Piezoelectric Shells

In this book, generic double-curvature piezoelectric shell theories are derived; generic distributed structural sensing, identification, energy harvesting and vibration control theories of a generic deep shell continuum are presented. Open and closed-loop dynamic system equations and state equations of piezoelectric structronic systems are formulated. Simple reduction procedures are proposed and applications to other common geometries and structures are demonstrated in case studies. The revised book not only corrected typos and minor mistakes, but also added new chapters on optimal control of parabolic shells and energy harvesting of shells, including both theoretical and experimental aspects. Furthermore, laboratory and experimental components are added to, almost, all chapters on distributed sensing, energy harvesting and control of shell and non-shell structures and structronic systems. Note that performances of piezoelectric sensors/harvesters/actuators are restricted by breakdown voltages, hysteresis effects, limited strain rates, etc. These material properties need to be further improved in order to enhance the sensor/actuator performance and efficiency. Also, laboratory experiments were carried out over time; different materials with various dielectric constants from different venders were used in various studies presented in newly added Chaps. 10–12. Extreme care should be taken when repeating those studies. It should be pointed out that all piezoelectric shell theories and distributed sensing/control and energy harvesting theories are based on a symmetrical hexagonal piezoelectric structure—class C_6v = 6 mm. Extension of these theories to more generic piezoelectric materials, such as a triclinic structure, would make them even more comprehensive and versatile. Besides, the temperature effect, e.g., the pyroelectricity and thermal induced stress/strains, is not considered in all studies; it should be considered when a working environment has significant temperature variations.

Active piezoelectric structures capable of self-adaptation (Tzou & Anderson, 1992) and high-precision manipulations (Tzou & Fukuda, 1992).

In this chapter, applications of the generic piezoelectric shell theories to a number of common piezoelectric continua were presented. A four-step reduction procedure was introduced and it was demonstrated in two geometries. The first case was a piezoelectric plate which includes 1) a thick plate and 2) a thin plate. The derived system equations of the thick piezoelectric plate were completely identical to published results (Tiersten, 1969). The second case was a piezoelectric shell of revolution which represents another class of shell continua e.g., piezoelectric spheres, cylinders, cones, etc., which were discussed in detail. Applications of the generic shell vibration theory to other piezoelectric continua can be further explored. Note that the theory was derived based on a symmetrical hexagonal piezoelectric structure—class \({\text{C}}_{6{\text{v}}} = 6\, {\text{mm}}\).

Distributed sensing and control of a generic distributed parameter system (DPS) or a generic smart structronic shell system, i.e., a deep elastic shell laminated with distributed piezoelectric sensor and actuator layers, was proposed and corresponding generic theories derived. Based on the direct piezoelectric effect, the distributed sensor can be used to monitor shell oscillations; the converse effect enables the distributed actuators to manipulate structural behaviors and to suppress structural vibrations. Two generic sensor/actuator design principles, i.e., the segmentation technique and the shaping technique, were also presented.

In this chapter, a theoretical development of a multi-layered thin shell distributed actuator is presented. The distributed actuator layers can be made of electromechanical sensitive materials which respond to externally supplied voltages and generate local control forces for active distributed vibration controls. Based on the assumptions, dynamic equations for the generic multi-layered thin shell actuator (with distributed control layers) were developed using Kirchhoff-Love’s theory and Hamilton’s principle. The system equations are generic and can be simplified to apply to many other common geometries and structures, such as plates (e.g., circular or rectangular), other conventional shells (e.g., cylindrical shell, spheres), beams, etc. The common geometries can be defined by the fundamental form, Lamé parameters, radii of curvatures, etc. It should be noted that the deformations resulting from transverse shears and rotatory inertias were neglected in the derivations.

Distributed control of a PVDF laminated cantilever beam was studied in this chapter. The laminated cantilever beam had a distributed piezoelectric sensor and a distributed actuator; both were surface bonded. Closed-loop feedback controls of the beam using the displacement and velocity signals were respectively evaluated and results compared. The results showed that the displacement feedback controls were insignificant and the velocity feedback controls were much more effective. In the velocity feedback control, the system damping increased to an ultimate value and then gradually dropped down as the feedback gain continuously increased. This was caused by the additional constraint imposed by the boundary control moment at the free-end. The free-end boundary condition was gradually changing to a sliding-roller boundary condition as proved by finite element analyses and laboratory experiments.

In the development of active piezoelectric/elastic structures, it was noted that a fully (symmetrically) distributed piezoelectric sensor/actuator could lead to minimum, or zero, sensing/control effects for anti-symmetrical modes of structures, especially with symmetrical boundary conditions. One method of improving the performance is to segment the symmetrically distributed sensor/actuator layers into a number of collocated sub-segments. In this chapter, mathematical models and analytical solutions of a simply supported plate with a single-piece distributed sensor/actuator and four-piece quarterly segmented sensors/actuators were derived. Modal sensitivities and modal feedback factors for the two sensor/actuator configurations are defined, and modal displacement and velocity feedbacks are formulated.

