Modelling fluid-film bearings in high-speed rotating machinery considering the prehistory of dynamic phenomena

Abstract

This article presents a concept for describing the dynamic states of a rotating machine, which accounts for the disruption of the continuity of the lubricating film in the slide bearings due to cavitation, and thus considers the time-varying boundaries of the positive hydrodynamic pressure region. Numerical procedures and an advanced computer programme have been proposed to account for the mutual couplings of the machine’s dynamic states at preceding and succeeding moments. This approach has been referred to as prehistory or continuous description. The article also presents an example of the practical application of the computational tools developed. The subject of the analysis was a 100 kW Organic Rankine Cycle (ORC) microturbine. The calculation results showed that at a nominal speed of 9,000 rpm, hydrodynamic instability may occur, with the amplitude of journal vibration reaching values close to the radial clearance. Based on the results of the theoretical research carried out using the developed programmes, a decision was made to modify the bearing system. The proposed method is particularly useful for large and rapid displacements of the journal within the lubricating gap, especially after the stability threshold of the rotor–bearing system has been exceeded. In such cases, differences in the assessment of the machine’s condition based on traditional research tools may be qualitative. This is demonstrated in the present work using the example of a 100 kW microturbine. Taking prehistory into account allows for a continuous, step-by-step mathematical description of the processes and represents a further refinement of the research tools being developed to capture the nature of the phenomena occurring in fluid-flow machines.

Introduction

The word “prehistory”, in its general technical sense, can refer to a description of the dynamic states of an object, taking into account the states preceding the moment under consideration. This is particularly important from a mathematical perspective, especially in relation to the equations of motion. The analysis of the influence of time on successive dynamic phenomena is particularly important here. The above issue can affect many objects, machines, and items of technical equipment in operation. In the same sense as ‘prehistory’, other researchers also use terms such as ‘memory effects’ and ‘time-coupled nonlinear modelling’. Examples of dynamic analyses are available in the literature, which confirm that the accurate representation of certain phenomena requires calculations in the time domain. Such an approach is used, among other applications, for the identification of rotor modal parameters^1,2, the determination of free vibration damping³, and for solving equations of motion of rotating systems with nonlinear⁴ or time-varying⁵ properties. It is not always justified to include prehistory, but when it is necessary, the issue generally becomes very complex. This will be illustrated using the example of slide journal bearings and rotors. Slide bearings are, in themselves, difficult to describe mathematically, and the level of complexity increases significantly when attempting to develop a unified description of the many possible dynamic states of the entire rotating machine. This is particularly true for large journal displacements in the bearing, and thus the need for analysis in a range where the force–displacement relationships are nonlinear, and often strongly nonlinear. In such cases, the inclusion of prehistory in the mathematical analysis becomes entirely justified, and sometimes even necessary to obtain correct results.

The inclusion of prehistory in the analysis of the state of a rotating machine can also be referred to as a “continuous description”, since time in the equations of motion is no longer a parameter but an independent variable. This makes solving this type of equation drastically more difficult.

Let us begin our considerations by presenting the slide bearing model, the rotor line, and the equations describing the properties of the rotating machine, particularly in the nonlinear range of its operation (i.e. with large journal displacements in the bearing). Large journal displacements in the bearing and nonlinear force–displacement relationships justify the use of prehistory.

Prehistory: problem formulation and the physical–numerical model of a rotating machine

