EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 Published online 17 May 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.691
Towards error-free hybrid simulation using mixed variables Tarek Elkhoraibi§ and Khalid M. Mosalam∗, †, ‡ Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720-1710, U.S.A.
SUMMARY Two procedures are developed and implemented in a hybrid simulation system (HSS) with the aim of enhancing the accuracy and reliability of the online, i.e. pseudo-dynamic, test results. The first procedure aims at correcting the experimental systematic error in executing the displacement command signal. The error is calculated as the difference between command and feedback signals and correlated to the actuator velocity using the least-squares method. A feed-forward error compensation scheme is devised leading to a more accurate execution of the test. The second procedure employs mixed variables with mode switching between displacement and force controls. The newly derived force control algorithm is evaluated using a parametric study to assess its stability and accuracy. The implementation of the mixed variables procedure is designed to adopt force control for high stiffness states of the structural response and displacement control otherwise, where the resolution of the involved instruments may favour this type of mixed control. A simple pseudo-dynamic experiment of steel cantilever members is used to validate the HSS. Moreover, two experiments as application examples for the two developed procedures are presented. The two experiments focus on the seismic response of (a) timber shear walls and (b) reinforced concrete frames with and without unreinforced masonry infill wall. Copyright q 2007 John Wiley & Sons, Ltd. Received 30 November 2006; Revised 15 March 2007; Accepted 15 March 2007 KEY WORDS:
error compensation; hybrid simulation; masonry infill walls; mixed displacement–force control; reinforced concrete frames; timber shear walls
INTRODUCTION Hybrid simulation (HS) continues to develop as a versatile method for evaluating the performance of structures under dynamic loading. Stimulated by the continuous technological advancements, more tools are available for researchers to exploit in expanding the range of applications and ∗ Correspondence
to: Khalid M. Mosalam, Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720-1710, U.S.A. † E-mail:
[email protected] ‡ Associate Professor. § Ph.D. Candidate. Contract/grant sponsor: NSF; contract/grant number: CMS0116005 Copyright q
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reliability of online testing. This paper introduces two procedures developed at the laboratory of the University of California, Berkeley (UCB), as part of the Network for Earthquake Engineering Simulation equipment site at UCB, namely nees@berkeley [1]. The first procedure is developed to predict and correct for the execution error of the displacement commands. Experimental errors especially systematic ones—as opposed to random ones—may greatly affect the accuracy of the test due to the nature of the online experimentation, and their cumulative effect [2–4]. The feed-forward correction scheme developed herein treats the experimental error due to the inability of the hybrid simulation system (HSS) to implement the command signal with perfect accuracy, thus having a difference between command and feedback signals. To the best of the authors’ knowledge, most available correction schemes associate the execution error with a variable or a constant time delay in executing the displacement command signal [5, 6]. In this study, the systematic (non-random) experimental error is correlated to the actuator velocity; therefore, offering the possibility for compensating the predicted error, and leading to a more accurate execution of the online experiment. The second procedure presents a novel mixed displacement–force control formulation and algorithm developed and implemented within the HSS of nees@berkeley. While for flexible behaviour, displacement commands can be implemented with adequate accuracy, it may be preferable in some stiffer states of behaviour to switch to force control and implement force commands which in these states are better suited for the control of the actuators. Such concerns were previously addressed by several researchers. Thewalt and Mahin [4] developed a new approach to solve the equations of motion partly in digital and partly in analog forms. This hybrid solution has the advantage of employing implicit integration methods without the need for iterations or for estimation of the tangent stiffness. They proposed an extension to this method to solve the equations of motion for force eliminating some problems in testing stiff structures. Sieble et al. [7–9] used elastomeric pads to develop the method of soft coupling, which increases the displacement control accuracy in the case of stiff degrees of freedom (DOF). While this technique provides an interesting solution, it is not ideal for DOF experiencing mixed stiff and flexible behaviours, or for higher amplitudes of loads, where the technique used to install the pads, employing a friction connection, becomes highly nonlinear and may affect the control accuracy. Pan et al. [10, 11] tested a high-damping rubber bearing under seismic loading, and used mixed displacement–force control to apply the bearing axial load. Force control was used for loading in compression, which is associated with a high elastic stiffness, and displacement control was used for loading in tension. While this solution is suitable for a linear elastic structure, for a structure with nonlinear stiffness, the solution is not valid. The procedure presented herein offers the possibility of conducting the online experiment in any of the two control modes, force or displacement, and switching back and forth based on the variation of the stiffness of the DOF in question. Several examples are presented for validation of the HSS of nees@berkeley and for demonstrating the two developed procedures.
HYBRID SIMULATION SYSTEM The HSS within nees@berkeley consists of several components interconnected to allow for an efficient implementation of the online experimental technique, and offers flexibility in designing the test set-up. Relevant details about this system with respect to the developed procedures and experimental validation are discussed in the following sections. Copyright q
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MatLab + xPC target (host) Command signals: force and displacement + control mode
Ethernet connection
Feedback signals: force and displacement
xPC target ScramNet Command signals: force and displacement + control mode
tor feedback signalss + Actuator l : force Command signals: and displacement
ScramNet STS Controller (host) Feedback signals: force and displacement
Instrumentation feedback signals
Actuator feedback signals: force and displacement
ScramNet Data Acquisition (host) Instrumentation feedback signals
Hydraulic command signal
Structural Laboratory Hydraulic
Physical
Actuator(s) t
Specimen(s)
Figure 1. Hybrid simulation system of nees@berkeley.
