Design Method for Angular Contact Ball Bearings with Low Vibration and High Rotational Accuracy

Abstract

This invention relates to a structural design method for angular contact ball bearings with low vibration and high rotational accuracy, comprising the following steps: S1, obtaining the structural parameters, operating parameters, and lubrication conditions of the angular contact ball bearing rotor system; S2, applying external force vectors to the rotor, and establishing the dynamic equations of the bearing rotor system based on the principles of torque balance and force balance; S3, obtaining the dynamic equations of the bearing inner ring, rolling elements, and cage; S4, establishing a dynamic model of the bearing-rotor system, solving the bearing-rotor system, calculating the dynamic behavior of the bearing-rotor system, and simultaneously obtaining the vibration and rotational accuracy of the system; S5, obtaining the variation law of the vibration and rotational accuracy of the bearing-rotor system, and determining the structural dimensions of the bearing with low vibration and high rotational accuracy. This invention can design the structural dimensions of angular contact ball bearings with low vibration and high rotational accuracy by predicting the vibration and rotational accuracy of the bearing rotor system based on the bearing rotor dynamic model.

Description

Technical Field

[0001] This invention relates to the field of high-end bearing design, and more specifically, to a structural design method for an angular contact ball bearing with low vibration and high rotational accuracy. Background Technology

[0002] Rolling bearings are crucial supporting components of rotor systems in rotating machinery. Their vibration and rotational accuracy significantly affect the operational precision, reliability, and lifespan of the machinery. Currently, the design of angular contact ball bearings with low vibration and high rotational accuracy mainly relies on theoretical calculations and simulation analysis, combined with extensive experimental testing. This process is time-consuming and costly.

[0003] Therefore, establishing a bearing rotor dynamics model, quantitatively analyzing bearing vibration, power consumption, and rotational accuracy, and determining the bearing structural dimensions for low vibration, low power consumption, and high rotational accuracy can effectively formulate experimental plans with low cost and short time.

[0004] Existing literature shows a considerable amount of research on bearing vibration and rotational accuracy. Wang et al. (Wang PF, Yang Y, Ma He et al. Vibration characteristics of rotor-bearing system with angular misalignment and cage fracture: Simulation and Experiment. Mechanical Systems and Signal Processing, 2023, 182: 109545) established a mechanical model of the bearing-rotor system and analyzed the impact of bearing misassembly and cage fracture on bearing vibration and rotational accuracy for specific bearing objects. Liu et al. (Liu Jing, A dynamic modelling method of arotor-roller bearing housing system with a localized fault including the additional excitation zone, Journal of Sound and Vibration, 2020, 469: 115144) used a stiffness-damping method to establish a bearing mechanical model and analyzed the vibration characteristics of the bearing-rotor system caused by bearing defects. However, this method neglected the influence of the dynamic behavior of bearing components on the vibration characteristics of the bearing-rotor system and also failed to study the influence of bearing structural dimensions on the vibration characteristics of the bearing-rotor system. Chen Yue et al. (Chen Yue, Qiu Ming, Du Hui, et al. Factors affecting the rotational accuracy of four-point contact ball bearings for robots. Chinese Journal of Mechanical Engineering, 2020, 31(14):1678-1685) established a bearing calculation model based on the kinematic and geometric relationships of the internal components of the bearing and studied the influence of the roundness error of bearing parts on the rotational accuracy of the bearing. However, they did not integrate the analysis of vibration and rotational accuracy. Yu et al. (Yu YJ, Li JS, Xue YJ. Influence of roundness errors of bearing components onrotational accuracy of cylindrical roller bearings. Scientific Reports, 2022, 12(1):6794) studied the influence of the roundness of the inner raceway, outer raceway and rollers on the rotational accuracy of the bearing. They used non-repeatable runout to evaluate the variation law of the rotational accuracy of the bearing. However, they also did not integrate the analysis of vibration and rotational accuracy.

[0005] The literature search results show that existing bearing rotor dynamics models cannot comprehensively analyze the vibration and rotational accuracy of the system, nor can they study the influence of bearing structural dimensions on the vibration and rotational accuracy of the bearing rotor system. Consequently, it is impossible to design angular contact ball bearing structural dimensions with low vibration and high rotational accuracy. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a structural design method for angular contact ball bearings with low vibration and high rotational accuracy, which can predict the vibration and rotational accuracy of the bearing rotor system based on the bearing rotor dynamics model to design the structural dimensions of angular contact ball bearings with low vibration and high rotational accuracy.

