Sharif Digital Repository

The nonlinear free and forced bending vibration of sandwich plates with incompressible viscoelastic core is investigated under the effects of different source of nonlinearities. For the core constrained between stiffer layers, the transverse shear strains, as well as the rotations are assumed to be moderate. The linear and quadratic displacement fields are also adopted for the in-plane and out-of-plane displacements of the core, respectively. The assumption of moderate transverse strains requires a nonlinear constitutive equation which is obtained from a single-integral nonlinear viscoelastic model using the assumed order of magnitudes for linear strains and rotations. The 5th-order method... 

An efficient hybrid modal-molecular dynamics method is developed for the vibration analysis of large scale nanostructures. Using the reduced order method, presented in this paper, linear and nonlinear vibrations of a suspended graphene nanoribbon (GNR) carrying an electric current in a harmonic magnetic field are investigated. The resonance frequencies as well as the nonlinear vibration response obtained by the present technique and direct molecular dynamic simulations are in very good agreement. Also, the obtained results illustrate the hardening behavior of nonlinear vibrations which is diminished by stretching the GNR  

Modified couple stress theory is a size-dependent theorem capturing the micro/nanoscale effects influencing the mechanical behaviors of the micro- and nanostructures. In this paper, it is applied to investigate the nonlinear vibration of carbon nanotubes under step DC voltage. The vibration, natural frequencies and dynamic pull-in characteristics of the carbon nanotubes are studied in detail. Moreover, the effects of various boundary conditions and geometries are scrutinized on the dynamic characteristics. The results reveal that application of this theory leads to the higher values of the natural frequencies and dynamic pull-in voltages  

Tension leg platforms (TLPs) are well-known structures for oil exploitation in deep water. One of the current issues in compliant structures in the sea is variation in frequency and structural response due to a nonlinear parameter in the equation of motion. Variation of frequency is important in fatigue life study of tethers. A perturbation method is used in contrast to the traditional methods. This method does not require a small parameter for finding surge motion of TLP. In this paper, homotopy perturbation method (HPM) is used to solve a highly nonlinear differential equation of surge motion. Calculated responses by HPM are compared with those obtained from both linear and nonlinear... 

In this paper, the strain gradient theory, a non-classical continuum theory able to capture the size effect happening in micro-scale structures, is employed in order to investigate the size-dependent nonlinear forced vibration of Euler-Bernoulli microbeams. The nonlinearities are caused by mid-plane stretching and nonlinear external forces such as van-der-Waals force. The nonlinear governing equations of the microbeams are solved analytically utilizing the perturbation techniques. The primary, super-harmonic and sub-harmonic resonances of a microbeam are studied and the size-dependency of the frequency responses is assessed. The results indicate that the nonlinear forced vibration behavior... 

The objective of this work is to create an analytical framework to study the non-linear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton's principal is utilized to derive the equations governing the nonlinear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these non-linear partial differential equations are reduced to two coupled non-linear ordinary differential equations. These equations are solved analytically using combination of the method of multiple time scales and homotopy perturbation analysis. Closed-form, parametric analytical expressions are presented for the time domain response of... 

In this paper, the nonlinear forced-vibration of Euler-Bernoulli beams with large deflections is investigated based on the modified couple stress theory, a non-classical theory capable of capturing size effects. The classical theory is unable to predict the size effects. In systems with the dimensions in order of microns and sub-microns the size effects are very significant. For some specific beams subjected to a concentrated force at its middle as the harmonic exciter, the size-dependent responses are investigated for primary, super-harmonic and sub-harmonic resonances. The results show that the frequency-responses of the system are highly size-dependent  

Variational principle is used to derive the nonlinear response of the floating roof of cylindrical liquid storage tanks due to harmonic base excitations. The formulation accounts for nonlinearity due to large deflections of the floating roof. The derived nonlinear governing equation for the sloshing response of the floating roof has a cubic nonlinear stiffness term similar to the well known Duffing equation. It is shown that accounting for large deflections could substantially reduce the wave elevation for near resonance harmonic excitations. Evaluating the response of the nonlinear model for increasing amplitudes of near resonance harmonic excitations gives rise to the appearance of sub and... 

This paper presents a nonlinear model to illustrate the effect of contact stress in the vibration behavior of mechanical components, especially rotating systems. The problem is considered in the case of the vertical vibration of a sphere on a plate. The Hertzian contact theory is used to obtain the relationship between contact force and the deflection of the mass center of the sphere. Modeling the system by a mass and a nonlinear spring, the vibration equation of the mass center of the sphere is derived. The method of LindstedtPoincar is implemented to solve the equation of motion, and obtain vibration characteristics under a compressive preload. The dependency of frequency on several... 

In this study, the nonlinear vibrations analysis of an inclined pinned-pinned self-weight Timoshenko beam made of linear, homogenous and isotropic material with a constant cross section and finite length subjected to a traveling mass/force with constant velocity is investigated. The nonlinear coupled partial differential equations of motion for the rotation of warped cross section, longitudinal and transverse displacements are derived using the Hamilton's principle. These nonlinear coupled PDEs are solved by applying the Galerkin's method to obtain dynamic responses of the beam. The dynamic magnification factor and normalized time histories of mid-point of the beam are obtained for various... 

