A method for modeling and analyzing vibration characteristics of rolling bearings
By establishing a rolling bearing dynamic model that considers cage flexibility and elastohydrodynamic lubrication, and combining it with time-varying displacement excitation for local faults, the problem of not being able to realistically simulate bearing operating conditions and identify excitation sources in existing technologies has been solved, enabling more refined vibration analysis and fault diagnosis.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- SUZHOU UNIV
- Filing Date
- 2023-03-21
- Publication Date
- 2026-05-19
AI Technical Summary
Existing dynamic analysis methods for rolling bearings fail to fully consider various factors such as lubrication, resulting in an inability to realistically simulate bearing operating conditions and failing to identify the main excitation sources in the dynamic model, thus affecting the accuracy of fault diagnosis.
A dynamic model of a rolling bearing considering cage flexibility, elastohydrodynamic lubrication, and time-varying displacement excitation during local faults is established. The time-varying displacement excitation during faults is described by a half-sine function, and the main excitation sources are identified by comparing the values and trends of the basic physical quantities in the dynamic model.
It achieves more refined and accurate bearing vibration simulation, provides a theoretical basis for fault diagnosis, and can simulate vibration response under different fault types and sizes, providing data support for intelligent fault diagnosis.
Smart Images
Figure CN116561904B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of health status assessment and fault diagnosis technology for mechanical equipment, specifically a method for dynamic modeling and vibration characteristic analysis of rolling bearings. Background Technology
[0002] Rolling bearings are widely used in rotating machinery, playing a vital role in supporting and transmitting power. However, they often operate in harsh environments and are prone to failure. Failure to address these failures promptly can lead to major accidents. Bearing performance directly affects the operational accuracy and service life of the entire equipment; therefore, bearing fault diagnosis is crucial. Dynamic modeling and vibration analysis of rolling bearings with localized defects provide a theoretical basis for analyzing fault causes and revealing the intrinsic relationship between system dynamic parameters and response signals during fault conditions. However, current dynamic analyses of rolling bearings often focus only on the influence of single factors such as lubrication on vibration response, lacking consideration of numerous factors during operation. Therefore, existing methods cannot realistically simulate the actual operating conditions of bearings, leaving significant room for improvement. Furthermore, current methods fail to identify the main excitation sources in the dynamic model, resulting in a lack of theoretical foundation for the vibration analysis of faulty bearings. Summary of the Invention
[0003] The purpose of this invention is to provide a method for dynamic modeling and vibration characteristic analysis of rolling bearings to solve the problems in the prior art.
[0004] To achieve the above objectives, the present invention provides the following technical solution: a method for dynamic modeling and vibration characteristic analysis of rolling bearings, comprising the following steps:
[0005] S1: Establish a healthy bearing dynamics model: The stiffness and damping of the bearing, the force between the rolling elements and the cage, and the force between the rolling elements and the raceway were calculated when the bearing was under elastohydrodynamic lubrication, and the basic physical quantities required in the healthy bearing dynamics model were determined.
[0006] S2: Establishing a bearing dynamics model with local faults: By introducing a half-sine function, the time-varying displacement excitation of the rolling element when passing through a local fault is described, and finally a bearing dynamics model with local faults is established.
[0007] S3: Identify the main excitation sources in a dynamic model with local faults: By comparing the numerical magnitudes and trends of the basic physical quantities in the dynamic model, the main excitation sources in the model are determined.
[0008] Preferably, S1.1: Calculate the bearing stiffness and damping based on elastohydrodynamic lubrication;
[0009] In the established dynamic model, the oil film in the Hertz contact area is hardened, and its oil film stiffness is much greater than the Hertz contact stiffness of the contact pair. Therefore, the oil film stiffness in the Hertz contact area is ignored.
[0010] The damping of the contact pair is composed of the structural damping of the bearing and the oil film damping of the Hertz region in series, and then in parallel with the viscous damping of the inlet region. Because the viscous damping of the oil film in the Hertz contact region is relatively small, the damping is negligible after being connected in series with the structural damping of the Hertz contact region. Therefore, the damping of the contact pair comes from the viscous damping of the oil film in the inlet region.
[0011] The contact stiffness of the contact pair can be expressed as:
[0012]
[0013] In the formula, E * Σρ is Young's modulus; μ is Poisson's ratio; Σρ is the raceway curvature; F is the first type of elliptic integral; E is the second type of elliptic integral; κ is the contact region ellipticity.
[0014] The viscous damping of the oil film in the inlet region can be expressed as:
[0015]
[0016] In the formula, η0 is the dynamic viscosity of the lubricant at room temperature; R x a is the equivalent radius of curvature along the rolling direction; nj The length of the semi-major axis of the contact ellipse;
[0017] The viscosity given in the ISO standard is kinematic viscosity, which can be converted to dynamic viscosity using the following formula:
[0018] η0=υ·ρ·10 -6 (Pa·s) (3)
[0019] For rolling bearings with elliptical point contact, the minimum oil film thickness and the oil film thickness at the center of the contact area are calculated using the Hamrock-Dowson film thickness formula:
[0020] h0 = 2.69R x U 0.67 G 0.53 W -0.067 (1-0.61e- 0.73κ (4)
[0021] In the formula, U, G and W are the dimensionless velocity parameter, dimensionless material parameter and dimensionless load parameter, respectively;
[0022] S1.2: Calculate the force between the rolling elements and the cage;
[0023] During operation, the rolling elements rotate around the bearing center; in the non-load-bearing area, the cage drives the rolling elements; in the load-bearing area, the rolling elements drive the cage; the contact stiffness between the rolling elements and the cage is calculated using Hertzian contact theory; due to the complexity of the cage structure, the stiffness between the cages is calculated using the finite element method.
