EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 Published online 4 June 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.722
Performance of a real-time pseudodynamic test system considering nonlinear structural response Rae-Young Jung1 , P. Benson Shing2, ∗, † , Eric Stauffer1 and Bradford Thoen3 1 Department
of Civil, Environmental & Architectural Engineering, University of Colorado, Boulder, CO 80309, U.S.A. 2 Department of Structural Engineering, University of California at San Diego, La Jolla, CA 92093, U.S.A. 3 MTS Systems Corporation, Eden Prairie, MN 55344, U.S.A.
SUMMARY This paper presents the implementation details of a real-time pseudodynamic test system that adopts an implicit time integration scheme. The basic configuration of the system is presented. Physical tests were conducted to evaluate the performance of the system and validate a theoretical system model that incorporates the dynamics and nonlinearity of a test structure and servo-hydraulic actuators, control algorithm, actuator delay compensation methods, and the flexibility of an actuator reaction system. The robustness and accuracy of the computational scheme under displacement control errors and severe structural softening are examined with numerical simulations using the model. Different delay compensation schemes have been implemented and compared. One of the schemes also compensates for the deformation of an actuator reaction system. It has been shown that the test method is able to attain a good performance in terms of numerical stability and accuracy. However, it has been shown that test results obtained with this method can underestimate the inelastic displacement drift when severe strain softening develops in a test structure. This can be attributed to the fact that the numerical damping effect introduced by convergence errors becomes more significant as a structure softens. In a real-time test, a significant portion of the convergence errors is caused by the time delay in actuator response. Hence, a softening structure demands higher precision in displacement control. Copyright q 2007 John Wiley & Sons, Ltd. Received 4 August 2006; Revised 28 April 2007; Accepted 1 May 2007 KEY WORDS:
real-time pseudodynamic tests; real-time hybrid tests; implicit time integration; time delay compensation; actuator dynamics; nonlinear structural response
∗ Correspondence
to: P. Benson Shing, Department of Structural Engineering, University of California at San Diego, 409 University Center, Rm 145, MC-0085, 9500 Gilman Drive, La Jolla, CA 92093-008, U.S.A. † E-mail:
[email protected] Contract/grant sponsor: National Science Foundation; contract/grant number: CMS 0086592 Contract/grant sponsor: University of Colorado at Boulder
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INTRODUCTION Since its development in early 1970s by Takanashi et al. [1], the pseudodynamic test method has been used and advanced by researchers worldwide for earthquake simulation studies [2–5]. This method involves the numerical solution of the equations of motion for a test structure using a step-by-step integration scheme. Numerical integration schemes can be classified into implicit and explicit types. A key advantage of an implicit scheme is that it can be unconditionally stable, while most explicit schemes are conditionally stable. However, an implicit scheme normally requires a Newton-type iterative method when the structure exhibits a nonlinear behaviour. For this reason, explicit schemes were used for pseudodynamic tests in the early years [1, 2] as well as for real-time pseudodynamic tests in a number of recent studies [6–9]. Numerical schemes that are unconditionally stable and yet do not require an iterative correction have been proposed, e.g. the OS method by Nakashima et al. [3], the ‘explicit’ scheme by Chang [10], and the predictor–correct method by Bonelli and Bursi [11]. Wu et al. [12] have extended the OS method and applied it to the real-time testing of nonlinear viscous fluid dampers. The OS method uses an explicit prediction–implicit correction approach, and in a pseudodynamic test, the displacements imposed on a test structure are computed with explicit prediction. The accuracy of this method is in general inferior to implicit methods with Newton-type iteration [12, 13]. This is an issue when the structural response is highly nonlinear. Similarly, the explicit scheme of Chang [10] is based on an explicit prediction using the initial elastic stiffness of a structure. For a linearly elastic structure, this method is essentially identical to the constant-average-acceleration method. However, for evaluating the nonlinear response of a structure, its accuracy has not been compared to that of the OS method or other implicit schemes. Recently, a real-time pseudodynamic test system has been developed for the Fast Hybrid Test facility at the University of Colorado, Boulder as a part of the George E. Brown, Jr Network for Earthquake Engineering Simulation. This system uses an unconditionally stable implicit timeintegration method for real-time tests and adopts a specially designed nonlinear solution strategy that combines a Newton-type iterative method with subincrementation [14]. The use of an iterative scheme allows a good control of numerical convergence errors. Furthermore, the iterative scheme allows a servo-hydraulic actuator to move in a smooth and continuous fashion. A similar approach has been used by Bayer et al. [15] for a particular type of substructure testing in which the experimental substructure is subjected to truly dynamic shaking. However, the system considered here retains the flexibility of the original pseudodynamic test concept and adopts a well-proven numerical integration strategy. The dynamics and performance of this system in testing linearly elastic structures have been investigated analytically using a system transfer function [16]. In this paper, the real-time pseudodynamic test system is described in detail. The performance of the system is demonstrated with physical tests. A simulation model that considers the dynamics and nonlinearity of a test structure and servo-hydraulic actuators, actuator control and delay compensation methods, and the flexibility of an actuator reaction system has been developed and validated by test results. The robustness and accuracy of the numerical scheme with displacement control errors and severe structural nonlinearity including strain softening are evaluated with the simulation model.
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REAL-TIME PSEUDODYNAMIC TEST SYSTEM In a pseudodynamic test, the equations of motion for the test structure are solved with a stepby-step integration scheme using the restoring forces and displacements measured from the test structure. For real-time testing, an efficient and robust computational strategy and a fast communication between the computation machine and actuator control system are essential. The system developed here, as shown in Figure 1, is configured with these requirements in mind. The basic platform consists of a computational machine, a digital controller integrated with a data-acquisition system, and servo-hydraulic actuators used to control the displacements of a test structure. The computational needs are served by a host computer and a target computer designated as the host– target pair 1 in Figure 1. The host computer is for program and model development, while the target is a real-time kernel that provides a deterministic environment for numerical computation and communication with the digital control/data-acquisition system in a synchronized fashion. The second host–target pair that is provided can be used for real-time simulations to simulate the dynamics of the servo-hydraulic actuator system and test structure. In a real-time test, this second pair is replaced by the actual hydraulic actuators and test structure. The dual test and simulation capabilities are symbolized by the switch in Figure 1. Digital controller and Data-acquisition System c
d SCRAMNet Real-time Target PC
dm , rm Controller PC
Host PC
Valve command,
Host-target pair 1
i
Sensors feedback d m,r m
SCRAMNet Switch Real-time Target PC
Host PC Host-target pair 2
Servo-hydraulic actuators and test structure
Figure 1. Real-time pseudodynamic test system.
