Energy control of inhomogeneous nonlinear lattices

Abstract

This paper considers energy control of an n -d.f. inhomogeneous nonlinear lattice with fixed–fixed and fixed–free ends. The lattice consists of dissimilar masses wherein each mass is connected to its nearest neighbour by a nonlinear or linear memoryless spring element. The potential functions of the nonlinear spring elements are assumed to be qualitatively different. Each potential is described by a twice continuously differentiable, strictly convex function, possessing a global minimum at zero displacement, with zero curvature possibly only at zero displacement. The energy control requirement is viewed from an analytical dynamics perspective and is recast as a constraint on the motion of the dynamical system. No linearizations and/or approximations of the nonlinear dynamical system or the controller are made. Given the set of masses at which control is to be applied, explicit closed form expressions for the nonlinear control forces are obtained. Global asymptotic convergence to any desired non-zero energy state is guaranteed provided that the first mass, or the last mass or, alternatively, any two consecutive masses in the lattice are included in the subset of masses that are controlled. Numerical simulations involving a 101-mass nonlinear lattice demonstrate the simplicity and efficacy of the approach.