CN110738003B - Time-varying reliability analysis method for heavy tractor PTO shell - Google Patents

Time-varying reliability analysis method for heavy tractor PTO shell Download PDF

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CN110738003B
CN110738003B CN201911011345.6A CN201911011345A CN110738003B CN 110738003 B CN110738003 B CN 110738003B CN 201911011345 A CN201911011345 A CN 201911011345A CN 110738003 B CN110738003 B CN 110738003B
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姜潮
刘凯
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Hunan University
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Abstract

The invention discloses a method for analyzing time-varying reliability of a PTO shell of a heavy tractor. The method first converts the PTO case time varying reliability problem into a time invariant system reliability problem by time discretizing a random process and a function. Secondly, reliability analysis is carried out on each unit by using a first order moment method (FORM), and a correlation coefficient matrix of the function after dispersion is calculated by a correlation method is given. And finally, acquiring the time-varying reliability of the PTO shell on the basis of the unit reliability result and the function correlation coefficient matrix. The method converts the complex time-varying problem into the conventional time-invariant problem for solving, avoids the calculation of the crossing rate, greatly simplifies the whole time-varying reliability analysis process and has higher efficiency, provides an effective calculation tool for the time-varying reliability analysis in the service period of the PTO shell, and can effectively ensure the safety and the economy of the PTO shell in the service process.

