CN111967204A - Anti-resonance reliability and sensitivity analysis method for flow transmission pipeline - Google Patents

Anti-resonance reliability and sensitivity analysis method for flow transmission pipeline Download PDF

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CN111967204A
CN111967204A CN202010805392.4A CN202010805392A CN111967204A CN 111967204 A CN111967204 A CN 111967204A CN 202010805392 A CN202010805392 A CN 202010805392A CN 111967204 A CN111967204 A CN 111967204A
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郭庆
陈冰茜
刘永寿
仝国军
文永帅
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Northwestern Polytechnical University
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Abstract

The invention relates to an anti-resonance reliability and sensitivity analysis method for a fluid delivery pipeline, which obtains a vibration control equation of the pipeline and realizes the solution of the natural frequency of the pipeline by adopting a dynamic stiffness method. The method comprises the steps of introducing uncertain random variables into a parametric model of the flow transmission pipeline, establishing a flow transmission pipeline anti-resonance function based on natural frequency and external excitation frequency of the pipeline, achieving efficient solving of anti-resonance failure probability through actively learning a Kriging method, further solving to obtain anti-resonance reliability variable sensitivity indexes and mode sensitivity indexes on the basis of the anti-resonance failure probability, providing a specific path for anti-resonance reliability and sensitivity evaluation of a flow transmission pipeline system through analysis of the sensitivity indexes, and indicating a direction for safety optimization design of the flow transmission pipeline.

Description

Anti-resonance reliability and sensitivity analysis method for flow transmission pipeline
Technical Field
The invention relates to the technical cross field of a flow transmission pipeline and reliability analysis, in particular to an anti-resonance reliability and sensitivity analysis method for the flow transmission pipeline.
Background
The flow transmission pipeline system is widely applied to large-scale mechanical equipment in the fields of aerospace, petrochemical industry, nuclear energy and ships, and often faces extremely complex vibration environments in service, and the complex alternating vibration environments often cause extremely destructive resonance failure to cause overall damage or failure of the equipment system, so that anti-resonance analysis of the flow transmission pipeline needs to be researched urgently. The inherent frequency of the pipeline is often influenced by the fluid pressure and flow velocity factors to change due to the fluid-solid coupling action of the fluid and the pipe wall in the fluid transmission pipeline, and in addition, a large number of uncertain factors are faced in the production, assembly, measurement and service life processes of the fluid transmission pipeline, so that the inherent frequency of the pipeline is uncertain. Most of the current patent documents of the fluid transmission pipeline are fluid-solid coupling characteristic prediction and vibration reduction methods and devices (CN109635500A, China; CN109740211A, China; CN109027425A, China; CN108763741A, China; CN108763628A, China), and attention to anti-resonance reliability research of the fluid transmission pipeline is lacked. In the anti-resonance reliability correlation paper of the flow transmission pipeline, a small amount of literature focuses on the research of anti-resonance reliability (the analysis of the resonance reliability of the two simply-branched flow transmission pipelines, vibration and impact, 2012,31 (12): 160-. Therefore, the invention introduces the uncertainty analysis idea into the anti-resonance analysis of the flow transmission pipeline, and provides a direction for anti-resonance optimization design by analyzing the influence degree of the traceable random variable on the pipeline resonance failure probability through sensitivity.
The above information disclosed in this background section is only for enhancement of understanding of the background of the invention and therefore it may contain information that does not constitute prior art that is already known to a person of ordinary skill in the art.
Disclosure of Invention
The technical problem solved by the invention is as follows: in order to solve the defects of the prior art, the invention aims to provide an analysis framework for the anti-resonance reliability and sensitivity of the flow pipeline under the influence of uncertain random variables, and provide theoretical guidance for the anti-resonance optimization design of the flow pipeline.
The technical scheme of the invention is as follows: a method for analyzing anti-resonance reliability and sensitivity of a flow transmission pipeline comprises the following steps:
step 1: establishing a flow transmission pipeline vibration control equation:
Figure BDA0002628929840000021
wherein, (EI)*Denotes the effective flexural rigidity, MfFluid mass per unit length, U fluid flow rate, p fluid pressure, AfIs the cross-sectional area of the fluid, m*The mass per unit length of the pipeline, w is the transverse displacement of the pipeline, x is the axial coordinate of the pipeline and t is the time;
step 2: solving the natural frequency of the flow transmission pipeline: solving a vibration control equation of the fluid delivery pipeline by adopting a dynamic stiffness method to obtain the natural frequency of each order of the fluid delivery pipeline, wherein the order is judged by an external excitation frequency range:
h(ω)＝det[Kcg]＝0
wherein: ω denotes the natural frequency of the flow conduit, KcgFor the overall stiffness matrix of the fluid delivery conduit, the subscript cg represents the matrix and vector of the additive boundary conditions, det [ ·]Representing determinant operations.
