CN109902377B - Method for analyzing contact stress of clearance revolute pair - Google Patents

Method for analyzing contact stress of clearance revolute pair Download PDF

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CN109902377B
CN109902377B CN201910138216.7A CN201910138216A CN109902377B CN 109902377 B CN109902377 B CN 109902377B CN 201910138216 A CN201910138216 A CN 201910138216A CN 109902377 B CN109902377 B CN 109902377B
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projection point
contact boundary
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皮霆
张云清
马冀
吕天启
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Huazhong University of Science and Technology
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Abstract

The invention relates to a method for improving the calculation precision of contact stress in finite element analysis of a clearance revolute pair. And aiming at the finite element model of the clearance revolute pair, carrying out grid division on the contact boundary, and setting numerical integration points on the contact boundary of the bearing. And searching a projection point of each numerical integration point on the contact boundary of the rotating shaft and the parameter coordinates thereof, and calculating the normal distance and the geometric measurement at the projection point. After the search of all the projection points and the calculation of related variables are completed circularly, the parameter coordinates, the normal spacing, the geometric measurement, the normal traction and the tangential traction of the equivalent projection points are calculated respectively aiming at each node of the contact boundary of the bearing. And after the calculation of equivalent variables of all the nodes is circularly finished, calculating contact force virtual work integral and submitting a system equation. The method can eliminate the system over-constraint caused by the increase of the number of the integration points and reduce the contact stress fluctuation in the clearance rotating pair.

Description

Method for analyzing contact stress of clearance revolute pair
Technical Field
The present invention relates to computer aided engineering analysis, and more particularly to a method for simulating gap revolute pair contact stresses in finite element analysis for time-marching engineering simulation to assist users in making decisions in improving engineering product (e.g., automobile, airplane) design.
Background
Revolute pairs in mechanical systems (e.g., automobiles, airplanes, machine tools, robots) inevitably have clearances due to design, manufacturing, machining, wear, and the like. The clearance makes bearing and pivot frequently take place contact collision in relative motion process, introduces phenomena such as vibration, noise, wearing and tearing in the system, seriously influences system operation precision, shortens system life-span. Therefore, the analysis and calculation of the dynamic response of the mechanism with the clearance kinematic pair have important academic significance and economic value.
The finite element method is the most accurate contact impact numerical simulation method at present. When the finite element method is adopted to calculate the contact force virtual work, a numerical method is needed to calculate the integral of the contact force virtual work. Using numerical integration methods generally requires the spatial distribution of several numerical integration points over the finite element parameters. From the perspective of applying contact constraint, too few integration points are easy to cause contact missing detection, and integration precision cannot be guaranteed, so a plurality of numerical integration points are generally required to be arranged. However, the number of nodes is limited after the finite element of the contact boundary is dispersed, and the excessive integration points can cause the over-constraint of a system equation, so that the stress fluctuation phenomenon occurs on the contact boundary, and the accuracy of numerical analysis is influenced.
Disclosure of Invention
The invention provides a method for eliminating system over-constraint caused by the increase of the number of integral points aiming at a finite element model of a clearance rotating pair, and the contact stress fluctuation in the clearance rotating pair can be reduced. The invention is realized by the following technical scheme:
a method of clearance revolute pair contact stress analysis, the method comprising the steps of:
s1, extracting finite element units of the bearing contact boundary, and setting Gaussian integration points for each unit;
s2, searching the projection point of each bearing contact boundary Gaussian integration point on the contact boundary of the rotating shaft and the parameter coordinates thereof;
s3, calculating the normal spacing and the geometric measurement of all the rotating shaft projection points;
s4, calculating the parameter coordinates, normal spacing and variation thereof, geometric measurement, normal and tangential traction of the equivalent projection points for each node of the contact boundary of the rotating shaft;
and S5, calculating the contact force virtual work.
Specifically, the extracting a bearing contact boundary finite element in the step S1 further includes: and carrying out finite element meshing on a rotating shaft and a bearing of the clearance kinematic pair.
