Stability of Interpolating Elastica.
Abstract
It is an old technique of draftsmen to use a mechanical spline to pass a smooth curve through a prescribed set of points in a plane. Curves which are obtained in this way (interpolating elastica, also called nonlinear spline curves) may be considered as the equilibrium positions of thin elastic beams which are constrained to pass through short, friction-less, freely rotating sleeve supports, anchored at the interpolation points. The strain energy of such a beam is given by the integral of the square of the curvature with respect to arc length, and equilibrium requires that the position be such that the energy be minimum for the given interpolation conditions. However, a global minimum cannot be attained (except in the trivial case of the unbent beam) since the energy can be made arbitrarily small by using sufficiently large loops between the supports. Instead one looks for local minima which guarantee stability against small perturbations. In this report some general stability criteria are established and some specific interpolating elastica are investigated for stability. Except for a few previous isolated observations these seem to be the first proven results on the stability of interpolating elastica.
Document Details
- Document Type
- Technical Report
- Publication Date
- May 01, 1978
- Accession Number
- ADA060716
Entities
People
- Michael Golomb
Organizations
- University of Wisconsin–Madison