CN107246944A - A kind of Structural Damage Identification theoretical based on statistical moment - Google Patents

A kind of Structural Damage Identification theoretical based on statistical moment Download PDF

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CN107246944A
CN107246944A CN201710223887.4A CN201710223887A CN107246944A CN 107246944 A CN107246944 A CN 107246944A CN 201710223887 A CN201710223887 A CN 201710223887A CN 107246944 A CN107246944 A CN 107246944A
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msub
mover
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CN107246944B (en
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阳洋
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Chongqing University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M5/00—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings
    • G01M5/0066—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings by exciting or detecting vibration or acceleration
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M5/00—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings
    • G01M5/0033—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings by determining damage, crack or wear

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Abstract

A kind of damnification recognition method theoretical based on statistical moment of present invention design.It includes：Structure occurs after damage, and structural dynamic feature can change, and include displacement, the change of acceleration of system；Further, it is described to displacement, the processing of acceleration, specifically include：Step one：The Fourth square M of theoretical displacementx,4(E), the rank statistical moment M of theoretical acceleration eighta,8(E) acquisition, acquisition process is as follows：For system with several degrees of freedom, it is considered to which viscous damping influences, its equation of motion is：Under ground seismic wave function, reinforced concrete frame structure enters after damage, can identify the microlesion that structure occurs using the present invention, and can preferably recognize the degree of injury of structure.

Description

A kind of Structural Damage Identification theoretical based on statistical moment
Technical field
A kind of Structural Damage Identification theoretical based on statistical moment of present invention design, especially proposes one kind and utilizes target Function constantly seeks the algorithm that optimal solution recognizes structure modulus of elasticity.
Background technology
In recent years, China and natural calamity takes place frequently all over the world, causes many buildings different degrees of damage occur, and With the corrosion of environment, the influence of the various factors such as material aging, the damage of structure is more serious, and ultimately results in the broken of structure It is bad.Therefore, carry out Study on Structural Damage Identification to be particularly important.
At present in terms of monitoring structural health conditions, damaging techniques always are the problem of a puzzlement engineers, and it is direct Have influence on the development of monitoring structural health conditions.Current damage identification technique is not also perfect, and some identification technologies can not be known well The damage of other structure, also can not evaluation structure operation state, to be walked from practicality also very long stretch.Based on vibration letter Number Damage Assessment Method technology receive the accreditation of domestic and international project teacher, with extensive scientific research prospect, it is related to a lot Knowledge theory, including theory of structural dynamics, random vibration theory, Genetic Algorithms Theory etc., therefore Damage Assessment Method system Need powerful theoretical foundation.
Judge structure whether damage be structural health detection basis, determine after structural damage, next be exactly find damage Hinder the position occurred, this is one of important research problem in damage identification technique, be also the difficult point of research.Know specific damage The degree of injury of position, then evaluation structure, is structural strengthening, is repaired and using there is provided foundation.Damage to structure at present Recognition methods is broadly divided into model-free non-destructive tests and has MODEL DAMAGE identification.Model-free non-destructive tests mainly have frequency shift Method, the vibration shape change method, flexibility matrix change method etc., and the advantage of these methods is need not to carry out finite element modeling to structure, its Non-destructive tests result is not also influenceed by FEM model precision；Can only typically judge damage whether occur and damage position really It is fixed, it is impossible to accurate judgement is made to degree of injury.And there is MODEL DAMAGE identification mainly to have the structural damage based on dynamic residual vector Recognition methods, feature are to Ling Dufa etc., and based on there is MODEL DAMAGE identification to distinguish, damage occurs, determines damage position and damage Degree, with preferable precision and good engineering practical value.
In having MODEL DAMAGE identification, Zhang et al. proposes the Structural Damage Identification based on statistic method, this Text by contrasting acceleration and displacement statistical moment, is proposed a kind of based on the rank of acceleration eight and displacement Fourth square on this basis Damnification recognition method.
