CN106560816B - Method for analyzing influence factors of dynamic stiffness of rolling bearing - Google Patents
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Abstract
The method for analyzing the influence factors of the dynamic stiffness of the rolling bearing is based on the following preconditions: the contact between the rolling bodies of the rolling bearings and the inner and outer ring raceways is considered to meet the Hertz contact theory, and the contact type between the rolling bodies and the inner and outer ring raceways of the bearings is point contact or/and line contact; the Hertz rigidity of the bearing meets the following requirements: under the action of a contact load Q, the total deformation delta of one rolling body of the rolling bearing is deltain+δouWherein δinThe amount of deformation of the rolling elements in the direction of the inner race, δouThe amount of deformation delta can be determined for the amount of deformation of the rolling elements in the direction of the outer ring, the hertzian contact stiffness a0Can also obtain; the deformation delta is used for measuring the dynamic stiffness of the rolling bearing. The invention carries out deep analysis aiming at the influence factors of the dynamic stiffness of the rolling bearing, provides a theoretical basis for reasonable selection of the bearing and reduces the vibration of the bearing during working.
Description
Technical Field
The invention carries out deep analysis and research aiming at a deep groove ball bearing of a transmission system, and belongs to the technical field of theoretical analysis and related application of dynamic rigidity of a rolling bearing; in particular to an analysis method for influence factors of dynamic stiffness of a rolling bearing.
Background
When the rolling bearing rotates, the rolling body contacts with the raceway to generate various rotating motions and friction, the rigidity, the bearing capacity and even the service life of the bearing mainly depend on the working condition of the bearing and the contact property between the rolling body and the raceway, so the research on the dynamic problem of the rolling bearing contact has become an important research direction for researchers (see the documents: bear Xiao jin, Zhang \40525, and the like; nonlinear finite element analysis of the rolling bearing contact [ J ]. test technical report, 2009, 23 (1): 23-27.). The dynamic stiffness of the rolling bearing refers to the instantaneous stiffness of the bearing in the motion process, which is represented by the comprehensive stiffness of each rolling body in contact with the raceway, and almost all bearing stiffness analysis is mostly static analysis. The application of finite element analysis and simulation in the field of rolling bearing dynamics has become widespread (see document 1: maskshire 22773, japan. finite element analysis of the rolling bearing contact problem [ J ] mechanical design and manufacture, 2010, 9: 8-9. document 2: muzzle, old man. rolling bearing elastic contact dynamic characteristics finite element analysis [ J ] mechanical strength, 2011, 33 (5): 708-.
The dynamic stiffness of the rolling bearing is influenced by various factors, except the structure and the size, the pre-tightening state, the external load property, the bearing rotating speed and the like influence the stiffness of the rolling bearing, so that the stiffness change rule of the rolling bearing is very complex and a plurality of nonlinear factors are involved (see document 3: Yanxian, Liuwenxiu. tapered roller bearing dynamic stiffness analysis [ J ] bearing, 2002, 2: 1-3. document 4: Chenzhuchu, crown warrior, Yanxuannian. cylindrical roller bearing dynamic stiffness analysis [ J ] bearing, 2007, 4: 1-5).
People hope to obtain an analysis method for the influence factors of the dynamic stiffness of the rolling bearing with excellent technical effect.
Disclosure of Invention
The invention aims to provide a method for analyzing influence factors of dynamic stiffness of a rolling bearing, which has excellent technical effect.
The invention provides a method for analyzing influence factors of dynamic stiffness of a rolling bearing, which is characterized by comprising the following steps of:
the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing is based on the following premises: the contact between the rolling elements of the rolling bearing and the inner and outer ring raceways is considered to satisfy the Hertz (Hertz) contact theory, and the contact type between the rolling elements and the inner and outer ring raceways of the bearing is point contact or/and line contact, wherein the point contact is adopted between the rolling elements of the ball bearing and the inner and outer rings, and the line contact is adopted between the cylindrical and tapered roller bearings; the relevant specific requirements of Hertz (Hertz) contact theory are as follows:
① ball bearing, the contact between the rolling element and the inner and outer races is point contact, under the action of load Q, the contact point gradually expands to an elliptical contact surface, and the maximum stress is sigma at the center point of ellipsemaxThe ellipse has a major axis length of 2a and a minor axis length of 2b, as shown in fig. 1, and has a stress distribution in the contact region, as shown in fig. 2. From the hertz contact theory:
in the formula, Q is the contact load of the rolling element and the inner and outer ring raceways; a is the contact ellipse area major semi-axis length; b is the short semi-axis length; sigmamaxIs the contact maximum stress; delta is the deformation; the equivalent elastic modulus E' is expressed as:
in the formula, E1，E2，μ1，μ2The elastic modulus and Poisson's ratio of the rolling element and the inner and outer ring raceways respectively;
② the principal curvature sum Σ ρ is the sum of the principal curvatures at the contact of the rolling elements and the raceways, i.e.:
∑ρ＝ρ11+ρ12+ρ21+ρ22(6)
maand mbMajor and minor axis coefficients, respectively, are expressed as:
wherein k is an ellipticity,
e is the eccentricity of the ellipse and the eccentricity of the ellipse,
k (e) and l (e) are the first and second types of integrals, respectively, associated with the eccentricity of the ellipse:
the ellipse eccentricity e is related to the ellipticity by:
the principal curvature function F (ρ) is expressed as:
also expressed as:
if the geometric dimensions of each part of the bearing are known, the principal curvature function F (rho) can be obtained from the formula (13) -the formula (17) and the formula (11); substituting the expressions (9) and (10) into the expression (12) to obtain k, and obtaining e by the expression (10), thereby measuring the bearing rigidity;
③ under the action of the contact load Q between the rolling body and the inner and outer rings, the coefficients of the major and minor semi-axes can be obtained by equations (7) and (8), and the maximum contact stress sigma of the contact zone can be obtained by equations (1) and (4)maxAnd a deformation amount δ;
for a point contact ball bearing, the main curvatures are:
rolling ball:
inner ring:
outer ring:
in the formula, DbThe diameter of the rolling ball;
α is contact angle, D is inner diameter of outer ring, D is outer diameter of inner ringmIs the diameter of the central circle of the rolling body; r ismIs the inner raceway radius; r isouIs the outer raceway radius; taking a deep groove ball bearing as an example, the main geometric parameters are shown in fig. 3.
