Due to the small linear region in electrorheological (ER) fluid, vibration analysis of the sandwich structure containing ER fluid should be investigated in nonlinear region where the material properties depend on frequency, amplitude and electric field. In present work, the nonlinear equations of motion have been obtained using finite element technique. The nonlinear amplitude dependent stiffness matrices in equations of motion have been previously expressed by B and N notations. In B-notation an asymmetric amplitude dependent stiffness matrix is achieved. On the other hand in N-notation a symmetric form of the nonlinear stiffness matrices is achieved. The main problem in nonlinear vibration analysis of structure using direct integration technique is the time-consuming integrations, which should be performed for several times throughout this method. Due to numerous degrees of freedom in sandwich shell/plate structures, the computational costs in finite element modeling of sandwich shell/plate structures becomes more expensive. In this study, by considering kinetic and potential energies attributed to the elastic and ER fluid layers and using Lagrange equations, nonlinear finite element formulation has been derived for the ER based sandwich shell structures. Also a new technique is developed to represent the equations of motion in a new notation referred to as H-notation which fundamentally reduces the computational costs in nonlinear vibration damping analysis of sandwich shell structure. Finally, parametric study is conducted to show the effect of small/ large displacement, electric field intensities and core thickness ratio on damping behavior of the ER based sandwich shell structures for different boundary conditions.

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