Specifically, the finite element unit for extracting the bearing contact boundary in the step S1 is a one-dimensional unit for respectively extracting the contact boundaries of the rotating shaft and the bearing contact body.
Specifically, the step S1 of setting gaussian integration points includes distributing a plurality of gaussian integration points on each unit of the bearing contact boundary according to a gaussian integration criterion, where the gaussian integration points are 2 times of the order of the interpolation function.
Specifically, the parameter coordinates corresponding to the projection point in the step S2
Figure BDA0001977672490000021
The following nonlinear equation is satisfied:
Figure BDA0001977672490000022
wherein f (-) represents a function, τ1Is a covariant base vector, x, at any point on the contact boundary of the rotating shaft1And x2Respectively are position vectors of any point on the contact boundary of the rotating shaft and the bearing,
Figure BDA0001977672490000023
and xi is the parameter coordinate of the contact boundary of the rotating shaft.
Specifically, the normal distance and the geometric metric at the projection point in the step S3 are calculated according to the following formula:
normal spacing between the gaussian integral point and the projection point:
Figure BDA0001977672490000031
wherein n is a projection point normal unit vector;
geometric measure at projection point
Figure BDA0001977672490000032
For measuring length, area, angle, etc. of the space geometry.
Specifically, the parameter coordinates of the equivalent projection point of each node a of the bearing contact boundary in the step S4
Figure BDA0001977672490000033
And its variation
Figure BDA0001977672490000034
The following equation is satisfied:
Figure BDA0001977672490000035
Figure BDA0001977672490000036
wherein Γ is the contact boundary, RAIs the NURBS interpolation function corresponding to node a.
Specifically, the contact force imaginary work δ W is calculated by the following formula:
Figure BDA0001977672490000037
wherein, tNANormal traction force, delta, for node A equivalent projection pointgNAIs the variation of the normal spacing at the equivalent projection point of the node A, tNAThe equivalent projected point tangential traction for node a.
Specifically, the normal spacing g at the equivalent projection pointNAAnd its variation deltagNACan be calculated as follows:
Figure BDA0001977672490000038
Figure BDA0001977672490000039
in particular, the normal traction force t at the equivalent projection pointNACan be calculated as follows:
tNA＝∈NgNA
wherein e isNA normal penalty parameter;
equivalence ofTangential traction force t of projection pointNACan be calculated as follows:
Figure BDA0001977672490000041
wherein, mu is a friction coefficient,
Figure BDA0001977672490000042
for a test node tangential traction force, the test node tangential traction force is defined as:
Figure BDA0001977672490000043
wherein the subscript n refers to the last time step, ∈TGeometric measures at equivalent projection points as tangential penalty parameters
Figure BDA0001977672490000044
Wherein m is11For geometric measures at the point M of projection corresponding to the point of Gaussian integration, i.e.
Figure BDA0001977672490000045
The values are calculated in step S3.
Compared with the prior art, the invention has the beneficial effects that:
the contact stress in the clearance rotating pair is calculated according to the algorithm disclosed by the invention, the contact force constraint number which is the same as the contact boundary node number can be obtained, the over-constraint problem caused by excessive integration points is effectively avoided, the contact stress fluctuation of the contact boundary is effectively reduced, and the contact stress calculation precision is improved.
Drawings
FIG. 1 is a schematic diagram of a typical slider-crank mechanism for an engine piston;
FIG. 2 is a schematic diagram of meshing of a slider-crank mechanism;
FIG. 3 is a schematic view of a gap rotary pair bearing model;
FIG. 4 is a schematic view of a gap revolute pair rotating shaft model;
FIG. 5 is a schematic view of the contact search and local kinematic geometry of the bearing and the rotating shaft;
FIG. 6 is a graph of the results of an analysis of the stress profile of a contact boundary region without the use of the method of the present invention;
FIG. 7 is a graph showing the results of an analysis of the stress profile of the contact boundary region using the method of the present invention.
