Experimental Validation of the Modified Continuously Variable Command Law of a Semi-Active Suspension Integrating a Magneto-Rheological (MR) Damper Said Boukerroum1,a, Nacer Hamzaoui2,b 1
Laboratoire de Mécanique Avancée, USTHB, BP 32 El Allia 16111, Bab Ezouar Alger, Algérie. Laboratoire Vibrations Acoustique, INSA, Bâtiment A de ST Exupéry, 25 bis avenue Jean Capelle 69621, Lyon, France a
[email protected],
[email protected]
2
Keywords: MR Damper , Semi-Active Suspension , Modified Bouc-Wen Model , Control Law.
Abstract: The present work consists of an experimental performances analysis of a suspension system with two degrees of freedom governed by a semi-active modified continuously variable command (MCVC) law. The internal dynamics of Magneto-Rheological (MR) damper used in this study is highlighted by the modified Bouc-Wen model in the mathematical modelling of the secondary suspension system. After the dynamic characterization of the MR damper, a comparison of performance obtained by this control scheme is carried out from the responses calculated using a numerical model and measured experimentally from a test bench of a semi-active suspension incorporating an MR damper and controlled by a dSPACE control chain. For a better representativeness of the modified Bouc-Wen numerical model, a rapprochement between the calculated and measured responses for the same dynamic characteristics of the test bench is possible by adjusting the most influential parameters of the numerical model. Through better management of the suspension during the low speeds, the modified Bouc-Wen model is more representative of the real behaviour of the MR damper, given its sensitivity at these low speeds during transitions between compression and expansion phases of the damper. Introduction In recent years, Magneto-Rheological dampers (MR) are the subject of special attention because of their characteristics including mechanical simplicity, high dynamic range, low power, high force capability and robustness. However, and in the early 2000s, researchers began to consider the development and use of these dampers in several areas, particularly where other technologies have begun to emerge to make their practical use possible. Microprocessors, sensors and controllers to increase the processing speed have created opportunities for control that did not exist before. A wide range of applications of MR dampers has allowed researchers to improve the performance of the latter, by focusing on their internal dynamics. Indeed, several mathematical models highlighting the physical and energetic properties (electric current, magnetic fields, flow, viscosity) MR fluids have enabled the development of several numerical models governing the internal dynamics of these dampers and the most recent is the model of Bouc-Wen modified developed by Spencer et al. [1] which is a modification of the Bouc-Wen model [2]. To validate the performances of these models, we use most often the experimental studies which can be very complex and expensive, whose interest is to adjust these models in real time [2]. To obtain a more representative model of the MR damper, which can substitute for an experimental model in other studies, it is necessary to validate and adjust from a confrontation 'calculation-test', highlighting the semi-active control schemes to better analyze the time of response and allow to perform the recalibration (readjustment), the main objective of this confrontation for a better representation of the numerical model [3].
Dynamic model Figure 1 shows the functional scheme of a semi-active suspension with two degrees of freedom reflecting the quarter-vehicle model incorporating an MR damper in the secondary suspension.
