Theoretical substantiation of the parameters of the vibration protection system for the workplaces of compressor plant operators

. The compressor module includes two units – the compressor itself and the energy source, that is, the engine, which come in two types: electric motors for low-power compressors and internal combustion engines. It should also be noted that in almost all sections of compressor stations, sound pressure levels exceed the permissible standard values, and the greater is the power of the compressor unit, the higher is the noise level. The purpose of the carried out theoretical research is to substantiate the parameters of the vibration protection system of the service personnel workplaces


Introduction
During the operation of compressor units in the technological mode, vibroacoustic factors arise at the workplaces of the service personnel, which affect the operability [1-4].In almost all sections of compressor stations, sound pressure levels exceed the permissible standard values.Noise reduction is achieved, among other things, by reducing vibrations by constructive and technological measures.Reducing vibrations at the workplaces of compressor station operators can technically be implemented in two ways [5-7]: -selection of shock absorbers with the required stiffness parameters; -designing a foundation with appropriate dimensions and dissipative properties.The first option is advisable to use for the conditions of compressor stations, when electric motors are used as an energy source.Indeed, for the conditions of such an arrangement of vibration sources, the levels of vibration velocity and vibration acceleration are significantly lower than with internal combustion engines.In addition, vibration dampers can be implemented for this option only for compressors, since their vibration levels are significantly higher than those of electric motors.
The theoretical study of the vibration levels of the foundation excited by the compressors was carried out according to the scheme shown in Fig. 1.The force effect on the foundation is set by the following dependence where   -vibration attenuation coefficient in the presence of vibration isolators;  = 9,81 m/s 2 -acceleration of gravity;  -circular speed of the crankshaft, rad/s; m1 -mass of the power plant, kg.
Then the dependence of the force action will take the form where  -s the vibrational energy loss coefficient of the vibration isolation system; n -s rotation frequency, rpm.
The validity of this calculation scheme is confirmed by the fact that the mass of the foundation and its geometric dimensions are disproportionately smaller than the mass of the floor and the area of the production room, therefore, in this case, the reaction from the floor to the foundation is not taken into account.
The mass of the foundation is defined as where  2 -the density of the material, kg/m3;  1 ,  2 -the length and width, m; ℎ -the thickness of the foundation The oscillatory system includes a foundation with a mass () and an engine operating at a rotational speed () and having a mass ( 1 ), which is mounted on vibration isolators with reduced rigidity ( ) 1 .The dynamic model of the oscillatory system is described by the differential equation: and to solve it, a system of equations is obtained in the form: Taking into account the above expressions and the introduced notation, we obtain: and denoting  =  •  и  =  •  .
The maximum value of the vibration velocity (V1) and the vibration velocity level (  ) are determined by the dependencies and vibration acceleration (  ) by dependence: Introducing the notation for the  coordinate − 1 2  1 (0,1) 2 +  1  1  (0,1) 4 −  1 2 (0,1) 2 = ;  1 (0,1) 5 = ; −2 1  2  1 (0,1) 6 For the foundation, the maximum value of the vibration velocity and vibration velocity level are determined as On the foundation, the level of vibration velocity should not exceed the allowable values.In this case Based on this ratio, the parameters of the foundation itself and the vibration isolation system are determined In the theory of vibrations, the quality of the vibration protection system is determined by the force transfer coefficient  в , which is the ratio between the amplitudes of the forces transmitted to the base   and the disturbing  0 .
Considering the main criteria for reducing the vibrations of a dynamic system, we use vibration protection methods of vibration isolation and vibration absorption.Based on the principles of equality of forces of action and reaction, the force transmitted to the base is equal to: Then, The force transfer coefficient (9), based on these transformations, is determined by the expression Thus, the vibration protection system includes methods of vibration absorption and vibration isolation, and the effectiveness of such a system is inversely proportional to the magnitude of the force transfer coefficient.Therefore, to improve the efficiency of the vibration protection system, the force transfer coefficient must be minimized, and its parameters are selected and justified on the basis of a system of restrictions [8-10].At the same time, it should be taken into account that it is impossible to vary the speed of the motor shaft or its mass, and variation in the mass of the foundation is in principle possible, but undesirable, therefore, the system of restrictions will take the following form: ≤  ≤     ≤  ≤   The range of stiffness variation is limited on the basis of the condition that the natural vibration frequencies of the engine on vibration dampers do not coincide with the engine shaft speed, and the drag coefficient () must be less than the critical one.As a result of such a selection, one of the parameters is on the boundary of limitations and a new approximation is set within technically permissible limits.
Determining the natural oscillation frequencies, it is allowed to consider the engine as a solid undeformed body, representing an oscillatory system with six degrees of freedom.However, such an assumption is possible when determining the natural vibration frequencies and the corresponding vibration levels in the frequency range from 4 to 125 Hz.
Thus, the initial data in the calculation of the vibration protection system are the mass of the engine and the moments of inertia about the main axes.Oscillations of an elastically suspended solid undeformed body are described by differential equations: 2   2 + М  = 0 where  -the mass of the engine, kg;   ,   ,   -the moments of inertia relative to the coordinate axes, N•m•s 2 ; θ,ψ,ξ -the angles of rotation relative to the coordinate axes; Mthe moments relative to the corresponding coordinate axes, N•m.
Taking into account the harmonic nature of the force action, we represent the possible increments of coordinates in the form where n is the engine speed.The system of equations for calculation (11) takes the form ( 1 −  2 ) = 0 ( 2 −  2 ) = 0 ( 3 −  2 ) = 0 ( 4 −  2   ) = 0 ( 5 −  2   ) = 0 ( 6 −  2   ) = 0 For ease of calculation, this system is written in matrix form as  When solving a system of differential equations, it is necessary that the determinant of the system be equal to 0, that is, | −  2 | = which makes it possible to determine six values of natural vibration frequencies.

Conclusion
The vibration protection system includes vibration isolation and vibration absorption methods, and the definition of its parameters has the following algorithm: -a dynamic model of the system is created, given by equations ( 3) and (4), and an equation is made with respect to z_1 (5); -the force transfer coefficient is determined, from expression (10) and in general form a system of restrictions is formed to minimize it (11); -to solve the minimization of the force transfer coefficient, an appropriate software package (MATLAB) is used.In addition, if the parameter values are at the limits of the restrictions, the calculation procedure should be repeated within technically acceptable limits; -when solving the task to determine the permissible values of vibration velocity (6) and vibration acceleration (7) during operation, their permissible values are substituted into the left side of the expression instead of the actual values of the vibration transmitted to the foundation.
The data obtained during the calculations make it possible to determine the natural vibration frequencies of the engine and the normalized frequency ranges in which they fall, as well as the vibration levels of the foundation at its attachment points.
Comparison of the values of vibration levels obtained by calculation with normalized values makes it possible to determine the required efficiency of vibration protection systems.

E3S
Web of Conferences 458, 04024 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345804024 K is the matrix of stiffness coefficients; M is the matrix of inertial coefficients; Q is the matrix of coordinates; p are natural oscillation frequencies, r/s. where