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Article

Methodology for Optimal Design of Active Fluid Film Bearings Considering Their Power Losses, Stability and Controllability: Theory and Experiment

by
Denis Shutin
*,
Alexander Fetisov
,
Maksim Litovchenko
,
Aleksey Rodichev
,
Yuri Kazakov
and
Leonid Savin
Department of Mechatronics, Mechanics and Robotics, Orel State University, 302026 Orel, Russia
*
Author to whom correspondence should be addressed.
Energies 2024, 17(23), 5879; https://doi.org/10.3390/en17235879
Submission received: 16 October 2024 / Revised: 12 November 2024 / Accepted: 19 November 2024 / Published: 23 November 2024
(This article belongs to the Special Issue Flow Control and Optimization in Power Systems)
Browse Figures
Figure 1
<p>Actively lubricated journal hybrid bearing.</p> "> Figure 2
<p>Sets of Pareto-optimal solutions obtained by MOGA and MOPSO algorithm.</p> "> Figure 3
<p>Sets of Pareto-optimal solutions obtained by MOGA and MMOPSO algorithm and solutions selected for experimental testing.</p> "> Figure 4
<p>Experimental rig for studying characteristics of rotor systems on AHBs.</p> "> Figure 5
<p>Structure of measuring and control facilities of experimental setup.</p> "> Figure 6
<p>Manufactured bearing samples: Case #1 (<b>left</b>); Case #3 (<b>right</b>).</p> "> Figure 7
<p>The rotor dimensions.</p> "> Figure 8
<p>A Campbell diagram for the rotor on infinite stiffness supports.</p> "> Figure 9
<p>Measured shaft loci in tested AHB samples.</p> "> Figure 10
<p>An analysis of the correlation between the shaft lift and the maximum control force.</p> "> Figure 11
<p>Experimental results for friction torque: (<b>a</b>) rotor runout diagrams; (<b>b</b>) calculated friction torque values.</p> "> Figure 12
<p>The parameters of the rotor response to a pulse force impact.</p> "> Figure 13
<p>Samples of the rotor response to a pulse force impact on the tested AHB configurations: (<b>a</b>) Case #1, ξ = 0.60; (<b>b</b>) Case #2, ξ = 0.81; (<b>c</b>) Case #3, ξ = 0.71; (<b>d</b>) Case #4, ξ = 0.69.</p> "> Figure 14
<p>Performance of considered AHB configurations at different loads applied: (<b>a</b>) imbalance load; (<b>b</b>) static radial force.</p> ">
Versions Notes

Abstract

:
This study addresses the problem of the automated synthesis of active fluid film bearings optimized for their adjustable design for new generations of turbomachines. The developed methodology proposes a criterion describing the ability of a bearing’s mechanical design to effectively implement control actions along with its energy efficiency and stability properties considered in a solved multi-objective optimization problem. The design process of actively lubricated journal bearings was investigated in the context of the proposed approach. A multi-objective optimization problem was solved with heuristic algorithms. An analysis of the results obtained with the MOGA and MOPSO algorithm revealed their shortcomings emerging in such problems. The MOPSO algorithm was improved to expand the range and uniformity of the distribution of solutions in the resulting Pareto set and to speed up calculations. Four bearing configurations with significantly different properties were selected from the obtained set of solutions, manufactured and experimentally tested, showing the good agreement between the actual parameters and those set during the design procedure. The results substantiate the applicability of the proposed theoretical and computational tools for designing active fluid film bearings with pre-specified properties to meet the comprehensive requirements of the energy efficiency, reliability and service life of turbomachines.

1. Introduction

1.1. Some Issues of Designing Conventional and Adjustable Fluid Film Bearings

Fluid film bearings are almost indispensable elements of turbomachines in a number of applications in the energy, aerospace, chemical and other industries due to their significant service life, load capacity, speed and relative simplicity. At the same time, they concentrate loads and energy flows in turbomachines, being some of their most energy-intensive and responsible elements. They directly affect the performance, reliability and efficiency of machines and therefore require a comprehensive approach to design.
Basic design procedures for fluid film bearings usually include several typical steps [1] and are based on simplified relations used to calculate a bearing’s geometric parameters for the known loads, materials used and some other system requirements [2]. As a result, new rotary equipment is usually able to operate normally within fairly narrow frequency and load ranges, determined by the bearing design chosen in this way.
The transition to adjustable rotor supports, including adjustable fluid film bearings, is discussed [3,4,5] since they allow for the expansion of the operating ranges of rotary machines due to their ability to reduce vibrations [6] and friction [7] and prevent the occurrence of undesirable nonlinear instabilities [8]. However, such changes also inevitably complicate design procedures since, at a minimum, they also require taking into account the variability in adjustable parameters.
Some adjustable fluid film bearing technologies, such as actively lubricated bearings [7,9], require the least changes compared to conventional passive bearings. Their mechanical design can essentially remain unchanged, while the main differences are concentrated in lubrication and control systems. However, conventional fluid film bearings are typically not optimized in terms of control efficiency, which requires some additions to their design procedures.
The overall design procedure for an active fluid film bearing can then be divided into two key parts. The first stage involves the synthesis of a mechanical design that best meets both the basic performance criteria of the bearing and takes into account the aspects of adjustable design. The obtained optimal mechanical solution is then used at the second stage, assuming the synthesis of an optimal controller. This study considers only the first of the abovementioned stages and does not touch upon the issues of the synthesis of controllers.

