Analysis of implicit HHT-α integration algorithm for real-time hybrid simulation
SUMMARY Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a succ...
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| Vydáno v: | Earthquake engineering & structural dynamics Ročník 41; číslo 5; s. 1021 - 1041 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Chichester, UK
John Wiley & Sons, Ltd
25.04.2012
Wiley |
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| ISSN: | 0098-8847, 1096-9845 |
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| Abstract | SUMMARY
Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a successful real‐time hybrid simulation. The implicit HHT α‐algorithm is a popular integration algorithm for conducting structural dynamic time history analysis because of its desirable properties of unconditional stability for linear elastic structures and controllable numerical damping for high frequencies. The implicit form of the algorithm, however, requires iterations for nonlinear structures, which is undesirable for real‐time hybrid simulation. Consequently, the HHT α‐algorithm has been implemented for real‐time hybrid simulation using a fixed number of substep iterations. The resulting HHT α‐algorithm with a fixed number of substep iterations is believed to be unconditionally stable for linear elastic structures, but research on its stability and accuracy for nonlinear structures is quite limited. In this paper, a discrete transfer function approach is utilized to analyze the HHT α‐algorithm with a fixed number of substep iterations. The algorithm is shown to be unconditionally stable for linear elastic structures, but only conditionally stable for nonlinear softening or hardening structures. The equivalent damping of the algorithm is shown to be almost the same as that of the original HHT α‐algorithm, while the period elongation varies depending on the structural nonlinearity and the size of the integration time‐step. A modified form of the algorithm is proposed to improve its stability for use in nonlinear structures. The stability of the modified algorithm is demonstrated to be enhanced and have an accuracy that is comparable to that of the existing HHT α‐algorithm with a fixed number of substep iterations. Both numerical and real‐time hybrid simulations are conducted to verify the modified algorithm. The experimental results demonstrate the effectiveness of the modified algorithm for real‐time testing. Copyright © 2011 John Wiley & Sons, Ltd. |
|---|---|
| AbstractList | SUMMARY
Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a successful real‐time hybrid simulation. The implicit HHT α‐algorithm is a popular integration algorithm for conducting structural dynamic time history analysis because of its desirable properties of unconditional stability for linear elastic structures and controllable numerical damping for high frequencies. The implicit form of the algorithm, however, requires iterations for nonlinear structures, which is undesirable for real‐time hybrid simulation. Consequently, the HHT α‐algorithm has been implemented for real‐time hybrid simulation using a fixed number of substep iterations. The resulting HHT α‐algorithm with a fixed number of substep iterations is believed to be unconditionally stable for linear elastic structures, but research on its stability and accuracy for nonlinear structures is quite limited. In this paper, a discrete transfer function approach is utilized to analyze the HHT α‐algorithm with a fixed number of substep iterations. The algorithm is shown to be unconditionally stable for linear elastic structures, but only conditionally stable for nonlinear softening or hardening structures. The equivalent damping of the algorithm is shown to be almost the same as that of the original HHT α‐algorithm, while the period elongation varies depending on the structural nonlinearity and the size of the integration time‐step. A modified form of the algorithm is proposed to improve its stability for use in nonlinear structures. The stability of the modified algorithm is demonstrated to be enhanced and have an accuracy that is comparable to that of the existing HHT α‐algorithm with a fixed number of substep iterations. Both numerical and real‐time hybrid simulations are conducted to verify the modified algorithm. The experimental results demonstrate the effectiveness of the modified algorithm for real‐time testing. Copyright © 2011 John Wiley & Sons, Ltd. Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when subject to dynamic loading. The integration algorithm used to solve the equations of motion has to be stable and accurate to achieve a successful real‐time hybrid simulation. The implicit HHT α‐algorithm is a popular integration algorithm for conducting structural dynamic time history analysis because of its desirable properties of unconditional stability for linear elastic structures and controllable numerical damping for high frequencies. The implicit form of the algorithm, however, requires iterations for nonlinear structures, which is undesirable for real‐time hybrid simulation. Consequently, the HHT α‐algorithm has been implemented for real‐time hybrid simulation using a fixed number of substep iterations. The resulting HHT α‐algorithm with a fixed number of substep iterations is believed to be unconditionally stable for linear elastic structures, but research on its stability and accuracy for nonlinear structures is quite limited. In this paper, a discrete transfer function approach is utilized to analyze the HHT α‐algorithm with a fixed number of substep iterations. The algorithm is shown to be unconditionally stable for linear elastic structures, but only conditionally stable for nonlinear softening or hardening structures. The equivalent damping of the algorithm is shown to be almost the same as that of the original HHT α‐algorithm, while the period elongation varies depending on the structural nonlinearity and the size of the integration time‐step. A modified form of the algorithm is proposed to improve its stability for use in nonlinear structures. The stability of the modified algorithm is demonstrated to be enhanced and have an accuracy that is comparable to that of the existing HHT α‐algorithm with a fixed number of substep iterations. Both numerical and real‐time hybrid simulations are conducted to verify the modified algorithm. The experimental results demonstrate the effectiveness of the modified algorithm for real‐time testing. Copyright © 2011 John Wiley & Sons, Ltd. |
| Author | Ricles, James M. Chen, Cheng |
| Author_xml | – sequence: 1 givenname: Cheng surname: Chen fullname: Chen, Cheng email: Cheng Chen, School of Engineering, San Francisco State University, San Francisco, CA 94132, USA., chcsfsu@sfsu.edu organization: School of Engineering, San Francisco State University, CA, 94132, San Francisco, USA – sequence: 2 givenname: James M. surname: Ricles fullname: Ricles, James M. organization: ATLSS Research Center, Department of Civil and Environmental Engineering, Lehigh University, 18015, Bethlehem, PAUSA |
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| Cites_doi | 10.1002/eqe.674 10.1002/eqe.451 10.1016/S0267-7261(97)00017-1 10.1061/(ASCE)0733-9445(1999)125:6(578) 10.1061/(ASCE)EM.1943-7889.0000083 10.1002/eqe.775 10.1098/rsta.2001.0877 10.1002/eqe.628 10.1002/eqe.4290210106 10.1061/(ASCE)0733-9399(2008)134:9(703) 10.1002/nme.135 10.1115/1.3153594 10.1061/(ASCE)0733-9399(2002)128:9(935) 10.1061/(ASCE)0733-9445(1985)111:7(1482) 10.1002/(SICI)1096-9845(199912)28:12<1541::AID-EQE880>3.0.CO;2-R 10.1061/(ASCE)0733-9399(2008)134:8(676) 10.1061/(ASCE)ST.1943-541X.0000124 10.1002/eqe.4290050306 10.1002/eqe.425 10.1002/(SICI)1096-9845(199904)28:4<393::AID-EQE823>3.0.CO;2-C 10.1002/eqe.838 |
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| Keywords | experimental studies real-time hybrid simulation algorithms integration algorithm high frequency simulation testing dynamic loading accuracy discrete transfer function transfer functions earthquake engineering performances stability |
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| References | Chen C, Ricles JM. (2008). Development of direct integration algorithms for structural dynamics using discrete control theory. Journal of Engineering Mechanics 2008; 134(8):676-683. Zhang YF, Sause R, Ricles JM and Naito CJ. Modified predictor-corrector numerical scheme for real-time pseudo dynamic tests using state-space formulation. Earthquake Engineering and Structural Dynamics 2005; 34:271-288. Chopra AK Dynamics of Structures: Theory and Applications to Earthquake Engineering, 2nd Edition, 2001, Prentice-Hall: New Jersey Chen C, Ricles JM. Stability analysis of direct integration algorithms applied to nonlinear structural dynamics. Journal of Engineering Mechanics 2008; 134(9):703-711. Chen C, Ricles JM. Stability analysis of direct integration algorithms applied to MDOF nonlinear structural dynamics. Journal of Engineering Mechanics 2010; 136(4):432-440. Wu B, Bao H, Ou J, Tian S. Stability and accuracy of the