Detailed Description
The invention is further described with reference to the following figures and specific examples.
The invention aims to provide a calculation method for reducing contact stress fluctuation caused by numerical integration point over-constraint in clearance rotating pair contact finite element analysis.
Crank-slider mechanisms are very common in mechanical systems, such as piston-connecting rod mechanisms of engines. Fig. 1 is a schematic view of a slider-crank mechanism according to the present embodiment. The crank-slider mechanism comprises a crank 10, a connecting rod 20, a slider 30, a bearing 40 and a rotating shaft 50. The distance between the two rotating shafts of the crank 10 is 1.2m, and the width is 0.2 m. The distance between the two rotating shafts of the connecting rod 20 is 1.6m, and the width is 0.3 m. The length and width of the slider 30 are 0.3 m. The inner diameter of the bearing 40 is 0.06 m. The diameter of the rotating shaft 50 is 0.057 m. The density of all parts is 7800kg/m3. The modulus of elasticity of the spindle 50 and the slider 30 is 207 GPa. The crank 10 and the connecting rod 20 have an elastic modulus of 20.7MPa and a Poisson's ratio of 0.29.
The method for analyzing the contact stress of the rotating shaft of the crank sliding block mechanism and the bearing clearance revolute pair comprises the following steps of:
s1, extracting a bearing contact boundary finite element model, and setting Gaussian integration points for each unit;
s2, searching the projection point of each bearing contact boundary Gaussian integration point on the contact boundary of the rotating shaft and the parameter coordinates thereof;
s3, calculating the normal spacing and the geometric measurement of all the rotating shaft projection points;
s4, calculating the parameter coordinates, normal spacing and variation thereof, geometric measurement, normal and tangential traction of the equivalent projection points for each node of the contact boundary of the rotating shaft;
and S5, calculating the contact force virtual work and submitting a system kinetic equation.
Specifically, first, finite element meshing is performed on the clearance kinematic pair and other structures. The present embodiment uses second order quadrilateral cells to mesh the mechanism, as shown in fig. 2-4. Wherein fig. 2 is a result of meshing the components of the entire slider-crank mechanism. Fig. 3-4 show the meshes of the bearing and the shaft portion, respectively, separately. The present embodiment is only an example of meshing for a slider-crank mechanism, and the proposed stress calculation method is not limited to the illustrated meshing.
In order to calculate the contact force imaginary work in the clearance kinematic pair, the rotating shaft and the bearing are respectively designated as a main contact body and a slave contact body. One-dimensional units, 8 units in the embodiment, on the contact boundary of the main contact body and the auxiliary contact body are respectively extracted, and a plurality of Gaussian integration points are distributed on each unit on the contact boundary of the bearing according to a Gaussian integration criterion. In this embodiment, 2 times of the order of the interpolation function is taken, i.e. 4 gaussian integration points are distributed inside each contact unit. As shown in fig. 5, S is a gaussian integral point on the bearing contact boundary, and in order to determine whether the point S contacts the rotating shaft, a projection point M of the point S on the rotating shaft contact boundary needs to be searched first. The parameter coordinate corresponding to the point M satisfies the following nonlinear equation:
Figure BDA0001977672490000061
wherein f (·) represents a function, ξ is the parameter coordinate of the contact boundary of the rotating shaft,
Figure BDA0001977672490000062
as parameter coordinates of projection point M, τ1Is a covariant base vector, x, at any point on the contact boundary of the rotating shaft1And x2Respectively are position vectors of any point on the contact boundary of the rotating shaft and the bearing,
Figure BDA0001977672490000063
the nonlinear equation is solved by using Newton implicit iteration, and the tangential stiffness matrix of the nonlinear equation is as follows:
Figure BDA0001977672490000064
wherein the content of the first and second substances,
Figure BDA0001977672490000065
geometric metric m11：＝τ1·τ1。
Determining projected point M parameter coordinates
Figure BDA0001977672490000066
Then, the normal spacing between the integral point S and the projected point M can be calculated according to equation (3):
Figure BDA0001977672490000067
wherein g isNIs the normal spacing between the integration point and the projection point, and n is the unit vector normal to point M, see fig. 3. Calculating a geometric metric at projection point M
Figure BDA0001977672490000068
The method is used for geometric measurement calculation at equivalent projection points of finite element nodes of the contact boundary, and measuring the length, the area, the angle and the like of space geometry.
