CN112541237A - Analytic-finite element method for calculating time-varying stiffness of flexible cylindrical roller bearing - Google Patents

Analytic-finite element method for calculating time-varying stiffness of flexible cylindrical roller bearing Download PDF

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CN112541237A
CN112541237A CN202011427157.4A CN202011427157A CN112541237A CN 112541237 A CN112541237 A CN 112541237A CN 202011427157 A CN202011427157 A CN 202011427157A CN 112541237 A CN112541237 A CN 112541237A
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马辉
徐宏阳
赵翔
俞昆
贺多
杨阳
罗忠
李旭
李响
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Northeastern University China
China North Vehicle Research Institute
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Abstract

The invention belongs to the technical field of mechanical dynamics, and discloses an analytic-finite element calculation method for time-varying rigidity of a flexible cylindrical roller bearing. On one hand, the method combines a finite element method and an analytic contact theory, provides a calculation method for radial rigidity of the cylindrical roller bearing considering flexibility of parts, considers flexible deformation of the bearing parts, and has higher calculation precision than that of a traditional analytic method. On the other hand, the method uses the analytic contact theory to calculate the contact deformation of the bearing part, and therefore is more efficient than a contact finite element method. The method makes up the vacancy of the calculation method of the time-varying rigidity of the flexible cylindrical roller bearing at the present stage.

Description

Analytic-finite element method for calculating time-varying stiffness of flexible cylindrical roller bearing
Technical Field
The invention belongs to the technical field of mechanical dynamics, and particularly relates to an analytic-finite element calculation method for time-varying rigidity of a flexible cylindrical roller bearing.
Background
The time-varying stiffness of the bearing is one of the main excitation sources of the rotor system vibrations, and therefore it is necessary to study the time-varying nature of the bearing stiffness that it possesses.
The cylindrical roller bearing generally comprises an outer ring, an inner ring, a rolling body and a retainer. Because of the advantages of small friction, high mechanical efficiency, easy start, compact structure, light weight, convenient disassembly and assembly and the like, the bearing is serialized, normalized and standardized at present, and special bearing manufacturers supply bearings of various types to the society. Cylindrical roller bearings are commonly used in the mechanical structure of simple daily appliances, such as bicycles, roller skates, fingertip gyros and the like, and rolling bearings are also indispensable in some very complex mechanical products, such as industrial gas turbines, various aircrafts, rolling mills and the like.
The bearing is easy to generate on the contact surface of the bearing parts due to the complex working condition of the bearing. When a certain bearing part contacts with other bearing parts, the contact force is changed violently, and impact is generated. The force excitation model mainly comprises a triangular pulse force model and a half-sine excitation force model. At present, the calculation method of the bearing rigidity mainly comprises an analytic method, a finite element method and an experimental method. A large number of researches show that when the rigidity of the bearing is calculated by using the traditional analytical method, the rigidity is calculated by using the bearing model established based on the rigid bearing part, so that the calculation result of the traditional analytical method is obviously different from that of an experimental method. However, the finite element results are well matched with the experimental results. Therefore, the flexibility of the bearing parts cannot be neglected in analyzing the contact characteristics and the stiffness of the bearing. While the finite element method takes into account the flexibility of the bearing component, it generally requires a denser grid at the joint, which is computationally inefficient.
Therefore, on the premise of considering the flexibility of the bearing part and ensuring high calculation accuracy, it is very important to find an effective bearing stiffness calculation method.
Disclosure of Invention
The invention aims to provide an analytic-finite element method for calculating time-varying rigidity of a flexible cylindrical roller bearing, which combines a finite element method and an analytic contact theory on the one hand, provides a method for calculating radial rigidity of the cylindrical roller bearing by considering flexibility of parts, considers flexible deformation of the bearing parts and has higher calculation precision than that of the traditional analytic method. On the other hand, the method uses the analytic contact theory to calculate the contact deformation of the bearing part, so that the method is more efficient than a contact finite element method.
