弹性半空间上EULER梁表面移动振源的虚拟反演
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摘要
城市轨道交通系统引起的环境振动日益成为一个严重的社会问题,振源特性的研究亦成为城市轨道交通研究领域的一个重要方面。为建立振源的特征模型,本文采用虚拟反演方法,给定振源参数,计算台阵各点的振动,再反演振源参数,比较结果和给定输入的差别,论证这个思路的可行性。
首先将地基-轨道系统简化为弹性半空间-Euler梁模型,将列车荷载简化为作用于Euler梁上的移动集中荷载,并且说明了如此简化可以获得足够精确度的合理性。接着引入了等效刚度的概念,并分四种不同的速度情况通过计算得到了数值结果,分析了相速度、梁宽、波数对等效刚度的影响。
推导了移动集中荷载作用下的Euler梁下位移解析解,并根据等效刚度数值计算了梁下反力。经分析,位移和反力分布形式不变,均随荷载同步向前运动,且分布形态与荷载速度密切相关。接着,叙述了半空间表面单位移动集中荷载作用下的表面一点竖向振动位移代数解的推导过程。在此基础上,通过在梁下反力分布荷载作用范围内沿Euler梁纵向积分得到了最终的梁上移动集中荷载作用下半空间表面竖向振动位移积分解,对其进行数值化实施,并编制具体FORTRAN程序,计算得到了地面一个密集台阵竖向振动位移的数值结果。在此基础上,用周期图法计算了密集台阵的单点自功率谱及两点互功率谱,并分析了位移时程的频谱特征。结果表明Euler梁表面作用移动荷载时引起的弹性半空间表面竖向振动位移的主要能量是低频成分产生的。
最后,给定弹性半空间上Euler梁模型、移动振源参数和台阵布置方案,利用微遗传算法虚拟反演了振源参数,分别计算了以台阵单点位移时程、两点位移时程联合、单点自功率谱以及两点互功率谱为目标的反演。结果均能正确、稳定地得到目标振源参数,表明遗传算法在振源特性研究领域的有效性与可靠性,验证了以台阵观测方法反演振源参数的思路的可行性。
Environmental vibration caused by urban track traffic is becoming an increasingly serious social problem, so study on vibration source characteristic is also playing an important role in the field of urban track traffic. In order to establish the vibration source characteristic model, the virtual inversion is adopted in this dissertation to calculate vibration at each point in an array from given vibration source parameters. Based on these vibrations, the parameters of the vibration source are inversed and are furthermore checked with the given ones to validate the feasibility of this idea.
In this dissertation, the foundation-track system was simplified to be an Euler beam on the elastic half-space, and the load caused by train was simplified to be a moving concentrated one on Euler beam based on the validation that the simplification above was precise enough. The definition of equivalent stiffness of elastic half-space was introduced, and the numerical results were calculated in four cases versus four different velocities, analyzing the influence of phase velocity, beam width, wave number to equivalent stiffness.
The analytical solution of displacements under the Euler beam from a moving concentrated load were derived, and the contact reaction between the beam and the surface of half-space was calculated numerically based on equivalent stiffness. the results show that the displacement and contact reaction were both found to be moving with the load simultaneously, which distributions are closely related to the load velocity. The algebraic solution were presented for the vertical vibration displacement produced at any point within an elastic half-space by a unit moving concentrated load over the surface. On the algebraic solution, integral solution of the vertical vibration displacement at the surface of half-space from a moving concentrated load on the Euler beam was obtained finally by means of integration in the range of contact reaction distribution along the beam longitudinal direction. Some numerical calculations based on the above solution were worked out by the related FORTRAN code, and the numerical result of vertical vibration displacement in a dense array on the ground was obtained, based on which the auto-power spectrum at single point and cross-power spectrum between a pair of points in the dense array were obtained by periodogram method, and hence the spectrum characteristic was analyzed. The results show that the major power of vertical vibration displacement at the surface of elastic half-space produced by a moving load on the Euler beam is generated by low frequency components.
Finally, Micro Genetic Algorithm was adopted in the virtual inversion of vibration source parameters with a given Euler beam model overlaid on elastic half-space, moving vibration source parameters and array layout. Inversions whose targets are the time history at single point, the time history joint between a pair of points, the auto-power spectrum at single point and the cross-power spectrum between a pair of points separately are preformed. Results show that correct and stable target vibration source parameters can be derived from all the four inversions, which indicates the validity and reliability of the genetic algorithm for the study on vibration source characteristic and the feasibility of the idea to inverse vibration source parameters by observed data in an array.