Description

Time-varying reliability analysis method for heavy tractor PTO shell
Technical Field
The invention relates to intelligent manufacturing of a heavy tractor, in particular to a method for analyzing time-varying reliability of a PTO shell of the heavy tractor.
Background
The PTO shell is used as a key part of power output of the heavy tractor, and is influenced by environmental change, random load, material performance degradation and the like in the long-term service process of the heavy tractor, and parameter uncertainty and structural reliability usually change along with the change of service duration, so that the time-varying reliability problem is more complex than the traditional time-invariant reliability problem. Taking into account the time-varying characteristics, the reliability of the PTO housing will decay with the period of service. Therefore, analysis of the reliability of the PTO housing taking into full account these time-varying characteristics is of great engineering importance to the safe operation of the heavy tractor and the personal safety of the user. Currently, the time-varying reliability analysis method for PTO housings is mainly a crossing rate method. The method comprises the steps of firstly calculating the crossing rate of a functional function crossing a threshold, and then converting the crossing rate into time-varying reliability based on a Poisson hypothesis model, a Markov hypothesis model and the like, thereby providing an important reference index for the safety design of a structure in the whole service period. However, "crossing rate" is conceptually relatively complex, requires a random process theory based on "esoteric", which greatly hinders its acceptance and understanding by engineers, and the analysis and calculation of crossing rate is relatively complex and time consuming. Accordingly, there is a need for a conceptually simpler, analytically and computationally efficient method of analyzing the time varying reliability of a heavy tractor PTO housing.
Disclosure of Invention
Aiming at the problems of complex structure, long time consumption and the like, the invention provides a time-varying reliability analysis method for a heavy tractor PTO shell, which can efficiently solve the time-varying reliability of the heavy tractor PTO shell and improve the economic benefit.
According to one aspect of the present invention, there is provided a method of time varying reliability assessment of a heavy tractor PTO housing, the method comprising the steps of:
the first step is as follows: according to the definition of the reliability of the use of the PTO casing of the heavy tractor, in the time period [0, T]Probability P of inner PTO housing being reliables(T) is:
Figure GDA0003646709300000011
wherein P {. cndot. } represents probability calculation, g (·) is a function, and X (t) ═ X1(t),X2(t),...,Xm(t)) is an m-dimensional random process vector, Y ═ Y1,Y2,...,Yn) Is an n-dimensional random vector, T is time, and T is a design reference period.
The second step is that: when the design reference period T in expression (1) is discretized into p equal time periods, the time-varying function is discretized into p time-invariant functions. If the PTO housing does not fail, the corresponding time invariant function at each moment must be greater than zero. So if the PTO housing is treated as different units at different times, the time-varying reliability of the PTO housing can be considered as the reliability of the time-invariant series system of these units, and equation (1) can then be expressed as:
Figure GDA0003646709300000021
In the formula Xi＝(X1(ti),X2(ti),...,Xm(ti) Is X (t) is subjected to process discretization at tiAnd (4) corresponding m-dimensional random vectors. Its covariance matrix COV (X)i) Can be obtained from the cross-correlation function matrix and the autocorrelation function vector of X (t).
The third step: when using FORM to analyze the reliability of the cell in equation (2), first, X is transformed using the total probabilityiAnd Y are transformed into a standard Gaussian vector U, respectivelyiAnd V:
Figure GDA0003646709300000022
Figure GDA0003646709300000023
in the formula of Ui,kAnd VkRespectively represent UiAnd the kth component of V,. phi-1(. cndot.) is the inverse of a standard Gaussian distribution function,
Figure GDA0003646709300000024
represents Xi,kThe distribution function of (a) is determined,
Figure GDA0003646709300000025
represents YkThe distribution function of (2).
After the above transformation, equation (2) may become:
Figure GDA0003646709300000026
wherein g' (U)i,V,ti) Corresponding to g (X)i,Y,ti) A function under a standard gaussian space. Then any function g' (U)i,V,ti) The Maximum Possible Point (MPP) of (c) can be obtained by solving the following optimization problem:
Figure GDA0003646709300000027
where T represents the vector transpose. The optimal solution of the above equation, i.e. MPP point MiExpressed as:
Figure GDA0003646709300000028
the reliability index betaiComprises the following steps:
βi＝||Mi|| (8)
the fourth step: all function functions g' (U)i,V,ti) 1,2, p performs a first order taylor expansion at its MPP, then equation (5) can be approximated as:
Figure GDA0003646709300000031
in the formula of alphaU,iIs an m-dimensional vector, corresponding to UiThe sensitivity of (c); alpha is alphaV,iIs an n-dimensional vector corresponding to the sensitivity of V, αU,iAnd alphaV,iCan be obtained by the following formula:
Figure GDA0003646709300000032
The fifth step: in the formula (9)
Figure GDA0003646709300000033
Because of UiAnd V are independent standard Gaussian vectors, and L can be obtained by the formulas (8) and (10)iObey a mean value of betaiGaussian distribution with standard deviation of 1, noted as Li～N(βi1), i ═ 1, 2. Equation (9) may be changed to:
Figure GDA0003646709300000034
any component of the matrix of correlation coefficients p, i.e. LiAnd LjCorrelation betweenCoefficient rhoi,jComprises the following steps:
Figure GDA0003646709300000035
in the formula
Figure GDA0003646709300000036
Are respectively LiAnd LjStandard deviation of (A), and
Figure GDA0003646709300000037
also consider UiAnd UjAnd V are generally independent of each other, equation (12) can be expressed as:
Figure GDA0003646709300000038
in the formula of alphaU,i,kIs alphaU,iThe kth component of (1).
And a sixth step: since X (t) components are independent of each other, the correlation coefficient ρ (U) is obtained when k ≠ k' in the above equationi,k,Uj,k′) When 0, then equation (13) becomes:
