Parameter identification of large-scale magnetorheological dampers in a benchmark building

Parameter identification of large-scale magnetorheological dampers in a benchmark building✩ Arash Bahara, Francesc Pozo∗,b, Leonardo Achob, Jos´e Rodellara, Alex Barbatc aCoDAlab, Departament de Matem`atica Aplicada III, Escola T`ecnica Superior d’Enginyers de Camins, Canals i Ports de Barcelona (ETSECCPB), Universitat Polit`ecnica de Catalunya (UPC), Jordi Girona, 1-3, 08034 Barcelona. Spain. bCoDAlab, Departament de Matem`atica Aplicada III, Escola Universit`aria d’Enginyeria T`ecnica Industrial de Barcelona (EUETIB), Universitat Polit`ecnica de Catalunya, Comte d’Urgell, 187, 08036 Barcelona, Spain. cCIMNE, Departament de Resist`encia de Materials i Estructures a l’Enginyeria, Escola T`ecnica Superior d’Enginyers de Camins, Canals i Ports de Barcelona, Universitat Polit`ecnica de Catalunya, Jordi Girona, 1-3, 08034 Barcelona, Spain. Abstract Magnetorheological (MR) dampers are devices that can be used for vibration reduction in structures. However, to use these devices in an effective way, a precise modeling is required. In this sense, in this paper we consider a modified parameter identification method of large-scale magnetorheological dampers which are represented using the normalized Bouc-Wen model. The main benefit of the proposed identification algorithm is the accuracy of the parameter estimation. The validation of the parameter identification method has been carried out using a black box model of an MR damper in a smart base-isolated benchmark building. Magnetorheological dampers are used in this numerical platform both as isolation bearings as well as semiactive control devices. ✩Supported by CICYT (Spanish Ministry of Science and Innovation) through grant DPI2008-06463-C02-01. ∗Corresponding author Email addresses: [email protected] (Arash Bahar), [email protected] (Francesc Pozo), [email protected] (Leonardo Acho), [email protected] (Jos´e Rodellar), [email protected] (Alex Barbat) URL: http://www-ma3.upc.es/codalab (Francesc Pozo), http://www.cimne.upc.es (Alex Barbat) Preprint submitted to Elsevier July 30, 2009 Key words: MR damper, parameter identification, benchmark building 1. Introduction Magnetorheological (MR) dampers are devices that change their mechanical properties when they are exposed to a magnetic field. The magnetorheological fluid of these actuators is characterized by a great ability to vary, in a reversible way, from a free-flowing linear viscous liquid to a semi-solid one within milliseconds [1]. Moreover, MR dampers have a low cost, low power requirements, large force capacity, robustness and can be controlled with a low voltage at the coils [1]. All these features make MR dampers very attractive and promising as actuators controlled by the voltage that can be used in different engineering fields, such as dampers and shock absorbers (pressure driven flow mode devices), as well as clutches, brakes, chucking, and locking devices (direct-shear mode devices) [9]. From a structural control point of view, MR dampers are usually employed as actuators operated by low voltages. In this respect, semi-active control systems seem to combine the best compromise between passive and active control: they offer the reliability of passive devices together with the versatility and adaptability of active systems [3, 5, 20]. However, the first step in the design of a semiactive control strategy is the development of an accurate model of the MR device. It is worth noting that the system-identification issue plays a key role in this control problem [19]. High-accuracy models can be designed using two different model families: semi-physical models [16, 20], and black box models [10, 23]. Some of the most known semi-physical models to describe the hysteretic behaviour of MR dampers are the Bingham model and its extended versions, the Bouc-Wen model, the Dahl model, the modified LuGre model and some other non-parametric models [12, 17]. It is important to remark that these models are not linear-in-parameters and, therefore, classical parameter identification methods, such as the gradient or the mean square algorithms, cannot be applied. Using a normalized version of the Bouc-Wen model, Ikhouane et al. [7] present an identification algorithm which is directly used for MR dampers in shear-mode [17]. However, this methodology can produce large parameter identification errors if the viscous friction is much smaller than the dry friction [17]. To cope with this drawback, a modified step was proposed by Rodr´ıguez et al. [17] and tested in a small-scale MR damper. When the 2 identification is applied to a large-scale MR