A UNIFORMLY CONVERGENT SECOND ORDER DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED SELF-ADJOINT ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, based on the idea of El-Mistikawy and Werle[1] we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.

Prediction of Three Dimensional Crack Path Brittle Fracture in Weldment under Dynamic Load

Three dimensional dynamic stress intensity factors are analyzed for a curved crack with a second order perturbation method. The method is extended to obtain an approximate representation of a three dimensional dynamic stress intensity factors at the tip of a curved crack. Due to three dimensional curved crack growth the dynamic energy release rate can be calculated by using the Irwin＇s formula. A three dimensional curved crack in materials with inhomogeneous fracture toughness are considered. Paths of a brittle three dimensional curved crack propagating along a welded joint are predicted via the present method, where the effects of dynamic applied stresses, residual stresses, and material deterioration due to welding are taken into considerations.

Nonlinear vibration analysis of a rotor supported by magnetic bearings using homotopy perturbation method

In this paper,the effects of nonlinear forces due to the electromagnetic field of bearing and the unbalancing force on nonlinear vibration behavior of a rotor is investigated.The rotor is modeled as a rigid body that is supported by two magnetic bearings with eightpolar structures.The governing dynamics equations of the system that are coupled nonlinear second order ordinary differential equations(ODEs)are derived,and for solving these equations,the homotopy perturbation method(HPM)is used.By applying HPM,the possibility of presenting a harmonic semi-analytical solution,is provided.In fact,with equality the coefficient of auxiliary parameter(p),the system of coupled nonlinear second order and non-homogenous differential equations are obtained so that consists of unbalancing effects.By considering some initial condition for displacement and velocity in the horizontal and vertical directions,free vibration analysis is done and next,the forced vibration analysis under the effect of harmonic forces also is investigated.Likewise,various parameters on the vibration behavior of rotor are studied.Changes in amplitude and response phase per excitation frequency are investigated.Results show that by increasing excitation frequency,the motion amplitude is also increases and by passing the critical speed,it decreases.Also it shows that the magnetic bearing system performance is in stable maintenance of rotor.The parameters affecting on vibration behavior,has been studied and by comparison the results with the other references,which have a good precision up to 2nd order of embedding parameter,it implies the accuracy of this method in current research.

The J Integral of a Slightly Curved Elasticity-plasticity Crack when Enduring Quasi Static Loads

The J Integral of an elastic-plastic curved crack under quasi-static loads has been mainly worked in the paper,and the J Integral has been calculated as a practical application of a second order perturbation method and theorem of surname KA where the effect of quasi-static applied stresses and normal and shear stresses on the boundaries of plasticity area are synthetically taken into consideration.A regular pattern of variations of the J Integral of an elastic-plastic curved crack with the variations of curved crack shape parameters under different quasi-static loads has been mainly carried out.Continuity and unity of a elastic-plastic fracture and linear elastic fracture has been demonstrated from the viewpoint of the mechanical parameter of the curved crack tip J integral.The elastic-plastic slightly curved crack tip J integral has been calculated approximately,an integral loop having been reasonably selected and continuity and unity of the fracture characteristic of centre penetration straight line crack and curved crack have been demonstrated.

Effeet of Hardening on Elastic-plastic Curved Crack Tip Displacement Under Dynamic Load

In this article, elastic-plastic curved crack tip opening displacement maximum in hardened material under dynamic loads has been studied, and curved crack tip opening displacement has been calculated as a practical application of a second order perturbation method and theorem of surname KA, where the effects of dynamic applied stresses and dynamic normal and shear stresses on the boundaries of plastic area are synthetically taken into considerations. Diagrams have been constructed to analyz the transformation relationships between the curved crack tip opening displacement and the work hardening exponents. Curved crack tip opening displacement will decrease with the increasing of hardening exponents in hardened material. The decrease extent of curved crack tip dynamic opening displacement will be more and more severe when hardening exponents increase evenly. Maximum dynamic opening displacement of curved crack tip will decrease when external load decrease with the same hardening exponents