CN111783243B - Metal structure fatigue crack propagation life prediction method based on filtering algorithm - Google Patents

Metal structure fatigue crack propagation life prediction method based on filtering algorithm Download PDF

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CN111783243B
CN111783243B CN202010557546.2A CN202010557546A CN111783243B CN 111783243 B CN111783243 B CN 111783243B CN 202010557546 A CN202010557546 A CN 202010557546A CN 111783243 B CN111783243 B CN 111783243B
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crack propagation
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CN111783243A (en
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靳慧
陈国良
丁克勤
王立彬
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Southeast University
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Abstract

The invention discloses a method for predicting the fatigue crack propagation life of a metal structure based on a filtering algorithm, which comprises the following steps: (1) improving the fatigue crack propagation two-parameter index model to obtain an improved fatigue crack propagation two-parameter index model; (2) constructing a state space evaluation model based on the improved fatigue crack propagation two-parameter index model; (3) calculating a state parameter X of the state space evaluation model constructed in the step (2) by using a filtering algorithmk+1(ii) a (4) Using the state parameters
Figure DDA0002544950300000011
And predicting the residual life of the crack propagation. The method considers the distribution conditions of the material parameters alpha and beta and the crack length a in the two-parameter index model, and can well predict the fatigue crack propagation life of the metal structure of the casting crane.

Description

Metal structure fatigue crack propagation life prediction method based on filtering algorithm
Technical Field
The invention relates to a method for predicting the fatigue crack propagation life of a metal structure, in particular to a method for predicting the fatigue crack propagation life of the metal structure based on a filtering algorithm.
Background
Since the work grade of the ladle crane structure is high and the ladle crane is often subjected to impact load in the hoisting process, fatigue failure is the main failure mode of the ladle crane. The fatigue crack is the main form of the fatigue damage of the ladle crane, the fatigue crack service life of the metal structure of the ladle crane is researched, and the method has important significance for the use and maintenance of the metal structure of the ladle crane.
The fatigue crack propagation life prediction method has various modes, and the double-parameter index formula has a simple structural form, so that the method is widely applied to the fatigue crack life prediction of the metal structure at present. Two important material parameters alpha and beta in the two-parameter index formula have great influence on the fatigue crack propagation life curve trend, while different structure forms, materials and slight changes in the structure manufacturing process all have influence on alpha and beta, so the two-parameter index formula material parameters alpha and beta are usually obtained by experiments. But the metal structure of the casting crane has complex structure form and dispersed crack propagation parts, and the single fatigue parameters alpha and beta are used for describing the crack propagation life error is large; although the use safety of the structure can be ensured by dividing a certain safety factor, the crack propagation life prediction mode is still different from the crack propagation real state of the structure when facing one crane or even a plurality of cranes.
Therefore, there is a need to solve the above problems.
Disclosure of Invention
The purpose of the invention is as follows: the invention aims to provide a metal structure fatigue crack propagation life prediction method based on a filter algorithm, which considers the distribution condition of material parameters alpha and beta in a two-parameter exponential model.
The technical scheme is as follows: in order to achieve the above purpose, the invention discloses a method for predicting the fatigue crack propagation life of a metal structure based on a filtering algorithm, which comprises the following steps:
(1) improving the fatigue crack propagation two-parameter index model to obtain an improved fatigue crack propagation two-parameter index model;
(2) constructing a state space evaluation model based on the improved fatigue crack propagation two-parameter index model;
(3) calculating a state parameter X of the state space evaluation model constructed in the step (2) by using a filtering algorithmk+1；
(4) Using the state parameters
Figure BDA0002544950280000011
And predicting the residual life of the crack propagation.
Wherein the two-parameter exponential model of fatigue crack propagation in the step (1) describes a fatigue crack propagation rate, and the formula is as follows:
Figure BDA0002544950280000021
where α and β are material parameters, γ is a coefficient relating to the loading pattern, crack shape and position, and Δ σ is stressAmplitude, a is the crack length; from the formula (1), when α, β are fixed parameters and β is<At 0, the crack propagation rate is increased continuously along with the growth of the crack length, and the increase is less than eα(ii) a When beta is>At 0, the crack propagation rate decreases with crack length growth and is not suitable for life prediction; thus, when β is<When the crack propagation rate is high, the alpha parameter needs to be changed greatly to achieve the prediction effect, and the practical application is not facilitated; therefore, the two-parameter exponential model is improved as follows:
Figure BDA0002544950280000022
preferably, the constructing of the state space estimation model in the step (2) specifically includes the following steps:
(2.1) constructing a state space parameter evaluation equation:
setting dN as 1, namely the crack propagation length under single cycle load; converting the improved fatigue crack propagation two-parameter exponential model into a discrete recursion form as follows:
Figure BDA0002544950280000023
because the frequent crack parts are dispersed and the stress amplitude on the crack surface is difficult to measure, the delta sigma is replaced by the stress amplitude obtained by finite element simulation; and because the stress amplitude obtained by simulation has calculation error, the stress amplitude delta sigma obtained by simulation is designed to obey normal distribution
Figure BDA0002544950280000024
To simulate the variance of equivalent response amplitude values with true values,
Figure BDA0002544950280000025
is the equivalent stress amplitude; thus ak+1＝f(akΔ σ) in
Figure BDA0002544950280000026
Is aligned with ak+1Performing first-order Taylor expansion to obtain:
Figure BDA0002544950280000027
let ak+1The process noise of (a) is:
Figure BDA0002544950280000028
in the improved fatigue crack propagation two-parameter exponential model, the partial derivative of the delta sigma is calculated as follows:
Figure BDA0002544950280000031
in that
Figure BDA0002544950280000032
In the case where it is known that,
Figure BDA0002544950280000033
and formula (6) are both definite values, and
Figure BDA0002544950280000034
thus, the
Figure BDA0002544950280000035
σaThe expression of (a) is:
Figure BDA0002544950280000036
the crack length a can be obtained from the formula (1), and when the structural characteristics, environment and working conditions cause the fatigue parameters alpha and beta to change, the crack length a is different; then the state space parameter evaluation equation of the two-parameter exponential model is:
Figure BDA0002544950280000037
in the formula (I), the compound is shown in the specification,
Figure BDA0002544950280000038
white gaussian noise of alpha, beta, as measured by standard practice fatigue tests; sigmaα,k、σβ,k、σa,kForming a system state parameter variance matrix Q;
(2.2) constructing a state space observation equation
The state space observation equation obtained from the fatigue crack length obtained by monitoring or detection, the corresponding cycle number and the error existing in the measurement process is as follows:
zk+1＝Hk+1xk+1+vk+1 (9)
in the formula, Hk+1Is a measurement matrix, here an identity matrix; x is the number ofk+1Is a measured value, here the crack length ak+1；vk+1To measure errors, and is subject to
Figure BDA0002544950280000039
The state space observation equation is as follows:
Zk+1＝ak+1+vk+1 (10)
wherein σr,kForming an observation variance matrix R.
