We study the apsidal precession of a physical symmetrical pendulum (PSP) (Allais’ precession) as a generalization of the precession corresponding to the ideal spherical pendulum (ISP) (Airy’s precession). Based on the Hamilton–Jacobi formalism and using the techniques of variation of parameters along with the averaging method, we obtain approximate analytical solutions, in terms of which the motion of both systems admits a simple geometrical description. The method developed in this paper is considerably simpler than the standard one in terms of elliptical functions, and the numerical agreement with the exact solutions is excellent. In addition, the present procedure permits to show clearly the origin of the Airy’s and Allais’ precession, as well as the effect of the spin of the physical pendulum on the Allais’ precession. Further, the method could be extended to the study of the asymmetrical pendulum in which an exact analytical solution is not possible anymore.
Study of the Apsidal Precession of the Physical Symmetrical Pendulum
Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 15, 2014; final manuscript received December 20, 2014; accepted manuscript posted December 31, 2014; published online January 13, 2015. Assoc. Editor: Alexander F. Vakakis.
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Maya, H. R., Diaz, R. A., and Herrera, W. J. (February 1, 2015). "Study of the Apsidal Precession of the Physical Symmetrical Pendulum." ASME. J. Appl. Mech. February 2015; 82(2): 021008. https://doi.org/10.1115/1.4029470
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