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WHEN considering a flat plate of finite width at a low angle of attack, it is observed that, downstream from the plate, where the air flow from the suction and pressure sides join, a transverse velocity appears. On the pressure side there is an outward flow toward the ends of the plate, while on the suction surface the component is inward. Since all streamlines have the same origin, in the case of steady flow the pressure where the flow over upper and lower surfaces joins must be the same and no velocity ‘jump’ is possible. It follows then that any velocity change must be purely transverse and that this will extend to the ends of the plate. The same phenomenon occurs with wings of finite span and the vortices are formed continually in flight. These vortices absorb energy, for which power has to be supplied continuously—in other words the induced drag caused solely by the vortex formation appears.

The purpose of this paper is to find an exact analytical expression for the periodic solutions of the double-hump Duffing equation and an expression for the period of these solutions.

The double-hump Duffing equation is presented as a Hamiltonian system and a phase portrait of this system has been found. On the ground of analytical calculations performed using Hamiltonian-based technique, the periodic solutions of this system are represented by Jacobi elliptic functions sn, cn and dn.

Expressions for the periodic solutions and their periods of the double-hump Duffing equation have been found. An expression for the solution, in the time domain, corresponding to the heteroclinic trajectory has also been found. An important element in various applications is the relationship obtained between constant Hamiltonian levels and the elliptic modulus of the elliptic functions.

The results obtained in this paper represent a generalization and improvement of the existing ones. They can find various applications, such as analysis of limit cycles in perturbed Duffing equation, analysis of damped and forced Duffing equation, analysis of nonlinear resonance and analysis of coupled Duffing equations.

Employs the integral transform method in the hybrid numerical‐analytical solution of fully developed laminar flow within a class of irregularly shaped ducts, with respect to the co‐ordinate system chosen to represent the geometry under consideration. A quite general formulation of a two‐dimensional steady‐state diffusion problem is initially considered, and a formal solution is provided. The original partial differential equation is analytically transformed into an infinite system of ordinary differential equations for the transformed velocity field in the flow direction. On truncation to a sufficiently large finite order, adaptively chosen to meet prescribed accuracy requirements, well‐established numerical schemes for boundary value problems are utilized, readily available in scientific subroutines libraries. Illustrates convergence rates for a few typical duct geometries and critically examines previously reported numerical solutions.

Metallic elements placed in the vicinity of coils and conducting rings exert an influence on the inductance of them. Authors' BIMS package based on the boundary‐integral approach leads up to determine the monopole and/or dipole surface densities of the ‘magnetic charge’ on the walls of simple metallic boxes. The special considerations have been performed in order to study the influence of these boundary magnetic charge densities on the magnetic flux of the coils. The general algorithms for computing the magnetic flux is presented.

Combined defects in ball bearings may be caused during the use or manufacturing process, which can significantly affect their vibration characteristics. The previous defect models in the literature can only describe single defects such as the surface waviness and localized defect. This paper aims to propose an in-depth understanding of radial vibrations of a ball bearing with the combined defect.

A dynamic model for a ball bearing with the combined defect including the surface waviness and localized defect on its races is proposed. The effects of the combined defect sizes on the radial bearing vibrations are investigated. The results from the proposed model considering the combined defect are compared with the available results from the previous methods considering the single defects.

The acceleration amplitude is significantly affected by the surface waviness, localized defect and the combined defect on its races. The effect of the combined defect on the acceleration amplitude is larger than that of the single defect. The amplitude and peak frequency of the spectrum of acceleration for the combined defect increases with the defect sizes. The RMS value of the accelerations for the combined defect increases with the combined defect sizes.

Consequently, the proposed model can predict more accurate and in-depth understanding of the radial vibrations caused by the combined defect in the ball bearing.

THE article derives expressions for the position of the neutral axis and the failing moment of resistance of symmetrical and unsymmetrical I‐sections and channel sections, angles, solid circular sections and thick and thin tubes for materials for which the stress‐strain curve is non‐linear and is different in tension and compression.