Hybrid simulation system components The main components of the used HSS are schematically illustrated in Figure 1 where data exchanged between the components are identified. These components are the following: Structural laboratory: Reconfigurable reaction walls (RRWs) are constructed from precast highstrength-reinforced concrete (RC) box-section units, and post-tensioned using high-strength steel rods to the strong floor. The stiffness and fundamental frequencies of the RRW were evaluated experimentally and confirmed no possible interaction with the dynamics of the test structures [12]. ScramNet: It is a real-time communications network, based on a replicated, shared memory concept. It is optimized for high-speed data transfer among real-time computers solving portions of a real-time problem. It is used in the present study as a link between components 3–5 of the HSS. It holds data available at all times for all involved parties, and is updated in real time. Structural test system (STS): This software is a digital control system acting as the interface for the hybrid controller hardware to link the physical laboratory (servo-valves, actuators, and test specimens) to the computational laboratory (xPC target and MatLab environment). The system includes the conventional PID actuator control loop and is responsible for converting digital command signals to analog signals (D/A), and analog feedback signals to digital signals (A/D). These signals are available in real time through ScramNet at interrupts of 1024 Hz. Moreover, STS can filter the feedback signals [13]. xPC target: It is an environment that uses a target PC, separate from a host PC, for running realtime applications. In the present study, xPC target is used as a link, running compiled Simulink models and operating in real time, between the STS controller and the MatLab environment. Copyright q
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The exchanged information includes command and feedback signals, displacement and/or force, as well as control modes. Moreover, xPC target performs computations related to the estimation of the secant stiffness as discussed in subsequent sections. Data acquisition system (DAQ): It is a receiver and a compiler of all the information available from the instruments installed on the test structure. The system has a capacity of 128 channels at a maximum sampling rate of 50 kHz. High-pass filters, at the desired frequency, are provided for all channels. The acquired information is stored on a host PC for extraction after completion of the experiment and is available on ScramNet during the online experiment for possible use within the online calculations. MatLab environment: It is used in the present study as the main site for algorithm computations. It has access to the STS controller feedback and the data acquisition information available through xPC target. An Ethernet connection links the PC operating MatLab to the xPC target. The numerical integration algorithm is programmed in this environment together with numerical substructuring computations [14, 15] and live displays for monitoring while the online experiment is in progress. Operation sequence in hybrid simulation The operation sequence during the ith numerical integration time step with time step duration td is discussed. Based on a prediction/correction scheme, using polynomial extrapolation/interpolation, designed to keep the actuators in continuous motions [16, 17], the sequence in Figure 2 is as follows: • Di−1 marks the end of execution, by the hydraulic system, of the command signal from the previous time step number i − 1. At this point, xPC target starts an extrapolation process td Di-1
Ai
Bi
MatLab & xPC host
Integration algorithm
xPC target
Extrapolation δt
STS
D/A conversion
time Di Ai+1
Ci Substructure + Displays computations
Interpolation
Integ. …
Extrapo. …
PID control δt A/D conversion
Hydraulic system + Actuators + Specimens δt DAQ
Data acquisition system
DAQ
Figure 2. Operation sequence in the hybrid simulation system. Copyright q
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Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 DOI: 10.1002/eqe
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of the command signals, and continues communicating with STS by sending and receiving command and feedback signals, respectively. At Ai , a new integration time step begins where MatLab receives the feedback from the previous time step through xPC target. This feedback is the latest received by xPC target from STS, and is used by MatLab to calculate the next command signal. Between Ai and Bi , MatLab performs its calculations, and xPC target continues extrapolating commands to send to keep the actuators in continuous motion. At Bi , MatLab is ready with the command signals for step number i, which are subsequently sent to xPC target. At Ci , xPC target receives the target commands, and immediately stops extrapolating and starts interpolating towards the current commands. Concurrently, MatLab performs other tasks, e.g. calculations related to substructuring or data displays. At Di , the actuators reach the commands of time step i ending interpolation and starting extrapolation in xPC target for the next time step i + 1 and repeating the above sequence.
