Optimization of clamped circular piezoelectric composite actuators

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Abstract

This paper addresses the design of clamped circular piezoceramic composite unimorph and bimorph configurations, specifically the conflicting requirements of maximum volume displacement for a prescribed bandwidth. An optimization problem is formulated that implements analytical solutions for unimorph and bimorph configurations using laminated plate theory, including the use of oppositely polarized piezoceramic patches. A range of actuator geometric parameters are studied, and bounds for volume displacement and natural frequency of optimal designs are determined and presented via design curves. In the selected design space, Pareto optimization results for unimorph and bimorph configurations show that optimal volume displacement is related to the bandwidth by a universal power law such that the product of the square of the natural frequency and the displaced volume, a “gain-bandwidth” product, is a constant. Characteristic trends are also described that are independent of the actuator radius for the Pareto optimal piezoceramic patch thickness and radius versus normalized bandwidth. The results are relevant, for example, in the design of zero-net mass-flux or synthetic jet actuators used in flow control applications.

Introduction

Piezoelectric composite plates are commonly used in many engineering applications, such as actuators for flow control applications (Glezer and Amitay [1]), transducers for acoustic applications (Horowitz et al. [2]), and in locomotion of robotic systems (Yumaryanto et al. [3]). Piezoelectric circular composite plates, in particular, are widely used for systems/applications where actuation and/or sensing are needed. Examples include, but are not limited to, mechanical actuation and sound generation or receiving devices (Chee et al. [4]), zero-net mass-flux or synthetic jets (Gallas et al. [5], [6]), micropumps (Morris and Foster [7]), energy harvesting (Kim et al. [8], [9], Horowitz el al. [10], and Liu et al. [11]) and active structural health monitoring applications (Liu et al. [12]).

Optimal system performance in such applications is dictated by the electromechanical characteristics of the piezoelectric composite plates, and therefore design optimization of the composite plate configurations is of great interest. In particular, it requires the determination of the electromechanical response when the design parameters are varied. Morris and Foster [7] studied optimization of a circular bimorph concept using the finite element method (FEM). However, FEM-based optimization is cumbersome because a new mesh may be required when the geometric design variables are varied. Furthermore, coupling with an external optimizer may be required. Hence, analytical solutions of piezoelectric composite plates for their electromechanical response are desirable in such design optimization problems. Coorpender et al. [13], for instance, noted the ease of their model and formulation to accommodate changing material parameters or geometry. They demonstrated the effects of inactive plate thickness as well as piezoceramic patch radius and thickness on the displacement for a fixed input voltage.

There are a number of published analytical studies on the determination of the electrically-driven transverse deflection of various piezoelectric composite actuators (see Fig. 1). Li et al. [14] studied the electromechanical behavior of PZT-brass rectangular and circular unimorphs where the PZT patch completely covered the brass plate. Ha and Kim [15] developed a model for an asymmetrical annular bimorph configuration. Prasad et al. [16] derived a static analytical model of a clamped axisymmetric piezoelectric unimorph transducer consisting of a piezoelectric inner disc perfectly bonded to a metal shim. They used lumped element modeling (LEM) to estimate the dynamic response.

