线弹性系统的分布动载荷识别理论与方法
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摘要
本文研究Euler-Bernoulli梁和弹性薄板上稳态和瞬态的分布动载荷的识别问题,期望找到简单却重要的性质或规律,并谋求建立新颖、简洁、有效的识别理论和方法。
由于实际测量所得数据总是空间上极为有限的离散信息,远比时间上获得的信息要少,通常达不到有限元法划分单元所需要的基本的节点数,因此应用有限元的方法不太合适;同时,线弹性系统上的分布动载荷识别问题本质上是无限维问题。因此,为近似识别出整个空间上动载荷的分布情况,需要应用模态方法进行空间坐标的变换,并应用有限维近似。由于测量数据为有限的离散信息,无法直接应用Galerkin投影法进行有限维近似,而需要将Galerkin投影法和配置投影法结合进行有限维近似,本文将其命名为“近似投影法”。
线弹性系统上的分布动载荷识别是不适定问题,而不适定问题存在信息不足的困扰,通常需要寻找附加信息。作者认为响应的时间信息和空间信息之间一定隐含着重要的关联信息,即使所获得的响应信息是部分而离散的。当Euler-Bernoulli梁承受单模态分布的简谐动载荷时,其动载荷和响应间呈现出简单而重要的时空关系,由此本文从中提出“缩放因子”的概念。
自然界生物的各种感知器官本质上都是识别系统,而其敏感区域总是有限却各不相同。结合这种自然而合理的有限性识别和“重要的事物一定会产生重要的影响”的想法,本文提出“识别有限性”的假设。融合这种假设和缩放因子概念,从而提出“模态选择法”并建立线弹性系统上稳态分布动载荷的识别理论。“近似投影法”和“模态选择法”实质上是对线弹性系统的分布动载荷识别这个无限维不适定问题,进行物理概念上的策略化的正则化处理;如有必要,之后还可以运用其它正则化方法。
在线弹性系统的固定边界附近,分布动载荷识别有很大难度。如果采用系统固有模态或其修改型为基函数来表达动载荷的空间分布,那么固定边界处的动载荷将无法识别。为此,作者提出动载荷空间分布的“一致性表述”概念,并初步尝试运用Legendre多项式进行表达,在固定边界处获得很好的效果。
小波变换具有分析局部信息的优势,但是对于动力学系统的输入、输出信息的变换没有简单好用的性质,因此应用起来很困难。作者提出“小波近似法”识别瞬态动载荷,获得不错的效果。同时,从非零初始条件的响应中识别瞬态动载荷一般都有困难,作者提出“分段识别”的想法,在小波近似法中获得很好的成功。
运用上述方法和概念,本文建立了Euler-Bernoulli梁和弹性薄板上稳态分布动载荷的识别理论,以及Euler-Bernoulli梁上的瞬态分布动载荷的识别理论。数值模拟显示新理论可给出很好的识别结果,并可揭示出若干新问题。
This dissertation deals with the reconstruction of distributed dynamic loads on Euler-Bernoulli beam and elastic thin plate, to find simple but important properties or principles underlying the problem, and to propose new, concise and effective reconstruction theory.
The finite element method cannot find its application in the situation, since the discrete data from engineering measurement is usually on only a small number of spatial points, which is much less than the data on time domain. However, the reconstruction of distributed dynamic loads on a linear elastic system is originally a problem involving infinite dimensions. Therefore, to approximately reconstruct the spatial distribution of the dynamic loads, both modal transformation and finite dimensional approximation are employed. Both Galerkin method and collocation method are applied together as projection methods to transform the original infinite dimensional problem into a finite dimensional one, and this coupling application is named approximate projection method.
The reconstruction of distributed dynamic loads on a linear elastic system is an ill-posed problem, which always needs additional information due to lack of information. The authors believe that some important connection lies between the spatial data and the temporal data in the partial discrete response data. When an Euler-Bernoulli beam is excited by a harmonic load with single mode-shape distribution, the simple and important relationship finally emerges between the spatial data and the temporal data during the transformation from load to response, and the concept of scale factor is thus proposed.
Animals’perceptual organs are naturally identification system, all of which are sensitive in finite but different bandwidth. Combing the naturally reasonable limit and the idea that important things will cause important effect, it is naturally to infer that finite reconstruction is also sensible. Based on the recognition and the concept of scale factor, a new reconstruction theory is found with the proposition of the mode selection method. The approximate projection method and the mode selection method are applied basically as physical regularization strategy to deal with the infinite-dimensional and ill-posed reconstruction problem, before applying other purely mathematical regularization methods.
Load reconstruction near the fixed spatial boundaries is usually hard. The loads near the fixed spatial boundaries are impossible to be correctly reconstructed, if the natural modal functions or their modified forms are applied as base functions to express the spatial distribution of the load as generalized Fourier series. To tackle this problem, the concept of consistent spatial expression is proposed, and Legendre polynomials are applied as the consistent spatial base functions, which result in good effect in numerical simulations.
Wavelet transform is not conveniently applied on the load reconstruction of a dynamic system, since it has no good and simple properties on the relationship between the input and output of a dynamic system, though it dose have profits on local analysis of signals. Nevertheless, the authors proposed the method of wavelet approximation to reconstruct the distributed dynamic loads on linear elastic system, and relatively good results are obtained. Meanwhile, as difficulties usually confront researchers identifying transient force from response with non-zero initial conditions, the method of fragment analysis is proposed to deal with this problem, and the method succeeds cooperating with the wavelet approximation method.
Based on abovementioned methods and concepts, the reconstruction theories of distributed dynamic loads on both Euler-Bernoulli beam and elastic thin plate are proposed. The numerical simulations show good accordance with the theories, and many new phenomena are disclosed.
