WO2019001025A1 - 一种用于结构局部变形与整体模态信息同时获取的传感器布设方法 - Google Patents

一种用于结构局部变形与整体模态信息同时获取的传感器布设方法 Download PDF

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WO2019001025A1
WO2019001025A1 PCT/CN2018/079405 CN2018079405W WO2019001025A1 WO 2019001025 A1 WO2019001025 A1 WO 2019001025A1 CN 2018079405 W CN2018079405 W CN 2018079405W WO 2019001025 A1 WO2019001025 A1 WO 2019001025A1
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displacement
strain
mode
matrix
freedom
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伊廷华
裴雪扬
李宏男
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大连理工大学
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/13—Architectural design, e.g. computer-aided architectural design [CAAD] related to design of buildings, bridges, landscapes, production plants or roads
    • 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
    • G01M5/0041—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings by determining deflection or stress
    • 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
    • G01M5/0083—Investigating the elasticity of structures, e.g. deflection of bridges or air-craft wings by measuring variation of impedance, e.g. resistance, capacitance, induction
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2111/00—Details relating to CAD techniques
    • G06F2111/10—Numerical modelling

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  • the invention belongs to the field of civil engineering structural health monitoring, and proposes a joint method of strain gauge and accelerometer for the local deformation of the structure and the acquisition of the overall modal information.
  • a good sensor layout solution should satisfy: 1) in a noisy environment, it is possible to obtain comprehensive and accurate structural parameter information using as few sensors as possible; 2) the measured structural response information should be able to be combined with the results of the numerical analysis. Corresponding; 3) capable of focusing on the vibration response data of interest through reasonable addition of sensors; 4) making the monitoring results have good visibility and robustness; 5) making equipment input, data transmission, and results of the monitoring system
  • the processing fee is the least.
  • strain gauges and accelerometers are used in large quantities, so it is of great practical value to study the joint layout method of the two.
  • the strain gauge and the accelerometer are jointly optimized to simultaneously acquire local deformation information and overall modal information of the structure.
  • the arrangement of the strain gauges not only requires consideration of large deformations of the structure, but also requires that the selected position contain as much displacement modal information as possible.
  • the obtained strain mode is used to estimate the structural displacement modes at other locations, and then the accelerometer's mathematical theory is increased according to the modal confidence criterion and the modal information redundancy to ensure that the acquired displacement modes are distinguishable and contain information redundancy. Less is less.
  • the strain gauge is mainly used to monitor the local deformation information of the structure, so it needs to be placed in the place where the structure has large deformation.
  • the bridge structure the strain gauge needs to be initially arranged at the position of the mid-span section.
  • Step 1 According to the finite element method, the structure is divided into individual units, each unit and node are numbered, and the section where the large deformation position of the structure is located is selected as the candidate position of the strain gauge.
  • strain strain mode
  • node displacement displacement mode
  • the subscript i indicates the number of the unit in which the strain is located; Indicates the strain displacement mode matrix corresponding to the position of the strain gauge in the unit; ⁇ i represents the node displacement mode matrix of the element, including the translational displacement mode and the rotational displacement mode in three directions; T i represents the strain mode The conversion relationship with the node displacement mode.
  • Each row of T i corresponds to one row of the strain mode matrix, corresponding to the position of one strain gauge; each column of T i corresponds to one row of the displacement mode matrix, that is, a displacement mode corresponding to one degree of freedom. Therefore, the amount of displacement modal information of each degree of freedom contained in the position of the strain gauge is determined by the magnitude of each variable in T i .
  • a certain variable in T i is 0, it means that the displacement modal information at the degree of freedom corresponding to the variable is not included in the strain mode.
  • the translational displacement mode is used more, so the selected strain gauge position needs to contain as much translational displacement modal information as possible. Therefore, it is necessary to ensure that the corresponding variable value in T i cannot be too small, and finally the position of the S1 strain gauges can be determined.
