text| Otani's brilliant notes
Edit| Otani's brilliant notes
●—≺ Preface ≻—●
With the rapid development of high-speed turbomachinery, the dynamic pressure bearings in application are also constantly updated, rigid surface dynamic bearings and elastic foil dynamic bearings are the representatives of the two major gas dynamic bearings.
In order to select suitable supporting components in turbomachinery, it is necessary to study and compare the static and dynamic characteristics of these two typical dynamic bearings to provide a theoretical reference for engineering applications.
Based on the central difference method, the compressible Reynolds equation is discretized and coupled with the rigid surface and elastic foil dynamic pressure bearings and different air film thickness equations, so as to solve it iteratively and obtain the static characteristics of the two bearings.
On the basis of static characteristics, the small disturbance method is applied to iteratively calculate, and the respective dynamic characteristic parameters - dynamic stiffness and dynamic damping.
For the rigid surface and elastic foil dynamic pressure bearings, the effects of eccentricity, speed and disturbance frequency on the static and dynamic characteristics were explored, and their advantages and disadvantages under different working conditions were analyzed.
●—≺ model information≻—●
Figure 1 is a schematic diagram of the structure of a rigid surface dynamic pressure bearing, and it can be seen that its structure is mainly composed of a bearing sleeve and a rotor, and a large amount of lubricating gas is filled between the two.
When the viscous gas is compressed in the gap between the bearing sleeve and the rotor, the lubricating gas will generate a bearing capacity to support the rotor.
Table 1 is the basic parameter table of dynamic pressure bearings, and the basic parameters of rigid surface dynamic pressure bearings and elastic foil dynamic pressure bearings are consistent.
Fig. 2 shows a schematic diagram of the structure of an elastic foil dynamic pressure bearing. Compared with the simple structure of rigid surface dynamic bearings, elastic foil dynamic pressure bearings introduce an elastic foil structure composed of wave foil and flat foil.
One end of the flat foil and the wave foil is spot welded to the bearing sleeve, and the other end is freely lapped and can move freely along the circumference.
The elastic foil structure composed of two kinds of foils can be deformed under pressure to generate reaction force, and together with the pressure of the lubricating air film, it provides the bearing rotor with a bearing capacity that balances gravity, so as to realize the high-speed suspension rotation of the rotor.
Table 2 shows the specific parameters of the elastic foil structure.
In order to iteratively solve the dynamic pressure Reynolds equation, it is necessary to analyze the gas film thickness equation.
Considering the actual situation of the elastic foil structure, it is assumed that due to the uniform distribution of the surface stiffness of the supporting wave foil, it can be regarded as a fixed value.
The flat foil shape variable between the adjacent two peaks is much smaller than the wave foil, so the flat foil deformation can be ignored and only its overall displacement with the wave foil can be considered.
The deformation caused by the load action is only related to the load at the action point.
Since the wave foil is equivalent to a linear spring support, without considering the stiffness characteristics of the flat foil and the friction damping effect in the foil structure, after analyzing the force of the foil structure, the relationship between the air film pressure and the foil structure can be written:
where p is the air film pressure (Pa), pa is the ambient pressure (Pa), A0 is the wave foil unit area (m2), kb is the equivalent stiffness of the wave foil (N/m), L is the bearing thickness (m) and u is the foil structural variable (m).
Dimensionless processing of foil structural variables yields:
where s is the length of the wave foil element (m), C is the radius gap (m), and pˉ is the dimensionless gas film pressure.
The dimensionless air film thickness of elastic foil dynamic pressure bearings is mainly composed of three parts: the original air film thickness, rotor eccentricity, and dimensionless foil structural variables:
where ε is eccentricity; θ is the circumferential angle.
The dimensionless air film thickness equation for rigid surface dynamic bearings does not need to consider the foil structural variant, and its mathematical expression is:
●—≺ dynamic pressure Reynolds equation ≻—●
Considering the gas flow in the gap of the dynamic bearing, the slip effect of the air flow between the layers is ignored, and the flow in the air film of each layer is assumed to be laminar flow.
Since the thickness of the air film is generally on the order of microns, the change of the air film pressure along the direction of the air film thickness can be ignored.
Since the thickness of the wedge gap forming the dynamic pressure gas film is much smaller than the curvature radius of the rotor in magnitude, the influence of the curvature radius of the rotor on the direction and shape of the air film can be ignored.
