# Evaluation of the minimum face clearance of a high-speed gas-lubricated bearing with Navier slip boundary conditions under random excitations

## Abstract

Motivated by ongoing developments in aero-engine technology, a model for a coupled gas-lubricated bearing is developed in terms of an extended dynamical system. A slip boundary condition, characterised by a slip length, is incorporated on the bearing faces which can be relevant for operation in non-ideal extreme conditions, notably where external vibrations or disturbances could destabilise the bearing. A modified Reynolds equation is formulated to model the gas flow, retaining the effects of centrifugal inertia which is increasingly important for high-speed operation, and is coupled to the structural equations; spring-mass-damper systems model the axial stator and rotor displacements. A novel model is developed corresponding to a bearing experiencing an external random force to evaluate the resulting induced displacements of the bearing components. The minimum face clearance is obtained from a mapping solver for the modified Reynolds equation and structural equations simultaneously. In the case of random excitations, the solver is combined with a Monte Carlo technique. Evaluation of the average value of the minimum gap and the probability of the gap reaching a prescribed tolerance are provided. Extensive insight is given on the effect of key bearing parameters on the corresponding bearing dynamics.

## Keywords

Coupled gas-lubricated bearing Monte Carlo method Navier slip boundary condition Random external force## 1 Introduction

Fluid-lubricated bearings typically utilise a thin fluid film to maintain a clearance between rotating and stationary components when an axial load is imposed. Increasingly industrial applications of bearings are characterised by high operating speeds, requiring low frictional losses and smaller clearances. Associated film lubrication technology is designed to enable further improvements in bearing efficiency. In this high-speed and small-clearance scenario, comprehensive and accurate predictions of variation in the face clearance are needed, especially when the bearing experiences a range of external loading and disturbances. Quantifying the possibility of contact between the bearing faces is essential.

The dynamics of an air bearing slider, which employs a thin lubricating air film to separate two non-parallel moving plates, has been studied in [1] when the bearing has relative normal motion between the two plates. The interaction of a compressible gas flow with a movable rigid surface was presented, together with linear stability analysis, showing separation of the plates can be maintained. Analytical methods were used in [2] to study the dynamics and stability of tapered air bearing sliders, providing insight into the slider behaviour.

Taylor and Saffman [3] performed an early theoretical study examining a coaxial parallel rotor and stator separated by a thin air film, experiencing normal motion. A model was derived from the compressible Navier–Stokes equations for negligible rotational inertia effects and small amplitude disturbances. For detailed predictions of the bearing dynamics, a coupled bearing model in which the fluid flow is coupled to the bearing structure is required, such that the air film is an integrated component of the bearing [4]. Etison [5] examined a coupled fluid-lubricated device in three dimensions, identifying hydrodynamic and hydrostatic components of the air film pressure; the hydrostatic component includes the squeeze film pressure. Results indicate that the squeeze film behaviour has a significant contribution to the dynamics and can potentially maintain the air film alone. For highly vibrating operational environments, Salbu [6] examined the effect of significant disturbances in the axial direction through theoretical analysis and experimental investigations. Describing the rotor–stator clearance by an oscillatory motion gives predictions of increasing pressure and force in the air film as the amplitude of axial oscillations increases.

The bearing gap dynamics in a coupled high-speed air-lubricated bearing with parallel faces were studied by Garratt et al. [7]. In this case, the axial displacement of one face has periodic axial oscillations of smaller amplitude than the equilibrium film thickness, and the other face moves axially in response to the film dynamics. Bailey et al. [8] considered a similar bearing configuration with the inclusion of a coned rotor and incompressible flow. In this case, an analytical solution of the corresponding Reynolds equation can be found, leading to an explicit analytical expression for the fluid force. The coupled bearing problem then reduces to a second-order non-autonomous ordinary differential equation for the bearing gap. For compressible flow, this direct approach is not feasible due to the nonlinearity of the resulting Reynolds equation. The effect of the rotor undergoing prescribed axial oscillations with amplitude larger than the equilibrium film thickness was examined. Results indicate that the fluid film can become very small, possibly invalidating the classical no-slip velocity condition. Comparison of a CFD model of an air-riding seal with experimental studies by Sayma et al. [9] identified that steady-state flow solutions could not be obtained for typical gaps of under 10 \(\mu \)m indicating more analysis is required at this smaller scale to comprehend the corresponding seal dynamics.

