An optimal gyroscopic stabilizer of a multidimensional vibration system

We study the stabilizing problem for a multidimensional vibration system using a gyroscopic stabilizer. A vibration system is given by a second order differential equation with symmetrical matrix coefficients: a positive definite stiffness matrix and an indefinite damping matrix; in general, such a system is unstable. We need find an optimal gyroscopic stabilizer for it represented by a skew-symmetric matrix coefficient in a speed term. The choice of this control type is dictated by the tendency to avoid additional vibrations caused by slip-slide friction. Its feature is a reduced order of the controller, regardless of the dimension of the system and the number of tunable parameter. Our goal is to elucidate the most important properties of the gyroscopic stabilizers variety. It is described by a polynomial equations system. The dimension of the general solution variety in the regular case is easily found and some of its points can be calculated numerically. We start with an example of dimension 3, which leads to a system of 6th order ordinary differential equations (ODE) and then a system of five polynomial equations with regard to six unknowns. Its general solution turns out to be a one-dimensional algebraic variety presented in a table form. The second example has dimension 5; it corresponds to a tenth-order system of differential equations and nine polynomial equations in fifteen unknowns. The dimension of the solution manifold is equal to six; we find a one-dimensional subvariety and some singular points. The main difficulty is the divergence of numerical calculations near the multiple poles of a closed system. One of the important properties, that manifested itself in both examples, was the presence of complex conjugate poles and occasionally multiples of real ones; thus, almost all solutions for an optimal gyroscopic stabilizer are made up of relatively rapidly decaying oscillations. In both examples, the variety of solutions consists mostly of simple poles and allows one to choose a stabilizer that does not create resonant effects.