conventional hybrid and real-time hybrid testing brian phillips 브라이언 필립스 university of...

Conventional Hybrid and

Real-Time Hybrid Testing

Brian Phillips브라이언 필립스

University of Illinois at Urbana-Champaign일리노이 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

Quasi-static

Shaking Table Hybrid

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

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

gx

Ramp Ramp

Hold Hold t

x

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

tFxxxRxRtxCtxM ENNN ,,

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)

Hybrid Test Flow of Information

D/A

A/D

PID

ServoController

Control Loop

Hybrid Testing Loop

LVDT

Load Cell

SpecimenActuator

Servovalvexc xc

xm

xm

Rm

Rm

i

Δt

1111 iiii FRxCxM

m

c, kgx

x = displacementR = forcei = current□c = commanded□m = measured

δt



Conclusions

Numerical Integration Discrete representation of equation of

motion

ti = iΔt, i = 1, …, n Smaller Δt increases accuracy as well as

computational demand

1111 iiii FxRxCxM

ti ti+1

xi xi+1

x

t

xi-1

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

Central Difference Method Explicit method Low computational cost Easily fits into hybrid testing framework Stability condition ωΔt ≤ 2

t

xxx iii

211

211 2

t

xxxx iiii

ti ti+1

xi xi+1

x

tti-1

xi-1

2Δt

CDM in Hybrid Testing Framework

Initial Conditions

NNN KCM ,,

Compute Velocity and Acceleration

11, ii xx

External Force

1iF

Impose Command on Actuator

1ix

Measure Restoring Force1, iER

Conditions at Step i

ixixix

Update

1ii

Compute Displacement

1ixiiii xtxtxx 2

1 2

1

11 2

1 iiii xxtxx

11,1,11 iiEiNiNiN FRRxCxM

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)

1

21 2

1iiiii xxtxtxx 11 1 iiii xxtxx

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

iiiiiii FFRxRxxCxCxM 1111 111

214

1 2212

1 031

Operator Splitting (OS) Method(Nakashima 1990) Predictor components

Based on previous steps only (explicit)

Corrector components Includes next step in formulation (implicit)

1

21 2

1iiiii xxtxtxx 11 1 iiii xxtxx

1~

ix 1~

ix

12

11~

iii xtxx 111

~ iii xtxx

Operator Splitting (OS) Method No iteration of command on specimen

Explicit formulation for inelastic portion Implicit formulation for elastic portion

1

~iR

R

x

0K

1ix1~

ix

11011~~

iiii xxKRR

FxxRxxRxCxM iiE

iiI

ii 111111~,~,

Unconditionally stable for softening type stiffness

Predictor Step

Corrector Step

Obtain restoring force at end of time step with no iteration

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

α-OS Method in Action

Compute Acceleration

11ˆˆ

ii FxM

External Force

1iF

Impose Command on Actuator

1~

ix

Compute Pseudo-Force1

ˆiF

Conditions at Step i

ixixix

Compute Correctors

1ix 1ixCompute Predictors

1~

ix 1~

ix

02 11ˆ KtCtMM

iiii xt

xtxx 212

~2

1

iii xtxx 1~1

iiiiii xCRRFFF ~~1

~1ˆ

111

12

11~

iii xtxx

111~

iii xtxx

Initial Conditions

0K M̂C



Conclusions

Errors in Hybrid Testing

xc

xm

Rm

Modeling Errors Numerical Integration Errors

1111 iiii FRxCxM

ti ti+1

xi xi+1

x

t

Experimental Errors

Experimental Error Sources

Flexibility of reaction frame Displacement control ofhydraulic actuators

x

t

Commanded

Measured

Intrinsic Noise

x

t

Instrumentation errors•Calibration errors•Noise

Precision errors•Range of instruments•Properties of specimen

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



Conclusions

Variations of Hybrid (Pseudodynamic, PsD) Testing

Conventional PsD Testing

Takanashi, et al., 1975

SubstructurePsD Testing

Dermitzakis and Mahin, 1985

ContinuousPsD Testing

Takanashi and Ohi, 1983

Real-TimeHybrid Testing

Nakashima, et al., 1992

Effective ForceTesting

Mahin, et al., 1985, 1989

DistributedSubstructure PsD

TestingWatanabe, et al., 2001

Distributed Continuous PsD

TestingMosqueda, et al., 2004

1

2

3 45

(Carrion, 2007)

Substructure PsD Testing

Experimental SubstructureNumerical Substructure

Structure of Interest

Distributed Substructure PsD Testing

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 Ramp

Hold Hold t

xPrediction Prediction

Correction Correction

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



Conclusions

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

NumericalCalculations

Apply Displacement

Measure RestoringForces

Δt = 10 – 20 msec

RTHT Hardware Restrictions Dynamically rated actuators

Double ended

Fast, dedicated computers xPC Target (Mathworks) dSpace (dSpace) CompactRIO (NI)

