Critical points of the clamped–pinned elastica

Abstract

We investigate equilibrium configurations of the clamped-pinned elastica where the pinned end can be displaced towards, and past, the clamped end. Solving the nonlinear ordinary differential equation for the clamped-pinned elastica for any mode in terms of elliptic integrals, we find sets of equations which govern the equilibrium configurations for given displacements. Equilibrium configurations for various displacements of the pinned end and any mode are obtained by numerically solving those sets of equations. A physical quantity, such as the force that arises in the elastica due to displacement of the pinned end, is taken to be a function of displacement. Although it is generally not possible to define a physical quantity as a function of displacement explicitly, an equation for the rate of change of this physical quantity with respect to displacement can be found by partial differentiation of the sets of equations which govern the equilibrium configurations. Setting that rate of change to zero provides a constraint equation for locating the critical points of that physical quantity. That constraint equation and the sets of equations which govern the equilibrium configurations are solved numerically to obtain the critical points of the physical quantity. Our procedure is demonstrated by finding local extrema on force-displacement plots (useful when analysing the stability of equilibrium configurations) and the maximum deflection of the elastica. Finally, we suggest how our procedure has scope for wider application.

Analytical modeling and simulation of a multifunctional segmented lithium ion battery unimorph actuator

“…The coupling mechanism for simulating the lithium-ion based actuation is discussed next. The constitutive equations are shown in equation (31) where it is assumed that due to the aspect ratio of a beam geometry, the only actuation strain that occurs is in the xx or 1 direction along the length. The actuation strain (the only nonzero term in the coupling matrix) is added to the elastic strain to achieve the total strain.…”

“…FEA can be computationally expensive, but can be high fidelity. Adomian decomposition [29], and elliptic integrals [31] are also methods used for the prediction of large deflection. Crawley and Anderson developed models for a beam structure based induced actuation specifically for piezoceramics [17].…”

A rod model for large bending and torsion of an elastic strip with a geometrical imperfection

“…Particularly at higher ratios L/H , the solution of the BVP (24) is not unique: various equilibria are possible. Most of them are, however, unstable, see Levyakov and Kuznetsov [19], Singh and Goss [26] for detailed discussions. The most straightforward option for solving is the numerical integration of an initial value problem by choosing an initial approximation for the curvature in the middle ϕ (0) as well as for the force Q 0 and then by iteratively seeking better approximations, which allow fulfilling the condition at s = L/2 and the integral one from (24) in the framework of the Newton method.…”