The global literature on rotor dynamics is extensive. Only selected publications in the field of nonlinear analysis of rotating machinery will be mentioned. Adiletta et al., in their theoretical and experimental studies, focused on the nonlinear dynamic phenomena occurring in a rigid rotor supported by slide journal bearings^6,7. The author of work⁸ has shown that turbocharger rotors supported on floating ring bearings exhibit strongly nonlinear properties, which become apparent, among other things, with changes in rotational speed and loads, as occurs, for example, during start-up. The correct representation of transient states occurring in all types of hydrodynamic and aerodynamic bearings requires the use of nonlinear methods^9,10. This is particularly relevant for the analysis of multi-support fluid-flow machinery (such as energy turbine sets), for which nonlinear models yield more accurate results¹¹. Nonlinear analysis is also necessary when subassemblies such as squeeze film dampers¹² are present in the rotating system, as well as defects such as transverse shaft cracks¹³. With nonlinear methods of analysis, the effects of flexural¹⁴ and torsional¹⁵ deformations of the shaft on the operation of the rotating system can be more accurately reproduced. A separate group of nonlinear issues involves cases where rotor parts rub against stationary components^15,16. Also, in cases where additional snubber rings are used to damp rotor vibrations, a correct representation of the dynamic phenomena can be achieved using a nonlinear approach¹⁷. It is worth noting that in all the works cited, time t in the differential equations is treated as a parameter, which implies a superposition of dynamic states rather than a continuous description that includes prehistory. In the following discussion, based on models previously developed by the authors of this article (enabling the determination of stiffness and damping coefficients of the lubricating film in bearings^18,19, the determination of the dynamic characteristics of the support structure^19–21, and the analysis of dynamic phenomena occurring in high-speed rotating systems^22,23, we will present in greater detail the issues related to the analysis of the possibility of a continuous description of a rotating machine.

The slide bearing is a highly complex object from the point of view of mathematical modelling. On the one hand, it is a powerful heat generator (particularly at high loads and rotor rotational speeds), and on the other, it is an object in which the continuity of the lubricating film is frequently interrupted, for example due to cavitation.

Determining the spatial distribution of the viscosity of the lubricant in the lubrication gap µ is not straightforward, and determining the time-varying boundaries of the cavitation zone ζ₁ and ζ₂, and thus the integration limits of the Reynolds equation (i.e. the determination of the positive hydrodynamic pressure region), is extremely difficult. The viscosity µ and the limits ζ₁ and ζ₂ are key parameters in describing the characteristics of a slide bearing. Since cavitation in fluid-film bearings is a highly complex phenomenon, it has received considerable attention in the literature on modelling such systems^24–30. The authors of²⁴ presented a mass-conserving model for the transient evolution of cavitation in lubrication analysis, together with practical examples of its application. A thermo-hydrodynamic analysis of a hydrodynamic journal bearing, using a new mass-conserving cavitation algorithm, is presented in²⁵. In this paper, the proposed algorithm uses a variable transformation for both pressure and mass fraction, formulated as a complementary condition. Artificial diffusion ensures stabilisation in both the streamwise and crosswind directions. In²⁶, an implicit finite element cavitation algorithm is presented, in which the pressure distribution in the non-cavitation zone and the density of the gas–liquid mixture in the cavitation zone are calculated simultaneously. In another article, the authors presented a transient finite element cavitation algorithm, which allows the determination of active and inactive lubricant film zones under non-stationary operating conditions²⁷. The model was successfully validated through comparisons with previously published studies. An investigation of transient rotor–bearing interaction, with a focus on cavitation, is also presented in²⁸. The authors proposed a new cavitation model as a modification of the Elrod–Adams model. The research results showed that the proposed model ensures stable and fast simulations, even under transient conditions. New lubrication models of double-film bearings, accounting for laminar/turbulent effects, cavitation, and other conditions, are presented in²⁹. Using the proposed models, the water film pressure, bearing load, and friction coefficient of double-film bearings under various operating conditions were investigated both theoretically and experimentally. The authors of paper³⁰ accounted for cavitation in their model of a hydrodynamic bearing lubricated with a composite magnetic fluid.

The remainder of this section describes how to model flow in a hydrodynamic transverse slide bearing, which was subsequently used to analyse phenomena occurring in bearings lubricated with a low-viscosity, low-boiling-point liquid. The main problem we currently face is determining the following relationship:

1

where y, z, and ζ are the coordinates along the thickness, width, and circumferential direction of the lubrication gap, respectively, and t is time.

This is illustrated in Fig. 1. The issue is somewhat simplified if the boundaries of the cavitation zone are stable, which may occur with constant loads on a rotating machine. Often, however, the boundaries of the cavitation zone vary over time, depending not only on the position of the journal and the rate of change of this position in the lubrication gap, but also on the physical processes occurring in the cavitation bubbles.

Fig. 1.

