stability and accuracy analysis of the central difference method for real-time substructure testing

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2005; 34:705–718Published online 17 January 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.451

Stability and accuracy analysis of the central di�erencemethod for real-time substructure testing

B. Wu∗;†, H. Bao, J. Ou and S. Tian

School of Civil Engineering; Harbin Institute of Technology; Harbin 150090; People’s Republic of China

SUMMARY

The central di�erence method (CDM) that is explicit for pseudo-dynamic testing is also believed tobe explicit for real-time substructure testing (RST). However, to obtain the correct velocity dependentrestoring force of the physical substructure being tested, the target velocity is required to be calculated aswell as the displacement. The standard CDM provides only explicit target displacement but not explicittarget velocity. This paper investigates the required modi�cation of the standard central di�erence methodwhen applied to RST and analyzes the stability and accuracy of the modi�ed CDM for RST. Copyright? 2005 John Wiley & Sons, Ltd.

KEY WORDS: real-time; substructure testing; central di�erence method; stability; accuracy

1. INTRODUCTION

Pseudo-dynamic testing (PDT) is an experimental technique for simulating the earthquake re-sponse of structures and structural components in the time domain. In this test, the structuralsystem is represented as a discrete spring–mass system, and its dynamic response to earth-quakes is solved numerically using direct integration. Unlike conventional direct integrationalgorithms, in the pseudo-dynamic test the restoring forces of the system are not modeledbut are directly measured from a test conducted in parallel with the direct integration. Inmany structures, the unpredictable non-linear behaviour that provides the motivation for lab-oratory testing is quite localized. In these circumstances a far more economical test can beperformed using the pseudo-dynamic testing with substructuring approach or pseudo-dynamicsubstructure testing (PST). The algorithms and implementations of PDT and PST are welldocumented, e.g. References [1, 2].

∗Correspondence to: B. Wu, School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, People’sRepublic of China.

†E-mail: [email protected]

Contract=grant sponsor: National Science Foundation of China; contract=grant number: 50338020Contract=grant sponsor: Ministry of Science and Technology of China; contract=grant number: 2001AA602015

Received 5 May 2004Revised 13 September 2004 and 27 October 2004

Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 27 October 2004

706 B. WU ET AL.

One of the critical prerequisites for conducting PDT is that the e�ect of the loading rate onthe restoring force of the structure should be of minor consequence, because the structure isloaded quasi-statically in the PDT. Lately, a variety of new types of structural components anddevices have been introduced in structures, particularly in connection with their vibration con-trol [3]. Many of them are very velocity dependent in vibration characteristics such as viscousdampers or viscoelastic dampers. To test the velocity dependent components incorporated instructures, real-time substructure testing (RST) was developed in the 1990s. The �rst reportedRST test [4] was performed on a viscous damper located at the base of a multi-story building.Only the damper was tested physically, with the isolated building modeled numerically.A key element of RST as well as PDT is the numerical algorithm that is used to perform

the stepwise integration of the equations of motion. Many numerical algorithms have beenused in RST such as the central di�erence method [4–9], linear acceleration method [10],backward Euler method [11], Tustin’s method [12], and �rst-order-hold discretization method[7]. The central di�erence method (CDM), which is explicit for PDT is also believed to beexplicit for RST [13]. However, to obtain the correct velocity dependent restoring force ofthe physical substructure being tested, the calculated target velocity is required as well as thedisplacement. The standard CDM provides only explicit target displacement but not explicittarget velocity. This paper will investigate the required modi�cation of the standard centraldi�erence method when applied to RST and analyze the stability and accuracy of the modi�edcentral di�erence method of RST.

