Hassan Askari

Multi-frequency excitation of microbeams supported by Winkler and Pasternak foundations

The multi-frequency excitation of a microbeam, resting on a nonlinear foundation, is investigated and the governing equation of motion of the microbeam system is developed. The viscoelastic-type foundation is considered by assuming nonlinear parameters for both Pasternak and Winkler coefficients. The well-known Galerkin approach is utilized to discretize the governing equation of motion and to obtain its nonlinear ordinary differential equations. The multiple time-scales method is employed to study the multi-frequency excitation of the microbeam. Furthermore, the resonant conditions due to the external excitation as well as the combination resonances for the first two modes are investigated. The influences of different parameters, namely the Pasternak and Winkler coefficients, the position of the applied force and the geometrical factors on the frequency response of the system are examined.

Nonlinear vibration of fluid-conveying carbon nanotube using homotopy analysis method

Nonlinear vibration analysis of a single-walled carbon nanotube, using the Eringen nonlocal elasticity and Euler-Bernoulli beam theories, is carried out. Pasternak-type foundation and the simply-supported boundary conditions are assumed for the carbon nanotube and the governing equation of motion is developed using those theories. The Galerkin method is utilized to obtain the nonlinear ordinary differential equation of vibration of the single-walled carbon nanotube and the homotopy analysis method is employed to determine its nonlinear natural frequency. A parametric sensitivity study is then carried out. Few of the parameters were the axial tension, nonlocal parameter, fluid velocity and the foundation stiffness. The parametric study is mainly focused on the nonlinear natural frequency of single-walled carbon nanotube. Finally, a numerical simulation is carried out to determine the accuracy of the obtained results. Furthermore, an elliptical integral is utilized to verify the nonlinear natural frequency, which was obtained using the homotopy analysis method.

Dynamic Behavior of Carbon Nanotubes Using Nonlocal Rayleigh Beam

Forced vibration of carbon nanotubes based on the Rayleigh beam theory in conjunction with Eringen’s nonlocal elasticity is investigated. The governing equation of vibration of carbon nanotube using the above theories is developed. The carbon nanotube is rested on a nonlinear Winkler and Pasternak foundation with the simply-supported boundary conditions. The Gelerkin procedure is utilized to find the nonlinear ordinary differential equation of vibration of system. The differential equation is solved using the multiple scales method in order to investigate the primary resonance of the considered system. The frequency response of the system is obtained and the effects of different parameters, such as the surface effect, position and magnitude of applied force and Pasternak and Winkler foundation, on the vibration behavior of the system are studied. The sensitivity of the amplitude of oscillation of carbon nanotube is depicted with respect to the surface effect. It is shown that the surface effect plays an important role in the forced vibration of the nano-scale structure.© 2014 ASME

Analysis of nonlinear oscillation of circular curved carbon nanotube

The nonlinear vibration of a circular curved carbon nanotube using Euler-Bernoulli beam theory is investigated. The governing equation of motion of the system is developed and the Galerkin method is utilized to obtain the nonlinear ordinary differential equation of the curved carbon nanotube. A quarter-circular curvature is considered for the nanotube and it is assumed to have a single wall with simply-supported boundary conditions. Two semi analytical approaches to study the behavior of the developed nonlinear differential equation are utilized and the frequency-amplitude relationship of the objective system is obtained. Subsequently, a parametric study is performed to study the importance of different parameters, such as the amplitude of oscillation and the curvature radius, on the nonlinear behavior of the system. Finally, numerical simulation is carried out to obtain the results and investigate the accuracy of the analytical solution methods applied.

Large Amplitude Free Vibration Analysis of Nanotubes Using Variational and Homotopy Methods

Linear theories are basically unable to model the dynamic behavior of nanotubes due to the large deflection/dimension ratios. In this paper the closed form expressions are obtained for the large-amplitude free vibration of nanotubes. The nonlinear governing differential-integral equation of motion is derived and solved using the Galerkin approach. The derived nonlinear differential equation is then solved using the Variational Approach (VA) and the Homotopy Analysis Method (HAM). The fundamental harmonic as well as higher-order harmonics are analytically obtained. The approximate solutions are compared with those of the numerical responses and accordingly a numerical analysis is carried out. A parametric sensitivity analysis is carried out and different effects of the physical parameters and initial conditions on the natural frequencies are examined. It is found that both the variational analysis and homotopy method are quite consistent and satisfactory techniques to analyze the vibration of nanotubes.Copyright