A slide bearing as a heat generator and as an object in which the continuity of the lubricating fluid in the lubrication gap is disrupted due to cavitation. This defines the main challenges in describing the bearing’s properties, namely: determining the spatial distribution of viscosity in the lubrication gap µ(y, z, ζ), and identifying the time-varying boundaries of the cavitation zone ζ₁(t) and ζ₂(t).

An unstable cavitation zone can form both when the rotating machine has not yet exceeded the formal stability limit of the entire system (e.g. due to rapid and large journal displacements), and when this limit has been exceeded (e.g. due to hydrodynamic instability—oil whirls and, in particular, oil whip).

A thermal model of a slide journal bearing, which takes into account the energy exchange processes in the lubricating fluid and the heat transfer in the sleeve, is described in detail in work¹⁹. In this article, we will present only the final equations of this model in the form of the so-called three-dimensional Reynolds Eq. (3D equation):

2

where

2a

whereas x and z are the geometric coordinates of the gap in the circumferential direction and across the width, at point y along its thickness h.

Relationships (2a) describe the viscosity of the lubricant at any point within the three-dimensional space of the lubrication gap (y, z, x). Assuming cylindrical coordinates appropriate for a journal bearing, i.e. setting x = ζ, we can write µ(y, z, ζ), which is consistent with the notation used in (1) and Fig. 1.

It is worth noting that (1) and (2a) define the left-hand side of the Reynolds Eq. (2) and represent all potential viscosity changes in the lubrication gap, caused not only by temperature variations but also by processes occurring in the cavitation zone.

Naturally, simpler forms of the three-dimensional Reynolds equation can be derived from Eq. (2). For example, assuming an adiabatic thermal model in which the walls bounding the lubrication gap do not transmit heat (which means that the dynamic viscosity along the y-axis is constant and µ = f(z, ζ)), the values of the integrals A and B from relation (2a) will, as can be easily calculated, take the following form:

While Eq. (2) for slide journal bearings will now take the familiar form:

The elastic and damping properties of a lubricating fluid can be determined by four stiffness coefficients, c_{i, k}, and four damping coefficients, d_{i, k}. Starting from Eq. (2), these can be determined using perturbation calculus and perturbation differential equations. This procedure is described in detail in the monograph¹⁹. For obvious reasons, we will not discuss it further here.

The effect of the rotor line, supports, and foundation on the slide bearings can also be determined by the stiffness (C_{m, n}) and damping (D_{m, n}) coefficients. However, their determination is only possible through experimental measurements or painstaking calculations using, for example, the finite element method (FEM).

It is worth noting that, while it is reasonable to assume constant values for the coefficients C_{m, n} and D_{m, n} for a particular support j (even for large displacements of the journal in the lubrication gap), a similar assumption for the coefficients c_{i, k} and d_{i, k} (for the same large displacements) is no longer possible. Thus:

3

This means that the elastic and damping properties of the lubricating fluid depend not only on the instantaneous position of the journal on the trajectory within the lubrication gap, ε, γ, but also on the rate of change of this position, . This also implies nonlinear, and sometimes even strongly nonlinear, properties of the lubricating film.

The issue is further complicated by the fact that the continuity of the lubricating film is interrupted due to processes occurring in the cavitation zone, as well as changes in the integration limits of the Reynolds equation, ζ₁(t) and ζ₂(t). These temporal dependencies naturally affect the values of c_{i, k} and d_{i, k}, which can be expressed symbolically as follows:

4

Time t in relation (4) is the independent variable, which makes the mathematical analysis extremely difficult. On the other hand, this fact is the essence of the continuous description and the impact of prehistory. This is illustrated in Fig. 2. The middle and right parts of this figure show a comparison of the new and classical approaches, illustrating the relationship between successive positions of the journal in the bearing. As shown in the figure on the right, each preceding position of the rotor affects the next. The successive operating states of the bearing are closely related (with consecutive areas of analysis overlapping) and are not considered independently.

Fig. 2.

The essence of continuous description incorporating prehistory³¹.

