CN114595526B - Method for reducing collision of rolling bodies of ball bearing without retaining - Google Patents

Method for reducing collision of rolling bodies of ball bearing without retaining Download PDF

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CN114595526B
CN114595526B CN202111610595.9A CN202111610595A CN114595526B CN 114595526 B CN114595526 B CN 114595526B CN 202111610595 A CN202111610595 A CN 202111610595A CN 114595526 B CN114595526 B CN 114595526B
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inner ring
rolling
ball bearing
outer ring
rolling body
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赵彦玲
周恩雯
张晓楠
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Harbin University of Science and Technology
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    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16C—SHAFTS; FLEXIBLE SHAFTS; ELEMENTS OR CRANKSHAFT MECHANISMS; ROTARY BODIES OTHER THAN GEARING ELEMENTS; BEARINGS
    • F16C33/00—Parts of bearings; Special methods for making bearings or parts thereof
    • F16C33/30—Parts of ball or roller bearings
    • F16C33/58—Raceways; Race rings
    • F16C33/583—Details of specific parts of races
    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
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Abstract

The invention discloses a method for reducing collision of a non-retainer ball bearing, which is characterized in that the inner surface of an outer channel of the non-retainer ball bearing is designed with discrete grooves, the kinetic energy of rolling bodies is changed to enable adjacent rolling bodies to generate space, and the purpose of reducing friction generated by collision between the rolling bodies is achieved; the method comprises the steps of establishing a spring-mass model of the ball bearing without the retainer, and establishing a differential equation of motion of a rolling body and an inner ring of the ball bearing without the retainer based on a Lagrange energy method; and carrying out numerical simulation solution and preliminary optimization of discrete groove parameters on the motion differential equation of the ball bearing without the retainer.

Description

Method for reducing collision of rolling bodies of ball bearing without retaining
Technical Field
The invention relates to the technical field of ball bearings without retainers, in particular to a method for reducing collision of rolling bodies of a ball bearing without retainers.
Background
In recent years, with the development of aero-engines, electric spindles and high-precision machine tools, the requirement for the rotating speed of a rolling bearing is higher and higher, and ceramic bearings are more and more emphasized by the advantages of higher limit rotating speed, high temperature resistance and the like. The mixed ceramic bearing is mainly applied in the engineering at the present stage, namely, a rolling body is made of ceramic materials, an inner ring and an outer ring are made of steel materials and contain a retainer, and the retainer is used for reducing resistance generated when the rolling body is in direct contact. However, in the case of high-speed operation, the friction between the cage and the rolling elements and races also generates a non-negligible resistance, causing the bearing to heat up, thereby limiting the limit rotational speed of the bearing. In order to avoid the frictional resistance caused by the retainer and realize the high-speed or ultrahigh-speed operation of the bearing, a ceramic ball bearing without the retainer is developed abroad in recent years. In a bearing having no cage and rolling elements in contact with each other, there are the following problems: (1) High contact surface pressure due to the mutual contact of the protrusions between adjacent rolling bodies; (2) The rapid sliding speeds in opposite directions due to the rotation of the rolling elements are liable to cause problems such as friction, wear, and seizure due to the oil film at the contact point being broken. Thus, the outer ring of the ball bearing without the retainer is provided with one or more discrete grooves, so as to achieve the purpose of avoiding the rolling bodies from contacting with each other
Disclosure of Invention
The technical problem to be solved by the invention is as follows: in order to solve the problem of reducing the collision of the rolling bodies of the existing retainer-free ball bearing, the invention provides a method for reducing the collision of the rolling bodies of the retainer-free ball bearing.
