Modeling of a Magnetorheological Actuator Including Magnetic Hysteresis

Transcription

1 Modeling of a Magnetorheological Actuator Including Magnetic Hysteresis JINUNG AN AND DONG-SOO KWON Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology 373-1, Gusong-dong, Yusong-gu, Taejon , Korea ABSTRACT: Magnetorheological (MR) actuators provide controlled torque through control of an applied magnetic field. Therefore knowledge of the relationship between the applied current and output torque is required. This paper presents a new nonlinear modeling of MR actuators considering magnetic hysteresis to determine the torque-current nonlinear relationship. Equations for transmitted torque are derived according to mechanical shear configurations of the MR actuator. Hodgdon s hysteresis model is used to capture the characteristics of hysteresis nonlinearity in the MR actuators. An MR actuator test setup has been constructed using a commercial MR brake to evaluate the proposed model. The measured torque shows hysteresis effects as the current increases and decreases. Using Hodgdon s hysteresis model of the magnetic circuit and Bingham model of the MR fluid, a novel nonlinear model of the MR actuator is obtained as a torque estimator for practical torque control purpose. The validity of the theoretical results is verified by comparison between experiments and simulations. Furthermore, the current vs. torque frequency response of the MR actuator is examined to evaluate its applicability to torque control. The bandwidth of the MR actuator is enough high for especially haptic applications.

2 1. INTRODUCTION Magnetorheological (MR) and Electrorheological (ER) fluids are controllable fluids with varying viscosity with applied magnetic and electric fields, respectively. The use of MR devices has several advantages over conventional actuation methods. Since an MR device can generate force to a system with an inherent stabilizing effect as a passive device, it can be superior for some applications such as hydraulic control valves, shock absorbers, clutches, and seismic control devices. Ashour et al. (1996) developed an MR fluid-based crossstepper exercise machine utilizing an MR throttle valve. They demonstrated that the modified cross-stepper outperformed comparable hydraulic fluid-based cross-steppers. Peel et al. (1997a, 1997b) reported an experimental study of an ER/MR long-stroke vibration damper for control of ground vehicles. Carlson et al. (1998) invented rotary MR fluid devices for controlling the force in exercise equipment. MR fluids typically exhibit the characteristics of a Bingham fluid when a magnetic field is applied. In a related study, Stanway et al. (1985, 1987) proposed an idealized mechanical model, denoted as the Bingham model. However, when linear stroke MR dampers are under harmonic loading conditions, the force-velocity relationship shows highly nonlinear hysteretic behavior that is somewhat different from that of the Bingham model (Spencer et al., 1996; Wereley et al., 1998a, 1998b). Based on the Bouc-Wen model for emulating hysteretic behavior of MR dampers, Spencer et al. (1996) proposed a phenomenological model to adequately characterize the force-velocity behavior of an MR damper. The model can predict the response of an MR damper over a wide range of operating conditions. Wereley et al. (1998a, 1998b) also successfully developed a

3 nonlinear viscoelastic-plastic model to predict the force-displacement and force-velocity damper behavior. The characteristics of MR actuators are influenced by this nonlinear behavior in nature. In addition, the nonlinear effects due to ferromagnetic material hysteresis in the process of magnetic field excitation considerably affect the performance of MR actuators (W. Magee et al., 1998). By applying a MR brake to a conventional motor system, it is possible to realize a high bandwidth hybrid actuator, assuring stability. Furusho et al. (1996) developed a robot arm with an ER damper in order to improve the stability. Takesue et al. (1999) applied the ER damper to the direct-drive motor system, improving the gain margin of the control system. Like this ER device, if the MR brake is to be treated as a torque transmitted element of the robotic arm system, it can be considered as a rotational actuator. We applied the MR actuator to the haptic device for improving stability and performance of haptic sensation and we proposed the hybrid haptic actuator using the motor and the MR actuator together (An et al., 00). Thus, our attention is focused on the properties of the actuator; i.e. the relationship between the input current and output torque of the MR actuator. In this case, the torque generated by the actuator is related to the magnetic field intensity due to coil currents. Usually, the MR actuator suffers from a nonlinear and time-varying relationship between the input current and output torque. Therefore, it is important to closely examine this relationship for effective control of the torque and reliable design of the MR actuator. While the transfer function of a conventional motor can be reasonably well approximated by a constant linear relationship between input current and the output torque, it is unfortunate that the relationship between

