含分数阶的无限地基交互作用时域阻尼抽取法
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摘要
以含分数阶的动力方程为研究背景,主要研究含分数阶的结构-地基交互作用的阻尼抽取法。首先推导含分数阶的阻尼抽取法结构-地基动力交互作用时域基本公式;其次,根据模态变换来解耦动力方程,并把其转换为传递函数的表达形式;然后,借助于Oustalop微分滤波器算法把分数阶传递函数近似表示为整数阶传递函数;最后转换为状态方程并组装得到系统的状态和输出方程,进而通过四阶龙格-库塔算法求出交界面作用力;最后给出一个计算实例并进行分析比较。
Based on the dynamic equations with fractional order(FO),the damping-extraction method for structure-media with FO is studied.Firstly,the time domain basic formula of the damping extraction method of dynamic structure-media interaction with FO is deduced.Secondly,the uncoupled dynamic equations of dynamic structure-media interaction with fractional order are determined and then transformed into transfer functions expression according to mode transform.Thirdly,the acquired FO transfer functions are expressed as integrated order ones approximately by Oustalop differential filter algorithm.Fourthly,the all transfer functions are written as state equations form and assembled as an entity state-output equations,the corresponding interaction forces can also be solved by the four-order Runge-Kutta algorithm.Finally,two examples are utilized to demonstrate the feasibility of the proposed method.
Based on the dynamic equations with fractional order(FO),the damping-extraction method for structure-media with FO is studied.Firstly,the time domain basic formula of the damping extraction method of dynamic structure-media interaction with FO is deduced.Secondly,the uncoupled dynamic equations of dynamic structure-media interaction with fractional order are determined and then transformed into transfer functions expression according to mode transform.Thirdly,the acquired FO transfer functions are expressed as integrated order ones approximately by Oustalop differential filter algorithm.Fourthly,the all transfer functions are written as state equations form and assembled as an entity state-output equations,the corresponding interaction forces can also be solved by the four-order Runge-Kutta algorithm.Finally,two examples are utilized to demonstrate the feasibility of the proposed method.
引文
[1]李建波,陈健云,林皋.求解非均匀无限地基相互作用力的有限元时域阻尼抽取法[J].岩土工程学报,2004,26(2):263-267.(LI Jianbo,CHEN Jianyun,LIN Gao.Finite element damping-extraction method for dynamic interaction time analysis of non-homogeneous unbounded rock[J].Chinese Journal of Geotechnical Engineering.2004,26(2):263-267.(Chinese))
[2]MCVAY M C,WASMAN S J.Dynamic soil-structure interaction of bridge substructure subject to vessel impact[J].Journal of Bridge Engineering,2009,14(1):7-16.
[3]STEHMEYER E H,RIZOS D C.Considering dynamic soil structure interaction(SSI)effects on seismic isolation retrofit efficiency and the importance of natural frequency ratio[J].Soil Dynamics and Earthquake Engineering,2008,28(6):468-479.
[4]GENES M C,KOCAK S.Dynamic soil-structure interaction analysis of layered unbounded media via a coupled finite element/boundary element/scaled boundary finite element model[J].International Journal for Numerical Methods in Engineering,2005,62(6):798-823.
[5]PADRON L A,AZNAREZ J J,MAESO O.Dynamic structure-soil-structure interaction between nearby piled buildings under seismic excitation by BEM-FEM model[J].Soil Dynamics and Earthquake Engineering,2009,29(6):1084-1096.
[6]YUN C B,CHANG S H,SEO C G.Dynamic infinite elements forsoil-structure interaction analysis in a layered soil medium[J].International Journal of Structural Stability and Dynamics,2007,7(4):693-713.
[7]陈少林,廖振鹏.多次透射公式在衰减波场中的实现[J].地震学报,2003,25(3):272-278.(CHEN Shaolin,LIAO Zhenpeng.Multi-transmitting formula for attenuating wave[J].Acta Seismologica Sinica,2003,25(3):272-278.(in Chinese))
[8]刘晶波,谷音,杜义欣.一致黏弹性人工边界及黏弹性边界单元[J].岩土工程学报,2006,28(9):1070-1075.(LIU Jingbo,GU Yin,DU Yixin.Consistent viscous-spring artificial boundaries and viscous-spring boundary elements[J].Chinese Journal of Geotechnical Engineering,2006,28(9):1070-1075.(in Chinese))
[9]李建波,陈健云,林皋.无限地基-结构动力相互作用分析的分区递归时域算法研究[J].工程力学,2005,22(3):88-96.(LI Jianbo,CHEN Jianyun,LIN Gao.A refined sub-regional reciprocal method in time domain for dynamic soil-structure interaction analysis[J].Engineering Mechanics,2005,22(3):88-96.(in Chinese))
[10]陈健云,李建波,林皋,等.结构-地基动力相互作用时域数值分析的显-隐式分区异步长递归算法[J].岩石力学与工程学报,2007,26(12):2481-2487.(CHEN Jianyun,LI Jianbo,LIN Gao,et al.A sub-regional explicit recursive method with mixed step-size strategy in time domain for dynamic structure-foundation interaction analysis[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(12):2481-2487.(in Chinese))
[11]LI J B,YANG J Y,LIN G.A stepwise damping-solvent extraction method for large-scale dynamic soil-structure interaction analysis in time domain[J].International Journal for Numerical and Analytical Methods in Geomechanics,2008,32(4):415-436.