The technical scheme adopted by the invention for solving the technical problem is as follows:
a method of reducing rolling element collision for a retainment-less ball bearing, comprising the steps of:
1) Establishing eta based on energy analysis i (i＝1,2,...,Z)、x in 、z in The differential equation of motion of (2) is specifically:
Figure RE-GDA0003616122230000021
wherein:
Figure RE-GDA0003616122230000022
in the formula, m b Is the mass of the rolling elements, η i Is the radial position of the rolling body, wherein i (i =1, 2.. Multidot., Z), Z being the number of rolling elements, g being 9.8N/kg,. Multidot. i (ψ i Not less than 0) is the position angle of the i-th rolling element, k in For the contact stiffness between the ith rolling element and the inner ring, k out For the contact stiffness between the i-th rolling element and the outer ring, R in Is the outer raceway radius, R, of the inner ring of the ball bearing without the retainer out Is the inner raceway radius R of the outer ring of the ball bearing without a retainer b Is the radius of the rolling body, xi i For deformation of the spring between the rolling body and the inner race, m in Is the inner ring mass, x in Is the displacement of the inner ring along the x-axis, k is, z in For displacement of the inner ring along the z-axis, F r Is a radial force, x, acting on the bearing system out Is the displacement of the outer ring along the x-axis, z out Displacement of the outer ring along the z-axis;
2) Respectively substituting the formulas (4), (5) and (6) into the formulas (1), (2) and (3), and solving eta by Runge-Kutta method i (i＝1,2,...,Z)、x in 、z in The motion differential equation is used for solving the speed and the displacement of the inner ring rolling body;
3) And optimizing discrete groove parameters according to the speed and the displacement of the inner ring rolling body.
Specifically, 11) regarding the ball bearing without the retainer as a spring-mass system, as shown in fig. 1, regarding contact forces of an inner ring and an outer ring of a rolling element as a nonlinear contact spring, solving the rigidity of the nonlinear spring based on the hertz elastic contact deformation theory, directly influencing kinetic energy and potential energy of each component of the bearing by the radial displacement of the centroid of the rolling element caused by discrete groove parameters, and establishing a group of lagrangian equations of an independent generalized coordinate system by using the lagrangian equation and the kinetic energy and potential energy expressions of each component of the ball bearing without the retainer:
Figure RE-GDA0003616122230000031
in the formula: t, V, p and f are kinetic energy, potential energy, a generalized degree of freedom coordinate vector and a generalized contact force vector respectively;
12 Assuming the outer ring is fixed, the total kinetic energy in the ball bearing without the cage is the sum of the kinetic energy of the rolling body and the kinetic energy of the inner ring, and the equation is as follows:
Figure RE-GDA0003616122230000032
in the formula: t is a unit of roller 、T in 、T out Respectively indicating rollsKinetic energy of the moving body, kinetic energy of the inner ring and kinetic energy of the outer ring;
13 The potential energy variation is caused by the deformation of the structure of the discrete grooves and the rolling elements along with the raceways, in the bearing system coordinate system, in the plane x 0 Oy 0 Potential energy calculation is carried out on a horizontal plane, namely the center of the bearing is selected as a gravitational potential energy zero position, the total potential energy in the ball bearing without the retainer is the sum of the potential energy of the rolling body, the potential energy of the inner ring, the potential energy of the outer ring and the potential energy of the spring, and the equation is as follows:
Figure RE-GDA0003616122230000033
in the formula: v roller 、V in 、V out Respectively representing the potential energy of the rolling body, the potential energy of the inner ring and the potential energy of the outer ring, V t Is potential energy generated by the contact of the rolling body and the nonlinear spring between the inner ring and the outer ring;
substituting the formulas (8) and (9) into the formula (7), and respectively aligning with eta i (i＝1,2,...,Z)、x in 、z in Derivation to obtain η i (i＝1,2,...,Z)、x in 、z in Differential equation of motion.
Specifically, in order to establish the kinetic energy and potential energy equations of the inner ring and the outer ring, a spring-mass model displacement analysis diagram of the ball bearing without the retainer is established, as shown in fig. 2, assuming that the center of the outer ring of the bearing is fixed and the outer ring is a rigid body,
Figure RE-GDA0003616122230000041
then it can be deduced
Figure RE-GDA0003616122230000042
ω out =0, the kinetic energy expression of the outer ring of the ball bearing without the retainer is as follows:
Figure RE-GDA0003616122230000043
the potential energy expression of the outer ring of the ball bearing without the retainer is as follows:
Figure RE-GDA0003616122230000044
in the formula, m out Representing the mass of the outer ring;
in addition to local deformation in contact, the inner ring is considered to be a rigid body, in x 0 Oz 0 The kinetic energy of the inner ring around the center of the bearing is calculated, and the expression of the kinetic energy of the inner ring is as follows:
Figure RE-GDA0003616122230000045
wherein: m is in Denotes inner circle mass, I in The specific value, omega, representing the moment of inertia of the inner ring is calculated by the mass and the radius of gyration in The rotational speed of the inner ring is represented,
Figure RE-GDA0003616122230000046
is shown in plane x 0 Oz 0 Variation of the displacement of the centre of the inner race relative to the coordinates (0, 0) of the bearing centre, i.e.