The FEM method was used to describe the dynamic states of the entire rotor-bearing system and, consequently, to derive the equations of motion for such a system. It allows the differential equations to be expressed in the form of global matrices. The procedure for constructing such matrices and the process of developing the corresponding numerical algorithms are presented in works¹⁹. Figure 3 shows a diagram of such an algorithm developed in the work cited above. It should be noted that, in addition to the Reynolds equation discussed above, the hydrodynamic slide bearing model also incorporates the energy equation. To some extent, it is also possible to analyse the thermal phenomena occurring in the slide bearing and to determine the deformation of the sleeve. As a result, a complex computer system called MESWIR was developed. The programmes included in this system have been developed over many years, and the purpose of this paper is not to present these procedures and algorithms in detail. The implicit Newmark method, often used for such problems, is employed to solve the nonlinear motion equation of the rotor–bearing system in the MESWIR programme. We will use these numerical tools for further considerations and, in particular, to analyse the impact of prehistory on the dynamics of the rotating machine. These issues have been explained using commonly employed variables and equations, enabling other researchers to familiarise themselves with the proposed method and to apply a similar approach in any programming environment.

Fig. 3.

MESWIR computer system. Diagram of the calculation algorithm¹⁹.

The nonlinear nature of the relations in (4) means that the global stiffness matrix (K) and damping matrix (D) of the entire rotor-bearing system are no longer constant (as is commonly assumed), but depend nonlinearly on the generalised displacements, the rates of change of these displacements, and time. The equation of motion of the rotor-bearing system becomes, in matrix notation, a set of multiple mutually coupled and nonlinear differential equations. This is described by the following equation:

5

where: M - global inertia matrix; D - global damping matrix; K - global stiffness matrix; , , - generalised displacement, velocity, and acceleration vectors; t – time.

Obviously, by neglecting temporal dependencies of the type (4) and thus disregarding prehistory, Eq. (5) will take the form commonly found in the literature:

However, let us return to Eq. (5). Time t, as an independent variable, defines both the left-hand side of the above equation (changes in the integration limits in the cavitation zone) and the right-hand side (arbitrary, periodic external excitation forces).

It is evident that such a complex Eq. (5) cannot have analytical solutions. We will use the MESWIR computer system for further analysis (see Fig. 3). However, it does not account for the temporal dependencies determining the processes occurring in the cavitation zone (symbolically represented by the time t on the left-hand side of Eq. (5)). It therefore becomes necessary to make additional assumptions and to use special algorithms to achieve the research objective and ensure the stability of the solutions.

In the classical approach to the problem (thus disregarding continuous description and prehistory), a stepwise method is used in which the dynamic state of a rotating machine is calculated for successive moments t_k, t_k+1, t_k+2,…,t_k+n. This means that, in such an approach, we are dealing with a sum of dynamic states that are independent of time and calculated separately at each time point (time becomes a parameter). The dynamic states of the machine are, therefore, essentially the sum of pseudo-static states.

The attempt to include temporal dependencies of the type (4), and consequently Eq. (5), in the description of the dynamics of a rotating machine leads to a mutual coupling of the machine states at particular moments t_k+1 with the states at preceding moments t_k. This is illustrated in Fig. 4.

Fig. 4.

Numerical approach to the problem. Graphical interpretation of the differences between a stationary description (small journal displacement trajectories at the static equilibrium point, with constant coefficients c_{i, k} and d_{i, k}) and a continuous description that includes prehistory (coefficients c_{i, k} and d_{i, k} are calculated separately at each point of the trajectory t_k, with sufficiently small intervals ∆x and ∆y defining their validity in this region).

The MESWIR system was verified on a large-scale test rig at the IMP PAN laboratory in Gdańsk (see Fig. 5, right). The test rig has the following parameters: a three-support rotor–bearing system, a journal diameter of 0.1 m, a disc diameter of 0.4 m, and plain oil-lubricated bearings with radial–axial clearance. Experimental research was carried out both within the stable operating range of the test rig and immediately after the stability threshold had been exceeded. Figure 5 (left) shows a fragment of the verification in the most interesting and hazardous range, namely after the system’s stability threshold had been exceeded. A satisfactory qualitative agreement is evident between the theoretical description and the results of the experimental measurements.

Fig. 5.

Multi-scale research rig used for experimental verification of the MESWIR system (right) and an example of verification after surpassing the stability threshold, in the region of oil whirls (left).