由于实际测量所得数据总是空间上极为有限的离散信息,远比时间上获得的信息要少,通常达不到有限元法划分单元所需要的基本的节点数,因此应用有限元的方法不太合适;同时,线弹性系统上的分布动载荷识别问题本质上是无限维问题。因此,为近似识别出整个空间上动载荷的分布情况,需要应用模态方法进行空间坐标的变换,并应用有限维近似。由于测量数据为有限的离散信息,无法直接应用Galerkin投影法进行有限维近似,而需要将Galerkin投影法和配置投影法结合进行有限维近似,本文将其命名为“近似投影法”。
线弹性系统上的分布动载荷识别是不适定问题,而不适定问题存在信息不足的困扰,通常需要寻找附加信息。作者认为响应的时间信息和空间信息之间一定隐含着重要的关联信息,即使所获得的响应信息是部分而离散的。当Euler-Bernoulli梁承受单模态分布的简谐动载荷时,其动载荷和响应间呈现出简单而重要的时空关系,由此本文从中提出“缩放因子”的概念。
自然界生物的各种感知器官本质上都是识别系统,而其敏感区域总是有限却各不相同。结合这种自然而合理的有限性识别和“重要的事物一定会产生重要的影响”的想法,本文提出“识别有限性”的假设。融合这种假设和缩放因子概念,从而提出“模态选择法”并建立线弹性系统上稳态分布动载荷的识别理论。“近似投影法”和“模态选择法”实质上是对线弹性系统的分布动载荷识别这个无限维不适定问题,进行物理概念上的策略化的正则化处理;如有必要,之后还可以运用其它正则化方法。
在线弹性系统的固定边界附近,分布动载荷识别有很大难度。如果采用系统固有模态或其修改型为基函数来表达动载荷的空间分布,那么固定边界处的动载荷将无法识别。为此,作者提出动载荷空间分布的“一致性表述”概念,并初步尝试运用Legendre多项式进行表达,在固定边界处获得很好的效果。
小波变换具有分析局部信息的优势,但是对于动力学系统的输入、输出信息的变换没有简单好用的性质,因此应用起来很困难。作者提出“小波近似法”识别瞬态动载荷,获得不错的效果。同时,从非零初始条件的响应中识别瞬态动载荷一般都有困难,作者提出“分段识别”的想法,在小波近似法中获得很好的成功。
运用上述方法和概念,本文建立了Euler-Bernoulli梁和弹性薄板上稳态分布动载荷的识别理论,以及Euler-Bernoulli梁上的瞬态分布动载荷的识别理论。数值模拟显示新理论可给出很好的识别结果,并可揭示出若干新问题。
This dissertation deals with the reconstruction of distributed dynamic loads on Euler-Bernoulli beam and elastic thin plate, to find simple but important properties or principles underlying the problem, and to propose new, concise and effective reconstruction theory.
The finite element method cannot find its application in the situation, since the discrete data from engineering measurement is usually on only a small number of spatial points, which is much less than the data on time domain. However, the reconstruction of distributed dynamic loads on a linear elastic system is originally a problem involving infinite dimensions. Therefore, to approximately reconstruct the spatial distribution of the dynamic loads, both modal transformation and finite dimensional approximation are employed. Both Galerkin method and collocation method are applied together as projection methods to transform the original infinite dimensional problem into a finite dimensional one, and this coupling application is named approximate projection method.
The reconstruction of distributed dynamic loads on a linear elastic system is an ill-posed problem, which always needs additional information due to lack of information. The authors believe that some important connection lies between the spatial data and the temporal data in the partial discrete response data. When an Euler-Bernoulli beam is excited by a harmonic load with single mode-shape distribution, the simple and important relationship finally emerges between the spatial data and the temporal data during the transformation from load to response, and the concept of scale factor is thus proposed.
Animals’perceptual organs are naturally identification system, all of which are sensitive in finite but different bandwidth. Combing the naturally reasonable limit and the idea that important things will cause important effect, it is naturally to infer that finite reconstruction is also sensible. Based on the recognition and the concept of scale factor, a new reconstruction theory is found with the proposition of the mode selection method. The approximate projection method and the mode selection method are applied basically as physical regularization strategy to deal with the infinite-dimensional and ill-posed reconstruction problem, before applying other purely mathematical regularization methods.
Load reconstruction near the fixed spatial boundaries is usually hard. The loads near the fixed spatial boundaries are impossible to be correctly reconstructed, if the natural modal functions or their modified forms are applied as base functions to express the spatial distribution of the load as generalized Fourier series. To tackle this problem, the concept of consistent spatial expression is proposed, and Legendre polynomials are applied as the consistent spatial base functions, which result in good effect in numerical simulations.
Wavelet transform is not conveniently applied on the load reconstruction of a dynamic system, since it has no good and simple properties on the relationship between the input and output of a dynamic system, though it dose have profits on local analysis of signals. Nevertheless, the authors proposed the method of wavelet approximation to reconstruct the distributed dynamic loads on linear elastic system, and relatively good results are obtained. Meanwhile, as difficulties usually confront researchers identifying transient force from response with non-zero initial conditions, the method of fragment analysis is proposed to deal with this problem, and the method succeeds cooperating with the wavelet approximation method.
Based on abovementioned methods and concepts, the reconstruction theories of distributed dynamic loads on both Euler-Bernoulli beam and elastic thin plate are proposed. The numerical simulations show good accordance with the theories, and many new phenomena are disclosed.
引文
[1] M. J. Stanek, N. Sinha, V. Ahuja and R. M. Birkbeck, Acoustics-compatible active flow control for optimal weapon separation, The 5th AIAA/CEAS aeroacoustics conference, Seattle, WA, 1999.
[2]许锋,陈怀海,鲍明.机械振动载荷识别研究的现状与未来[J].中国机械工程,2002,(6).
