# Mathematical Modeling on Statics and Dynamics of Aerostatic Thrust Bearing with External Combined Throttling and Elastic Orifice Fluid Flow Regulation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design Scheme of the Bearing

_{s}through a throttling diaphragm of radius r

_{p}of the elastic orifice 3 is supplied to the chamber 4 of volume v

_{p}, where the pressure becomes equal to p

_{p}< p

_{s}. Then, through damping annular diaphragms of diameter d

_{k}evenly spaced along a circle of radius r

_{1}, gas under pressure p

_{k}< p

_{p}enters the bearing layer of the bearing, having overcome it and then flows into the environment at pressure p

_{a}.

_{k}, then a pressure p

_{p}, as a result of which the elastic orifice is deformed and the radius r

_{p}of the throttling diaphragm changes, as shown in Figure 2. As a result, an additional change in gas flow through the elastic orifice occurs. By a targeted choice of the elasticity of the elastic orifice, it is possible to obtain the desired shape of the load characteristic and bearing stiffness in the calculation area.

## 3. Mathematical Modeling of Elastic Orifice Deformation

#### 3.1. Elastic Orifice Deformation Model

_{p}

_{0}is the radius of the undeformed orifice, r is the current radius, ψ = ψ(r) is the deflection function, u = u (r) is the tensile function of the middle surface of the elastic orifice.

_{s}is the outer radius of the elastic orifice, E is the elastic modulus of the orifice material, ν is Poisson’s ratio [26].

_{p}of the deformed elastic orifice by the formula (1).

^{−4}is accepted).

#### 3.2. Analysis of Elastic Orifice Calculation Results

_{p}

_{0}= 0.125 mm, δ = 2.64 mm; 2—for r

_{p}

_{0}= 0.19 mm, δ = 1.14 mm. The comparison results are shown in Figure 3.

^{2}, ν = 0.5) at p

_{s}= 238 kPa. The pressure difference p

_{s}− p

_{p}is plotted along the x-axis, and the mass flow q of gas through the orifice is plotted along the y-axis. The calculations used the formula

_{p}= 0.85 the empirical coefficient of an orifice diaphragm [17], R

_{0}= 287 m

^{2}/(c

^{2}K) is the gas constant, T

_{0}= 293 K is the absolute temperature, Γ = 4.313 is the adiabatic constant of air [17].

_{p}substantially depends on Δ and ε and only slightly depends on ν.

_{p}is an almost linear function of the differential pressure, therefore, it can be approximated with the expression

_{e}is the elastic ratio of the orifice.

_{e}. For the curves corresponding to ε = 15, we have approximate Equation (18), which implies that K

_{e}≈ 0.95. For the curves corresponding to ε = 30, a similar equation has the form $0.0119\approx 0.02\left(1-1.2{K}_{e}\right),$ whence it follows that K

_{e}≈ 0.34.

## 4. Bearing Mathematical Model

#### 4.1. Bearing Static Model

_{p}through the elastic orifice and Q

_{k}through the damping annular diaphragms, the flow rates Q

_{k}and Q

_{h}at the entrance to the bearing layer of the dimensionless thickness H, as well as the equation of force equilibrium of the movable element 2, connecting the bearing capacity W and the load F to the movable element

_{p}

_{s}, radius R

_{c}, the tuning factors of the combined external throttling system χ, and the elastic ratio of the elastic orifice K

_{e}.

- –
- determination of the “calculated point” parameters corresponding to the dimensionless thickness of the bearing layer H = 1,
- –
- calculation of flow rate Q(F) and load H(F) characteristics of the bearing.

_{k}, A

_{p}of expenses (21), (22) are determined. First, we calculate the coefficient A

_{h}and determine the pressure P

_{k}, P

_{p}, corresponding to the value of the calculated clearance H = 1

_{p}pressure values ${P}_{p}\in \left[1,{P}_{s}\right]$ according to the formula ${P}_{p}=1+{i}_{p}\left({P}_{s}-1\right)/{n}_{p},\hspace{0.17em}$ ${i}_{p}=0,1,\dots ,{n}_{p}.$ For each of its values, we can find the flow Q

_{p}according to formula (19).

_{e}, at which the bearing reaches zero compliance (K = 0).

_{k}, ΔP

_{p}, quantities H, P

_{k}, P

_{p}, respectively.

