CN106021186B - A kind of multiple dimensioned alternative manner of Efficient Solution large-scale nonlinear Random Structural Systems state - Google Patents
A kind of multiple dimensioned alternative manner of Efficient Solution large-scale nonlinear Random Structural Systems state Download PDFInfo
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Abstract
The invention discloses a kind of multiple dimensioned alternative manners of Efficient Solution large-scale nonlinear Random Structural Systems state, including：Establish the moment function equation of nonlinear random structures system mode response；Using Gauss Truncation Technique, the closing moment function equation of condition responsive is established；Establish the multiple dimensioned discrete segment in time domain；Using Runge Kutta methods, in the closing moment function equation of multiple dimensioned section inner iteration solving state response, the final statistical nature for obtaining the response of nonlinear random structures system mode.The multiple dimensioned alternative manner of the present invention can not only save a large amount of computer storage resource, and greatly improve computational efficiency to avoid three stochastic differential equations are solved simultaneously.In addition, iterations can be automatically adjusted by control errors and initial value setting, the acceleration convergence of solution procedure is realized.The present invention provides efficient method for solving for large-scale nonlinear Random Structural Systems state analysis, has practical engineering application value.
Description
Technical field
The present invention is mainly the Efficient Solution suitable for large-scale nonlinear Random Structural Systems condition responsive statistical nature
It calculates, and in particular to a kind of multiple dimensioned alternative manner of Efficient Solution large-scale nonlinear Random Structural Systems state.
Background technology
Earthquake, wind load and marine stormy waves are common environmental loads.In the case where being acted on for a long time by environmental load, artificial knot
The dynamic response of structure can influence function and the safety of structure.Therefore, the Dynamic Response of the structure under environmental load effect
It is particularly important.It is random process since environmental load is usually handled, needs to establish and solve in the Dynamic Response random micro-
Divide equation.
Fokker-Planck-Kolmogorov (FPK) method, which is widely used in, establishes the nonlinear stochastic system differential equation,
The equation that this method is established substantially is equivalent toStochastic differential equation.However, in steady related excitation, only when
Random external excitation is represented by white noise filter, and FPK methods can use.Although by stochastic averaging methods and filtering side
Method promotes the application of FPK methods, however this method still has two aspect disadvantages：On the one hand, random external excitation is necessary for
Delta correlations random process is represented by white noise filter；On the other hand, this method increase the number of states of system.
In this case, some scholars more pay attention to the research of statistical moment functional based method.For example, Lutes etc. has studied in delta phases
The dynamic response of nonlinear system under arbitrary excitation is closed, and proposes a kind of foundations response moment function equation and accumulates flow function side
The relatively direct expression formula of journey.Di Paola etc. are encouraged using the gaussian filtering of polynomial form as random external, are had studied
The response of nonlinear system obtains the statistical nature of non-gaussian response.As mentioned above, many research work concentrate on
Nonlinear Dynamic Response is analyzed.However, the improvement that most theoretical researches have ignored moment function equation numerical value solution efficiency is ground
Study carefully.Obviously, inefficient numerical solution algorithm can cause Nonlinear Dynamic Response analysis method to be met difficulty in practical application, special
It is not large-scale nonlinear Random Structural Systems.In addition, delta correlation arbitrary excitations are typically used to establish nonlinear system
Moment function equation, but this random process can not accurate description environmental load feature.In fact, Qiu Zhi equalitys have been ground
Study carefully by the dynamical system that environmental load acts on and has pointed out to establish the moment function equation of condition responsive very using exact propability
Effectively, however, the research work is only applicable to linear stochastic structure system problem.
The present invention is directed to the nonlinear random structures system acted on by environmental load, and state is established using exact propability
The moment function equation of response.The multiple dimensioned discrete segment in time domain is established, in the response of multiple dimensioned section inner iteration solving state
Close moment function equation, the final statistical nature for obtaining condition responsive.The present invention is large-scale nonlinear Random Structural Systems shape
State analysis provides efficient method for solving, has practical engineering application value.
Invention content
It is an object of the invention to overcome the deficiencies of the prior art and provide a kind of solution large-scale nonlinear random structure systems
The multiple dimensioned alternative manner of system state avoids solving three stochastic differential equations simultaneously, realizes the rapidly and efficiently meter of solution procedure
It calculates, effective computational methods is provided for large-scale nonlinear Random Structural Systems state analysis, there is the engineer application of reality
Value.
