Controlling the finite rotation of an elastic system from one state to another with vibration suppression at the final moment of operation

The problem of passive control of an arbitrary elastic system that performs a finite rotation in the general case with acceleration or breaking about the unmovable axis and small nonstationary vibration under the impact of an arbitrarily distributed load proportional to some unknown finite function has been considered. The equations of motion of the system have been written in the normal coordinates which represent the eigenmodes of the free rotating system. In this case, the finite rotation of the system as an absolutely rigid body is represented by the zero mode. It is required that, at the end of the system rotation at the given angle and for the given time, elastic oscillations are eliminated for several lower eigenmodes. An unknown control function (control law) is sought for the considered time interval in the form of a series of sinuses (and also cosines) with unknown coefficients. On the basis of the exact solution of the equations in normal coordinates with initial and final conditions, the problem reduces to a system of linear algebraic equations for unknown coefficients. As an example, a roll rotation on a finite angle from one state of rest to another of spacecraft with two symmetrical multi-link solar panels has been considered. Calculations have been carried out for various numbers of eigenmodes to be eliminated with comparisons relative to the numerical solutions of the equations in generalized coordinates under the found control actions. It has been shown that in order to obtain a practically acceptable accuracy, it is sufficient to eliminate the vibrations of not more than two or three the lowest eigenmodes.