_{e}, at which the bearing has zero compliance (K = 0). Most simply, this can be done for the "calculated point" mode H = 1. Using (19), (20), (22), (29), we find

_{e}, the bearing can lose stiffness. This is the case when the denominator (29) vanishes. It is easy to verify that this occurs when

_{e}calculated by formula (32) occurs when R

_{p}> 0, that is, when the orifice is still able to pass gas during deformation. It closes only when the pressure drop is too large and ${K}_{e}\ge \frac{1}{{P}_{s}-{P}_{p}}.$

#### 4.2. Static Characteristics of the Bearing and Their Discussion

_{e}. All curves intersect at one point corresponding to the “calculated point” mode H = 1.

_{e}, the curvature of the lines changes significantly. When K

_{e}= 0, that is for the rigid orifice 3, the compliance of the bearing K is always positive. With increasing K

_{e}in some parts of the load characteristic, the compliance decreases to zero (K = 0), which corresponds to infinite stiffness, and even a negative value (K < 0). So, at K

_{e}= 0.22 and F ≈ 1.5, the bearing has zero compliance. At K

_{e}= 0.22, one can find the region 0.35 < F < 4.4, in which the bearing has negative static compliance (K < 0).

_{e}, the nature of the curves changes sharply. Comparison of graphs Figure 6 and Figure 7 shows that the less the compliance of the bearing, the lower the mass flow rate of air.

_{s}− P

_{p}is large, and as follows from formulas (18), (19) for K

_{e}> 0, the diaphragm radius R

_{p}will be small, and, therefore, the flow rate Q

_{p}will be small. As the load F increases, the pressure difference P

_{s}− P

_{p}decreases and, consequently, the radius R

_{p}and the flow rate Q

_{p}increase, thereby providing an additional supply of lubricant to the bearing, which contributes to a more intensive decrease in bearing compliance K.

_{e}

_{0}on the coefficient ς of the tuning of the damping annular diaphragms for the “calculated point” mode H = 1, at which the bearing has zero compliance. For smaller χ and ς, smaller K

_{e}= K

_{e}

_{0}are required. At the same time, when setting to too small χ, there may be no modes in which the bearing reaches zero compliance.

_{ed}(ς), at which the bearing loses its static stiffness, and, consequently, stability, are shown in Figure 10.

_{ed}values determined by formula (31) give the upper boundary on the admissible values of the elastic ratio K

_{e}= K

_{ed}. For K

_{e}> K

_{ed}, the bearing is also statically unstable.

## 5. Bearing Dynamic Model

_{p}, P

_{p}are static parameters.

#### 5.1. Method for Determining the Transfer Function Coefficients of Dynamic Bearing Compliance

^{3}(hereinafter, the order of complexity of the computational method means the time complexity of the algorithm that implements it [30,31]). For large n and m, this can entail a significant expenditure of computer time in the process of multi-parameter optimization of a dynamic system.

^{2}.

_{n}≠ 0 and b

_{m}≠ 0, then the infinite limit

_{j}, (j = 1, 2, …, k) in sequence (35), we obtain a system of linear equations for unknown coefficients (33)

^{−1}by (37), we bring the system (35) to the form

^{−1}and M cells are mutually inverse matrices of the discrete Fourier transform, therefore, their product will give the identity matrix E. Elements of the block of cells are obtained by multiplying the rows of the matrix Φ

^{−1}and the columns of the matrix M, which are also elements of the direct and inverse matrices Fourier transform. The sums of their products, giving off-diagonal elements of the identity matrix, will be zeros by analogy with the way this holds for the zero elements of the block E located above them.

^{3}, and using special fast methods taking into account the features of Equation (39) and having complexity proportional to m

^{2}. The latter include the methods described in [33,34].

#### 5.2. Quality Criteria for Bearing Dynamics

- –
- degree of stability η = Max Re{s
_{i}}, where s_{i}are the zeros of the characteristic polynomial of the dynamical system, which is the polynomial denominator of the TF (33), - –
- damping of oscillations over a period $\mathsf{\xi}=100[1-Exp\left(-\left|2\mathsf{\pi}\mathsf{\beta}/\mathsf{\eta}\right|\right)]\hspace{0.17em}\%$, where β is the imaginary part of the root of the characteristic equation with the largest real part.

