Dynamics of nonlinear disordered structures by means of nonlinear normal modes

Randomness can affect structural systems by several points of view; in particular, randomness can affect the forcing process that a system undergoes, or might as well influence the structural parameters. The study of disordered systems, as such struc-tures are usually referred to, is a point of concern in specialized literature: a fair amount of methods to investigate the dynamics of disordered structures can in fact be found. One of the most common methods is taken into consideration in the present work: the first applica-tion is described in Liu et al., 1986, and subsequent-ly enhanced in Chiostrini and Facchini, 1999. It can be classified as a perturbation method and makes use of sensitivity vectors to evaluate the first two mo-ments of the response. Unfortunately, a severe drawback of the method is that the number of degrees of freedom of the ex-amined structure grows rapidly for increasing num-ber of random parameters, thus leading to the solu-tion of very large (non) linear systems. The idea that is introduced in the present work is to investigate the possibility to compute and apply the concept of nonlinear modal shapes in order to reduce the number of degrees of freedom. It has re-cently been developed by Rizzo (2007) in his Ph.D. dissertation. It is possible to define the normal non-linear (NNMs) modes only as particular synchronous peri-odic solutions of the non-linear motion equations without looking for any link of such motions to the (linear) principle of superposition. For free vibration problems one uses system modes to construct reduced order models, and these techniques have been well developed for both linear and nonlinear systems by Vakakis (1997) and by Vakakis et al. (1996) . One such technique, introduced by Shaw and Pierre (1991, 1993, 1993), defines the normal mode of a nonlinear oscillatory system in terms of invariant manifolds in the phase space that are tangent to the linear eigen-modes at the equilibrium point. In such a formulation, a master mode is selected, and the nor-mal mode is constructed by a formulation in which the remaining linear modes of the system, i.e., the slave modes, depend on the master mode in a man-ner consistent with the system dynamics. This de-pendence defines the invariant manifold for the non-linear normal mode (NNM). By studying the dynamics of the reduced-order model, it is possible to recover the associated modal dynamics of the original nonlinear system.