Summary of the invention
Technical matters: the purpose of this invention is to provide a kind of progressive health monitoring method based on cable system in the Cable Structure of cable force monitoring, at Cable Structure (particularly large-scale Cable Structure, for example large-scale cable-stayed bridge, suspension bridge) in the health monitoring problem of cable system, monitor by Suo Li Cable Structure, disclose and a kind ofly set up ladderingly and analyzed the method that concerns between the health status of rope force data that monitoring record obtains, and then can monitor the method for health status of the cable system of Cable Structure rationally and effectively with cable system.
Technical scheme: the present invention is made up of the two large divisions.Be respectively: one, set up the required knowledge base of the health monitoring systems of cable system of Cable Structure and the method for parameter, and based on the health status appraisal procedure of the cable system of knowledge base (containing parameter) and real measured data; Two, the software and hardware part of health monitoring systems.
First of the present invention: set up the method for required knowledge base of cable system health monitoring and parameter, and based on the health status appraisal procedure of the cable system of the Cable Structure of knowledge base (containing parameter) and real measured data.Can be successively circularly as follows, laddering carrying out, to obtain the health status assessment of cable system more accurately.
The first step: during circulation beginning each time, the cable system initial damage vector d when at first needing to set up or set up this circulation beginning
^{i} _{o}(i=1,2,3 ...) and the Mechanics Calculation benchmark model A of structure
^{i}(finite element benchmark model for example, i=1,2,3 ...), subscript i represents the i time circulation.
The cable system initial damage vector that needs during the i time circulation beginning is designated as d
^{i} _{o}(as the formula (1)), use d
^{i} _{o}Cable Structure when representing this time circulation beginning is (with Mechanics Calculation benchmark model A
^{i}The health status of cable system expression).
${d}_{o}^{i}={\left[\begin{array}{cccccc}{d}_{o1}^{i}& {d}_{o2}^{i}& \·\·\·& {d}_{\mathrm{oj}}^{i}& \·\·\·& {d}_{\mathrm{oN}}^{i}\end{array}\right]}^{T}---\left(1\right)$
D in the formula (1)
^{i} _{Oj}(i=1,2,3, J=1,2,3 ...., when N) the i time circulation of expression begins, Mechanics Calculation benchmark model A
^{i}In the initial damage value of j root rope of cable system, d
^{i} _{Oj}Being to represent j root rope not damaged at 0 o'clock, is to represent that this rope thoroughly lost load-bearing capacity at 100% o'clock, represents the load-bearing capacity of j root rope forfeiture corresponding proportion in the time of between 0 and 100%.
If total N root rope in the cable system is set up cable system initial damage vector during circulation beginning for the first time and (is designated as d according to formula (1)
^{1} _{o}) time, the data of utilizing the Non-Destructive Testing data etc. of rope can express the health status of rope are set up cable system initial damage vector d
^{1} _{o}If when not having the data of the Non-Destructive Testing data of rope and other health status that can express rope, perhaps can think when the structure original state is the not damaged state vectorial d
^{1} _{o}Each element numerical value get 0.
The i time (i=2,3,4,5,6 ...) the cable system initial damage vector d of needs when circulation begins
^{i} _{o}, be preceding once (promptly the i-1 time, i=2,3,4,5,6 ...) the preceding calculating acquisition of loop ends, concrete grammar is described below.
The Mechanics Calculation benchmark model that need set up during the i time circulation beginning or the Mechanics Calculation benchmark model of having set up are designated as A
^{i}
The Mechanics Calculation benchmark model of the Cable Structure of setting up during circulation beginning for the first time is designated as A
^{1}, A
^{1}It is design drawing according to Cable Structure, the measured data of as-constructed drawing and Cable Structure (generally comprises the rope force data, the structure angle-data, shape data, spatial data, strain data, measured datas such as structural modal data, to cable-stayed bridge, suspension bridge and the rope force data of Yan Shiqiao, the bridge type data, strain data, angle-data, measured datas such as the modal data of bridge) etc. the data of original state when the reflection Cable Structure is built up, or the data of reflection Cable Structure current state when health monitoring systems is installed, utilize mechanics method (for example finite element method) to set up the Mechanics Calculation benchmark model of this structure (for example finite element benchmark model), the Structure Calculation data that calculate based on this calculating benchmark model are (to cable-stayed bridge, suspension bridge and the rope force data of Yan Shiqiao, the bridge type data, strain data, measured datas such as angle-data) must be very near its measured data, error generally must not be greater than 5%.Can guarantee like this to calculate the measured data of data (for example rope force data, strain data, angle-data, planform computational data and spatial coordinates calculation data etc.) when truly taking place near analog case reliably under the analog case of calculating gained on the benchmark model at this.
The i time (i=2,3,4,5,6 ...) the Mechanics Calculation benchmark model A of needs when circulation begins
^{i}The time, be preceding once (promptly the i-1 time, i=2,3,4,5,6 ...) the preceding calculating acquisition of loop ends, concrete grammar is described below.
Existing Mechanics Calculation benchmark model A
^{1}With cable system initial damage vector d
^{1} _{o}After, model A
^{1}In the damage of each rope by vectorial d
^{1} _{o}Express.At A
^{1}The basis on, the damage of all ropes is changed to 0, mechanical model A
^{1}The damage that is updated to all ropes all is that 0 mechanical model (is designated as A
^{0}), mechanical model A
^{0}Be actually the mechanical model of intact Cable Structure correspondence.Might as well claim model A
^{0}Not damaged model A for Cable Structure
^{0}
The monitored rope force data of structure can be described by the Suo Li of all N root ropeway carrying-ropes of Cable Structure, and the variation of structure rope force data is exactly the variation of the Suo Li of all N root ropes.Each total N cable force measurement value or calculated value characterize structure Suo Li.Use vectorial C
^{i} _{o}(i=1,2,3 ...) expression the i time (i=1,2,3,4,5,6 ...) initial value (referring to formula (2)) of the monitored amount (Suo Li of all N root ropes) of all appointments in when beginning circulation Cable Structure, C
^{i} _{o}Be called " the initial value vector of the i time monitored amount of circulation ".
