Semi-analytical method and system for foil gas dynamic pressure bearing stiffness and damping coefficients

Abstract

This invention discloses a semi-analytical method and system for calculating the stiffness and damping coefficients of foil gas dynamic bearings, relating to the field of air conditioning and clothing temperature control technology. It aims to address the shortcomings of traditional Reynolds equation methods in terms of accuracy and universality, as well as the low computational efficiency and mesh distortion issues of existing CFD methods. The method includes: constructing a fluid-structure interaction (FSI) computational model to calculate and obtain the static equilibrium position of the journal; applying an eccentricity perturbation at the static equilibrium position and performing an additional steady-state FSI calculation to obtain the four stiffness coefficients of the foil bearing using a first set of analytical formulas; applying small-amplitude simple harmonic vibrations along the x and y directions to the journal and performing two transient FSI calculations to obtain the four damping coefficients of the foil bearing using a second set of analytical formulas. This invention combines high accuracy and high computational efficiency, providing an efficient and reliable analytical tool for evaluating the dynamic characteristics of foil bearings and designing the stability of the rotor systems they support.

Description

Technical Field

[0001] This invention relates to the field of high-speed rotating machinery design and rotor dynamics analysis, specifically to a semi-analytical method and system for the stiffness damping coefficient of a foil gas dynamic bearing. Background Technology

[0002] Foil gas hydrodynamic bearings are widely used in high-speed rotating machinery such as micro gas turbines, air cycle machines, and flywheel energy storage systems due to their advantages of oil-free lubrication, high-speed adaptability, and low power consumption. Their dynamic characteristics are defined by eight stiffness-damping coefficients (k...). xx k xy k yx k yy c xx c xy c yx c yy These are key input parameters for predicting the critical speed, stability, and unbalanced response of a rotor system. Currently, the main methods for obtaining these dynamic coefficients include: 1. Full analytical / numerical methods based on the Reynolds equations, which calculate by solving simplified Reynolds equations and combining them with foil structural mechanical models (such as compliance coefficient models and spring-linkage models). This method suffers from high model simplification, poor universality, complexity in developing dedicated solver programs, and limited accuracy under complex flow conditions. 2. Semi-analytical methods based on general-purpose computational fluid dynamics (CFD) software (e.g., CN111666644A), which use commercial software (such as ANSYS Fluent) to solve the complete set of fundamental fluid dynamics equations and obtain film pressure and foil deformation through fluid-structure interaction (FSI) simulation.

[0003] However, existing CFD methods have certain drawbacks: 1. The static balance position search is inefficient; it requires two-variable parameter scanning or time-consuming transient simulation to find the journal's static balance position, resulting in high computational costs. 2. The dynamic coefficient identification process is complex; it usually requires a complete transient eddy simulation, which consumes huge computational resources, and the process of decoupling and identifying stiffness and damping coefficients from the coupled motion response is cumbersome. 3. Mesh distortion problem: When simulating the combined motion of journal rotation and linear vibration, severe distortion of the moving mesh is likely to occur, leading to calculation failure or decreased accuracy. Summary of the Invention

[0004] To address the shortcomings of traditional Reynolds equation methods in terms of accuracy and universality, as well as the low computational efficiency and easy generation of mesh distortion in existing CFD methods, a semi-analytical method and system for calculating the stiffness damping coefficient of foil gas hydrodynamic bearings is proposed. This method maintains high accuracy and universality while significantly improving computational efficiency.

[0005] To achieve the above objectives, the present invention employs one of the following technical solutions: a semi-analytical method for determining the stiffness damping coefficient of a foil gas hydrodynamic bearing, comprising the following steps: S1. Construct a fluid-structure interaction calculation model to calculate and obtain the static equilibrium position of the journal under a given rotational speed and static load. S2, apply an eccentric disturbance at the static equilibrium position, perform an additional steady-state fluid-structure interaction calculation, and calculate the four stiffness coefficients of the foil bearing using the first set of analytical formulas. S3. At the static equilibrium position and rotational speed, small-amplitude simple harmonic vibrations along the x and y directions are applied to the journal. Two transient fluid-structure interaction calculations are performed, and the four damping coefficients of the foil bearing are calculated by the second set of analytical formulas.

[0006] In this invention, firstly, a single-parameter steady-state fluid-structure interaction calculation is performed by changing the eccentricity to efficiently determine the static equilibrium position of the journal; secondly, a single small eccentric disturbance is applied based on this position, and steady-state calculation is performed again, and the four stiffness coefficients are directly solved using the derived first set of analytical formulas; finally, simple harmonic motion is applied to the journal for transient calculation, and the composite motion is controlled by a user-defined function to avoid mesh distortion, and the four damping coefficients are directly obtained based on the second set of analytical formulas.