c1 k c k3
Figure 1. A two degrees of freedom suspension incorporating a MR damper with a modified Bouc-Wen model
Figure 2. A Bouc-Wen modified model by Spencer [1]
Such a suspension differs from a conventional passive suspension by the presence of an MR damper whose internal dynamics is governed by the modified Bouc-Wen model (Fig. 2). The variation of the damping coefficient in the MR damper is obtained by modification of the oil viscosity mixed with ferromagnetic nanoparticles controlled by a DC. This model, equivalent to a quarter of vehicle consists of a secondary suspension, formed by the mass m 2 (of the chassis), suspended by a k2 spring stiffness and connected in parallel with the c variable damping coefficient of the MR damper. The unsprung mass m1 (of the axles), is related to tire stiffness k1, constitutes the primary suspension. The displacements z1 and z2, respectively of sprung and unsprung masses, characterizes the vehicle dynamics excited by vertical profile of the input road z e. As from the measurement of acceleration by sensors placed on the two vibrating masses, we can control in real time the damping force developed by the MR damper, using a control algorithm deduced of the ‘’MCVC’’ control law [4] used in this study. The elements of the internal dynamics of the numerical model of MR damper are described below. The physical characteristics of such a suspension system are presented in table 1. Table 1. Quarter model parameters Parameters Notation Sprung mass m2 Unsprung mass m1 Equivalent stiffness of the secondary suspension k2 Equivalent stiffness of the primary suspension k1 cmax Damping coefficients of secondary suspension cmin
Value 50 kg 140 kg 38 103 N/m 400 103 N/m 6517 N.s/m 590 N.s/m
Numerical model To better predict the response of the MR damper, a modified version (Fig. 1) of the Bouc-Wen model was developed by Spencer et al. Taking into account the parameters introduced by this model, the vibrational behaviour of the two degrees of freedom passive suspension showing in figure 1 will be governed by the following differential equations:
M1z1 (k 2 k 3 )(z1 z 2 ) k1 (z1 z e ) c1 (z 1 z 3 ) 0 M 2z2 (k 2 k 3 )(z 2 z1 ) k (z 2 z 3 ) c(z 2 z 3 ) z 0. k (z z ) c(z z ) c (z z ) z 0 3 2 1 3 1 3 2
(1)
To obtain the equations corresponding to this model, only the upper section is considered. is the coefficient of stiffness associated with the hysteresis function z. The forces on either side of the rigid bar are equivalent:
c1 (z 3 z 1 ) kz 2 z3 c(z 2 z 3 ) z.
(2)
Its allows us to present the variable evolution, given by the following equation:
z 3
1 k z z cz 2 c1z 1 z. c c1 2 3
(3)
The total damping force generated by the system is then the sum of the forces on upper and lower sections of the system of figure 1:
Fd cz 2 z 3 kz2 z3 k3 (z2 z1 ) z.
(4)
Or :
Fd k 3 (z 2 z1 ) c1 (z 3 z 1 ). Evolutionary variable z is given by [3]:
z γ z 2 z 3 z z
n 1
β(z 2 z 3 ) z A(z 2 z 3 ). n
(5)
(6)
Where c, k , β, , γ, n and A, are the characteristic parameters of the Bouc-Wen model that can be controlled to adjust the linearity and smoothing the transition periods pre and post-stress threshold. In this model, the accumulator rigidity is represented by k3 and viscous attenuation observed at higher speeds is represented by the coefficient c. A damping coefficient c1 is included in the model to produce the decrease observed in the experimental data at low speed [3]. The coefficient k is introduced for controlling the rigidity at high speeds. z20 is the initial displacement of the spring k3 due to the accumulator linked to the nominal force of the damper. Characterization of the MR damper The RD-1005-3 is a small compact monotube damper by Lord Corporation [5], length 208mm in the extended position, it incorporates an electromagnet at the piston, powered by a low voltage of 12 volts (DC) and low electric current can be varied within a range of 0-2 A. It has a 25mm stroke and develops an intensity maximum force 4448N. The current setting by the controller is used to vary the magnetic field at the coil of the piston and therefore modify in real time the damping coefficient and thus the force. Response time does not exceed 20ms (time to reach 90% of the maximum level for step input of 0A to 1A at the speed of 51mm/s) [5].