And (II) the Hertz rigidity of the bearing meets the following requirements:
under the action of a contact load Q, the total deformation delta of one rolling body of the rolling bearing is deltain+δouWherein δinThe amount of deformation of the rolling elements in the direction of the inner race, δouThe amount of deformation of the rolling elements in the outer race direction is given by equation (4):
the amount of deformation δ is then expressed as:
where K is the contact stiffness coefficient, then:
the contact load Q is expressed as:
formula (21) is a formula for calculating the contact force between a single rolling element and the inner and outer races of the bearingWhere δ is the total contact deformation of the contact and Q is the contact force, the Hertz contact stiffness a0Expressed as:
where Σ ρinThe main curvature sum of the contact of the rolling body and the inner ring; sigma rhoouIs the sum of the main curvatures of the outer ring;
and
the coefficients of the long axes of the inner and outer rings are respectively; k (e)inAnd K (e)ouA first type of integral which is related to eccentricity of the inner and outer rings;
here, the deformation δ is used as a measure of the dynamic stiffness of the rolling bearing.
The method for analyzing the influence factors of the dynamic stiffness of the rolling bearing is characterized by comprising the following steps:
the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing further meets the following requirements: calculating the dynamic stiffness under the action of the dynamic radial load, establishing a three-dimensional finite element model of the bearing, setting reasonable boundary conditions, applying the radial load, establishing a relational expression of the bearing load and the displacement, and further solving the dynamic stiffness of the bearing;
specifically, the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing analyzes and expresses the dynamic stiffness of the rolling bearing through the dynamic stiffness of the bearing under a sinusoidal excitation force; the specific requirements are as follows:
under the condition of elastic contact, the relation between the radial constant load and the radial displacement in the load direction of the bearing is studied in the allowable radial load range without considering the bending deformation of the bearing retainer and the shaft, so as to represent the change of the radial rigidity of the bearing; specifically, the dynamic stiffness of the bearing is a ratio of a variation of an external load applied to the bearing to a variation of a bearing displacement, and therefore, in order to calculate the dynamic stiffness of the bearing, a relationship between the load applied to the bearing and the displacement must be established, and the relationship is expressed by a formula:
δ ═ f (Q) or Q ═ f (δ) (23)
Where δ is the amount of deformation in mm; q is the contact load of the rolling body and the inner and outer ring raceways, and the unit is N; the dynamic stiffness of the bearing is then expressed as:
the dynamic characteristics of a rolling bearing as a single-degree-of-freedom system are expressed as follows (see the literature: Li pure, Hongjun, Zhang Xinhua, and the like; experimental research on dynamic stiffness of angular contact ball bearings [ J ]. school report of Sian traffic university, 2013, 47 (7): 68-72.):
Z(ω)＝K(ω)-mω2+iC(ω)ω (25)
in formula (25): k (ω) is the rolling bearing equivalent stiffness: m is the equivalent vibration mass of the rolling bearing; omega2An inertia term which is the equivalent stiffness of the rolling bearing; omega is the angular frequency of rotor operation; c (omega) is the equivalent damping coefficient of the rolling bearing;
when the rolling bearing is in operation, a radial load synchronous with the rotating speed is directly applied, and corresponding displacement generated by the center of the bearing is measured to obtain:
in the formula: q is a rotating radial force which acts on the inner ring of the bearing and is synchronous with the running speed of the bearing; delta is the radial displacement of the bearing center;
is the phase difference between the displacement and the force vector.
The method for analyzing the influence factors of the dynamic stiffness of the rolling bearing is characterized by comprising the following steps: the first problem to be solved by kinetic analysis of rolling bearings is contact analysis. The rolling bearing simulation dynamics method is based on a contact theory, but the boundary condition of a semi-infinite space of the contact theory is only suitable for analyzing objects with simple shapes and cannot meet the conditions of complex structures and complex loads. Furthermore, if the geometry of the contact body is too small or the applied load is too large, the assumption of the contact theory that the ratio of the size of the contact surface to the radius of curvature of the surface of the contact body is small cannot be satisfied, thereby limiting its range of use (see article: Pijiasen. Rolling bearing dynamic load distribution and stress expression research [ D ]. Wuhan: Wuhan university of science 2012.). The dynamic stiffness of the deep groove ball bearing of the input shaft of the transmission is calculated under the action of dynamic radial load by adopting an explicit finite element method, a three-dimensional finite element model of the bearing is established, reasonable boundary conditions are set, radial load is applied, a relational expression of bearing load and displacement is established, and further the dynamic stiffness of the bearing is obtained. Table 1 shows the dimensional parameters of the deep groove ball bearing entity, and table 2 shows the material parameters of each part of the bearing.
Establishing a three-dimensional finite element model of the bearing according to the geometric dimensions in the table 1, as shown in fig. 4;
the axis of the bearing coincides with a z bearing of the whole Cartesian coordinate system, and the xy plane is a symmetrical plane of the bearing; linear hexahedron units are selected, and the bearing rolling elements are subjected to grid refinement, wherein the total number of model units is 23034; taking the static friction coefficient between the rolling body and the inner and outer ring raceways as 0.002, the dynamic friction coefficient as 0.0018, the whole model adopting the properties of linear elastic material, the bearing material GCr15 steel, the elastic modulus as 2.07 x 105MPa, Poisson's ratio of 0.3, static friction coefficient of 0.1, and dynamic friction coefficient of 0.06;
according to the practical use condition of the bearing, the complete constraint is applied to the outer surface of the outer ring of the bearing to describe the limiting effect of the bearing seat on the bearing, a rigid shaft and a rigid retainer are built on the inner ring of the bearing, the rotational freedom degree of the retainer around the axial direction is constrained, and an axial force F is applied to the rigid shaft of the inner ring of the bearingaApplying radial sine excitation load F on the rigid shaft of the bearing inner ringrAs shown in fig. 5, if the loading frequency is f:
wherein omega is the bearing inner ring corner frequency, the variable stiffness characteristic under the dynamic excitation of the bearing is simulated by adopting an explicit nonlinear finite element, the total calculation time is 0.1s, and the final contact related parameters are determined by comprehensively considering two factors of calculation time and calculation result precision;
the analysis takes the calculation time into consideration, and controls the penetration amount to be within 5% of the total displacement by adjusting the parameters so as to obtain a more accurate calculation result. The cloud chart of the maximum contact stress of the bearing at 4.6801ms is shown in fig. 6, and it can be seen that the maximum stress is 1222MPa when the bearing inner ring is at the uppermost position and the bearing is loaded at the maximum at the peak value of the sinusoidal radial excitation force.
Model loading frequency F =50Hz, FaWhen the amplitude of the radial force is 18567N, the radial load of the rolling bearing forms a hysteresis loop with the radial displacement due to the geometric nonlinearity of the distribution of the rolling bodies and the nonlinear influence of the contact of the rolling bodies and the raceways, and the radial displacement of the bearing has obvious nonlinear hysteresis characteristics.