Shore WesternSeries 91 Actuator

Real Time

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

tΔt Δt

extrapolation interpolation

δtδt

PIDδt

ServoController

x

tΔtNumericalIntegration

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)

Δt Δt

x

t

ni

i

i

d

d

d

1

21

11

...

ti-1 ti ti+1

δtδt = Δt / n

Quadratic Curves

Actual Commands

δt δt δt



Conclusions

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

0 0.5 1 1.5 2

-1

-0.5

0

0.5

1

Time (sec)

Digital - Analog Conversion

Digital SignalAnalog Signal

Time Lags Finite response time of actuators Control-structure interaction (Dyke et al., 1995)

Dynamic coupling of actuator and specimen

Frequency dependent

Actuator FRF

0 10 20 30 40 500

0.5

1

1.5

Frequency (Hz)

Ma

gn

itud

e

0 10 20 30 40 50

-250

-200

-150

-100

-50

0

Frequency (Hz)

Ph

ase

(d

eg

)

0 10 20 30 40 50-14

-13

-12

-11

-10

Frequency (Hz)

Tim

e la

g (

ms)

Effects of Time Delays and Time Lags

Td

Imposedxm

Commandedxc

MeasuredResponse

ActualResponse

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

t

x

x

R

xcxm

Rm



Conclusions

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

0 10 20 30 40 500

0.5

1

1.5

Ma

gn

itud

e

Frequency (Hz)

ExperimentalModel

0 10 20 30 40 500

0.5

1

1.5

Frequency (Hz)

ExperimentalModel

0 10 20 30 40 500

0.5

1

1.5

Frequency (Hz)

ExperimentalModel

3-Pole Model 4-Pole Model 5-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

pexnumpexnumdF KKsCCMssG

2

1

sG

snSS

sdKsCMs

sdsG

xfxu

L

Fxunumnum

xudF 12

Actuator Dynamics

sd

snsG

xu

xuxu

FdKKdCCdM pexnumpexnum

sG

sG

S

SKsCMs

sG

xf

xu

L

Fnumnum

dF

2

1

pexpex

xf kscmssG

2

1Experimental Component

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

Structure

k2

m1

k1 c1

FRFζ = 5%

0 1 2 3 4 5 6 7 8 9 1010

-5

10-4

10-3

10-2

10-1

100

Frequency (Hz)

Mag

nitu

de

Exact

RTHT

0 1 2 3 4 5 6 7 8 9 10-180

-160

-140

-120

-100

-80

-60

-40

-20

0

Frequency (Hz)

Pha

se ( )

Exact

RTHT

FRF Magnitude FRF Phase

Negative DampingNegative Damping

ζ = 5% > ζth = 3.54%

Pole-Zero Map ζ = 5%

-250 -200 -150 -100 -50 0-400

-300

-200

-100

0

100

200

300

4000.050.110.180.250.340.48

0.64

0.86

0.050.110.180.250.340.48

0.64

0.86

50

100

150

200

250

300

350

400

50

100

150

200

250

300

350

400

Pole-Zero Map

Real Axis

Imag

inar

y A

xis

Exact

RTHT Pole

RTHT Zero

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0-8

-6

-4

-2

0

2

4

6

80.0040.0090.0140.0210.030.042

0.065

0.14

0.0040.0090.0140.0210.030.042

0.065

0.14

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

Pole-Zero Map

Real Axis

Imag

inar

y A

xis

Exact

RTHT Pole

Pole-Zero Map Pole-Zero Map Zoom

Additional RTHT Poles and Zeros

Dominant Poles

ζ = 5% > ζth = 3.54%

Impulse Response ζ = 5%

0 1 2 3 4 5 6 7 8 9 10-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Time (sec)

Dis

p (in

)

Exact

RTHT

Negative Damping

ζ = 5% > ζth = 3.54%

Structure

k2

m1

k1 c1

0 1 2 3 4 5 6 7 8 9 1010

-5

10-4

10-3

10-2

10-1

100

Frequency (Hz)

Mag

nitu

de

Exact

RTHT

0 1 2 3 4 5 6 7 8 9 10-180

-160

-140

-120

-100

-80

-60

-40

-20

0

Frequency (Hz)

Pha

se ( )

Exact

RTHT

FRFζ = 3%

FRF Magnitude FRF Phase

Negative DampingNegative Damping

ζ = 3% < ζth = 3.54%

-250 -200 -150 -100 -50 0 50-400

-300

-200

-100

0

100

200

300

4000.060.130.210.30.40.54

0.7

0.9

0.060.130.210.30.40.54

0.7

0.9

50

100

150

200

250

300

350

400

50

100

150

200

250

300

350

400

Pole-Zero Map

Real Axis

Imag

inar

y A

xis

Exact

RTHT Pole

RTHT Zero

-0.4 -0.2 0 0.2-8

-6

-4

-2

0

2

4

6

80.0060.0130.0210.0320.044

0.065

0.1

0.2

0.0060.0130.0210.0320.044

0.065

0.1

0.2

1

2

3

4

5

6

7

8

1

2

3

4

5

6

7

8

Pole-Zero Map

Real Axis

Imag

inar

y A

xis

Exact

RTHT Pole

Pole-Zero Map ζ = 3%

Pole-Zero Map Pole-Zero Map Zoom

Additional RTHT Poles and Zeros

Dominant Poles

ζ = 3% < ζth = 3.54%

0

Impulse Response ζ = 3%

0 1 2 3 4 5 6 7 8 9 10-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Time (sec)

