Abstract. This research paper is aimed to investigating rotor dynamics of multistage centrifugal machines with ball bearings by using the computer programs "Critical frequencies of the rotor" and "Forced oscillations of the rotor," which are implemented the mathematical model based on the use of beam finite elements. Free and forces oscillations of the rotor for the multistage centrifugal oil pump NPS 200-700 are observed by taking into account the analytical dependence of bearing stiffness on rotor speed, which is previously defined on the basis of results' approximation for the numerical simulation in ANSYS by applying 3D finite elements. The calculations found that characteristic and constrained oscillations of rotor and corresponded to them forms of vibrations, as well as the form of constrained oscillation on the actual frequency for acceptable residual unbalance are determined. IntroductionThe existing numerical methods for calculation of rotor dynamics, which are based on the use of modern programs on the basis of finite element method, use beam elements with taking into account the Bernoulli's Hypothesis or 3D finite elements [1]. In this connection, appliance of 3D models, for example in ANSYS, needs carrying out relatively long operations and computational time [2]. The appliance of beam models [3] is not a long-time process for the initial data preparation, as well as it makes the computational process faster. However, the problem of determination of critical frequencies of rotor systems by using existing programs for the advanced determined dependence of the bearing stiffness and gap seals on rotor speed leads to drawing the Campbell diagram [4] that complicates the computational process with additional graphic constructions. Besides, the determination of rotor oscillations by taking into account the system of imbalances and nonlinear bearing stiffness is the high-complicated scientific problem [5].In accordance with the abovementioned, the aim of this research is a development of methods for determining eigenfrequencies and critical frequencies, as well as characteristics of rotor oscillations by applying a combined using 3D and beam finite elements on the example multistage centrifugal machines. To achieve this aim the following problems are set: 1) determination of the bearing stiffness with taking into account rotor speed by using ANSYS; 2) development of the mathematical model of rotor dynamics with the predetermined dependence of the bearing stiffness on rotor speed;

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This paper is aimed at refinement of the computational model of the turbopump rotor systems associated taking into consideration the effect of rotation of moving parts and compliance of bearing supports elements. The up-to-date approach for investigation of non-linear reactions in rotor's bearing supports is proposed for turbo-pump units for liquid rocket engines. Five models for modelling contact interaction are investigated, and comparative bearing stiffness characteristics are given. The geometry of the housing and corresponding design scheme are set for each support due to the assembly drawing of the turbopump unit. Rotation of the shaft is taking into account by applying corresponding inertial forces to the inner cage of the bearing. Experimental points of the dependence "loaddisplacement" as the diagram "Fv" are built by the calculated points as an array of numerical simulation data, obtained by the ANSYS software. As a result of numerical simulation, including loading of the bearing support on the scheme "remote force" in a wide range of rotor speeds, the corresponding displacements are determined. The brandnew approach for evaluation of bearing stiffness coefficients is proposed based on the linear regression procedure. As a result, the obtained values of coefficients are summarized and approximated by the quadratic polynomials.

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This article dials with the refinement of the mathematical and computational models of the oxidizer turbopump rotor considering bearing gaps, axial preloading, compliance of the housing parts and the effect of rotation. The loading scheme consists of four substeps is proposed considering preliminary displacement of the outer cage, axial displacement as a result of the support deformation due to the axial preloading force, radial displacement due to the support deformation, as well as centrifugal forces of inertia caused by rotation of the rotor with an inner cage. Modelling of contacts interactions using ANSYS software is carried out according to the appropriate models of contact behaviour. The contact areas between the rolling elements, inner and outer cases are obtained. The contact angle is determined. Isosurfaces of axial and radial displacements for the bearing supports are built. Nonlinear stiffness of bearing supports is determined as the tangent of the angle of inclination for the curve "radial loadradial displacement". The proposed approach, which used for designing turbopump units for liquid rocket engines, will allow refining the reliable mathematical and computational models of rotor dynamics for turbopump units and providing appropriate computer simulation of forced oscillations of the rotor systems for given permissible residual imbalances considering nonlinear stiffness characteristics of bearing supports.

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