CN111159948B - Reliability analysis method of joint bearing considering random uncertainty - Google Patents
Reliability analysis method of joint bearing considering random uncertainty Download PDFInfo
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Abstract
The disclosure relates to the technical field of reliability analysis, and provides a reliability analysis method of a joint bearing considering random uncertainty, which comprises the following steps: establishing a reliability analysis model of the joint bearing; determining a response surface approximate model of the joint bearing according to the reliability analysis model; and carrying out reliability analysis on the joint bearing according to the response surface approximate model. The reliability analysis method and the reliability analysis device can analyze the reliability of the joint bearing.
Description
Technical Field
The disclosure relates to the technical field of reliability analysis, in particular to a reliability analysis method of a joint bearing considering random uncertainty.
Background
As a sliding bearing, the joint bearing has the characteristics of flexible rotation, compact structure, easy assembly and disassembly and the like, can meet the requirements of heavy load and long service life, and is widely applied to various mechanical equipment such as mines, metallurgy, electric power, traffic, aerospace, textile and the like.
The joint bearing mainly comprises an inner ring with an outer spherical surface and an outer ring with an inner spherical surface, can bear larger load, and can bear radial load, axial load or combined load existing in both radial and axial directions according to different types and structures of the joint bearing. Under these loads, the spherical plain bearing is prone to failure. However, the prior art lacks a method for analyzing the reliability of the joint bearing.
It is to be noted that the information disclosed in the above background section is only for enhancement of understanding of the background of the present disclosure, and thus may include information that does not constitute prior art known to those of ordinary skill in the art.
Disclosure of Invention
The purpose of the present disclosure is to provide a reliability analysis method for a joint bearing, which is capable of analyzing the reliability of the joint bearing, in consideration of random uncertainty.
According to an aspect of the present disclosure, there is provided a reliability analysis method of a joint bearing considering random uncertainty, including:
establishing a reliability analysis model of the joint bearing;
determining a response surface approximate model of the joint bearing according to the reliability analysis model;
and carrying out reliability analysis on the joint bearing according to the response surface approximate model.
In an exemplary embodiment of the present disclosure, the joint bearing includes an inner ring and an outer ring, the outer ring surrounds the inner ring, and a reliability analysis model of the outer ring is:
g out ＝σ b-out -σ out (E out ，E in ，F r ，F a )；
wherein, g out As a function value, σ, of the outer ring b-out As ultimate strength of the outer ring, σ out Is the maximum stress of the outer ring, E out Is the modulus of elasticity of the outer ring, E in Is the elastic modulus of the inner ring, F r Radial loads to which the articulated bearing is subjected, F a Axial loads to which the joint bearing is subjected.
In an exemplary embodiment of the present disclosure, the response surface approximation model is:
in an exemplary embodiment of the present disclosure, the joint bearing includes an inner ring and an outer ring, the outer ring surrounds the inner ring, and a reliability analysis model of the inner ring is:
g in ＝σ b-in -σ in (E out ，E h ，F r ，F a )；
wherein, g in As a function value of the inner circle, σ b-in Is the ultimate strength of the inner ring, σ in Is the maximum stress of the inner ring, E out Is the modulus of elasticity of the outer ring, E in Is the elastic modulus of the inner ring, F r Radial loads to which the articulated bearing is subjected, F a Axial loads to which the joint bearing is subjected.
In an exemplary embodiment of the present disclosure, the response surface approximation model is:
in an exemplary embodiment of the present disclosure, the E out The same as described in the above in The above-mentioned F r And said F a Are normally distributed.
In an exemplary embodiment of the present disclosure, the σ b-out 1960 MPa.
In an exemplary embodiment of the present disclosure, the σ b-in Is 1340 MPa.
In an exemplary embodiment of the present disclosure, the establishing of the reliability analysis model of the joint bearing includes:
establishing a finite element model of the joint bearing;
and establishing a reliability analysis model of the joint bearing according to the finite element model.
In an exemplary embodiment of the disclosure, the reliability analysis model is an implicit extreme state function and the response surface approximation model is an explicit extreme state function.
According to the reliability analysis method of the joint bearing considering the random uncertainty, the reliability analysis model of the joint bearing is established, and the response surface approximate model is determined according to the reliability analysis model, so that the reliability analysis of the joint bearing can be carried out by utilizing the response surface approximate model; meanwhile, the response surface approximate model is expressed in an explicit mode, so that the problem that reliability analysis is difficult to perform due to the fact that the existing reliability analysis model is expressed in an implicit mode is solved.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the disclosure.
