Dynamics of Gyroelastic Continua

Abstract

This work is concerned with the theoretical development of dynamic equations for gyroelastic systems which are dynamic systems with four basic types of continuous mechanical influences, i.e. inertia, elasticity, damping, and gyricity or stored angular momentum. Assuming unrestricted or large attitude changes for the axes of the gyros and utilizing two different theories of elasticity, i.e. the classical and micropolar theories of elasticity, the energy expressions and equations of motion for the undamped classical and micropolar gyroelastic continua are derived. Whereas the micropolar gyroelastic continuum model with extra coefficients and degrees of freedom is primarily developed to account for the asymmetric elasticity, it also proves itself to be more comprehensive in describing the actual gyroscopic system or structure.

The dynamic equations of the general three-dimensional gyroelastic continua are reduced to the case of a one-dimensional gyroelastic continua in the three-dimensional space, i.e. three-dimensional gyrobeams. Two different gyrobeam models are developed, one based on the classical beam torsion and bending theories and one based on the simplified micropolar beam torsion and bending theories. Finite element models corresponding to the classical and micropolar gyrobeams are built in MATLAB and used for numerical analysis.

The classical and micropolar gyrobeam models are analyzed and compared, against the earlier gyrobeam models developed by other authors and also against each other, through numerical examples. It is shown that there are significant differences between the developed unrestricted classical gyrobeam model and the previously derived zero-order restricted classical gyrobeam models. These differences are more pronounced in the shorter beams and for the transverse gyricity case. The results also indicate that the unrestricted classical and micropolar gyrobeam models behave very diversely in a wide range of micropolar elastic constants even where the classical and micropolar elasticity models coincide.

As a foundation for development of the above-mentioned theories, the correct approach for simplification of the micropolar elasticity to the classical elasticity, the simple torsion and bending theories for micropolar beams, and the correct approximation of infinitesimal rotations or microrotations are derived and presented.