[0024] The force between the j-th rolling element and the cage can be expressed as:
[0025]
[0026]
[0027] Where, ψ r and ψ c These are the rotational angular displacements of the rolling elements and the cage around the bearing center, respectively. n It is the load deformation index, which is 1 within a small range of elastic deformation; K rc It is the connection stiffness between the rolling elements and the cage; R m It is the diameter of the joint;
[0028] The friction between the j-th rolling element and the cage can be expressed as:
[0029]
[0030]
[0031] Where, μ c It is the coefficient of friction;
[0032] The internal forces between the cages are expressed as:
[0033]
[0034]
[0035]
[0036] In the formula, K cc and C CC These are the connection stiffness and connection damping between the cages, respectively; and the angular velocity of the cage rotating around the bearing center.
[0037] S1.3: Calculate the force between the rolling element and the raceway; based on Hertz's nonlinear contact theory, the contact force between the rolling element and the inner / outer raceway is expressed as:
[0038]
[0039]
[0040] In the formula, λ is a symbolic function, expressed as:
[0041]
[0042] The contact deformation between the j-th rolling element and the raceway is expressed as:
[0043]
[0044]
[0045] In the formula, and r j These represent the radial displacements of the inner ring, outer ring, and rolling element at the j-th rolling element, respectively; e represents the radial clearance. It is the displacement excitation caused by the j-th rolling element entering the defect;
[0046] The radial displacements of the inner and outer rings at the j-th rolling element are expressed as:
[0047]
[0048]
[0049] The angular position of the j-th rolling element is represented as:
[0050]
[0051] In the formula, N b Represents the number of rolling elements;
[0052] The frictional force between the j-th rolling element and the raceway is expressed as:
[0053]
[0054]
[0055] The coefficient of friction between the rolling elements and the raceway is calculated using the following formula:
[0056]
[0057] In the formula, The relative sliding velocity between the j-th rolling element and the raceway is obtained through the following calculation:
[0058] ΔV i j =V ir -V ri (twenty three)
[0059]
[0060]
[0061] In the formula, and R represents the circumferential velocity and rotational velocity of the j-th rolling element, respectively. r This represents the radius of the rolling element.
[0062] Preferably, S2 includes 2.1: describing a local fault in the bearing raceway; for early-stage bearing defects, the fault width is small, the displacement of the rolling element as it passes the defect is smaller than the fault depth, and in the state where the rolling element passes the defect, L is the fault width, B is the fault depth, and H... max For the maximum displacement excitation;
[0063] When the rolling element enters and passes through the defect, it remains in contact with side I. When it passes side II, it has already left the defect area. This passage pattern uses a half-sine function to describe the time-varying excitation of the displacement. The maximum displacement excitation H of the rolling element... max for:
[0064]
[0065] The time-varying displacement excitation function of a rolling element passing over a defect can be represented by the following function:
[0066]
[0067]
[0068] In the formula, Represents the defect angle; The initial angle representing the defect; n = i, o; This can be expressed by the following formula:
[0069]
[0070] in, The corner position representing the defect;
[0071] Based on the above analysis, the dynamic equation of the bearing is as follows:
[0072] The dynamic equation of the inner circle horizontal motion:
[0073]
[0074] The dynamic equation of the vertical motion of the inner circle:
[0075]
[0076] The dynamic equation of the cage's circular motion:
[0077]
[0078] The dynamic equation of the circular motion of a rolling element:
[0079]
[0080] The dynamic equation of the rotational motion of the rolling element:
[0081]
[0082] The dynamic equation of the radial motion of the rolling element:
[0083]
[0084] In the formula, the centrifugal force of the j-th rolling element can be expressed as:
[0085] F wj =m r R m w oj 2 (36)
[0086] Preferably, step S3 includes the following steps:
[0087] S3.1: To verify the accuracy of the model, an experiment was conducted using a bearing fault simulation test bench to obtain experimental signals. Simultaneously, simulation was performed using MATLAB software, and the ode45 solver was used to obtain simulated signals. A qualitative comparison of the envelope spectrum characteristics between the experimental and simulated signals was conducted. The bearing used was an SKF6205-2RS, with grooves machined on both the inner and outer rings via wire cutting. The bearing was driven by a motor at a speed of 900 r / min, and the sampling frequency was set to 10 kHz. The acceleration of the bearing's outer ring in the Y direction was used as the original signal for the dynamic model. The envelope spectrum of the original signal was obtained through Fourier transform and Hilbert transform.
