Sharif Digital Repository

In this work a new method for analyzing nanostructured materials has been proposed to accelerate the simulations for solid crystalline materials. The proposed Structural Approximation Method (SAM) is based on Molecular Dynamics (MD) and the accuracy of the results can also be improved in a systematic manner by sacrificing the simulation speed. In this method a virtual material is used instead of the real one, which has less number of atoms and therefore fewer degrees of freedom, compared to the real material. The number of differential equations that must be integrated in order to specify the state of the system will decrease significantly, and the simulation speed increases. To generalize... 

This technical note addresses the free vibration problem of an elastically restrained Euler–Bernoulli beam with rotational spring-lumped rotary inertia system at its mid-span hinge. The governing differential equations and the boundary conditions of the beam are presented. Special attention is directed toward the conditions of the intermediate spring-mass system which plays a key role in the solution. Sample frequency parameters of the beam system are solved and tabulated. Mode shapes of the beam are also plotted for some spring stiffnesses  

A semi-analytical solution for rotating axisymmetric disks made of functionally graded materials was previously proposed by Hosseini Kordkheili and Naghdabadi [1]. In the present work the solution is employed to study thermoelastic creep behavior of the functionally graded rotating disks with variable thickness in to the time domain. The rate type governing differential equations for the considered structure are derived and analytically solved in terms of rate of strain as a reduced to a set of linear algebraic equations. The advantage of this method is to avoid simplifications and restrictions which are normally associated with other creep solution techniques in the literature. The thermal... 

In this paper, a size-dependent formulation is developed for Timoshenko beams made of functionally graded materials (FGMs). The developed formulation is based on the strain gradient theory; a non-classical continuum theory able to capture the size-effect in micro-scaled structures. Five new equivalent length scale parameters are introduced as functions of the constituents' length scale parameters. It is shown that the size-dependent static and dynamic behavior of FG micro-beams can be described using these equivalent length scales. The governing differential equations of motion and both classical and non-classical sets of boundary conditions are derived for the proposed strain gradient FG... 

In this paper, a size-dependent formulation is presented for mechanical analyses of inhomogeneous micro-plates based on the modified couple stress theory. The plate properties can arbitrarily vary through the thickness. The governing differential equations of motion are derived for functionally graded (FG) plates with arbitrary shapes utilizing a variational approach. Moreover, the boundary conditions are provided at smooth parts of the plate periphery and also at the sharp corners of the periphery. Utilizing the derived formulation, the free-vibration behavior as well as the static response of a rectangular FG micro-plate is investigated  

A composite beam with single delamination traveled by a constant amplitude moving force is modeled accounting for the Poisson's effect, shear deformation and rotary inertia. The mechanical behavior between the delaminated surfaces is modeled using a piecewise-linear spring foundation. The governing differential equations of motion for such system are derived. Primarily, eigen-solution technique is used to obtain the natural frequencies and their corresponding mode shapes of such beam. Then, the Ritz method is employed to derive the dynamic response of the beam due to the moving force. The obtained results for the free and forced vibrations of beams are verified against reported similar... 

This paper presents the dynamic response of a delaminated composite beam under the action of a moving oscillating mass. In this analysis the Poisson's effect is considered for the first time. Moreover, the effects of rotary inertia and shear deformation are incorporated. In our modeling linear springs are used between delaminated surfaces to simulate the dynamic interaction between sub-beams. To solve the governing differential equations of motion using modal expansion series, eigen-solution technique is used to obtain the natural frequencies and their corresponding mode shapes necessary for forced vibration analysis. The obtained results for the free and forced vibrations of beams are... 

In this article, the dynamic responses of a Timoshenko beam subjected to a moving mass, and a moving sprung mass are analyzed. By making recourse to Hamilton's principle, governing differential equations for beam vibration are derived. By using the modal superposition method, the partial differential equations of the system are transformed into a set of Ordinary Differential Equations (ODEs). The resulted set of ODEs is represented in state-space form, and solved by means of a numerical technique. The accuracy of the results has been ascertained through comparing the results of our approach with those available from previous studies; moreover, a reasonable agreement has been obtained. The... 

In this paper, a size-dependent formulation is developed for Timoshenko beams made of functionally graded materials (FGM). The developed formulation is based on the strain gradient theory;a non-classical continuum theory able to capture the size-effect in micro-scaled structures. Considering the material length scale parameters of the FG beams vary through the thickness, the new equivalent length scale parameters are proposed as functions of the constituents' length scale parameters to describe the size-dependent static and dynamic behavior of FG microbeams. The governing differential equations of equilibrium and both classical and nonclassical sets of boundary conditions are derived for the... 

