Bull Earthquake Eng (2016) 14:89–114 DOI 10.1007/s10518-015-9816-0 ORIGINAL RESEARCH PAPER
Comparison of explicit integration algorithms for real-time hybrid simulation Fei Zhu1 • Jin-Ting Wang1 • Feng Jin1 • Yao Gui2
Received: 15 January 2015 / Accepted: 13 September 2015 / Published online: 21 September 2015 Springer Science+Business Media Dordrecht 2015
Abstract Real-time hybrid simulation (RTHS) combines physical experimentation with numerical simulation to evaluate dynamic responses of structures. The inherent characteristics of integration algorithms change when simulating numerical substructures owing to the response delay of loading systems in physical substructures. This study comprehensively investigates the effects of integration algorithms on the delay-dependent stability and accuracy of multiple degrees-of-freedom RTHS systems. Seven explicit integration algorithms are considered; and the discrete-time root locus technique is adopted. It is found that the stability of RTHS system is mainly determined by the time delay rather than the integration algorithms, whereas its accuracy mainly depends on the accuracy characteristic of the applied integration algorithm itself. An unconditionally stable integration algorithm cannot always guarantee good stability performance; and the inherent accuracy or numerical energy dissipation of integration algorithms should be taken into account in RTHSs. These theoretical findings are well verified by RTHSs. Keywords Real-time hybrid simulation Integration algorithm Delay-dependent stability and accuracy Delay compensation Discrete-time root locus
& Jin-Ting Wang
[email protected] Yao Gui
[email protected] 1
State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China
2
Changjiang Institute of Survey, Planning, Design and Research, Wuhan 430010, China
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1 Introduction Real-time hybrid simulation (RTHS) is a method for evaluating the dynamic response of structures under earthquake by combining physical experimentation with numerical simulation. In RTHS, the response time delay, caused by the dynamics of the shaking table (or actuator) and the associated control system, and the time taken to data transfer, results in inaccurate response and even system instability. The delay-dependent stability of RTHS systems has been extensively studied over the past decades. Some studies are based on the assumption of a continuous-time system. Horiuchi et al. (1999) used energy balance method and demonstrated that the delayed response is equivalent to introducing negative damping into RTHS systems. Horiuchi and Konno (2001) studied the stability conditions of a mass-block system by checking the gain of the open-loop system transfer function of RTHS systems. Wallace et al. (2005) analyzed the delay-dependent stability by using the delay differential equation. Mercan and Ricles (2007, 2008) applied the pseudo-delay technique to find critical stability conditions of RTHS systems with multiple delay sources. Chi et al. (2010) studied the delay-dependent stability conditions and the instability mechanism of single degree-of-freedom (SDOF) structures by using continuous-time root locus (RL) technique. In these studies, integration algorithms and delay compensation methods on the basis of integration algorithms cannot be considered because of the continuous-time assumption. Meanwhile, some other researchers have adopted the discrete-time system assumption to conduct stability analysis, considering the discrete-time characteristic of RTHS systems. Wu et al. (2005) found that the stability of the operator-splitting algorithm (Wu et al. 2006; Verma and Rajasankar 2012) for RTHS declined because of the actuator delay. Chen and Ricles (2008b) investigated the stability of an SDOF RTHS system by checking the locations of poles of the discrete transfer function. The effects of Newmark explicit method and CR algorithm (Chen and Ricles 2008a) on stability were presented. Zhu et al. (2015) used the discrete-time RL technique to analyze the stability of multiple degrees-of-freedom (MDOF) RTHS systems with one or more physical substructures. They comprehensively considered the CR algorithm and different delay compensation methods. This paper aims at evaluating the effects of different integration algorithms on the stability and accuracy of MDOF RTHS systems. The application of integration algorithms to solve numerical substructures has been extensively investigated, and explicit integration algorithms have gained the most attention because of their highly computational efficiency. The most popular time integration algorithms for RTHS may be the central difference method (CDM) (Horiuchi et al. 1999; Nakashima et al. 1992), and the Newmark explicit method (Igarashi et al. 2000). Chang (2002) proposed an explicit algorithm [named Chang (2002) in this paper] which has excellent stability performance and exhibits no numerical dissipation. Chen and Ricles (2008a) utilized the pole mapping method to develop an unconditionally stable CR algorithm in which the displacement and velocity in each integration step can be explicitly calculated. Gui et al. (2014) developed a family of explicit algorithms with a controlled parameter k (named Gui-k method in this paper), in which certain subfamilies are found to be unconditionally stable for any system state (linear elastic, stiffness hardening or softening). Kolay and Ricles (2014) proposed the explicit KR-a method with controllable numerical energy dissipation on the basis of the generalized-a method (Chung and Hulbert 1993). However, these studies focus more on the characteristics of the algorithm itself than on its effect on the performance of RTHS systems (Bonnet et al. 2008; Gui et al. 2014).
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In this paper, six unconditionally/conditionally stable integration algorithms without numerical damping and the newly-proposed explicit KR-a method with controllable numerical damping are considered. The analytical model on the basis of the discrete-time RL technique (Zhu et al. 2015) is adopted to obtain stability limitations and compare accuracy differences. The stability performance of RTHS system in both pure time delay and delay compensation with the 3rd-order polynomial method conditions is investigated. The accuracy of RTHS systems is also analyzed with numerical simulation and by comparing the shapes of RL plots. RTHSs are conducted to verify the correctness of the theoretical conclusions.
2 Analytical model for delay-dependent analysis 2.1 Analytical model Following a previous investigation (Zhu et al. 2015), an M-storey shear frame shown in Fig. 1a is considered as the model structure used to analyze the stability and accuracy of MDOF RTHS system, in which m, c, and k are the mass, damping and stiffness coefficients, respectively. The entire structure is idealized to be a lumped-mass system (Fig. 1b) and then partitioned into two parts (Fig. 1c): the upper part, as the physical substructure, is experimented on a shaking table; and the lower part, as the numerical substructure, is numerically simulated. The linear equations of motion of the entire structure can be formulated as ⎡ m1 ⎢ m2 ⎢ ⎢ ⎢ M NS ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣
⎡c1 + c2 ⎢ −c 2 ⎢ x NS ,i ⎢ ⎥⎪ ⎪ ⎢ C ⎥ ⎪ xb ⎪ + ⎢ NS ⎥ ⎨ xb+1 ⎬ ⎢ ⎪ ⎪ ⎢ ⎥ ⎥ ⎪ xb+ 2 ⎪ ⎢ ⎥ x PS ,i ⎢ ⎥ ⎢ mM ⎦⎥ ⎩⎪ xM ⎭⎪ ⎢⎣ ⎤ ⎧ x1 ⎫ ⎥⎪ x ⎪ 2
mb mb+1 mb+2 M PS
cb + cb+1 −cb+1 −cb+1 cb+1 + cb+2 −cb+2 −cb+2 C PS
xi
M
⎡ k1 + k2 ⎢ −k 2 ⎢ ⎢ ⎢ K NS +⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣
−c2 c2 + c3 −c3 −c3
C
− k2 k2 + k3 −k3 −k3 kb + kb+1 −kb+1 kb+1 + kb+2 −kb+2 −kb+1 −kb+ 2 K PS K
kM −1 + kM −kM
cM −1 + cM −cM
⎤ ⎧ x1 ⎫ ⎥⎪ x ⎪ ⎥⎪ 2 ⎪ x NS ,i ⎥ ⎪ xb ⎪ ⎥ ⎨ xb+1 ⎬ ⎪ ⎥⎪ ⎥ ⎪ xb+ 2 ⎪ ⎪ ⎪ −cM ⎥ x ⎥ PS ,i c ⎦⎥ ⎩⎪ x ⎭⎪ M
M
xi
⎤ ⎧ x1 ⎫ ⎥⎪ x ⎪ ⎥⎪ 2 ⎪ x NS ,i ⎥⎪ ⎪ ⎥ ⎪ xb ⎪ = F = ⎧FNS ,i ⎫ ⎨ ⎬ i ⎥ ⎨ xb+1 ⎬ ⎩FPS ,i ⎭ ⎪ ⎥⎪ ⎪ ⎪ x ⎥ b+ 2 −kM ⎥ x PS ,i ⎥ kM ⎦⎥ ⎩⎪ xM ⎭⎪
xi
ð1Þ where x is the displacement vector and divided into xNS and xPS which represented the displacement vector of numerical and physical substructures, respectively; and the superposed dot denotes the derivative with respect to time; M, C, K are mass, damping and stiffness matrices of entire structure, respectively; MNS, CNS, KNS are the mass, damping and stiffness of numerical substructure, respectively; MPS, CPS, KPS are the mass, damping
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(a)
Bull Earthquake Eng (2016) 14:89–114
(b)
(c)
Fig. 1 Conceptual view of substructure splitting for MDOF structure system: a M-storey shear frame, b idealized M-DOF system, c split system
and stiffness of physical substructure, respectively. F is the earthquake force vector which is also divided into FNS and FPS; the subscript i denotes the ith time step. Considering the interaction between two substructures, the equations of motion of each substructure can be derived from Eq. 1 as 8 9 > > : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} > ; T1 9 8 > > = ; : |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} > T2
ð2Þ Considering the error caused by the effect of time delay or delay compensation when loading the physical substructure, the actual displacements of the shaking table, x0b ; is usually not equal to xb : Hence, xb and x_b should be replaced by x0b and x_0b ; respectively, in T1 and T2 in the right-hand side of Eq. 2. Finally, rewriting the modified Eq. 2 according to the form of Eq. 1, the equations of motion of the entire structure in RTHS can be given as ð3Þ M€ xi þ C1 x_i þ K1 xi þ C2 x_0i þ K2 x0i ¼ Fi T where x0 ¼ x01 ; . . .x0b ; . . .; x0M ; x0b is the actual displacement that the shaking table achieves, and the remainder elements in x0 are all auxiliary variables and without any physical sense; K1 ? K2 = K, and C1 ? C2 = C; K2 and C2 are the stiffness and damping matrices, respectively, related to the restoring force at the interface. It should be noted that the splitting of K and C here is different with that in Shao et al. (2011). K1, K2, C1, and C2 can be expressed as
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Bull Earthquake Eng (2016) 14:89–114 ⎡ k1 + k2 ⎢ −k 2 ⎢ ⎢ ⎢ K1 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎢ ⎡ c1 + c2 ⎢ ⎢ −c2 ⎢ ⎢ C1 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣ ⎡ 0 ⎢ 0 ⎢ ⎢ ⎢ C2 = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎢
93
−k2 k 2 + k3
− k3
− kb
kb
−kb +1
0
kb +1 + kb + 2
− kb + 2
−kM −1
kM −1 + kM −kM
−kM kM
−c2 c2 + c3
−c3
−cb
cb 0
−cb +1 cb +1 + cb + 2
−cb + 2
−cM −1
cM −1 + cM
−cM
− cM
cM
0
0 cb +1
0
−cb +1
0
0
0
0 0
0 0
0
0
kb +1 − kb +1
0 0
0
0
0 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ 0 ⎦⎥
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦⎥
0 0
⎤ ⎡ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ K =⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎣⎢ ⎦⎥
According to the control theory (Benjami and Farid 2003), the RTHS system in Fig. 1 can be simplified as a discrete-time single-loop control system, as shown in Fig. 2, where Fi(z), Pi(z), Ri(z), and Xi(z) are the z transform of the earthquake force Fi, excitation force Pi, feedback force Ri and displacement xi, respectively; G(z) is the discrete transfer function of the integration algorithm utilized in the hybrid test that relates the displacement xi to the excitation force Pi; and H(z) is the discrete transfer function that relates the feedback force Ri to the displacement xi. The dynamic performance of the discrete-time control system is determined by the closed-loop transfer function Gcl ðzÞ as Gcl ðzÞ ¼
Xi ðzÞ GðzÞ ¼ Fi ðzÞ 1 þ GðzÞHðzÞ
ð4Þ
where G(z)H(z) is the open-loop transfer function of the system. The characteristic equation of the closed-loop control system is given in a polynomial form as follows: Input Fi(z) +
∑
Pi(z)
-
Feedback Ri(z)
Numerical Substructure G(z)
Output Xi(z)
Physical Substructure H(z)
Fig. 2 Block representation of the closed-loop control system
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Qn0 ðz zi Þ ¼0 1 þ GðzÞHðzÞ ¼ 1 þ lm Qmi¼1 0 ðz pj Þ j¼1
ð5Þ
where zi and pj are the zeros and poles of the open-loop transfer function in the z-plane, respectively; m0 and n0 are positive integers, and lm is the system gain. For the emulated system, the z transform of Eq. 3 can be written as h i €i ðzÞ þ C1 X_ i ðzÞ þ K1 Xi ðzÞ ¼ Fi ðzÞ C2 X_ 0 ðzÞ þ K2 X0 ðzÞ ð6Þ MX i i Applying integration algorithms to solve Eq. 6, the corresponding characteristic equation can be obtained; and its roots (also named closed-loop poles) can be sketched. The RL plot can then be drawn by assigning lm ranging from zero to infinity with the use of MATLAB (Mathworks Inc 2006).
2.2 Numerical integration algorithms Numerous time integration algorithms have been developed for RTHS. In this paper, seven explicit integration algorithms are considered to solve numerical substructures, as follows: (1) (2) (3) (4)–(6) (7)
The CDM; The Newmark explicit method; The Chang (2002); Three subfamilies of Gui-k method with k = 2, 4, and 11.5; and The subfamily with q? = 0.2 of the explicit KR-a method.
The accuracy of these integration algorithms in RTHS has been investigated by comparing with conventional shaking table tests (e.g., Wang et al. 2015; Mosalam and Gu¨nay 2014) as well as pure numerical simulations (e.g., Nakashima et al. 1992; Kolay et al. 2015). It is noted that the value of integration parameters (k and q?) should be predetermined according to the characteristics of the entire RTHS system. However, in this paper, the varied values of structural parameters and time delays of RTHS systems are studied. In order to simplify, specific integration parameters (k = 2, 4, and 11.5, q? = 0.2) resulting in unique properties of integration algorithms are chose. The comparative presentation of these algorithms is tabulated in Table 1. The CDM, the Newmark explicit method, and the Gui-k method (k = 11.5) are conditionally stable, whereas the four other algorithms are unconditionally stable. Furthermore, the Gui-k method (k = 4) is the same as the CR algorithm. The first six algorithms possess no numerical energy dissipation. The KR-a method is a family of unconditionally stable explicit method with controllable numerical energy dissipation. The amount of numerical damping is controlled by a single parameter, q?, ranging from 0 to 1. In this paper, q? is selected as 0.2 to ensure enough numerical damping. More details on the parameters for the KR-a method can be referred to Kolay and Ricles (2014).