The relative movement of the viscous gas along the rotor contact surface is regarded as translational motion, and its speed is equivalent to the tangential velocity of the rotor contact surface.
Since the inertial force of the air film is much smaller than its viscous force, the action of inertial force and volumetric force can be ignored.
The magnitude of the gas film thickness is very small, assuming that the gas film viscosity and density do not change in the direction of the gas film thickness. On the basis of the above assumptions, the Reynolds equation for dynamic pressure of compressible gases can be given:
where x is the circumferential coordinate, z is the axial coordinate, p is the gas film pressure (Pa), h is the gas film thickness (m), μ is the gas-dynamic viscosity (Pa·s), v is the circumferential motion speed of the rotor (m/s), and t is the time variable.
When the heat dissipation characteristics of the gas dynamic pressure bearing are good, the dynamic viscosity of the lubricating gas can be regarded as a constant, and there is no need to consider the influence of temperature increase on the pressure of the lubricating gas film.
Therefore, for a constant ideal gas that does not change with time in terms of gas dynamic viscosity and working temperature, its dynamic pressure Reynolds equation can be dimensionless:
The dimensionless parameters in the equation are:
where θ is the dimensionless circumferential coordinate, λ is the dimensionless axial coordinate, C is the radius clearance (m), L is the bearing thickness (m), and R is the radius of the rotor (m).
ω is the rotor angular velocity (rad/s), h is the dimensionless air film thickness, pˉ is the dimensionless air film pressure, and Λ is the number of bearings, which reflects the operating conditions and performance parameters of the bearing.
Based on the central difference method, the gas film thickness equation and the compressible gas Reynolds equation are iteratively solved by fluid-structure interaction.
In order to verify the correctness of the method and procedure, the results of rigid surface and elastic foil gas dynamic bearings are verified for examples.
Figure 3 shows the comparison of the air film pressure and distribution curve obtained by the self-programmed program simulation. The solution results obtained by comparing the two figures are consistent, which verifies the correctness of the method and the self-programmed program.
Under the working conditions of eccentricity of 0.5 and speed of 5×104r/min, the full-size air film pressure and thickness distribution of rigid surface dynamic pressure bearings can be obtained by applying the central differential method and the coupling iterative method (see Figure 4).
It is clear from the figure that rigid surface dynamic bearings have a support area greater than the ambient pressure and a negative pressure area that is less than the ambient pressure.
"In gas lubrication, the gas film pressure is always greater than the ambient pressure, because the surrounding gas can freely enter the gap".
But this leads to the problem of non-conservation of mass flow. In a physical sense, it is normal for the air film pressure to be less than the ambient pressure in the local area of the bearing.
In addition, the thickness of the air film of the rigid surface dynamic pressure bearing is uniform and unchanged, which is because the air film is in direct contact with the bearing sleeve, which will also have higher requirements for the material properties and heat resistance and wear resistance of the bearing sleeve.
Figure 5 shows the change curve of the air film pressure of rigid surface dynamic pressure bearing with eccentricity at 5×104r/min speed.
Figure 6 shows the change curve of the air film pressure of rigid surface dynamic pressure bearing with speed under the condition of 0.5 eccentricity.
The information in the analysis figure shows that with the increase of eccentricity or speed, the air film pressure of the rigid surface dynamic pressure bearing will increase as a whole, and the high-pressure bearing area will expand.
This is due to the increase in eccentricity and the increase in rotor misalignment, resulting in the gas in the gap being further compressed and increasing the pressure of the gas film.
The speed increase will cause the same rotor, the density of the viscous gas rotating at high speed will be reduced, and the incompressibility will be enhanced, which can provide greater gas film pressure.
Figure 7 shows the dynamic characteristic parameters of rigid surface dynamic bearings, which change with the disturbance frequency.
It can be seen from the figure that their direct dynamic stiffness Kxx and Kyy show an upward trend with the increase of disturbance frequency, and the trend gradually flattens and approaches their respective limit values.
The dynamic cross stiffness Kxy and Kyx, on the other hand, increase their respective trends with the increase of disturbance frequency, and are also approaching a certain limit value.
Except for the cross-dynamic damping Dyx, the remaining dynamic damping coefficients decreased with the increase of the disturbance frequency, but all the dynamic damping parameters increased with the disturbance frequency and approached the zero value.
This shows that as the disturbance frequency increases, the rigid surface dynamic pressure bearing resists displacement, and the ability of disturbance is improved, but the rate of dissipation disturbance capacity is reduced.