To achieve the above object, a kind of Efficient Solution large-scale nonlinear Random Structural Systems state provided by the invention
Multiple dimensioned alternative manner, this method realize that steps are as follows：
Step 1: being based on nonlinear random structures system state equation, nonlinear system shape is established by exact propability
The moment function equation of state response；
Step 2: being blocked to the non-close item in condition responsive moment function equation using Gauss Truncation Technique, establish
The closing moment function equation of condition responsive；
Step 3: by section of entire analysis time it is discrete be multiple thick scale time intervals, then in each thick scale time
Thin scale time discrete is carried out in section, finally establishes the multiple dimensioned discrete segment in time domain；
Step 4: in thick scale time interval [ti,ti+1] on, closed shape is solved using Runge-Kutta methods simultaneously
The mean-square value moment function equation of the mean value moment function equation and closed condition responsive of state response, obtains condition responsive mean valueWith condition responsive mean-square value
Step 5: judging condition responsive mean valueWith condition responsive mean-square valueConvergence, if result receive
It holds back, jumps to step 7；If result does not restrain, step 6 is continued to execute；
Step 6: the condition responsive mean value that will be obtained in step 4With condition responsive mean-square valueSubstitute into envelope
The Correlation Moment functional equation for the condition responsive-excitation closed calculates condition responsive-excitation correlation moment functionEnable k=k+
1, jump to step 4；
Step 7: judging whether time conditions meet t≤T, if meeting condition, i=i+1 and k=1 are enabled, jumps to step
Rapid four；If being unsatisfactory for condition, output state responds mean valueCondition responsive mean-square valueSwash with condition responsive-
Encourage related moment functionIteration ends.
Wherein, the moment function equation of non-linear system status response is established in step 1 using exact propability, including：Shape
The mean value moment function equation of state response, the mean-square value moment function equation moment function side related to condition responsive-excitation of condition responsive
Journey is embodied as：
The mean value moment function equation of condition responsive：
In formula, A1, A3For sytem matrix, B is input matrix, mx(t) it is the mean function of condition responsive, mu(t) it is outside
The mean function of arbitrary excitation, Rxοxοx(t) it is the joint moment function of condition responsive, symbol " ο " is two vectorial Hadamard
Product indicates that each corresponding element of two vectors is multiplied, and result of product remains as the identical vector of dimension.
The mean-square value moment function equation of condition responsive：
In formula, Rxx(t, t) is the mean-square value function of condition responsive,For the phase of condition responsive-excitation
Moment function is closed, remaining meaning of parameters is same as above.
Condition responsive-excitation Correlation Moment functional equation：
In formula, Rux(t, t) is the related moment function of condition responsive-excitation, Ruu(t, t) is the square of random external excitation
Value function, remaining meaning of parameters are same as above.
Wherein, the closing moment function equation of condition responsive is established in step 2 using Gauss Truncation Technique, including：It is closed
The mean value moment function equation of condition responsive, the mean-square value moment function equation of closed condition responsive and closed condition responsive-swash
The Correlation Moment functional equation encouraged, is embodied as：
The mean value moment function equation of closed condition responsive：
In formula, parameters meaning is same as above.
The mean-square value moment function equation of closed condition responsive：
In formula, parameters meaning is same as above.
The Correlation Moment functional equation of closed condition responsive-excitation：
In formula, parameters meaning is same as above.
Wherein, the multiple dimensioned discrete segment in time domain is established using discrete way at equal intervals in step 3.It first, will be entire
Analysis time section [t0, T] and discrete for multiple thick scale time interval [ti,ti+1], it is embodied as：
Dcoarse={ ti:t0=t0<t1<…<tI=T }
In formula, t0For initial time, tI=T is end time, ti, i=0,1 ..., I is the discrete of thick scale time interval
Time point, I are thick scale time interval number.
Secondly, in each thick scale time interval [ti,ti+1] in carry out thin scale time discrete, when obtaining thin scale
Between section [ti,j,ti,j+1], it is embodied as：
In formula, ti=ti,0For the initial time in the thick scale time interval of i+1,For the thick scale of i+1
End time in time interval, ti,j, j=0,1 ..., JiFor the discrete time point of thin scale time interval, JiFor i+1
Thin scale time interval number in thick scale time interval.
