The Design Implications of Chaotic and Near-chaotic Vibrations in Machines

Accurate performance prediction is key to the design of high performance machines. It is shown here that connection clearance and component flexibility can result in machine dynamic behaviors that are hypersensitive to small variations of system design parameters and operating conditions. These hypersensitivities, which can limit the usefulness of computer dynamic simulations for design, are associated with chaotic and near chaotic behavior. The dynamic behaviors of two systems, an Impact Beam System and a Spatial Slider Crank, are studied. The chaotic vibration of these systems is confirmed numerically and experimentally. Hypersensitivity is shown for both chaotic and periodic response regions, in which case actual dynamic behavior of such machines could be very different in practice from that predicated by design simulation studies. Design guidelines are developed for evaluating the fatigue life and the reliability of machines under these conditions. INTRODUCTION The effects of non-ideal characteristics of machines, such as connection clearances and component flexibility, often degrade dynamic performance by causing impacts, vibration and noise, component fatigue, and poor precision (Johnson, 1963; Dubowsky and Freudenstein, 1971). During the past thirty years, a number of computer-based analytical design models have been developed to predict the dynamic performance of machines with such non-ideal characteristics (Dubowsky and Gardner, 1975; Grant and Fawcett, 1978 and 1979; Dubowsky et al., 1987). Recently, it has been found that the dynamic responses of machine systems with clearance connections could exhibit large variations in their dynamic behavior with very small changes in their parameters and operating conditions (Deck and Dubowsky, 1994). These results suggest that the use of computer dynamic simulations of certain machines for design analysis is limited and should be used with great care. It is argued that a real machine, when manufactured and used, is somewhat different from its nominal design, due to normal variations in material properties and component characteristics. Hence, the dynamic behavior of a machine “as built” could be quite different from the behavior predicted by simulations used in its design. Similarly, it has been shown that the dynamic response of some machines with clearances may vary greatly with only very small changes in operating conditions, such as system operating speed. Sensitivity to small variations in the parameters may indicate the existence of chaotic behavior in a system (Moon, 1987). In fact, chaotic behavior has been found in the simulations of systems with non-ideal elements such as clearance connections (Moon and Shaw, 1983; Shaw, 1985b; Shaw and Rand, 1989; Li et al., 1990; Peterka and Vacík, 1992; Lin and Ewins, 1993). Most of this research focuses on the chaos itself and is usually concerned with demonstrating the existence of the chaos, rather than with the effects of chaotic behavior on machine performance. Yet the ability to predict accurately a real system’s performance should be the goal of design analysis. As shown in this paper, such sensitivity can occur in these systems, even when they are exhibiting periodic non-chaotic behavior. It has been found, both analytically and experimentally, that under periodic excitation, even a simple impact system can exhibit complex dynamics such as subharmonic and chaotic vibrations (Moore and Shaw, 1990; Shaw et al., 1989.) Harmonic, subharmonic and chaotic vibrations were found to exist in a very simple bilinear model of an asymmetrical clearance (Shaw and Holmes, 1983a). The forced vibrations of a nonlinear elastic beam were shown numerically and experimentally to exhibit chaotic behavior for a sinusoidal excitation (Moon and Shaw, 1983; Shaw, 1985a). Structures