- Step 1.
- Put i= 1 and m = 1, η
_{0}= inf, ξ_{0}= inf, where inf is a large number (for example, inf = 10^{10}), set the accuracy of determining the degree of stability ε_{η}and the damping of oscillations for the period ε_{ξ}. - Step 2.
- Calculate n = p + m and, after performing rational interpolation find the vector a of the CP coefficients.
- Step 3.
- Determine the roots of the characteristic equation, find among them the root with the largest real part and calculate the criteria η
_{i}and ξ_{i}. - Step 4.
- Verify that the iterative process converges to a solution$$\left|{\mathsf{\eta}}_{i}-{\mathsf{\eta}}_{i-1}\right|<{\epsilon}_{\eta},\left|{\mathsf{\xi}}_{i}-{\mathsf{\xi}}_{i-1}\right|<{\mathsf{\epsilon}}_{\mathsf{\xi}}.$$
- Step 5.
- If conditions (12) are fulfilled, then the quality criteria of the system dynamics are determined with the required accuracy, otherwise the process should be continued. To do this, increase the values of the iteration counter i and degree m by one and go to step 2.

#### 5.3. Bearing Dynamic Characteristics and Discussion

_{p}[23]. The influence of these parameters is of particular interest, because they are a resource to optimize the quality of a dynamic system.

_{p}) whose values deliver the maximum degree of stability η. Optimization was carried out for the values of other input parameters: P

_{s}= 5, R

_{c}= 0.5, χ = 0.48, ς = 0.3, H = 1. The parameter iK was varied, with the help of which the static compliance K was calculated by the formula

_{0}is the static compliance of the bearing corresponding to hard orifice 3 (K

_{e}= 0). Corresponding K values of the coefficient K

_{e}were calculated by the formula (30).

_{p}that deliver the maximum performance criterion η. From Table 1 it follows that with a decrease in compliance, the optimal σ tends to increase, while the optimal V

_{p}tends to decrease. With an increase in K

_{e}, the performance of the bearing decreases somewhat, but it remains stable even with negative values of K that are in absolute value superior to those of a conventional bearing (K

_{e}= 0).

_{p}on the degree of stability is given by the graph curves of Figure 11, which are constructed for the same initial data and K

_{e}= 0.32, at which the bearing has zero static compliance (K = 0).

_{p}) are extreme. It can be seen that for the regime of zero compliance, it is necessary to choose the values σ > σ

_{min}, where σ

_{min}is the value of the parameter σ at which the system reaches the stability boundary (η = 0).

_{min}, the system is unstable, and for σ > σ

_{min}the stability region is divided into two parts σ

_{min}< σ < σ

_{opt}and σ > σ

_{opt}, where σ

_{opt}is the value of the parameter σ at which the function η(σ) reaches its maximum.

_{p}< V

_{popt}, the criterion ξ indicates the oscillatory nature of the transients (ξ < 100%), where V

_{p}opt are the values of the parameter Vp at which the function η(V

_{p}) reaches its maximum. The graphs show that for V

_{p}< V

_{p opt}all the curves, regardless of the value of σ, indicate the vibration nature of the transients (ξ < 100%), for V

_{p}> V

_{popt}the transients become aperiodic (ξ = 100%). It is also seen that with an increase in V

_{p}, the range of aperiodicity of the transient characteristics expands, that is, an increase in the volume of V

_{p}contributes to an increase in the dynamic stability margin of the bearing as a dynamic system.

_{p}. The values of σ and V

_{p}should be considered the best, which are slightly larger than the optimal σ

_{opt}and V

_{popt}. For example, for the mode of zero static compliance K = 0, σ = 14 and V

_{p}= 7 will be the best. In this case, the bearing will have close to maximum speed and an aperiodic nature of the transition characteristics.

## 6. Conclusions

_{p}of the throttle chamber, the bearing can provide high speed and a guaranteed stability margin, while keeping the transition characteristics non-oscillatory.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviation

CP | characteristic polynomial |

E | elastic modulus of the orifice material |

f, F | dimensional and dimensionless bearing loads |

h, H | dimensional and dimensionless gaps |

K | dimensionless bearing compliance |

K_{e} | dimensionless elastic ratio |

n, m | transfer function polynomial orders |

p_{k}, P_{k} | dimensional and dimensionless air pressures at the exit of annular orifice plates |

p_{p}, P_{p} | dimensional and dimensionless air pressures at the chamber 4 |

p_{s}, P_{s} | dimensional and dimensionless air supply pressures |

Q_{h}, Q_{p} | dimensionless mass flow rates |

r, R | dimensional and dimensionless radii |

r_{1}, R_{1} | dimensional and dimensionless radii of arrangement of annular diaphragms |

r_{p}, R_{p} | dimensional and dimensionless radii of elastic orifice diaphragm |

r_{p0}, R_{p0} | dimensional and dimensionless radii of elastic orifice |

s | Laplace transform variable |

TF | transfer function |

u,U | dimensional and dimensionless tensile functions of the middle surface of the elastic orifice |

v_{p}, V_{p} | dimensional and dimensionless values of chamber 4 |

$\overline{\Delta H},\text{}\overline{\Delta F}$ | Laplace transformants of the deviation of dynamic functions from their static values |