The method for analyzing the influence factors of the dynamic stiffness of the rolling bearing is characterized by comprising the following steps:
the dynamic stiffness of the bearing under odd pressure and even pressure loading meets the following requirements:
the bearing rotates through the loaded area in sequence, and undergoes the processes of maximum deformation from contact to contact and then deformation recovery, namely two typical load bearing conditions are presented as shown in fig. 9 and fig. 10 in sequence: in odd-pressure and even-pressure states, the rollers are symmetrically distributed according to radial loads;
during the rotation of the rolling body, the loading state of the rolling body changes along with the position of the rolling body, so that alternating elastic force is caused. When acting in a direction by a radial load, periodic vibration is induced. The vibration frequency is:
f＝Zfc(28)
wherein Z is the number of rolling elements; f. ofcFor turning the inner race of the bearingA dynamic frequency; when the bearing rotates, the vibration can be regarded as that the inner ring is switched between an odd pressure state and an even pressure state under the action of radial load, so that the inner ring is caused to reciprocate in the direction of the radial load;
FIG. 11 is a curve of load and radial displacement of a bearing under two loading modes of odd pressure and even pressure; the radial displacement of the bearing is gradually increased along with the increase of the external load, and when the external load is equal to the external load, the radial displacement of the bearing under the odd-pressure condition is smaller than that generated under the even-pressure condition; the load displacement relation of the bearing under the two loading modes of odd pressure and even pressure can be obtained according to the data in FIG. 11:
δ＝aQb(29)
in the formula, delta is the displacement of the inner ring of the bearing, Q is the radial load of the bearing, and a and b are corresponding fitting coefficients respectively; the expression of the rigidity of the bearing under two loading forms is deduced according to the formula (24), the radial rigidity of the bearing under the odd-pressure form is larger than that under the even-pressure form, and the difference of the rigidity values is larger when the load is larger, so that the dynamic behavior of the bearing is influenced importantly.
In the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing, the influence of different excitation factors on the dynamic stiffness of the bearing is considered, and the following requirements are met:
the study of dynamic stiffness of bearings is much more complex than static stiffness, which was conducted in the early 70 s by the american advanced air force laboratory (AFAPL), and a large number of bearings were plotted to show the bearing stiffness of different bearings under various load conditions for easy review by designers of high speed rotor systems. In the domestic research of rotor-rolling bearing systems, no similar such detailed work has been seen. However, the stiffness of the bearing in a specific situation is also shown in the large number of curves, and the bearing stiffness is not considered to be influenced by factors such as temperature, centrifugal force, axial force and initial play during actual operation (see the literature: Rottus, Sunxider, Wulinfeng. the bearing stiffness of the rolling bearing in any direction [ J]Journal of the university of avigation, Nanjing 1992, 24 (3): 248-256.). For analyzing variable stiffness characteristics of rolling bearing of transmission drive systemEstablishing a rigid shaft and a rigid retainer on the inner ring of the bearing by adopting an explicit dynamic finite element method, and applying an axial force F in the radial direction of the bearingaApplying sinusoidal excitation load F in radial direction of bearingrAnd f, analyzing the influence of the radial load loading frequency and the axial force on the dynamic stiffness of the rolling bearing when the radial clearance of the deep groove ball bearing at the input front end of the transmission is 0 by adopting the bearing dynamic stiffness correlation analysis method under the sine exciting force.
Fig. 12 is a curve of the variation of the radial dynamic stiffness of the bearing under different axial forces of the input front end bearing of the transmission along with the loading frequency, and it can be seen that the radial stiffness is increased along with the increase of the loading frequency, and the smaller the axial force is, the larger the radial stiffness of the bearing is. The stiffness of the bearing 1 under 25Hz excitation is compared with the static stiffness of the bearing 1 in the second chapter, and it can be seen that the dynamic stiffness is 22.103X 104N/mm, specific static stiffness 19.118X 104N/mm is 26% larger.
In the analysis method for the influence factors of the dynamic stiffness of the rolling bearing, one or a combination of the following three factors is also considered:
first, the influence of temperature
The elastic modulus of the material at the temperature of 20 ℃ (293K) is 207000MPa, and the bearing metal material (GCr15 steel) has the relation between the temperature and the linear expansion coefficient at the temperature of 0-200 ℃ at the low temperature (see the document: xi with heptyl. inorganic material thermophysics [ M ]. Shanghai: Shanghai science publisher, 1981. and the document: Pan stay, Jongqing, Du Xiao Yong. the relation between the linear expansion coefficient and Young modulus of the common alloy material at the high temperature [ J ]. Nature science report of Hunan university, 2000, 23 (2): 47-51.):
E＝E0(1-25αT) (30)
in the formula, E0The elastic modulus of the bearing material at 0 ℃, T is the Kelvin temperature of the bearing, α is the linear expansion coefficient of the bearing material, and α is 13.3 multiplied by 10 at low temperature-6From an elastic modulus E at 20 DEG C20207000MPa, giving a modulus of elasticity E of the bearing at 0 DEG C0229343.2MPa, the elastic modulus of the bearing at different temperatures is obtained by the formula (30);
in order to study the dynamic stiffness of the bearing at different temperatures, the dynamic stiffness change conditions of the bearing at the temperatures t of 20 ℃, 50 ℃, 100 ℃, 150 ℃ and 200 ℃ are respectively calculated, wherein the initial clearance of the bearing is 0, the loading frequency of the radial load is 50Hz, the amplitude of the radial force is 18567N, and the axial force is 1617N.
Fig. 13 shows the dynamic stiffness of the bearing at different temperatures along with the variation of the loading frequency, and it can be seen that the radial dynamic stiffness of the rolling bearing at normal temperature (t 20 ℃) increases along with the increase of the radial loading frequency, the radial loading frequency at higher temperature (t 50 ℃, 100 ℃, 150 ℃, 200 ℃) takes 75Hz as an inflection point, the temperature of the bearing decreases along with the increase of the radial loading frequency, and then increases along with the increase of the loading frequency, and the radial dynamic stiffness of the bearing decreases along with the increase of the temperature.
Second, influence of radial play
Fig. 14 shows the variation of the radial dynamic stiffness of the bearing with the radial load frequency in three types of clearances, namely, the clearances of the rolling bearing are respectively 0 μm, 10 μm and 20 μm, and it can be seen that the radial dynamic stiffness of the rolling bearing is increased with the increase of the radial load frequency in all three types of clearances, and the radial stiffness of the rolling bearing is gradually reduced with the increase of the radial clearance of the rolling bearing.
Third, the effect of the finite element mesh on the results
In order to analyze the influence of the number of meshes of the bearing finite element model on the simulation result, the bearing is divided into 3 number of finite element mesh models, as shown in FIGS. 15-17. The load working conditions of different rolling bearing grid models are as follows: bearing clearance of 0, axial load F a3000N, radial load Fr18567N, radial loading frequency 150 Hz.
