A method for calculating dynamic stiffness and damping coefficient of water-lubricated bearing

Abstract

The application discloses a kind of water-lubricated bearing dynamic stiffness and damping coefficient calculation method, comprising the following steps: first, based on the contact interface topography characteristics of water-lubricated bearing under mixed lubrication state, the calculation of water-lubricated bearing dynamic stiffness and damping coefficient is divided into water film bearing area dynamic stiffness and damping coefficient, micro asperity contact area dynamic stiffness and damping coefficient two parts, then parallel equivalent water film bearing area dynamic stiffness and damping coefficient, micro asperity contact area dynamic stiffness and damping coefficient, obtain water-lubricated bearing overall dynamic contact stiffness and damping coefficient.The application can calculate the dynamic characteristics of water-lubricated bearing under any working condition, fill the blank of overall dynamic characteristics calculation under the existing water-lubricated bearing mixed lubrication state.

Description

Technical Field

[0001] This invention relates to the field of sliding bearing technology, and specifically to a method for calculating the dynamic stiffness and damping coefficient of a water-lubricated bearing. Background Technology

[0002] The dynamic characteristic coefficients of bearings, namely stiffness and damping coefficients, are important parameters for studying the vibration characteristics of water-lubricated bearings. These parameters are closely related to the contact lubrication morphology of water-lubricated bearings. The interfacial behavior of the friction pairs in water-lubricated bearings is complex, especially under low-speed, heavy-load conditions. Due to the low viscosity of water, hydrodynamic lubrication cannot be achieved, and the interface is in a mixed lubrication state, failing to form elastohydrodynamic lubrication. Furthermore, the stiffness and damping coefficients of the bearing are highly interface-dependent. The stiffness and damping coefficients can easily generate severe excitation in the bearing system, leading to abnormal stick-slip vibrations. Therefore, extensive theoretical calculations and experimental measurements of dynamic characteristics have been conducted both domestically and internationally. However, existing methods for calculating the stiffness and damping coefficients of water-lubricated bearings are mainly based on full-film lubrication results, neglecting the changes caused by micro-protrusion contact under mixed lubrication conditions, and thus failing to provide accurate dynamic parameters. Summary of the Invention

[0003] To address the above problems, this invention provides a method for calculating the dynamic stiffness and damping coefficient of water-lubricated bearings.

[0004] This invention employs the following technical solution: a method for calculating the dynamic stiffness and damping coefficient of a water-lubricated bearing.

[0005] Step 1: Based on the contact interface morphology characteristics of water-lubricated bearings under mixed lubrication conditions, calculate the dynamic stiffness K1 and damping coefficient D1 of the water film bearing area, and the dynamic stiffness K2 and damping coefficient D2 of the micro-protrusion contact area.

[0006] Step 2: Based on the average Reynolds equation, contact model, and film thickness equation, obtain the static average Reynolds equation and the dynamic average Reynolds equation; according to the static average Reynolds equation, obtain the mixed lubrication parameters of the water-lubricated composite bearing; according to the dynamic average Reynolds equation, obtain the partial derivatives of the dynamic water film pressure and film thickness, and then substitute them into the static Reynolds equation, combined with the film thickness equation and elastic deformation equation, to obtain the dynamic stiffness K1 and damping coefficient D1 of the water film bearing area;

[0007] Step 3: Construct the normal contact model and tangential contact model of the micro-convex body; Based on the normal contact model and tangential contact model of the micro-convex body, combined with the eccentricity and offset angle parameters of the water-lubricated bearing, obtain the dynamic stiffness K2 and damping coefficient D2 of the micro-convex body contact area;

[0008] Step 4: Connect the dynamic stiffness K1 and damping coefficient D1 of the water film bearing area described in Step 2 and the dynamic stiffness K2 and damping coefficient D2 of the micro-protrusion contact area described in Step 3 in parallel to obtain the overall dynamic contact stiffness K and damping coefficient D of the water-lubricated bearing.