The content of the invention
It is an object of the present invention in view of the above-mentioned problems, by contrasting acceleration and displacement statistical moment, propose that one kind is based on The rank of acceleration eight and the damnification recognition method of displacement Fourth square, this method are calculated simply in time domain, and with higher Accuracy of identification.
To achieve the above object, the technical solution adopted by the present invention is：A kind of structural damage theoretical based on statistical moment is known Other method, including：
Structure occurs after damage, and structural dynamic feature can change, and include displacement, the change of acceleration of system；Enter One step, it is described to displacement, the processing of acceleration, specifically include：
Step one：The Fourth square M of theoretical displacementx,4(E), the rank statistical moment M of theoretical acceleration eighta,8(E) acquisition, is obtained Take process as follows：
For system with several degrees of freedom, it is considered to which viscous damping influences, its equation of motion is：
Wherein：M, C, K are respectively the mass matrix of structure, damping matrix and stiffness matrix,To accelerate Degree, speed and dynamic respond, f (t)=[f1(t),f2(t),...,fN(t)] it is external drive.Obtained by solving in the time domain The dynamic respond x of structurej=[x1j,x2j,...,xNs,j] and acceleration responsiveWherein xjFor j-th The displacement of measuring point,For the acceleration of j-th of measuring point, NsFor sampled point.It is understood that its statistical moment can be expressed as:
Step 2: obtaining the Fourth square M of theoretical displacementx,4(E), the rank statistical moment M of theoretical acceleration eighta,8(E) it is and real The Fourth square of displacementThe rank statistical moment of actual measureed value of acceleration eightDifference F between them1And F (E)2(E)。
And F (E)=[F1(E), F2(E)], its amount of variability | | F (E) | |2For residual error.
Step 3: solving | | F (E) | |2Minimum is reached, the E now obtained is the true estimate of structure of calculating.Obtain Process is as follows：
The computational methods are to utilize object function | | F (E) | |2=0 constantly seeks optimal solution identification structure modulus of elasticity, I.e.：
First order Taylor series expansion is pressed respectively to formula (1.6) and (1.7), and takes E0For initial value, obtain
If it is considered that external drive encourages for the load of zero-mean, then it can obtain：
σx,2Represent displacement second-order statisticses square, σa,2Acceleration second-order statisticses square is represented, can according to the relation between statistical moment To obtain, Mx,4=3 σx,2 2, Ma,8=105 σa,2 2。
The statistical moment can be obtained after seeking elastic modulus E local derviation：
Formula (1.12) is substituted into formula (1.8) respectively with formula (1.13) can obtain with (1.9)：
Here we takeThen:
E1=E0+ΔE (1.16)
By continuous iteration, until what is obtained | | F (E) | |2Minimum is reached, the E now obtained is the structure of calculating True estimate.
Other features and advantages of the present invention will be illustrated in the following description, also, partly be become from specification Obviously.
Below by embodiment, technical scheme is described in further detail.
Brief description of the drawings
Fig. 1 is the structural model of selected embodiment；
Fig. 2 is the non-destructive tests result based on displacement Fourth-order moment of selected embodiment；
Fig. 3 is for the present invention based on the theoretical Structural Damage Identification of statistical moment to selected embodiment non-destructive tests knot Really；
Fig. 4 is that the present invention is 40 to the signal to noise ratio of selected embodiment based on the theoretical Structural Damage Identification of statistical moment Non-destructive tests result；
Fig. 5 is that the present invention is 30 to the signal to noise ratio of selected embodiment based on the theoretical Structural Damage Identification of statistical moment Non-destructive tests result；
The lower Floor 12 displacement temporal response contrast of Fig. 6 white noises effect；
1 each layer of beam bending rigidity of Fig. 7 operating modes compares recognition result；
9 each layers of beam bending rigidity of Fig. 8 operating modes compare recognition result；
16 each layers of beam bending rigidity of Fig. 9 operating modes compare recognition result.
