# Application of the Lagrange multipliers to the construction of three new methods for the study of mechanical systems

Shervani Kh Soltakhanov, Mikhail P. Yushkov, Sergei A. Zegzhda

Research outputpeer-review

## Abstract

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. The second method suggested is connected with the study of the dynamics of system of rigid bodies. In this case the Lagrange multipliers are introduced for the abstract constraints taking into account that the number of introduced coordinates, by which the kinetic energy of rigid body has a simple form, is excessive. In this case the elimination of the Lagrange multipliers leads to a new special form of equations of motion of rigid body. This form is utilized to describe a motion of a dynamic stand, which lets us to imitate the state of a pilot in the cabin in extremal situations. The third method is used in the problem of vibration suppression (damping) of mechanical systems. It is shown that formulation of such problems is equivalent to imposing a high-order constraint on the motion of system. This makes it necessary to solve a mixed problem of dynamics. It turns out, that the Pontryagin maximum principle chooses from the possible class of mixed problems that one in which a control force is given by a series in natural frequencies of system. In the suggested method of vibration suppression (damping) the generalized Gauss principle is used, which makes it possible to find the control force as a polynomial in time. The computational results obtained by the Pontryagin maximum principle and the generalized Gauss principle are compared.

Original language English Mechanics of non-holonomic systems A New Class of control systems Shervani Soltakhanov, Sergei Zegzhda, Mikhail Yushkov 149-191 43 https://doi.org/10.1007/978-3-540-85847-8_6 Published - 2009

### Publication series

Name Foundations in Engineering Mechanics 1612-1384

## Scopus subject areas

• Mechanics of Materials
• Mechanical Engineering