CN107609221A - It is a kind of that hinged structure nonlinear parameter recognition methods is contained based on genetic algorithm - Google Patents
It is a kind of that hinged structure nonlinear parameter recognition methods is contained based on genetic algorithm Download PDFInfo
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Abstract
Hinged structure nonlinear parameter recognition methods is contained based on genetic algorithm the invention discloses a kind of, including describing the nonlinear characteristic containing hinge in hinged structure with a kind of nonlinear model, the kinetic model of hinge arrangement is established, obtains M, K, C equation containing hinged structure；By M, K, C containing hinged structure it is equations turned be nonlinear Algebraic Equation set；Obtained Algebraic Equation set is solved using the Newton iterative methods in numerical analysis, obtains the frequency response data containing hinged structure；Object function is built according to the frequency response data of solution and the frequency response data that experiment obtains, globalization search is carried out by the GA genetic algorithms in Matlab, finally identifies optimal nonlinear parameter value.The recognition methods containing hinged structure nonlinear parameter of the present invention, can identify the nonlinear characteristic containing hinged structure, overcoming conventional method needs offer experience initial value and can only ensure the limitation of local optimum.
Description
Technical field
It is non-in hinged structure more particularly to a kind of containing for genetic algorithm the present invention relates to the recognition methods of the structural parameters containing hinge
Linear dimensions recognition methods.
Background technology
With developing rapidly for aerospace industry, deployable structure is widely applied in space tasks, such as solar sail, too
Positive energy cell array and space antenna supporting mechanism etc..
Connected between each part of deployable structure by hinge, but the construction of hinge is complicated, due to the presence in gap, outside
In the presence of boundary's load, the non-linear phenomena such as it can contact with each other, rub and collide between part, make the dynamics of structure special
Property has stronger nonlinear characteristic.Therefore, influence of the hinge to structural dynamic characteristics is very important, it is necessary to power to hinge
Characteristic spread research is learned, reliable evaluation and reference are provided for design.
In order to study the nonlinear characteristic containing hinged structure, it is necessary to obtain accurate nonlinear parameter value.It is many in engineering
Non-linear factor can not directly obtain the non-linear ginseng, it is necessary to according to the anti-selection structure of test data with theoretical or experiment method
Numerical value.The Reverse Problem of dynamic system --- parameter identification is the key link of modern scientist and optimization design, therefore this year
Carry out the extensive concern by domestic and foreign scholars, and propose various parameters recognition methods in succession, but these methods exist mostly
The experience initial value of nonlinear parameter is provided, and is easily absorbed in the trap of local minimum and " endless loop " phenomenon occurs so that iteration
It can not carry out, and the limitations such as parameter value local optimum can only be ensured.
The content of the invention
Goal of the invention：There is provided it is a kind of can effectively identify the nonlinear parameter containing hinge in hinged structure, overcome conventional method
The existing limitation for needing to provide experience initial value and local optimum can only be ensured based on genetic algorithm containing hinged structure it is non-thread
Property parameter identification method.
Technical scheme：It is a kind of that hinged structure nonlinear parameter recognition methods is contained based on genetic algorithm, comprise the following steps：
(1) nonlinear characteristic containing hinge in hinged structure is described with a kind of nonlinear model, establishes the dynamic of hinge arrangement
Mechanical model, obtain M, K, C equation containing hinged structure；
(2) by harmonic wave equilibrium method to non-linear containing the displacement of each free degree, speed, acceleration and hinge in hinged structure
Restoring force carry out first harmonic expansion, by M, K, C containing hinged structure it is equations turned be nonlinear Algebraic Equation set；
(3) Algebraic Equation set obtained in step (2) is solved using the Newton iterative methods in numerical analysis, obtained
To the frequency response data containing hinged structure；
(4) object function is built according to the frequency response data solved in step (3) and the frequency response data that experiment obtains, passed through
GA genetic algorithms in Matlab carry out globalization search, finally identify optimal nonlinear parameter value.