And (3) circularly searching the projection points of all the integral points of 8 units on the bearing contact boundary, and calculating the normal spacing and the geometric measurement of the projection points, and then calculating the parameter coordinates of equivalent projection points, the normal spacing and the variation thereof, the geometric measurement, the normal traction and the tangential traction aiming at each finite element node of the bearing contact boundary.
The method comprises the following specific steps:
for any node A on the bearing boundary, the parameter coordinates of the equivalent projection point
Figure BDA0001977672490000069
And its variation
Figure BDA00019776724900000610
Can be calculated according to the formulas (4) to (5):
Figure BDA00019776724900000611
Figure BDA0001977672490000071
wherein Γ is the contact boundary, RAIs the NURBS interpolation function corresponding to node a.
Normal spacing g at equivalent projection pointNAAnd its variation deltagNACan be calculated according to the following equations (6) to (7):
Figure BDA0001977672490000072
Figure BDA0001977672490000073
geometric metric m at equivalent projection point11ACan be calculated according to equation (8):
Figure BDA0001977672490000074
wherein m is11For geometric measures at the point M of projection corresponding to the point of Gaussian integration, i.e.
Figure BDA0001977672490000075
The calculation is respectively carried out, so that repeated calculation is avoided, and the calculation efficiency is high.
Normal traction force t at equivalent projection pointNACan be calculated according to equation (9):
tNA＝∈NgNA (9)
wherein e isNIs a normal penalty parameter.
In order to calculate the node equivalent projected point tangential traction force, the tangential contact state (static friction or sliding friction) needs to be distinguished. The tangential traction force of a test node shown by the formula (10) is defined:
Figure BDA0001977672490000076
wherein the subscript n refers to the last time step, ∈TIs a tangential penalty parameter. According to the value of the tangential traction of the test node, the tangential traction of the equivalent projection point can be divided into two cases of static friction and sliding friction according to the formula (11):
Figure BDA0001977672490000077
where μ is the coefficient of friction.
After the calculation of the parameter variable of the equivalent projection point of each node is circularly finished, calculating the contact force virtual work delta W:
Figure BDA0001977672490000081
and submitting the contact force virtual work to a system equation set for calculation for system dynamics analysis for later use.
Fig. 6-7 compare the effect of using the method of the present invention on the stress fluctuations in the contact area of an interstitial kinematic pair at a given time. FIG. 6 shows the result of the method without using the method, and it can be seen from the figure that when the penalty factor e is larger thanNRespectively take 104And 106The contact boundary region stress produced significant fluctuations. Fig. 7 shows the stress of the contact boundary region after the method is used, and the black dots show the stress of the contact boundary region without the method. It can be seen from the figure that the stress fluctuation is significantly reduced with respect to the results in fig. 6 (i.e., the results indicated by the black dots in fig. 7), illustrating the effect of the present method on improving the accuracy of the contact stress calculation.
According to the method, the contact boundary is subjected to grid division aiming at a finite element model of the clearance revolute pair, and numerical integration points are arranged on the contact boundary of the bearing. And searching a projection point of each numerical integration point on the contact boundary of the rotating shaft and the parameter coordinates thereof, and calculating the normal distance and the geometric measurement at the projection point. After the search of all the projection points and the calculation of related variables are completed circularly, the parameter coordinates, the normal spacing, the geometric measurement, the normal traction and the tangential traction of the equivalent projection points are calculated respectively aiming at each node of the contact boundary of the bearing. And after the calculation of equivalent variables of all the nodes is circularly finished, calculating contact force virtual work integral and submitting a system equation.
