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Displacement Amplifier Mechanism for Piezoelectric Actuators Design

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Displacement Amplifier Mechanism for Piezoelectric Actuators Design Displacement amplifier mechanism for piezoelectric actuators design using SIMP topology optimization approach Thomas Schlinquer, Abdenbi Mohand-Ousaid and Micky Rakotondrabe, member, IEEE Abstract— Due to their inherent crystalline properties piezo- non-compact actuators. The second configuration combines electric actuators have a limited deformation. This intrinsic two piezoelectric layers. It provides a larger displacement drawback deprives to exploit the potential of these actuators than piezostack however the provided force is low. Another such as, high bandwidth and high resolution in applications that require large displacement range. To overcome this limitation, classical approach employs external mechanism to magnify classical as well as systematic approaches were proposed to the deformation of the piezoelectric actuator. Known as am- design amplification mechanisms. The classical approach leads plifying mechanism, several structures have been proposed to empirical mechanisms which are not trivial and needs much in the literature. The widespread one is the flexural lever experience and intuition. In contrast, systematic approach uses structure [4]. Flextensional mechanism such as Moonie [5] topology optimization method which permits to automatically derive optimal designs that can satisfy specified performances [6], Cymbal [7] and nested rhombus structure [3] are also and imposed constraints simultaneously, this with a reasonable investigated. However, the design of such mechanisms is not time and cost. trivial because much experience and intuition are required. This paper proposes the design of a mechanism devoted to In contrast to the above mentioned approach, systematic amplify the displacement of a piezoelectric actuators (PEA). approaches including topology optimization [8], interval [9]– Based on the SIMP topology optimization method, the approach permits to derive a design with a displacement amplification [11] and blocs methods [12] can be used. Topology optimiza- ratio of 4:5, which is higher than with the existing method of tion [13] [14] [15] [8] is particularly suitable to systematize Rhombus mechanism. Both finite element (FE) simulation and the design of such mechanisms. It aims to propose a system- experimental results confirm and demonstrate the efficiency of atic methodology for optimal designs and it can find its appli- the approach. cation in a wide variety of field, from designing a lightweight Index Terms— piezoelectric actuators, optimal design, com- city bus, bone remodelling for prosthesis design or frame op- pliant structure, SIMP topology optimization timization design for crashworthiness of cars as presented in [8]. Considering the amplification mechanisms as monolithic I. INTRODUCTION compliant structures, this method can be applied directly to Nowadays, the interest of piezoelectric actuators is con- master the trade-off between the mass, the volume and the siderable and well established [1]. As they provide high stiffness of the structure. The literature review shows that displacement resolution, large output force, high dynamic various studies attempted to utilize this approach to design response and high energy density, they have been used mechanical amplifier. In 1997, Sigmund was the first to apply in various applications where high precision is required, topology optimization to design an inverting displacement for instance: precise alignment, scanning force microscopy, amplifier based on maximum mechanical advantage [14]. micro/nano assembly and micro/nano manipulation and in- Subsequently, Canfield et al. proposed a design where the terferometry. Nevertheless, one of their main drawback is overall stroke amplification and mechanical efficiency are the low relative deformation restricting their range of mo- considered as objective functions [16]. Silva et al. applied the tion. In fact, larger strain would increase the performances homogenization formulation of topology optimization to de- and range of applications of piezoelectric actuators and is sign flextensional actuator and piezoelectric transducer [17] therefore desirable. To deal with this limitation, several [18]. Unlike these studies, another formulation is proposed solutions have been proposed to increase the stroke of in [2] [19] to design an amplification mechanism under these actuators. As reviewed in [2] and [3] classical as dynamic motion. Other designs have been also presented well as systematic approaches were investigated. The trivial in [20] [21]. Although the mentioned studies describe in- approach that operates on the shape, the polarity and the teresting solutions, their interpretation remains difficult and geometric