The bearing finite element model adopts C3D8R linear hexahedral units, the grid model shown in FIG. 15 is a sparse grid, 18316 units are used in total, the bearing rigidity k obtained through simulation is 152853.81N/mm, and the simulation time of the model in one loading period is 21 min; the grid model shown in fig. 16 is a medium-scale grid, 23034 cells in total, and the obtained bearing stiffness k is 236664.44N/mm, and the simulation time of one loading cycle is 35 min; the grid model shown in fig. 17 is a large-scale grid with 46498 cells, and the bearing stiffness k is 235781.91N/mm, and the simulation time of one loading cycle is 108 min. The model shown in fig. 16 is a mesh model used in the calculation, and it can be seen that the number of cells of the mesh model shown in fig. 15-17 is gradually increased, the radial stiffness of the bearing obtained by the mesh model shown in fig. 15 and 16 is 83810.63N/mm, and the relative error is 35.4%, the radial stiffness of the bearing obtained by the mesh model shown in fig. 16 and 17 is 882.53N/mm, and the relative error is 0.37%, and the relative errors are small, but the calculation time of the model shown in fig. 17 is much longer than that of the model shown in fig. 16, and therefore the mesh model shown in fig. 17 used herein is reasonable.
In the related problems of the invention, due to the influence of geometric nonlinearity of the distribution of rolling elements of the rolling bearing and contact nonlinearity of the rolling elements and a raceway, the radial displacement of the rolling bearing has obvious hysteresis nonlinearity; the factors influencing the dynamic stiffness of the rolling bearing mainly comprise radial load frequency, axial load, bearing temperature, bearing initial clearance and the like, the radial dynamic stiffness is increased along with the increase of the radial load frequency and is reduced along with the increase of the axial load, meanwhile, the dynamic stiffness is generally reduced along with the increase of the bearing temperature and is reduced along with the increase of the radial clearance, and research results provide theoretical basis for reasonable selection of the bearing so as to reduce vibration of the bearing during working. The related principle of the invention can be popularized to all the related technical fields which can be transplanted. The method has potential huge economic value and social value.
Drawings
The invention is described in further detail below with reference to the following figures and embodiments:
FIG. 1 is a schematic diagram of a principle that a ball bearing gradually expands from point contact to elliptical contact between a rolling element and inner and outer ring raceways under the action of a load Q;
FIG. 2 is a schematic view of the stress distribution in the contact zone corresponding to FIG. 1;
FIG. 3 is a view of the geometric parameters of the ball bearing;
FIG. 4 is a deep groove ball bearing finite element model;
FIG. 5 is a radial load curve;
FIG. 6 is a cloud of bearing contact stresses;
FIG. 7 is a radial displacement curve of a bearing inner race;
FIG. 8 is a radial force versus displacement curve;
FIG. 9 is a schematic diagram of an odd pressure loading mode;
FIG. 10 is a schematic diagram of the even pressure loading mode;
FIG. 11 is a schematic representation of bearing radial displacement versus radial load;
FIG. 12 is a schematic view of bearing radial stiffness versus loading frequency;
FIG. 13 is a graph showing the variation of radial dynamic stiffness of a bearing with loading frequency at different temperatures;
FIG. 14 is the variation of the dynamic stiffness of the rolling bearing with the radial load frequency under different radial play;
FIG. 15 is a sparse mesh model of a rolling bearing;
FIG. 16 is a medium mesh model of a rolling bearing;
fig. 17 is a large sparse mesh model of a rolling bearing.
Detailed Description
Example 1
A method for analyzing influence factors of dynamic stiffness of a rolling bearing,
the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing is based on the following premises: the contact between the rolling elements of the rolling bearing and the inner and outer ring raceways is considered to satisfy the Hertz (Hertz) contact theory, and the contact type between the rolling elements and the inner and outer ring raceways of the bearing is point contact or/and line contact, wherein the point contact is adopted between the rolling elements of the ball bearing and the inner and outer rings, and the line contact is adopted between the cylindrical and tapered roller bearings; the relevant specific requirements of Hertz (Hertz) contact theory are as follows:
① ball bearing, the contact between the rolling element and the inner and outer races is point contact, under the action of load Q, the contact point gradually expands to an elliptical contact surface, and the maximum stress is sigma at the center point of ellipsemaxThe ellipse has a major axis length of 2a and a minor axis length of 2b, as shown in fig. 1, and has a stress distribution in the contact region, as shown in fig. 2. From the hertz contact theory:
in the formula, Q is the contact load of the rolling element and the inner and outer ring raceways; a is the contact ellipse area major semi-axis length; b is the short semi-axis length; sigmamaxIs the contact maximum stress; delta is the deformation; the equivalent elastic modulus E' is expressed as:
in the formula, E1，E2，μ1，μ2The elastic modulus and Poisson's ratio of the rolling element and the inner and outer ring raceways respectively;
② the principal curvature sum Σ ρ is the sum of the principal curvatures at the contact of the rolling elements and the raceways, i.e.:
∑ρ＝ρ11+ρ12+ρ21+ρ22(6)
maand mbMajor and minor axis coefficients, respectively, are expressed as:
wherein k is an ellipticity,
e is the eccentricity of the ellipse and the eccentricity of the ellipse,
k (e) and l (e) are the first and second types of integrals, respectively, associated with the eccentricity of the ellipse:
the ellipse eccentricity e is related to the ellipticity by:
the principal curvature function F (ρ) is expressed as:
also expressed as:
if the geometric dimensions of each part of the bearing are known, the principal curvature function F (rho) can be obtained from the formula (13) -the formula (17) and the formula (11); substituting the expressions (9) and (10) into the expression (12) to obtain k, and obtaining e by the expression (10), thereby measuring the bearing rigidity;
③ under the action of the contact load Q between the rolling body and the inner and outer rings, the coefficients of the major and minor semi-axes can be obtained by equations (7) and (8), and the maximum contact stress sigma of the contact zone can be obtained by equations (1) and (4)maxAnd a deformation amount δ;
for a point contact ball bearing, the main curvatures are:
rolling ball:
inner ring:
outer ring:
in the formula, DbThe diameter of the rolling ball;α is contact angle, D is inner diameter of outer ring, D is outer diameter of inner ringmIs the diameter of the central circle of the rolling body; r ismIs the inner raceway radius; r isouIs the outer raceway radius; taking a deep groove ball bearing as an example, the main geometric parameters are shown in fig. 3.
And (II) the Hertz rigidity of the bearing meets the following requirements:
under the action of a contact load Q, the total deformation delta of one rolling body of the rolling bearing is deltain+δouWherein δinThe amount of deformation of the rolling elements in the direction of the inner race, δouThe amount of deformation of the rolling elements in the outer race direction is given by equation (4):
the amount of deformation δ is then expressed as:
where K is the contact stiffness coefficient, then:
the contact load Q is expressed as:
formula (21) is a formula for calculating the contact force between a single rolling element and the inner and outer rings of the bearing, wherein delta is the total contact deformation of the contact, Q is the contact force, and the Hertz contact stiffness a0Expressed as:
where Σ ρinThe main curvature sum of the contact of the rolling body and the inner ring; sigma rhoouIs the sum of the main curvatures of the outer ring;
andthe coefficients of the long axes of the inner and outer rings are respectively; k (e)inAnd K (e)ouA first type of integral which is related to eccentricity of the inner and outer rings; here, the deformation δ is used as a measure of the dynamic stiffness of the rolling bearing.
