Application Of The Lagrange Multipliers To The Construction Of Three New Methods For The Study Of Mechanical Systems

The Lagrange multipliers are used to construct three new methods for the study of mechanical systems. The first of them corresponds to the problem of determining the normal frequencies and normal forms of oscillations of elastic system, consisting of the elements with the known normal frequencies and normal forms. In this method the conditions of connection of elastic bodies to one another are regarded as holonomic constraints. Their reactions equal to the Lagrange multipliers are the forces of interaction between the bodies of system. Using the equations of constraints, the system of linear uniform equations with respect to the amplitudes of the Lagrange multipliers for normal oscillations is obtained. By the solution of this system the normal frequencies and normal forms of complete system are expressed in terms of the normal frequencies and normal forms of its elements. An approximate algorithm for determining the normal frequencies and normal forms, based on a quasistatic account of higher forms of its elements, is proposed.