  • Step 2 According to the unit number of the strain section position obtained in step 1, the size of each variable in the T i matrix is checked according to formula (1). If the variable value is too small, fine tune the strain position to include as much displacement modal information as possible.
  • the position of the strain gauge obtained from steps 1 and 2 can ensure that the monitoring position is a large deformation position of the structure, and the strain gauge can contain as much structural displacement modal information as possible, which is very advantageous for the subsequent displacement mode estimation. From equation (1), the relationship between the strain mode of all strain gauge positions in the structure and the displacement modes of all nodes of the structure can be derived.
  • the strain mode corresponding to the position of the strain gauge can be calculated from the strain monitoring data. Due to the limitation of the strain count, the strain modal matrix The number of rows is smaller than the number of rows of the displacement modal matrix ⁇ , so it is not feasible to directly estimate the displacement modes of all nodes. At this time, only the displacement mode ⁇ r of some nodes can be estimated, r represents the degree of freedom corresponding to the selected displacement mode, and ⁇ r is the displacement mode matrix corresponding to the r degrees of freedom.
  • Step 3 According to the node displacement modal matrix of partial degrees of freedom, formula (2) can be rewritten as:
  • T r represents the r-column vector corresponding to the selected displacement modal degree of freedom in T
  • T nr is composed of the remaining nr column vectors in T
  • ⁇ nr is composed of the remaining nr row vectors in ⁇
  • n represents The number of rows of the ⁇ matrix is also the total number of degrees of freedom of the displacement mode.
  • w the error
  • each column w (i) is zero mean
  • Step 4 When the number of rows of the T r matrix is greater than the number of columns, the multiplicative multiple least squares method can be used to estimate the displacement mode at the selected degree of freedom.
  • Each column can be expressed as:
  • the covariance matrix can be written as:
  • Step 5 Each diagonal element in the covariance matrix indicates that the estimated displacement mode of the order corresponds to the magnitude of the estimation error at each degree of freedom, and therefore the trace of the covariance matrix can be used to represent the magnitude of the estimation error.
  • trace indicates the trace of the matrix
  • the magnitude of the error of all order displacement modal estimation is composed of the estimation errors of the estimated displacement modes of each order:
  • N represents the order of the displacement mode.
  • Equation (9) can be further written as:
  • the error magnitude of the estimated displacement modality corresponding to the selected degree of freedom is mainly determined by T r .
  • Different degrees of freedom selection correspond to different transformation matrices T r
  • different transformation matrices T r correspond to different estimation error sizes.
  • the degrees of freedom corresponding to the minimum estimation error are selected, and the displacement modes on these degrees of freedom are estimated from the strain modes.
  • the structural displacement modes obtained from the structural health monitoring system need to be distinguishable.
  • the modal confidence criterion (MAC) can be used to measure the distinguishability of structural displacement modes:
  • ⁇ *, i and ⁇ *, j are the i-th order displacement modal vector and the jth modal vector corresponding to the selected measuring point; the numerical magnitude of MAC i, j corresponds to the two-order modal vector Distinguishable.
  • MAC i,j If the value of MAC i,j is close to 0, it means that the two-order modal vectors are easily distinguished; if the value of MAC i,j is close to 1, it means that the two-order modal vectors are not easily distinguishable. In actual engineering, it is necessary to ensure that the values of the variables in the MAC matrix are as small as possible, generally less than 0.2.
  • a structural redundancy factor is defined to measure the degree of modal redundancy between displacement mode locations.
  • ⁇ i,j represents the redundancy coefficient between the i-th position and the j-th position in the finite element structure
  • F represents the Frobenius norm.
  • Step 1 Set a redundancy threshold h.
  • Step 2 Calculate the displacement mode estimated by the strain mode With the modal redundancy coefficient of the remaining position, the position corresponding to the coefficient exceeding the threshold is deleted.