The method for analyzing the influence factors of the dynamic stiffness of the rolling bearing further meets the following requirements: calculating the dynamic stiffness under the action of the dynamic radial load, establishing a three-dimensional finite element model of the bearing, setting reasonable boundary conditions, applying the radial load, establishing a relational expression of the bearing load and the displacement, and further solving the dynamic stiffness of the bearing;
specifically, the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing analyzes and expresses the dynamic stiffness of the rolling bearing through the dynamic stiffness of the bearing under a sinusoidal excitation force; the specific requirements are as follows:
under the condition of elastic contact, the relation between the radial constant load and the radial displacement in the load direction of the bearing is studied in the allowable radial load range without considering the bending deformation of the bearing retainer and the shaft, so as to represent the change of the radial rigidity of the bearing; specifically, the dynamic stiffness of the bearing is a ratio of a variation of an external load applied to the bearing to a variation of a bearing displacement, and therefore, in order to calculate the dynamic stiffness of the bearing, a relationship between the load applied to the bearing and the displacement must be established, and the relationship is expressed by a formula:
δ ═ f (Q) or Q ═ f (δ) (23)
Where δ is the amount of deformation in mm; q is the contact load of the rolling body and the inner and outer ring raceways, and the unit is N; the dynamic stiffness of the bearing is then expressed as:
the dynamic characteristics of a rolling bearing as a single-degree-of-freedom system are expressed as follows (see the literature: Li pure, Hongjun, Zhang Xinhua, and the like; experimental study on dynamic stiffness of angular contact ball bearings [ J ]. school newspaper of Sian traffic university, 2013, 47 (7): 68-72):
Z(ω)＝K(ω)-mω2+iC(ω)ω (25)
in formula (25): k (omega) is the equivalent stiffness of the rolling bearing; m is the equivalent vibration mass of the rolling bearing; omega2An inertia term which is the equivalent stiffness of the rolling bearing; omega is the angular frequency of rotor operation; c (omega) is the equivalent damping coefficient of the rolling bearing;
when the rolling bearing is in operation, a radial load synchronous with the rotating speed is directly applied, and corresponding displacement generated by the center of the bearing is measured to obtain:
in the formula: q is a rotating radial force which acts on the inner ring of the bearing and is synchronous with the running speed of the bearing; delta is the radial displacement of the bearing center;
is the phase difference between the displacement and the force vector.
The first problem to be solved in the kinetic analysis of rolling bearings is the contact analysis. The rolling bearing simulation dynamics method is based on a contact theory, but the boundary condition of a semi-infinite space of the contact theory is only suitable for analyzing objects with simple shapes and cannot meet the conditions of complex structures and complex loads. Furthermore, if the geometric dimensions of the contact body are too small or the applied load is too large, the assumption of the contact theory that the ratio of the dimension of the contact surface to the radius of curvature of the surface of the contact body is small cannot be satisfied, thereby limiting the range of use thereof (see the literature: caucasian. research on dynamic load distribution and stress expression of rolling bearings [ D ]. wuhan: university of wuhan science 2012). In the embodiment, an explicit finite element method is adopted to calculate the dynamic stiffness of a deep groove ball bearing of an input shaft of a certain transmission under the action of dynamic radial load, a three-dimensional finite element model of the bearing is established, reasonable boundary conditions are set, radial load is applied, a relational expression of bearing loading and displacement is established, and further the dynamic stiffness of the bearing is obtained. Table 1 shows the dimensional parameters of the deep groove ball bearing entity, and table 2 shows the material parameters of each part of the bearing.
Establishing a three-dimensional finite element model of the bearing according to the geometric dimensions in the table 1, as shown in fig. 4;
the axis of the bearing coincides with a z bearing of the whole Cartesian coordinate system, and the xy plane is a symmetrical plane of the bearing; linear hexahedron units are selected, and the bearing rolling elements are subjected to grid refinement, wherein the total number of model units is 23034; taking the static friction coefficient between the rolling body and the inner and outer ring raceways as 0.002, the dynamic friction coefficient as 0.0018, the whole model adopting the properties of linear elastic material, the bearing material GCr15 steel, the elastic modulus as 2.07 x 105MPa, Poisson's ratio of 0.3, static friction coefficient of 0.1, and dynamic friction coefficient of 0.06;
according to the practical use condition of the bearing, the complete constraint is applied to the outer surface of the outer ring of the bearing to describe the limiting effect of the bearing seat on the bearing, a rigid shaft and a rigid retainer are built on the inner ring of the bearing, the rotational freedom degree of the retainer around the axial direction is constrained, and an axial force F is applied to the rigid shaft of the inner ring of the bearingaApplying radial sine excitation load F on the rigid shaft of the bearing inner ringrSuch asAs shown in fig. 5, if the loading frequency is f:
wherein omega is the bearing inner ring corner frequency, the variable stiffness characteristic under the dynamic excitation of the bearing is simulated by adopting an explicit nonlinear finite element, the total calculation time is 0.1s, and the final contact related parameters are determined by comprehensively considering two factors of calculation time and calculation result precision;
the analysis takes the calculation time into consideration, and controls the penetration amount to be within 5% of the total displacement by adjusting the parameters so as to obtain a more accurate calculation result. The maximum contact stress cloud diagram of the bearing at 4.6801ms is shown in fig. 6, and it can be seen that the peak value of the sinusoidal radial excitation force is at the moment, the bearing inner ring is at the uppermost position, the bearing bears the maximum load, and the maximum stress is 1222 MPa.
The loading frequency of the model is F-50 Hz, FaWhen the amplitude of the radial force is 18567N, the radial load of the rolling bearing forms a hysteresis loop with the radial displacement due to the geometric nonlinearity of the distribution of the rolling bodies and the nonlinear influence of the contact of the rolling bodies and the raceways, and the radial displacement of the bearing has obvious nonlinear hysteresis characteristics.
