ORIGINAL CONTRIBUTION 2D Free Vibration Solution of the Hybrid Piezoelectric Laminated Beams Using Extended Kantorovich Method Agyapal Singh 1 • Poonam Kumari 1 • Prabhakar Bind 1 Received: 14 August 2018 / Accepted: 22 May 2019 Ó The Institution of Engineers (India) 2019 Abstract Analytical two-dimensional (2D) piezoelasticity free vibration solution is presented for the beams under different combinations of support conditions, using the multi-term multi-field extended Kantorovich method (MMEKM). Piezoelasticity-based extended Hamilton principle with the mixed variational field is applied to derive the governing equations in terms of stresses, dis- placements along with electric displacements and electric potential. Therefore, boundary conditions, both natural and essential, are satisfied in an exact manner at all points. By employing MMEKM, the first-order differential–algebraic system of 8n equations is obtained along the z-direction (thickness) for each layer and another set along the x-di- rection (in-plane). The final solution of these first-order ODEs is obtained in closed form. The numerical results are verified by comparing against the exact 2D solution available in the literature for the simply supported bound- ary condition case and with 2D finite element results for other support conditions. New benchmark results for free vibration are presented for piezoelectric beams subjected to arbitrary boundary conditions. Keywords Extended Kantorovich method  Piezoelectric beam  Free vibration  2D piezoelasticity  Analytical  Energy harvesting List of Symbols x, z Coordinates in axial and thickness directions a, h Span length, thickness of beam u; w Displacement along x and z, respectively r i , e i Normal stresses and normal strains s ij , c ij Shear stresses and shear strains E i , D i , / Electric field, electric displacements and electric potential Y i , G ij , m ij Young’s moduli, shear moduli and Poisson’s ratio  e ij , g ij Constant stress field dielectric permittivities, constant strain dielectric permittivities  s ij ,  d ij , q Transformed elastic compliances, piezoelectric strain constants and density x n ,  x Natural frequency, non-dimensionalized frequency parameter Introduction Laminates integrated with piezoelectric layers are known as ‘smart structures.’ Piezoelectric material layers are used to regulate the static and dynamic behavior of the structures through actuation and sensing and also used to control the vibration generated during operation which increases safety, usability and durability of structures [1]. Thus, the application of piezoelectric composite laminates in struc- tural components has increased extensively in the field of civil engineering, automobile, aeronautics and medical fields. In engineering and medical fields, we use load cells, pressure sensors, accelerometers, gyroscopes and ultra- sonic transducers in which piezoelectric beams or disks act as sensing and actuating mechanism. Piezoelectric beams are also used in small-scale energy harvesting devices, which convert vibrational energy into an electrical voltage or DC power with the help of an electronic circuit. & Poonam Kumari kpmech@iitg.ac.in 1 Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India 123 J. Inst. Eng. India Ser. C https://doi.org/10.1007/s40032-019-00518-w