[0009] Furthermore, in step 2:

[0010] The dynamic stiffness K1 and damping coefficient D1 of the water film bearing area are:

[0011]

[0012]

[0013] The stiffness coefficient is represented by K, and the damping coefficient is represented by D. The first subscript indicates the direction of force change, and the second subscript indicates the direction of displacement change.

[0014] Furthermore, the specific steps for calculating the dynamic stiffness K2 and damping coefficient D2 of the micro-convex contact area in step 3 are as follows:

[0015] Based on equivalent viscous damping, the normal dynamic damping coefficient of the micro-convex body contact area is obtained according to the following formula:

[0016]

[0017] ω0 is the angular frequency of the simple harmonic excitation force, E N Let X be the total energy consumption in the normal direction. Nm Y Nm These are the normal and tangential displacement amplitudes;

[0018] The normal dynamic contact stiffness of the micro-convex contact area is:

[0019]

[0020] tanβ N ,tanγ N This refers to the dynamic damping coefficient;

[0021] Similarly, the tangential dynamic contact stiffness K of the micro-convex body contact area is obtained according to the calculation method of the normal contact model. τN ,K ττ and damping coefficient D τN D ττ ;

[0022]

[0023] ω0 is the angular frequency of the simple harmonic excitation force, E τ For the total tangential energy consumption, X τm Y τm These are the normal and tangential displacement amplitudes;

[0024] The normal dynamic contact stiffness of the micro-convex contact area is:

[0025]

[0026] ω0 is the angular frequency of the simple harmonic excitation force, tanβ τ ,tanγ τ This is the tangential loss coefficient;

[0027] By combining the dynamic contact stiffness and damping coefficient of the normal contact area and the dynamic contact stiffness and damping coefficient of the tangential contact area, the dynamic stiffness K2 and damping coefficient D2 of the micro-convex body contact area are obtained.

[0028]

[0029] The stiffness coefficient is represented by K, and the damping coefficient is represented by D. The first subscript indicates the direction of force change, and the second subscript indicates the direction of displacement change.

[0030] Furthermore, in step 4, the overall dynamic contact stiffness K and damping coefficient D of the lubricated bearing are the sum of the water film region and the micro-protrusion contact region;

[0031]

[0032] This invention offers the following advantages: Addressing the shortcomings of existing research, which only employs dry friction contact models or single-film lubrication steady-state analyses and neglects the interaction between the water film and micro-asperities, this invention proposes a novel method for calculating the stiffness and damping coefficients of water-lubricated bearings that simultaneously considers water film load-bearing and micro-asperity off-center contact. This allows for the calculation of more accurate dynamic parameters, providing a guarantee for constructing dynamic models of water-lubricated bearing systems and offering theoretical support for improving the vibration reduction and noise reduction design of water-lubricated bearing systems. Attached Figure Description

[0033] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings of the embodiments will be briefly described below. Obviously, the drawings described below only relate to some embodiments of the present invention and are not intended to limit the present invention.

[0034] Figure 1 This is a flowchart of the calculation process of the present invention;

[0035] Figure 2 A simplified diagram of a water-lubricated bearing;

[0036] Figure 3 This is a schematic diagram of the interface morphology under mixed lubrication conditions;

[0037] Figure 4 A schematic diagram of the off-center contact of a single micro-protrusion;

[0038] Figure 5This is a parallel equivalent model of the stiffness / damping coefficients of the water film bearing area and the micro-protrusion contact area. Detailed Implementation

[0039] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the described embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0040] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0041] like Figure 1 As shown, a method for calculating the dynamic stiffness and damping coefficient of a water-lubricated bearing includes the following steps:

[0042] Step 1: Based on the contact interface morphology characteristics of water-lubricated bearings under mixed lubrication conditions, the calculation of dynamic stiffness and damping coefficient of water-lubricated bearings is divided into two parts: dynamic stiffness / damping of the water film bearing area and dynamic stiffness / damping coefficient of the micro-protrusion contact area.