Secondary suspension
Sprung mass
Guide column
Excitation point MR damper (RD-1005-3)
MR damper Primary suspension
Figure 3. Dynamic characterization of the MR damper on the MTS 810 Machine
Figure 4. Experimental test bench of two degrees of freedom suspension
The characterization of the damper RD 1005-3 is necessary for the dynamic data relating to its functioning such as ratios force/displacement and force/velocity and the saturation electrical current. The MR damper has been tested on a dynamic testing machine MTS brand (Fig. 3) which can generate sinusoidal cyclic inputs, with adjustable frequency and amplitude. The measured performance of this damper as a function of input current, are the temporal evolution of the force in function of the current (Fig. 5), for a sinusoidal input frequency of 1Hz, a peak to peak displacement of 15mm and intensity of the current delivered to the shock 0A, 0.4A, 0.8A, 1.2A, 1.4A and 1.6A. A beginning of the damper saturation from a current 0.8A is observed. The effects of the amplitude of displacement imposed on the damper and the excitation frequency are shown in figure 6. As shown in this figure, an increase of the force amplitude produced by an MR damper, is not only a function of the amperage current, but also a function of frequency and the displacement amplitude that are imposed [6]. 2500
1800
0A 0.4 A 0.8 A 1.2 A 1.4 A 1.6 A
Force (N)
900 450
0 A, 1 Hz ; 0 A, 2.5 Hz; 0 A, 5 Hz ;
2000 1500
1.5 A, 1 Hz 1.5 A, 2.5 Hz 1.5 A, 5 Hz
1000
Force (N)
1350
0 -450
500 0 -500 -1000
-900
-1500
-1350 -1800
Frequency:1 Hz peak-peak displacement: 15 mm 3,0
3,5
4,0
4,5
5,0
Time (s)
Figure 5. Time evolution of the damper force versus current for frequency excitation 1Hz and a displacement of 7.5 mm
-2000 -2500
peak-peak displacement: 15 mm 0,4
0,6
0,8
1,0
1,2
1,4
1,6
Time (s)
Figure 6. Time evolution of the damper force in function of the excitation frequency and the current for a displacement of 7.5 mm
Indeed the equivalent damping coefficient ceq directly related to the energy dissipated per cycle depends on the current and the input frequency and the amplitude of displacement. Assuming there is a linear passive damper with an ideal damping at each operating condition, we can find ceq by the following equation [7]: E c eq . (7) fx 02
Where E is the energy dissipated per cycle and f is the sine input frequency in Hz and amplitude x0. To validate the force evolution developed by the MR damper with respect displacement and velocity as a function of current and displacement imposed, A comparison between the curves illustrating the ratios of force/displacement and force/velocity obtained from the numerical model by the modified Bouc-Wen model and measured experimentally, is given by the figure 7 and figure 8 respectively. 2000
2000
experimental (15mm); experimental (25mm);
1500
modifed Bouc-Wen model (15mm) modifed Bouc-Wen model (25mm)
modifed Bouc-Wen model (15mm) modifed Bouc-Wen model (25mm)
1000
Damping Force (N)
1000
Damping Force (N)
experimental (15mm); experimental (25mm);
1500
500 0 -500 -1000
Frequency: 1 Hz Current: 1 A
-1500
500 0 -500 -1000
Frequency: 1 Hz Current: 1 A
-1500 -2000
-2000 -14
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
14
-80
-60
-40
-20
Figure 7. Evolution of the ratio force/displacement calculated and measured according to the displacement
0
20
40
60
80
Velocity (mm/s)
Displacement (mm)
Figure 8. Evolution of ratio force/velocity calculated and measured according to the displacement
It is seen from these ratios of force/displacement and force/velocity that highlight the hysteretic behaviour (visco-elastic behaviour of the MR fluid), an increase in the damper force amplitude according to the amplitude of the imposed displacement for a frequency and amperage current fixed now to 1Hz and 1A respectively. Semi-active control of the sprung mass By applying at the semi-active secondary suspension a modified continuously variable control law, the total damping force generated by the passive system and given by the equation (4) obeys now at the condition function below: c d (z 2 y ) k(z 2 - y) k 3 (z 2 - z1 ) αz Fd c min (z 2 y ) k(z 2 - y) k 3 (z 2 - z1 ) αz
if z 2 (z 2 z 1 ) > 0 if z 2 (z 2 z 1 ) 0
Where cd is the damping coefficient of the MR damper, which can take the following values: z 2 > c max if c s c max z z 1 2 z 2 z2 c max . c d c s if c p < c s z 2 z 1 z 2 z 1 z 2 c p c p if c s z 2 z 1 Where : cmax : is the maximum value of the semi-active damping coefficient system, cmin : is the minimum value of the semi-active damping coefficient system, cs : is the optimum value of the skyhook damping coefficient system, cp : is the optimum value of the passive damping coefficient system.