Further, the step (3) of calculating the state parameter by using the particle filter includes the following specific steps:
initializing state parameters, inputting initial value X of state parameters0＝[a0、α0、β0]And the corresponding variance Q ═ σa,σα,σβ]；
Sampling a set of particles, collecting the set of particles using a probability distribution according to alpha, beta, a
Figure BDA0002544950280000041
Generation of state function samples using state space parameter estimation equationsThis collection
Figure BDA0002544950280000042
k +1 is the number of state function steps,
Figure BDA0002544950280000043
is a normal function matrix;
calculating importance weight according to Z in state observation equationk+1Calculating a weight value:
Figure BDA0002544950280000044
wherein the content of the first and second substances,
Figure BDA0002544950280000045
is composed of
Figure BDA0002544950280000046
In the presence of Zk+1The probability of occurrence;
Figure BDA0002544950280000047
is composed of
Figure BDA0002544950280000048
The probability of occurrence;
Figure BDA0002544950280000049
is Zk+1Under the conditions of occurrence
Figure BDA00025449502800000410
The probability of occurrence;
normalizing the weight:
Figure BDA00025449502800000411
random resampling: weighted value according to importance
Figure BDA00025449502800000412
The sample is re-drawn among the N particles,
Figure BDA00025449502800000413
the large number of the distributed particles is more,
Figure BDA00025449502800000414
smaller direct deletion, last by difference
Figure BDA00025449502800000415
Generating new N particles by specific gravity; and reassign the weight value to each particle
Figure BDA00025449502800000416
Obtaining N expected values of the newly generated particles by the formula (8);
Figure BDA00025449502800000417
moreover, the calculating the state parameter by using the extended kalman filter in the step (3) includes the following specific steps:
inputting initial value X of state parameter0＝[a0、α0、β0]And the corresponding variance Q ═ σa,σα,σβ]；
Kalman filtering is the optimal analytic solution of the recursive Bayesian formula, assuming the equation of state Xk+1Observation equation Zk+1The following were used:
Xk+1＝AXk+Wk+1 (13)
Zk+1＝HXk+1+Vk+1 (14)
in the formula: a is a state transition matrix, Wk+1Predicting noise for a system state; h is an observation matrix, Vk+1To observe noise;
the equations (8) and (10) are generally nonlinear functions, and the core idea of the extended Kalman filter algorithm is to linearize the nonlinear function and to estimate the state of the function Xk+1And an observation function Zk+1Surrounding the filtered value
Figure BDA00025449502800000418
And performing Taylor series expansion and neglecting the influence of second-order and higher-order terms, thereby forming an approximate linearized state parameter estimation model as follows:
Figure BDA0002544950280000051
wherein, it is made
Figure BDA0002544950280000052
Figure BDA0002544950280000053
Figure BDA0002544950280000054
Figure BDA0002544950280000055
The equation (22) and the equation (23) are Jacobian matrixes of the state parameter estimation model and the observation model respectively, and the prediction result of the step k and the equation (21) can be obtained respectively
Figure BDA0002544950280000056
And
Figure BDA0002544950280000057
the extended kalman filter equation can then be expressed as:
Figure BDA0002544950280000058
Figure BDA0002544950280000059
since the predicted value is calculated for the equation (13), the influence of the second order term and higher order terms is ignored after the taylor series expansion, and therefore, the equation (20) has a certain error, and the calculation is performed
Figure BDA00025449502800000510
Calculating by using the formula (13) and the formula (14);
Figure BDA00025449502800000511
in the formula (I), the compound is shown in the specification,
Figure BDA00025449502800000512
predicting value in the k step, and setting the k as 0 to be initial value of state parameter;
after new state parameters are obtained through prediction, the correlation among the state parameters is changed, the system covariance is changed, and the covariance matrix of the prior state estimation is as follows:
Figure BDA00025449502800000513
wherein Q is a system state parameter variance matrix;
and (3) updating: obtaining an observation parameter Zk+1Then, the state prediction value can be corrected, and the correction equation is as the following formula,
Figure BDA0002544950280000061
kalman gain matrix:
Kk+1＝p'k+1HT(Hp'k+1HT+R)-1 (24)
and (4) updating the predicted value:
Figure BDA0002544950280000062
covariance update estimation:
pk+1＝(I-Kk+1H)p'k+1 (26)。
preferably, the calculating the state parameter by using unscented kalman filtering in step (3) includes the following specific steps:
inputting initial value X of state parameter0＝[a0、α0、β0]And the corresponding variance Q ═ σa,σα,σβ](ii) a Covariance matrix p0；
Obtaining a group of sampling points, a Sigma point set and a corresponding weight value by using a formula (27) and a formula (28); calculating 2n +1 Sigma sampling points and n finger-shaped state function variable dimensions;
Figure BDA0002544950280000063
wherein the content of the first and second substances,
Figure BDA0002544950280000064
column i showing the square root of the matrix;
calculating the weight of the sampling point,
Figure BDA0002544950280000065
wherein, subscript m is mean value, c is covariance;
Figure BDA0002544950280000066
and
Figure BDA0002544950280000067
weights representing the mean and covariance of the corresponding points; parameter λ ═ α2(n + κ) -n is a scaling parameter; the value of parameter alpha is [0,1 ]]Determining the distribution of sampling points, and adjusting the sampling points and the mean value
Figure BDA0002544950280000071
The distance of (d); kappa is a candidate parameter for ensuring that the matrix (n + lambda) p is a semi-positive definite matrix when n is more than or equal toK is 0 when 3, and n-3 when n < 3; parameter beta>0 can adjust the equation high-order term dynamic difference, and beta is taken to be 2 when the equation is normally distributed;
first of all generated to
Figure BDA0002544950280000072
Sigma point set as mean:
Figure BDA0002544950280000073
substituting Sigma point set into state parameter estimation function fk(Xk) Obtaining a set of state parameter estimation sample points
Figure BDA0002544950280000074
Sampling points
Figure BDA0002544950280000075
To correspond to
Figure BDA0002544950280000076
The weighted sum can obtain the mean and variance of the corresponding predicted values of the state parameters, and the specific formula is as follows:
Figure BDA0002544950280000077
Figure BDA0002544950280000078
according to the predicted value
Figure BDA0002544950280000079
A set of samples is obtained again using equations (27) and (28):
Figure BDA00025449502800000710
the predicted Sigma point set is substituted into observation equation (10) to obtain a predicted observation value, i ═ 1,2, …,2n + 1.