${C}_{o}^{i}={\left[\begin{array}{cccccc}{C}_{o1}^{i}& {C}_{o2}^{i}& \·\·\·& {C}_{\mathrm{ok}}^{i}& \·\·\·& {C}_{\mathrm{oN}}^{i}\end{array}\right]}^{T}---\left(2\right)$
C in the formula (2)
^{i} _{Ok}(i=1,2,3, K=1,2,3 ...., N; ) be k monitored amount (Suo Li) in the i time when beginning circulation, the Cable Structure, T represents the transposition (back with) of vector.Vector C
^{i} _{o}Be to be formed according to certain series arrangement by the monitored amount of previously defined N, this is put in order there is no specific (special) requirements, only require all associated vector of back also in this order array data get final product.
During circulation beginning for the first time, the initial value vector C of monitored amount
^{1} _{o}(seeing formula (2)) is made up of measured data, because according to model A
^{1}The initial value of calculating the monitored amount of gained approaches corresponding measured value reliably, in the narration of back, will represent this calculated value composition of vector and measured value composition of vector with prosign.
The i time (i=2,3,4,5,6 ...) the initial value vector C of the monitored amount of needs when circulation begins
^{i} _{o}, be preceding once (promptly the i-1 time, i=2,3,4,5,6 ...) the preceding calculating acquisition of loop ends, concrete grammar is described below.
Second step: circulation each time needs to set up " unit damage monitored numerical quantity transformation matrices ", and " unit damage monitored numerical quantity transformation matrices " that the i time circulation set up is designated as Δ C
^{i}(i=1,2,3 ...).
The Cable Structure " unit damage monitored numerical quantity transformation matrices " that circulation is for the first time set up is designated as Δ C
^{1}Set up Δ C
^{1}Process as follows:
Mechanics Calculation benchmark model A in Cable Structure
^{1}The basis on carry out several times and calculate, equal the quantity of all ropes on the calculation times numerical value.(original damage can be 0 in original damage to have only a rope in the cable system of calculating hypothesis each time, can not be 0 also) the basis on increase unit damage (the monitored numerical quantity that unit damage should be less and it causes changes and can accurately be identified by sensor, for example gets 10% damage and is unit damage) again.Calculate for convenient, when setting unit damage in the circulation each time can all be structural health conditions during this time circulation beginning as being healthy fully, and set on this basis unit damage (in subsequent step, damage numerical value that calculate, rope---be called name damage d
^{i} _{c}(i=1,2,3 ...), all with respect to this time when beginning circulation, with the health status of rope as being healthy fully speech, the name that the formula that therefore must foundation hereinafter provides will calculate is damaged and is converted into true damage.)。The rope of appearance damage was different from the rope that appearance damages in other time calculating during a round-robin calculated each time together, and supposed that each time the unit damage value of the rope that damage is arranged can be different from the unit damage value of other ropes, with " the vectorial D of nominal unit damage
^{i} _{u}" (as the formula (3)) write down the unit damage of the supposition of all ropes in each time circulation, circulation time is designated as D for the first time
^{1} _{u}Calculate each time all utilize mechanics method (for example finite element method) calculate Cable Structure, the current calculated value of the N of appointment monitored amount in front, the current calculated value that calculates gained N monitored amount is each time formed one " the current numerical value vector of the calculating of monitored amount ", and (when hypothesis j root rope had unit damage, available formula (4) was represented the current numerical value vector of the calculating C of N monitored amount of all appointments
^{1} _{Tj}); The current numerical value vector of the calculating of the monitored amount that calculates each time deducts the initial value vector C of monitored amount
^{1} _{o}, the gained vector is exactly that " the numerical value change vector of monitored amount " of (is mark with the position of rope that unit damage is arranged or numbering etc.) (when j root rope has unit damage, used δ C under this condition
^{1} _{j}The numerical value change vector of representing monitored amount, δ C
^{1} _{j}Definition see formula (5), formula (6) and formula (7), formula (5) deducts after the formula (2) again divided by vectorial D for formula (4)
^{1} _{u}J element D
_{Uj}Gained), the numerical value change of monitored amount vector δ C
^{1} _{j}Each element representation since when calculating supposition the unit damage (D for example of the Na Gensuo (for example j root rope) of unit damage is arranged
_{Uj}), and the numerical value change amount of the pairing monitored amount of this element that causes is with respect to the unit damage D of supposition
_{Uj}Rate of change; There is N root rope that N " the numerical value change vector of monitored amount " just arranged, the numerical value change vector of each monitored amount has N element, forms " the unit damage monitored numerical quantity transformation matrices Δ C that N * N element arranged successively by this N " the numerical value change vector of monitored amount "
^{1}" (the capable N row of N), each vectorial δ C
^{1} _{j}(j=1,2,3 ...., N) be matrix Δ C
^{1}One row, Δ C
^{1}Definition as the formula (8).
${D}_{u}^{i}={\left[\begin{array}{cccccc}{D}_{u1}^{i}& {D}_{u2}^{i}& \·\·\·& {D}_{\mathrm{uj}}^{i}& \·\·\·& {D}_{\mathrm{uN}}^{i}\end{array}\right]}^{T}---\left(3\right)$
Nominal unit damage vector D in the formula (3)
^{i} _{u}Element D
^{i} _{Uj}(i=1,2,3, J=1,2,3 ...., N) the unit damage numerical value of the j root rope of supposition in the i time circulation of expression, vectorial D
^{i} _{u}In the numerical value of each element can be the same or different.
${C}_{\mathrm{tj}}^{i}={\left[\begin{array}{cccccc}{C}_{\mathrm{tk}1}^{i}& {C}_{\mathrm{tk}2}^{i}& \·\·\·& {C}_{\mathrm{tjk}}^{i}& \·\·\·& {C}_{\mathrm{tjN}}^{i}\end{array}\right]}^{T}---\left(4\right)$
Elements C in the formula (4)
^{i} _{Tjk}(i=1,2,3 ...; J=1,2,3 ...., N; K=1,2,3 ...., N) the i time circulation of expression be owing to j root rope when unit damage is arranged, according to the current numerical value of calculating of the monitored amount of pairing k the appointment of coding rule.
