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The main objective of this research is to study the nonlinear vibrations of a single walled carbon nanotube. For this purpose, the lattice structure of carbon nanotube is replaced with a continuum structure using nanoscale continuum mechanics. Firstly, each carbon-carbon bond is replaced with an equivalent beam element and then the whole discrete structure of carbon nanotube is replaced with a virtual continuum medium representing hollow cylinder. Then, governing equations for vibrations is obtained taking into account geometric nonlinearity arisen from stretching of a mid-plane due to bending. Perturbation technique is used to analyze the nonlinear vibrations of carbon nanotubes. Frequency responses of carbon nanotubes for free vibrations and force vibrations in both primary and secondary resonance cases are studied. Obtained results are in a very good agreement with numerical integration technique. The results imply on hardening behavior of carbon nanotube. Moreover, nonlinear bifurcation and nonlinear jump phenomena are observed.

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In this paper, the homotopy analysis method is used to nonlinear free vibration analysis of a mechanical and thermal loaded functionally graded beam on nonlinear elastic foundation. At first, the governing partial differential equation of the problem has been derived based on the Euler-Bernoulli theory and the Von-Karman strain-displacement relationship. Then, it was reduced to a nonlinear ordinary differential equation via the Galerkin method. The homotopy analysis method which has high accuracy was implemented in order to obtain a closed form solution and study the problem parametrically. The accuracy of the proposed method is verified by those available in literatures. The numerical results demonstrate that proposed method yields a very rapid convergence of the solution as well as low computational effort. Finally, the effects of different parameters such as amplitude, linear and nonlinear elastic foundation, thermal and mechanical loads and boundary conditions were investigated on the beam vibration and their results are presented for future work.

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In this paper, nonlinear free vibration of a Nano plates rectangular has been investigated. For this purpose, first, the equations of motion for Nano plate which is considered as a continuous system have been derived using Hamilton principle based on classical plate theory. Then, by definition an stress function and using Galerkin method the equations converted to an ordinary nonlinear equation and a compatible equations. Using multiple time scale method this equation has been solved and analytical relations for first nonlinear natural frequency and nonlinear mode shaped have been derived. Then for example, these relations have been studied for Graphen sheet in Armchair and Zigzag structure and the effect of aspect ratio and nonlocal elasticity parameter on natural frequency have been investigated.

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The accurate evaluation and experimental investigation of the gear dynamic response have indicated some interesting nonlinear phenomena such as bifurcation and chaotic behavior on some system parameters. The chaotic motion is an unusual and unpredictable behavior and has been considered as an undesirable phenomenon in the nonlinear gear vibration systems. Therefore, in order to design and develop an optimal gear transmission system, it is important to control and eliminate these nonlinear phenomena. This paper presents the design of a gear system in order to control and suppress the chaos. A generalized nonlinear dynamics model of a spur gear pair including the backlash and the static transmission error is formulated. The idea behind the design of this control system is applying an additional control excitation force to the driver gear. The parameter spaces of the control excitation force are obtained analytically by using the Melnikov approach. The numerical simulations including the bifurcation diagram, the phase portrait, and the time history are used to confirm the analytical predictions and show effectiveness of the proposed control system for chaos suppression in nonlinear gear systems.

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The present study deals with nonlinear vibration analysis of the plate having at least one free edge. the plate has a part-through central crack with arbitrary limit length which is parallel to one side of the plate. Due to complexity of the governing equation of motion, the Galerkin method is used for solving the problem. Therefore, the appropriate admissible functions satisfying free edge conditions in the cracked plate, must be employed. The beam functions can not satisfy free edge conditions in the plate, nevertheless these functions used in many researches in the literature, which lead to high numerical errors in computing the frequency and mode shapes. Therefore, in this research, a new admissible functions is proposed which can obviate incapability of the beam functions to accurately estimate natural frequency of the intact and cracked plate and reduce the related computational errors. The effect of changes in thickness and non-dimensional crack length on natural frequency of the cracked plate are investigated using proposed functions, and frequency response curves indicating the dependence of frequency on amplitude is derived for different boundary conditions. Also, the influence of crack length on the changes in the nonlinear behavior of the cracked plate are investigated. The results, are compared with those of the available results in the literature.

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Abstract In this work, nonlinear electromagnetic vibration energy harvesting from cantilever beam under base harmonic oscillation is investigated and the effects of electromagnetic parameters on behavior of system is considered. For modeling assumed mode method is used, and beam is modeled in according to Timoshenko theory, which includes shear deformation and rotary inertia. In energy harvesting, the frequency response of the system is very important because it shows the best areas for energy harvesting and it is a good criterion for designing energy harvesters, hence a semi analytical method is used to find simply the amplitude of oscillation in terms of excitation frequency. In this method, at first equations of motion are solved with complex averaging method and obtained equations are solved with continuation method. For validation, comparison between results obtained from numerical and semi analytical method is given. Also comparison between linear and nonlinear system, and stability of periodic response and their bifurcations are given. In addition, in order to the effect of number of mode shapes and convergence of solution, frequency response of one, two, three modes and four modes cases are compared with together.

