As a result of environmental and accidental actions, damage occurs in structures. The early detection of any defect can be achieved by regular inspection and condition assessment. In this way, the safety and reliability of structures can be increased. This paper is devoted to propose a new and effective method for detecting, locating, and quantifying beam-like structures. This method is based on Rayleigh-Ritz approach and requires a few numbers of natural frequencies and mode shapes associated with the undamaged and damaged states of the structure. The great advantage of the proposed approach against the other methods is that it considers all kinds of boundary and damping effects. To detect damage using the penalty method, this article considers lumped rotational and translational springs for determining the boundary conditions. Result will show that the proposed method is an effective and reliable tool for damage detection, localization, and quantification in the beam-like structures with different boundary conditions even when the modal data are contaminated by noise.

In this paper, the Ritz method has been employed to analyze the free inplane vibration of heterogeneous (non-uniform) rectangular nanoplates corresponding to Eringen’ s nonlocal elasticity theory. The non-uniformity is taken into account using combinations of linear and quadratic forms in the thickness, material density and Young’ s modulus. Two-dimensional boundary characteristic orthogonal polynomials are applied in the Ritz method in order to examine the nonlocal effect, aspect ratio, length of nanoplate and non-uniformity parameters on the vibrational behaviors of the nanoplate. Results are verified with the available published data and good agreements are observed. The outcomes confirm apparent dependency of inplane frequency of nanoplate on the small scale effect, non-uniformity, aspect ratio and boundary conditions. For instance, frequency parameter decreases by increasing the nonlocal parameter in all vibration modes; the frequency parameters increase with length and aspect ratio of nanoplates. Furthermore, the effect of nonlocal parameters on the frequency parameter is more prominent at the higher aspect ratios. Finally, the effect of nonlocal parameter on the in-plane modes is also presented in this analysis.

This work presents forced vibration responses of a cantilever beam made of functionally graded material under a harmonic load. The material properties of beam vary along the axial direction. The kinematics of the beam are considered within Timoshenko beam theory. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem the Ritz method is used and algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method is used in the time history. In this study, free and forced vibration responses of the axially functionally graded beam are investigated in detail. In the numerical examples, the effects of material graduation, geometric and dynamic parameters on the free and forced vibration response of axially graded beam are investigated.

In this paper, the Ritz-Galerkin method in Bernstein polynomial basis is applied for solving the nonlinear problem of the magnetohydrodynamic (MHD) flow of third grade fluid between the two plates. The properties of the Bernstein polynomials together with the Ritz-Galerkin method are used to reduce the solution of the MHD Couette flow of non-Newtonian fluid in a porous medium to the solution of algebraic equations.

Bending, buckling and vibration behaviors of nonlocal Timoshenko beams are investigated in thisresearch using a variational approach. At first, the governing equations of the nonlocal Timoshenko beams areobtained, and then the weak form of these equations is outlined in this paper. The Ritz technique is selected toinvestigate the behavior of nonlocal beams with arbitrary boundary conditions along them. To find the equilibriumequations of bending, buckling, and vibration of these structures, an analytical procedure is followed. In order toverify the proposed formulation, the results for the nonlocal Timoshenko beams with four classical boundaryconditions are computed and compared wherever possible. Since the Ritz technique can efficiently model the nanosizedstructures with arbitrary boundary conditions, two types of beams with general boundary conditions areselected, and new results are obtained.

‎This paper presents a numerical approach for reconstructing the leading coefficient in an inverse heat conduction problem (IHCP)‎. ‎We consider a one-dimensional heat equation with known input data‎, ‎including the initial condition‎, ‎a supplementary temperature measurement at the final time‎, ‎and two integral observations‎. ‎By incorporating the terminal condition‎, ‎the unknown spatially dependent coefficient is eliminated‎, ‎reducing the problem to a nonclassical parabolic equation‎. ‎The unknown temperature distribution and its derivatives are approximated and applied to the modified governing equation‎, ‎which is then discretized using operational matrices of differentiation‎. ‎To ensure stable derivative estimation‎, ‎the method is coupled with a regularization technique‎. ‎A least squares scheme is employed to formulate a nonlinear system of algebraic equations‎, ‎which is solved using Newton’s method‎. ‎The reliability of the proposed solution is demonstrated through several numerical examples‎.

In this article, we solve control problems with the conditions of parabolic partial differentialequations. In order to solve this type of mentioned problems, we use an iterative method based on basic polynomials and generating function. In this method, we first consider the solution of the problem based on the given partial differential equation, and then we approximate this solution based on the basic polynomials and the generating function (the function that applies to the initial conditions of the problem). which apply precisely in the given boundary conditions. After that, by placing this solution in the given sub-function and optimality conditions, we reach a system of nonlinear equations. By solving this device of unknown coefficients, we get the approximate answer. In fact, an approximation of the answer is made in a finitedimensional space. We have also shown the convergence of the proposed method for this type of problem.In order to check the effectiveness of the method, we have solved some examples using it and presented the obtained numerical results.

In this work, nonlocal vibration of double of carbon nanotubes (CNTs) system conveying fluid coupled by visco-Pasternak medium is carried out based on nonlocal elasticity theory where CNTs are placed in uniform temperature change and magnetic field. Considering Euler-Bernoulli beam (EBB) model and Knudsen number, the governing equations of motion are discretized and Ritz method is applied to obtain the frequency of coupled CNTs system. The detailed parametric study is conducted, focusing on the remarkable effects of the Knudsen number, aspect ratio, small scale, thermo-magnetic fields, velocity of conveying fluid and visco-Pasternak medium on the stability of coupled system. The results indicate that magnetic field has significant effect on stability of coupled system. Also, it is found that trend of figures have good agreement with the previous researches. Results of this investigation could be applied for optimum design of nano/micro mechanical devices for controlling stability of coupled systems conveying fluid under thermo-magnetic fields.

This paper is intended to analyse the post buckling behavior of annular plate under the effects of symmetric uniform loading using Rayleigh-Ritz method. This is done first by formulating the problem in large defection using Von-Karman nonlinear theory. The geometry in natural coordinate system using the correct and useful mapping and the development of displacement fields in natural coordinate system using Hierarchic, Hermitian and Lagrange shape functions respectively in interpolation of out-of-plane and in-plane displacement field of plates are also considered. Finally, we deal with the presentation of a proper Hookian displacement field for controlling post buckling behavior of circular plates. To solve the problems, this methid enjoys a number of advantages including continuity in stress and strain, few numbers of degree of freedom and application of bounder easy conditions.

In this article, a class of variable-order differential equations is solved by the Ritz-approximation. Firstly, the unknown function is estimated using the Ritz-approximation and Genocchi polynomials as the basis functions. Then, by collocation method and preference of Genocchi roots as the node points, a set of algebraic equations is ob-tained. This system of nonlinear equations is solved by Mathematica 10 software. Finally, by solving some numerical examples and comparing the achieved results with other methods, the validity and efficiency of the presented method are shown.