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Analysis of proper fluctuations of technological systems

The article describes the principles of building dynamic models of technological systems with concentrated parameters with a finite number of degrees of freedom. Dynamic properties of technological system are considered taking into account elastic and inertial parameters of its elements. The model of three-mass torsional chain system and the model with progressively moving masses are used for estimation of vibrational processes in drives of the technological equipment. According to the proposed mathematical model the analysis is carried out and the forms of natural oscillatory movements in the system are revealed. Oscillations of zero, first and second eigenfrequencies are determined, for each of which sets of form of the coefficients are found that describe the system configuration in the process of free oscillations.


Introduction
In modern conditions, the intensification of production processes is observed, which eventually leads to an increase in the operating speeds of technological machines. There is a need for deeper consideration of dynamic factors affecting the accuracy of production equipment [1,2].
Any technological machine consists of several functional parts, namely, the engine, the mechanical system and the motion control system. The mechanical system of the machine carries out the transformation of relatively simple movements of the input links into complex movements of the actuators, which ensure the performance of given technological operations. Mechanical system components are primarily subject to different types of vibration. Occurring oscillations lead to essential distortions of the set functions of program movement of working elements, reduce accuracy in processing of products and worsen other conditions of normal operation of the equipment [5, 6,8,14,15].
The study of the conditions of the emergence and analysis of vibrations during machining using modern high-performance CNC machines is of particular importance. Here, significant dynamic loads arise in the operation of main motion drives, as well as feed drives that provide the formation of a complex tool path along the surface to be machined. Dynamic errors caused by free oscillations in the unsteady modes of the machine tools many times exceed the static errors of positioning of actuators. The attenuation time of these vibrations can be commensurate with the time of program movements. In this case, accuracy is inevitably reduced, deviations from the given geometric shape appear, and the quality of the surface deteriorates [3,4,9,16].
The analysis of torsional oscillations in the main drive and feed drives reveals two areas of intense oscillation. The first is due to the dynamics of the main drive, and the second is due to the spectrum of the natural frequencies of the machine's support parts. Increase of rigidity of the mechanical system "machine-tool-detail", reduction of friction forces in feed drives, selection of optimal cutting modes for areas with reduced vibration levels, application of software algorithms and design solutions for vibration

Materials and methods
For engineering calculations dynamic models with concentrated (i.e. discretely set) parameters are often used, in which the number of degrees of freedom is finite. Construction of such models is based on the following principles [8,9].
1. Inertial properties of a mechanical system are displayed by masses or moments of inertia, which are concentrated in separate points or sections.
2. These points or sections are connected by elastic, dissipative and geometric (kinematic) bonds without inertial properties. Dissipative bonds take into account the resistive forces that cause mechanical energy to dissipate and partially transfer it to other types of energy.
To simplify the dynamic model, the most massive elements and the most flexible (i.e. the least rigid) parts of the kinematic chain are singled out in the machine drive and its mechanisms [11,12,13].
The application of a sophisticated model can lead to errors due to inaccurate rigidity and dissipative factors in the system. To increase the reliability of the analysis it is necessary to choose the simplest dynamic models, which, nevertheless, are able to adequately reflect the vibrational processes under study in the technological system.
Many works related to the problem of limiting the elastic vibrations in technological systems take into account only the elastic properties of kinematic links and mechanical transmissions of motion. In this case, a two-mass design scheme is used to simulate the motion of the actuators by individual degrees of mobility. However, for the analysis of dynamic properties of high-performance equipment it is necessary to take into account also inertia of system elements and to use multi-mass design schemes, among which three-mass vibrating systems are of the greatest interest [6,10,11,12,13].