1.2. Research Background in Field of Optimal Design of Fluid Film Bearings

The use of numerical models of rotor-bearing systems allows for an increase in the accuracy of design in comparison with simplified engineering methods [10]. Conventional parametric analysis is considered as a basic approach to improve some specific bearing characteristics, e.g., texturing parameters [11], throttling devices [12], the shape of a bearing’s surface [13,14] or even the whole bearing design [15]. However, this problem becomes too complicated when the number of considered factors, design variables and/or objective functions increases. Applying optimization algorithms is a way to overcome the noted disadvantage. Researchers have reported their application not only to various types of bearings (rolling [16] and fluid film [17]; journal [18] and thrust [19]; foil and other gas-lubricated bearings [20]) but also to other components of rotor systems, including dampers [21], seals [22] and even entire rotor systems [23,24].
The most advanced studies solve such optimization problems in a multi-criteria formulation due to the complexity and multiphysical nature of rotor-bearing systems. Zhao et al. [25] investigated the tribological properties and optimized the design of camshaft bearings in order to reduce friction and increase the minimum oil film thickness. Shi et al. [20] investigated and optimized the load capacity and energy consumption of gas foil bearings. Cukurel et al. [23] presented a solution to the problem of the multi-objective optimization of a microturbine in terms of energy efficiency and the bearings’ life. Zhang et al. [26] proposed a new hybrid approach to the multi-criteria optimization of hydrodynamic bearings in order to minimize the lateral lubricant leakage and energy losses.
The choice of a specific set of objective functions in an optimization problem may differ depending on the type and application features of the bearing. At the same time, they ultimately concern complex characteristics such as the performance, reliability and energy efficiency of the designed machine.
Bearing load capacity is one of the key parameters of rotor supports and is often considered as an objective function in studies. It can be used as the only design criterion [27] but is often combined with others [19,28,29]. Ye et al. [28] increased the load capacity and reduced the lubricant consumption of an aerostatic bearing. A similar problem was solved by Li et al. [29] for porous gas bearings for linear compressors. Michalec et al. [19] optimized the energy losses in a hydrostatic bearing pad along with the two abovementioned parameters. The minimum film thickness, as well as related parameters such as eccentricity or the Sommerfeld number, characterizes the load capacity margins and thus, to some extent, the system’s reliability. Anjos et al. [30] used the Sommerfeld number as an objective function for optimizing the chevron textures of a hydrodynamic bearing and increased the minimum film thickness. A similar approach was also demonstrated by Jamali et al. in [18], where the minimum film thickness was one of the objective functions.
Load capacity in adjustable bearings is, in general, a variable parameter depending on the control signals. The ability of active bearings to effectively generate a carrying force in a certain range has not been considered previously in the context of optimization problems. This study fills this gap by proposing a new formulation for the corresponding objective function reflecting the control action margin in a rotor-bearing system.
The friction torque in fluid film is often used as a basic measure of the energy efficiency in optimization problems. Hu et al. [17] optimized the design and reduced friction in a hydrodynamic journal bearing. Roy et al. [31] applied a genetic algorithm to reduce friction in three- and four-lobe bearings. Ravindra et al. [32] created an ANN-based model of a fluid film bearing to obtain solutions with specified properties, including the friction torque. Khatri et al. [33] optimized the texturing parameters in vein-bionic hydrodynamic bearings and reduced friction by almost half. Zhang et al. [28], Roy et al. [31] and Michalec et al. [19] considered the lubricant flow rate together with the bearing friction value as objective functions. An even more comprehensive approach to assessing energy efficiency was implemented by Zhang et al. [34] in relation to a straight-line conjugate internal meshing gear pump, where a number of mechanical and hydrodynamic processes were taken into account at once.
Energy losses due to friction and the pumping of the lubricant in adjustable bearings also depend on the operation of the control system. Actively lubricated bearings directly affect the lubricant flow during operation, making it difficult to use this parameter as an objective function. In addition, some new studies also demonstrated the possibility of the active control of friction in such bearings by optimizing the distributed parameters of the lubricant film along with the shaft positioning [7,35]. Therefore, the proposed approach considers the friction torque as a single basic measure of the system’s energy efficiency.
Finally, the dynamic characteristics of the designed bearings can also be used to assess the stability and reliability of the system. One of the most obvious approaches to considering rotor dynamics in optimization problems is directly assessing vibration amplitudes. Beris in [24] optimized the stiffness and damping of bearings to reduce vibration amplitudes in resonant modes. Another approach is to estimate the bearings’ damping capacity. Saruhan [36] considered the damping decrement in journal oscillations in three-lobe bearings as a measure of bearing stability. Ribeiro et al. [37] also reduced rotor vibrations by 68% using the damping decrement as an objective function.
The use of bifurcation theory is an alternative approach with a more comprehensive description of the rotor dynamic behavior. Chasalevris et al. [7] applied bifurcation analysis for a gas foil bearing and optimized the dynamic coefficients of the bearing in order to eliminate bifurcations preceding the appearance of limit cycles with large amplitudes. P. Zeise et al. [15] also used elements of bifurcation theory to select dynamically stable configurations of air ring bearings using parametric analysis. The previously mentioned Khatri et al. [33] used bifurcation diagrams to evaluate the optimization results of textured hydrodynamic journal bearings. Thus, bifurcation analysis is used primarily for the analytical evaluation of optimal design results but not among objective functions in optimization problems, probably due to the complexity of the automated processing of the results.
In general, the direct analysis of the aspects of dynamic rotor behavior significantly increases the computational complexity of optimization problems [24] and therefore is used relatively rarely. This study proposes to use the bearing damping capacity as a measure of system stability and an objective function that provides a balance between computational complexity and the completeness of the solution of the optimization problem.
Summarizing this review, we can note the variety of components and properties of rotor systems that may be improved by solving optimization problems. However, the adjustable design of fluid film bearings has practically not been considered in the context of optimal design procedures by researchers; only several studies partly address this issue. Ganesha [38] applied the Jaya algorithm to determine the settings of adjustable bidirectional pads in multi-pad bearings that optimize their static performance. Chasalevris in [7] used optimization algorithms to find the configuration of active gas foil bearing actuators that provides the required stability and load capacity for the given operating conditions. Thus, the issues of the comprehensive methodology of the automated optimal design of the mechanical part of active fluid film bearings have not been raised before. This leads to the focus and objectives of the present study, described in the next section.

1.3. Aim, Scope and Structure of the Present Study

The present study aims to highlight previously almost unconsidered issues of designing fluid film bearings, taking into account their active design. The main result of this study is a basic comprehensive methodology for the procedures of the automated optimal design of fluid film bearings, which underwent testing and experimental validation.
This study focuses on the problem of synthesizing the mechanical design of journal actively lubricated hybrid bearings (AHBs) by solving a multi-criteria optimization problem. It addresses the issues of the selection of optimal solutions with the required properties based on the proposed set of objective functions. The performance of the algorithms used for calculations is also assessed, resulting in the creation of a novel modification of the MOPSO algorithm with an improved solution selection mechanism. The pre-set properties of several synthesized solutions are confirmed by both experimental and numerical studies, thus confirming the validity of the proposed comprehensive methodology for the automated mechanical design of active fluid film bearings. The results create a basis for the further development of the methodology and also contribute to the improvement in the reliability, energy efficiency and service life of turbomachines.

2. Models and Methods

2.1. Model of Rotor-Bearing System

This study considers a rotor-bearing system including a symmetrical rotor supported by two actively lubricated hybrid bearings (AHBs). The load capacity of such bearings is created by a combination of hydrodynamic and hydrostatic effects. The hydrostatic component is adjustable due to the operation of electrohydraulic valves included in each lubricant supply. The valves adjust the pressure of the lubricant supply to the bearing depending on the control system signals. The bearing sleeve contains 4 rectangular lubricant supply pockets with capillary restrictors placed in a centered position. The bearing structure and parameters are illustrated in Figure 1.
The bearing model is based on the main provisions of the hydrodynamic lubrication theory [39,40], namely on the numerical solution of the modified Reynolds equation:
∂ ∂ x h 3 μ ∂ p ∂ x + ∂ ∂ z h 3 μ ⋅ ∂ p ∂ z = 6 ∂ ∂ x U h + 12 V ,
where x and z are the Cartesian coordinates, h is the radial clearance function, µ is the dynamic viscosity of the lubricant, p is the distribution of the lubricant pressure in the bearing gap, U is the circumferential (sliding) velocity and V is the radial (film squeezing) velocity. Water is considered as the lubricant in this study because it is used in the experimental equipment described in Section 4.
The influence of the hydrostatic effect on the pressure distribution in the fluid film was taken into account by a joint solution of the Reynolds Equation (1) with the flow balance equation, taking into account the hydraulic resistance in the bearing gap and the throttling effect in the capillary restrictors [35,41]:
Q = ∑ i = 1 N H Q H i = ∑ i = 1 N H π d H 4 128 l H p 0 − p H ρ K H μ ,
where Q is the lubricant flow through the bearing, QHi is the lubricant flow through a certain restrictor, NH = 4 is the number of restrictors in the bearing, p 0 is the lubricant pressure at the inlet of a restrictor (directly adjusted by valves), p H is the pressure after a restrictor and in a hydrostatic pocket and K H is a coefficient for taking into account the reduction in the flow rate due to the turbulence in a restrictor [42,43]:
K H = R e H R e * 3 4 ,
where R e H is the Reynolds number characterizing the lubricant flow in a restrictor; Re* = 2000 is the Reynolds number limit for laminar flow.
Equations (2) and (3) are used to calculate the pressure drop in feed chambers relative to the inlet pressure p0. The inlet pressure p0 is adjustable separately for each hydrostatic pocket in the range from 0 to p m a x due to the operation of a corresponding valve in a lubrication channel. The pressure is assumed to be almost the same over the entire area of a hydrostatic pocket due to the appropriate choice of its depth within 20 film thicknesses [44] and the limited area compared to the full bearing’s surface area.
The Reynolds Equation (1) is solved numerically using the finite difference method with a computational grid of size 31 × 31 elements. Pressure values in the hydrostatic pockets calculated by means of Equations (2) and (3) are used as boundary conditions during the iterative procedure of solving Equation (1).
The fluid film forces are calculated by the numerical integration of the pressure distribution obtained after solving Equations (1)–(3):
  F x = ∫ 0 2 π R ∫ 0 L p   sin ( α ) dxdz , F y = ∫ 0 2 π R ∫ 0 L p   cos ( α ) dxdz .
The friction torque in the fluid film occurring due to viscous forces is as follows:
T f r = D 2 ∬ S h 2 ∂ p ∂ x + U μ h d S .
The modeled rotor has a symmetrical design, so its weight is evenly distributed over two bearings. The rotor overall mass is 9 kg, including a centrally located load disk. As will be shown in Section 4, the rotor operates in a subcritical mode in the considered conditions. It is represented by a rigid single-mass model [45]; its motion is described by the following equations:
m r d V x / d t d V y / d t = F x F y + F x e x t F y e x t + m d ω 2 cos ⁡ ω t sin ⁡ ω t + m 0 g ,
where t is time, m is the rotor mass, d is the rotor imbalance, g is free-fall acceleration and F i e x t are external forces applied to the rotor (besides imbalance forces). The shaft misalignments in bearings are assumed to be insignificant in the presented model and therefore not taken into account.
The problem of calculating the rotor motion in bearings was thus solved by means of an iterative procedure for the joint solution of Equations (1)–(4) and (6) using the Runge–Kutta method.
The simulation model of the rotor-bearing system was verified using experimental data from several studies. The pressure distribution calculations in the hydrodynamic and hybrid lubrication modes were verified by comparing the calculations with the data presented by Mansoor et al. [46] and Foss et al. [47], respectively. The calculations of friction in the fluid film and the bearing lubricant flow rate at different values of lubricant pressure and temperature were verified using the data presented by Bouyer et al. [48] and Yi et al. [49]. The rotor motion model was verified by comparing its results with the experimental results of Yi et al. [49], namely with the measured loci of the rotor on hydrostatic bearings at different lubricant supply pressures. The detailed results of the verification procedures are omitted in this paper and can be found in [50]. The verification showed good qualitative and quantitative agreement between the simulated and experimental data, which allows us to conclude that the model is adequate and valid for further calculations.