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In a real-time test, the displacement commands d c generated in each computation step by a target computer are sent to the digital controller to control the displacements of the structural specimen. The displacements d m and restoring forces r m measured from the structural specimen are then acquired by the data-acquisition system and sent back to the target computer for the computation of the structural response in the next increment. In this system, the high-speed communication between the target computer and digital control/data-acquisition system is served by an optical network called Shared Common RAM Network (SCRAMNet® ), and the synchronization of the two processors is achieved through ‘interrupts’ generated by the controller.
COMPUTATIONAL SCHEME The approach proposed here can be implemented with different implicit integration schemes. However, the -method [17] has been chosen for the study presented here. This scheme is implicit and unconditionally stable and has a second-order convergence. Its implementation, accuracy, and error-propagation characteristics for non-real-time pseudodynamic tests were investigated in prior studies [18, 19]. This section describes its implementation for real-time pseudodynamic tests, which require a special nonlinear solution strategy. In the -method, the time-discretized equations of motion and the displacement and velocity approximations are expressed as follows: Mai+1 +(1 + )Cvi+1 −Cvi +(1 + )ri+1 −ri = (1 + )fi+1 − fi
(1)
di+1 = di +tvi +t 2 [( 12 −)ai +ai+1 ]
(2)
vi+1 = vi + t[(1 − )ai + ai+1 ]
(3)
in which M and C are the intended mass and damping matrices of the test structure that is modelled numerically; di , vi , ai , and ri are the nodal displacement, velocity, acceleration, and restoring force vectors at time step i; fi is the vector of applied forces; t is the integration time interval; and , , and are parameters that govern the numerical properties of the integration scheme. For unconditional stability and second-order convergence, it is required that − 13 0, = 12 (1 − 2), and = 14 (1 − )2 . For a nonlinear structure, the -method requires a Newton-type iterative procedure to solve the equations of motion. Since it is generally difficult to evaluate the tangent stiffness of a structure in a reliable manner during a test, a modified Newton approach using the initial structural stiffness has to be adopted. With Equations (1)–(3), a solution strategy based on the modified Newton method can be formulated for pseudodynamic testing as follows. First, construct the effective stiffness K∗ (k) and residual vector Ri+1 with Equations (4) and (5), respectively K∗ =
M + (1 + )Kini t 2
(4)
(k)
M ˆ m(k) m(k) [di+1 − di+1 ] − (1 + )ri+1 t 2
(5)
Ri+1 = Copyright q
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where M = M + (1 + )tC
(6)
dˆ i+1 = di + tvi + t 2 ( 12 − )ai −1
+ t 2 M
[(1 + )fi+1 − fi − Cvi − (1 + )(1 − )tCai + ri ]
(7)
(k)
Then, solve Equation (8) for di+1 (k)
(k)
K ∗ di+1 = Ri+1
(8)
(k)
With di+1 computed, update the displacements as follows: (k+1)
m(k)
(k)
di+1 = di+1 + di+1 m(k)
(9)
m(k)
In the above equations, di+1 and ri+1 denote the actually realized displacements and corresponding restoring forces measured from the test structure at the beginning of iteration k in time step (i + 1), and Kini is the initial stiffness of the structure. In a non-real-time pseudodynamic test, (k+1) Equation (8) is solved in each iteration and the displacements di+1 updated with Equation (9) are sent as command signals to an actuator controller to control the structural displacements with m(k+1) servo-hydraulic actuators. The actually realized displacements di+1 , which are normally slightly (k+1)
different from di+1
because of control errors in the servo-hydraulic loading system, and the rem(k+1)
(k+1)
are measured. A new residual Ri+1 is then computed sulting structural restoring forces ri+1 with Equation (5) and used to update the response in the next iteration. This process is repeated until a specified convergence tolerance has been attained. After convergence, the acceleration and velocity response is updated with Equations (2) and (3), and the solution is advanced to the next time step. This method has been successfully used for non-real-time pseudodynamic tests where actuators move slowly following a linear ramp function, stop at the end of the ramp, and wait for the commands from the next iteration [18]. Real-time iteration strategy In a real-time pseudodynamic test, the actuators controlling structural displacements have to move continuously with a smooth velocity, and the displacement commands to the actuators have to be sent at the sampling frequency of the digital controller, which is 1024 Hz for the system used here. However, with the iterative solution scheme described above, the displacement corrections, (k) di+1 , will be progressively reduced as the trial displacements approach a converged solution. (k+1)
Hence, updating the displacement commands directly with the trial displacements di+1 will lead to undesired velocity fluctuations from one time step to the next. In addition, the number of iterative corrections required in each time step will vary depending on the degree of nonlinearity developed in the structure. This uncertainty is not acceptable for a real-time test where a converged solution has to be attained within the designated time interval t. To circumvent the aforementioned problem, a special iterative method that has a fixed number of iterations in each time step is proposed. As shown in Figure 2, this method relies on an interpolation technique to assure a smooth motion of an actuator during iteration. Each iteration consists of two Copyright q
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(n) di+1
Displacement
d id+(1n)
(k+1) (2) di+1 di+1
Quadratic Curves
(1) di+1
d(2)
d(1)
di+1
di+1
Actual Commands
d id ( n) d id−(1n)
δτ
(i−1)∆τ
δτ (= ∆τ / n)
i∆τ Time
(i+1)∆τ
Figure 2. Command generation scheme. (k+1)
steps: (1) the trial displacement di+1 for each degree of freedom is evaluated as in a conventional Newton-type iterative process; and (2) a fraction of the trial displacement, which is termed the d(k+1) desired displacement di+1 , is then computed with a quadratic interpolation and imposed on the structural specimen. Except in the first time step, three points are used for the construction of the (k+1) interpolation function: the trial displacement di+1 in the current iteration, and the displacements d(n)
d(n)
di−1 and di , which are computed in the very last iteration of the two previous time steps, with n denoting the total number of iterations specified in each time step and k varying from 0 to (n − 1). In the first time step, the interpolation will be based on the initial velocity and displacement. The mathematical expressions for calculating the displacement commands are given below (k+1)
d(k+1)
= m 2 di+1
d(k+1)
= 12 (m 2 − m)di−1 + (1 − m 2 )di
di+1 di+1
+ t (m − m 2 )vini + (1 − m 2 )dini (first time step) d(n)
d(n)
(k+1)
+ 12 (m 2 + m)di+1
(other time steps)
(10) (11)
in which dini and vini are the initial displacement and velocity and m = (k + 1)/n. d(n) (n) It should be noted that in the very last iteration, di+1 = di+1 . The timing of each iteration is
d(k+1) controlled by the ‘interrupts’ issued by the digital controller so that a desired displacement di+1 is computed by the target computer and sent to the controller at the same frequency as the control signals are generated. As shown in Figure 2, t and denote the time frames of the earthquake ground motion and the actual test, respectively, with /t representing the time-scaling factor for a test. When this factor is 1, the test is conducted in real time. This factor can be much greater than one for a non-real time test. The number of iterations used in each integration time step is governed by the desired rate of loading and the update rate of the digital controller. If the update time interval of the controller is , the number of iterations in each time integration interval is n = /.
