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Axially moving beams are extensively involved in various industries and have significant importance in many mechanical engineering problems. In this paper, the nonlinear forced vibrations of axially moving beam under harmonic force and thermal environment have been studied. In order to considering the effects of transverse shear deformation and rotary inertia, the Timoshenko beam theory has been used to model the axially moving beam. The nonlinear governing equations are derived with the help of Hamilton’s principle. Then the equations and boundary conditions are discretized through generalized differential quadrature method (GDQ) and its differential matrix operators, and accordingly the partial differential equations are converted into the ordinary differential equations. To study the frequency response of the system, the harmonic balance method is used. Also the time responses of the axially moving beam are obtained by the Runge-Kutta method. In a case study, the effects of various parameters such as the axial speed, transverse force acting on the beam, damping coefficient and temperature change on the frequency responses of the axially moving beam with both end simply supported boundary conditions are discussed. The results show that the dynamic behavior of system is significantly affected by any of the mentioned factors.

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In the present study the free vibration analysis of functionally graded rectangular nanoplates in thermal environment is investigated. The modified coupled stress theory based on the first order shear deformation theory has been used to obtain the natural frequencies of the nanoplate. Modified coupled stress theory is a non-classical theory. In this theory material length scale parameter is applied to capture the size effect of the microstructures which the earlier classical plate theories were not able to explain these effects. The functionally graded material properties are varied continuously and smoothly along the thickness. The Poisson’s ratio of the FGM plate is assumed to be constant in the whole plate. In order to validate the present method, the natural frequencies of the both functionally graded rectangular plate and rectangular nanoplates are compared with those are reported in the literature, separately. Finally, the effect of various parameters such as; the power law index, the thickness to length scale parameter ratio, aspect ratio, thickness ratio on the natural frequencies of plates in thermal environments with different temperatures are presented and discussed in detail.