${\mathrm{\δC}}_{j}^{i}=\frac{{C}_{\mathrm{tj}}^{i}-{C}_{o}^{i}}{{D}_{\mathrm{uj}}^{i}}---\left(5\right)$
The subscript i of each amount in the formula (5) (i=1,2,3 ...) the i time circulation of expression, subscript j (j=1,2,3 ...., N) expression j root rope has unit damage, D in the formula
^{i} _{Uj}Be vectorial D
^{i} _{u}In j element.Vector δ C
^{i} _{j}Definition as the formula (6), δ C
^{i} _{j}K (k=1,2,3 ...., N) individual element δ C
^{i} _{Jk}Represent in the i time circulation, set up matrix Δ C
^{i}The time, supposition j root rope calculates the unit damage D of the change amount of a gained k monitored amount with respect to supposition when unit damage is arranged
^{i} _{Uj}Rate of change, it defines as the formula (7).
${\mathrm{\δC}}_{j}^{i}={\left[\begin{array}{cccccc}\mathrm{\δ}{C}_{j1}^{i}& {\mathrm{\δC}}_{j2}^{i}& \·\·\·& {\mathrm{\δC}}_{\mathrm{jk}}^{i}& \·\·\·& {\mathrm{\δC}}_{\mathrm{jN}}^{i}\end{array}\right]}^{T}---\left(6\right)$
${\mathrm{\δC}}_{\mathrm{jk}}^{i}=\frac{{C}_{\mathrm{tjk}}^{i}-{C}_{\mathrm{ok}}^{i}}{{D}_{\mathrm{uj}}^{i}}---\left(7\right)$
The definition of each amount has been previously described in the formula (7).
${\mathrm{\ΔC}}^{i}=\left[\begin{array}{cccccc}{\mathrm{\δC}}_{1}^{i}& \mathrm{\δ}{C}_{2}^{i}& \·\·\·& {\mathrm{\δC}}_{j}^{i}& \·\·\·& {\mathrm{\δC}}_{N}^{i}\end{array}\right]---\left(8\right)$
Vectorial δ C in the formula (8)
^{i} _{j}(i=1,2,3 ....,, j=1,2,3 ...., N) in the i time circulation of expression, because j root rope has unit damage D
^{i} _{Uj}Cause, the relative value of all monitored amounts changes.Matrix Δ C
^{i}The coding rule of row (subscript j) and front vector d
^{i} _{o}The coding rule of subscript j of element identical.
The 3rd step: the current health status of identification cable system.Detailed process is as follows.
I (i=1,2,3 ...) in the inferior circulation, cable system " current (calculating or actual measurement) numerical value vector C of monitored amount
^{i}" " the initial value of monitored amount vector C together
^{i} _{o}", " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" and " the vectorial d of current name damage
^{i} _{c}" between linear approximate relationship, shown in (9) or formula (10).
${C}^{i}={C}_{o}^{i}+{\mathrm{\ΔC}}^{i}\·{d}_{c}^{i}---\left(9\right)$
${C}^{i}-{C}_{o}^{i}={\mathrm{\ΔC}}^{i}\·{d}_{c}^{i}---\left(10\right)$
Current (calculating or actual measurement) numerical value vector C of monitored amount in formula (9) and the formula (10)
^{i}Definition be similar to the initial value vector C of monitored amount
^{i} _{o}Definition, see formula (11); The vectorial d of the current name damage of cable system
^{i} _{c}Definition see formula (12).
${C}^{i}={\left[\begin{array}{cccccc}{C}_{1}^{i}& {C}_{2}^{i}& \·\·\·& {C}_{k}^{i}& \·\·\·& {C}_{N}^{i}\end{array}\right]}^{T}---\left(11\right)$
Elements C in the formula (11)
^{i} _{k}(i=1,2,3 ....; K=1,2,3 ...., N) be the i time circulation time Cable Structure, according to the current numerical value of the monitored amount of the pairing k of being numbered of coding rule.
${d}_{c}^{i}={\left[\begin{array}{cccccc}{d}_{c1}^{i}& {d}_{c2}^{i}& \·\·\·& {d}_{\mathrm{cj}}^{i}& \·\·\·& {d}_{\mathrm{cN}}^{i}\end{array}\right]}^{T}---\left(12\right)$
D in the formula (12)
^{i} _{Cj}(i=1,2,3 ....; J=1,2,3 ..., N) be the current nominal impairment value of cable system j root rope in the i time circulation, vectorial d
^{i} _{c}The coding rule and the matrix Δ C of subscript j of element
^{i}The coding rule of row identical.
When the rope actual damage was not too big, because the Cable Structure material still is in the linear elasticity stage, the distortion of Cable Structure was also less, and the represented a kind of like this linear relationship of formula (9) or formula (10) is less with the error of actual conditions, and error can be used error vector e
^{i}(formula (13)) definition, the error of linear relationship shown in expression (9) or the formula (10).
${e}^{i}=\mathrm{abs}({\mathrm{\ΔC}}^{i}\·{d}_{c}^{i}-{C}^{i}+{C}_{o}^{i})---\left(13\right)$
Abs () is the function that takes absolute value in the formula (13), and each element of the vector of trying to achieve in the bracket is taken absolute value.
Because there are certain error in formula (9) or the represented linear relationship of formula (10), therefore can not be simply according to formula (9) or formula (10) and " current (actual measurement) numerical value vector C of monitored amount
^{i}" directly find the solution and obtain the vectorial d of Suo Dangqian name damage
^{i} _{c}If done like this, the damage vector d that obtains
^{i} _{c}In element in addition bigger negative value can appear, just negative damage, this obviously is irrational.Therefore obtaining rope damages vectorial d
^{i} _{c}Acceptable separating (promptly have reasonable error, but can be more accurately from cable system, determine the position and the degree of injury thereof of damaged cable) become a rational solution, available formula (14) is expressed this method.
$\mathrm{abs}({\mathrm{\ΔC}}^{i}\·{d}_{c}^{i}-{C}^{i}+{C}_{o}^{i})\≤{g}^{i}---\left(14\right)$
Abs () is the function that takes absolute value in the formula (14), vectorial g
^{i}Description departs from the reasonable deviation of ideal linearity relation (formula (9) or formula (10)), is defined by formula (15).
${g}^{i}={\left[\begin{array}{cccccc}{g}_{1}^{i}& {g}_{2}^{i}& \·\·\·& {g}_{k}^{i}& \·\·\·& {g}_{N}^{i}\end{array}\right]}^{T}---\left(15\right)$
G in the formula (15)
^{i} _{k}(i=1,2,3 ...; K=1,2,3 ...., N) maximum allowable offset that departs from the ideal linearity relation shown in formula (9) or the formula (10) in the i time circulation has been described.Vector g
^{i}Can be according to the error vector e of formula (13) definition
^{i}Tentative calculation is selected.
