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In this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution. Crown  

We consider the nonlinear Euler differential equation t2 x″ + g (x) = 0. Here g (x) satisfies x g (x) > 0 for x ≠ 0, but is not assumed to be sublinear or superlinear. We present implicit necessary and sufficient condition for all nontrivial solutions of this system to be oscillatory or nonoscillatory. Also we prove that solutions of this system are all oscillatory or all nonoscillatory and cannot be both. We derive explicit conditions and improve the results presented in the previous literature. We extend our results to the extended equation t2 x″ + a (t) g (x) = 0. © 2006 Elsevier Inc. All rights reserved  

This paper studies undamped oscillations of fractional-order Duffing system. Stability theorems for fractional order systems are used to determine the characteristic polynomial of the system in order to find the parametric ranges for undamped oscillations in this system. We also derive relations for estimating the frequency and the amplitude of the oscillations in this system using a describing function method. Finally numerical simulation results are provided to justify the analysis  

Recent research works demonstrated that the interaction between the loads and the carrying structure's boundary which is related to the inertia of the load is an influential factor on the dynamic response of the structure. Although effects of the inertia in moving loads were considered in many works, very few papers can be found on the inertial effects of the stationary loads on structures. In this paper, an elastodynamic formulation was employed to investigate the dynamic response of a homogeneous isotropic elastic half-space under an inertial strip foundation subjected to a time-harmonic force. Fourier integral transformation was used to solve the system of Poisson-type partial... 

We study the concept and the calculus of Non-convex self-dual (Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to... 

The interinjection locking of microwave oscillators is investigated using differential equations governing the locking behaviour of two mutually coupled oscillators. The behaviour of the oscillators before and after locking is described, and the common oscillation frequencies, locking range and transient time constant are calculated. The spectra of the oscillators in the unlocked condition are also investigated, with particular attention to the spacing and amplitude variation of the spectral lines. Experimental results in connection with the theoretical investigations are also presented  

A closed-loop control methodology is investigated for stabilization of a vibrating non-classical micro-scale Euler-Bernoulli beam with nonlinear electrostatic actuation. The dimensionless form of governing nonlinear Partial Differential Equation (PDE) of the system is introduced. The Galerkin projection method is used to reduce the PDE of system to a set of nonlinear Ordinary Differential Equations (ODE). In non-classical micro-beams, the constitutive equations are obtained based on the non-classical continuum mechanics. In this work, proper control laws are constructed to stabilize the free vibration of non-classical micro-beams whose governing PDE is derived based on the modified strain... 

The stochastic behavior of a point reactor is modeled with a system of Ito stochastic differential equations. This new approach does not require computing the square root of a matrix which is a great computational advantage. Moreover, the derivation procedure clearly demonstrates the mathematical approximations involved in the final formulation. Three numerical benchmarks show the accuracy of this model in predicting the mean and variance of the neutron and precursor population in a point reactor  

Vibrations of electrostatically-Actuated microbeams are investigated. Effects of electrostatic actuation, axial stress and midplane stretching are considered in the model. Galerkin's decomposition method is utilized to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. Homotopy perturbation method (i.e. a special and simpler case of homotopy analysis method) is utilized to find analytic expressions for natural frequencies of predeformed microbeam. Effects of increasing the voltage, midplane stretching, axial force and higher modes contribution on natural frequency are also studied. The anayltical results are in good agreement with the... 

In this study vibration amplitude, frequency and damping of a microbeam is controlled using a RLC block containing a capacitor, resistor and inductor in series with the microbeam. Applying this method all of the considerable characteristics of the oscillatory system can be determined and controlled with no change in the geometrical and physical characteristics of the microbeam. Euler-Bernoulli assumptions are made for the microbeam and the electrical current through the microbeam is computed by considering the microbeam deflection and its voltage. Considering the RLC block, the voltage difference between the microbeam and the substrate is calculated. Two coupled nonlinear partial... 

The existing methods in attitude control of satellites are based on using the estimate of satellite attitude, which is usually generated by using multiple vector measurements. In this paper we propose an output feedback controller that directly uses a single vector measurement and does not use an attitude estimator. The output feedback gain is computed by solving a generalized Riccati differential equation (GRDE). The existence of a solution to the GRDE depends on a uniform controllability condition  

In this paper, employing the limit analysis theorem, critical loading on functionally graded (FG) circular plate with simply supported boundary conditions and subjected to an arbitrary rotationally symmetric loading is determined. The material behavior follows a rigid-perfectly plastic model and yielding obeys the von-Mises criterion. In the homogeneous case, the highly nonlinear ordinary differential equation governing the problem is analytically solved using a variational iteration method. In other cases, numerical results are reported. Finally, the results are compared with those of the FG plate with Tresca yield criterion and also in the homogeneous case with those of employing the... 

The existing methods in attitude control of satellites are based on employing the estimate of satellite attitude which is usually generated by using multiple vector measurements. In this paper we propose an output feedback control law that directly uses a single vector measurement and gyro, and without any need for estimating the satellite attitude. The output feedback gain is computed by solving a generalized Riccati time varying differential equation. We assume the moment-of-inertia matrix of satellite is unknown. The controller guarantees asymptotic convergence of the attitude to its desired value. A realistic simulation is presented where a magnetometer is used to provide the single... 

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... 

This paper contains analysis and simulation of recent dynamic models, describing human emotion. The pharmacological discussions lead to a new model of drug taking, which also have a better performance for description of psychological and psychiatric phenomena. Studying the effects of the order of fractional model, obtains an advantage of the fractional order model over the integer order one  

Deployment inertial effects of a spacecraft appendage on its flexible dynamics are investigated. The Euler-Bernoulli beam theory and the actual deployment profile, in which appendage axial motion accelerates from static state and then decelerates to end at zero velocity and acceleration, are employed. The study is concentrated on the arm dynamic stiffness introduced by inertial effects of the arm deployment, and the resultant effects on the arm flexible motions. Lagrange's equations and some appropriate shape functions in the series approximation method are employed to study the arm lateral elastic displacements. Finally a system of ordinary differential equations with time varying... 

An accurate approximate closed-form solution is presented for bending of thin skew plates with clamped edges subjected to uniform loading using the extended Kantorovich method (EKM). Successive application of EKM together with the idea of weighted residual technique (Galerkin method) converts the governing forth-order partial differential equation (PDE) to two separate ordinary differential equations (ODE) in terms of oblique coordinates system. The obtained ODE systems are then solved iteratively with very fast convergence. In every iteration step, exact closed-form solutions are obtained for two ODE systems. It is shown that some parameters such as angle of skew plate have an important... 

In this study, several analytical expressions are obtained for the vibrational characteristics of a Jeffcott model for micro-rotor systems based on the strain gradient theory to investigate the small-scale effects on the model. The Jeffcott model consists of a massless microrotating shaft and a disk as a rotor with eccentricity. The disk is mounted on the middle of the shaft. Two second-order differential equations associated with the oscillating motion of the rotor in the plane perpendicular to the longitudinal axis are presented and transformed into a complex form. The stiffness of the system is determined by obtaining the deflection of a strain-gradient-based nonrotating microbeam...