American Journal of Mechanics and Applications 2016; 4(1): 1-9 http://www.sciencepublishinggroup.com/j/ajma doi: 10.11648/j.ajma.20160401.11 ISSN: 2376-6115 (Print); ISSN: 2376-6131 (Online) Modeling of MEMS Resonator Piezoelectric Disc Partially Covered with Electrodes Ismail Naciri 1, * , Lahoucine Elmaimouni 1 , Jean-Etienne Lefebvre 2 , Faniry Emilson Ratolojanahary 3 , Mohamed Rguiti 4 , Tadeusz Gryba 2 1 Laboratoire Sciences Ingénierie et Energie, Energie Renouvelable, Microsystèmes Acoustique et Micanique, Polydisciplinary Faculty of Ouarzazate, Ibn Zohr University, Morocco 2 The Institute of Electronics, Microelectronics and Nanotechnology, Opto-Acousto-Electronic Department, University of Valenciennes, France 3 Laboratory of Applied Physics, Fianarantsoa University, Madagascar 4 Laboratoire des Matériaux Céramiques et Procédés Associés, Université de Valenciennes, Maubeuge, France Email address: nacirismail@gmail.com (I. Naciri), la_elmaimouni@yahoo.fr (L. Elmaimouni) * Corresponding author To cite this article: Ismail Naciri, Lahoucine Elmaimouni, Jean-Etienne Lefebvre, Mohamed Rguiti, Faniry Emilson Ratolojanahary, Tadeusz Gryba. Modeling of MEMS Resonator Piezoelectric Disc Partially Covered with Electrodes. American Journal of Mechanics and Applications. Vol. 4, No. 1, 2016, pp. 1-9. doi: 10.11648/j.ajma.20160401.11 Received: August 31, 2016; Accepted: September 26, 2016; Published: October 19, 2016 Abstract: The Legendre polynomial method has been extended to the modeling of MEMS resonator disc partially covered with electrodes. The disc has been divided into two areas: one with electrodes and the other without electrodes. For each area, The Maxwell equations and the piezoelectric constitutive equations of motion are studied and solved to yield a frequency response and electrical behavior of the MEMS resonator applying a semi analytical method based on a Legendre polynomials series and trigonometric functions. However, the method allows incorporating the boundary conditions directly into the governing equations by assuming position-dependent of elastic constants, mass density and delta functions. The alternating electrical source is described by specific terms which are also introduced into the equation of motion. The formalism has been developed which allows for both harmonic and modal analyses. In order to validate our polynomial approach, numerical results are presented such as resonant and anti-resonant frequencies, electric input admittance, electromechanical coupling coefficient and field profiles of fully and partially metallized PZT5A resonator discs. The results obtained were compared with those obtained by an approximated analytical method. The developed software proves to be very efficient to retrieve the contour modes of all orders. Keywords: MEMS Resonators, Legendre Polynomial Approach, Centralized Metallization, Piezoelectric Resonator Disc, Electrical Admittance, Resonant, Anti-resonant Frequencies 1. Introduction When a piezoelectric material is subjected to a mechanical strain, electrical charges were generated and conversely. This phenomenon was widely exploited in various engineering applications such as Micro-Electro-Mechanical systems (MEMS) technology. MEMS technology has been obtained significant growth in its field of application such as in electro-optic modulators, ultrasonic detectors, accelerometers, transducers, oscillators, electromechanical sensors and actuators [1-6]. Although, the vibration characteristics of piezoelectric materials are extracted from the piezoelectric constitutive equations, linear piezoelectricity and the Maxwell equations [7-8]. Several methods allow calculating vibration characteristics of piezoelectric devices. In 1967, Eer Nisse [9] presented a vibrational method to analyze the vibrational behavior of piezoelectric disks. C.H.Huang and C.C Ma used Electronic speckle pattern interferometry method to study vibration