Small in-plane oscillations of a slack catenary using assumed modes

Abstract: In this paper we study a problem in oscillations wherein the assumed modes method offers some analytical and theoretical peculiarities. Specifically, we study small in-plane oscillations of a slack catenary, or a sagging inextensible chain fixed at both endpoints. The horizontal and vertical displacements cannot be approximated independently because of pointwise inextensibility in the chain. Moreover, the potential energy is a linear function of the generalized coordinates, and does not directly cause oscillations. Using assumed modes for the vertical displacements only, integrating from one endpoint to compute the required horizontal displacements, treating the horizontal fixity at the distal end as an added scalar constraint, and obtaining linearized equations, we construct an eigenvalue problem which contains a Lagrange multiplier. For generic assumed modes, the Lagrange multiplier is determined by enforcing equilibrium in the undeflected shape. However, when the modes thus determined are reinserted in the assumed mode expansion and the calculation done afresh, then the Lagrange multiplier is indeterminate at first order. Upon retaining terms at the next order, the distal end fixity constraint introduces quadratic terms into a Lagrangian without constraints. Results from these two approaches match perfectly. Our approach offers nontrivial insights into both oscillations and Lagrangian mechanics. It is also potentially applicable to other problems with inextensibility in one-dimensional slender members.