The reduced scale of micro- and nanofluidics devices results in the fluid flow being characterised by the confined fluid at micro- and nanoscales, where surface effects begin to dominate over volume-related phenomena. Mathematical models of thin gas flow regimes are usually classified by the Knudsen number \(Kn=l/{\hat{h}}_0\), where *l* is the mean free path (for air at atmospheric conditions \(l =68\) nm) and \({\hat{h}}_0\) is the characteristic film thickness. Knudsen numbers in the range \(10^{-3}\le Kn \le 10^{-1}\), represent flow at micro- and nano-scales, where a continuum flow model with slip boundary condition is usually employed. The existence of velocity slip was first predicted by Navier [10], who proposed a constant slip model with a linear relationship between the fluid–wall velocity difference and the tangential shear rate, i.e. the velocity slip is proportional to the derivatives of the surface fluid velocity, with the slip length as the proportionality constant (a first-order model). Using Maxwell’s [11] classical theory, the slip coefficient is taken as proportional to \((2-f)/f\) and is usually of the order of the mean free path; *f* is the momentum accommodation coefficient.

Use of a slip flow condition has been previously incorporated into models of various bearing geometries. A gas-lubricated inclined plane slider bearing has been studied in the slip flow regime using a first-order slip model [12], second-order slip model [13] and modified second-order slip model incorporating additional physical considerations, referred to as a 1.5 slip model [14]. For each case, the corresponding Reynolds equation for compressible flow was developed and analytical results were obtained.

A non-axisymmetric thrust bearing with slip flow and foil pads on the rotor face is examined by Park et al. [15], where the gas flow modelled by a classical Reynolds equation is coupled to the bearing structure. Bearing dynamic predictions are studied for small amplitude rotor displacement using perturbation analysis in the case of a no-slip and slip condition imposed on the bearing faces. A slip condition results in a reduced hydrodynamic pressure and thus load carrying capacity when compared with a no-slip condition, as well as decreased stiffness and damping coefficients in the case of axial perturbations. Incorporating a slip boundary condition in the analysis of a coupled high-speed fluid-lubricated parallel bearing with small face clearance in the limit of incompressible flow [16] shows that face contact does not formally occur when the rotor undergoes prescribed periodic oscillations but the bearing gap can become very small. An increased lubrication force on the stator can be attained by the use of a coned rotor with stability studies identifying an optimal coning angle [5]. However, detailed analysis of a coned bearing in the slip regime under extreme operating conditions gives predictions of possible face contact [17]. A similar trend is also identified for the case of a coupled high-speed fluid-lubricated bearing in the case of compressible flow with slip boundary conditions under extreme operating conditions [18]. Possible external disturbances imposed on the rotor can cause the rotor to be displaced by magnitudes larger than the initial equilibrium film thickness. Detailed results show a parallel rotor and stator maintain a face clearance, although it can become very small, where as a coned bearing configuration permits contact between the rotor and stator.

For increasingly more compact designs at higher rotational speeds, there exists a requirement for greater predictability of the dynamic behaviour in industrial bearing designs. Sources of uncertainty in a bearing model typically arise from the random axial motion of the rotor due to external excitations. In [16] and [17], the effect of external excitations on the bearing is studied by prescribing the axial motion of the rotor as periodic oscillations and utilising a comprehensive deterministic mathematical model for the bearing dynamics. The probability of bearing face contact is correspondingly examined using the method of derived distributions to incorporate the lack of precise knowledge in the amplitude of rotor oscillations [19]. The probability density function (transforming functions) and cumulative distribution function of the minimum bearing gap are readily determined when the amplitude of rotor oscillations is described by a specified probability distribution function. Utilising the deterministic relationship between the amplitude of rotor oscillations and minimum value of the bearing gap subsequently leads to the evaluation of the probability of contact.