Electrostatically actuated beams are fundamental blocks of many different nano and micro electromechanical devices. Accurate design of these devices strongly relies on recognition of static and dynamic behavior and response of mechanical components. Taking into account the effect of internal forces between material particles nonlocal theories become highly important. In this paper nonlinear vibration of a microano doubly clamped and cantilever beam under electric force is investigated using nonlocal continuum mechanics theory. Implementing differential form of nonlocal constitutive equation the nonlinear partial differential equation of motion is reformulated. The equation of motion is... 

In this paper, simple analytical expressions are presented for geometrically non-linear vibration analysis of thin nanobeams with both simply supported and clamped boundary conditions. Gurtin-Murdoch surface elasticity together with Euler-Bernoulli beam theory is used to obtain the governing equations of motions of the nanobeam with surface effects consideration. The governing nonlinear partial differential equation is reduced to a single nonlinear ordinary differential equation using Galerkin technique. He's variational approach is employed to obtain analytical solution for the resulted nonlinear governing equation. The effects of different parameters such as vibration amplitude, boundary... 

In this paper, nonlinear vibration of a micro cantilever exposed to a constant velocity flow is studied. In order to obtain vibration frequency and time response of the micro beam the variational iteration method is used as a novel tool for solving nonlinear differential equations. Results of the analytical solution are compared with those obtained by Runge-Kutta method which shows very good agreement between them. Results confirm that frequency of vibration depends on the flow velocity. Also, the high sensitivity of the vibration frequency to the flow velocity means that it can be an effective indicator of velocity  

In this study, simple analytical expressions are presented for large amplitude free vibration and post-buckling analysis of functionally graded beams rest on nonlinear elastic foundation subjected to axial force. Euler-Bernoulli assumptions together with Von Karman's strain-displacement relation are employed to derive the governing partial differential equation of motion. Furthermore, the elastic foundation contains shearing layer and cubic nonlinearity. He's variational method is employed to obtain the approximate closed form solution of the nonlinear governing equation. Comparison between results of the present work and those available in literature shows the accuracy of this method. Some... 

The non-linear flexural vibration for a dagger-shaped atomic force microscope cantilever has been investigated using the Timoshenko beam theory. In this article, the normal and tangential tip-sample interaction forces are found from Hertzian contact model and the effects of the geometry, normal and lateral contact stiffness, height of the tip, thickness of the beam, the angle between the cantilever and the sample surface and breadth and height taper ratios on the non-linear frequency to linear frequency ratio have been studied. The differential quadrature method (DQM) is employed to solve the non-linear differential equations of motion. The results show that the softening behaviour is seen... 

In this paper, we investigate the influence of nonideal boundary conditions on the nonlinear vibration of damped Euler-Bernoulli beams subjected to harmonic loads. These nonidealities allow for small deflections and/or moments at the supports of the beam. Using the iteration perturbation method, analytical expressions for the case of simply supported beams with immovable end conditions are provided. The results reveal that the first order of approximation obtained by the proposed method is more accurate than the perturbation solutions. Moreover, compared with the previous studies, some interesting phenomenon is predicted. We have shown that in some special combinations of the nonidealities... 

In this study, the homotopy analysis method (HAM) is used to obtain an approximate analytical solution for geometrically non-linear vibrations of thin laminated composite plates resting on non-linear elastic foundations. Geometric non-linearity is considered using von Karman's straindisplacement relations. Then, the effects of the initial deflection, ply properties, aspect ratio of the plate and foundation parameters on the non-linear free vibration is studied. Comparison between the obtained results and those available in the literature demonstrates the potential of HAM for the analysis of such vibration problems, whose governing differential equations include the quadratic and cubic... 

The non-linear flexural vibration for a double-tapered atomic force microscope cantilever has been investigated by using the Timoshenko beam theory. In this article, the normal and tangential tip-sample interaction forces are found from the Hertzian contact model, and the effects of the contact position, normal and lateral contact stiffness, height of the tip, thickness of the beam, angle between the cantilever and the sample surface, and breadth and height taper ratios on the non-linear frequency to linear frequency ratio have been studied. The differential quadrature method is employed to solve the non-linear differential equations of motion. The results show that the softening behaviour... 

In this paper, homotopy perturbation and modified Lindstedt-Poincare methods are employed for nonlinear free vibrational and buckling analysis of simply supported and double-clamped beams subjected to axial loads. Mid-plane stretching effect has also been accounted in the model. Galerkin's decomposition technique is implemented to convert the dimensionless equation of the motion to nonlinear ordinary differential equation. Homotopy and modified Lindstedt-Poincare (HPM) are applied to find analytic expressions for nonlinear natural frequencies and critical axial loads of the beams. Effects of design parameters such as axial load and slenderness ratio are investigated. The analytic expressions... 

Nonlinear vibration of a microbeam actuated by a suddenly applied voltage with considering the effect of voltage distribution on the beam due to electrical resistivity of beam is investigated. Homotopy perturbation method is implemented to solve the coupled nonlinear partial differential equations of motion. The vibration frequency variation and damping at various resistivities is studied. Considering resistivity, effect of applied voltage and beam length on the frequency shift and damping ratio is analyzed. Findings indicate there exists a jump in frequency shift and damping ratio at a critical resistivity. Variation of critical resistivity with respect to modulus of elasticity and beam...