首先将地基-轨道系统简化为弹性半空间-Euler梁模型,将列车荷载简化为作用于Euler梁上的移动集中荷载,并且说明了如此简化可以获得足够精确度的合理性。接着引入了等效刚度的概念,并分四种不同的速度情况通过计算得到了数值结果,分析了相速度、梁宽、波数对等效刚度的影响。
推导了移动集中荷载作用下的Euler梁下位移解析解,并根据等效刚度数值计算了梁下反力。经分析,位移和反力分布形式不变,均随荷载同步向前运动,且分布形态与荷载速度密切相关。接着,叙述了半空间表面单位移动集中荷载作用下的表面一点竖向振动位移代数解的推导过程。在此基础上,通过在梁下反力分布荷载作用范围内沿Euler梁纵向积分得到了最终的梁上移动集中荷载作用下半空间表面竖向振动位移积分解,对其进行数值化实施,并编制具体FORTRAN程序,计算得到了地面一个密集台阵竖向振动位移的数值结果。在此基础上,用周期图法计算了密集台阵的单点自功率谱及两点互功率谱,并分析了位移时程的频谱特征。结果表明Euler梁表面作用移动荷载时引起的弹性半空间表面竖向振动位移的主要能量是低频成分产生的。
最后,给定弹性半空间上Euler梁模型、移动振源参数和台阵布置方案,利用微遗传算法虚拟反演了振源参数,分别计算了以台阵单点位移时程、两点位移时程联合、单点自功率谱以及两点互功率谱为目标的反演。结果均能正确、稳定地得到目标振源参数,表明遗传算法在振源特性研究领域的有效性与可靠性,验证了以台阵观测方法反演振源参数的思路的可行性。
Environmental vibration caused by urban track traffic is becoming an increasingly serious social problem, so study on vibration source characteristic is also playing an important role in the field of urban track traffic. In order to establish the vibration source characteristic model, the virtual inversion is adopted in this dissertation to calculate vibration at each point in an array from given vibration source parameters. Based on these vibrations, the parameters of the vibration source are inversed and are furthermore checked with the given ones to validate the feasibility of this idea.
In this dissertation, the foundation-track system was simplified to be an Euler beam on the elastic half-space, and the load caused by train was simplified to be a moving concentrated one on Euler beam based on the validation that the simplification above was precise enough. The definition of equivalent stiffness of elastic half-space was introduced, and the numerical results were calculated in four cases versus four different velocities, analyzing the influence of phase velocity, beam width, wave number to equivalent stiffness.
The analytical solution of displacements under the Euler beam from a moving concentrated load were derived, and the contact reaction between the beam and the surface of half-space was calculated numerically based on equivalent stiffness. the results show that the displacement and contact reaction were both found to be moving with the load simultaneously, which distributions are closely related to the load velocity. The algebraic solution were presented for the vertical vibration displacement produced at any point within an elastic half-space by a unit moving concentrated load over the surface. On the algebraic solution, integral solution of the vertical vibration displacement at the surface of half-space from a moving concentrated load on the Euler beam was obtained finally by means of integration in the range of contact reaction distribution along the beam longitudinal direction. Some numerical calculations based on the above solution were worked out by the related FORTRAN code, and the numerical result of vertical vibration displacement in a dense array on the ground was obtained, based on which the auto-power spectrum at single point and cross-power spectrum between a pair of points in the dense array were obtained by periodogram method, and hence the spectrum characteristic was analyzed. The results show that the major power of vertical vibration displacement at the surface of elastic half-space produced by a moving load on the Euler beam is generated by low frequency components.
Finally, Micro Genetic Algorithm was adopted in the virtual inversion of vibration source parameters with a given Euler beam model overlaid on elastic half-space, moving vibration source parameters and array layout. Inversions whose targets are the time history at single point, the time history joint between a pair of points, the auto-power spectrum at single point and the cross-power spectrum between a pair of points separately are preformed. Results show that correct and stable target vibration source parameters can be derived from all the four inversions, which indicates the validity and reliability of the genetic algorithm for the study on vibration source characteristic and the feasibility of the idea to inverse vibration source parameters by observed data in an array.