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This paper deals with the nonlinear transient vibration of composite sandwich plates with an electrorheological (ER) fluid core. The initial excitation is a distributed transverse load or the flutter instability due to supersonic airflow. The Bingham plastic model is adopted to accurately model the post-yield behavior of the ER material. . The first order piston theory is used for evaluating the aerodynamic forces. The von Karman strain-displacement relations are employed to account for moderately large deflection. The Hamilton’s principle is applied in conjunction with the finite element method to derive the equations of motion. The solution is then obtained through use the Newmark time integration scheme. Numerical investigations are conducted to study the effect of ER core layer on the vibration characteristics of the sandwich plate. The influence of the electric filed strength, ER core thickness, initial excitation and the boundary conditions on the settling time of transient vibration are also examined. The results show that the damping of transient vibration is significantly improved as the electric field applied to the ER layer, but the amplitude of post-flutter oscillations remains unchanged.

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The objective of this paper is to analyze the nonlinear vibrations of simply supported pseudoelastic shape memory alloy (SMA) cylindrical shell under harmonic internal pressure based on Donnell-type classical deformation shell theory. The pressure is a function of time and space. The behavior of pseudoelastic SMA is simulated via the Boyd–Lagoudas constitutive model numerically implemented by the Convex Cutting Plane Mapping algorithm. The Hamilton’s principle is employed to obtain the equations of motion. Differential Quadrature Method (DQM) and Newmark time integration scheme are applied to get the time and frequency responses of the cylinder. Also, the natural frequencies of the shell are obtained for the case of pure austenitic phase to compare the frequency response of the present nonlinear system (phase transformation –induced material nonlinearity) with the linear one around them. Results indicate that the strength of the material will decrease during the phase transformation. This fact is proved by the softening behavior observed in the frequency response of the system due to the phase transformation. Further, the pure austenitic phase shell is simulated in ABAQUS to verify the results. A good agreement is found between two outcomes.

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In this paper, the nonlinear free vibrations of a bimorph piezoelectric nanoactuator is studied based on nonlocal elasticity. The Euler-Bernoulli beam theory and Hamilton’s principle are used to derive the equation of motion of the actuator. In order to obtain the reduced-order form of equations, the Galerkin method is used. The mode shapes of a multi-span beam are used for a faster convergence. The nonlinear natural frequencies are obtained by using He’s variational approach. Equations are solved for clamped-clamped boundary conditions, and the effects of values of DC voltage, actuator length and thickness, length of piezoelectric layers and nonlocal parameter on the nonlinear natural frequencies are studied. The results show that applying a DC voltage induces a static deflection and an increase in the stiffness of the actuator. Therefore, the natural frequency increases. Moreover, increasing the nonlocal parameter decreases the rate of change in frequency variation. An increase in the nonlocal parameter or the length of the actuator increases the nonlinear to nonlinear natural frequency ratio. Finally, the effect of the middle layer material of the actuator on the frequency ratio is studied.

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The present study probes the nonlinear free vibrations of viscoelastic polymeric composite plate reinforced by carbon nanotubes. For this purpose, Kelvin-Voigt model is utilized. Moreover, the equations of motion are extracted by the Hamilton principle and taking into account Von Karman nonlinearity. In order to solve and analyze nonlinear free vibrations, the researchers utilized multiple scales method. Thanks to this method, the normal nonlinear frequencies of the system were obtained, and as well, the impact of various factors such as dampness coefficient, material viscosity and carbon nanotubes volume fraction were investigated. Besides, the thickness-dimension ratio of the plate and its impact on the normal frequency was also studied. The findings of the study highlighted that an increase in the ratio of plate’s thickness to its length causes an increase in the normal nonlinear frequency of the plate. Additionally, as the volume fraction of the carbon nanotubes increases, system’s normal nonlinear frequency increases as well. Finally, the impact of different distribution of carbon nanotubes on the normal nonlinear frequency and system’s time response was also probed. As it could be vividly observed, nonlinear frequency for FGO distribution was reported to be further than uniform distribution, but the trend was in reverse for FGX distribution.

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Conical shells are widely used in various engineering applications such as mechanical, civil and aerospace engineering. In the present paper, based on the first order shear deformation theory (FSDT) of shells, the nonlinear vibration behavior of truncated conical shells with different boundary conditions is investigated using a numerical approach. To this end, the governing equations of motion and corresponding boundary conditions are derived by the use of Hamilton's principle. After catching the dimensionless form of equations, the generalized differential quadrature (GDQ) method is employed to obtain a discretized set of nonlinear governing equations. Thereafter, a Galerkin-based scheme is applied to achieve a time-varying set of ordinary differential equations and a method called periodic grid discretization is used to discretize the equations on the time domain. The pseudo arc-length continuation method is finally applied to obtain the frequency-amplitude response of conical shells. Selected numerical results are presented to examine the effects of different parameters such as thickness-to-radius ratio, small-to-large edge radius ratio, semi-vertex angle of cone, circumferential wave number and boundary conditions. It is concluded that the changes of the vibrational mode shapes and circumferential wave number have significant effects on the nonlinear vibration characteristics and hardening effects.