  • Step 3 randomly select an accelerometer position from the remaining measuring points, add the existing sensor arrangement scheme of the structure, calculate the MAC matrix of the displacement modal matrix after adding the position, and obtain the maximum off-diagonal MAC max in the MAC matrix. , select the corresponding position with the minimum MAC max value.
  • Step 4 Verify that there is still a location to be selected, if yes, go back to step 2; if not, go to the next step.
  • Step 5 Verify the magnitude of the displacement modal MAC max and the number of selected positions corresponding to the resulting sensor arrangement. If the MAC max is less than 0.2 and the selected location is more, then return to step 1 to reduce the redundancy threshold h; if the condition is not met, the S2 accelerometer position is finally selected according to the magnitude of the MAC max value.
  • Step 6 The S1 strain gauges determined by the strain gauge selection process and the S2 accelerometers determined by the accelerometer selection process together form the final sensor arrangement.
  • the dual target sensor joint layout method proposed by the invention can monitor the strain information of the large deformation position of the structure, and can obtain the overall displacement mode matrix of the structure for other analysis. Therefore, the information of the strain gauge is fully utilized, and it can monitor the strain at the large deformation position, and can also be used to estimate the displacement mode of other node positions corresponding to the position strain mode.
  • the arrangement of the accelerometer makes the resulting displacement modal matrix have good distinguishability and low displacement modal information redundancy, ensuring the quality of the resulting displacement modal matrix.
  • Figure 1 is a schematic diagram of a finite element model of a bridge.
  • Figure 2 is a joint arrangement diagram of an accelerometer and a strain gauge.
  • the method utilizes a two-span highway bridge reference model for verification calculations.
  • Figure 1 shows the finite element diagram of the bridge model.
  • the model has a total of 177 nodes. Each node considers the translational displacement and rotational displacement of six degrees of freedom, namely x, y, and z.
  • the beam section is an I-beam section, model number S3 ⁇ 5.7.
  • the Euler beam element is used to simulate the structure, and the relationship between the structural strain mode and the displacement mode is analyzed. After the relationship between the strain mode and the displacement mode is determined, the method of assembling the strain gauge and the accelerometer proposed by the present invention can be used.
  • the first step uses the strain gauges in the invention to select the corresponding steps to determine the position of the strain gage: firstly, the four mid-span positions on the main beam are used as the cross-sectional position of the strain gauge arrangement; then the transformation matrix of the strain mode and the displacement mode is utilized. Adjust the position of the strain gauge; finally, a total of 16 strain gauges are arranged at the four corners of the four mid-sections. These positions correspond to the large deformation position of the structure, and also ensure that these positions contain as many displacement modes as possible. information.
  • the second step uses the accelerometer in the invention to select the corresponding step to determine the position of the accelerometer. After several calculations, it is finally determined that the redundancy threshold h is 0.5, and a total of 7 accelerometer positions are selected to ensure that the MAC max value is as small as possible.
  • Figure 2 shows the combined results of the final accelerometer and strain gage, where the hollow rectangle represents the accelerometer position and the specific position of the strain gage on the I-beam section is indicated by a solid rectangle.