2.2. Optimization Problem

As stated in Section 1.1, the main aim of this study is to develop and test a comprehensive methodology for the automated parametric synthesis of the mechanical design of journal AHBs, taking into account the variability in its controlled parameters, in order to provide an optimal platform for a further synthesis of the controller. Taking into account the arguments described in Section 1.2 regarding the design criteria for adjustable fluid film bearings, three objective criteria were formulated for setting and solving a multi-criteria optimization problem.
1. Since one of the key properties of the considered adjustable bearing is the ability to produce a controllable force affecting the rotor motion, a maximum force Fmax that can be generated by the AHB due to the control system was adopted as a controllability criterion. The maximum control force Fmax is calculated by the simulation model for the centered shaft position in the bearing and the maximum positive control signal value set for both control loops, i.e., along the X and Y axes. This corresponds to an AHB’s state when the lubricant supply pressure in two adjacent channels is set to the maximum, while in the other two opposite channels, it is set to the minimum [51].
The dimensionless form of this parameter is the ratio of the Fmax value to a characteristic load in the rotor system:
F ¯ = F m a x W ,
where W is the rotor weight per bearing. The parameter F ¯ in this formulation also characterizes the margin in the magnitude of the control action for a certain bearing configuration, which allows us to choose solutions that are more suitable and efficient from this point of view. The control force margin F ¯ is subject to maximization in the optimization problem solved.
2. The friction torque T in the fluid film is used as a measure of the energy efficiency of the designed system, while other parameters also relating to power losses in a rotor system, such as the lubricant flow rate, are directly affected by the control system of the AHB. Therefore, T is considered as an objective function in the optimization problem solved explicitly and is generally subject to minimization. The friction torque T is calculated according to Equation (5) at the rotor’s steady state in the operation mode and in the absence of control actions, i.e., when the lubricant is supplied to all bearing inputs at the default pressure p0.
3. The damping coefficients of the bearing are used as a basic measure of the stability of the bearing as they partly characterize the bearing’s ability to resist vibrations in the rotor system. The damping capacity of a fluid film bearing is described by a set of direct and cross-coefficients. The minimum value among the direct damping coefficients C = min (CXX, CYY) is considered as an objective function in this study. This parameter generally corresponds to the meaning of the damping decrement in oscillations considered as objective functions in the previously mentioned studies [36,37]. At the same time, the calculation of the damping coefficient by the small perturbation method [52] requires fewer calculations and thus accelerates the solution of the optimization problem. The parameter C is also used in the optimization problem solved explicitly and is generally subject to maximization.
Since the present study only addresses the mechanical design of AHBs, the geometric parameters describing the bearing sleeve (see Figure 1) are considered as design variables. A list of design variables as well as their limits considered as constraints is shown in Table 1. The ranges of the variables are determined by the generally accepted designing, technological and operational limitations and recommendations for fluid film bearings introduced by engineering and the scientific literature along with standards [40,41,53]. A list of fixed parameters describing the specified operating modes of the rotor system is presented in Table 2.
The adequacy of the design variable selection was verified by analyzing the sensitivity of the objective functions to their variations. The sensitivity analysis assumed an evaluation of the change in objective functions in response to changes in design variables within the specified ranges. At each step of the analysis, the impact of changes in only one variable was assessed, while the others were fixed at the middle of their range. Thus, the sensitivity was estimated for each of the objective functions as follows:
Δ = max ( F ( x i ) ) − min ( F ( x i ) ) min ( F ( x i ) ) .
Expression (8) allows us to estimate the range of change in each objective function in response to a change in a design variable within the limits shown in Table 1. Table 3 shows the calculated sensitivity values.
The sensitivity evaluation results show that most of the design variables significantly affect (more than 50%, i.e., ∆ > 0.5) at least one objective function. The bearing gap and length have the most pronounced effect on the considered bearing characteristics among all the variables. On the contrary, the objective functions are a little sensitive to changes in the restrictor length, the sensitivity ratio of which is several orders of magnitude fewer than that for other variables. Taking into account the results of the sensitivity analysis, the restrictor length was excluded from the list of design variables. Thus, the optimization problem can be formulated as follows:
min ⁡ f G = T , − C , − F ¯ s u b j e c t   t o 0.5 < L D < 1.5 40   μ m < h 0 < 80   μ m 0.5   m m < d h < 4   m m 5   m m < l h < 12   m m 5 % < W p < 60 % 5 ° < L p < 40 ° where ⁡ G = L ,   h 0 , d h , l h , W p , L p .

2.3. Optimization Algorithms

The relationships between the parameters of fluid film bearings, considered within the optimization problem, are complex and nonlinear. The use of a numerical model of the rotor support system also complicates the application of non-zero-order optimization methods due to limitations in calculating the gradients of the corresponding functions. In most studies considered in Section 1.2, zero-order algorithms were also most often used for the problems of comparable complexity, primarily heuristic algorithms. The multi-criteria formulation of the problem further narrows the choice of available methods. In this regard, the multi-objective genetic optimization algorithm (MOGA) [54] and multi-objective particle swarm optimization (MOPSO) algorithm [55] were primarily used to solve the optimization problem.
Further, as will be shown in Section 3, each of the mentioned algorithms has its own advantages and disadvantages in the context of the problem. The MOGA provides a relatively wide range and uniformity of the obtained solutions, but it is inferior to the MOPSO algorithm in terms of performance due to differences in the mechanisms for selecting optimal solutions. The disadvantage of the MOPSO algorithm is its tendency to group solutions and their smaller range compared to those of the MOGA. Given the significant computational cost of rotor dynamic calculation procedures, the advantage in performance turns out to be a significant factor. In order to compensate for the shortcomings of the MOPSO algorithm, it was partially improved in terms of the mechanisms for selecting solutions during the calculation process. The main changes made affect the principle of choosing optimal solutions and selecting the leading global solution, as well as the method of calculation of the particles’ speed.
Before the main loop starts, the algorithm randomly generates a set of solutions containing an array of design variables, estimated values of the objective functions and a personal best solution. The optimal solutions are determined as follows:
f o r   e a c h   p a r t i c l e : i f   F i → ≤ F j →   &   a n y ( F i k < F j k ) → P a r t i c l e   i   ∈   R a n k m i n , i f   C D i R m i n > C D j R m i n   → P a r t i c l e   i   ∈ S e t   o f   l e a d e r s ,
where F i → is the vector of the objective functions’ values of the i-th particle in the population; F j → is the vector of the objective functions’ values of the remaining particles in the population; F i k is the value of the k-th objective function of the i-th particle in the population; F j k is the value of the k-th objective function of the remaining particles in the population; C D i is the crowding distance of the i-th particle of the minimum rank; C D j is the crowding distance of other particles of the minimum rank.
The entire population of particles is divided into ranks based on Pareto optimality. Accordingly, a particle will be a rank-one particle if it contains a solution that is not Pareto-dominant relative to all other solutions in the population. Once all the solutions of rank one are determined, the selected particles are no longer taken into account when determining the optimality of all solutions, and a rank one higher than the previous one is assigned to a new set of solutions of the minimum rank. This operation is repeated until all particles are assigned ranks. In the next step, the crowding distance is determined for all solutions, showing how far the particle is from all other particles of the population of the current rank on the Pareto frontier:
C D i = F i + 1 − F i − 1 F 1 − F e n d ,
where F i + 1 , F i − 1 are the vectors of the objective functions’ values of neighboring particles of a rank;   F 1 , F e n d are the vectors of the objective functions’ values of the end particles of a rank.
The leading solutions are selected after the rank and crowding distance are determined as follows. The first rank solutions are selected from the set of solutions, where solutions with higher crowding distance values are entered into the repository first, since the greater the crowding distance, the greater the diversity a particle brings to the diversity of the set of leading solutions. The main loop begins with the definition of the leading global solution, which is randomly selected from the repository. All particles move relative to the global leading solution and the personal best solution:
v i t i = w · v i t − 1 i + k 1 P p e r B e s t i − P i t − 1 i + k 2 P g l B e s t i − P i t − 1 i ,
where v i t − 1 i , v i t i are the velocity of a particle at the previous iteration and the velocity of the current movement of the i-th particle, respectively; w is the inertia weight needed to improve the accuracy of particle movement in subsequent iterations; P i t − 1 i is the vector of design variables at the previous iteration; P p e r B e s t i is the personal best solution of the i-th particle in all previous iterations; P g l B e s t i is the global leading solution.
Also, an additional reduction in the inertia weight is introduced for cases when, at new iterations after all movements, the position of the particle takes on boundary values:
i f   N e w P o s i t i o n   ∉ L o w B o u n d s ; U p p e r B o u n d s : v i t i = v i t i ⋅ A d d W , e l s e v i t i = v i t i .
A new set of design variables is determined based on particle velocity using the following formula:
P i t i = P i − 1 i t + v i t i .
The new set of solutions is sorted by optimality using the procedure described above. The new most optimal solutions are entered into the repository, after which the optimality sorting is re-applied to the updated repository. If the repository overflows beyond the established capacity, the solutions with the smallest crowding distance are removed from the repository first. The algorithm execution is terminated upon reaching the specified number of iterations or upon reaching the convergence criterion.