Further, in step (1), comprise the steps of：
(11) suitable nonlinear model is selected to describe the nonlinear characteristic containing hinge in hinged structure according to actual conditions,
Wherein, according to hinge nonlinear restoring force and hinge member relative displacement xjRelation, with backlash nonlinearity model, cube non-thread
Property model or piecewise linear model describe the nonlinear characteristic of hinge, wherein relative displacement xjRefer to the hinge both sides free degree
Displacement difference；
(12) to carrying out Dynamic Modeling containing hinged structure, its matrix form kinetics equation is obtained：
Wherein, M is the mass matrix containing hinged structure, and C is the damping matrix containing hinged structure, and K is the rigidity square containing hinged structure
Battle array, q is each free degree motion vector containing hinged structure,For q first derivative, i.e., each free degree velocity vector containing hinged structure,For q second dervative, i.e., each free degree vector acceleration containing hinged structure, FNLThe Nonlinear Vector introduced for hinge, F is sharp
Encourage broad sense force vector.
Further, step (11) intermediate gap nonlinear model, due to gap be present in hinge, not yet occurring
Rigidity is not present during contact, nonlinear restoring force and displacement show as free movement characteristic, then the nonlinear restoring force table of hinge
It is up to formula：
Wherein, f is nonlinear restoring force, xjFor relative displacement, δ is the gap of hinge, and k is that the linear contact of hinge is firm
Degree.
Further, cube nonlinear model in the step (11), hinge is non-linear as caused by nonlinear contact
Restoring force has odd function characteristic, and has hard spring characteristic, therefore the power displacement of hinge is described using Nonlinear Cubic model
Relation, then expression formula be：
Wherein, FNLFor nonlinear restoring force, xjFor relative displacement, kcsFor stiffness coefficient.
Further, piecewise linear model in the step (11), due to gap in hinge be present, the contact between component
The change of rigidity can be caused, can be described with piecewise linear model, then expression formula is：
Wherein, f is nonlinear restoring force, and ε is variation rigidity threshold values, K1And K2To be segmented contact stiffness, when displacement is less than or equal to
During ε, contact stiffness K1, when displacement is more than ε, contact stiffness K2。
Further, the step (2) includes：
(21) theoretical based on harmonic balance, the nonlinear restoring force of response and hinge to system carries out first harmonic point
Solution, the nonlinear restoring force of displacement corresponding to the n frees degree, speed, acceleration and hinge is expressed as：
qn=an sinωt+bn cosωt
Wherein, anAnd bnSinusoidal coefficients and cosine coefficient, j corresponding to each free degree respectively after harmonic expansionpAnd jqRespectively
For the sinusoidal coefficients and cosine coefficient after hinge nonlinear restoring force harmonic expansion, f is hinge nonlinear restoring force, qn、WithDisplacement, speed, acceleration corresponding to the n frees degree are represented respectively, and ω represents frequency.
(22) bring the formula in step (21) kinetics equation of step (12) into, and assume that external drive power is Fsin
ω t, it is equal according to equation both sides sinusoidal coefficients and cosine coefficient difference, the kinetics equation in step (12) is converted into algebraically
Equation group：
Wherein, CpFor matrix, C corresponding to the sinusoidal coefficients after hinge nonlinear restoring force harmonic expansionqIt is non-linear for hinge
Matrix corresponding to cosine coefficient after restoring force harmonic expansion, it is respectively n to make the hinged free degree1And n2, have：
Wherein, A is sinusoidal coefficients vector corresponding to each free degree, and B is cosine coefficient vector corresponding to each free degree, is had：
A=[a1 a2 … an], B=[b1 b2 … bn]
Further, the step (3) includes：
(31) Algebraic Equation set obtained in step (22) is subjected to an equation conversion, obtains following formula：
(32) equation in step (31) can be considered as f (x)=0, whereinFor as corresponding to each free degree just
The column vector of m-cosine composition, it is possible thereby to which the equation in step (31) is configured into Newton Iterations：
Wherein, it represents iterations, and operation efficiency is integrated during iterative and computational accuracy selects suitable iteration time
Number, i.e. it values.