[0088] After verifying the accuracy of the established dynamic model through the above experiments, the changes of key physical quantities in the established dynamic model over a certain period of time were shown in the form of images. Finally, by analyzing and comparing the similarity of the physical quantities with the vibration response of the bearing in terms of numerical magnitude and trend, it was determined that the main excitation source in the model is the contact force between the rolling element and the raceway.
[0089] Compared with existing technologies, the advantages of this invention are as follows: Taking rolling bearings as the research object, a bearing dynamic model considering cage flexibility, elastohydrodynamic lubrication, and time-varying displacement excitation caused by local faults is established. This model is more refined and can more accurately simulate the vibration mechanism of bearings. By comparing the vibration response of the established dynamic model with experimental signals, the accuracy of the model is verified. Through comparative analysis, the excitation source in the model is determined, providing a theoretical basis for bearing vibration response analysis. Simultaneously, the established fault model can simulate the vibration response of bearings under different fault types and sizes, providing a data source for intelligent fault diagnosis of rolling bearings based on big data. Attached Figure Description
[0090] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings:
[0091] Figure 1 This refers to the contact model in the bearing dynamics model;
[0092] Figure 2 A schematic diagram of the force analysis for the cage;
[0093] Figure 3 This is a diagram showing the state of the rolling element after passing through a defect.
[0094] Figure 4 This shows the changes in the main physical quantities in the dynamic model when the fault is located in the inner ring of the bearing.
[0095] Figure 5 This shows the changes in the main physical quantities in the dynamic model when the fault is located in the outer ring of the bearing.
[0096] Figure 6 Vibration response and some physical quantities when the fault is located in the inner ring of the bearing. The changes;
[0097] Figure 7 Vibration response and some physical quantities when the fault is located in the outer ring of the bearing. The changes;
[0098] Figure 8 For bearing dynamics model;
[0099] Figure 9 This is a comparison of the envelope spectra of simulated and experimental signals under different fault conditions. Detailed Implementation
[0100] 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 a part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to represent selected embodiments of the invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0101] Please see Figure 1-9 In this embodiment of the invention, a method for dynamic modeling and vibration characteristic analysis of rolling bearings includes the following steps:
[0102] S1: Establish a healthy bearing dynamics model: The stiffness and damping of the bearing, the force between the rolling elements and the cage, and the force between the rolling elements and the raceway were calculated when the bearing was under elastohydrodynamic lubrication, and the basic physical quantities required in the healthy bearing dynamics model were determined.
[0103] S2: Establishing a bearing dynamics model with local faults: By introducing a half-sine function, the time-varying displacement excitation of the rolling element when passing through a local fault is described, and finally a bearing dynamics model with local faults is established.
[0104] S3: Identify the main excitation sources in a dynamic model with local faults: By comparing the numerical magnitudes and trends of the basic physical quantities in the dynamic model, the main excitation sources in the model are determined.
[0105] Preferably, S1.1: Calculate the bearing stiffness and damping based on elastohydrodynamic lubrication;
[0106] In the established dynamic model, the oil film in the Hertz contact area is hardened, and its oil film stiffness is much greater than the Hertz contact stiffness of the contact pair. Therefore, the oil film stiffness in the Hertz contact area is ignored.
[0107] The damping of the contact pair is composed of the structural damping of the bearing and the oil film damping of the Hertz region in series, and then in parallel with the viscous damping of the inlet region. Because the viscous damping of the oil film in the Hertz contact region is relatively small, the damping is negligible after being connected in series with the structural damping of the Hertz contact region. Therefore, the damping of the contact pair comes from the viscous damping of the oil film in the inlet region.
[0108] The contact stiffness of the contact pair can be expressed as:
[0109]
[0110] In the formula, E * Σρ is Young's modulus; μ is Poisson's ratio; Σρ is the raceway curvature; F is the first type of elliptic integral; E is the second type of elliptic integral; κ is the contact region ellipticity.
[0111] The viscous damping of the oil film in the inlet region can be expressed as:
[0112]
[0113] In the formula, η0 is the dynamic viscosity of the lubricant at room temperature; R x a is the equivalent radius of curvature along the rolling direction; nj The length of the semi-major axis of the contact ellipse;
[0114] The viscosity given in the ISO standard is kinematic viscosity, which can be converted to dynamic viscosity using the following formula:
[0115] η0=υ·ρ·10 -6 (Pa·s) (3)
[0116] For rolling bearings with elliptical point contact, the minimum oil film thickness and the oil film thickness at the center of the contact area are calculated using the Hamrock-Dowson film thickness formula:
[0117] h0 = 2.69R x U 0.67 G 0.53 W -0.067 (1-0.61e -0.73κ (4)
[0118] In the formula, U, G and W are the dimensionless velocity parameter, dimensionless material parameter and dimensionless load parameter, respectively;
[0119] S1.2: Calculate the force between the rolling elements and the cage;
[0120] During operation, the rolling elements rotate around the bearing center; in the non-load-bearing area, the cage drives the rolling elements; in the load-bearing area, the rolling elements drive the cage; the contact stiffness between the rolling elements and the cage is calculated using Hertzian contact theory; due to the complexity of the cage structure, the stiffness between the cages is calculated using the finite element method.