Drawings
The above and other features and advantages of the present disclosure will become more apparent by describing in detail exemplary embodiments thereof with reference to the attached drawings. It is to be understood that the drawings in the following description are merely exemplary of the disclosure, and that other drawings may be derived from those drawings by one of ordinary skill in the art without the exercise of inventive faculty.
FIG. 1 is a flow chart of a method of reliability analysis of a spherical plain bearing accounting for random uncertainties in accordance with an embodiment of the present disclosure;
fig. 2 is a frequency histogram of the response amount in the embodiment of the present disclosure.
Detailed Description
Example embodiments will now be described more fully with reference to the accompanying drawings. Example embodiments may, however, be embodied in many different forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the concept of example embodiments to those skilled in the art. The described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided to give a thorough understanding of embodiments of the disclosure. One skilled in the relevant art will recognize, however, that the embodiments of the disclosure can be practiced without one or more of the specific details, or with other methods, materials, devices, etc. In other instances, well-known technical solutions have not been shown or described in detail to avoid obscuring aspects of the present disclosure.
Furthermore, the drawings are merely schematic illustrations of the present disclosure and are not necessarily drawn to scale. The same reference numerals in the drawings denote the same or similar parts, and thus their repetitive description will be omitted. The terms "a" and "the" are used to indicate the presence of one or more elements/components/etc.; the terms "comprising" and "having" are intended to be inclusive and mean that there may be additional elements/components/etc. other than the listed elements/components/etc.
In the related art, the joint bearing is generally used for swinging motion (i.e., angular motion) with low speed, and since the sliding surface is spherical, the joint bearing can also perform tilting motion (i.e., aligning motion) within a certain angle range, and can still normally operate when the concentricity of the supporting shaft and the shaft housing hole is large. The joint bearing can be divided into a radial joint bearing, an angular contact joint bearing and a thrust joint bearing according to the main stress form. The knuckle bearing has a large swing angle and aligning performance, can realize rotation, swing and aligning, has a simple structure, is beneficial to improving the flexibility of a structural part, and is widely applied to aerospace machinery. The reliability of the mechanism is very important for guaranteeing the safety of the structural mechanism product, the reliability of the structural mechanism is developed on the basis of the reliability of electronic components, and the development of the reliability of the structural mechanism is lagged behind compared with the reliability of the electronic components. This is mainly expressed in two aspects, one is that the data required for the reliability analysis of the structural mechanism is relatively lacking, because the structural mechanism is relatively low in standardization; secondly, the reliability analysis of the structural mechanism and the development of the design theory method are lagged behind, in particular to the reliability analysis and design problem of the complex structural mechanism system. Meanwhile, in the engineering practical problem, the uncertainty of the parameters widely exists, so that the analysis of the process of transferring the uncertainty of the input parameters to the uncertainty of the output response of the model in the engineering practical problem is necessary.
For the joint bearing structure, at present, an analytic method is difficult to solve, analysis can be performed only by means of finite element software, and because the extreme state equation at the moment is implicitly represented, the traditional reliability analysis method has difficulty to overcome.
The embodiment of the disclosure provides a reliability analysis method of a joint bearing considering random uncertainty. As shown in fig. 1, the reliability analysis method of the joint bearing considering random uncertainty may include steps S110 to S130, in which:
and step S110, establishing a reliability analysis model of the joint bearing.
And step S120, determining a response surface approximate model of the joint bearing according to the reliability analysis model.
And S130, carrying out reliability analysis on the joint bearing according to the response surface approximate model.
According to the reliability analysis method of the joint bearing considering the random uncertainty, the reliability analysis model of the joint bearing is established, and the response surface approximation model is determined according to the reliability analysis model, so that the reliability analysis of the joint bearing can be carried out by using the response surface approximation model; meanwhile, the response surface approximate model is expressed in an explicit mode, so that the problem that reliability analysis is difficult to perform due to the fact that the existing reliability analysis model is expressed in an implicit mode is solved.
The following describes in detail the steps of the disclosed embodiments:
in step S110, a reliability analysis model of the joint bearing is established.
The joint bearing comprises an inner ring and an outer ring, wherein the outer ring surrounds the inner ring, and the reliability analysis model of the outer ring is as follows:
g out ＝σ b-out -σ out (E out ，E in， F r ，F a )，
wherein, g out Function value of outer circle, σ b-out Is the ultimate strength of the outer ring, σ out Is the maximum stress of the outer ring, E out Is the modulus of elasticity of the outer ring, E in Is insideModulus of elasticity of the ring, F r Radial loads to which the joint bearing is subjected, F a Is the axial load to which the joint bearing is subjected. E out ,E in ,F r ,F a Are subject to a normal distribution. Wherein E is out ,E in ,F r ,F a The distribution parameters of (a) are shown in table 1:
TABLE 1
In addition, the reliability analysis model of the inner ring may be:
g in ＝σ b-in -σ in (E out ，E in ，F r ，F a )；
wherein, g in Function value of inner circle, σ b-in Ultimate strength of the inner ring, σ in Is the maximum stress of the inner ring. The reliability analysis model g out And g in Is an implicit extreme state function.
