Известия Томского политехнического университета: Инжиниринг георесурсов (May 2019)
Form and structure of differential equations of motion and process of auto-balancing in the rotor machine with auto-balancers
Abstract
The relevance of work is conditioned by a need of investigation of the process of equilibration by auto-balancers of rotating machines in equipment of the extraction and transportation facilities of geo-resources, particularly, in mine ventilators, in gas turbines for natural gas transportation. The main aim of the study is to ascertain the structure and to specify the form of differential equations that describe the motion of a rotary machine with auto-balancers with many corrective weights and differential equations that describe the auto-balancing of rotor. The methods used in the study: elements of theoretical mechanics, Lyapunov stability theory, theories of rotary machines. The results. In the framework of a simplified theory of rotary machines with auto-balancers with many corrective weights the authors ascertained the structure and specified the form of systems of differential equations that describe the movement of a rotary machine and the process of balancing of the rotor by auto-balancers. It was determined that the rotary machine conditionally consists of several interacting parts - a rotor (rotor in corps) and unbalanced auto-balancers. Unbalanced auto-balancers act on the rotor with the forces that apply to the point of suspension of auto-balancers and are equal to the second derivative by time of the vectors of the total imbalances. The rotor affects the movement of the corrective weights in auto-balancers by forces of moving space that are proportional to the acceleration of points of suspension of auto-balancers. The system of differential equations describing the motion of a rotary machine was drawn up with respect to the generalized coordinates of the machine. It is composed of two or more of the associated subsystems. The first - describes the motion of the rotor. It can always be written relatively to the generalized coordinates that describe the motion of the rotor and total imbalances of the rotor and auto-balancer in each correction plane. Thus, if the rotor is mounted with rotation around its longitudinal axis in the corps which is held by pliant supports then the rotor and the corps form a conditioned composite rotor (more elongated and massive than the rotor) and the equations are made for it. The number of other subsystems equals to the number of auto-balancers which counterbalance the rotor. Thus, the subsystem, corresponding to j-th auto-balancer, has a standard form and describes the motion of the corrective weights in this auto-balancer. It consists of nj differential equations, where nj - the number of corrective weights in j-th auto-balancer. The system of differential equations that describes the process of auto-balancing of the rotary machine is compiled relatively of generalized coordinates of the rotor and of projections of the total imbalances of the rotor and auto-balancer in each correction plane. It is designed to investigate the stability of families of basic movements and the behavior of transients at auto-balancing. This system also consists of two or more of the associated subsystems. The first is obtained from the subsystem, describing the motion of the rotor if we write it relatively to the generalized coordinates of the rotor and total imbalances. The number of other subsystems also equals to the number of auto-balancers. Each of these subsystems has a standard form and consists of two equations that are obtained by combination of the equations of motion of corrective weights of corresponding auto-balancer. Rules of composition of differential equations describing the motion of the rotary machine and the process of auto-balancing are formulated. They are applicable for any kinematics of the rotor motion (the rotor, placed in the corps); for any number of auto-balancers; for any number and different types of corrective weights in auto-balancer. The type of differential equations of the first subsystem is confirmed using the basic theorems of dynamics. The formulated rules were applied to the rotary machine consisting of the rotor placed in the corps with the possibility to be rotated, which is held by pliant supports, and of two auto-balancers.
