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Elena A. Ivanova St. Petersburg State Polytechnical University | |
Rigid body dynamics
The lecture course is based on the monography:
Zhilin P.A. Theoretical mechanics. Fundamental laws of mechanics. St. Petersburg: SPbGPU. 2003. 340 p. (In Russian)
The course is a presentation of the rigid body dynamics, focused on the mechanics of continuous and discrete media, as well as the dynamics of the discrete-continuous systems. The lecture course provides an overview of the methods of rigid body dynamics and the areas of science and engineering practice where there are the problems associated with rigid body dynamics. The course includes the tensor methods for describing the spin (rotational) motions, which are developed by P.A.Zhilin, and methods for deriving the equations of motion of rigid body and system of bodies, finding discontinuous solutions and their interpretation, integrating the equations of small oscillations of rigid bodies, finding the equilibrium positions and stationary movements of rigid bodies and analyzing their stability. The advantage of tensor methods is that they can be used both to solve the rigid body dynamics problems and to solve the problems of joint dynamics of rigid bodies and solids. Such problems take place in engineering practice when there are rapid rotations and high-frequency oscillations (e.g. the problems of ultracentrifuge dynamics). In addition, the study of tensor methods of rigid body dynamics considerably simplifies the study of continuum models with rotational degrees of freedom: the rod theory, the shell theory, and a number of three-dimensional models based on the Cosserat continuum. The tensor methods of rigid body dynamics are useful for describing discrete media, if such media consist of body-points (rigid bodies occupying the zero volume). Such models are used, for example, when describing complex crystal lattices. The crystal lattices are modeled by using both the discrete media and continua. If we want to compare these models, it is important to use the same mathematical formalism for the description of discrete and continuous models.
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