Wherein, the time step of Runge-Kutta methods is thin scale time interval [t in step 4i,j,ti,j+1], state
Response-excitation correlation moment function value in the mean-square value moment function equation of response isIt is embodied as：
In formula, parameters meaning is same as above.
Wherein, the constringent condition of judging result includes that condition responsive convergence in the mean condition and condition responsive are equal in step 5
Side's value condition of convergence, is embodied as：
In formula, ε1For condition responsive mean value allowable error, ε2For condition responsive mean-square value allowable error.
Wherein, with thin scale time interval [t in step 6i,j,ti,j+1] it is time step, using Runge-Kutta methods
The Correlation Moment functional equation of closed condition responsive-excitation is individually solved, the condition responsive mean value in Correlation Moment functional equation takes
Value isCondition responsive mean-square value value in Correlation Moment functional equation isIt is embodied as：
In formula, parameters meaning is same as above.
The advantages of the present invention over the prior art are that：The present invention provides a kind of Efficient Solution large-scale nonlinear with
The multiple dimensioned alternative manner of machine structured system condition, major advantage are as follows：
(1) present invention establishes the moment function equation of nonlinear random structures system mode response using exact propability, no
It needs to consider the complicated calculations that FPK methods are related to.The condition responsive statistical nature of system can according to primary condition and it is external with
The statistical nature of machine excitation directly obtains, and environmental load is encouraged equally applicable；
(2) present invention proposes the new algorithm for the moment function equation for solving the response of nonlinear random structures system mode, will
Discrete analysis time section is thick scale time interval and thin scale time interval, and moment function side is solved in multiple dimensioned section inner iteration
Journey avoids while solving three stochastic differential equations, saves a large amount of computer storage resource；
(3) present invention can automatically adjust iterations by control errors and initial value setting according to the condition of convergence, realize
The acceleration of solution procedure restrains, and greatly improves computational efficiency, especially suitable for solving large-scale nonlinear random structure system
System state analysis problem has practical engineering application value.
Description of the drawings
Fig. 1 is the multiple dimensioned alternative manner implementation flow chart of the present invention；
Fig. 2 is Transmission tower in present example；
Fig. 3 is the mean value of 3 end vertical speed of Transmission tower in present example；Wherein Fig. 3 (a) is that 3 ends are vertically fast
Spend the time history of mean value；Fig. 3 (b) is the partial enlarged view of Fig. 3 (a)；
Fig. 4 is the mean value of 3 end vertical displacement of Transmission tower in present example；Wherein Fig. 4 (a) is the vertical position in 3 ends
Move the time history of mean value；Fig. 4 (b) is the partial enlarged view of Fig. 4 (a)；
Fig. 5 is the solution time of multiple dimensioned alternative manner and conventional method comparison in present example.
Specific implementation mode
Below in conjunction with the accompanying drawings and specific embodiment further illustrates the present invention.
As shown in Figure 1, the present invention proposes a kind of more rulers of Efficient Solution large-scale nonlinear Random Structural Systems state
Alternative manner is spent, specific implementation step is：
(1) it is based on nonlinear random structures system state equation, non-linear system status is established using exact propability and is rung
The moment function equation answered, including：The mean value moment function equation of condition responsive, the mean-square value moment function equation and state of condition responsive
Response-excitation Correlation Moment functional equation.
The governing equation of one n degree of freedom Nonlinear Cubic Random Structural Systems is expressed as：
In formula, M is mass of system matrix, C1, C3For system damping matrix, K1, K3For system stiffness matrix, Q (t) is outer
Portion's arbitrary excitation, y (t),WithRespectively system motion vector, velocity vector and vector acceleration, symbol " ο " are two
The Hadamard products of a vector indicate that each corresponding element of two vectors is multiplied, and result of product remains as the identical vector of dimension.
