Features of differential and variational-differential formulations of the problem of the longitudinally cross bend of the core from inertia forces

The variational-differential method of calculation of rectilinear cores stability on axial inertial loadings is developed. The formulations of the boundary value problem of longitudinal cross bending movements are calculated: differential formulation with a final and differential approximation of the allowing equations and variational-differential one. The task is reduced to the generalized problem of own numbers Ax = λBx - for an non-trivial vector x it is required to identify its own number λ, here A is a matrix rigidity, B is a matrix of internal forces of inertia. In considering the differential formulation of the task the main particularity of the inertial loads is that the discrete matrix B gets values on the main diagonal (the rows of matrix can degenerate). Another feature is associated with the approximation of the differential equations by the method of grids, which forms the matrix B asymmetrical about the main diagonal. The generalized problem has no decision like its feedback form Bx = λ *Ax where λ * = 1 / λ. Brining to the problem of eigenvalues AB -1x = λEx and BA -1x = λ *Ex where A -1 and B -1 are the inverse matrix, E is the identity matrix, doesn’t give any result. Therefore transition from the differential formulation of a task to the variation formulation with sampling by a variational and differential method is executed. The algorithm of formation of matrixes A and B is developed for this approach, which is based on uniform properties of variations of functional. Here the matrix B is always symmetric to the main diagonal and is positively defined. Zeros on the main diagonal were presented because it is a feature of loading, however rows don’t degenerate. The technique of the solution of a task is shown. Examples of calculation of own values and forms of stability loss are given. The critical axial accelerations lose their stability and critical angular speeds for the cores rotating in a drum of the centrifuge when the core is fixed from both sides. Investigated the convergence of solutions from condensation of a finite-difference grid. Purpose: to develop a method of calculation of cores on inertial loadings.