And step 3: establishing an anti-resonance function, and obtaining the resonance failure probability P of the flow transmission pipeline according to the natural frequency and the external excitation frequency of the flow transmission pipeline obtained in the step 2
Figure BDA0002628929840000022
Wherein, gi(X)＝Zi(X)-q，
Figure BDA0002628929840000023
gi (X) is a function of the ith order resonance failure, Zi(X) represents the closeness degree of the ith order natural frequency and the exciting force frequency, q is the threshold value of resonance failure, and omegaiIs the ith order natural frequency, and S is the exciting force frequency;
and 4, step 4: analyzing the anti-resonance reliability of the flow transmission pipeline: solving the anti-resonance failure probability of the flow transmission pipeline in the step 3 by adopting an active learning Kriging model so as to obtain the anti-resonance reliability of the flow transmission pipeline, wherein the expression of the Kriging model is
g(x)＝β+z(x)
Wherein g (x) is a function to be fitted, beta represents global approximation of a model, z (x) provides local deviation for a Gaussian random process, and the Kriging model comprises a Gaussian random process and can provide a predicted value mug(x) Sum variance
Figure BDA0002628929840000031
I.e., g (x) N (mu)g(x),σg(x) ). Using U-shaped learning equations
Figure BDA0002628929840000032
Adding points, thereby greatly improving the prediction capability of the Kriging model;
and 5: analyzing anti-resonance sensitivity of the flow transmission pipeline: based on the Kriging model established in the step 4, the anti-resonance sensitivity analysis of the flow transmission pipeline is further carried out, and the variable global sensitivity index formula is
Figure BDA0002628929840000033
Wherein:
Figure BDA0002628929840000034
i.e. the variable X of the flow transmission pipelineiSensitivity index of (1), PfThe probability of the failure of the resonance of the fluid transmission pipeline is shown,
Figure BDA0002628929840000035
represents when variable XiFailure probability under fixed time flow pipe resonance condition, IFRepresenting a function indicative of resonant failure of the pipe,
Figure BDA0002628929840000036
represents when variable XiAnd E represents an expected operation. The overall sensitivity index of the resonance variable of the flow transmission pipeline can fully reflect the influence degree of the input variable on the resonance failure probability of the flow transmission pipeline, and the importance sequence of the input variable obtained through overall sensitivity analysis provides important reference for engineering designers to carry out anti-resonance optimization design on the flow transmission pipeline.
The mode sensitivity index is formulated as
Figure BDA0002628929840000037
Figure BDA0002628929840000038
Wherein, PfjRepresents the j-th order resonance failure probability of the pipeline, PfRepresenting the probability of resonant failure, P, of the pipe systemfj/PfThe ratio is a factor of the probability difference, Pr denotes the probability operation, IFRepresenting a conduit resonance failure indication function, IjRepresenting the j-th order resonant failure indication function of the pipeline. By definition, the first mode sensitivity index MIFjThe influence of the j-th order resonance failure mode on the failure domain and the failure probability of the flow transmission pipeline is measured.
The further technical scheme of the invention is as follows: and 5, fully reflecting the input variable and the influence degree on the resonance failure probability of the flow transmission pipeline by the overall sensitivity index of the resonance variable of the flow transmission pipeline in the step 5, and obtaining the importance sequence of the input variable through overall sensitivity analysis.
The further technical scheme of the invention is that: MI in said step 5FjThe larger the jth order resonance failure mode, the more important the failure of the fluid delivery conduit. Second mode sensitivity index MISjThe influence of the j-th order resonance failure mode on the safety of the flow transmission pipeline is quantified. MISjThe larger the influence of the j-th order resonance failure mode on the safety domain of the flow transmission pipeline is.