In this chapter, generic distributed piezoelectric shell convolving sensors and actuators were proposed and detailed electromechanical behaviors (sensor and actuator electromechanics) were analyzed. It was observed that the sensor output is contributed by membrane strains and bending strains experienced in the sensor layer. Two sensor sensitivities: 1) a transverse modal sensitivity and 2) a membrane modal sensitivity can be defined accordingly. In general, the transverse modal sensitivity is defined for out-of-plane transverse natural modes and the membrane modal sensitivity for in-plane natural modes. Proper design of distributed sensor shape and convolution can provide modal filtering to prevent observation spillover in distributed structural control systems.

In this chapter, distributed sensors and actuators for cylindrical shells were designed and their spatially distributed sensing/control effects were analyzed. Mathematical model and analytical solutions suggest that the fully distributed shell sensor is sensitive only to all odd modes and insensitive to all even modes. This is due to signal cancellations of positive and negative signals in opposite strain regions. The diagonal stripe sensor is sensitive only to the m = n modes and insensitive to the m ≠ n modes. Three sensor sensitivities, i.e., transverse, in-plane longitudinal x and in-plane circumferential θ, were defined for each sensor and their normalized sensitivities evaluated. It was observed that the in-plane sensitivities are insensitive to thickness variations of elastic shells because the in-plane strains remain identical regardless of the thickness change. However, the transverse sensitivity increases as the shell becomes thicker due to an increase of bending strains. Furthermore, control effects of a fully distributed actuator and a diagonal strip actuator are evaluated.

Open parabolic cylindrical shells are important to radial signal collection, reflection and/or transmission applied to radar antennas, space reflectors, solar collectors, etc. The spatially distributed microscopic modal control effectiveness induced by piezoelectric actuators laminated on a simply-supported parabolic cylindrical shell panel was investigated in this study. Distinct distributed modal actuation behaviors of transverse vibrations of the shell were analyzed based on a newly-formulated mode shape function. The expression of modal control force induced by an actuator patch was derived. The spatially distributed microscopic actuation effectiveness induced by an infinitesimal actuator element was also derived to precisely illustrate the spatial distribution behavior.

Energy harvesting based on distributed piezoelectric laminated structures has been proposed and extensively investigated for over a decade. The objective of this study is to develop a generic distributed piezoelectric shell energy harvester theory based on a generic linear/nonlinear double-curvature shell, which can be simplified to account for many linear/nonlinear shell and non-shell type distributed energy harvesters. Distributed electromechanical coupling mechanism of the energy harvester was discussed; voltage and power output across the external resistive load of the shell energy harvester were evaluated. Those equations were explicitly expressed in terms of design parameters and modes. Once the intrinsic Lamé parameters and the curvature radii of the selected host structure are specified, one can simplify the piezoelectric energy harvesting equations to account for common shell and non-shell harvester structures. To demonstrate the simplifications, the generic piezoelectric shell energy harvesting mechanism was applied to a cantilever beam, a circular ring and a conical shell in cases studies. Again, the generic piezoelectric energy harvesting formulations derived from a double-curvature shell can be applied to many shell, e.g., ring shells, cylindrical shell, conical shells, paraboloidal shells, etc., and non-shell, e.g., plates, beams, etc., structures using two Lamé parameters and two curvature radii of the specified structures. Besides, these shell and non-shell structures can be either linear or nonlinear with the von Karman geometric nonlinearity. With given boundary conditions and external loading forces, generated voltage and power across the resistive load in the closed-circuit condition can be estimated for the distributed piezoelectric laminated structure.

This chapter involved energy harvesting of a simply supported tubular (circular) cylindrical shell laminated with piezoelectric patches. The distributed modal energy generations using different energy harvester patch sizes (i.e., (1 mm,3.6°) in Case 1, (10 mm,30°) in Case 2, and (20 mm,60°) in Case 3) at various mode numbers were evaluated in case studies. Analytical and simulation results suggest that the maximum magnitude of the spatially distributed modal energies changes at various modes in two cases, due to the patch size enlarged or the number of energy harvester patches in the circumferential direction decreased. It should be noted that the signal averaging effects on energy harvester patches become more significant when the patch size continuously increasing. Additionally, the bending energy components are much smaller than the circumferential membrane energy component, and they increase when mode number increases. Furthermore, the maximum magnitude of the (m, n)th modal energy, in general, increases when energy harvester’s thickness h_p or shell’s thickness h increases, but decreases when the shell radius R increases. A tubular shell energy harvesting system was designed and tested in the StrucTronics and Control Laboratory at Zhejiang University. Experimental results suggest that there is an optimal external loading resistance leading to the maximal power output. Both analytical predictions and experimental data were compared favorably. These data evaluated in this study can be used as guidelines to design the optimum piezoelectric energy harvester in practical engineering applications.

Conventional elastic structures are “passive” in nature, i.e., they do not possess any inherent self-sensation and action/reaction capabilities. Thus, development of new-generation active structures with integrated sensors, actuators, and control electronics, i.e., so called the structronic system, has received an increasing attention and interest in recent years (Tzou & Anderson, 1992). This chapter presents a finite element development and analysis of integrated distributed piezoelectric sensor/actuator structures—active distributed parameter systems (DPS’s) or structronic systems.