Figure RE-GDA0003616122230000047
Assuming that the center of the outer ring of the bearing is fixed, the displacement change of the center of the inner ring relative to the center of the outer ring is known
Figure RE-GDA0003616122230000048
Then the
Figure RE-GDA0003616122230000049
Can be expressed as:
Figure RE-GDA00036161222300000410
can also be expressed as:
Figure RE-GDA00036161222300000411
the center of the bearing outer ring is fixed, so
Figure RE-GDA00036161222300000412
Therefore, equation (24) is simplified to:
Figure RE-GDA00036161222300000413
the potential energy expression of the inner ring of the ball bearing without the retainer is as follows:
V in ＝m in g(z out -z in_out ) (14)
in the formula: z is a radical of in_out Representing the relative coordinates of the center of the inner ring relative to the center of the outer ring;
the sum of kinetic energies generated by Z rolling bodies in the bearing is as follows:
Figure RE-GDA0003616122230000051
the position of the mass center of the rolling body relative to the center of the outer ring is determined, so that the kinetic energy of the ith rolling body is as follows:
Figure RE-GDA0003616122230000052
in the formula, m b As to the mass of the rolling elements,
Figure RE-GDA0003616122230000053
the ith rolling element mass center displacement vector is expressed by the expression:
Figure RE-GDA0003616122230000054
bonding of
Figure RE-GDA0003616122230000055
Differentiating the time yields:
Figure RE-GDA0003616122230000056
due to the fact that
Figure RE-GDA0003616122230000057
And
Figure RE-GDA0003616122230000058
therefore, it is
Figure RE-GDA0003616122230000059
The kinetic energy expression of the ith rolling body is as follows:
Figure RE-GDA00036161222300000510
assuming no slip, the relative transition speeds of the outer ring and the balls must be the same and opposite, and therefore the contact equation of the ith rolling element with the outer ring is:
Figure RE-GDA00036161222300000511
wherein, the first and the second end of the pipe are connected with each other,
Figure RE-GDA00036161222300000512
since the outer ring is stationary, it is not necessary to provide a separate seal
Figure RE-GDA00036161222300000513
To sum up, the kinetic energy equation generated by the Z rolling elements in the bearing is:
Figure RE-GDA0003616122230000061
in the formula: m is the mass of the rolling body, R o Is the radius of the contact point of the outer ring and the rolling body, I i Is the moment of inertia of the inner ring; in the formula: m is in Denotes inner circle mass, I in Representing the moment of inertia of the inner ring, omega in Representing the inner ring rotation speed;
the potential energy equation generated by Z rolling bodies in the bearing is as follows:
Figure RE-GDA0003616122230000062
in the formula: z is a radical of out A Z-axis coordinate representing the center of the outer ring;
according to the spring-mass model of the ball bearing without the retainer in FIG. 2, the discrete groove causes the radial displacement m of the center of mass of the rolling element lsc Directly influencing the radial position eta of the rolling body in the kinetic energy equation and the potential energy equation of the rolling body i The relation is as follows:
Figure RE-GDA0003616122230000063
the contact between the rolling body and the inner ring and the outer ring is regarded as a nonlinear spring, the rigidity of the nonlinear spring is obtained by the Hertz elastic theory, and the expression of potential energy caused by the contact deformation of the spring is as follows:
Figure RE-GDA0003616122230000064
in the formula: k is a radical of in Is the nonlinear rigidity caused by the Hertz contact effect between the rolling body and the inner ring out Is the nonlinear rigidity caused by the Hertz contact effect between the rolling body and the outer ring, delta in For contact deformation of the i-th rolling element with the inner race, δ out For the ith rolling body to contact with the outer ringShaping; if (R) in +R b )＞ξ i The spring between the rolling body and the inner race is subjected to a compressive or restoring force, delta in ＝[(R in +R b )-ξ i ]If (r) in +R b )≤ξ i Then δ in =0; if R is out ＜(η i +R b ) The spring between the rolling body and the outer ring is compressed or restored to force delta out ＝[R out -(η i +R b )](ii) a If R is out ≥(η i +R b ) Then δ out ＝0。
Specifically, the contact rigidity between the i-th rolling element and the inner ring is:
Figure RE-GDA0003616122230000065
the contact rigidity between the ith rolling body and the outer ring is as follows:
Figure RE-GDA0003616122230000071
wherein R is ni Effective contact radius of the i-th rolling element and the inner ring, R wi Is the effective contact radius of the i-th rolling element and the outer ring, R out Is the radius of an inner raceway of an outer ring of a ball bearing without a retainer, L 10 、L 20 The lengths of the springs for balancing the rolling bodies with the inner ring contact spring model and the outer ring contact spring model are respectively.