[3] L. A. Lifschitz and C. E. D'Attellis, Input force reconstruction using wavelets with applications to a pulsed plasma thruster, Mathematical and Computer Modelling, 2005, 41(4-5): 361-369.
[4] J. Liu, C. Ma, I. Kung and D. Lin, Input force estimation of a cantilever plate by using a system identification technique, Computer Methods in Applied Mechanics and Engineering, 2000, 190(11-12): 1309-1322.
[5] D. C. Kammer, Input force reconstruction using a time domain technique, AIAA-96-1201-CP.
[6] S. S. Law, S. Q. Wu and Z. Y. Shi, Moving load and prestress identification using wavelet-based method, Journal of Applied Mechanics-Transactions of the Asme, 2008, 75(2).
[7] L. Yu, T. H. T. Chan and J. H. Zhu, A MOM-based algorithm for moving force identification: Part I - Theory and numerical simulation, Structural Engineering and Mechanics, 2008, 29(2): 135-154.
[8] L. Yu, T. H. T. Chan and J. H. Zhu, A MOM-based algorithm for moving force identification: Part II - Experiment and comparative studies, Structural Engineering and Mechanics, 2008, 29(2): 155-169.
[9] S. S. Law, J. Q. Bu, X. Q. Zhu and S. L. Chan, Moving load identification on a simply supported orthotropic plate, International Journal of Mechanical Sciences, 2007, 49(11): 1262-1275.
[10] T. H. T. Chan and D. B. Ashebo, Theoretical study of moving force identification on continuous bridges, Journal of Sound and Vibration, 2006, 295(3-5): 870-883.
[11] X. Q. Zhu and S. S. Law, Moving load identification on multi-span continuous bridges with elastic bearings, Mechanical Systems and Signal Processing, 2006, 20(7): 1759-1782.
[12] L. Yu and T. H. T. Chan, Moving force identification from bridge dynamic responses, Structural Engineering and Mechanics, 2005, 21(3): 369-374.
[13] L. Yu and T. H. T. Chan, Moving force identification based on the frequency-time domain method, Journal of Sound and Vibration, 2003, 261(2): 329-349.
[14] L. Yu and T. H. T. Chan, Moving force identification from bending moment responses of bridge,Structural Engineering and Mechanics, 2002, 14(2): 151-170.
[15] X. Q. Zhu and S. S. Law, Practical aspects in moving load identification, Journal of Sound and Vibration, 2002, 258(1): 123-146.
[16] T. H. T. Chan, L. Yu, S. S. Law and T. H. Yung, Moving force identification studies, I: Theory, Journal of Sound and Vibration, 2001, 247(1): 59-76.
[17] T. H. T. Chan, L. Yu, S. S. Law and T. H. Yung, Moving force identification studies, II: Comparative studies, Journal of Sound and Vibration, 2001, 247(1): 77-95.
[18] S. S. Law, T. H. T. Chan, Q. X. Zhu and Q. H. Zeng, Regularization in moving force identification, Journal of Engineering Mechanics-Asce, 2001, 127(2): 136-148.
[19] S. S. Law and Y. L. Fang, Moving force identification: optimal state estimation approach, Journal of Sound and Vibration, 2001, 239(2): 233-254.
[20] T. H. T. Chan, S. S. Law and T. H. Yung, Moving force identification using an existing prestressed concrete bridge, Engineering Structures, 2000, 22(10): 1261-1270.
[21] T. H. T. Chan, L. Yu and S. S. Law, Comparative studies on moving force identification from bridge strains in laboratory, Journal of Sound and Vibration, 2000, 235(1): 87-104.
[22] S. S. Law and X. Q. Zhu, Study on different beam models in moving force identification, Journal of Sound and Vibration, 2000, 234(4): 661-679.
[23] T. H. T. Chan, S. S. Law, T. H. Yung and X. R. Yuan, An interpretive method for moving force identification, Journal of Sound and Vibration, 1999, 219(3): 503-524.
[24] S. S. Law, T. H. T. Chan and Q. H. Zeng, Moving force identification - A frequency and time domains analysis, Journal of Dynamic Systems Measurement and Control-Transactions of the Asme, 1999, 121(3): 394-401.
[25] H. Sekine and S. Atobe, Identification of locations and force histories of multiple point impacts on composite isogrid-stiffened panels, Composite Structures, 2009, 89(1): 1-7.
[26] N. S. Vyas and A. L. Wicks, Reconstruction of turbine blade forces from response data, Mechanism and Machine Theory, 2001, 36(2): 177-188.
[27] M. C. Djamaa, N. Ouelaa, C. Pezerat and J. L. Guyader, Reconstruction of a distributed force applied on a thin cylindrical shell by an inverse method and spatial filtering, Journal of Sound and Vibration, 2007, 301(3-5): 560-575.
[28] Yi Liu and W. S. Shepard Jr., An improved method for the reconstruction of a distributed force acting on a vibrating structure, Journal of Sound and Vibration, 2006, 291(1-2): 369-387.
[29] N. Sehlstedt, A well-conditioned technique for solving the inverse problem of boundary tractionestimation for a constrained vibrating structure, Computational Mechanics, 2003, 30(3): 247-258.
[30] C. Pezerat, J. L. Guyader, Force analysis technique: Reconstruction of force distribution on plates, Acta Acustica, 2000, 86: 322-332.
[31] S. Granger and L. Perotin, An inverse method for the identification of a distributed random excitation acting on a vibrating structure—part I: theory, Mechanical Systems and Signal Processing, 1999, 13(1): 53-65.
[32]秦远田,张方.具有连续分布梁模型动载荷的识别技术研究[J].振动与冲击,2005,(2).
[33]张方,唐旭东,秦远田,邓吉宏.结构连续分布的动态随机载荷识别方法研究[J].振动与冲击,2006,(2).
[34]张方,秦远田,邓吉宏.复杂分布动载荷识别技术研究[J].振动工程学报,2006,(1).