Figure GDA0003646709300000041
in the formula Ck(. is) Xk(t) autocorrelation coefficient function. If Xk(t) instead of a gaussian process, the correlation between random variables will change after the total probability transformation, and the correlation coefficients can be modified using a Nataf transform:
ρ(Ui,k,Uj,k)＝Nataf(Ck(ti,tj)) (15)
wherein Nataf (. cndot.) denotes a Nataf transformation. When X is presentkWhen (t) is a Gaussian process, Nataf (C)k(ti,tj))＝Ck(ti,tj). Equation (14) can be expressed in a more general form:
Figure GDA0003646709300000042
in the formula Ni,jIs a diagonal matrix:
Figure GDA0003646709300000043
the seventh step: the correlation coefficient of any two post-discretization function functions can be calculated by equation (16), and thus a correlation coefficient matrix ρ can be established. By combining the reliability index vector β obtained by equation (8), we can solve the final time-varying reliability by the following equation:
Ps(T)＝Φp(β,ρ) (18)
In the formula phip(. beta.) is a standard Gaussian distribution function of dimension p, beta ═ beta1,β2,...,βp) As a function of g (X)i,Y,ti) I 1,2, and p corresponds to a reliability indicator vector. Rho is g (X)i,Y,ti) 1,2, a matrix of correlation coefficients for p.
The invention has the advantages and beneficial effects that:
1. the method of the invention converts the time-varying reliability problem into the traditional time-invariant reliability problem to solve by dispersing the random process and the function. The calculation of the crossing rate is avoided, the whole analysis process is greatly simplified and easy to understand, and the calculation process is more convenient and efficient.
2. The method can be expanded to the problem of system reliability, and a system time-varying reliability analysis method is established to process more complex multiple failure modes.
Description of the figures
FIG. 1 is a block flow diagram of the method of the present invention.
Fig. 2PTO housing model.
Detailed Description
The invention provides a time-varying reliability analysis method for a PTO shell of a heavy tractor based on process dispersion. Compared with the traditional reliability calculation method, the method has the advantages that the complicated time-varying reliability problem is converted into the conventional time-invariant reliability problem to be solved, and the calculation of the crossing rate is avoided; meanwhile, the method is expanded to the important reliability problem of a time-varying system, and has the double advantages of high precision and high efficiency.
The invention is described in further detail below with reference to the following figures and specific examples:
the first step is as follows: the PTO casing was modeled using a finite element method, and the entire model comprised 58674 hexahedral elements. A Latin hypercube method is adopted to obtain 60 samples, a finite element model is called for analysis, and a second-order response surface model of stress is constructed on the basis that:
Figure GDA0003646709300000051
in a time period [0, T]Probability P of inner PTO housing being reliables(T) is:
Figure GDA0003646709300000052
wherein g (·) is a second-order response surface model of stress, x (T) ═ p (T) is a one-dimensional random process vector, Y ═ E (v) is a two-dimensional random vector, T is time, and T is a design reference period.
The second step is that: from expert experience and limited information available, the uncertainty parameter distribution for a heavy tractor PTO housing is shown in table 1:
TABLE 1 heavy tractor PTO casing uncertainty parameter distribution chart
Figure GDA0003646709300000053
The third step: the design reference period T is 10 years, and is discretized into 50 equal periods, and the time-varying function is discretized into 50 time-invariant functions. If the PTO housing does not fail, the corresponding time invariant function at each moment must be greater than zero. Considering the PTO housing as different units at different times, the time-varying reliability problem of the PTO housing can be translated into a time-invariant series system reliability problem, as shown in the following equation:
Figure GDA0003646709300000054
The fourth step: the reliability of the cell in equation (20) is solved by using the FORM method:
Figure GDA0003646709300000055
in the formula
Figure GDA0003646709300000061
And VTRespectively represent XiAnd the transpose of the standard Gaussian vector of Y, g' (U)i,V,ti) Corresponding to g (X)i,Y,ti) A function under a standard gaussian space. The optimal solution of the above equation, i.e. MPP point MiExpressed as:
Figure GDA0003646709300000062
reliability index betaiComprises the following steps:
βi＝||Mi|| (8)
the fifth step: all function functions g' (U)i,V,ti) 1,2, p performs a first order taylor expansion at its MPP, then equation (20) can be approximated as:
Figure GDA0003646709300000063
in the formula of alphaU,iIs an m-dimensional vector, corresponding to UiThe sensitivity of (2); alpha is alphaV,iIs an n-dimensional vector corresponding to the sensitivity of V, αU,iAnd alphaV,iCan be obtained by the following formula:
Figure GDA0003646709300000064
note the book
Figure GDA0003646709300000065
Because of UiAnd V are independent standard Gaussian vectors, and L can be obtained by the formulas (8) and (10)iObey a mean value of betaiAnd a gaussian distribution with a standard deviation of 1. L isiAnd LjCoefficient of correlation between pi,jComprises the following steps:
Figure GDA0003646709300000066
in the formula
Figure GDA0003646709300000067
Are respectively LiAnd LjStandard deviation of (A), and
Figure GDA0003646709300000068
also consider UiAnd UjAnd V are generally independent of each other, then equation (12) may be changed to:
Figure GDA0003646709300000069
in the formula of alphaU,i,kIs alphaU,iThe kth component of (1).
And a sixth step: by the obtained correlation coefficient matrix rho and the reliability index vector beta, the reliability of the PTO shell at different moments can be solved by the following formula:
Ps(T)＝Φp(β,ρ) (18)
finally, the time-varying reliability result of the PTO housing in 1 to 10 years is solved, in order to verify the accuracy of the method, monte carlo simulation analysis (MCS) is performed on the stress response surface of the PTO housing by using 1e6 samples, and the result is shown in table 1. The table shows that the method of the patent is very close to the MCS result, has higher precision, and illustrates the effectiveness of the patent.