[0043] Step 2: Based on the average Reynolds equation, contact model, and film thickness equation, the static and dynamic average Reynolds equations are derived using the small perturbation method. The contact model is used to calculate the contact pressure, and the film thickness equation is used to calculate the water film thickness. By incorporating structural and operating parameters, the mixed lubrication contact parameters of the water-lubricated composite bearing are calculated, and the static solution of the water-lubricated bearing is obtained. Furthermore, the partial derivatives of the dynamic water film pressure and film thickness are obtained based on the dynamic average Reynolds equation. The real and imaginary parts of the equations are solved separately and then integrated to obtain the dynamic stiffness K1 and damping coefficient D1 of the water film.

[0044] The average Reynolds equation is:

[0045]

[0046] R is the bearing radius, p is the water film pressure, θ is the circumferential angle, z is the axial coordinate, t is the time term, h is the film thickness, and φ is the bearing radius. s For shear flow factor, φ θ φ z η is the pressure-flow factor, η is the viscosity coefficient, U1 and U2 are the motion velocities of the two surfaces, σ is the overall surface roughness, and h is the viscosity coefficient. T The average film thickness;

[0047] The most important characteristic of hybrid lubrication is micro-forehead contact. According to the LEE-REN model, the average film thickness h in the contact model... T and contact pressure The relationship is:

[0048]

[0049] Actual contact area S r and nominal contact area S nom The relationship is:

[0050]

[0051] Where γ is the surface texture parameter, H Y It is a parameter for the surface hardness of the material. It is the average contact pressure, [G] i ]、[A i The matrix is ​​the parameter matrix obtained from the LEE-REN model.

[0052] like Figure 2 As shown, the film thickness equation h considering the elastic deformation of the composite bearing can be expressed by the following formula:

[0053] h(θ,z,t)=h g (θ,z,t)+v(θ,z,t) (4);

[0054] h g Here, v is the geometric equation, and v is the elastic deformation equation.

[0055] Geometric equation h of water-lubricated composite bearing g It can be represented as:

[0056]

[0057] C is the bearing clearance, ε is the eccentricity, and ψ0 is the static balance misalignment angle; for example... Figure 2 As shown, the groove region of the bearing consists of three circular arcs, Δh ij It is a function of the radius of the arc in the groove region R2, R3, R4 and θ;

[0058] The total elastic deformation v can be expressed as:

[0059]

[0060] Where ξ is the circumferential displacement, ζ is the axial displacement, and t is time;

[0061] Calculation of dynamic stiffness and damping coefficient of the water film bearing zone:

[0062] The dynamic center position of the journal is determined by the eccentricity ε and the offset angle ψ. The position of the journal at any given time is:

[0063]

[0064] E, Ψ periodic small perturbation, E0 perturbation eccentricity amplitude, Ψ0 perturbation offset angle amplitude, ε0 and ψ0 represent the journal position at static equilibrium;

[0065] Using the small perturbation method, let the dynamic water film pressure and water film thickness be:

[0066] P = P0 + Q0e iT (8);

[0067]

[0068] Where P0 is the static water film pressure, Q0 is the dynamic water film pressure, and H0 is the static film thickness. For dynamic film thickness;

[0069] Substituting into the mean Reynolds equation, we obtain the static mean Reynolds equation and the dynamic mean Reynolds equation, respectively.

[0070] The static average Reynolds equation is

[0071]

[0072] φ θ φ z For pressure-flow factor, φ c Where D is the contact factor, R is the inner diameter, P0 is the bearing radius, H0 is the static water film pressure, L is the bearing length, θ is the circumferential angle, z is the axial coordinate, and p is the static water film pressure. s Where C is the water supply pressure, C is the bearing clearance, and ω is the bearing rotation speed;

[0073] The dynamic average Reynolds equation is

[0074]

[0075] Q0 is the dynamic film thickness, and Q0 is the dynamic water film pressure; ω is the bearing rotation speed. j For the dimensionless parameter of the time term;

[0076] Increment of water film thickness during dynamic conditions:

[0077]

[0078] H gd0 The dynamic geometric clearance increment caused by journal disturbance:

[0079] H gd0 =E0cos(θ-ψ0)+ε0Ψ0sin(θ-ψ0) (13);

[0080] V d0Dynamic elastic deformation of composite bearing bushes caused by dynamic force disturbance:

[0081]

[0082] ξ is the circumferential displacement, ζ is the axial displacement, and t is time;

[0083] The dynamic average Reynolds equation includes disturbances E0 and Ψ0. To obtain the water film stiffness and damping of the water-lubricated composite alloy bearing, let...