引文
1蔡翔宇.基于交通荷载随机特性的环境振动评价方法研究.浙江大学硕士学位论文.2007:3～6
2 Bata M. Effects on Buildings of vibrations caused by Traffic [J] Building Science, 1971, 6: 221～246
3陶夏新.国家自然科学基金项目《城市轨道交通引起的环境振动及传播规律》计划书.国家自然科学基金委员会.2005:5～17
4黄义,何芳社.弹性地基上的梁、板、壳.北京:科学出版社,2007,3～15
5 H. Lamb. On the propagation of tremors over the surface of an elastic solid[J]. Philosophical Transactions of the Royal Society(London). 1904, A203: 1~42
6 J. T. Kenny. Steady-state vibrations of beam on elastic foundation for moving load. Journal of Applied Mechanics. 1954,21(4):359～364
7 Eringen A C, Suhubi E. S. Elastodynamics,Volume 2, Linear Theory. New York: Academic Press, 1975
8 Miklowitz J. Elastic Waves and Wave guides. Amsterdam: North-Holland Publishing Company, 1978
9 H. P. Verhas. Prediction of the propagation of train-induced ground vibration, Journal of Sound and Vibration, 1979, 66(3):371～376
10 H. G. Georgiadis, Barber J. R. Steady-state transonic motion of a line load over an elastic half-space(The corrected Cole-Huth solution). J Appl Mech, ASME, 1993, 60, 772～774
11 V. V. Krylov and C. Ferguson. Calculation of low frequency ground vibrations from railway trains. Applied Acoustics, 1994, 42: 199～213
12 H. D. Dieterman and A. V. Metrikine. The equivalent stiffness of a half-space interacting with a beam. Critical velocities of a moving load along the beam. European Journal of Mechanics. A/Solids. 1996, 15(1): 67~90
13 H. D. Dieterman A. V. Metrikine. Steady-state displacements of a beam on an elastic half-space due to a uniformly moving constant load. European Journal ofMechanics. A/Solids. 1996, 16(2):295~306
14 A. V. Metrikine and H. D. Dieterman.Three-dimensional vibrations of a beam on an elastic half-space:Resonance interaction of vertical-longitudinal and lateral beam waves. Journal of Applied Mechanics. 1997, 64(4): 951～956
15 A. T. Peplow,C. J. C.Jones,M.Petyt. Surface vibration propagation over a layered elastic half-space with an inclusion.Applied Acoustics.1999,56:283～296
16 H. H. Hung,Y. B. Yang. Elastic waves in visco-elastic half-space generated by various vehicle loads[J]. Soil Dynamics and Earthquake Engineering,2001,21:1～17
17郑小平,王尚文,陈百屏.弹性地基无限长梁动力问题的一般解[J].应用数学和力学,1991,12(7):593～597
18孙璐,邓学钧.匀速运动的线源荷载激励下无限长梁动力分析[J].应用数学和力学,1998,19(4):341～347
19蒋建群,周华飞,张土乔.弹性半空间体在移动集中荷载作用下的稳态响应.岩土工程学报,2004,26(4):440～444
20周仁义,黄茂松,金波.高速移动荷载下单层弹性体的动力响应.西部开发中的岩土工程问题:岩土工程系列学术研讨会之九论文集.上海:同济大学出版社,2005
21边学成.列车荷载作用下轨道和地基的动力响应分析.力学学报.2005,37(4):476～484
22周华飞,蒋建群.高速移动荷载下粘弹性半空间体的动力响应.力学学报. 2007,39(4):545～553
23郑鑫.弹性半空间单一覆盖层表面振源的虚拟反演.哈尔滨工业大学硕士学位论文,2007
24周华飞.移动荷载作用下结构与地基动力响应特性研究.浙江大学博士学位论文,2005
25王常晶.列车移动荷载作用下地基的动应力及饱和软粘土特性研究.浙江大学博士学位论文,2006:109～121
26王胜.弹性半空间上欧拉梁表面移动荷载引起的位移场.哈尔滨工业大学硕士学位论文,2007
27 Yun-min Chen, Chang-jing Wang. Steady-state response of a Timoshenko beam on an elastic half-space under a moving load. Acta Mechanica Solida Smica, 2006, 19(1), 26～39
28王彦飞.反演问题的计算方法及应用.北京:高等教育出版社,2007,1～2
29王秉忱译.陈恕行校.不适定问题的解法.地质出版社,1979
30陶夏新,师黎静.利用地脉动台阵观测推断场地速度结构的虚拟反演.《新世纪地震工程与防灾减灾-庆贺胡聿贤院士八十寿辰》论文集.北京:地震出版社.2002:443~452
31师黎静,陶夏新.地脉动台阵方法的有效性分析[J].岩石力学与工程学报,2006,25(8):1683~1690
32邢强.轨道交通引起的环境振动的振源参数虚拟反演.中国地震局工程力学研究所硕士学位论文,2007