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In this paper stability analysis of a nonlinear micro rotating shaft near the primary resonances by considering the modified couple stress theory and micro inertia effect is investigated. The geometric nonlinearities due to classical and non-classical theory (the modified couple stress theory) are considered. Using Hamilton principle, the nonlinear equations of motion are obtained. In order to solve the equations of motion the multiple scales method are used and an analytical expression is presented for forward and backward frequencies which can be seen the effects of modified couple stress theory and micro inertia effect. The frequency response curves, amplitude versus damping coefficient, amplitude versus total eccentricities, etc. are reported. It is seen that due to the modified couple stress theory and micro inertia effect the amplitude of the system is decreased and the loci of bifurcation points is changed. Symmetrical micro-shaft in the presence of classical theory and without micro inertia effects becomes completely stable in the least damping coefficient and asymmetrical micro-shaft in the presence of classical theory and without micro inertia effects becomes completely stable in the most damping coefficient. Symmetrical micro-shaft in the presence of modified couple stress theory and with micro inertia effects becomes completely stable in the least total eccentricity and asymmetrical micro-shaft in the presence of classical theory and without micro inertia effects becomes completely stable in the most total eccentricity. So, considering the small-scale effects due to strain and velocity gradients for analysis of the system is mandatory.

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In this paper free and forced vibrations analysis of a viscoelastic nonlinear nano rotating beam by considering surface effects is investigated. Using Hamilton principle and Gurtin Murdoch theory, the equations of motion are obtained and discretized by Galerkin method. Using the multiple time scales method the equations of motion are solved. In free vibrations analysis, the analytical expressions for amplitude and phase are obtained. In forced vibrations analysis the steady state solution are obtained. The effect of surface effect, damping coefficients, dimensions of cross section area, external excitation amplitude etc. on frequency response curves are investigated. It is seen that in free vibrations, by increasing surface stress the amplitude of the system decreased, and by increasing surface density or elasticity it is increased. Also, by increasing internal and external damping coefficients free vibration amplitude is decreased. In forced vibrations, it is seen that considering surface effect the amplitude of the system is decreased and the first bifurcation point is obviously changed. By increasing internal and external damping coefficients the amplitude is decreased and the first bifurcation point occur in frequencies near the natural frequency. It is seen that for two different dimensions of cross section with same area, amplitude and the loci of the bifurcation points are changed. By increasing the amplitude of external excitation the amplitude of response is increased the bifurcation points occur in frequencies far away from natural frequency. So, considering the surface effects for free and forced vibrations analysis of the nano rotating beams is mandatory.

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In this article, for the first time, the effect of non-uniformity of microbeam cross section and various boundary conditions on the nonlinear vibration of microbeam is investigated considering the size dependent behavior based on modified couple stress theory. Using the Hamilton’s principle, the governing equation of Euler–Bernoulli microbeam with von Karman geometric nonlinearity based on the modified couple stress theory is derived. The nonlinear vibration governing equation is then solved using the Generalized Differential Quadrature method (GDQ) and direct iterative method to obtain the nonlinear natural frequencies. In this step, the Galerkin method is used to reduce the nonlinear PDE governing the vibration into a time-dependent ODE of Duffing-type. The time domain is then discretized via spectral differentiation matrix operators which are defined based on the derivatives of a periodic base function. Next, the nonlinear parametric equation is solved using pseudo arc-length method and the frequency–response curves of microbeam nonlinear forced vibration is obtained. Finally, nonlinear natural frequency and frequency response of microbeam with various non-uniformity of cross sections and boundary conditions are obtained. Present results show that, the nonlinear free and forced vibration of microbeam is size dependent. Moreover, this size dependency is more significant for non-uniform microbeam and is deferent for various boundary conditions. The result of present method for simple case including uniform section and simply supported boundary condition is validated with that of exact method and have good agreement.

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In this research, nonlinear transverse vibrations of a fluid conveying microtube under parametric magnetic axial resonance condition is studied. For this purpose, nonlinear governing equations of transverse motion of beam-like microtube are derived using Reddy’s first-order shear deformation theory with considering the effect of fluid viscosity and fluid centripetal acceleration. In this model, nonlinear terms of Hetenyi foundation and nonlinear geometric terms of the Von-Karman theory under magnetic excitations in the presence of fluid flow beyond the flutter instability is considered. In the following, the effects of foundation parameters on the linear flutter specifications of fluid conveying magnetizable microtubes are studied. Then, the nonlinear system behavior for fluid flow velocities more than critical velocity corresponding to the coupling of the first and second vibration modes is studied using multiple scales method. Nonlinear response curves in velocities above critical velocity are obtained and effects of variations of various system parameters including flow velocity, amplitude, and frequency of the magnetic field, Hetenyi foundation stiffness constants, viscosity, and dimensions ratio on the nonlinear response of the system are investigated. Some results indicate that increasing the values of shear stiffness parameter of the Hetenyi foundation has an unstable effect so that with its increasing, the flutter instability occurs at lower frequencies.