The MESWIR system has been successfully employed in tests of 200 MW turbines conducted by ALSTOM at power plants in Poland and Lithuania. It is a computer system, developed in-house by the authors of this article, which is not intended for commercialisation.

At this point, it is worth noting that the MESWIR system is intended exclusively for the analysis of fluid-flow machines employing hydrodynamic plain bearings. The authors of this article employed readily available commercial tools to analyse systems mounted on rolling bearings.

Modification of the MESWIR system: concept and main simplifying assumptions

The fundamental question is this: how can the concept of prehistory, illustrated in Figs. 1 and 2, and 4, and most notably in Eq. (5), be incorporated into practical calculations? In other words, how can the influence of time t, associated with the displacement of the boundaries of lubricating film continuity, be represented in numerical computations, and which boundary conditions should be adopted when integrating the Reynolds equation? This problem is illustrated in Fig. 6.

Fig. 6.

Theoretical (p_t) and actual (p_r) pressure distributions in the cavitation zone. Interpretation of the Reynolds boundary condition for the theoretical distribution (p_t).

A common practice in determining the pressure distribution in the circumferential direction of the bearing is to use the so-called Reynolds boundary condition, that is, to assume that the positive hydrodynamic pressure p ends at the boundary ξ₂, at which both the pressure value and its derivative are equal to zero. An analogous assumption is made for the boundary ξ₁, which defines the beginning of the zone of positive hydrodynamic pressure. Thus, we can write:

6

Condition (6) can be satisfied during the calculations using a relatively simple numerical trick. As a result, we obtain a pressure distribution p_t as shown in Fig. 6 (which we will conventionally refer to as the theoretical distribution). The pressure distribution p_t can therefore be determined quite easily.

However, the actual boundaries of lubricating film continuity (cavitation zones) may differ. We have defined these in Fig. 6 as ζ₁, ζ₂, and the pressure distribution as p_r (the actual distribution). A fundamental question now arises concerning the adequacy of the theoretical pressure distribution p_t in relation to the actual distribution p_r. As it turns out, the Reynolds boundary condition (6) and the resulting pressure distribution p_t are experimentally confirmed in many cases and are therefore often used in practice^11,31.

The actual boundaries of the cavitation zone, due to the physical processes occurring in this zone and the implosions of the bubbles, are extremely difficult to determine. In the absence of more reliable data, we make the following key assumption:

7

This means that, in the further analysis, we will consider the displacements of the boundaries ∂ζ₁/∂t and ∂ζ₂/∂t, determined from the Reynolds boundary condition, as shown in Fig. 6.

The most important question now remains: how to account for the influence of the dynamic state of the rotating machine at the moment t_k−1, preceding the moment t_k under consideration, in the next calculation step t_k+1, i.e. how to account for the mutual coupling of these states (Fig. 4).

The authors of this article propose using the relaxation method and the following notation in the stepwise procedure:

8

where δ denotes the relaxation factor. It is worth noting that for δ = 0, we have t_k+1 = t_k−1, and for δ = 1, we get t_k+1 = t_k. During the calculations, the reaction of the lubricating film in the bearing is determined at each time step, and the stiffness (c_{i, k} ) and damping (d_{i, k} ) coefficients are updated accordingly. For calculations at the next time step, k + 1, the coefficients determined from solving the Reynolds equation at the previous time step, k, are not used directly. Instead, the ‘relaxed’ values according to formula (8) in the article are employed, taking into account the coefficients from step k − 1.

Thus, in the concept presented above, it is the relaxation factor δ that defines the interaction between the machine’s dynamic states, i.e. how prehistory is accounted for in the analysis. Unfortunately, determining this factor is not straightforward. Its value is determined through painstaking numerical experimentation, experience, and the appropriate choice of algorithms. Ultimately, the goal must be to achieve stable solutions.

The authors’ experience indicates that the optimal value of the relaxation coefficient δ, necessary to achieve numerical stability, ranges from 0.01 to 0.5, depending on the case under consideration. For smaller and slower changes in the position of the journal within the bearing, a value of δ = 0.5 was sufficient, ensuring rapid calculations and, consequently, faster attainment of stable solutions. However, for more challenging cases, that is, for large and rapid displacements of the journal within the bearing, it was necessary to use a smaller relaxation coefficient and conduct multiple trials with different values of δ. This naturally resulted in more time-consuming calculations. In each case, the final result of the calculations had to be independent of the chosen value of δ. The relaxation coefficient serves only to ensure the numerical stability required to achieve full convergence of the final result.

