![](bg2.png)
Table 1: Comparison of MR damper models. Power col-
umn shows if the research work studied the generated power
(
), or did not ( ). Energy column same meaning as
power column. Finally, the reference are cited.
Power Energy SSS+C training patterns
X X (Spencer et al., 1996)
X X (Li et al., 2000)
X (Wang and Kamath, 2006)
(Shivaram and Gangadharan, 2007)
X X (Guo et al., 2006)
X X (Nino-Juarez et al., 2008)
Power Energy BWN+C training patterns
X (Burton et al., 1996)
(Wang and Liao, 2001)
(Savaresi et al., 2005)
Power Energy BWN+BWN training patterns
(Wang and Liao, 2001)
(Chang and Zhou, 2002)
X X (Du et al., 2006)
Sinusoidal Sweep Signal (SSS) with specific fre-
quency and a Constant electric current. This is a typ-
ical training input for MR damper models. Exploiting
this pattern, both Energy (E) and Power (P) are suc-
cessfully simulated. The number of experiments is
high. The obtained model accuracy is high (5% er-
ror). There are not electrical current transients, this
compromises the use of the model. Table 1 shows six
representative works with this type of training inputs.
Section two (BWN+C). The displacement is a
bandwidth BandWidth Noise (BWN) pattern and a
Constant electric current (C). This training input pat-
terns has the same features as SSS+C signals. How-
ever, the information richness due to the magnitude of
displacement is decreased. Power and Energy simu-
lation are not achieved.
Section three (BWN+BWN), both displacement
and electric current follow a bandwidth noise BWN
pattern. Power and energy simulation are not
achieved, except if a displacement greater than 10 mm
is generated, (Du et al., 2006). The pattern requires
shorter training inputs. The fitting error this pattern is
low (3%).
The BandWidth (BW) in all the reviewed works
was lower than 6 Hz, except in (Savaresi et al., 2005)
and (Nino-Juarez et al., 2008). Hence, the MR damper
response to high frequency has not been explored.
Neither, hard non-linearities due to the broken mag-
netic bounds of metallic particles (because of high
frequency and displacements). There are missing
analysis of power and energy responses in automo-
tive applications for frequencies around 10-15 Hz and
displacement greater than 10 mm.
There are research works with other type of pat-
terns, such as, Amplitude Pseudo Random Binary
Signal APRBS in electric current (Savaresi et al.,
2005). (Wang and Liao, 2005). explored electric cur-
rent with sinusoidal wave signals.
There are not a standard definition of training pat-
terns in order to identify the power and energy fea-
tures. The overuse of the MR damper due to long ex-
perimental exploration at electric current greater than
3 amperes could give a skewed model. Therefore,
more research of training patterns for MR damper
modelling is needed.
2.2 Modeling Approaches
Several models have been developed with differ-
ent approaches. These models could be: phe-
nomenological (P), semi-phenomenological (SP) and
black-box (BB) (neural network, fuzzy, non-linear
ARX, polynomial among others). The training pat-
tern will be tested with both non-linear with Auto
Regressive eXogenous inputs (NARX) and a Semi-
Phenomenological models. A brief review of these
models will be included for completeness.
Table 2: Description of variables for DoE.
Variable Description
x
k
or x Damper piston displacement
I
k
or I Electrical current
˙x
k
or ˙x Damper piston velocity
f
MRk
or f
MR
Damping force
a
j
j-esime modeling coefficient
d time delays
q
1
, q
2
Electrical current exponents
ESR Error-Signal-to-noise Ratio
k Discrete sample, discrete time
j Subindex
The MR damper model based on a non-linear
ARX structure is a lineal combination of a vector
of delayed inputs multiplied for their parameters. If
electric current is not an input, all the parameters have
a polynomial dependence on it.
In (Nino-Juarez et al., 2008), a non-linear ARX
model of nine parameters (1) achieves high precision
simulation of power and energy. Table 2 defines the
parameters of this equation.
f
MRk
= a
1
f
MRk−1
+ a
2
f
MRk−2
+ a
3
f
MRk−3
+a
4
x
k−1
+ a
5
x
k−2
+ a
6
x
k−3
+a
7
˙x
k−1
+ a
8
˙x
k−2
+ a
9
˙x
k−3
(1)
By the side of Semi-Phenomenological (SP) ap-
proaches, the bi-viscous and hysteretic behavior are
shaped with smooth and concise forms. The instanta-
neous force is delivered without taking into account
BUILDING TRAINING PATTERNS FOR MODELLING MR DAMPERS
157