Claims (2)

1. A method for analyzing time-varying reliability of a heavy tractor PTO shell is characterized by comprising the following steps:
the first step is as follows: according to the definition of the use reliability of the PTO shell of the heavy tractor, modeling the PTO shell by adopting a finite element method, wherein the whole model comprises 58674 hexahedron units, obtaining 60 samples by adopting a Latin hypercube method, calling the finite element model for analysis, and constructing a second-order response surface model of stress on the basis:
Figure FDA0003646709290000011
at time period [0, T]Probability P of inner PTO housing being reliables(T) is:
Figure FDA0003646709290000012
in the formula, P {. is a probability calculation, g (·) is a second-order response surface model of stress, X (T) ═ P (T) is a one-dimensional random process vector, Y ═ E, v is a two-dimensional random vector, T is time, and T is a design reference period;
the second step is that: when the design reference period T in the formula (1) is discretized into p equal periods, the time-varying function is discretized into p time-invariant function functions, and if the PTO housing does not fail, the time-invariant function corresponding to each moment must be greater than zero, so if the PTO housing is regarded as different units in different periods, the time-varying reliability of the PTO housing can be regarded as the reliability of the time-invariant series system formed by the units, and then the formula (1) can be expressed as:
Figure FDA0003646709290000013
In the formula Xi＝(X1(ti),X2(ti),...,Xm(ti) Is X (t) is subjected to process discretization at tiCorresponding m-dimensional random vector, its covariance matrix COV (X)i) Can be obtained by the cross-correlation function matrix and the self-correlation function vector of X (t);
the third step: when using FORM to analyze the reliability of the cell in equation (2), first, X is transformed using the total probabilityiAnd Y are transformed into a standard Gaussian vector U, respectivelyiAnd V:
Figure FDA0003646709290000014
Figure FDA0003646709290000015
in the formula of Ui,kAnd VkRespectively represent UiAnd the kth component of V,. phi-1(. cndot.) is the inverse of a standard Gaussian distribution function,
Figure FDA0003646709290000016
represents Xi,kThe distribution function of (a) is determined,
Figure FDA0003646709290000017
represents YkThe distribution function of (a) is determined,
after the above transformation, equation (2) may become:
Figure FDA0003646709290000021
in the formula g' (U)i,V,ti) Corresponding to g (X)i,Y,ti) Function in standard Gaussian space, then any function g' (U)i,V,ti) The Maximum Possible Point (MPP) of (c) can be obtained by solving the following optimization problem:
Figure FDA0003646709290000022
in the formula
Figure FDA0003646709290000023
And VTRespectively represent XiTransposition of the standard Gaussian vector of Y, the optimal solution of the above equation, i.e. MPP point MiExpressed as:
Figure FDA0003646709290000024
the reliability index betaiComprises the following steps:
βi＝||Mi|| (8)；
the fourth step: all function functions g' (U)i,V,ti) 1,2, p performs a first order taylor expansion at its MPP, then equation (5) can be approximated as:
Figure FDA0003646709290000025
in the formula of alphaU,iIs an m-dimensional vector, corresponding to UiThe sensitivity of (c); alpha is alphaV,iIs an n-dimensional vector corresponding to the sensitivity of V, α U,iAnd alphaV,iCan be obtained by the following formula:
Figure FDA0003646709290000026
the fifth step: in the formula (9)
Figure FDA0003646709290000027
Because of UiAnd V are independent standard Gaussian vectors, and L can be obtained by the formulas (8) and (10)iObey mean value of betaiGaussian distribution with standard deviation of 1, noted as Li～N(βi,1),i＝1,2,...,p，Equation (9) may be changed to:
Figure FDA0003646709290000028
any component of the matrix of correlation coefficients p, i.e. LiAnd LjCoefficient of correlation between pi,jComprises the following steps:
Figure FDA0003646709290000029
in the formula
Figure FDA0003646709290000031
Are respectively LiAnd LjStandard deviation of (A), and
Figure FDA0003646709290000032
also consider UiAnd UjAnd V are generally independent of each other, equation (12) can be expressed as:
Figure FDA0003646709290000033
in the formula of alphaU,i,kIs alphaU,iThe kth component of (a);
and a sixth step: since X (t) components are independent of each other, the correlation coefficient ρ (U) is obtained when k ≠ k' in the above equationi,k,Uj,k′) When 0, then equation (13) becomes:
Figure FDA0003646709290000034
in the formula Ck(. is) Xk(t) autocorrelation coefficient function if Xk(t) instead of a gaussian process, the correlation between random variables will change after the total probability transformation, and the correlation coefficients can be modified using a Nataf transform:
ρ(Ui,k,Uj,k)＝Nataf(Ck(ti,tj)) (15)
wherein Nataf (. cndot.) denotes the Nataf transformation, when XkWhen (t) is a Gaussian process, Nataf (C)k(ti,tj))＝Ck(ti,tj) Equation (14) can be expressed in a more general form:
Figure FDA0003646709290000035
in the formula Ni,jIs a diagonal matrix:
Figure FDA0003646709290000036
the seventh step: by calculating the correlation coefficient of any two discrete function functions through the formula (16), a correlation coefficient matrix p can be established, and by combining the reliability index vector beta obtained through the formula (8), the final time-varying reliability can be solved through the following formula:
Ps(T)＝Φp(β,ρ) (18)
In the formula phip(. beta.) is a standard Gaussian distribution function of dimension p, beta ═ beta1,β2,...,βp) As a function of g (X)i,Y,ti) 1,2, p, and ρ is g (X)i,Y,ti) 1,2, a matrix of correlation coefficients for p.
2. The method of claim 1, wherein the third step further comprises: designing a reference period T as 10 years, dispersing the reference period T into 50 equal periods, dispersing the time-varying function into 50 time-invariant functions, if the PTO shell does not fail, the time-invariant function corresponding to each moment needs to be larger than zero, and regarding the PTO shell as different units in different periods, the time-varying reliability problem of the PTO shell can be converted into the time-invariant reliability problem of a series system, as shown in the following formula:
Figure FDA0003646709290000041
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