Effects of the invention
The invention has the technical effects that: the invention can provide guidance for anti-resonance optimization design of the flow transmission pipeline system. According to the concrete pipe type, the vibration control equation of the pipeline is obtained by referring to the existing literature data, and the solution of the natural frequency of the pipeline is realized by adopting a dynamic stiffness method. The method comprises the steps of introducing uncertain random variables into a flow transmission pipeline parameterized model, establishing an anti-resonance function based on natural frequency and external excitation frequency of a pipeline, achieving efficient solving of anti-resonance failure probability through actively learning a Kriging (ALK) method, and further solving to obtain anti-resonance reliability moment independent global sensitivity indexes on the basis of the anti-resonance failure probability, wherein the anti-resonance reliability moment independent global sensitivity indexes comprise variable sensitivity indexes and mode sensitivity indexes. And judging the risk degree of the resonance failure of the pipeline system according to the anti-resonance failure probability, and providing a criterion for further anti-resonance optimization design. And sorting according to the variable sensitivity indexes, screening out random variables with larger influence degree on the resonance of the pipeline system, controlling the uncertainty of the random variables, considering the variables with small sensitivity values and neglecting the influence of the uncertainty, and achieving the purpose of reducing the dimension of the anti-resonance optimization design. And identifying each order of frequency with large influence on the resonance failure of the flow transmission pipeline system based on mode sensitivity index sequencing, and pertinently optimizing the order of natural frequency and external excitation frequency to guide the anti-resonance optimization design of the pipeline system. The invention completely realizes the analysis processes of theoretical solution of the fluid transmission pipeline, uncertainty introduction, establishment of a resonance failure function, solution of the resonance failure probability and solution of the variable sensitivity and the mode sensitivity, and provides a specific path for anti-resonance reliability and sensitivity evaluation of the fluid transmission pipeline system.
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FIG. 1 is a flow chart of the overall method of the present invention
FIG. 2m is a schematic view of node displacement across cells
FIG. 3 is a schematic diagram of a transient-loaded two-span functionally gradient material flow pipeline: (a) loading; (b) variable sensitivity index values (numbers 1 to 12 represent the variable ρ) when the load position map 4U is 20m/sf，ρi，ρo，Ro，T，Ei，Eo，S1，S2U, p and n)
Detailed Description
Referring to fig. 1-4, the vibration control equation of the fluid pipeline is determined according to the factors of the material, structure, working environment and fluid property of the fluid pipeline system;
in an exemplary embodiment of the disclosure, the vibration control equation of the multi-span functional gradient fluid delivery pipeline is
Figure BDA0002628929840000051
Wherein, (EI)*Denotes the effective flexural rigidity, MfFluid mass per unit length, U fluid flow rate, p fluid pressure, AfIs the cross-sectional area of the fluid, m*The mass per unit length of the pipeline, w is the transverse displacement of the pipeline, x is the axial coordinate of the pipeline and t is the time;
solving the inherent frequency of each order of the flow pipeline by adopting a dynamic stiffness method according to a flow pipeline control equation, wherein the order is judged by an external excitation frequency range;
in an exemplary embodiment of the present disclosure, let general solution be w (x, t) ═ w (x) eiωtSubstituting the obtained node displacement into the local coordinate system of the m-span unit
Figure BDA0002628929840000052
The corresponding matrix vector is in the form of Wm＝Ym(ω)wm. Corresponding nodal forces of
Figure BDA0002628929840000061
The corresponding matrix vector form is Fm＝Xm(ω)wm. After the multi-span unit dynamic stiffness matrix is assembled, the relation K of node force and node displacement under the global coordinate system can be obtainedg(ω)Wg＝Fg。
The natural frequency of the pipeline can pass h (omega) det (K)cg]The value is determined as 0.
Establishing an anti-resonance reliability function of the flow transmission pipeline according to the natural frequency and the external excitation frequency of the flow transmission pipeline, and determining a random variable which has a large influence on the resonance failure probability;
in an exemplary embodiment of the present disclosure, the anti-resonance reliability function is
Figure BDA0002628929840000062
Wherein, gi(X)＝Zi(X)-q，
Figure BDA0002628929840000063
ωiIs the ith order natural frequency.
Solving the anti-resonance failure probability by adopting an active learning Kriging model according to the flow transmission pipeline parameterized model and the anti-resonance reliability function;
in an exemplary embodiment of the present disclosure, the expression of the Kriging model is
g(x)＝β+z(x)
Where g (x) is the function to be fitted, β represents the model global approximation, and z (x) provides the local bias for the gaussian random process.
And obtaining a variable sensitivity index value and a mode sensitivity index value, a variable importance sequence and a mode importance sequence based on the variable global sensitivity index and the mode sensitivity index formula according to the established ALK model and the anti-resonance failure probability.