Response update Once iteration is completed in a time step, the response needs to be updated so that the computation for the next time step can begin. Since the number of iterations in each time step is fixed in this Copyright q
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real-time test method, the convergence errors are expected to be larger than those with a conventional Newton-type iteration. In addition, the convergence errors can be exacerbated by the time delays in actuator response and in the data exchange between the controller/data-acquisition processor and the target computer. In particular, the synchronization of the two processors using ‘interrupts’ as mentioned previously will introduce an inevitable time delay of . To minimize the effect of these errors and enforce equilibrium at the end of each time step, the following approximate correction is introduced for updating the displacements and forces in the last iteration d(n)
di+1 = di+1
m(n−1)
ri+1 = ri+1
(12) d(n)
m(n−1)
+ Kini [di+1 − di+1
]
(13)
in which di+1 and ri+1 are the updated displacements and forces that are treated as the converged d(n) solutions, di+1 is the desired displacement vector computed in the nth iteration, which is the m(n−1)
m(n−1)
and ri+1 are the displacements and restoring forces measured at the last iteration, and di+1 beginning of the last iteration. This correction method has been successfully used for non-real-time pseudodynamic tests with the implicit integration scheme [18]. Based on Equations (2) and (3), the velocity and acceleration are then updated as follows: 1 1 ai+1 = 2 di+1 − di − tvi − t2 − ai (14) 2 t vi+1 = vi + t[(1 − )ai + ai+1 ]
(15)
The computational procedure and its interaction with the digital controller via the SCRAMNet® are shown in Figure 3. It must be mentioned that the pseudodynamic test method was originally built upon the assumption that the inertia and damping characteristics of the test structure are numerically accounted for in Equation (1) and only the static restoring forces are measured from the test structure. However, in a real-time test, this assumption may introduce an error in that the restoring forces measured will include the actual inertia and damping forces, which are not anticipated in the numerical computation. This error can be significant when the mass of the physical test structure is not negligible as compared to the model mass in Equation (1). Possible remedies for this problem include numerically removing the inertia effect from the measured forces [16] based on measured accelerations or modifying the inertia and damping properties in the equations of motion to account for these effects [20]. Even though these methods have been used with some success, each method has its advantages and disadvantages. While the first approach is simple in concept, it could be impacted by the noise in acceleration measurements. The second approach generalizes the pseudodynamic test concept and unifies dynamic and pseudodynamic test methods. Nevertheless, it has a consistency issue with respect to the velocities and accelerations actually attained in a test as compared to those assumed in a numerical integration scheme [20]. Hence, this is a topic that deserves further investigation. The integration scheme presented here was successfully used for real-time substructure tests conducted on a linearly elastic structure [21] and the substructure tests of a steel braced frame [22], which had a highly nonlinear behaviour both in the analytical substructure and the test specimen due to brace buckling. The braced frame had 45 degrees of freedom. Because of the Copyright q
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i+1
4 Acquire d i+m1( k ) and ri+m1( k ) measured from test structure. 5 Compute Ri+( k1) by means of equation (5). Solve equation (8) for
d i+( k1) .
Calculate trial displacements d i(+k1+1) with equation (9). 6 Compute d id+(1k +1) by means of equation (10) or (11). 7 Send d id+(1k +1) to actuator controller. 8 If k
(n 1) , set k = k + 1 and go to Step 4.
Controller/ Data Acquisition System 1/1024 second
3 Set k = 0. Input excitation f i+1 . Compute dˆ by means of equation (7).
Send valve commands to servo-valve Acquire Sensor data
2 Set i = 0.
Exchange information
1 Set the number of iterations, n . Evaluate M and K * with equations (6) and (4). Initialize d 0 , v0 , r0 , and a0 .
Record to SCRAMNet
Computational Procedure
Read from SCRAMNet ® SCRAMNet
R.-Y. JUNG ET AL.
9 Update the solution using equations (12) through (15). 10 Set i = i + 1 and go to Step 3.
Figure 3. Computational procedure.
severe geometric and material nonlinearities developed in the analytical substructure and the large amount of data writing during a test, these tests were run 60 times slower than a real-time test with the use of a Pentium 4 2-GHz computer for numerical computation. However, if the analytical substructure had only mild nonlinearity and numerical data were written on a hard drive after a test, a real-time test could be conducted for a structure with as many as 35 degrees of freedom using the same computer. In any respect, the nonlinearity of a test specimen will not affect the computational effort except for the fact that the time step may have to be reduced to assure convergence if severe strain softening occurs. Convergence The convergence and error-propagation characteristics of this integration scheme have been investigated by Shing et al. [18] and Shing and Vannan [19]. It has been shown that time-lag errors in actuator response will introduce a damping effect in the numerical solution, and, therefore, will not destabilize the numerical solution as long as the initial stiffness of the structure used in the iterative solution and response update (Equation (13)) is higher than the actual stiffness, Ka , i.e. (Kini − Ka ) being positive definite. Furthermore, convergence is assured as long as [M/(t 2 ) + (1 + )Ks ] is positive definite, where Ks is the incremental secant stiffness of the structure. Hence, for a structure exhibiting severe softening, a sufficiently small time step will be required to assure convergence [22]. Copyright q
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r
ry
−dy
1
1
KE
KE dy
d
−ry
Figure 4. Bilinear load–displacement relation.
In general, it has been found that accurate results can be obtained by having 10 iterations in each time step. This is demonstrated here with numerical examples using inelastic, single storey, shear frames. The frames have a bilinear load–displacement relation as that shown in Figure 4. The initial stiffness and mass of the frames are assumed to be 21 kips/in (3681 kN/m) and 0.2 kip s2 /in (35 030 kg), respectively. A damping of 5% is assumed. The NS component of the 1940 El Centro ground motion with the peak acceleration scaled to 0.1g is used. The time interval for integration is set to 0.01 s and the numerical parameters , , and are set to 0, 0.25, and 0.50, respectively. The results are compared with the ‘exact’ numerical solution obtained with the central difference method (CDM) using a very small integration time step of 0.001 s. Results from the worst-case scenario of a strain-softening structure are shown in Figure 5. For this frame, the yield displacement, dy , and the post-yield stiffness in the softening range are selected to be 1.45 in (3.68 cm) and −10.5 kips/in (−1840.5 kN/m), respectively. To investigate the influence of the convergence errors, analyses are performed with the number of iterations set to 10 and 50, respectively. The error in the maximum displacement response obtained is 4.3% with 10 iterations and 2.4% with 50 iterations. For elastic–perfectly plastic structures, the accuracy is higher in general. However, these examples do not consider the time delays introduced by a physical test system, which will be examined next.
MODEL FOR A REAL-TIME PSEUDODYNAMIC SYSTEM A Simulink model has been developed to simulate the behaviour of a complete real-time pseudodynamic test system. The model accounts for the nonlinearity and dynamics of the test structure and servo-hydraulic actuator system. The modelling of the main components of the servo-hydraulic actuator system including the proportional integral derivative (PID) controller is briefly summarized below. Readers are referred to other publications, e.g. Williams et al. [8] and Zhao et al. [9], for a more detailed coverage of this subject. The digital controller used in the test system adopts a PID control supplemented with a feedforward scheme for actuator delay compensation. The feed-forward compensation scheme is based on the rate of variation of the displacement command and the anticipated time delay. The PID Copyright q
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3
CDM ∆t = 0.001 RTI ∆t = 0.01, N = 10 RTI ∆t = 0.01, N = 50
2.5
Displacement (inch)
2 1.5 1 0.5 0 -0.5 -1 -1.5
0
5
10
15 Time (sec)
(a) 30
20
25
30
0 -100 Residual Force Error (kips)
20
Force (kips)
10
0
-10 CDM ∆t = 0.001 RTI ∆t = 0.01, N = 10 RTI ∆t = 0.01, N = 50
-20
-30 -1
(b)
-0.5
0
0.5 1 1.5 Disp. (inch)
2
2.5
-200 -300 -400 -500 -600 -700 -800 -900 5.42
3
RTI ∆t = 0.01, N = 10 RTI ∆t = 0.01, N = 50
(c)
5.43 Time (sec)
5.44
Figure 5. SDOF shear frame with strain-softening behaviour (1 in = 24.5 mm; 1 kip = 4.45 kN): (a) displacement time histories; (b) load–displacement hystereses; and (c) residual force error.
control scheme with feed-forward compensation can be expressed as follows: de(t) d + kff dpc (t) i(t) = kp e(t) + ki e(t) dt + kd dt dt
(16)
in which i is the control signal sent by the controller to a servo-valve, e is the feedback error, which is the difference between the commanded and measured displacements, kp , ki , kd , and kff are control parameters that are called the proportional, integral, derivative, and feed-forward gains, respectively, and dpc is the displacement command. The spool displacement in a servo-valve is determined by the control signal i. The relation between the control signal and spool displacement is expressed as a second-order differential equation as follows: x¨s (t) + 2s s x˙s (t) + 2s xs (t) = 2s i(t) Copyright q
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(17)
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in which s and s are the equivalent damping ratio and natural frequency of the servo-valve system, and xs (t) is the spool displacement. The spool displacement obtained from the above equation changes the orifice opening, and the volumetric oil flow rate from side i to side j through the orifice is described by the square-root orifice law as follows: Pi − P j qi j (t) = kv Ao |Pi − P j | |Pi − P j |
(18)
in which kv is the discharge coefficient which is related to the geometry of the orifice, Ao is the opening area of the orifice, which is a function of x s (t), and Pi and P j are the oil pressures at sides i and j, respectively. From Equation (18), the rate of oil flow qA into or out of each chamber of an actuator can be determined by replacing Pi and P j by the corresponding chamber pressure and the supply or return pressure, PS or PR , respectively. The oil flow qA into a chamber of an actuator has to satisfy the equation of continuity qA (t) = Ap
PA · Ap ddp (t) ddp (t) VA dPA (t) + + dt E dt E dt
(19)
in which Ap is the cross-sectional area of the actuator chamber or piston area, E is the effective bulk modulus of the oil, dp is the displacement of the piston, PA is the chamber oil pressure, and VA is the oil volume in the chamber. The piston force rp resulting from the loading pressure in the servo-hydraulic actuator will drive the test structure towards the desired position. The response of the structure to the piston force rp can be represented by a second-order differential equation Mt d¨t + Ct d˙t + K t dt = rp
(20)
in which Mt , Ct , and K t are the mass, damping, and stiffness coefficients, and dt is the displacement of the test structure. Since the inertia effect of the test structure is accounted for in the computer model, the actual structural mass Mt can be much smaller than the theoretical mass M shown in Equation (1). The damping coefficient Ct is used to approximate the damping properties of the test structure, which can be introduced by friction or hysteretic damping, for example. This is in addition to what is assumed in Equation (1). The structural stiffness K t may or may not be exactly the same as the initial stiffness used for the modified Newton iteration depending on how accurate it can be measured. For a nonlinear structure, the term K t dt will be replaced by a restoring force rt , which depends on dt and the deformation history of the structure. Error compensation schemes The numerical computation in a pseudodynamic test relies on the displacement and force measurements, whose accuracy depends very much on the quality of the displacement control. The dynamics of an actuator and test structure as discussed above can introduce a significant time delay in actuator response. To mitigate this problem, two compensation schemes are considered here in addition to the feed-forward scheme presented previously. One is a discrete feed-forward compensation (DFC) scheme, which is based on the assumption that the displacement control errors in a time step are more or less the same as those in the previous step. Thus, the updated displacement (k+1) di+1 , computed with Equation (9) in each iteration, is modified as follows to compensate for the Copyright q
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anticipated error: c(k+1)
di+1
(k+1)
= di+1
c(n)
+ kDFC · [di
c(n)
m(n)
− di
]
(21)
m(n)
are the commanded and measured displacements in which kDFC is a gain factor, and di and di c(n) in the last iteration of the previous time step. The computed displacement di is used in place of (k+1) di+1 in Equations (10) and (11). One distinct advantage of the DFC method is that it can also correct for displacement errors introduced by the deformation of the reaction frame supporting the actuator when the PID control is based on the actuator displacement feedback and the structural displacement is independently measured with respect to a fixed reference frame. The other compensation scheme is a first-order phase-lead compensator (PLC), which is used quite extensively in mechanical control to improve system response. A PLC is usually constructed in the form of a first-order transfer function as follows [23]: PLC(s) =
Td s + 1 d Td s + 1
(22)
where d and Td are parameters for the phase-lead network. The maximum phase angle introduced by the above compensator is decided by the constant d . In general, a single-lead compensator can contribute a maximum of 60◦ to the phase [24]. This method has been proposed by Zhao et al. [9] for real-time structural testing. They have suggested that Td be calculated as the inverse of the roll-off frequency obtained from a sine-sweep test and d be equal to 0.1. The idea of using a feed-forward compensation is to project how a system will function in the future and to make corrections based on the projection. In the feed-forward scheme considered here, corrections are introduced by multiplying the velocity of the desired command by the feedforward gain. Since feed-forward control works outside of the PID control loop, it does not directly affect the stability of the system. However, excessive values of the feed-forward gain can cause overshoot, which can destabilize the numerical computation [18]. Simulink model of the real-time test system A Simulink program has been developed to model a real-time pseudodynamic test system using the formulations presented previously. Figure 6 shows the block diagram of an entire physical test system that consists of a controller, servo-valve, hydraulic actuator, and test structure. The delay block e−s shown in the figure is to simulate the time delay in generating a command signal in a digital controller that has a sampling frequency . For the system used here, is 1/1024 s. Figure 7 shows the block diagram for the entire real-time test system including the time integration scheme. As shown, the computation scheme consists of two feedback loops. One is an inner loop for the iterative correction that is executed at an update rate of 1/. The other is an outer loop for updating the response at the end of the last iteration in each time step t. In Figure 7, the shaded block represents the physical test system that is shown in Figure 6. The hydraulic actuator and test specimen in Figure 6 can be either a Simulink model or the actual physical set-up. A e−2 (2 ms) delay block is inserted in front of the physical test system to represent the time delays introduced by the communication between the target computer that performs the computation and the digital servo-controller. For real-time simulations using the actual control system, as depicted in Figure 1, the delay blocks in Figures 6 and 7 are not needed. For a multi-degree-of-freedom test structure, the simulation model can be expanded accordingly. Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
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PERFORMANCE OF A REAL-TIME PSEUDODYNAMIC TEST SYSTEM
Feed-forward loop
d dt
k
rm +
dpc
Σ
+ -
e
PID
+
Controller
Σ
e− s δτ -
Delay
rm
i Servo-valve and hydraulic actuator
dm
Test specimen
k ∆p Ap pressure feedback displacement response
Figure 6. Block diagram of the physical test system.
Figure 7. Simulink program for the real-time pseudodynamic test system.
Physical tests and model validation Physical tests were performed on single- and two-degree-of-freedom (SDOF and 2-DOF) structures. The test specimen was a W14 × 120 cantilever beam, which was set up with two actuators, X1 and X2, as shown in Figure 8. The beam was supported by a steel frame. Actuator X1 had a load capacity of 220 kips (100 ton) with a three-stage 250 gpm (947 lpm) servo-valve while X2 had a capacity of 110 kips (50 ton) with a three-stage 180 gpm (682 lpm) servo-valve. The beam was tested as SDOF and 2-DOF structures by using one and two actuators, respectively. Its mass is assumed to be concentrated at points where the actuators were attached to. For the SDOF tests, the right end of the cantilever beam was attached to actuator X2 with the other actuator (X1) removed. For the 2-DOF tests, both actuators were attached to the cantilever beam. In spite of the fact that Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
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Figure 8. Specimen and set-up for real-time tests (1 in = 25.4 mm). Table I. Parameters of hydraulic actuator model. Parameter
Value
Servo-valve Damping ratio, s Natural frequency, s , rad/s Discharge coefficient, kv , in4 /s lb0.5 (m4 /s N0.5 ) Supply pressure, PS , psi (MPa) Return pressure, PR , psi (MPa) Actuator Piston area, Ap , in2 (m2 ) Total volume, V, in3 (m3 ) Effect bulk modulus, E , psi (MPa)
0.7 817 21.91 (4.325 × 10−3 ) 3000 (20.68) 50 (0.345) 38.48 (2.483) 423.3 (693.6) 100 000 (690)
the beam remained linearly elastic, iteration was required with the proposed numerical solution scheme. Ten iterations were used for each integration time step. SDOF test A SDOF test is presented to demonstrate the performance of the system with the DFC scheme and to validate the Simulink model described above. The steel support frame had some flexibility. The resulting stiffness of the SDOF cantilever structure was measured to be 13 kips/in (2277 kN/m). The mass concentrated at the right end was assumed to be 82.3 lb s2 /in (1.41 × 104 kg) for the numerical model so that the structure had a natural frequency of 2.0 Hz in the actual and simulated pseudodynamic tests. However, the actual mass (Mt ) of the system was estimated to be 3.9 lb s2 /in (683 kg), considering the weight of the steel beam and other attached pieces. A damping of 5% was assumed for the numerical model in both the test and simulation. In the Simulink model, the additional damping (Ct ) actually developed by the test specimen was assumed to be 5%. This damping value was indirectly deduced from a real-time free-vibration test. The parameters chosen for the hydraulic actuator and servo-valve model are shown in Table I. These values were based on the properties and performance of actuator X2. In both the test and simulation, Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
1799
PERFORMANCE OF A REAL-TIME PSEUDODYNAMIC TEST SYSTEM
1.2 Exact Numerical Sol. Real Time Simulation Real Time Test
Displacement (inch)
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3.5
Figure 9. Displacement responses of a linearly elastic SDOF structure (1 in = 24.5 mm): (a) comparison of measured displacements and (b) zoomed-in measured displacements.