[0121] The force between the j-th rolling element and the cage can be expressed as:
[0122]
[0123]
[0124] Where, ψr and ψ c These represent the rotational angular displacements of the rolling elements and cage around the bearing center, respectively; n is the load deformation exponent, which is 1 within a small range of elastic deformation; K rc It is the connection stiffness between the rolling elements and the cage; R m It is the diameter of the joint;
[0125] The friction between the j-th rolling element and the cage can be expressed as:
[0126]
[0127]
[0128] Where, μ c It is the coefficient of friction;
[0129] The internal forces between the cages are expressed as:
[0130]
[0131]
[0132]
[0133] In the formula, K cc and C CC These are the connection stiffness and connection damping between the cages, respectively; and the angular velocity of the cage rotating around the bearing center.
[0134] S1.3: Calculate the force between the rolling element and the raceway; based on Hertz's nonlinear contact theory, the contact force between the rolling element and the inner / outer raceway is expressed as:
[0135]
[0136]
[0137] In the formula, λ is a symbolic function, expressed as:
[0138]
[0139] The contact deformation between the j-th rolling element and the raceway is expressed as:
[0140]
[0141]
[0142] In the formula, r i j , and r jThese represent the radial displacements of the inner ring, outer ring, and rolling element at the j-th rolling element, respectively; e represents the radial clearance. It is the displacement excitation caused by the j-th rolling element entering the defect;
[0143] The radial displacements of the inner and outer rings at the j-th rolling element are expressed as:
[0144]
[0145]
[0146] The angular position of the j-th rolling element is represented as:
[0147]
[0148] In the formula, N b Represents the number of rolling elements;
[0149] The frictional force between the j-th rolling element and the raceway is expressed as:
[0150]
[0151]
[0152] The coefficient of friction between the rolling elements and the raceway is calculated using the following formula:
[0153]
[0154] In the formula, The relative sliding velocity between the j-th rolling element and the raceway is obtained through the following calculation:
[0155] ΔV i j =V ir -V ri (twenty three)
[0156]
[0157]
[0158] In the formula, and R represents the circumferential velocity and rotational velocity of the j-th rolling element, respectively. r This represents the radius of the rolling element.
[0159] Preferably, S2 includes 2.1: describing a local fault in the bearing raceway; for early-stage bearing defects, the fault width is small, the displacement of the rolling element as it passes the defect is smaller than the fault depth, and in the state where the rolling element passes the defect, L is the fault width, B is the fault depth, and H...max For the maximum displacement excitation;
[0160] When the rolling element enters and passes through the defect, it remains in contact with side I. When it passes side II, it has already left the defect area. This passage pattern uses a half-sine function to describe the time-varying excitation of the displacement. The maximum displacement excitation H of the rolling element... max for:
[0161]
[0162] The time-varying displacement excitation function of a rolling element passing over a defect can be represented by the following function:
[0163]
[0164]
[0165] In the formula, Represents the defect angle; The initial angle representing the defect; n = i, o; This can be expressed by the following formula:
[0166]
[0167] in, The corner position representing the defect;
[0168] Based on the above analysis, the dynamic equation of the bearing is as follows:
[0169] The dynamic equation of the inner circle horizontal motion:
[0170]
[0171] The dynamic equation of the vertical motion of the inner circle:
[0172]
[0173] The dynamic equation of the cage's circular motion:
[0174]
[0175] The dynamic equation of the circular motion of a rolling element:
[0176]
[0177] The dynamic equation of the rotational motion of the rolling element:
[0178]
[0179] The dynamic equation of the radial motion of the rolling element:
[0180]
[0181] In the formula, the centrifugal force of the j-th rolling element can be expressed as:
[0182] F wj =m r R m w oj 2 (36)
[0183] Preferably, step S3 includes the following steps:
[0184] S3.1: To verify the accuracy of the model, an experiment was conducted using a bearing fault simulation test bench to obtain experimental signals. Simultaneously, simulation was performed using MATLAB software, and the ode45 solver was used to obtain simulated signals. A qualitative comparison of the envelope spectrum characteristics between the experimental and simulated signals was conducted. The bearing used was an SKF6205-2RS, with grooves machined on both the inner and outer rings via wire cutting. The bearing was driven by a motor at a speed of 900 r / min, and the sampling frequency was set to 10 kHz. The acceleration of the bearing's outer ring in the Y direction was used as the original signal for the dynamic model. The envelope spectrum of the original signal was obtained through Fourier transform and Hilbert transform.
[0185] The changes in the main physical quantities in the dynamic model proposed in this invention are as follows: Figure 4 , Figure 5 As shown, the frictional force between the rolling element and the inner rolling element can be clearly seen from the amplitude. Friction between the rolling element and the outer rolling element Friction between the rolling element and the preceding cage Friction between the rolling element and the subsequent cage Much smaller than the contact force between the rolling element and the inner raceway Contact force between rolling elements and outer raceways Contact force between the rolling element and the preceding cage Contact force between the rolling element and the subsequent cage This indicates that, and It plays a dominant role in influencing bearing vibration response.