2. CENTRAL DIFFERENCE METHOD FOR RST

For RST, the equations of motion may be written in matrix form as

MN �X+RN(X; X) +RE(X; X; �X)=F (1)

where MN is the mass matrix of the numerical substructure, RN is the restoring force vectorof the numerical substructure, RE is the restoring force vector of the physical substructure(test specimen), X is the vector of nodal displacements, F is the vector of external excitationforces, and the dots represent di�erentiation with respect to time. In many substructure teststhe mass of the specimens can be ignored and the properties of the specimens are not relatedto acceleration so that the restoring forces take the form of RE(X; X). We also assume thatthe numerical substructure is with linear damping force and displacement dependent restoringforce. Then we get

MN �X+CNX+RN(X) +RE(X; X)=F (2)

where CN is the damping coe�cient of the numerical substructure. Using the CDM, thevelocity and acceleration in step i are approximated by

Xi =Xi+1 −Xi−1

2�t(3)

�Xi =Xi+1 − 2Xi +Xi−1

�t2(4)

Copyright ? 2005 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2005; 34:705–718

ANALYSIS OF THE CENTRAL DIFFERENCE METHOD FOR RST 707

where �t is a time interval. Substituting Equations (3) and (4) into Equation (2) at the i-thstep, we obtain

Xi−1 =(MN

�t2+CN2�t

)−1 [Fi −RN(Xi) + 2MN

�t2Xi −

(MN

�t2− CN2�t

)Xi−1 −RE(Xi ; Xi)

](5)

From the above equation we see that the calculation of X1 involves X0 and X−1. WithX0, X0, �X0 known (given X0 and X0, then �X0 is calculated using Equation (2)), the relationsin Equations (3) and (4) can be used to obtain X−1 as [14]:

X−1 =X0 −�tX0 + �t2

2�X0 (6)

In conventional PDT, the calculated target displacement is imposed upon the specimenand then the rate-independent restoring force can be measured. However, in a RST withthe physical substructure non-linearly dependent on velocity, an iteration, or a linearizationprocess that requires online estimation of damping of the specimen, is inevitable for solvingthe non-linear equation about Xi+1 (see Equation (5) with substitution of Equation (3)) withinthe framework of standard CDM. In this regard, the standard CDM is actually an implicitalgorithm for RST and it is obviously inadequate for standard CDM to be used as an explicitalgorithm for RST. For an explicit RST algorithm, the velocity of the next step (i.e. stepi + 1) must also be explicitly calculated and imposed on the specimen to obtain correctlythe restoring force dependent on velocity. Since the standard CDM is inherently implicitif non-linear damping is considered, any extra explicit formulation of the velocity wouldfundamentally change the basic assumptions, hence the stability and accuracy, of the CDM.To the writers’ knowledge, the issue of the velocity calculation method in RST and particularlyits consequence on the stability and accuracy have not been discussed theoretically by otherresearchers.To achieve su�cient accuracy in both displacement and velocity control, a digital servo-

mechanism was used in the RST by Nakashima et al. [4]. The mechanism interpolates thetarget displacement signal Xi+1 linearly into a set of displacement signals: 1Xi+1; 2Xi+1; : : :,jXi+1; : : : ; j0Xi+1 with jXi+1 =Xi+j× �t× Xi+1 and j0Xi+1 =Xi+1. In other words, the followingadditional assumption for the target velocity was implied in Nakashima et al.’s [4] test:

Xi+1 =Xi+1 −Xi�t

(7)

With the above equation, CDM becomes explicit for velocity as well as for displacement.It is worth addressing how the target velocity expressed by Equation (7) is realized by

the actuator with the current practice of doing RST. With the state-of-the-practice of servo-hydraulic control, it is hard to simultaneously realize velocity and displacement feedbackcontrol even if an explicit target velocity as well as displacement is available. Therefore, thetarget velocity cannot be realized by directly being imposed onto the actuator. The ideal thingis that the target velocity is automatically reached when the target displacement is achievedon the real-time base. Whether or not the target velocity in Equation (7) can be achievedautomatically by the actuator depends very much on the dynamics of the actuator and the wayto achieve the target displacement in real time. To cope with time delays inherent in the testinghardware, a forward extrapolation of position beyond the next time step [9] is commonly used.