3. Numerical Results

The numerical model of the rotor-bearing system and the optimization algorithms were implemented in the MATLAB R2020b software package. The Global Optimization Toolbox was used to implement calculations with the basic MOGA and MOPSO algorithm. The software ran on a personal computer with a CPU 11th Gen Intel(R) Core (TM) i5-11,600 K @ 3.90 GHz 3.91 GHz and 16 GB of Samsung RAM.
The following hyperparameters were set for the MOGA: a Crossover Fraction of 0.8, Elite Count of 10, Initial Penalty of 10, Max Stall Generations of 20, Migration Fraction of 0.2, Migration Interval of 10 and a Plot Interval of 1. Migration was set in both directions; an adaptive mutation function was used.
For the MOPSO algorithm, the personal training coefficient was 1.4, social training coefficient was 2.2, inertial weight was 0.5, mutation coefficient was 0.1 and the inertial weight damping coefficient was 0.95.
The population size in the MOGA and the repository size in the MOPSO algorithm were both set to 200 elements. The relative tolerance used as the convergence threshold was 10−5. The limit number of generations was 40 for both algorithms.
The presented optimization problem was solved by the MOGA after reaching the maximum number of generations, and the calculation by the MOPSO algorithm was completed after 26 iterations. The obtained results are presented in Figure 2. The result is a set of Pareto-optimal solutions, since the objective functions used are conflicting.
The analysis of the results shows that both sets of solutions partially intersect and partially complement each other, also producing a quite smooth surface. At the same time, the MOPSO algorithm demonstrates a tendency to excessively group solutions and unify them into clusters. The unevenness of the solutions’ distribution results and the presence of the low-density region do not enable the possibility of sufficiently smooth transitions between the properties of neighboring solutions. At the same time, faster calculation convergence in the considered problem is an obvious advantage of the MOPSO algorithm over the MOGA since it requires fewer calls to the calculation function and less solving time. Therefore, the original MOPSO algorithm was improved by modifying the method of selecting solutions for the repository in order to obtain a more uniform distribution of solutions over the calculation domain. The changes made are described in Section 2.3. The set and values of hyperparameters for the improved MMOPSO algorithm are the same as those for the original MOPSO algorithm. The calculation results using the MMOPSO algorithm in comparison with the results of the MOGA are presented in Figure 3.
The results show that the tendency to excessively group solutions is partially overcome by the MMOPSO algorithm. The MMOPSO algorithm also demonstrates an increase in the uniformity of the distribution of solutions compared to the original MOPSO algorithm, as well as the expansion of the range of objective functions’ values covered by them. As in the case of the MOPSO algorithm, the surface formed by the set of solutions fits well with the surface obtained with the MOGA. These sets complement each other and form an extended range of solutions compared to those obtained by each algorithm separately. Such a distribution of solutions indicates, firstly, the adequacy of the proposed modifications of the MOPSO algorithm and, secondly, the possibility of combining the obtained results and considering them as a single solution to the problem as a whole. In addition, the convergence rates of the MMOPSO algorithm remained close to those of the original, which follows from the data on the number of calls to the calculation function shown in Table 4. Both algorithms provided approximately 40% less solution time compared to the MOGA. Thus, the desired result in terms of improving the MOPSO algorithm can be considered achieved.
Considering the combined solution of the problem, the values of objective functions T, C and F ¯ can be conventionally considered as reflecting the properties of the energy efficiency, stability and controllability of a certain solution. The corresponding designations are added to the coordinate axes in Figure 3 with an indication of the direction of increase and decrease in each property. Specific solutions to the AHB design problem can be selected from the resulting Pareto set, taking into account the values of the objective functions and the corresponding criteria.
Using the diagram obtained in this way, three solutions with significantly different properties were selected for further manufacturing and experimental study in order to confirm the differences established for them. Thus, Case #1 is characterized by a high value of stability and controllability, but the energy loss due to the friction torque in the fluid film is also the highest. Case #2 partially compensates for this drawback and is characterized by a relatively low friction torque, but at the same time, it has reduced controllability and the greatest decrease in stability. Case #4 is characterized by the lowest energy consumption and average stability but the worst controllability.
The choice of configuration in Case #3 involved the intermediate values of all properties and was obtained using the weighted sum of objective functions as follows. The weighted sum method can be considered as an alternative method for selecting a single solution with the required properties [18] in multi-objective problems. The required severity of each objective function is specified by a weight coefficient. A weighted sum value is calculated for the considered optimization problem as follows:
N = ω 1 ⋅ C C ¯ + ω 2 ⋅ T T ¯ + ω 3 ⋅ F F ¯ ,  
where ω i are the weighting coefficients for each objective function, ∑ ω i = 1 .
The normalization of the values of the objective functions in Equation (14) is performed to balance their contribution to the total sum. The choice of weighting coefficients allows for a shift in the focus of a single solution obtained in order to follow the requirements that are the most significant for a specific rotor system configuration. The problem thus moves from a multi-objective to a single-objective problem and can be solved by single-objective optimization algorithms.
Therefore, the values of the weighting coefficients of the objective functions were chosen to be equal, i.e., w = [0.33 0.33 0.33], to obtain the most balanced solution enumerated in Case #3. This part of the problem was solved with the basic genetic algorithm from the Global Optimization Toolbox. The location of the four obtained solutions relative to the remaining ones is graphically presented in Figure 3, and the corresponding values of the objective functions and the design variables are shown in Table 5.
The hydrostatic pocket depth was chosen as 1.2 mm for all the presented solutions based on the considerations of balance between the uniform distribution of the lubricant pressure over the pocket and the factor of turbulence in the lubricant flows in it, developing with increasing pocket depth.
The presented set of AHB samples was manufactured and tested using the tools and methods described in Section 4 in order to experimentally evaluate and confirm the specified properties.

