New models in the frame of the dynamics of rigid bodies

We know the role which is played by a usual oscillator in the Newtonian mechanics. In the Eulerian mechanics, the analogous role is played by a rigid body on an elastic foundation. This system can be named a rigid body oscillator. The last one is necessary when constructing the dynamics of multipolar media, but in its general case it is not investigated neither even described in the literature. Of course, its particular cases were considered, for instance, in the analysis of the nuclear magnetic resonance, and also in many applied works, but for infinitesimal angles of rotation. A new statement of the problem of the dynamics of a rigid body on a nonlinear elastic foundation [1, 3, 6] is proposed (1997). The general definition of the potential torque is introduced. Some examples of problem solutions are given.

For the first time (1997) the mathematical statement for the problem of a two-rotor gyrostate on an elastic foundation is given [2, 4, 5]. The elastic foundation is determined by setting of the strain energy as a scalar function of the rotation vector. Finally, the problem is reduced to the integration of a system of nonlinear differential equations having a simple structure but a complex nonlinearity. The difference of these equations from those traditionally used in the dynamics of rigid bodies is that when writing them down it is not necessary to introduce any artificial parameters of the type of Eulerian angles or Cayley-Hamilton parameters. The solutions of concrete problems are considered. A new method of integration of basic equations is described in application to a particular case. The solutions is obtained in quadratures for the isotropic nonlinear elastic foundation.

The model of a rigid body is generalised (2003) for the case of a body consisting not of the mass points, but of the point-bodies of general kind [7]. There was considered a model of a quasi-rigid body, consisting of the rotating particles, with distances between them remaining constant in the process of motion.

  1. Zhilin P.A. Dynamics and stability of equilibrium positions of a rigid body on an elastic foundation // /Proc. of XXIV Summer School - Seminar “Nonlinear Oscillations in Mechanical Systems” St. Petersburg. 1997. P. 90-122. (In Russian).
  2. Zhilin P.A., Sorokin S.A. Multi-rotor gyrostat on a nonlinear elastic foundation // IPME RAS. Preprint N~140. 1997. 83 p. (In Russian).
  3. Zhilin P.A. A General Model of Rigid Body Oscillator // “Nonlinear Oscillations in Mechanical Systems”: Proc. of the XXV-XXIV Summer Schools. Vol. 1. St.-Petersburg. 1998. P. 288-314.
  4. Zhilin P.A., Sorokin S.A. The Motion of Gyrostat on Nonlinear Elastic Foundation // ZAMM. Z. Angew. Math. Mech. 78, (1998), S2. S. 837-838.
  5. Zhilin P.A. Dynamics of the two rotors gyrostat on a nonlinear elastic foundation // ZAMM. Z. angew. Math. Mech. 79, (1999), S2. S. 399-400.
  6. Zhilin P.A. Rigid body oscillator: a general model and some results // Acta Mechanica. Vol. 142. P. 169-193. (2000).
  7. Zhilin P.A. Theoretical mechanics. Fundamental laws of mechanics. Tutorial book. St. Petersburg State Polytechnical University. 2003. 340 p. (In Russian).