708 B. WU ET AL.

Figure 1. Computation schematic of a linear SDOF system in RST.

A typical forward extrapolation based on a cubic �t to the previous few displacement pointsmathematically implies a quadratic variation in velocity. But whether this type of velocityvariation could be realized during the calculation time interval also depends on the dynamicsof the actuator. It can be proved that the target velocity formulation expressed with forwarddi�erence by Equation (7) complies with the actual velocity of the actuator in common practiceof RST, if it is assumed that the actuator is with bilinear step response characteristic, timedelay is �xed, and complete delay compensation is achieved. With the same assumptions, italso follows that the constant velocity pro�le is attained between current and next calculationinstants even if the cubic displacement extrapolation is used. It is not known whether thenumerical behaviour of the CDM for RST (CDM–RST) would become better or not if thevelocity formulation is given by considering a more precise model of the actuator dynamics.If it becomes better, the associated method will become an improved CDM–RST over theone proposed by the authors; if not, then the target velocity formulation proposed by theauthors should be tried to achieve although it may not be easy. One way to achieve the targetvelocity through displacement feedback control is to interpolate (very possibly non-linearly)the target displacement based on the actuator dynamics into a series of sub-displacements,extrapolate the sub-displacements beyond the next step into a set of displacement signals tocompensate the time delay of the testing system, and then successively send them to theactuator.Of course, other target velocity formulations are also possible and may be better, but they

are beyond the scope of this paper. In the following section, we will investigate the stabilityand accuracy of CDM–RST with the assumption about target velocity in Equation (7).

3. STABILITY AND ACCURACY ANALYSIS OF CDM–RST

To analyze the stability and accuracy we consider a single-degree-of-freedom (SDOF) systemwith linear numerical and physical substructures (see Figure 1), i.e.

RN(X ) =KNX (8)

RE(X; X ) =CE X + KEX (9)

in which KN is the sti�ness of the numerical substructure, and CE and KE are the dampingcoe�cient and sti�ness of the physical substructure, respectively. Substituting Equations (8)



and (9) into (5) and then (5) into (7), we obtain

Xi+1 =(MN

�t2+CN2�t

)−1 [Fi −

(KN+KE − 2MN

�t2

)Xi−

(MN

�t2− CN2�t

)Xi−1 − CE Xi

](10)

X i+1 =(MN

�t+CN2

)−1 [Fi −

(KN + KE−MN

�t2+CN2�t

)Xi −

(MN

�t2− CN2�t

)Xi−1 − CE Xi

]

(11)

The stability and accuracy can be evaluated with the free vibration solution succinctly writtenin the recursive form [15]:

Yi+1 =AYi (12)

whereYi=[Xi; Xi−1;�tX t] (13)

and the ampli�cation matrix of CDM–RST, A, is expressed as

A=

2−�21 + �N�

�N�− 11 + �N�

−2�E�1 + �N�

1− �N�−�21 + �N�

�N�− 11 + �N�

−2�E�1 + �N�

(14)

in which �=�t!=�t√(KN + KE)=MN, �N =CN=(2MN!), �E =CE=(2MN!). The stability

and accuracy of an algorithm depend upon the eigenvalues of the ampli�cation matrix.