In particular, when the rolling bodies are located in discrete grooves, i.e.
Figure RE-GDA0003616122230000072
The rolling body is separated from the inner ring, so that the moment xi i And =0. From FIG. 2, ξ i The expression of (a) is:
Figure RE-GDA0003616122230000073
the deformation of the spring between the rolling body and the inner ring is as follows:
Figure RE-GDA0003616122230000074
in the formula, # i (ψ i Not less than 0) is the position angle of the i-th rolling element, x in For displacement of the inner ring along the x-axis, z in For displacement of the inner ring along the z-axis, x out Is the displacement of the outer ring along the x-axis, z out Displacement of the outer ring along the z-axis;
xi is i To eta i (i＝1,2,...,Z)，x in ，z in The partial derivatives are respectively calculated to obtain:
Figure RE-GDA0003616122230000075
according to the study of Farhang K and Mehra K, the position angle of the ith rolling element is as follows:
Figure RE-GDA0003616122230000076
wherein:
Figure RE-GDA0003616122230000077
in the formula: omega i At inner ring rotation speed, R b Is the radius of the rolling body, α i The contact angle of the rolling element with the inner ring, d m The pitch circle diameter of the ball bearing without the retainer takes the center of the bearing as the center of a circle and the distance from the center of mass of the rolling body of the conventional outer ring raceway to the center of the bearing as the radius, which is called the pitch circle of the ball bearing without the retainer, Z is the number of the rolling bodies, t is time, psi 0 For the initial position angle of the rolling elements, generally psi 0 ＝0。
The invention has the beneficial effects that: the invention provides a method for reducing collision of rolling bodies of a retainer-free ball bearing, which avoids collision friction between the rolling bodies and rolling and provides the method for reducing collision of the retainer-free ball bearing.
Drawings
FIG. 1: an analysis chart of contact relation between the rolling bodies and the inner and outer rings;
FIG. 2 is a schematic diagram: a spring-mass model displacement analysis chart of the ball bearing without the retainer;
FIG. 3: an elliptical discrete slot schematic;
FIG. 4 is a schematic view of: cage-free ball bearing motion equation solving analysis chart
FIG. 5 is a schematic view of: the discrete groove is positioned in an inner ring phase diagram of the bearing area;
FIG. 6: the elliptical discrete groove is positioned in an inner ring phase diagram of the non-bearing area;
FIG. 7: an inner ring phase diagram of two elliptical discrete grooves;
FIG. 8: an inner ring phase diagram of three elliptical discrete grooves;
FIG. 9: a rolling body speed-displacement phase diagram of a conventional outer ring raceway;
FIG. 10: a rolling body speed-displacement phase diagram positioned in the elliptical discrete groove;
FIG. 11: and a poincare graph of an inner ring of the oval discrete groove bearing positioned in the bearing area.
Detailed Description
The present invention will now be described in further detail.
The invention discloses a method for reducing collision of rolling bodies of a non-retaining ball bearing, which comprises the following steps:
1) Establishing eta based on energy analysis i (i＝1,2,...,Z)、x in 、z in The differential equation of motion of (2) is specifically:
Figure RE-GDA0003616122230000091
wherein:
Figure RE-GDA0003616122230000092
in the formula, m b Is the mass of the rolling elements, eta i Is the radial position of the rolling body, wherein i (i =1, 2.. Multidot.Z), Z being the number of rolling elements, g being 9.8N/kg,. Multidot. i (ψ i Not less than 0) is the position angle of the i-th rolling element, k in For the contact stiffness between the ith rolling element and the inner ring, k out For the contact stiffness between the i-th rolling element and the outer ring, R in The radius of an inner ring outer raceway of the ball bearing without the retainer, R out Is the inner raceway radius R of the outer ring of the ball bearing without a retainer b Is the radius of the rolling body, xi i M is deformation of the spring between the rolling body and the inner race in Is the inner ring mass, x in Is the displacement of the inner ring along the x-axis, k is, z in For displacement of the inner ring along the z-axis, F r For radial forces acting on the bearing system, x out Is the displacement of the outer ring along the x-axis, z out Displacement of the outer ring along the z-axis;
2) Respectively substituting the formulas (4), (5) and (6) into the formulas (1), (2) and (3), and solving eta by Runge-Kutta method i (i＝1,2,...,Z)、x in 、z in The motion differential equation is used for solving the speed and the displacement of the inner ring rolling body;
3) And optimizing discrete groove parameters according to the speed and the displacement of the inner ring rolling body.