In an exemplary embodiment of the present disclosure, the variable global sensitivity index is formulated as
Figure BDA0002628929840000064
The mode sensitivity index is formulated as
Figure BDA0002628929840000065
Figure BDA0002628929840000071
Wherein:
Figure BDA0002628929840000072
i.e. the variable X of the flow transmission pipelineiSensitivity index of (1), PfThe probability of the failure of the resonance of the fluid transmission pipeline is shown,
Figure BDA0002628929840000073
represents when variable XiFailure probability under fixed time flow pipe resonance condition, IFRepresenting a function indicative of resonant failure of the pipe,
Figure BDA0002628929840000074
represents when variable XiAnd E represents an expected operation. The overall sensitivity index of the resonance variable of the flow transmission pipeline can fully reflect the influence degree of the input variable on the resonance failure probability of the flow transmission pipeline, and the importance sequence of the input variable obtained through overall sensitivity analysis provides important reference for engineering designers to carry out anti-resonance optimization design on the flow transmission pipeline.
For each step, the following section 4 is detailed below:
1. solution of natural frequency
The natural frequency of the fluid transmission pipeline is solved by adopting a dynamic stiffness method. Taking the solution of the multi-span functional gradient fluid transmission pipeline as an example. The differential equation of motion of the multi-span functional gradient flow transmission pipeline under the pressure action is taken as
Figure BDA0002628929840000075
Wherein, (EI)*，Mf，U，p，Af，m*And w, x and t respectively represent effective bending stiffness, fluid mass per unit length, fluid flow rate, fluid pressure, fluid cross-sectional area, pipeline mass per unit length, pipeline lateral displacement, pipeline axial coordinate and time.
Let us assume the general solution of the differential equation of motion as
w(x,t)＝W(x)eiωt (2)
Where w (x) represents a solution of the lateral displacement in the frequency domain, ω represents the natural circular frequency, and i is an imaginary unit.
Substituting the general solution into an equation to obtain a vibration equation in a frequency domain
Figure BDA0002628929840000076
Further, the solution of the displacement in the frequency domain is set as
W(x)＝ceikx (4)
Wherein c represents a undetermined constant, and k represents a wave number.
Substituting the equation to obtain a scattering equation
(EI)*(1+iωα)k4-(MfU2+pAf)k2-2ωMfUk-ω2(Mf+m*)＝0 (5)
From the above equation, it can be seen that there are four solutions to the wave number k, and the solution of the displacement in the frequency domain can be set as
Figure BDA0002628929840000081
According to the Euler-Bernouli theory, the solution of the corner, the bending moment and the shearing force of the functional gradient pipeline in the frequency domain is
Figure BDA0002628929840000082
Figure BDA0002628929840000083
Figure BDA0002628929840000084
The node displacement of the mth span unit of the multi-span functional gradient pipeline is shown in fig. 2, and according to fig. 2, the relationship between the node displacement and the node degree of freedom at the left and right ends of the mth span unit can be expressed as
Figure BDA0002628929840000085
Mml＝-M(0),Qml＝-Q(0),Mmr＝M(lm),Qmr＝Q(lm) (11)
Wherein lmDenotes the length of the mth cross-pipe unit, and l and r denote the left and right ends of the unit, respectively.
In order to obtain the dynamic stiffness matrix of the mth span unit of the fluid transmission pipeline, the node displacement of the unit is as follows in a local coordinate system
Figure BDA0002628929840000086
Wherein λ isj＝ikj(j＝1,2,3,4)。
Writing the above formula into a matrix vector form
Wm＝Ym(ω)wm (13)
Wherein, WmAnd wmRespectively representA displacement vector and a coefficient vector.
Then, the node force vector of the mth cross unit is
Figure BDA0002628929840000091
Wherein the content of the first and second substances,
Figure BDA0002628929840000092
j＝1,2,3,4。
accordingly, the matrix vector form of the above formula is
Fm＝Xm(ω)wm (15)
Wherein, FmA nodal force matrix is represented.
Therefore, the relationship between the node force vector and the node displacement vector of the mth cross unit is
Fm＝Km(ω)wm (16)
Wherein, KmRepresenting the dynamic stiffness matrix, K, of the mth span unitm(ω)＝Xm(ω)Ym(ω)-1。
According to the method, the dynamic stiffness matrix of other spans can be established, then the unit dynamic stiffness matrix is assembled by referring to a finite element method, and the relation between the node displacement and the node force of the multi-span functional gradient fluid transmission pipeline in a global coordinate system is established as
Kg(ω)Wg＝Fg (17)
Wherein, Kg、WgAnd FgRespectively representing a dynamic stiffness matrix, a displacement vector and a node force vector under a global coordinate system. For integral rigidity matrix KgAdding boundary conditions to
Kcg(ω)Wcg＝Fcg (18)
Where the subscript cg represents the matrix and vector to which the boundary condition is added.