damper, the parameter identification errors increase [18]. The aim of this paper is to improve the accuracy of the identification algorithm. This is based on augmenting the normalized Bouc-Wen model with an additional term. The validation of this modified parameter identification method has been carried out using a black box model of an MR damper in a smart base-isolated benchmark building [14]. This model is unknown for the user of this benchmark and for the designer of control systems. The benchmark platform is then considered as a virtual laboratory experiment. The numerical results show that the proposed modified method is able to improve significantly the accuracy of the parameter identification. The paper is organized as follows. In Section 2, the magnetorheological damper model is presented. In Section 3, the key points of the modified identification method are discussed. In Section 4, the application of the proposed identification method to a large-scale MR damper in a benchmark building is considered. Finally, some concluding remarks are stated in Section 5. 2. The magnetorheological damper model The normalized version of the Bouc-Wen model [7] is an equivalent representation of the original Bouc-Wen model [23]. For MR dampers in shear mode it takes the form: Φn( ˙x, w)(t) = κ˙x(v) ˙x(t) + κw(v)w(t),(1) ˙w(t) = ρ( ˙x(t)−σ|˙x(t)||w(t)|n−1w(t) + (σ−1) ˙x(t)|w(t)|n),(2) where Φn( ˙x, w) is the output force of the MR damper, ˙x(t) and vare the velocity and voltage inputs, respectively. The voltage input vis the applied voltage at the coil of the MR damper. The system parameters, which are voltage-dependent, are κ˙x(v)>0, κw(v)>0, ρ > 0, σ > 1/2, and n≥1. These parameters control the shape of the hysteresis loop and their meaning can be found in [6]. The state variable w(t) has not a physical meaning so that it is not accessible to measurements. Since the normalized Bouc-Wen representation described in equations (1)-(2) is not a linear-in-parameter model, classical parameter identification methods cannot be applied. In this regard, a new parameter identification algorithm has been proposed in [7, p. 38], which is based on a physical understanding of the device along with a black box description. Rodr´ıguez et 3 al. [18] used this methodology for a large-scale MR damper. The method is based on applying a periodic input velocity ˙x(t) at a constant voltage coil vand observing the periodic steady-state force response of the MR damper. Nonetheless, large relative errors in the identification process can be observed when the MR damper has a viscous friction (κ˙x(v) ˙x(t)) small enough with respect to the dry friction (κw(v)w(t)). To cope with this drawback, when the displacement is large enough, an alternative method based on the plastic region of the force-velocity diagram of the MR damper has been proposed in [17]. However, the model in equations (1)-(2) may not give an accurate representation of large-scale MR dampers which do not belong to the shear-type category. For instance, consider the black box model of an MR damper in the smart base-isolated benchmark building [14]. Figure 1 contains the forcedisplacement and force-velocity diagrams when this MR damper is excited by a sinusoidal displacement and velocity. In order to test the goodness of the Bouc–Wen model, this figure also contains the response of the dynamic model in equations (1)-(2) with some appropriate parameters. It can be observed that the resulting plastic branch in the force-velocity diagram is wider that the corresponding branch of the Bouc–Wen model. In the literature, the same type of cycles have been experimentally reported, for instance, in [4, Figure 4(b)], [11, Figure 9] and [13, Figure 7]. Therefore, it can be derived that this kind of hysteretic loops are unable to be reproduced with the original Bouc–Wen model. To improve the accuracy of the model representation and, consequently, the accuracy of the parameter identification, we use the following extended Bouc-Wen model: Φe(x, ˙x, w)(t) = κx(v)x(t) + κ˙x(v) ˙x(t) + κww(t),(3) ˙w(t) = ρ( ˙x(t)−σ|˙x(t)||w(t)|n−1w(t) + (σ−1) ˙x(t)|w(t)|n),(4) where the term κx(v)x(t), which represents a linear elastic force, has been added. We consider that the coefficient κxis voltage-dependent, as the other parameters. The effect of this term can be observed not only in the forcevelocity diagram, but also in the force-displacement: the resulting plot is inclined. This feature has also been experimentally reported in, for instance, [2, Figure 8], [11, Figure 9] and [13, Figure 7]. 