Reference motion in deformable bodies under rigid body motion and vibration. Part I: theory

This paper examines the motions of reference systems linked to deformable bodies under simultaneously vibration and large translations and rotations. These motions depend on the particular type of linkage between the moving reference system and the deformable body, which is defined by the so-called reference conditions. When using the Rayleigh–Ritz method, the reference conditions also dictate the boundary conditions to be fulfilled by the shape functions used to describe the bodyʹs elasticity. This paper analyses three different types of reference conditions, namely: free linkage, rigid linkage and two-point linkage. It is shown that, moving reference frames only evolve at a constant velocity in the absence of external forces when the free linkage is used. The reference velocities for systems with a free linkage are designated rigid body equivalent velocities for the deformable body here. Such velocities can also be calculated under other types of reference conditions and are usually functions of the elastic and reference co-ordinates, and also of their derivatives. Rigid body equivalent velocities are useful for purposes such as estimating the trajectory of deformable bodies moving freely in space without the need to examine the deformations they undergo. Also, their calculation is required with a view to determining the kinematic restitution coefficient for deformable body collisions, which is dealt within Part II of this series.