Obviously, a sufficient condition for the equation to take a non-zero solution is that the determinant of the coefficient matrix is zero, i.e. the equation takes a non-zero solution
h(ω)＝det[Kcg]＝0 (19)
The natural frequency of the multi-span flow transmission pipeline can be obtained through the formula.
2. Establishing an anti-resonance function
Assuming that the natural frequency of the fluid transmission pipeline is RR and the exciting force frequency is S (considering the influence of external and self modes), the traditional vibration design specification considers that when 1-k is1<RR/S<1+k2The vibration design is risky, where the parameter k1、k2Distributed between 0 and 0.3, k1And k2The corresponding values are different according to different materials. Accordingly, it is considered that when the natural frequency of the fluid transmission pipe and the frequency of the exciting force approach to a certain level, the structure is destroyed by resonance. Consider first the case of a first order resonance failure, with Z1To indicate the proximity of the natural frequency to the frequency of the exciting force, Z1Is expressed as
Figure BDA0002628929840000101
Wherein, ω is1Denotes a first order natural frequency, X denotes a random input variable, and X ═ X1,X2,…,XnAnd n is the variable dimension. Note that under the influence of random variables of the pipeline structure and fluid-solid coupling effect, the natural frequency RR of the pipeline at this time is a random variable, and the corresponding Z1And also becomes a random variable.
Assuming that the threshold for resonance failure is q, then the functional function g of the first order resonance reliability1(X) is
g1(X)＝Z1(X)-q (21)
Further considering the broadband characteristic of the frequency of the exciting force, resonance may occur at the multi-order natural frequency, and g is1And (5) popularization is carried out. Suppose thatjjFunction g of order resonance reliabilityjj(X) is
gjj(X)＝Zjj(X)-q (22)
Wherein the content of the first and second substances,
Figure BDA0002628929840000102
ωjjis the jj th order natural frequency.
It is readily appreciated that too close a common frequency of any one order to the excitation frequency will cause a resonant failure, gjjThe relation is logical OR, so that the total function g (X) under the condition of multi-frequency resonance failure of the flow transmission pipeline is established as
Figure BDA0002628929840000103
The probability P of the failure of the resonance of the fluid transmission pipeline is
Figure BDA0002628929840000104
3. Anti-resonance reliability analysis of flow transmission pipeline
The part adopts an active learning Kriging method (ALK) to carry out anti-resonance reliability analysis.
The expression of the Kriging model is
g(x)＝β+z(x) (25)
Where g (x) is the function to be fitted, β represents the model global approximation, and z (x) provides the local bias for the gaussian random process.
Given DoEs containing m sample points, the prediction mean value and variance of a Kriging model at an unknown point to be measured are
Figure BDA0002628929840000111
Figure BDA0002628929840000112
Wherein the content of the first and second substances,1is a m-dimensional unit column vector, and r (theta, x) is a representative trainingAn m-dimensional column vector of point correlations, R (theta) is a correlation matrix of an arbitrary pair of training points,
Figure BDA0002628929840000113
σ2and θ is the Kriging model parameter. The parameter theta has the largest influence on the prediction accuracy of the Kriging model. The estimate of θ should be obtained by solving the following optimization problem
Figure BDA0002628929840000114
And solving a global optimal solution of theta by adopting a global optimization algorithm DIRECT strategy.
To improve the prediction capability of the Kriging model in reliability analysis, the most likely mispredicted sample is selected and added to the DoE. Therefore, the basic idea of the ALK model is as follows: (1) establishing an initial Kriging model by using a small amount of samples; (2) and searching the training point with the maximum symbol prediction error risk through an active learning equation. The label being x*And adds it to the DoEs. If the convergence criterion of the learning equation is met, stopping; (3) calculating a function at x*Updating Kriging model and returning to step (2).
The key to the ALK method is a learning-plus-pointing strategy. The Kriging model comprises a Gaussian random process and can provide a predicted value mug(x) Sum variance
Figure BDA0002628929840000115
I.e., g (x) N (mu)g(x),σg(x) ). Adding points by using a U-shaped learning equation with the expression of
Figure BDA0002628929840000116
The U-shaped learning function focuses on sample points near the extreme states. Obviously, the value of the U-shaped learning function is directly determined by the predicted value and variance of the Kriging model. The point with the minimum U-shaped learning function value is added to the DoE, so that the prediction capability of the Kriging model is greatly improved.
