Abstract A one dimensional finite element model is presented to assess the effect of position and size of the piezoelectric layer of a hybrid beam. The efficient layerwise (zigzag) theory is used for making the finite element model. The 1D beam element has eight mechanical and a variable number of electrical degrees of freedom. The codes are developed in Matlab based on the FE formulation. The beams are also modelled in 2D planar modelling space as a deformable shell using FE package ABAQUS for comparison of results. An 8-noded piezoelectric quadrilateral element is used for piezo layers and an 8-noded quadrilateral element with reduced integration is used for the elastic layers of hybrid beams for making the finite element mesh in ABAQUS. The accuracy of the used elements are assessed for static response. Cantilever hybrid beams with a piezoelectric layer bonded on top of the elastic substrate are considered for the analysis. The beams are subjected to electromechanical loading. A detailed study is conducted to highlight the influence of positon and size of piezoelectric layer on the deflection profiles, tip deflections and through the thickness distribution of displacements and stresses of hybrid composite/sandwich beams. The shape control using various numbers of piezoelectric patches is also studied. The 1D-FE results are compared with the 2D-FE results.

Abstract A 1D Finite Element model for static response and free vibration analysis of functionally graded material (FGM) beam is presented in this work. The FE model is based on efficient zig-zag theory (ZIGT) with two noded beam element having four degrees of freedom at each node. Linear interpolation is used for the axial displacement and cubic hermite interpolation is used for the deflection. Out of a large variety of FGM systems available, Al/SiC and Ni/Al2O3 metal/ceramic FGM system has been chosen. Modified rule of mixture (MROM) is used to calculate the young's modulus and rule of mixture (ROM) is used to calculate density and poisson's ratio of FGM beam at any point. The MATLAB code based on 1D FE zigzag theory for FGM elastic beams is developed. A 2D FE model for the same elastic FGM beam has been developed using ABAQUS software. An 8-node biquadratic plane stress quadrilateral type element is used for modeling in ABAQUS. Three different end conditions namely simply-supported, cantilever and clamped- clamped are considered. The deflection, normal stress and shear stress has been reported for various models used. Eigen Value problem using subspace iteration method is solved to obtain un-damped natural frequencies and the corresponding mode shapes. The results predicted by the 1D FE model have been compared with the 2D FE results and the results present in open literature. This proves the correctness of the model. Finally, mode shapes have also been plotted for various FGM systems.

AbstractIn this paper, modelling and simulation of a multi-body system for its vibration control using sensors and actuators is presented. The model developed on MSC ADAMS consists of six legs connecting moving and base plates using spherical joints with axial rotation of legs constrained. It contains linear actuator collocated with force sensor. Mathematical model of the mechanism representing the Kinematic and Dynamic analysis of model is presented. The Leg Length variation obtained using MATLAB is obtained in order to validate the model developed in MSC ADAMS. Natural frequencies for six different mode shapes are obtained from vibration excitation analysis. A decentralized force feedback controller which uses PID control law with single gain is used for actively attenuating the vibration, which is coming from base platform. The transfer function, defined as the ratio of Laplace transformation of the acceleration of moving plate and base plate, is considered as a measure of isolation. Effect of proportional, derivative and integral feedback is studied separately and together as well. A significant isolation in the resonance transmissibility and natural frequency of the mechanism is demonstrated.

A one-dimensional finite element model for buckling analysis of hybrid piezoelectric beams under electromechanical load is presented in this work. The coupled zigzag theory is used for making the model. The inplane displacement is approximated as a combination of a global third order variation across the thickness with an additional layer wise linear variation. The longitudinal electric field is also taken into account. The deflection field is approximated to account for the transverse normal strain induced by electric fields. Two nodded elements with four mechanical and a variable number of electric degrees of freedom at each node are considered. To meet the convergence requirements for weak integral formulation, cubic Hermite interpolation function is used for deflection and electric potential at the sub-layers and linear interpolation function is used for axial displacement and shear rotation. The expressions for the variationally consistent stiffness matrix and load vector are derived and evaluated in closed form using exact integration. The present 1D-FE formulation of zigzag theory is validated by comparing the results with the analytical solution for simply-supported beam and 2D-FE results obtained using ABAQUS. The finite element model is free of shear locking. The critical buckling parameters are obtained for clamped-free and clamped-clamped hybrid beams. The obtained results are compared with the 2D-FE results to establish the accuracy of the zigzag theory for above boundary conditions. The effect of lamination angle on critical buckling load is also studied.

Vibration suppression of smart beams using the piezoelectric patch structure is presented in the present work. The smart system consists of a beam as the host structure and piezoceramic patches as the actuation and sensing elements. An experimental set-up has been developed to obtain the active vibration suppression of smart beam. The set-up consists of a smart cantilever beam, the data acquisition system and a LabView based controller. Experiments are performed for different beam specimen. The coupled effcient layerwise (zigzag) theory is used for theoretical finite element modeling. The finite element model is free of shear locking. The beam element has two nodes with four mechanical and a variable number of electric degrees of freedom at each node. In the thickness direction, the electric field is approximated as piecewise linear across an arbitrary number of sub-layers in the piezoelectric layers. Cubic Hermite interpolation is used for the deflection, and linear interpolation is used for the axial displacement and the shear rotation. Undamped Natural Frequencies are obtained by solving the Eigen Value problem using Subspace Iteration method for cantilever beam. A state space model characterizing the dynamics of the physical system is developed from experimental results using PID approach for the purpose of control law design. The experimental results obtained by using the active vibration control system have demonstrated the validity and effciency of PID controller. Experiments are conducted to compare the controlling of various cantilever beams of different sizes. It shows that the present actuator and sensor based control method is effective and the LabView control plots for various beams can be used as a benchmark for analytical work. The results are compared with ABAQUS software and 1D Finite element formulation based on zigzag theory.