[0007] The present invention is further configured such that: the construction of the fluid-structure interaction calculation model includes: based on the physical structure of the foil gas dynamic bearing, establishing a fluid-structure interaction calculation model including the journal, the gas film fluid domain and the foil structure.

[0008] The present invention is further configured such that: the calculation and obtaining of the static balance position of the journal includes: under a given rotational speed and static load, adjusting the journal eccentricity by a single parameter, and performing a series of steady-state fluid-structure interaction calculations until the air film force and the external load are balanced, and finally determining the static balance position of the journal.

[0009] The present invention is further configured such that step S3 includes: applying a preset eccentricity disturbance at the static equilibrium position, performing an additional steady-state fluid-structure interaction calculation, and directly calculating the four stiffness coefficients of the foil bearing based on the air film force components and geometric relationships before and after the disturbance through the first set of analytical formulas.

[0010] The present invention is further configured such that step S4 includes: applying simple harmonic motion along the x and y directions to the journal at the static equilibrium position and rotational speed, performing two transient fluid-structure interaction calculations, controlling the composite motion of the air film boundary through a user-defined function, and directly calculating the four damping coefficients of the foil bearing through the second analytical formula group based on the relationship between the air film force increment and velocity when the journal passes through the equilibrium position.

[0011] The present invention is further configured such that: the first set of analytical formulas is obtained by constructing two equivalent disturbance modes with constant static load in the horizontal direction and constant static load in the vertical direction and solving the equation set; the four stiffness coefficients are all equal to the ratio of the force increment to the displacement disturbance combination, the denominator is the Jacobian determinant formed by the displacement disturbance components of the two disturbance modes, and the numerator is the difference between the disturbed force and the static equilibrium force.

[0012] The present invention is further configured such that: the four damping coefficients in the second analytical formula group are all ratios of the damping force component to the maximum velocity in that direction, the damping force component is the difference between the static equilibrium air film force and the instantaneous actual air film force, and the maximum velocity is the product of the vibration angular frequency and the amplitude in that direction.

[0013] The present invention is further configured such that the preset eccentricity disturbance Δd satisfies the ratio of Δd to C being less than 0.05, where C is the bearing radius clearance.

[0014] The present invention is further configured such that: the amplitude A of the simple harmonic motion satisfies the ratio of A to C being less than 0.01, where C is the bearing radius clearance, and the vibration angular frequency ω of the simple harmonic motion is equal to the rotational angular velocity Ω of the journal.

[0015] A semi-analytical system for calculating the stiffness and damping coefficient of a foil gas hydrodynamic bearing, applicable to the aforementioned semi-analytical method for calculating the stiffness and damping coefficient of a foil gas hydrodynamic bearing, comprising: The preprocessing and modeling module constructs a fluid-structure interaction computational model; The steady-state solution and equilibrium search module determines the static equilibrium position through single-parameter scanning; The stiffness coefficient analytical calculation module calculates the stiffness coefficient using the first set of analytical formulas. The transient solution and motion control module calculates the damping coefficient using the second set of analytical formulas. The output module outputs and stores the static equilibrium position, stiffness coefficient, and damping coefficient.

[0016] The system of this technical solution mainly includes a preprocessing and modeling module, a steady-state solution and equilibrium search module, a stiffness coefficient analytical calculation module, a transient solution and motion control module, and an output module. The preprocessing and modeling module, the steady-state solution and equilibrium search module, the stiffness coefficient analytical calculation module, the transient solution and motion control module, and the output module are connected in sequence to jointly complete the above steps S1 to S3.

[0017] The present invention can bring the following beneficial effects: 1. Significantly improved computational efficiency: The static equilibrium position is determined by single-parameter steady-state scanning, and the stiffness coefficient is calculated by a single steady-state disturbance, avoiding the huge computational overhead of traditional multivariable scanning or full transient simulation. 2. Good accuracy and universality; based on the complete set of fluid dynamics equations, fluid-structure interaction simulation is performed, preserving flow details, and the model has high accuracy and wide applicability; 3. Strong numerical robustness; The proposed damping calculation strategy achieves boundary motion control through user-defined functions, effectively avoiding mesh distortion caused by compound motion and improving the stability and success rate of the calculation. 4. High engineering practical value: The eight dynamic coefficients obtained can be directly applied to rotor dynamics analysis, providing an efficient and reliable theoretical tool for the dynamic design, stability prediction and optimization of foil bearings and their rotor systems. Attached Figure Description

[0018] Figure 1 This is a schematic diagram of the overall structure of a foil gas dynamic bearing.

[0019] Figure 2 This is a schematic diagram showing the details of the foil layer structure.

[0020] Figure 3 This is a schematic diagram showing the static equilibrium position and forces acting on the journal.

[0021] Figure 4 This is a schematic diagram of a disturbance mode used for stiffness calculation.

