Chaos Analysis and Control in AFM and MEMS Resonators

For years, chaotic phenomena have been mainly studied from a theoretical point of view. In the last two decades, considerable developments have occurred in the control, prediction and observation of chaotic behaviour in a wide variety of dynamical systems, and a large number of applications have been discovered and reported (Moon & Holmes, 1999; Endo & Chua, 1991; Kennedy, 1993). Chaotic behaviour can only be observed in particular nonlinear dynamical systems. In recent years, nonlinearity is known as a key characteristic of micro resonant systems. Such devices are used widely in variety of applications, including sensing, signal processing, filtering and timing. In many of these applications some purely electrical components can be replaced by micro mechanical resonators. The benefits of using micro mechanical resonators include smaller size, lower damping, and improved the performance. Two examples of micro mechanical resonators that their complex behaviour is described briefly in this chapter are atomic force microscopy (AFM) and micro electromechanical resonators. AFM has been widely used for surface inspection with nanometer resolution in engineering applications and fundamental research since the time of its invention in 1986 (Hansma et al., 1988). The mechanism of AFM basically depends on the interaction of a micro cantilever with surface forces. The tip of the micro cantilever interacts with the surface through a surface–tip interaction potential. One of the performance modes of an AFM is the so called “tapping mode”. In this mode, the micro-cantilever is driven to oscillate near its resonance frequency, by a small piezoelectric element mounted in the cantilever. In this chapter it will be shown that micro-cantilever in tapping mode may exhibit chaotic behaviour under certain conditions. Such a chaotic behaviour has been studied by many researchers (Burnham et al. 1995; Basso et al., 1998; Ashhab et al., 1999; Jamitzky et al., 2006; Yagasaki, 2007). In section 3, the chaotic behaviour of micro electromechanical resonators is studied. Micro electromechanical resonant systems have been rapidly growing over recent years because of their high accuracy, sensibility and resolution (Bao, 1996). The resonators sensing application concentrate on detecting a resonance frequency shift due to an external perturbation such as accreted mass (Cimall et al., 2007). The other important technological applications of mechanical resonators include radiofrequency filter design (Lin et al., 2002) and scanned probe microscopy (Garcia et al., 1999). Many researchers have tried to analyze nonlinear behaviour in micro electeromechanical systems (MEMS) (Mestrom et al., 2007;