Annex of the variation and differential method for calculation of the longitudinally cross bend of plates by inertia forces

The variation and differential method of calculation of stability of thin plates on the inertial loadings operating in the basic plane is developed. The regional task is given to the generalized problem of own numbers. Thus matrixes are formed: the first, a rigidity matrix - is based on the biharmonic equation of S. Germaine, and the second matrix, represents changes of the internal tension or internal efforts arising in a plate. The annex of a method of final differences to a differential problem definition shows the following. The rigidity matrix always symmetric is also positively defined for the fixed plate. The second matrix to - a matrix of internal efforts at approximation of derivative functions with application of the central differences, from actions of forces of inertia is formed asymmetrical rather main diagonal, also lines of this matrix - such is feature of inertial loadings can also degenerate. Despite possibility of formation of systems of the equations certain difficulties if the plate has free edges arise by method of final differences of big dimensions; it is necessary to exclude the second row the zakonturnykh of knots of a grid that complicates procedure of calculations, especially in corners of a plate, external and internal. Therefore transition from the differential formulation of a task to the integrated formulation with sampling by a variation and differential method is executed. At this approach, when forming a matrix of rigidity at free edges there is no second row the zakonturnykh of knots, in corners there are no additional knots of a grid; - the matrix of internal efforts is always symmetric, such is feature of the appendix of integrated approach. The matrix of internal efforts can be badly caused, however this factor doesn’t influence the solution of a problem of definition of own numbers. The set of theoretical researches and solutions of practical problems of calculation of stability of designs, including by calculation of a longitudinally cross bend of thin plates is given in literature. However, in a bigger measure, these tasks have positively certain operators. There is a certain search and research operation of application of a variation and differential method to calculation of stability of designs on various loadings is performed. The differential formulation of a regional task is transformed to the variation formulation; criteria of stability are given and issues of approximation of differential operators for a discrete task with final number of variables are resolved. The algorithms are developed for mathematical Maple system and programs of calculation are made. Two examples of calculation of plates are given. The plate which is rigidly fixed on all parties is considered; operating forces of inertia change under the linear law. Also the plate which is rigidly fixed on one party, and on three other parties is considered - the plate is free from fixing. Values of critical accelerations are received. The purpose of the work is to develop the method of calculation of plates on inertial loadings.