[0084]

[0085] Q E Q is the partial derivative of the dynamic water film pressure with respect to the disturbance eccentricity. θ H is the partial derivative of the dynamic water film pressure with respect to the disturbance offset angle. E H is the partial derivative of the dynamic film thickness with respect to the perturbation eccentricity. θ The partial derivative of the dynamic film thickness with respect to the disturbance offset angle;

[0086] Taking the partial derivative of the dynamic average Reynolds equation (11) with respect to E0 and substituting it into the static equation, we get:

[0087]

[0088]

[0089] H E The partial derivative of the dynamic film thickness with respect to the perturbation eccentricity is given by θ, where θ is the circumferential angle and z is the axial coordinate; D is the inner diameter, and p... s Where C is the water supply pressure, E' is the bearing clearance, ξ is the equivalent elastic modulus, and ζ is the circumferential and axial displacement, respectively.

[0090] Solving equations (15) and (16) simultaneously, we can obtain Q. E .

[0091] Similarly, taking the partial derivative of the dynamic Reynolds equation (11) with respect to Ψ0, we get Q. θ .

[0092] Furthermore, Q E and Q θ Substituting these values ​​into the following formula, we can obtain the stiffness and damping coefficient of the water film bearing area:

[0093]

[0094]

[0095] The stiffness coefficient is represented by K, and the damping coefficient is represented by D. The first subscript indicates the direction of force change, and the second subscript indicates the direction of displacement change. and For dimensionless dynamic stiffness and damping coefficient;

[0096] The dimensionless dynamic stiffness and damping coefficient are respectively:

[0097]

[0098]

[0099] Obtain the dynamic stiffness K1 and damping coefficient D1 of the water film:

[0100]

[0101]

[0102] Step 3: The dynamic stiffness and damping coefficient of the micro-forehead contact area calculated in Step 1 are based on the elastoplastic deformation and adhesive slip behavior of the contact interface. By introducing off-center contact and contact angle, the normal / tangential contact model of the micro-forehead is obtained. Based on the normal / tangential contact model of the micro-forehead, combined with the Gauss probability density function and parameters such as contact load and contact area in the mixed lubrication analysis, the calculation model of dynamic stiffness K2 and damping coefficient D2 of the micro-forehead contact area is obtained according to the equivalent viscous damping theory.

[0103] Furthermore, the normal load is analyzed. First, a force analysis is performed on the micro-protrusion with offset contact, decomposing the normal contact force on the micro-protrusion into normal and tangential components. The dynamic load can be expressed as two parts: stable and dynamic: F = F0 + f n (sinω0t) When the dynamic part of ω0t is in the interval -π / 2≤ω0t≤π / 2, the micro-convex body is in the loading state; when the dynamic part of ω0t is in the interval π / 2≤ω0t≤π3 / 2, the micro-convex body is in the unloading state.

[0104] Furthermore, based on Hertz contact theory, KKE model, etc., the normal deformation state of micro-protrusions in the elastic, elastoplastic and plastic stages is analyzed respectively. Considering the changes in contact interface morphology and material with the friction process, expressions for normal force and probabilistic contact area in different stages are obtained.

[0105] Loading phase:

[0106]

[0107] δ in S is the amount of deformation of a micro-convex body under a normal load in the normal direction. m It is the maximum contact pressure factor.

[0108] Uninstallation phase:

[0109]

[0110]

[0111] δ inr δ inm It represents the residual deformation and maximum deformation of a micro-convex body under normal load in the normal direction, n. F ,n S These are the indices of unloading force and unloading area, respectively.

