1. Conventional Hybrid and Real-Time Hybrid Testing Brian Phillips 브라이언 필립스 University of Illinois at Urbana-Champaign 일리노이 대학교 - 어바나 샴페인 For 2008 Asia-Pacific Summer School in Smart Structures Technology at KAIST

2. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

3. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

4. Experimental Testing  Experimental evaluation of components required when Response not well understood Difficult to model numerically Model development stage  Outcomes help improve Understanding of dynamic response Computational models and constitutive relationships Design methods and codes

5. Seismic Evaluation of Structures Quasi-static Shaking TableHybrid

6. Hybrid versus Quasi-Static Shore Western Series 92 Actuator  Quasi-Static Predefined loading path  Hybrid Loading path depends on structural response  Similar Qualities Provide structural capacity Hardware Controller

7. Hybrid versus Shake Table  Shake Table Dynamic loading rate Directly account for rate dependent behavior Model entire structure, usually scaled Predefined loading path  Conventional Hybrid Quasi-static loading rate Rate dependent behavior included numerically Continual observation and monitoring of experiment Pause and resume test Substructuring Loading path depends on structural response Ramp Hold t x

8. Hybrid Test Method  Combination of Experimental testing Analytical simulation  Concept proposed in late 1960’s (Hakuno et al., 1969)  Developed in the mid 1970’s (Takanashi et al. 1975) Incorporated digital computers Discrete systems  quasi-static loading  Also known as Hybrid Simulation Pseudodynamic test method (PsD) Computer-actuator on-line test Virtual prototyping

9. Basis of Method  Equation of Motion  M N = mass (numerical)  C N = viscous damping (numerical)  F = effective external force  R N = restoring force (numerical)  R E = restoring force (experimental) Represents stiffness, damping, and inertial forces in experimental structure

10. R E in Conventional Hybrid Testing  R E (x)  R E = K∙x(t) for linear elastic  Rate dependent behavior included numerically  Experiment conducted arbitrarily slowly Actuator dynamics become insignificant Larger actuators can be easily accommodated Full scale specimens

11. Hybrid Testing Components  System inputs Earthquake record Analytical model of structure (M N, C N, K N )  Numerical integration scheme Calculate displacements (x) at discrete points in time  Experimental setup Apply displacements (x) to specimen Usually applied at 100 to 1000 time scale Measure specimen restoring force (R E )  Numerical integration of next time step

12. Required Equipment (Shopping List)  Servo-hydraulic system Servo-controller Servo-valve Hydraulic actuators  Instrumentation Displacement transducer Load cell  Strong floor and reaction wall  On-line computer Numerical integration Generate command signal (D/A conversion) Read restoring force (A/D conversion)

13. Hybrid Test Flow of Information D/A A/D PID Servo Controller Control Loop Hybrid Testing Loop LVDT Load Cell SpecimenActuator Servovalve xcxc xcxc xmxm xmRmxmRm RmRm i ΔtΔt m c, k x = displacement R = force i = current □ c = commanded □ m = measured δtδt

14. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

15. Numerical Integration  Discrete representation of equation of motion  t i = iΔt, i = 1, …, n  Smaller Δt increases accuracy as well as computational demand titi t i+1 xixi x i+1 x t x i-1 t i-1

16. Numerical Integration Schemes  Explicit Displacement solution at t i +1 is based on previous steps (t i, t i-1, etc.) Computationally efficient Conditionally stable solution  Related to natural frequencies of structure and Δt  Implicit Displacement solution at t i+1 is based on previous and current steps (t i+1, t i, t i-1, etc.) Iterative procedure for nonlinear behavior Some implicit methods are unconditionally stable  Beneficial to stiff and MDOF structures

17. Central Difference Method  Explicit method  Low computational cost  Easily fits into hybrid testing framework  Stability condition ωΔt ≤ 2 titi t i+1 xixi x i+1 x t t i-1 x i-1 2Δt2Δt

18. CDM in Hybrid Testing Framework Initial Conditions Compute Velocity and Acceleration External Force Impose Command on Actuator Measure Restoring Force Conditions at Step i Update Compute Displacement