Analysis of dynamical systems and bearing models with uncertainty arising from external excitations use computational methods which are typically based on Monte Carlo techniques [20]. The Monte Carlo method samples directly from the random input parameter and the deterministic model is evaluated for each sample, giving the corresponding output. This robust procedure is able to evaluate complex cases and a large number of uncertain parameters; however, an extensive number of code executions may be needed to generate sufficiently accurate results. To minimise the computational cost and increase efficiency, other computational approaches for uncertainty quantification have been developed; examples include polynomial chaos, Gaussian process emulator and a Quasi-Monte Carlo method, see, e.g. [21, 22, 23].

The dynamic response and vibration characteristics of two in-line rotor bearing systems are examined by Li et al. [24] to quantify the effects of uncertainty. A model for the rotor system is derived using stochastic modelling based on the polynomial chaos expansion technique for uncertainty in the damping and nonlinear support stiffness. The results obtained are reported to have good agreement with Monte Carlo simulation. Response statistics demonstrate that uncertainty in the nonlinear support stiffness and damping has a significant effect on the predicted dynamic response of the rotor system.

Increasingly environmental and efficiency considerations are leading to the next generation of bearing and sealing technologies having an extremely small gap, resulting in an almost contact design. In this study, we outline a model for a gas-lubricated bearing incorporating a slip boundary condition. The compressible flow model is coupled to the structural dynamics through the axial force of the fluid on the bearing faces. Previous analysis is extended to incorporate operation in a non-ideal environment; external vibrations or disturbances could act to destabilise the bearing which are modelled as a random external force applied to the rotor. In Sect. 2, the coupled governing equations are formulated where a Reynolds equation for compressible flow is derived that retains the effects of centrifugal inertia for high-speed operation and incorporates slip boundary conditions, characterised by a slip length parameter, on the bearing faces. The axial stator and rotor displacements are modelled by spring-mass-damper systems. Of significant importance is the analysis of a random external force imposed on the rotor, modelled as a continuous, stationary, bounded process with finite modes of frequencies in its spectrum. The generality of the proposed random force is potentially transferable to other models of physical phenomena beyond this particular application. To numerically solve the coupled bearing model, the modified Reynolds equation is solved simultaneously with the structural equations via a mapping solver, details of which are given in Sect. 3. The Monte Carlo technique is implemented to determine the cumulative distribution density function of the minimum gap (face clearance) due to the imposed random destabilising external force, and also to find the average of the minimum gap and probability that it reaches a prescribed tolerance. Results are presented in Sect. 4 examining the effect of key model parameters.

## 2 Mathematical model

A simplified mathematical model is derived for a gas-lubricated bearing with a thin fluid film modelled as compressible flow and a slip boundary condition imposed on the bearing faces. The coaxial, axisymmetric annular rotor and stator can move axially, giving the respective displacement heights as \({\hat{h}}_r\) and \({\hat{h}}_s\). The rotor also has a prescribed rotational speed \(\hat{\varOmega }\) and the bearing has imposed pressures at the inner and outer radius. This configuration is motivated by aerospace applications and similar technology in the literature is referred to as air-riding [7], film-riding [25], oil-free [26] or noncontacting [4].

The rotor experiences an external axial force *N*(*t*), which is assumed to be axisymmetric and imposed at \(r=a\). It is taken to be a random process, which represents external excitations on the bearing that could act to destabilise the bearing operation. For the case of non-axisymmetric forcing, for example, point loading, the bearing faces may become misaligned and experience tilt and/or swash. In turn, this may lead to coning angle instabilties over time. A dynamic analysis for a noncontacting face seal with coned faces has been examined in three dimensions under the assumption of assembly face misalignment and rotor axial run-out. Optimal conditions to minimise face contact and maximise the stability are addressed in [27, 28, 29].

A key quantity in the bearing dynamics is the time-dependent minimum face clearance (MFC), defined by the variable \(g(t)=h_s(t)-h_r(t)\), evaluated at the outer radius for a NCB and inner radius for a PCB; for a parallel bearing \(g(t)=h(t)\). If the MFC always remains positive, i.e. \(g>0\), there is no bearing face contact and a fluid film is maintained.