Figure BDA00025449502800000711
Obtaining the mean value and the variance of system prediction through weighted summation according to the obtained prediction observation value;
Figure BDA00025449502800000712
Figure BDA00025449502800000713
Figure BDA00025449502800000714
kalman gain matrix calculation:
Figure BDA00025449502800000715
updating state parameters and covariance to obtain real observation parameters Zk+1Updating the post-state parameters:
Figure BDA0002544950280000081
and (3) covariance updating:
Figure BDA0002544950280000082
further, the step (4) specifically includes the steps of: after the state parameter estimation is carried out by utilizing the step (3), the parameter of the K +1 step can be obtained
Figure BDA0002544950280000083
Predicting the fatigue crack extension life, and obtaining the crack length of the (k + n) th step by only substituting the parameters into a discrete recursion equation to carry out recursion calculation, wherein the smaller the n is, the higher the precision is, certainly, because the state parameters are changed at any time; the concrete prediction formula of the discrete recursive equation is as follows;
Figure BDA0002544950280000084
has the advantages that: compared with the prior art, the invention has the following remarkable advantages: the method aims at the influence of the impact load, environment, structural form, material and other uncertainties on the fatigue crack propagation process in the hoisting process of the ladle crane; establishing an improved double-parameter index model, forming a discretized state parameter evaluation model, and estimating fatigue performance parameters and a crack propagation trend by using a particle filter algorithm, an extended Kalman filter algorithm or an unscented Kalman filter algorithm; finally, predicting the residual life of the fatigue crack of the metal structure of the casting crane according to the crack propagation trend; the comparison shows that the prediction results of the particle filter algorithm, the extended Kalman filter algorithm and the unscented Kalman filter algorithm are superior to those of the prior art, and the maximum values of the relative absolute errors of the predicted values of the extended Kalman filter algorithm, the unscented Kalman filter algorithm and the particle filter algorithm are respectively 2.5%, 2% and 11% when the crack is expanded in the first stage; the prediction requirement of the crack propagation life of the casting crane is met; during the second stage of crack propagation, the maximum values of relative absolute errors of predicted values of an extended Kalman filter algorithm, an unscented Kalman filter algorithm and a particle filter algorithm are respectively 8.5%, 15% and 13.5%; the prediction requirement of the crack propagation life of the casting crane is also met; therefore, the combination of the particle filter algorithm, the extended Kalman filter algorithm and the unscented Kalman filter algorithm with the improved two-parameter index model can well predict the fatigue crack extension life of the metal structure of the casting crane.
Drawings
FIG. 1 is a schematic diagram of a single-beam solid-shell coupling model according to the present invention;
FIG. 2 is a schematic view of a fatigue crack propagation curve of the diaphragm and the lower cover plate according to the present invention;
FIG. 3 is a first stage simulation crack propagation life prediction diagram in accordance with the present invention;
FIG. 4 is a diagram of the second stage simulated crack propagation life prediction in the present invention;
FIG. 5 is a graph of relative absolute error of the first stage of the present invention;
FIG. 6 is a diagram of relative absolute error at the second stage of the present invention;
FIG. 7 is a diagram of the first stage and the true value variance in the present invention;
FIG. 8 is an enlarged schematic view of FIG. 7;
FIG. 9 is a graph of the first stage and mean variance of the present invention;
FIG. 10 is an enlarged schematic view of FIG. 9;
FIG. 11 is a second stage and true variance plot of the present invention;
FIG. 12 is an enlarged schematic view of FIG. 11;
FIG. 13 is a graph of the second stage and mean variance of the present invention;
fig. 14 is an enlarged schematic view of fig. 13.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
Example 1
The invention discloses a method for predicting the fatigue crack propagation life of a metal structure based on a filtering algorithm, which is characterized by comprising the following steps of:
(1) improving the fatigue crack propagation two-parameter index model to obtain an improved fatigue crack propagation two-parameter index model; wherein the fatigue crack propagation two-parameter exponential model describes the fatigue crack propagation rate, and the formula is as follows:
Figure BDA0002544950280000091
where α and β are material parameters, γ is a coefficient related to the loading pattern, crack shape and position, Δ σ is the stress amplitude, and a isCrack length; from the formula (1), when α, β are fixed parameters and β is<At 0, the crack propagation rate is increased continuously along with the growth of the crack length, and the increase is less than eα(ii) a When beta is>At 0, the crack propagation rate decreases with crack length growth and is not suitable for life prediction; thus, when β is<When the crack propagation rate is high, the alpha parameter needs to be changed greatly to achieve the prediction effect, and the practical application is not facilitated; therefore, the two-parameter exponential model is improved as follows:
Figure BDA0002544950280000101
(2) constructing a state space evaluation model based on the improved fatigue crack propagation two-parameter index model;
the method for constructing the state space evaluation model specifically comprises the following steps:
(2.1) constructing a state space parameter evaluation equation:
setting dN as 1, namely the crack propagation length under single cycle load; converting the improved fatigue crack propagation two-parameter exponential model into a discrete recursion form as follows:
Figure BDA0002544950280000102
because the frequent crack parts are dispersed and the stress amplitude on the crack surface is difficult to measure, the delta sigma is replaced by the stress amplitude obtained by finite element simulation; and because the stress amplitude obtained by simulation has calculation error, the stress amplitude delta sigma obtained by simulation is designed to obey normal distribution
Figure BDA0002544950280000103
To simulate the variance of equivalent response amplitude values with true values,
Figure BDA0002544950280000104
is the equivalent stress amplitude; thus, theak+1＝f(akΔ σ) in
Figure BDA0002544950280000105
To ak+1Performing first-order Taylor expansion to obtain:
Figure BDA0002544950280000106
let ak+1The process noise of (a) is:
Figure BDA0002544950280000107
in the improved fatigue crack propagation two-parameter exponential model, the partial derivative of the delta sigma is calculated as follows:
Figure BDA0002544950280000108
in that
Figure BDA0002544950280000109
In the case where it is known that,
Figure BDA00025449502800001010
the formula (6) is a definite value, and
Figure BDA00025449502800001011
thus, the
Figure BDA00025449502800001012
σaThe expression of (a) is:
Figure BDA00025449502800001013
the crack length a can be obtained from the formula (1), and when the structural characteristics, environment and working conditions cause the fatigue parameters alpha and beta to change, the crack length a is different; then the state space parameter evaluation equation of the two-parameter exponential model is:
Figure BDA0002544950280000111
in the formula (I), the compound is shown in the specification,