Other configurations and boundary conditions have also been studied recently. In a preliminary version of the present work, Gallas et al. [5] developed analytical models for clamped annular and inner disc axisymmetric configurations designed to maximize the volume displacement of the actuator (see Fig. 1). Li and Chen [17] obtained a solution for both clamped and simply-supported boundary conditions for a valve-less micropump. Their model is valid for a disc in pure bending loaded with an axisymmetric moment at the edge. Fox et al. [18] extended this approach to an annular configuration under clamped and simply-supported boundary conditions. Chang and Lin [19] studied a piezoelectric ring, which consists of an isotropic elastic ring laminated between two identical piezoelectric rings (termed a “trimorph” configuration in their paper but referred to here as a “bimorph” configuration). They developed an electroelastic laminated plate theory to analyze its dynamic behavior, such as electric current response and resonant frequencies. They also considered several boundary conditions, namely clamped–free, free–clamped, and clamped–clamped at the inner-outer radius. In an extension of their earlier work, Prasad et al. [20] presented a two-port electroacoustic model which provided the solution for the transverse static deflection field as a function of pressure and voltage loading. Classical laminated plate theory (CLPT) was used to derive the equations of equilibrium of clamped circular laminated plates containing a piezoelectric layer. Closed-form expressions for the static deflection field as a function of the applied uniform pressure and/or the uniform electric field across the piezoelectric layer were obtained. Another recent effort on modeling circular composite actuators is by Mo et al. [21]. Their solution was solely for electrical loading but considered different boundary conditions as well as full and partial coverage piezoelectric disc configurations. Experimental validation was also reported. Dong et al. [22], similar to Prasad et al. [20], also determined transverse deformation shape of a circular axisymmetric piezoelectric-metal composite unimorph actuator for both electrical and distributed mechanical loading (such as uniform pressure). Using their analytical solution, they performed a parameteric study of the piezoelectric-to-metal thickness ratio and presented design curves. The most recent work to the authors’ knowledge is the analytical solution by Deshpande and Saggere [23], which mainly focused on a unimorph configuration with a central piezoelectric layer. Their approach, however, permits an arbitrary number of layers, which makes it versatile enough to investigate other configurations (e.g., bimorph).

Despite the availability of diverse analytical solutions, reported implementations in design optimization are sorely lacking. This study is a continuation of the initial assessment of different configurations summarized in Gallas et al. [5] that used LEM in conjunction with analytical models of a clamped piezoelectric composite plate to optimize the performance of unimorph-driven synthetic jets for flow control applications. Here, we incorporate analytical solutions for both unimorph and bimorph configurations to find optimal designs for maximum volume displacement over a prescribed bandwidth, which is desirable in pump and flow control applications. The bandwidth of the device is limited to frequencies from dc to near the natural frequency of the composite plate, which can be easily obtained from a static solution using LEM without having to resort to FEM-based methods (Rossi [24]). Similar to an amplifier, which has a gain-bandwidth limitation, a piezoceramic composite plate will have a tradeoff between high dc or static displacement (i.e., related to its “gain”) and a large bandwidth. Hence, optimal unimorph and bimorph designs subject to a prescribed natural frequency constraint are found and compared. The tradeoffs between the volume displacement and the natural frequency are investigated within the framework of Pareto optimization (Belegundu and Chandrupatla [25]) as a function of actuator geometry. The Pareto curves and their opposite ends, which correspond to the conflicting objectives of maximum volume displacement and maximum natural frequency, are computed. Finally, empirical design curves as a function of overall actuator radius are provided to determine the expected bounds on the objective for a specified actuator size, and an optimal design algorithm with examples is provided.

Section snippets

Description of actuator and design parameters

Piezoelectric circular unimorph and bimorph composite plates subject to electrical and differential pressure loads were studied in this work. Fig. 2 shows the unimorph configuration subject to differential pressure loading. The clamped composite plate consists of PZT patch(es) perfectly bonded to a brass shim. The shim and piezoelectric layer material properties are summarized in Table 1. Note that, in practice, clamping the PZT patch is not desirable as it may fracture due to stress

Design optimization

The next step is to optimize the design of piezoelectric circular plate actuators. The specific goal of this study is to find the optimal piezoelectric composite circular plate configuration and design for maximized volume displacement and bandwidth. These conflicting tradeoffs are investigated via Pareto optimization. This section first defines the design variables, objective function, and the constraints for the optimization. A formal problem formulation is then provided, followed by the

Results

Several different size unimorph and bimorph configurations were optimized using the MATLAB optimization toolbox. Optimum piezoelectric patch dimensions of the actuators at the selected radii were determined for maximum volume displacement corresponding to application of the coercive electric field. A discussion of the results is delayed until the next section.

