Solutions of Euler’s Dynamic Equations for the Motion of a Rigid Body
Publication: Journal of Aerospace Engineering
Volume 30, Issue 4
Abstract
The aim of this work is to investigate the analytical solutions for the equations of motion of a rigid body about a fixed point through the process of decoupling Euler’s dynamic equations. This body is acted upon by a gyrostatic torque about the axes of rotation, and in the presence of a moment about the same axes, it depends on an external loading in which its components have been expressed as a harmonic function of time. The achieved analytical solutions for the equations of motion are obtained under conditions consistent with the physical nature of the body, and the uniqueness of the solution is proved. Some new theoretical applications are presented when the body is symmetric about one of its principal axes and when the body is in complete symmetry. The graphical representations for the motion of the body are represented to show the effectiveness of the physical parameters of the body. Moreover, the phase plane plots are given to ensure that the considered motion is free of chaos.
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References
Akulenko, L. D., Kozachenko, T. D., and Leshchenko, D. D. (2001). “Perturbed rotary motions of the solid body under the action of the non-stationary restoring moment, depending on the nutation angle.” J. Mekh. Tverd. Tela, 31, 57–62.
Amer, T. S. (2008). “On the motion of a gyrostat similar to Lagrange’s gyroscope under the influence of a gyrostatic moment vector.” Nonlinear Dyn., 54(3), 249–262.
Calvo, M., Franco, J. M., Montijano, J. I., and Randez, L. (2011). “A note on some exact analytical solutions of the rotation of a rigid body with a external torque.” Monografias de la Real Academia de Ciencias de Zaragoza, 35, 27–39.
Elfimov, V. S. (1978). “Existence of periodic solutions of equations of motion of a solid body similar to the Lagrange gyroscope.” J. Appl. Math. Mech., 42(2), 262–269.
Gilat, A. (2013). Numerical methods for engineers and scientists, Wiley, Hoboken, NJ.
Gorr, G. V., and Kovalev, A. M. (2013). The motion of a gyrostat, Naukova Dumka, Kiev, Ukraine.
Gradshteyn, I. S., and Ryzhik, I. M. (1965). Table of integrals, series and products, 4th Ed., Academic Press, New York.
Ismail, A. I., Amer, T. S., and Shaker, M. O. (1998). “Perturbed motions of a rotating symmetric gyrostat.” Eng. Trans., 46(3–4), 271–289.
Ismail, A. I., Sperling, L., and Amer, T. S. (2000). “On the existence of periodic solutions of a gyrostat similar to Lagrange’s gyroscope.” Technishe Mechanik, 20(4), 295–304.
Kushpil, T. A., Leshchenko, D. D., and Timoshenko, I. A. (2000). “Some problems of evolution of rotations of a rigid body under the action of perturbed elements.” J. Mekh. Tverd. Tela, 30, 119–125.
Leimanis, E. (1965). The general problem of the motion of coupled rigid bodies about a fixed point, Springer, New York.
Leshchenko, D. D. (1999). “On the evolution of rigid body rotations.” Int. Appl. Mech., 35(1), 93–99.
Leshchenko, D. D., and Sallam, S. N. (1990). “Perturbed rotational motions of a rigid body similar to regular precession.” J. Appl. Math. Mech., 54(2), 183–190.
Magnus, K. K. (1971). Theorie und Anvendungen, Springer, New York.
Mark, L. G., and Phillip, A. G. (2000). “Abel’s differential equations.” Houston J. Math., 28(2), 329–351.
Merkin, D. R., and Smolnikov, B. A. (2003). “Applied problems of rigid body dynamics.” St. Petersburg Univ., St. Petersburg, Russia.
Nickalls, R. W. (1993). “A new approach to solving the cubic: Cardan’s solution revealed.” Math. Gazette, 77(480), 354–359.
Panayotounakos, D. E., Rizou, I., and Theotokoglou, E. E. (2011). “A new mathematical construction of the general nonlinear ODEs of motion in rigid body dynamics (Euler’s equations).” Appl. Math. Comput., 217(21), 8534–8542.
Panayotounakos, D. E., and Theocaris, P. S. (1990). “On the decoupling and the solutions of the Euler dynamic equations governing the motion of a gyro.” ZAMM, 70(11), 489–500.
Romano, M. (2008a). “Exact analytic solution for the rotation of a rigid body having spherical ellipsoid of inertia and subjected to a constant torque.” Celest. Mech. Dyn. Astr., 100(3), 181–189.
Romano, M. (2008b). “Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque.” Celest. Mech. Dyn. Astr., 101(4), 375–390.
Simon, J. A. (2014). “An introduction to Lagrangian and Hamiltonian mechanics.” Heriot-Watt Univ., Edinburgh, U.K.
Smol’nikov, B. A., and Stepanova, M. V. (1970). “The motion of a nonsymmetric self-exciting gyrostat.” J. Appl. Math. Mech., 34(4), 567–574.
Tsiotras, P., and Longuski, J. M. (1991). “A complex analytical solution for the attitude motion of a near-symmetric rigid body under body-fixed torques.” Celest. Mech. Dyn. Astr., 51(3), 281–301.
Tsiotras, P., and Longuski, J. M. (1996). “Analytical solution of Euler’s equations of motion for an asymmetric rigid body.” J. Appl. Mech, 63(1), 149–155.
Wittenburg, J. (1977). Dynamics of systems of rigid bodies, B. G. Teubner, Stuttgart, Germany.
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©2017 American Society of Civil Engineers.
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Received: Apr 15, 2016
Accepted: Jan 12, 2017
Published online: Mar 15, 2017
Published in print: Jul 1, 2017
Discussion open until: Aug 15, 2017
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