Figure BDA0002544950280000112
white gaussian noise of alpha, beta, as measured by standard practice fatigue tests; sigmaα,k、σβ,k、σa,kForming a system state parameter variance matrix Q;
(2.2) constructing a state space observation equation
The state space observation equation obtained from the fatigue crack length obtained by monitoring or detection, the corresponding cycle number and the error existing in the measurement process is as follows:
zk+1＝Hk+1xk+1+vk+1 (9)
in the formula, Hk+1Is a measurement matrix, here an identity matrix; x is the number ofk+1Is a measured value, here the crack length ak+1；vk+1To measure errors, and is subject to
Figure BDA0002544950280000113
The state space observation equation is as follows:
Zk+1＝ak+1+vk+1 (10)
wherein σr,kForming an observation variance matrix R;
(3) calculating a state parameter X of the state space evaluation model constructed in the step (2) by using a filtering algorithmk+1；
The method for calculating the state parameters by using the particle filter comprises the following specific steps:
initializing state parameters, inputting initial value X of state parameters0＝[a0、α0、β0]And the corresponding variance Q ═ σa,σα,σβ]；
Sampling a set of particles using probability scores based on α, β, aCollection of particles by cloth
Figure BDA0002544950280000114
Generating a set of state function samples using a state space parameter estimation equation
Figure BDA0002544950280000115
k +1 is the number of state function steps,
Figure BDA0002544950280000116
is a normal function matrix;
calculating importance weight according to Z in state observation equationk+1Calculating a weight value:
Figure BDA0002544950280000117
wherein the content of the first and second substances,
Figure BDA0002544950280000118
is composed of
Figure BDA0002544950280000119
In the presence of Zk+1The probability of occurrence;
Figure BDA00025449502800001110
is composed of
Figure BDA00025449502800001111
The probability of occurrence;
Figure BDA00025449502800001112
is Zk+1Under the conditions of occurrence
Figure BDA00025449502800001113
The probability of occurrence;
normalizing the weight:
Figure BDA0002544950280000121
random resampling: according to importance rightsValue of
Figure BDA0002544950280000122
The sample is re-drawn among the N particles,
Figure BDA0002544950280000123
the large number of the distributed particles is more,
Figure BDA0002544950280000124
smaller direct deletion, last by difference
Figure BDA0002544950280000125
Generating new N particles by specific gravity; and reassign the weight value to each particle
Figure BDA0002544950280000126
Obtaining N expected values of the newly generated particles by the formula (8);
Figure BDA0002544950280000127
(4) using the state parameters
Figure BDA0002544950280000128
Predicting the residual life of crack propagation, wherein the method specifically comprises the following steps: after the state parameter estimation is carried out by utilizing the step (3), the parameter of the K +1 step can be obtained
Figure BDA0002544950280000129
Predicting the fatigue crack extension life, and obtaining the crack length of the (k + n) th step by only substituting the parameters into a discrete recursion equation to carry out recursion calculation, wherein the smaller the n is, the higher the precision is, certainly, because the state parameters are changed at any time; the concrete prediction formula of the discrete recursive equation is as follows;
Figure BDA00025449502800001210
example 2
The invention discloses a method for predicting the fatigue crack propagation life of a metal structure based on a filtering algorithm, which is characterized by comprising the following steps of:
(1) improving the fatigue crack propagation two-parameter index model to obtain an improved fatigue crack propagation two-parameter index model; wherein the fatigue crack propagation two-parameter exponential model describes the fatigue crack propagation rate, and the formula is as follows:
Figure BDA00025449502800001211
in the formula, alpha and beta are material parameters, gamma is a coefficient related to a loading mode, a crack shape and a crack position, delta sigma is a stress amplitude value, and a is a crack length; from the formula (1), when α, β are fixed parameters and β is<At 0, the crack propagation rate is increased continuously along with the growth of the crack length, and the increase is less than eα(ii) a When beta is>At 0, the crack propagation rate decreases with crack length growth and is not suitable for life prediction; thus, when β is<When the crack propagation rate is high, the alpha parameter needs to be changed greatly to achieve the prediction effect, and the practical application is not facilitated; therefore, the two-parameter exponential model is improved as follows:
Figure BDA0002544950280000131
(2) constructing a state space evaluation model based on the improved fatigue crack propagation two-parameter index model;
the method for constructing the state space evaluation model specifically comprises the following steps:
(2.1) constructing a state space parameter evaluation equation:
setting dN as 1, namely the crack propagation length under single cycle load; converting the improved fatigue crack propagation two-parameter exponential model into a discrete recursion form as follows:
Figure BDA0002544950280000132
because the frequent crack parts are dispersed and the stress amplitude on the crack surface is difficult to measure, the delta sigma is replaced by the stress amplitude obtained by finite element simulation; and because the stress amplitude obtained by simulation has calculation error, the stress amplitude delta sigma obtained by simulation is designed to obey normal distribution
Figure BDA0002544950280000133
To simulate the variance of equivalent response amplitude values with true values,
Figure BDA0002544950280000134
is the equivalent stress amplitude; thus ak+1＝f(akΔ σ) in
Figure BDA0002544950280000135
Is aligned with ak+1Performing first-order Taylor expansion to obtain:
Figure BDA0002544950280000136
let ak+1The process noise of (a) is:
Figure BDA0002544950280000137
in the improved fatigue crack propagation two-parameter exponential model, the partial derivative of the delta sigma is calculated as follows:
Figure BDA0002544950280000138
in that
Figure BDA0002544950280000139
In the case where it is known that,
Figure BDA00025449502800001310
the formula (6) is a definite value, and
Figure BDA00025449502800001311
thus, the
Figure BDA00025449502800001312
σaThe expression of (a) is:
Figure BDA00025449502800001313
the crack length a can be obtained from the formula (1), and when the structural characteristics, environment and working conditions cause the fatigue parameters alpha and beta to change, the crack length a is different; then the state space parameter evaluation equation of the two-parameter exponential model is:
Figure BDA0002544950280000141
in the formula (I), the compound is shown in the specification,
Figure BDA0002544950280000142
white gaussian noise of alpha, beta, as measured by standard practice fatigue tests; sigmaα,k、σβ,k、σa,kForming a system state parameter variance matrix Q;
(2.2) constructing a state space observation equation
The state space observation equation obtained from the fatigue crack length obtained by monitoring or detection, the corresponding cycle number and the error existing in the measurement process is as follows:
zk+1＝Hk+1xk+1+vk+1 (9)
in the formula, Hk+1Is a measurement matrix, here an identity matrix; x is the number ofk+1Is a measured value, here the crack length ak+1；vk+1To measure errors, and is subject to
Figure BDA0002544950280000143
The state space observation equation is as follows:
Zk+1＝ak+1+vk+1 (10)
wherein σr,kForming an observation variance matrix R;
