A superposition principle for nonlinear systems

Nonlinear systems fail to satisfy the principle of linear superposition because when multiple external forces are applied, nonlinearities allow the forces to interact with one another and with the initial conditions of the system. Since these interactions create coupling between the linear and nonlinear dynamics, it is difficult to experimentally characterize nonlinear systems and identify accurate models of nonlinear dynamic behavior. This paper introduces a new principle of superposition for nonlinear systems that decouples the linear and nonlinear dynamics. The principle is based on a spatial perspective of nonlinearities as internal forces, which feedback through the analytical/experimental response degrees of freedom. From this perspective nonlinear systems are closed loop systems and linear systems are the corresponding open loop systems. The superposition principle is used to derive a new formulation for the frequency response function matrices of nonlinear systems and powerful multiple degree of freedom characterization and identification techniques for nonlinear systems. A companion paper presents applications of these new techniques.