[0022] Figure 5 This is a schematic diagram of disturbance mode two used for stiffness calculation.

[0023] Figure 6 This is a schematic diagram of simple harmonic motion of the journal used for damping calculation.

[0024] Figure 7 This is the overall flowchart of the semi-analytical calculation method.

[0025] Figure 8 This is a trajectory diagram showing the change of the static balance position of the journal at different speeds.

[0026] Figure 9 This is a graph showing the stiffness coefficient as a function of rotational speed.

[0027] Figure 10 This is a graph showing the damping coefficient as a function of rotational speed.

[0028] Figure label: 1. Outer flat foil 2. Inner flat foil 3. Corrugated foil. Detailed Implementation

[0029] Example 1 To address the shortcomings of traditional Reynolds equation methods in terms of accuracy and universality, as well as the low computational efficiency and susceptibility to mesh distortion in existing CFD methods, this paper refers to... Figure 7This embodiment proposes a semi-analytical method for determining the stiffness damping coefficient of a foil gas dynamic bearing, which includes the following steps.

[0030] Step S1, Model Construction and Static Equilibrium Position Determination: Construct a fluid-structure interaction calculation model. After constructing the above model, under a given rotational speed and static load, obtain the static equilibrium position of the journal through a series of calculations.

[0031] In more detail, the model construction process involves establishing a fluid-structure interaction calculation model based on the physical structure of the foil gas dynamic bearing, including the journal, the gas film fluid domain, and the foil structure.

[0032] The process of determining the static equilibrium position is described in more detail as follows: under a given rotational speed and static load, the journal eccentricity is adjusted by a single parameter and a series of steady-state fluid-structure interaction calculations are performed until the air film force and the external load are balanced, thus determining the static equilibrium position of the journal. The static equilibrium position is defined by the eccentricity d0 and the offset angle ψ0.

[0033] Among them, the single parameter is the eccentricity d0 between the journal center and the bearing center. Its search direction is fixed, and only its magnitude is changed to find the resultant force of the air film force (F0=W) that satisfies the force balance condition. Here, W represents the static load.

[0034] Step S2, stiffness coefficient calculation: Apply an eccentric disturbance at the static equilibrium position, perform an additional steady-state fluid-structure interaction calculation, and calculate the four stiffness coefficients of the foil bearing using the first set of analytical formulas.

[0035] More specifically, in step S2 above, at the static equilibrium position, an additional steady-state fluid-structure interaction calculation is performed by applying a pre-set eccentricity disturbance (Δd). Based on the gas film force components and geometric relationships before and after the disturbance, the four stiffness coefficients of the foil bearing are directly calculated using the first set of analytical formulas. The four stiffness coefficients are represented as k. xx k xy k yx and k yy .

[0036] In this embodiment, the value obtained by dividing the pre-set eccentricity disturbance Δd by C is less than 0.05; C is the bearing radius clearance.

[0037] The first analytical formula mentioned above is obtained by constructing two equivalent disturbance modes with constant static load in the horizontal direction and constant static load in the vertical direction, and solving the system of equations. The four stiffness coefficients are all equal to the ratio of the force increment to the combination of displacement disturbances. The combination of displacement disturbances is the Jacobian determinant formed by the displacement disturbance components of the two disturbance modes. The force increment is the difference between the force after disturbance and the static equilibrium force.

[0038] Specifically, k xx equals F x 'With F x0 The difference is then multiplied by Δy j Then divide by D; k xy equals F x0 With F x The difference is then multiplied by Δx j Then divide by D; k yx equals F y0 With F y The difference is then multiplied by Δy j ', then divide by D; k yy equals F y With F y0 The difference is then multiplied by Δx j Then divide by D.

[0039] Where D equals Δx j Δy j 'Subtract Δx j 'Δy j F x0 F y0 For the film force component at static equilibrium position, Δx j Δy j and Δx j '、Δy j ' represents the journal center displacement disturbance component under two disturbance modes.

[0040] Step S3, Calculation of damping coefficients: At the static equilibrium position and rotational speed, small-amplitude simple harmonic vibrations along the x and y directions are applied to the journal. Two transient fluid-structure interaction calculations are performed, and the four damping coefficients of the foil bearing are calculated by the second set of analytical formulas.

[0041] More specifically, for step S3 above, at the static equilibrium position and rotational speed, small-amplitude simple harmonic vibrations along the x and y directions are applied to the journal, and two transient fluid-structure interaction calculations are performed. The composite motion of the air film boundary is controlled by a user-defined function to avoid mesh distortion. Based on the relationship between the air film force increment and velocity when the journal passes through the equilibrium position, the four damping coefficients of the foil bearing are directly calculated using the second set of analytical formulas. The four damping coefficients are expressed as c... xx c xy c yx and c yy .

