Zhilin P.A. “Rigid body oscillator: a general model and some results” // Acta Mechanica. Vol. 142. P. 169-193. (2000).

Rigid body oscillator

The nonlinear (linear) oscillator is the most important model of classical physics. An investigation of many physical phenomena and a development of many methods of nonlinear mechanics had arisen due to this model. At the same time, the necessity of construction of models with new properties was recognized. Especially, it was important in quantum mechanics where many authors pointed out that a new model must be something like a rigid body on an elastic foundation. However, such a model was not created up to now. At the present time, two huge branches of mechanics, i.e. continuum mechanics and rigid body dynamics, are existing without close contacts. While, maybe, rigid body dynamics does not need in methods of continuum mechanics, the same can not be said with respect to continuum mechanics. This is clear from the end of the last century. The theories of rods and shells, the theory of Cosserat continuum, the theory of liquid crystals, the theory of ferromagnetic media, and other theories involve ideas from rigid body dynamics. In the theory of liquid crystals, each point of the medium is a rigid body. In the theory of multi-polar continuum, each point is a gyrostat with many rotors inside. Thus, it is clear that the theory of multi-polar continuum can not be constructed without basic ideas of rigid body dynamics. In linear theories, there is no problem. In this case, continuum mechanics and rigid body dynamics use the same language. However, rotations of particles of media are not small in many cases. Therefore, we have to use nonlinear dynamics. In nonlinear theories, the difference between methods of rigid body dynamics and continuum mechanics is essential. Rigid body dynamics uses matrix methods or quaternion methods which are not suited for aims of continuum mechanics. As a matter of fact, the only language which can be used in continuum mechanics is the tensor calculus. Thus, if we are going to apply the methods of rigid body dynamics to continuum mechanics, it is necessary to describe rigid body dynamics in terms of tensors.

A rigid body on an elastic foundation will be called the rigid body oscillator in the following. A general model of such an object can be used in many cases, e.g. in mechanics of multi-polar continuum. For the construction of the model, three new elements are required: the turn-vector, the integrating tensor, and the potential moment. Let us briefly discuss these concepts.

An unusual situation takes place with the turn-vector. On the one side, the wellknown Euler theorem proves that any turn of the body can be realized as the turn around an unit vector n by an angle θ: Thus, the turn can be described by a vector θ = θn. This fact can be found in many works on mechanics. On the other side, the same works claim that the vector θn is not a vector, and a description of a turn in terms of a vector is impossible. Maybe by this reason, the turn-vector has not found great acceptance in conventional rigid body dynamics. However, namely the turn-vector plays a major role in dynamics of a rigid body on an elastic foundation. In classical mechanics, the linear differential form vdt is the total differential of the vector of position, i.e. vdt = dR: This is not true for rotations. If the vector ω is a vector of angular velocity, then the linear differential form ω dt is not a total differential of the turn-vector. However, it can be proved that there exists an integrating tensor Z that transforms the linear differential form ω dt into the total differential dθ of the turn-vector θ. The integrating tensor Z plays the decisive role for an introduction of a potential moment which expresses an action of the elastic foundation on the rigid body. Thus, it is an essential element of a general model of a rigid body oscillator. The basic equations of dynamics of a rigid body oscillator contain a strong nonlinearity, but their form is rather simple. These equations give a very interesting object for applying methods of nonlinear mechanics. In the paper, some simple examples are considered. In particular, the basic equations are integrated explicitly in the case of the simplest model.