[12]殷德顺,任俊娟,和成亮,等.一种新的岩土流变模型元件[J].岩石力学与工程学报,2007,26(9):1899-1903.(YIN Deshun,REN Junjuan,HE Chengliang,et al.A new rheological model element for geomaterials[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(9):1899-1903.(in Chinese))
[13]MAIONE G.Continued fractions approximation of the impulse response of fractional-order dynamic systems[J].IET Control Theory and Applications,2008,2(7):564-572.
[14]CAI X,LIU F.Numerical simulation of the fractional-order control system[J].Journal of Applied Mathematics and Computing,2007,23(1/2):229-241.
[15]OUSTALOUP A,LEVRON F.Frequency-band complex noninteger differentiator:characterization and synthesis[C]//IEEE Transaction on Circuit and Systems I:Fundamental Theory and Applications.[S.l.]:[s.n.],2000:25-39.
[2]MCVAY M C,WASMAN S J.Dynamic soil-structure interaction of bridge substructure subject to vessel impact[J].Journal of Bridge Engineering,2009,14(1):7-16.
[3]STEHMEYER E H,RIZOS D C.Considering dynamic soil structure interaction(SSI)effects on seismic isolation retrofit efficiency and the importance of natural frequency ratio[J].Soil Dynamics and Earthquake Engineering,2008,28(6):468-479.
[4]GENES M C,KOCAK S.Dynamic soil-structure interaction analysis of layered unbounded media via a coupled finite element/boundary element/scaled boundary finite element model[J].International Journal for Numerical Methods in Engineering,2005,62(6):798-823.
[5]PADRON L A,AZNAREZ J J,MAESO O.Dynamic structure-soil-structure interaction between nearby piled buildings under seismic excitation by BEM-FEM model[J].Soil Dynamics and Earthquake Engineering,2009,29(6):1084-1096.
[6]YUN C B,CHANG S H,SEO C G.Dynamic infinite elements forsoil-structure interaction analysis in a layered soil medium[J].International Journal of Structural Stability and Dynamics,2007,7(4):693-713.
[7]陈少林,廖振鹏.多次透射公式在衰减波场中的实现[J].地震学报,2003,25(3):272-278.(CHEN Shaolin,LIAO Zhenpeng.Multi-transmitting formula for attenuating wave[J].Acta Seismologica Sinica,2003,25(3):272-278.(in Chinese))
[8]刘晶波,谷音,杜义欣.一致黏弹性人工边界及黏弹性边界单元[J].岩土工程学报,2006,28(9):1070-1075.(LIU Jingbo,GU Yin,DU Yixin.Consistent viscous-spring artificial boundaries and viscous-spring boundary elements[J].Chinese Journal of Geotechnical Engineering,2006,28(9):1070-1075.(in Chinese))
[9]李建波,陈健云,林皋.无限地基-结构动力相互作用分析的分区递归时域算法研究[J].工程力学,2005,22(3):88-96.(LI Jianbo,CHEN Jianyun,LIN Gao.A refined sub-regional reciprocal method in time domain for dynamic soil-structure interaction analysis[J].Engineering Mechanics,2005,22(3):88-96.(in Chinese))
[10]陈健云,李建波,林皋,等.结构-地基动力相互作用时域数值分析的显-隐式分区异步长递归算法[J].岩石力学与工程学报,2007,26(12):2481-2487.(CHEN Jianyun,LI Jianbo,LIN Gao,et al.A sub-regional explicit recursive method with mixed step-size strategy in time domain for dynamic structure-foundation interaction analysis[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(12):2481-2487.(in Chinese))
[11]LI J B,YANG J Y,LIN G.A stepwise damping-solvent extraction method for large-scale dynamic soil-structure interaction analysis in time domain[J].International Journal for Numerical and Analytical Methods in Geomechanics,2008,32(4):415-436.
[12]殷德顺,任俊娟,和成亮,等.一种新的岩土流变模型元件[J].岩石力学与工程学报,2007,26(9):1899-1903.(YIN Deshun,REN Junjuan,HE Chengliang,et al.A new rheological model element for geomaterials[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(9):1899-1903.(in Chinese))
[13]MAIONE G.Continued fractions approximation of the impulse response of fractional-order dynamic systems[J].IET Control Theory and Applications,2008,2(7):564-572.
[14]CAI X,LIU F.Numerical simulation of the fractional-order control system[J].Journal of Applied Mathematics and Computing,2007,23(1/2):229-241.
[15]OUSTALOUP A,LEVRON F.Frequency-band complex noninteger differentiator:characterization and synthesis[C]//IEEE Transaction on Circuit and Systems I:Fundamental Theory and Applications.[S.l.]:[s.n.],2000:25-39.
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