[0112] According to the KKE model, the strain energy consumption of a micro-convex body during one loading and unloading cycle is:

[0113]

[0114] The strain energy consumption of a micro-convex body under normal stress at different stages for one cycle of loading and unloading was obtained:

[0115]

[0116] Furthermore, based on the CM adhesive slip theory and EPB theory, the tangential component force and frictional energy consumption of the micro-convex body are calculated. The relationship between the loading / unloading forces and the displacement is as follows:

[0117]

[0118]

[0119] δ iτ δ iτm It represents the deformation and maximum deformation of a micro-convex body under normal load in the tangential direction.

[0120] The surface friction coefficient of the micro-protrusion contact surface can be obtained from the BKE model:

[0121]

[0122] The frictional energy consumption of the micro-protrusion during one loading and unloading cycle can be obtained:

[0123]

[0124] δ iτm It is δ iτ Maximum deformation;

[0125] By combining the normal and tangential components of the normal contact force, the resultant normal force and energy loss during the loading and unloading phases can be obtained:

[0126] Loading status:

[0127]

[0128] Uninstallation status:

[0129]

[0130] F iNun ,F iτun Normal load and normal / tangential unloading force;

[0131] Normal contact energy loss:

[0132] E i =E iN +E iτ (33);

[0133] Furthermore, the interface roughness is characterized based on the statistical theory of the Gaussian probability density function:

[0134]

[0135] σ0, σ z This represents the variance of the rough surface and the variance of the micro-convexity height.

[0136] Calculate the off-contact angle of the micro-convexity based on the Gorbatikh model:

[0137]

[0138] α represents the contact angle of the micro-protrusion, and n represents the number of contact cycles of the micro-protrusion.

[0139] The normal contact force across the entire interface, including all micro-protrusions, is obtained by combining the contact load and contact area of ​​the micro-protrusions.

[0140] Loading phase:

[0141] Elastic phase

[0142] Elastic-plastic stage

[0143]

[0144] The resultant force of the normal contact in the micro-convex contact area during the loading stage is:

[0145] F I =F e +F ep +F p (37);

[0146] Uninstallation phase:

[0147] Elastic phase

[0148] Elastic-plastic stage

[0149]

[0150] The resultant force of the normal contact in the micro-convex contact area during the unloading phase is:

[0151] F Iun =F eun +F epun +F pun (39);

[0152] Therefore, according to formula (21), the normal energy loss in the contact area of ​​the micro-convex body is:

[0153]

[0154] Based on the concept of equivalent viscous damping, the normal dynamic damping coefficient of the contact area of ​​the micro-convex body can be calculated using the following formula:

[0155]

[0156] X Nm Y Nm Indicates the magnitude of normal and tangential displacement.

[0157] The phase difference between the dynamic component and the deformation can be expressed as:

[0158]

[0159] F Nm This indicates the amplitude of the normal force in the contact area.

[0160] The normal dynamic contact stiffness of the micro-convex contact area is:

[0161]

[0162]

[0163] tanβ N ,tanγ N This represents the damping loss factor.

[0164] Further analysis of tangential contact is conducted. The analysis and calculation method are the same as in the normal section above, yielding the normal dynamic contact stiffness K of the micro-convex body contact area. τN ,K ττ and damping D τN D ττ .

[0165] Furthermore, the models of normal contact force and tangential contact force are combined, and... Figure 2 By analyzing parameters such as eccentricity and offset angle of water-lubricated composite bearings, the dynamic contact stiffness and damping coefficient of the entire interface micro-protrusion contact area can be obtained.

[0166]

[0167]

[0168] Step 4: Connect the dynamic stiffness K1 and damping coefficient D1 of the water film described in Step 2 and the dynamic stiffness K2 and damping coefficient D2 of the micro-protrusion contact area described in Step 3 in parallel to obtain the overall dynamic contact stiffness K and damping coefficient D of the water-lubricated bearing.