2.3 Conceptual illustration of the performance of delayed RTHS system In this section, variations in the stability and accuracy of RTHS systems with the time delay and integration algorithms are conceptually illustrated. A 2-DOF structure is taken as an example, as shown in Fig. 3. The upper DOF is set as the physical substructure, and the
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KR-a method (q? = 0.2)
Gui-k methods (k = 2, 4 and 11.5)
Chang (2002)
Newmark explicit
CDM
Integration algorithms
xi ¼ xi1 þ Dtx_i1 þ aDt2 x€i1 x_i ¼ x_i1 þ ax€i1
where
8 € ¼ ðI a3 Þ€ xi þ a3 x€i1 x^ > > < i x_iaf ¼ ð1 af Þx_i þ af x_i1 where > > xiaf ¼ ð1 af Þxi þ af xi1 : Fiaf ¼ ð1 af ÞFi þ af Fi1
1
8 > 1 1 2 1 1 1 > 1 > I þ DtM1 C < b1 ¼ I þ DtM C þ Dt M K 2 4 2 1 > 1 1 1 > 1 2 1 > : b2 ¼ I þ DtM C þ Dt M K 2 2 4
where a ¼ 2k½2kM þ kDtC þ 2Dt2 K M
xi ¼ xi1 þ b1 Dtx_i1 þ b2 Dt2 x€i1 Dt x_i ¼ x_i1 þ ðx€i1 þ x€i Þ 2
x_i ¼ x_i1 þ a1 x€i1 xi ¼ xi1 þ Dtx_i1 þ a2 Dt2 x€i1 Mx^€i þ Cx_iaf þ Kxiaf ¼ Fiaf
(
8 1 > < x_i ¼ ðx_iþ1 x_i1 Þ 2Dt > : x€i ¼ 1 ðxiþ1 2xi þ xi1 Þ Dt2 8 1 > < xi ¼ xi1 þ Dtx_i1 þ Dt2 x€i1 2 > : x_i ¼ x_i1 þ Dt ðx€i1 þ x€i Þ 2
Formula in time domain
Table 1 Comparative presentation of characteristics of numerical integration algorithms
None
Nonep (kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2, 4) Dt 4k=ðk 4Þ xn (k = 11.5)
Yes
No
No
No
Dt \ 2/xn
None
No
Numerical dissipation
Dt \ 2/xn
Stability condition
Bull Earthquake Eng (2016) 14:89–114 95
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Fig. 3 Schematic representation of the emulated 2-DOF structure
k2 c2
m1
k1 c1
Physical substructure Interface Numerical substructure
lower DOF is simulated as the numerical substructure. The mass, stiffness, and damping matrices are given by 0 m1 0 k1 k2 k2 c1 c2 ; C1 ¼ ; K1 ¼ ; K2 ¼ ; M¼ 0 k2 0 c2 0 m2 k2 0 0 c2 C2 ¼ c2 0 where kI ¼ mI x2I and cI ¼ 2mI fI xI (I = 1, 2); xI and fI (I = 1, 2) are the circular frequency and damping ratio of each DOF in the emulated structure, respectively. As an example, the CDM is adopted to solve Eq. 6, the characteristic equation can be expressed in the following form: 1 þ GðzÞHðzÞ ¼ 1 þ lm
hðzÞ ðn4 z4 þ n3 z3 þ n2 z2 þ n1 z1 þ n0 Þ ¼0 d4 z4 þ d3 z3 þ d2 z2 þ d1 z1 þ d0
ð7Þ
where lm = m2/m1 is the system gain that is increased; and hðzÞ is the transfer function between the measured and desired displacement. It should be noted that since the time delay in servo-hydraulic dynamics is complicated and hard to capture accurately, the pure time delay model is adopted to simulate the time delay effect and the amplitude error is neglected here. When time delay s occurs, which is assumed to be integral times of Dt, i.e., s = jDt (j = 1, 2, …), h(z) is equal to z-j. If s disappears, h(z) is reduced to be a proportional gain of 1. In the cases that s is not equal to integral times of Dt, a interpolation operation is applied to predict the actual displacement x0i to obtain h(z). The coefficients of Eq. 7 are given in Table 2. According to the theory of RL design, a general effect of the addition of open-loop poles is to lower system’s stability (Benjami and Farid 2003). For pure time delay condition, for example, hðzÞ = z-j (j = 1, 2, …), substituting the expression of hðzÞ into Eq. 7 results in additional open-loop poles, which indicates that the time delay will negatively affect stability. To investigate the system accuracy, Eq. 4 can be rewritten as Xi ðzÞ ¼
GðzÞ Fi ðzÞ 1 þ GðzÞHðzÞ
ð8Þ
The earthquake force Fi is generally predetermined; hence, Fi ðzÞ in Eq. 8 could be assumed constant. For a given structure system, GðzÞ and HðzÞ are independent on the structural properties, which can be expressed in the form of GðzÞ ¼ GðDt; zÞ; HðzÞ ¼ HðDt; s; zÞ: Considering Dt or s varying, writing Eq. 8 with the incremental form leads to dXi ðzÞ ¼ and
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Fi ðzÞ 2
½1 þ GðzÞHðzÞ
dGðzÞ
Fi ðzÞGðzÞ ½1 þ GðzÞHðzÞ2
dHðzÞ
ð9Þ
Bull Earthquake Eng (2016) 14:89–114
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Table 2 Coefficients of the characteristic equation Numerator n4
Denominator
f2 x2 Dt 2
x22
n3
Dt
n2
2Dt2 x22 2
x22
Dt
n0
f2 x2 Dt
d1 d0
f1 f2 x1 x2 Dt2 ðf1 x1 þ f2 x2 ÞDt þ 1
d3
2f2 x2 Dt
n1
1 þ f1 x1 Dt þ f2 x2 Dt þ f1 f2 x1 x2 Dt2
f2 x21 x2 Dt3 þ f1 x1 x22 Dt3 þ x21 þ x22 Dt2 2ðf1 x1 þ f2 x2 ÞDt 4
x21 x22 Dt4 2 x21 þ x22 Dt2 2f1 f2 x1 x2 Dt2 6
f2 x21 x2 Dt3 f1 x1 x22 Dt3 þ x21 þ x22 Dt2 þ 2ðf1 x1 þ f2 x2 ÞDt 4
d4 d2
þ 2f2 x2 Dt
dXi ðzÞ dGðzÞ dHðzÞ ¼ Xi ðzÞ GðzÞ½1 þ GðzÞHðzÞ ½1 þ GðzÞHðzÞ
ð10Þ
The term on the left-hand side of Eq. 10 represents the rate of change of displacement. Eq. 10 indicates that the variation of Dt or s in G(z) or H(z) causes error in the result of Xi ðzÞ; which leads to accuracy variation of the RTHS system. Hence, it can be preliminarily deduced that the performance of integration algorithms in the RTHS system is closely related to the integration time step and time delay, which should be investigated in detail.
3 Stability analysis 3.1 Stability analysis considering pure time delay condition This section investigates the stability performance of a 2-DOF RTHS system with pure time delay condition. The emulated 2-DOF structure is the same as that in Fig. 3. When conducting the RL analysis, lm, defined as the mass ratio between the physical and numerical substructures, is set as the system gain. Based on this definition, lm is ranging from zero to infinity. For convenience, the mass proportional factor qm = lm/(1 ? lm), the mass proportion of the physical substructure in the entire structure (varying from 0 to 1), is further defined to be the index to evaluate the stability state of RTHS systems. The value of qm at the critical stable state is represented by qcr m : It can be indicated that if the practical mass ratio qm is larger than qcr m ; the RTHS will be induced to be unstable. In addition, qcr m ¼ 1 represents that lm is infinity and the emulated system is absolutely stable. Firstly, the effects of structural properties and time parameters on the stability of the 2-DOF system are investigated. Each DOF has the same structural properties as follows: damping ratio f1 = f2 = f, and circular frequency x1 = x2 = xn, where xn varies from 0 to 314 rad/s. Figure 4 shows the stability results of the emulated system with f = 0.05, Dt = 0.01 s, and s = 0–0.02 s. In Fig. 4a, the stability of the system is reduced to that of the integration algorithm itself when the time delay s is zero. The stability limitation qcr m is identical to 1 for xn = 0–314 rad/s when the Chang (2002), the Gui-k methods (k = 2 and 4), and the KR-a method are used, which means that these four integration algorithms are unconditionally stable. However, the stability limitation qcr m is lower than 1 when the CDM, the Newmark explicit method, and the Gui-k method (k = 11.5) are performed, which shows their conditional stability. Moreover, the stability limitation qcr m based on the Gui-k method (k = 11.5) is significantly higher than those based on the CDM and the Newmark explicit method, indicating that the former algorithm is more stable.