Embodiment
Based on the theoretical Structural Damage Identification of statistical moment to implement process as follows：
Step one：Structure is vibrated under noise incentive action, acceleration and displacement by sensor record structure Time-histories data.Using each layer summit acceleration and displacement temporal response is obtained, obtain it is each layer by layer between acceleration and displacement, pass through public affairs Formula (1.10), (1.11) obtain the acceleration and displacement statistical moment of structureThe statistical moment is estimation statistical moment；
Step 2：One initial stiffness matrix K (E of given structure0), by the mass matrix M of structure, damping matrix C, when Obtained in domain structure it is assumed that initial stiffness K (E0) under acceleration and displacement theory statistical moment Ma,8(E)、Mx,4(E), the system It is theoretical statistical moment to count square；
Step 3：By methods described herein, iteration is constantly updated, until meeting residual error | | F (E) | |2Untill minimum, so Output makes residual error afterwards | | F (E) | |2Stiffness matrix under minimal condition(under the conditions of structural damage), it is real close to structure Rigidity, referred to as estimates stiffness matrix；
4th step：Stiffness matrix of the structure under the conditions of lossless can be obtained according to design or known conditionsOr most open Beginning structure input white noise when lossless, repeat preceding step can obtain structure it is lossless under estimation stiffness matrix
5th step：Contrast the estimation rigidity value that third and fourth two step is obtainedAnd each floor of structure is damaged Identification, i.e. damage reason location and degree of injury estimation
Below by taking one 12 layers of framework as an example, the damnification recognition method of the present invention is done with reference to accompanying drawing and further retouched in detail State.
The 12 layer plane frame model structural models that the present embodiment is used are as shown in figure 1, wherein：Set a roof beam in place, the springform of post Amount is 3 × 1010N·m2, each layer height is 3m, and the span of Vierendeel girder is 6m, and damping ratio takes ξi=0.05 (i=1,2, 3...N), beam line density isThe line density of the left and right post of framework isThe model exists White Gaussian noise and do not consider to carry out numerical simulation under noise conditions, simulation operating mode is as follows：
Operating mode 1：Structure is not damaged；
Operating mode 2：1st layer of beam element damage 10% is set；
Operating mode 3：1st layer of beam element damage 20% is set；
Operating mode 4：The 1st layer of beam and the 2nd layer of beam is set to damage 20%；
Operating mode 5：The 1st layer of beam and the 3rd layer of beam is set to damage 20%.
1. the non-destructive tests simulation based on displacement Fourth square
Non-destructive tests are carried out using displacement Fourth square index first herein, recognition result is as shown in Figure 2.Abscissa table Show the different operating modes of simulation, ordinate represents the ratio between recognition result and theoretical value, from figure 2 it can be seen that in the situation of operating mode 2 Under, preferably, worst error is no more than 3% to non-destructive tests effect；In the case of operating mode 3, partial loss (i.e. first can only be identified Layer beam identification of damage 14%), error reaches 6%；In the case of operating mode 4, first layer beam can only identify 11% damage, error Reach 9%；In the case of operating mode 5, structure first layer and third layer beam equally can only also identify 8% and 12% damage, and And second layer beam position occurs in that damage erroneous judgement (identifying second layer beam damage 11%)；In terms of comprehensive, under time domain, pass through Structure is identified displacement Fourth-order moment, and recognition effect is undesirable, and occurs in that the operating mode of erroneous judgement, has much room for improvement.
On this basis, it is used as damage criterion pair set forth herein comprehensive displacement Fourth square and the rank statistical moment of acceleration eight It is as shown in Figure 3 that structure carries out identification rigidity and the ratio between theoretical stiffness under non-destructive tests, each operating mode：
From figure 3, it can be seen that the non-destructive tests based on displacement and acceleration statistical moment, are damaged at the list of operating mode 2~3 In the case of, recognition result has good precision, with assuming that operating mode fits like a glove, in the case of the panels with multiple site damage of operating mode 4~5, most Big error is also less than 1%.