Initial value vector C in monitored amount
^{i} _{o}(survey or calculate), Cable Structure unit damage monitored numerical quantity transformation matrices Δ C
^{i}The current numerical value vector C of (calculating) and monitored amount
^{i}When (actual measurement obtains) is known, can utilize suitable algorithm (for example multi-objective optimization algorithm) to find the solution formula (14), obtain the vectorial d of the current name damage of cable system
^{i} _{c}Acceptable separating, the current actual damage of cable system vector d
^{i}The element of (formula (16) is seen in definition) can calculate according to formula (17), has just obtained Suo Dangqian actual damage vector d
^{i}Thereby, can be by d
^{i}Determine the position and the degree of injury of damaged cable, just realized the health monitoring of cable system.
${d}^{i}={\left[\begin{array}{cccccc}{d}_{1}^{i}& {d}_{2}^{i}& \·\·\·& {d}_{j}^{i}& \·\·\·& {d}_{N}^{i}\end{array}\right]}^{T}---\left(16\right)$
D in the formula (16)
^{i} _{j}(i=1,2,3, J=1,2,3 ...., N) the actual damage value of j root rope in the i time circulation of expression, formula (17), d are seen in its definition
^{i} _{j}Being to represent j root rope not damaged at 0 o'clock, is to represent that this rope thoroughly lost load-bearing capacity at 100% o'clock, represents the load-bearing capacity of j root rope forfeiture corresponding proportion in the time of between 0 and 100%, vectorial d
^{i}The coding rule of element and formula (1) in vectorial d
^{i} _{o}The coding rule of element identical.
${d}_{j}^{i}=1-(1-{d}_{\mathrm{oj}}^{i})(1-{d}_{\mathrm{cj}}^{i})---\left(17\right)$
D in the formula (17)
^{i} _{Oj}(i=1,2,3,4, J=1,2,3 ...., N) be vectorial d
^{i} _{o}J element, d
^{i} _{Cj}Be vectorial d
^{i} _{c}J element.
The 4th step: judge whether to finish this (the i time) circulation, if, then finish the preceding tailing in work of this loop ends, for next time (promptly the i+1 time, i=1,2,3,4 ...) circulation preparation Mechanics Calculation benchmark model and necessary vector.Detailed process is as follows.
In this (the i time) circulation, try to achieve the vectorial d of current name damage
^{i} _{c}After, at first, set up mark vector F according to formula (18)
^{i}, formula (19) has provided mark vector F
^{i}The definition of j element; If mark vector F
^{i}Element be 0 entirely, then in this circulation, continue health monitoring and calculating to cable system; If mark vector F
^{i}Element be not 0 entirely, then finish subsequent step after, enter next time circulation.So-called subsequent step is: at first, according to formula (20) calculate next time (promptly the i+1 time, i=1,2,3,4 ...) the required initial damage vector d of circulation
^{I+1} _{o}Each element d
^{I+1} _{Oj}The second, at Mechanics Calculation benchmark model A
^{i}(i=1,2,3,4 ...) or the not damaged model A of Cable Structure
^{0}The basis on, the health status situation that makes rope is d
^{I+1} _{o}The back upgrade and to obtain next time (the i+1 time, i=1,2,3,4 ...) the required Mechanics Calculation benchmark model A of circulation
^{I+1}At last, by to Mechanics Calculation benchmark model A
^{I+1}The initial value that calculates monitored amount, by its form next time (promptly the i+1 time, i=1,2,3,4 ...) required " the initial value vector C of monitored amount of circulation
^{I+1} _{o}" (i=1,2,3,4 ...).
${F}^{i}={\left[\begin{array}{cccccc}{F}_{1}^{i}& {F}_{2}^{i}& \·\·\·& {F}_{j}^{i}& \·\·\·& {F}_{N}^{i}\end{array}\right]}^{T}---\left(18\right)$
Mark vector F in the formula (18)
^{i}Subscript i represent the i time the circulation, its element F
^{i} _{j}(j=1,2,3 ..., subscript j N) represents the damage characteristic of j root rope, can only get 0 and 1 two amount, concrete value rule is seen formula (19).
${F}_{j}^{i}=\left\{\begin{array}{ccc}0,& \mathrm{if}& {d}_{\mathrm{cj}}^{i}<{D}_{\mathrm{uj}}^{i}\\ 1,& \mathrm{if}& {d}_{\mathrm{cj}}^{i}\≥{D}_{\mathrm{uj}}^{i}\end{array}\right.---\left(19\right)$
Element F in the formula (19)
^{i} _{j}Be mark vector F
^{i}J element, D
^{i} _{Uj}Be nominal unit damage vector D
^{i} _{u}J element (seeing formula (3)), d
^{i} _{Cj}Be the vectorial d of the current name damage of cable system
^{i} _{c}J element (seeing formula (12)), they all represent the relevant information of j root rope.
${d}_{\mathrm{oj}}^{i+1}=1-(1-{d}_{\mathrm{oj}}^{i})(1-{D}_{\mathrm{uj}}^{i}{F}_{j}^{i})---\left(20\right)$
D in the formula (20)
^{i} _{Uj}Be nominal unit damage vector D
^{i} _{u}J element (seeing formula (3)), d
^{i} _{Cj}Be the vectorial d of the current name damage of cable system
^{i} _{c}J element (seeing formula (12)).
Second portion of the present invention: the software and hardware part of health monitoring systems.Hardware components comprises cable force monitoring system (for example comprising the signal conditioner of cable tension sensor, correspondence etc.), signal picker and computing machine.The general Suo Li of each root rope of monitoring in real time that requires.Software should the following function of tool: software section should be finished the process that first of the present invention sets, i.e. the data in real time that at first transmits according to monitoring system or quasi real time analyze the current numerical value vector C that obtains monitored amount
^{i}, the unit damage monitored numerical quantity transformation matrices Δ C of the Cable Structure of reading pre-stored then
^{i}, monitored amount initial value vector C
^{i} _{o}With nominal unit damage vector D
^{i} _{u}, find the solution formula (14) according to suitable algorithm (for example multi-objective optimization algorithm), obtain the vectorial d of current name damage of cable system
^{i} _{c}Noninferior solution, obtain the current actual damage of cable system vector d according to formula (17)
^{i}, just have reasonable error but can be more exactly from cable system, determine the position of damaged cable and separating of degree of injury thereof.Software section also will be according to the mark vector F that circulates and try to achieve each time
^{i}Concrete condition judge whether to enter next time circulation.