3.1. Stability

The stability condition of an integration method is [15]:

�(A)61 (15)

where �(A) is the spectral radius of A which is de�ned as max |�i|, and �i are the eigenvaluesof A. For the matrix A in Equation (14), we have

�1;2 =A± iB and �3 = 0 (16)

where

A=2−�2 − 2�E�2(1 + �N�)

; B=�

√−�2 − 4�E� + 4− 4�2N − 4�2E − 8�E�N

2(1 + �N�)(17)

when

�¡√4− 4�2N − 8�N�E − 2�E (18)


710 B. WU ET AL.

Figure 2. Spectral radii of the ampli�cation matrix.

or

�1;2 =A± B and �3 = 0 (19)

where

A=2−�2 − 2�E�2(1 + �N�)

; B=�

√�2 + 4�E�− 4 + 4�2N + 4�2E + 8�E�N

2(1 + �N�)(20)

when

�¿√4− 4�2N − 8�N�E − 2�E (21)

Based on Equations (16) and (19), the spectral radius of A is plotted against � in Figure 2for various values of �E with �N =0.From inequalities or Equations (15)–(21), the stability criteria for CDM–RST can be ob-

tained as

�¿√4 + 4�2E − 2�E; Unstable (22a)

√4− 4�2N−8�N�E−2�E6�6

√4+4�2E−2�E; Stable (two real and one zero eigenvalues)

(22b)

�¡√4− 4�2N − 8�N�E − 2�E; Stable (two complex conjugate and one zero eigenvalues)

(22c)

From inequality (22a), we see that unlike the standard CDM for PDT (CDM–PDT) [16],the upper limit of � for a stable CDM–RST is not constant. The relationship of the stabil-ity limit of � and damping ratio of the physical substructure is plotted in Figure 3. From



Figure 3. Stability limit of �.

Figure 3 we see that the stability limit of � decreases with increasing damping ratio of thephysical substructure, which means the stability of CDM–RST deteriorates with higher damp-ing ratio of the specimen. When the damping ratio of the physical substructure is zero, thestability limit becomes 2 which is the same as the result of the CDM–PDT.It might appear surprising that the stability of the algorithm decreases with increasing

damping in the physical substructure. However, the stability of a structure in the real worldand that of a numerical algorithm are two di�erent issues. Although the increasing dampingin the physical substructure will of course increase the stability of the structure as in thereal world, that does not necessarily mean the stability of computation will be improved.A similar case can be seen in Reference [16], where the stability limit of the standard CDMis not increased with increasing damping of the structure. For the speci�c case of CDM–RSTin this paper, an explanation of why the stability of the algorithm decreases with increasingdamping of the physical substructure is given in Appendix A.

3.2. Accuracy

If the condition in inequality (18) is satis�ed, then the eigenvalues in Equation (16) can beexpressed as

�1;2 = exp �(−�± i) �� (23a)

where

�=− ln (A2 + B2)2 ��

(23b)

��= arctan(BA

)(23c)

Thus, by de�ning �!= ��=�t, the free vibration displacement response can be expressedas [15]:

Xn = exp (−� �!n�t)(c1 cos �!n�t + c2 sin �!n�t) (24)


712 B. WU ET AL.

where the cj value, are constants determined from the initial conditions. The above equa-tion can be rewritten as the closed-form solution for a viscously underdamped free-vibrationresponse:

Xn = exp (−� �!nn�t)(c1 cos �!d�tn+ c2 sin �!dn�t) (25)

where

�=�√1 + �2

(26a)

�!d = �! (26b)

�!n =�!d√1− �2

(26c)

From Equations (25) and (26), we can consider � as a damping ratio consisting of threeparts: those of the numerical and physical substructures and that due to numerically inducedenergy dissipation, and �!d as the distorted frequency of the numerical solution for the dampedvibration. The numerical damping ratio �� is de�ned as

��= �− �N − �E (27)

and period distortion for the damped system as

�TdTd

=Td − �T dTd

= 1− !d�!d= 1− �

��

√1− (�N + �E)2 (28)

The numerical damping ratio and the ratio of period distortion are generally considered asmeasures of numerical inaccuracies.Based on Equation (27), the value of �� is plotted against � in Figure 4 for di�erent

