传感器布设中有效独立法的简捷快速算法
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摘要
本文提出一个通过正交三角分解快速计算有效独立法系数的方法。有效独立法是传感器布设中影响最广泛的一种方法,它使目标模态尽可能线性独立。传统的有效独立法计算需要对信息阵进行特征值分解或者计算矩阵逆,计算量较大。本文在笔者以前得出的有效独立法与模态动能法关系的基础上,提出先通过对模态矩阵进行正交三角分解(QR),然后比较其行范数即可得到有效独立法的系数,进而对各待选传感器位置进行排序并迭代依次删除。在待选传感器位置比感兴趣的模态数较多时,该法的计算效率明显提高。同时,采用正交三角分解删除一行后的更新算法,进一步提高了迭代计算的效率。最后通过I-40桥的算例表明该法的有效性。
A new simple and fast computation algorithm of the Effective Independence for sensor placement is presented.It is based on downdating of the QR decomposition of the reduced modal matrix.The Effective Independence method,which tries to make the target modes as linearly independent as possible,is significant and influential in the field of sensor placement.Traditional computation of the Effective Independence requires eigenvalue decomposition or inversion of the Fisher Information matrix,and both are costly.By virtue of the connection between the Modal Kinetic Energy and Effective Independence,we can compare the norms of the orthonormal rows after the QR decomposition of the modal matrix to compute the Effective Independence.Moreover,new downdating steps of the QR decomposition based on the combination of modified Gram-Schmidt and Householder transformations are implemented,which further improved its efficiency by eliminating unnecessary expensive QR decompositions in the iterations.Finally,numerical test on the I-40 Bridge shows that the new algorithm computes far more quickly with fewer computation steps,has the advantages of simpleness and clarity in physical significance,and is quite stable.
A new simple and fast computation algorithm of the Effective Independence for sensor placement is presented.It is based on downdating of the QR decomposition of the reduced modal matrix.The Effective Independence method,which tries to make the target modes as linearly independent as possible,is significant and influential in the field of sensor placement.Traditional computation of the Effective Independence requires eigenvalue decomposition or inversion of the Fisher Information matrix,and both are costly.By virtue of the connection between the Modal Kinetic Energy and Effective Independence,we can compare the norms of the orthonormal rows after the QR decomposition of the modal matrix to compute the Effective Independence.Moreover,new downdating steps of the QR decomposition based on the combination of modified Gram-Schmidt and Householder transformations are implemented,which further improved its efficiency by eliminating unnecessary expensive QR decompositions in the iterations.Finally,numerical test on the I-40 Bridge shows that the new algorithm computes far more quickly with fewer computation steps,has the advantages of simpleness and clarity in physical significance,and is quite stable.
引文
[1]李宏男,李东升.土木工程结构安全性评估、健康监测及诊断述评[J].地震工程与工程振动,2002,22(3):82-90.Li H N,Li D S.Safety assessment,health monitoringand damage diagnosis for structures in civil engineer-ing[J].Earthquake Engineering and Engineering Vi-bration,2002,22(3):82-90.
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[5]De Clerk J P,Avitabile P.Development of severalnew tools for modal pre-test evaluation[C]∥Proceed-ings of the 10thInternational Modal Analysis Confer-ence.Schenectady,NY:Union College Press,1996.1272-1277.
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[7]Kammer D C.Sensor placement for on-orbit Modal i-dentification and correlation of large space structures[J].Journal of Guidance,Control and Dynamics,1991,14(9):251-259.
[8]Golub G H.Van Loan C P.Matrix Computation[M].Baltimore,MD:The Johns Hopkins University Press,1996.
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[11]Farrar C R,Baker W E,Bell T,et al.Dynamic charac-terization and damage detection in the I-40 Bridge overthe Rio Grande[R].Los Alamos National Laboratoryreport LA-12767-MS,1994.
[2]Kubrusly C S,Malebranche H.Sensors and con-trollers location in distributed systems——a survey[J].Automatica,1985,(21):117-128.
[3]刘福强,张玲弥.作动器/传感器优化配置的研究进展[J].力学进展,2000,25(4):506-516.Liu F Q,Zhang L M.Advances in optimal placementof actuators and sensors[J].Advances in mechanics,2000,25(4):506-516.
[4]刘效尧,蔡健,刘晖,桥梁损伤诊断[M].北京:人民交通出版社,2002.Liu X Y,Cai J,Liu H.The Damage Diagnosis ofBridges[M].Beijing:China Communications Press,2002.
[5]De Clerk J P,Avitabile P.Development of severalnew tools for modal pre-test evaluation[C]∥Proceed-ings of the 10thInternational Modal Analysis Confer-ence.Schenectady,NY:Union College Press,1996.1272-1277.
[6]Li D S,Li HN,Fritzen C P.The connection betweenEffective Independence and Modal Kinetic Energymethods for sensor placement[J].Journal of Soundand Vibration,2007,305(4-5):945-955.
[7]Kammer D C.Sensor placement for on-orbit Modal i-dentification and correlation of large space structures[J].Journal of Guidance,Control and Dynamics,1991,14(9):251-259.
[8]Golub G H.Van Loan C P.Matrix Computation[M].Baltimore,MD:The Johns Hopkins University Press,1996.
[9]Bjǒrck.Numerical Methods for Least SquaresProblems[M].Philadelphia:SIAM.1996.
[10]Yoo K,Park H.Accurate downdating of a modifiedGram-Schmidt QR decomposition[J].BIT NumericalMathematics,1996,36(1):166-181.
[11]Farrar C R,Baker W E,Bell T,et al.Dynamic charac-terization and damage detection in the I-40 Bridge overthe Rio Grande[R].Los Alamos National Laboratoryreport LA-12767-MS,1994.
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