurcation. For each of the subplots, experimentally measured data points are represented by dots.

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