In the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing, the dynamic stiffness of the bearing under odd pressure and even pressure loading meets the following requirements:
the bearing rotates through the loaded area in sequence, and undergoes the processes of maximum deformation from contact to contact and then deformation recovery, namely two typical load bearing conditions are presented as shown in fig. 9 and fig. 10 in sequence: in odd-pressure and even-pressure states, the rollers are symmetrically distributed according to radial loads;
during the rotation of the rolling body, the loading state of the rolling body changes along with the position of the rolling body, so that alternating elastic force is caused. When acting in a direction by a radial load, periodic vibration is induced. The vibration frequency is:
f＝Zfc(28)
wherein Z is the number of rolling elements; f. ofcIs the bearing inner ring rotation frequency; when the bearing rotates, the vibration can be regarded as that the inner ring is switched between an odd pressure state and an even pressure state under the action of radial load, so that the inner ring is caused to reciprocate in the direction of the radial load; FIG. 11 is a curve of load and radial displacement of a bearing under two loading modes of odd pressure and even pressure; the radial displacement of the bearing is gradually increased along with the increase of the external load, and when the external load is equal to the external load, the radial displacement of the bearing under the odd-pressure condition is smaller than that generated under the even-pressure condition; the load displacement relation of the bearing under the loading modes of odd pressure and even pressure can be obtained according to the data in FIG. 11:
δ＝aQb(29)
in the formula, delta is the displacement of the inner ring of the bearing, Q is the radial load of the bearing, and a and b are corresponding fitting coefficients respectively; the expression of the rigidity of the bearing under two loading forms is deduced according to the formula (24), the radial rigidity of the bearing under the odd-pressure form is larger than that under the even-pressure form, and the difference of the rigidity values is larger when the load is larger, so that the dynamic behavior of the bearing is influenced importantly.
In the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing, the influence of different excitation factors on the dynamic stiffness of the bearing is considered, and the following requirements are met:
the study of dynamic stiffness of bearings is much more complex than static stiffness, which was conducted in the early 70 s by the american advanced air force laboratory (AFAPL), and a large number of bearings were plotted to show the bearing stiffness of different bearings under various load conditions for easy review by designers of high speed rotor systems. In the domestic research of rotor-rolling bearing systems, no similar such detailed work has been seen. However, the stiffness of the bearing in a specific situation is also shown in the large number of curves, and the bearing stiffness is not considered to be influenced by factors such as temperature, centrifugal force, axial force and initial play during actual operation (see the literature: Rottus, Sunxider, Wulinfeng. the bearing stiffness of the rolling bearing in any direction [ J]Journal of the university of avigation, Nanjing, 1992,24(3): 248-256.). In order to analyze the variable stiffness characteristic of a rolling bearing of a transmission system, a rigid shaft and a rigid retainer are established on an inner ring of the bearing by adopting an explicit dynamics finite element method, and an axial force F is applied to the radial direction of the bearingaApplying sinusoidal excitation load F in radial direction of bearingrAnd f, analyzing the influence of the radial load loading frequency and the axial force on the dynamic stiffness of the rolling bearing when the radial clearance of the deep groove ball bearing at the input front end of the transmission is 0 by adopting the bearing dynamic stiffness correlation analysis method under the sine exciting force.
Fig. 12 is a curve of the variation of the radial dynamic stiffness of the bearing under different axial forces of the input front end bearing of the transmission along with the loading frequency, and it can be seen that the radial stiffness is increased along with the increase of the loading frequency, and the smaller the axial force is, the larger the radial stiffness of the bearing is. The stiffness of the bearing 1 under 25Hz excitation is compared with the static stiffness of the bearing 1 in the second chapter, and it can be seen that the dynamic stiffness is 22.103X 104N/mm, specific static stiffness 19.118X 104N/mm is 26% larger.
In the analysis method for the influence factors of the dynamic stiffness of the rolling bearing, one or a combination of the following three factors is also considered:
first, the influence of temperature
The elastic modulus of the material at the temperature of 20 ℃ (293K) is 207000MPa, and the bearing metal material (GCr15 steel) has the relation between the temperature and the linear expansion coefficient at the temperature of 0-200 ℃ at the low temperature (see the document: xi with heptyl. inorganic material thermophysics [ M ]. Shanghai: Shanghai science publisher, 1981. and the document: Pan stay, Jongqing, Du Xiao Yong. the relation between the linear expansion coefficient and Young modulus of the common alloy material at the high temperature [ J ]. Nature science report of Hunan university, 2000, 23 (2): 47-51.):
E＝E0(1-25αT) (30)
in the formula, E0The elastic modulus of the bearing material at 0 ℃, T is the Kelvin temperature of the bearing, α is the linear expansion coefficient of the bearing material, and α is 13.3 multiplied by 10 at low temperature-6From an elastic modulus E at 20 DEG C20207000MPa, giving a modulus of elasticity E of the bearing at 0 DEG C0When 229343.2MPa, different temperatures are obtained from the formula (30)Lower bearing modulus of elasticity;
in order to study the dynamic stiffness of the bearing at different temperatures, the dynamic stiffness change conditions of the bearing at the temperatures t of 20 ℃, 50 ℃, 100 ℃, 150 ℃ and 200 ℃ are respectively calculated, wherein the initial clearance of the bearing is 0, the loading frequency of the radial load is 50Hz, the amplitude of the radial force is 18567N, and the axial force is 1617N.
Fig. 13 shows the dynamic stiffness of the bearing at different temperatures along with the variation of the loading frequency, and it can be seen that the radial dynamic stiffness of the rolling bearing at normal temperature (t 20 ℃) increases along with the increase of the radial loading frequency, the radial loading frequency at higher temperature (t 50 ℃, 100 ℃, 150 ℃, 200 ℃) takes 75Hz as an inflection point, the temperature of the bearing decreases along with the increase of the radial loading frequency, and then increases along with the increase of the loading frequency, and the radial dynamic stiffness of the bearing decreases along with the increase of the temperature.
Second, influence of radial play
Fig. 14 shows the variation of the radial dynamic stiffness of the bearing with the radial load frequency in three types of clearances, namely, the clearances of the rolling bearing are respectively 0 μm, 10 μm and 20 μm, and it can be seen that the radial dynamic stiffness of the rolling bearing is increased with the increase of the radial load frequency in all three types of clearances, and the radial stiffness of the rolling bearing is gradually reduced with the increase of the radial clearance of the rolling bearing.
Third, the effect of the finite element mesh on the results
In order to analyze the influence of the number of meshes of the bearing finite element model on the simulation result, the bearing is divided into 3 number of finite element mesh models, as shown in FIGS. 15-17. The load working conditions of different rolling bearing grid models are as follows: bearing clearance of 0, axial load F a3000N, radial load Fr18567N, radial loading frequency 150 Hz.
The bearing finite element model adopts C3D8R linear hexahedral units, the grid model shown in FIG. 15 is a sparse grid, 18316 units are used in total, the bearing rigidity k obtained through simulation is 152853.81N/mm, and the simulation time of the model in one loading period is 21 min; the grid model shown in fig. 16 is a medium-scale grid, 23034 cells in total, and the obtained bearing stiffness k is 236664.44N/mm, and the simulation time of one loading cycle is 35 min; the grid model shown in fig. 17 is a large-scale grid with 46498 cells, and the bearing stiffness k is 235781.91N/mm, and the simulation time of one loading cycle is 108 min. The model shown in fig. 16 is a mesh model used in the calculation, and it can be seen that the number of cells of the mesh model shown in fig. 15-17 is gradually increased, the radial stiffness of the bearing obtained by the mesh model shown in fig. 15 and 16 is 83810.63N/mm, and the relative error is 35.4%, the radial stiffness of the bearing obtained by the mesh model shown in fig. 16 and 17 is 882.53N/mm, and the relative error is 0.37%, and the relative errors are small, but the calculation time of the model shown in fig. 17 is much longer than that of the model shown in fig. 16, and therefore the mesh model shown in fig. 17 used herein is reasonable.
