In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of generalized differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

More recently, Wazwaz [An analytic study of Fisher’s equation by using Adomian decomposition method, Appl. Math. Comput.154 (2004) 609–620] employed the Adomian decomposition method (ADM) to obtain exact solutions to Fisher’s equation and to a nonlinear diffusion equation of the Fisher type. In this paper, He’s homotopy perturbation method is employed for the generalized Fisher’s equation to overcome the difficulty arising in calculating Adomian polynomials.

This paper investigates on control and stabilization of a new hyperchaotic system. The hyperchaotic system is stabilized using a new technique which called Generalized Backstepping Method (GBM). Because of its similarity to Backstepping approach, this method is called GBM. But, this method is more applicable in comparison with conventional Backstepping. Backstepping method is used only for systems with strictly feedback form, but GBM works for a wide range form of the nonlinear dynamical systems. In Design procedure, two cases is considered that their difference is in the number of control inputs. Numerical simulation results are presented to show the effectiveness of the proposed controller.

In recent years the interest in studying the absolute value equation has been of great interest both theoretically and practically. The main reason for this is that various optimization problems, such as the complementary linear programming problem, can be written in the form of a robust equation that is easier to solve. The main purpose of this paper is to present a duplicate method for solving the equations of magnitude. Actually, in this paper, by introducing a scalar matrix, an improved generalized Newton method is proposed to solve the Absolute value equation. This new method is based on the methods of Mangserin [1] and Li [2]. In fact, if in the matrix A + α I-D, the value of the identity matrix coefficient is equal to zero, the Mangserin method and if the coefficient of the same matrix to one, the Li method is obtained. when all the singular values of the system matrix exceed one.

There are several methods to solve generalized eigenvalue problem, AX=lBX but the number of computed eigenvalues by these methods might be less or more than expected. There are also methods which can lead us to the amount of results needed. In this paper we describe free inverse method for computing the generalized eigenvalue. Then, by combination this method and projection method, we develop two new iterative method for solving the generalized eigenvalue problem. The implementation of the algorithm has been tested by numerical examples, the results show that the algorithm converges fast and works with high accuracy.