[0169] Adopting such Figure 5 Parallel equivalent model:

[0170] K = K1 + K2 (45);

[0171] D = D1 + D2

[0172] The calculated dynamic contact stiffness / damping coefficients of the water film bearing area and the micro-protrusion contact area are combined in parallel for equivalent calculation to obtain the overall dynamic contact stiffness / damping coefficient of the water-lubricated bearing.

[0173]

[0174]

[0175] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some modifications or alterations to the above-disclosed technical content to create equivalent embodiments without departing from the scope of the present invention. Any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention shall still fall within the scope of the present invention.

Claims

1. A method for calculating the dynamic stiffness and damping coefficient of a water-lubricated bearing, characterized in that, Includes the following steps: Step 1: Based on the contact interface morphology characteristics of water-lubricated bearings under mixed lubrication conditions, calculate the dynamic stiffness K1 and damping coefficient D1 of the water film bearing area, and the dynamic stiffness K2 and damping coefficient D2 of the micro-protrusion contact area. Step 2: Based on the average Reynolds equation, contact model, and film thickness equation, obtain the static average Reynolds equation and the dynamic average Reynolds equation; according to the static average Reynolds equation, obtain the mixed lubrication parameters of the water-lubricated composite bearing; according to the dynamic average Reynolds equation, obtain the partial derivatives of the dynamic water film pressure and film thickness, and then substitute them into the static average Reynolds equation, combined with the film thickness equation and elastic deformation equation, to obtain the dynamic stiffness K1 and damping coefficient D1 of the water film bearing area; Step 3: Construct the normal contact model and tangential contact model of the micro-convex body; Based on the normal contact model and tangential contact model of the micro-convex body, and combined with the eccentricity and offset angle parameters of the water-lubricated bearing, obtain the dynamic stiffness K2 and damping coefficient D2 of the micro-convex body contact area; The specific steps for calculating the dynamic stiffness K2 and damping coefficient D2 of the micro-protrusion contact area in step 3 are as follows: Based on equivalent viscous damping, the normal dynamic damping coefficient of the micro-convex body contact area is obtained according to the following formula: (3)； The angular frequency of the simple harmonic excitation force. For the total energy consumption in the normal direction, These are the normal and tangential displacement amplitudes; The normal dynamic contact stiffness of the micro-convex contact area is: （4）； This refers to the dynamic damping coefficient; Similarly, the tangential dynamic contact stiffness of the micro-convex body contact area is obtained based on the tangential contact model calculation method. and damping coefficient ; (5)； The angular frequency of the simple harmonic excitation force. For total tangential energy consumption, These are the normal and tangential displacement amplitudes; The tangential dynamic contact stiffness of the micro-convex contact area is: （6）； The angular frequency of the simple harmonic excitation force. This is the tangential loss coefficient; By combining the dynamic contact stiffness and damping coefficient of the normal contact area and the dynamic contact stiffness and damping coefficient of the tangential contact area, the dynamic stiffness K2 and damping coefficient D2 of the micro-convex body contact area are obtained. （7）； The stiffness coefficient is represented by K, and the damping coefficient is represented by D. The first subscript indicates the direction of force change, and the second subscript indicates the direction of displacement change. Step 4: Connect the dynamic stiffness K1 and damping coefficient D1 of the water film bearing area described in Step 2 and the dynamic stiffness K2 and damping coefficient D2 of the micro-protrusion contact area described in Step 3 in parallel to obtain the overall dynamic contact stiffness K and damping coefficient D of the water-lubricated bearing.

2. The method for calculating the dynamic stiffness and damping coefficient of a water-lubricated bearing according to claim 1, characterized in that, In step 2: The dynamic stiffness K1 and damping coefficient D1 of the water film bearing area are: (1) (2)， The stiffness coefficient is represented by K, and the damping coefficient is represented by D. The first subscript indicates the direction of force change, and the second subscript indicates the direction of displacement change.

3. The method for calculating the dynamic stiffness and damping coefficient of a water-lubricated bearing according to claim 1, characterized in that, In step 4, the overall dynamic contact stiffness K and damping coefficient D of the water-lubricated bearing are the sum of the water film region and the micro-protrusion contact region. （8）。

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