Dis

p (in

)

Exact

RTHT

Negative DampingUnstable Response

ζ = 3% < ζth = 3.54%

Structure

k2

m1

k1 c1



Conclusions

Delay/Lag Compensation

Td

ti+1 ti+1+Td

x(ti)

x(ti+1)

xm(ti+1)

imposed

current

calculated

Td

ti+1 ti+1+Td

x(ti)

x(ti+1)

xm(ti+1)

imposedxp(ti+1+Td)

current

calculated

predicted

x(ti+1) ≠ xm(ti+1) x(ti+1) ≈ xm(ti+1)

t

x

Uncompensated

t

x

Compensated

Delay/lag compensation is a critical component of RTHT

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

0 10 20 30 40 500

0.5

1

1.5

Frequency (Hz)

Ma

gn

itud

e

0 10 20 30 40 50

-250

-200

-150

-100

-50

0

Frequency (Hz)

Ph

ase

(d

eg

)

0 10 20 30 40 50-14

-13

-12

-11

-10

Frequency (Hz)

Tim

e la

g (

ms)

Td

1

Td ≈ 12.5 msec

Response PredictionPolynomial Extrapolation (Horiuchi 1996) Most widely used method Send command based on command desired after Td Predicted displacement based on current and previous

time steps 3rd order provides balance of speed and accuracy Accuracy and stability concern when Td is large

compared to smallest period of structure

xc

Td

x0x1x2x3

x

t

Response PredictionModel Based Can estimate response of system after Td(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

Td

1ix1ˆ ix

Δt

1

1

1

i

i

i

R

F

x

1ˆ ixModel-Based

PredictorM, C, K

uncompensated target displacement(initial condition)

restoring forceCompensated target

displacement(send to controller)

ix

ΔtΔtt

input force

Model BasedFeedforward 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

GFF(s) Gxu(s)

Feedforward Experimental Setup

d u x

d≈x

Feedforward Compensation

K

pssG

n

iixu

FF

1

,

n

iixu

n

iixu

nFF

ps

pssG

1,

1,

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

Mag

nitu

de

Experimental

Model

0 5 10 15 20 25 30 35 40 45 50

-200

-100

0

Pha

se (

deg)

0 5 10 15 20 25 30 35 40 45 50-14

-12

-10

Frequency (Hz)

Tim

e la

g (m

s)

Actuator FRF with 3 Pole Model Feedforward FRF

n

iixu

xu

ps

KsG

1,

Not proper system, unstable

Modified inverse dynamics

Proper system, stable, α > 1

Feedforward Numerical Simulation

3 Pole Model of Actuator FRFResponse to Unit

Step Displacement Input

0 20 40 60 80 1000.94

0.96

0.98

1

1.02

1.04

1.06

Frequency (Hz)

Ma

gn

itud

e

Gxu

GFF

GxFFu

0 0.01 0.02 0.03 0.04 0.05 0.060

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Dis

pla

cem

en

t (in

)

Without FFWith FF

Feedback Compensation Closed loop compensation Error between desired and measured displacement used

to modify command Minimize e = d - x

Example controller is GFB(s) = KFB

Slower than feedforward compensation Not effective at reducing actuator lag

GFB(s) Gxu(s)

Feedback Experimental System

xd uFB

+-

+

+e u

d

x

d≈x

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

GFB(s) Gxu(s)

GFF(s)

Feedforward

Feedback Experimental System

xd

uFF

uFB

+-

+

+e u

x

d≈x

Experimental Comparison of Delay Compensation Methods

0.00

0.25

0.50

0 5 10 15

Frequency (Hz)

Err

or

no

rm

No Delay Comp.Pol. ExtrapolationModel Based 0.00

0.25

0.50

0 5 10 15 20

Frequency (Hz)

Err

or

no

rm

Pol. Extrapolation

Model Based

(Carrion, 2007)

Linear Ky/Ke = 1.0 Nonlinear Ky/Ke = 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

k2

m1

k1 c1



Conclusions

Real-Time Hybrid Testing Applications

Experimental Substructure

DamperActuator

Dampers

Base Isolation Devices

m2

m1

k1

k2

Numerical Substructure

Numerical Substructure

m2

m1

k1

k2

gx

Structure

k1

k2

m1

m2

gx

Structure

k1

k2

m1

m2

gx

Actuator

Gravity load

Base Isolation Device

Experimental Substructure



Conclusions

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

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.

conventional hybrid and real-time hybrid testing brian phillips 브라이언 필립스 university of...

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