BULLETIN OF APPLIED MECHANICS 4(15), 88-91 (2008)

AbstractThe paper deals with the creation of the phenomenological damper model using the neuro-fuzzy identification algorithm LOLIMOT. The models are identified based on the measured data of passenger car shock absorber. The different variants of model inputs are tested. The damper force of the final model depends statically on relative velocity and relative acceleration inputs. The identified model is implemented into simulation model of the car suspension. The acceleration is resolved from the implicit nonlinear equations in order to preserve causality of differential equations solution.

Index TermsIdentification, shock absorber model, neuro-fuzzy method, global damper characteristics.

INTRODUCTION HE reliable simulation model of the shock absorber must be set up based on the suitable set of identification

experiments. This is true for the model structure as well as for the model parameters. Both are tuned in order to finish with the best possible agreement between model output and the measured damper data. Because of the typical nonlinearities and complex dynamics of shock absorbers also the robustness and numerical stability of model is of great interest. The final choice of the model is always compromise, the scale of possible models is very broad [1]. Two basic concepts or directions of modeling can be applied.

Physical modeling using knowledge of shock absorber inner structure (e.g. [2]). Phenomenological modeling based on the input-output relations for important damper regimes (e.g. [3]). The physical shock absorber models are characterized

typically by equations for flow of oils through the throttle-valves and pressure changes in separate chambers. Nevertheless such models also contain some phenomenological parts, for example a simple regression relations between flow and pressure difference. Therefore the bound between the phenomenological and physical modeling is more or less fuzzy. The first type of the phenomenological models is based on the replacement of hydrodynamic system by simple mechanical model combining basic idealized components like a spring, damping element, Coulomb friction elements etc. The well known example of the models of such type is described in [3]. The second type of the phenomenological models use the general input-output modeling. Such type of modeling is presented e.g. in [4] or in [5]. The usage of the phenomenological models is impossible

in context of the shock absorber design including changing of the damper structure. On the other hand the advantage of the phenomenological models is their simple adaptability for dampers of different physical principle and for different operating conditions. Globally the shock absorber model must fulfill several basic properties.

The shock absorber is the dissipative device. The basic characteristics of the shock absorber is the nonlinear force-velocity dependence. The shock absorber has an important hysteretic behaviour. The internal damper hysteresis is typically increased by the silent block mounting.

The presented paper deals with the creation of the phenomenological damper model using the static identification based on the neuro-fuzzy identification algorithm LOLIMOT [6], [7]. The different variants of model inputs are tested, the final model is successfully implemented into simulation model. The method can be useful also for the controllable shock absorbers [8], [9], [10]. This will be topic of the future development.

LOLIMOT IDENTIFICATION APROACH

The identification approach is based on general (black-box) input-output models. There has been used the local linear neuro-fuzzy modeling approach. It is an adaptive sequence of linear models that approximates the nonlinear behaviour of the identified system by linear models valid especially in particular sub-regions of the whole region. The identification algorithm is called LOLIMOT and was developed in [6]. The new implementation was done at CTU [7]. The mathematical models of the system identified with the LOLIMOT are called LOLI-models. They are created with multiple inputs but single output. For systems with multiple outputs the LOLI-models are determined for each output exclusively. The main idea is to approximate the generally non-linear multivariable function of system output with scalar product of vector of linear functions and vector of so-called validity functions. The input space is stepwise splitted into sub-regions. Roughly speaking, each linear function approximates the system output in sub-region determined by relevant validity function. The complexity of the resulting model is given by number of members of these vectors. The output of the model can be then written generally for dynamic models including quantities in different discrete time points.

I.

T

II.

ika Z., Bauma V., Valek M., Blkovsk, A., Kukula P., Steinbauer P.

Identification of Phenomenological Shock Absorber Model Using Neuro-Fuzzy Method

BULLETIN OF APPLIED MECHANICS 4(15), 88-91 (2008)

( ) ( ) ( ) ( )(( ) ( ) ( )

( ) ( )( ) ( )) ( )

Liypnnnnipnnni

pppnnnippnnni

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M

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M

iLii

nkywkywnkuwkuw

nkuwkuwkuw

nkuwkuwkuwwyy

ypp

pp

u

u

+++++++

+++++

+++++= =

++++++++++

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=

+=

...,1...,

...,...,

222,23,22,

1111,12,11,0,

1

2121

21121

2111

1

...1......