Throughout the online experiment, xPC target is constantly sending command values to STS at time intervals t = 1/1024 s, whether deduced by interpolation or by extrapolation, and receiving feedbacks. Provided that the time needed to reach Ci is small, each time step is performed in the prescribed td . However, in case of a delay in one or more of the previously discussed operations, the hydraulic system goes into a ‘slow’ mode, i.e. slower than prescribed to reach the target command signals in td , followed by a system ‘hold’ mode [17] if needed for long delays, causing the experiment to run slower than desired, a particularly advantageous feature for geographically distributed testing [18, 19]. The DAQ is operating in parallel to the previously discussed sequence. In this operation, data are extracted from the sensors installed directly on the test structure or in the attached actuators at the selected sampling rate with time interval tDAQ , not necessarily the same as t. However, this information is readily available online for all components of the HSS through ScramNet. Validation of hybrid simulation system To evaluate the performance of the HSS, an online (pseudo-dynamic) experiment is designed with a numerically reproducible structural behaviour of the test structures where test results can be verified against pure numerical simulation (NS). Two steel cantilever columns, C1 and C2 , constitute the two physical substructures, and are tested simultaneously where C1 is a simple cantilever and C2 is a stiffened cantilever by a shorter cantilever adjacent to it when an adjustable initial gap closes in the push or the pull loading directions, Figure 3. This results in a two DOF system with linear and stiffening bilinear behaviours for C1 and C2 , respectively, with a fundamental natural period of 0.31 s. The structural parameters used for the online HS of the two columns are summarized in Table I. A numerically simulated linear spring with high stiffness connects the two DOF of C1 and C2 in parallel, thus forcing them to be fully coupled, i.e. subjecting them to similar displacement time history. Note that although the linear spring is of high stiffness, the difference in the displacement between the two DOF is not to be entirely neglected and was a subject of study in itself. That is, the HS validation system is intended to mimic the system of test structure II, discussed later, where the two DOF of the infilled and bare frames are tied together by the numerically simulated RC slab with high stiffness. The two DOF validation test structure is excited by the strong motion, Northridge, Tarzana station, 1994 earthquake scaled to 17% of the original record. Using the evaluated stiffness of the columns from the test results, Copyright q
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Dimensions in mm
(b) Physical substructures mC2
mC1 kC1
(a)
kC2
(c)
Figure 3. Validation test structures for the hybrid simulation system: (a) details of steel cantilever column C2 (C1 is similar but without the cantilever stiffener); (b) photograph of column C2 ; and (c) idealized model for the hybrid simulation validation structure.
Table I. Evaluated parameters for the HSS validation test using two DOF steel structure.
Parameter Value
Parameter Value
Linear stiffness, kC1 [kN/mm (kip/in)] 0.613 (3.50)
S I Initial stiffness, kC2 Secondary stiffness, kC2 S /k I [kN/mm (kip/in)] [kN/mm (kip/in)] Stiffness ratio, kC2 C2
0.536 (3.06)
Mass for C1 Mass for C2 [kN s2 /mm(kip s2 /in)] [kN s2 /mm(kip s2 /in)] 0.0017 (0.0096)
0.0011 (0.0061)
2.011 (11.48)
3.75
Damping ratio (%)
Numerical integration time step (s)
5
0.01
Figure 4(a), pure NS validates the experimental results with good accuracy, as illustrated in Figure 4(b), where representative numerical samples are shown along the displacement time history with a maximum error of 6.8% relative to the peak recorded displacement. The HS results are obtained using displacement controlled HS with the -method [20] with an operator splitting (OS) scheme [21] as discussed later. Copyright q
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Displacement [mm] 15.2 30.4 35.6 Bilinear 0
Force [kip]
Linear
4
4
17.8
0
0
0
-4
-4
-8 -1.2
Displacement [in]
(a)
-0.6 0 0.6 Displacement [in]
1.2
-17.8
-8 -1.2
0.6
-35.6 1.2
-0.6 0 0.6 Displacement [in]
1.03
HS results
0
0 -0.6 -1.2
-0.46 0
5
-15.2
-0.64
10 Time [sec]
1.02
0.6
15
-30.4 20
Pure NS results 0.3
0.37
30.4 15.2 0
0 -0.6 -1.2
30.4 15.2
0.28
0.37
1.2 Displacement [in]
1.2
-0.48 0
5
(b)
-15.2
-0.68
10 Time [sec]
Force [kN]
-30.4 -15.2 8
15
-30.4 20
Displacement [mm]
15.2 30.4
0
Displacement [mm]
Displacement [mm] -30.4 -15.2 8
1503
Figure 4. Validation of the hybrid simulation system of nees@berkeley: (a) recorded structural behaviour for the two DOF steel test structure and (b) hybrid test versus pure numerical simulation results.
TEST STRUCTURES Two test structures are presented herein and used in subsequent sections to demonstrate different aspects of the developed procedures. The test structures have the common feature of being large substructures of shaking table experiments conducted at UCB for different structural systems. Test structure I: A full-scale two-storey wood house over garage, with 1 × 12 shiplap siding (3–8d common nails per stud crossing) shear walls reinforced by V-shaped 2 × 4 diagonal blocking between studs (2–16d toe common nails) represented the prototype tested on the UCB seismic simulator, Figures 5 and 6(a). In the HS experiment, Figure 6(b), the prototype structure with Copyright q
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N (a)
Bottom of joists 18’-6 1/4”
Bottom of joists 9’-0”
(b)
Garage curb 0’-0” Shaking table -0’-8”
FF 9’-6 1/4”
Diagonal blocking between studs
Test structure I for HS
(c)
Figure 5. Prototype shaking table structure defining test structure I (1 = 304.8 mm, 1 = 25.4 mm): (a) first floor plan (numbers in boxes indicate designations of openings); (b) east wall elevation; and (c) north wall elevation.
Copyright q
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(a)
(b) Displacement [mm] -127 10
-63.5
0
63.5
127 44.5
22.2
0
0
-22.2
-5
-10 -5
Force [kN]
Force [kip]
Stiff regions 5
-44.5 -2.5
(c)
0 Displacement [in]
2.5
5
Figure 6. Test structure I: First storey longitudinal shear walls of a two-storey wood house over garage: (a) two-storey structure on the shaking table; (b) physical substructure in hybrid simulation; and (c) typical force–deformation response from HS corresponding to UBE input motion.
soft first storey is idealized as a single DOF system, with the dynamic properties summarized in Table II. The equivalent inertia mass is taken as that of the entire building, and mass proportional damping is selected with 5% damping ratio to match the pull-back results from the shaking table structure. The initial stiffness listed in Table II is for the undamaged structure and is estimated from preliminary low level shaking table tests. The test structure for the HS is excited by Copyright q
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Table II. Structural parameters for test structure I (refer to Figure 5). Parameter Value
Total mass, m [kN s2 /mm (kip s2 /in)]
Damping ratio, (%)
Initial stiffness, k [kN/mm (kip/in)]
0.011 (0.065)
5
2.6 (15)
a uni-directional motion parallel to the E–W direction where the N direction is identified in Figure 5(a). Since the second storey behaved very closely to a rigid mass atop the first storey shear walls, the two longitudinal (north and south) shear walls (5.94 m × 2.74 m (19 –6 × 9 ) with 2 × 4 studs at 406 mm (16 ) spacing) are constructed at full-scale and treated in HS as the physical substructure. The studs are end nailed (2–16d common) to double 2 × 4 top plates and toe nailed (3–16d common) to 3 × 6 sill plate with 13 mm × 305 mm (1/2 × 12 ) anchor bolts at 1.22 m (4 ) spacing with 38 mm (1–1/2 ) cut washers and square nuts to connect the sill plates to RC wall footings. All lumber is Douglas Fir-Larch. A rigid steel horizontal frame connects the two shear walls to a single dynamic actuator as identified in Figure 6(b). The seismic motions applied on the HS test structure are the measured accelerations on the shaking table corresponding to Loma Prieta, Los Gatos station, 1989 (LPR1) earthquake. This ground motion is scaled to match design spectra of a site in Richmond district in San Francisco. Prior to the HS experiments, the shaking table motion is filtered to a cut-off frequency of 25 Hz to eliminate any high frequency, introduced mainly by electrical noise from the connecting cables, which has no structural significance in the test results. The fundamental frequency of the undamaged test structure I is estimated to be 2.44Hz. The integration time step adopted is t = 0.005s corresponding to the sampling frequency of the recorded shaking table motion. This selected t comfortably satisfies the requirements for capturing the fundamental dynamic response of test structure I. A typical force–deformation response from the HS experiment is presented in Figure 6(c) for the input motion corresponding to the upper bound earthquake (UBE) level with 10% probability of being exceeded in 100 years in the shaking table experiment. It is obvious that during loading, a rather flexible behaviour is recorded. On the other hand, the circled unloading regions are much stiffer and are suitable for force control as discussed later. Test structure II: The shaking table test structure consists of a reduced-scale one-storey RC moment-resisting frame structure with unreinforced masonry (URM) infill wall, Figure 7(a). The 3/4-scale shaking table structure represents the first-storey middle bays (A1–A2, B1–B2, and C1– C2) of a five-storey RC prototype structure designed based on the requirements of ACI318-02 [23] and NEHRP [24] recommendations in seismic regions. URM infill walls are assumed in the interior frames only. Moreover, a computationally determined additional mass is placed on the RC slab of the shaking table test structure with the objective of matching the base shear of this test structure with that of the prototype building when subjected to the design-level ground motion [22]. This additional mass together with the mass of the shaking table test structure itself represent the total mass considered numerically in the present HS experiments. In this design, the centre-to-centre span of each frame is 4.11 m (13 –6 ) with a total height of 3.28 m (10 –9 ). Column sections are 305 mm × 305 mm (12 × 12 ) with 8–19 mm diameter (#6) longitudinal reinforcing bars and an unbounded 32 mm diameter (1 14 ) post-tensioning central rod to represent column axial loads from the not modelled upper stories of the prototype building. The transverse reinforcing bars of the columns consist of 10 mm diameter @95 mm (#3@3 34 ) over 610 mm (24 ) from the face of Copyright q
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C2 B0
(a)
A0
A1
A2
B1
B2
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C3
B3
A3
Physical substructures
m inertia mass k stiffness BF bare frame IF infilled frame S slab
mBF kBF
kS kIF
loading direction mIF kS
mBF
kBF
(b)
(c)
Figure 7. Test structure II: one-storey RC frames with and without URM infill wall: (a) RC prototype and one-storey shaking table structure with URM infill in the middle frame [22]; (b) idealized model for the hybrid simulation test structure; and (c) hybrid simulation experimental set-up on the laboratory strong floor.