### 2.1 Air flow model

The air flow model is derived from the compressible Navier–Stokes momentum equations and conservation of mass equation for a compressible flow in axisymmetric coordinates; the flow is assumed to be an isothermal ideal gas. Thermal effects in similar configurations have been studied in the absence of external forcing by numerous authors, for example see [30, 31], suggesting thermal deformation of the faces is more significant than viscous heating. A study by Minikes and Bucher [32] concluded that temperature variations were less than five per-cent of the ambient temperature and consequently the flow could be an isothermal process. Correspondingly, the thermal effects are neglected in the paper as they are secondary to the effects of the dynamic motion.

*Fr*is defined as \(Fr={\hat{U}}({\tilde{g}}{\hat{h}}_0)^{-1/2}\), where \(\widetilde{g}\) denotes the acceleration due to gravity and parametrises the importance of gravitational effects relative to the radial flow. Gravity is neglected because taking the Froude number of

*O*(1), ensuring consistency with lubrication theory, leads to \(Re_U \delta _0Fr^{-2}\ll 1\). Classical lubrication theory neglects inertia when the reduced Reynolds number \(Re_U \delta _0\ll 1\); however, in the case of high-speed bearing operation, an additional term corresponding to the ratio of the Reynolds numbers \(Re^*\) must be considered, which is not always negligible. A measure of the importance of the viscous time scale is given by \(\kappa ={{\hat{h}}_0}^2/\nu {\hat{T}}\), where \(\kappa =O(1)\) gives the time scale of the order of the time it takes viscosity effects to diffuse over the characteristic film thickness. Thus giving the acceleration term of the same order as the viscous term. In the current study, the time scale is taken to be of \(O({\hat{h}}_0{\hat{r}}_0/\nu )\), resulting in \(\kappa =O(\delta _0)\), justifying a quasi-static approximation of the momentum equations.

*R*is the gas constant and

*M*is the molar mass.

*h*, subsequently yielding the flow characteristics when solved. Dependence on the coning angle is given implicitly in the rotor–stator clearance

*h*. Taking \(\lambda =0\) results in the centrifugal inertia effects being neglected;however, the rotor and stator still experience relative rotational motion due to the velocity boundary conditions.

### 2.2 Structural dynamics

*N*(

*t*). In dimensionless form, the stator equation is

The form of the random external force *N*(*t*) contrasts with Langevin-type equations associated with white noise models that are typically unbounded, discontinuous and aperiodic [33, 34]. Boundedness of the noise and frequency in the current work, together with continuity, is important for a realistic model of an external force in aero-applications. Correspondingly, a realisation of *N*(*t*) from (11), determined from a realisation of \(A_i\), \(B_i\), \(\phi _i\), provides a periodic function of time. Noting that \(\phi _i\) is a uniform random variable on \([0,\, 2\pi ]\), then *N*(*t*) is a stationary stochastic process [35, 36]. Since \(A_i\) are bounded random variables and *n* is finite, *N*(*t*) is bounded which is a natural requirement for modelling a physical force. The properties of the proposed random force (11), namely that it is continuous, stationary, boundedness in physical space and has a finite number of frequencies in its spectrum, is potentially transferable to other models; the proposed random force may be useful when modelling other physical phenomena which experience random fluctuations.

## 3 Numerical methods

*J*(

*T*) are found by solving similar systems to the auxiliary system of equations (16), but which are the derivative with respect to each of the initial guess value \(\tilde{\mathbf {g}}_0\); corresponding initial conditions are used.

An alternative method, as the modified Reynolds equation (13) does not depend explicitly on the rotor height \(h_r\), is to solve directly for the pressure and face clearance only and find the rotor height in post processing using *F*(*t*) and *N*(*t*). This will eliminate the second and fourth equations in (16) and reduce the Jacobian (19) by removing the second and fourth rows and columns. However, if the number of radial points \(\xi \) is large, the reduction will be marginal.

The stroboscopic map is combined with the Monte Carlo technique, where a large number of independent random realisations of the stroboscopic map are computed, each with a randomly generated external force imposed on the rotor.