4. Experimental Study

4.1. Experimental Facilities

A test rig for the experimental study of the manufactured samples of bearings is shown in Figure 4. It includes a rotor with a central load disk mounted on two actively lubricated bearings, each in a separate housing. The design of the housings allows us to replace bearing bushings, fix them and connect lubricant supply hoses. The alignment of the bearings and the motor is ensured during assembly using the laser alignment tester BALTECH Quantum-LM (Saint Petersburg, Russia).
The rotor is driven by a frequency-controlled 1.1 kW induction motor AIR71 with a default rotation speed of 3000 rpm at 50 Hz AC. The rotor is connected to the motor with a gear coupling with the possibility of quick disconnection for conducting experiments with free runout. The rotation speed is measured by an infrared sensor connected to an Owen TH-01 tachometer.
Water at a temperature of 16–18 °C is supplied to the lubrication system by a centrifugal pump, which provides a maximum pressure of at least 0.3 MPa for all bearing configurations. Water from a collector is supplied to separate lubrication channels, each of which includes a Neptun Bugatti Pro 12 V valve with an electric actuator and a PPT-G-ST-016-0-10-1-1 pressure sensor located after the valve to provide feedback in the pressure control loop. Lubricant is supplied to the bearings in parallel and then returns from the housings to the tank, from where it is pumped out periodically as the tank fills. Bearing housings are equipped with contactless air seals that prevent lubricant leakage from the internal volume due to excess air pressure between the shaft and the seal, created by an air compressor. Contactless seals allow us to avoid the influence of additional sources of friction in the system when measuring the parameters of the rotor free runout, thus measuring only the friction directly in the fluid film of the bearings.
The lateral displacements of the shaft are measured by Sick IMA 12-06BE3ZC0K (Waldkirch, Germany) eddy current displacement sensors, mounted in housings directly near the bearing bushings, mutually perpendicularly, in the direction of the X and Y axes. The resolution of the displacement sensors according to the datasheet is less than 1 μm; the repeatability of measurements is within 10 μm.
The Global Test (Sarov, Russia) AU002 modal hammer with the Global Test AC21 force sensor is used to create measurable pulsed force impacts on the rotor and experimentally determine the dynamic characteristics of the bearings.
Signals from the displacement and pressure sensors, as well as from the tachometer and modal hammer, are concentrated in the National Instruments (Austin, TX, USA) cDAQ-9178 USB chassis with a set of corresponding input–output modules under the control of the LabVIEW v.16.0 environment, which is used as the main measuring and control tool in the experimental system. The virtual instrument in the LabVIEW environment provides parallel data reception from the sensors at a frequency of 10,000 measurements per second and their subsequent recording. Third-order low-pass infinite impulse response Butterworth filters with a cutoff frequency of 200 Hz were implemented in the measurement system to remove noise.
The control loops for the water supply pressure through the lubrication channels to the bearings are controlled by pulse PID controllers implemented on several Arduino boards with an ATmega328 (Shanghai, China) controller. The setpoint value for each channel is set automatically or manually in the LabVIEW virtual instrument and transmitted to the controller boards for execution. The structure of the main measuring and control equipment in the experimental complex is shown in Figure 5.
The main purpose of this experimental study was to obtain the practical confirmation of the presence of the specified properties in the manufactured bearing samples, including the presence of the specified differences between them. The presence of such experimental confirmation would allow us to consider the developed methodology and software suitable for the problems of the automated synthesis of the mechanical design of adjustable fluid film bearings. Accordingly, the experimental tasks included the assessment and comparison of the calculated and measured characteristics of the AHB samples by the selected objective functions, namely as follows: the maximum force action created by the bearing as a result of applying control signals; the viscous friction torque in the fluid film; and the bearing damping capacity. A description of the methods and the results is given below in the corresponding Section 4.3.
Four configurations of AHBs and a rotor were manufactured for the experimental testing of their accordance to the parameters given in Table 5. The bearing material is bronze CuAl8Fe3, and the rotor material is steel 40Cr with a hardness of HRC 34-38. The manufactured rotor has a balance quality grade of G6.3 for general purpose machinery according to ISO 21940-11 [56]. The manufactured samples are presented in Figure 6.
The measurement of the manufactured bearings samples assembled with the rotor showed that the actual value of the radial clearance h0 increased relative to the design value. The actual clearance was 86–88 µm for all samples instead of 70 µm according to the calculation results shown in Table 5. This difference is most likely due to the tolerances adopted for the production of parts, primarily the rotor, the actual diameter of which turned out to be at the lower limit. Other geometric parameters corresponded to the geometric dimensions established during the design. A list of the actual parameters of the manufactured samples of the AHBs is presented in Table 6.
Taking into account the comparatively low viscosity of water as a lubricant, the Reynolds number characterizing the flow regime in the fluid film was calculated for the actual obtained dimensions of the bearings and the rotor. The Re value for the actual parameters of the rotor system is Re < 490 at a rotation speed of 3000 rpm, which is significantly less than the critical value Re* = 2000 [57]. That indicates a laminar regime of the lubricant flow in bearings and thus the applicability of the Reynolds Equation (1) for calculations. Together with the fact that the clearance changed, the proportionality and uniformity of the changes that appeared for the manufactured AHB samples allowed us to draw a conclusion based on the theory of similarity about their applicability for conducting an experiment with the goals described above.
Eigenfrequencies were calculated for the designed experimental rotor to ensure the operation of the subcritical system and the possibility of using a rigid rotor model. The design of the rotor is shown in Figure 7. The APM shaft module of APM engineering software [58] was used to perform a modal analysis and determine the rotor eigenfrequencies, also taking into account the dynamic characteristics of the AHB samples.
A Campbell diagram for the designed rotor with infinite stiffness supports is presented in Figure 8. The first and second eigenfrequencies are observed at a rotation speed of 12,960 rpm (216 Hz), which is significantly higher than the operating frequency of the system under study which is 3000 rpm.
Linearized stiffness and damping coefficients were calculated for the obtained configurations of AHBs by the small perturbation method [39]. Table 7 presents the calculated values for the four experimental configurations both with the initial and actual values of the clearance for the manufactured samples. The rotor eigenfrequencies were also recalculated, taking into account the calculated dynamic coefficients, and the corresponding first critical speed values are also presented in Table 7.
The increased bearing clearance value resulted in a proportional and even decrease in direct damping coefficients that were used for calculating the corresponding objective function. The first critical speed also decreased for all the configurations. Its lowest value among the configurations (Case 4) decreased from 4831 to 4257 rpm, so the lowest margin for the critical speed is 29%, considering an operating speed of 3000 rpm. This margin value corresponds to typical recommendations for designing rotor systems considering critical frequencies [39], and the operating speed still does not exceed the first critical speed. This confirms the adequacy of the rigid rotor model used in the calculations and the correctness of the experiment as a whole.

4.2. Maximum Control Action Evaluation

The maximum control action that can be implemented by the AHBs was studied indirectly due to the difficulty of directly measuring the bearing force. The magnitude of the bearing force was estimated by the value of the shaft hydrostatic lift in the bearing, when the control system creates the maximum control action in the vertical direction. The measurements were made without shaft rotation in order to avoid the negative influence of imbalance and hydrodynamic forces on the results obtained. Since the steady-state shaft position is recorded during this experiment, the hydrostatic lift force is equal to the rotor weight, and the lift value characterizes the margin in the magnitude of the control action.
The methodology of this part of the experiment included measuring the shaft locus in bearings under four system states created by the control system:
(1)
The initial shaft position H0, when the lubricant is not supplied to the bearing, and the shaft takes the lowest possible position; this position is the reference for determining the shaft lift in other states.
(2)
Basic lift H1, when the control signals in both control loops are neutral (uX = uY = 0), and, accordingly, the lubricant supply pressure in all lubrication channels p0 = 0.2 MPa.
(3)
Lift H2, when the configuration of the control signals corresponds to the previous state, but in the vertical control loop, the pressure in the upper channel decreases to the minimum (pmin = 0 MPa), which leads to an increase in the hydrostatic lift.
(4)
Lift H3, when the control signal in the horizontal control loop is neutral (uX = 0), and in the vertical loop, it is set to the maximum (uY = max). In this case, the lubricant pressure in the lower channel becomes the maximum possible for this lubrication system (pmax = 0.3 MPa), and in the upper channel, it becomes the minimum (pmin = 0 MPa), so the maximum possible hydrostatic lift is achieved.
The intermediate position H2 was introduced into the experimental methodology to possibly increase the representativeness of its results. The neutral level of the control signal in the horizontal circuit (uX = 0) ensures the most centered position of the shaft, which simplifies the analysis of the results.
The locus measurements were made by the multiple cyclic sequential switching of states from H0 to H3 during the experiment. The average values of the measured shaft lift values for all the considered AHB configurations are presented in Table 8 and are also presented in graphic form in Figure 9.
The obtained results demonstrate the expected trend for lifting the shaft in the vertical direction with changes in the control pressure combinations in the AHB supply channels. The diagram in Figure 10 presents the results of a comparative analysis of the shaft lift values and the values of the maximum control force F ¯ for the considered bearing configurations. Thus, the values H2 and H3 and the difference dH = H3−H1 are considered as parameters characterizing the shaft lift value. The H1 parameter is excluded from the analysis, since it is determined to a greater extent by the set of all bearing geometry parameters, while the other specified values precisely characterize the ability of a given bearing configuration to create control actions in response to control signals. Thus, the analysis includes several parameters at once describing the shaft lift, since this analysis is indirect, and determining the most representative parameter is also a question. In addition, due to the heterogeneity of the compared physical quantities, namely forces and displacements, their values were preliminarily normalized by correlation with the maximum value of each of the parameters.
The correlation coefficient was used as a quantitative metric for evaluating the comparison results. The calculated values are given in the legend of Figure 10, and the calculation formula is as follows:
r = ∑ ( x − x ¯ ) ( y − y ¯ ) ∑ ( x − x ¯ ) 2 ∑ ( y − y ¯ ) 2   ,
where x and y are comparable quantities, and x ¯ and y ¯ are their average values for the sample under consideration.
The results of the analysis show a positive correlation r ≥ 0.956 for all combinations, but it is most pronounced (r = 0.981) for the dH parameter, which covers a larger range of control action changes. The results obtained allow us to conclude that there is an unambiguous relationship between the calculated value of the maximum control force F ¯   and the shaft lift value, thereby confirming the power characteristics set during the synthesis procedure. Thus, among the considered configurations, it is Case #1 that would most effectively convert the controller’s signals to control force actions and thereby affect the shaft position and other related parameters and aspects of the system’s behavior. Thus, we can also conclude that the considered methodology is adequate in terms of the power characteristics of the designed AHBs.