(3) calculating a state parameter X of the state space evaluation model constructed in the step (2) by using a filtering algorithmk+1；
The method for calculating the state parameters by using the extended Kalman filter comprises the following specific steps:
inputting initial value X of state parameter0＝[a0、α0、β0]And the corresponding variance Q ═ σa,σα,σβ]；
Kalman filtering is the optimal analytic solution of the recursive Bayesian formula, assuming the equation of state Xk+1Observation equation Zk+1The following were used:
Xk+1＝AXk+Wk+1 (13)
Zk+1＝HXk+1+Vk+1 (14)
in the formula: a is a state transition matrix, Wk+1Predicting noise for a system state; h is an observation matrix, Vk+1To observe noise;
the equations (8) and (10) are generally nonlinear functions, and the core idea of the extended Kalman filter algorithm is to linearize the nonlinear function and to estimate the state of the function Xk+1And an observation function Zk+1Surrounding the filtered value
Figure BDA0002544950280000144
And performing Taylor series expansion and neglecting the influence of second-order and higher-order terms, thereby forming an approximate linearized state parameter estimation model as follows:
Figure BDA0002544950280000151
wherein, it is made
Figure BDA0002544950280000152
Figure BDA0002544950280000153
Figure BDA0002544950280000154
Figure BDA0002544950280000155
The equation (22) and the equation (23) are Jacobian matrixes of the state parameter estimation model and the observation model respectively, and the prediction result of the step k and the equation (21) can be obtained respectively
Figure BDA0002544950280000156
And
Figure BDA00025449502800001512
the extended kalman filter equation can then be expressed as:
Figure BDA0002544950280000157
Figure BDA0002544950280000158
since the predicted value is calculated for the equation (13), the influence of the second order term and higher order terms is ignored after the taylor series expansion, and therefore, the equation (20) has a certain error, and the calculation is performed
Figure BDA0002544950280000159
Calculating by using the formula (13) and the formula (14);
Figure BDA00025449502800001510
in the formula (I), the compound is shown in the specification,
Figure BDA00025449502800001511
predicting value in the k step, and setting the k as 0 to be initial value of state parameter;
after new state parameters are obtained through prediction, the correlation among the state parameters is changed, the system covariance is changed, and the covariance matrix of the prior state estimation is as follows:
p'k+1＝FKpkFk T+Q (22)
wherein Q is a system state parameter variance matrix;
and (3) updating: obtaining an observation parameter Zk+1Then, the state prediction value can be corrected, and the correction equation is as the following formula,
Figure BDA0002544950280000161
kalman gain matrix:
Kk+1＝p'k+1HT(Hp'k+1HT+R)-1 (24)
and (4) updating the predicted value:
Figure BDA0002544950280000162
covariance update estimation:
pk+1＝(I-Kk+1H)p'k+1 (26)
(4) using the state parameters
Figure BDA0002544950280000163
Predicting the residual life of crack propagation, wherein the method specifically comprises the following steps: after the state parameter estimation is carried out by utilizing the step (3), the parameter of the K +1 step can be obtained
Figure BDA0002544950280000164
Predicting fatigue crack propagation life by only substituting and dispersing parametersThe crack length of the (k + n) th step can be obtained by carrying out recursion calculation in a recursion equation, and the smaller the n, the higher the precision is, of course, because the state parameters are changed at any time; the concrete prediction formula of the discrete recursive equation is as follows;
Figure BDA0002544950280000165
example 3
The invention discloses a method for predicting the fatigue crack propagation life of a metal structure based on a filtering algorithm, which is characterized by comprising the following steps of:
(1) improving the fatigue crack propagation two-parameter index model to obtain an improved fatigue crack propagation two-parameter index model; wherein the fatigue crack propagation two-parameter exponential model describes the fatigue crack propagation rate, and the formula is as follows:
Figure BDA0002544950280000166
in the formula, alpha and beta are material parameters, gamma is a coefficient related to a loading mode, a crack shape and a crack position, delta sigma is a stress amplitude value, and a is a crack length; from the formula (1), when α, β are fixed parameters and β is<At 0, the crack propagation rate is increased continuously along with the growth of the crack length, and the increase is less than eα(ii) a When beta is>At 0, the crack propagation rate decreases with crack length growth and is not suitable for life prediction; thus, when β is<When the crack propagation rate is high, the alpha parameter needs to be changed greatly to achieve the prediction effect, and the practical application is not facilitated; therefore, the two-parameter exponential model is improved as follows:
Figure BDA0002544950280000171
(2) constructing a state space evaluation model based on the improved fatigue crack propagation two-parameter index model;
the method for constructing the state space evaluation model specifically comprises the following steps:
(2.1) constructing a state space parameter evaluation equation:
setting dN as 1, namely the crack propagation length under single cycle load; converting the improved fatigue crack propagation two-parameter exponential model into a discrete recursion form as follows:
Figure BDA0002544950280000172
because the frequent crack parts are dispersed and the stress amplitude on the crack surface is difficult to measure, the delta sigma is replaced by the stress amplitude obtained by finite element simulation; and because the stress amplitude obtained by simulation has calculation error, the stress amplitude delta sigma obtained by simulation is designed to obey normal distribution
Figure BDA0002544950280000173
To simulate the variance of equivalent response amplitude values with true values,
Figure BDA0002544950280000174
is the equivalent stress amplitude; thus alphak+1＝f(akΔ σ) in
Figure BDA0002544950280000175
Is aligned with ak+1Performing first-order Taylor expansion to obtain:
Figure BDA0002544950280000176
let ak+1The process noise of (a) is:
Figure BDA0002544950280000177
in the improved fatigue crack propagation two-parameter exponential model, the partial derivative of the delta sigma is calculated as follows:
Figure BDA0002544950280000178
in that
Figure BDA0002544950280000179
In the case where it is known that,
Figure BDA00025449502800001710
and formula (6) are both definite values, and
Figure BDA00025449502800001711
thus, the
Figure BDA0002544950280000181
σaThe expression of (a) is:
Figure BDA0002544950280000182
the crack length a can be obtained from the formula (1), and when the structural characteristics, environment and working conditions cause the fatigue parameters alpha and beta to change, the crack length a is different; then the state space parameter evaluation equation of the two-parameter exponential model is:
Figure BDA0002544950280000183
in the formula (I), the compound is shown in the specification,
Figure BDA0002544950280000184
white gaussian noise of alpha, beta, as measured by standard practice fatigue tests; sigmaα,k、σβ,k、σa,kForming a system state parameter variance matrix Q;
(2.2) constructing a state space observation equation
The state space observation equation obtained from the fatigue crack length obtained by monitoring or detection, the corresponding cycle number and the error existing in the measurement process is as follows:
zk+1＝Hk+1xk+1+vk+1 (9)
in the formula, Hk+1Is a measurement matrix, here an identity matrix; x is the number ofk+1Is a measured value, here the crack length ak+1；vk+1To measure errors, and is subject to