[0007] The technical solution adopted by this invention to solve its technical problem is: a method for constructing a structural design of an angular contact ball bearing with low vibration and high rotational accuracy, comprising the following steps:

[0008] S1. Obtain the structural parameters, operating parameters, and lubrication conditions of the angular contact ball bearing rotor system;

[0009] S2. An external force vector is applied to the rotor, and the rotor experiences a load vector F. s ={F sx ,F sy ,F sz M sy M sz The function of} is to generate the displacement vector d of the inner raceway. s ={δ sx ,δ sy ,δ sz ,θ sy ,θ sz The left bearing generates a nonlinear force vector F. L ={F L x ,F L y ,F L z M L y M L z} and displacement vector d L ={δ L x ,δ L y ,δ L z ,θ L y ,θ L z The right bearing generates a nonlinear force vector F. R ={F R x,F R y ,F R z M R y M R z} and displacement vector d R ={δ R x ,δ R y ,δ R z ,θ R y ,θ R z The outer raceway is fixed inside the bearing housing, and the rotor moves at an angular velocity ω. i Rotation; establish the dynamic equations of the bearing rotor system based on the principles of torque balance and force balance;

[0010] S3. Obtain the dynamic equations of the bearing inner ring, rolling elements, and cage;

[0011] S4. Establish a dynamic model of the bearing-rotor system based on the dynamic equations of the bearing-rotor system, the dynamic equations of the rolling elements, and the dynamic equations of the cage. Solve the bearing-rotor system to calculate its dynamic behavior and obtain the vibration and rotational accuracy of the system.

[0012] S5. By changing the bearing structure dimensions, the variation law of vibration and rotational accuracy of the bearing-rotor system is obtained, and the bearing structure dimensions with low vibration and high rotational accuracy are determined.

[0013] According to the above scheme, in step S2, the dynamic equation of the bearing rotor system is:

[0014]

[0015] In the formula, m r The rotor mass, bearing contact force, and torque are respectively expressed as F. ix ,F iy ,F iz M iy and M iz l0 represents the distance from the point of force application to the left bearing, and l1 and l2 represent the distances from the left and right bearings to the rotor's center of mass, respectively.

[0016] According to the above scheme, in step S3, the dynamic equation of the bearing inner ring is:

[0017]

[0018] According to the above scheme, in step S3, the dynamic equation of the rolling element is:

[0019]

[0020] According to the above scheme, in step S3, the cage dynamics equation is:

[0021]

[0022] According to the above scheme, in step S4, the formula for calculating the rotor rotation accuracy is:

[0023]

[0024] According to the above scheme, in step S4, the Runge-Kutta method is used to solve the dynamic equations of the bearing rotor system, the rolling elements, and the cage. The initial value of the angular velocity about the x-direction is the bearing rotational angular velocity.

[0025] The structural design method for angular contact ball bearings with low vibration and high rotational accuracy according to the present invention has the following beneficial effects:

[0026] This invention comprehensively considers bearing structural dimensions and bearing loads to evaluate the vibration and rotational accuracy of the bearing-rotor system. By coordinating bearing loads and structural dimensions, it designs angular contact ball bearings with low vibration and high rotational accuracy. This method can effectively guide experimental design, reduce experimental costs, and shorten the experimental cycle. Furthermore, this method can also determine reasonable operating conditions through comprehensive analysis of the vibration and rotational accuracy of the bearing-rotor system. It can also be used for comprehensive analysis of the vibration and rotational accuracy of different operating conditions and specific bearing-rotor system objects, thereby matching the operating conditions with the bearing-rotor system. Moreover, similar or identical applications are within the scope of protection of this patent. Attached Figure Description

[0027] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0028] Figure 1 This is a schematic diagram of the process of the present invention;

[0029] Figure 2 A simplified diagram of the bearing rotor system;

[0030] Figure 3 The rotational trajectory of the rotor center under the combination of preload and number of rolling elements: (a) Pr = 300 N, (b) Pr = 350 N, (c) Pr = 400 N, (d) Pr = 450 N;

[0031] Figure 4The speed response of the rotor center when considering the combination of preload and number of rolling elements: (a) P r =300N time domain, (b)P r =300N frequency domain, (c)P r =350N time domain, (d)P r =350N frequency domain, (e)P r =400N time domain, (f)P r =400N frequency domain, (g)P r =450N time domain, (h)P r =450N frequency domain;

[0032] Figure 5 (a) represents the average contact load of the rolling elements of the left bearing, and (b) represents the average contact load of the rolling elements of the right bearing. Detailed Implementation

[0033] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0034] Reference Figure 1 As shown, this invention provides a structural design method for angular contact ball bearings with low vibration and high rotational accuracy, which includes the following steps:

[0035] S1. Based on the actual situation, the rotor is simplified into a rigid body with back-to-back bearings to establish a mathematical model of the bearing rotor system, such as... Figure 2 As shown.

