CN105956710A - Structural dynamic system response prediction method of considering parameter indeterminacy - Google Patents

Structural dynamic system response prediction method of considering parameter indeterminacy Download PDF

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CN105956710A
CN105956710A CN201610325963.8A CN201610325963A CN105956710A CN 105956710 A CN105956710 A CN 105956710A CN 201610325963 A CN201610325963 A CN 201610325963A CN 105956710 A CN105956710 A CN 105956710A
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hermite
system response
response
formula
stochastic variable
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肖斌
宋宗彪
高超
张艾萍
曹丽华
李亚轩
金建国
孙斌
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Northeast Electric Power University
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Northeast Dianli University
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Abstract

A structural dynamic system response prediction method of considering parameter indeterminacy is characterized by comprising the steps of carrying out the parameter description on the indeterminacy of a system by random variables, and giving out a distribution law of the random variables; transforming the multivariate random variables of abnormal distribution into the multivariate normal distribution variables of mutually statistics independence; expanding based on a multivariate Fourier-Hermite polynomial, and expressing the structural dynamic system response as a multivariate normal distribution variable polynomial function format of mutually statistics independence; utilizing the Gauss-Hermite numerical integration to calculate the undetermined coefficients of the polynomial expansion, and calculating the mean response; back substituting the obtained polynomial coefficients and mean response in the Fourier-Hermite polynomial to obtain the displayed polynomial function of the system response; based on the system response polynomial function, embedding local MonteCarlo simulation to obtain the statistics characteristic of the structural dynamic system response.