The technical scheme of the invention is that an analytic-finite element method for calculating the time-varying stiffness of the bearing with the flexible cylindrical roller comprises the following specific steps:
step 1. calculate Linear Global flexibility
Step 1-1, establishing a finite element model of the cylindrical roller bearing in an MATLAB environment by using the unit and node information, and calculating the flexible deformation of the bearing part. And calculating the azimuth angle of the roller according to the motion relation of the bearing parts. For cylindrical roller bearings, the azimuth angle of the jth roller
Figure BDA0002825418080000024
Expressed as:
Figure BDA0002825418080000021
wherein
Figure BDA0002825418080000022
Is the initial azimuth angle of the roller; n is a radical ofbIs the number of rollers; t is the running time of the bearing; f. ofcIs the frequency of the cage, expressed as:
Figure BDA0002825418080000023
wherein f isiAnd foThe rotational frequencies of the inner and outer races, respectively; ri、RoAnd RrRespectively representing the radius of an inner ring, the radius of an outer ring and the radius of a roller;
step 1-2, in order to obtain the flexibility of the bearing part, the bearing part needs to be restrained. And constraining all degrees of freedom of the outer node of the outer ring, the inner node of the inner ring and the central node of the roller. And (4) constraining the circumferential freedom of all nodes on the roller contact point connecting line. It is to be noted that the above-mentioned purpose of constraining the bearing part is only to extract the compliance of the bearing part, and the actual constraining method of the bearing part is different from this method. After the restraint is completed, unit force is applied to contact points on the inner ring, the outer ring and the roller respectively along the radial direction. In order to prevent local deformation distortions caused by the direct action of unit forces on the nodal points and to take into account the global flexural deformations and local contact deformations of the bearing parts, respectively, rigid zones are established at the contact locations of the bearing parts.
Since the size of the rigid region is related to the contact deformation, the contact half-width is used to define the size of the rigid region. When no radial play exists, a radial force is applied to the inner ring of the cylindrical roller bearing, and the maximum load of a load distribution area is QmaxExpressed as:
Figure BDA0002825418080000031
for cylindrical roller bearings with radial play, the maximum load QmaxExpressed as:
Figure BDA0002825418080000032
assuming that there is a virtual roller at the maximum load, the half width of the virtual roller contacting the inner ring and the outer ring is expressed as:
Figure BDA0002825418080000033
Figure BDA0002825418080000034
in the above formula, L is the effective contact line length; ee is the equivalent modulus of elasticity expressed as:
Figure BDA0002825418080000035
in the above formula E1And E2The elastic modulus of two objects in contact with each other; v is1V and v2Respectively the poisson's ratio of two objects in contact with each other;
the radius of the rigidified area of the roller at the contact points with the inner and outer races is expressed as:
Figure BDA0002825418080000036
Figure BDA0002825418080000037
and (3) considering unit forces on the local rigid area and the contact node, and obtaining an integral deformation vector U of the bearing part based on a finite element method:
Figure BDA0002825418080000038
where K is the global stiffness matrix taking into account the local stiffening regions; f is a nodeA force vector; n is a radical ofnIs the total number of nodes.
U is a 2N radicalnA vector of dimensions containing the deformations of all nodes in the x and y directions. And according to the integral deformation vector U, the deformation of the contact nodes on the inner ring, the outer ring and the roller of the bearing is found. Since the force applied to the contact node is a unit force, the total deformation of the contact node in the radial direction is its compliance. Assuming that the node at which the compliance is extracted is j, the node at which the unidirectional force is applied is k, and the number of rollers that may be contacted is n, both j and k are in the range of 1 to n. When the contact node is located at the inner circle, the flexibility of the contact node is expressed as:
Figure BDA0002825418080000041
compliance of the contact node when the contact node is located in the outer race
Figure BDA0002825418080000042
Expressed as:
Figure BDA0002825418080000043
when the contact node is on the roller, the nodes imposed by the unit force are always the same as the nodes of extraction compliance, and are all nodes k. Therefore, the compliance of the roller with the inner and outer ring contact nodes is expressed as:
Figure BDA0002825418080000044
Figure BDA0002825418080000045
superscripts i and o indicate that the contact points are located on the inner ring and the outer ring, respectively; the superscript ri indicates the node point on the roller and in contact with the inner ring. Superscript ro denotes the node point on the roller and in contact with the outer race; λ represents compliance;
Figure BDA0002825418080000046
and
Figure BDA0002825418080000047
respectively, the azimuth angle of the jth and kth rollers. It should be noted that the azimuth of the contact node corresponds one-to-one to the azimuth of the roller.
When a unit force is applied to a certain contact node of the inner ring, the flexibility of all the contact nodes of the inner ring is extracted. Finally, all the compliances are assembled into one matrix. The compliance matrix of the outer ring is obtained in a similar way to the inner ring. Because each roller is independent of the other, the compliance matrix extraction of the rollers is relatively simple. A unit force is applied to the contact nodes of each roller in turn and the compliance of all the contact nodes is extracted and assembled into a matrix. Compliance matrix lambda of inner ring, outer ring and rolleri,，λoAnd λrRespectively expressed as:
Figure BDA0002825418080000048
Figure BDA0002825418080000051
Figure BDA0002825418080000052
as seen from the above equation, the compliance matrix of the inner and outer rings has a cross term λjk(j ≠ k). Thus, the model presented herein takes into account the cross effect between the contact nodes of the inner and outer rings. However, since each roller is independent of the other, there are no cross terms in the compliance matrix of the rollers.
