Conventional Hybrid and Real-Time Hybrid Testing

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

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

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

Seismic Evaluation of Structures Shaking Table Hybrid Quasi-static

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

x Hold Hold t Ramp Ramp 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

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

Basis of Method • Equation of Motion • MN = mass (numerical) • CN = viscous damping (numerical) • F = effective external force • RN = restoring force (numerical) • RE = restoring force (experimental) • Represents stiffness, damping, and inertial forces in experimental structure

RE in Conventional Hybrid Testing • RE(x) • RE = 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

Hybrid Testing Components • System inputs • Earthquake record • Analytical model of structure (MN, CN, KN) • 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 (RE) • Numerical integration of next time step

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)

m c, k PID Servo Controller Hybrid Test Flow of Information Hybrid Testing Loop x = displacement R = force i = current □c = commanded □m = measured Control Loop i xc Servovalve xc D/A xm Rm Actuator Specimen LVDT A/D xm δt Load Cell Rm Δt

x t Numerical Integration • Discrete representation of equation of motion • ti = iΔt, i = 1, …, n • Smaller Δt increases accuracy as well as computational demand xi xi-1 xi+1 ti-1 ti ti+1

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

x xi xi-1 xi+1 2Δt t ti-1 ti ti+1 Central Difference Method • Explicit method • Low computational cost • Easily fits into hybrid testing framework • Stability condition ωΔt ≤ 2

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

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)

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

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

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

α-OSMethod (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

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

Numerical Integration Errors Modeling Errors x xi xi+1 t ti ti+1 Experimental Errors Errors in Hybrid Testing xc xm Rm

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

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

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

Variations of Hybrid (Pseudodynamic, PsD) Testing Conventional PsD Testing Takanashi, et al., 1975 5 1 3 4 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 2 Distributed Substructure PsD Testing Watanabe, et al., 2001 Distributed Continuous PsD Testing Mosqueda, et al., 2004 (Carrion, 2007)

Substructure PsD Testing Structure of Interest Numerical Substructure Experimental Substructure

Distributed Substructure PsD Testing

Correction Correction Prediction Prediction x t Hold Hold Ramp Ramp 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

m 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)

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 Δt = 10 – 20 msec Apply Displacement Measure Restoring Forces

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

x PID δt t Δt Servo Controller x Numerical Integration Δt t 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 extrapolation interpolation δt δt Δt

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) x δt = Δt / n δt δt δt δt Quadratic Curves t Δt Δt ti ti+1 ti-1 Actual Commands

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

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

Effects of Time Delays and Time Lags • Inaccuracies that propagate throughout experiment • Introduces negative damping into system • ceq = -kTd for SDOF • Problems arise with • structures with low damping • experiments with large hydraulic actuators R x Actual Response Imposed xm Rm xm xc t x Commanded xc Td Measured Response

System Modeling in theFrequency Domain • Measure frequency response function (FRF) from command (xc) to measured response (xm) • Determine number of poles and zeros based on theoretical models • Create system model to match experimental data 3-Pole Model 5-Pole Model 4-Pole Model

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 Experimental Component Actuator Dynamics

m1 k1 k2 c1 Structure 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 • ζth = 3.54% stability threshold

FRFζ = 5% FRF Magnitude FRF Phase Negative Damping Negative Damping ζ = 5% > ζth = 3.54%

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

Conventional Hybrid and Real-Time Hybrid Testing