[0186] Figure 6 This demonstrates the vibration response and key physical quantities of the bearing inner ring when the fault is located within the inner ring. The changes in [the value]. From the perspective of numerical magnitude and the pattern of change, [it is related to...]. compared to, The vibration response of the inner ring has a significant impact. Therefore, it can be determined that the contact force between the rolling elements and the raceway is the main physical quantity affecting the bearing vibration response, which is the excitation source in the proposed dynamic model. When the fault is located in the outer ring of the bearing, the vibration response of the inner ring and the main physical quantities are... Changes such as Figure 7 As shown in the figure, the same conclusion can be drawn from the trend of change in the figure: the contact force between the rolling element and the raceway is the main physical quantity affecting the bearing vibration response, which is the excitation source in the proposed dynamic model.
[0187] The dynamic model of the bearing is as follows Figure 8 As shown in the figure. This invention uses a bearing fault simulation test bench to conduct experiments, obtain experimental signals, and then uses Hilbert transform to obtain the envelope of the original experimental signals. Finally, a Fourier transform is performed on the envelope to obtain the envelope spectrum of the experimental signals, as shown in the figure. Figure 9 As shown in Table 1, the theoretical characteristic frequencies of the bearing are shown in Table 2. The parameters used in the dynamic model are shown in Table 2. This invention considers two defect scenarios: localized failure of the outer ring and localized failure of the inner ring.
[0188] Table 1. Fault characteristic frequencies of SKF6205 deep groove ball bearings
[0189]
[0190] Table 2 SKF6205 Deep Groove Ball Bearing Parameters
[0191]
[0192]
[0193] The invention will be described in detail below with reference to simulation signal analysis. The invention includes the following steps:
[0194] Step 1: Establish a healthy bearing dynamics model. This step calculates the bearing stiffness and damping, the forces between the rolling elements and the cage, and the forces between the rolling elements and the raceway when the bearing is under elastohydrodynamic lubrication, thus determining the basic physical quantities required for a healthy bearing dynamics model.
[0195] Step 2: Establish a bearing dynamics model with local faults. By introducing a half-sine function, the time-varying displacement excitation of the rolling element when passing through a local fault is described, and finally, a bearing dynamics model with local faults is established.
[0196] Step 3: After verifying the accuracy of the model through simulation signal analysis, the main excitation sources in the dynamic model with local faults are identified. This step determines the main excitation sources in the model by comparing the simulation signals with experimental signals to confirm the model's accuracy and by comparing the numerical magnitudes and trends of the basic physical quantities in the dynamic model.
[0197] Step 1 includes:
[0198] Step 1.1: Calculate the bearing stiffness and damping based on elastohydrodynamic lubrication. In the dynamic model established in this invention, the contact model is as follows: Figure 1 As shown. Due to the hardening of the oil film in the Hertz contact region, its oil film stiffness is much greater than the Hertz contact stiffness of the contact pair. Therefore, the oil film stiffness in the Hertz contact region can be ignored. The damping of the contact pair consists of the bearing's structural damping and the Hertz region oil film damping connected in series, and then connected in parallel with the viscous damping in the inlet region. Because the viscous damping of the Hertz contact region oil film is relatively small, its damping after being connected in series with the Hertz contact region structural damping can be ignored. Therefore, the damping of the contact pair mainly comes from the viscous damping of the inlet region oil film. The contact stiffness of the contact pair can be expressed as:
[0199]
[0200] In the formula, E * Σρ is Young's modulus; μ is Poisson's ratio; Σρ is the raceway curvature; F is the first-type elliptic integral; E is the second-type elliptic integral; κ is the contact region ellipticity.
[0201] The viscous damping of the oil film in the inlet region can be expressed as:
[0202]
[0203] In the formula, η0 is the dynamic viscosity of the lubricant at room temperature; R x a is the equivalent radius of curvature along the rolling direction; nj Let be the length of the semi-major axis of the contact ellipse. The viscosity given in the ISO standard is the kinematic viscosity, which can be converted to dynamic viscosity using the following formula:
[0204] η0=υ·ρ·10 -6 (Pa·s) (3)
[0205] For rolling bearings with elliptical point contact, the most common formula for calculating the minimum oil film thickness and the oil film thickness at the center of the contact area is the Hamrock-Dowson film thickness formula:
[0206] h0 = 2.69R x U 0.67 G 0.53 W -0.067 (1-0.61e-0.73κ (4)
[0207] In the formula, U, G, and W are the dimensionless velocity parameter, dimensionless material parameter, and dimensionless load parameter, respectively.
[0208] Step 1.2: Calculate the forces between the rolling elements and the cage. During bearing operation, the rolling elements rotate around the bearing center. In the non-load-bearing region, the cage drives the rolling elements; in the load-bearing region, the rolling elements drive the cage. The contact stiffness between the rolling elements and the cage is calculated using Hertzian contact theory. Due to the complexity of the cage structure, the stiffness between the cage elements is calculated using the finite element method. The force analysis of the cage is as follows: Figure 2 As shown.
[0209] The force between the j-th rolling element and the cage can be expressed as:
[0210]
[0211]
[0212] Where, ψ r and ψ c These represent the angular displacements of the rolling elements and cage around the bearing center, respectively. n is the load deformation exponent, which is 1 within a small range of elastic deformation. K rc It refers to the connection stiffness between the rolling elements and the cage. R m It is the diameter of the joint.