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Abstract

一种针对结构局部变形和整体模态信息同时获取的传感器布设方法，包括：(1)将应变计布置在结构的大变形位置用于监测结构局部变形信息，并调整应变计所在位置使其包含尽可能多的重要位移模态信息；(2)使用应变计位置的应变模态估计结构的位移模态，并增加加速度计以改善估计所得位移模态的可区分性，同时减少获得的位移模态间的信息冗余；其中应变计的布置方案既能给出结构关键位置的局部变形信息又能得到精确的结构位移模态信息，加速度计的布置方案改善应变计数据估计所得的位移模态信息。该方法充分利用了应变计和加速度计所包含的不同结构信息，得到高质量的结构整***移模态信息。

Description

一种用于结构局部变形与整体模态信息同时获取的传感器布设方法 技术领域
本发明属于土木工程结构健康监测领域，针对结构局部变形和整体模态信息的获取，提出了一种应变计和加速度计联合布设方法。
背景技术
结构健康监测***的建立首先需要进行传感器的选型与优化布设，不适当的传感器布设将影响参数识别的精度；而且传感器本身也需要一定的成本，与其配套使用的数据采集与处理设备的代价也都较高，从经济方面考虑，工程师希望采用尽可能少的传感器达到监测的目的。一种好的传感器布设方案应满足：1)在含噪音的环境中，能够利用尽可能少的传感器获取全面而精确的结构参数信息；2)测得的结构响应信息应能够与数值分析的结果相对应；3)能够通过合理添加传感器对感兴趣的振动响应数据进行重点采集；4)使得监测结果具有良好的可视性和鲁棒性；5)使监测***的设备投入、数据传输、结果处理的费用最少。
在一套完整的结构健康监测***中，应变计和加速度计被大量使用，因此研究两者的联合布设方法，具有重要的实用价值。
发明内容
本发明中应变计和加速度计被共同优化布设，来同时获取结构的局部变形信息和整体模态信息。应变计的布置位置不仅需要考虑结构的大变形，还需要使选择的位置包含尽可能多的位移模态信息。获取的应变模态被用来估计其他位置的结构位移模态，然后依据模态置信准则和模态信息冗余度增加加速度计的数理，以保证获取的位移模态的可区分且包含信息冗余较少。
一、应变计位置的选择
在结构健康监测***中，应变计主要被用于监测结构的局部变形信息，因此需要将其布置在结构出现大变形的地方。例如桥梁结构，应变计一开始需要选择布置在跨中截面位置处。
步骤1：依据有限元方法，将结构划分为各个单元，对各单元和节点进行编号，选取结构大变形位置所在截面作为应变计的待选位置。
由有限元方法可知，结构的应变(应变模态)和该应变所在单元处的节点位移(位移模态)有着一定的转换关系，由下式表示：
Figure PCTCN2018079405-appb-000001
式中：下标i表示应变所在单元的编号；
Figure PCTCN2018079405-appb-000002
表示该单元内应变计位置所对应的应变位移模态矩阵；φ i表示该单元的节点位移模态矩阵，包含三个方向的平动位移模态和转动位移模态；T i表示应变模态与节点位移模态之间的转换关系。
T i的每一行对应应变模态矩阵的一行，对应着一个应变计的位置；T i的每一列对应位移模态矩阵的一行，也就是对应一个自由度的位移模态。因此，应变计位置处所包含的各个自由度的位移模态信息的多少，是由T i中各个变量的数值大小决定的。当T i中某个变量为0时，即表示该变量所对应的自由度处的位移模态信息没有被包在应变模态中。在结构整体模态测试时，平动位移模态使用较多，所以所选的应变计位置需要包含尽可能多的平动位移模态信息。因此，需要保证T i中对应的变量数值不能过小，最终可确定出S1个应变计的位置。
步骤2：根据步骤1所得的应变截面位置所在的单元编号，依据公式(1)验算T i矩阵中各个变量的大小。如果变量数值过小，微调应变位置，使其包含尽可能多的位移模态信息。
由步骤1和步骤2所得的应变计位置，既能确保监测位置是结构的大变形 位置，又能使应变计包含尽可能多的结构位移模态信息，对后面的位移模态估计十分有利。由公式(1)，可以推出结构中所有应变计位置的应变模态和结构所有节点位移模态的关系。
Figure PCTCN2018079405-appb-000003
