The use of principal coordinates in the problem of oscillation suppresion of a trolley with two pendulums

Abstract

The Pontryagin maximum principle and the generalized Gauss principle can be applied constructively to the problem of oscillation suppression of a trolley with two pendulums in the case when a system of equations describing the controlled motion of a trolley with two pendulums is written in principal coordinates. The paper shows that these equations can be obtained without lengthy transformations connected with representation of the kinetic and potential energies of the considered mechanical system in principal coordinates. At first the Lagrange equations of the second kind are composed in the simplest Lagrange coordinates which are the horizontal displacement of a trolley and the angles of rotation of pendulums. Then the displacement of a trolley is eliminated from them. The remaining two equations in the angles of rotation of pendulums make it possible to determine two nonzero natural frequencies and corresponding to them vibration modes of the mechanical system in question. If one knows these frequencies then one will know how in natural oscillation the angles of rotation of pendulums are connected with principal coordinates. Going to principal coordinates in the two equations in angles of rotation of pendulums we get linearly independent combinations of the unknown equations in principal coordinates. This makes it possible to define these equations fairly simply. Completing them with the equation of motion of the centre of mass and going to dimensionless variables in all equations we obtain a system of three equations in principal coordinates written in the simplestform

Keywords:

control theory, oscillation suppression, Lagrange equations, principal coordinates