Figure BDA0002544950280000185
The state space observation equation is as follows:
Zk+1＝ak+1+vk+1 (10)
wherein σr,kForming an observation variance matrix R;
(3) calculating a state parameter X of the state space evaluation model constructed in the step (2) by using a filtering algorithmk+1；
The method for calculating the state parameters by using the unscented Kalman filtering comprises the following specific steps:
inputting initial value X of state parameter0＝[a0、α0、β0]And the corresponding variance Q ═ σa,σα,σβ](ii) a Covariance matrix p0；
Obtaining a group of sampling points, a Sigma point set and a corresponding weight value by using a formula (27) and a formula (28); calculating 2n +1 Sigma sampling points and n finger-shaped state function variable dimensions;
Figure BDA0002544950280000191
wherein the content of the first and second substances,
Figure BDA0002544950280000192
the ith column representing the square root of the matrix;
calculating the weight of the sampling point,
Figure BDA0002544950280000193
wherein, subscript m is mean value, c is covariance;
Figure BDA0002544950280000194
and
Figure BDA0002544950280000195
weights representing the mean and covariance of the corresponding points; parameter λ ═ α2(n + κ) -n is a scaling parameter; the value of parameter alpha is [0,1 ]]Determining the distribution of sampling points, and adjusting the sampling points and the mean value
Figure BDA0002544950280000196
The distance of (d); k is a candidate parameter for ensuring that a matrix (n + lambda) p is a semi-positive definite matrix, wherein k is 0 when n is larger than or equal to 3, and k is n-3 when n is smaller than 3; parameter beta>0 can adjust the equation high-order term dynamic difference, and beta is taken to be 2 when the equation is normally distributed;
first of all generated to
Figure BDA0002544950280000197
Sigma point set as mean:
Figure BDA0002544950280000198
substituting Sigma point set into state parameter estimation function fk(Xk) Obtaining a set of state parameter estimation sample points
Figure BDA0002544950280000199
Sampling points
Figure BDA00025449502800001910
Corresponding to
Figure BDA00025449502800001911
The weighted sum can obtain the mean and variance of the corresponding predicted values of the state parameters, and the specific formula is as follows:
Figure BDA00025449502800001912
Figure BDA00025449502800001913
according to the predicted value
Figure BDA00025449502800001914
A set of samples is obtained again using equations (27) and (28):
Figure BDA00025449502800001915
the predicted Sigma point set is substituted into observation equation (10) to obtain a predicted observation value, i ═ 1,2, …,2n + 1.
Figure BDA0002544950280000201
Obtaining the mean value and the variance of system prediction through weighted summation according to the obtained prediction observation value;
Figure BDA0002544950280000202
Figure BDA0002544950280000203
Figure BDA0002544950280000204
kalman gain matrix calculation:
Figure BDA0002544950280000205
updating state parameters and covariance to obtain real observation parameters Zk+1Updating the post-state parameters:
Figure BDA0002544950280000206
and (3) covariance updating:
Figure BDA0002544950280000207
(4) using the state parameters
Figure BDA0002544950280000208
Predicting the residual life of crack propagation, wherein the method specifically comprises the following steps: after the state parameter estimation is carried out by utilizing the step (3), the parameter of the K +1 step can be obtained
Figure BDA0002544950280000209
Predicting the fatigue crack extension life, and obtaining the crack length of the (k + n) th step by only substituting the parameters into a discrete recursion equation to carry out recursion calculation, wherein the smaller the n is, the higher the precision is, certainly, because the state parameters are changed at any time; the concrete prediction formula of the discrete recursive equation is as follows;
Figure BDA00025449502800002010
in order to verify the accuracy of the prediction method of the above embodiment, the crack propagation life is predicted according to the result of the Abaqus simulation shown in fig. 1 and 2.
Initial crack length a in improved two-parameter index model0The geometry factor γ is 1.125 at 16mm, and the assumptions of the initial distribution are as in table 1 below.
TABLE 1 simulation parameter set based on improved two-parameter exponential model
Figure BDA0002544950280000211
Note: delta sigma is the equivalent stress amplitude; alpha is a fatigue parameter; beta is a fatigue parameter; and N is a normal distribution function.
The simulated crack life prediction is as shown in fig. 3 and 4, and the parameter estimation of the crack propagation in the first stage and the second stage and the crack propagation life prediction are accurate.
Selecting evaluation parameters obtained in the first and second stages of ABAQUS simulation results under the cycle times (41000, 42000, 43000, 44000, 45000, 46000, 47000, 48000, 49000, 50000, 61000, 62000, 63000 and 64000)
Figure BDA0002544950280000212
AREk+1、Vk+1、
Figure BDA0002544950280000213
And comparing and analyzing the advantages and disadvantages of the three methods in crack life prediction. The respective estimated values, the mean, the relative absolute error, and the variance of the predicted values are shown in fig. 5 to 14.
As shown in fig. 5 and 6, in the parameter evaluation stage, the crack propagation length can be well described by combining the improved biparametric index formula with the extended kalman filter algorithm, the unscented kalman filter algorithm and the particle filter algorithm, and during the crack propagation in the first stage, the maximum values of the relative absolute errors of the predicted values of the extended kalman filter algorithm, the unscented kalman filter algorithm and the particle filter algorithm are respectively 2.5%, 2% and 11%; the prediction requirement of the crack propagation life of the casting crane is met; during the second stage of crack propagation, the maximum values of relative absolute errors of predicted values of an extended Kalman filter algorithm, an unscented Kalman filter algorithm and a particle filter algorithm are respectively 8.5%, 15% and 13.5%; and the prediction requirement of the crack propagation life of the casting crane is also met. And when the service life prediction order is reached, the crack prediction by the extended Kalman filtering algorithm and the unscented Kalman filtering algorithm is more accurate than that by the particle filtering algorithm.
Three algorithms were evaluated:
1) as shown in fig. 7 and 8, when the crack propagates in the first stage, the estimation results of the unscented kalman filter algorithm and the particle filter algorithm are both better than those of the extended kalman filter algorithm. In the prediction stage, the variance of the extended Kalman filtering algorithm is smaller than that of the particle filtering algorithm and the unscented Kalman filtering algorithm. Therefore, in terms of result accuracy, the first-stage life prediction method uses an extended Kalman filtering and improved two-parameter exponential model combined algorithm.
2) As shown in fig. 9 and 10, in the first stage of crack propagation prediction, the convergence of the particle filter algorithm and the extended kalman filter algorithm is better than that of the unscented kalman filter algorithm. In the parameter estimation stage, the convergence of unscented Kalman filtering and particle algorithm is superior to that of the extended Kalman filtering algorithm, but the service life prediction of the extended Kalman filtering algorithm is not influenced by the small divergence of the extended Kalman filtering algorithm.