[0042] The user-defined function described above can achieve both rotational speed around the axis and simple harmonic oscillation speed along the excitation direction at the boundary of the air film inner wall.

[0043] In this embodiment, the amplitude A of the simple harmonic motion divided by C is less than 0.01, and the vibration angular frequency ω of the simple harmonic motion is equal to the rotational angular velocity Ω of the journal.

[0044] For the second set of analytical formulas, its four damping coefficients c xx c xy c yx and c yy Both are equal to the ratio of the damping force component to the maximum velocity in that direction. The damping force component is the difference between the static equilibrium air film force and the instantaneous actual air film force, and the maximum velocity is the product of the vibration angular frequency and the amplitude in that direction.

[0045] Specifically, c xx equals F x0 Subtract F x Then divide by ωX; c yx equals F y0 Subtract F y Then divide by ωX; c xy equals F x0 Subtract F x Then divide by ωY; c yy equals F y0 Subtract F y Then divide by ωY.

[0046] Where ω is the angular frequency of vibration, X and Y are the amplitudes in the x and y directions, respectively, and F x F y This is the instantaneous air film force component when the journal passes through the equilibrium position.

[0047] This invention addresses the shortcomings of traditional Reynolds equation methods in terms of accuracy and universality, as well as the low computational efficiency and mesh distortion issues of existing CFD methods. It proposes a hybrid strategy combining computational fluid dynamics and analytical derivation. First, single-parameter steady-state fluid-structure interaction calculations are performed by changing the eccentricity to efficiently determine the journal's static equilibrium position. Second, a single small eccentric disturbance is applied to this position, and steady-state calculations are performed again, with the four stiffness coefficients directly calculated using derived analytical formulas. Finally, simple harmonic motion is applied to the journal for transient calculations, and the composite motion is controlled by a user-defined function to avoid mesh distortion. The four damping coefficients are then directly obtained using another set of analytical formulas. This invention combines high accuracy with high computational efficiency, providing an efficient and reliable analytical tool for evaluating the dynamic characteristics of foil bearings and designing the stability of the rotor systems they support.

[0048] This embodiment also proposes a semi-analytical system for the stiffness and damping coefficient of a foil gas dynamic bearing, including a preprocessing and modeling module, a steady-state solution and equilibrium search module, a stiffness coefficient analytical calculation module, a transient solution and motion control module, and an output module. The preprocessing and modeling module, the steady-state solution and equilibrium search module, the stiffness coefficient analytical calculation module, the transient solution and motion control module, and the output module are connected in sequence to jointly complete the above steps S1 to S3.

[0049] The preprocessing and modeling module mainly performs the function of constructing a fluid-structure interaction calculation model; for details, please refer to step S1 above.

[0050] The main function of the steady-state solution and equilibrium search module is to determine the static equilibrium position through single-parameter scanning.

[0051] The main function of the stiffness coefficient analytical calculation module is to calculate the stiffness coefficient using the first set of analytical formulas; for details, please refer to step S2 above.

[0052] The main function of the transient solution and motion control module is to calculate the damping coefficient using the second set of analytical formulas; for details, please refer to step S3 above.

[0053] The main function of the output module is to output and store the static equilibrium position, stiffness coefficient, and damping coefficient for subsequent processing and viewing.

[0054] This embodiment also proposes a computer-readable storage medium storing a computer program that, when executed by a processor, can implement the aforementioned semi-analytical method for the stiffness and damping coefficient of a foil gas dynamic bearing.

[0055] Example 2 Based on Example 1, this example takes a certain type of radial wave foil gas hydrodynamic bearing as the research object, and its specific structural parameters are shown in Table 1. The semi-analytical method for determining the stiffness and damping coefficient of the foil gas hydrodynamic bearing is implemented as follows: Table 1. Structural parameters of foil bearings .

[0056] Step 1: Construct a two-way fluid-structure interaction simulation model of the air film fluid domain and the foil structure.

[0057] refer to Figure 1 and Figure 2A physical model of the bearing was created in 3D modeling software, mainly consisting of an extremely thin film fluid domain, an inner flat foil 2, a wave foil 3, and an outer flat foil 1, forming a foil structure. The model was then imported into the ANSYS Workbench platform to establish a two-way system coupling between "static structure" and "fluid flow (Fluent)".

[0058] Solid domain (foil structure) settings: The foil structure is meshed with a mesh size of 1 mm. Material properties are set according to Table 1. Boundary conditions are as follows: the outer surface of the outer flat foil and the slotted portions at both ends of the foil are set as fixed supports; the contact surfaces between the corrugated foil and the inner and outer flat foils are set as frictional contact; the inner surface of the inner flat foil (the surface in contact with the gas film) is set as a fluid-structure interaction interface.