the column-footing and beam–column joints and 10 mm diameter @152 mm (#3@6 ) elsewhere. Long-direction single-span beam sections are 267 mm × 343 mm (10 12 × 13 12 ) with 3–19 mm diameter (#6) top and bottom longitudinal reinforcing bars. Beam transverse reinforcing bars are 10 mm diameter @70 mm (#3@2 34 ) over 711 mm (28 ) from the face of the beam–column joint and 10 mm diameter @203 mm (#3@8 ) elsewhere. Design details and more information on the prototype structure are documented in [22]. In the case of the test structure for HS, only two frames are constructed to the same 34 -scale, with one of the frames having URM infill wall and the other one bare. Each of the frames is considered as a separate physical substructure represented by one DOF, while the connecting RC slab is simulated numerically by means of a linear spring, and the third RC frame is simulated by symmetry to the physically simulated bare frame. The resulting idealized structural model is presented in Figure 7(b), and the experimental set-up in nees@berkeley laboratory is shown in Figure 7(c). The strong motion applied on the test structure in the present study is Loma Prieta, Bran station, 1989 (LPR2) earthquake. The structural parameters of test structure II are summarized in Table III where the recorded data, i.e. mass, damping ratio (), Copyright q
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Table III. Structural parameters for test structure II (refer to Figure 7(b)).
Parameter
m BF [kN s2 /mm (kip s2 /in)]
m IF [kN s2 /mm (kip s2 /in)]
Value
0.012 (0.07)
0.019 (0.11)
kBF
kIF
ks
7.2 (41)
122.6 (700)
289.0 (1650)
(%) [kN/mm (kip/in)] [kN/mm (kip/in)] [kN/mm (kip/in)] 6.2
and initial stiffness values, correspond to their preliminary estimates from the shaking table test structure. The fundamental frequency of the undamaged test structure II is estimated to be 7.46 Hz. As for test structure I, the integration time step t = 0.005 s is adopted for test structure II.
FEED-FORWARD ERROR COMPENSATION A feed-forward error prediction-correction scheme is applied to the actuators displacement command signals, with the aim of minimizing the experimental execution error. The employed error prediction method is closely linked to the rate of loading of the test structures. Test rate As mentioned above, LPR1 and LPR2 motions are used to test structures I and II, respectively. These motions are applied to the respective test structures at different rates. The test rate TR = td /t is defined as the ratio of the execution time of one integration time step td to t; e.g. TR = 10 indicates that the test is run 10 times slower than real time. The OS scheme [21] is chosen as the numerical integration algorithm for this part of the study, and is treated in more details in a following section. Several test rates ranging from 50 to 6.25 times slower than real time are applied on test structures I and II. Faster rates could not be executed without considerable hold periods. The monitored execution periods of the different operations within one integration time step show that the total time needed for numerical integration, operations within Simulink, and signal transfer between Siumilnk and MatLab is insignificant. However, the data transfer time through the Ethernet connection, Figure 1, between the host Simulink PC and xPC target (Di−1 –Ai and Bi – Ci in Figure 2) is the limiting operation in this process. A minimum time of ∼ 0.02 s is needed to execute one integration time step, which translates into a limiting TR ≈ 4 in the case of t = 0.005 s. Note that a by-pass of this Ethernet connection is possible if the tasks executed in the MatLab environment are instead implemented in the Simulink model, which would lift this time limitation by increasing the communication speed and allowing the HSS to achieve faster test rates [25]. In this study, since developing real-time HS is not set as one of the sought objectives and to allow efficient and detailed monitoring of the HS during the development of the algorithms for the two procedures discussed in this paper, the HSS is used with the architecture illustrated in Figure 1 and previously described. Error prediction and compensation The execution error of the displacement command is measured in all runs, and is defined as Ericc = (dic − dim )/d max for time step i, where dic is the command displacement, dim is the measured feedback displacement, and Ericc is the calculated error in dic when dic is applied. This error is Copyright q
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normalized for simplicity by the maximum actuator stroke d max = 254 mm (10 in). The actuator c )/t is calculated for an assumed constant value of t throughout the velocity viAct = (dic − di−1 d d entire test. This assumption is valid if the prescribed rate is within reach of the HSS, i.e. four times slower than real time or larger in the present study. Figures 8(a) and (b) show the relationships between Ericc and viAct for test structures I and II, respectively. At high actuator velocities, linear correlations between the two quantities are obtained for both test structures using least-squares method, resulting in the so-called ‘hockey-stick’ model. A generic relationship, that is valid for all considered test rates, can be represented by Ericc = 0,
|viAct |a
Ericc = sign(viAct ) × b(|viAct | − a),
|viAct |>a
(1)
where a and b are two positive calibration parameters. Therefore, the error carries the same sign as the actuator velocity implying that the measured feedback is lagging behind the command and the value of a is the threshold of the actuator velocity after which the systematic error occurs. Based on the above error relationship after calibrating its parameters, and noting that viAct can be calculated before applying dic , an error compensation scheme is devised to eliminate the predicted error. Therefore, an adjusted command signal dia is sent to the actuator in lieu of dic , where dia = dic , dia =
|viAct |a
a /t ) + sign(v Act ) × a] dic − bd max [(di−1 d i
1 − (bd max /td )
,
|viAct |>a
(2)
This modified command results in the execution of the desired command dic with a much smaller error Erica than Eriaa , where Erica is the error in dic when dia is applied, and Eriaa is the error in dia when dia is applied analogous to Ericc when error compensation is not accounted for. It is to be stressed that an online estimation of the parameters a and b, that would allow the error prediction and compensation to be implemented during the same test run, would alleviate the need for a separate run for the purpose of calibration and would also make the procedure suitable for situations where these parameters may change during testing. This is an important endeavour to be pursued in future enhancements of the HSS. Application examples The same motion and test rates applied as described in a previous section are repeated using the feed-forward error compensation scheme on test structure I with a = 5.18 mm/s (0.2040 in/s) and b = 7.874 × 10−5 s/mm (0.0021 s/in), Figure 8(a). Figure 8(c) shows the two error quantities calculated in the same run, i.e. Eriaa and Erica , at TR = 6.25. The latter quantity is much lower than the former, though not zero but close to the resolution of the measuring sensors, which is approximately ±0.025mm (±0.001in). It is worth noting that the error compensation scheme treats the system (hydraulics, actuator, test structure, etc.) as a black box, and thus a and b in Equations (1) and (2) need to be recalibrated for a different set of PID control tuning parameters, a different actuator, or a different test structure. However, based on observations from two very different test structures, I and II as further discussed in the next paragraph, and their associated different PID control tuning parameters, it is anticipated that the form of the hockey-stick relationship between the execution error and the actuator velocity and the effectiveness of the error compensation scheme for other test structures will hold. Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 DOI: 10.1002/eqe
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T. ELKHORAIBI AND K. M. MOSALAM
Actuator velocity, v Act [mm/sec] -63.5
0
63.5
127
-127 1
Recorded Predicted
0.5
Error value [%]
Error value [%]
-127 1
Actuator velocity, v Act [mm/sec]
0 -0.5 -1 -5
(a)
-2.5 0 2.5 Actuator velocity, v Act [in/sec]
63.5
127
0.5 0 -0.5 -1 -5
5
0
-63.5 Recorded Predicted
(b)
-2.5 0 2.5 Actuator velocity, v Act [in/sec]
5
Correlation between error and actuator velocity 1
Eri
0 -0.5 -1
(c)
TR = 6.25 0
5
10 Time [sec]
15
Eri
Eri
0.5
Error value [%]
Error value [%]
Eri
1
aa ca
0.5 0 -0.5 -1
20
aa ca
TR TR= =4040 TR=40 0
(d)
5
10
15
Time [sec]
Feed-forward error compensation
Figure 8. Error prediction and compensation: (a) test structure I; (b) test structure II; (c) test structure I; and (d) test structure II.
In order to confirm the statement at the end of the above paragraph, the same procedure described above is applied to test structure II. In addition to the different test structure and excitation motion, the PID control tuning parameters for the hydraulic actuator are also different from test structure I. Using the least-squares method, one obtains a hockey-stick error model with a = 0.51 mm/s (0.0202 in/s) and b = 1.181 × 10−4 s/mm (0.0030 s/in), Figure 8(b). Figure 8(d) shows the error compensation performed on the DOF corresponding to the bare frame at TR = 40. Clearly, the reduction of the execution error with feed-forward compensation for test structure II is as effective as that for test structure I.
MIXED DISPLACEMENT–FORCE CONTROL The numerical integration algorithms, used in the present study, are discussed. Moreover, a parametric study is conducted to evaluate the new implicit force control algorithm. Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 DOI: 10.1002/eqe
TOWARDS ERROR-FREE HYBRID SIMULATION USING MIXED VARIABLES
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Displacement control (DC) A variation of the implicit Newmark’s method, namely the -method [26], as presented by Shing et al. [20] is used to solve the equations of motion. In addition, an OS scheme is used [21] where the stiffness is split into numerical and experimental parts. While the numerical part may be solved implicitly, the experimental part is solved using an explicit predictor-corrector method, thus eliminating the need for an iterative procedure, which is not suitable for the experimental part. At time integration step i + 1, the equations of motion are discretized in the form Mu¨ i+1 + (1 + ) Cu˙ i+1 − Cu˙ i + (1 + )Ri+1 − Ri = (1 + ) Pi+1 − Pi
(3)
¨ u, ˙ and u are, respectively, the where M and C are, respectively, the mass and damping matrices, u, displacement, velocity, and acceleration vectors, P is the excitation force vector; R = RE +RN is the restoring force vector split into experimental (E) and numerical (N ) parts; and is the integration parameter. u˜ i+1 = ui + t u˙ i + t 2 (1/2 − )u¨ i and u˜˙ i+1 = u˙ i + t (1 − ) u¨ i are the predicted displacement and velocity, respectively, where = (1 − )2 /4 and = 1/2 − . The corrected displacements and velocities are ui+1 = u˜ i+1 + t 2 u¨ i+1 and u˙ i+1 = u˜˙ i+1 + tu¨ i+1 , respectively. E =R ˜ E + KEI (ui+1 − u˜ i+1 ) where R ˜ E is the vector of measured Using the OS scheme, Ri+1 i+1 i+1 restoring forces from the experimental substructure subjected to the predicted displacements u˜ i+1 ˜E + and KEI is the initial stiffness of the experimental substructure. It follows that Ri+1 = R i+1 N EI K (ui+1 − u˜ i+1 ) + Ri+1 . Finally, the equations of motion can be formulated as the following equivalent problem ˆ u¨ i+1 = Pˆ i+1 M
(4)
ˆ = M + t (1 + )C + t 2 (1 + )KEI M
(5)
˜ i+1 Pˆ i+1 = (1 + ) Pi+1 − Pi − (1 + )Cu˜˙ i+1 + Cu˜˙ i − (1 + )R ˜ i + (tC + t 2 KEI )u¨ i + R
(6)
The implementation of the above numerical integration algorithm is shown in Figure 9. Implicit force control (FC) An implicit integration algorithm that can utilize force control (FC) is formulated. From the equations of motion, Equation (3), and using the definitions of predicted and corrected displacement and velocity, one obtains the equivalent static problem ∗ K∗ ui+1 + (1 + )Ri+1 = Pi+1 j
j
K∗ =
(7)
(1 + )C M + 2 t t
(8)
ui+1 = ui + ui+1 = ui + [Ki+1 ]−1 (Ri+1 − Ri ) j
j
j
j
∗ = (1 + ) Pi+1 −Pi −(1 + ) Cu˜˙ i+1 +Cu˙ i + K∗ u˜ i+1 + Ri Pi+1
Copyright q
2007 John Wiley & Sons, Ltd.
(9) (10)
Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 DOI: 10.1002/eqe
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T. ELKHORAIBI AND K. M. MOSALAM
Figure 9. Implementation of numerical integration algorithm in displacement control. j
where Ki+1 is the estimated secant stiffness at integration time step i + 1 and iteration j, with j
1 = Klast . This estimation is iteratively acquired such that the secant stiffness k Ki+1 i i+1 for every DOF is j
j−1
j−1
ki+1 = (Ri+1 − Ri )/(u i+1 − u i )
(11) j
The individual stiffness for all DOF are then assembled in the stiffness matrix Ki+1 . Note that Equation (11) is only valid for physical substructures with a diagonal stiffness matrix, i.e. systems that may be represented as in Figure 10(a) or (c), where the linking springs kCi , i = 1, 2, . . . in Figure 10(c) are numerically simulated, similar to test structure II. While for systems as in Figure 10(b), the computation of the stiffness matrix is possible but involves solving simultaneous equations, systems as in Figure 10(d), where the linking springs kPi , i = 1, 2, . . . , are modelled physically, are not suited for this integration method and are out of the scope of this study. Rewriting Equation (7), using Equation (9), one obtains ∗ {K∗ [Ki+1 ]−1 + (1 + ) I}Ri+1 = Pi+1 + K∗ [Ki+1 ]−1 Ri − K∗ ui j
j
j
(12)
where I is the identity matrix. From Equation (12), the restoring force vector is obtained and applied to the test structure under FC. The above algorithm is illustrated in Figure 11 where i max and jmax are the maximum number of time steps and iterations, respectively. The tolerance to be checked for convergence of the iterative solution R is in terms of the change of the restoring force. The same integration algorithm in FC can be extended to mixed variables by allocating Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 DOI: 10.1002/eqe
TOWARDS ERROR-FREE HYBRID SIMULATION USING MIXED VARIABLES
(a)
(b)
(c)
(d)
1513
Figure 10. Lumped mass systems.
Figure 11. Implementation of numerical integration algorithm in force control.
Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 DOI: 10.1002/eqe
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T. ELKHORAIBI AND K. M. MOSALAM
DOF to either DC and proceeding as in Figure 9 iteratively to satisfy global equilibrium, or FC and proceeding as in Figure 11. Parametric study of FC algorithm To evaluate the performance of the formulated implicit FC algorithm, a numerical parametric study is conducted. For linear systems, the FC algorithm is equivalent to the corresponding DC algorithm, thus it has the same characteristics such as period distortion and numerical damping effects. This parametric study aims to evaluate the ability of the FC algorithm, in the case of nonlinear systems, to detect the correct stiffness value through the proposed iterative solution and to converge to the ‘exact’ solution. Two values ( = 0 and −0.3) are adopted in this study, spanning the practical range of the integration parameter and presenting two different cases of artificial numerical damping. For = 0, no numerical damping is introduced, and the algorithm reduces to the average acceleration method with = 14 and = 12 . A constant impulse force is applied to the considered system, and the response is evaluated using both the well-known implicit DC -method algorithm and the developed implicit FC algorithm. For this purpose, undamped ( = 0%) and damped ( = 5%) single DOF (SDOF) nonlinear elastic systems with unit mass and unit initial natural period are considered. The force–deformation relationship is based on the following nonlinear elastic Menegotto–Pinto model [27], R=u +
(1 − )u (1 + |u|n )1/n
(13)
where is the ratio of the secondary to initial stiffness, and the parameter n controls the sharpness of transition between these two stiffness values. In this study, n = 1, and is varied through a large range from 0.2 to 5.0 to capture both softening (1.0) behaviours. To evaluate the deviation from the exact response, the following error quantity is evaluated for the considered cases: Alg |S − S Exact | Alg Er (%) = i i Exacti × 100 (14) | i |Si where Si is the strain energy at time step i and superscript ‘Alg’ refers to the response of the algorithm in question. The summations are carried over all time steps. The ‘Exact’ response is the converged solution obtained using the implicit average acceleration method with a very small integration time step t = 0.0002 s. The secant stiffness is calculated numerically as in Equation (11), the convergence tolerances for displacement and force are u = R = 10−10 in DC and FC, respectively, with a maximum of 10 iterations. Note that u is compared to the iterative displacement increment analogous to R for force, Figure 11. Figure 12 shows two identical curves (solid for FC and dashed for DC), for each studied case, designating the calculated responses using the implicit FC method and the DC -method, thus confirming the ability of the new FC algorithm to produce as accurate solution as the DC algorithm.
IMPLEMENTATION OF MIXED VARIABLES ALGORITHM IN HS EXPERIMENTATION The different operations and strategies needed for the implementation of the mixed variables HS procedure are presented. This includes the procedure of stiffness estimation, criteria and decisions Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 DOI: 10.1002/eqe
1515
TOWARDS ERROR-FREE HYBRID SIMULATION USING MIXED VARIABLES α =0 FC DC
15 ∆t = 0.02
α = - 0 .3
20
ζ = 0%
Error value, Er Alg [%]
Error value, Er Alg [%]
20
ζ = 5%
10 ∆t = 0.005
5
FC DC
∆t = 0.02
ζ = 5%
10 ∆t = 0.005
5
ζ = 0% 0 -1 10
ζ = 0%
ζ = 5% 0
10
θ
ζ = 0%
15
1
10
ζ = 5%
0 -1 10
0
10
θ
1
10
Figure 12. Parametric study of FC and DC algorithms using nonlinear elastic SDOF systems.
on control mode switching between force and displacement, and a practical iterative approach. Finally, a demonstration example using test structure I is presented. Secant stiffness estimation The developed FC algorithm relies on the estimation of the secant stiffness during the online experiment. The most direct way of calculating the stiffness is by using the displacement and force feedback signals at the end of each iteration within a time step, i.e. Equation (11). However, the resolutions of the measuring sensors for force and displacement in addition to the random noise associated with the signals can considerably affect this estimation. A procedure is therefore developed to minimize these effects and to obtain an improved accuracy in estimating the secant stiffness. Noting that the feedback signals are available at a very high frequency in the xPC target (1024 Hz) through ScramNet, average values over the last N Avg feedbacks in one time step for each of the quantities in Equation (11) are used instead, e.g. for the application example discussed later with td = 2 s, N Avg = 100 out of 1024 td points. Subsequently, the averaged signals are passed j to the numerical integration algorithm in MatLab to calculate an average stiffness (Ki+1 )Avg , refer j
j
to Figure 13(a). Figure 13(b) shows a more reliable (Ki+1 )Avg when compared to Ki+1 for a typical HS of test structure I. Although yielding similar values in most of the shown plot, where the restoring force R and the displacement increments become smaller at the peak, an unrealistic large negative stiffness K is estimated, while K Avg does not suffer from this anomalous fluctuation. Note that for faster rates, if N Avg needs to be reduced to the point where the averaging effect would be lost, other techniques may be implemented such as the least-squares method to evaluate a linear secant stiffness over the whole integration time step. Mode switching decision and implementation To decide on the control mode in which any integration time step i will be executed, a switching criterion is needed, and is selected in the current implementation as the secant stiffness. Note that Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1497–1522 DOI: 10.1002/eqe
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T. ELKHORAIBI AND K. M. MOSALAM
N Avg
j −1
(Ki+j 1 ) Avg
R i +1
j
Ki +1 Ri Avg Ri
N ui ui (a)
Avg
Avg
(uij+−11)
Avg
(200) 35.0
j −1
(b)
K Avg
(100) 17.5
(5) 22.2 0
0 K -17.5 (-100) -35.0 (-200) 900
ui +1
(10) 44.4
-22.2 (-5)
R
950 Time step
Measured force [kN (kip)]
Avg
Stiffness estimates [kN/mm (kip/in)]
( Ri +j −11)
-44.4 (-10) 100
Figure 13. Secant stiffness estimation in a hybrid simulation of test structure I: (a) schematic representation at time step i + 1 and (b) example of one loading/unloading cycle.
Figure 14. Mode switch decision-making scheme.
more than one criterion may be adopted based on the knowledge of the test structure, such as applying tension or compression load as in [10, 11]. Both DC and FC integration algorithms are performed in parallel during the full time history, and at the end of each time step, a decision is made whether to stay in the current control mode (e.g. DC) or to switch to the other control mode (e.g. FC). Figure 14 shows the decision-making scheme implemented using as an example two criteria for switching, namely the secant stiffness K and the restoring force R with measured values K i and Ri , respectively. Two threshold values are needed for each switching criterion, e.g. K d and K f for K , where K d