In the problems related to the embodiment, due to the influence of geometric nonlinearity of the distribution of rolling elements of the rolling bearing and the contact nonlinearity of the rolling elements and the raceway, the radial displacement of the rolling bearing has obvious hysteresis nonlinearity; the factors influencing the dynamic stiffness of the rolling bearing mainly comprise radial load frequency, axial load, bearing temperature, bearing initial clearance and the like, the radial dynamic stiffness is increased along with the increase of the radial load frequency and is reduced along with the increase of the axial load, meanwhile, the dynamic stiffness is generally reduced along with the increase of the bearing temperature and is reduced along with the increase of the radial clearance, and research results provide theoretical basis for reasonable selection of the bearing so as to reduce vibration of the bearing during working.
Claims (6)
1. The method for analyzing the influence factors of the dynamic stiffness of the rolling bearing is characterized by comprising the following steps:
the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing is based on the following premises: the contact between the rolling bodies of the rolling bearings and the inner and outer ring raceways is considered to meet the Hertz contact theory, and the contact type between the rolling bodies and the inner and outer ring raceways of the bearings is point contact or/and line contact; the relevant specific requirements of hertzian contact theory are as follows:
① ball bearing, the contact between the rolling element and the inner and outer races is point contact, under the action of load Q, the contact point gradually expands to an elliptical contact surface, and the maximum stress is sigma at the center point of ellipsemaxThe major axis of the ellipse is 2a, the minor axis is 2b, and is connected by HertzThe theory of touch knows that:
in the formula, Q is the contact load of the rolling element and the inner and outer ring raceways; a is the contact ellipse area major semi-axis length; b is the short semi-axis length; sigmamaxIs the contact maximum stress; delta is the deformation; the equivalent elastic modulus E' is expressed as:
in the formula, E1，E2，μ1，μ2The elastic modulus and Poisson's ratio of the rolling element and the inner and outer ring raceways respectively;
② the principal curvature sum Σ ρ is the sum of the principal curvatures at the contact of the rolling elements and the raceways, i.e.:
∑ρ＝ρ11+ρ12+ρ21+ρ22(6)
maand mbMajor and minor axis coefficients, respectively, are expressed as:
wherein k is an ellipticity,
e is the eccentricity of the ellipse and the eccentricity of the ellipse,
k (e) and l (e) are the first and second types of integrals, respectively, associated with the eccentricity of the ellipse:
the ellipse eccentricity e is related to the ellipticity by:
the principal curvature function F (ρ) is expressed as:
also expressed as:
if the geometric dimensions of each part of the bearing are known, the principal curvature function F (rho) can be obtained from the formula (13) -the formula (17) and the formula (11); substituting the expressions (9) and (10) into the expression (12) to obtain k, and obtaining e by the expression (10), thereby measuring the bearing rigidity;
③ under the action of the contact load Q between the rolling body and the inner and outer rings, the coefficients of the major and minor semi-axes can be obtained by equations (7) and (8), and the maximum contact stress sigma of the contact zone can be obtained by equations (1) and (4)maxAnd a deformation amount δ;
for a point contact ball bearing, the main curvatures are:
rolling ball:
inner ring:
outer ring:
in the formula, DbThe diameter of the rolling ball;
α is contact angle, D is inner diameter of outer ring, D is outer diameter of inner ringmIs the diameter of the central circle of the rolling body; r isinIs the inner raceway radius; r isouIs the radius of the outer raceway;
and (II) the Hertz rigidity of the bearing meets the following requirements:
under the action of a contact load Q, the total deformation delta of one rolling body of the rolling bearing is deltain+δouWherein δinThe amount of deformation of the rolling elements in the direction of the inner race, δouThe amount of deformation of the rolling elements in the outer race direction is given by equation (4):
the amount of deformation δ is then expressed as:
where K is the contact stiffness coefficient, then:
the contact load Q is expressed as:
formula (21) is a formula for calculating the contact force between a single rolling element and the inner and outer races of the bearing, wherein delta is the deformation, Q is the contact load between the rolling element and the raceways of the inner and outer races, and the Hertz contact stiffness a0Expressed as:
where Σ ρinThe main curvature sum of the contact of the rolling body and the inner ring; sigma rhoouIs the sum of the main curvatures of the outer ring;
and
the coefficients of the long axes of the inner and outer rings are respectively; k (e)inAnd K (e)ouA first type of integral which is related to eccentricity of the inner and outer rings;
here, the deformation δ is used as a measure of the dynamic stiffness of the rolling bearing.
2. The method for analyzing the influence factors on the dynamic stiffness of the rolling bearing according to claim 1, wherein:
the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing further meets the following requirements: calculating the dynamic stiffness under the action of the dynamic radial load, establishing a three-dimensional finite element model of the bearing, setting boundary conditions, applying the radial load, establishing a relational expression of the bearing loading and the displacement, and further solving the dynamic stiffness of the bearing;
specifically, the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing analyzes and expresses the dynamic stiffness of the rolling bearing through the dynamic stiffness of the bearing under a sinusoidal excitation force; the specific requirements are as follows:
the method is characterized in that the relation between the radial load and the radial displacement in the load direction of the bearing is studied in the allowable radial load range under the elastic contact condition without considering the bending deformation of a bearing retainer and a shaft, so as to represent the change of the radial rigidity of the bearing; specifically, the dynamic stiffness of the bearing is a ratio of a variation of an external load applied to the bearing to a variation of a bearing displacement, and therefore, in order to calculate the dynamic stiffness of the bearing, a relationship between the load applied to the bearing and the displacement must be established, and the relationship is expressed by a formula:
δ ═ f (Q) or Q ═ f (δ) (23)
Where δ is the displacement of the center of the bearing inner race in mm; q is the contact load of the rolling body and the inner and outer ring raceways, and the unit is N; the dynamic stiffness of the bearing is then expressed as:
the dynamic characteristics of the rolling bearing as a single-degree-of-freedom system are expressed as follows:
Z(ω)＝K(ω)-mω2+iC(ω)ω (25)
in formula (25): k (omega) is the equivalent stiffness of the rolling bearing; m is the equivalent vibration mass of the rolling bearing; omega2An inertia term which is the equivalent stiffness of the rolling bearing; omega is the angular frequency of rotor operation; c (omega) is the equivalent damping coefficient of the rolling bearing;
when the rolling bearing is in operation, a radial load synchronous with the rotating speed is directly applied, and corresponding displacement generated by the center of the bearing is measured to obtain:
in the formula: q is a rotating radial force which acts on the inner ring of the bearing and is synchronous with the running speed of the bearing; delta is the radial displacement of the bearing center;
is the phase difference between the displacement and the force vector.
