THE GYROSCOPIC FORCES INFLUENCE ON THE OSCILLATIONS OF THE ROTATING SHAFTS

The results of numerical investigation of shafts transverse oscillations with account of gyroscopic inertia forces are presented. It is shown what the action and how the gyroscopic forces influence on the transverse oscillations of the shafts during rotation. The study has been done with computer program with a graphical interface that is developed by authors. The process of numerical solution of the differential equations of oscillations of rotating rods using the method of numerical differentiation of rod's bend forms by polynomial spline-functions and the Houbolt time integration method is described. A general block diagram of the algorithm is shown. This algorithm describes the process of repeated (cyclical) solving the system of differential equations of oscillations for every point of mechanical system in order to find the new coordinates of positions of these points in each next point of time t+t. The computer program in which the shown algorithm is realized allows to monitor for the behavior of moving computer model, which demonstrates the process of oscillatory motion in rotation. Moreover, the program draws the graphics of oscillations and changes of angular speeds and accelerations in different coordinate systems. Defines the dynamic stability fields and draw the diagrams of found fields. Using this program, the dynamics of a range of objects which are modeled by long elastic rods have been studied. For some objects is shown that on special rotational speeds of shafts with different lengths, in the rotating with shaft coordinate system, the trajectories of center of the section have an ordered character in the form of n-pointed star in time interval from excitation to the start of established circular oscillation with amplitude that harmoniously changes in time. It is noted that such trajectories are fact of the action of gyroscopic inertia forces that arise in rotation.

Introduction. The tasks of dynamics of elastic shafts' systems that rotate in fields of inertia forces have actuality while structural elements of machines and devices are designed. The rotating shafts are responsible elements in the constructions of engines, turbines, wind and hydropower plants, other machines. For these objects, in many cases, the cause of oscillations is the periodic changes of the gyroscopic inertia forces of system per time.
When the shaft rotates and begins to bend under the action of external loads, the gyroscopic forces start to transmit energy in direction that is perpendicular to the plane of bending. After it the shaft begins to oscillate in two mutually perpendicular planes of the coordinate system that rotates with it.
The behavior of rotating elastic systems that consist of shafts, rods and rotors is described by complex differential equations systems with partial derivatives with account of gyroscopic inertia forces. Low rigidity, large length, relatively high values of the excitation intensity parameters, in which the structural elements are used, all of these make the necessary to analysis of oscillatory motion around of critical and overcritical rotational speeds. Also, these make the necessary to search the natural oscillation frequencies range and range of critical rotational speeds with account of gyroscopic forces, and stability analysis in the study of different motion modes.
In recent years, the dynamic tasks of oscillations of shafts and rotating rods were investigated in works of many authors. The task of rotating shaft with influence of axial loads to the propagation characteristics of the elastic waves is studied in paper [14]. The shaft is viewed with non-uniform cross-sections per length. Axial loads considered with constant values.
The paper [8] presents the study of problems with elastic stabilization and long-term strength of the system under cyclically changing external impacts that are appearing because of eccentricities. Task is considered taking into account the gyroscopic loads, in linear statement. The paper [7] presents the results of study of space bending oscillations of horizontal rod that is rotating around its axis. Rod is under the action of periodic harmonic force of self-weight per length. The task is considered with account of the gyroscopic loads, too.
Questions about the transverse oscillations of the rods under the action of axial periodic loads, also the tasks of longitudinal-transverse oscillations under the action of beat loads are considered in papers [5,6]. But in them the investigated rods don't rotate.
The analysis of presented in scientific literature results shows that the task of investigation of dynamics and strength of rotating elastic systems, with account of gyroscopic forces, is actual. Many authors are paid attention to calculation of critical rotational speeds and natural oscillation frequencies by different system parameters. But the nature of the oscillation process itself and how the certain parameters of the system influence on the development of oscillations almost is not considered. Therefore, it is interesting to study the dynamic behavior of the consider systems and define what kind of effects the gyroscopic forces generate.
Problem statement. In the process of oscillation of rotating shafts or rods with considerable lengths with different physical, geometric and dynamic parameters, the various bend forms that change in time are possible. As a dynamic model is considered a rod with length l (Fig. 1) that can be exposed by action of an axial load P(t). The rod is rotated on angular speed ω around the rectilinear axis O 1 X 1 of the stationary coordinate system O 1 X 1 Y 1 Z 1 . The rotating coordinate system OXYZ is tied to the rod and rotates with it. The direction of OX axis coincides with  (1) direction of O 1 X 1 axis. Axis of rod in deformed state is coincided with the OX and O 1 X 1 axis. The oscillatory motion of the rod in the OXYZ coordinate system is characterized by y(x,t) and z(x,t) displacements of the points, that belong to the axis of rod in the OY and OZ coordinate axes' direction, respectively.
The oscillations of rotating rod in space are described by the corresponding system of differential equations, which taking into account the geometric nonlinearity and the axial force [2,3] have a form: where E -elastic modulus of rod's material; I 1 , I 2 -inertia moments of rod section in mutually perpendicular planes; r -radius of gyration; m -mass of unit per length;  -rotational speed of rod around the axis that coincides with the axis of rod in undeformed state; P(t) -periodic axial force; 1 1  , 2 1 main curvatures of rod's axis in mutually perpendicular planes.
Technique. The solving of differential equations of rotating rods oscillations for searching their geometric position in space in the process of oscillation and analysis of dynamic behavior is carried out using the method of numerical differentiation described in papers [10,11], and the Houbolt time integration method [13] in form: of rod axis for next point of time t+t (Fig. 2).
To apply the Houbolt method need to know at least the first three points of time integration. To find these points, it is advisable to use the finite differences method with initial conditions, namely, to make the assumption that at time t=0, when the system start out of equilibrium, the initial deviation value for each point of the rod axis is known: , where a n is random deviation, and values t n t n y y . The solving of the dynamic tasks of the oscillatory motion for rotating shafts and rods, based on described technique, has been done with computer program with a graphical interface. The general block diagram's algorithm of the program is shown in Figure 3. This algorithm describes the process of repeated (cyclical) solving the system of differential equations of oscillations for every point of rod elastic line in order to find the new coordinates for these points in each next point of time t+t. This is performed with the show of current calculations' results as a moving computer model, which displays the oscillating process of rotating rod in real time.
Results. Using specified program, the dynamics of a range of objects which are modeled by long elastic rods have been studied. For research objects is shown that on special rotational speeds of shafts with different lengths the trajectories of center of the section have an ordered character. For example, for a transmission shaft with outer diameter D=0.1 m, inner diameter d=0.06 m, length l=3 m, on rotational speed =31.79 s -1 the motion trajectory of center of the section on the middle of shaft, in rotating coordinate system, it will look like a five-pointed star (Fig. 4). On rotational speed =52.95 s -1 in rotating coordinate system the motion trajectory will look like a tree-pointed star (Fig. 5). On rotational speed =79.35 s -1 in rotating coordinate system the motion trajectory will look like a four-pointed star (Fig. 6). Herewith, the trajectories of motion in a stationary coordinate system will have a similar character (Fig. 7, 8, 9). The same trajectories of motion are observed for other objects with different parameters, but on other rotational speeds.
The shown below on diagrams trajectories due to the action of gyroscopic inertia forces that are arisen in rotation. Лізунов П.П., Недін В.О.