For convenience of demonstration, the shape parameters of the grooves are defined as representative symmetrical shape ellipses, the differential equations (1), (2) and (3) of the motion of the ball bearing without the retainer are solved numerically by adopting a fourth-order Runge-Kutta method, and the radial speed and displacement of the rolling elements in two directions of the x axis and the z axis of the center of the bearing inner ring with different discrete groove parameters are obtained. The parameter values to be selected by the cage-free ball bearing model are shown in the table 1-1:
TABLE 1-1 model parameters for a cage-less ball bearing
Figure RE-GDA0003616122230000101
In order to facilitate the comparative analysis of discrete grooves with different planar projection shapes, m is taken lsc The maximum value of the rolling body center of mass is 1mm, the radial displacement of the rolling body center of mass caused by the oval discrete groove (as shown in figure 3) is constantly changed along with time, in order to distinguish the slight difference of the discrete grooves with different shapes, the radial displacement value of the rolling body at a certain position caused by the oval discrete groove is 0.8mm respectively, and meanwhile, when the discrete groove is positioned in a bearing area, the initial position of the inner ring center is (0, 0); when the discrete slot is positioned in the bearing area, the initial position of the center of the inner ring is taken as (0, 0.5), and the solving time is 0.1s.
The influence of the positions and the numbers of the discrete grooves on the motion stability of each part in the bearing is obtained by comparing and analyzing the speed-displacement phase diagrams of the center of mass and the center of the inner ring of the rolling element of the ball bearing without the retainer with the discrete groove parameters, and a specific solving method and a program solving flow chart are shown in fig. 4.
As shown in FIG. 5, the velocity-displacement phase diagram of the center of the inner ring of the ball bearing without the cage is also a closed elliptic curve with a certain width, and the velocity range of the center of the inner ring along the x-axis direction is-1.5 multiplied by 10 4 ～1.5×10 4 Mu m/s, displacement range of 0-1.9 mu m, speed range along the z-axis direction of-6000 mu m/s, and displacement range of 0.5-1.38 mu m.
The speed-displacement phase diagram of the discrete-groove ball bearing without the retainer in the directions of the x axis and the z axis is a closed elliptic curve with a certain width, which shows that the motion of the inner ring of the ball bearing is stable quasi-periodic motion. In fig. 5, the velocity-displacement phase diagram in the x-axis direction is almost the same, however, when the velocity-displacement phase diagram in the z-axis direction is observed, it is found that, when the number of the discrete grooves of the cage-free ball bearing with the oval discrete grooves is one and is located in the bearing area, the discrete grooves cause the inner ring of the cage-free ball bearing to vibrate little, and the motion stability is good.
When the ball bearing system without the retainer has a longer period, it is difficult to judge whether the system is in periodic motion or chaotic motion only through the track of a phase diagram, so that a more effective method is adopted for judgment, namely a Poincare section method. The initial parameters are kept unchanged, the solving time is prolonged to be 1s, 50 intersection points are provided between the motion track of the inner ring and a certain section according to the rotating speed of the inner ring of 3000r/min, and therefore 50 points exist on the drawn Poincare section. As shown in fig. 11, whether the motion of the inner ring of the ball bearing without the cage, which has one discrete groove located in the bearing area, in the z-axis direction is periodic motion or chaotic motion is judged by a poincare section method.
If the Poincare section diagram is a single point or a limited number of concentrated points, the inner ring movement is a single period movement or a multiple period movement; if the points displayed on the poincare section diagram are closely distributed on a closed curve, the inner ring motion is quasi-periodic motion; if infinite points exist on the Poincare sectional diagram and no rule exists, the motion of the inner ring is chaotic. According to the characteristics of the poincare section view in fig. 11, the motion of the inner ring of the ball bearing without the retainer along the z-axis direction is quasi-periodic motion. Observing fig. 11, it can be seen that when the retainer-less ball bearing has an elliptical discrete groove in the bearing area, the inner ring has high running accuracy and stability.