[0059] Fluid domain (film gas) setup: The film gas domain is finely meshed with a mesh size of 0.4 mm, and divided into 10 layers in the thickness direction to ensure computational accuracy. Boundary conditions are (e.g.) Figure 1 As shown): The inner surface of the air film (corresponding to the journal surface) is set as a rotating wall to simulate journal rotation; the outer surface of the air film (corresponding to the inner surface of the inner flat foil) is set as a stationary wall and designated as a fluid-structure interaction interface to exchange pressure and displacement data with the solid domain; the two end faces of the air film are set as pressure outlets with gauge pressure values ​​of ambient atmospheric pressure.

[0060] Coupling settings: In the system coupling component, the fluid-structure interaction surface of the solid domain is connected to the fluid-structure interaction surface of the fluid domain to achieve bidirectional data exchange (force and displacement).

[0061] Step 2: Determine the static balance position of the journal.

[0062] In this embodiment, the dynamic coefficient is calculated as an example when the rotational speed n is 20000 r / min and the static load W is 4.96 N.

[0063] Single-parameter search strategy: Reference Figure 3 Define the bearing center as O b The journal center is O j The static equilibrium position is uniquely determined by the eccentricity d0 and the displacement angle ψ0. This embodiment employs an efficient single-parameter search: the direction of the eccentricity d0 is taken on the +e axis and remains unchanged, and only the magnitude of the eccentricity d0 is systematically changed to perform a series of steady-state fluid-structure interaction calculations.

[0064] Balance determination: For each set d0, perform a steady-state coupling calculation to obtain the two components F of the film force. e0 F θ0The resultant force F0; the criterion for static equilibrium is that the resultant force of the air film force is balanced with the external static load, i.e., F0 = W. By continuously adjusting d0, this equilibrium condition is satisfied. The corresponding d0 and ψ0 at this point are the static equilibrium positions at that rotational speed. The offset angle ψ0 is the angle between the -y direction (the negative direction of the resultant force F0) and the +e direction, i.e., ψ0 equals -arctan(F0 / W). θ0 / F e0 ).set up Figure 3 The journal center O shown j It is in static equilibrium position, and its coordinate system O b If the coordinates of x and y are x0 and y0 respectively, then x0 is equal to d0 multiplied by sinψ0 and y0 is equal to -d0 multiplied by cosψ0.

[0065] Example of results: Through calculation, under the condition of 20000 r / min, the static equilibrium position parameters are obtained as d0 = 19.6 μm and ψ0 = 17.8°. Repeating this process yields the static equilibrium position at different speeds, and the trajectory of change is as follows... Figure 8 As shown, it exhibits a "balanced semicircle" characteristic that approximates a circular arc, meaning that as the rotational speed increases, the journal center moves closer to the bearing center.

[0066] Step 3: Calculate the four stiffness coefficients.

[0067] Based on the obtained static equilibrium position (eccentricity d0, displacement angle ψ0), the four stiffness coefficients k are solved through a first steady-state disturbance calculation and analytical derivation. xx k xy k yx and k yy The core innovation of this step lies in the fact that all four stiffness coefficients can be calculated using the constructed analytical model by applying only one physical perturbation, thereby greatly improving computational efficiency.

[0068] When the journal center stabilizes at a new static equilibrium position after being disturbed, the magnitude of the film force includes F. x and F y F x equals F x0 Subtract k xx Δx j Subtract k xy Δy j F y equals F y0 Subtract k yx Δx j Subtract k yy Δy j .

[0069] Among them, F x F yThese are the components of the film force in the x and y directions, respectively; the subscript 0 indicates the value of the variable at the original equilibrium position; k xx k xy k yx k yy Δx is the stiffness coefficient of the bearing. j Δy j The journal center O j Displacement disturbances in the x and y directions.

[0070] When the static load in the horizontal direction remains unchanged: Assuming the increase in air film force is caused by the increase in static load in the vertical direction (y-direction), while the static load in the horizontal direction (x-direction) remains unchanged at 0, then... Figure 4 As shown, the eccentricity becomes d, which equals d0 plus Δd (the eccentricity increment Δd lies on the +e axis), and the +y direction is determined by the resultant film force F. The film force increment includes ΔF. x and ΔF y ; ΔF equals -k xx Δx j With k xy Δy j The opposite of the sum is also equal to F. x Subtract F x0 That is, equal to 0, ΔF y equals -(k) yx Δx j With k yy Δy j The opposite of the sum is also equal to F. y Subtract F y0 .

[0071] Journal center O j coordinates (x) j y j ), where x j Equals dsinψ, y j Equal to -dcosψ, compared with the original static equilibrium position (d0, ψ0), the journal center O is obtained. j Coordinate change Δx j and Δy j , Δx j Δy equals dsinψ minus d0sinψ0. j It equals d0cosψ0 minus dcosψ.