3) As shown in fig. 11 and 12, in the second stage of crack propagation, in the parameter estimation stage, the estimated value of the unscented kalman filter algorithm is closest to the true value. And in the life prediction stage, the predicted value of the particle filter algorithm is closest to the real result, then the extended Kalman filter algorithm is used, and finally the unscented Kalman filter algorithm is used.
4) As shown in fig. 13 and 14, in the second stage of crack propagation, in the parameter estimation stage, the unscented kalman filter algorithm has the best convergence, and in the life prediction stage, the predicted value of the particle filter algorithm has the best convergence, and then the extended kalman filter algorithm and finally the unscented kalman filter algorithm are performed.
In summary, when the improved two-parameter index model is fused with the filter algorithm, when the crack propagation rate is slow and the crack length is short (i.e. the crack propagation in the first stage), the prediction result of the extended kalman filter algorithm is superior to that of the particle filter algorithm and the unscented kalman filter algorithm. The crack propagation speed is high, the crack length is long (namely, the crack propagation in the second stage), and the prediction result is superior to the unscented Kalman filter algorithm and the extended Kalman filter algorithm by adopting the particle filter algorithm.

Claims (6)

1. A metal structure fatigue crack propagation life prediction method based on a filtering algorithm is characterized by comprising the following steps:
(1) improving the fatigue crack propagation two-parameter index model to obtain an improved fatigue crack propagation two-parameter index model;
wherein the fatigue crack propagation two-parameter exponential model describes the fatigue crack propagation rate, and the formula is as follows:
Figure FDA0002833395320000011
in the formula, alpha and beta are material parameters, gamma is a coefficient related to a loading mode, a crack shape and a crack position, delta sigma is a stress amplitude value, and a is a crack length; when alpha and beta are fixed parameters and beta is less than 0, the crack propagation rate is increased along with the increase of the crack length, and the increase is less than eα(ii) a When β > 0, the crack propagation rate decreases with increasing crack length and is not suitable for life prediction; therefore, when the beta is less than 0, the formula (1) can be used for fatigue crack propagation prediction, but is restricted by the alpha parameter, and when the crack propagation rate is high, the alpha parameter needs to be changed greatly to achieve the prediction effect, so that the method is not beneficial to practical application; therefore, the two-parameter exponential model is improved as follows:
Figure FDA0002833395320000012
(2) constructing a state space evaluation model based on the improved fatigue crack propagation two-parameter index model;
(3) and (3) constructing a state estimation function X by using a filtering algorithm on the state space evaluation model constructed in the step (2)k+1；
(4) Using the state estimation function Xk+1And predicting the residual life of the crack propagation.
2. The method for predicting the fatigue crack propagation life of the metal structure based on the filter algorithm is characterized by comprising the following steps of: the method for constructing the state space evaluation model in the step (2) specifically comprises the following steps:
(2.1) constructing a state space parameter evaluation equation:
setting dN as 1, namely the crack propagation length under single cycle load; converting the improved fatigue crack propagation two-parameter exponential model into a discrete recursion form as follows:
Figure FDA0002833395320000013
because the frequent crack parts are dispersed and the stress amplitude on the crack surface is difficult to measure, the delta sigma is replaced by the stress amplitude obtained by finite element simulation; and because the stress amplitude obtained by simulation has calculation error, the stress amplitude delta sigma obtained by simulation is designed to obey normal distribution
Figure FDA0002833395320000021
Figure FDA0002833395320000022
To simulate the variance of equivalent response amplitude values with true values,
Figure FDA0002833395320000023
is the equivalent stress amplitude; thus ak+1＝f(akΔ σ) in
Figure FDA0002833395320000024
Is aligned with ak+1Performing a first order Taylor expansion to obtain:
Figure FDA0002833395320000025
let ak+1The process noise of (a) is:
Figure FDA0002833395320000026
in the improved fatigue crack propagation two-parameter exponential model, the partial derivative of the delta sigma is calculated as follows:
Figure FDA0002833395320000027
in that
Figure FDA0002833395320000028
In the case where it is known that,
Figure FDA0002833395320000029
and formula (6) are both definite values, and
Figure FDA00028333953200000210
thus, the
Figure FDA00028333953200000211
σaThe expression of (a) is:
Figure FDA00028333953200000212
the crack length a can be obtained from the formula (1), and when the structural characteristics, environment and working conditions cause the fatigue parameters alpha and beta to change, the crack length a is different; then the state space parameter evaluation equation of the two-parameter exponential model is:
Figure FDA00028333953200000213
in the formula (I), the compound is shown in the specification,
Figure FDA00028333953200000214
white Gaussian noise of alpha and beta, sigma can be measured by a fatigue test in standard practiceα，k、σβ，k、σa，kForming a system state parameter variance matrix Q;
(2.2) constructing a state space observation equation
The state space observation equation obtained from the fatigue crack length obtained by monitoring or detection, the corresponding cycle number and the error existing in the measurement process is as follows:
zk+1＝Hk+1xk+1+vk+1 (9)
in the formula, Hk+1Is a measurement matrix, here an identity matrix; x is the number ofk+1Is a measured value, here the crack length ak+1；vk+1To measure errors, and is subject to
Figure FDA0002833395320000031
The state space observation equation is as follows:
Zk+1＝ak+1+vk+1 (10)
wherein σr，kForming an observation variance matrix R.