central difference method for real-time substructure testing. Earthquake Engineering and Structural Dynamics 2005; 34:705-718. Jung RY, Shing PB, Stauffer E, Thoen B. Performance of a real-time pseudodynamic test system considering nonlinear structural response. EESD 2007, 36(12):1785-1809. Wen YK. Equivalent linearization for hysteretic systems under random excitation. Journal of Applied Mechanics, Transaction of ASME, 1980; 47:150-154. Chang SY. Explicit Pseudodynamic Algorithm with Unconditional Stability. Journal of Engineering Mechanics 2002; 128(9):935-947. Wu B, Wang Q, Shing PB, Ou J. Equivalent force control method for generalized real-time substructure testing with implicit integration. EQ Eng. and Struc. Dynamics 2007; 36(9):1127-1149. Buonopane SG, White RN. Pseudodynamic testing of masonry infilled reinforced concrete frame. Journal of Structural Engineering (ASCE) 1999; 125(6):578-589. Nakashima M, Kato H, Takaoka E. Development of real-time pseudodynamic testing. Earthquake Engineering and Structural Dynamics 1992; 21:79-92. Mahin SA, Shing PB. Pseudodynamic method for seismic testing. Journal of Structural Engineering (ASCE) 1985; 111(7):1482-1503. Bonnet PA, Lim CN, William MS, Blakeborough A, Neild SA, Stoten DP, Taylor CA. Real-time hybrid experiments with Newmark integration, MCSmd outer-loop control and multi-tasking strategies. Earthquake Engineering and Structural Dynamics 2005; 36(1):119-141. Chen C and Ricles JM. Stability analysis of SDOF real-time hybrid testing systems with explicit integration algorithms and actuator delay. EQ Eng. and Structural Dynamics 2008; 37(4):597-613. Ogata K Discrete-Time Control Systems, 2nd Edition, 1995, Prentice-Hall: New Jersey. Chen C, Ricles JM. Tracking Error Based Servo-Hydraulic Actuator Adaptive Compensation for Real-Time Hybrid Simulation. Journal of Structural Engineering 2010, 136(4):432-440. Nakashima M, Masaoka N. Real-time on-line test for MDOF systems. Earthquake Engineering and Structural Dynamics 1999; 28:393-420. Molina FJ, Verzeletti G, Magonette G, Buchet P. and Geradin M. Bidirectional pseudodynamic tet of a full-size three-storey building.Earthquake Eng. and Structural Dynamics 1999; 28:1541-1566. Bass BJ Christenson R, 2007. System identification of a 200kN Magneto-Rheological fluid damper for structural control in large-scale smart structures; Proceedings, 2007 American Control Conference, New York City, pp. 2690-2695. Blakeborough A, Williams MS, Darby AP, Williams DM. The development of real-time substructure testing. Philosophical Transactions of the Royal Society of London 2001; 359:1869-1891. Chen C, Ricles JM, Marullo T, Mercan O. Real-time hybrid testing using the unconditionally stable explicit CR integration algorithm. Earthquake Eng. and Structural Dynamics 2008; 38:23-44. Mugan A, Hulbert GM. Frequency Domain Analysis of Time Integration Methods for Semidiscrete Finite Element Equations, Part II. Hyperbolic and Parabolic-Hyperbolic Problems. International Journal of Numerical Methods in Engineering 2001; 51:351-376. Jung RY, Shing PB. Performance evaluation of a real-time pseudodynamic test system. Earthquake Engineering and Structural Dynamics 2006; 25(4):333-355. Combescure D, Pegon P. α-operator splitting time integration technique for pseudodynamic testing error propagation analysis. Soil Dynamics and Earthquake Engineering 1997; 16:417-443. Hilber HM, Hughes TJR, Taylor RL. Improved numerical dissipation for time integration algorithms in structural mechanics. Earthquake Eng. and Structural Dynamics 1977; 5:283-292. Franklin GF, Powell JD, Naeini AE. 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| References_xml | – reference: Nakashima M, Kato H, Takaoka E. Development of real-time pseudodynamic testing. Earthquake Engineering and Structural Dynamics 1992; 21:79-92. – reference: Chen C, Ricles JM. (2008). Development of direct integration algorithms for structural dynamics using discrete control theory. Journal of Engineering Mechanics 2008; 134(8):676-683. – reference: Wen YK. Equivalent linearization for hysteretic systems under random excitation. Journal of Applied Mechanics, Transaction of ASME, 1980; 47:150-154. – reference: Combescure D, Pegon P. α-operator splitting time integration technique for pseudodynamic testing error propagation analysis. Soil Dynamics and Earthquake Engineering 1997; 16:417-443. – reference: Wu B, Wang Q, Shing PB, Ou J. Equivalent force control method for generalized real-time substructure testing with implicit integration. EQ Eng. and