Introduction

Reduced-order models are very useful to study nonlinear dynamics of fluid and solid systems. In fact, the study of nonlinear vibrations and dynamics of systems described by too many degrees of freedom, like those obtained by using commercial finite-element programs, is practically impossible. In fact, not only is the computational time needed to obtain a solution by changing a system parameter too long, as the excitation frequency around a resonance, but also spurious solutions and lack of convergence are easily obtained for large-dimension systems. For this reason, techniques to reduce the number of degrees of freedom in nonlinear problems are an important and challenging research area.

The two most popular methods used to build reduced-order models (ROMs) are the proper orthogonal decomposition (POD) and the nonlinear normal modes (NNMs) methods. The first method (POD, also referred to as the Karhunen-Loève method) uses a cloud of points in the phase space, obtained from simulations or from experiments, in order to build the reduced subspace that will contain the most information (Zahorian and Rothenberg 1981; Sirovich 1987; Aubry et al. 1988; Breuer and Sirovich 1991; Georgiou et al. 1999; Azeez and Vakakis 2001; Amabili et al. 2003; Kerschen et al. 2003, 2005; Sarkar and Païdoussis 2003, 2004; Georgiou 2005; Amabili et al. 2006). This method is, in essence, linear, as it furnishes the best orthogonal basis, which decorrelates the signal components and maximizes variance.