3. The method for predicting the fatigue crack propagation life of the metal structure based on the filter algorithm is characterized by comprising the following steps of: the step (3) of calculating the state parameters by using particle filtering comprises the following specific steps:
initializing state parameters, inputting initial value X of state parameters0＝[a0、α0、β0]And the corresponding variance Q' ═ σa，σα，σβ]；
Sampling a set of particles, collecting the set of particles using a probability distribution according to alpha, beta, a
Figure FDA0002833395320000032
Generating a set of state function samples using a state space parameter estimation equation
Figure FDA0002833395320000033
k +1 is the number of state function steps,
Figure FDA0002833395320000034
is a normal function matrix;
calculating importance weight according to state space observation equation Zk+1Calculating a weight value:
Figure FDA0002833395320000035
wherein the content of the first and second substances,
Figure FDA0002833395320000036
is composed of
Figure FDA0002833395320000037
In the presence of Zk+1The probability of occurrence;
Figure FDA0002833395320000038
is composed of
Figure FDA0002833395320000039
The probability of occurrence;
Figure FDA00028333953200000310
is Zk+1Under the conditions of occurrence
Figure FDA00028333953200000311
The probability of occurrence;
normalizing the weight:
Figure FDA00028333953200000312
random resampling: weighted value according to importance
Figure FDA00028333953200000313
The sample is re-drawn among the N particles,
Figure FDA00028333953200000314
the large number of the distributed particles is more,
Figure FDA00028333953200000315
smaller direct deletion, last by difference
Figure FDA00028333953200000316
Generating new N particles by specific gravity; and reassign the weight value to each particle
Figure FDA00028333953200000317
Obtaining N expected values of the newly generated particles by the formula (8);
Figure FDA00028333953200000318
4. the method for predicting the fatigue crack propagation life of the metal structure based on the filter algorithm is characterized by comprising the following steps of: the step (3) of calculating the state parameters by using the extended kalman filter includes the following specific steps:
inputting initial value X of state parameter0＝[a0、α0、β0]And the corresponding variance Q' ═ σa，σα，σβ]；
Kalman filtering is the optimal analytic solution of the recursive Bayesian formula, assuming a state estimation function Xk+1Equation of state space observation Zk+1The following were used:
Xk+1＝AXk+Wk+1 (13)
Zk+1＝HXk+1+Vk+1 (14)
in the formula: a is a state transition matrix, Wk+1Predicting noise for a system state; h is an observation matrix, Vk+1To observe noise;
the equations (8) and (10) are generally nonlinear functions, and the core idea of the extended Kalman filter algorithm is to linearize the nonlinear function and to estimate the state of the function Xk+1And the state space observation function Zk+1Surrounding the filtered value
Figure FDA0002833395320000041
And performing Taylor series expansion and neglecting the influence of second-order and higher-order terms, thereby forming an approximate linearized state parameter estimation model as follows:
Figure FDA0002833395320000042
wherein, it is made
Figure FDA0002833395320000043
Figure FDA0002833395320000044
Figure FDA0002833395320000045
Figure FDA0002833395320000046
The equation (16) and the equation (17) are Jacobian matrixes of the state parameter estimation model and the observation model respectively, and the prediction result of the step k and the equation (21) can be obtained respectively
Figure FDA0002833395320000047
And
Figure FDA0002833395320000048
the extended kalman filter equation can then be expressed as:
Figure FDA0002833395320000051
Figure FDA0002833395320000052
since the predicted value is calculated for the equation (13), the influence of the second order term and higher order terms is ignored after the taylor series expansion, and therefore, the equation (20) has a certain error, and the calculation is performed
Figure FDA0002833395320000053
Calculating by using the formula (13) and the formula (14);
Figure FDA0002833395320000054
in the formula (I), the compound is shown in the specification,
Figure FDA0002833395320000055
predicting value in the k step, and setting the k as 0 to be initial value of state parameter;
after new state parameters are obtained through prediction, the correlation among the state parameters is changed, the system covariance is changed, and the covariance matrix of the prior state estimation is as follows:
Figure FDA0002833395320000056
wherein Q is a system state parameter variance matrix;
and (3) updating: obtaining an observation parameter Zk+1The predicted state value can then be corrected, and the correction equation is as follows:
Figure FDA0002833395320000057
kalman gain matrix:
Figure FDA0002833395320000058
and (4) updating the predicted value:
Figure FDA0002833395320000059
covariance update estimation:
Figure FDA00028333953200000510
5. the method for predicting the fatigue crack propagation life of the metal structure based on the filter algorithm is characterized by comprising the following steps of: the calculating of the state parameters by using the unscented kalman filter in the step (3) comprises the following specific steps:
inputting initial value X of state parameter0＝[a0、α0、β0]And the corresponding variance Q ═ σa，σα，σβ](ii) a Covariance matrix p0；
Obtaining a group of sampling points, a Sigma point set and a corresponding weight value by using a formula (27) and a formula (28); 2n +1 Sigma sampling points are calculated, n is the state estimation function variable dimension:
Figure FDA0002833395320000061
wherein the content of the first and second substances,
Figure FDA0002833395320000062
the ith column representing the square root of the matrix;
calculating the weight of the sampling point,
Figure FDA0002833395320000063
wherein, subscript m is mean value, c is covariance;
Figure FDA0002833395320000064
and
Figure FDA0002833395320000065
weights representing the mean and covariance of the corresponding points; parameter λ ═ α2(n + κ) -n is a scaling parameter; the value of parameter alpha is [0,1 ]]Determining the distribution of sampling points, and adjusting the sampling points and the mean value
Figure FDA0002833395320000066
The distance of (d); k is a candidate parameter for ensuring that a matrix (n + lambda) p is a semi-positive definite matrix, wherein k is 0 when n is larger than or equal to 3, and k is n-3 when n is smaller than 3; the parameter beta is more than 0, the dynamic difference of a high-order term of an adjustable equation is adjusted, and beta is taken to be 2 when the normal distribution is carried out;
first of all generated to
Figure FDA0002833395320000067
Sigma point set as mean:
Figure FDA0002833395320000068
substituting Sigma point set into state parameter estimation function fk(Xk) Obtaining a set of state parameter estimation sample points
Figure FDA0002833395320000069
Sampling points
Figure FDA00028333953200000610
To correspond to
Figure FDA00028333953200000611
The weighted sum can obtain the mean and variance of the corresponding predicted values of the state parameters, and the specific formula is as follows:
Figure FDA00028333953200000612
Figure FDA00028333953200000613
according to the predicted value
Figure FDA00028333953200000614
A set of samples is obtained again using equations (27) and (28):
Figure FDA00028333953200000615
substituting the predicted Sigma point set into an observation equation (10) to obtain a predicted observation value, i is 1,2, …,2n + 1;
Figure FDA0002833395320000071
obtaining the mean value and the variance of system prediction through weighted summation according to the obtained prediction observation value;
Figure FDA0002833395320000072
Figure FDA0002833395320000073
Figure FDA0002833395320000074
kalman gain matrix calculation:
Figure FDA0002833395320000075
status parameter update andupdating the covariance to obtain the true observation parameter Zk+1Updating the post-state parameters:
Figure FDA0002833395320000076
and (3) covariance updating:
Figure FDA0002833395320000077
6. the method for predicting the fatigue crack propagation life of the metal structure based on the filter algorithm according to any one of claims 3 to 5, wherein: the step (4) specifically comprises the following steps: after the state parameter estimation is carried out by utilizing the step (3), the parameter of the K +1 step can be obtained
Figure FDA0002833395320000078
Predicting the fatigue crack extension life, and obtaining the crack length of the (k + n) th step by only substituting the parameters into a discrete recursion equation to carry out recursion calculation, wherein the smaller the n is, the higher the precision is, certainly, because the state parameters are changed at any time; the concrete prediction formula of the discrete recursive equation is as follows;
Figure FDA0002833395320000079
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