It introduces 2n dimension state vectors and 2n dimension random external excitation vectors is expressed as：
The governing equation (1) of nonlinear random structures system can be rewritten into non-linear probabilistic system posture equation, specifically
It is expressed as：
In formula, A1, A3For sytem matrix, B is input matrix, and x (t) is system mode vector, and u (t) encourages for random external
Vector is embodied as：
Mathematic expectaion is taken to formula (3), can be obtained：
The mathematic expectaion expression formula of definition status response：
Formula (5) can be rewritten as the mean value moment function equation of condition responsive, be embodied as：
In formula,
According to the definition of random vector function mean-square value, can obtain：
Formula (3) and its transposition expression formula are substituted into formula (9), can be obtained：
Mathematic expectaion is taken to formula (10), the mean-square value moment function equation of condition responsive can be obtained, be embodied as：
In formula, Rxx(t, t) is the mean-square value function of condition responsive,For condition responsive-excitation
Related moment function, is embodied as：
Similar, according to the definition of the Correlation Moment of condition responsive-excitation, can obtain：
The transposition of formula (3) is substituted into formula (13), can be obtained：
Mathematic expectaion is taken to formula (14), the Correlation Moment functional equation of condition responsive-excitation can be obtained, be embodied as：
In formula, Rux(t, t) is the related moment function of condition responsive-excitation, Ruu(t, t) is the square of random external excitation
Value function is embodied as：
(2) Gauss Truncation Technique is used to establish the closing moment function equation of condition responsive, including：Closed condition responsive
Mean value moment function equation, closed condition responsive mean-square value moment function equation and closed condition responsive-excitation Correlation Moment
Functional equation.
Formula (7), the high-order non-close item in formula (11) and formula (15) is respectively Rxοxοx(t), R(xοxοx)x(t, t) and
Ru(xοxοx)(t, t) can obtain high-order non-close item application Gauss Truncation Technique：
Formula (17) is substituted into formula (7) respectively, in formula (11) and formula (15), can be obtained：
The mean value moment function equation of closed condition responsive：
The mean-square value moment function equation of closed condition responsive：
The Correlation Moment functional equation of closed condition responsive-excitation：
(3) the multiple dimensioned discrete segment in time domain is established.First, by section [t of entire analysis time0, T] average discrete be etc.
Interlude section [ti,ti+1], referred to as thick scale time interval is embodied as：
Dcoarse={ ti:t0=t0<t1<…<tI=T } (21)
In formula, t0For initial time, tI=T is end time, ti, i=0,1 ..., I is the discrete of thick scale time interval
Time point, I are thick scale time interval number.
Secondly, then by each thick scale time interval [ti,ti+1] average discrete be equally spaced subinterval [ti,j,
ti,j+1], referred to as thin scale time interval is embodied as：
In formula, ti=ti,0For the initial time in the thick scale time interval of i+1,For the thick scale of i+1
End time in time interval, ti,j, j=0,1 ..., JiFor the discrete time point of thin scale time interval, JiFor i+1
Thin scale time interval number in thick scale time interval.
(4) in thick scale time interval [ti,ti+1] on, it solves closed state simultaneously using Runge-Kutta methods and rings
The mean-square value moment function equation of the mean value moment function equation and closed condition responsive answered.
The time step of Runge-Kutta methods is taken as thin scale time interval length Δ t=tI, j+1-tI, j, carrying out the
When k iterative calculation, response-excitation correlation moment function value in the mean-square value moment function equation of condition responsive isIt is embodied as：
In thick scale time interval [ti,ti+1] on carry out when the 1st iterative calculation (k=1), response-excitation correlation
Moment functionValue is response-excitation Correlation Moment R of the time interval initial timeux(ti,ti), i.e.,
(5) judge condition responsive mean valueWith condition responsive mean-square valueConvergence, judge constringent item
Part is embodied as：
In formula, ε1For condition responsive mean value allowable error, ε2For condition responsive mean-square value allowable error.
If adjacent iteration result twice meets the inequality condition of formula (24), result convergence jumps to step (7)；
Otherwise result does not restrain, and continues to execute step (6).
(6) according to the condition responsive mean value obtained in step (4)With condition responsive mean-square valueIn thick ruler
Spend time interval [ti,ti+1] on the Correlation Moment letter of closed condition responsive-excitation individually solved using Runge-Kutta methods
Number equation.
The time step of Runge-Kutta methods is taken as thin scale time interval length △ t=ti,j+1-ti,j, carrying out the
When k iterative calculation, the condition responsive mean value in the Correlation Moment functional equation of condition responsive-excitation and condition responsive mean-square value
Value is respectivelyWithIt is embodied as：
After the completion of formula (25) solution, k=k+1 is enabled, jumps to step (4), continues next iteration and solves.
