Active damping of vibrations of large-size structures

Large-sized systems that carry a payload, usually concentrated at its end, are subject to undesirable vibrations. The task of damping such vibrations is very relevant. The article examines a similar mechanical system equipped with a special unit designed for active vibration damping. As a carrying mechanical system, a long beam is assumed, at the end of which the payload is placed. Structures of this type are used, for example, in the composition of spacecraft. In such devices, as a rule, the payload, in the form of equipment for remote sensing of the Earth or telescopes for observing the stars, is carried out on a long rod. The mass of the equipment exceeds the mass of the rod and such a design oscillate due to vibrations associated with the activity of the spacecraft itself or due to the operation of the engines for orbit correction. This leads to blurring of the images obtained from the equipment. To improve the image quality, it is necessary to extinguish these unwanted oscillations of the equipment. It is proposed to do this with a special device for active vibration damping. It consists of a guide placed perpendicular to the load-bearing beam at its end. Along this guide, under the action of an electric motor, a damper of a certain mass moves. By moving the damper, it is possible to achieve damping of vibrations of the beam with the equipment. To do this, you need to control the movement of this vibration damper. The control is based on linear feedback. Four parameters are taken into account: the speed of the equipment and the damper, and the position of the equipment and the dampener. Depending on these parameters, the linear feedback coefficients are formed in the law of motion control of the damper. With some reasonable assumptions, the equations of motion of such a mechanical system, including the control force, are written out. This control force is to be determined depending on the requirements for the transition process. One can suggest such a requirement as maximizing the stability of the entire system during the transition process. Then, analyzing the roots of the characteristic polynomial, we can establish a relationship between the feedback coefficients, and then express the three coefficients through the fourth. The choice of this fourth coefficient can be made for geometric reasons. Namely, assuming the symmetry of the guide and, accordingly, the equality of the displacements of the dampener relative to the beam. It turns out that with this control mode, the displacement of the dampener will be significant. In order to reduce the geometric dimensions of the vibration damping unit, a different control law can be proposed. This law will provide a zero solution to the equations of motion (the equilibrium position) so that it is asymptotically stable. Of course, the transition process is assumed to be stable, and the displacements of the damper are symmetric. It is found that in this mode, the displacement of the dampener is less, and the transition time is longer than in the previously proposed control mode. To achieve such a transition process, in which the displacement of the dampener and the time will be less than in the first and second control modes, respectively, a combined control law can be proposed. It will consist in the fact that during the transition process, the feedback coefficients will switch from the set for the first control mode to the set for the second mode. It is possible to propose a control law that contains only one switching moment. The proposed combined control can be characterized as a piecewise linear control with a stepby-step switching of the feedback coefficients. This study can be used to determine the limiting possibilities of damping unwanted vibrations of large-sized beam-type structures by moving the internal mass (dampener).