Vibration analysis of a new type of compliant parallel mechanism with flexible intermediate links is investigated. The application of the Timoshenko beam theory to the mathematical modeling of the intermediate flexible link is described, and the equations of motion of the flexible links are obtained by using Lagrange's equation of motion. The equations of motion are obtained in the form of a set of ordinary differential equations by using assumed mode method theory. The governing differential equations of motion are solved using perturbation method. The assumed mode shapes and frequencies are to be obtained based on clamped-clamped boundary conditions. Comparing perturbation method with... 

The dynamic response of a delaminated composite beam under the motion of an oscillatory mass moving with a constant velocity has been studied. The delaminated composite beam is modeled as four interconnected sub-beams using the delamination limits as their boundaries. The constrained model is used to model the delamination region. The continuity and equilibrium conditions are forced to be satisfied between the adjoining beams. A set of derived governing differential equations along with those obtained by imposing boundary conditions are simultaneously solved in a closed form manner. The results for the response of the delaminated beam were compared with those of the intact beam. Furthermore,... 

Rectangular plates on distributed elastic foundations are widely employed in footings and raft foundations of variety of structures. In particular, mounted columns and single footings may partially occupy the rectangular plate of any kind. This study deals with free vibration problem of thin rectangular plates on Winkler and Pasternak elastic foundation model which is distributed over a particular arbitrary area of the plate. Closed form solutions are developed through solving the governing differential equations of plates. Moreover, a novel mathematical approach is proposed to find the exact analytical solution of free vibration of plates with mixed or fully-clamped boundary conditions.... 

In this paper, distributions of stress and strain components of rotating disks with non-uniform thickness and material properties subjected to thermo-elasto-plastic loading are obtained by semi-exact method of Liao's homotopy analysis method (HAM) and finite element method (FEM). The materials are assumed to be elastic-linear strain hardening and isotropic. The analysis of rotating disk is based on Von Mises' yield criterion. A two dimensional plane stress analysis is used. The distribution of temperature is assumed to have power forms with the hotter point located at the outer surface of the disk. A mathematical technique of transformation has been proposed to solve the homotopy equations... 

The geometrically nonlinear governing differential equations of motion and the corresponding boundary conditions are derived for the mechanical analysis of Timoshenko microbeams with large deflections, based on the strain gradient theory. The variational approach is employed to achieve the formulation. Hinged-hinged beams are considered as an important practical case, and their nonlinear static and free-vibration behaviors are investigated based on the derived formulation  

In this paper, strain gradient elasticity formulation for analysis of FG (functionally graded) micro-cylinders is presented. The material properties are assumed to obey a power law in radial direction. The governing differential equation is derived as a fourth order ODE. A power series solution for stresses and displacements in FG micro-cylinders subjected to internal and external pressures is obtained. Numerical examples are presented to study the effect of the characteristic length parameter and FG power index on the displacement field and stress distribution in FG cylinders. It is observed that the characteristic length parameter has a considerable effect on the stress distribution of FG... 

The governing differential equation of motion of a thin rectangular plate excited by a moving mass is considered. The moving mass is traversing on the plate's surface at arbitrary trajectories. Eigenfunction expansion method is employed to solve the constitutive equation of motion for various boundary conditions. Approximate and exact expressions of the inertial effects are adopted for the problem formulation. In the approximate formulation, only the vertical acceleration component of the moving mass is considered while in the exact formulation all the convective acceleration components are included in the problem formulation as well. Parametric studies are carried out to investigate the... 

The aim of this study is to carry out the numerical computation of nonlinear normal modes for rotating pretwisted composite thin-walled beam in axial-torsional vibrations. The structural model considered here, incorporates a number of non-classical effects such as primary and secondary warping, non-uniform torsional model, rotary inertia and pretwist angle. Ignoring the axial inertia term leads to differential equation of motion in terms of angle of twist in the case of axially immovable beam ends. The governing differential equations of motion are derived using Hamilton's principle and the reduced model around the static equilibrium position is obtained using 2-mode Galerkin discretization... 

The governing differential equation and both classical and non-classical boundary conditions of strain gradient bars are derived using variational approach. A closed-form analytical solution is obtained for static torsion and the characteristic equation, which gives the natural frequencies, is derived and analytically solved for the free torsional vibrations of the strain gradient microbars. A fixed-fixed microbar is considered as a specific case to investigate the torsional size-dependent static and free-vibration behavior of strain gradient microbars. The results of the current model are compared to those of the modified couple stress and classical theories  

A semi-analytical method is presented to calculate the dynamic responses of a rectangular plate due to a moving oscillator. In previous analytical solutions of the moving oscillator problem, the elastic distributed structure has usually been modeled by an elastic beam structure. This restrictive assumption is removed in this study by assuming a general plate as two-dimensional elastic distributed structure. The method can be applied for any arbitrary path on the plate. A combination of the Fourier and Laplace transformation as well as the convolution theorem is used to solve the governing differential equations of the problem. A modified integration technique is then presented to solve the... 

In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained...