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2013, 2013(special): 105-113. doi: 10.3934/proc.2013.2013.105

New class of exact solutions for the equations of motion of a chain of $n$ rigid bodies

1. 

Department of Mathematics, Engineering, and Computer Science, City University of New York, LaGCC, Long Island City, NY 11101, United States

Received  September 2012 Published  November 2013

In this paper we construct a new class of nonstationary exact solutions for the equations of motion of a classical model of multibody dynamics -- a chain of $n$ heavy rigid bodies that are sequentially coupled by ideal spherical hinges. We establish sufficient conditions for the existence of the solutions and show how the equations of motion can be reduced to quadratures in the case when these conditions are fulfilled.
Citation: Dmitriy Chebanov. New class of exact solutions for the equations of motion of a chain of $n$ rigid bodies. Conference Publications, 2013, 2013 (special) : 105-113. doi: 10.3934/proc.2013.2013.105
References:
[1]

D. Bobylev, On a certain particular solution of the differential equations of rotation of a heavy rigid body about a fixed point, Trudy Otdel. Fiz. Nauk Obsc. Estestvozn., 8 (1896), 21-25.

[2]

A.V. Borisov and I.S. Mamaev, "Dynamics of a rigid body. Hamiltonian methods, integrability, chaos," Institute of Computer Science, Moscow - Izhevsk, 2005.

[3]

D.A. Chebanov, On a generalization of the problem of similar motions of a system of Lagrange gyroscopes, Mekh. Tverd. Tela, 27 (1995), 57-63.

[4]

D.A. Chebanov, New dynamical properties of a system of Lagrange gyroscopes, Proceedings of the Institute of Applied Mathematics and Mechanics, 5 (2000), 172-182.

[5]

D. Chebanov, Exact solutions for motion equations of symmetric gyros system, Multibody Syst. Dyn., 6 (2001), 39-57.

[6]

D.A. Chebanov, A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the systems's skeleton, Mekh. Tverd. Tela, 41 (2011), 244-254.

[7]

L. Euler, Du mouvement de rotation des corps solides autour d'un axe variable, Memoires de l'Academie des Sciences de Berlin, XIV (1765), 154-193.

[8]

G.V. Gorr, Precessional motions in rigid body dynamics and the dynamics of systems of coupled rigid bodies, J. Appl. Math. Mech., 67 (2003), 511-523.

[9]

G.V. Gorr, L.V. Kudryashova, and L.A. Stepanova, "Classical problems in the theory of solid bodies. Their development and current state," Naukova dumka, Kiev, 1978.

[10]

G.V. Gorr and V.N. Rubanovskii, A new class of motions of a system of heavy freely connected rigid bodies, J. Appl. Math. Mech., 52 (1988), 551-555.

[11]

P.V. Kharlamov, The equations of motion of a system of rigid bodies, Mekh. Tverd. Tela, 4 (1972), 52-73.

[12]

P.V. Kharlamov, Some classes of exact solutions of the problem of the motion of a system of Lagrange gyroscopes, Mat. Fiz., 32 (1982), 63-76.

[13]

E.I. Kharlamova, A survey of exact solutions of problems of the motion of systems of coupled rigid bodies, Mekh. Tverd. Tela, 26 (1998), 125-138.

[14]

S.V. Kovalevskaya, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232.

[15]

J.L. Lagrange, "Mechanique Analitique," Veuve Desaint, Paris, 1815.

[16]

E. Leimanis, "The General Problem of the Motion of Coupled Rigid Bodies about Fixed Point," Springer-Verlag, Berlin, 1965.

[17]

L. Lilov and N. Vasileva, Steady motion of a system of Lagrange gyroscopes with a tree structure, Teoret. Prilozhna Mekh., XV (1984), 24-34.

[18]

A.Ya. Savchenko and M.E. Lesina, Particular solution for motion equations of Lagrange gyroscopes system, Mekh. Tverd. Tela, 5 (1973), 27-30.

[19]

N. Sreenath, Y.G. Oh, P.S. Krishnaprasad, and J.E. Marsden, The dynamics of coupled planar rigid bodies. Part I: Reduction, equilibria and stability, Dynam. Stability Systems, 3 (1988), 25-49.