式中：
Figure PCTCN2018079405-appb-000004
为所有应变计位置对应的应变模态；φ为结构的所有节点的位移模态；T表示应变模态与位移模态之间的转换关系矩阵。
应变计位置所对应的应变模态可由应变监测数据计算，由于应变计数量的限制，应变模态矩阵
Figure PCTCN2018079405-appb-000005
的行数小于位移模态矩阵φ的行数，因此直接估计所有节点的位移模态并不可行。这时只能估计部分节点的位移模态φ r，r表示选择的位移模态对应的自由度，φ r是这r个自由度所对应的位移模态矩阵。
步骤3：根据部分自由度的节点位移模态矩阵，公式(2)可改写为：
Figure PCTCN2018079405-appb-000006
式中：T r表示T中对应于选择的位移模态自由度的r列向量；T n-r则由T中剩余的n-r列向量组成；φ n-r则由φ中的剩余n-r行向量组成；n代表着φ矩阵的行数，也是位移模态的自由度总数。
考虑在实际工程中，通过应变监测数据计算得到的应变模态有时会和结构实际应变模态有所不同，即存在一定的误差。误差来源主要是测量误差和结构的模型误差所示。为此，公式(3)可以进一步写成：
Figure PCTCN2018079405-appb-000007
式中：w表示误差，一般可假定为平稳高斯噪声，每一列w (i)均为零均值，协方差为Cov(w (i))＝σ iI。
步骤4：当T r矩阵的行数大于列数时，即可利用多元多重的最小二乘法来估计所选自由度处的位移模态。
Figure PCTCN2018079405-appb-000008
式中：
Figure PCTCN2018079405-appb-000009
为所选自由度处的估计所得位移模态。
Figure PCTCN2018079405-appb-000010
的每一列可以表示为：
Figure PCTCN2018079405-appb-000011
式中：
Figure PCTCN2018079405-appb-000012
表示
Figure PCTCN2018079405-appb-000013
矩阵的第i列，也代表着第i阶模态。
Figure PCTCN2018079405-appb-000014
符合多元正态分布，协方差矩阵可以写成：
Figure PCTCN2018079405-appb-000015
步骤5：协方差矩阵中的各个对角元，表示该阶估计位移模态对应各个自由度处估计误差的大小，因此可用协方差矩阵的迹来表示估计误差的大小。
Figure PCTCN2018079405-appb-000016
式中：trace表示对矩阵求迹；
Figure PCTCN2018079405-appb-000017
表示i阶位移模态的估计误差大小。
所有阶数位移模态估计误差的大小，由各阶估计位移模态的估计误差组合而成：
Figure PCTCN2018079405-appb-000018
式中：N表示位移模态的阶数。
式(9)可以进一步写成：
Figure PCTCN2018079405-appb-000019
由式(10)可以看出，所选自由度对应的估计位移模态的误差大小主要由T r决定。不同的自由度选择对应着不同的转换矩阵T r，不同的转换矩阵T r对应着不同的估计误差大小。最终选择对应最小估计误差的自由度，这些自由度上的位移模态由应变模态估计得出。
二、加速度计位置的选择
从结构健康监测***中获得的结构位移模态需要具有一定的可区分性，模态置信准则(MAC)可被用于衡量结构位移模态的可区分性：
Figure PCTCN2018079405-appb-000020
式中：φ *,i和φ *,j分别为选取测点对应的第i阶位移模态向量和第j阶模态向量；MAC i,j的数值大小对应着这两阶模态向量的可区分度。
如果MAC i,j的值接近于0，则表示这两阶模态向量容易被区分；如果MAC i,j的值接近于1，则表示这两阶模态向量不容易区分。在实际工程中，需要保证MAC矩阵中的各个变量的数值要尽量小，一般小于0.2即可。