4 applied current and torque for a MR actuator is more difficult to model. Brookfield et al. (1996) modeled an ER clutch as a rotary actuator through a linear transfer function between input voltage and output torque. However, little work has been done to compare the torque versus current relationship to the actual performance of the MR actuators. Furthermore, the hysteretic torque-current behavior has been deemed insignificant. An MR actuator surely comprises not only MR fluids in the cavity but also the ferromagnetic material to form the flux path. The ferromagnetic material has experienced the magnetic hysteresis in nature. On the other hand, MR fluids exhibit approximately linear magnetic properties in their magnetization and little or no hysteresis can be observed in their B-H curve (Jolly et al., 1998). Thus the magnetic hysteresis of the ferromagnetic material is a major factor in the nonlinearity of the output torque of the MR actuator. In this article, a new nonlinear model of a MR actuator considering a magnetic hysteresis of the ferromagnetic material and a nonlinear Bingham model of the MR fluids is proposed for obtaining a torque estimator for the practical control purpose. If an MR actuator is used as a passive actuator in torque control, the proposed nonlinear model of the MR actuator can exactly estimate the output torque for arbitrary input current. Hysteresis effects are hereby found to cause significant variations in the measured torques. The experimental results demonstrate that the proposed model is useful to illustrate the characteristics of an MR actuator. To further explore and identify the behavior of an MR actuator in terms of frequency response, an experimental analysis of torque versus current is presented. The remainder of this article is organized as follows. On the basis of the Bingham model and the parallel disk shearing, the mechanical model of an MR actuator is presented in Section. The magnetic model of the

5 MR actuator comprising magnetic circuit design and magnetic hysteresis model is described in Section 3. Using the mechanical model and magnetic model of the MR actuator, the novel nonlinear model of the MR actuator is proposed in Section 4. To evaluate the effectiveness of the proposed model, it is applied to an MR actuator and Section 5 demonstrates the experimental results. Finally, conclusions are summarized in Section 6.. MR ACTUATOR MECHANICAL MODELING Magnetorheological fluids are well known for changing their apparent viscosity in a magnetic field. A simple Bingham plastic model effectively describes the magnetic field dependent fluid characteristics (Kordonsky, 1993; Weiss et al., 1994). In this model, total yield stress τ is given by τ = τ sgn & γ + ηγ& y (1) where τ y is the dynamic yield stress caused by the applied field, γ& is the shear strain rate, and η is the field independent plastic viscosity defined as the ratio of the shear stress versus the shear strain rate relationship. The schematic of an MR actuator is shown in Figure 1. In this configuration, the shear operation occurs at the two sides surface of the rotor. The shears in the two side gaps are the major torques due to magnetic yield strength. When the MR fluids are sheared between two parallel plates as shown in Figure, the shear strain rate is a function of r alone and simply expressed by (Bird et al., 1987; Carreau et al., 1997) rω γ& = () g Substituting Equation () into Equation (1) leads to

6 () r rω = τ ysgn Ω + η g τ (3) The torque transmitted by a differential donut element is expressed as dt τ () r πr dr = (4) Using Equation (3) to integrate (4) results in the torque transmitted by shear in the gap: T one side 3 3 Ω 4 4 ( R R ) sgn Ω + ( R R ) = y o i πη 3 g πτ o i (5) Since the shear operation occurs at the two sides surface of the rotor, thus the torque of the MR actuator is consequently modeled by T 3 3 Ω 4 4 ( R R ) sgn Ω + ( R R ) 4 = y o i πη 3 g πτ (6) o i 3. MR ACTUATOR MAGNETIC MODELING The dynamic yield stress given in the Bingham model of Equation (1) varies with magnetic induction for the MR fluid, as shown in Figure 3. It can be fitted reasonably with a third-order polynomial of the form y 3 1 f + KB f K3B f τ K B + = (7) where K 1 = , K = , and K 3 = are constant, and B f is the magnetic induction for the MR fluid. Using the principal of Continuity of magnetic flux, the magnetic induction of the steel (B) throughout the flux (Φ) conduit can be determined (Hayt, 1981) as