(7) judge whether time conditions meet t≤T, if meeting condition, enable i=i+1 and k=1, jump to step
(4), continue the first time iterative solution of next thick scale time interval；If being unsatisfactory for condition, output state responds mean valueCondition responsive mean-square valueMoment function related to condition responsive-excitationIteration ends.
Embodiment：
1. structural parameters and model introduction
The characteristics of in order to more fully understand the invention and its to the actual applicability of engineering, the present invention is with shown in Fig. 2
Illustrate multiple dimensioned alternative manner Efficient Solution large-scale nonlinear Random Structural Systems condition responsive for Transmission tower
Validity.Pylon structure in Fig. 4 is mainly made of tubing, and section is divided into two kinds, is main column, rectangular section on four edges
For 120 × 120 × 4mm of, remaining tube section is 50 × 5mm of Φ.The elasticity modulus of material is 2.06 × 105MPa, Poisson's ratio
It is 0.3.Pylon is fixed at bottom everywhere pedestal, and cantilever end bears downward concentrfated load effect F (t)=- psin (4 at 6
π t), wherein p is random parameter, and mean value and standard deviation are respectively μp=1960N and σp=100.Assuming that the rigidity right and wrong of structure
Linear, the nonlinear Control differential equation of structural system is expressed as：
In formula, M and K are respectively linear quality matrix and linear stiffness matrix, and F (t) encourages for random external.
2. multiple dimensioned iterative program and running environment
According to multiple dimensioned alternative manner, in conjunction with existing certainty program of finite element, using Matlab devise with
Machine program of finite element.In this procedure, the geometrical model and FEM mesh of structure are by certainty finite element analysis
Program generates.In the present embodiment, the linear quality matrix of Transmission tower and linear stiffness matrix use MSC/Nastran
Finite element software is established, and using the DMAP extractions in Nastran.The statistics characteristic analysis of system mode response is in Matlab
It is executed on R2012, the processor of computer is tetra- core processors of 3.40GHz Intel (R) Core (TM) i7.
3. the statistical nature of system mode response
Based on nonlinear random structures system state equation, it can be obtained using exact propability and be made in external arbitrary excitation
The statistical nature of structured system condition response under.Fig. 3 and Fig. 4 be set forth vertical speed mean value at cantilever end 3 when
Between course and vertical displacement mean value time history.For the validity of comparison result, Monte Carlo are given simultaneously in figure
Simulate the result obtained.In stochastic structure analysis, Monte Carlo simulation results are considered as precise results.From Fig. 3 (a)
Have with Monte Carlo simulation results with can see the condition responsive statistical nature that direct probability method obtains in Fig. 4 (a)
Good consistency.It can be clearly seen that the fine difference of two methods result from partial enlarged view, this species diversity derives from
Truncated error caused by Gauss intercept method.For engineering problem, this small difference can be ignored completely, therefore, directly
Connecing the condition responsive statistical nature that probabilistic method obtains has certain engineering application value.
4. multiple dimensioned alternative manner and conventional method comparative analysis
Non-linear system status can be improved to avoid three stochastic differential equations are solved simultaneously using multiple dimensioned alternative manner
The solution efficiency for responding statistical nature, to provide efficient calculating for large-scale nonlinear Random Structural Systems state analysis
Method promotes the engineer application of stochastic structure analysis method.
In order to illustrate the computational efficiency of multiple dimensioned alternative manner, the present embodiment considers the pylon structure of five kinds of different scales
Finite element model.The finite element unit quantity of five kinds of models increases to 920 from 488, and corresponding degree of freedom quantity increases from 1500
To 2844.In the multiple dimensioned alternative manner of the present embodiment, thick scale time value is △ tcoarse=0.05s, thin scale time
Value is △ tfine=0.0125s.Table 1 list different scales problem multiple dimensioned alternative manner and conventional method solution when
Between and two methods take the ratio between.From table it can be seen that, with the increase of Solve problems scale, when the calculating of conventional method
Between from 510.544s to 633.114s, and the calculating time of multiple dimensioned alternative manner is from 409.021s to 487.245s.Compared to biography
System method, multiple dimensioned alternative manner take the less calculating time.In addition, the multiple dimensioned alternative manner in table 1 and tradition side
Method take the ratio between about between 76% to 80%.It can be seen that multiple dimensioned alternative manner saves a large amount of calculating time,
The time of especially large-scale engineering problem, saving is very significant.According to the data in table 1, Fig. 5 gives multiple dimensioned change
For the curve graph for solving time comparison of method and conventional method.