When the number of the discrete grooves is one and the discrete grooves are located in a non-bearing area, the shape of the plane projection of the discrete grooves has influence on the movement of the inner ring, the displacement range of the central shaft direction of the inner ring of the ball bearing without the retainer of the discrete grooves is slightly smaller, and when the number of the discrete grooves is one and the discrete grooves are located in the non-bearing area, the elliptical discrete grooves enable the inner ring of the ball bearing without the retainer to vibrate less and the movement stability is better.
When the discrete grooves are located in the bearing area and the non-bearing area, the number of the rolling bodies bearing radial loads is different, and the positions of the discrete grooves inevitably have different influences on the axial movement of the inner ring of the ball bearing without the cage, for example, fig. 6 is a speed-displacement phase diagram of the ball bearing without the cage, which has one discrete groove located in the non-bearing area and is located in the direction of the central axis of the inner ring.
As shown in FIG. 7, when there are two discrete grooves with a position angle interval of π in the ball bearing without cage, the velocity-displacement phase diagram of the center of the inner ring in both the x-axis and z-axis directions is shown, wherein one discrete groove is located in the bearing region and the other discrete groove is located in the non-bearing regionThe line width. Meanwhile, the speed range of the center of the inner ring along the direction of the x axis is-1.5 multiplied by 10 4 ～1.5×10 4 Mu m/s, a displacement range of 0-2 mu m, a speed range of-7000 mu m/s along the z-axis direction, and a displacement range of 0.5-1.45 mu m.
The number of the discrete grooves is two, one of the discrete grooves is located in a bearing area, the other discrete groove is located in a non-bearing area, speed-displacement phase diagrams of the center of the inner ring of the ball bearing without the retainer are in the x-axis direction and the z-axis direction, the speed-displacement phase diagrams are closed curves, so that the movement of the inner ring has obvious periodicity, the vibration of the inner ring of the ball bearing without the retainer is small, and the movement stability is good.
Fig. 8 shows a velocity-displacement phase diagram of the center of the inner ring with three elliptical discrete grooves in a cage-less ball bearing. As can be seen from the figure, speed-displacement phase diagrams in the x-axis direction and the z-axis direction of the center of the inner ring of the ball bearing without the retainer are closed elliptic curves with certain width, and the width of the curve in the z-axis direction is slightly larger than that of the curve in the x-axis direction. Meanwhile, the speed range of the center of the inner ring along the direction of the x axis is-1.5 multiplied by 10 4 ～1.5×10 4 μ m/s, displacement range of 0-2.3 μm, velocity range along z-axis of-1 × 10 4 ～1×10 4 Mu m/s, and the displacement range is 0.5-1.7 mu m.
The number of the discrete grooves is three, one discrete groove is located in the bearing area, the other discrete groove is located in the non-bearing area, the speed-displacement phase diagram of the center of the inner ring of the ball bearing without the retainer along the x-axis direction is a closed circular curve, and the speed-displacement phase diagram along the z-axis direction is a closed elliptic curve with a certain width.
Regarding the preferred position of the discrete groove, when the number of the discrete groove is one, the speed and the displacement variation range of the inner ring along the z-axis direction are smaller when the discrete groove is positioned in the bearing area than when the discrete groove is positioned in the non-bearing area. Therefore, when a discrete groove located in the bearing area exists, the inner ring of the ball bearing without the retainer has stronger motion periodicity and more stable motion.
The radial displacement and speed ranges of the center of mass of the rolling body are different when the rolling body is positioned in the conventional raceway and the elliptical discrete groove, as shown in fig. 9 and 10, the radial displacement and speed phase diagram of the center of mass of the rolling body is when the rolling body is positioned in the conventional raceway and the elliptical discrete groove respectively. When the rolling bodies are positioned on the conventional outer ring raceway, as shown in FIG. 9, the phase diagram of the speed-displacement of the rolling bodies in the ball bearing without the retainer is a plurality of closed elliptic curves, and the speed range is-8 multiplied by 10 7 ～8×10 7 Mu m/s, and the displacement range is-100 to 100 mu m. When the rolling bodies are positioned in the oval discrete grooves, as shown in fig. 10, the phase diagram of the speed-displacement of the rolling bodies in the ball bearing without the retainer is a plurality of closed oval curves, and the speed range is-3 multiplied by 10 7 ～3×10 7 Mu m/s, and the displacement range is-70 to 60 mu m.
