Finite Element Modelling of Laminated Piezo-Elastic Structures: Lyapunov Stability Analysis

Dynamic stability analysis of finite element modeling of piezoelectric composite plates

“…(27) is prohibitively expensive and numerically formidable for large problems in general. In fact, this numerical calculation procedure is not needed for a feedback control system that satisfies the LyapunovÕs second stability criteria strictly (Kekana, 2002;Kwon and Bang, 1997). Therefore, such a robust controller would be more desirable in practice and is thus presented next.…”

“…To minimize such non-linearities,, in practice, two separate, but very close to each other, pieces of piezoelectrics are often used and even considered as an approximate collocation (Ang et al, 2002;Wang, 2002). As a further approximation, piezoelectric sensor/ actuator pairs bonded at the top and bottom surfaces of the host structure, respectively, have also been considered as collocated by many researchers in their models (Hanagud et al, 1992;Hwang and Park, 1993;Hwang et al, 1994;Samanta et al, 1996;Lam et al, 1997;Saravanos et al, 1997;Lam and Ng, 1999;Liu et al, 1999;Kekana, 2002). Stability investigation of such approximations is therefore of practical importance.…”

“…The classical negative-velocity feedback control theory requires the sensors and actuators to be collocated to guarantee the dynamic stability (Kwon and Bang, 1997), although in practice nonperfectly collocated sensors/actuators are often used to avoid the non-linearity that arises from the interactions between actuating and sensing signals (Chen et al, 1996) when the piezoelectric materials are used for both sensing and actuating to meet the collocation requirement. In fact, many researchers have considered a sensor bonded at the top surface and an actuator bonded at the same location but on the bottom surface of the host plates as being collocated (Tzou, 1993;Hwang and Park, 1993;Hwang et al, 1994;Lam et al, 1997;Liu et al, 1999;Lim et al, 1999;Kekana, 2002). This is actually different from the perfect collocation requirement and the question of whether such imperfect collocation, which leads to an asymmetric active damping matrix (Lim et al, 1999), can guarantee the dynamic stability has not been properly answered.…”

“…(11) by ð _ e U U e U UÞ T and integrating it with respect to the time t and omitting the external forces F, which can be considered as bounded (Kekana, 2002;Kwon and Bang, 1997) and thus does not affect the stability analysis, an energy flux equation is obtained as follows:…”

“…The governing equations of motion for a piezoelectric composite plate were formulated in (Wang et al, 2001), assuming that: (a) the composite plate is thin or moderately thick and piezoelectric sensors/actuators are integrated into the laminated composite substrate as plies; (b) the laminate is perfectly bonded, elastic and orthotropic in behavior (Kekana, 2002) with small strains and displacements; (c) piezoelectric sensors/ actuators are made of homogenous and isotropic dielectric materials (Cheng, 1989) and high electric fields and cyclic fields are not involved (Ehlers and Weisshaar, 1990).…”