4.3. Viscous Friction Evaluation

The friction in the bearings was studied by analyzing the deceleration of the rotor in the free runout mode, which allows us to indirectly assess the friction torque in the fluid film. The measurement algorithm included the following steps: (1) accelerating the rotor to a rotation speed of 3100 rpm; (2) mechanically disconnecting the motor from the shaft by opening the gear coupling and (3) stopping the motor and recording the shaft rotation speed until the rotor came to a complete stop. Eight measurements were performed during the experiment for each of the tested bearing configurations.
The inertial forces and viscous friction forces in the sliding bearings mainly act on the rotor during the free runout mode. The calculated rotor’s moment of inertia was I = 1.347·10−2 kg·m2. The symmetry of the rotor ensures an almost uniform distribution of its weight between the bearings, so the friction torque in them can also be considered equal. The friction torque can be calculated based on the rotor motion data as follows:
M = − π I 60 d n d t
where dn is the change in the rotor speed over a period of time dt.
The results of measurements and further calculations using Equation (16) are shown in Figure 11. It should be noted that the data on the rotation speed obtained from the tachometer required preliminary processing due to the low frequency of the update of the output signal of the Owen TH-01 tachometer with a period of about 1 s. Therefore, the recorded data in Figure 11a were initially approximated by a third-degree polynomial function with a coefficient of determination of R2 > 0.985. The friction torque was calculated for the 30 s of the rotor runout in each experiment, since the reference values were obtained during solving the optimization problem at a rotation speed of 3000 rpm. A summary of the results is also presented in Table 9.
The analysis of the results shows good qualitative agreement between the sets of the experimental and calculated values of the friction torque with a correlation coefficient of r ≥ 0.99. At the same time, the discrepancy in absolute values is about 30–32%, which indicates the presence of constant unaccounted-for factors causing additional friction in the system. First of all, these include the presence of “cone-to-cone” mechanical contact between the ends of the rotor and the motor shaft due to the design features of the experimental setup. The residual misalignment of bearings can also possibly affect friction. Both noted factors can be attributed to a systematic error that can be excluded from consideration when analyzing the results. In this case, we can conclude that the proposed methodology can be considered adequate in terms of the tribological characteristics of the designed AHB.

4.4. Evaluation of Damping Capacity

The damping capacity of the AHB samples as a measure of their stabilizing properties was studied by assessing the response parameters to an impulse force impact. The corresponding impact was exerted on the rotor by means of a Global Test AU002 modal hammer, which allows us to obtain data on the parameters of the force impact.
This experiment was performed by initially setting the bearing control loops to a neutral state (uX = uY = 0), at which the lubricant supply pressure in all lubrication channels was p0 = 0.2 MPa. A series of impacts with a modal hammer were applied to the load disk in the vertical direction during this experiment, while the parameters of the impacts and shaft displacement in the bearings were recorded.
Impacts with an uneven pulse distribution between two bearings were determined during data pre-processing by the difference in the corresponding displacements in the vertical direction of more than 20% and were removed from the data set and designated as erroneous.
The damping capacity of the tested bearing samples was further determined based on the analysis of the response parameters to the impacts. For this purpose, the rotor response graphs along the Y axis were parameterized, as shown in Figure 12, which depicts a typical shape of the transient process.
The primary verification of the correctness of the measured data was carried out by calculating the correlation coefficients between the applied force F and the amplitude of the shaft displacement in the bearing Amax caused by it. The values of the correlation coefficients for all samples were in the range r = (0.88–0.95), which allows us to consider the reliability of the obtained data sufficient.
The damping capacity of the bearing was estimated based on the typical method of determining the damping coefficient of damped oscillations. However, due to the complex shape of the transient process, a decrease in the full amplitude of oscillations for several successive half-periods was assessed instead of one-sided amplitude in order to avoid introducing additional errors. The resulting formula for the calculated damping factor is as follows:
ξ = A 2 / A 1   ,
where A1 and A2 are the amplitudes measured, as shown in Figure 12.
Finally, the damping factor ξ values were calculated as the mean of the results of 15 experiments (impacts) for each bearing configuration. The data on the calculated values, including the data on sample variance, as well as their correlation with the calculated damping coefficients, are given in Table 10. The typical shapes of the response for the four tested samples are illustrated in Figure 13.
The value of the correlation coefficient between the calculated values of the damping factor ξ and the values of the damping coefficient obtained during the synthesis procedures was r = 0.981, which allows us to conclude that an almost unambiguous relationship is present.
An additional qualitative analysis of the typical shapes of the transient processes, like those shown in Figure 13, also allows us to confirm that there is a relationship with the properties set for the AHB’s configurations selected based on the set of Pareto-optimal solutions. Thus, the Case #1 configuration is distinguished by the most stable behavior as a whole, including oscillations near the equilibrium position and the fastest return of the shaft center to the initial position after the final attenuation of oscillations. The least stable configuration, Case #2, on the contrary, is characterized by the longest duration of the transient process as well as the drift of the oscillation center around the initial equilibrium position. The remaining configurations demonstrate intermediate behavior, also determined by the combination of their dynamic characteristics. The combination of analyzed factors allows us to draw a conclusion about the adequacy of the methodology for synthesizing the mechanical design of AHBs in terms of their dynamic characteristics and stabilizing properties.