η | degree of stability |

ν | Poisson’s ratio |

ξ | damping of oscillations over a period |

Π | Prandtl expiration function |

σ | “compression number” |

ς | coefficient of resistance adjustment of damping annular diaphragms |

φ, Φ | dimensional and dimensionless Airy stress functions |

χ | elastic orifice resistance adjustment coefficient |

ψ, Ψ | dimensional and dimensionless deflection functions |

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**Figure 3.**Comparison of the calculated (solid lines) and experimental (dashed lines) of the mass air flow dependences q through elastic openings from pressure difference p

_{s}− p

_{p}at r

_{s}= 3.175 mm, 1—for r

_{p}

_{0}= 0.125 mm and δ = 2.64 mm, 2—for r

_{p}

_{0}= = 0.19 mm and δ = 1.14 mm.

**Figure 4.**Dependences of R

_{p}on P

_{s}− P

_{p}for various Δ; ε = 25, ν = 0.5, R

_{p}

_{0}= 0.02, ν = 0.5.

**Figure 5.**Dependences of R

_{p}on P

_{s}− P

_{p}for various values of ε; ν = 0.5, Δ = 0.4, R

_{p}

_{0}= 0.02, solid lines— theory, dashed lines—experiment.

**Figure 6.**Load characteristics H(F) for various values of the elastic ratio K

_{e}; P

_{s}= 5, R

_{c}= 0.5, χ = 0.48, ς = 0.3.

**Figure 7.**Flow rate characteristics Q(F) for various values of the elastic ratio K

_{e}; P

_{s}= 5, R

_{c}= 0.5, χ = 0.48, ς = 0.3.

**Figure 8.**The dependences of the static compliance K on the coefficient ς for various values of the elastic ratio K

_{e}; P

_{s}= 5, R

_{c}= 0.5, χ = 0.48.

**Figure 9.**Dependences of the elastic parameter K

_{e}

_{0}on the coefficient ς for various values of the tuning coefficient χ; P

_{s}= 5, R

_{c}= 0.5.

**Figure 10.**Dependences of the critical values of the elastic ratio K

_{ed}on the coefficient ς for various values of the tuning coefficient χ; P

_{s}= 5, R

_{c}= 0.5.

**Figure 11.**Dependences of the degree of stability η on the “compression number” σ for various values of the volume V

_{p}.

**Figure 12.**Dependences of criterion ξ on the “compression number” σ for various values of the volume V

_{p}.

iK | K | K_{e} | η | σ | V_{p} |
---|---|---|---|---|---|

1.0 | 0.127 | 0 | 0.301 | 11.6 | 15.4 |

0.5 | 0.064 | 0.342 | 0.288 | 13.2 | 5.9 |

0 | 0 | 0.320 | 0.278 | 13.2 | 6.6 |

−0.5 | −0.064 | 0.382 | 0.272 | 13.9 | 4.2 |

−1.0 | −0.127 | 0.378 | 0.274 | 14.0 | 4.3 |

−1.5 | −0.264 | 0.401 | 0.264 | 15.3 | 3.2 |

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**MDPI and ACS Style**

Kodnyanko, V.; Shatokhin, S.; Kurzakov, A.; Pikalov, Y. Mathematical Modeling on Statics and Dynamics of Aerostatic Thrust Bearing with External Combined Throttling and Elastic Orifice Fluid Flow Regulation. *Lubricants* **2020**, *8*, 57.
https://doi.org/10.3390/lubricants8050057

**AMA Style**

Kodnyanko V, Shatokhin S, Kurzakov A, Pikalov Y. Mathematical Modeling on Statics and Dynamics of Aerostatic Thrust Bearing with External Combined Throttling and Elastic Orifice Fluid Flow Regulation. *Lubricants*. 2020; 8(5):57.
https://doi.org/10.3390/lubricants8050057

**Chicago/Turabian Style**

Kodnyanko, Vladimir, Stanislav Shatokhin, Andrey Kurzakov, and Yuri Pikalov. 2020. "Mathematical Modeling on Statics and Dynamics of Aerostatic Thrust Bearing with External Combined Throttling and Elastic Orifice Fluid Flow Regulation" *Lubricants* 8, no. 5: 57.
https://doi.org/10.3390/lubricants8050057