�E values with �N =0. It can be seen that, for a relatively low damping ratio of the phys-ical substructure, �� increases with increasing � until �� reaches a peak value and then itdecreases until � reaches the upper limit within which the solution with two complex con-jugate eigenvalues exists (see inequality (22c)); for high damping ratio of the physical sub-structure, �� keeps increasing until � reaches the upper limit of oscillation solution rangeand there is no peak occurring within the oscillation solution range. For the same rela-tively low damping ratios of physical substructure, say 0.3, 0.25, 0.2 and 0.1 as shown inFigure 4, the numerical damping ratio reduces to a negative value before � reaches the above-mentioned limit. The numerical damping ratio is zero when �E =0, which coincides with theresult of CDM–PDT. This can be understood if we replace RE(Xi ; X1) with RE(X1) in Equa-tion (5), which results in the solution form of CDM–PDT so that the additional assumptionfor the velocity by CDM–RST, i.e. Equation (7), will not a�ect the displacement response.From Figure 4, we also see that �� increases with �E for a �xed � when � is not near theoscillation solution limit. The numerical damping ratio with 5% damping in the numericalsubstructure is shown in Figure 5. From comparison of Figures 4 and 5, it is seen that a typ-ical 5% damping of the numerical substructure has a minor in�uence on the accuracy of theCDM–RST.



Figure 4. Numerical damping ratios (�N =0).

Figure 5. Numerical damping ratios (�N =0:05).

Period distortion is plotted against � in Figure 6 for di�erent �E values with �N =0. FromFigure 6 we see that except for a very high damping ratio of the physical substructure(e.g. �E =0:6 or 0.8 in Figure 6), the period distortion increases with increasing �; theperiod distortion increases with physical damping ratio for the �xed � value. For the casewith �E =0, Td is equal to Tn and the �Td=Td − � relationship is the same as that byCDM–PDT [15].As well as the numerical damping ratio and period distortion, another inaccuracy of CDM–

RST that should be noted is that it may change the initial condition of the vibration system.From Equations (6) and (10), we obtain

X1 =

(1− 1

2�2 + �N�− 1

2�N�3)X0 + (�t − �E��t − �2E�2�t − �N�E�2�t)X0

1 + �N�(29)


714 B. WU ET AL.

Figure 6. Period distortion.

Using the known initial displacement X0 and the displacement of the �rst step, X1, calculatedfrom Equation (29), we can determine c1 and c2 in Equation (24):

c1 = X0 (30a)

c2 =X1 − X0 cos �� exp (−� ��)

sin �� exp (−� ��)(30b)

c1 and c2 in the closed-form solution (Equation (25)) can be determined with initial conditionas [17]:

c1 =D0 (31a)

c2 =V0 + � �!nD0

�!d(31b)

in which D0 and V0 are equivalent initial displacement and velocity, respectively, in thesolution with closed-form. Comparing Equations (30) and (31), we get

D0 = X0 (32a)

V0 = �1(�; �N; �E) �!X0 + �2(�; �N; �E)X 0 (32b)

where �1 and �2 are the functions about �, �N and �E, which are determined by comparingEquations (30) and (31). Apparently �1 �=0, �2 �=1 except for very special values of �, �Nand �E satisfying �1 = 0 and �2 = 1. The wrong initial velocity V0 by Equation (32a) is plottedagainst � in Figure 7, in which the real initial velocity X0 is selected to be zero and Tn = 1 s,�N =0, �E =0:25, X0 = 1. From Figure 7, the initial velocity error increases with increasing� and �E.



Figure 7. Initial velocity error.

Figure 8. Free-vibration of a linear SDOF system.

4. NUMERICAL EXAMPLE

The simulated displacement responses of the free vibrations with Tn = 1 s, �N =0,�E =0:25, X0 = 1, X0 = 0 and various �, together with the exact solution, are shown inFigure 8.Numerical instability, period shrinkage and amplitude increase due to initial velocity change

can be easily observed from Figure 8. Because of the small numerical damping ratio in thisexample, its e�ect on the displacement response is too small to observe.