19. Newmark Beta Method (Newmark, 1959)  β and γ determine the stability and accuracy of method  Popular variations β = 0 and γ = 1/2  Central Difference Method (explicit) β = 1/4 and γ = 1/2  Constant Average Acceleration (implicit) β = 1/6 and γ = 1/2  Linear Acceleration Method (implicit)  γ controls numerical damping γ = 1/2  No numerical damping (second order accurate) γ < 1/2  Negative numerical damping (first order accurate) γ > 1/2  Positive numerical damping (first order accurate)

20. Alpha Method (α-HHT) (Hibler et al., 1977)  Modification of the Newmark method  Properties Unconditionally stable α alters numerical damping α = 0  Constant average acceleration method Maintains second-order accuracy for any γ Favorable dissipation in higher modes (potentially spurious) with little affect on lower modes

21. Operator Splitting (OS) Method (Nakashima 1990)  Predictor components Based on previous steps only (explicit)  Corrector components Includes next step in formulation (implicit)

22. Operator Splitting (OS) Method  No iteration of command on specimen Explicit formulation for inelastic portion Implicit formulation for elastic portion R x Unconditionally stable for softening type stiffness Predictor Step Corrector Step Obtain restoring force at end of time step with no iteration

23. α-OS Method (Combescure and Pegon, 1997)  Combination of α-HHT and OS Methods  Allows alpha method to be implemented without iterating commands on the specimen  Unconditionally stable for softening nonlinearities  Accuracy of higher modes affected by severe stiffness degradation, lower modes remain accurate

24. α-OS Method in Action Compute Acceleration External Force Impose Command on Actuator Compute Pseudo-Force Conditions at Step i Compute Correctors Compute Predictors Initial Conditions

25. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

26. Errors in Hybrid Testing xcxc xmxm RmRm Modeling Errors Numerical Integration Errors titi t i+1 xixi x i+1 x t Experimental Errors

27. Experimental Error Sources Flexibility of reaction frame Displacement control of hydraulic actuators x t Commanded Measured Intrinsic Noise x t Instrumentation errors Calibration errors Noise Precision errors Range of instruments Properties of specimen

28. Experimental Errors in Hybrid Testing  Method is sensitive to experimental errors  Closed loop experiment Errors accumulate throughout entire test System instability Undesired damage to specimen  Quasi-static and shake table test methods are less sensitive to experimental error Predefined command history

29. Experimental Error Types  Systematic errors Actuator overshooting and undershooting Actuator lag Can lead to system instabilities  Random errors High frequency noise in instrumentation Less severe than systematic errors Can be controlled using dissipative integration algorithms  Relaxation of restoring forces Can be reduced by minimizing or eliminating hold period  Rate effects Can increase speed to fast or real-time hybrid testing

30. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

31. Variations of Hybrid (Pseudodynamic, PsD) Testing Conventional PsD Testing Takanashi, et al., 1975 Substructure PsD Testing Dermitzakis and Mahin, 1985 Continuous PsD Testing Takanashi and Ohi, 1983 Real-Time Hybrid Testing Nakashima, et al., 1992 Effective Force Testing Mahin, et al., 1985, 1989 Distributed Substructure PsD Testing Watanabe, et al., 2001 Distributed Continuous PsD Testing Mosqueda, et al., 2004 1 2 34 5 (Carrion, 2007)

32. Substructure PsD Testing Experimental Substructure Numerical Substructure Structure of Interest

33. Distributed Substructure PsD Testing

34. Continuous PsD Testing  Provides continuous actuator movement No hold phase Avoids force relaxation Can be conducted for both slow and fast rates  Prediction and correction phases Ramp Hold t x Prediction Correction

35. Effective Force Testing  Convert earthquake ground motion into equivalent inertial forces at each DOF Independent of stiffness and damping Force controlled actuators  Force commands known prior to experiment No substructuring Full mass and damping must be included in specimen  Control-structure interaction limits ability to apply force control around natural frequencies (Dyke et al., 1995) Must apply accurate compensation (challenging) m

36. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

37. Real-Time Hybrid Testing (RTHT)  1:1 time scaling  Accurately test rate dependent components (i.e. dampers, friction devices, and base isolation)  Cycles must be performed very quickly  System dynamics become important Time delays: computation and communication Time lags: lag in response of actuator to command Numerical Calculations Apply Displacement Measure Restoring Forces Δt = 10 – 20 msec

38. RTHT Hardware Restrictions  Dynamically rated actuators Double ended  Fast, dedicated computers xPC Target (Mathworks) dSpace (dSpace) CompactRIO (NI) Shore Western Series 91 Actuator Real Time

39. RTHT Restrictions on Explicit Numerical Integration  Controller sampling rate δt smaller than typical Δt of numerical integration  Separate signal generation (δt) and response analysis (Δt), (Nakashima and Masaoka, 1999)  Signal generation based on polynomial extrapolation and interpolation x t ΔtΔt ΔtΔt extrapolationinterpolation δtδt δtδt PID δt Servo Controller x t ΔtΔt Numerical Integration

40. RTHT Restrictions on Implicit Numerical Integration  Actuators must move with smooth velocity  Iteration of implicit schemes unpredictable Fix number of iterations n Interpolate commands (δt) between time steps (Δt) based on each subsequent iteration (Jung and Shing 2007) ΔtΔtΔtΔt x t t i-1 titi t i+1 δtδt δt = Δt / n Quadratic Curves Actual Commands δtδtδtδtδtδt

41. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

42. Time Delays  Data acquisition and communication D/A conversion of command signal A/D conversion of measured signals Communication delays  Computer, controller, DAQ system  Computation time Numerical integration strategy Complexity of numerical model  Constant throughout test

43. Time Lags  Finite response time of actuators  Control-structure interaction (Dyke et al., 1995) Dynamic coupling of actuator and specimen  Frequency dependent Actuator FRF

44. Effects of Time Delays and Time Lags TdTd Imposed x m Commanded x c Measured Response Actual Response  Inaccuracies that propagate throughout experiment  Introduces negative damping into system c eq = -kT d for SDOF  Problems arise with structures with low damping experiments with large hydraulic actuators t x x R xcxc xmxm RmRm

45. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

46. System Modeling in the Frequency Domain  Measure frequency response function (FRF) from command (x c ) to measured response (x m )  Determine number of poles and zeros based on theoretical models  Create system model to match experimental data 3-Pole Model 4-Pole Model 5-Pole Model

47. Effect of Actuator Dynamics on RTHT  Exact system FRF for SDOF has 2 poles, no zeros  RTHT system FRF includes additional number of poles and zeros equal to the order of the actuator FRF Actuator Dynamics Experimental Component

48. Effect of Actuator Dynamics on RTHT  Examine using numerical simulation SDOF model, 1 Hz natural frequency  Exact system: 2 poles 4 pole model of actuator dynamics  RTHT system: 6 poles and 4 zeros  Actuator dynamics add negative damping Characterize stability based on structural damping ζ ζ th = 3.54% stability threshold Structure k2k2 m1m1 k1k1 c1c1

49. FRF ζ = 5% FRF Magnitude FRF Phase Negative Damping ζ = 5% > ζ th = 3.54%

50. Pole-Zero Map ζ = 5% Pole-Zero Map Pole-Zero Map Zoom Additional RTHT Poles and Zeros Dominant Poles ζ = 5% > ζ th = 3.54%

51. Impulse Response ζ = 5% Negative Damping ζ = 5% > ζ th = 3.54% Structure k2k2 m1m1 k1k1 c1c1

52. FRF ζ = 3% FRF Magnitude FRF Phase Negative Damping ζ = 3% < ζ th = 3.54%

53. Pole-Zero Map ζ = 3% Pole-Zero Map Pole-Zero Map Zoom Additional RTHT Poles and Zeros Dominant Poles ζ = 3% < ζ th = 3.54% 0

54. Impulse Response ζ = 3% Negative Damping Unstable Response ζ = 3% < ζ th = 3.54% Structure k2k2 m1m1 k1k1 c1c1

55. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

56. Delay/Lag Compensation TdTd t i+1 t i+1 +T d x(t i ) x(t i+1 ) x m (t i+1 ) imposed current calculated TdTd t i+1 t i+1 +T d x(t i ) x(t i+1 ) x m (t i+1 ) imposed x p (t i+1 +T d ) current calculated predicted x(t i+1 ) ≠ x m (t i+1 )x(t i+1 ) ≈ x m (t i+1 ) t x Uncompensated t x Compensated  Delay/lag compensation is a critical component of RTHT