Table 1
For Transmission tower, the closing moment function side of condition responsive has been solved using the multiple dimensioned alternative manner of the present invention
Journey obtains the statistical nature of condition responsive.This method avoid three stochastic differential equations are solved simultaneously, realizes and solved
The rapidly and efficiently calculating of journey.Above example demonstrates the method for the present invention Efficient Solution large-scale nonlinear Random Structural Systems state
The feasibility and superiority of response.
The specific steps that the above is only the present invention, are not limited in any way protection scope of the present invention.
Part of that present invention that are not described in detail belong to the well-known technology of those skilled in the art.
Claims (7)
1. a kind of multiple dimensioned alternative manner of Efficient Solution large-scale nonlinear Random Structural Systems state, it is characterised in that：It should
Method is directed to Transmission Tower, and this method realizes that steps are as follows：
Step 1: being based on nonlinear random structures system state equation, non-linear system status is established by exact propability and is rung
The moment function equation answered, including：The mean value moment function equation of condition responsive, the mean-square value moment function equation of condition responsive and
Condition responsive-excitation Correlation Moment functional equation；
Step 2: being blocked to the non-close item in condition responsive moment function equation using Gauss Truncation Technique, state is established
The closing moment function equation of response, including：The mean value moment function equation of closed condition responsive, closed condition responsive
The Correlation Moment functional equation of mean-square value moment function equation and closed condition responsive-excitation；
Step 3: by section of entire analysis time it is discrete be multiple thick scale time intervals, then in each thick scale time interval
It is interior to carry out thin scale time discrete, finally establish the multiple dimensioned discrete segment in time domain；
Step 4: in thick scale time interval [ti,ti+1] on, it solves closed state simultaneously using Runge-Kutta methods and rings
The mean-square value moment function equation of the mean value moment function equation and closed condition responsive answered, obtains condition responsive mean valueWith
Condition responsive mean-square value
Step 5: judging condition responsive mean valueWith condition responsive mean-square valueConvergence, if result restrain,
Jump to step 7；If result does not restrain, step 6 is continued to execute；
Step 6: the condition responsive mean value that will be obtained in step 4With condition responsive mean-square valueSubstitute into closed shape
The Correlation Moment functional equation of state response-excitation calculates condition responsive-excitation correlation moment functionK=k+1 is enabled, is redirected
To step 4；
Step 7: judging whether time conditions meet t≤T, if meeting condition, i=i+1 and k=1 is enabled, step 4 is jumped to；
If being unsatisfactory for condition, output state responds mean valueCondition responsive mean-square valueIt is related to condition responsive-excitation
Moment functionIteration ends；Application process is：
(1) structural parameters and model introduction
Transmission tower includes tubing, and section is divided into two kinds, is main column on four edges, and rectangular section is 120 × 120
× 4mm, remaining tube section are 50 × 5mm of Φ, and the elasticity modulus of material is 2.06 × 105MPa, Poisson's ratio 0.3, pylon exists
Bottom is fixed at pedestal everywhere, and cantilever end bears downward concentrfated load effect F (t)=- psin (4 π t) at 6, and wherein p is
Random parameter, mean value and standard deviation are respectively μp=1960N and σp=100, it is assumed that the rigidity of structure is nonlinear, structure
The nonlinear Control differential equation of system is expressed as：
In formula, M and K are respectively linear quality matrix and linear stiffness matrix, and F (t) encourages for random external；
(2) multiple dimensioned iterative program and running environment
According to multiple dimensioned alternative manner, in conjunction with existing certainty program of finite element, devised using Matlab is had at random
Finite element analysis program, in this procedure, the geometrical model and FEM mesh of structure are by certainty program of finite element
It generating, the linear quality matrix and linear stiffness matrix of Transmission tower are established using MSC/Nastran finite element softwares, and
Using the DMAP extractions in Nastran；
(3) statistical nature of system mode response
Based on nonlinear random structures system state equation, can be obtained under the effect of external arbitrary excitation using exact propability
Structured system condition response statistical nature；It is equal to obtain the time history of vertical speed mean value and vertical displacement at cantilever end 3
The time history of value.
