In this research, the frequency characteristic equation for some Non-uniform and homogeneous beams based on Euler-Bernoulli beam theory is presented in closed analytical form. At both ends, the beam carries objects with deviations from the center, mass moment of inertia and linear and rotational elastic constraints. Therefore, the closed analytical characteristic equation has the ability to provide frequency parameters for a wide range of Non-classical boundary conditions. Solving the governing differential equation and applying boundary conditions leads to the solution of an eigenvalue problem. Since the beam is Non-uniform, the exact solution of the governing differential equation depends on finding a closed analytical solution for the beam deflection. Therefore, a limited type of Non-uniform beams can be accurately solved. In order to verify the validity and accuracy of the presented relationships, the results of the method presented in this research have been compared with the existing results for uniform beams. Also, the mode shape of the deflection are given for a beam sample.

This work presents free vibration analysis of laminated rectangular plates with classical boundary conditions based on the classical plate theory (CPT). The first involves an S-FGM plate with different boundary conditions. Then involves a two-layer plate in which an S-FGM layer is coated on a homogeneous substrate with classical boundary conditions, simply called an S-FGM-coated plate; the other involves a three-layer plate in which an S-FGM employed for intermedium layer and different homogeneous materials are in top and bottom layers with classical boundary conditions; this is called an S-FGM-undercoated plate. The Young’s modulus of FGM plates is assumed to vary in the thickness direction and the poison’s ratio remain constant throughout the FGM plate. The material gradations in FGM portion of laminate structures follow sigmoid functions; therefore these plates are called S-FGM. For multi-layer structures, the calculations of the quantities A11, B11, C11, Q11, S11 defined herein become complex, therefore, a method is used to easily and efficiently calculating them in this investigation. The results are first compared with the ABAQUS software, showing an excellent agreement. Then frequency parameters of three plates for different boundary conditions and aspect ratios are graphically presented for a wide range of gradient index.

In this paper the wave equation in some Non-classic cases has been studied. In the first case boundary conditions are Non-local and Non- periodic. At that case the associated spectral problem is a self-adjoint problem and consequently the eigenvalues are real. But in the second case the associated spectral problem is Non-self-adjoint and consequently the eigenvalues are complex numbers, in which two cases, the solutions of the problem are constructed by the Fourier method. By compatibility conditions and asymptotic expansions of the Fourier coefficients, the convergence of series solutions are proved.Finally, series solutions are established and the uniqueness of the solution is proved by a special way which has not been used in classic texts.

In this article, for the first time, the effect of Non-uniformity of microbeam cross section and various boundary conditions on the Nonlinear vibration of microbeam is investigated considering the size dependent behavior based on modified couple stress theory. Using the Hamilton’ s principle, the governing equation of Euler– Bernoulli microbeam with von Karman geometric Nonlinearity based on the modified couple stress theory is derived. The Nonlinear vibration governing equation is then solved using the Generalized Differential Quadrature method (GDQ) and direct iterative method to obtain the Nonlinear natural frequencies. In this step, the Galerkin method is used to reduce the Nonlinear PDE governing the vibration into a time-dependent ODE of Duffing-type. The time domain is then discretized via spectral differentiation matrix operators which are defined based on the derivatives of a periodic base function. Next, the Nonlinear parametric equation is solved using pseudo arc-length method and the frequency– response curves of microbeam Nonlinear forced vibration is obtained. Finally, Nonlinear natural frequency and frequency response of microbeam with various Non-uniformity of cross sections and boundary conditions are obtained. Present results show that, the Nonlinear free and forced vibration of microbeam is size dependent. Moreover, this size dependency is more significant for Non-uniform microbeam and is deferent for various boundary conditions. The result of present method for simple case including uniform section and simply supported boundary condition is validated with that of exact method and have good agreement.

Singular perturbation problems have been studied by many mathematicians. Since the approximate solutions of these problems are as the sum of internal solution (boundary layer area) and external ones, the formation or Non-formation of boundary layers should be specified. This paper, investigates this issue for a singular perturbation problem including a first order differential equation with general Non-local boundary condition. It needs to say that it is simple for local boundary conditions and there is no difficulty. However, the formation of boundary layers for Non-local case is not as straight forward as local case. To tackle this problem generalized solution of differential equation and some necessary conditions are used.

In this paper we study the boundary layer problems in which boundary conditions are Non-local. Here we try to find the necessary conditions by the help of fundamental solution to the given adjoint equation. By getting help from these conditions, at first the boundary condition is changed from Non-local to local. The main aim of this paper is to identify the location of the boundary layer. In other words, at which point the boundary layer is formed.

In this papear, we produce the method for formation and recognizing boundary layers in singular perturbation problems. This method involves four step for localization of Non-local boundary conditions to local case. For the given problem some sufficient and necessary conditions are given for formation and Non formation of boundary layers. Since the existence of boundary layers and their places has a direct relation with the structure of approximate solutions and uniform solutions, therefore the main purpose of this paper is recognition and formation of boundary layers in singular perturbation problems with Non-local boundary conditions. This process will be done by using fundamental solution of adjoint given differential equation and necessary conditions. In fact by using these necessary conditions and given boundary conditions, we make an algebraic system. By solving this algebraic system by Cramer rule we obtain boundary values of unknown function. These values of unknown function are local boundary conditions. The mathematical model for this kind of problem usually is in the form of either ordinary differential equations (O.D.E) or partial differential equations (P.D.E) in which the highest derivative is multiplied by some powers of as a positive small parameter.