Nonlinear Vibration of Nanobeam With Quadratic Rational Bezier Arc Curvature

Nonlinear vibration of nanobeam with the quadratic rational Bezier arc curvature is investigated. The governing equation of motion of the nanobeam based on the Euler-Bernoulli beam theory is developed. Accordingly, the non-uniform rational B-spline (NURBS) is implemented in order to write the implicit form of the governing equation of the structure. The simply-supported boundary conditions are assumed and the Galerkin procedure is utilized to find the nonlinear ordinary differential equation of the system. The nonlinear natural frequency of the system is found and the effects of different parameters, namely, the waviness amplitude, oscillation amplitude, aspect ratio, curvature shape and the Pasternak foundation coefficient are fully investigated. The hardening and softening responses of the natural frequency of structure are detected for variations of the shape and amplitude of the curvature waviness. It is revealed that the ratio of nonlinear to linear frequency increases with an increase in the oscillation amplitudes. It is found that by increasing the Pasternak foundation coefficient, the ratio of nonlinear to linear frequency decreases.Copyright

Nonlinear Forced Vibration of Carbon Nanotubes Considering Thermal Effects

Nonlinear forced vibration of carbon nanotubes is investigated. The Euler-Bernoulli beam theory in conjunction with Eringen’s theory is considered and the thermal effect is incorporated into the formulation of the governing equation. The Winkler model is assumed for the foundation of carbon nanotube and the Galerkin method is performed to find the nonlinear ordinary differential equation of system based on the assumed boundary conditions. The multiple times scale is applied to investigate the forced vibration of carbon nanotubes. The effect of different parameters, namely, temperature variations and carbon nanotube length changes on the amplitude of oscillation of carbon nanotube are studied. It is found that the linear natural frequency of system increases by increasing the temperature and subsequently, the oscillation amplitude will decrease.Copyright

Periodic solutions for nonlinear oscillations of nanowires using variational iteration method

Nonlinear oscillation of nanowires, based on the Timoshenko beam theory, are investigated. The analytical solution of the nanowire structure using the Galerkin technique together with the variational iteration method is developed. A coupled nonlinear differential equation is obtained utilizing the Galerkin technique. Subsequently, the frequency-amplitude relationships are found for the nonlinear vibration of the nanowires using the variational iteration method. The main objective of this work is to obtain the frequency-amplitude relationships of the nanowires and also to investigate the effects of different parameters, such as the surface effect, transverse shear deformation and the aspect ratio, on their dynamic behaviors. Furthermore, the influence of varying amplitude on the nonlinear oscillations of the nanowires is analyzed.

Nonlinear Longitudinal Vibration Solutions of an Elastic Rod

Nonlinear longitudinal vibration of an elastic rod is studied. The motion of a uniform elastic rod is described by a nonlinear partial differential equation, which has a cubic nonlinear term and a Winkler elastic force that acts along the longitudinal axis of the rod. Galerkin method is used to develop the nonlinear differential equation of elastic rod, which resembles similarity with the Duffing equation. Three different types of robust analytical methods are chosen to solve the nonlinear differential equation and obtain the natural frequency of the system. These are the Homotopy analysis method (HAM), Energy balance method (EBM) and Hamiltonian approach (HA). Subsequently, the analytical results are compared with the numerical solution of the exact equation in order to evaluate the correctness of the applied approaches. Moreover, the effects of the constant coefficients of the elastic force on the ratio of the nonlinear to the linear frequencies are studied. The singular points of the nonlinear differential equation of the elastic rod are extracted and the Jacobian matrix is constructed to recognize their types. Finally, phase-plane trajectories of the system are constructed in order to verify the results obtained from the Jacobian matrix.Copyright

Nonlinear Vibration of Carbon Nanotube Resonators Considering Higher Modes

Nonlinear forced vibration of the carbon nanotubes based on the Euler-Bernoulli beam theory is studied. The Euler-Bernoulli beam theory is implemented to find the governing equation of the vibrations of the carbon nanotube. The Pasternak and Nonlinear Winkler foundation is assumed for the objective system. It is supposed that the system is supported by hinged-hinged boundary conditions. The Galerkin procedure is employed in order to find the nonlinear ordinary differential equation of the vibration of the objective system considering two modes of vibrations. The primary and secondary resonant cases are developed for the objective system employing the multiple scales method. Influence of different factors such as length, thickness, position of applied force, Pasternak and Winkler foundation are fully shown on the primary and secondary resonance of the system.Copyright