Evaluation of rotor axial vibrations in a turbo pump unit equipped with an automatic unloading machine

The article presents forced axial vibrations of the rotor with an automatic unloading machine in an oxidizer pump. A feature of the design is the use in the autoloading system of slotted throttles with mutually inverse throttling. Their conductivity is determined by a numerical experiment in the ANSYS CFX software package.


Introductory remarks
The article presents ( Figure 1) a simplified rotor design of the oxidizer pump equipped with an automatic unloading machine (AUM), in which, unlike in traditional devices [1], inverse slotted throttles are used. Static calculation of the system is given in the article "Static calculation of the rotor unloading automatic machine for a high-pressure centrifugal pump" (V A Martsynkovskyy, A Deineka, A Korczak and G Peczkis), published in this issue of the journal.
Unit rotor together with the automatic unloading machine is a complex dynamic system with distributed parameters that is a subject to periodic external perturbations. Each cross section of the rotor performs interrelated radial, angular and axial oscillations [2]. In regards to reliability, primarily axial vibrations can be dangerous, since their relatively large amplitudes limit the operating life of ball bearings wherein rotor is located. In this paper, as a first approximation, a simplified problem is considered: a rotor is studied as a solid body with an automatic loading system which performs onedimensional axial oscillations along the support axis. Such a simplified model enables to obtain static and dynamic characteristics in an analytical way and accurately describe the main regularities of the system oscillations.

Derivation of the axial oscillation equation
Expression of the total axial force acting on the impeller, obtained in the article "Static calculation of the rotor unloading automatic machine for a high-pressure centrifugal pump", has the form: where 0 Т -is the residual axial force to be balanced;  (1) is the pressure force conditioned by the average angular velocity of liquid in the chambers. Capital letters indicate the flat annular sections shown in Figure 1.
At axial oscillations, the force of viscous resistance z с  (c is the damping coefficient) acts on the rotor, therefore, on the basis of the 2nd Newton's law, the equation of axial oscillations will take the form: т -indicated rotor weight. Having devided these equation term-by-term on п Ар , we obtain the dimensionless form of the equation: The equation (3) includes an unknown pressure 2 р in the chamber 2, which depends on the gap size and is determined from the balance equation of the flow rate passing through the slotted throttles of the equilibrium system. For automodeling section of the turbulent flow, the flow rate through the upper and lower throttles is i.e. decrease in pressure at the lower radius of the corresponding chamber due to the inertial effect is taken into account. In static calculation According to the results of the numerical experiment, conductance almost linearly depends on the gap: And for the unit under consideration The obtained first-order differential equation relating to the unknown 2 p is nonlinear. We linearize it in a neighborhood of the equilibrium position, passing to the variational equations. It should be taken into account that inertial forces in the corresponding chambers are: Further, we go into the quasistatic change in the rotation frequency, therefore , , , 0 and variational equations are (7): 10 20 0 20 10 10 20 To obtain the equation in a short form, it is necessary to divide the last equation term-by- To reduce the recording, variation sign will further be omitted, keeping in mind, however, that this is not about the absolute values of the variables, but about their small deviations from the established values. The latter are marked with an additional zero index and are determined by static calculation. Having denoted the time constants and transmission coefficients as: we obtain the standard form of the linearized flow rate balance: Pressure in the chamber can be considered as a controlling action, and equation (10) we can derive the control influence of it: proper operator of the controller and operator of the actions by error, respectively.
Relation of the controller reaction to the stimulus is the transfer function of the controller. If the stimulus is a harmonic function, then an operator of time differentiation is replaced by an imaginary operator  i :     , so this condition of static stability is fulfilled at the whole range of changes in external influences.
The real part 2 U characterizes the controller's stiffness itself, and the imaginary part 2 V -its "contribution" to damping [2]. In the absence of external damping c = 0, axial oscillations decay, if This condition with some margin for the stabilizing effect of external damping can be considered as a condition for rotor axial stability.
Having substituted expression of the regulating effect (11) into the equation of axial oscillations (3), we obtain motion equation of the system "impeller-balancing unit". Preliminary equation (3) must be linearized, i.e. move to variations. Herein the last term on the right-hand part is . After some transformations, we obtain: Having groupped the terms in powers of the differentiation operator in time, we obtain final form of the system equation: Static rotor displacement relative to the equilibrium position, caused by the deviations of external influences, can be determined using the equation (15), assuming . Using coefficients of the proper operator one can determine stability of the axial oscillations. According to the Routh-Hurwitz criterion, a third-order system is stable if the following condition is fulfilled: is the equation of free axial vibrations of the impeller. Here we confine ourselves to analyzing forced oscillations and stability. A similar problem of rotor axial oscillations of a shaftless motor pump with the traditional design of an automatic unloading machine was considered in the article [5].
We will assume that variations of external influences  We single out real and imaginary parts in the proper operator (16) and the action operators (17), replacing  i s  : Having substituted these expressions into formulas (21), we obtain frequency transfer functions in the form of complex numbers, for example: Now the amplitude and phase By analogy, frequency response characteristics according to the influence of