[0213] The frictional force between the j-th rolling element and the cage can be expressed as:
[0214]
[0215]
[0216] Where, μ c It is the coefficient of friction.
[0217] The internal forces between the cages can be expressed as:
[0218]
[0219]
[0220]
[0221] In the formula, K cc and C CC These are the connection stiffness and connection damping between the cages, respectively; and the angular velocity of the cage rotating around the bearing center.
[0222] Step 1.3: Calculate the force between the rolling element and the raceway. Based on Hertz's nonlinear contact theory, the contact force between the rolling element and the inner / outer raceway can be expressed as:
[0223]
[0224]
[0225] In the formula, λ is a symbolic function, which can be expressed as:
[0226]
[0227] The contact deformation between the j-th rolling element and the raceway can be expressed as:
[0228]
[0229]
[0230] In the formula, r i j , and r j These represent the radial displacements of the inner ring, outer ring, and rolling element at the j-th rolling element, respectively; e represents the radial clearance. It is the displacement excitation caused by the j-th rolling element entering the defect, which will be explained in detail in step 2.
[0231] The radial displacements of the inner and outer rings at the j-th rolling element can be expressed as:
[0232]
[0233]
[0234] The angular position of the j-th rolling element can be represented as:
[0235]
[0236] In the formula, N b This represents the number of rolling elements.
[0237] The frictional force between the j-th rolling element and the raceway can be expressed as:
[0238]
[0239]
[0240] The coefficient of friction between the rolling elements and the raceway can be calculated using the following formula:
[0241]
[0242] In the formula, The relative sliding velocity between the j-th rolling element and the raceway is obtained through the following calculation:
[0243] ΔV i j =V ir -V ri (twenty three)
[0244]
[0245]
[0246] In the formula, and R represents the circumferential velocity and rotational velocity of the j-th rolling element, respectively. r This represents the radius of the rolling element.
[0247] Step 2 includes:
[0248] Step 2.1: Describe localized faults in the bearing raceway. This invention primarily studies early-stage bearing defects, characterized by a small width and a displacement of the rolling element less than the depth of the defect. The state of the rolling element passing through the defect is as follows: Figure 3 As shown in the figure, L represents the width of the fault, B represents the depth of the fault, and H represents the width of the fault. max This is the maximum displacement excitation.
[0249] Depend on Figure 3 As can be seen, when the rolling element enters and passes through the defect, it remains in contact with side I. When the rolling element passes side II, it has already left the defect area. This passage pattern can be described by a half-sine function to represent the time-varying excitation of the displacement. The maximum displacement excitation H of the rolling element... max for:
[0250]
[0251] The time-varying displacement excitation function of a rolling element passing over a defect can be represented by the following function:
[0252]
[0253]
[0254] In the formula, Represents the defect angle; The initial angle representing the defect; n = i, o; This can be expressed by the following formula:
[0255]
[0256] in, The corner position representing the defect.
[0257] Based on the above analysis, the dynamic equation of the bearing can be obtained as follows:
[0258] The dynamic equation of the inner circle horizontal motion:
[0259]
[0260] The dynamic equation of the vertical motion of the inner circle:
[0261]
[0262] The dynamic equation of the cage's circular motion:
[0263]
[0264] The dynamic equation of the circular motion of a rolling element:
[0265]
[0266] The dynamic equation of the rotational motion of the rolling element:
[0267]
[0268] The dynamic equation of the radial motion of the rolling element:
[0269]
[0270] In the formula, the centrifugal force of the j-th rolling element can be expressed as:
[0271] F wj =m r R m w oj 2 (36)
[0272] Step 3 includes:
[0273] Step 3.1: To verify the accuracy of the model, an experiment was conducted using a bearing fault simulation test bench to obtain experimental signals. Simultaneously, simulation was performed using MATLAB software, and the ode45 solver was used to obtain simulated signals. A qualitative comparison of the envelope spectrum characteristics between the experimental and simulated signals was then performed. The bearing used was an SKF6205-2RS, with grooves machined on both the inner and outer rings via wire cutting. The bearing was driven by a motor at a speed of 900 r / min, and the sampling frequency was set to 10 kHz. The acceleration of the bearing's outer ring in the Y direction was used as the original signal for the dynamic model. The envelope spectrum of the original signal was obtained through Fourier transform and Hilbert transform.
[0274] This invention considers two defect scenarios: localized faults in the outer ring and localized faults in the inner ring. A comparison of simulated and experimental signals under these two defect scenarios is shown below. Figure 9 As shown. When the bearing is in the case of an inner ring failure, the envelope spectra of the simulated and experimental signals of the bearing are as follows. Figure 9 As shown in (a) and (b). From Figure 9 In (a), it is clear that the fault characteristic frequency Bpfi and its harmonics of the inner ring are modulated by the frequency conversion fs. From Figure 9 A similar situation can also be clearly seen in (b).
[0275] Figure 9 (c) and (d) describe the envelope spectra of the simulated and experimental signals under the condition of a local fault in the outer ring, respectively. The fault characteristic frequency Bpfo and its harmonics are clearly visible in both the simulated and experimental signals.