(8)
(9)
The total force Fd generated by the system is then the sum of forces on upper and lower sections of the system of figure 1. The damping can take three values depending on the condition based on the control law given by (9). For the semi-active system incorporating the modified Bouc-Wen model as internal dynamics, a damping coefficient c can be under the control law four values: cmin, cmax=6517N.s/m, cs=4596N.s/m, cp=1164N.s/m is considered. For a current supplied to the MR damper I null, cmin=590Ns/m, stiffness coefficient α=38240N/m and a c1=4730Ns/m are introduced by the modified Bouc-Wen model. The coefficients of the hysteresis function z: А=10.013, β=3.044mm-2, γ=0.103 mm-2 and k=1.121N/mm. The parameters α, c and c1, were derived from polynomial functions below, expressed as a function of the current I input [8].
(I) 826.67 I3 905.14I 2 412.521I 38.24
(10)
C(I) 1173I3 10.51I 2 11.02I 0.59
(11)
C1 (I) 54.40I3 57.03I 2 64.57 I 4.73
(12)
Experimental test bench The experimental test bench shown in figure 4 was mounted in LVA of INSA de Lyon. It reproduces the elements of the dynamic model of two degrees of freedom in figure 1, and includes a MR damper type: RD-1005-3 of Lord Corporation [5]. The latter is controlled by a current controller of the test bench, composed of a dSPACE control system, equipped with the software and ControlDesk running under Matlab/Simulink. The diagram of real-time control of the test bench secondary suspension is shown in figure 9.
Amplifier power PC TOSHIBA P 1,7 Hz, R 512Mo
Matlab/Simulink ControlDesk
Expansion box dSPACE
PCMCIA DS815
Carte DS1103
Shaker V650
Low pass filter -
Connector panel
LED CLP1103
I Controler RD RD- 3002-3
Controler
ICP
- Accelerometer - Force sensor
MR damper RD-1003-03
Figure 9. Flowchart acquisition and control of the secondary suspension The dSPACE system is used for acquisition and control of closed control law governing the damping force generated by the MR damper via the current controller loop. To manage the inputoutput parameters via the acceleration and force sensors using a interface with graphics and indicators measurable and controllable parameters is constructed in ControlDesk of dSPACE from a generated computer code. Results and discussion Figures 10 and 11 show respectively the experimental curves of the damping force and the absolute acceleration of the test bench sprung mass. These responses measured from the system controlled by ‘’MCVC’’ law and excited by a sinusoidal function of frequency 5Hz and peak to peak amplitude of 3cm, are obtained for current intensities: 0A, 0.1A, 0.2A and 0.6A.
25,0
6000
D am ping Force (N )
4000
Sprung masse acceleration (m/s2)
0A 0.1 A 0.2 A 0.6 A
2000
0
-2000
-4000
0A 0.1 A 0.2 A 0.6 A
12,5
0,0
-12,5
-25,0
-6000 6,0
6,1
6,2
6,3
6,4
7,2
6,5
7,3
7,4
7,5
7,6
7,7
7,8
7,9
8,0
Time (s)
T im e (s)
Figure 10. Damping force measured of MR damper according to the current
Figure 11. Sprung mass absolute acceleration measured according to the current
It should be noted that the increase in the intensity of the input current increases the damping force by producing a phase shift between the time responses of the damper up to saturation at 0.6A, which no longer obeys to the control law (Fig. 10). This increase in current reduced normally the sprung mass acceleration to a current of 0.2A (Fig. 11). In deed, beyond this intensity, the effect of the current on the acceleration is reversed, causing an increase of the sprung mass absolute acceleration. The figures below show a comparison between the calculated responses of the secondary suspension controlled by the ‘MCVC’ law incorporating the modified Bouc-Wen digital model and those measured from the test bench. These responses presented in terms of the sprung mass absolute acceleration (Fig. 12) and the damping force (Fig. 13), represent the control parameters of the command law used in our work. 6000
m odified B ouc-W en m odel experim ental
modified Bouc-Wen model experimental
8
4000
4
2000
D am ping Force (N )
2
Sprung mass acceleration (m/s )
12
0
-4
0
-2000
-4000
-8
-6000
-12 7,2
7,3
7,4
7,5
7,6
Time (s)
Figure 12. Sprung mass acceleration measured and calculated by the modified Bouc-Wen model
9,4
9,5
9,6
9,7
9,8
T im e (s)
Figure 13. Damping force measured and calculated by the modified Bouc-Wen model
The figure 12 shows calculated and measured sprung mass acceleration curve controlled by ''MCVC'' law, with a 5 Hz sinusoidal excitation frequency and peak to peak amplitude of 3cm. Qualitatively, the response curve of the absolute acceleration calculated by the modified Bouc-Wen model approaches the measured acceleration, where the adjustment of the parameters of the numerical model, allows a rapprochement between the two curves in terms amplitude. However, perturbations and discontinuities found on the measured curve, this are due to friction in the test bench guide columns, which disturb the measurement of the absolute acceleration by the acceleration sensor disposed on the suspended mass.