Sigma in the reliability analysis model out And σ in Are all composed of E out ,E in ,F r ,F a Implicit functions are represented. Assume an input variable E out ,E in ,F r ,F a Is x ═ { x1, x2, x3, x 4 Is then g out Can be expressed as g out (x)，g in Can be expressed as g in (x)。
For example, establishing a reliability analysis model of the joint bearing includes: establishing a finite element model of the joint bearing; and establishing a reliability analysis model of the joint bearing according to the finite element model. Wherein σ above out And σ in May be determined by finite element modeling.
In step S120, a response surface approximation model of the joint bearing is determined from the reliability analysis model.
The response surface approximation model is an explicit extreme state function. With reliability analysis model equal to g out (x) For example, the response surface approximation model may be:
the response surface approximate model comprises n +1 undetermined coefficients b ═ b 0 ,b 1 ,…,b n } T The undetermined coefficient b can be obtained by extracting m (m is more than or equal to n +1) sample points x i ＝{x i1 ,x i2 ,…,x in (i ═ 1,2, …, m), and the coefficient to be determined b is solved using the least squares method shown below:
b＝(a T a) -1 a T y；
wherein y ═ g out (x 1 ),g out (x 2 ),…,g out (x m )} T Is an array of response values corresponding to the sample points,
is a matrix of regression coefficients of order m (n +1) composed of m sample points.
Further, the present disclosure may employ statistical concepts of weighted regression to solve for vector b. By giving | g out (x 1 ) Experimental sample point x with smaller | i Larger weight w in regression analysis i Let | g out (x 1 ) Points where | is smaller are determined
Plays a more important role in making
Can better approximate g out (x)。
If with w i (i-1, …, m) represents the weight of each sample point, then
An m × m diagonal matrix formed by the weights of the m sample points is called a weight matrix, and the weight of each sample point in the regression analysis is considered to obtainThe calculation formula of the weighted least square method of the undetermined coefficient vector b is as follows:
b＝(a T Wa) -1 a T Wy；
to better approximate g out (x) When fitting a response surface, | g is desired out (x 1 ) Sample point x with smaller | i The more important the role, the weight of each sample point and the corresponding weight matrix can be constructed as follows:
where diag (·) represents the diagonal elements of the matrix. The weights given by the above formula are less likely to cause ill-conditioning of the regression matrix than exponential weights.
In other embodiments of the present disclosure, determining a response surface approximation model for a spherical plain bearing from a reliability analysis model includes:
step S1201, selecting linear response surface function
To approximate g out (x)。
Step S1202, selecting a sample point by Bucher design, namely surrounding
Sample points were selected as follows:
for the first iteration, the sampling center point is selected as the mean point, i.e. the
The other 2n points are as follows:
wherein
And
are respectively the ith basic variable x i The mean value and the standard deviation of (a),
is the mean vector, N is the interpolation coefficient, and the superscript (k) represents the kth iteration of the response surface method.
Step S1203, if k is equal to 1, may perform this step, i.e. compare
And
where i is 2,3, …, n +1, j is n +2, n +3, …,2n + 1. Discarding the sample points with large absolute values, and reserving the points with small absolute values to obtain n +1 sample points, and constructing the weight matrix W of the sample points according to the first relational expression.
Step S1204, selecting sample point
The undetermined coefficient matrix and the response surface function of the kth iteration are obtained by applying the first relational expression
Step S1205, the AFOSM method
Design point of
And a reliability index beta (k) 。
Step S1206, using the sample point (mu) x ,g out (μ x ) ) and
linear interpolation is performed to obtain
Of the next iteration of the sampling center
Ith coordinate of point
The calculation can be as follows:
step S1207, repeating step S1204, step S1205, and step S1206 until the reliability index | β calculated twice before and after (k) -β (k-1) |<ξ, ξ are predetermined accuracy criteria. It can be known that, by using a response surface method and combining advanced finite element software, an explicit expression which can be converged to a real implicit limit state function in probability can be obtained through a small amount of calculation, and then a traditional reliability analysis method can be used for reliability analysis.