[0276] The above comparison shows that, under different defect conditions, the main characteristics of the simulated signal remain consistent with those of the experimental signal, thus proving the effectiveness of the proposed model.
[0277] Step 3.2: The changes in the main physical quantities in the dynamic model proposed in this invention are as follows: Figure 4 , Figure 5 As shown, the frictional force between the rolling element and the inner rolling element can be clearly seen from the amplitude. Friction between the rolling element and the outer rolling element Friction between the rolling element and the preceding cage Friction between the rolling element and the subsequent cage Much smaller than the contact force between the rolling element and the inner raceway Contact force between rolling elements and outer raceways Contact force between the rolling element and the preceding cage Contact force between the rolling element and the subsequent cage This indicates that, and It plays a dominant role in influencing bearing vibration response.
[0278] Figure 6 This demonstrates the vibration response and key physical quantities of the bearing inner ring when the fault is located within the inner ring. The changes in [the value]. From the perspective of numerical magnitude and the pattern of change, [it is related to...]. compared to, The vibration response of the inner ring has a significant impact. Therefore, it can be determined that the contact force between the rolling elements and the raceway is the main physical quantity affecting the bearing vibration response, which is the excitation source in the proposed dynamic model. When the fault is located in the outer ring of the bearing, the vibration response of the inner ring and the main physical quantities are... Changes such as Figure 7 As shown in the figure, the same conclusion can be drawn from the trend of change in the figure: the contact force between the rolling element and the raceway is the main physical quantity affecting the bearing vibration response, which is the excitation source in the proposed dynamic model.
[0279] To address the challenge of traditional models failing to accurately reflect the real-world operating conditions of bearings, this invention focuses on rolling bearings and establishes a bearing dynamics model that considers cage flexibility, hydrodynamic lubrication, and time-varying displacement excitation caused by local faults. This more refined model can more accurately simulate the bearing's vibration mechanism. By comparing the vibration response of the established dynamics model with experimental signals, the model's accuracy is verified. Through comparative analysis, the excitation source in the model is identified, providing a theoretical basis for bearing vibration response analysis. Simultaneously, the established fault model can simulate the bearing's vibration response under different fault types and sizes, providing a data source for intelligent fault diagnosis of rolling bearings based on big data.
[0280] Figure 6 Vibration response and some physical quantities when the fault is in the inner ring of the bearing Variations: (a, c) Contact force between the rolling element and the inner / outer raceway in the horizontal direction (b, d) Contact forces between the rolling element and the inner / outer raceway in the vertical direction (e, g) Contact forces between the rolling element and the preceding / following cage in the horizontal direction (f, h) Contact forces between the rolling element and the preceding / following cage in the vertical direction. (i) Vibration response of the inner ring in the horizontal direction; (j) Vibration response of the inner ring in the vertical direction.
[0281] Figure 7 Vibration response and some physical quantities when the fault is in the outer ring of the bearing Variations: (a, c) Contact force between the rolling element and the inner / outer raceway in the horizontal direction (b, d) Contact forces between the rolling element and the inner / outer raceway in the vertical direction (e, g) Contact forces between the rolling element and the preceding / following cage in the horizontal direction (f, h) Contact forces between the rolling element and the preceding / following cage in the vertical direction. (i) Vibration response of the inner ring in the horizontal direction; (j) Vibration response of the inner ring in the vertical direction.
[0282] Figure 9Comparison of the envelope spectra of analog and experimental signals under different fault conditions: (a) Envelope spectrum of analog signal under inner ring fault condition; (b) Envelope spectrum of experimental signal under inner ring fault condition; (c) Envelope spectrum of analog signal under outer ring fault condition; (d) Envelope spectrum of experimental signal under outer ring fault condition.
[0283] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.