5. Discussion

The synthesized configurations of the AHB with the experimentally proved characteristics were further tested in numerical simulations with several scenarios of loading and control system operation in order to evaluate and compare their overall performance. A fairly simple proportional controller described in some previous works [35], with a gain of KP = 10, was used as the AHB controller in simulations. High-response controlling servo valves, comparable in performance to MOOG valves used in [5] with a time constant of 4 ms, were modeled in the simulations.
The first of the simulated scenarios implied the occurrence of an excessive rotor imbalance of 2 g∙m when the system was in a steady state. The control system was turned on after several oscillation cycles at 0.15 s; the setpoint was close to the steady-state shaft position when the controller was turned off. Only for Case #4 was the setpoint set closer to the bearing center to avoid significant nonlinearities in oscillations and simplify the analysis. The simulation results are shown in Figure 14a.
The second scenario implied the application of a constant radial force of four rotor weights to the rotor in the vertical direction after the control system was turned on at 0.15 s. The controller setpoint was chosen to be the same as that in the previous scenario, and the controller aimed to minimize shaft displacement. The simulation results are shown in Figure 14b.
The obtained results correspond well to the characteristics set for the tested AHB samples during the design process. Thus, the Case #1 configuration demonstrates the smallest amplitudes of shaft displacements in all modes, as well as the largest vibration reduction ratio. Case #4, selected to demonstrate the case with the lowest stability and controllability, also demonstrates the confirmation of these properties. The largest displacement amplitudes and the smallest decrease in vibrations under the action of the control system are observed for this configuration. Cases #2 and Case #3 demonstrate intermediate behavior, which also reflects the properties set within the design procedures as well as the differences between their characteristics. Thus, the results of this part of this study also allow us to consider the presented methodology and calculation facilities valid for the problems of the optimal design of active fluid film bearings.
The introduced parameter F ¯ characterizing the control action margin also adequately reflects the ability of an AHB to perform control strategies associated with reducing the shaft deviation from the setpoint, as in the considered case, with a certain degree of correction for the remaining dynamic characteristics of the obtained solutions.
Several significant points should also be noted regarding the present study and the obtained results. The presented methodology concerns a fairly simple rotor system configuration in terms of its structure and operating modes, since these are the first attempts to solve design problems in this way and for systems such as active fluid film bearings. More complex active bearing configurations, such as tilting-pad [59] or foil bearings [60], will also require a further sophistication of the approaches, in particular, an increase in the number of design variables and/or complexity of the objective functions, although the general approach may remain the same as that presented in this paper.
In addition, this study did not take into account nonlinear effects and instabilities in rotor dynamics. If it is necessary to take such factors into account, then the corresponding changes can be made to the formulation of the optimization problem, both in objective functions and in constraints. For example, ensuring a margin of critical frequencies would require including data on the boundaries of dynamic coefficients to the constraint set. At the same time, such calculations can lead to a sharp increase in the volume of required calculations, although even in the considered formulation, a typical calculation can take tens of hours. More complex bearing models considering additional physical effects, e.g., when solving elastic–hydrodynamic problems, will also lead to an increase in the volume of calculations. This challenge requires further study, on the one hand, on finding the most relevant optimization algorithms and improving their performance and, on the other hand, on other methods for improving the performance of the models themselves, for example, reducing their dimension [61], using surrogate models [62,63], etc. Advanced techniques can also be useful for the assessment of the nonlinear stability of rotor systems, such as the abovementioned bifurcation methods. However, they also require additional efforts to parameterize the results in order to use them as part of objective functions.
Also, other control strategies, different from the tracking control implemented in the considered AHBs, may require the use of other criteria describing the ability of the bearing to implement control actions. For example, the control strategy in active gas foil bearings described by Chasalevris et al. in [7] implies the static formation of the shape of the bearing surface to expand the system’s stability margin. In this case, the force generated by the bearing does not play a decisive role in itself and thus may not be taken into account, while the criterion for the location of the stability threshold becomes a priority. However, such a problem formulation also already touches on the issues of optimal control, not just the mechanical design, where the results of the present study are mostly applicable.
Finally, the issue of the performance of optimization algorithms also requires some discussion. As is shown by the results of solving the multi-criteria optimization problem, different algorithms have their own characteristics that affect the quality of the results obtained. The shortcomings may be associated not only with an insufficient uniformity of the distribution of solutions obtained but also with an insufficient range of their variations in the coordinate system of objective functions. Simply increasing the volume of calculations within a single algorithm, like increasing the population size in the genetic algorithm, can improve the distribution of solutions but may also have little effect on the range they cover. Measures taken to improve the quality of decisions should address all the aspects mentioned. So, modifying optimization algorithms in terms of increasing the uniformity of the distribution of solutions is also a justified measure, which was also demonstrated in this study.
At the same time, the ranges of solutions provided by different algorithms can not only intersect but also complement each other, expanding the area for making a decision on the choice of bearing design. This also makes it justifiable to combine the results provided by different algorithms to improve the overall quality of solutions.
In summary, the presented methodology can be considered as a means for more fully exploiting the potential of adjustable fluid film bearings. Its provisions can and should be adapted to the specific design features of specific rotary machines and bearing designs, but in general, they can be used as a basis for formulating modified optimization problems. The optimization of the mechanical design of active fluid film bearings contributes to obtaining more efficient solutions for rotary machines in general and therefore to a potential increase in their overall operating characteristics.

6. Conclusions

This study presents a basic comprehensive methodology for the optimal design of adjustable fluid film bearings. The proposed design procedure focuses on the bearing’s mechanical design and is based on solving a multi-criteria optimization problem. Its formulation takes into account for the first time the adjustable design of fluid film bearings considering their controllability along with stability and energy efficiency. The problem was solved for journal actively lubricated bearings using standard and modified evolutionary algorithms. Four bearing configurations with significantly different properties of controllability, stability and energy efficiency were selected, manufactured and experimentally tested. This experimental study confirmed the properties of the samples set during the automated design procedure, with high correlation coefficient values > 0.98, and allowed us to draw several key conclusions.
  • The results of the study allow us to consider the presented methodology and calculation facilities valid for the automated solution of the problems of the optimal design of active fluid film bearings with pre-specified properties. Due to this, further controller synthesis procedures can be based on already-optimized mechanical designs, taking into account the key requirements for rotor-bearing systems.
  • The proposed parameter describing the control action margins of the active bearing represents its ability to follow the setpoints under various disturbances well and thus is suitable to represent a solution’s controllability within the optimal design procedures.
  • The presented methodology can be considered as a starting point for solving the problems of the synthesis of designs of various types of active fluid film bearings and/or other operating modes of the rotor system and can be adapted, first of all, by making changes to the constraints and objective functions.
  • However, as the models used and the problem statement become more complex, especially when considering rotor dynamic problems, an increase in the computational cost can become a challenge, requiring comprehensive solutions, including attention to the optimization algorithms used. Using a combination of several of them can be a way to expand the range of solutions obtained, in case the result is a Pareto-optimal solution set. The modification of optimization algorithms from the point of view of increasing the uniformity of the solution distribution is also a justified measure, which was demonstrated by the proposed modification of the MOPSO algorithm. At the same time, simply increasing the number of desired solutions within a single algorithm can improve the distribution of solutions but also may have little effect on the range they cover, thereby not allowing for expanding the choice of solutions.
The presented methodology can be considered as a means for obtaining a more complete use of the capabilities of adjustable fluid film bearings, thus contributing to the improvement in the reliability, energy efficiency and service life of turbomachines. The proposed solution can be considered as a basis for solving any problems in modified, expanded or complicated formulations, depending on the type and operating conditions of the designed rotor-bearing systems.

Author Contributions

Conceptualization, L.S. and D.S.; Formal analysis, D.S., A.F. and M.L.; Funding acquisition, L.S.; Investigation, D.S., A.F. and Y.K.; Methodology, D.S., A.F. and L.S.; Project administration, L.S. and D.S.; Resources, L.S. and A.R.; Software, A.F., D.S. and Y.K.; Supervision, L.S. and D.S.; Validation, D.S. and A.F.; Visualization, A.F., M.L. and D.S.; Writing—original draft, D.S., A.F., M.L., A.R. and Y.K.; Writing—review and editing, L.S. and D.S. All authors have read and agreed to the published version of the manuscript.

Funding

This study was supported by the Russian Science Foundation grant No. 22-19-00789, https://rscf.ru/en/project/22-19-00789/, accessed on 15 October 2024.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