In summary, it can be stated that relations (6), (7), and (8) constitute the main simplifying assumptions when solving Eq. (5) using the MESWIR computer system (Fig. 3). They therefore represent a fairly significant modification of the MESWIR system, but, above all, present a new conceptual approach to the issue discussed in this article.

The authors of this article acknowledge that the assumptions made may be subject to debate. Nevertheless, due to the complexity of the problem and the lack of proprietary data or data available in the literature, the proposed concept constitutes a novel approach to the dynamics of rotating machinery and provides an alternative perspective on the still poorly understood issue of prehistory.

In rotor dynamics, many issues remain open and unresolved. Some of these issues are mentioned in work³¹, which highlights the difficulties involved in the dynamic analysis of rotating systems with nonlinear characteristics, particularly when the stability limit is exceeded. Other researchers, in their publications, have also identified areas of rotor dynamics that require further investigation. These include, among other aspects, the growing use of active bearings^30,32 and the development of oil-free bearing systems³³. Nonlinear methods for modelling rolling bearings are also under further development³⁴. Modern diagnostic systems³⁵ also face significant challenges, as they must process increasing volumes of data and are used to monitor and diagnose rotating machinery operating at increasingly higher rotational speeds³⁶. Protecting the environment from vibrations generated during the operation of rotating machinery is also an important issue³⁷, as is reducing external dynamic influences on the operational performance of precision mechanisms with rotating components^38,39. In the context of these challenges, the research presented in this article seeks to address a gap in the literature and contributes to the advanced modelling of slide bearings, which exhibit strongly nonlinear properties under certain conditions.

Example of the application of research tools

An example will now be presented to illustrate how the developed research tools are applied in practice. As part of a project at the Institute of Fluid-Flow Machinery (IMP PAN), a 100 kW ORC microturbine with a rated speed of 9,000 rpm was developed. The working medium of the microturbine was a low-boiling fluid designated MDM (silicone oil), consisting of pure, clear, colourless polydimethylsiloxanes. The flow system of the microturbine consists of seven axial stages, with all rotor discs positioned between two bearing supports spaced approximately 800 mm apart. The overall length of the microturbine rotor is approximately 1,020 mm, and the outer diameter of the largest rotor disc in the final stage is approximately 300 mm. The weight of the complete rotor is 56 kg. Torque from the microturbine shaft is transmitted to the shaft of the low-speed generator via a single-stage belt transmission⁴⁰. The calculations assume that the slide bearings have a cylindrical clearance, with a diameter of 60 mm and a width of 30 mm. The nominal radial clearance was 0.04 mm. The density of the MDM lubricant was 900 kg/m³, and its viscosity at 40 °C was approximately 0.0008 N s/m². The transverse force due to the belt tension in the belt drive was not taken into account in calculations. In the model, we only considered the torque acting on the end of the rotor where the pulley was mounted. A photograph of the microturbine, along with the FEM model and the results of preliminary kinetostatic calculations, is shown in Fig. 7. The FEM model of the microturbine shaft consisted of 30 Timoshenko beam elements with four degrees of freedom at each node. The model included the mass and moments of inertia of the rotor discs. The stiffness and damping of the bearings at the rotor support points were determined as previously discussed at each calculation step.

Fig. 7.

Test object: a 100 kW microturbine using an ORC low-boiling working fluid with a rated speed of 9,000 rpm, designed at IMP PAN. FEM discretisation of the system. The vibration-exciting force is the residual unbalance R, defined in accordance with the ISO standards applicable to this class of machinery.