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(a)
(b)
1.5
1.5
Chang (2002) KRand 4
CDM / Newmark explicit Chang (2002) , 4 and 11.5
cr
CDM / Newmark explicit
m
1.0
m
cr
1.0
KR-
0.5
0.5
0.0 0
200
0.0
300
(rad/s)
n
0
100
(d)
1.5
200
300
(rad/s)
n
1.5
1.0
1.0 cr
m
CDM / Newmark explicit Chang (2002) , 4 and 11.5
0.5
0.5
KR-
KR-
0.0 0
100
200
(rad/s)
n
CDM / Newmark explicit Chang (2002) , 4 and 11.5
m
(c)
cr
100
300
0.0 0
100
200
300
(rad/s) n
Fig. 4 Stability limitations for different integration algorithms with Dt = 0.01 s, f = 0.05: a s = 0, b s = 0.005 s, c s = 0.01 s, d s = 0.02 s
The stability limitations decrease sharply with the time delay s increasing from 0.005 to 0.02 s, as shown in Fig. 4b–d, which indicates that the time delay is a decisive factor of delay-dependent stability. With s at 0.005 s, as shown in Fig. 4b, the stability limitations of the system with the seven integration algorithms decrease rapidly and become lower than 1 regardless of the value of xn. This observation means that the emulated system is conditionally stable no matter which integration algorithm is applied because of the time delay. The emulated system with the six integration algorithms without numerical damping possesses similar stability limitations with the variation of xn. The KR-a method achieves almost the same stability limitations as those six integration algorithms in the low-frequency range (approximately 0–50 rad/s), but larger stability limitations in the moderateto high-frequency range (approximately 50–314 rad/s). Similar results can also be observed in Fig. 4c, d. Figure 5 shows the effects of f on the stability limitations. Two cases with f = 0.01 and 0.10 are investigated. It is shown that qcr m increases when a higher damping ratio is presented, indicating that the structural damping improves the stability when time delay occurs. In addition, the stability limitation based on the KR-a method is always greater than that based on the six other integration algorithms in the moderate- to high-frequency range regardless of the value of f. It is also observed that the stability limitations of the six integration algorithms without numerical damping (f = 0.10) in Fig. 5b remain lower than the stability limitation of the KR-a method with f = 0.05 in Fig. 4d, particularly in the high-frequency range. To confirm this observation, an additional case with Gui-k method (k = 4) and a larger
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(b)
1.5
CDM / Newmark explicit Chang (2002) , 4 and 11.5
0.5
1.0 cr
m
cr
1.0
0.0 100
200
CDM / Newmark explicit Chang (2002) , 4 and 11.5 KR-
0.5
KR-
0
1.5
m
(a)
99
0.0
300
0
100
200
300
(rad/s) n
(rad/s) n
Fig. 5 Stability limitations considering different damping ratios (Dt = 0.01 s, s = 0.02 s): a f = 0.01, b f = 0.10
1.5
,
= 0.40
m
cr
1.0
0.5
KR- ,
= 0.01
(244,0.1) 0.0 0
100
200
300
(rad/s)
n
Fig. 6 Comparison of stability limitations between the KR-a method and Gui-k method (k = 4) with different damping ratios (Dt = 0.01 s, s = 0.02 s)
inherent damping of f = 0.40 is performed. The result is shown in Fig. 6, plotted against the stability limitation of the KR-a method with f = 0.01 from Fig. 4d. It is shown that the two cases have an intersection point when xn & 244 rad/s, obtaining the same stability limitations. This observation proves that the KR-a method possesses large amount of numerical damping into the system, especially for structures with high natural frequency. This finding agrees well with the conclusion in Kolay and Ricles (2014) that choosing an appropriate value of q? can adequately damp out spurious high modes. Thus, the numerical energy dissipation of integration algorithms is important to improve the stability performance of RTHS systems. Previous analysis showed that all the seven integration algorithms provide similar stability limitations in the low-frequency range. Hence, detailed RL analysis of special structures with low natural frequency is carried out to further investigate the instability mechanism. For a 2-DOF emulated system without time delay, two pairs of conjugate branches in an RL plot, corresponding to the two order modes of the structure, are called inherent modes (Chi et al. 2010; Zhu et al. 2015); in time delay condition, the inherent modes will distort and the added modes caused by the time delay will be introduced in RL plot. The stability condition of the system can be determined by analyzing the relationship
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between inherent modes and the unit circle in the z-plane. Moreover, the accuracy of the system also can be qualitatively characterized by evaluating the degree of distortion of the inherent modes, which is discussed in Sect. 4.2. Parameters of the emulated system are selected as x1 = 12.57 rad/s, f1 = 0.05, x2 = 18.85 rad/s and f2 = 0.03. The following three cases are analyzed: Case A-1: Dt = 0.01 s, s = 0; Case A-2: Dt = 0.01 s, s = 0.01 s; Case A-3: Dt = 0.001 s, s = 0.01 s where Cases A-1 and A-2 are performed with the same Dt but different s to illustrate how time delay affects instability; Cases A-2 and A-3 are compared to verify the effect of Dt on stability. For Case A-1, the RL plots are shown in Fig. 7. The inherent modes with the seven integration algorithms show significantly different shapes, reflecting the difference in the stability property of the seven integration algorithms. Specifically, the inherent mode 2 crosses from the inside to the outside of the unit circle in Fig. 7a, f, while remains in the inside the unit circle in Fig. 7b–e. This result indicates that the CDM, the Newmark explicit method, and the Gui-k method (k = 11.5) are conditionally stable, whereas the four other algorithms are unconditionally stable. Figure 8 presents the RL plots for Case A-2. All the second inherent modes distort severely and cross from the inside to the outside of the unit circle, thereby inducing the system instability. The stability limitations, qcr m ; and corresponding instability frequencies, xcr m ; in the crossover points are similar, with values of about 0.36 and 25 rad/s, respectively. This observation means that the stability of the delayed RTHS systems appears nearly the same, although seven integration algorithms with different stability characteristics are adopted. To evaluate the effect of Dt on the stability limitation, Case A-3 (Dt = 0.001 s, s = 0.01 s) is further investigated. Corresponding stability results are listed in Table 3, cr compared with the two other cases. With a small Dt = 0.001 s, all qcr m and xm for the seven integration algorithms are the same. As the value of time delay in Case A-3 is cr identical to that in Case A-2, both qcr m and xm show little difference in these two cases, which indicates insignificant effect of Dt on stability performance. The above results demonstrate that the delayed RTHS system could not be absolutely stable no matter what kind of integration algorithm is applied. The six integration algorithms without numerical damping provide almost the same stability limitations, whereas the KR-a method with controllable numerical dissipation provides larger stability limitations in the moderate to high-frequency range, indicating the introduced numerical damping strengthens the stability performance of delayed RTHS systems.
3.2 Stability analysis considering delay compensation condition The aforementioned results confirm that the existence of the time delay significantly destabilizes RTHS systems, and compensation methods should be utilized to overcome possible instability. This section conducts the stability analysis of MDOF RTHS systems when considering delay compensation. Since this investigation focuses on the comparison of the stability performance for different integration algorithms, only the 3rd-order polynomial method, a commonly-used method for delay compensation, is adopted herein.