2. consider influence of the different noise levels to structure
Consider the noise level of varying strength respectively under different operating modes, it recognizes rigidity and such as Fig. 4 of theoretical stiffness And Fig. 5.Figure 4, it is seen that identification influence of the noise on this method be not obvious, in the case of signal to noise ratio 40, to various The recognition effect of operating mode is all relatively good, can come out the non-destructive tests of each unit, and worst error occurs in operating mode 4, is no more than 3%；In fig. 5 it can be seen that in the case of signal to noise ratio 30, worst error occurs in operating mode 5, no more than 6%, illustrates the party Method has certain noise immunity.
3. Floor 12 model framework shaketalle test
Tongji University's civil engineering National Key Laboratory's shaketalle test room of taking precautions against natural calamities has carried out 12 floor standard in 2003 Frame vibration platform is tested, and test model is 12 layers of reinforced concrete frame structure of single span, using 1/10 ratio scaled model.Model Material is microconcrete and galvanized wire.Practical structures finishing and 50% mobile load are considered, onboard with mass counterweight.Standard On layer 19.7kg counterweights are arranged on every layer of arrangement 19.4kg counterweight, roofing layer.Four kinds of seismic wave has been selected in experiment：El Centro ripples, Kobe ripples, the artificial ripple in Shanghai, Shanghai basement rock ripple；Basic top surface, 2 layers, 4 layers, 6 layers, 8 layers, 10 layers, 12 layers Single Pin Vierendeel girders center is respectively arranged X to sensor (principal direction), this experiment comprising X to, X, Y be two-way and X, Y, Z three-dimensional The input of seismic wave, by increasing the size of seismic amplitude, completes the input of 62 secondary earthquake ripples, and each seismic input wave it Afterwards, observation structure has no problem and recorded.
The typical condition of this experimental observation result description is as follows：(seven are born under preceding 7 operating modes equivalent to prototype system Spend frequently occurred earthquake), do not find any crack.After the 9th artificial ripple in operating mode Shanghai (equivalent to prototype system with bearing seven degree Shake), occur trickle from top to bottom and from bottom to top developing first in the beam-ends of 4 layers of Vierendeel girder parallel to X direction of vibration Vertical fracture, slit width is less than 0.05mm.After the 16th operating mode, have parallel to the beam-ends of 4~6 layers of Vierendeel girder of X direction of vibration Crack is not observed in vertical fracture, slit width about 0.08mm, each post.
3.1 theoretical models and test model data comparison
X-direction (principal direction) areal model is set up using consistent Mass Matrix and consistent-mass matrix herein, analysis is obtained First three order frequency of theoretical model coincide substantially with practical frequency, by modeling white noise and earthquake wave excitation, can be with The displacement temporal response and measured displacements time-histories data for carrying out structure are contrasted, and Fig. 6 contrasts for white noise bottom offset time-histories data, and two Person's result is substantially suitable, shows the reasonability of simplified areal model.
3.2 have carried out 62 operating modes altogether based on the theoretical Structural Damage Identification verification experimental verification experiment of statistical moment, the It is once white noise vocal input, i.e. operating mode 1.
In operating mode 1, the displacement temporal response of correspondence floor sensors is obtained by surveying response analysis, foregoing reason is utilized Acquisition EI discre values, each layer beam recognition result are optimized by the beam to 12 each layers of layer plane framework, the moment of flexure rigidity of post component See Fig. 7.Abscissa represents floor number, and ordinate represents the ratio between recognition result and theoretical value, can from the recognition result in figure Go out, theoretical stiffness and the optimum results of structure coincide substantially, and worst error is no more than 5%, in the range of engineering precision, identification knot Fruit is basically identical with experimental phenomena.