the following control gains were used: kp = 5 V/in (200 V/m), ki = 0.1 V/in s (4 V/m s), kd = 0, and kp = 0. The DFC scheme was used with kDFC = 0.8 to reduce phase-lag errors. The NS component of the 1940 El Centro earthquake with the peak ground acceleration scaled to 0.16g was used as the input excitation. During the test, the structure remained linearly elastic. The integration time step t = 0.01 s, and , , and were set to −0.25, 0.391, and 0.75, respectively. In Figure 9, results of the real-time simulation and test are compared to the ‘exact’ numerical solution obtained with the CDM using a very small time step of 0.001 s. To account for the additional damping introduced by the test structure, damping of 6% was used in the exact numerical solution. In addition, the initial stiffness of the structure used in the modified Newton iteration and response correction was set to be 5% higher than the actual stiffness to account for the uncertainty in the actual stiffness of the structure and avoid negative damping that could otherwise be introduced by experimental errors if the assumed stiffness was less than the actual stiffness [18]. It can be observed from Figure 9 that both the test and simulation results match the ‘exact’ solution well and that the Simulink model was able to reproduce the response of the test system reasonably well. A small mismatch of the test and simulation results with the exact numerical solution was Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
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caused by the inertia force that was introduced into the restoring force measured from the test structure. As mentioned before, this discrepancy can be reduced or eliminated with two possible methods [16, 20]. 2-DOF tests The stiffness of the 2-DOF test structure was measured by loading the beam with actuators X1 and X2 independently. The stiffness coefficients measured were K 11 = 322 kips/in (5.64 × 104 kN/m), K 12 = K 21 = −107 kips/in (−1.87 × 104 kN/m), and K 22 = 49 kips/in (8.58 × 104 kN/m), with DOF 1 associated with actuator X1. The mass at each degree of freedom was assumed to be 326 lb s2 /in (5.59 × 104 kg) for the numerical model so that the structure had natural frequencies of 1.0 and 5.3 Hz. Damping was assumed to be 5% of the critical for each mode. Tests were conducted to evaluate the effectiveness of the three error-compensation schemes presented previously. For these tests, the proportional, integral, derivative, and delta pressure gains were set to 8.7 V/in (340 V/m), 0.13 V/in s (5 V/m s), 0, and 0, for actuator X1; and 10 V/in (390 V/m), 0.1 V/in s (4 V/m s), 0, and 0 for actuator X2. With the DFC scheme, kDFC was set to 0.8. With the PLC, the time constant Td was set to 15 ms, which was obtained from a sine-sweep test, and d = 0.1 for both degrees of freedom. The feed-forward gains were set to 0.17 and 0.14 for actuators X1 and X2, respectively. These and all control gain values used here were obtained by tuning the controller with harmonic displacement commands. The same integration scheme and time interval used in the SDOF test were used here. The initial stiffness of the structure used in the modified Newton iteration and response correction was again set to be 5% higher than the actual stiffness. The NS component of the 1940 El Centro earthquake was scaled to 0.1g. The structure remained linearly elastic. The responses obtained at DOF 1 from the different tests are shown and compared to the ‘exact’ (CDM) result in Figure 10. Damping was assumed to be 6% for each mode in the exact result. It can be seen that the test results match the ‘exact’ response reasonably well except for a slight mismatch of the response frequency probably due to the inertia forces developed by the specimen. All of the compensation schemes showed a positive influence as compared to the PID control alone. The amplitude error is a little larger than that of the SDOF case probably due to the synchronization errors in the two actuators, which had slightly different performances because of the difference in load and servo-valve capacities.
SYSTEM EVALUATION WITH REAL-TIME SIMULATIONS A structure that exhibits a nonlinear behaviour with severe strain softening cannot be tested multiple times and expected to yield a consistent and repeatable behaviour. Furthermore, the potential instability of a large physical specimen as a result of strain softening imposes a significant risk in high-speed tests. Hence, to evaluate the performance of the real-time system in testing severely nonlinear structures, real-time simulations are conducted with the Simulink model that has been presented and validated in a previous section. Structural models For the evaluation study, SDOF and 2-DOF shear frame structures that have bilinear strain hardening and softening load–deformation relations similar to that shown in Figure 4 are considered. The SDOF structure is assumed to have an elastic stiffness of 13 kips/in (2277 kN/m), and a Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
1801
PERFORMANCE OF A REAL-TIME PSEUDODYNAMIC TEST SYSTEM
0.5
CDM ∆t = 0.001 PID PID + DFC PID + PLC PID + FF
0.4 Displacement (inch)
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5
0
5
10
(a)
15
20
25
0.5
CDM ∆t = 0.001 PID PID + DFC PID + PLC PID + FF
0.4 0.3 Displacement (inch)
30
Time (sec)
0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 3.5
4
(b)
4.5
5
5.5
Time(sec)
Figure 10. Displacement responses at degree-of-freedom 1 of 2-DOF linearly elastic structure (1 in = 24.5 mm): (a) comparison of measured displacements and (b) zoomed-in measured displacements.
mass of 82.3 lb s2 /in (1.41 × 104 kg), which result in a natural frequency of 2.0 Hz. Damping is assumed to be 5% of the critical. The actual mass (Mt ) of the physical specimen is assumed to be 3.9 lb s2 /in (683 kg). For the 2-DOF structure, the elastic stiffness coefficients are K 11 = 37 kips/in (6.48 × 103 kN/m), K 12 = K 21 = −17 kips/in (−2.98 × 103 kN/m), and K 22 = 17 kips/in (2.98 × 103 kN/m). The mass at each degree of freedom of the model is assumed to be 82.3 lb s2 /in (1.41 × 104 kg), while that of the physical structure is assumed to be 3.9 lb s2 /in (683 kg). Damping is assumed to be 5% of the critical for each mode. In modelling the physical specimen, 5% damping is assumed for each mode to account for the additional damping Ct . The NS component of the 1940 El Centro earthquake is used with the peak ground acceleration scaled to 0.16 and 0.1g for the SDOF and 2-DOF structures, respectively. The time interval for integration is 0.01 s with , , and set to −0.25, 0.391, and 0.75. Ten iterations are used for each integration time step. To investigate the effectiveness of the error-compensation schemes, real-time simulations are performed with the DFC, PLC, and simple feed-forward schemes presented previously. The parameters shown in Table I are used for the servo-valve and hydraulic actuator model for all cases. By a trial-and-error method, it has been found that the optimal value of the proportional gain Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