7 Φ = Φ f = Φ Φ = BA = B f s A f (8) where Φ f and Φ s are fluxes of the MR fluid and steel, respectively. A and A f are effective pole areas of the steel and the gap, which contains MR fluid, respectively. Generally the magnetic properties of MR fluids vary significantly from the properties of most bulk ferromagnetic properties in that little or no hysteresis can be observed in the magnetic induction curves of the MR fluids. This superparamagnetic behavior is a consequence of the magnetically soft properties of the iron used as particulate material in these fluids and the mobility of this particulate phase (Jolly et al., 1998). The MR actuators include not only MR fluids, but also ferromagnetic material to form the flux path. Ferromagnetic materials typically have nonlinear properties as characterized by the hysteresis loop mentioned in Figure 4. The performance of MR actuators also depends upon the nonlinear characteristics of their magnetic circuits. This nonlinearity is undesirable because it leads to distortion in performance of the MR actuators. Therefore, it is important to develop a magnetic hysteresis model that both captures all of the essential characteristics relevant to MR actuators and is applicable for the MR actuator design. Hysteresis in ferromagnetic materials has been studied extensively, and two distinct types of models have been proposed to capture the observed hysteretic characteristics. The first is derived from Preisach model (Bertotti, 1998; Mayergoyz, 1986, 1988). Preisach model is based on the assumption that any hysteresis can be expressed as a sum of elementary hysteresis loops. The distribution function of the elementary hysteresis is determined fully from the measured BH loops (Igarashi et al., 1998). Preisach model and its generalization are

8 not simple to understand, nor to implement: the experimental data needed are very numerous and delicate to interpolate and the control of the memorized turning points must also be done properly (Ossart et al., 1990). The other, proposed by Hodgdon (Coleman and Hodgdon, 1986; Hodgdon, 1988), assumes a constitutive relation between the magnetic field (H) and the magnetic induction (B). In Hodgdon s model, magnetic hysteresis is described by a differential equation derived from physical insight into the magnetization process. Therefore Hodgdon s model is much simpler to understand and to implement (Ossart et al., 1990). Generally Due to the fact that Preisach model uses much more data, Preisach model follows better the experimental behavior of the magnetic material than Hodgdon s model. On the other hand, all of parameters in the Hodgdon s model are easily determined from the major BH loops. In addition, Hodgdon s model is efficient to implement with good accommodation of minor loops. Thus Hodgdon s model is used as the magnetic hysteresis model in this study. The work by Hodgdon (1988) shows the following differential equation, [ f ( B) H ] Bg & ( B) H & = α B& + (9) which relates the time rate of change of the magnetic induction (B) to that of the magnetic field (H), along with a set of constraints on α and on the material functions f and g represented in Equation (9). f g ( B) ( B) A tan 1 = A tan 1 A tan 1 f = f ' ' ( B) ( B) ( A B) ( A B ) + ( B B )/ s s ( A B ) + ( B + B ) 1 A 3 s A B 4 exp Bs B s µ s / µ s B B B > B s B < B s B B s B > B s s (10)