Integral compliance matrix lambda of cylindrical roller bearingbExpressed as: lambda [ alpha ]b＝λi+λo+λr
Step 2, calculating the nonlinear local contact flexibility
Step 2-1, neglecting the boundary effect of the line contact, and respectively obtaining the contact deformation delta between the roller and the inner ring and the outer ring according to the analytical formula of the finite length line contactiAnd deltao：
Figure BDA0002825418080000053
Figure BDA0002825418080000054
Wherein Q is the contact load on the contact line; l is the effective contact line length; eeIs the equivalent modulus of elasticity.
Step 2-2, the contact flexibility between the rollers and the inner and outer races is expressed as:
Figure BDA0002825418080000055
Figure BDA0002825418080000056
and 2-3, each loaded roller has two contact pairs. They are the contact pair between the roller and the inner ring and the contact pair between the roller and the outer ring. If n rollers are loaded, the contact compliance matrix lambdacExpressed as:
Figure BDA0002825418080000057
step 3, calculating radial rigidity calculation
Step 3-1, in order to calculate the load distribution and radial deformation of the cylindrical roller bearing under the radial load, introducing a deformation coordination equation:
Figure BDA0002825418080000061
q is a contact force vector, Q ═ Q1,Q2,…,Qj,…,Qn]T；
Figure BDA0002825418080000062
Is the cosine vector of the azimuth angle that may carry the roller,
Figure BDA0002825418080000063
In×1is an n-dimensional unit column vector; deltarIs the relative radial displacement between the inner ring and the outer ring of the cylindrical roller bearing; srIs a radial play;
step 3-2, the force balance equation of the cylindrical roller bearing in the radial direction is expressed as:
Figure BDA0002825418080000064
and 3-3, providing an iterative equation of the contact analysis of the bearing based on the deformation coordination equation and the force balance equation:
Figure BDA0002825418080000065
and 3-4, obtaining contact force vectors of all contact points and radial displacement of the cylindrical roller bearing based on the equation. The radial stiffness of the cylindrical roller bearing is calculated using a finite difference method:
Figure BDA0002825418080000066
wherein Kr(Fr) Denotes a radial force FrRadial stiffness of the cylindrical roller bearing; Δ FrIs a slight increment of radial force; deltar(Fr) Denotes a radial force FrRadial deformation of the cylindrical roller bearing occurs.
Since the contact force at each roller contact point is usually notAnd are equal, so the initial contact force cannot satisfy the deformation coordination condition. In an iterative process, if Qj<0, the contact point j is regarded as a false contact point, namely the contact point j is not in a contact state; deleting the rows and columns of the matrix where the false contact points are located, and then performing the next iteration; let the contact force vector solved by the kth iteration reach the convergence criterion | | Q (k) -Q (k-1) | non-charging<ε Q, ε Q is the convergence tolerance, then the iteration stops.
The method has the beneficial effects that the flexibility of the bearing part is considered, and the method for calculating the time-varying rigidity of the cylindrical roller bearing is provided. The method uses finite element theory to calculate the overall flexible deformation of the bearing part, and uses analytic contact theory to calculate the local contact deformation of the bearing part. The method has higher calculation precision than the traditional analytic method and higher calculation efficiency than a finite element method.
Drawings
FIG. 1 is a flow chart of an analytic-finite element method for calculating time-varying stiffness of a flexible cylindrical roller bearing according to the present invention.
FIG. 2 is a finite element model of a cylindrical roller bearing in a MATLAB environment, where (a) is the constraint state at the time of extracting compliance, and (b) is each unit force FuThe application state of (1).
Fig. 3 shows the comparison of the radial stiffness of the present invention with the literature method under different radial forces, where (a) is the result of the proposed method of the present invention and (b) is the result in the literature.
Detailed Description
In order to explain the present invention more clearly, the following detailed description is made with reference to the accompanying drawings.
Fig. 1 is a flowchart of a method for calculating time-varying radial stiffness of a cylindrical roller bearing disclosed in the present invention, and in this embodiment, the parameters of the cylindrical roller bearing are shown in table 1:
TABLE 1 bearing parameters
Figure BDA0002825418080000071
Assuming that the outer race is fixed in the bearing housing, the inner race is connected to the shaft. The time that a roller rolls through one roller spread angle is considered as a roller pass period. In this case, only the results of two roller pass cycles were selected for comparison and discussion.
