The Reverse Spaghetti Problem: Drooping Motion of an Elastica Issuing from a Horizontal Guide

The nonlinear equations of motion of an elastica that moves out of a horizontal guide at a constant velocity are expressed in terms a dimensionless weight-to-stiffness ratio and a dimensionless velocity. The equations are written in horizontal-vertical directions rather than tangential-normal directions to minimize algebraic complexities. The introduction of deformation potentials allows each of the linear momentum equations to be integrated once. This simplifies the remaining equations. A series solution of the equations, useful for small motions—and perhaps useful for design—is given. To facilitate numerical solution, the triangular space-time domain of the problem is transformed into a square domain in pseudo space-time. Finally, some solutions based on the finite element method are presented for typical values of the dimensionless weight-to-stiffness and velocity parameters.