9 where the values for the material constants A 1 through A 4 can be calculated from the values as shown in Figure 4. A 4 = B r B B r s H µ A A 3 s A cl 1 = H sin 1 = 1 A A s ( B ) s A ( B A ) 1 + αh µ c ( A B ) 1 cos ln A3 A1 A A 1 cot 3 r s = 0 c 1 + αa1 tan µ r ( A B ) r (11) Actually, magnetic materials comprised in the electromagnetic devices have experienced demagnetization, which gives rise to a magnetic field in a direction opposite to that of the magnetization because of their open circuits. This field, called the demagnetizing field (H d ), is proportional to the intensity of magnetization (M). H N M = d d (1) where N d is the demagnetizing factor which depends mainly on the shape of the body. But it can be calculated exactly only for an ellipsoid. Values of demagnetizing factors have selected by Bozorth (1951) from a number of investigations. The intensity of the magnetic field produced by a coil is proportional to the electric current, which follows in the coil H a = Ci (13) where C is the coil constant, which depends on the shape of the coils and on the number of turns in the windings. For a multi-layer solenoid of finite length, the coil constant is given in research done by Bozorth (1951) and Cullity (197). The applied field (H a ) due to the solenoid must compensate for the demagnetizing effect to

10 obtain a correct true field (H). H = H a H d (14) The magnetic induction (B) can be easily measured rather than the intensity of magnetization (M) as follows: B = H + 4πM (15) From Equations (1) to (15), the intensity of magnetization and the true field become (Cullity, 197) M B H = π N a 4 (16) d H = H a B H ( 4 / N ) 1 d a π (17) 4. PROPOSED NONLINEAR MODEL OF THE MR ACTUATOR In the previous Section and 3, the MR actuator mechanical modeling and the magnetic modeling were described. In this section, combining the MR actuator magnetic modeling with the mechanical modeling, the nonlinear model of the MR actuator is designed and its applicability to practical torque control use is introduced. In Section 3, we described Hodgdon s model of the ferromagnetic material in Equation (9). To show the effectiveness of Hodgdon s model, we simulated the magnetic hysteresis of various ferromagnetic materials (Figure 5). In Table 1, the required parameters of the materials are presented. The measurements of the material constants are based on test using measurement standards traceable to the national primary standard of the Korea Research Institute of Standards and Science (KRISS). Figure 5(c) shows the comparison of the measured

11 hysteresis loop from KRISS measurement with the simulation result of Hodgdon s model. From this figure, it is evident that the simulation result of Hodgdon s model is very close to the measured hysteresis loop. And also comparing Figure 5 with Table 1, Hodgdon s model shows a good prediction of the magnetic material hysteresis. To simply show how the MR actuator components are interconnected, its block diagram is represented in Figure 6. The block diagram models the new nonlinear MR actuator. It clearly describes the input current(i)- output torque(t) of the MR actuator. To understand the proposed nonlinear MR actuator model, we explain the composition and interconnection of the model in detail. The initial magnetic intensity (H a ) generated by the solenoid is proportional to the input current (i) supplied by the current amplifier as shown in Equation (13). As previously mentioned in Equation (14) in Section 3, due to the demagnetization of the steel (H d ), the magnetic intensity (H) becomes lower than the initial magnetic intensity (H a ). When the steel is placed in oscillatory magnetic field, the steel has experienced the hysteresis between the magnetic intensity (H) and the magnetic induction (B). To formulate this nonlinear relationship, we make use of Hodgdon s model shown in Equation (9). Since the relationship between the magnetic induction (B f ) and magnetic field for a MR fluid is similar to that for the steel (B) in the magnetic circuit shown in Equation (8), the magnetic induced yield stress (τ) shown in Equation (7) is numerically estimated from the magnetic induction-the yield stress curve (Figure 3). Finally, from the Equation (6), the Bingham model produces the output torque of the MR actuator (T). The proposed nonlinear MR actuator model can be used as a torque estimator for practical torque control purpose. If we consider the torque control case using an MR actuator as a passive actuator, it is very difficult to accurately predict the output torque under the oscillatory input current environment because of the current-torque

12 hysteretic behavior. Then, if this hysteretic behavior can be modeled, we can estimate the output torque at the present input current state. Because the above suggestion is beyond the scope of this article, the torque estimator problem remain areas open to further study. 5. SIMULATION AND EXPERIMENTAL RESULTS Figure 7 shows the MR actuator experimental setup used to verify the proposed nonlinear model of the MR actuator. The experimental setup consists of an MR actuator, a geared AC servomotor, and a torque sensor. The geared AC servomotor is used as a driving source. A rotary type torque transducer measures the output torques of the MR actuator. The rotary MR actuator (MRB107-3) discussed in this paper is shown in Figure 8. This MR actuator provides variable resistive torques according to the amount of external current. It yields 6Nm maximum torque in accordance with a 1Ampere external current for speeds up to 1000RPM. The bi-directional current amplifier is constructed to supply the current into the MR actuator. Actuator components (both stator and rotor) are made of cold-wrought steel, AISI1L14, which has high relative permeability, about Its B- H curve is shown in Figure 9 (Carlson et al., 1998; Lord Corporation, 1999). Figure 10 shows a plot of the experimentally measured static torque versus current for three constant velocity levels, 60 rpm, 150 rpm, and 40 rpm. From Figure 10, the MR actuator torque is proportional to the current supplied and independent of the rotational speed. The hysteresis behavior of an MR actuator produces a variation in the torque between an increasing and decreasing magnetic field generated by the current. Thus in a dynamic environment such as oscillatory current

13 excitation, uncertainty is introduced in any calculated torque. All hysteresis tests were conducted under 1.0 Hz sinusoidal current excitation at a rotational speed of 150 rpm. Currents of varying amplitudes were applied in the range of 0 ~ 1.0 A. The sample set of experimental data is shown in Figure 11. The lower excitation current provides that the torque/current relationship is nearly linear. At a higher current, however, the hysteretic behavior is revealed more significantly. These hysteretic behaviors quite resemble ferromagnetic hysteresis in shape and phase. Thus, we infer that this nonlinear torque/current behavior of the MR actuator can originate from general magnetic hysteresis. In order to show the performance of the proposed nonlinear model of the MR actuator, a comparison between the predicted hysteresis behavior in the MR actuator and the corresponding experimental data is provided in Figure 1. Simulations were performed in MATLAB and SIMULINK and the parameters for the model are numerically calculated or adjusted by the modeling procedure as mentioned above. The simulation result shows that the proposed model predicts the hysteretic behavior of the MR actuator reasonably. Therefore, the proposed nonlinear MR actuator model seems to effectively estimate the actuator dynamics in terms of input currents and output torques. To evaluate the feasibility for a torque estimator in control use, the current-torque frequency test of the actuator has been conducted. Input is a sinusoidal perturbation current with a cycle frequency between 1 Hz and 0 Hz with a bias current of 0.5A and the rotational speed is 150 rpm. The results of the frequency test shown in Figure 13. It indicates that simulated frequency responses closely resemble experimented results and the actuator has a bandwidth of approximately 0 Hz enough high for torque control.

14 6. CONCLUSIONS A novel nonlinear model of the MR actuator is proposed to determine the toque-current relationship of the MR actuator. The proposed model uses both Bingham model for MR fluids and Hodgdon s model for the magnetic hysteresis of the steel. The graphical approach is used to achieve the integrated model, which has complicated interconnections between mechanical composition and magnetic composition. With the proposed model the undesirable hysteretic nonlinearity can be estimated and mitigated in the design of the MR actuator in advance. Thus, the proposed nonlinear model of the MR actuator can be used as a torque estimator for practical torque control applications. The nonlinear effects of magnetic hysteresis and MR dynamics have been thus quantified. Hysteresis tests have been conducted to quantify the magnitude of actuator torque variation due to material hysteresis. The torque depends on the specific current trajectory and the amplitude of perturbation currents. Through the simulation and experiment, the performance of the proposed nonlinear model of the MR actuator is demonstrated and the bandwidth of the MR actuator is enough high for torque control applications. ACKNOWLEDGEMENTS The authors wish to show their appreciation for support provided by MOST, National R&D Program (Critical Technology 1), Program ID: 99-J A (Development of Service Robot Technology).