Further, the step (4) includes：
(41) test value and the difference of harmonic wave equilibrium method calculated value for the dynamic respond that residual error item is structure are defined：
R (p)=xe(ω)-xa(ω,p)
Wherein, p represents hinge nonlinear parameter to be identified, xeAnd xaTest value and the calculating of dynamic respond are represented respectively
Value, in experiment, the analytical expression of amplitude frequency curve can not be obtained, but the discrete data point represented with resolution ratio, it is assumed that
The Frequency point that experiment is chosen is ω1,ω2,...,ωn, and according to least square method principle, the frequency response data to calculate gained
Can best fit result of the test, objective function is：
(42) GA genetic algorithms are utilized, global search are carried out to nonlinear parameter, until identifying optimal non-linear ginseng
Numerical value paSo that object function R (p) minimalizations, then paFor the exact value of nonlinear parameter.
Beneficial effect：Compared with prior art, the recognition methods of the invention containing hinged structure nonlinear parameter, can be accurate
Nonlinear parameter value of the identification containing hinged structure, because genetic algorithm has the advantage of global optimizing, overcome conventional method need
Experience initial value is provided and the limitation of local optimum can only be ensured.
Brief description of the drawings
Fig. 1 is to contain hinged structure nonlinear parameter identification process figure based on genetic algorithm；
Fig. 2 is backlash nonlinearity model schematic；
Fig. 3 is Nonlinear Cubic model schematic；
Fig. 4 is piecewise linear model schematic diagram；
Fig. 5 is hinge girder construction kinetic model schematic diagram；
Fig. 6 is nonlinear parameter value n iteration situation schematic diagrams；
Fig. 7 is the model and test structure amplitude frequency curve figure after identification.
Embodiment
Technical solution of the present invention is described in detail below in conjunction with the accompanying drawings.
As shown in figure 1, a kind of of the present invention contains hinged structure nonlinear parameter recognition methods, the present invention based on genetic algorithm
Illustrated exemplified by cutting with scissors girder construction, wherein, beam section is circle, and diameter D is 0.02m, every beam length leFor 1m, total length
L is 2m, and material is aluminium, elastic modulus E 70Gpa, density p 2700kg/m3, it is proportional damping to make the damping in girder construction,
Mass matrix damped coefficient CMFor 0.0001, stiffness matrix damped coefficient CKFor 0.0001, generalized displacement corresponding to hinge is rotates
The free degree, the influence of different nonlinear parameters and amplitude of exciting force to structure amplitude versus frequency characte is studied for convenience, makes hinge stiffness
Coefficient kcs=nEI/le, exciting force amplitude Fc=0.5EI/L2。
This method comprises the following steps：
Step 1：It is assumed that a kind of nonlinear model describes the nonlinear characteristic containing hinge in hinged structure, hinge arrangement is established
Kinetic model, and finally give M, K, C equation containing hinged structure.Wherein it is determined that hinge girder construction kinetics equation include with
Lower step：
Step 1.1：Select suitable nonlinear model according to actual conditions describe containing in hinged structure hinge it is non-linear
Characteristic, usually, according to hinge nonlinear restoring force and hinge member relative displacement xjRelation, can have three kinds of nonlinear models
Type describes the nonlinear characteristic of hinge；Backlash nonlinearity model, Nonlinear Cubic model or piecewise linear model, wherein relatively
Displacement xjRefer to the displacement difference of the hinge both sides free degree.
(1) backlash nonlinearity model：Due to gap be present in hinge, rigidity is not present when being not yet in contact, it is non-thread
Property restoring force and displacement show as free movement characteristic, as shown in Fig. 2 then the nonlinear restoring force expression formula of hinge is：
Wherein, f is nonlinear restoring force, xjFor relative displacement, δ is the gap of hinge, and k is that the linear contact of hinge is firm
Degree.
(2) Nonlinear Cubic model：The hinge nonlinear restoring force as caused by nonlinear contact has odd function characteristic, and
There is hard spring characteristic, therefore the power displacement relation of hinge is described using Nonlinear Cubic model, as shown in figure 3, then expression formula
For：
Wherein, f is nonlinear restoring force, xjFor relative displacement, kcsFor stiffness coefficient.
