THE COMPLICATED DYNAMICS OF HEAVY RIGID BODIES ATTACHED TO DEFORMABLERODS

We study the motion in space of light nonlinearly elastic and viscoela stic rods with heavy rigid attachments. The rods, which can suffer fle xure, extension, torsion, and shear, are described by a general geomet rically exact theory. We pay particular attention to the leading term of the asymptotic expansion of the governing equations as the inertia of the rod goes to zero. When the rods are elastic and weightless, and when they have appropriate initial conditions, they move irregularly through a family of equilibrium states parametrized by time; the motio n of the rigid body is governed by an interesting family of multivalue d ordinary differential equations. These ordinary differential equatio ns for a heavy mass point attached to an elastica undergoing planar mo tion are explicitly treated. These problems illuminate such phenomena as snap-buckling. On the other hand, when the rods are viscoelastic an d weightless, the rigid body is typically not governed by ordinary dif ferential equations, but, as we show, the motion of the system is well -defined for arbitrary initial conditions. This analysis relies critic ally on the careful use of our properly invariant constitutive hypothe ses.