During the design phase of this machine, the attractive idea of utilising the turbine’s working medium to lubricate the slide bearings emerged. The entire unit would have been a hermetically sealed structure, which was desirable for safety reasons. However, before making a decision, it was important to verify whether the proposed slide bearings would do the job. The modified MESWIR system presented in the previous chapter was used to analyse the dynamics of the microturbine. Due to the lack of experimental data in the literature on microturbines with slide bearings of similar geometry, lubricated with a low-boiling liquid, we have not included a typical experimental verification of the modified model in this article. It was used to analyse a newly designed microturbine, in which various alternative bearing systems were considered, including slide bearings lubricated with a low-boiling-point liquid, which is not commonly employed as a bearing lubricant.

It was accepted that the residual unbalance, R (with a value of 140 g·mm), as determined by ISO standards for this class of machinery, represented the dynamic excitation force acting on the rotor-bearing system in question. Of course, the static load Q, resulting from the mass of the rotor, also acted on the system. Based on this data, the amplitude-frequency characteristics of the machine, shown in Fig. 8, were constructed. It can be seen from these that, at nominal speed (N = 9,000 rpm), both bearings operate in a range of strongly developed hydrodynamic instability.

Fig. 8.

Amplitude-frequency characteristics for the chosen test object.

To gain further insight into the cause of this condition, the last two rotor revolutions were analysed. If we denote the angular measure in the circumferential direction as T = ωt, where ω denotes the angular velocity of the rotor and t denotes the time, recording two revolutions means recording the interval T = 0°–720°, i.e. T = 0°–360° and T = 360°–720° for each of the two revolutions separately.

Figures 9, 10 and 11 show the results of the numerical analysis of the adopted rotor-bearing system using the research tools presented in this article. It can be seen from Fig. 9 that, at the rated speed (N = 9,000 rpm), the journal trajectories in the bearing are identical for the last two revolutions and are extremely large, and therefore hazardous, being limited only by the bearing clearance. This is an obvious symptom of developed hydrodynamic instability, also referred to as medium whip.

Fig. 9.

Trajectories (within the clearance circle) and vibration spectra calculated for stable (N = 8000 rpm), transient (N = 8500 rpm), and unstable (N = 9000 rpm) operating ranges. Registration of the last two rotor revolutions: T = 0°–720° (TAL represents T in the figure). Q – static load, R – residual unbalance as an external vibration-exciting force. Blue dots (left) mark the journal centre positions during successive revolutions; spectral graphs (right) show displacements (µm) in the horizontal (yellow) and vertical (blue) directions.

Fig. 10.

Pressure distribution in the bearing lubrication gap for the stable operating range of the system (N = 8000 rpm) and the last two rotor revolutions: T = 0°–360° (upper part) and T = 360°–720° (lower part).

Fig. 11.

Pressure distribution in the bearing lubrication gap for the unstable operating range of the system (N = 9000 rpm) and the last two rotor revolutions: T = 0°–360° (upper part) and T = 360°–720° (lower part).

A different picture emerges from the analysis of Figs. 10 and 11. These figures provide interesting information. They refer to the pressure distributions in the bearing lubrication gap, calculated for both the stable and unstable operating ranges of the system, and recorded for the last two rotor revolutions. In both cases, the same forces act on the system: the static load Q and the residual unbalance R (with a period of T = 360°), serving as an external force inducing the vibration.

The result shown in Fig. 10 is evident. Under stable operating conditions, excitation forces (R) with a period of 360° produce a system response with the same period. Hence, the pressure distributions for the ranges T = 0°–360° and T = 360°–720° coincide exactly. The situation is different in Fig. 11. Although the same forces (static load Q and residual unbalance R) continuously act on the system, the system’s response is completely different. The pressure distributions for the last two revolutions do not coincide and differ significantly; not only does the pressure position shift relative to the R vector, but its maximum value (P_max) is more than 2.3 times higher than in the case of stable system operation. These results were achieved through the use of an advanced nonlinear model of slide bearings. If simple linear models were used across the entire analysed speed range, we would achieve stable rotor operation, as is observed up to 8,000 rpm. The considerably greater capabilities of the developed nonlinear model proved crucial in this case, enabling the detection of problems related to hydrodynamic instability.