Under this heading are published regularly abstracts of all Reports and Memoranda of the Aeronautical Research Council, Reports and Technical Memoranda of the United States National Advisory Committee for Aeronautics and publications of other similar Research Bodies as issued.

A. Conformal Transformation Methods A standard analytic method for the determination of field and potential distributions uses the Schwarz‐Christoffel (SC) conformal transformation integral. When applied to configurations such as the capacitor with fringing accounted for or a metal stripe separated by a dielectric from a ground plane, it leads to complicated expressions containing elliptic integrals, and when applied to a metal disc separated from a ground plane, Hankel transforms are also involved. Since elliptic integrals must be evaluated numerically in practice, it is desirable to replace these complicated analytic processes with one that is numerical from the start. Such a method has been published by Foster, Anderson, and Warner. It is based on the standardization of a two‐step conformal transform; Step 1 takes the original geometry and lays it out along the real axis and Step 2 converts this arrangement—using a reverse Schwarz‐Christoffel transform—into a rectangular structure from which the field lines and equipotentials can be determined by inspection. Step 1 is different for each problem but Step 2 is common to all problems, and represents one of several advantages of this procedure. The simplest example given by the originators is the tri‐plate strip line of Figure 1, which—by symmetry—can be reduced to the quadrant of Figure 2. The SC

This paper aims to present new analytical expressions for the mutual inductance and forces between non-coaxial co-planar circular thin-wall air coils.

The expressions are based on an integration method found in the literature, so far used only to describe mutual inductances. It is new to apply this method as a starting point to get the forces between non-coaxial co-planar rings and coils. The approach is further extended to include non-coaxial co-planar thin-wall cylindrical coils.

This new method enables interaction modelling between coils by solving integrals numerically, covering fully, partially and non-overlapping coils in a single form. The expressions are verified by comparing the results with alternatives methods.

The forces and mutual inductances of non-coaxial co-planar circular coils are obtained with analytical expressions, fitted well for optimization studies. The study is limited to coils in free air.

A typical application is the interaction between coils in a wireless energy transfer system (as applied for battery loading for mobile phones and automotive). The expressions can be used to also predict the forces between two non-coaxial disk or ring magnets as used in magnetic levitation systems.

Maxwell described the coupling between two coaxial co-planar rings, and steadily more and more equations describing the interaction between circular coils became available in the past decades. The target of this study is to obtain compact equations for non-coaxial co-planar circular coils. This is realized with the combination of existing literature and mathematical modelling tools.

The purpose of this paper is to propose a Chebyshev collocation method (CCM) for Hallén’s equation of thin wire antennas.

Since the current induced on the thin wire antennas behaves like the square root of the distance from the end, a smoothed current is used to annihilate this end effect. Then the CCM adopts Chebyshev polynomials to approximate the smoothed current from which the actual current can be quickly recovered. To handle the difficulty of the kernel singularity and to realize fast computation, a decomposition is adopted by separating the singularity from the exact kernel. The integrals including the singularity in the linear system can be given in an explicit formula while the others can be evaluated efficiently by the fast cosine transform or the fast Fourier transform.

The CCM convergence rate is fast and this method is more efficient than the other existing methods. Specially, it can attain less than 1 percent relative errors by using 32 basis functions when a/h is bigger than 2×10−5 where h is the half length of wire antenna and a is the radius of antenna. Besides, a new efficient scheme to evaluate the exact kernel has been proposed by comparing with most of the literature methods.

Since the kernel evaluation is vital to the solution of Hallén’s and Pocklington’s equations, the proposed scheme to evaluate the exact kernel may be helpful in improving the efficiency of existing methods in the study of wire antennas. Due to the good convergence and efficiency, the CCM may be a competitive method in the analysis of radiation properties of thin wire antennas. Several numerical experiments are presented to validate the proposed method.