Description

A kind of Forecasting Methodology of the Structural Dynamic System response considering parameter uncertainty
Technical field
The present invention relates to the Forecasting Methodology of a kind of system response, especially relate to a kind of structure considering parameter uncertainty The Forecasting Methodology of dynamical system response.
Background technology
Systematic uncertainty is the problem certainly existed in dynamical system.Owing to systematic parameter has uncertainty so that Natural frequency, system response etc. become random quantity, and its result of calculation places one's entire reliance upon the statistical nature of random parameter, and it realized Journey is extremely complex.
Over nearly 20 years, dynamical system response prediction all achieves a lot of achievement in research theoretical in application.But it is at present Only, most of dynamical system response prediction broadly falls into Deterministic Methods, neither considers included in structural parameters and response Uncertainty, thus constrain the effective application on labyrinth of the dynamical system response prediction to a great extent, this is also It is dynamical system response prediction theoretical developments to institute's problem demanding prompt solution after certain phase.Additionally, dynamical system response prediction It is primarily directed to non-linear, the structural system of low frequency, and to high-frequency percussion, Nonlinear Large Deformation, coupling and random phenomenon For the situation of feature, due in now structural system and experiment containing the most uncertain so that traditional dynamical system is rung Should predict and cannot effectively be applied.When considering system randomness, response distribution and the response statistical property such as average, variance thereof Analysis is then the hot research problem that stochastic system is analyzed.The function of many variables for any one uncertain system respond Say, when the random parameter variable that it produces impact is more, utilize conventional method to solve difficulty that this response truly solves and multiple Miscellaneous degree is quite big, and error is the biggest.Therefore, use approximation by polynomi-als to obtain system response best fit approximation, now need Study the Forecasting Methodology of a kind of dynamical system response, it is achieved Random Dynamical Systems vibration analysis, and analyze acquisition sound further Answer and statistical nature.
Summary of the invention
It is an object of the invention to provide the Forecasting Methodology of a kind of Structural Dynamic System response considering parameter uncertainty, mesh Be the statistical nature obtaining Structural Dynamic System response, it was predicted that the explicit function expression of response, there is precision of prediction high, The suitability is strong, the advantage that effect is good.
It is an object of the invention to realize by following technical scheme: a kind of structural dynamic system considering parameter uncertainty The Forecasting Methodology of system response, is characterized in that, it comprises the following steps:
Step 1: by the uncertainty of system, by stochastic variable parameter X=[xi,...,xn]TIt is described, is given each The distribution law of individual stochastic variable isJudging stochastic variable whether Normal Distribution, if meeting, the most directly carrying out Step 3；If being unsatisfactory for, carry out step 2；
Step 2: utilize numerical value transformation approach, at RNOnThe stochastic variable of nonnormal distribution is converted to The stochastic variable of Normal Distribution, and provide distribution law u of each stochastic variablei～N (μi,σi 2),
That is:
Step 3: based on multivariate Fourier-Hermite polynomial expansion, Structural Dynamic System response is expressed as phase The multivariate normal distributions variable polynomial function form of statistical iteration mutually:
(3.1) any structure dynamical system response y is described as the multivariate normal distributions variable of mutual statistical independenceExplicit polynomial function form,
That is:
Wherein: N is stochastic variable number,For the model response component dominated by s stochastic variable；y0 Being model response average, f () is the function representation of system response；
(3.2) willLaunch based on multivariate Fourier-Hermite multinomial, based on formula (3) Calculate:
y i 1 , ... , i s ( u i 1 , ... , u i s ) = Σ j s = 1 ∞ ... Σ j 1 = 1 ∞ C i 1 ... i s j 1 ... j s Π k = 1 s H j k ( u i k ) - - - ( 3 )
In formula:For multinomial undetermined coefficient；Hj() is jth rank Hermite multinomials；
Step 4: utilize undetermined coefficient that Gauss-Hermite numerical integration evaluator launches and calculate average and ring Should:
(4.1) Gauss-Hermite numerical integration is introduced,
That is:
(4.2) estimate integral node number n for (4) formula, and determine integration weights Ak；
(4.3) Hermite multinomial order m is selected, normalization Hermite multinomial,
That is:
(4.4) Gauss-Hermite numerical integration evaluator undetermined coefficient is utilized
And calculate average response
Step 5: the average response that the coefficient obtained after (6) formula gained multinomial coefficient is substituted into (3) formula and (7) formula obtain Together bring in (2) formula, obtain the display polynomial function of system response；
Step 6: respond polynomial function based on system, embeds local Monte Carlo simulation, it is thus achieved that Structural Dynamic System The statistical nature of response:
(6.1) Monte Carlo simulation is utilized to generate the stochastic variable u meeting Gauss distributioni, form polynary random change Measure M sample
(6.2) by step (6.1) gained multiple random variable sample UMIn (3) formula of substitution, obtain Hermite multinomial {Hi}MSample；Then this sample is substituted in (2) formula and obtain system response prediction sample yk(k=1 ..., M):
(6.3) results based on step (6.2), utilize system response prediction sample yk, estimate its each rank statistical moment, analyze Obtain the statistical nature of Structural Dynamic System response.
The present invention is a kind of consider parameter uncertainty Structural Dynamic System response Forecasting Methodology compared with prior art, Have an advantage in that: during system response prediction, consider the parameter uncertainty of system, based on multivariate Fourier- Hermite polynomial expansion, it is proposed that a kind of Structural Dynamic System response prediction method considering parameter uncertainty, by structure Dynamical system response is expressed as explicit polynomial function form, can be used for black box system；Not only statement is simple, and calculates Measure little, utilize local Monte Carlo simulation to substantially increase theoretical precision and efficiency, the system met with a response also by analysis Meter feature.Having precision of prediction high, the suitability is strong, the advantage that effect is good.
Accompanying drawing explanation
Fig. 1 is the modeling procedure figure of the stochastic model of the embodiment of the present invention；
Fig. 2 is the flowchart of the embodiment of the present invention.
Detailed description of the invention
See figures.1.and.2, the prediction side of a kind of Structural Dynamic System response considering parameter uncertainty of the present invention Method, comprises the following steps:
Step 1: by the parameter uncertainty of system, by stochastic variable parameter X=[xi,...,xn]TIt is described, gives The distribution law going out each stochastic variable isJudge stochastic variable whether Normal Distribution, if meeting, the most directly Carry out step 3；If being unsatisfactory for, carry out step 2；
Step 2: utilize numerical value transformation approach, at RNOnThe stochastic variable of nonnormal distribution is converted to The stochastic variable of Normal Distribution, and provide distribution law u of each stochastic variablei～N (μi,σi 2),
That is:
Step 3: based on multivariate Fourier-Hermite polynomial expansion, Structural Dynamic System response is expressed as phase The multivariate normal distributions variable polynomial function form of statistical iteration mutually:
(3.1) any structure dynamical system response y is described as the multivariate normal distributions variable of mutual statistical independenceExplicit polynomial function form,
That is:
Wherein: N is stochastic variable number,For the model response component dominated by s stochastic variable；y0 Being model response average, f () is the function representation of system response；
(3.2) willLaunch based on multivariate Fourier-Hermite multinomial, based on formula (3) Calculate:
y i 1 , ... , i s ( u i 1 , ... , u i s ) = Σ j s = 1 ∞ ... Σ j 1 = 1 ∞ C i 1 ... i s j 1 ... j s Π k = 1 s H j k ( u i k ) - - - ( 3 )
In formula:For multinomial undetermined coefficient；Hj() is jth rank Hermite multinomials；
Step 4: utilize undetermined coefficient that Gauss-Hermite numerical integration evaluator launches and calculate average and ring Should:
(4.1) Gauss-Hermite numerical integration is introduced,
That is:
(4.2) estimate integral node number n for (4) formula, and determine integration weights Ak；
(4.3) Hermite multinomial order m is selected, normalization Hermite multinomial,
That is:
(4.4) Gauss-Hermite numerical integration evaluator undetermined coefficient is utilized
And calculate average response
Step 5: the average response that the coefficient obtained after (6) formula gained multinomial coefficient is substituted into (3) formula and (7) formula obtain Together bring in (2) formula, obtain the display polynomial function of system response；
Step 6: respond polynomial function based on system, embeds local Monte Carlo simulation, it is thus achieved that Structural Dynamic System The statistical nature of response:
(6.1) Monte Carlo simulation is utilized to generate the stochastic variable u meeting Gauss distributioni, form polynary random change Measure M sample
(6.2) by step (6.1) gained multiple random variable sample UMIn (3) formula of substitution, obtain Hermite multinomial {Hi}MSample；Then this sample is substituted in (2) formula and obtain system response prediction sample yk(k=1 ..., M):
(6.3) results based on step (6.2), utilize system response prediction sample yk, estimate its each rank statistical moment, analyze Obtain the statistical nature of Structural Dynamic System response.
Fourier-Hermite is Fourier-Hermite；Gauss-Hermite is Gauss-Hermite； MonteCarlo is Meng Te-Caro.
The embodiment of the present invention is only used for that the present invention is further illustrated, not exhaustive, is not intended that claim The restriction of protection domain, the enlightenment that those skilled in the art obtain according to case study on implementation of the present invention, without creative work just It is conceivable that other replacement being substantially equal to, all in scope.