[0072] When the static load in the vertical direction remains unchanged: Assuming the increment of the film force is caused by the increment of the static load in the horizontal direction, we have F x 'Not equal to 0, let F' y 'equals Fy0 That is, the static load in the vertical direction remains unchanged. For example... Figure 5 As shown, the +y' direction is from F y If determined, then the air film force F x 'equals (F) 2 -F y0 2 ) 1 / 2 At this time, the increase in film force includes ΔF x 'and ΔF y ',ΔF x 'equals k xx Δx j 'with k xy Δy j The opposite of the sum of ' is also equal to F. x 'Subtract F x0 ;ΔF y 'equals k yx Δx j 'with k yy Δy j The opposite of the sum of ' is equal to 0; the position angle Ψ' is equal to -acrtan(F) θ / F e Subtract arctan(F) x ' / F y0 ), journal center O j coordinates (x) j ',y j '), where x j 'equals dsinψ', y j 'Equal to -dcosψ', compared with the original static equilibrium position (d0, ψ0), the journal center O is obtained. j Coordinate change Δx j ' and Δy j ',Δx j 'equals dsinψ' minus d0sinψ0, Δy j 'Equals d0cosψ0 minus dcosψ'.

[0073] By rearranging the equations, we can obtain a system of equations for four stiffness coefficients, namely k. xx Δx j Add k xy Δy j equals 0, k xx Δx j 'Add k xy Δy j 'equals F x0 Subtract F x ',k yx Δx j Add k yy Δy jequals F y0 Subtract F y k yx Δx j 'Add k yy Δy j 'Equals 0.'

[0074] Ultimately, we can obtain expressions for four stiffness coefficients with respect to the air film force and disturbance displacement, namely k xx equals F x 'With F x0 The difference is then multiplied by Δy j Then divide by D; k xy equals F x0 With F x The difference is then multiplied by Δx j Then divide by D; k yx equals F y0 With F y The difference is then multiplied by Δy j ', then divide by D; k yy equals F y With F y0 The difference is then multiplied by Δx j Then divide by D.

[0075] Where D equals Δx j Δy j 'Subtract Δx j 'Δy j F x0 F y0 For the film force component at static equilibrium position, Δx j Δy j and Δx j '、Δy j ' represents the journal center displacement disturbance component under two disturbance modes.

[0076] Results and Verification: Based on experience, the disturbance displacement Δd should satisfy Δd divided by C being less than 0.05 (where C is the radius clearance). In this example, Δd is selected as 0.2 μm. The calculated stiffness coefficient varies with rotational speed as follows: Figure 9 As shown. Principal stiffness coefficient k xx It decreases slightly with increasing rotational speed, while the other principal stiffness coefficient k yy It decreases significantly with increasing rotational speed; cross stiffness k yx The variation with rotational speed is not obvious, while the other cross stiffness k xy The cross stiffness increases significantly with increasing rotational speed. At high speeds, the cross stiffness values ​​are all greater than the principal stiffness values. The variation of the four stiffness coefficients with rotational speed (cross stiffness exceeds principal stiffness at high speeds) is a key characteristic for evaluating the high-speed stability of foil bearing-rotor systems.

[0077] Step 4: Calculate the four damping coefficients.

[0078] After obtaining the static equilibrium position (d0, ψ0) and the stiffness coefficient (k) xx k xy k yx k yy After that, the four damping coefficients c are directly solved through two preset simple harmonic transient simulations and analytical methods. xx c xy c yx and c yy The core of this step lies in proposing a direct calculation method for the damping coefficient based on specific phase analytical extraction, and fundamentally avoiding the mesh distortion problem through motion control strategies.

[0079] Setting the simple harmonic vibration excitation: To identify the damping coefficient, a small-amplitude simple harmonic excitation is applied to the journal. The excitation amplitude A must satisfy the condition that A divided by C is less than 0.01 (where C is the radial clearance) to ensure the response is within the linear range; in this example, A is taken as 0.1 μm. The excitation angular frequency ω is typically taken as the journal's rotational angular velocity Ω (i.e., 2πn / 60°) to simulate actual vortex conditions. The identification of the damping coefficient is performed in two independent transient calculations.

[0080] First calculation: Let the journal undergo simple harmonic motion only along the x-direction. Its equation of motion is expressed as: x j (t) equals x0 plus Xsin(ωt), where x j The journal center is O j The x-axis represents the static equilibrium position; x0 is the x-axis of the position; X is the amplitude; and ω is the angular frequency of the vibration.

[0081] Then O j The velocity in the x-direction is x j '(t) is ωXcos(ωt), O j When the motion passes through the equilibrium position for an integer number of cycles, the increase in the film force caused by the film stiffness is 0, and the increase in the film force is caused only by damping, i.e., F x equals F x0 Subtract c xx x j '(t), F y equals F y0 Subtract c yx x j '(t). O j The velocity reaches its maximum value ωX when passing the equilibrium position, from which two damping coefficients, c, can be calculated. xx equals F x0 Subtract F x Then divide by ωX; c yx equals Fy0 Subtract F y Then divide by ωX.