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......1

......1~~

, (1) where iy

~

is the output of i-th local linear model (LLM), [ ( ) ( ) ( ) ( )

( ) ( ) ( ) ]TyppL

nkykynkukunkukuku

=

,...,1,

...,,,...,1, 21111u

is the vector of inputs, ( )Li u is the validity function for i-th LLM, w = [ pnnnniii ypwww +++++ ...,1,0, 21,...,, ]

T are the

parameters of i-th LLM, and M is the number LLMs. The vector of the inputs generally contains not only actual dynamic system inputs but also previous values of inputs and also the previous values of the output to reflect the dynamic order of the system. The normalized orthogonal Gaussian functions are chosen as the validity functions.

The usage of dynamic nonlinear LOLI models for long time simulation brings typically serious stability problems. They are mainly suitable for predictive models working on-line in regulators with relatively short time horizon. This is the reason for static identification (input and output quantities are taken only in actual time point) presented within the paper.

III. PHENOMENOLOGICAL NEURO-FUZZY MODEL OF DAMPER

The LOLIMOT algorithm enables systematic creation of the generalized version of the phenomenological models proposed in [5] and especially in [4] as so called global damper characteristics. The global damper characteristics is given by general static multi-dimensional dependency of damper force as a function of damper kinematical quantities.

The used training signal is given in Fig. 1. It consists on harmonic excitations with frequency from 0.1 Hz to 10 Hz with different amplitudes typical for damper working circumstances. The velocity and acceleration are reconstructed from the length measurement. This is the reason why the stochastic excitation has been omitted for the time being. The complexity of the model can be controlled by the setting of the input space splitting into sub-regions as mentioned in paragraph III. The 20 splitting sub-regions have been used (Fig. 5) based on experiments. Further splitting brings only small improvement of error between model and experiment results.

0.5 1 1.5 2 2.5 3 3.5x 104

-500

0

500

1000

1500

time [ms]

Dam

per fo

rce [N

]

Fig. 1 Model identification training signal

1.8 2 2.2 2.4 2.6x 104

-200

-100

0

100

200

300

400

500

600

700

time [ms]

Da

mpe

r fo

rce

[N

]

processmodelerror

Fig. 2 Approximation )(vFF d= for low frequencies

3.62 3.64 3.66 3.68 3.7 3.72x 104

-1000

-500

0

500

1000

1500

2000

time [ms]

Dam

per

forc

e [N

]

processmodelerror

Fig. 3 Inferior properties of )(vFF d= for 10 Hz

BULLETIN OF APPLIED MECHANICS 4(15), 88-91 (2008)

3.62 3.64 3.66 3.68 3.7 3.72x 104

-1000

-500

0

500

1000

1500

2000

time [ms]

Dam

per

forc

e [N

]

processmodelerror

Fig. 4 Approximation ),( avFF d= for 10 Hz

0 5 10 15 200

2

4

6

8

10

12 x 108

splitting

Ove

rall e

rror

F=F(v)F=F(l,v)F=F(v,a)F=F(l,v,a)

Fig. 5 Dependency of overall model error for different approximations

The figures show, that the superior LOLIMOT static approximation is reached using the velocity and acceleration inputs together (Fig. 4). The addition of the length ( )),,( avlFF d= doesnt significantly improve the response quality (Fig. 5). The further model response refinement can be obtained using direct acceleration measurement. For the time being the acceleration has been reconstructed from the length measurement, which isnt optimum. The pure velocity approximation (Fig. 2, Fig. 3) completely fails especially for the higher excitation frequencies (see 10 Hz case in Fig. 3), where the hysteresis is very significant.

CAUSAL IMPLEMENTATION OF PHENOMENOLOGICAL MODEL The main target of the prepared damper model is its usage

within the simulation model. The problem of its straightforward implementation consists in the significance of the relative acceleration input as shown in the previous chapter. The dependency ),( avFd is recommended also by other mentioned authors [5], [4], nevertheless the problem of model usage for simulation remains unanswered. The dependency cant be implemented directly to Newton Euler equations of mechanisms because of acceleration on both sides of equations. First the acceleration must be resolved from the basically implicit equations in order to preserve causality of differential equations solution.

Fig. 6 Quarter car model with damper

The simulation model implementation has been tested on the quarter car model of vehicle suspension. The basic equations for sprung and unsprung masses (Fig. 6) are

)(),( 1212121222 zzkzzzzFzm d = &&&&&&&& (2) ).()(),( 01011212121211 zzkzzkzzzzFzm d += &&&&&&&&

The acceleration of the system centroid is influenced only by external forces

)()( 010121 zzkzmm S =+ && , (3) where 221121 )( zmzmzmm S &&&&&& +=+ . The relative damper acceleration is given as 12 zzld &&&&&& = and from (2) can be written as

( ).

