Dynamics, Structures and Design Group
Chebyshev Polynomials Fits for Non-parametric Model
of Magneto-rheological Damper
Hassan A. Metered1, Dr Philip Bonello2 and Dr S.O. Oyadiji2
1
Ph. D student, The University of Manchester, Manchester, UK.
2Mech., Aero.
and Civil Engineering School, University of Manchester, Manchester, UK.
Introduction
The magneto-rheological (MR) damper is one of the most promising new devices for
vehicle vibration reduction. Because of its mechanical simplicity, high dynamic range,
low power requirements, large force capacity and robustness, this device offers a
compromise solution for the two conflicting requirements of ride comfort and vehicle
handling. In the present work a new approach for studying the dynamical behaviour of
MR damper is presented. It consists of a procedure for the application of Chebyshev
orthogonal polynomial fits for damping force generated by numerically solving the
Bouc-Wen model as a function of displacement and velocity at constant voltage. Also,
a three dimensional fit is proposed for expressing the damping force as a function of
displacement, velocity and input voltage. For all cases, the use of the Chebyshev
polynomials fits expression is seen to be very efficient and precise for MR damper.
Then, use the function “griddata” available in MATLAB to construct the surface plot as
shown in Fig.1 & 2. Finally, substitute in the two equations of Ck,l,z and F respectively.
a
1500
Chebyshev
Bouc-Wen
1000
Force (N)
500
MR Damper Model
0
-500
Spencer Jr. et al. [1] proposed a modified Bouc-Wen model (see Fig. 1), which avoided
-1000
most of the problems observed in the other models.
-1500
0
1
2
3
Time (sec)
4
5
6
b
In this model, the damping force F
1000
Force (N)
Fig. 2 Surface plot according to set 1
produced by the MR device is given by
  k1 ( x  xo )
F  c1 y
500
0
-500
-1000
-0.01
where y is governed by the following equations
-0.005
0
Displacement (m)
c
Fig.1 Modified Bouc-Wen model
0.005
0.01
1000
1
[z  co x  ko ( x  y )]
co  c1
y
 z
z   x
n 1
y
z
z   x
n
Force (N)
y 
0
-500
-1000
y
)
 A( x
-0.1
Fig. 3 Surface plot according to set 2
In the above equations, parameters α, c1 and co are directly related to the voltage u
applied to the MR damper in a manner shown below,
   (u )   a   bu, c1  c1 (u )  c1a  c1bu,
500
-0.05
0
0.05
Velocity (m/sec)
0.1
0.15
Fig. 4 Comparison between Bouc-Wen model and
Chebyshev polynomials fit (30 * 30 terms)
according to set 1 and 2
co  co (u )  coa  cobu
where u is given as the output from a first-order filter as follows
   (u  v )
u
in which, v is the command voltage signal sent to the current driver.
Fig. 5 Some surface plots according to set 3 at different voltages
The corresponding parameters of the MR model that used in this research are obtained
4500
The main idea of this section is based on the remark made above. As functions of three
variables, the damping force can be approximated as triple series involving variables
displacement (x), velocity (x ) , and input voltage (v).
ˆ ( x, x
 , v)  F
 , v) 
F ( x, x
K , L,Z
C
k ,l , z  0
 )Tz (v)
Tk ( x)Tl ( x
klz
where Cklz is Chebyshev coefficients that governed by the following equation [3]
C k ,l , z 
1500
1500
0
-1500
-3000
-3000
-4500
-4500
0
1
2
3
Time (sec)
4
5
6
Fig. 6 Comparison between Bouc-Wen model and
Chebyshev polynomials fit (20 * 20 * 20 terms)
according to set 3
0
1
2
3
Time (sec)
4
5
6
F ig. 7 Comparison between Bouc-Wen model and
Chebyshev polynomials fit (20 * 20 * 20 terms)
according to set 4
Conclusions and Recommendation for Future Work
The Chebyshev polynomials fits method for characterizing the non-linear dynamical
behaviour of MR damper was presented. Displacement, velocity, and voltage
dependency of the MR damper are highlighted by Chebyshev polynomials fit very
effectively, and being a fast system identification method. The non-parametric results
based on Chebyshev for MR damper are presented for all cases in table 1. These results
are compared with those of Bouc-Wen model. An excellent match is observed in the
MR damper behavior for all cases. Extended work with Chebyshev polynomials fit to
express voltage as a function of displacement, velocity, and damping force.
Q Q
Results and Discussion
Hassan Metered
0
-1500
8
(1   k 0 )(1   l 0 )(1   z 0 )Q Q Q
A nonlinear simulation technique using MATLAB 7.1 and SIMULINK is used to solve
the Modified Bouc-Wen model and compare it with Chebyshev polynomials fits. In
order to validate the non-parametric study that proposed in this study, a series of
validation data sets are defined in table 1. Predicted damping forces for the MR fluid
damper using the Chebyshev Polynomials Fit model and the Modified Bouc–Wen
model with the first 2 sets are shown in Fig. 4, in which the damping force versus time,
displacement, and velocity are plotted. To apply the Chebyshev polynomials fit, firstly,
a surface plot must be plotted to get the damping force as a function of displacement
and velocity (2-dimensional problem) by solving modified Bouc-Wen model.
Force (N)
3000


 2s  1 
2 j 1 
ˆ ( ,  , ) cos 2i  1  * cos




* F
*
cos






i 1 j 1 s 1
 2Q 
 2Q 
 2Q 
Q
Chebyshev
Bouc-Wen
3000
Force (N)
from ,Wang and Liao 2005, [2].
Chebyshev Polynomials Fits
4500
Chebyshev
Bouc-Wen
References
1-
Spencer, B. F. Jr., Dyke, S. J., Sain, M. K. & Carlson, J. D.,” Phenomenological Model for Magneto-Rheological
Dampers” , 1997, Journal of Engineering Mechanics, Volume: 123, pp 230–238
2-
Wang, D. H., & Liao, W. H.,” Testing and Steady State Modeling of a Linear MR Damper Under Sinusoidal Loading”, 2005,
Journal of Smart Materials and Structures, Volume: 9, pp 95-102
3-
F. A. Rodrigues, F. Thouverez, C. Gibert, & L. Jezequel,” Chebyshev Polynomials Fits for Efficient Analysis of Finite Length
Squeeze Film Damped Rotors”, 2003, Journal of Engineering for Gas Turbines and Power, Volume: 125, pp 175-183