Operating mode 9 is the artificial ripple input in Shanghai, is the operating mode for observing crack for the first time.Using the above method to each layer beam column Bending rigidity be identified, the ratio between each layer beam identification rigidity and theoretical stiffness are as shown in Figure 8.By calculating the identification knot of operating mode 9 The average value of fruit, in the range of 5% engineering precision, is considered as lossless, the as damage more than 5%；It can be drawn from result of calculation, Occurs obvious Stiffness degradation in the 4th layer of beam, degree of injury is close to 10%, and other positions are in the range of engineering precision, substantially It is intact, with experimental phenomena (the beam-ends of 4 layers of Vierendeel girder parallel to X direction of vibration occur first it is trickle from top to bottom and from The vertical fracture of lower and upper development, the small 0.05mm of slit width) it coincide substantially.
Operating mode 16 is that white noise third time is inputted, and is the operating mode that crack is increased.Each layer beam identification rigidity and theoretical stiffness it Than as shown in Figure 9；It is as lossless in the range of 5% engineering precision by calculating the average value of recognition result, from result of calculation It can draw, the degree of injury of the 4th layer of beam is substantially increased, degree of injury is close to 24%, 5,6 layers of beam also produce different degrees of damage Wound, wherein the 5th layer of beam degree of injury is close to 19%, the 6th layer of beam degree of injury close to 16%, other positions are in engineering precision model It is substantially intact in enclosing, (there are vertical fracture, slit width parallel to the beam-ends of 4~6 layers of Vierendeel girder of X direction of vibration with experimental phenomena Crack is not observed in about 0.08mm, each post) it coincide substantially.
In summary, embodiment demonstrates feasibility of the present invention in time domain, under ground seismic wave function, armored concrete Frame structure enters after damage, can identify the microlesion that structure occurs using the present invention, and can preferably recognize The degree of injury of structure.
There is certain applicability using the present invention to the non-destructive tests of RC three-dimensional frame structures, the application to Practical Project There is certain directive significance.
What is finally illustrated is：The preferred embodiments of the present invention are the foregoing is only, are not intended to limit the invention, to the greatest extent The present invention is described in detail with reference to the foregoing embodiments for pipe, for those skilled in the art, and it still can be with Technical scheme described in foregoing embodiments is modified, or equivalent substitution is carried out to which part technical characteristic.It is all Within the spirit and principles in the present invention, any modification, equivalent substitution and improvements made etc., should be included in the guarantor of the present invention Within the scope of shield.

Claims (1)

1. a kind of Structural Damage Identification theoretical based on statistical moment, it is characterised in that this method comprises the following steps：
Step 1, the Fourth square M of theoretical displacementx,4(E), the rank statistical moment M of theoretical acceleration eighta,8(E) acquisition, was obtained Journey is as follows：
For system with several degrees of freedom, it is considered to which viscous damping influences, its equation of motion is：
<mrow> <mi>M</mi> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>K</mi> <mi>x</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.1</mn> <mo>)</mo> </mrow> </mrow>
Wherein：M, C, K are respectively the mass matrix of structure, damping matrix and stiffness matrix,For acceleration, speed Degree and dynamic respond, f (t)=[f1(t),f2(t),...,fN(t)] it is external drive.Structure is obtained by solving in the time domain Dynamic respond xj=[x1j,x2j,...,xNs,j] and acceleration responsiveWherein xjFor j-th of measuring point Displacement,For the acceleration of j-th of measuring point, NsFor sampled point.It is understood that its statistical moment can be expressed as:
<mrow> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>N</mi> <mi>s</mi> </msub> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>s</mi> </msub> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mover> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mn>4</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.2</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>N</mi> <mi>s</mi> </msub> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>s</mi> </msub> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>-</mo> <mover> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>j</mi> </msub> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mn>8</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.3</mn> <mo>)</mo> </mrow> </mrow>
Step 2, the Fourth square M according to theoretical displacementx,4(E), the rank statistical moment M of theoretical acceleration eighta,8And measured displacements (E) Fourth squareThe rank statistical moment of actual measureed value of acceleration eightObtain the difference F between them1And F (E)2(E), i.e.,：
<mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.4</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.5</mn> <mo>)</mo> </mrow> </mrow>
And F (E)=[F1(E), F2(E)], its amount of variability | | F (E) | |2For residual error.
