This article deals with the free in-plane vibration analysis of a frame with four arbitrary inclined members by differential transform method. Based on four differential equations and sixteen boundary and compatibility conditions, the related structural eigenvalue problem will be analytically formulated. The frequency parameters and mode shapes of the frame will be calculated for various values of the structural properties, such as joint angles, springs' stiffness and flexural rigidity of members. Finally, the obtained solution by the proposed method will be verified by authors' finite element program.

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.

Atomic force microscope (AFM) has been developed at first for topography imaging; in addition, it is used for characterization of mechanical properties. Most researches have been primarily focused on rectangular single-beam probes to make vibration models simple. Recently, the U-shaped AFM probe is employed to determine sample elastic properties and has been developed to heat samples locally. In this study, a simplified analytical model of these U-shaped AFM is described and three beams have been used for modelling this probe. This model contains two beams are clamped at one end and connected with a perpendicular cross beam at the other end. The beams are supposed only in bending flexure and twisting, but their coupling allows a wide variety of possible dynamic behaviors. In the present research, the natural frequency and sensitivity of flexural and torsional vibration for AFM probes have been analyzed considering influence of scale effect. For this purpose, governing equations of dynamic behavior of U-shaped AFM probe are extracted based on Eringen's theory using Euler–Bernoulli beam theory and an analytical method is employed to solve these equations. The results in this paper have been extracted for different values of nonlocal parameters; it is shown that for a special case, there is a good agreement between reported results in available references and our results. The obtained results show that the frequencies of U-shaped AFM decrease with increasing the nonlocal parameter.

Thermo-elastic damping is a significant dissipation mechanism in high quality factor microstructures. In this paper, thermo-elastic damping of the in-plane vibration of fully clamped rectangular micro-plates has been studied. The governing equation of the micro-plate motion and heat conduction equation were derived. Then, The Galerkin method has been used to solve the coupled heat-displacement equations. Eventually, considering the micro-plate of various materials, the effects of geometrical parameters including the length and width of micro-plate and also ambient temperature on the thermo-elastic damping quality factor have been investigated.

In this study, thermo-nonlocal vibration of double bonded graphene sheet (DBGS) subjected to 2D-magnetic field under biaxial in-plane pre-load are presented. The elastic forces between layers of graphene sheet (GS) are taken into account by Pasternak foundation and the classical plate theory (CLPT) and continuum orthotropic elastic plate are used. The nonlocal theory of Eringen and Maxwell’s relations are employed to incorporate the small-scale effect and magnetic field effects, respectively, into the governing equations of the GSs. The differential quadrature method (DQM) is used to solve the governing differential equations for simply supported edges. The detailed parametric study is conducted, focusing on the remarkable effects of the angle and magnitude of magnetic field, different type of loading condition for couple system, tensile and compressive in-plane pre-load, aspect ratio and nonlocal parameter on the vibration behavior of the GSs. The result of this study can be useful to design of micro electro mechanical systems and nano electro mechanical systems.