●—≺ elastic foil dynamic pressure bearing≻—●
Under the same working conditions with an eccentricity of 0.5 and a speed of 5×104 r/min, the full-size air film pressure and thickness distribution of elastic foils and dynamic bearings can be iteratively solved (see Figure 8).
It can be seen from the figure that the thickness of the air film of the elastic foil dynamic pressure bearing, compared with the rigid surface dynamic pressure bearing, has obvious concave inside, which is due to the wave foil in the elastic foil structure, there is a deformation effect, which can bear a part of the supporting force, which also makes the supporting force provided by the lubricating air film effectively reduced.
This explains why under the same working conditions, the maximum air film pressure of elastic foil dynamic pressure bearings is less than that of rigid surface dynamic pressure bearings, and the minimum air film thickness at the middle section is greater than the latter.
In order to study the advantages and disadvantages of rigid surface dynamic bearings and elastic foil dynamic bearings in engineering applications, it is necessary to compare and analyze their static and dynamic characteristics.
Figure 12 is a visual comparison of the air film pressure distribution, bearing capacity and declination angle of the two dynamic bearings under the same working conditions.
The analysis of the information in these three figures shows that the overall air film pressure of rigid surface dynamic pressure bearings is higher than that of elastic foil dynamic pressure bearings, but the area of its high-pressure bearing area is not as good as the latter, and the maximum negative pressure is relatively larger.
The load carrying capacity of rigid surface dynamic bearings is better than that of elastic foil dynamic pressure bearings, in contrast, the offset angle of the former is significantly greater than that of the latter.
It can be seen that rigid surface dynamic bearings obtain better load-bearing performance at the cost of higher material property requirements, but the introduction of elastic foil can reduce the offset angle of the bearing, which helps the stability of the rotor during normal operation.
Fig. 13 shows a comparison of the dynamic characteristic parameters of rigid surface dynamic bearings and elastic foil dynamic pressure bearings, with the change curve of disturbance frequency.
Fig. 14 shows a comparison of the dynamic characteristic parameters of rigid surface dynamic bearings and elastic foil dynamic pressure bearings, with eccentricity curves.
At the small eccentricity, the difference in dynamic characteristic parameters between the two is not obvious, but when the eccentricity increases, the direct dynamic stiffness Kyy and direct dynamic damping Dyy of rigid surface dynamic pressure bearings rise rapidly, while the dynamic parameter values of elastic foil dynamic pressure bearings change little.
This shows that the rigid surface dynamic pressure bearing is greatly affected by eccentricity, and under the large eccentricity condition, its dynamic pressure air film resistance to displacement disturbance is rapidly enhanced, and the dissipation disturbance energy rate is also accelerated.
However, the dynamic characteristic parameters of elastic foil dynamic pressure bearings are less affected by eccentricity, and the value changes little, which has good stability.
Fig. 15 shows a comparison of the dynamic characteristics of rigid surface dynamic bearings and elastic foil dynamic pressure bearings, and the parameters change curve with speed.
Comparing the difference between the solid line and the dashed line, it can be seen that there is no significant difference in the dynamic damping coefficient between rigid surface dynamic pressure bearings and elastic foil dynamic pressure bearings.
The dynamic stiffness of the former is better than the latter at all speed conditions, which shows that the dissipation disturbance energy velocity of the two dynamic bearings has no obvious difference with the speed change.
The air film formed by rigid surface dynamic pressure bearings, when the eccentricity is fixed, its resistance to displacement disturbance is stronger than that of elastic foil dynamic pressure bearings.
Rigid surface dynamic bearings obtain better load-bearing performance at the cost of higher material property requirements, but the introduction of elastic foil can reduce the offset angle of dynamic pressure bearings, which contributes to the stability of the rotor during normal operation.
The difference between the dynamic characteristic parameters of the two with the change of disturbance frequency is not obvious, but under the working conditions of large eccentricity and large speed, the ability of rigid surface dynamic pressure bearings to resist displacement disturbance is stronger than that of elastic foil dynamic pressure bearings.
Rigid surface dynamic bearings, due to the lack of protection of elastic foil structure, during the start-stop process, the rotor and bearing sleeve are easy to rub against each other, resulting in wear and reducing service life.
Therefore, rigid surface dynamic bearings are not suitable for high-speed turbomachinery with frequent start and stop.