(3) piecewise linear model：Due to gap in hinge be present, the contact between component can cause the change of rigidity, can be with
Described with piecewise linear model, as shown in figure 4, then expression formula is：
Wherein, f is nonlinear restoring force, and ε is variation rigidity threshold values, K1And K2To be segmented contact stiffness, when displacement is less than or equal to
During ε, contact stiffness K1, when displacement is more than ε, contact stiffness K2。
The hinge nonlinear model that the present embodiment uses is Nonlinear Cubic model.
Step 1.2：Using Fig. 5 hinge girder construction as object, to carrying out Dynamic Modeling containing hinged structure, its matrix form is obtained
Dynamics equations be：
Wherein, M is the mass matrix containing hinged structure, and C is the damping matrix containing hinged structure, and K is the rigidity square containing hinged structure
Battle array, q is each free degree motion vector containing hinged structure,For q first derivative, i.e., each free degree velocity vector containing hinged structure,For q second dervative, i.e., each free degree vector acceleration containing hinged structure, FNLThe Nonlinear Vector introduced for hinge, i.e. by
The Nonlinear Vector that hinge nonlinear restoring force f is introduced, its expression formula are：F is excitation broad sense force vector,
What is represented is the column vector formed by acting on the exciting force amplitude above each free degree.
The element stiffness matrix and element mass matrix of beam element be：
Wherein EI be beam bending rigidity, leFor element length, ρ is the density of material, and rigidity assembling is carried out to it, can be with
Obtain Bulk stiffness matrix and total quality matrix is：
Damping matrix can be obtained by formula below：
C=CM×M+CK×K (9)
As shown in fig. 5, it is assumed that exciting force active position is 6DOF, the frequency domain at 6DOF is solved with the above method
Respond, corresponding corner should be 4 in the non-linear Moment-rotation Relationship of hinge, and the rotation displacement of 5DOF is poor, i.e. displacement amplitudeThe frequency response data of hypothesis test are amplitude of exciting force 0.5Fc, hinge stiffness 0.1kcsFeelings
Obtained under condition.
Step 2：By harmonic wave equilibrium method to non-containing the displacement of each free degree, speed, acceleration and hinge in hinged structure
Linear restoring power carry out first harmonic expansion, by M, K, C containing hinged structure it is equations turned be nonlinear Algebraic Equation set.Its
In, harmonic wave equilibrium method solves kinetics equation and comprised the steps of：
Step 2.1 is based on harmonic balance theory, and the nonlinear restoring force of response and hinge to system carries out first harmonic
Decompose, what the response of system referred to is containing the displacement of each free degree, speed and acceleration in hinged structure.By corresponding to the n frees degree
Displacement, speed, the nonlinear restoring force of acceleration and hinge are expressed as：
Wherein, anAnd bnSinusoidal coefficients and cosine coefficient, j corresponding to each free degree respectively after harmonic expansionpAnd jqRespectively
For the sinusoidal coefficients and cosine coefficient after hinge nonlinear restoring force harmonic expansion, f is hinge nonlinear restoring force, qn、WithDisplacement, speed, acceleration corresponding to the n frees degree are represented respectively, and ω represents frequency.
The first harmonic expansion coefficient of hinge Nonlinear Cubic model is：
Step 2.2：Bring formula (12) kinetics equation of formula (11) into, and assume that external drive power is Fsin ω t, bring into
Formula (4) is equal according to equation both sides sinusoidal coefficients and cosine coefficient difference, and formula (4) kinetics equation is converted into algebraic equation
Group：
Wherein, CpFor matrix, C corresponding to the sinusoidal coefficients after hinge nonlinear restoring force harmonic expansionqIt is non-linear for hinge
Matrix corresponding to cosine coefficient after restoring force harmonic expansion, it is respectively n to make the hinged free degree1And n2, have：
Wherein, A is sinusoidal coefficients vector corresponding to each free degree, and B is cosine coefficient vector corresponding to each free degree, is had：
A=[a1 a2 … an], B=[b1 b2 … bn] (16)
So far, dynamics equations group is converted for Groebner Basis.