4. Anti-resonance sensitivity analysis of flow pipeline
For the flow transmission pipeline resonance reliability model Y ═ g (X), the unconditional failure probability of Y is PfWhen the unconditional probability density function of the output response is fY(y) when a certain basic variable XiFixed at the realized value
Figure BDA0002628929840000121
When, the conditional failure probability value of Y is
Figure BDA0002628929840000122
The conditional probability density function at this time is
Figure BDA0002628929840000123
According to the idea of moment-independent global sensitivity analysis proposed by borgnonovo, the effect of input uncertainty variables on output responses is characterized by the cumulative effect of the areas of difference of the unconditional and conditional distribution densities of the responses. According to the thought, an independent global sensitivity measure index of the resonance reliability moment of the flow transmission pipeline based on the failure probability is established
Figure BDA0002628929840000124
The index measures the input variable XiThe average influence degree on the resonance failure probability in the distribution domain is specifically defined as:
Figure BDA0002628929840000125
wherein E (·) represents the desired operator, F represents the failure domain of the function, F ═ X: g (X)<0}, here XiCan represent a basic random variable XiCan also be expressed as a set of basic random variables
Figure BDA0002628929840000126
Wherein (1 ≦ i)1≤i1≤…≤il≤n)。
Figure BDA0002628929840000127
Represents the variable xiThe joint probability density function of (a).
The absolute value operation in the formula (30) is difficult to realize, the absolute value operation is equivalently converted into square operation, and the converted global sensitivity index based on the resonance failure probability is set as
Figure BDA0002628929840000128
In a specific form of
Figure BDA0002628929840000129
Since the function of the resonance reliability of the flow transmission pipeline is implicit, the specific distribution of the response quantity Y is difficult to obtain. The failure domain indication function I is introduced hereFSolving failure probability P through Monte Carlo simulationf。
The failure domain indicator function is
Figure BDA00026289298400001210
Randomly generating N sample points, calculating the number of the sample points which enable the resonance failure probability of the flow transmission pipeline to be less than zero, and further estimating the unconditional failure probability PfIs composed of
Figure BDA0002628929840000131
Accordingly, when a certain input variable X is orderediGet the realized value
Figure BDA0002628929840000132
Then, the conditional failure probability can be obtained
Figure BDA0002628929840000133
Is composed of
Figure BDA0002628929840000134
Wherein the content of the first and second substances,
Figure BDA0002628929840000135
a function representing a conditional fail field indication function,
Figure BDA0002628929840000136
substituting equations (33) and (34) into the global sensitivity index
Figure BDA0002628929840000137
In (3), an input variable X can be obtainediVariable global sensitivity index to probability of resonance failure
Figure BDA0002628929840000138
Is composed of
Figure BDA0002628929840000139
Variable global sensitivity index of flow transmission pipeline
Figure BDA00026289298400001310
Can fully reflect the input variable XiThe influence degree on the resonance failure probability of the flow transmission pipeline is sorted according to the importance of the input variables obtained by the global sensitivity analysis, and an important reference is provided for engineering designers to carry out anti-resonance optimization design on the flow transmission pipeline.
To quantify the impact of each individual failure Mode on failure and safety in a pipeline, we introduced two reliability-based Mode Sensitivity Analysis (MSA) indicators. Defining the first MSA index of the jth order resonant failure mode as the failure probability of the jth order resonant failure mode
Figure BDA00026289298400001311
Wherein P isfj/PfThe ratio is a factor of the probability difference. By definition, MIFjThe influence of the j-th order resonance failure mode on the failure domain and the failure probability of the flow transmission pipeline is measured. MIFjThe larger the jth order resonance failure mode, the more important the failure of the fluid delivery conduit. MIFjIs determined by the location of the jth order resonant failure mode and the failure probability. When Pr (I)F＝1|Ij＝1)≥Pr(IF＝1|Ij0) when there is MIFjNot less than 0, further can be deduced as:
Figure BDA00026289298400001312
when P is presentfjPr(IF＝0|Ij＝1)/(1-Pfj) Not less than 0 in the presence of MIFj≤Pr(Ij＝1|IF1 ≦ 1). Therefore MSA index MIFjMI of 0. ltoreqFjLess than or equal to 1. From equation (35), MI Fj0 means Pr (I)j＝1|IF1, namely, the failure of the jth order resonance failure mode does not change the resonance failure probability of the flow transmission pipeline; from equation (36), MI Fj1 implies Pr (I)j＝0|IF1-0 and Pr (I)j＝1|IF1, that is, if the system fails, the jth order resonance failure mode must fail, and the failure domain F of the flow pipeline is contained in the failure domain F of the jth order resonance failure modejIn (1).
