Research Papers

Study of the Apsidal Precession of the Physical Symmetrical Pendulum

[+] Author and Article Information
Héctor R. Maya

Departamento de Física,
Universidad de Córdoba,
Montería, Colombia;
Departamento de Física,
Universidad Nacional de Colombia,
Bogotá, Colombia
e-mail: hrmaya@hotmail.com

Rodolfo A. Diaz

Departamento de Física,
Universidad Nacional de Colombia,
Bogotá, Colombia
e-mail: radiazs@unal.edu.co

William J. Herrera

Departamento de Física,
Universidad Nacional de Colombia,
Bogotá, Colombia
e-mail: jherreraw@unal.edu.co

From the initial conditions that we shall use, we have θ¯ ≲ 10−1 and for the dimensionless momenta Pa ≲ 10–3. Hence, it is logical to consider H1 as a perturbation with respect to H0.

This is a reasonable ansatz since those variables are constant when we use the nonperturbed Hamiltonian H0. The time evolution of these variables arises from the introduction of the (much smaller) Hamiltonian H1.

In addition, it can be seen from Eqs. (A2a) and (A2b) that only the functions cosϕ and sinϕ are important but not ϕ by itself, and such functions are smooth.

We are using the convention of positive direction of the Z-axis in the direction of the gravitational field (i.e., downward).

1In the most usual scenario of the symmetrical top, the spin is the dominant part of its motion.

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.

J. Appl. Mech 82(2), 021008 (Feb 01, 2015) (12 pages) Paper No: JAM-14-1313; doi: 10.1115/1.4029470 History: Received July 15, 2014; Revised December 20, 2014; Accepted December 31, 2014; Online January 13, 2015

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.

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Drumheller, D. S., 2012, “Barometric Compensation of a Pendulum,” ASME J. Appl. Mech., 79(6), p. 061009. [CrossRef]
Schmidt, B. A., and McDowell, D. G., 1992, “Analysis of a Pendulum on a Rotating Arm,” ASME J. Appl. Mech., 59(1), pp. 233–234. [CrossRef]
Rivas-Cambero, I., and Sausedo-Solorio, J. M., 2012, “Dynamics of the Shift in Resonance Frequency in a Triple Pendulum,” Meccanica, 47(4), pp. 835–844. [CrossRef]
Mehidi, N., 2007, “Averaging and Periodic Solutions in the Plane and Parametrically Excited Pendulum,” Meccanica42(4), pp. 403–407. [CrossRef]
Sado, D., and Gajos, K., 2003, “Note on Chaos in Three Degree of Freedom Dynamical System With Double Pendulum,” Meccanica, 38(6), pp. 719–729. [CrossRef]
Dragani, R., and Sarra, M., 1983, “The Centrifugal Pendulum as a Nonlinear Torsional Vibration Absorber,” Meccanica, 18(2), pp. 67–70. [CrossRef]
Sanyal, A. K., and Goswami, A., 2013, “Dynamics and Balance Control of the Reaction Mass Pendulum: A Three-Dimensional Multibody Pendulum With Variable Body Inertia,” ASME J. Dyn. Syst. Meas. Control, 136(2), p. 021002. [CrossRef]
Masoud, Z. N., and Alhazza, K. A., 2013, “Frequency-Modulation Input Shaping Control of Double-Pendulum Overhead Cranes,” ASME J. Dyn. Syst. Meas. Control, 136(2), p. 021005. [CrossRef]
Lima, F. M. S., 2010, “Analytical Study of the Critical Behavior of the Nonlinear Pendulum,” Am. J. Phys.78(11), pp. 1146–1151. [CrossRef]
Giltinan, D. A., Wagner, D. L., and Walkiewicz, T. A., 1996, “The Physical Pendulum on a Cylindrical Support,” Am. J. Phys.64(2), pp. 144–146. [CrossRef]
Olsson, M. G., 1981, “Spherical Pendulum Revisited,” Am. J. Phys.49(6), pp. 531–534. [CrossRef]
Ø. Grøn, 1983, “A Tidal Force Pendulum,” Am. J. Phys.51(5), pp. 429–431. [CrossRef]
Livesey, D. L., 1987, “The Precession of Simple Pendulum Orbits,” Am. J. Phys.55(7), pp. 618–621. [CrossRef]
Valéry Roy, R., 1994, “Averaging Method for Strongly Non-Linear Oscillators With Periodic Excitations,” Int. J. Non-Linear Mech., 29(5), pp. 737–753. [CrossRef]
Ben Lemlih, A., and Ellison, J. A., 1985, “Method of Averaging and the Quantum Anharmonic Oscillator,” Phys. Rev. Lett.55(19), pp. 1950–1953. [CrossRef] [PubMed]
Lagrange, J. L., 1788, Méchanicque Analitique, Veuve Desaint, Paris.
Poisson, S. D., 1813, “Sur un cas particulier du mouvement de rotation des corps pesans,” J. I’École Polytech., 9(16), pp. 247–267.
Golubev, V. V., 1960, Lectures on Integration of Equations of Motion of a Rigid Body About a Fixed Point, Israeli Program for Scientific Translations, Israel.
Leimanis, E., 1965, The General Problem of the Motion of Coupled Rigid Bodies About a Fixed Point (Springer Tracts in Natural Philosophy, Vol. 7), Springer-Verlag, Berlin, Germany.
Johansen, K. F., and Kane, T. R., 1969, “A Simple Description of the Motion of a Spherical Pendulum,” ASME J. Appl. Mech., 36(3), pp. 76–82. [CrossRef]
Miles, J. W., 1962, “Stability of Forced Oscillations of a Spherical Pendulum,” Q. Appl. Mater., 20, pp. 21–32.
Hemp, G. W., and Sethna, P. R., 1964, “The Effect of High-Frequency Support Oscillation on the Motion of a Spherical Pendulum,” ASME J. Appl. Mech., 31(2), pp. 351–354. [CrossRef]
Airy, G. B., 1851, “On the Vibration of a Free Pendulum in an Oval Differing Little From a Straight Line,” R. Astron. Soc.XX, pp. 121–130.
Olsson, M. G., 1978, “The Precessing Spherical Pendulum,” Am. J. Phys., 46(11), pp. 1118–1119. [CrossRef]
Synge, J., and Griffith, B., 1959, Principles of Mechanics, McGraw-Hill, New York, 1959.
Gusev, A. V., and Vinogradov, M. P., 1993, “Angular Velocity of Rotation of the Swing Plane of a Spherical Pendulum With Anisotropic Suspension,” Meas. Tech., 36(10), pp. 1078–1082. [CrossRef]
Goldstein, H., Poole, C., and Safko, J., 2002, Classical Mechanics, 3rd ed., Addison-Wesley, New York.
Allais, M., 1999, “The ‘Allais Effect’ and My Experiments With the Paraconical Pendulum 1954–1960,” (A memoir prepared for NASA), available at: http://www.allais.info/alltrans/nasareport.pdf
Goodey, T. J., Pugash, A. F., and Olenici, D., 2010, “Correlated Anomalous Effects Observed During the August 1st 2008 Solar Eclipse,” J. Adv. Res. Phys., 1(2), p. 021007.
Saxl, E. J., and Allen, M., 1971, “1970 Solar Eclipse as Seen by a Torsion Pendulum,” Phys. Rev. D, 3(4), pp. 823–825. [CrossRef]
Bogolyubov, N., and Mitropolsky, Y., 1961, Asymptotic Methods in Theory on Nonlinear Oscillations, Gordon and Breach, New York.
Brizard, A. J., 2007, “A Primer on Elliptic Functions With Applications in Classical Mechanics,” e-print arXiv:0711.4064.
Lima, F. M. S., 2008, “Simple Log Formulae for Pendulum Motion Valid for Any Amplitude,” Eur. J. Phys., 29(5), pp. 1091–1098. [CrossRef]


