Sharif Digital Repository

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 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... 

We prove under quite general assumptions the existence of a bounded positive solution of the semilinear Schrödinger equation Δ u + f (x, u) = 0 in a two-dimensional exterior domain. Our results are independent of the behavior of f (x, u) when u is sufficiently small or sufficiently large and just require some knowledge about the nonlinearity f (x, u) for a ≤ u ≤ b, for some a, b > 0. We obtain solutions with a prescribed positive lower bound. © 2007 Elsevier Ltd. All rights reserved  

Approximate solutions for vibrations of flexible beam-type appendages subjected to tip mass are studied while uniform and exponential profiles for arm deployment are simulated. Applying an equivalent dynamical system and following Lagrangian approach, the equations of motion of the system are derived as nonlinear ordinary differential equations (ODEs) (with time-varying coefficients), in which the effect of the tip mass can be considered as some nonlinearity added to the 'no tip mass' case dynamics. The approximate closed-form solutions are obtained through a novel methodology using a computer algorithm, in which the solutions of the 'no tip mass' case are expanded by imposing quadratic... 

In this paper a saturable absorber medium is employed as an optical limiter to reduce the spot size to the range of several tens of nanometres. The characteristics of a Gaussian beam are theoretically analysed upon propagation through the saturable absorber medium. Based on Maxwell equations a system of coupled nonlinear ordinary differential equations for intensity, beam radius and beam curvature is obtained. Theoretical analyses and numerical results show that the behaviour of a Gaussian beam in a saturable absorber medium strongly depends on the initial characteristics of the laser beam. Numerical results indicate that, depending on the initial conditions and a suitable saturable absorber... 

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... 

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... 

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... 

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... 

Prior to the design and fabrication of MEMS/NEMS devices, analysis of static and dynamic behaviors of these systems is necessary. In the present study, the nonlinear dynamic behavior of micro- and nano-mechanical resonators is investigated and classified based on the resonator’s physical parameters for first time. The Galerkin method is used to convert the distributed-parameter model to a nonlinear ordinary differential equation where mid-plane stretching, axial stress, DC electrostatic and AC harmonic voltages are taken into account. To obtain the analytical frequency response of the micro resonator near its primary resonance, the second order multiple scales method is applied to the... 

In this paper, nonlinear dynamical behavior of a rectangular plate traveled by a moving mass as well as an equivalent concentrated force with non-constant velocity is studied. The nonlinear governing coupled partial differential equations (PDEs) of motion are derived by energy method using Hamilton's principle based on the large deflection theory in conjuncture with the von-Karman strain-displacement relations. Then Galerkin's method is used to transform the equations of motion into a set of three coupled nonlinear ordinary differential equations (ODEs) which then is solved in a semi-analytical way to get the dynamical response of the plate. Also, by using the Finite Element Method (FEM)... 

In this paper, primary resonance of a double-clamped microbeam has been investigated. The Microbeam is predeformed by a DC electrostatic force and then driven to vibrate by an AC harmonic electrostatic force. Effects of midplane stretching, axial loads and damping are considered in modeling. Galerkin's approximation is utilized to convert the nonlinear partial differential equation of motion to a nonlinear ordinary differential equation. Afterward, a combination of homotopy perturbation method and the method of multiple scales are utilized to find analytic solutions to the steady-state motion of the microbeam, far from pull-in. The effects of different design parameters on dynamic behavior... 

The upper and lower bound principals of limit analysis are employed to determine the critical loading on solid circular plate with simply supported boundary conditions and subjected to any distributed loading with rotational symmetry. In this study, material behavior follows a rigid perfectly plastic model and yielding obeys the von-Mises criterion. Homotopy analysis method is employed to achieve the analytical solution to the high nonlinear ordinary differential equations governing the problem. This analytic solution has been obtained in terms of convergent series with easily computable terms. The results are verified with the Tresca yield criterion and presented as plots to show the... 

A semi-analytical method is proposed to solve the equations governing a bi-directionally pumped Raman amplifier through continuous-spectrum radiation. The governing equations are systems of uncountable Nonlinear Ordinary Differential Equation (NODE). By applying the moment method, the uncountable system of NODE is reduced to a system of finite number of NODEs. This system of equations is solved numerically and the results are compared with that of the full numerical method. It was shown that the moment method is a powerful and efficient technique for the analysis of a bi-directionally pumped Raman amplifier through continuous-spectrum radiation  

Prior to the design and fabrication of MEMS/NEMS devices, analysis of static and dynamic behaviors of these systems is necessary. In the present study, the nonlinear dynamic behavior of micro- and nano-mechanical resonators is investigated and classified based on the resonator’s physical parameters for first time. The Galerkin method is used to convert the distributed-parameter model to a nonlinear ordinary differential equation where mid-plane stretching, axial stress, DC electrostatic and AC harmonic voltages are taken into account. To obtain the analytical frequency response of the micro resonator near its primary resonance, the second order multiple scales method is applied to the... 

In this paper, primary resonance of a double-clamped microbeam has been investigated. The Microbeam is predeformed by a DC electrostatic force and then driven to vibrate by an AC harmonic electrostatic force. Effects of midplane stretching, axial loads and damping are considered in modeling. Galerkin's approximation is utilized to convert the nonlinear partial differential equation of motion to a nonlinear ordinary differential equation. Afterward, a combination of homotopy perturbation method and the method of multiple scales are utilized to find analytic solutions to the steady-state motion of the microbeam, far from pull-in. The effects of different design parameters on dynamic behavior... 

In this study, nonlinear oscillations of microbeams, actuated by suddenly applied electrostatic force, are investigated. Effects of electrostatic actuation, residual stress, midplane stretching and fringing fields are considered in modeling. Galerkin's decomposition method is utilized to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. Homotopy analysis method is used to find semi-analytic solutions to the vibrations of microbeams. Convergence regions of the solution series are determined. Influences of increasing the voltage and midplane stretching on the frequency of vibrations are also studied. Results are in good agreement with... 

Nonlinear vibrations of viscoelastic thin cylindrical shells are studied in this paper. The viscoelastic properties are modeled using the Kelvin–Voigt fractional-order constitutive relationship. Based on the nonlinear Love thin shell theory, the structural dynamics of the cylindrical shell is modeled by using the Newton’s second law, and the Galerkin method is used to discretize the nonlinear partial differential equations into the set of nonlinear ordinary differential equations. The method of multiple scales is used to solve the nonlinear ordinary differential equations, and the amplitude–frequency and phase–frequency equations are extracted. The obtained results are verified with... 

In this paper, the vibration and primary resonance of electrostatically actuated microbridges are investigated, with the effects of electrostatic actuation, axial stress, and mid-plane stretching considered. Galerkin's decomposition method is adopted to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. The homotopy perturbation method (a special case of homotopy analysis method) is then employed to find the analytic expressions for the natural frequencies of predeformed microbridges, by which the effects of the voltage, mid-plane stretching, axial force, and higher mode contribution on the natural frequencies are studied. The primary...