Claims (1)

1. considering a Forecasting Methodology for the Structural Dynamic System response of parameter uncertainty, it is characterized in that, it includes following step Rapid:
Step 1: by the uncertainty of systematic parameter, by stochastic variable parameter X=[xi,...,xn]TIt is described, is given each The distribution law of individual stochastic variable isJudging stochastic variable whether Normal Distribution, if meeting, the most directly carrying out Step 3；If being unsatisfactory for, carry out step 2；
Step 2: utilize numerical value transformation approach, at RNOnThe stochastic variable of nonnormal distribution is converted to obey The stochastic variable of normal distribution, and provide distribution law u of each stochastic variablei～N (μi,σi 2),
That is:
Step 3: based on multivariate Fourier-Hermite polynomial expansion, be expressed as mutually uniting by Structural Dynamic System response Count independent multivariate normal distributions variable polynomial function form:
(3.1) any structure dynamical system response y is described as the multivariate normal distributions variable of mutual statistical independence Explicit polynomial function form,
That is:
Wherein: N is stochastic variable number,For the model response component dominated by s stochastic variable；y0It it is mould Type response average, f () is the function representation of system response；
(3.2) willLaunch based on multivariate Fourier-Hermite multinomial, calculate by formula (3):
y i 1 , ... , i s ( u i 1 , ... , u i s ) = Σ j s = 1 ∞ ... Σ j 1 = 1 ∞ C i 1 ... i s j 1 ... j s Π k = 1 s H j k ( u i k ) - - - ( 3 )
In formula:For multinomial undetermined coefficient；Hj() is jth rank Hermite multinomials；
Step 4: utilize undetermined coefficient that Gauss-Hermite numerical integration evaluator launches and calculate average response:
(4.1) Gauss-Hermite numerical integration is introduced,
That is:
(4.2) estimate integral node number n for (4) formula, and determine integration weights Ak；
(4.3) Hermite multinomial order m is selected, normalization Hermite multinomial,
That is:
(4.4) Gauss-Hermite numerical integration evaluator undetermined coefficient is utilized
And calculate average response
Step 5: the average that the coefficient obtained after (6) formula gained multinomial coefficient is substituted into (3) formula and (7) formula obtain responds together Bring in (2) formula, obtain the display polynomial function of system response；
Step 6: respond polynomial function based on system, embeds local Monte Carlo simulation, it is thus achieved that Structural Dynamic System responds Statistical nature:
(6.1) Monte Carlo simulation is utilized to generate the stochastic variable u meeting Gauss distributioni, composition multiple random variable M Sample
(6.2) by step (6.1) gained multiple random variable sample UMIn (3) formula of substitution, obtain Hermite multinomial { Hi}MSample This；Then this sample is substituted in (2) formula and obtain system response prediction sample yk(k=1 ..., M):
(6.3) results based on step (6.2), utilize system response prediction sample yk, estimate its each rank statistical moment, analysis obtains The statistical nature of Structural Dynamic System response.
CN201610325963.8A 2016-05-17 2016-05-17 Structural dynamic system response prediction method of considering parameter indeterminacy Pending CN105956710A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110941881A (en) * 2019-10-16 2020-03-31 北京航空航天大学 Mixed uncertainty structure fatigue life analysis method based on chaos polynomial expansion
CN110941923A (en) * 2019-11-22 2020-03-31 汉思科特(盐城)减震技术有限公司 Method for determining sensitive parameters of air spring structure

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110941881A (en) * 2019-10-16 2020-03-31 北京航空航天大学 Mixed uncertainty structure fatigue life analysis method based on chaos polynomial expansion
CN110941923A (en) * 2019-11-22 2020-03-31 汉思科特(盐城)减震技术有限公司 Method for determining sensitive parameters of air spring structure

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