arrangement of the actuator leads to piezostack, thus their application to new physical systems design is not bimorph and rainbow configurations. In the first one, several straightforward. This is mainly due to the formulation of the piezoelectric layers are stacked together in serial. In this topology optimization based on homogenization method and case, the output displacement is the sum of the displacement the variable thickness method which are purely mathematical of each actuation layer. However to produce reasonable and which are far to physical signification. Furthermore, the output displacement, this configuration leads to long and problem resolution leads often to designs that require a post- processing step to extract a realistic structure. Finally there Authors are with FEMTO-ST Institute and the Univ. Bourgogne Franche- is no study focusing on how the supports and the volume of Comte´ / UFC / CNRS UMR-6174 / 24, rue Alain Savary, 25000 Besancon France fthomas.schlinquer,abdenbi.mohand, the amplification mechanisms influence the optimal solution. [email protected] To address the first matter, we propose here to use SIMP kout ξ4y ξ3y 4 3 ξ4x ξ3x fin ξ1x ξ2x kin 1 2 ξ1y ξ2y Ωvoid Ωmat Fig. 2. Support modeling where each node have springs attached in X and Y direction. Γsup optimization algorithm and the computation of sensitivities Fig. 1. Mechanical modeling of a topology optimization problem. are presented. For more details readers can refer to [8]. A. Modeling (Solid Isotropic Material Penalization) formulation of topol- Consider a mechanical element occupying a domain Wmat ogy optimization to design a novel amplification mechanism. as illustrated in Fig. 1 included in a larger reference domain This formulation uses penalization power law to make ma- W in R2. This domain is modeled as a plane-stress solid. terial intermediate density unattractive and therefore avoids There is also a fixed domain Wvoid ⊂ Wmat without material. the 0-1 problem. Otherwise, it is mathematically well-posed Furthermore a number of supports are to be set on a fixed and easy to implement. Indeed, a Matlab implementation boundary Gsup: two fixed input ports lie on the Gvoid bound- including 99 code lines was proposed by Sugmund in [22]. ary and one fixed output port lies on Gmat nGsup boundary. The To address the second matter we propose to substitute the OC piezoelectric actuator which is a stack (piezostack) placed at (Optimality Criteria) method originally implemented in [22] the input port is assumed to have a linear behavior. It has by MMA method (Method Moving Asymptotes) in order to a lamped model characterized by a spring of stiffness kin. take into account more than one constraint. Finally once the The driving voltage will be represented by an input force optimization is realized under Matlab software, we propose fin. From these parameters, the free (un-loaded) actuator an automatic procedure to extract the resulted structures. displacement is given by: uin = fin=kin, as demonstrated in This step is important to convert the structures to a format [8] for further details. compatible with CAO software’s. To resume, this paper deals B. Method formulation with the optimal design of an amplification mechanism for piezoelectric actuator. The contributions of the paper are: In this paper we propose to use the SIMP method (eq. 2c- d) [8] to perform the optimization under finite element (FE) - an improvement of the 99 code of topology optimization linear elasticity theory with Hooke’s law (eq. 2b). The clas- presented in [22] by using MMA method, sical FE approach is used with 4-node bilinear quadrilateral - an analysis of the influence of the supports and the elements mapping the whole design domain. The objective volume as design variables on the amplification mech- function is set as the maximum output displacement u , anism, out reformulated as a minimization (eq. 2a), with a constraint on - an automatic extraction of the designed structure under the maximum volume (eq. 2-f). In these equations, r, K and Matlab, realization and experimental validation of the K are the density, the global and the element wise stiffness optimal amplification prototype. e matrix respectively; U and F are the displacement and the The remainder of the paper is organized as follows. force vector respectively; N is the total number of element of First, section-II provides the SIMP topology optimization the mesh, re is the element wise density, p is the penalization method adapted to a mechanism design. In section-III, the factor, and finally rmin stands for the minimum bound for specification of the amplification mechanism are formulated the density and Vf rac the normalized maximum allowed in order to further apply SIMP method. The influence of the volume between 0% and 100%. Regarding the optimization supports and the volume on the optimal designs is analyzed
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