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Imaginary Axis
1 0.5
0.74
0.6
0.42
(b)
0.22
0.84
ρ m = 0.991, ζ m = 0
0.91
ωm = 314.16 rad s
1.4
1.2
1
0.8
0.6
0.4
First inherent mode
Second inherent mode
0.2
0.99
-0.5
0.96
0.5
0.42
0.22
0.96 0.99 1.4
0
1.2
1
0.8
0.6
0.4
0.2
0.99 0.96
-0.5
0.91
-1
0.6
0.84 0.91
0.96 0.99
0
0.74
1
Imaginary Axis
(a)
101
0.91
Unit circle
0.84 0.74
-1.5
0.6
-1
0.42
0
0.5
1
0.84
-1
0.22
-0.5
1.5
0.74
-1.5
0.6
0.42
-1
0.22
-0.5
0
(c) 1
0.74
0.6
0.42
(d)
0.22
1
0.84
Imaginary Axis
Imaginary Axis
0.96 0.99
0
1.4
1.2
1
0.8
0.6
0.4
0.2
0.99
-0.5
0.96
0.5
0.74
0.6
0.42
0.74
-1.5
1.4
1.2
1
0.8
0.6
0.4
0.6
-1
0.42
-0.5 -1
0.22
-0.5
0
0.5
1
0.74
0.6
0.42
0.74
(f)
0.22
1
Imaginary Axis
Imaginary Axis
0.96
1.2
1
0.8
0.6
0.4
0.2
0.99
-0.5
0.96
0.5
0.6
-1
0.42
0
0.5
0.74
0.6
0.42
0.22
0.84
ρ m = 0.994, ζ m = 0
0.91
ωm = 314.16 rad s
0.96
1.4
1.2
1
0.8
0.6
0.4
0.2
0.99
-0.5
0.96 0.91
0.84
-1.5
0.22
-0.5
0.99
0
0.91
-1
1.5
0.84
Real Axis
0.84
1.4
1
0.96
-1.5
1.5
0.99
0
0.2
0.99
0.91
0.5
1.5
0.96
Real Axis 1
1
0.22
0.84
0.91
0.84
(e)
1.5
0.99
0
0.91
-1
1
0.91
0.91
0.5
0.5
Real Axis
Real Axis
0.74
0.6
-1
0.42
-0.5
-1
0.22
0
Real Axis
0.5
1
1.5
0.84
-1.5
0.74
0.6
-1
0.42
-0.5
0.22
0
0.5
Real Axis
Fig. 7 RL plots for different integration algorithms for Case A-1: a CDM/Newmark explicit, b Chang (2002), c KR-a method, d k = 2, e k = 4, f k = 11.5
The effect of delay compensation method on the stability of the 2-DOF structures with structural properties of f1 = f2 = f = 0.05, and x1 = x2 = xn = 0–314 rad/s is investigated. Figure 9 shows the results of the stability limitations. In Fig. 9a, when s = 0.01 s, the stability limitations of the RTHS system with the six integration algorithms without numerical damping show slight difference in the moderatefrequency range (approximately 50–100 rad/s). The stability limitation values for the unconditionally stable algorithms [Chang (2002), Gui-k methods (k = 2 and 4)] are generally greater than those for the conditionally stable algorithms [CDM, Newmark explicit method and Gui-k method (k = 11.5)]. When s = 0.02 s, the difference in the
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Imaginary Axis
1
0.74
0.6
0.42
ρ = 0.359, ζ cr m
0.91
0.5
(b)
0.22
0.84
cr m
1
=0
ωmcr = 24.99 rad s
0.96
First inherent mode
0.99
0 1.4
1.2
1
0.8
0.6
0.4
0.99
-0.5
Second inherent mode
0.2
Added mode
0.96
0.5
0.74
0.42
-1
0.22
-0.5
0
0.5
1
1.5
0.74
0.6
0.42
ρ = 0.363, ζ
=0
ωmcr = 24.61rad s
0.96
1.2
1
0.8
0.6
0.4
0.2
0.99 0.96
0.5
0.74
-1
0.42
0
0.5
1
-0.5
0.6
-1
0.42
0.22
-0.5
0
0.5
1
1.5
0.74
0.6
0.42
1
1.5
1
1.5
0.22
0.84
ρ mcr = 0.355, ζ mcr = 0 ωmcr = 24.53 rad s
0.96
1.2
1
0.8
0.6
0.4
0.2
0.96
0.74
0.6
0.42
cr m
=0
ω = 24.72 rad s cr m
0.96
1.2
1
0.8
0.6
0.4
0.2
0.99 0.96
0.42
0.22
-0.5
0.74
0.6
0.42
0.84
0
0.5
0.5
0.22
ρ mcr = 0.355, ζ mcr = 0
0.91
ωmcr = 24.84 rad s
0.96 0.99
0 1.4
1.2
1
0.8
0.6
0.4
0.2
0.99
-0.5
0.91
0.96 0.91
0.84 0.74
-1.5
0.6
-1
Real Axis 1
ρ = 0.355, ζ
0.74
(f)
0.22
cr m
0.84
-1.5
1.5
0.99
-1
0.74
0.99
-1
-0.5
0.91
-0.5
0.2
0.99
0.22
Imaginary Axis
Imaginary Axis
0.6
0.84
0 1.4
0.4
0.91
Real Axis
0.5
0.6
0.84
0 1.4
0.84
-1.5
1
0.8
0.91
0.91
(e)
1
Real Axis 1
cr m
0.99
-1
ωmcr = 24.80 rad s
0.96
(d)
0.22
cr m
0.91
-0.5
1.2
-1.5
Imaginary Axis
Imaginary Axis
0.6
-1
0.84
0 1.4
ρ mcr = 0.360, ζ mcr = 0
0.91
Real Axis
0.5
0.22
0.99
-0.5
-1.5
1
0.42
0.99
0.84
(c)
0.6
0.96
0 1.4
0.91
-1
0.74 0.84 0.91
Imaginary Axis
(a)
0.6
-1
0.42
-0.5
-1
0.22
0
Real Axis
0.5
1
1.5
0.84
-1.5
0.74
0.6
-1
0.42
-0.5
0.22
0
0.5
Real Axis
Fig. 8 RL plot for different integration algorithms for Case A-2: a CDM/Newmark explicit, b Chang (2002), c KR-a method, d k = 2, e k = 4, f k = 11.5
stability limitation among these six integration algorithms disappears, as shown in Fig. 9b. For the KR-a method with numerical damping, the corresponding stability limitation is slightly higher than that obtained from the six other integration algorithms in the moderateto high-frequency range (50–314 rad/s) in both Fig. 9a, b. In addition, comparing Fig. 9a with Fig. 4c (or Fig. 9b with Fig. 4d), the 3rd-order polynomial method significantly increases the stability limitation of RTHS systems with all the seven integration algorithms in the low- to moderate-frequency range, but shows slightly negative effects on that in the high-frequency range. In this range, the 3rd-order polynomial method particularly decreases the stability limitation of the RTHS system with the KR-a method. To show the instability mechanism of the emulated system when the 3rd-order polynomial method is employed to compensate for the time delay, Case A-2 with parameters of x1 = 12.57 rad/s, f1 = 0.05, x2 = 18.85 rad/s, and f2 = 0.03 is performed again. The RL
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Table 3 Stability limitations for different integration algorithms in Cases A-1 to A-3 Cases
CDM/Newmark explicit
Chang (2002)
KR-a method
1 –
Gui-k method k=2
k=4
1
1
1
–
–
–
k = 11.5
A-1 qcr m
0.991
xcr m (rad/s)
314.16
0.994 314.16
A-2 qcr m
0.359
xcr m (rad/s)
24.99
0.360
0.363
0.355
0.355
24.80
24.61
24.53
24.72
0.355 24.84
A-3 qcr m
0.364
xcr m (rad/s)
(a)
25.03
0.364
0.364
0.364
0.364
25.02
25.02
25.02
25.03
(b)
1.5
CDM / Newmark explicit Chang (2002) KRm
cr
m
cr
CDM / Newmark explicit Chang (2002) KR-
1.0
1.0
0.5
0.5
0.0
1.5
0.364 25.03
0
100
200
300
0.0
0
(rad/s) n
100
200
300
(rad/s) n
Fig. 9 Stability limitations considering the 3rd-order polynomial compensation (f = 0.05): a Dt = 0.01 s, s = 0.01 s, b Dt = 0.01 s, s = 0.02 s
plots for the seven integration algorithms are presented in Fig. 10. All the modes and locations of the stability critical point of the RL plots are nearly similar. The two inherent modes of the emulated system remain inside the unit circle. However, the instability is caused by an added mode crossing the boundary of the unit circle. All qcr m are about 0.90, which is larger than that in pure time delay condition. Table 4 presents the results.
4 Accuracy analysis The accuracy of integration algorithms has been extensively investigated, while the accuracy of RTHS systems with the application of integration algorithms has been rarely discussed. Considering the disturbance of time delay and delay compensation in RTHS systems, the accuracy performance should be reassessed.
