Introduction to Experimental Nonlinear Dynamics A Case Study in Mechanical Vibration

Introduction

All of the preceding chapters have dealt with single-degree-of-freedom oscillators, generally with a periodic excitation and, hence, a threedimensional phase space. Many real dynamical systems are continuous, modeled by partial differential equations with both space and time as independent variables. Despite the fact that the dynamics often take place on a relatively low-order subspace of the (infinite) phase space of the full system, there are still many situations in which an analysis in a high-order space is necessary. For continuous systems such as beams and plates (see Chapter 4), modal analysis has proved to be a powerful technique for extracting the dominant dynamic characteristics from complex systems, especially for linear systems. In a theoretical context, Galerkin's method can be used to reduce a partial differential equation into a set of coupled ordinary differential equations, which can then be analyzed using standard techniques. The success with which a reducedorder model captures the full range of behavior is a very complicated issue (especially in fluid mechanics (Lorenz, 1963; Ruelle and Takens, 1971)), but, for example, a continuous beam excited close to its fundamental natural frequency will display behavior dominated by the lowest mode, and hence a lumped parameter model will likely be good enough in an engineering context.

Some of the earliest studies in chaos were generated by the consideration of thin beams, which under certain circumstances could be very successfully modeled by Duffing's equation (see (Moon, 1992) and Chapter 4). The presence of multiple equilibria and periodic excitation provided conditions under which a wide range of nonlinear behavior could be observed and measured. In this appendix, we take a brief look at a continuous (i.e., high-order) experimental system that displays behavior that is qualitatively similar to the single-degree-of-freedom examples encountered earlier. The practical context for this example occurs in certain aerospace systems where thin metal panels are subject to intense acoustic excitation and are often in a postbuckled equilibrium configuration owing to thermal effects (Tauchert, 1991).

The theoretical treatment is rather involved, and the reader is referred to Refs. (Murphy, Virgin, and Rizzi, 1996a; Murphy, Virgin, and Rizzi, 1997; Murphy, 1994) for more details. Here, we shall concentrate on experimental results and try to contrast the similarities and differences with some of the (low-order) results presented earlier in this book.