3. Model parameter identification This section is concerned with the computation of the parameters of the extended model in equations (3)-(4). The proposed algorithm will be divided 4 −200 −150 −100 −50 0 50 100 150 200 250 Force (kN) −2 −1 0 1 2 −400 −300 −200 −100 0 100 200 300 400 Force (kN) −250 Displacement (m) Velocity (m/s) 0−0.2−0.4 0.40.2 plastic branch Figure 1: Force-displacement (left) and force-velocity (right) diagrams of the black box model of the MR damper when it is excited by a sinusoidal displacement and velocity (dashed). The response of the dynamic Bouc–Wen model in equations (1)-(2) under the same excitation is represented by a solid line. in two steps: (a) the estimation of the value of κxand (b) the estimation of the rest of the parameters based on the identification algorithm in [18]. At constant voltage, the computation of the parameter κx(v) can be performed graphically by considering the force-displacement diagram of the MR damper. When this device is excited by a sinusoidal displacement with a large enough amplitude, the average inclination of the resulting plot gives an estimation of this parameter. As an example, consider the black box model of an MR damper in the smart base-isolated benchmark building. When this MR damper is driven with zero coil command voltage, we obtain the force-displacement diagram in Figure 2. The estimated value of κxis then computed as κx=82.8 0.4= 207 kN. To estimate the rest of the parameters, we use the knowledge of the parameter κxand the fact that Φn( ˙x, w)(t) = Φe(x, ˙x, w)(t)−κx(v)x(t). Figure 3 depicts (in solid line) the force-velocity diagram for the resulting output force Φeof the MR damper minus the linear elastic force κx(v)x(t), when this device is excited by a sinusoidal displacement and velocity. It can be recognized from this figure that this new cycle has the same shape as 5 the force-velocity cycle of the model in equations (1)-(2), which is plotted in Figure 1 right . As a result, for the identification of the rest of the parameters, that is, the parameters of the model Φn( ˙x, w)(t) = κ˙x(v) ˙x(t) + κw(v)w(t), ˙w(t) = ρ( ˙x(t)−σ|˙x(t)||w(t)|n−1w(t) + (σ−1) ˙x(t)|w(t)|n), we can follow basically the same idea as in [18]. The details of this method are omitted here but can be found in the Appendix. −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −400 −300 −200 −100 0 100 200 300 400 Displacement (m) Force (kN) 82.8 kN Figure 2: The average inclination of the force-displacement diagram, when the MR damper is excited by a sinusoidal displacement, gives an estimation of the parameter κx. 4. Application to a benchmark building This section is concerned with the application of the proposed method on a virtual MR damper. More precisely, the proposed identification algorithm is tested using a black box model of an MR damper, which is a part of a smart base-isolated benchmark building problem [14]. Consequently, we use this numerical platform as a virtual laboratory test (Figure 4). To validate 6 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −250 −200 −150 −100 −50 0 50 100 150 200 250 Velocity (m/s) Force (kN) Figure 3: Force-velocity diagram for the resulting output force Φeof the MR damper (dashed) and the force Φn= Φe−κx(v)x(t) (solid). the results, the output forces of the virtual device and the identified one will be compared using seven predefined earthquake records of the benchmark problem with their corresponding fluctuating voltage during full simulations. The MR damper is used in this context as a semi-active device to reduce the structural response of the building. 4.1. Smart base-isolated benchmark building The smart base-isolated benchmark building [14] is employed as an interesting and more realistic example to further investigate the effectiveness of the proposed design approach. This benchmark problem is recognized by the American Society of Civil Engineers (ASCE) Structural Control Committee as a state-of-the-art model developed to provide a computational platform for numerical experiments of seismic control attenuation [15, 21]. The benchmark structure is an eight-storey frame building with steelbraces, 82.4 m long and 54.3 m wide, similar to existing buildings in Los 7 MR damper x ˙x v Φ Figure 4: Input-output variables of the virtual MR damper. Angeles, California. Stories one to six have an L-shaped plan while the higher floors have a rectangular plan. The superstructure rests on a rigid concrete base, which is isolated from the ground by an isolator layer, and consists of linear beam, column and bracing elements and rigid slabs. Below the base, the isolation layer consists of a variety of 92 isolation bearings. The isolators are connected between the drop panels and the footings below, as shown in Figure 5. Figure 5: Elevation view with devices 4.2. Identification results In order to implement the identification procedure in Section 2 it is necessary to apply a periodic excitation displacement and observe the corre8 sponding MR damper force. Figure 6 illustrates these two signals for a zero voltage. A set of experiments have been performed for different voltages in the range [0,1] volts. 