Claims
1. A method for dynamic modeling and vibration characteristic analysis of rolling bearings, characterized in that: Includes the following steps: S1: Establish a healthy bearing dynamics model: The stiffness and damping of the bearing, the force between the rolling elements and the cage, and the force between the rolling elements and the raceway were calculated when the bearing was under elastohydrodynamic lubrication, and the basic physical quantities required in the healthy bearing dynamics model were determined. S2: Establishing a bearing dynamics model with local faults: By introducing a half-sine function, the time-varying displacement excitation of the rolling element when passing through a local fault is described, and finally a bearing dynamics model with local faults is established. S3: Identify the main excitation sources in a dynamic model with local faults: By comparing the numerical magnitudes and trends of the basic physical quantities in the dynamic model, the main excitation sources in the model were determined. S1.1: Calculate the bearing stiffness and damping based on elastohydrodynamic lubrication; In the established dynamic model, the oil film in the Hertz contact area is hardened, and its oil film stiffness is much greater than the Hertz contact stiffness of the contact pair. Therefore, the oil film stiffness in the Hertz contact area is ignored. The damping of the contact pair is composed of the structural damping of the bearing and the oil film damping of the Hertz region in series, and then in parallel with the viscous damping of the inlet region. Because the viscous damping of the oil film in the Hertz contact region is relatively small, the damping is negligible after being connected in series with the structural damping of the Hertz contact region. Therefore, the damping of the contact pair comes from the viscous damping of the oil film in the inlet region. The contact stiffness of the contact pair is expressed as: (1) In the formula, Young's modulus; Poisson's ratio; For the rolling track curve; For a complete elliptic integral of the first kind, This is a complete elliptic integral of the second kind; The ellipticity of the contact area; The viscous damping of the oil film in the inlet region is expressed as: (2) In the formula, The dynamic viscosity of the lubricant at room temperature; Let be the equivalent radius of curvature along the rolling direction; The length of the semi-major axis of the contact ellipse; The viscosity given in the ISO standard is kinematic viscosity, which can be converted to dynamic viscosity using the following formula: (3) For rolling bearings with elliptical point contact, the minimum oil film thickness and the oil film thickness at the center of the contact area are calculated using the Hamrock-Dowson film thickness formula: (4) In the formula, , and These are dimensionless velocity parameters, dimensionless material parameters, and dimensionless load parameters, respectively. S1.2: Calculate the force between the rolling elements and the cage; During operation, the rolling elements of the bearing rotate around the center of the bearing; in the non-load-bearing area, the cage drives the rolling elements to move. In the load-bearing region, the rolling elements drive the cage motion; the contact stiffness between the rolling elements and the cage is calculated using Hertzian contact theory; due to the complexity of the cage structure, the stiffness between the cages is calculated using the finite element method. The force between the j-th rolling element and the cage is expressed as: (5) (6) in, and These are the rotational angular displacements of the rolling elements and the cage around the bearing center, respectively. It is the load deformation index, which is 1 within a small range of elastic deformation. It refers to the connection stiffness between the rolling elements and the cage; It is the pitch diameter; The friction between the j-th rolling element and the cage is represented as: (7) (8) in, It is the coefficient of friction; The internal forces between the cages are expressed as: (9) (10) (11) In the formula, and These are the connection stiffness and connection damping between the cages; It is the angular velocity of the cage rotating around the center of the bearing; S1.3: Calculate the force between the rolling element and the raceway; based on Hertz's nonlinear contact theory, the contact force between the rolling element and the inner / outer raceway is expressed as: (12) (13) In the formula, It is a symbolic function, represented as: (14) The contact deformation between the j-th rolling element and the raceway is expressed as: (15) (16) In the formula, , and These represent the radial displacements of the inner ring, outer ring, and rolling element at the j-th rolling element, respectively; Represents radial clearance; It is the displacement excitation caused by the j-th rolling element entering the defect; The radial displacements of the inner and outer rings at the j-th rolling element are expressed as: (17) (18) The angular position of the j-th rolling element is represented as: (19) In the formula, Represents the number of rolling elements; The frictional force between the j-th rolling element and the raceway is expressed as: (20) (21) The coefficient of friction between the rolling elements and the raceway is calculated using the following formula: (22) In the formula, The relative sliding velocity between the j-th rolling element and the raceway is obtained through the following calculation: (23) (24) (25) In the formula, and These are the circumferential velocity and rotational velocity of the j-th rolling element, respectively. This represents the radius of the rolling element.
2. The method for dynamic modeling and vibration characteristic analysis of rolling bearings according to claim 1, characterized in that: S2 includes 2.1: describing local faults in the bearing raceway; for early-stage bearing defects, the fault width is small, the displacement of the rolling element as it passes the defect is smaller than the depth of the fault, and the rolling element is in a state of passing the defect... The width of the fault. The depth of the fault. For the maximum displacement excitation; When the rolling element enters and passes through the defect, it remains in contact with side I. When it passes side II, it has already left the defect area. This passage pattern uses a half-sine function to describe the time-varying excitation of the displacement; the maximum displacement excitation of the rolling element... for: (26) The time-varying displacement excitation function of the rolling element when it passes over a defect is represented by the following function: (27) (28) In the formula, Represents the defect angle; The initial angle representing the defect; n = i, o; This can be expressed by the following formula: (29) in, The corner position representing the defect; Based on the above analysis, the dynamic equation of the bearing is as follows: The dynamic equation of the inner circle horizontal motion: (30) The dynamic equation of the vertical motion of the inner circle: (31) The dynamic equation of the cage's circular motion: (32) The dynamic equation of the circular motion of a rolling element: (33) The dynamic equation of the rotational motion of the rolling element: (34) The dynamic equation of the radial motion of the rolling element: (35) In the formula, the centrifugal force of the j-th rolling element is expressed as: (36)。 3. The method for dynamic modeling and vibration characteristic analysis of rolling bearings according to claim 1, characterized in that: S3 includes the following steps: S3.1: To verify the accuracy of the model, an experiment was conducted using a bearing fault simulation test bench to obtain experimental signals. Simultaneously, simulation was performed using MATLAB software, and the ode45 solver was used to obtain simulated signals. A qualitative comparison of the envelope spectrum characteristics between the experimental and simulated signals was conducted. The bearing used was an SKF6205-2RS, with grooves machined on both the inner and outer rings via wire cutting. The bearing was driven by a motor at a speed of 900 r / min, and the sampling frequency was set to 10 kHz. The acceleration of the bearing's outer ring in the Y direction was used as the original signal for the dynamic model. The envelope spectrum of the original signal was obtained through Fourier transform and Hilbert transform.