Based on a finite element method, a finite element model of the cylindrical roller bearing is established. The constraint method of the bearing parts under ideal conditions is as follows: the outer node of the outer ring is fixed; the inner nodes of the inner ring are rigidly coupled; constraining the circumferential degree of freedom of the nodes on the roller contact point connecting line; a contact constraint is imposed on the contact surface of the bearing component. A radial force is applied to the inner race and the radial stiffness of the cylindrical roller bearing is obtained as shown in fig. 3. The proposed method calculates the resulting time-varying stiffness (see fig. 3(a)) taking into account the radial play. By comparing the results of the proposed method with the results of the literature method (see fig. 3(b)), it was found that the time-varying stiffness results calculated by both methods fit well.
In summary, the analytic-finite element method for calculating the time-varying stiffness of the flexible cylindrical roller bearing has the following advantages:
the method calculates the overall flexible deformation of the bearing part based on a finite element theory, and calculates the local contact deformation of the bearing part by using an analytic contact theory. Compared with the traditional analytic method, the method has higher calculation precision due to the consideration of the flexibility of the bearing part; compared with a contact finite element method, the method has higher calculation efficiency because the contact deformation is considered by analyzing the contact theory. The method makes up the vacancy of the calculation method of the time-varying rigidity of the flexible cylindrical roller bearing at the present stage.

Claims (1)

1. An analytic-finite element method for calculating time-varying stiffness of a flexible cylindrical roller bearing is characterized by comprising the following steps:
step 1, calculating linear global compliance;
step 1-1, establishing a finite element model of the cylindrical roller bearing by using the unit and node information, and calculating the flexible deformation of a bearing part; method for calculating roller according to motion relation of bearing partsAn azimuth angle; for cylindrical roller bearings, the azimuth angle of the jth roller
Figure FDA0002825418070000011
Expressed as:
Figure FDA0002825418070000012
wherein
Figure FDA0002825418070000013
Is the initial azimuth angle of the roller; n is a radical ofbIs the number of rollers; t is the running time of the bearing; f. ofcIs the frequency of the cage, expressed as:
Figure FDA0002825418070000014
wherein f isiAnd foThe rotational frequencies of the inner and outer races, respectively; ri、RoAnd RrRespectively representing the radius of an inner ring, the radius of an outer ring and the radius of a roller;
in the step 1-2, the step of the method,
in order to respectively consider the whole flexible deformation and the local contact deformation of the bearing part, a rigid area is established at the contact position of the bearing part, and the size of the rigid area is defined by adopting the half-width of contact; when no radial play exists, a radial force is applied to the inner ring of the cylindrical roller bearing, and the maximum load of a load distribution area is QmaxExpressed as:
Figure FDA0002825418070000015
for cylindrical roller bearings with radial play, the maximum load QmaxExpressed as:
Figure FDA0002825418070000016
and a virtual roller exists at the maximum load, and the contact half width of the virtual roller with the inner ring and the outer ring is represented as follows:
Figure FDA0002825418070000017
Figure FDA0002825418070000021
in the above formula, L is the effective contact line length; eeEquivalent modulus of elasticity, expressed as:
Figure FDA0002825418070000022
in the above formula E1And E2The elastic modulus of two objects in contact with each other; v is1V and v2Respectively the poisson's ratio of two objects in contact with each other;
the radius of the rigidified area of the roller at the contact points with the inner and outer races is expressed as:
Figure FDA0002825418070000023
Figure FDA0002825418070000024
the concrete operations for rigidizing the rigidized area are as follows: increasing radius R of stiffened regionrigidThe modulus of elasticity of the cell within the determined range; considering a rigidization area, and based on a finite element method, grouping to obtain an overall stiffness matrix K:
Figure FDA0002825418070000025
before calculating the integral deformation, all degrees of freedom of an outer node of a bearing outer ring, an inner node of an inner ring and a central node of a roller need to be restrained; specifically, the rigidity element k corresponding to the degree of freedom of the node is usedpqThe elements in the row and column are rewritten to obtain:
Figure FDA0002825418070000026
considering the unit force at the contact node, the overall deformation vector U of the bearing part is obtained by the following formula:
Figure FDA0002825418070000031