The radial displacement-speed phase diagram of the rolling body shows that the phase diagram is not repeated and is full of phase space, so that the motion of the rolling body is chaotic. In the phase diagram, the speed variation range of the rolling body positioned in the discrete groove is much smaller than that of the rolling body positioned in the conventional outer ring raceway, and the reason is that the rolling body enters the discrete groove from the conventional outer ring raceway and is separated from the inner ring, so that the kinetic energy is reduced. According to the comparison between the radial displacement of the rolling body and the speed range in the figure, the elliptical discrete grooves are beneficial to inhibiting the chaotic motion of the rolling body in the radial direction.
When the ball bearing without the retainer is provided with one or two discrete grooves, the speed change range of the inner ring along the x-axis direction is consistent and is slightly smaller than the speed change range of the inner ring along the x-axis direction when the ball bearing is provided with three discrete grooves. The displacement variation range of the inner ring of the ball bearing without the retainer along the x-axis direction, the speed along the z-axis direction and the displacement variation range are increased along with the increase of the number of the discrete grooves. Therefore, the theoretically preferable result of the number of discrete grooves is also one.
When a discrete groove located in a bearing area exists in the ball bearing without the retainer, the inner ring has higher motion precision and stronger stability.
In light of the foregoing description of the preferred embodiment of the present invention, many modifications and variations will be apparent to those skilled in the art without departing from the spirit and scope of the invention. The technical scope of the present invention is not limited to the content of the specification, and must be determined according to the scope of the claims.

Claims (7)

1. A method for reducing collision of rolling bodies of a ball bearing without a retainer comprises the following steps:
1) Establishing eta based on energy analysis i ，i＝1,2,...,Z、x in 、z in The differential equation of motion of (2) is specifically:
Figure FDA0003747954810000011
wherein:
Figure FDA0003747954810000012
in the formula, m b Is the mass of the rolling elements, eta i Is the radial position of the rolling elements, wherein i =1, 2.. Multidot.Z, Z is the number of rolling elements, and g is 9.8N/kg,. Multidot. i ，ψ i Not less than 0 is the position angle of the i-th rolling element, k in For the contact stiffness between the ith rolling element and the inner ring, k out For the contact stiffness between the i-th rolling element and the outer ring, R in Is the outer raceway radius, R, of the inner ring of the ball bearing without the retainer out Is the inner raceway radius R of the outer ring of the ball bearing without a retainer b Is the radius of the rolling body, xi i M is deformation of the spring between the rolling body and the inner race in Is the inner ring mass, x in For displacement of the inner ring along the x-axis, z in For displacement of the inner ring along the z-axis, F r For radial forces acting on the bearing system, x out Is the displacement of the outer ring along the x-axis, z out Displacement of the outer ring along the z-axis;
2) Respectively substituting the formulas (4), (5) and (6) into the formulas (1), (2) and (3), and solving eta by Runge-Kutta method i ，i＝1,2,...,Z、x in 、z in The motion differential equation is used for solving the speed and the displacement of the inner ring rolling body;
3) And optimizing discrete groove parameters according to the speed and the displacement of the inner ring rolling body.
2. A method of reducing rolling element collisions for a retainment-less ball bearing, as claimed in claim 1, wherein: establishing eta i ，i＝1,2,...,Z、x in 、z in The motion differential equation comprises the following specific steps:
11 Lagrange equations to build a set of independent generalized coordinate systems:
Figure FDA0003747954810000021
in the formula: t, V, p and f are kinetic energy, potential energy, generalized degree of freedom coordinate vectors and generalized contact force vectors respectively;
12 Assuming the outer ring is stationary, the total kinetic energy in the ball bearing without the cage is the sum of the kinetic energy of the rolling elements and the kinetic energy of the inner ring, and the equation is:
Figure FDA0003747954810000022
in the formula: t is roller 、T in 、T out Respectively representing kinetic energy of the rolling body, kinetic energy of the inner ring and kinetic energy of the outer ring;
13 The bearing center is selected as a gravitational potential energy zero position, the total potential energy in the ball bearing without the retainer is the sum of the potential energy of the rolling body, the potential energy of the inner ring, the potential energy of the outer ring and the potential energy of the spring, and the equation is as follows:
Figure FDA0003747954810000023
in the formula: v roller 、V in 、V out Respectively represents the potential energy of the rolling body, the potential energy of the inner ring and the potential energy of the outer ring, V t Is potential energy generated by the contact of the rolling body and the nonlinear spring between the inner ring and the outer ring;
substituting the formulas (8) and (9) into the formula (7), and respectively aligning with eta i ，i＝1,2,...,Z、x in 、z in Derivation to obtain η i ，i＝1,2,...,Z、x in 、z in Differential equation of motion.