Step 3, by continuous iteration, until what is obtained | | F (E) | |2Minimum is reached, the E now obtained is the structure of calculating True estimate.Acquisition process is as follows：
The computational methods are to utilize object function | | F (E) | |2=0 constantly seeks optimal solution identification structure modulus of elasticity, i.e.,：
<mrow> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.6</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.7</mn> <mo>)</mo> </mrow> </mrow>
First order Taylor series expansion is pressed respectively to formula (1.6) and (1.7), and is taken as initial value, is obtained
<mrow> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>E</mi> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </msub> <mo>*</mo> <mi>&amp;Delta;</mi> <mi>E</mi> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.8</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>E</mi> </mrow> </mfrac> <msub> <mo>|</mo> <mrow> <mi>E</mi> <mo>=</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> </mrow> </msub> <mo>*</mo> <mi>&amp;Delta;</mi> <mi>E</mi> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.9</mn> <mo>)</mo> </mrow> </mrow>
If it is considered that external drive encourages for the load of zero-mean, then it can obtain：
<mrow> <msub> <mi>&amp;sigma;</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>N</mi> <mi>s</mi> </msub> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>s</mi> </msub> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.10</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>&amp;sigma;</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>N</mi> <mi>s</mi> </msub> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>s</mi> </msub> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.11</mn> <mo>)</mo> </mrow> </mrow>
σx,2Represent displacement second-order statisticses square, σa,2Acceleration second-order statisticses square is represented, can be obtained according to the relation between statistical moment Arrive, Mx,4=3 σx,2 2, Ma,8=105 σa,2 2。
The statistical moment can be obtained after seeking elastic modulus E local derviation：
<mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>E</mi> </mrow> </mfrac> <mo>=</mo> <mn>6</mn> <mo>&amp;CenterDot;</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;sigma;</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mfrac> <mn>1</mn> <msub> <mi>N</mi> <mi>s</mi> </msub> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>s</mi> </msub> </munderover> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>(</mo> <mi>E</mi> <mo>)</mo> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>E</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.12</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <mi>E</mi> </mrow> </mfrac> <mo>=</mo> <mn>210</mn> <mo>&amp;CenterDot;</mo> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;sigma;</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mfrac> <mn>1</mn> <msub> <mi>N</mi> <mi>s</mi> </msub> </mfrac> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>s</mi> </msub> </munderover> <mi>d</mi> <mi>i</mi> <mi>a</mi> <mi>g</mi> <mrow> <mo>(</mo> <mn>2</mn> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>i</mi> </msub> <mo>(</mo> <mi>E</mi> <mo>)</mo> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;&amp;CenterDot;</mo> </mover> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>E</mi> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.13</mn> <mo>)</mo> </mrow> </mrow>
Formula (1.12) is substituted into formula (1.8) respectively with formula (1.13) can obtain with (1.9)：
<mrow> <msub> <mi>&amp;Delta;E</mi> <mn>1</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>(</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>)</mo> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>*</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>M</mi> <mrow> <mi>x</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>E</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.14</mn> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>&amp;Delta;E</mi> <mn>2</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mo>(</mo> <msub> <mi>E</mi> <mn>0</mn> </msub> <mo>)</mo> <mo>-</mo> <msub> <mover> <mi>M</mi> <mo>^</mo> </mover> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mo>)</mo> </mrow> <mo>*</mo> <msup> <mrow> <mo>(</mo> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>M</mi> <mrow> <mi>a</mi> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mo>&amp;part;</mo> <mi>E</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1.15</mn> <mo>)</mo> </mrow> </mrow>
Here we takeThen:
E1=E0+ΔE (1.16)
By continuous iteration, until what is obtained | | F (E) | |2Minimum is reached, the E now obtained is that the structure of calculating is truly estimated Evaluation.
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