[35]秦远田,张方,陈国平.二维分布动载荷识别的频域方法[J].振动工程学报,2007,(5).
[36]马晨明.基于有限元方法的Kirchhoff板上动态分布载荷的辨识[J].计算物理,2007,(4).
[37] M. De Araújo, J. Antunes and P. Piteau, Remote identification of impact forces on loosely supported tubes—Part 1: basic theory and experiments, Journal of Sound and Vibration, 1998, 215(5): 1015-1041.
[38] H. W. Kim and S. K. Lee, Estimation of impact load in thick plate by using theoretical green function and experimental measurement of vibration, International Journal of Modern Physics B, 2008, 22(9-11): 1377-1382.
[39] S. J. Kim and S. K. Lee, Identification of impact force in thick plates based on the elastodynamics and time-frequency method (II) - Experimental approach for identification of the impact force based on time frequency methods, Journal of Mechanical Science and Technology, 2008, 22(7): 1359-1373.
[40] S. K. Lee, Identification of impact force in thick plates based on the elastodynamics and time-frequency method (I) - Theoretical approach for identification the impact force based on elastodynamics, Journal of Mechanical Science and Technology, 2008, 22(7): 1349-1358.
[41] S. K. Lee, S. Banerjee and A. Mal, Identification of impact force on a thick plate based on the elastodynamic and higher-order time-frequency analysis, Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science, 2007, 221(11): 1249-1263.
[42] S. K. Lee, S. U. Hwang and J. H. Gu, Identification of impact force for base on higher order wigner distribution, Damage Assessment of Structures Vi, 2005, 293-294: 111-118.
[43] H. Fukunaga, T. Umino and N. Hu, Impact force identification of CFRP stiffened panel under multiple loading, Structural Health Monitoring 2007: Quantification, Validation, and Implementation, Vols 1 and 2, 2007: 177-184.
[44] S. Matsumoto and H. Fukunaga, A. Tajima, Impact force identification of plates using PZT piezoelectric sensors, Structural Health Monitoring and Intelligent Infrastructure, Vols 1 and 2, 2007: 837-845.
[45] H. Fukunaga and N. Hu, Experimental impact force identification of composite structures, Proceedings of the Third European Workshop Structural Health Monitoring, 2006: 840-847.
[46] N. Hu, H. Fukunaga, S. Matsumoto, B. Yan, et al., An efficient approach for identifying impact force using embedded piezoelectric sensors, International Journal of Impact Engineering, 2007, 34(7): 1258-1271.
[47] N. Hu, H. Fukunaga and B. Yan, An efficient approach for identifying impact force using embedded piezoelectric sensors and Chebyshev polynomial - art. no. 60412Z, ICMIT 2005: Information Systems and Signal Processing, 2005, 6041, Z412-Z412.
[48] H. Inoue, K. Kishimoto, T. Shibuya and K. Harada, Regularization of numerical inversion of the Laplace transform for the inverse analysis of impact force, Jsme International Journal Series a-Solid Mechanics and Material Engineering, 1998, 41(4): 473-480.
[49] A. Voropai and E. Yanyutin, Identification of several impulsive loads on a plate, International Applied Mechanics, 2007, 43(7): 780-785.
[50] E. G. Yanyutin and I. V. Yanchevsky, Identification of an impulse load acting on an axisymmetrical hemispherical shell, International Journal of Solids and Structures, 2004, 41(13): 3643-3652.
[51] E. G. Yanyutin and A. V. Voropai, Identification of the impulsive load on an elastic rectangular plate, International Applied Mechanics, 2003, 39(10): 1199-1204.
[52] C. Kaczmarek and Z. Kaczmarek, Sensors of impulsive force and pressure with one point and two point strain measurement applied in tasks of reconstruction, Joint International Conference Imeko Tc3/Tc5/Tc20, 2002, 1685: 193-198.
[53] Z. Kaczmarek, C. Kaczmarek and V. Nichoga, A modified impulsive force and pressure sensor intended for waveform reconstruction purposes, Ieee Transactions on Instrumentation and Measurement, 2002, 51(1): 102-106.
[54] Z. Kaczmarek and A. Drobnica, Waveform reconstruction of impulsive force and pressure by means of deconvolution, Non-Linear Electromagnetic Systems, 1998, 13: 438-441.
[55] M. Wiklo and J. Holnicki-Szulc, Impact load identification based on local measurements, Damage Assessment of Structures Vi, 2005, 293-294: 159-166.
[56] M. T. Martin and J. F. Doyle, Impact force identification from wave propagation responses, International Journal of Impact Engineering, 1996, 18(1): 65-77.
[57] M. T. Martin and J. F. Doyle, Impact force location in frame structures, International Journal of Impact Engineering, 1996, 18(1): 79-97.
[58] G. Yan and L. Zhou, Impact load identification of composite structure using genetic algorithms, Journal of Sound and Vibration, 2009, 319(3-5): 869-884.
[59] G. Yan, L. Zhou and F. G. Yuan, Identification of impact load for composites using genetic algorithms, Structural Health Monitoring 2007: Quantification, Validation, and Implementation, Vols 1 and 2, 2007: 185-192.
[60] K. Min-Soo, L. Sang-Kwon and K. Sung-Jong, Identification of impact force on the gas pipe based on analysis of the acoustic wave, International Journal of Modern Physics B, 2008, 22(9-11): 1039-1044.
[61] R. Hashemi and M. H. Kargarnovin, Vibration Base Identification of Impact Force Using Genetic Algorithm, Proceedings of World Academy of Science, Engineering and Technology, Vol 26, Parts 1 and 2, 2007, 26: 624-630.
[62] A. A. Cardi, D. E. Adams and S. Walsh, Ceramic body armor single impact force identification on a compliant torso using acceleration response mapping, Structural Health Monitoring-an International Journal, 2006, 5(4): 355-372.