Through calculation, the response surface approximation model of the joint bearing can be as follows:
with reliability analysis model equal to g in (x) For example, the response surface approximation model can be determined by the same method as follows:
because the joint bearing structure comprises two failure modes of inner ring failure and outer ring failure, the response surface approximate model of the joint bearing can be written into
Verification of response surface approximation model
The present disclosure adopts MCS, LS, SubSet _ MCMC and SubSet _ The IS four reliability analysis methods determine the failure probability of the joint bearing, and the results are shown in Table 2:
TABLE 2
| Method | MCS | LS | SubSet_MCMC | SubSet_IS |
| Number of throw points | 10 7 | 100 | 10000 | 10000 |
| Probability of failure | 0.18323 | 0.18094 | 0.18673 | 0.18369 |
| Error (%) | —— | 1.25 | 1.91 | 0.25 |
As can be seen from table 2, the failure probability of the joint bearing was about 0.183.
The statistical moments (first to fourth orders) of the response calculated by the monte carlo method using the function value as the response and the response surface approximation model of the present disclosure are shown in table 3:
TABLE 3 statistical moments of the orders of the response
| Joint bearing structure | |
| Mean value | 0.88233 |
| Standard deviation of | 0.9904 |
| Deflection degree | -0.088825 |
| Kurtosis | 3.141 |
The histogram of the frequency of the response quantity is shown in fig. 2, and is obtained from each order of statistical moment of the response quantity: the mean value is 0.88233, which corresponds to the case shown in the histogram; the standard deviation is 0.9904, which accords with the condition shown in the histogram; the deviation value is-0.088825, the absolute value of which is small and should be distributed symmetrically, which is quite consistent with the situation shown in the histogram; the kurtosis is 3.141, which is very close to the kurtosis value (3) in the case of a normal distribution. Therefore, it is feasible to perform reliability analysis by using the response surface approximation model of the present disclosure.
Other embodiments of the disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the disclosure. This application is intended to cover any variations, uses, or adaptations of the disclosure following, in general, the principles of the disclosure and including such departures from the present disclosure as come within known or customary practice within the art. It is intended that the specification and examples be considered as exemplary only, with a true scope and spirit of the disclosure being indicated by the following claims.
Claims (9)
1. A method for analyzing reliability of a joint bearing considering random uncertainty, comprising:
establishing a reliability analysis model of the joint bearing;
determining a response surface approximate model of the joint bearing according to the reliability analysis model;
carrying out reliability analysis on the joint bearing according to the response surface approximate model;
the joint bearing comprises an inner ring and an outer ring, the outer ring surrounds the inner ring, and a reliability analysis model of the outer ring is as follows:
g out ＝σ b-out -σ out (E out ,E in ,F r ,F a )；
wherein, g out As a function value, σ, of the outer ring b-out As ultimate strength of the outer ring, σ out Is the maximum stress of the outer ring, E out Is the modulus of elasticity of the outer ring, E in Is the modulus of elasticity of the inner ring，F r Radial loads to which the articulated bearing is subjected, F a Axial loads to which the joint bearing is subjected.
2. The method of analyzing reliability of a joint bearing considering random uncertainty according to claim 1, wherein the response surface approximation model is:
3. the reliability analysis method of a joint bearing considering random uncertainty according to claim 1, wherein the joint bearing comprises an inner ring and an outer ring, the outer ring surrounds the inner ring, and a reliability analysis model of the inner ring is:
g in ＝σ b-in -σ in (E out ,E in ,F r ,F a )；
wherein, g in As a function value of said inner circle, σ b-in Is the ultimate strength of the inner ring, σ in Is the maximum stress of the inner ring, E out Is the modulus of elasticity of the outer ring, E in Is the elastic modulus of the inner ring, F r Radial loads to which the articulated bearing is subjected, F a Axial loads to which the joint bearing is subjected.
4. The method of analyzing reliability of a spherical plain bearing considering random uncertainty as claimed in claim 3, wherein said response surface approximation model is:
5. reliability analysis of a joint bearing considering random uncertainties according to any of claims 1 to 4Method, characterized in that said E out The same as described in the above in The above-mentioned F r And said F a Are subject to a normal distribution.
6. The method of claim 1, wherein σ is the reliability analysis of the spherical plain bearing considering random uncertainty b-out 1960 MPa.
7. Method for the reliability analysis of spherical plain bearings considering the random uncertainty according to claim 3, characterized in that said σ is b-in Is 1340 MPa.
8. The method of claim 1, wherein establishing the reliability analysis model of the oscillating bearing considering random uncertainty comprises:
establishing a finite element model of the joint bearing;
and establishing a reliability analysis model of the joint bearing according to the finite element model.
9. The method of claim 1, wherein the reliability analysis model is an implicit extreme state function and the response surface approximation model is an explicit extreme state function.
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