[0082] Second calculation: Let the journal undergo simple harmonic motion only along the y-direction; its equation of motion is expressed as: y j (t) equals y0 plus Ysin(ωt), where y j The journal center is O j The vertical axis is y0, which is the vertical axis of its static equilibrium position; Y is the amplitude; and ω is the angular frequency of vibration.

[0083] O j The velocity in the y-direction is y j '(t) equals ωYcos(ωt), O j When the air film force passes through the equilibrium position after an integer number of motion cycles, the air film force is: F x equals F x0 Subtract c xy y j '(t), F y equals F y0 Subtract c yy y j '(t); O j The velocity reaches its maximum value ωX when passing the equilibrium position, from which the other two damping coefficients can be calculated: c xy equals F x0 Subtract F x Then divide by ωY; c yy equals F y0 Subtract F y Then divide by ωY.

[0084] Transient Calculation and Mesh Control: Performing transient fluid-structure interaction calculations. The key is defining the motion of the inner wall boundary of the air film using the user-defined function (UDF) macro DEFINE_CG_MOTION provided by ANSYS Fluent. This UDF gives the boundary a vibrational velocity along the x or y direction. Rotational motion is achieved by the moving wall surface of the boundary conditions in the ANSYS FLUENT transient flow field. This method directly specifies the boundary motion velocity, rather than relying on mesh deformation, thus completely avoiding the problem of dynamic mesh distortion.

[0085] Implementation examples and result verification: Taking a rotational speed n=20000 r / min as an example, the above process is implemented. The trends of the four damping coefficients calculated by this method with rotational speed are as follows: Figure 10 As shown. The results indicate that the principal damping coefficient c xx and c yy The cross-damping coefficient c is positive and decreases with increasing rotational speed. xy and c yxIt is a negative value, and its absolute value decreases as the rotational speed increases.

[0086] Step 4 has the following advantages: 1) High computational efficiency: Only two target-specific transient calculations are required, avoiding lengthy random vibration or complex eddy simulations; 2) Direct results: By accurately selecting the physical moment when the displacement is zero, the damping force and stiffness force are naturally decoupled, without the need for complex parameter identification algorithms; 3) Strong robustness: The innovative UDF motion control method fundamentally solves the problem of mesh distortion under composite motion, ensuring the stability and success rate of the calculation.

[0087] Step 5: Experimental verification and analysis.

[0088] To verify the correctness and engineering applicability of the semi-analytical method for the stiffness and damping coefficient of the foil gas hydrodynamic bearing involved in this embodiment, a dynamic characteristic experimental platform for foil bearings was built for comparative testing. Since directly measuring the damping coefficient is extremely difficult experimentally, this verification mainly focuses on the stiffness coefficient. For ease of assembly and measurement, the bearing radius clearance C was adjusted to 69 μm. The test speed n was selected as 10000 r / min. Due to the large bearing clearance in the experiment, the generated gas film force is small; therefore, the applied static load W was set to 1 N. Table 2 shows the experimental fitting values ​​and errors of the stiffness coefficient.

[0089] Table 2. Experimental fitting values ​​and errors of stiffness coefficients .

[0090] Experimental results show that the two principal stiffness coefficients k xx and k yy The calculated values ​​agree very well with the experimental values, and the relative errors are all within the acceptable range for engineering applications. This fully demonstrates that the method for determining the static equilibrium position based on single-parameter search proposed in this invention is accurate and effective; and the method for calculating the stiffness coefficient based on first-order steady-state disturbance and analytical formula is reliable, with calculation results possessing practical physical meaning and engineering accuracy.

[0091] Although the experiment could not fully verify all eight coefficients, the successful verification of the stiffness coefficient, as the most important dynamic characteristic parameter, provides strong indirect support for the overall semi-analytical method of this invention (including the calculation of the damping coefficient). The semi-analytical method proposed in this invention has high calculation accuracy and can efficiently and reliably obtain all eight stiffness and damping coefficients of the foil gas hydrodynamic bearing, providing key inputs for the dynamic design and stability analysis of the rotor system.

[0092] Through the specific implementation process described above, the semi-analytical method for calculating the stiffness and damping coefficients of foil gas dynamic bearings of this invention is fully demonstrated, from model establishment, parameter determination, coefficient calculation to experimental verification. The results show that this method effectively overcomes the three major shortcomings of traditional methods pointed out in the background art: low computational efficiency, complex process, and susceptibility to mesh problems. It achieves a balance between high precision, high efficiency, and high robustness. The calculated dynamic coefficients can directly serve the dynamic design, stability analysis, and life prediction of rotor systems, and have significant engineering application value.