Mathematical description and analysis of the three-massvibrating system
Let us consider the oscillating torsional chain system shown in Figure 1. There are three inertial discs on the elastic shaft with moments of inertia J1, J2 and J3. The torsional stiffness coefficients of the shaft sections are indicated via C1 and С2. The mechanical drive system includes the motor rotor 1, gear 2 and implement 3. The inertia moments of the links shown in the diagram are calculated relative to the natural rotation axes. Motor torque М1 is applied to the engine rotor, and the working bodytechnological resistance torque М3. Viscous friction in the system is not considered in this article. The mechanical system has three degrees of freedom. Let us choose as generalized coordinates the rotation angles of disks φ1, φ2 and φ3, counting them from any position of the system at the undeformed shaft. Let us record the kinetic energy of the system: The potential energy is determined by the expression: . 22 Before we begin to integrate the system (4), let us consider which movements are described by this system of equations. Having summed up all the differential equations, we obtain: The obtained equation is convenient to write in the form of a theorem about the change in the kinetic moment of the system: It is more convenient to estimate the possible system movements described by equation (7)   Let us imagine the motion of the system bodies as complex and consisting of motion together with the center of inertia and relative to the center of inertia. Then the velocities of bodies can be written as follows:  x In this motion there is no deformation of elastic elements. The other three terms in (10) determine the motion of bodies relative to the system considered to be a rigid whole. In this case, the elastic elements undergo deformation. Consequently, the last three terms in (10) reflect the oscillations of the bodies superimposed on the motion of the system together with the center of inertia. From the comparison (8) and (10) (11) From (11), it follows that: 1. If the initial conditions are such that const ≠ 0, the system moves as a rigid body with a speed equal to that of the body:   3. The number of system movements in case of oscillations is zero. The results obtained are fully applicable to the differential equation (7). In general, when const ≠ 0, this equation defines the rotation of the system as a rigid whole with a constant angular velocity ω0: 12) where φ0 and ω0 are from the initial conditions.
In addition to solution (12), the system of equations (4) also allows for oscillations of the discs near the undeformed state of the shaft, described by the functions: The functions (13) define the motion of the system in which the discs oscillate at the same frequency k and with the same starting phase α, but with different amplitudes. Substitute (13) in equation (4) and after the dividing by sin(kt+ α), we obtain a system of algebraic equations relative to the unknown amplitudes: The homogeneous system of equations (14) has non-zero solutions if the system determinant is equal to zero. Let us solve the system of equations (14) in a matrix way: Expanding the determinant, we obtain the frequency equation: (16) One of the roots of this equation is zero: k0 = 0. This root corresponds to the rotation of the system as a rigid whole. In addition to zero, equation (16) defines two more roots: the first k1 and the second k2 natural frequencies of the mechanical system.
Mechanical systems whose frequency equation has one or more zero-roots are called semi-defined. These roots correspond to a form of motion in which the deformation of the elastic links is absent and the system moves as a rigid whole. Thus, the torsional system shown in Figure 1 is semi-defined.  (14) is zero, the equations themselves are not independent. This means that the system (14) allows only the amplitude ratios to be determined, not the amplitudes themselves. Therefore, let us introduce the form coefficients: ; .
Using the first and second equations (14), the following is obtained: .
(18) One must note that μ1=1. Substituting the values of the roots k0, k1 and k2 of the frequency equation into these equations, we will find a set of form coefficients corresponding to each eigenfrequency: 10 20 30 , .
Each set of form factors describes the configuration of the system at with amplitude deviations from the equilibrium position in the process of free oscillations with one of the natural frequencies. This configuration is called as its own form. Thus, the torsional system under consideration (Figure 1) has three own forms. The form of motion of the system with zero frequency k0 = 0 is not oscillatory, but determines the motion in which the discs rotate at the same constant angular speed ω0.This movement can be conventionally called as the frequency k0 oscillation.

Conclusion
Modern technological equipment can be considered as a complex oscillating system consisting of many elements with inertial and elastic parameters. An unlimited number of degrees of freedom in such system requires the construction of a relatively simple and adequate dynamic model to solve the problem of analyzing the natural oscillations in the system.
In the paper mathematical models of three-mass vibrating torsional system and system with progressively moving masses are considered. The solution of differential equations of motion revealed the forms of natural vibrational movements in the system. The frequency equation is obtained, three roots of which determine the oscillations of the zero, first and second natural frequencies of the mechanical system.
For each eigenfrequency, we find a set of form of the coefficients describing the configuration of the system in the process of free oscillations.