3. A method of reducing rolling element collisions for a retainment-less ball bearing, as claimed in claim 2, wherein: the kinetic energy of the rolling body, the kinetic energy of the inner ring and the kinetic energy of the outer ring are as follows:
Figure FDA0003747954810000024
in the formula: m is the mass of the rolling body, R o Is the radius of the contact point of the outer ring and the rolling body, I i Is the moment of inertia of the inner ring;
Figure FDA0003747954810000031
in the formula: m is in Denotes inner circle mass, I in Representing the moment of inertia of the inner ring, omega in Representing the inner ring rotation speed;
T out ＝0 (12)。
4. a method of reducing rolling element collision for a retainfree ball bearing as set forth in claim 2, wherein: the rolling body potential energy, the inner ring potential energy, the outer ring potential energy and the spring potential energy are specifically as follows:
Figure FDA0003747954810000032
in the formula: z is a radical of out A Z-axis coordinate representing the center of the outer ring;
V in ＝m in g(z out -z in_out ) (14)
in the formula: z is a radical of in_out Representing the relative coordinates of the center of the inner ring relative to the center of the outer ring;
Figure FDA0003747954810000033
in the formula: m is out Representing the mass of the outer ring;
Figure FDA0003747954810000034
in the formula: k is a radical of in Is the nonlinear rigidity caused by the Hertz contact effect between the rolling body and the inner ring out Is the nonlinear rigidity caused by the Hertz contact effect between the rolling body and the outer ring, delta in For the i-th rolling element to deform in contact with the inner race, δ out The ith rolling body is deformed by contacting with the outer ring; if (R) in +R b )＞ξ i The spring between the rolling body and the inner race is subjected to a compressive or restoring force, delta in ＝[(R in +R b )-ξ i ]If (R) in +R b )≤ξ i Then δ in =0; if R is out ＜(η i +R b ) The spring between the rolling body and the outer race is subjected to a compressive or restoring force, delta out ＝[R out -(η i +R b )](ii) a If R is out ≥(η i +R b ) Then δ out ＝0。
5. A method of reducing rolling element collisions for a retainment-less ball bearing, as claimed in claim 1, wherein: the contact rigidity between the ith rolling body and the inner ring is as follows:
Figure FDA0003747954810000041
the contact rigidity between the ith rolling body and the outer ring is as follows:
Figure FDA0003747954810000042
wherein R is ni Effective contact radius of the i-th rolling element and the inner ring, R wi Effective contact radius of the i-th rolling element with the outer ring, R out Is without a retainerRadius of raceway in outer ring of ball bearing, L 10 、L 20 The lengths of the springs for balancing the rolling bodies with the inner ring contact spring model and the outer ring contact spring model are respectively.
6. A method of reducing rolling element collisions for a retainment-less ball bearing, as claimed in claim 1, wherein: the deformation of the spring between the rolling body and the inner ring is as follows:
Figure FDA0003747954810000043
in the formula, # i ,ψ i Not less than 0 is the position angle of the ith rolling element, x in For displacement of the inner ring along the x-axis, z in For displacement of the inner ring along the z-axis, x out Is the displacement of the outer ring along the x-axis, z out Is the displacement of the outer ring along the z-axis.
7. A method of reducing rolling element collisions for a retainment-less ball bearing, as claimed in claim 1, wherein: the position angle of the ith rolling body is as follows:
Figure FDA0003747954810000044
wherein:
Figure FDA0003747954810000045
in the formula: omega i At inner ring rotation speed, R b Is the radius of the rolling body, α i The contact angle of the rolling element with the inner ring, d m The pitch circle diameter of the ball bearing without the retainer takes the center of the bearing as the center of a circle and the distance from the center of mass of the rolling body of the conventional outer ring raceway to the center of the bearing as the radius, which is called the pitch circle of the ball bearing without the retainer, Z is the number of the rolling bodies, t is time, psi 0 For the initial position angle of the rolling elements, generally psi 0 ＝0。
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Patent Citations (2)

* Cited by examiner, † Cited by third party
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