A Newtonian (vectorial) approach is used to develop the governing differential equations of motion for a Three layer sandwich beam in which the uniform distribution of mass and stiffness is dealt with exactly. The model allows for each layer of material to be of unequal thickness and the effects of coupled bending and longitudinal motion are accounted for. This results in an eighth order ordinary differential equation whose closed form solution is developed into an exact dynamic member stiffness matrix (exact finite element) for the beam. Such beams can then be assembled to model a variety of structures in the usual manner. However, such a formulation necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick-Williams algorithm, whose implementation is discussed in detail. The algorithm enables any desired natural frequency to be converged upon to any required accuracy with the certain knowledge that none have been missed. The accuracy of the method is then confirmed by comparison with five sets of published results together with a further example that indicates its range of application. A number of further issues are considered that arise from the difference between sandwich beams and uniform single material beams, including the accuracy of the characteristic equation, co-ordinate transformations, modal coupling and the application of boundary conditions.

Being inspired by nature and moving towards clean energy has become a very necessary and indispensable objective these days. By observing a collection of plants/vegetation growing underneath the rivers, oceans, and seas, the idea of designing a plant farm being capable of absorb fluid energy was born. The farm comprises numerous aluminum cantilever beams equipped with a piezoelectric layer in different shapes. The field has elements of different sizes and natural frequencies, where the maximum voltage can be obtained using the resonance phenomenon. Frequency of the fluid flow is considered, because it follows the oscillatory behavior of the wakes and vortices when the fluid passes through the obstacle. The Strouhal number helps to obtain the fluid frequency according to the fluid velocity and the characteristic length of the barrier. In this research, it was observed that the increase in the length of the aluminum layer caused a rise in the voltage generated. The triangular model produced a higher amount of voltage compared with the other Three-dimensional models i.e. the rectangular, trapezoidal and triangular ones. Therefore, it was concluded that providing the farm with triangular elements with different dimensions is considered to be an effective and reasonable measure. Actually, having a variety of sizes of these elements covers a wide range of natural frequencies causing a higher number of them to be excited.

This research presents a semi analytical solution of finitely long, simply supported, orthotropic, piezoelectric, radically polarized, shell panel under pressure and electrostatic excitation. The general solution of the governing partial differential equations is obtained by the method of separation of variables. The displacements and electric potential are expanded in appropriate trigonometric Fourier series in the circumferential and axial coordinate to satisfy the boundary conditions at the simply-supported circumferential and axial edges. The governing ordinary differential equations are solved by the Galerkin finite element method. In this procedure, the quadratic shape function is used in each element. Numerical examples are provided for typical external pressure on outer .I'll/face of a single layer piezoelectric panel.

The current article investigates the free vibrations of a Three-layer beam. The middle layer of this structure is selected from porous material. For modeling the porous layer, linear pro-elasticity relationships are applied, while Young's modulus and its density vary along the thickness. The upper and lower layers of the structure are reinforced with graphene nanoplates and can take different configurations as Parabolic, linear, and uniform. In this study, with the help of Halpin-Tsai modified theory, equivalent composite coefficients will be extracted. The equations of motion in Three layers are derived with the help of third order shear theory, energy method and Hamilton's Principle. Among the significant results of this article, we can mention the effect of amplifiers in improving the vibration behavior of the beam, the effect of pore pressure and volume fraction of reinforcement on the frequency of vibrations. The results of this research can be applied in marine, aerospace, and civil industries. ©,2023 IAU, Arak Branch. All rights reserved.