Set Of Coordinate Functions Research Articles

A Homotopy Method for Calculating Cylindrical Waveguide Guided Waves with an Impedance Boundary

A mathematical model for a rectangular waveguide with an impedance boundary condition has been formulated and substantiated. The model is based on the application of the Galerkin boundary method. It is assumed that the surface impedance is not a constant, but is a function of the coordinates on the surface. The solution is represented as a linear combination of coordinate functions, each of which exactly satisfies Maxwell’s equations inside a cylindrical domain. The set of coordinate functions at the border forms a complete system. The coefficients are determined from the orthogonality condition for the surface residual to the system of projection functions. Because of the use of the Galerkin method, the projection functions coincide with the system of coordinate functions. To calculate the guided waves of a rectangular waveguide with an impedance boundary, a homotopy method has been proposed and justified. The decomposition of the solution into a power series in a small parameter has also been constructed.

Large Amplitude Forced Vibration Analysis of Stiffened Plates Under Harmonic Excitation

Large amplitude forced vibration behaviour of stiffened plates under harmonic excitation is studied numerically incorporating the effect of geometric non-linearity. The forced vibration analysis is carried out in an indirect way in which the dynamic system is assumed to satisfy the force equilibrium condition at peak excitation amplitude. Large amplitude free vibration analysis of the same system is carried out separately to determine the backbone curves. The mathematical formulation is based on energy principles and the set of governing equations for both forced and free vibration problems derived using Hamilton’s principle. Appropriate sets of coordinate functions are formed by following the two dimensional Gram-Schmidt orthogonalization procedure to satisfy the corresponding boundary conditions of the plate. The problem is solved by employing an iterative direct substitution method with an appropriate relaxation technique and when the system becomes computationally stiff, Broyden’s method is used. The results are furnished as frequency response curves along with the backbone curve in the dimensionless amplitude-frequency plane. Three dimensional operational deflection shape (ODS) plots and contour plots are provided in a few cases.

The multiple-reciprocity method. A new approach for transforming BEM domain integrals to the boundary

The application of the boundary element method for solving boundary value problems with body forces, time dependent effects or certain class of non-linearities generally leads to an integral equation which contains domain integrals (e.g. Ref. 1). Although these integrals do not introduce any new unknowns they detract from the elegance of the formulation and affect the efficiency of the method as integrations over the whole volume are required. Hence, a substantial amount of research has been carried out to find a general and efficient method of transforming volume integrals into equivalent boundary ones. The approaches which have so far been proposed can be divided into three groups. The first approach started in 1982 and is called the Dual Reciprocity Method, This approach was developed by Nardini and Brebbia 2'3 and afterwards extended for a variety of problems by Brebbia and WrobeP and by Loeffler and Mansur 5. It has been proved that the Dual Reciprocity Method (e.g. Refs 6, 7, 8) permits to solve a wide range of problems and is accurate. Another group of approaches is based on the expansion of the source term into the Fourier series. The method was proposed by Tang 9 to deal with potential and elasticity problems and has also been successfully applied for solving neutron diffusion problems by Itagaki and Brebbia 1 o. The third class of methods is the use of particular solutions which can convert domain integrals into boundary integrals. This technique offers some advantages versus the previous approaches in some specific applications as demonstrated by Azevedo and Brebbial 1. It has also been pointed out in Ref. 11 that the use of global particular solutions can be thought of as a special case of Dual Reciprocity. In this paper a new method of transforming domain integrals into boundary integrals is proposed. The Multiple Reciprocity Method (MRM) can be thought of as an extension of the idea of Dual Reciprocity. However, instead of approximating the source term by the set of coordinate functions, a sequence of functions related to the fundamental solution is introduced. These constitute a set of higher order fundamental solutions which permit the second Green's identity to be applied to each term of

The influence of tip mass on the stability of a rotating cantilever subjected to dissipative and transverse follower forces

The state of stability of a rotating viscoelastic cantilever beam subjected to a transverse follower force applied at its free end in the plane of rotation is determined by a method of approximation that is based upon an adjoint variational principle. Particular attention is devoted to the determination of the dependence of the critical flutter load of the system on the transverse, twisting, and rotary inertia properties of a mass capping the free end of the beam. The equations of motion are derived from a conservation law, the adjoint boundary value problem is introduced, and an approximate stability determinant is developed from the variational principle upon assuming a set of coordinate functions which satisfy a selected set of boundary conditions. The stability determinant is solved numerically for a variety of choices of values for the internal damping, the hub radius, tip mass inertia, the rotational speed, and warping rigidity parameters, and several graphs are presented to show the influence of these parameters upon the value of the critical flutter load.