考虑到结构的空间连续性，当两个结构位移模态位置太过于接近时，这两处的位移模态会非常类似。这就表示这两个位置包含着近似的模态信息，造成了模态信息的冗余。过多的冗余模态信息显然会造成获取信息的浪费，这是需要避免的。这里，定义了一个结构冗余系数来衡量位移模态位置之间的模态冗余程度。
Figure PCTCN2018079405-appb-000021
式中：γ i,j代表在有限元结构中，第i个位置和第j个位置间的冗余度系数，下标F表示Frobenius范数。当γ i,j的值接近于1时，表示两个位置之间的模态冗余度很大，包含着几乎一样的位移模态信息。此时，这两个位置没有必要同时存在，需要删除一个位置。实际操作时，可设置一个合适的冗余度阈值h，如果冗余度系数大于该冗余度阈值，则对应的测点位置将被删除。
步骤1：设定一个冗余度阈值h。
步骤2：计算由应变模态估计所得的位移模态
Figure PCTCN2018079405-appb-000022
与剩余位置的模态冗余度系数，将超过阈值的系数对应的位置删除。
步骤3：从剩余测点中随机选择一个加速度计位置，加入结构已有的传感器布置方案中，计算加入该位置后位移模态矩阵的MAC矩阵，得出MAC矩阵中的最大非对角MAC max，选择与最小MAC max数值时的相应位置。
步骤4：检验是否还存在待选测点位置，如果有，返回步骤2；如果没有，进入下一步。
步骤5：检验最终所得的传感器布置所对应的位移模态MAC max的大小和所选的位置数量。如果MAC max小于0.2，且所选位置较多，则返回步骤1，减小冗余度阈值h；如果不符合条件，则依据MAC max数值的大小，最终选择S2加速度计位置。
步骤6：由应变计选择过程确定的S1个应变计和由加速度计选择过程确定的S2个加速度计，共同构成了最终的传感器布置方案。
本发明的有益效果：
本发明提出的双目标传感器联合布设方法，可以监测结构大变形位置的应变信息，且能得到结构的整***移模态矩阵以用于其他分析。因此，应变计的信息得到了充分利用，既可以监测大变形位置的应变，也可用于对应位置应变模态来估计其他节点位置的位移模态。此外，加速度计的布设使得最终所得的位移模态矩阵具有良好的可区分性和较低的位移模态信息冗余度，确保所得位移模态矩阵的质量。
附图说明
图1是桥梁有限元模型示意图。
图2是加速度计与应变计联合布置图。
具体实施方式
以下结合附图和技术方案，进一步说明本发明的具体实施方式。
实施例
本方法利用一个两跨公路桥梁基准模型来进行验证计算。图1给出了桥梁模型的有限元图，模型共有177个节点，每个节点考虑6个自由度即x，y，z三个方向的平动位移和转动位移。梁截面为工字钢截面，型号为S3×5.7。采用欧拉梁单元来模拟结构，分析结构应变模态与位移模态的关系。应变模态与位移模态的关系被确定后，既可采用本发明提出的应变计与加速度计联合布设方法。
第一步利用发明中的应变计选择对应步骤来确定应变计的位置：首先采用主梁上的四个跨中位置作为应变计布置的截面位置；然后利用应变模态和位移模态的转换矩阵调整应变计位置；最终在四个跨中截面的四个边角处共布置了16个应变计，这些位置既对应了结构的大变形位置，也保证了这些位置包含尽可能多的位移模态信息。
第二步利用发明中的加速度计选择对应步骤来确定加速度计的位置。经过多次计算，最终确定了冗余度阈值h的大小为0.5，共7个加速度计位置被选择保证MAC max数值尽量小。
图2给出了最终的加速度计与应变计的联合布设结果，其中空心矩形表示加速度计位置，应变计在工字钢截面上的具***置由实心矩形表示。

Claims (1)

  1. 一种用于结构局部变形与整体模态信息同时获取的传感器布设方法，其特征在于，步骤如下：
    步骤1：依据有限元方法，将结构划分为各个单元，对各单元和节点进行编号，选取S1个结构大变形位置所在截面作为应变计的待选位置；
    结构的应变即应变模态和该应变所在单元处的节点位移即位移模态存在如下转换关系：
    Figure PCTCN2018079405-appb-100001
    式中：下标i表示应变所在单元的编号；
    Figure PCTCN2018079405-appb-100002
    表示该单元内应变计位置所对应的应变位移模态矩阵；φ i表示该单元的节点位移模态矩阵；T i表示应变模态与节点位移模态之间的转换关系；T i的每一行对应应变模态矩阵的一行，对应着一个应变计的位置；T i的每一列对应位移模态矩阵的一行，也就是对应一个自由度的位移模态；