Struc. Dynamics 2007; 36(9):1127-1149. – reference: Mahin SA, Shing PB. Pseudodynamic method for seismic testing. Journal of Structural Engineering (ASCE) 1985; 111(7):1482-1503. – reference: Chen C, Ricles JM. Tracking Error Based Servo-Hydraulic Actuator Adaptive Compensation for Real-Time Hybrid Simulation. Journal of Structural Engineering 2010, 136(4):432-440. – reference: Chen C and Ricles JM. Stability analysis of SDOF real-time hybrid testing systems with explicit integration algorithms and actuator delay. EQ Eng. and Structural Dynamics 2008; 37(4):597-613. – reference: Nakashima M, Masaoka N. Real-time on-line test for MDOF systems. Earthquake Engineering and Structural Dynamics 1999; 28:393-420. – reference: Chen C, Ricles JM. Stability analysis of direct integration algorithms applied to nonlinear structural dynamics. Journal of Engineering Mechanics 2008; 134(9):703-711. – reference: Chen C, Ricles JM, Marullo T, Mercan O. Real-time hybrid testing using the unconditionally stable explicit CR integration algorithm. Earthquake Eng. and Structural Dynamics 2008; 38:23-44. – reference: Mugan A, Hulbert GM. Frequency Domain Analysis of Time Integration Methods for Semidiscrete Finite Element Equations, Part II. Hyperbolic and Parabolic-Hyperbolic Problems. International Journal of Numerical Methods in Engineering 2001; 51:351-376. – reference: Ogata K Discrete-Time Control Systems, 2nd Edition, 1995, Prentice-Hall: New Jersey. – reference: Chang SY. Explicit Pseudodynamic Algorithm with Unconditional Stability. Journal of Engineering Mechanics 2002; 128(9):935-947. – reference: Chen C, Ricles JM. Stability analysis of direct integration algorithms applied to MDOF nonlinear structural dynamics. Journal of Engineering Mechanics 2010; 136(4):432-440. – reference: Jung RY, Shing PB. Performance evaluation of a real-time pseudodynamic test system. Earthquake Engineering and Structural Dynamics 2006; 25(4):333-355. – reference: Chopra AK Dynamics of Structures: Theory and Applications to Earthquake Engineering, 2nd Edition, 2001, Prentice-Hall: New Jersey – reference: Molina FJ, Verzeletti G, Magonette G, Buchet P. and Geradin M. Bidirectional pseudodynamic tet of a full-size three-storey building.Earthquake Eng. and Structural Dynamics 1999; 28:1541-1566. – reference: Zhang YF, Sause R, Ricles JM and Naito CJ. Modified predictor-corrector numerical scheme for real-time pseudo dynamic tests using state-space formulation. Earthquake Engineering and Structural Dynamics 2005; 34:271-288. – reference: Bass BJ Christenson R, 2007. System identification of a 200kN Magneto-Rheological fluid damper for structural control in large-scale smart structures; Proceedings, 2007 American Control Conference, New York City, pp. 2690-2695. – reference: Jung RY, Shing PB, Stauffer E, Thoen B. Performance of a real-time pseudodynamic test system considering nonlinear structural response. EESD 2007, 36(12):1785-1809. – reference: Hilber HM, Hughes TJR, Taylor RL. Improved numerical dissipation for time integration algorithms in structural mechanics. Earthquake Eng. and Structural Dynamics 1977; 5:283-292. – reference: Buonopane SG, White RN. Pseudodynamic testing of masonry infilled reinforced concrete frame. Journal of Structural Engineering (ASCE) 1999; 125(6):578-589. – reference: Wu B, Bao H, Ou J, Tian S. Stability and accuracy of the central difference method for real-time substructure testing. Earthquake Engineering and Structural Dynamics 2005; 34:705-718. – reference: Blakeborough A, Williams MS, Darby AP, Williams DM. The development of real-time substructure testing. Philosophical Transactions of the Royal Society of London 2001; 359:1869-1891. – reference: Bonnet PA, Lim CN, William MS, Blakeborough A, Neild SA, Stoten DP, Taylor CA. 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Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices... Real‐time hybrid simulation is a viable experiment technique to evaluate the performance of structures equipped with rate‐dependent seismic devices when... |
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| SubjectTerms | accuracy discrete transfer function Earth sciences Earth, ocean, space Earthquakes, seismology Engineering and environment geology. Geothermics Engineering geology Exact sciences and technology integration algorithm Internal geophysics real-time hybrid simulation stability |
| Title | Analysis of implicit HHT-α integration algorithm for real-time hybrid simulation |
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