Discussion

First, optimal unimorph versus bimorph configurations are compared. Table 3, Table 4 indicate that, as expected, the bimorph configuration provides larger volume displacement. The natural frequencies of optimal unimorph and bimorph designs coincide when they are not constrained. For a fair comparison of the volume displacement, however, one might consider the percentage gain in volume displacement or volume displacement/voltage versus the percentage increase of PZT material use. When

Concluding remarks

Clamped circular composite piezoceramic actuators were optimized for volume displacement at a prescribed bandwidth using classical laminated plate theory. Unimorph and bimorph configurations, including oppositely polarized PZT patches, were considered. The results indicate that the optimized bimorph configuration is an ideal choice to maximize volume displacement when no constraint on bandwidth is imposed. As the bandwidth of the piezoceramic increases, the performance advantage of the bimorph

Acknowledgment

The authors gratefully acknowledge financial support from NASA Grant NNX07AD94A, monitored by Brian G. Allan and The Scientific and Technological Research Council of Turkey – TÜBİTAK Grant 106M364.

Melih Papila received the B.S. and M.S. degrees in aeronautical engineering from the Middle East Technical University, Ankara, Turkey, in 1990 and 1995, respectively. He received the Ph.D. degree in aerospace engineering from the University of Florida, Gainesville, in 2001, as a member of Multidisciplinary and Structural Optimization group. He is an assistant professor in the Materials Science and Engineering Program at the Sabancı University (SU), Istanbul, Turkey. Prior to joining SU in 2004,

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    Melih Papila received the B.S. and M.S. degrees in aeronautical engineering from the Middle East Technical University, Ankara, Turkey, in 1990 and 1995, respectively. He received the Ph.D. degree in aerospace engineering from the University of Florida, Gainesville, in 2001, as a member of Multidisciplinary and Structural Optimization group. He is an assistant professor in the Materials Science and Engineering Program at the Sabancı University (SU), Istanbul, Turkey. Prior to joining SU in 2004, he was a Postdoctoral Associate jointly in Interdisciplinary Microsystems and Multidisciplinary and Structural Optimization Groups at the Department of Aerospace and Mechanical Engineering of University of Florida, from 2002 to 2004. His current research focuses on the electroactive polymers and composites for sensors and actuators, design and optimization of smart/composite materials and structures (http://people.sabanciuniv.edu/∼mpapila).

    Mark Sheplak received the B.S., M.S., and Ph.D. degrees in mechanical engineering from Syracuse University, Syracuse, NY, in 1989, 1992, and 1995, respectively. During his Ph.D. studies, he was a GSRP Fellow at NASA-Langley Research Center, Hampton, VA, from 1992 to 1995. He is an associate professor in the Department of Aerospace and Mechanical Engineering and an affiliate associate professor of Electrical and Computer Engineering at the University of Florida (UF). Prior to joining UF in 1998, he was a postdoctoral associate at the Massachusetts Institute of Technology's Microsystems Technology Laboratories, Cambridge, from 1995 to 1998. His current research focuses on the design, fabrication, and characterization of high-performance, instrumentation-grade, MEMS-based sensors and actuators that enable the measurement, modeling, and control of various physical properties (http://www.img.ufl.edu).

    Louis N. Cattafesta III is currently an associate professor in the Department of Mechanical and Aerospace Engineering at the University of Florida. His primary research interests are experimental fluid dynamics, particularly active flow control, and aeroacoustics, particularly airframe noise. Prior to joining UF in April of 1999, he was a senior research scientist at High Technology Corporation in Hampton, VA, where he was the group leader of the Experimental and Instrumentation Group. He received a B.S. degree in Mechanical Engineering in 1986 from Penn State University, a M.S. degree in Aeronautics from MIT in 1988, and a Ph.D. degree in Mechanical Engineering in 1992 from Penn State University. In 1992, he joined High Technology Corporation as a Research Scientist at NASA Langley Research Center. His research at NASA Langley focused on supersonic laminar flow control and pressure- and temperature-sensitive paint measurement techniques. At that time, he became involved in active control of flow-induced cavity oscillations, which provoked his current research interests in active flow control and aeroacoustics.

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