2. a kind of multiple dimensioned iteration of Efficient Solution large-scale nonlinear Random Structural Systems state according to claim 1
Method, it is characterised in that：The moment function equation of non-linear system status response is established in step 1 using exact propability, wherein
Including：
The mean value moment function equation of condition responsive：
In formula, A1, A3For sytem matrix, B is input matrix, mx(t) it is the mean function of condition responsive, mu(t) it is random external
The mean function of excitation,For the joint moment function of condition responsive, symbolFor two vectorial Hadamard products,
Indicate that each corresponding element of two vectors is multiplied, result of product remains as the identical vector of dimension；
The mean-square value moment function equation of condition responsive：
In formula, Rxx(t, t) is the mean-square value function of condition responsive,For the Correlation Moment of condition responsive-excitation
Function, remaining meaning of parameters are same as above；
Condition responsive-excitation Correlation Moment functional equation：
In formula, Rux(t, t) is the related moment function of condition responsive-excitation, Ruu(t, t) is the mean-square value letter of random external excitation
Number, remaining meaning of parameters are same as above.
3. a kind of multiple dimensioned iteration of Efficient Solution large-scale nonlinear Random Structural Systems state according to claim 2
Method, it is characterised in that：The closing moment function equation for establishing condition responsive in step 2 using Gauss Truncation Technique, wherein wrapping
It includes：
The mean value moment function equation of closed condition responsive：
In formula, parameters meaning is same as above；
The mean-square value moment function equation of closed condition responsive：
In formula, parameters meaning is same as above；
The Correlation Moment functional equation of closed condition responsive-excitation：
In formula, parameters meaning is same as above.
4. a kind of multiple dimensioned iteration of Efficient Solution large-scale nonlinear Random Structural Systems state according to claim 3
Method, it is characterised in that：The multiple dimensioned discrete segment in time domain is established using discrete way at equal intervals in step 3, it first, will
Entire analysis time section [t0, T] and discrete for multiple thick scale time interval [ti,ti+1], it is embodied as：
Dcoarse={ ti:t0=t0＜ t1＜ ... ＜ tI=T }
In formula, t0For initial time, tI=T is end time, ti, i=0,1 ..., I is the discrete time of thick scale time interval
Point, I are thick scale time interval number；
Secondly, in each thick scale time interval [ti,ti+1] in carry out thin scale time discrete, obtain thin scale time interval
[ti,j,ti,j+1], it is embodied as：
In formula, ti=ti,0For the initial time in the thick scale time interval of i+1,For the i+1 thick scale time
End time in section, ti,j, j=0,1 ..., JiFor the discrete time point of thin scale time interval, JiFor the thick ruler of i+1
Spend the thin scale time interval number in time interval.
5. a kind of multiple dimensioned iteration of Efficient Solution large-scale nonlinear Random Structural Systems state according to claim 4
Method, it is characterised in that：The time step of Runge-Kutta methods is thin scale time interval [t in step 4i,j,ti,j+1],
Response-excitation correlation moment function value in the mean-square value moment function equation of condition responsive isIt is embodied as：
In formula, parameters meaning is same as above.
6. a kind of multiple dimensioned iteration of Efficient Solution large-scale nonlinear Random Structural Systems state according to claim 5
Method, it is characterised in that：The constringent condition of judging result includes that condition responsive convergence in the mean condition and state are rung in step 5
The mean-square value condition of convergence is answered, is embodied as：
In formula, ε1For condition responsive mean value allowable error, ε2For condition responsive mean-square value allowable error.
7. a kind of multiple dimensioned iteration of Efficient Solution large-scale nonlinear Random Structural Systems state according to claim 6
Method, it is characterised in that：With thin scale time interval [t in step 6i,j,ti,j+1] it is time step, using Runge-Kutta
Method individually solves the Correlation Moment functional equation of closed condition responsive-excitation, wherein condition responsive mean value value isCondition responsive mean-square value value isIt is embodied as：
In formula, parameters meaning is same as above.
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