4.1 Analysis on the basis of numerical simulation Generally, the accuracy of an integration algorithm is characterized by analyzing the amplitude decay and period error on the basis of the solution of a free-vibration problem of
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0.74
1
0.6
0.42
(b)
0.22
0.84
Imaginary Axis
0.91
0.5
cr m
cr m
0.905,
cr m
175.50 rad s
0.96 0.99
0
1.4
1.2
1
0.8
0.6
0.4
0.2
0.99 0.96
-0.5
1
0
0.5
0.74
0.6
0.42
1
-1
-0.5
0
0.74
0.6
0.42
1
cr m
0.917,
cr m
-0.5
1.2
159.92 rad s
1
0.8
0.6
0.4
0.2
0.99
-0.5
0.96
0.74
0.6
-1
0.42
0.5
(e)
-0.5
0
0.74
0.6
0.42
1
cr m
0.892,
cr m
-0.5
0.6
0.4
0.2
0.74
0.6
0.42
-1
0.22
-0.5
0
0.5
1
1.5
0.74
0.6
0.42
0.22
0.84
cr m
0.900,
cr m
cr m
171.12 rad s
0
0.96
1.4
1.2
1
0.8
0.6
0.4
0.2
0.96
1.2
1
0.8
0.6
0.4
0.2
0.96
0.5
0.74
0.6
-1
0.42
0.22
-0.5
0
0.5
1
1.5
0.74
0.6
0.42
0.22
0.84
cr m
0.887,
0.91
cr m
cr m
171.19 rad s
0
0.96 0.99
0
1.4
1.2
1
0.8
0.6
0.4
0.2
0.99
-0.5
0.91
0.96 0.91
0.84 0.74
-1.5
0.84
Real Axis
171.16 rad s
0.99
-1
0.8
0.99
1
0
0.99
-0.5
1
0.84
(f) cr m
0.96
1.4
0
0.96
-1.5
1.5
Imaginary Axis
Imaginary Axis
0.5
0.22
0.84
0.91
0
cr m
0.99
0
-1
0.22
Real Axis
0.5
175.75 rad s
0.91
0.84
-1.5
1
1.2
0.91
0.91
-1
0.917,
cr m
Real Axis 1
0
0.96
1.4
1.4
(d) cr m
0.99
0
cr m
0.96
-1.5
1.5
Imaginary Axis
Imaginary Axis
0.5
0.22
0.84
0.91
0.5
0.22
0.99
-1
0.22
Real Axis
(c)
0.42
0.91
0.84
-1.5
0.6
0.99
0
0.91
-1
0.74 0.84
0.91
Imaginary Axis
(a)
0.6
-1
0.42
-0.5
-1
0.22
0
Real Axis
0.5
1
1.5
0.84
-1.5
0.74
0.6
-1
0.42
-0.5
0.22
0
0.5
1
1.5
Real Axis
Fig. 10 RL plots for different integration algorithms for Case A-2 with the 3rd-order polynomial compensation: a CDM/Newmark explicit, b Chang (2002), c KR-a method; d k = 2, e k = 4; f k = 11.5
SDOF structure. However, the factor of time delay or delay compensation is difficult to be considered from a mathematical point of view in analyzing the accuracy of RTHS systems with the application of integration algorithms. Hence, in this section, numerical simulations in MATLAB/Simulink are presented to compare the accuracy of RTHS systems as the seven integration algorithms are applied, respectively. The time delay s is assumed as integral times of Dt. A 2-DOF structure in Fig. 3 is taken as an example, with the following parameters: m1 = 15.7 kg, m2 = 1 kg, f1 = f2 = 0.05, and x1 = x2 = 62.8 rad/s. The natural frequency of the entire structure is 8.82 and 11.34 Hz. Such values for m1 and m2 are chosen to ensure stability. The flowchart of the numerical simulation is presented in
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Table 4 Stability limitations based on the 3rd-order polynomial method for delay compensation CDM/Newmark explicit qcr m
Chang (2002)
0.905
xcr m (rad/s)
Earthquake input
0.917
175.50
175.75
Numerical substructure m1 x1 c1 x1 k1 x1 m1u g T
Time delay
KR-a method 0.917 159.92
Compensator
Gui-k method k=2 0.900 171.12
k=4 0.892 171.16
k = 11.5 0.887 171.19
Physical substructure m2 x2 c2 x2 k2 x2 m2u g c2 x1 k 2 x1
Measured force T =c2 ( x2 x1 ) k2 ( x2 x1 )
Fig. 11 Flowchart of numerical simulation
Fig. 11. The physical substructure is also virtually simulated and the time delay is modeled by a time-delay block in Simulink. Five cases are performed as follows: Case B-1: Dt = 0.001 s, s = 0; Case B-2: Dt = 0.01 s, s = 0; Case B-3: Dt = 0.001 s, s = 0.01 s; Case B-4: Dt = 0.01 s, s = 0.01 s; Case B-5: Dt = 0.01 s, s = 0.01 s, the 3rd-order polynomial method for delay compensation. Among these five cases, Cases B-1 and B-2 are conducted with different integration time steps Dt and without time delay to compare the effect of Dt on accuracy. The effect of s on accuracy is determined by comparing Case B-3 with Case B-1 (or Case B-4 with Case B-2). Case B-5 is conducted to show the effect of delay compensation on accuracy. Fig 12 graphically compares the displacement responses of the five cases under random excitation. Given that the integration time step Dt = 0.001 s in Case B-1 is far smaller than the minimum structural vibration period (1/11.34 s & 0.09 s), the response of the emulated structure by using the Gui-k method (k = 11.5) could serve as the ‘‘numerical exact’’ solution. In Case B-1, each integration algorithm achieves the same level of accuracy, as shown in Fig. 12a. With a relatively large Dt of 0.01 s in Case B-2, the loss of accuracy for the seven integration algorithms is observed, as illustrated in Fig. 12b. The Gui-k method (k = 2) presents the worst accuracy, whereas the Gui-k method (k = 11.5) presents the best accuracy in this case. Under the time delay condition of s = 0.01 s, for Case B-3 with Dt = 0.001 s, each integration algorithm still obtains the same level of accuracy, but shows a slight difference with the exact solution, as shown in Fig. 12c. When Dt increases to 0.01 s in Case B-4, the accuracy of RTHS systems with the seven integration algorithms becomes different again, as shown in Fig. 12d. For Case B-5 with the 3rd-order polynomial method for delay compensation, as shown in Fig. 12e, the responses based on the seven
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(b)
6
CDM / Newmark explicit Chang (2002) KR- method
4
=2 =4 = 11.5 / Numerical exact
Displacement (mm)
Displacement (mm)
(a)
Bull Earthquake Eng (2016) 14:89–114
2 0 -2
-1
-4
-2
-6
-3
-8
-4 2.10
0
1
2
2.15
3
2.20
CDM / Newmark explicit Chang (2002) KR- method
4
0 -2
-1
-4
-2
-6
-3
5
-4 2.10
0
1
2
4
=2 =4 = 11.5
2 0 -2
-1
-4
-2
-6
-3
-8
-4 2.10
0
1
2
3
6
CDM / Newmark explicit Chang (2002) KR- method
Numerical exact
Displacement (mm)
Displacement (mm)
CDM / Newmark explicit Chang (2002) KR- method
2.15
2.20
4
4
Displacement (mm)
6
=2 =4 = 11.5
2.25
5
Numerical exact
0 -2
-1
-4
-2
-6
-3
-8
2.25
5
-4 2.10
0
1
2
3
2.15
2.20
4
2.25
5
Time (s) CDM / Newmark explicit Chang (2002) KR- method
4
2.20
4
2
Time (s)
(e)
2.15
3
Time (s)
(d)
6
Numerical exact
2
Time (s)
(c)
=2 =4 = 11.5
-8
2.25
4
6
=2 =4 = 11.5
Numerical exact
2 0 -2
-1
-4
-2
-6
-3
-8
-4 2.10
0
1
2
3
2.15
2.20
4
2.25
5
Time (s)
Fig. 12 Accuracy comparison on the basis of numerical simulations: a Case B-1 (Dt = 0.001 s, s = 0), b Case B-2 (Dt = 0.01 s, s = 0), c Case B-3 (Dt = 0.001 s, s = 0.01 s), d Case B-4 (Dt = 0.01 s, s = 0.01 s), e Case B-5 (Dt = 0.01 s, s = 0.01 s, delay compensation)
integration algorithms appear closer to the exact solution, although slight amplitude amplification occurs.