where K is the global stiffness matrix taking into account the local stiffening regions; f is the node force vector; n is a radical ofnIs the total number of nodes; u is a 2N radicalnA vector of dimensions containing the deformations of all nodes in the x and y directions; setting a node for extracting flexibility as j, a node for applying unidirectional force as k, the number of rollers possibly contacted as n, and the j and k are in the range of 1 to n; when the contact node is located at the inner circle, the flexibility of the contact node is expressed as:
Figure FDA0002825418070000032
compliance of the contact node when the contact node is located in the outer race
Figure DEST_PATH_BDA0002825418080000042
Expressed as:
Figure FDA0002825418070000034
when the contact node is on the roller, the nodes imposed by the unit force are always the same as the nodes of extraction compliance, and are all nodes k; the flexibility of the roller and the contact node of the inner ring and the outer ring is respectively expressed as:
Figure FDA0002825418070000035
Figure FDA0002825418070000036
superscripts i and o indicate that the contact points are located on the inner ring and the outer ring, respectively; the superscript ri denotes the node point on the roller and in contact with the inner ring; superscript ro denotes the node point on the roller and in contact with the outer race; λ represents compliance;
Figure FDA0002825418070000037
and
Figure FDA0002825418070000038
azimuth angles of the jth and kth rollers, respectively; the azimuth angles of the contact nodes correspond to the azimuth angles of the rollers one by one;
1-3, sequentially applying unit force to contact nodes of each roller, extracting the flexibility of all the contact nodes, and assembling the contact nodes into a matrix; compliance matrix lambda of inner ring, outer ring and rolleri,，λoAnd λrRespectively expressed as:
Figure FDA0002825418070000039
Figure FDA0002825418070000041
Figure FDA0002825418070000042
step 1-4, the overall flexibility matrix lambda of the cylindrical roller bearingbExpressed as:
λb＝λi+λo+λr
step 2, calculating nonlinear local contact flexibility;
step 2-1, neglecting the boundary effect of the line contact, and respectively obtaining the contact deformation delta between the roller and the inner ring and the outer ring according to the analytical formula of the finite length line contactiAnd deltao：
Figure FDA0002825418070000043
Figure FDA0002825418070000044
Wherein Q is the contact load on the contact line; l is the effective contact line length; eeIs the equivalent modulus of elasticity;
step 2-2, the contact flexibility between the rollers and the inner and outer races is expressed as:
Figure FDA0002825418070000045
Figure FDA0002825418070000046
2-3, each loaded roller is provided with two contact pairs; a contact pair between the roller and the inner ring and a contact pair between the roller and the outer ring, respectively; assuming n rollers are loaded, the contact compliance matrix λcExpressed as:
Figure FDA0002825418070000047
step 3, calculating the radial stiffness;
step 3-1, in order to calculate the load distribution and radial deformation of the cylindrical roller bearing under the radial load, introducing a deformation coordination equation:
Figure FDA0002825418070000051
q is a contact force vector, Q ═ Q1,Q2,…,Qj,…,Qn]T；
Figure FDA0002825418070000052
Is the cosine vector of the azimuth angle that may carry the roller,
Figure FDA0002825418070000053
In×1is an n-dimensional unit column vector; deltarIs the relative radial displacement between the inner ring and the outer ring of the cylindrical roller bearing; srIs a radial play;
step 3-2, the force balance equation of the cylindrical roller bearing in the radial direction is expressed as:
Figure FDA0002825418070000054
and 3-3, providing an iterative equation of the contact analysis of the bearing based on a deformation coordination equation and a force balance equation:
Figure FDA0002825418070000055
and 3-4, calculating the radial rigidity of the cylindrical roller bearing by using a finite difference method:
Figure FDA0002825418070000056
wherein Kr(Fr) Denotes a radial force FrRadial stiffness of the cylindrical roller bearing; Δ FrIs a slight increment of radial force; deltar(Fr) Denotes a radial force FrRadial deformation of the cylindrical roller bearing;
since the contact force of each roller contact point is generally not equal, the initial contact force cannot satisfy the deformation coordination condition; in an iterative process, if Qj<0, the contact point j is regarded as a false contact point, namely the contact point j is not in a contact state; deleting the rows and columns of the matrix where the false contact points are located, and then performing the next iteration; let the contact force vector solved by the kth iteration reach the convergence criterion | | Q(k)-Q(k-1)||<εQ，εQIs a convergence tolerance, the iteration stops.
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CN115713024A (en) * 2023-01-05 2023-02-24 西南交通大学 Bearing load-temperature-stress equivalent method based on symbolic regression algorithm
CN115713024B (en) * 2023-01-05 2023-03-28 西南交通大学 Bearing load-temperature-stress equivalent method based on symbolic regression algorithm

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