We define the second MSA index for the jth failure mode as the probability of the jth order resonant failure mode, i.e.:
Figure BDA0002628929840000141
by definition, the MSA index MISjThe influence of the j-th order resonance failure mode on the safety of the flow transmission pipeline is quantified. MISjThe larger the influence of the j-th order resonance failure mode on the safety domain of the flow transmission pipeline is. In the formula (38), it is apparent that Pr (I)F＝0|Ij＝0)≥Pr(IF＝0|Ij1) and thus MISjIs more than or equal to 0. Further, MISjCan be expressed as:
Figure BDA0002628929840000142
the second order of formula (39) is not less than 0, and MI can be obtainedSj≤Pr(Ij＝0|IF0) is less than or equal to 1. Therefore, MISjAlso satisfies 0. ltoreq. MISjLess than or equal to 1. From formula (39), MI Sj0 implies Pr (I)F＝0|Ij＝0)＝Pr(IF＝0|Ij1). That is, the state change of the jth failure mode does not affect the reliability of the fluid delivery pipeline; MI in formula (39)Sj1 represents Pr (I)j＝1|IF0 ═ 0 and Pr (I)j＝0|IF0) 1, that is, when the structural system works, the jth resonant failure mode cannot fail, and the security domain of the jth resonant failure mode must include the security domain of the fluid transmission pipeline. MI for each failure mode for a series architectureSjAre all 1.
The following shows the implementation by a specific example.
The mechanical properties of the functionally gradient material flow pipeline with the outer layer of SiC and the inner layer of Ti-6Al-4V are shown in Table 1, and the schematic diagram of the pipeline is shown in FIG. 3. It can be seen that this example duct is a two-span duct, which is loaded by a vertically downward concentrated force in the center of the right span, with a magnitude and duration of 100N and 0.1s, respectively. The length, the outer diameter, the inner diameter, the thickness and the fluid density of the pipeline are respectively equal to 6m and Ro＝50mm，Ri＝48mm，T＝2mm，ρf＝1000kg/m3。
TABLE 1 mechanical Properties of functionally graded materials
Figure BDA0002628929840000151
Considering the two-order resonance failure, the random variable information is shown in table 2, i.e., the uncertainty input information for the piping system.
TABLE 2 distribution types and distribution parameters of input variables
Figure BDA0002628929840000152
Under the condition that four working conditions of flow rate are considered, the first two-step natural frequency of the flow pipeline is calculated by a dynamic stiffness method and is shown in table 3 along with the change of the flow rate of the liquid. It has been found that as the flow rate of the liquid in the pipe increases, the first two natural frequencies of the pipe tend to decrease.
TABLE 3 inherent frequency of the pipe at different flow rates (p ═ 0; n ═ 100)
Figure BDA0002628929840000161
The anti-resonance failure probabilities at different flow rates calculated by the ALK method are shown in Table 4. The result shows that the anti-resonance failure probability of the pipeline is gradually reduced along with the increase of the flow velocity, in addition, the resonance failure probability is calculated by adopting an ALK method, the function calling frequency is far lower than that of the last ten thousand times of common Monte Carlo simulation, the calculation efficiency can be greatly improved, and the engineering application feasibility is improved.
TABLE 4 probability of antiresonance failure at different flow rates (p 0; n 100)
Figure BDA0002628929840000162
The first item of the function calling times is an initial training point of a Kriging model, and the second item is a training point which is selected by a learning equation and added into DoEs.
Fig. 4 shows the flow line variable sensitivity index calculated by the ALK method when the flow rate is 20 m/s. From the results in FIG. 4, it can be found that p, S2And ρfThe influence on the failure probability of resonance is the greatest, and the changes are important to pay attention to in the anti-resonance optimization designControl of uncertainty of the quantity. In addition, the variable with the sensitivity value close to 0 can be ignored, thereby achieving the purpose of designing the dimension reduction of the variable.
The measure of importance of the fluid delivery pipeline pattern at different flow rates is shown in table 5. The resonance failure of the flow transmission pipeline system is mainly influenced by the first-order resonance failure, the anti-resonance failure probability is gradually reduced along with the gradual distance of the first-order natural frequency from the external excitation frequency, and the sensitivity index value of the corresponding first-order resonance failure mode is gradually reduced.