    步骤2：根据步骤1所得的应变截面位置所在的单元编号，依据公式(1)验算T i矩阵中各个变量的大小；如果变量数值过小，则在应变计原来的位置附近重新选择，使其包含尽可能多的位移模态信息；
    由公式(1)，推出结构中所有应变计位置的应变模态和结构所有节点位移模态的关系：
    Figure PCTCN2018079405-appb-100003
    式中：
    Figure PCTCN2018079405-appb-100004
    为所有应变计位置对应的应变模态；φ为结构的所有节点的位移模态；T表示应变模态与位移模态之间的转换关系矩阵；
    应变计位置所对应的应变模态由应变监测数据计算，由于应变计数量的限制，应变模态矩阵
    Figure PCTCN2018079405-appb-100005
    的行数小于位移模态矩阵φ的行数，直接估计所有节点的位移模态并不可行；这时只能估计部分节点的位移模态φ r，r表示选择的位移模 态对应的自由度，φ r是这r个自由度所对应的位移模态矩阵；
    步骤3：根据部分自由度的节点位移模态矩阵，公式(2)改写为：
    Figure PCTCN2018079405-appb-100006
    式中：T r表示T中对应于选择的位移模态自由度的r列向量；T n-r则由T中剩余的n-r列向量组成；φ n-r则由φ中的剩余n-r行向量组成；n代表着φ矩阵的行数，也是位移模态的自由度总数；
    考虑在实际工程中，通过应变监测数据计算得到的应变模态和结构实际应变模态有所不同，即存在一定的误差；误差来源主要是测量误差和结构的模型误差所示，为此，公式(3)进一步写成：
    Figure PCTCN2018079405-appb-100007
    式中：w表示误差，假定为平稳高斯噪声，每一列w (i)均为零均值，协方差为Cov(w (i))＝σ iI；
    步骤4：当T r矩阵的行数大于列数时，利用多元多重的最小二乘法来估计所选自由度处的位移模态
    Figure PCTCN2018079405-appb-100008
    式中：
    Figure PCTCN2018079405-appb-100009
    为所选自由度处的估计所得位移模态；
    Figure PCTCN2018079405-appb-100010
    的每一列表示为：
    Figure PCTCN2018079405-appb-100011
    式中：
    Figure PCTCN2018079405-appb-100012
    表示
    Figure PCTCN2018079405-appb-100013
    矩阵的第i列，也代表着第i阶模态；
    Figure PCTCN2018079405-appb-100014
    符合多元正态分布，协方差矩阵写成：
    Figure PCTCN2018079405-appb-100015
    步骤5：用协方差矩阵的迹来表示估计误差的大小：
    Figure PCTCN2018079405-appb-100016
    式中：trace表示对矩阵求迹；
    Figure PCTCN2018079405-appb-100017
    表示i阶位移模态的估计误差大小；
    所有阶数位移模态估计误差的大小，由各阶估计位移模态的估计误差组合而成：
    Figure PCTCN2018079405-appb-100018
    式中：N表示位移模态的阶数；
    式(9)进一步写成：
    Figure PCTCN2018079405-appb-100019
    由式(10)看出，所选自由度对应的估计位移模态的误差大小主要由T r决定；不同的自由度选择对应着不同的转换矩阵T r，不同的转换矩阵T r对应着不同的估计误差大小；最终选择对应最小估计误差的自由度，自由度上的位移模态由应变模态估计得出。
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