Grahic Jump Location
Fig. 1

Inertial system XYZ and the spherical pendulum. The angular coordinates θ and ϕ are shown, as well as the weight and tension.

Grahic Jump Location
Fig. 2

(a) Graphical form of the approximate and exact solutions (dashed and solid lines, respectively) of the ISP for θ with 300 ≤ τ ≤ 310. These graphics cannot be distinguished from each other. (b) Graphical form of the approximate values of the angle ϕ coming from Eq. (40) and the exact values (dashed and solid lines, respectively), for 500 ≤ τ ≤ 510. The approximate and exact solutions are superposed. In addition, the straight line corresponds to the linear approximation given by Eq. (43).

Grahic Jump Location
Fig. 3

Projection of the trajectory of the ISP on the XY plane for 30nT ≤ t ≤ (30n + 1)T with n = 0, 1, and 2, using the approximate solution (47) and the exact one (dashed and solid lines, respectively). The approximate and exact solutions cannot be distinguished.

Grahic Jump Location
Fig. 4

PSP hung on an edge. θ, Φ, and ψ are the Euler angles used in the description of the motion. The spherical coordinate ϕ is shown as well as its relation with the Euler angle Φ.

Grahic Jump Location
Fig. 5

Plot of the approximate solution (88) and the exact one (dashed and solid lines, respectively) for the coordinate of nutation θ¯(sinθ) of the PSP as a function of time, for 500 ≤ t ≤ 504 s. The exact solution is superposed to the approximate one.

Grahic Jump Location
Fig. 6

(a) Approximate and exact solutions (dashed and solid lines, respectively) for the azimuthal angle Φ of the PSP, for 500 ≤ t ≤ 504 s, (b) approximate and exact solutions (dashed and solid lines, respectively) for the spin angle ψ of the PSP, for 500 ≤ t ≤ 504 s. The straight lines correspond to the linear approximations for each one of these angles. The exact solutions cannot be distinguished from the approximate ones.

Grahic Jump Location
Fig. 7

Projections of the trajectory of the CM of the PSP on the XY plane, obtained with the approximate and exact solutions (dashed and solid lines, respectively), for time intervals of a period T, (a) for 0 ≤ t ≤ T: (T). (b) For 29 T ≤ t ≤ 30 T: (30 T). (c) for 59 T ≤ t ≤ 60 T: (60 T). In all cases, the exact solutions are superposed to the approximate ones.




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