[63] M. L. Lee and W. K. Chiu, Comparative study on impact force prediction on a railway track-like structure, Structural Health Monitoring-an International Journal, 2005, 4(4): 355-376.
[64] B. T. Wang and C. H. Chiu, Determination of unknown impact force acting on a simply supported beam, Mechanical Systems and Signal Processing, 2003, 17(3): 683-704.
[65] E. S. Shin, Real-time recovery of impact force based on finite element analysis, Computers & Structures, 2000, 76(5): 621-635.
[66] L. Gaul and S. Hurlebaus, Determination of the impact force on a plate by piezoelectric film sensors, Archive of Applied Mechanics, 1999, 69(9-10): 691-701.
[67] H. Huang, J. Pan and P. G. McCormick, Prediction of impact forces in a vibratory ball mill using an inverse technique, International Journal of Impact Engineering, 1997, 19(2): 117-126.
[68] J. F. Doyle, Impact force identification for plate structures, Proceedings of the 1995 Sem SpringConference on Experimental Mechanics, 1995: 452-457.
[69] J. C. Briggs and M.-K. Tse, Impact force identification using extracted modal parameters and pattern matching, International Journal of Impact Engineering, 1992, 12(3): 361-372.
[70] S. Odeen and B. Lundberg, Prediction of impact force by impulse response method, International Journal of Impact Engineering, 1991, 11(2): 149-158.
[71] P. E. Hollandsworth and H. R. Busby, Impact force identification using the general inverse technique, International Journal of Impact Engineering, 1989, 8(4): 315-322.
[72] J. F. Doyle, A wavelet deconvolution method for impact force identification, Experimental Mechanics, 1997, 37(4): 403-408.
[73]赵玉成,袁树清,李舜酩,许庆余.动态载荷的小波正交算子变换识别法[J].机械强度,1998,(2).
[74] J. C. Santamarina, D. Fratta, Discrete signals and inverse problems: an introduction for engineers and scientists, Wiley, 2005.
[75]肖庭延,于慎根,王彦飞.反问题的数值解法.北京:科学出版社,2003.
[76]刘继军.不适定问题的正则化方法及应用.北京:科学出版社,2005.
[77] A. Tarantola, Inverse problem theory and methods for model parameter estimation, Philadelphia: SIAM, 2005.
[78] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes: the art of scientific computing, Cambridge University Press, 1992.
[79] P. C. Hansen, Regularization tools: a matlab package for analysis and solution of discrete ill-posed problems, Numerical Algorithms, 1994, 6: 1-35.
[80] F. E. Gunawan, H. Homma and Y. Kanto, Two-step B-splines regularization method for solving an ill-posed problem of impact-force reconstruction, Journal of Sound and Vibration, 2006, 297(1-2): 200-214.
[81] H. Lee and Y. Park, Error analysis of indirect force determination and a regularization method to reduce force determination error, Mechanical Systems and Signal Processing, 1995, 9(6): 615-633.
[82] K. Mosegaard and A. Tarantola, Monte Carlo sampling of solutions to inverse problems, Journal of Geophysical Research, 1995, 100(B7): 12,431–12,447.
[83] H. G. Choi, A. N. Thite, D. J. Thompson, Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination, Journal of Sound and Vibration, 2007, 304: 894-917.
[84] H. G. Choi, A. N. Thite and D. J. Thompson, A threshold for the use of Tikhonov regularization in inverse force determination, Applied Acoustics, 2006, 67(7): 700-719.
[85] J. W. Hilgers and B. S. Bertram, Comparing different types of approximators for choosing the parameters in the regularization of ill-posed problems, Computers and Mathematics with Applications, 2004, 48: 1779-1790.
[86] D. Calvetti, S. Morigi, L. Reichel, F. Sgallari, Tikhonov regularization and the L-curve for large discrete ill-posed problems, Journal of Computational and Applied Mathematics, 2000, 123: 423-446.
[87] S. H. Yoon and P. A. Nelson, Estimation of acoustic source strength by inverse methods—part II: experimental investigation of methods for choosing regularization parameters, Journal of Sound and Vibration, 2000, 233(4): 669-705.
[88] J. L. Castellanos, S. Gomez and V. Guerra, The triangle method for finding the corner of the L-curve, Applied Numerical Mathematics, 2002, 43: 359-373.
[89] P. C. Hansen, T. K. Jensen and G. Rodriguez, An adaptive pruning algorithm for the discrete L-curve criterion, Journal of Computational and Applied Mathematics, 2007, 198: 483-492.
[90] D. Reichel and H. Sadok, A new L-curve for ill-posed problems, Journal of Computational and Applied Mathematics, 2008, 219(2): 493-508.
[91] C. W. Groetsch, The theory of Tikhonov regularization for Fredholm equations of the first kind, Boston: Pitman, 1984.
[92]柳重堪.正交函数及其应用.北京:国防工业出版社,1982.
[93]崔锦泰.小波分析导论(程正兴译).西安:西安交通大学出版社,1995.
[94] D. F. Mix, K. J. Olejniczak.小波基础及应用教程(杨志华,杨力华译).北京:机械工业出版社,2006.
[95]丁康,钟舜聪.通用的离散频谱相位差校正方法[J].电子学报,2003,(1).
[96]丁康,朱小勇,谢明,钟舜聪,罗江凯.离散频谱综合相位差校正法[J].振动工程学报,2002,(1).
[97] F. Kuhnent.广义逆矩阵与正则化方法(陈杰译).北京:高等教育出版社,1985.
[98]孙继广.矩阵扰动分析(第二版).北京:科学出版社,2001:157-166.
[2]许锋,陈怀海,鲍明.机械振动载荷识别研究的现状与未来[J].中国机械工程,2002,(6).
[3] L. A. Lifschitz and C. E. D'Attellis, Input force reconstruction using wavelets with applications to a pulsed plasma thruster, Mathematical and Computer Modelling, 2005, 41(4-5): 361-369.
[4] J. Liu, C. Ma, I. Kung and D. Lin, Input force estimation of a cantilever plate by using a system identification technique, Computer Methods in Applied Mechanics and Engineering, 2000, 190(11-12): 1309-1322.
[5] D. C. Kammer, Input force reconstruction using a time domain technique, AIAA-96-1201-CP.
[6] S. S. Law, S. Q. Wu and Z. Y. Shi, Moving load and prestress identification using wavelet-based method, Journal of Applied Mechanics-Transactions of the Asme, 2008, 75(2).
[7] L. Yu, T. H. T. Chan and J. H. Zhu, A MOM-based algorithm for moving force identification: Part I - Theory and numerical simulation, Structural Engineering and Mechanics, 2008, 29(2): 135-154.
[8] L. Yu, T. H. T. Chan and J. H. Zhu, A MOM-based algorithm for moving force identification: Part II - Experiment and comparative studies, Structural Engineering and Mechanics, 2008, 29(2): 155-169.
[9] S. S. Law, J. Q. Bu, X. Q. Zhu and S. L. Chan, Moving load identification on a simply supported orthotropic plate, International Journal of Mechanical Sciences, 2007, 49(11): 1262-1275.
[10] T. H. T. Chan and D. B. Ashebo, Theoretical study of moving force identification on continuous bridges, Journal of Sound and Vibration, 2006, 295(3-5): 870-883.
[11] X. Q. Zhu and S. S. Law, Moving load identification on multi-span continuous bridges with elastic bearings, Mechanical Systems and Signal Processing, 2006, 20(7): 1759-1782.
[12] L. Yu and T. H. T. Chan, Moving force identification from bridge dynamic responses, Structural Engineering and Mechanics, 2005, 21(3): 369-374.
[13] L. Yu and T. H. T. Chan, Moving force identification based on the frequency-time domain method, Journal of Sound and Vibration, 2003, 261(2): 329-349.
[14] L. Yu and T. H. T. Chan, Moving force identification from bending moment responses of bridge,Structural Engineering and Mechanics, 2002, 14(2): 151-170.
[15] X. Q. Zhu and S. S. Law, Practical aspects in moving load identification, Journal of Sound and Vibration, 2002, 258(1): 123-146.
[16] T. H. T. Chan, L. Yu, S. S. Law and T. H. Yung, Moving force identification studies, I: Theory, Journal of Sound and Vibration, 2001, 247(1): 59-76.
[17] T. H. T. Chan, L. Yu, S. S. Law and T. H. Yung, Moving force identification studies, II: Comparative studies, Journal of Sound and Vibration, 2001, 247(1): 77-95.
[18] S. S. Law, T. H. T. Chan, Q. X. Zhu and Q. H. Zeng, Regularization in moving force identification, Journal of Engineering Mechanics-Asce, 2001, 127(2): 136-148.
[19] S. S. Law and Y. L. Fang, Moving force identification: optimal state estimation approach, Journal of Sound and Vibration, 2001, 239(2): 233-254.
[20] T. H. T. Chan, S. S. Law and T. H. Yung, Moving force identification using an existing prestressed concrete bridge, Engineering Structures, 2000, 22(10): 1261-1270.
[21] T. H. T. Chan, L. Yu and S. S. Law, Comparative studies on moving force identification from bridge strains in laboratory, Journal of Sound and Vibration, 2000, 235(1): 87-104.
[22] S. S. Law and X. Q. Zhu, Study on different beam models in moving force identification, Journal of Sound and Vibration, 2000, 234(4): 661-679.
[23] T. H. T. Chan, S. S. Law, T. H. Yung and X. R. Yuan, An interpretive method for moving force identification, Journal of Sound and Vibration, 1999, 219(3): 503-524.
[24] S. S. Law, T. H. T. Chan and Q. H. Zeng, Moving force identification - A frequency and time domains analysis, Journal of Dynamic Systems Measurement and Control-Transactions of the Asme, 1999, 121(3): 394-401.
[25] H. Sekine and S. Atobe, Identification of locations and force histories of multiple point impacts on composite isogrid-stiffened panels, Composite Structures, 2009, 89(1): 1-7.
[26] N. S. Vyas and A. L. Wicks, Reconstruction of turbine blade forces from response data, Mechanism and Machine Theory, 2001, 36(2): 177-188.
[27] M. C. Djamaa, N. Ouelaa, C. Pezerat and J. L. Guyader, Reconstruction of a distributed force applied on a thin cylindrical shell by an inverse method and spatial filtering, Journal of Sound and Vibration, 2007, 301(3-5): 560-575.
[28] Yi Liu and W. S. Shepard Jr., An improved method for the reconstruction of a distributed force acting on a vibrating structure, Journal of Sound and Vibration, 2006, 291(1-2): 369-387.
[29] N. Sehlstedt, A well-conditioned technique for solving the inverse problem of boundary tractionestimation for a constrained vibrating structure, Computational Mechanics, 2003, 30(3): 247-258.
[30] C. Pezerat, J. L. Guyader, Force analysis technique: Reconstruction of force distribution on plates, Acta Acustica, 2000, 86: 322-332.
[31] S. Granger and L. Perotin, An inverse method for the identification of a distributed random excitation acting on a vibrating structure—part I: theory, Mechanical Systems and Signal Processing, 1999, 13(1): 53-65.
[32]秦远田,张方.具有连续分布梁模型动载荷的识别技术研究[J].振动与冲击,2005,(2).
[33]张方,唐旭东,秦远田,邓吉宏.结构连续分布的动态随机载荷识别方法研究[J].振动与冲击,2006,(2).
[34]张方,秦远田,邓吉宏.复杂分布动载荷识别技术研究[J].振动工程学报,2006,(1).
[35]秦远田,张方,陈国平.二维分布动载荷识别的频域方法[J].振动工程学报,2007,(5).
[36]马晨明.基于有限元方法的Kirchhoff板上动态分布载荷的辨识[J].计算物理,2007,(4).
[37] M. De Araújo, J. Antunes and P. Piteau, Remote identification of impact forces on loosely supported tubes—Part 1: basic theory and experiments, Journal of Sound and Vibration, 1998, 215(5): 1015-1041.
[38] H. W. Kim and S. K. Lee, Estimation of impact load in thick plate by using theoretical green function and experimental measurement of vibration, International Journal of Modern Physics B, 2008, 22(9-11): 1377-1382.
[39] S. J. Kim and S. K. Lee, Identification of impact force in thick plates based on the elastodynamics and time-frequency method (II) - Experimental approach for identification of the impact force based on time frequency methods, Journal of Mechanical Science and Technology, 2008, 22(7): 1359-1373.
[40] S. K. Lee, Identification of impact force in thick plates based on the elastodynamics and time-frequency method (I) - Theoretical approach for identification the impact force based on elastodynamics, Journal of Mechanical Science and Technology, 2008, 22(7): 1349-1358.
[41] S. K. Lee, S. Banerjee and A. Mal, Identification of impact force on a thick plate based on the elastodynamic and higher-order time-frequency analysis, Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science, 2007, 221(11): 1249-1263.
[42] S. K. Lee, S. U. Hwang and J. H. Gu, Identification of impact force for base on higher order wigner distribution, Damage Assessment of Structures Vi, 2005, 293-294: 111-118.
[43] H. Fukunaga, T. Umino and N. Hu, Impact force identification of CFRP stiffened panel under multiple loading, Structural Health Monitoring 2007: Quantification, Validation, and Implementation, Vols 1 and 2, 2007: 177-184.
[44] S. Matsumoto and H. Fukunaga, A. Tajima, Impact force identification of plates using PZT piezoelectric sensors, Structural Health Monitoring and Intelligent Infrastructure, Vols 1 and 2, 2007: 837-845.
[45] H. Fukunaga and N. Hu, Experimental impact force identification of composite structures, Proceedings of the Third European Workshop Structural Health Monitoring, 2006: 840-847.
[46] N. Hu, H. Fukunaga, S. Matsumoto, B. Yan, et al., An efficient approach for identifying impact force using embedded piezoelectric sensors, International Journal of Impact Engineering, 2007, 34(7): 1258-1271.
[47] N. Hu, H. Fukunaga and B. Yan, An efficient approach for identifying impact force using embedded piezoelectric sensors and Chebyshev polynomial - art. no. 60412Z, ICMIT 2005: Information Systems and Signal Processing, 2005, 6041, Z412-Z412.
[48] H. Inoue, K. Kishimoto, T. Shibuya and K. Harada, Regularization of numerical inversion of the Laplace transform for the inverse analysis of impact force, Jsme International Journal Series a-Solid Mechanics and Material Engineering, 1998, 41(4): 473-480.
[49] A. Voropai and E. Yanyutin, Identification of several impulsive loads on a plate, International Applied Mechanics, 2007, 43(7): 780-785.
[50] E. G. Yanyutin and I. V. Yanchevsky, Identification of an impulse load acting on an axisymmetrical hemispherical shell, International Journal of Solids and Structures, 2004, 41(13): 3643-3652.
[51] E. G. Yanyutin and A. V. Voropai, Identification of the impulsive load on an elastic rectangular plate, International Applied Mechanics, 2003, 39(10): 1199-1204.
[52] C. Kaczmarek and Z. Kaczmarek, Sensors of impulsive force and pressure with one point and two point strain measurement applied in tasks of reconstruction, Joint International Conference Imeko Tc3/Tc5/Tc20, 2002, 1685: 193-198.
[53] Z. Kaczmarek, C. Kaczmarek and V. Nichoga, A modified impulsive force and pressure sensor intended for waveform reconstruction purposes, Ieee Transactions on Instrumentation and Measurement, 2002, 51(1): 102-106.
[54] Z. Kaczmarek and A. Drobnica, Waveform reconstruction of impulsive force and pressure by means of deconvolution, Non-Linear Electromagnetic Systems, 1998, 13: 438-441.
[55] M. Wiklo and J. Holnicki-Szulc, Impact load identification based on local measurements, Damage Assessment of Structures Vi, 2005, 293-294: 159-166.
[56] M. T. Martin and J. F. Doyle, Impact force identification from wave propagation responses, International Journal of Impact Engineering, 1996, 18(1): 65-77.
[57] M. T. Martin and J. F. Doyle, Impact force location in frame structures, International Journal of Impact Engineering, 1996, 18(1): 79-97.
[58] G. Yan and L. Zhou, Impact load identification of composite structure using genetic algorithms, Journal of Sound and Vibration, 2009, 319(3-5): 869-884.
[59] G. Yan, L. Zhou and F. G. Yuan, Identification of impact load for composites using genetic algorithms, Structural Health Monitoring 2007: Quantification, Validation, and Implementation, Vols 1 and 2, 2007: 185-192.
[60] K. Min-Soo, L. Sang-Kwon and K. Sung-Jong, Identification of impact force on the gas pipe based on analysis of the acoustic wave, International Journal of Modern Physics B, 2008, 22(9-11): 1039-1044.
[61] R. Hashemi and M. H. Kargarnovin, Vibration Base Identification of Impact Force Using Genetic Algorithm, Proceedings of World Academy of Science, Engineering and Technology, Vol 26, Parts 1 and 2, 2007, 26: 624-630.