Step 3：The Algebraic Equation set obtained in step (2) is asked using the Newton iterative methods in numerical analysis
Solution, obtains the frequency response data containing hinged structure.Newton solution by iterative method Groebner Basis comprises the steps of：
Step 3.1：The Algebraic Equation set obtained in step (22) is subjected to an equation conversion, obtains following formula：
Above formula is configured to Newton Iterations and utilizes Newton solutions by iterative method, finally tries to achieve the frequency response number of structure
According to.
Step 3.2：Equation in formula (17) can be considered as f (x)=0, whereinFor as corresponding to each free degree
The column vector of sine and cosine vector composition, it is possible thereby to which the equation of formula (17) is configured into Newton Iterations：
Wherein, it represents iterations, and operation efficiency is integrated during iterative and computational accuracy selects suitable iteration time
Number, i.e. it values.
Step 4：The comparison result of the frequency response data obtained according to the frequency response data of Numerical Methods Solve with experiment, passes through
GA genetic algorithms in Matlab carry out globalization search, finally identify optimal nonlinear parameter value.
GA genetic algorithms carry out global optimum's search and comprised the steps of：
Step 4.1：In order to identify the nonlinear parameter value containing hinged structure, that is, cut with scissors beam rigidity value kcs=nEI/leIn n, this
The nonlinear parameter n at place is specific nonlinear parameter value in the present embodiment, defines fitness function, i.e. object function, has：
Step 4.2：Based on GA genetic algorithms, the individual using nonlinear parameter n as GA is fixed in order to reduce search time
Adopted individual search bound [0.05,0.15], global search minimum fitness value, individual now be identify it is non-linear
Parameter value.Fig. 6 and Fig. 7 is respectively the model and test structure amplitude frequency curve after nonlinear parameter value iterativecurve figure and identification,
It will be appreciated from fig. 6 that ordinate is " nonlinear parameter value ", abscissa is " algebraically ", iterations is represented, because genetic algorithm is mould
The optimized algorithm that imitative biological heredity obtains, therefore be accustomed to iterations " algebraically ", similar to parent inside biology and son
Generation.It can be seen that after the oscillation search in 520 generations, nonlinear parameter value gradually tends to a stationary value, that is, the hinge after identifying
Chain nonlinear parameter value, it is 0.1kcs, while the result after identification is contrasted with result of the test, as shown in fig. 7, can see
It is coincide to two results fine, illustrates the accuracy of the parameter value after identification.
It can be seen that disclosed by the invention can effectively identify containing hinge in hinged structure containing hinged structure nonlinear parameter recognition methods
Nonlinear parameter value.
Claims (8)
1. a kind of contain hinged structure nonlinear parameter recognition methods based on genetic algorithm, it is characterised in that comprises the following steps：
(1) nonlinear characteristic containing hinge in hinged structure is described with a kind of nonlinear model, establishes the dynamics of hinge arrangement
Model, obtain M, K, C equation containing hinged structure；
(2) by harmonic wave equilibrium method to containing the displacement of each free degree, speed, acceleration and the non-linear recovery of hinge in hinged structure
Power carry out first harmonic expansion, by M, K, C containing hinged structure it is equations turned be nonlinear Algebraic Equation set；
(3) Algebraic Equation set obtained in step (2) is solved using the Newton iterative methods in numerical analysis, contained
The frequency response data of hinged structure；
(4) object function is built according to the frequency response data solved in step (3) and the frequency response data that experiment obtains, passes through Matlab
In GA genetic algorithms carry out globalization search, finally identify optimal nonlinear parameter value.
A kind of hinged structure nonlinear parameter recognition methods, its feature are contained based on genetic algorithm 2. according to claim 1
It is：In step (1), comprise the steps of：
(11) suitable nonlinear model is selected to describe the nonlinear characteristic containing hinge in hinged structure according to actual conditions, its
In, according to hinge nonlinear restoring force and hinge member relative displacement xjRelation, with backlash nonlinearity model, Nonlinear Cubic
Model or piecewise linear model describe the nonlinear characteristic of hinge, wherein relative displacement xjRefer to the hinge both sides free degree
Displacement difference；
(12) to carrying out Dynamic Modeling containing hinged structure, its matrix form kinetics equation is obtained：
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Wherein, M is the mass matrix containing hinged structure, and C is the damping matrix containing hinged structure, and K is the stiffness matrix containing hinged structure, q
For each free degree motion vector containing hinged structure,For q first derivative, i.e., each free degree velocity vector containing hinged structure,For
Q second dervative, i.e., each free degree vector acceleration containing hinged structure, FNLThe Nonlinear Vector introduced for hinge, F are wide for excitation
Adopted force vector.