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for the PID controller is 10 V/in (390 V/m) for both the SDOF and 2-DOF cases. The integral, derivative, and delta pressure gains are selected to be 0.1 V/in s (4 V/m s), 0, and 0, respectively. The ‘measured’ displacement responses are compared with the ‘exact’ numerical solutions, which are obtained using the CDM with an integration time step of 0.001 s. For the exact numerical solution, damping is set to be 6% of the critical to account for the additional damping introduced by the physical test structure. As before, the initial stiffness of the structural model used in the iteration and response correction is set to be 5% larger than the actual initial stiffness. SDOF structure For the real-time simulations with error compensation, kDFC and kff are set to 0.8 and 0.16, respectively, and the value of Td for the PLC is determined to be 24 ms based on a sine-sweep test, and d is set to 0.1. Figure 11(a) and (b) shows the measured displacements and the ‘exact’ numerical solution for the SDOF structure with a softening load–displacement relation. For this simulation, dy and are set to 0.55 in (14 mm) and −0.3, respectively. It can be observed that the simulation results match the exact response well. The three compensation schemes are all effective in improving the performance of the system. In particular, the DFC and feed-forward schemes provide similar performance. 2-DOF structure Real-time simulations are also performed on an inelastic 2-DOF shear frame structure. For the DFC and feed-forward compensation schemes, kDFC and kff are set to 0.8 and 0.15, respectively, for both degrees of freedom. The values of Td for the PLC as determined from a sine-sweep test are 30.9 and 24.2 ms for the first and second degrees of freedom, respectively, and d is set to 0.1. In the sine-sweep test, two chirp signals that have the same frequency but different amplitudes are used for the first and second degrees of freedom. The ratio of the amplitudes of the two signals applied is the average ratio of the two displacements obtained from the ‘exact’ numerical solution. Figure 12 shows the displacement responses at the first storey obtained from real-time simulations and the exact numerical solution. In this example, a strain-hardening model is used and the values of dy and are set to 0.30 in (7.6 mm) and 0.3, respectively, for both stories. It can be observed that the system provides a reasonably good performance, and both the DFC and feed-forward schemes are effective in reducing the phase-lag errors. However, the performance of the PLC is not as good as the other two. Next, a 2-DOF structure that has a softening-type load–displacement relation is considered. For this example, dy and are set to 0.4 in (10 mm) and −0.4 for the first storey, and 0.3 in (7.6 mm) and −0.3 for the second storey. Figure 13 shows the displacement responses at the top floor obtained from real-time simulations. It can be seen that the accuracy of the displacement responses is not as good as that for the strain-hardening case. This can be attributed to the fact that the initial stiffness used in the iteration and response update is very different from the actual tangent stiffness of the structure in the softening regime, which results in a more significant damping effect and, therefore, reduces the displacement drift as shown in Figure 13. This is verified by the fact that the simulated response matches the ‘exact’ solution well when the time delay due to the dynamics of the actuator is removed as shown by the curve labelled ‘RTI w/o actuator’ in the figure. The small discrepancy that remains is due to the numerical convergence errors introduced in the iterative scheme. Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
1803
PERFORMANCE OF A REAL-TIME PSEUDODYNAMIC TEST SYSTEM
1.2 CDM ∆t = 0.001 PID PID + DFC PID + PLC PID + FF
0.9 Displacement (inch)
0.6 0.3 0 -0.3 -0.6 -0.9 -1.2
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15 Time (sec)
(a)
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1 CDM ∆t = 0.001 PID PID + DFC PID + PLC PID + FF
0.8 0.6 Displacement (inch)
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4.6
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5 Time (sec)
8
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2 0 -2
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PID+ DFC
0 -2 -4
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5.2
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10
CDM ∆t = 0.001
Force (kips)
Force (kips)
8
4.9
-8 -10 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Displacement (inch)
Figure 11. Effectiveness of error-compensation schemes for SDOF structure with strain softening (1 in = 24.5 mm; 1 kip = 4.45 kN): (a) comparison of measured displacements; (b) zoomed-in measured displacements; and (c) load–displacement hystereses.
Flexibility of reaction frame Normally, for safety reasons, only the displacement of an actuator is directly controlled in a realtime test. However, a reaction frame supporting an actuator has a finite stiffness. When this stiffness Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
1804
R.-Y. JUNG ET AL.
Displacement (inch)
0.6
CDM ∆t = 0.001 PID PID + DFC PID + PLC PID + FF RTIw/o Actuator
0.3
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CDM∆t =0.001
6
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Force(kips)
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Time (sec) 8
0 -2
0 -2
-4
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-6
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-8 -0.5
(c)
30
0 Displacement (inch)
0.5
-8 -0.5
0 Displacement (inch)
0.5
Figure 12. Effectiveness of error-compensation schemes for 2-DOF structure with strain hardening (1 in = 24.5 mm; 1 kip = 4.45 kN): (a) comparison of measured displacements at the first storey; (b) zoomed-in measured displacements; and (c) first-storey load–displacement hystereses.
is too low, large displacement errors will be introduced [16]. The schematic of a test system that has a flexible reaction frame supporting a servo-hydraulic actuator is shown in Figure 14, in which the variables with a subscript t represent the properties of the test structure while those with a subscript Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
1805
PERFORMANCE OF A REAL-TIME PSEUDODYNAMIC TEST SYSTEM
1.2 CDM ∆t = 0.001 PID PID + DFC PID + PLC PID + FF RTIw/o Actuator
0.9 Displacement (inch)
0.6 0.3 0 -0.3 -0.6 -0.9 -1.2
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(a)
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0.9 Displacement (inch)
0.6 0.3 0 -0.3 -0.6 -0.9 -1.2 12
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CDM ∆t = 0.001
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Figure 13. Effectiveness of error-compensation schemes for 2-DOF structure with strain softening (1 in = 24.5 mm; 1 kip = 4.45 kN): (a) comparison of measured displacements at the top storey; (b) zoomed-in measured displacements; and (c) top-storey load–displacement hystereses.
r are those of the reaction frame, and d m is measured with a linear variable differential transformer (LVDT) mounted inside the hydraulic actuator. The measured displacement d m , therefore, reflects the piston movement, which is the sum of the displacements of the test structure and reaction Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
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R.-Y. JUNG ET AL.
Figure 14. Diagram of a test system with flexible actuator support.