TABLE 5 model importance measure index at different flow rates
Figure BDA0002628929840000171

Claims (3)

1. A method for analyzing anti-resonance reliability and sensitivity of a fluid transmission pipeline is characterized by comprising the following steps:
step 1: establishing a flow transmission pipeline vibration control equation:
Figure FDA0002628929830000011
wherein, (EI)*Denotes the effective flexural rigidity, MfFluid mass per unit length, U fluid flow rate, p fluid pressure, AfIs the cross-sectional area of the fluid, m*The mass per unit length of the pipeline, w is the transverse displacement of the pipeline, x is the axial coordinate of the pipeline and t is the time;
step 2: solving the natural frequency of the flow transmission pipeline: solving a vibration control equation of the fluid delivery pipeline by adopting a dynamic stiffness method to obtain the natural frequency of each order of the fluid delivery pipeline, wherein the order is judged by an external excitation frequency range:
h(ω)＝det[Kcg]＝0
wherein: ω denotes the natural frequency of the flow conduit, KcgFor the overall stiffness matrix of the fluid delivery conduit, the subscript cg represents the matrix and vector of the additive boundary conditions, det [ ·]Representing a determinant operation;
and step 3: establishing an anti-resonance function, and obtaining the resonance failure probability P of the flow transmission pipeline according to the natural frequency and the external excitation frequency of the flow transmission pipeline obtained in the step 2
Figure FDA0002628929830000012
Wherein, gi(X)＝Zi(X)-q，
Figure FDA0002628929830000013
gi(X) is a function of the failure of the ith order resonance, Zi(X) represents the closeness degree of the ith order natural frequency and the exciting force frequency, q is the threshold value of resonance failure, and omegaiIs the ith order natural frequency, and S is the exciting force frequency;
and 4, step 4: analyzing the anti-resonance reliability of the flow transmission pipeline: solving the anti-resonance failure probability of the flow transmission pipeline in the step 3 by adopting an active learning Kriging model so as to obtain the anti-resonance reliability of the flow transmission pipeline, wherein the expression of the Kriging model is
g(x)＝β+z(x)
Wherein g (x) is a function to be fitted, beta represents global approximation of a model, z (x) provides local deviation for a Gaussian random process, and the Kriging model comprises a Gaussian random process and can provide a predicted value mug(x) Sum variance
Figure FDA0002628929830000014
I.e., g (x) N (mu)g(x),σg(x) ); using U-shaped learning equations
Figure FDA0002628929830000021
Adding points, thereby greatly improving the prediction capability of the Kriging model;
and 5: analyzing anti-resonance sensitivity of the flow transmission pipeline: based on the Kriging model established in the step 4, the anti-resonance sensitivity analysis of the flow transmission pipeline is further carried out, and the variable global sensitivity index formula is
Figure FDA0002628929830000022
Wherein:
Figure FDA0002628929830000023
i.e. the variable X of the flow transmission pipelineiSensitivity index of (1), PfThe probability of the failure of the resonance of the fluid transmission pipeline is shown,
Figure FDA0002628929830000024
represents when variable XiFailure probability under fixed time flow pipe resonance condition, IFRepresenting a function indicative of resonant failure of the pipe,
Figure FDA0002628929830000025
represents when variable XiA fixed-time flow transmission pipeline resonance condition failure indication function, wherein E represents an expected operation;
the mode sensitivity index is formulated as
Figure FDA0002628929830000026
Figure FDA0002628929830000027
Wherein, PfjRepresents the j-th order resonance failure probability of the pipeline, PfRepresenting the probability of resonant failure, P, of the pipe systemfj/PfThe ratio is a factor of the probability difference, Pr denotes the probability operation, IFRepresenting a conduit resonance failure indication function, IjRepresenting a j-th order resonance failure indication function of the pipeline; by definition, the first mode sensitivity index MIFjThe influence of the j-th order resonance failure mode on the failure domain and the failure probability of the flow transmission pipeline is measured.
2. The method for analyzing anti-resonance reliability and sensitivity of a fluid delivery pipeline according to claim 1, wherein the global sensitivity index of the resonant variables of the fluid delivery pipeline in the step 5 fully reflects the input variables and the degree of influence on the probability of resonant failure of the fluid delivery pipeline, and the importance of the input variables is ranked by the global sensitivity analysis.
3. The method of claim 1, wherein the MI in step 5 is performedFjThe larger the jth order resonance failure mode is, the more important the failure of the flow transmission pipeline is; second mode sensitivity index MISjThe influence of the j-th order resonance failure mode on the safety of the flow transmission pipeline is quantified; MISjThe larger the influence of the j-th order resonance failure mode on the safety domain of the flow transmission pipeline is.
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