基于Rosenbrock的耦合积分方法及其在实时子结构试验中的应用
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摘要
作为一种集计算机模拟和物理试验于一体的新型混合试验方法,实时子结构试验在过去20年得到迅速发展。该试验方法的关键在于如何保证数值子结构和试验子结构的实时耦联。对于复杂结构来说,更需要高效的数值积分方法以确保每步计算在一个采样步长内完成。鉴于此,本文在Rosenbrock实时积分方法的基础上,提出了一种具有完全并行计算格式的耦合积分方法,并基于单自由度分离质量模型分析了该方法的收敛性;再通过对三自由度分离质量模型的数值模拟,验证了该方法的收敛性;最后,在多自由度试验平台上完成了两自由度结构的实时子结构试验。理论分析、数值模拟及实时子结构试验表明,该方法具有良好的稳定性和2阶精度,与直接积分方法相比更适用于复杂结构的实时子结构试验。
As a hybrid computational-experimental technique,real-time substructure testing(RTST) was under rapid development in the last two decades.The challenge of RTST is to ensure that the numerical substructure and the experimental substructure interact in real time.For complex structures,this requires efficient numerical integration methods in order that the computation of each step completes during a sampling time step.With this in mind,based on the Rosenbrock real-time integrator,a coupled time integration method is proposed,which renders completely parallel computation.By virtue of a Single-DoF split-mass test problem,convergence analysis is performed theoretically for the coupled method.Moreover,numerical simulations of a Three-DoF split-mass system are conducted to validate the convergence of the coupled method.Last but not least,experimental validation of a Two-DoF system is conducted within a framework of Multi-DoF test-rig.The theoretical analyses,the numerical simulations and the experimental results reveal that the method exhibits favourable stability and second order accuracy.With respect to direct time integration methods,the proposed method is more desirable for RTST tests of complicated structures.
As a hybrid computational-experimental technique,real-time substructure testing(RTST) was under rapid development in the last two decades.The challenge of RTST is to ensure that the numerical substructure and the experimental substructure interact in real time.For complex structures,this requires efficient numerical integration methods in order that the computation of each step completes during a sampling time step.With this in mind,based on the Rosenbrock real-time integrator,a coupled time integration method is proposed,which renders completely parallel computation.By virtue of a Single-DoF split-mass test problem,convergence analysis is performed theoretically for the coupled method.Moreover,numerical simulations of a Three-DoF split-mass system are conducted to validate the convergence of the coupled method.Last but not least,experimental validation of a Two-DoF system is conducted within a framework of Multi-DoF test-rig.The theoretical analyses,the numerical simulations and the experimental results reveal that the method exhibits favourable stability and second order accuracy.With respect to direct time integration methods,the proposed method is more desirable for RTST tests of complicated structures.
引文
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[22]Lamarche C P,Bonelli A,Bursi O S,et al.A Rosenbrock-w method for real-time dynamic substructuring and pseudo-dynamic testing[J].Earth-quake Engineering&Structural Dynamics 2009,38(9):1071-1092.
[2]Wagg D J,Stoten D P.Substructuring of dynamical systems via the adaptive minimal control synthesis algorithm[J].Earthquake Engineering andStructural Dynamics,2001,30(6):865-877.
[3]Wu B,Wang Q,Shing P B,et al.Equivalent force control method for generalized real-time substructure testing with implicit integration[J].Earthquake Engineering and Structural Dynamics,2007,36(9):1127-1149.
[4]Bonnet P A,Williams M S,Blakeborough A,et al.Real-time hybrid experiments with Newmark integration,MCSmd outer-loop control andmulti-tasking strategies[J].Earthquake Engineering and Structural Dynamics,2007,36(1):119–141.