33 H. Yamamaka,H. Ishida. Application of genetic algorithms to an inversion of surface-wave dispersion data. Bulletin of Seismological Society of America[J]. 1996,86:436~444
34 Timoshenko, S. P. Method of analysis of statical and dynamical stress in rails. Proceedings, 2nd International Congress of Applied Mechanics. 1926,407,418,Zurich, Switzerland
35 Kenney, J.T. Steady-state vibrations of beam on elastic foundation for moving load. Journal of Appl. Mech. 76, 1954, 359～364
36 Yin, J. H. Closed-form solution for reinforced Timoshenko beam on elastic foundation.J. Engng. Mech. 2000, 126(8): 868～874
37 Yin, J. H.Comparative modeling study of reinforced beam on elastic foundation. J. Geotech. Geoenvir. Engng. 2000, 126(3): 265～271
38 G. Eason. The stresses produced in a semi-infinite solid by a moving surface force. International Journal of Engineering Science, 1965,Vo1.2, 581-609
39 D. L. Lansing. The displacements in an elastic half-space due to a moving concentrated normal load. NASA technical report, TR R-238, 1966
40杨位钦,顾岚.时间序列分析与动态数据建模[M].北京:北京理工大学出版社.1988:203~225
41王彦飞.反演问题的计算方法及研究.北京:高等教育出版社,2007,1～2
42姚姚.蒙特卡洛非线性反演方法及应用.北京:冶金工业出版社,1997,1～21
43 Holland J H Adaptation in Natural and Artificial Systems, Ann Arbor, MI: The University of Michigan Press, 1975
44王正志,薄涛.进化计算.第一版.北京:国防科技大学出版社,2000:3～5
45陈国良,王熙法,庄镇泉等.遗传算法及其应用.第一版.北京:人民邮电出版社,1996:3～5
46 Michalewicz, Z. Genetic Algorithms+Data Structrue=Evolutionary Program,Spring-Verlag. Berlin, Heidelberg, 2nd Edition, 1994
47 Krishnakumar K. Micro-Genetic Algorithms for Stationary Non-stationary Function Optimization. SPIE: Intelligent Control and Adaptive Systems, Philadelphia, PA, 1196, 1989, 289
2 Bata M. Effects on Buildings of vibrations caused by Traffic [J] Building Science, 1971, 6: 221～246
3陶夏新.国家自然科学基金项目《城市轨道交通引起的环境振动及传播规律》计划书.国家自然科学基金委员会.2005:5～17
4黄义,何芳社.弹性地基上的梁、板、壳.北京:科学出版社,2007,3～15
5 H. Lamb. On the propagation of tremors over the surface of an elastic solid[J]. Philosophical Transactions of the Royal Society(London). 1904, A203: 1~42
6 J. T. Kenny. Steady-state vibrations of beam on elastic foundation for moving load. Journal of Applied Mechanics. 1954,21(4):359～364
7 Eringen A C, Suhubi E. S. Elastodynamics,Volume 2, Linear Theory. New York: Academic Press, 1975
8 Miklowitz J. Elastic Waves and Wave guides. Amsterdam: North-Holland Publishing Company, 1978
9 H. P. Verhas. Prediction of the propagation of train-induced ground vibration, Journal of Sound and Vibration, 1979, 66(3):371～376
10 H. G. Georgiadis, Barber J. R. Steady-state transonic motion of a line load over an elastic half-space(The corrected Cole-Huth solution). J Appl Mech, ASME, 1993, 60, 772～774
11 V. V. Krylov and C. Ferguson. Calculation of low frequency ground vibrations from railway trains. Applied Acoustics, 1994, 42: 199～213
12 H. D. Dieterman and A. V. Metrikine. The equivalent stiffness of a half-space interacting with a beam. Critical velocities of a moving load along the beam. European Journal of Mechanics. A/Solids. 1996, 15(1): 67~90
13 H. D. Dieterman A. V. Metrikine. Steady-state displacements of a beam on an elastic half-space due to a uniformly moving constant load. European Journal ofMechanics. A/Solids. 1996, 16(2):295~306
