Forced vibration of smart laminated viscoelastic plates by RPT finite element approach
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RESEARCH PAPER
Forced vibration of smart laminated viscoelastic plates by RPT finite element approach Jafar Rouzegar1 · Maryam Davoudi1 Received: 5 November 2019 / Revised: 24 March 2020 / Accepted: 15 May 2020 © The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this study, a finite element formulation based on the four-variable refined plate theory (RPT) is presented for forced vibration analysis of laminated viscoelastic composite plates integrated with a piezoelectric layer. To the best of the authors’ knowledge, this is the first time that the proposed approach is extended for study of the dynamic behavior of the smart viscoelastic plate. The utilized RPT which works for both thick and thin plates predicts a parabolic variation for transverse shear stresses across the plate thickness. Considering a linear viscoelastic model for the substrate material, the relaxation module is predicted by the Prony series. Using Hamilton’s principle, the weak form equation is constructed and a four-node rectangular plate element is utilized for discretizing the domain. The Newmark scheme is employed for advancing the solution in time. A MATLAB code is developed based on the formulations and several benchmark problems are solved. Comparing the findings with existing results in previous studies confirms the accuracy and efficiency of the proposed method. The dynamic response of the smart viscoelastic plates under various electromechanical loads is investigated and the results show that the vibration can be passively controlled by adding and actuating the piezoelectric layer. The damping effects of viscoelastic parameters on the results are investigated, too. Keywords Forced vibration · Refined plate theory · Viscoelastic material · Piezoelectricity · Smart structure · Finite element method List of symbols ̇ Ȧ s A matrix defined in vector form of weak A, As , A, form governing equation a Plate length ̇ Ḃ s A matrix defined in vector form of weak B, Bs , B, form governing equation b Plate width C Damping Cij Material matrix components ̇ Ḋ s A matrix defined in vector form of weak D, Ds , D, form governing equation d Electrical displacement vector D𝟏 to D𝟔 Differential operator vectors e Piezoelectric constant matrix e𝟏xp to e𝟑xp Piezoelectric constant vectors Ei Elastic modulus Ė i Rate of elastic modules * Jafar Rouzegar [email protected] 1
Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran
F Force vector Fq Sub-vectors for the force vector Feff Effective force vector f Defined function in displacement field Gij Shear modulus Ġ ij Rate of shear modules g Defined function in strain tensor s Hs , Ḣ A matrix defined in vector form of weak form governing equation h Substrate thickness hp Piezoelectric layer thickness ht Total plate thickness I0 to I5 Inertia terms defined in vector form of weak form governing equation K Stiffness matrix K ij Elastic sub-matrices
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