[5]吴斌,保海娥.实时子结构实验Chang方法的稳定性和精度[J].地震工程与工程振动,2006,26(2):41-48.WU Bin,BAO Haie.Stability and accuracy of Chang algorithm for real time substructure testing[J].Journal of Earthquake Engineering and Engi-neering Vibration,2006,26(2):41-48.(in Chinese)
[6]迟福东,王进廷,金峰.实时耦联动力试验的时滞稳定性分析[J].工程力学,2010,27(9):12-16.CHI Fudong,WANG Jinting,JIN Feng.Delay-dependent stability analysis of real-time dynamic hybrid testing[J].Engineering Mechanics,2010,27(9):12-16.(in Chinese)
[7]袁涌,熊世树,朱宏平.加载速率对速度控制型实时子结构拟动力实验结果的影响[J].东南大学学报:自然科学版,2008,5:784-787.YUAN Yong,XIONG Shishu,ZHU Hongping.Effect of loading ratio on results of real-time substructure hybrid experiment based on velocity con-trol[J].Journal of Southeast University:Natural Science Edition,2008,5:784-787.(in Chinese)
[8]Akira I,Fernando S,Hirokazu I,et al.Real-Time Hybrid Testing of laminated rubber dampers for seismic retrofit of bridges[C]//3rd Interna-tional Conference on Advances in Experimental Structural Engineering.15-16,October,2009,San Francisco,California,USA:http://peer.berkeley.edu/events/2009/icaese3/.
[9]Belytschko T,Yen H-J,Mullen R.Mixed methods for time integration[J].Computer Methods in Applied Mechanics and Engineering,1979,17(2):259-275.
[10]Saouma V,Sivaselvan M V.Hybrid Simulation-Theory,implementation and applications[M].London:Taylor&Francis,2008.
[11]Nakshatrala K B,Hjelmstad K D,Tortorelli D A.A FETI-based domain decomposition technique for time-dependent first-order systems based ona DAE approach[J].International Journal for Numerical Methods in Engineering,2008,75(12):1385-1415.
[12]Pegon P,Magonette G.Continuous PSD testing with non-linear substructuring:Presentation of a stable parallel inter-field procedure[R].Techni-cal Report I.02.167,E.C.,JRC,ELSA,Ispra,Italy,2002.
[13]Burgermeister B,Arnold M,Esterl B.DAE time integration for real-time applications in multibody dynamics[J].Zeitschrift Angewandte Math-ematik Mechanik,2006,86(10):759-771.
[14]Hairer E,Wanner G.Solving ordinary differential equations II.stiff and differential-algebraic problems[M].Berlin:Springer,1996.
[15]Bursi O S,Jia C,Vulcan L,et al.Rosenbrock-based algorithms and subcycling strategies for real-time non-linear substructure testing[J].Earth-quake Engineering and Structural Dynamics,2011,40(1):1-19.
[16]Farhat C,Crivelli L,Geradin M.Implicit time integration of a class of constrained hybrid formulations-part i:Spectral stability theory[J].Com-puter Methods in Applied Mechanics and Engineering,1995,125(1):71-107.
[17]JIA C.Monolithic and partitioned Rosenbrock-based time integration methods for dynamic substructure tests[D].Ph.D.Thesis,University ofTrento,Italy 2010.
[18]Mosqueda G,Ahmadizadeh M.Iterative implicit integration procedure for hybrid simulation of large nonlinear strucutures[J].Earthquake Engi-neering and Structural Dynamics,2011,40(9):945-960.
[19]Prakash A,Hjelmstad K D.A FETI-based multi time-step coupling method for Newmark schemes in structural dynamics[J].International Journo-al for numerical methods in engineering,2004,61(13):2183-2204.
[20]Daniel W J T.A study of the stability of subcycling algorithms in structural dynamics[J].Computer Methods in Applied Mechanics and Engineer-ing,1998,156(4):1-13.
[21]Morari M,Zariou E.Robust process control[M].New Jersey:Prentice Hall 1989.
[22]Lamarche C P,Bonelli A,Bursi O S,et al.A Rosenbrock-w method for real-time dynamic substructuring and pseudo-dynamic testing[J].Earth-quake Engineering&Structural Dynamics 2009,38(9):1071-1092.
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