The authors express their gratitude to the Russian Science Foundation for the provided financial support and to the reviewers for their help in improving the quality of this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Actively lubricated journal hybrid bearing.
Figure 1. Actively lubricated journal hybrid bearing.
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Figure 2. Sets of Pareto-optimal solutions obtained by MOGA and MOPSO algorithm.
Figure 2. Sets of Pareto-optimal solutions obtained by MOGA and MOPSO algorithm.
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Figure 3. Sets of Pareto-optimal solutions obtained by MOGA and MMOPSO algorithm and solutions selected for experimental testing.
Figure 3. Sets of Pareto-optimal solutions obtained by MOGA and MMOPSO algorithm and solutions selected for experimental testing.
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Figure 4. Experimental rig for studying characteristics of rotor systems on AHBs.
Figure 4. Experimental rig for studying characteristics of rotor systems on AHBs.
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Figure 5. Structure of measuring and control facilities of experimental setup.
Figure 5. Structure of measuring and control facilities of experimental setup.
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Figure 6. Manufactured bearing samples: Case #1 (left); Case #3 (right).
Figure 6. Manufactured bearing samples: Case #1 (left); Case #3 (right).
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Figure 7. The rotor dimensions.
Figure 7. The rotor dimensions.
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Figure 8. A Campbell diagram for the rotor on infinite stiffness supports.
Figure 8. A Campbell diagram for the rotor on infinite stiffness supports.
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Figure 9. Measured shaft loci in tested AHB samples.
Figure 9. Measured shaft loci in tested AHB samples.
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Figure 10. An analysis of the correlation between the shaft lift and the maximum control force.
Figure 10. An analysis of the correlation between the shaft lift and the maximum control force.
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Figure 11. Experimental results for friction torque: (a) rotor runout diagrams; (b) calculated friction torque values.
Figure 11. Experimental results for friction torque: (a) rotor runout diagrams; (b) calculated friction torque values.
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Figure 12. The parameters of the rotor response to a pulse force impact.
Figure 12. The parameters of the rotor response to a pulse force impact.
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Figure 13. Samples of the rotor response to a pulse force impact on the tested AHB configurations: (a) Case #1, ξ = 0.60; (b) Case #2, ξ = 0.81; (c) Case #3, ξ = 0.71; (d) Case #4, ξ = 0.69.
Figure 13. Samples of the rotor response to a pulse force impact on the tested AHB configurations: (a) Case #1, ξ = 0.60; (b) Case #2, ξ = 0.81; (c) Case #3, ξ = 0.71; (d) Case #4, ξ = 0.69.
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Figure 14. Performance of considered AHB configurations at different loads applied: (a) imbalance load; (b) static radial force.
Figure 14. Performance of considered AHB configurations at different loads applied: (a) imbalance load; (b) static radial force.
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Table 1. Design variables and their ranges.
Table 1. Design variables and their ranges.
#VariableLower ValueUpper Value
1Bearing length L, mm4080
2Bearing clearance h0, µm4080
3Restrictor diameter dh, mm0.54
4Restrictor length lh, mm512
5Hydrostatic pocket width Wp, % of bearing length560
6Hydrostatic pocket length Lp, degrees540
Table 2. The constant parameters of the rotor-bearing system.
Table 2. The constant parameters of the rotor-bearing system.
#Variable Value
1Lubricant (water) dynamic viscosity µ, mPa·s1.14
2Lubricant (water) density ρ, kg/m31000
3Lubricant temperature T, °C15
4Rotation speed n, rpm3000
5Lubricant supply operating (default) pressure p0, MPa0.2
6Maximum lubricant supply pressure pS, MPa0.8
7Rotor mass m, kg9
8Hydrostatic pocket depth Hp, mm1.2
Table 3. The sensitivity of the objective functions to the design variables.
Table 3. The sensitivity of the objective functions to the design variables.
Design VariableSensitivity Ratio ∆
Damping
C		Friction Torque
T		Control Force Margin
F ¯
Bearing length, L3.252.042.51
Bearing gap, h011.051.930.00141
Restrictor diameter, dh4.870.820.63
Restrictor length, lh0.00040.0660.002
Hydrostatic pocket width, Wp5.150.9250.75
Hydrostatic pocket length, Lp5.310.4310.42
Table 4. Number of function calls by tested algorithms.
Table 4. Number of function calls by tested algorithms.
AlgorithmMOGAMOPSOMMOPSO
Number of function calls
to obtain final solution		800046544649
Table 5. Parameters of synthesized configurations of AHBs.
Table 5. Parameters of synthesized configurations of AHBs.
ParameterCase #1Case #2Case #3Case #4
Objective functions
Control force margin F ¯ , [18]22.410.99.32.8
Friction torque T , N·mm18.19.710.28.7
Damping C, N·s/m × 10311.83.115.864.95
Design variables
Bearing length L, mm673436.730.8
Radial clearance h0, µm696867.568
Restrictor diameter dh, mm0.80.90.650.55
Hydrostatic pocket width Wp, % of L5852.755.551
Hydrostatic pocket length Lp, degrees18.819.718.711.65
Hydrostatic pocket depth, mm1.2
Table 6. The actual parameters of the manufactured AHB samples.
Table 6. The actual parameters of the manufactured AHB samples.
ParameterCase #1 Case #2Case #3Case #4
Objective functions
Control force margin, F ¯ , [57]13.56.43.371.17
Friction torque T , N·mm13.57.58.27.8
Damping C, N·s/m × 10311.83.115.864.95
Design variables
Bearing length L, mm67343731
Radial clearance h0, µm87878787
Restrictor diameter dh, mm0.80.90.650.55
Hydrostatic pocket width, mm4018.52015
Hydrostatic pocket length Lp, mm8888
Hydrostatic pocket depth, mm1.2
Table 7. Dynamic parameters for different configurations of rotor-bearing system considering designed and actual clearance values.
Table 7. Dynamic parameters for different configurations of rotor-bearing system considering designed and actual clearance values.
Config. #Clearance h0, µm Stiffness ,   ( N / m ) K x x K x y K y x K y y × 10 6 Damping ,   ( N · s / m ) C x x C x y C y x C y y × 10 3 1st Critical Speed, rpm
Case #170 4.64 − 1.02 0.88 4.42 12.31 1.92 0.24 11.81 8358
87 4.43 − 0.31 0.11 4.36 7.38 0.72 − 0.02 7.25 8310
Case #270 1.89 − 0.23 0.05 1.63 3.33 0.94 0.18 3.11 5739
87 1.9 0.02 − 0.22 1.69 1.68 0.26 0.04 1.63 5761
Case #370 3.01 − 0.49 0.15 2.85 5.89 0.71 − 0.07 5.86 6919
87 1.71 − 0.23 − 0.01 1.82 2.82 0.22 − 0.03 3 5478
Case #470 1.57 − 0.96 0.94 1.31 4.95 1.63 0.84 5.45 4831
87 0.82 − 0.3 0.73 1.04 2.95 1.67 1.35 4 4257
Table 8. Experimentally obtained values of hydrostatic shaft lift in tested AHB samples.
Table 8. Experimentally obtained values of hydrostatic shaft lift in tested AHB samples.
ConfigurationBasic Lift H1, µmH2 Lift (0.2 MPa), µmH3 Lift (0.2 MPa), µm
Case #167.895.4108.1
Case #251.467.176.5
Case #342.545.052.9
Case #427.129.634.9
Table 9. Comparison of calculated and experimental values of friction torque in one bearing.
Table 9. Comparison of calculated and experimental values of friction torque in one bearing.
Case #1Case #2Case #3Case #4
T calculated (at 3000 rpm), N·mm13.57.58.27.8
T experimental (at 3000 rpm), N·mm19.510.712.111.4
Discrepancy, %30.729.932.231.5
Table 10. The experimental values of the damping factor and its correlation with the calculated damping coefficients.
Table 10. The experimental values of the damping factor and its correlation with the calculated damping coefficients.
Config. #Damping
Factor, ξ 		Variance,
σ2 (ξ)		Damping Coef.,
N·s/m × 103		Correlation
Coefficient, r
Case 10.5950.00417.250.981
Case 20.7960.00291.63
Case 30.7050.00672.82
Case 40.7120.00112.95
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Shutin, D.; Fetisov, A.; Litovchenko, M.; Rodichev, A.; Kazakov, Y.; Savin, L. Methodology for Optimal Design of Active Fluid Film Bearings Considering Their Power Losses, Stability and Controllability: Theory and Experiment. Energies 2024, 17, 5879. https://doi.org/10.3390/en17235879

AMA Style

Shutin D, Fetisov A, Litovchenko M, Rodichev A, Kazakov Y, Savin L. Methodology for Optimal Design of Active Fluid Film Bearings Considering Their Power Losses, Stability and Controllability: Theory and Experiment. Energies. 2024; 17(23):5879. https://doi.org/10.3390/en17235879

Chicago/Turabian Style

Shutin, Denis, Alexander Fetisov, Maksim Litovchenko, Aleksey Rodichev, Yuri Kazakov, and Leonid Savin. 2024. "Methodology for Optimal Design of Active Fluid Film Bearings Considering Their Power Losses, Stability and Controllability: Theory and Experiment" Energies 17, no. 23: 5879. https://doi.org/10.3390/en17235879

APA Style

Shutin, D., Fetisov, A., Litovchenko, M., Rodichev, A., Kazakov, Y., & Savin, L. (2024). Methodology for Optimal Design of Active Fluid Film Bearings Considering Their Power Losses, Stability and Controllability: Theory and Experiment. Energies, 17(23), 5879. https://doi.org/10.3390/en17235879

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