Application of Method of Consecutive Approximations to Problems of Stability and Oscillations of Elastic Systems

摘要

An approximate method of reduction of problem of the eigenvalues of differential equations with variable coefficients to an algebraic equation relative to the unknown parameter is presented. This method is distinguished from many methods of approximations by the fact that, first, it can be applied completely formally, i.e., without connection with preliminary approximation on unknown functions. Secondly, it reduces problem to an algebraic equation, allowing to determine whole spectrum of eigenvalues with any required accuracy. Third, it gives a solution with bilateral estimate that allows the accuracy of the result obtained to be judged. The given method differs from the method of successive approximations further by the fact that in the process of each consecutive approximation it is not necessary to satisfy boundary conditions, which are satisfied only once for general solution, built on definite rules of differential equation. The described method can be used to solve a number of complicated problems of structural mechanics.