The inventive method specifically comprises:
A. establish total N root rope, at first determine the coding rule of rope, with rope numberings all in the Cable Structure, this numbering will be used to generate the vector sum matrix in subsequent step by this rule;
B. the data of utilizing the Non-Destructive Testing data etc. of rope can express the health status of rope are set up cable system initial damage vector d
^{1} _{o}If when not having the data of the Non-Destructive Testing data of rope and other health status that can express rope, perhaps can think when the structure original state is the not damaged state vectorial d
^{1} _{o}Each element numerical value get 0.
C. setting up cable system initial damage vector d
^{1} _{o}The time, directly measure the monitored amount of all appointments that calculate Cable Structure,, form " the initial value vector C of monitored amount even the initial rope force data of ropeway carrying-rope is arranged
^{1} _{o}";
D. setting up cable system initial damage vector d
^{1} _{o}Initial value vector C with monitored amount
^{1} _{o}The time, actual measurement obtains the initial geometric data of Cable Structure;
E. according to the above-mentioned measured data of design drawing, as-constructed drawing and the Cable Structure of Cable Structure, set up the mechanics model of Cable Structure, computational data based on this Model Calculation gained is approaching more good more with above-mentioned measured data, difference therebetween generally must not be greater than 5%, and this moment, this model was called as the Mechanics Calculation benchmark model A of structure
^{1}
F. at Mechanics Calculation benchmark model A
^{i}The basis on carry out the several times Mechanics Calculation, by calculate obtaining " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" and " nominal unit damage vector D
^{i} _{u}".Wherein i represents cycle index, and back i and subscript i represent cycle index, i=1, and 2,3 ...;
G. actual measurement obtain Cable Structure all specify the current measured value of monitored amount, form " the current numerical value vector C of monitored amount
^{i}".When numbering to the element of the institute's directed quantity that occurred before this step and this step, should use same coding rule, can guarantee the element each vector, that numbering is identical that occurs before this step and this step like this, represent same monitored amount, corresponding to vectorial defined relevant information under this element;
H. define the vectorial d of the current name damage of cable system
^{i} _{c}With current actual damage vector d
^{i}, the element number of damage vector equals the quantity of rope, is one-to-one relationship between the element of damage vector and the rope, and the element numerical value of damage vector is represented the degree of injury or the health status of corresponding rope;
I. according to " the current numerical value vector C of monitored amount
^{i}" " the initial value of monitored amount vector C together
^{i} _{o}", " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" and " the vectorial d of current name damage
^{i} _{c}" between the linear approximate relationship that exists, this linear approximate relationship can be expressed as formula 1, removes d in the formula 1
^{i} _{c}Other outer amount is known, finds the solution formula 1 and just can calculate the vectorial d of current name damage
^{i} _{c}
${C}^{i}={C}_{o}^{i}+{\mathrm{\ΔC}}^{i}\·{d}_{c}^{i}$ Formula 1
J. the current actual damage vector d that utilizes formula 2 to express
^{i}With initial damage vector d
^{i} _{o}With the vectorial d of current name damage
^{i} _{c}Element between relation, calculate current actual damage vector d
^{i}All elements.
${d}_{j}^{i}=1-(1-{d}_{\mathrm{oj}}^{i})(1-{d}_{\mathrm{cj}}^{i})$ Formula 2
J=1 in the formula 2,2,3 ..., N.
Because current actual damage vector d
^{i}Element numerical value represent the degree of injury of corresponding rope, so according to current actual damage vector d
^{i}Just can define the impaired and degree of injury of which rope, promptly realize the health monitoring of cable system in the Cable Structure; If the numerical value of a certain element of current actual damage vector is 0, represent that the pairing rope of this element is intact, do not damage; If its numerical value is 100%, represent that then the pairing rope of this element has completely lost load-bearing capacity; If its numerical value between 0 and 100%, is then represented this rope and has been lost the load-bearing capacity of corresponding proportion.
K. try to achieve the vectorial d of current name damage
^{i} _{c}After, set up mark vector F according to formula 3
^{i}, formula 4 has provided mark vector F
^{i}The definition of j element;
${F}^{i}={\left[\begin{array}{cccccc}{F}_{1}^{i}& {F}_{2}^{i}& \·\·\·& {F}_{j}^{i}& \·\·\·& {F}_{N}^{i}\end{array}\right]}^{T}$ Formula 3
${F}_{j}^{i}=\left\{\begin{array}{ccc}0,& \mathrm{if}& {d}_{\mathrm{cj}}^{i}<{D}_{\mathrm{uj}}^{i}\\ 1,& \mathrm{if}& {d}_{\mathrm{cj}}^{i}\≥{D}_{\mathrm{uj}}^{i}\end{array}\right.$ Formula 4
Element F in formula 3, the formula 4
^{i} _{j}Be mark vector F
^{i}J element, D
^{i} _{Uj}Be nominal unit damage vector D
^{i} _{u}J element, d
^{i} _{Cj}Be the vectorial d of the current name damage of cable system
^{i} _{c}J element, they all represent the relevant information of j root rope.J=1 in two formulas, 2,3 ..., N.
If mark vector F l.
^{i}Element be 0 entirely, then get back to g step and continue this circulation; If mark vector F
^{i}Element be not 0 entirely, then enter next step, i.e. m step.
M. according to formula 5 calculate next time, i.e. the i+1 time required initial damage of circulation vector d
^{I+1} _{o}Each element d
^{I+1} _{Oj}
${d}_{\mathrm{oj}}^{i+1}=1-(1-{d}_{\mathrm{oj}}^{i})(1-{D}_{\mathrm{uj}}^{i}{F}_{j}^{i})$ Formula 5
D in the formula 5
^{i} _{Uj}Be nominal unit damage vector D
^{i} _{u}J element, d
^{i} _{Cj}Be the vectorial d of the current name damage of cable system
^{i} _{c}J element, F
^{i} _{j}Be mark vector F
^{i}J element.J=1 in the formula 5,2,3 ..., N.
