In rigid body dynamics, the co-ordinates of three points on the rigid body can be used to completely define the location and orientation of the rigid body co-ordinate system. Similarly, in flexible body dynamics, the body co-ordinate system and the component modes cannot be arbitrarily or independently selected. The shape functions are defined in a co-ordinate system and, therefore, the expression of these functions define the nature of the deformable body co-ordinate system. This note demonstrates that resonance conditions are not absolute in the sense that different resonance frequencies can be obtained for the same system if the deformation is defined in different co-ordinate systems that can have arbitrary rigid body displacements. As a consequence, speaking of the resonance phenomenon must be associated with the selection of the co-ordinate system of the elastic body. Only geometric conditions are required to define the co-ordinate system and, as such, natural force conditons are of much less significance in defining the resonance conditions. The analysis presented in this note clearly explains why two different sets of mode shapes and two different sets of co-ordinate systems can be used to obtain approximately the same displacement solutions in multi-body simulations. In view of the analysis presented in this note, the fundamental relationship between the selection of the component modes and the deformable body co-ordinate system is discussed, and it is demonstrated that a proper selection of the deformable body co-ordinate system not only leads to a consistent formulation, but also leads to a proper definition of the resonance conditions.
In most existing rigid multi-body codes, a centroidal body fixed co-ordinate system is used in order to simplify the inertia matrix and the centrifugal forces. In flexible multi-body dynamics, body fixed or floating co-ordinate systems are often used to define the deformation of the elastic bodies. Conventional mode shapes as defined in standard vibration texts [10][11][12] are described in co-ordinate systems defined by the geometric boundary conditions. For example, cantilever mode shapes of a beam are defined in a body fixed co-ordinate system the origin of which is rigidly attached to one of the beam ends. Free-free modes, on the other hand, define a floating frame of reference the origin of which is not rigidly attached to a material point on the elastic beam. The relationship between the selected modes of vibration and the co-ordinate system is fundamental in multi-body dynamics, since improper selection of the co-ordinate system and the associated modes of deformation not only leads to inconsistent formulation and numerical problems, but also leads to the wrong resonance conditions.
The mode shapes and the co-ordinate systems of deformable bodies that undergo large rigid body displacements cannot be selected arbitrarily or independently. For instance, the mode shapes obtained for a chassis of a vehicle using free-free boundary conditions do not define the deformation in a body fixed co-ordinate system. In principle, however, any