4.2 Analysis on the basis of RL technique This section conducts the accuracy analysis on the basis of the RL technique. As mentioned in Sect. 3.1, the shape of RL not only demonstrates the stability behavior, but also indirectly reveals the accuracy characteristics of RTHS systems. The 2-DOF structure with the same parameters of f1 = f2 = 0.05, and x1 = x2 = 62.8 rad/s in Sect. 4.1 is taken as the example again. The mass ratio, lm = m2/m1, is used as the system gain when conducting RL analysis. The inherent mode 1 is plotted only for comparison because it corresponds to the first-order mode of the emulated structure (Zhu et al. 2015).
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As mentioned in Sect. 2.1, each point in an RL plot represents a closed-loop pole corresponding to a special value of lm; therefore, the RL plots in this section will contain the closed-loop poles of the emulated structure in Sect. 4.1 due to the fixed lm = m2/m1 = 1/15.7. As lm = 1/15.7 is very small, the position of its corresponding closedloop poles in RL will be extremely closed to the open-loop poles with lm = 0. Hence, the accuracy of the emulated system in Sect. 4.1 can also be reflected by focusing on the difference of the section of trajectory which approaches to the open-loop poles. Corresponding RL plots are shown in Fig. 13. For Case B-1, all the first inherent modes have the same shape, as presented in Fig. 13a. This finding indicates that the seven integration algorithms provide the same solution in this condition, which agrees well with the numerical results in Fig. 12a. With the increase of Dt in Case B-2, however, the shapes of the first inherent modes appear to be different among one another, as shown in Fig. 13b, which indicates an appreciable difference in the accuracy among the seven integration algorithms for RTHSs. The trajectories, closed to the positions of the open-loop poles (labeled with ‘‘X’’ in Fig. 13), show the largest difference, which demonstrates significant difference in the accuracy of the emulated system with small lm. This finding matches well with the numerical results in Fig. 12b. For Case B-3, all the first inherent modes corresponding to the seven integration algorithms in Fig. 13c coincide well with one other, but distort significantly when compared with those in Fig. 13a. This observation demonstrates the negative effect of time delay on the accuracy of RTHS systems and matches with the conclusion of the numerical results in Fig. 12c. For Case B-4, as the time delay exists, the first inherent modes in Fig. 13d distort and reduce to accuracy falling, which can also be verified by the numerical results in Fig. 12d. Finally, for Case B-5 shown in Fig. 13e, the distorted modes for the seven integration algorithms in Case B-4 are corrected with the delay compensation and exhibit almost the same shape as that in Fig. 13b. This observation shows that the accuracy can be improved by delay compensation, which is well consistent with the results of the numerical simulations in Fig. 13e. The analysis above provides a general understanding of accuracy of delayed RTHS system. In fact, the period elongation (PE) and equivalent damping ratio (feq) can be obtained based on RL plots to quantitatively evaluate the accuracy of delayed RTHS. In this way, effects of time delay or delay compensation methods may be straightforwardly deal with. For the special case in this section with lm = 1/15.7, the PE and feq are calculated on the basis of the root locus plots in Fig. 13 using the following equations: 8 lnðr2 þ e2 Þ > > < feq ¼ 2tan1 ðe=rÞ ð11Þ x0 D t > > : PE ¼ 1 tan1 ðe=rÞ where r and e are the real and imaginary coordinates of the poles, respectively; x0 is the circular frequency of structures. Since the 2-DOF structure is studied in this section, x0 is referred to the first-order circular frequency of the structure which results to be x0 = 55.39 rad/s. Figure 14 presents the PE and equivalent damping ratio feq. It can be seen from Fig. 14a that the PE values in Case B-1 are almost the same among the seven algorithms because of quite small Dt. However, in Case B-2, the PE shows greatly difference: KR-a method has the largest PE value; Gui-k method (k = 4) and Chang (2002) have the same PE value; Gui-k method (k = 11.5) has the smallest absolute PE value; and the CDM/Newmark
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0.08 CDM/Newmark explicit Chang (2002) KR- method =2 =4 =11.5
Imaginary Axis
0.04 0.02 0.00
0.064 0.062 0.060
-0.04
-0.08
0.2 0.60
0.0
0.56
-0.2
0.52 0.48
0.056 0.994
0.995
-0.4
0.996
1.000
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CDM/Newmark explicit Chang (2002) KR- method =2 =4 =11.5
0.00
1.0
0.062 0.060
-0.08
0.995
1.000
1.005
0.0
1.5
0.60 0.56
-0.2
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1.010
-0.6 0.7
0.76 0.78 0.80 0.82 0.84
Unit circle
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0.9
Real Axis
(e)
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0.2
-0.4
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1.3
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1
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1.1
CDM/Newmark explicit Chang (2002)
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Real Axis
(d)
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0.02
-0.6 0.7
1.010
Real Axis
0.04
1
0.76 0.78 0.80 0.82 0.84
Unit circle
Unit circle
0.995
0.06
Imaginary Axis
KR- method
0.058
-0.06
(c)
0.4
0.066
-0.02
0.6 CDM/Newmark explicit Chang (2002)
Imaginary Axis
0.06
Imaginary Axis
(a)
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1.0
1.1
1.2
1.3
1.4
1.5
Real Axis
0.6 CDM/Newmark explicit Chang (2002)
Imaginary Axis
0.4
KR- method
0.2 0.60
0.0
0.56
-0.2
0.52 0.48
-0.4
0.76 0.78 0.80 0.82 0.84
Unit circle
-0.6 0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
Real Axis Fig. 13 Accuracy comparison of RTHS systems with the seven integration algorithms on the basis of RL plots: a Case B-1 (Dt = 0.001 s, s = 0), b Case B-2 (Dt = 0.01 s, s = 0), c Case B-3 (Dt = 0.001 s, s = 0.01 s), d Case B-4 (Dt = 0.01 s, s = 0.01 s), e Case B-5 (Dt = 0.01 s, s = 0.01 s, delay compensation)
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(b) 0.010
Case B-1
Case B-3
PE
0.005 0.000 0.10
Case B-2
Case B-4
Case B-5
0.05 0.00 CDM/
Chang
KR
=2
=4
=11.5
quivelent damping ratio
(a)
eq
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0.15
Case B-1 Case B-3 Case B-5
0.10
Case B-2 Case B-4
0.05
0.00
CDM/
Chang
KR
=2
=4
=11.5
Newmark explicit
Newmark explicit
Integration algorithms
Integration algorithms
Fig. 14 Accuracy estimation of RTHS systems with the seven integration algorithms (lm = 1/15.7): a PE, b equivalent damping ratio feq
explicit have negative PE value. These results agree well with the accuracy results in Gui et al. (2014) and Kolay and Ricles (2014). When time delay occurs in Cases B-3 and B-4, the PE values are larger than those in Cases B-1 and B-2, respectively. With delay compensation in Case B-5, the PE values show slight decrease, which indicates the positive effect of the delay compensation method. In Fig. 14b, when Dt equals to 0.001 s in Cases B-1 and B-3, the feq values are almost the same for all the seven integration algorithms. When Dt is enlarged to 0.01 s in Cases B-2, B-4 and B-5, the feq values are almost the same for all the integration algorithms except for KR-a method; it is obvious that the KR-a method has the largest feq values in these three cases because of its numerical damping effect. In addition, for Cases B-3 and B-4 with pure time delay, the feq values generally increase. This increase is possible because the feq presented here corresponds to the firstorder frequency of the 2-DOF structure, which has been observed in Zhu et al. (2015). In addition, the feq values approach to those in Case B-2 when the delay compensation method is applied. In sum, the results on the basis of RL plots analysis agree well with the results of numerical simulations. It demonstrates that both the accuracy characteristic of the applied integration algorithm itself and the time delay negatively affects on the solution accuracy of RTHS systems. The former is the decisive factor. In addition, the delay compensation method also improves the accuracy.