0 2 4 6 8 10 −80 −60 −40 −20 0 20 40 60 80 Time (s) Force (kN) 0 2 4 6 8 10 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 Time (s) Displacement (m) Figure 6: Response of the MR damper model in the benchmark building platform. The resulting values of the parameters of the model in equations (3)-(4) are listed in Table 1. Figure 8 plots these parameters as a function of the voltage. To find an accurate voltage-dependent relation of these parameters, and according with the functional dependence in Figure 8, we consider that κx(v) is constant, κ˙x(v) is linear and n(v), ρ(v) and σ(v) are exponential: κx(v) = κx(5) κ˙x(v) = κ˙x,a +κ˙x,bv(6) n(v) = na+nbexp(−13v) (7) ρ(v) = ρa+ρbexp(−14v) (8) σ(v) = σa+σbexp(−14v) (9) Because of the importance of the parameter κwdue to its great influence in the resulted force (the range of its magnitude is, approximately, from 50 kN to 1000 kN, as can be seen in Table 1), its voltage dependence function has been estimated in three different regions κw(v) =    κw1+κw2v1.15, v ≤0.3 κw3+κw4sin(π(v−0.3) 0.8) + κw5sin(3π(v−0.3) 0.8),0.3≤v≤0.7 κw6+κw7v+κw8v3+κw9v5,0.7≤v , (10) based on the variation of the resulted values (Figure 7). 9 0 5 10 15 20 25 30 −1000 −500 0 500 1000 Time (s) Damper Force Error (kN) 0 5 10 15 20 25 30 −1000 −500 0 500 1000 Time (kN) Damper Force Error (kN) Figure 11: Generated damper force errors for proposed model (above), and original method [17] (below) , under Kobe ground motion (FP-y). identification method has been tested using the MR damper as a semi-active device under time-varying voltage and earthquake excitation. A. Appendix The parameter identification in [18] departs from the next shear-mode model: Φn( ˙x)(t) = κ˙x(v) ˙x(t) + κw(v)w(t) (13) ˙w(t) = ρ( ˙x(t)−σ|˙x(t)||w(t)|n−1w(t) + (σ−1) ˙x(t)|w(t)|n) (14) where κ˙x>0, κw>0, ρ > 0, σ > 1/2, and n≥1. All of these parameters can be voltage or current dependent (here the case of voltage dependency is under consideration, as emphasized for κ˙xand κw). It has been shown in [7] that this model is meaningful in the sense that the limit cycle depends 16 directly on the parameters that appear in the normalized form, and thus depends only indirectly on the parameters of the standard form as ρ=A Dz0 >0, σ=β β+γ≥0, κ˙x=αk > 0, κw= (1 −α)Dkz0>0, where A, D, α, β, γ and kcomes from the standard Bouc–Wen model ΦBW(x)(t) = αkx(t) + (1 −α)Dkz(t), ˙z=D−1A˙x−β|˙x|z|z|n−1−γ˙x|z|n. For parameter identification, a T-periodic input ˙x(t) (see Figure 12) is applied to the Bouc-Wen system under constant voltage v. It has been proved [7] that the output force of the Bouc-Wen model goes asymptotically to a periodic steady-state so that a limit cycle is obtained. The identification method assumes the knowledge of the relation ¯w(x) that describes this cycle as illustrated in Figure 13. The whole identification process can be summarized as follows. The parameter κ˙xis first determined using the plastic region ( ¯w≈1) of the hysteresis loop by a linear regression for each constant voltage: ¯ F(τ) = κ˙x(v) ˙x(τ) + κw(v). To continue with parametric estimation, a function θis computed as: θ(x(τ)) = ¯ F(x(τ)) −κ˙x dx(τ) dτ , τ ∈[0, T+],(15) which has a unique zero, i.e, there exists a time instant τ∗∈[0, T+], and a corresponding value x∗=x(τ∗)∈[Xmin, Xmax], such that the function θis zero. Because θis known, then ˙x∗is also known. Define the quantity a=dθ(x) dx x=x∗ .(16) 17 0T+TmT mT +T+(m+ 1)T Xmin Xmax Input signal x Time Figure 12: A sample T-wave periodic signal Then, the parameter nis determined as: n= log (dθ(x) dx )x=x∗2 −a (dθ(x) dx )x=x∗1 −a log θx=x∗2 θx=x∗1(17) where x∗2> x∗1> x∗are design parameters. Define b= a−dθ(x) dx x=x∗2 θ(x∗2)n.(18) Then, the parameters κwand ρare computed as follows: κw=n ra b,(19) ρ=a κw .(20) 18 Figure 13: Symmetry property of the hysteresis loop of the normalized Bouc–Wen model. The function ¯w(x) can be computed as: ¯w(x) = θ(x) κw .(21) Finally, the remaining parameter σis determined as: σ=1 2   (d¯w(x) dx )x=x∗3 ρ−1 (−¯w(x∗3)n)+ 1  (22) where x∗3is a design parameter such that x∗3< x∗. References [1] G. Bossis , P. Khuzir, S. Lacis, and O. Volkova, Yield behavior of magnetorheological suspensions, Journal of Magnetism and Magnetic Materials,258-259:456-458, 2003. 19 [2] W.W. Chooi, S.O. 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