[0036] S2. Obtain the structural parameters, operating parameters, and lubrication parameters of the bearing rotor system. The structural parameters mainly include bearing parameters, material parameters, rotor geometric parameters, and bearing and rotor position parameters. The operating parameters include rotational speed and external force vector. The lubrication parameters include lubrication viscosity and lubrication density.

[0037] S3. An external force vector is applied to the rotor, and the rotor experiences a load vector F. s ={F sx ,F sy ,F sz M sy M sz The function of} is to generate the displacement vector d of the inner raceway. s ={δ sx ,δ sy ,δ sz ,θ sy ,θ sz Therefore, the left bearing generates a nonlinear force vector F. L ={F L x ,F Ly ,F L z M L y M L z} and displacement vector d L ={δ L x ,δ L y ,δ L z ,θ L y ,θ L z}, while the right bearing is F R ={F R x ,F R y ,F R z M R y M R z} and d R ={δ R x ,δ R y ,δ R z ,θ R y ,θ R z The outer raceway is fixed inside the bearing housing, and the rotor moves at an angular velocity ω. i Rotation. Based on the principles of torque balance and force balance, the dynamic equations of the bearing rotor system are established:

[0038]

[0039] In the formula, m r The rotor mass, bearing contact force, and torque are respectively expressed as F. ix ,F iy ,F iz M iy and M iz l0 represents the distance from the point of force application to the left bearing, and l1 and l2 represent the distances from the left and right bearings to the rotor's center of mass, respectively.

[0040] S4. The dynamic balance of the rotor is inseparable from the force action of the inner rings of the left and right bearings. The dynamic equation of the inner rings of the bearings is as follows:

[0041]

[0042] S5. The dynamic behavior of the bearing inner ring is closely related to the dynamics of the rolling elements. The dynamic equations of the rolling elements are as follows:

[0043]

[0044] S6. The dynamics of the rolling elements are closely influenced by the dynamics of the cage. The cage dynamics equations are as follows:

[0045]

[0046] S7. Establish a dynamic model of the bearing-rotor system, use the variable step size Runge-Kutta method to solve the bearing-rotor system, calculate the dynamic behavior of the bearing-rotor system, and obtain the vibration and rotation accuracy of the system.

[0047] S8. By changing the bearing structure dimensions, the variation law of vibration and rotational accuracy of the bearing-rotor system is obtained, and the bearing structure dimensions with low vibration and high rotational accuracy are determined.

[0048] In the above scheme, the formula for calculating the rotor rotation accuracy in step S7 is as follows:

[0049]

[0050] In the above scheme, the Runge-Kutta method is used in step S7 to solve the dynamic equations (1)-(4), and the initial value of the angular velocity around the x direction is the bearing rotation angular velocity.

[0051] To facilitate understanding and avoid omissions, all formula parameters involved in this embodiment are as follows:

[0052] X 1j X 2j This is an auxiliary quantity used in intermediate processes and has no specific definition.

[0053] δ ij The deformation of the j-th rolling element in contact with the inner ring is an unknown parameter to be determined; δ oj Let be the deformation of the j-th rolling element in contact with the outer ring, and be an unknown parameter to be determined.

[0054] K oj K ij : Load displacement constant; Dimensionless contact displacement; ∑ρ i ,∑ρ o curvature and

[0055] M gj : Rolling body gyro torque;

[0056] J: Rolling element inertial torque;

[0057] w is the inner angular velocity;

[0058] w mj Let j be the orbital speed of the j-th rolling element;

[0059] w Rj Let be the rotational speed of the j-th rolling element;

[0060] α ij The contact angle between the j-th rolling element and the inner ring is an unknown parameter to be determined; α oj Let be the contact angle between the j-th rolling element and the outer ring, and be an unknown parameter to be determined.

[0061] β j Ball attitude angle, calculated using formula;

[0062] θ j : Rolling element position angle, ω c Maintain the frame angular velocity, w is the angular velocity of the inner ring, i.e., the rotor.