)(

)(),(11

1

0101

121221

m

zzk

zzkllFmm

l dddd

+

++

+= &&&&&

(4)

The system dynamics can be alternatively described by centroid acceleration Sz&& and the relative damper acceleration

dl&& , which must be resolved from the implicit nonlinear dynamic equation.

IV.

BULLETIN OF APPLIED MECHANICS 4(15), 88-91 (2008)

( ).0)(

)(),(11

1

0101

121221

=

++

+

m

zzk

lzzkllFmm

dddd&&&&&

(5)

After resolving dl&& the differential equations can be integrated either in Sz&& and dl&& or in mass accelerations

dSdS lmm

mzzl

mm

mzz &&&&&&&&&&&&

21

21

21

12 , +

=

++= . The example

of simulation results for 2Hz and 5 Hz road excitations of quarter car model are in Fig.7 and Fig. 8.

-0.06 -0.04 -0.02 0 0.02 0.04 0.06-150

-100

-50

0

50

100

150

200

250

dld/dt

F d

Fig. 7 Damper simulation for 2 Hz 5 mm road profile

-0.3 -0.2 -0.1 0 0.1 0.2-400

-200

0

200

400

600

800

dld/dt

F d

Fig. 8 Damper simulation for 5 Hz 5 mm road profile

V. CONCLUSION The phenomenological shock absorber model has been

created using the neuro-fuzzy identification algorithm LOLIMOT. The models have been identified based on the experimental data obtained by hydraulic stand tests with standard passenger car damper. The different variants of model input structure have been tested. The damper force of the final model depends statically on relative velocity and relative acceleration inputs. The model structure corresponds to so called global damper characteristics known from the literature. The usage of LOLIMOT identification brings the possibility of automated model training and testing. The static variant of identification has been chosen because of instability danger of dynamic neuro-fuzzy models used for long time simulations. The further model refinement can be obtained using direct acceleration measurement. The identified model has been implemented into simulation model of the car suspension. The acceleration is resolved from the implicit nonlinear equations in order to preserve causality of differential equations solution. The presented methodology will be further extended for controllable shock absorbers.

ACKNOWLEDGMENT

The authors appreciate the support by the project Algorithms for computer simulation and application in engineering MSM 6840770003.

REFERENCES [1] S. Duym, R. Stiens, K. Reybrouck, Evaluation of Shock

Absorbers Models, Vehicle System Dynamics 27 (2), 1997, pp. 109-127.

[2] S. Duym, K. Reybrouck, Physical Characterization of Nonlinear Shock Absorber Dynamics, European Journal of Mechanical and Environmental Engineering, Vol. 43(4), 1998, pp. 181-188.

[3] B.F. Spencer, S.J. Dyke, M.K., Sain, J. Carlson, Phenomenological Model of a Magnetorheological Damper, ASCE Journal of Engineering Mechanics, Vol. 123(3), 1997, pp. 230-238.

[4] R. Votrubec, Global damper characteristics, TU of Liberec, Faculty of Mechatronics, Informatics and Interdisciplinary Studies, Liberec 2005 (in Czech).

[5] S. Duym, Simulation Tools, Modelling and Identification, for an Automotive Shock Absorber in the Context of Vehicle Dynamics, Vehicle System Dynamics 33 (4), 2000, pp. 261-285.

[6] O. Nelles, Nonlinear system identification with local linear fuzzy-neuro models. Automatisierungs technik. Shaker Verlag, Aachen 1999.

[7] M. tefan, CTU-LOLIMOT, Research Report U2052-02-45, FME CTU in Prague, Prague 2002.

[8] R. Votrubec, Fuzzy Control of Damper with Magnetorheological Fluid, In: EUROMECH Colloquium 455 on Semi-Active Vibration Suppression, Prague, July 2004.

[9] M. Valek, Z. ika, J. Kejval, J. Honc, P.Steinbauer, Force Control of Variable Shock Absorber, Proceedings of AVEC 02, Tokyo 2002, Vol. 1, pp 141-146.

[10] M. Valek, Z. ika, J. Kejval, J. Honc, P.Steinbauer, Measurement of Controllable Shock Absorber for Dynamic Identification and Evaluation of Its Force Control, In: Proc. of ICOVP 2003, Liberec 2003, pp. 67-74.