N. at Mechanics Calculation benchmark model A
^{i}The basis on, the health status that makes rope is d
^{I+1} _{o}The back renewal obtains next time, required Mechanics Calculation benchmark model A promptly circulates for the i+1 time
^{I+1}
O. pass through Mechanics Calculation benchmark model A
^{I+1}Calculate corresponding to model A
^{I+1}The numerical value of all monitored Suo Li of structure, these numerical value are formed next time, the vectorial C of initial value of the required monitored amount that promptly circulates for the i+1 time
^{I+1} _{o}
P. get back to the f step, beginning is circulation next time.
In step f, obtain " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" concrete grammar be:
F1. at the Mechanics Calculation benchmark model A of structure
^{i}The basis on carry out the several times Mechanics Calculation, equal the quantity of all ropes on the calculation times numerical value, there is N root rope that N calculating is just arranged, calculating each time in the hypothesis cable system has only a rope to increase unit damage again on the basis of original damage, the rope that occurs damage in calculating each time is different from the rope that occurs damage in other time calculating, and supposition each time has the unit damage value of the rope of damage can be different from the unit damage value of other ropes, uses " nominal unit damage vector D
^{i} _{u}" write down the unit damage of the supposition of all ropes, calculate each time that all specify the current numerical value of monitored amount in the Cable Structure, the current numerical value of the monitored amount of all that calculate is formed one " the current numerical value vector of the calculating of monitored amount " each time.When hypothesis j root rope has unit damage, available C
^{i} _{Tj}" the current numerical value vector of the calculating of monitored amount " that expression is corresponding.When giving each vectorial element numbering in this step, should use same coding rule with other vector among the present invention, can guarantee any one element in each vector in this step like this, with element in other vector, that numbering is identical, expressed the relevant information of same monitored amount or same target;
F2. that calculates each time " the current numerical value of the calculating of monitored amount vector C
^{i} _{Tj}" deduct " initial value of monitored amount vector C
^{i} _{o}" obtain a vector, during all calculating divided by this, each element that again should vector obtains " numerical value change vector δ a C of monitored amount after the unit damage value of supposition
^{i} _{j}"; There is N root rope that N " the numerical value change vector of monitored amount " just arranged;
F3. form " the unit damage monitored numerical quantity transformation matrices Δ C that the N row are arranged successively by this N " the numerical value change vector of monitored amount "
^{i}"; Each row of " unit damage monitored numerical quantity transformation matrices " are corresponding to one " the numerical value change vector of monitored amount " in other words.The coding rule of the row of " unit damage monitored numerical quantity transformation matrices " and the vectorial d of current name damage
^{i} _{c}With current actual damage vector d
^{i}The element coding rule identical.
Beneficial effect: system and method disclosed by the invention is having under the simultaneously impaired condition of more rope (for example more than 30 ropes or more than 30% rope) monitoring and evaluation very exactly go out the health status (position and the degree of injury that comprise all damaged cables) of cable system.This is because " the current numerical value vector C of monitored amount
^{i}" " the initial value of monitored amount vector C together
^{i} _{o}", " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" and " the vectorial d of current name damage
^{i} _{c}" between linear relationship be similar to; be actually nonlinear relation; when particularly more the or extent of damage is big at damaged cable; the nonlinear characteristic of the relation between the above-mentioned amount is more obvious; for overcoming this obstacle, the invention discloses a kind of health monitor method that approaches this nonlinear relationship in the minizone with linear relationship.In fact the present invention has used the method with linear relationship piecewise approximation nonlinear relationship, big interval is divided into minizone one by one, in each minizone internal linear relation all is enough accurately, the health status of the cable system that obtains according to its judgement also is reliably, and therefore system and method disclosed by the invention is very useful to effective health monitoring of cable system.
Embodiment
At the health monitoring of the cable system of Cable Structure (particularly large-scale Cable Structure, for example large-scale cable-stayed bridge, suspension bridge), the invention discloses a kind of system and method for health status of each root rope of the cable system that can monitor Cable Structure rationally and effectively.The following describes of embodiments of the invention in fact only is exemplary, and purpose never is to limit application of the present invention or use.
The algorithm that the present invention adopts is used for monitoring the health status of the cable system (all ropes) of Cable Structure (for example large-scale cable-stayed bridge, suspension bridge).During concrete enforcement, the following step is a kind of in the various steps that can take.
The first step: determine type, position and the quantity of monitored amount, and numbering.Detailed process is:
At first determine the coding rule of rope, all ropes are numbered by this rule.This numbering will be used to generate the vector sum matrix in subsequent step.
If total N root ropeway carrying-rope in the cable system, the monitored amount of structure just is meant the rope force data of this N root rope among the present invention, and the variation of monitored amount data is exactly the variation of the Suo Li of this N root rope.Each total N cable force measurement value or calculated value characterize the monitored amount of structure.
Second step: the data of utilizing the Non-Destructive Testing data etc. of rope can express the health status of rope are set up cable system initial damage vector d
^{1} _{o}If when not having the data of the Non-Destructive Testing data of rope and other health status that can express rope, perhaps can think when the structure original state is the not damaged state vectorial d
^{1} _{o}Each element numerical value get 0.
The 3rd step: set up cable system initial damage vector d
^{1} _{o}The time, directly measure the monitored amount of all appointments that calculate Cable Structure, form " the initial value vector C of monitored amount
^{1} _{o}".For example can pass through the vibration of acceleration transducer survey record rope, and obtain initial rope force data by the analysis to vibration signal, this is ripe measuring technique.
The 4th step: setting up cable system initial damage vector d
^{1} _{o}Initial value vector C with monitored amount
^{1} _{o}The time, can adopt ripe measuring method to carry out strain measurement, measurement of angle and volume coordinate measures, calculate the original geometric form data (is exactly its initial bridge type data for cable-stayed bridge) of Cable Structure after directly measuring or measuring, the original geometric form data of Cable Structure can be the spatial datas that the spatial data of the end points of all ropes adds a series of point on the structure, and purpose is just can determine according to these coordinate datas the geometric properties of Cable Structure.For cable-stayed bridge, the original geometric form data can be the spatial datas that the spatial data of the end points of all ropes adds some points on the bridge two ends, so-called bridge type data that Here it is.