3. according to it is according to claim 2 it is a kind of hinged structure nonlinear parameter recognition methods is contained based on genetic algorithm, its
It is characterised by, step (11) intermediate gap nonlinear model, due to gap be present in hinge, is not deposited when being not yet in contact
In rigidity, nonlinear restoring force and displacement show as free movement characteristic, then the nonlinear restoring force expression formula of hinge is：
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Wherein, f is nonlinear restoring force, xjFor relative displacement, δ is the gap of hinge, and k is the linear contact rigidity of hinge.
A kind of hinged structure nonlinear parameter recognition methods, its feature are contained based on genetic algorithm 4. according to claim 2
It is, cube nonlinear model in the step (11), hinge nonlinear restoring force has strange letter as caused by nonlinear contact
Characteristic is counted, and has hard spring characteristic, therefore the power displacement relation of hinge is described using Nonlinear Cubic model, then expression formula
For：
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Wherein, f is nonlinear restoring force, xjFor relative displacement, kcsFor stiffness coefficient.
A kind of hinged structure nonlinear parameter recognition methods, its feature are contained based on genetic algorithm 5. according to claim 2
It is, piecewise linear model in the step (11), due to gap in hinge be present, the contact between component can cause the change of rigidity
Change, can be described with piecewise linear model, then expression formula is：
Wherein, f is nonlinear restoring force, and ε is variation rigidity threshold values, K1And K2To be segmented contact stiffness, when displacement is less than or equal to ε,
Contact stiffness is K1, when displacement is more than ε, contact stiffness K2。
A kind of hinged structure nonlinear parameter recognition methods, its feature are contained based on genetic algorithm 6. according to claim 1
It is, the step (2) includes：
(21) theoretical based on harmonic balance, the nonlinear restoring force of response and hinge to system carries out first harmonic decomposition, by n
Displacement corresponding to the free degree, speed, the nonlinear restoring force of acceleration and hinge are expressed as：
qn=an sinωt+bn cosωt
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Wherein, anAnd bnSinusoidal coefficients and cosine coefficient, j corresponding to each free degree respectively after harmonic expansionpAnd jqRespectively cut with scissors
Sinusoidal coefficients and cosine coefficient after chain nonlinear restoring force harmonic expansion, f are hinge nonlinear restoring force, qn、WithPoint
Not Biao Shi displacement, speed, acceleration corresponding to the n frees degree, ω represent frequency.