*m*; \(m=1,\ldots ,M\). The Monte Carlo error is

*M*of independent realisations. We note that the estimate \(\hat{g}_\mathrm{{min}}\) has bias \(E[g_\mathrm{{min}}]-E[\bar{g}_\mathrm{{min}}]\), corresponding to the error of the numerical approximation \(\bar{g}(t)\) of the stroboscopic map

*g*(

*t*). The numerical solution was compared to an analytical asymptotic solution in the case of small face clearance in Bailey et al. [8]. Results showed the proposed numerical solution has negligible error of the minimum face clearance compared to the asymptotic solution, indicating that the bias of the estimate \(\hat{g}_\mathrm{{min}}\) can be considered to be negligible. An estimate for the variance of \(g_\mathrm{{min}}\) is computed as in (22), and its Monte Carlo error is evaluated as

## 4 Results and discussion

A coupled gas bearing is examined to investigate the effects of key parameters of the bearing dynamics and to identify possible face contact corresponding to a generic aero-engine industrial application. The bearing configuration is taken to be a narrow bearing \(a=0.8\) with \(\sigma =0.821\) and ambient pressure imposed at the outer radius \(p_O=p_a=1\). Initially ambient pressure is considered at the inner radius \(p_I=1\), the bearing faces taken to be parallel \(\beta =0\) and the external force (11) has parameters \(n=20\), \(\bar{B}=10\), and \(\bar{A}=20\); the effect of these parameters on potential face contact is examined. The speed parameter is considered to be \(\lambda =0.0029\), dimensionless slip length to be \(l_s=0.1\) and the ideal gas constant as \(K_s=1\). The structural parameters are given by \(K_z=12.2\), \(D_a=1.5\) and the force coupling parameter by \(\alpha =1.22\).

Evaluation of the parallel bearing is examined for increasing internal pressure imposed at the inner radius; for \(p_I=1\) an ambient pressure is imposed across the bearing, \(p_I=0.5\) corresponds to external pressurisation and \(p_I=1.5\) to internal pressurisation. The mean and variance of \(g_\mathrm{{min}}\) are given in Fig. 7, together with the corresponding Monte Carlo error, for a parallel bearing with increasing pressure at the inner radius \(p_I\). In general, as the pressure at the inner radius is increased, the average value of \(g_\mathrm{{min}}\) increases. It is noted that the case of \(p_I=1.1\) has a value less than \(p_I=1\), however with the Monte Carlo error this case fits with the overall trend of the mean value of \(g_\mathrm{{min}}\) increasing as the value of \(p_I\) increases. In all cases, the variance of \(g_\mathrm{{min}}\) is small; \(\sigma ^2\) is of the order of \(10^{-2}\).

*N*(

*t*); the bound is finite due to the structural limitation of the bearing geometry and its local environment. To test our numerical solution, values \(20\le \bar{A} \le 100\) are examined; the mean value of \(g_\mathrm{{min}}\) is similiar for all cases examined, and the variance has a small value of the order of \(10^{-2}\) in all cases. The probability of \(g_\mathrm{{min}}\) reaching a given tolerance \(g_\mathrm{{tol}}\) remains similiar when the values of \(\bar{A}\) increase, with negligible probability of face contact, i.e. \(g_\mathrm{{tol}}\sim 0\).

Mean and variance of \(g_\mathrm{min}\) for a given number of realisations *M* together with the probability of \(g_\mathrm{min}\) reaching a given tolerance \(g_\mathrm{tol}\) in the case of a parallel bearing with increasing number of frequencies in the forcing term \(10\le n \le 30\); \(\bar{A}=20\), \(\beta =0\) and \(p_I=1\)

| | \(\mu \) | \(\sigma ^2\) | \(g_\mathrm{{tol}}\) | |||
---|---|---|---|---|---|---|---|

0.1 | 0.01 | 0.001 | 0 | ||||

10 | 600 | \(0.623\,\pm \,0.0151\) | \(0.0342\,\pm \, 0.00810\) | \(0.00932\,\pm \,0.00785\) | \(0.00119\,\pm \,0.00281\) | \(0.00101\,\pm \,0.00259\) | \(0\,\pm \,0\) |