The method for analyzing the contact stress of the clearance rotating pair can obtain the contact force constraint number which is the same as the number of the contact boundary nodes, effectively avoid the over-constraint problem caused by excessive integration points, and effectively reduce the contact stress fluctuation of the contact boundary.

Claims (7)

1. A method for analyzing contact stress of a clearance revolute pair is characterized by comprising the following steps: the method comprises the following steps:
s1, extracting finite element units of the bearing contact boundary, and setting Gaussian integration points for each unit;
s2, searching a projection point M of each bearing contact boundary Gaussian integration point on the contact boundary of the rotating shaft and parameter coordinates thereof;
s3, calculating the normal spacing and the geometric measurement of all the rotating shaft projection points M;
s4, calculating the parameter coordinates and the variation of the equivalent projection point, the normal spacing and the variation of the equivalent projection point, the geometric measurement, the normal traction and the tangential traction for each node A of the contact boundary of the rotating shaft;
s5, calculating the contact force virtual work;
the parameter coordinates of the equivalent projection point of each node A of the bearing contact boundary in the step S4
Figure FDA0002682041490000016
And its variation
Figure FDA0002682041490000015
The following equation is satisfied:
Figure FDA0002682041490000011
Figure FDA0002682041490000012
wherein Γ is the contact boundary, RAFor the NURBS interpolation function corresponding to node a,
Figure FDA0002682041490000017
parameter coordinates of the projection point M;
normal spacing g at equivalent projection pointNAAnd its variation δ gNACalculated according to the following formula:
Figure FDA0002682041490000013
Figure FDA0002682041490000014
wherein, gNThe normal distance between the Gaussian integral point and the projection point;
normal traction force t at equivalent projection pointNACalculated as follows:
tNA＝∈NgNA
wherein e isNA normal penalty parameter;
tangential traction force t at equivalent projection pointTACalculated as follows:
Figure FDA0002682041490000021
wherein, mu is a friction coefficient,
Figure FDA0002682041490000024
for a test node tangential traction force, the test node tangential traction force is defined as:
Figure FDA0002682041490000022
wherein the subscript n refers to the last time step, ∈TIn order to be a tangential penalty parameter,
geometric measurement at equivalent projection point
Figure FDA0002682041490000025
Wherein m is11For the geometric measure at the above-mentioned projection point M, i.e.
Figure FDA0002682041490000026
2. The method of claim 1, wherein the step of extracting bearing contact boundary finite element in S1 further comprises: and carrying out finite element meshing on a rotating shaft and a bearing of the clearance kinematic pair.
3. The method as claimed in claim 2, wherein the step of extracting finite element elements of the bearing contact boundary in S1 is extracting one-dimensional elements on the contact boundary of the rotating shaft and the bearing contact body respectively.
4. The method according to any one of claims 1 to 3, wherein the step of setting Gaussian integration points in S1 includes distributing Gaussian integration points on each cell of the bearing contact boundary according to a Gaussian integration criterion, wherein the Gaussian integration points are 2 times of the order of the interpolation function.
5. The method according to any one of claims 1 to 3, wherein the parameter coordinate corresponding to the projection point M in the step S2 is set to
Figure FDA0002682041490000027
The following nonlinear equation is satisfied:
Figure FDA0002682041490000023
wherein f (-) represents a function, τ1Is a covariant base vector, x, at any point on the contact boundary of the rotating shaft1And x2Respectively are position vectors of any point on the contact boundary of the rotating shaft and the bearing,
Figure FDA0002682041490000033
and xi is the parameter coordinate of the contact boundary of the rotating shaft.
6. The method of claim 5, wherein the normal spacing and the geometric measure at the projection point M in the step S3 are calculated according to the following formula:
normal spacing between the gaussian integral point and the projection point:
Figure FDA0002682041490000031
wherein n is a projection point M normal unit vector;
geometric measurement at projection point M
Figure FDA0002682041490000034
7. The method of claim 6, wherein the contact force imaginary work δ W is calculated by:
Figure FDA0002682041490000032
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