(22) bring the formula in step (21) kinetics equation of step (12) into, and assume that external drive power is Fsin ω t,
It is equal according to equation both sides sinusoidal coefficients and cosine coefficient difference, the kinetics equation in step (12) is converted into algebraic equation
Group：
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<mrow>
<mo>-</mo>
<mi>C</mi>
<mi>&omega;</mi>
<mo>-</mo>
<msub>
<mi>C</mi>
<mi>q</mi>
</msub>
<mi>&omega;</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>C</mi>
<mi>&omega;</mi>
<mo>+</mo>
<msub>
<mi>C</mi>
<mi>q</mi>
</msub>
<mi>&omega;</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>K</mi>
<mo>-</mo>
<msup>
<mi>M&omega;</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msub>
<mi>C</mi>
<mi>p</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>A</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>B</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>F</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
Wherein, CpFor matrix, C corresponding to the sinusoidal coefficients after hinge nonlinear restoring force harmonic expansionqFor the non-linear recovery of hinge
Matrix corresponding to cosine coefficient after power harmonic expansion, it is respectively n to make the hinged free degree1And n2, have：
Wherein, A is sinusoidal coefficients vector corresponding to each free degree, and B is cosine coefficient vector corresponding to each free degree, is had：
A=[a1 a2 … an], B=[b1 b2 … bn]
7. according to claim 1 contain hinged structure nonlinear parameter recognition methods, it is characterised in that step (3) bag
Include：
(31) Algebraic Equation set obtained in step (22) is subjected to an equation conversion, obtains following formula：
<mrow>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mrow>
<mi>K</mi>
<mo>-</mo>
<msup>
<mi>M&omega;</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msub>
<mi>C</mi>
<mi>p</mi>
</msub>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>-</mo>
<mi>C</mi>
<mi>&omega;</mi>
<mo>-</mo>
<msub>
<mi>C</mi>
<mi>q</mi>
</msub>
<mi>&omega;</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>C</mi>
<mi>&omega;</mi>
<mo>+</mo>
<msub>
<mi>C</mi>
<mi>q</mi>
</msub>
<mi>&omega;</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>K</mi>
<mo>-</mo>
<msup>
<mi>M&omega;</mi>
<mn>2</mn>
</msup>
<mo>+</mo>
<msub>
<mi>C</mi>
<mi>p</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>A</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>B</mi>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mfenced open = "[" close = "]">
<mtable>
<mtr>
<mtd>
<mi>F</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mo>&lsqb;</mo>
<mn>0</mn>
<mo>&rsqb;</mo>
</mrow>
(32) equation in step (31) can be considered as f (x)=0, whereinFor the sine and cosine as corresponding to each free degree
The column vector of vector composition, it is possible thereby to which the equation in step (31) is configured into Newton Iterations：
<mrow>
<msub>
<mi>x</mi>
<mrow>
<mi>i</mi>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>i</mi>
<mi>t</mi>
</mrow>
</msub>
<mo>-</mo>
<mfrac>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>i</mi>
<mi>t</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mo>&prime;</mo>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>x</mi>
<mrow>
<mi>i</mi>
<mi>t</mi>
</mrow>
</msub>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mo>...</mo>
<mo>)</mo>
</mrow>
</mrow>
Wherein, it represents iterations, and operation efficiency is integrated during iterative and computational accuracy selects suitable iterations, i.e.,
It values.
A kind of hinged structure nonlinear parameter recognition methods, its feature are contained based on genetic algorithm 8. according to claim 1
It is, the step (4) includes：
(41) test value and the difference of harmonic wave equilibrium method calculated value for the dynamic respond that residual error item is structure are defined：
R (p)=xe(ω)-xa(ω,p)
Wherein, p represents hinge nonlinear parameter to be identified, xeAnd xaThe test value and calculated value of dynamic respond are represented respectively, are tried
In testing, the analytical expression of amplitude frequency curve can not be obtained, but the discrete data point represented with resolution ratio, it is assumed that experiment choosing
The Frequency point taken is ω1,ω2,...,ωn, and according to least square method principle, can be most to the frequency response data for calculating gained
Good fitting result of the test, objective function are：
<mrow>
<mi>R</mi>
<mrow>
<mo>(</mo>
<mi>p</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</munderover>
<msup>
<mrow>
<mo>(</mo>
<msup>
<mi>x</mi>
<mi>e</mi>
</msup>
<mo>(</mo>
<msub>
<mi>&omega;</mi>
<mi>i</mi>
</msub>
<mo>)</mo>
<mo>-</mo>
<msup>
<mi>x</mi>
<mi>a</mi>
</msup>
<mo>(</mo>
<mrow>
<msub>
<mi>&omega;</mi>
<mi>i</mi>
</msub>
<mo>,</mo>
<mi>p</mi>
</mrow>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
(42) GA genetic algorithms are utilized, global search are carried out to nonlinear parameter, until identifying optimal nonlinear parameter value
paSo that object function R (p) minimalizations, then paFor the exact value of nonlinear parameter.
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| WO2019033771A1 (en) * | 2017-08-15 | 2019-02-21 | 东南大学 | Genetic algorithm-based method for identifying nonlinear parameter of hinge structure |
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| CN107609221B (en) | 2018-07-17 |
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