From the confrontation represented by the graph in figure 13 between the curves of the damping force calculated from the numerical model and that measured on the test bench of the system controlled by ‘’MCVC’’ law, a rapprochement between the shapes of the curves calculated and measured for the same excitation and adjustment parameters of the numerical model conditions is to be noted. Indeed, by adjusting the MR damper digital model parameters, it is possible to approach the response of the damping force calculated that measured for a better representation of the numerical model. These performances obtained by imposing a ''MCVC'' semi-active control scheme, may well be improving in terms of response time if we consider a preview control of the suspension system [9]. Conclusion The results of this analysis showed the interest to take into account internal dynamic in numerical model of Magneto-Rheological (MR) damper to predict real behaviour of this damper. Indeed, through the semi-active ''MCVC'' strategy verify the effectiveness of the modified Bouc-Wen model developed by Spencer et al., which corrects the deficiencies of Bouc-Wen model at the low speeds, during the transition between the phases of compression and expansion of the shock absorber MR, and in the case where the acceleration and velocity are of opposite sign. Indeed, by adjusting the parameters β, γ, n, and A, of the numerical model, we can control the linearity in the transition periods (pre-stress and post-stress). References [1] B. F. Spencer, S. J. Dyke, S. J. Sain and J. D. Carlson, (1997) “Phenomenological Model of a Magneto-Rheological Damper”. Journal of Engineering Mechanics, 123, pp. 230-238. [2] Y. K. Wen, (1976) “Method for Random Vibration of Hysteresis Systems”, Journal of engineering mechanics division. ASCE, vol 102, No EM2, pp 249-263. [3] S. Boukerroum, N. Hamzaoui and N. Ouali, “Experimental Validation and Adjustment of the Semi-Active Suspension Numerical Model Incorporating a MR Damping”, Applied Mechanics and Materials Vol. 232 (2012) pp 828-835. [4] B. J. Chan and C. Sandru, (2003) “A Ray-Tracing Approach to Simulation and Evaluation of a Real-time Quarter Car Model with Semi-Active Suspension System Using Matlab”, Proceedings of DETC’03, Chicago, Lilinois USA, September 2-6. [5] Rheonetic RD-1005-3 MR damper. Lord Corporation product bulletin, 2001, www.lord.com [6] Li Pang, Kamath G M, Wereley N M, “Dynamic Characterisation And Analysis Of Magnetorheological Damper Behaviour”, SPIE Conference on Passive Damping and Isolation SPIE Vol. 3327, pp 284-302, 1998. [7] W. L. Ang, W. H. Li and H. Du, “Experimental and Modelling Approaches of a MR Dampers Performance under Harmonic Loading”, Journal of The Institution of Engineers, Singapore, Vol. 44 Issue 4 2004. [8] M. T. Braz-Cézar and R. C. Barros, “Experimental and Numerical Analysis of MR Dampers”, 4th ECCOMAS, Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering, Kos Island, Greece, 12-14 June 2013. [9] R.S. Prabakar, C. Sujatha and S. Narayanan, “Optimal semi-active preview control response of a half car vehicle model with magnetorheological dampe”, Journal of Sound and Vibration 326 (2009) 400–420.
