Smart Composite Plate Shape Control Using Piezoelectric Materials II. Optimization

Abstract

For the aircraft industry, the ability to change and control the shape of the structures has been a challenging problem. In the present work the shape control of fiberreinforced composite plate with embedded piezoelectric actuators and sensors is investigated. For shape control and shape optimization, an optimization algorithm based on finite element technique, is presented which formulate the error between the actual and the desired shape. The error (objective) function is the root mean square of the error between the point in the actual surface and the corresponding point in the desired surface for each element of the finite element grid. The algorithm is able to compute the optimal amount of voltage applied to each actuator to achieve minimum error function, and determine the value of the error function for different actuator(s) position for a specified amount of the applied voltage.

Finite Element Model of Smart Beams with Distributed Piezoelectric Actuators

“…For the piezoelectric element the electric field is treated as the electric degree of freedom like a generalized displacements degrees of freedoms (Elshafei, 1996;Elshafei et al, 1997). The electric potential function takes the form (Cook et al, 1974):…”

“…where M uu is the global mass matrix of the structure and {U} ¼ {{u}{w}} is the global nodal generalized displacements coordinates vector, {u} is the global nodal generalized electric coordinates vector describing the applied voltages at the actuators (Elshafei, 1996), {F} is the applied mechanical load vector, and {Q} is the electric excitation vector. This model is transferred to a computer program where the program input are the applied mechanical and/or electrical loads and the geometric properties of the structure such as the dimensions, the moment of inertia of beam, the adhesive layer, the piezoelectric layers, number of layers, ply orientation angle, and the material properties of the structure subsystem.…”

“…They also showed that: (i) the cantilever beam had frequencies significantly higher than simply supported or clamped beams subjected to actuating voltages, and (ii) the effect of in-plane stress on the overall stiffness was high compared to the bending stress with the application of electric loading. A finite element model for a composite plate with piezoelectric actuators has been developed (Elshafei, 1996;Elshafei et al, 1997). In this model, the electrical degrees of freedom were assumed to be constant along the plate and varied linearly through the thickness of the piezoelectric layers.…”

Vibration reduction of composite plates by piezoelectric patches using a modified artificial bee colony algorithm

“…Piezoelectric materials generate an electrical signal when subjected to mechanical strains and vice versa. These effects are known as direct and inverse piezoelectric effects and can be written in the form of two constitutive equations (Elshafei 1996):…”

Finite Element Model of a Smart Isotropic Beam With Distributed Pizoelecetric Actuators

“…is the global nodal generalized displacement coordinates, { } ϕ is the global nodal generalized electric coordinates vector describing the applied voltage at the actuators [16], { } F is the applied mechanical load vector, and { } Q is the electric excitation vector. The global mass matrix, stiffness matrix and applied mechanical vector could be shown in Appendix A.…”