Design drawing according to Cable Structure, the measured data of as-constructed drawing and Cable Structure (can comprise structure original geometric form data, the initial angle coordinate data, data such as the initial Suo Li of all ropes, to cable-stayed bridge, suspension bridge and the bridge type data of Yan Shiqiao, the angle coordinate data, data such as rope force data), utilize mechanics method (for example adopting finite element method) to set up the Mechanics Calculation benchmark model of this structure (for example finite element benchmark model), computational data based on this Model Calculation gained is approaching more good more with above-mentioned measured data (comprising initial rope force data), difference therebetween generally must not be greater than 5%, and this moment, this model was called as the Mechanics Calculation benchmark model A of structure
^{1}
The 5th step: the hardware components of pass line structural healthy monitoring system.Hardware components comprises at least: cable force monitoring system (for example containing cable tension sensor, signal conditioner etc.), signal (data) collector, the computing machine and the panalarm of communicating by letter.Can use acceleration transducer to monitor the vibration of rope in real time, can get the rope force data by analysis of vibration signal.The monitored amount of each appointment all must arrive by monitored system monitoring; Monitoring system is monitored the monitored amount of each appointment, and transfers signals to signal (data) collector; Signal is delivered to computing machine through signal picker; Computing machine then is responsible for the health monitoring software of the cable system of operation Cable Structure, comprises the signal that the transmission of tracer signal collector comes; When monitoring rope when damage is arranged, the computer control communication panalarm to monitor staff, owner and (or) personnel of appointment report to the police.
The 6th step: establishment and the cable system health monitoring systems software of pass line structure on supervisory control comuter.All move this software at circulation time each time, this software is all the time in operation in other words.This software comprises following several functional module:
1. finish this step and be other calculating, control and monitoring work in steps, and can be according to the condition of setting (for example damage reaches a certain value), notice or prompting monitor staff notify specific technician to finish necessary evaluation work automatically.
2. read " the initial value vector C of monitored amount the data file on being stored in hard disc of computer
^{i} _{o}" and Cable Structure " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" wait all call parameters.Express cycle index with i among the present invention, i=1 when carrying out this step for the first time, back i and subscript i represent cycle index, i=1,2,3 ...; Express the relevant information of j root rope with j, j=1,2,3 ..., N.
3. the signal that transmits by signal picker of (or trigger-type) at random record regularly.
4. the signal to record carries out signal Processing, calculates the current numerical value of all monitored amounts to be measured, and the current numerical value of all monitored amounts is formed " the current numerical value vector C of monitored amount
^{i}".
5. according to " current (calculating or actual measurement) numerical value vector C of monitored amount
^{i}" " the initial value of monitored amount vector C together
^{i} _{o}", " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" and " the vectorial d of current name damage
^{i} _{c}" linear approximate relationship (seeing formula (9)) that exists between (the current nominal amount of damage by all ropes is formed), calculate the vectorial d of the current name damage of cable system according to multi-objective optimization algorithm
^{i} _{c}Noninferior solution.
The multi-objective optimization algorithm that can adopt has a variety of, for example: based on the multiple-objection optimization of genetic algorithm, based on the multiple-objection optimization of artificial neural network, based on the multi-objective optimization algorithm of population, multiple-objection optimization, leash law (Constrain Method), weighted method (Weighted Sum Method), goal programming method (Goal Attainment Method) or the like based on ant group algorithm.Because various multi-objective optimization algorithms all are conventional algorithms, can realize easily that this implementation step is that example provides and finds the solution the vectorial d of current name damage with the goal programming method only
^{i} _{c}Process, the specific implementation process of other algorithm can realize in a similar fashion according to the requirement of its specific algorithm.
According to the goal programming method, formula (9) can transform the multi-objective optimization question shown in an accepted way of doing sth (21) and the formula (22), γ in the formula (21)
^{i}Be a real number, R is a real number field, and area of space Ω has limited vectorial d
^{i} _{c}Span (the present embodiment requirements vector d of each element
^{i} _{c}Each element be not less than 0, be not more than 1).The meaning of formula (21) is to seek the real number γ of an absolute value minimum
^{i}, make formula (22) be met.G (d in the formula (22)
^{i} _{c}) by formula (23) definition, weighing vector W in the formula (22)
^{i}With γ
^{i}Product representation formula (22) in G (d
^{i} _{c}) and vectorial g
^{i}Between the deviation that allows, g
^{i}Definition referring to formula (15), its value will the 8th the step calculate.Vector W during actual computation
^{i}Can with vectorial g
^{i}Identical.The concrete programming of goal programming method realizes having had universal program directly to adopt.Just can be according to the goal programming method in the hope of the vectorial d of current name damage
^{i} _{c}
minimize γ
^{i}
γ
^{i}∈R，
${d}_{c}^{i}\∈\mathrm{\Ω}---\left(21\right)$
$G\left({d}_{c}^{i}\right)-{W}^{i}{\mathrm{\γ}}^{i}\≤{g}^{i}---\left(22\right)$
$G\left({d}_{c}^{i}\right)=\mathrm{abs}({\mathrm{\ΔC}}^{i}\·{d}_{c}^{i}-{C}^{i}+{C}_{o}^{i})---\left(23\right)$
Try to achieve the vectorial d of current name damage
^{i} _{c}After, can be according to the vectorial d of the current actual damage that formula (17) obtain
^{i}Each element, current actual damage vector d
^{i}Have reasonable error exactly but can be more exactly from all ropes, determine the position of damaged cable and separating of degree of injury thereof.If the current actual damage vector d that solves
^{i}The numerical value of a certain element be 0, represent that the pairing rope of this element is intact, not damage; If its numerical value is 100%, represent that then the pairing rope of this element has completely lost load-bearing capacity; If its numerical value between 0 and 100%, is then represented this rope and has been lost the load-bearing capacity of corresponding proportion.
6. data systematic function.Can regularly or by the personnel operation health monitoring systems generate cable system health condition form.
7. warning function.Under specified requirements, automatically the operation communication panalarm to monitor staff, owner and (or) personnel of appointment report to the police.
