Shaping of Piezoelectric Sensors/Actuators for Vibrations of Slender Beams: Coupled Theory and Inappropriate Shape Functions

Abstract

Flexural vibrations of smart slender beams with integrated piezoelectric actuators and sensors are considered. A spatial variation of the sensor/actuator activity is achieved by shaping the surface electrodes and/or varying the polarization profile of the piezoelectric layers, and this variation is characterized by shape functions. Seeking shape functions for a desired purpose is termed a shaping problem. Utilizing the classical lamination theory of slender composite beams, equations for shaped sensors and actuators are derived. The interaction of mechanical, electrical and thermal fields is taken into account in the form of effective stiffness parameters and effective thermal bending moments. Self-sensing actuators are included. From these sensor/actuator equations, shaping problems with a practical relevance are formulated and are cast in the form of integral equations of the first kind for the shape functions. As a practical interesting aspect of these inverse problems, shape functions which fail to measure or to induce certain structural deformations are investigated in the present paper. Such inappropriate shape functions are termed nilpotent solutions of the shaping problems. In order to derive an easy-to-obtain class of such nilpotent solutions, the homogeneous versions of the integral equations for the shaping problems are compared to orthogonality relations valid for redundant beams. Hence, by analogy, the presented nilpotent solutions are shown to correspond to solutions of the basic theory of thermoelastic structures, namely to thermally induced static bending moment distributions. This result beautifully reflects the close connection between the theory of thermally loaded structures and the theory of smart structures. A particular result for a nilpotent shape function previously investigated in the literature is explained in the context of the present theory, and examples of nilpotent shape functions for various structural systems are presented.

Computational aspects of structural shape control

“…the solution of the force-displacements u i(F) , i = 1,2,3, by means of the principle of virtual forces, see Ziegler [27], is given in the form of the volume integral, a complementary GreenÕs formula,…”

“…The latter produce stress but no deformations and thus have no influence on shape control. Classical examples are known in thermoelasticity, Ziegler and Irschik [26], or have been encountered in the case of flexural vibrations of piezoelectric beams and attributed to an electric field without deflection, Irschik et al [27]. When properly defined, they can be used to redistribute the load stresses without influencing shape control.…”

Natural frequency of beams with embedded piezoelectric sensors and actuators

“…The elastic strain energy of the piezoelectric inclusion is obtained using the method presented in Mura (1987) ε is the eigenstrain due to the electromechanical coupling of the piezoelectric actuator or the eigenstrain actuation (Irschik, et al, 1998). Applying Mura's (1987) energy of inhomogeneous inclusion, Equation 10 can be expressed as…”

“…The piezoelectric problem was decoupled into an elastic inclusion problem and a dielectric inclusion problem connected by some eigenstrain and eigen-electricfield. Irschik et al (1998) analyzed the piezoelectric actuation for vibration and shape control of structures as an eigenstrain actuation. An eigenstrain technique was presented by Dasgupta (1993a, 1993b) for the vibration of beams with embedded arrays of piezoelectric sensors and actuators.…”

Bending moment tracking and the reduction of the axial stress in vibrating beams by piezoelectric actuation

“…A literature review on this topic is presented by Irschik [14]. Further discussions on shape control can be found in Hubbard and Burke [15], Hafka and Adelman [16], Irschik et al [17,18], Nader [19], Irschik and Pichler et al [20], Schoeftner et al [21][22][23], Austin et al [24] and Agrawal and Treanor [25]. Displacement tracking is a feed-forward control method, where one asks for the eigenstrain sources in order to obtain a certain deflection.…”