[0063] m: mass of the rolling element; n: number of rolling elements;

[0064] d m : Pitch circle radius;

[0065] D is the diameter of the rolling element;

[0066] Δ c This is the amount of radial clearance reduction after installation;

[0067] B is the total curvature;

[0068] δ a Let be the total axial deformation of the bearing, and be the unknown parameter to be determined; θ be the angular deformation of the bearing, and be the unknown parameter to be determined; F a F is the axial force acting on the bearing. r M is the radial force acting on the bearing; M′ is the torque acting on the bearing;

[0069] r i r is the radius of curvature of the inner raceway of the bearing. o The radius of curvature of the outer raceway of the bearing;

[0070] FF (i,o) The first type of complete elliptic integral, EE (i,o) The second kind of complete elliptic integral, R x(i,o) R is the equivalent axial contact radius between the rolling element and the inner and outer rings of the bearing; y(i,o) The equivalent radial contact radius between the rolling element and the inner and outer rings of the bearing;

[0071] u ij ,u oj : Sucking speed;

[0072] E pEquivalent modulus, v1 and v2 are the Poisson's ratios of the two contacting materials; E1 and E2 are the elastic moduli of the two contacting materials;

[0073] η o η is the viscosity of the lubricating oil. or Reference viscosity, α1 is the viscosity-pressure coefficient of the lubricating oil;

[0074] k (i，o)j Hertzian contact stiffness; k S(i，o)j Oil film stiffness

[0075] w (i，o)j Contact load;

[0076] A: The distance between the centers of curvature of the raceway grooves;

[0077] F p α is the axial preload; p : Contact angle generated by preload; δ ep : Axial displacement caused by preload;

[0078] K j(L，R) Equivalent contact stiffness; δ jL δ jR Normal displacement; C j(L，R) The damping of the elastohydrodynamic lubricating oil film; δ′ j(L，R) The derivative of the normal displacement;

[0079] One point represents the first derivative, and two points represent the second derivative;

[0080] M is the mass of the rotor, and C is the system damping, which mainly considers the Hertzian contact deformation of the rolling elements and the interaction between the outer ring and the bearing housing.

[0081] f0 is a coefficient related to the bearing type, which can be obtained by looking up a table;

[0082] v0 is the kinematic viscosity of the lubricating oil, which can be found in the lubricating oil table.

[0083] N i This refers to the shaft speed;

[0084] f1 is a coefficient related to the bearing structure and load.

[0085] The following example further illustrates the structural design method of the angular contact ball bearing with low vibration and high rotational accuracy according to the present invention. This example is not intended to limit the invention. The steps are as follows:

[0086] 1) Obtain such Figure 2 The structural parameters, operating parameters, and lubrication parameters of the bearing rotor system are shown in Tables 1 and 2:

[0087] Table 1 Rotor Parameters

[0088]

[0089] Table 2 Parameters for Bearing B7008C and Lubrication

[0090]

[0091]

[0092] 1) Using the bearing rotor dynamics model established in steps 2) to 6) based on the parameters in step 1), the dynamic behavior, vibration and rotational accuracy of the bearing rotor system are calculated using the variable step size Runge-Kutta method in step 7).

[0093] from Figure 3 It can be seen that under the preload P r At 300N, as the number of rolling elements gradually increases, the trajectory of the rotor center becomes increasingly non-repetitive. At P r At P=350N, the repeating trajectory still exists with different numbers of rolling elements, but the diameter of the trajectory increases as the number of rolling elements increases. r At P=400N, as the number of rolling elements increases, the repeatability of the rotor trajectory gradually improves, and the repeatability increases. r When the torque is 450 N, the rotor trajectory with good repeatability appears when the number of rolling elements Z is 21 and 22. Based on this, the runout of the rotor center's rotation trajectory under the combination of preload and number of rolling elements is shown in Table 3.

[0094] Table 3. Rotor runout values ​​(μm) for different combinations of preload and number of rolling elements.

[0095]