Claims

1. A semi-analytical method for the stiffness and damping coefficient of a foil gas hydrodynamic bearing, characterized in that, Includes the following steps: S1. Construct a fluid-structure interaction calculation model to calculate and obtain the static equilibrium position of the journal under a given rotational speed and static load. S2, apply an eccentric disturbance at the static equilibrium position, perform an additional steady-state fluid-structure interaction calculation, and calculate the four stiffness coefficients of the foil bearing using the first set of analytical formulas. S3. At the static equilibrium position and rotational speed, small-amplitude simple harmonic vibrations along the x and y directions are applied to the journal. Two transient fluid-structure interaction calculations are performed, and the four damping coefficients of the foil bearing are calculated by the second set of analytical formulas.

2. The semi-analytical method for determining the stiffness and damping coefficient of a foil gas dynamic bearing according to claim 1, characterized in that, The construction of the fluid-structure interaction calculation model includes: establishing a fluid-structure interaction calculation model based on the physical structure of the foil gas dynamic bearing, which includes the journal, the gas film fluid domain, and the foil structure.

3. The semi-analytical method for determining the stiffness and damping coefficient of a foil gas dynamic bearing according to claim 1 or 2, characterized in that, The calculation and determination of the static equilibrium position of the journal includes: adjusting the journal eccentricity by a single parameter under a given rotational speed and static load, and performing a series of steady-state fluid-structure interaction calculations until the air film force and the external load are balanced, and finally determining the static equilibrium position of the journal.

4. The semi-analytical method for determining the stiffness and damping coefficient of a foil gas dynamic bearing according to claim 1 or 2, characterized in that, Step S3 includes: applying a preset eccentricity disturbance at the static equilibrium position, performing an additional steady-state fluid-structure interaction calculation, and directly calculating the four stiffness coefficients of the foil bearing based on the air film force components and geometric relationships before and after the disturbance through the first set of analytical formulas.

5. The semi-analytical method for determining the stiffness and damping coefficient of a foil gas hydrodynamic bearing according to claim 1 or 2, characterized in that, Step S4 includes: applying simple harmonic motion along the x and y directions to the journal at the static equilibrium position and rotational speed, performing two transient fluid-structure interaction calculations, controlling the composite motion of the air film boundary through a user-defined function, and directly calculating the four damping coefficients of the foil bearing through the second analytical formula set based on the relationship between the air film force increment and velocity when the journal passes through the equilibrium position.

6. The semi-analytical method for determining the stiffness and damping coefficient of a foil gas dynamic bearing according to claim 4, characterized in that, The first set of analytical formulas is obtained by constructing two equivalent disturbance modes with constant static load in the horizontal direction and constant static load in the vertical direction and solving the equations. The four stiffness coefficients are all equal to the ratio of the force increment to the displacement disturbance combination. The denominator is the Jacobian determinant formed by the displacement disturbance components of the two disturbance modes, and the numerator is the difference between the disturbed force and the static equilibrium force.

7. The semi-analytical method for determining the stiffness and damping coefficient of a foil gas dynamic bearing according to claim 5, characterized in that, The four damping coefficients in the second set of analytical formulas are all ratios of the damping force component to the maximum velocity in that direction. The damping force component is the difference between the static equilibrium air film force and the instantaneous actual air film force, and the maximum velocity is the product of the vibration angular frequency and the amplitude in that direction.

8. The semi-analytical method for determining the stiffness and damping coefficient of a foil gas dynamic bearing according to claim 5, characterized in that, The preset eccentricity disturbance Δd satisfies that the ratio of Δd to C is less than 0.05, where C is the bearing radius clearance.

9. The semi-analytical method for determining the stiffness and damping coefficient of a foil gas dynamic bearing according to claim 5, characterized in that, The amplitude A of the simple harmonic motion satisfies that the ratio of A to C is less than 0.01, where C is the bearing radius clearance, and the vibration angular frequency ω of the simple harmonic motion is equal to the rotational angular velocity Ω of the journal.

10. A semi-analytical system for determining the stiffness and damping coefficient of a foil gas hydrodynamic bearing, applicable to the semi-analytical method for determining the stiffness and damping coefficient of a foil gas hydrodynamic bearing according to any one of claims 1-9, characterized in that, include The preprocessing and modeling module constructs a fluid-structure interaction computational model; The steady-state solution and equilibrium search module determines the static equilibrium position through single-parameter scanning; The stiffness coefficient analytical calculation module calculates the stiffness coefficient using the first set of analytical formulas. The transient solution and motion control module calculates the damping coefficient using the second set of analytical formulas. The output module outputs and stores the static equilibrium position, stiffness coefficient, and damping coefficient.

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