716 B. WU ET AL.

5. CONCLUSIONS

To maintain the explicit form of CDM both for velocity and for displacement, some modi�-cation on the algorithm is required when it is applied to RST. The stability and accuracy ofa modi�ed CDM for RST with additional assumption about target velocity are investigated.The main results are concluded as follows.

(1) The stability of CDM–RST deteriorates as the damping ratio of the physical substructureincreases.

(2) The numerical damping ratio is positive and increases both with increasing !�t andwith increasing damping ratio of the physical substructure for most !�t in the stablerange; a minor negative numerical damping ratio occurs when !�t is near the stablelimit for the cases with relatively low damping ratio of physical substructure.

(3) Period distortion increases with increasing !�t and damping ratio of physical substruc-ture except for a very high damping ratio of the physical substructure.

(4) There is error in the initial velocity and the amount of the error increases with increasing!�t and damping ratio of the physical substructure.

It should be noted that, although the above conclusions indicate that the velocity formulation(Equation (7)) makes CDM–RST fairly di�erent from standard CDM, it is valid as long asthe system considered is lightly damped by the physical substructure and the integration timeinterval is set at a small value to ensure accuracy, which can be easily seen from Figures 2–8.In this paper, investigation is restricted to CMD modi�ed with the forward di�erence for-

mulation of velocity. However, the same velocity formulation can also be applied to RSTcombined with other explicit pseudo-dynamic algorithms such as the operator-splitting methodproposed by Nakashima et al. [18] and the one developed by Chang [19]. Unlike CDM, thosetwo algorithms are both unconditionally stable. So it appears more promising if the velocityformulation is used together with them in RST. The relevant stability and accuracy issues areworth researching.

APPENDIX A

For the analytical research on the numerical integration method, we apparently need a mathe-matical model of the physical part of the hybrid testing system. In fact, in the above analysisof CDM–RST, the physical part of the hybrid system has already been represented by a lin-ear mathematical model and we assume that the mathematical model precisely describes themechanical property of the physical substructure. To concentrate on researching the behaviourof the numerical integration method itself, we also assume that the mathematical model ofthe numerical substructure exactly represents its counterpart in the real world. Then the onlysource of error will be the numerical integration method itself. So the stability of the nu-merical method of the hybrid system seems to be more a mathematical issue than a physicalissue. Perhaps a mathematical explanation is more suitable and easier, which is given basedon the mathematical approximation as follows.From Equations (12)–(14), the displacement and velocity at the i+1 step are obtained as

Xi+1 =2−�21 + �N�

Xi +�N�− 11 + �N�

Xi−1 +−2�E�1 + �N�

�tXi (A1)



Xi−1 =1− �N�−�2(1 + �N�)�t

Xi +�N�− 1

(1 + �N�)�tXi−1 +

−2�E�1 + �N�

Xi (A2)

For the �xed �N, � and non-zero Xi, we always can �nd a large enough �E such that theabsolute value of the third term on the right hand side of Equations (Al) and (A2) is muchlarger than the �rst two terms. Neglecting these two terms, the displacement and velocity atthe i + 1 step are approximated by

Xi+1 ≈ −2�E�1 + �N�

�tXi (A3)

X i+1 ≈ −2�E�1 + �N�

Xi (A4)

Successively replacing i with i−1; : : : ; j in Equation (A4) and substituting Equation (A4) into(A3), we have

Xi+1 ≈( −2�E�1 + �N�

)i+1−j�tXj= �i+1−j�tXj (A5)

where

�=−2�E�1 + �N�

(A6)

For the case with non-zero initial condition, we always can �nd by using Equations (A1) and(A2) some j such that Xj �=0. It is easy to �nd a large enough �E such that |�|¿1, hence Xi+1goes to in�nity as i approaches in�nity. In other words, for the CDM–RST with the �xed �Nand �, the stability of the numerical method will be reduced but not be strengthened if thedamping of the physical structure is increased largely enough. This may help to understandwhy the stability of the CDM–RST becomes worse with increasing damping ratio of thephysical structure.