5 RTHS of a two-storey shear frame In this section, several RTHSs are carried out to experimentally validate the theoretical stability and accuracy conclusions. The RTHS system with dual target computers (Zhu et al. 2014) is applied for experimentation, as shown in Fig. 15. The system comprises three components: the distributed real-time calculation system, the shaking table loading system, and the data acquisition and transmission system. The MATLAB/xPC-Target toolbox (Mathworks Inc 2006) is applied as the solution in the distributed real-time calculation system. The shaking table loading system consists of the MTS 469D digital controller, two MTS shaking tables, and an oil source. The data acquisition and transmission system uses PXI and LabVIEW to construct the data acquisition platform. SCRAMNet cards are also utilized to ensure real-time data transmission.
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Distributed Real-Time Calculation System
Shaking Table Loading System
Host Computer
Physical Substructure
TCP/IP
RS232
Controller Reference
Shaking Shaking Table Table #1 #1
Shaking Table #2 Target Computer 2
Target Computer 1 SCRAMNet Card 1
SCRAMNet Card 2
Optical Fiber
SCRAMNet Card 3
Feedback
Optical Fiber
Data Acquisition and Transmission System
Data Acquisition System
Fig. 15 Outline of the RTHS system with dual target computers
In the following RTHSs, the integration time step is selected as Dt = 1/2048 s, and the time delay of the servo hydraulic system is approximately s = 22/2048 s. Three subfamilies of the Gui-k method with k = 2, 4 and 11.5 are chosen to analyze the numerical substructure and the 3rd-order polynomial method is applied to compensate for the time delay. The displacement control is used to drive the shaking tables. The investigated structure system is a two-storey shear frame, as shown in Fig. 16. The upper storey is served as the physical substructure and the lower storey is served as the numerical substructure. The mass, natural frequency and damping ratio of the physical substructure are m2 = 5.28 kg, f2 = 6.2 Hz, and f2 = 0.042, respectively. For the numerical substructure, the natural frequency f1 and damping ratio f1 are selected as f1 = f2, f1 = f2, respectively. The mass m1 is determined according to the theoretical mass stability limitation qcr m : The theoretical stability limitations for Cases C-1 to C-3 are listed in Table 5. Corresponding to Cases C-1 to C-3 in Table 5, six tests (defined as Tests 1–6 in Table 6) are conducted to demonstrate the real stability performance. The real mass ratio qm is selected as 0.575 for Tests 1–3 and 0.667 for Tests 4–6. The external excitation is a sinusoidal ground motion with a frequency of 2.5 Hz and amplitude of 0.15 g. To compare the accuracy of the RTHS results, the numerical exact solutions of the dynamic response are obtained by numerical simulations of the entire structure with Gui-k method (k = 11.5) with the same integration time step and without any time delay. Given that the instability of acceleration is always faster than that of displacement during testing, only acceleration response results at the interface of physical and numerical substructures are presented. Figure 17 shows the dynamic responses in time domain for Tests 1–3. All the acceleration responses remain stable during the entire testing process, as shown in Fig. 17a. A close-up view in Fig. 17b shows that the acceleration response in Test 3 with the Gui-k method (k = 11.5) exhibits the best accuracy, compared with the numerical exact result. This observation can also be verified by the Fourier spectrum of the acceleration responses and their partial enlargement in Fig. 18. Figure 19 shows the dynamic responses for Tests 4–6. In Fig. 19a, the acceleration responses in the three tests diverge rapidly at the beginning of the tests, advancing beyond the predetermined acceleration limit of the controller and causing the shaking table to stop. From the Fourier spectrum of the acceleration responses in Fig. 19b, the unstable frequencies for Tests 4–6 are approximately 130.50 rad/s (20.77 Hz), 130.56 rad/s
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(a)
111
(b)
m2
m2 k2
Measured force
c2
(c)
k2 c2 Command displacement
Shaking table
m1 k1
Physical substructure
m1
c1 k1 c1
Numerical substructure
Fig. 16 Test setup: a entire structure, b substructure splitting, c photograph of the physical substructure
Table 5 Theoretical stability limitations for Cases C-1 to C-3 Cases
Integration algorithm
Delay compensation 3rd-order polynomial method
qcr m
xcr m (rad/s)
C-1
k=2
0.625
141.90
C-2
k=4
0.625
141.90
C-3
k = 11.5
0.625
141.90
Table 6 Description for Tests 1–6 Integration algorithm
Compensation method
qm
1
k=2
3rd-order polynomial method
0.575
2
k=4
Tests
3
k = 11.5
4
k=2
5
k=4
6
k = 11.5
0.667
(20.78 Hz), and 128.81 rad/s (20.50 Hz), respectively, in agreement with xcr m ¼ 141:90 rad/s in Table 6. It is noted that the unstable frequencies for the three tests are not exactly the same and do not perfectly agree with the theoretical values. This is attributed to the simplification of the complex RTHS system and the assumption of reducing the time-varying time delay to be constant in RTHSs. Nevertheless, the stability and accuracy conclusions achieved from the theoretical analysis are well proven by the RTHSs.
6 Conclusions The delay-dependent performance of MDOF RTHS systems with the application of different explicit integration algorithms is extensively compared in this paper. The analytical model on the basis of the discrete-time RL technique is adopted to evaluate the stability and accuracy of the systems. Six explicit integration algorithms without numerical damping [the CDM, the Newmark explicit method, the Chang (2002) and three subfamilies
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(b) Test 1, = 2 Test 2, = 4 Test 3, = 11.5 Numerical exact
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2
4
6
8
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12
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0.8 0.4 0.0 -0.4 5.0
5.5
6.0
Time (s)
Time (s)
Fig. 17 Acceleration results for stable RTHSs: a entire time history, b close-up view
(b) Test 1, = 2 Test 2, = 4 Test 3, = 11.5 Numerical exact
0.3 0.2 0.1
Amplitude (g)
Amplitude (g)
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0.0 0
5
10
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25
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0.25
Test 1, = 2
0.20 2.40
30
2.45
Frequency (Hz)
2.50
2.55
Frequency (Hz)
Fig. 18 Acceleration results for stable RTHSs: a Fourier spectrum, b close-up view
(b)
1.2
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0.8 0.4
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0
5
10
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15
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Stable frequency (2.5 Hz)
Unstable frequency (20.78 Hz)
0.05
(20.50 Hz)
0.00
0
5
10
15
(20.77 Hz)
20
25
30
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Fig. 19 Acceleration results for unstable RTHSs: a time history, b Fourier spectrum (0–2 s)
of the Gui-k method] and the recently-proposed KR-a method with controllable numerical dissipation are considered. For the delay-dependent stability, the six integration algorithms without numerical damping provide a similar solution for mass stability limitations. The KR-a method provides larger mass stability limitation in the moderate- to high-frequency range because of the characteristic of numerical dissipation. This conclusion indicates that the time delay
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rather than the integration algorithm determines the stability performance, and the numerical damping of an integration algorithm benefits the stability of RTHS systems. The accuracy of RTHS systems is analyzed through numerical simulations as well as RL analysis with a group of special cases. Both numerical results and analysis of RL plots demonstrate that the solution accuracy of RTHS systems mainly depends on the accuracy characteristic of the applied integration algorithm itself rather than the time delay, although the latter also contributes to loss of accuracy. In addition, the 3rd-order polynomial method for delay compensation helps improve the accuracy. Finally, the stability and accuracy of RTHS systems are verified via real RTHSs. The experimental results show good agreement with the theoretical conclusions. Consequently, it is recommended that an integration algorithm with better accuracy (less period elongation and amplitude decay) or numerical energy dissipation rather than one with unconditional stability may be more suitable for RTHSs. It should be noted that considering the damage risk of instability on the RTHS system as well as many uncertain factors during testing, the structure studied in this paper is relatively simple. Future work will focus on the verification of the theoretical conclusions with complex structures. Acknowledgments This research is financially supported by the National Natural Science Foundation of China (Nos. 51179093, 91215301, 41274106), the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130002110032) and Tsinghua University Initiative Scientific Research Program (No. 20131089285). The authors express their sincerest gratitude for the support.
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