frame. In a real-time pseudodynamic test, this measured displacement value is used in two places. At every sampling point, d m is sent back to the digital controller for the PID control. In addition, the time integration scheme uses d m and the corresponding r m to calculate the trial displacement in each iteration step and for the response update in the last iteration. Therefore, if the reaction frame supporting the actuator is flexible, errors are introduced in the displacement control and force feedback, which will affect the numerical result. The above problem can be circumvented by using an external LVDT mounted between a test structure and a fixed reference frame to obtain d m . However, the use of an external LVDT for the PID control is highly risky. One solution to this problem is to use a dual displacement control strategy [18] in which d m is measured with an external LVDT for calculating the trial displacements and updating the response in the last iteration while the displacement measured with the internal LVDT is used for the PID control. This approach will recover the actual stiffness of the test structure by using a consistent pair of d m and r m in the numerical computation. It will, nevertheless, still result in a displacement control error that corresponds to the deformation of the reaction frame. To remove this error, the DFC scheme can be adopted and the displacement measured by the external LVDT is used to compute the displacement command in Equation (21). To evaluate the effectiveness of the aforementioned method, the Simulink model has been modified to account for the flexibility of a reaction frame. Results of real-time simulations performed on a SDOF elastic–perfectly plastic structure are shown in Figure 15. The mass and elastic properties of the structure are the same as those of the SDOF structures considered previously. The yield displacement is 0.3 in (7.62 mm). The mass and stiffness of the reaction frame are assumed to be 8 times those of the test structure. The NS component of the 1940 El Centro earthquake with the peak ground acceleration scaled to 0.16g is used as the input excitation. The actuator model and the integration scheme and time interval are the same as those in the previous examples. In the simulations, the DFC and simple feed-forward delay compensation schemes are used with the gains set to 0.8 and 0.2, respectively. In both cases, a dual displacement control with two LVDTs is used. It can be observed from Figure 15 that the result obtained with the PID control alone has a small discrepancy with the result obtained with PID control and a rigid reaction frame. This is caused by the displacement error introduced by the support deformation. However, both results do not compare too well with the ‘exact’ numerical solution obtained with the CDM because of the actuator delay error. As shown, with the presence of the reaction frame deformation, this situation cannot be effectively corrected by the simple feed-forward scheme. On the other hand, the DFC scheme provides a very accurate result. Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
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PERFORMANCE OF A REAL-TIME PSEUDODYNAMIC TEST SYSTEM
1 0.8 Displacement (inch)
0.6 0.4 0.2 CDM ∆t = 0.001 PID PID + DFC PID + FF PID with Rigid Support
0 -0.2 -0.4
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(a)
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0 -0.2 -0.4 4
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5
(b)
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6 Time (sec)
6.5
7
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Figure 15. Displacement responses of an elastic–perfectly plastic structure with dual displacement control (1 in = 24.5 mm; 1 kip = 4.45 kN): (a) comparison of measured displacements and (b) zoomed-in measured displacements.
CONCLUSIONS This paper presents a real-time pseudodynamic test system that adopts an implicit time integration method for solving the governing equations of motion for a test structure. Several delay compensation schemes have been considered for reducing phase-lag errors in actuator response. To evaluate the performance of the system and the effectiveness of the compensation schemes, a Simulink model has been developed for an entire test system considering the nonlinearity and dynamics of the test structure and servo-hydraulic actuators and the deformation of the reaction system supporting the actuators. The accuracy of the model has been validated with real-time tests. The performance of the system for testing SODF and 2DOF structures that have inelastic load–displacement relations has been examined by means of real-time simulations using the Simulink model and the actual control system. It has been observed that with the delay compensation schemes, the system can deliver a good performance. However, the performance of the system can be lowered when a Copyright q
2007 John Wiley & Sons, Ltd.
Earthquake Engng Struct. Dyn. 2007; 36:1785–1809 DOI: 10.1002/eqe
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structure exhibits a severe strain-softening-type nonlinearity. In this situation, the test result may underestimate the inelastic displacement drift. This can be attributed to the increased numerical damping effect introduced by the actuator delay errors and structural softening. This damping effect is an inherent property of the numerical scheme used here. Furthermore, it has been shown that the DFC scheme presented here can also be used to mitigate the errors introduced by the deformation of the actuator reaction system. ACKNOWLEDGEMENTS
This study was conducted with financial support from the George E. Brown, Jr. Network for Earthquake Engineering Simulation program of the National Science Foundation under the Cooperative Agreement No. CMS 0086592 and cost sharing from the University of Colorado at Boulder. However, opinions expressed in this paper are those of the authors and do not necessarily represent those of the sponsors. REFERENCES 1. Takanashi K et al. Non-linear earthquake response analysis of structures by a computer actuator on-line system, part 1—details of the system. Transactions of the Architectural Institute of Japan 1975; 229:77–83. 2. Mahin SA, Shing PB. Pseudodynamic method for seismic testing. Journal of Structural Engineering (ASCE) 1985; 111:1482–1503. 3. Nakashima M, Kaminosono T, Ishida M. Integration techniques for substructure pseudodynamic test. Proceedings of the 4th US National Conference on Earthquake Engineering 1990; II:515–524. 4. Shing PB, Nakashima M, Bursi O. Application of pseudodynamic test method to structural research. Earthquake Spectra 1996; 12(1):29–56. 5. Pinto AV, Pegon P, Magonette G, Tsionis G. Pseudo-dynamic testing of bridges using non-linear substructuring. Earthquake Engineering and Structural Dynamics 2004; 33:1125–1146. 6. Nakashima M, Masaoka N. Real-time on-line test for MDOF systems. Earthquake Engineering and Structural Dynamics 1999; 28:393–420. 7. Horiuchi T, Nakagawa M, Sugano M, Konno T. Development of a real-time hybrid experimental system with actuator delay compensation. Proceedings of the 11th World Conference on Earthquake Engineering, Acapulco, Mexico, 1996, Paper No. 660. 8. Williams DM, Williams MS, Blakeborough A. Numerical modeling of a servo-hydraulic testing system for structures. Journal of Engineering Mechanics 2001; 127(8):816–827. 9. Zhao J, French C, Shield C, Posberg T. Considerations for the development of real-time dynamic testing using servo-hydraulic actuation. Earthquake Engineering and Structural Dynamics 2003; 32:1773–1794. 10. Chang SY. Explicit pseudodynamic algorithm with unconditional stability. Journal of Engineering Mechanics 2002; 128(9):935–947. 11. Bonelli A, Bursi OS. Generalized- methods for seismic structural testing. Earthquake Engineering and Structural Dynamics 2004; 33:1067–1102. 12. Wu B, Xu G, Wang Q, Williams MS. Operator-splitting method for real-time substructure testing. Earthquake Engineering and Structural Dynamics 2006; 35(3):293–314. 13. Bursi O, Shing PB. Evaluation of some implicit time-stepping algorithms for pseudodynamic tests. Earthquake Engineering and Structural Dynamics 1996; 25:333–355. 14. Jung R-Y. Development of real-time hybrid test system. Doctoral Thesis, University of Colorado at Boulder, Boulder, CO, 2005. 15. Bayer V, Dorka UE, F¨ullekrug U, Gschwilm J. On real-time pseudodynamic sub-structure testing: algorithm, numerical and experimental results. Aerospace Science and Technology 2005; 9:223–232. 16. Jung R-Y, Shing PB. Performance evaluation of a real-time pseudodynamic test system. Earthquake Engineering and Structural Dynamics 2006; 35(7):789–810. 17. Hughes TJR. Analysis for transient algorithms with particular reference to stability behaviour. In Computational Methods for Transient Analysis, Belyschko T, Hughes TJR (eds). North-Holland: Amsterdam, 1983. 18. Shing PB, Vannan MT, Carter E. Implicit time integration for pseudodynamic tests. Earthquake Engineering and Structural Dynamics 1991; 20(6):551–576. Copyright q
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