The 7th step: at Mechanics Calculation benchmark model A
^{i}Carry out the several times Mechanics Calculation on the basis, equal the quantity of all ropes on the calculation times numerical value, there is N root rope that N calculating is just arranged, calculating each time in the hypothesis cable system has only a rope to increase unit damage again on the basis of original damage, the rope that occurs damage in calculating each time in this circulation is different from the rope that occurs damage in other time calculating, and supposition each time has the unit damage value of the rope of damage can be different from the unit damage value of other ropes, uses " nominal unit damage vector D
^{i} _{u}" write down the unit damage of the supposition of all ropes; calculate (for example adopting finite element method) each time and obtain that all specify the current numerical value of monitored amount in the Cable Structure, the current numerical value of the monitored amount of all that calculate is formed one " the current numerical value vector of the calculating of monitored amount " each time.When hypothesis j root rope has unit damage, available C
^{i} _{Tj}" the current numerical value vector of the calculating of monitored amount " that expression is corresponding.That that calculates " the current numerical value of the calculating of monitored amount vector C each time
^{i} _{Tj}" deduct " initial value of monitored amount vector C
^{i} _{o}" obtain a vector, during all calculating divided by this, each element that again should vector obtains " numerical value change vector δ a C of monitored amount after the unit damage value of supposition
^{i} _{j}"; There is N root rope that N " the numerical value change vector δ C of monitored amount just arranged
^{i} _{j}" (j=1,2,3 ..., N).Form " the unit damage monitored numerical quantity transformation matrices Δ C that the N row are arranged successively by this N " the numerical value change vector of monitored amount "
^{i}"; " unit damage monitored numerical quantity transformation matrices Δ C in other words
^{i}" each row (for example j row) corresponding to one " the numerical value change vector of monitored amount " (δ C for example
^{i} _{j}).The coding rule of the row of " unit damage monitored numerical quantity transformation matrices " and the vectorial d of current name damage
^{i} _{c}With current actual damage vector d
^{i}The element coding rule identical.
Reach in this step when giving each vectorial element numbering thereafter, should use same coding rule with other vector among the present invention, can guarantee any one element in each vector in this step like this, with element in other vector, that numbering is identical, expressed the relevant information of same monitored amount or same target.
The 8th step: set up linear relationship error vector e
^{i}With vectorial g
^{i}Utilize data (" the initial value vector C of monitored amount of front
^{i} _{o}", " unit damage monitored numerical quantity transformation matrices Δ C
^{i}"), when the 7th step calculated each time, promptly in calculating each time, have only in the hypothesis cable system increase unit damage again on the basis of rope in original damage in, calculate each time and form a vectorial d of damage
^{i} _{t}, damage vectorial d
^{i} _{t}Element number equal the quantity of rope, vectorial d
^{i} _{t}All elements in have only the numerical value of an element to get to calculate each time in hypothesis increase the unit damage value of the rope of unit damage, d
^{i} _{t}The numerical value of other element get 0, that is not numbering and the supposition of 0 the element corresponding relation that increases the rope of unit damage, be identical with the element of the same numbering of other vectors with the corresponding relation of this rope; With C
^{i} _{Tj}, C
^{i} _{o}, Δ C
^{i}, d
^{i} _{t}Bring formula (13) into, obtain a linear relationship error vector e
^{i}, calculate a linear relationship error vector e each time
^{i}There is N root rope that N calculating is just arranged, N linear relationship error vector e just arranged
^{i}, with this N linear relationship error vector e
^{i}Obtain a vector after the addition, the new vector that each element of this vector is obtained after divided by N is exactly final linear relationship error vector e
^{i}Vector g
^{i}Equal final error vector e
^{i}With vectorial g
^{i}Be kept on the hard disc of computer of operation health monitoring systems software, use for health monitoring systems software.
The 9th step: will " the initial value vector C of monitored amount
^{i} _{o}" and " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" etc. parameter be kept on the hard disc of computer of operation health monitoring systems software in the mode of data file.
The tenth step: the cable system health monitoring systems system (containing hardware and software) of operation Cable Structure, finish following several function:
1. read " the initial value vector C of monitored amount the data file on being stored in hard disc of computer
^{i} _{o}" and Cable Structure " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" wait all call parameters.
2. the signal that transmits by signal picker of (or trigger-type) at random record regularly.
3. the signal to record carries out signal Processing, calculates the current numerical value of all monitored amounts to be measured, and the current numerical value of all monitored amounts is formed " the current numerical value vector C of monitored amount
^{i}".When numbering to the element of the institute's directed quantity that occurred before this step and this step, should use same coding rule, can guarantee the element each vector, that numbering is identical that occurs before this step and this step like this, represent same monitored amount, corresponding to vectorial defined relevant information under this element.
4. according to " current (calculating or actual measurement) numerical value vector C of monitored amount
^{i}" " the initial value of monitored amount vector C together
^{i} _{o}", " unit damage monitored numerical quantity transformation matrices Δ C
^{i}" and " the vectorial d of current name damage
^{i} _{c}" linear approximate relationship (seeing formula (9)) that exists between (the current nominal amount of damage by all ropes is formed), calculate the vectorial d of the current name damage of cable system according to multi-objective optimization algorithm
^{i} _{c}Noninferior solution.
5. data systematic function.Can regularly or by the personnel operation health monitoring systems generate cable system health condition form.
6. warning function.Under specified requirements, the operation communication panalarm is reported to the police to personnel such as monitoring automatically.
The 11 step:, promptly try to achieve the vectorial d of current name damage in the i time circulation in this circulation
^{i} _{c}After, set up mark vector F according to formula (18), formula (19)
^{i}
The 12 step: if mark vector F
^{i}Element be 0 entirely, then got back to for the tenth step and continue this circulation; If mark vector F
^{i}Element be not 0 entirely, then enter next step, i.e. the 13 step.
The 13 step: according to the initial damage vector d that formula (20) calculates next time, i.e. the i+1 time circulation is required
^{I+1} _{o}Each element d
^{I+1} _{Oj}
The 14 step: at Mechanics Calculation benchmark model A
^{i}The basis on, the health status that makes rope is the vectorial d that previous step calculates
^{I+1} _{o}After, obtain new Mechanics Calculation benchmark model, next time promptly the required Mechanics Calculation benchmark model A of (the i+1 time) circulation
^{I+1}
The 15 step: by to Mechanics Calculation benchmark model A
^{I+1}Calculate corresponding to model A
^{I+1}The numerical value of all monitored Suo Li of structure, these numerical value are formed next time, required vectorial C promptly circulates for the i+1 time
^{I+1} _{o}, i.e. the initial value vector of monitored amount.
The 16 step: got back to for the 7th step, beginning is circulation next time.