3. The method for analyzing the influence factors on the dynamic stiffness of the rolling bearing according to claim 2, wherein:
establishing a three-dimensional finite element model of the bearing according to the geometric dimension, wherein the axis of the bearing is coincided with a z bearing of an integral Cartesian coordinate system, and an xy plane is a symmetrical plane of the bearing; selecting a linear hexahedral unit, and carrying out grid refinement on a bearing rolling body;
the method is characterized in that complete constraint is applied to the outer surface of the bearing outer ring to describe the limiting effect of the bearing seat on the bearing, a rigid shaft and a rigid retainer are built in the bearing inner ring, the rotational freedom degree of the retainer around the axial direction is constrained, and an axial force F is applied to the rigid shaft of the bearing inner ringaApplying radial sine excitation load F on the rigid shaft of the bearing inner ringrAnd if the loading frequency is f, then:
wherein omega is the bearing inner ring corner frequency, the variable stiffness characteristic under the dynamic excitation of the bearing is simulated by adopting an explicit nonlinear finite element, the total calculation time is 0.1s, and the final contact related parameters are determined by comprehensively considering two factors of calculation time and calculation result precision;
during analysis, the penetration amount is controlled within 5 percent of the total displacement by adjusting the parameters while considering the calculation time, so that a more accurate calculation result is obtained.
4. The method for analyzing the influence factors on the dynamic stiffness of the rolling bearing according to claim 2, wherein:
the dynamic stiffness of the bearing under odd pressure and even pressure loading meets the following requirements:
the bearing sequentially passes through the loaded area in the rotation process, and undergoes the processes of largest deformation from contact to contact and then deformation recovery, namely two typical load bearing conditions are presented in sequence: in odd-pressure and even-pressure states, the rollers are symmetrically distributed according to radial loads;
in the rotating process of the rolling body, the load state of the rolling body changes along with different positions of the rolling body, so that alternating elastic force is caused; causing periodic vibration when acting in a direction by a radial load; the vibration frequency is:
f＝Zfc(28)
wherein Z is the number of rolling elements; f. ofcIs the bearing inner ring rotation frequency; when the bearing rotates, the vibration can be regarded as that the inner ring is switched between an odd pressure state and an even pressure state under the action of radial load, so that the inner ring is caused to do reciprocating motion in the direction of the radial load;
the radial displacement of the bearing is gradually increased along with the increase of the external load, and when the external load is equal to the external load, the radial displacement of the bearing under the odd-pressure condition is smaller than that generated under the even-pressure condition; obtaining a load displacement relation of the bearing under two loading modes of odd pressure and even pressure:
δ＝aQb(29)
in the formula, delta is the displacement of the inner ring of the bearing, Q is the radial load of the bearing, and a and b are corresponding fitting coefficients respectively; and (3) deducing an expression of the rigidity of the bearing under two loading forms according to the formula (24), wherein the radial rigidity of the bearing under the odd-pressure form is larger than that under the even-pressure form, and the difference of rigidity values is larger when the load is larger.
5. The method for analyzing the influence factors on the dynamic stiffness of the rolling bearing according to claim 2, wherein: in the method for analyzing the influence factors of the dynamic stiffness of the rolling bearing, the influence of different excitation factors on the dynamic stiffness of the bearing is considered, and the following requirements are met:
in order to analyze the variable stiffness characteristic of a rolling bearing of a transmission system, a rigid shaft and a rigid retainer are established on an inner ring of the bearing by adopting an explicit dynamics finite element method, and the radial direction of the bearing isApplying an axial force FaApplying sinusoidal excitation load F in radial direction of bearingrAnd f, analyzing the influence of the radial load loading frequency and the axial force on the dynamic stiffness of the rolling bearing when the radial clearance of the deep groove ball bearing at the input front end of the transmission is 0.
6. The method for analyzing the influence factors on the dynamic stiffness of the rolling bearing according to claim 2, wherein: in the analysis method for the influence factors of the dynamic stiffness of the rolling bearing, one or a combination of the following three factors is also considered:
first, the influence of temperature
The elastic modulus of the material at the temperature of 293K is 207000MPa, and the bearing metal material has the relationship between the temperature and the linear expansion coefficient at the low temperature of 0-200 ℃:
E＝E0(1-25αT) (30)
in the formula, E0The elastic modulus at 0 ℃ of the bearing material, T is the Kelvin temperature of the bearing, α is the linear expansion coefficient of the bearing material, and the elastic modulus at 20 ℃ E20Obtaining the elastic modulus E of the bearing at 0 DEG C0Then obtaining the elastic modulus of the bearing at different temperatures according to the formula (30);
second, influence of radial play
The rigidity of the bearing is increased along with the increase of the radial load frequency, and the radial rigidity of the rolling bearing is gradually reduced along with the increase of the radial clearance of the rolling bearing;
third, the effect of the finite element mesh on the results
In order to analyze the influence of the number of the bearing finite element model grids on the simulation result, the bearing is divided into 3 number of finite element grid models, and the load working conditions of different rolling bearing grid models are as follows: bearing clearance of 0, axial load Fa3000N, radial load Fr18567N, radial loading frequency 150 Hz.
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Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2014061699A1 (en) * | 2012-10-17 | 2014-04-24 | Ntn株式会社 | Bearing element, rolling bearing and process for producing bearing element |
| CN103927414A (en) * | 2014-04-04 | 2014-07-16 | 北京工业大学 | Vibration response simulation analyzing method for rolling bearing with single-point failures |
| CN104239654A (en) * | 2014-10-13 | 2014-12-24 | 中国科学院光电技术研究所 | Bearing simplification method in finite element simulation analysis |
-
2016
- 2016-02-02 CN CN201610084829.3A patent/CN106560816B/en not_active Expired - Fee Related
Patent Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| WO2014061699A1 (en) * | 2012-10-17 | 2014-04-24 | Ntn株式会社 | Bearing element, rolling bearing and process for producing bearing element |
| CN103927414A (en) * | 2014-04-04 | 2014-07-16 | 北京工业大学 | Vibration response simulation analyzing method for rolling bearing with single-point failures |
| CN104239654A (en) * | 2014-10-13 | 2014-12-24 | 中国科学院光电技术研究所 | Bearing simplification method in finite element simulation analysis |
Non-Patent Citations (3)
| Title |
|---|
| 《滚动轴承_偏置转子系统涡摆耦合动力学特性研究》;梁明轩等;《振动与冲击》;20141231;第35-41页 * |
| 《滚动轴承_柔性碰摩转子系统非线性动力学响应分析》;梁明轩等;《动力学与控制学报》;20130630;第153-158页 * |
| 《角接触球轴承动刚度的实验研究》;李纯洁等;《西安交通大学学报》;20130731;第68-72，123页 * |
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