[0096] As can be clearly seen from Table 3, in P r At P=300N, as the number of rolling elements gradually increases, the rotor runout increases significantly, indicating a significant deterioration in the rotor trajectory. r At P=350N, as the number of rolling elements increases, the rotor runout increases slightly, indicating a slight deterioration in the rotor trajectory. r At 400N, as the number of rolling elements increases, the rotor trajectory reverses from before and gradually improves. Furthermore, when the number of rolling elements is 21 or 22, the runout initially improves and then worsens. Specifically, smaller runout amounts occur in (Z, P) r It appears in the combination of {(19,350N),(22,400N)}. From Figure 4 It can be seen that in P rAt P=300N, the rotor speed gradually increases, and the speed fluctuations become more pronounced with the increase in the number of rolling elements. Particularly when the number of rolling elements is 21 and 22, the periodicity of the fluctuations becomes increasingly severe. The speed spectrum shows that the main peaks of both low and high frequencies gradually strengthen with the increase in the number of rolling elements, confirming that the number of rolling elements affects the rotor speed. r At 350N, the magnitude and fluctuation of the rotor speed compared to P r =300N decreases; it is worth noting that when the number of rolling elements is 21 and 22, the periodicity of the fluctuation is relatively smaller than P. r =300N significantly improved. In P r At 400N, the periodicity of the oscillation is evident regardless of the number of rolling elements. The magnitude of the speed and the oscillation gradually decrease with the increase of the number of rolling elements, which is consistent with P. r The situation is reversed when P = 350N. r When = 400N, except for (Z,P) r Besides the velocity response of (22,350N), the intensity and P r =350N is almost the same, and in (Z,P) r The main peak of the fluctuation at (19,350N) and (Z,P) r The main peaks at (22, 400 N) are almost identical. At P r At 450N, the magnitude and fluctuation of the velocity are greater than P. r =400N. Specifically, the smaller vibration of the bearing-rotor system is within (Z, P) r It appears in the combination of {(19,350N),(22,400N)}.

[0097] from Figure 5 It can be seen that in (Z,P) r The average contact load of the rolling element at (19, 350 N) is approximately 57 N, while at (Z, P) r The average contact load of the rolling elements at (Z, P) is approximately 57 N. This indicates that both excessively high and insufficient average contact loads of the rolling elements are detrimental to improving the vibration and rotational accuracy of the bearing-rotor system. Furthermore, 22 rolling elements have 3 more rolling elements than 19 rolling elements, resulting in greater frictional power consumption during bearing rotation compared to bearing with 19 rolling elements. Therefore, choosing (Z, P) is preferable. r The preload and number of rolling elements of the bearing-rotor system are designed using (19, 350 N) to achieve a low-vibration, high-rotational-precision angular contact ball bearing structure design.

[0098] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A design method of a structure of an angular contact ball bearing with small vibration and high rotation accuracy, characterized by, Comprise the following steps: S1, obtaining the structure parameters, working condition parameters and lubrication conditions of the angular contact ball bearing rotor system; S2, the external force vector acts on the rotor, the rotor is subjected to load vector F s ={F sx , F sy , F sz , M sy , M sz} action, and the inner raceway displacement vector d s ={δ sx , δ sy , δ sz , θ sy , θ sz} is generated; the left bearing generates nonlinear force vector F L ={F L x , F L y , F L z , M L y , M L z} and displacement vector d L ={δ L x , δ L y , δ L z , θ L y , θ L z}, the right bearing generates nonlinear force vector F R ={F R x , F R y , F R z , M R y , M R z} and displacement vector d R ={δ R x , δ R y , δ R z , θ R y , θ R z}; the outer raceway is fixed in the bearing seat, and the rotor rotates at angular velocity ω i ; the bearing rotor system dynamics equation is established according to the principle of moment balance and force balance; S3, obtaining the dynamic equation of the bearing inner ring, the dynamic equation of the rolling body and the dynamic equation of the cage; S4, establishing the bearing-rotor system dynamics model according to the bearing rotor system dynamics equation, the dynamic equation, the dynamic equation of the rolling body and the dynamic equation of the cage, solving the bearing-rotor system, calculating the dynamic behavior of the bearing-rotor system, and obtaining the vibration and rotation accuracy of the system; S5, changing the bearing structure size, obtaining the change rule of the bearing-rotor system vibration and rotation accuracy, and determining the bearing structure size with small vibration and high rotation accuracy.

2. The angular contact ball bearing structure design method of claim 1, wherein In the step S2, the bearing rotor system dynamics equation is: （1） where m r represents the rotor mass, and the contact force and moment of the bearing are represented as F ix , F iy , F iz , M iy and M iz , l 0 represents the distance from the force application point to the left bearing, l 1 and l 2 represent the distances from the left and right bearings to the center of mass of the rotor, respectively.

3. The structure design method of an angular contact ball bearing with low vibration and high rotation accuracy according to claim 1, characterized in that, In the step S4, the formula for calculating the rotor rotation accuracy is: (5)。 4. The structure design method of an angular contact ball bearing with low vibration and high rotation accuracy according to claim 1, characterized in that, In the step S4, the Runge-Kutta method is used to solve the bearing rotor system dynamics equation, the dynamic equation, the dynamic equation of the rolling body and the dynamic equation of the cage, and the initial value of the angular velocity around the x direction is the bearing rotation angular velocity.

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