20 | 1500 | \(0.440\,\pm \,0.0109\) | \(0.0435\,\pm \, 0.00682\) | \(0.0680\,\pm \,0.0130\) | \(0.0147\,\pm \,0.00621\) | \(0.00329\,\pm \,0.00296\) | \(0\,\pm \,0\) |

30 | 3000 | \(0.318\,\pm \,0.00776\) | \(0.0450\,\pm \, 0.00890\) | \(0.191\,\pm \,0.0218\) | \(0.0607\,\pm \,0.0132\) | \(0.0193\,\pm \,0.00763\) | \(0\,\pm \,0\) |

Mean and variance of \(g_\mathrm{{min}}\) for given number of realisations *M* in the case of a NCB (\(p_I=0.5\)) and PCB (\(p_I=1.5\)); \(n=20\) and \(\bar{A}=20\)

\(\beta \) | | \(\mu \) | \(\sigma ^2\) |
---|---|---|---|

0 | 1900 | \(0.398\pm 0.00975\) | \(0.0450\pm 0.00728\) |

\(-0.1\) | 1800 | \(0.375\pm 0.00930\) | \(0.0390\pm 0.00483\) |

\(-0.2\) | 1900 | \(0.369\pm 0.00923\) | \(0.0404\pm 0.00555\) |

\(-0.3\) | 2200 | \(0.340\pm 0.00850\) | \(0.0400\pm 0.00561\) |

0 | 1400 | \(0.488\pm 0.0120\) | \(0.0504\pm 0.00854\) |

0.1 | 1300 | \(0.500\pm 0.0123\) | \(0.0492\pm 0.00988\) |

0.2 | 1500 | \(0.485\pm 0.0121\) | \(0.0547\pm 0.00788\) |

0.3 | 1200 | \(0.493\pm 0.0123\) | \(0.0456\pm 0.00862\) |

Probability of \(g_\mathrm{{min}}\) reaching a prescribed tolerance \(g_\mathrm{{tol}}\) in the case of a NCB (\(p_I=0.5\)) and PCB (\(p_I=1.5\)); \(n=20\) and \(\bar{A}=20\)

\(\beta \) | \(g_\mathrm{{tol}}\) | |||
---|---|---|---|---|

0.1 | 0.01 | 0.001 | 0 | |

0 | \(0.106\,\pm \,0.0141\) | \(0.0248\,\pm \,0.00714\) | \(0.00946\,\pm \,0.00444\) | \(0\,\pm \,0\) |

\(-0.1\) | \(0.111\,\pm \,0.0148\) | \(0.0347\,\pm \,0.00863\) | \(0.0140\,\pm \,0.00554\) | \(0.0130\,\pm \,0.00534\) |

\(-0.2\) | \(0.120\,\pm \,0.0149\) | \(0.0375\,\pm \,0.00872\) | \(0.0154\,\pm \,0.00565\) | \(0.0146\,\pm \,0.00550\) |

\(-0.3\) | \(0.153\,\pm \,0.0153\) | \(0.0446\,\pm \,0.00880\) | \(0.0363\,\pm \,0.00798\) | \(0.0338\,\pm \,0.00771\) |

0 | \(0.0602\,\pm \,0.0127\) | \(0.00932\,\pm \,0.00514\) | \(0.00207\,\pm \,0.00243\) | \(0\,\pm \,0\) |

0.1 | \(0.0513\,\pm \,0.0122\) | \(0.00822\,\pm \,0.00501\) | \(0.00311\,\pm \,0.00309\) | \(0.00216\,\pm \,0.00258\) |

0.2 | \(0.0633\,\pm \,0.0126\) | \(0.0116\,\pm \,0.00553\) | \(0.00823\,\pm \,0.00467\) | \(0.00811\,\pm \,0.00463\) |

0.3 | \(0.0506\,\pm \,0.0127\) | \(0.0160\,\pm \,0.00724\) | \(0.0124\,\pm \,0.00639\) | \(0.00903\,\pm \,0.00546\) |