14 A. V. Metrikine and H. D. Dieterman.Three-dimensional vibrations of a beam on an elastic half-space:Resonance interaction of vertical-longitudinal and lateral beam waves. Journal of Applied Mechanics. 1997, 64(4): 951～956
15 A. T. Peplow,C. J. C.Jones,M.Petyt. Surface vibration propagation over a layered elastic half-space with an inclusion.Applied Acoustics.1999,56:283～296
16 H. H. Hung,Y. B. Yang. Elastic waves in visco-elastic half-space generated by various vehicle loads[J]. Soil Dynamics and Earthquake Engineering,2001,21:1～17
17郑小平,王尚文,陈百屏.弹性地基无限长梁动力问题的一般解[J].应用数学和力学,1991,12(7):593～597
18孙璐,邓学钧.匀速运动的线源荷载激励下无限长梁动力分析[J].应用数学和力学,1998,19(4):341～347
19蒋建群,周华飞,张土乔.弹性半空间体在移动集中荷载作用下的稳态响应.岩土工程学报,2004,26(4):440～444
20周仁义,黄茂松,金波.高速移动荷载下单层弹性体的动力响应.西部开发中的岩土工程问题:岩土工程系列学术研讨会之九论文集.上海:同济大学出版社,2005
21边学成.列车荷载作用下轨道和地基的动力响应分析.力学学报.2005,37(4):476～484
22周华飞,蒋建群.高速移动荷载下粘弹性半空间体的动力响应.力学学报. 2007,39(4):545～553
23郑鑫.弹性半空间单一覆盖层表面振源的虚拟反演.哈尔滨工业大学硕士学位论文,2007
24周华飞.移动荷载作用下结构与地基动力响应特性研究.浙江大学博士学位论文,2005
25王常晶.列车移动荷载作用下地基的动应力及饱和软粘土特性研究.浙江大学博士学位论文,2006:109～121
26王胜.弹性半空间上欧拉梁表面移动荷载引起的位移场.哈尔滨工业大学硕士学位论文,2007
27 Yun-min Chen, Chang-jing Wang. Steady-state response of a Timoshenko beam on an elastic half-space under a moving load. Acta Mechanica Solida Smica, 2006, 19(1), 26～39
28王彦飞.反演问题的计算方法及应用.北京:高等教育出版社,2007,1～2
29王秉忱译.陈恕行校.不适定问题的解法.地质出版社,1979
30陶夏新,师黎静.利用地脉动台阵观测推断场地速度结构的虚拟反演.《新世纪地震工程与防灾减灾-庆贺胡聿贤院士八十寿辰》论文集.北京:地震出版社.2002:443~452
31师黎静,陶夏新.地脉动台阵方法的有效性分析[J].岩石力学与工程学报,2006,25(8):1683~1690
32邢强.轨道交通引起的环境振动的振源参数虚拟反演.中国地震局工程力学研究所硕士学位论文,2007
33 H. Yamamaka,H. Ishida. Application of genetic algorithms to an inversion of surface-wave dispersion data. Bulletin of Seismological Society of America[J]. 1996,86:436~444
34 Timoshenko, S. P. Method of analysis of statical and dynamical stress in rails. Proceedings, 2nd International Congress of Applied Mechanics. 1926,407,418,Zurich, Switzerland
35 Kenney, J.T. Steady-state vibrations of beam on elastic foundation for moving load. Journal of Appl. Mech. 76, 1954, 359～364
36 Yin, J. H. Closed-form solution for reinforced Timoshenko beam on elastic foundation.J. Engng. Mech. 2000, 126(8): 868～874
37 Yin, J. H.Comparative modeling study of reinforced beam on elastic foundation. J. Geotech. Geoenvir. Engng. 2000, 126(3): 265～271
38 G. Eason. The stresses produced in a semi-infinite solid by a moving surface force. International Journal of Engineering Science, 1965,Vo1.2, 581-609
39 D. L. Lansing. The displacements in an elastic half-space due to a moving concentrated normal load. NASA technical report, TR R-238, 1966
40杨位钦,顾岚.时间序列分析与动态数据建模[M].北京:北京理工大学出版社.1988:203~225
41王彦飞.反演问题的计算方法及研究.北京:高等教育出版社,2007,1～2
42姚姚.蒙特卡洛非线性反演方法及应用.北京:冶金工业出版社,1997,1～21
43 Holland J H Adaptation in Natural and Artificial Systems, Ann Arbor, MI: The University of Michigan Press, 1975
44王正志,薄涛.进化计算.第一版.北京:国防科技大学出版社,2000:3～5
45陈国良,王熙法,庄镇泉等.遗传算法及其应用.第一版.北京:人民邮电出版社,1996:3～5
46 Michalewicz, Z. Genetic Algorithms+Data Structrue=Evolutionary Program,Spring-Verlag. Berlin, Heidelberg, 2nd Edition, 1994
47 Krishnakumar K. Micro-Genetic Algorithms for Stationary Non-stationary Function Optimization. SPIE: Intelligent Control and Adaptive Systems, Philadelphia, PA, 1196, 1989, 289