The laws of conservation of linear and angular momentum are shown to impose certain integral conditions on the body force density in an elastic dielectric which is in static equilibrium.An equation of state, quadratic in the variables Di, Di;j, Ei, Ei;j, is postulated for the body force density. In the case of nonpiezoelectric cubic material symmetry it is found that, within the quadratic approximation, the integral condition imposed by the law of conservation of linear momentum is satisfied only if the body force of this form vanishes.
This paper deals with numerical solutions of singular integral equations in interaction problems of elliptical inclusions under general loading conditions. The stress and displacement fields due to a point force in infinite plates are used as fundamental solutions. Then, the problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknowns are the body force densities distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. To determine the unknown body force densities to satisfy the boundary conditions, four auxiliary unknown functions are derived from each body force density. It is found that determining these four auxiliary functions in the range 0≦φk≦π/2 is equivalent to determining an original unknown density in the range 0≦φk≦2π. Then, these auxiliary unknowns are approximated by using fundamental densities and polynomials. Initially, the convergence of the results such as unknown densities and interface stresses are confirmed with increasing collocation points. Also, the accuracy is verified by examining the boundary conditions and relations between interface stresses and displacements. Randomly or regularly distributed elliptical inclusions can be treated by combining both solutions for remote tension and shear shown in this study.
AbstractThe Maxwell stress tensor (MST) method is commonly used to accurately compute the global efforts, such as electromagnetic torque ripple and unbalanced electromagnetic forces in electrical machines. The MST has been extended to the estimation of local magnetic surface force for the vibroacoustic design of electrical machines under electromagnetic excitation. In particular, one common air-gap surface force (AGSF) method based on MST is to compute magnetic surface forces on a cylindrical shell in the air gap. However, the AGSF distribution depends on the radius of the cylindrical shell. The main contribution of this study is to demonstrate an analytic transfer law of the AGSF between the air gap and the stator bore radius. It allows us to quantify the error between the magnetic surface force calculated in the middle of the air gap and the magnetic force computed on the stator teeth. This study shows the strong influence of the transfer law on the computed tangential surface force distribution through numerical applications with induction and synchronous electrical machines. Finally, the surface force density at stator bore radius is more accurately estimated when applying the new transfer law on the AGSF.
Analytical study of air-gap surface force – application to electrical machines