Juxtaposing these results with those in Fig. 9, it is worth noting that the journal trajectories for the case of unstable operation (N = 9,000 rpm) coincide for the last two revolutions, unlike the pressure distributions. This can be explained by the specific balance between the inertial forces of the rotating masses and the hydrodynamic load capacity under conditions of high system instability. The time-varying boundaries of the continuous lubricating film zone, ξ₁(t) and ξ₂(t), which determine the hydrodynamic load capacity, may be located in parts of the lubrication gap quite different from what one might intuitively expect, and are the result of the mutual coupling of successive dynamic states. In such cases, the concept proposed by the authors of this article—describing the dynamics of a rotating machine while taking prehistory into account—can be very useful.

The final results of the numerical analysis of the object shown in Fig. 7 led to the conclusion that the slide bearings, at the intended rated speed of 9,000 rpm and using the proposed lubricating medium—a low-boiling liquid—would always operate within an unstable range. Therefore, this rules out their suitability for this particular fluid-flow machine. Rolling ball bearings were used in the final version of the microturbine.

The research object shown in Fig. 7 was also a prototype machine, designed for a specific customer and for subsequent implementations. The power of 100 kW and the rated speed of 9,000 rpm were so high that conducting experimental research beyond the stability limit was excessively hazardous and could have damaged the machine. Direct verification of the research tools used on this machine, analogous to that performed on a test rig (see Fig. 5), was therefore impossible. However, it is possible to make an indirect assessment of the impact of prehistory on the results of the analysis conducted.

For small displacements and speeds of the journal within the bearing, the influence of prehistory is negligible, so the results of calculations that account for prehistory or ignore it are similar, as shown in Fig. 10. In all other cases, accounting for prehistory may even introduce qualitative changes in the results obtained, as shown in Fig. 11.

Final remarks

The considerations presented in this article reveal the extent of the difficulty in describing the dynamic state of a rotating machine, taking into account the loss of lubricating film continuity in the slide bearings and the time-dependent variation of the boundaries of the hydrodynamic positive pressure zone. This problem has been identified as the effect of prehistory, commonly known as the memory effect. The authors propose a conceptualisation of the coupling between the dynamic states of a rotating machine over time, which may also be described as a continuous representation or as time-coupled modelling.

The proposed nonlinear slide bearing model was integrated into the MESWIR computer system, which has been in development for many years. This allowed a dynamic analysis of the rotor of a 100 kW axial-flow microturbine with a rated speed of 9,000 rpm to be carried out. The results of the calculations showed that the proposed bearing system, which uses slide bearings lubricated by a low-boiling fluid in liquid form, operates within the hydrodynamic instability range at rated speed. This was confirmed through a detailed analysis of the calculation results. The amplitudes of the bearing journal vibrations at nominal speed exceeded 0.03 mm, thus approaching values close to the radial clearance of the bearing. Given the high-speed rotor of the microturbine and the slide bearings lubricated with a low-boiling, low-viscosity fluid, this would be especially hazardous. Unfavourable dynamic phenomena detected by the computational method could lead to rapid failure in a real machine. We were able to achieve such a result thanks to an advanced time-dependent model of the bearings and rotor, in which successive dynamic phenomena are taken into account.

Rotor dynamics is an ever-evolving branch of knowledge, yet many issues remain unresolved. These clearly require further research. Some of these unresolved issues were noted in the previously cited publications.

The authors of this article lead teams focused on the further development of microturbines and tools for assessing their dynamic condition, as well as on applications of these machines in distributed electricity generation systems. This pertains to their use in low-power installations (e.g., domestic cogeneration power plants) and in applications utilising waste heat and renewable energy sources (RES). Distributed generation using RES is currently a key component of the energy transition, particularly in the European Union and Poland, where the concept of decarbonisation is of special significance.

Author contributions

Conceptualisation, J.K.; methodology, J.K.; software, J.K.; validation, J.K. and G.Z.; formal analysis, J.K. and G.Z.; investigation, J.K. and G.Z.; resources, J.K.; data curation, J.K. and G.Z.; writing—original draft preparation, J.K. and G.Z.; writing—review and editing, J.K. and G.Z.; visualisation, J.K. and G.Z.; supervision, J.K. All authors read and approved the final manuscript.

Funding

The authors declare that no funds, grants, or other external support were received during the preparation of this manuscript.

Data availability

All data generated and analysed during this study are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.