REFERENCES

1. Mahin SA, Shing PB. Pseudodynamic method for seismic testing. Journal of Structural Engineering (ASCE)1985; 111:1482–1503.

2. Takanashi K, Nakashima M. Japanese activities on online testing. Journal of Engineering Mechanics (ASCE)1987; 113:1014–1032.

3. Soong TT, Spencer Jr BF. Supplemental energy dissipation: state-of-the-art and state-of-the-practice. EngineeringStructures 2002; 24:243–259.

4. Nakashima M, Kato H, Takaoka E. Development of real-time pseudo dynamic testing. Earthquake Engineeringand Structural Dynamics 1992; 21:79–92.

5. Nakashima M, Masaoka N. Real-time on-line test for MDOF systems. Earthquake Engineering and StructuralDynamics 1999; 28:393–420.

6. Darby AP, Blakeborough A, Williams MS. Real-time substructure tests using hydraulic actuator. Journal ofEngineering Mechanics (ASCE) 1999; 125:1133–1139.

7. Darby AP, Blakeborough A, Williams MS. Improved control algorithm for real-time substructure testing.Earthquake Engineering and Structural Dynamics 2001; 30:431–448.

8. Horiuchi T, Inoue M, Konno T, Namita Y. Real-time hybrid experimental system with actuator delaycompensation and its application to a piping system with energy absorber. Earthquake Engineering andStructural Dynamics 1999; 28:1121–1141.


718 B. WU ET AL.

9. Horiuchi T, Konno T. A new method for compensating actuator delay in real-time hybrid experiments.Philosophical Transactions of the Royal Society of London A 2001; 359:1893–1909.

10. Horiuchi T, Inoue M, Konno T. Development of a real-time hybrid experimental system using a shakingtable. Proceedings of the 12th World Conference on Earthquake Engineering, Auckland, New Zealand, 2000;paper no. 0843.

11. Igarashi A, Iemura H, Tanaka H. Development of substructure hybrid shake table test method and applicationto veri�cation tests of vibration control devices. China–Japan Workshop on Vibration Control and HealthMonitoring of Structures and Third Chinese Symposium on Structural Vibration Control, Shanghai, China,December 2002, paper no. C007.

12. Blakeborough A, Williams MS, Darby AP, Williams DM. The development of real-time substructure testing.Philosophical Transactions of the Royal Society of London A 200l; 359:1869–1891.

13. Williams MS, Blakeborough A. Laboratory testing of structures under dynamic loads: an introductory review.Philosophical Transactions of the Royal Society of London A 2001; 359:1651–1669.

14. Bathe KJ. Finite Element Procedures. Prentice-Hall: Englewood Cli�s, New Jersey, p 770, 1996.15. Shing PB, Mahin SA. Computational aspects of a seismic performance test method using on-line computer

control. Earthquake Engineering and Structural Dynamics 1985; 13:507–526.16. Nakashima M. Stability and accuracy behavior of pseudodynamic response. Part 1: relationship between

integration time interval and response stability in pseudo dynamic testing. Journal of Structural andConstruction Engineering 1985; 353:29–34.

17. Chopra AK. Dynamics of Structures. Prentice-Hall: Englewood Cli�s, New Jersey, p 45, 1995.18. Nakashima M, Kaminosono T, Ishida M, Ando K. Integration technique for substructure pseudodynamic test.

Proceedings of the 4th U.S. National Conference on Earthquake Engineering, vol. 2, Palm Springs, CA, 1990,pp. 515–524.

19. Chang SY. Explicit pseudodynamic algorithm with unconditional stability. Journal of Engineering Mechanics(ASCE) 2002; 128:935–947.


stability and accuracy analysis of the central difference method for real-time substructure testing

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