[20]

V.A. Steklov, A certain case of motion of a heavy rigid body having a fixed point, Trudy Otdel. Fiz. Nauk Obsc. Lyubit. Estestvozn., 8 (1896), 19-21.

[21]

J. Wittenburg, "Dynamics of Multibody Systems," Springer-Verlag, Berlin, 2008.

show all references

References:
[1]

D. Bobylev, On a certain particular solution of the differential equations of rotation of a heavy rigid body about a fixed point, Trudy Otdel. Fiz. Nauk Obsc. Estestvozn., 8 (1896), 21-25.

[2]

A.V. Borisov and I.S. Mamaev, "Dynamics of a rigid body. Hamiltonian methods, integrability, chaos," Institute of Computer Science, Moscow - Izhevsk, 2005.

[3]

D.A. Chebanov, On a generalization of the problem of similar motions of a system of Lagrange gyroscopes, Mekh. Tverd. Tela, 27 (1995), 57-63.

[4]

D.A. Chebanov, New dynamical properties of a system of Lagrange gyroscopes, Proceedings of the Institute of Applied Mathematics and Mechanics, 5 (2000), 172-182.

[5]

D. Chebanov, Exact solutions for motion equations of symmetric gyros system, Multibody Syst. Dyn., 6 (2001), 39-57.

[6]

D.A. Chebanov, A new class of nonstationary motions of a system of heavy Lagrange tops with a non-planar configuration of the systems's skeleton, Mekh. Tverd. Tela, 41 (2011), 244-254.

[7]

L. Euler, Du mouvement de rotation des corps solides autour d'un axe variable, Memoires de l'Academie des Sciences de Berlin, XIV (1765), 154-193.

[8]

G.V. Gorr, Precessional motions in rigid body dynamics and the dynamics of systems of coupled rigid bodies, J. Appl. Math. Mech., 67 (2003), 511-523.

[9]

G.V. Gorr, L.V. Kudryashova, and L.A. Stepanova, "Classical problems in the theory of solid bodies. Their development and current state," Naukova dumka, Kiev, 1978.

[10]

G.V. Gorr and V.N. Rubanovskii, A new class of motions of a system of heavy freely connected rigid bodies, J. Appl. Math. Mech., 52 (1988), 551-555.

[11]

P.V. Kharlamov, The equations of motion of a system of rigid bodies, Mekh. Tverd. Tela, 4 (1972), 52-73.

[12]

P.V. Kharlamov, Some classes of exact solutions of the problem of the motion of a system of Lagrange gyroscopes, Mat. Fiz., 32 (1982), 63-76.

[13]

E.I. Kharlamova, A survey of exact solutions of problems of the motion of systems of coupled rigid bodies, Mekh. Tverd. Tela, 26 (1998), 125-138.

[14]

S.V. Kovalevskaya, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232.

[15]

J.L. Lagrange, "Mechanique Analitique," Veuve Desaint, Paris, 1815.

[16]

E. Leimanis, "The General Problem of the Motion of Coupled Rigid Bodies about Fixed Point," Springer-Verlag, Berlin, 1965.

[17]

L. Lilov and N. Vasileva, Steady motion of a system of Lagrange gyroscopes with a tree structure, Teoret. Prilozhna Mekh., XV (1984), 24-34.

[18]

A.Ya. Savchenko and M.E. Lesina, Particular solution for motion equations of Lagrange gyroscopes system, Mekh. Tverd. Tela, 5 (1973), 27-30.

[19]

N. Sreenath, Y.G. Oh, P.S. Krishnaprasad, and J.E. Marsden, The dynamics of coupled planar rigid bodies. Part I: Reduction, equilibria and stability, Dynam. Stability Systems, 3 (1988), 25-49.

[20]

V.A. Steklov, A certain case of motion of a heavy rigid body having a fixed point, Trudy Otdel. Fiz. Nauk Obsc. Lyubit. Estestvozn., 8 (1896), 19-21.

[21]

J. Wittenburg, "Dynamics of Multibody Systems," Springer-Verlag, Berlin, 2008.

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