ENERGY CONTROL OF NONLINEAR NON-HOMOGENEOUS CHAINS

In this paper, we consider the problem of energy control of an n-degrees-of-freedom system consisting of a chain of masses wherein each of the masses is connected to its neighboring mass with the help of a nonlinear memoryless spring element. The qualitative nature of the nonlinearity in each of the spring elements can, in general, be different. Both fixed-fixed and fixed-free boundary conditions are considered. The energy control problem is approached from a new perspective that of constrained motion. For a given set of masses at which the control is to be applied, explicit closed form expressions for the nonlinear control forces are derived by using the fundamental equation of mechanics. Through the use of the invariance principle, we show that these control forces provide global asymptotic convergence to any 'given' non-zero energy state provided that the first mass, or the last mass, or alternately, any two consecutive masses in the chain are included in the subset of masses that are controlled. The results obtained in this paper are, in general, applicable to any finite degrees-of-freedom, fixed-fixed or fixed-free nonlinear chain whose spring potentials are described by a class of twice continuously differentiable, strictly convex functions, which possess a global minimum at zero displacement, with zero curvature possibly only at zero displacement.