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19 Ω R i R o R e g Figure 1. The Schematic of an MR actuator

20 R o R i Ω r g Figure. Mechanical shear mechanism of an MR actuator

21 Figure 3. Characteristics of the MR: Shear stress vs. magnetic induction

22 B B s B r B r : remanence H c : coercive force B s : closure magnetization H s: closure intensity H c H s H µ c : permeability at H c µ r : permeability at B r µ cl : permeability after B cl µ s : permeability at (H s, B s) Figure 4. A graphical illustration of typical ferromagnetic material hysteresis

23 0 Low Carbon Magnetic Iron : S15C 0 Rolled Staniless Steel : SS41 Magnetic induction [KGauss] Magnetic induction [KGauss] Magnetic field [Oe] Magnetic field [Oe] (a) (b) Magnetic induction [KGauss] Stainless Steel Bars : SUS430 Measurement by KRISS Modeling Result Magnetic induction [KGauss] Cold Wrought Steel : AISI 1L Magnetic field [Oe] (c) Magnetic field [Oe] (d) Figure 5. The results of magnetic hysteresis modeling for various magnetic materials: (a) Hysteresis model for low carbon steel-s15c (JIS G 4051), (b) for rolled stainless steel-ss41 (JIS G 3101), (c) for stainless steel bar-sus430 (JIS G 4303), and (d) for cold wrought steel-aisi 1L14 (ASTM A576)

24 Equation (13) Equation (9) Equation (8) Equation (7) Equation (6) Input i Coil Constant H a H Magnetic Hysteresis Model B Flux Ratio B f MR Dynamics τ y Bingham Model Output T Velocity Ω H d Demagnetization Equation (14)~(17) Figure 6. Block diagram of the proposed nonlinear model of the MR actuator

25 MR Actuator Torque Sensor AC Motor Figure 7. The MR actuator experimental setup

26 Figure 8. MRB107-3 MR actuator

27 15 Magnetic induction [KGauss] M agnetic field [Oe] Figure 9. Graphical illustration of the B-H Curves for AISI1L14

28 torque [Nm] rpm 150 rpm 40 rpm current [A] Figure 10. Measured quasi-steady torque versus current curves for MRB107-3 MR actuator

29 6 Amplitude = 0.05 A 6 Amplitude = 0.1 A Torque [Nm] 4 Torque [Nm] Current [A] (a) Current [A] (b) 6 Amplitude = 0.15 A 6 Amplitude = 0. A Torque [Nm] 4 Torque [Nm] Current [A] (c) Current [A] (d) 6 Amplitude = 0. A 6 Amplitude = 0.5 A Torque [Nm] 4 Torque (Nm) Current [A] Current (A) (e) (f) 6 Amplitude = 0.4 A 6 Amplitude = 0.5 A Torque [Nm] 4 Torque [Nm] Current [A] (g) Current [A] (h) Figure 11. A sample set of experimental hysteresis data for different current amplitudes: (e) Torque vs. current for amplitude 0.A duplicated (d) with different current scale

30 6 Modeling Result Measured Result torque [Nm] current [A] Figure 1. Comparison of the hysteresis model results and the experimental data for testing conditions of current, I=0.5A at a frequency of 1.0Hz and rotational speed, 150rpm

31 1 magnitude 0.1 Simulated FRS of MRB107-3 Experimented FRS of MRB frequency (Hz) Figure 13. Frequency response of MRB107-3 MR actuator

32 Table 1. Required hysteresis data and values for the material constants in Equation (1). The measurements of the material constants shaded values are based on test using measurement standards traceable to the national primary standard of the Korea Research Institute of Standards and Science (KRISS). Units here are KGauss for magnetic induction and Oersteds for magnetic fields. All the data for AISI 1L14 are estimated from Figure 3. S 15 C SS 41 SUS 430 AISI 1L14 B cl H cl B r H c µ r µ c µ s µ cl α A A A A