The cumulative distribution function (CDF) is examined which gives the probability that \(g_\mathrm{{min}}\) is less than or equal to a given value of \(g_\mathrm{{min}}\), i.e. \(F_{g_\mathrm{{min}}}(x)=\text {Prob}(g_\mathrm{{min}} \le x)\). The cases of \(\beta =0\) and \(\mid \beta \mid = 0.3\) for a NCB and PCB are shown in Fig. 11 with \(95\%\) upper and lower confidence bounds; the probability of contact is given by the CDF when \(g_\mathrm{{min}}=0\). The probability that \(g_\mathrm{{min}}\) becomes equal to or less than a prescribed tolerance \(g_\mathrm{{tol}}\) is shown in Table 3; note that the parallel bearing is examined under different pressurisations for comparison. Both scenarios have predictions for a parallel bearing having negligible probability of contact. In contrast, a coned bearing has a non-zero probability of contact for all the cases considered. A NCB has an increase in the probability of \(g_\mathrm{{min}}\) reaching a given tolerance as the magnitude of the coning angle increases. However, in the case of a PCB this situation occurs only for \(g_\mathrm{{tol}}=0.001\) and \(g_\mathrm{{tol}}=0\); there is a non-monotonic trend otherwise. A NCB has a larger probability of \(g_\mathrm{{min}}\) reaching the prescribed tolerance than the corresponding PCB, resulting in a higher probability of contact at the outer radius of the bearing than at the inner radius (see geometry in Fig. 3). The asymmetry between the behaviour of a NCB and PCB is noted; a NCB has a monotonic trend always, whereas a PCB can have a non-monotonic trend as seen in Tables 2 and 3. This type of analysis can be used to identify safe operating conditions or limitations on bearing geometries under operating conditions.

## 5 Conclusions

A dynamical model for a gas-lubricated bearing with a very small face clearance is derived, which is capable of predicting the bearing dynamics under destabilising operating conditions. A modified compressible Reynolds equation is formulated to model a gas film, which is then coupled to the bearing structure through an axial force of the gas on the bearing faces. The governing compressible Navier–Stokes equations are reduced through using an axisymmetric lubrication approximation, but maintain leading-order effects of centrifugal inertia, relevant for high-speed flows. Modified surface boundary conditions corresponding to a slip length formulation are incorporated, relevant to the case of very small face clearance. The axial displacement of the rotor and stator are modelled as spring-mass-damper systems. Typically the rotor experiences an external random force due to fluctuations in the bearings surrounding environment. The generality of the proposed random force, which is continuous, stationary, bounded in physical space and has a finite number of frequencies in its spectrum, is potentially transferable to other models of physical phenomena beyond this particular application.

To numerically solve the coupled bearing problem, the modified Reynolds equation and structural equations are simulated simultaneously within a period. Computed results directly give the face clearance and rotor height, whilst post-processing calculations are used to determine the force of the gas on the bearing faces and axial displacement of the stator. Monte Carlo simulations are combined with a derived stroboscopic map solver to evaluate approximations of the minimum face clearance \(g_\mathrm{{min}}\). The probability associated with a specified minimum face clearance \(g_\mathrm{{min}}\) is examined, with face contact corresponding to a zero value of \(g_\mathrm{{min}}\).

Utilising the bearing dynamics allows an investigation into the effect of the external force on the face clearance. Axial displacement of the rotor is due to the imposed external force, however axial displacement of the stator arises from the rotor being forced into close proximity of it. In this case, a peak in the force of the fluid causes the faces to move apart in order to maintain a fluid film. Results are given for an increasing number of realisations in a Monte Carlo simulation, yielding the mean and variance of the minimum face clearance.

The effect of key parameters have been examined, including the pressure gradient across the bearing, value of the bounds of the random variables describing the amplitude of the random force components and the coning angle of the rotor. Outcomes of this type of analysis provide insight to a given bearing geometry to identify safe operating conditions and inform ideal bearing geometries such as in the case of establishing a maximum coning angle for safe operation under given working conditions.

## Notes

### Acknowledgements

This work was supported by funding from the EPSRC Doctoral Prize Grant No. EP/M506588/1.

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