Finite element modelling to assess the effect of surface mounted piezoelectric patch size on vibration response of a hybrid beam

Vibration response analysis of a hybrid beam with surface mounted patch piezoelectric layer is presented in this work. A one dimensional finite element (1D-FE) model based on efficient layerwise (zigzag) theory is used for the analysis. The beam element has eight mechanical and a variable number of electrical degrees of freedom. The beams are also modelled in 2D-FE (ABAQUS) using a plane stress piezoelectric quadrilateral element for piezo layers and a plane stress quadrilateral element for the elastic layers of hybrid beams. Results are presented to assess the effect of size of piezoelectric patch layer on the free and forced vibration responses of thin and moderately thick beams under clamped-free and clamped-clamped configurations. The beams are subjected to unit step loading and harmonic loading to obtain the forced vibration responses. The vibration control using in phase actuation potential on piezoelectric patches is also studied. The 1D-FE results are compared with the 2D-FE results.


Introduction
Many light weight structural components are made of composite and sandwich laminates because of their high stiffness to weight ratio and similar other properties. Most of these components are generally modelled as smart multi-layered beams. These have found varied applications in the aerospace, robotics, defense equipments, measurement devices, satellites, automobile and ship building industries. Precise analysis of hybrid laminates require systematic and precise approximation of the displacement and the potential field across the thickness. Hodges et al. [1] presented methods for predicting the vibration frequencies of composite beams and their mode shapes. They proposed different methods to determine the sectional elastic constants and ways to solve the equations of motion. Banergee [2] used dynamic stiffness matrix method to carry out the free vibration analysis of composite Timoshenko beams loaded in axial direction. The theory includes the coupling between the bending and torsional modes of deformations. Lee et al. To whom any correspondence should be addressed. beams using micromechanics approach. Chevallier et al. [8] presented a 3D FE simulation for free vibration of thick beams and plates with a symmetric pair of surface mounted piezoceramic patches. Rahman and Alam [9][10][11] presented 1D-FE modelling using layerwise (zigzag) theory for dynamic analysis of smart laminated beam. Giunta et al. [12] studied free-vibration analysis of cross-ply beams under simply supported configuration. They used various higher-order and classical theories for the analysis and approximated the through the thickness displacement field in the form of a polynomial expansion. Koutsawa et al. [13] presented 1D finite elements to attain the free vibration response of composite smart beams. They approximated displacements and electric potential using Lagrange's polynomials. Shang et al. [14] used Generalized FE Method to carry out the dynamic analyses of 1D bar and beam problems. They compared their results with conventional FE formulation to prove efficiency of the method. Sayyad A S and Ghugal [15] presented a critical review of literature on bending, buckling and vibration analysis of laminated beams based on various theories. He et al. [16] proposed a new model for the two-layer composite beam with partial interaction by modifying Reddy's higher order beam theory. They used Hamilton's principle for formulation of governing equations for free vibration and buckling.
A one dimensional finite element (1D-FE) model based on efficient layerwise (zigzag) theory is presented for the vibration response analysis of a hybrid beam with surface mounted patch piezoelectric layers. The size of the piezoelectric patches are varied and its effects on free and forced vibration responses are studied. Results are presented for thin and moderately thick beams under clamped-free and clamped-clamped configurations. Unit step pressure loading and harmonic pressure loading are applied to set forced vibration in beam. The vibration control using in phase actuation potential on piezoelectric patches is also studied. The 1D-FE results are compared with the 2D-FE results.

Finite element model of hybrid beam with patch piezoelectric layers
Consider a hybrid beam with surface mounted piezoelectric patches. The beam is divided into elastic and hybrid segments due to the existence of these patches. Using coupled zigzag theory approximations [5], the potential and displacement field variables  , x u and z u are assumed as are the electric field components. Let b q be the pressure on the bottom surface and t q on the top surface of the beam. The variational equation, using the extended Hamilton's principle [17] for the beam, is obtained as: where , II are inertia matrices;   (9) where, I is beam inertia matrix; D is the generalized stiffness matrix of the beam; The contributions from all the elements are summed up to obtain the system equation as where M , C and K are assembled from element inertia, damping and stiffness matrices , ee MC and e K respectively. d, P are the assembled displacement and load vectors respectively. Equation (10) can be partitioned and the damping matrix and load terms are set as zero to obtain the undamped free vibration response.The emanating eigenvalue problem is solved using subspace iteration method to acquire first n undamped natural frequencies n  .

Results and discussion
A composite substrate (b) and a sandwich substrate (c) are considered for the analysis. The substrate (b) is a symmetric graphite epoxy composite laminate of four layers with layup (0ᴼ/ 90ᴼ/90ᴼ/0ᴼ) and thicknesses (0.25h/0.25h/0.25h/0.25h). The substrate (c) is a symmetric sandwich laminate of three layers. The thickness of top and bottom face sheets is taken as 0.08h and the thickness of soft core is taken as 0.84h. Here h is the total thickness of the elastic substrate for both the beams. Piezoelectric patch (PZT-5A) is bonded on top surface of the elastic substrate at fixed end for cantilever configuration and at one of the clamped ends for clamped-clamped configuration. The width of the beam and piezo patch are modelled as unity and the thickness of piezo-patches is taken as 0.1h. The length of PZT-5A patch,     Table 1 and table 2 show comparison of first three normalised natural frequencies for beam b and beam c respectively with different piezo patch sizes for span to thickness ratios, 10, 20 S  . The values of frequencies increase as the beam becomes thinner for both the end conditions. The results are in compliance with the 2D-FE results obtained using ABAQUS for both the hybrid beams.   The beams vibrate at the first natural frequency in case of unit step loading and at the forcing frequency in case of harmonic loading. It is observed that the size of piezo-patch layer has no significant effect on amplitude of vibration for closed circuit condition i.e. for 0 0   . The controlled tip deflection is obtained with applied in phase actuation potential on top surface of piezo-patch layer and the amplitude of deflection can be minimised by using suitable value of the actuation potential. Further for the same actuation potential, the deflection amplitude can be restricted to a lower value with a larger size of piezo patch layer.

Conclusions
The size of piezoelectric patch layers in smart beams is significant as it results in different structural response. A 1D-FE model to assess the effect of size of piezoelectric patch layer on vibration response of a hybrid beam is presented. The results are obtained for clamped-free and clamped-clamped end conditions. With increase in size of piezo patch layer, the free vibration frequencies decrease for all the vibration modes. This decrease is less significant for lower modes but more significant for higher modes. In case of forced vibrations, the size of piezo-patch layer has no significant effect on amplitude of vibration for closed circuit condition i.e. for 0 0   . The controlled tip deflection is obtained with applied in phase actuation potential on top surface of piezo-patch layer. The piezoelectric patch actuators of large size placed nearer to the fixed end with the application of appropriate actuation potential results in better control of tip deflection in cantilever beams. These observations may be used as a basis for experimental investigation and design for vibration control system for composite and sandwich hybrid structures.