[62] A. A. Cardi, D. E. Adams and S. Walsh, Ceramic body armor single impact force identification on a compliant torso using acceleration response mapping, Structural Health Monitoring-an International Journal, 2006, 5(4): 355-372.
[63] M. L. Lee and W. K. Chiu, Comparative study on impact force prediction on a railway track-like structure, Structural Health Monitoring-an International Journal, 2005, 4(4): 355-376.
[64] B. T. Wang and C. H. Chiu, Determination of unknown impact force acting on a simply supported beam, Mechanical Systems and Signal Processing, 2003, 17(3): 683-704.
[65] E. S. Shin, Real-time recovery of impact force based on finite element analysis, Computers & Structures, 2000, 76(5): 621-635.
[66] L. Gaul and S. Hurlebaus, Determination of the impact force on a plate by piezoelectric film sensors, Archive of Applied Mechanics, 1999, 69(9-10): 691-701.
[67] H. Huang, J. Pan and P. G. McCormick, Prediction of impact forces in a vibratory ball mill using an inverse technique, International Journal of Impact Engineering, 1997, 19(2): 117-126.
[68] J. F. Doyle, Impact force identification for plate structures, Proceedings of the 1995 Sem SpringConference on Experimental Mechanics, 1995: 452-457.
[69] J. C. Briggs and M.-K. Tse, Impact force identification using extracted modal parameters and pattern matching, International Journal of Impact Engineering, 1992, 12(3): 361-372.
[70] S. Odeen and B. Lundberg, Prediction of impact force by impulse response method, International Journal of Impact Engineering, 1991, 11(2): 149-158.
[71] P. E. Hollandsworth and H. R. Busby, Impact force identification using the general inverse technique, International Journal of Impact Engineering, 1989, 8(4): 315-322.
[72] J. F. Doyle, A wavelet deconvolution method for impact force identification, Experimental Mechanics, 1997, 37(4): 403-408.
[73]赵玉成,袁树清,李舜酩,许庆余.动态载荷的小波正交算子变换识别法[J].机械强度,1998,(2).
[74] J. C. Santamarina, D. Fratta, Discrete signals and inverse problems: an introduction for engineers and scientists, Wiley, 2005.
[75]肖庭延,于慎根,王彦飞.反问题的数值解法.北京:科学出版社,2003.
[76]刘继军.不适定问题的正则化方法及应用.北京:科学出版社,2005.
[77] A. Tarantola, Inverse problem theory and methods for model parameter estimation, Philadelphia: SIAM, 2005.
[78] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes: the art of scientific computing, Cambridge University Press, 1992.
[79] P. C. Hansen, Regularization tools: a matlab package for analysis and solution of discrete ill-posed problems, Numerical Algorithms, 1994, 6: 1-35.
[80] F. E. Gunawan, H. Homma and Y. Kanto, Two-step B-splines regularization method for solving an ill-posed problem of impact-force reconstruction, Journal of Sound and Vibration, 2006, 297(1-2): 200-214.
[81] H. Lee and Y. Park, Error analysis of indirect force determination and a regularization method to reduce force determination error, Mechanical Systems and Signal Processing, 1995, 9(6): 615-633.
[82] K. Mosegaard and A. Tarantola, Monte Carlo sampling of solutions to inverse problems, Journal of Geophysical Research, 1995, 100(B7): 12,431–12,447.
[83] H. G. Choi, A. N. Thite, D. J. Thompson, Comparison of methods for parameter selection in Tikhonov regularization with application to inverse force determination, Journal of Sound and Vibration, 2007, 304: 894-917.
[84] H. G. Choi, A. N. Thite and D. J. Thompson, A threshold for the use of Tikhonov regularization in inverse force determination, Applied Acoustics, 2006, 67(7): 700-719.
[85] J. W. Hilgers and B. S. Bertram, Comparing different types of approximators for choosing the parameters in the regularization of ill-posed problems, Computers and Mathematics with Applications, 2004, 48: 1779-1790.
[86] D. Calvetti, S. Morigi, L. Reichel, F. Sgallari, Tikhonov regularization and the L-curve for large discrete ill-posed problems, Journal of Computational and Applied Mathematics, 2000, 123: 423-446.
[87] S. H. Yoon and P. A. Nelson, Estimation of acoustic source strength by inverse methods—part II: experimental investigation of methods for choosing regularization parameters, Journal of Sound and Vibration, 2000, 233(4): 669-705.
[88] J. L. Castellanos, S. Gomez and V. Guerra, The triangle method for finding the corner of the L-curve, Applied Numerical Mathematics, 2002, 43: 359-373.
[89] P. C. Hansen, T. K. Jensen and G. Rodriguez, An adaptive pruning algorithm for the discrete L-curve criterion, Journal of Computational and Applied Mathematics, 2007, 198: 483-492.
[90] D. Reichel and H. Sadok, A new L-curve for ill-posed problems, Journal of Computational and Applied Mathematics, 2008, 219(2): 493-508.
[91] C. W. Groetsch, The theory of Tikhonov regularization for Fredholm equations of the first kind, Boston: Pitman, 1984.
[92]柳重堪.正交函数及其应用.北京:国防工业出版社,1982.
[93]崔锦泰.小波分析导论(程正兴译).西安:西安交通大学出版社,1995.
[94] D. F. Mix, K. J. Olejniczak.小波基础及应用教程(杨志华,杨力华译).北京:机械工业出版社,2006.
[95]丁康,钟舜聪.通用的离散频谱相位差校正方法[J].电子学报,2003,(1).
[96]丁康,朱小勇,谢明,钟舜聪,罗江凯.离散频谱综合相位差校正法[J].振动工程学报,2002,(1).
[97] F. Kuhnent.广义逆矩阵与正则化方法(陈杰译).北京:高等教育出版社,1985.
[98]孙继广.矩阵扰动分析(第二版).北京:科学出版社,2001:157-166.