57. Traditional Delay/Lag Compensation  Delays and lags are combined to create a total time delay Actuator lags are actually frequency dependent Single delay may be inadequate for MDOF Actuator FRF with time delay model TdTd 1 T d ≈ 12.5 msec

58. Response Prediction Polynomial Extrapolation (Horiuchi 1996)  Most widely used method  Send command based on command desired after T d  Predicted displacement based on current and previous time steps  3 rd order provides balance of speed and accuracy  Accuracy and stability concern when T d is large compared to smallest period of structure xcxc TdTd x0x0 x1x1 x2x2 x3x3 x t

59. Response Prediction Model Based  Can estimate response of system after T d (ω) based on available system information M, C, K, F Known prior to testing or at onset of experiment  Model may be updated as necessary TdTd ΔtΔt Model-Based Predictor M, C, K uncompensated target displacement (initial condition) restoring force Compensated target displacement (send to controller) ΔtΔtΔtΔt t input force

60. Model Based Feedforward Compensation  Open loop compensation  Sends a command to test setup that is a best guess to produce the desired response  Ideally completely cancels actuator dynamics  No added stability issues G FF (s) G xu (s) FeedforwardExperimental Setup d ux d≈x

61. Feedforward Compensation Actuator FRF with 3 Pole ModelFeedforward FRF Not proper system, unstable Modified inverse dynamics Proper system, stable, α > 1

62. Feedforward Numerical Simulation 3 Pole Model of Actuator FRF Response to Unit Step Displacement Input

63. Feedback Compensation  Closed loop compensation  Error between desired and measured displacement used to modify command Minimize e = d - x  Example controller is G FB (s) = K FB  Slower than feedforward compensation Not effective at reducing actuator lag G FB (s) G xu (s) FeedbackExperimental System x d u FB + - + + e u d x d≈x

64. Combined Feedfoward and Feedback Compensation  Feedforward controller ideally cancels actuator dynamics  Feedback controller eliminates errors due to Inaccuracies in modeling of feedforward controller Added dynamics to make feedforward controller stable Changes in specimen during experiment G FB (s) G xu (s) G FF (s) Feedforward FeedbackExperimental System x d u FF u FB + - + + e u x d≈x

65. Experimental Comparison of Delay Compensation Methods (Carrion, 2007) Linear K y /K e = 1.0 Nonlinear K y /K e = 0.02  SDOF System ζ = 2%, CDM, 2δt = Δt = 0.0062 sec  Model based approach allows testing of structure with twice the natural frequency as polynomial extraction Structure k2k2 m1m1 k1k1 c1c1

66. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

67. Real-Time Hybrid Testing Applications Experimental Substructure Damper Actuator DampersBase Isolation Devices m2m2 m1m1 k1k1 k2k2 Numerical Substructure m2m2 m1m1 k1k1 k2k2 Structure k1k1 k2k2 m1m1 m2m2 k1k1 k2k2 m1m1 m2m2 Actuator Gravity load Base Isolation Device Experimental Substructure

68. Presentation Outline  Introduction and Motivation  Numerical Integration Schemes  Errors in Hybrid Testing  Variations of Hybrid Testing  Real-Time Hybrid Testing Basics Time Delays and Time Lags System Modeling Compensation Techniques Applications  Conclusions

69. Conclusions  Hybrid testing is an appealing structural testing method Similar equipment as quasi-static testing Time scale may be extended  Substructuring Facilitates full-scale testing  Real-Time Hybrid Testing Accurately test rate dependent components Time delays and lags can undermine experiment Model based compensation techniques are a powerful alternative

70. Acknowledgements  Dr. B.F. Spencer and Dr. J. Carrion for their advice and support  Dr. C.B. Yun for his invitation to provide a lecture for the APSS program. 감사합니다 ! This material is based upon work supported under a National Science Foundation Graduate Research Fellowship. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.