A parametric damper model validated on a track Xabier ... - CiteSeerX

Int. J. Heavy Vehicle Systems, Vol. x, No. x, xxxx

A parametric damper model validated on a track Xabier Carrera Akutain* TECNUN (University of Navarra), P.Manuel de Lardizabal 13, San Sebastián (E-20018), Spain Fax: +34 943 311442 E-mail: [email protected] *Corresponding author

Jordi Viñolas and Joan Savall CEIT and TECNUN (University of Navarra), P.Manuel de Lardizábal 15, San Sebastián (E-20018), Spain Fax: +34 943 213076 E-mail: [email protected] E-mail: [email protected]

Jorge Biera AP Amortiguadores, S.A., Ctra. Irurzun, s/n, (E-31171) Ororbia, Spain Fax: +34 948 322353 E-mail: [email protected] Abstract: This paper presents the design and experimental validation of an explicit parametric model for mono tube dampers. The aim is to develop a model with few physical parameters, in order to make it both easy to manage and computationally light. Firstly, the model is validated with the help of a dynamometer. In addition, advanced experimental work has been developed, which makes the model match the desired results in an acceptable manner during real driving maneuvers with a single seater sports car on a track. A good correlation has been achieved in real time, so it allow us to have a good feeling about its further use as subroutine on full model simulations or on semi-active suspension control systems. This study shows custom built experimental tools and techniques used to validate the model as well as results illustrated on the figures presented. Keywords: fast explicit model; experimental tools; validation on a track. Reference to this paper should b made as follows: Carrera Akutain, X., Viñolas, J., Savall, J. and Biera, J. (xxxx) ‘A parametric damper model validated on a track’, Int. J. Heavy Vehicle Systems, Vol. x, No. x, pp.xxx–xxx. Biographical notes: X. Carrera Akutain is a Research Student and PhD candidate in the school of Mechanical Engineering at Tecnun (University of Navarra). He received his MS degree from Tecnun in 2001 and has been Assistant Professor of Machine Elements since 2002. His research interests are in the areas of Vehicle Dynamics, Data-acquisition and Mechatronics.

Copyright © 200x Inderscience Enterprises Ltd.

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X. Carrera Akutain, J. Viñolas, J. Savall and J. Biera J. Viñolas is at present the Head of the Applied Mechanics Department at CEIT (Centro de Estudios e Investigaciones Técnicas) and Associate Professor at the Faculty of Engineering at Tecnun (University of Navarra). He obtained his Ph.D. in 1991. The title of the thesis was A New Experimental Methodology for Testing Active and Conventional Active Suspensions. He has been involved in different research projects related with vehicle dynamics, noise and vibration. J. Savall is a Research Staff Member at CEIT and Assistant Professor of Machine Theory at Tecnun (University of Navarra) Engineering School. He received his MS degree in Mechanical Engineering from the University of Navarra in 1995. He was a Visiting Researcher in 1996 at PMA in Belgium, where he started out in Mechanical Design for Mechatronics. He has several patents pending. His research interests include Mechanical Design, Robotics and Mechatronics. J. Biera received his MS degree in Mechanical Engineering from the University of Navarra in 1992 and his PhD degree in Mechanical Engineering from the University of Navarra in 1997. He has been on the engineering staff of AP Amortiguadores, SA since 1998 and is the Manager of the Research and Development department since 2000.

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Introduction

Dynamics of dampers are known to strongly influence vehicle ride comfort, handling and the level of noise, vibration and harshness (NVH). If the aim is to get a good handling and braking, tyre-road forces need to be stable. For a better riding comfort, vibrations induced by road profiles should cause minimal discomfort to passengers. These two features are often antagonistic and dampers are one of the most critical components that can be used to reach a satisfactory compromise. Dampers, also called shock absorbers, behave in a non-linear and time-variant manner and consequently models are not straightforward. Damper designers need to introduce non-linear characteristics in their models in order to satisfy the conflicting requirements of comfort and vehicle handling. Force-velocity diagrams are the most powerful method to represent damper dynamics. However, the behaviour of the car on a road relates very much not only to some absolute values of these diagrams but also to the shape of the curves, which are influenced by the disturbing frequency and amplitude of the imposed motion and by temperature. Moreover, hysteresis loops show up due to the compressibility of the damper fluid and the state changes of entrapped gases, resulting in a finite area enclosed within the curves. This leads to avoidance of linear or multi-linear approaches and to the use of force-displacement diagrams and force history (force-time) plots if a good characterisation and understanding of the damper is the goal. Models can be implemented on full car simulators in order to analyse car dynamics with a virtual damper, as well as on board on a car with an active or semi-active suspension. They need to respond to load disturbances in real time, so very complex models are not suitable because of their needs of memory and computational time. Most detailed damper models are those that relate their parameters directly to the physical properties of each damper element. These mechanistic models help to improve our understanding of the damper’s behaviour. Manufacturers can study how shock absorbers react to changes in physical parameters, both in the design, and in the testing

A parametric damper model validated on a track

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phases. The most careful theoretical study of an absorber is that of Lang (1977). His detailed physical model was based on the processes of the oil/gas flow through the various internal chambers. He described the behaviour of the damper in a broad range of operating conditions. Even though the results of the model with 87 parameters showed good agreement with experimentation, it was uneconomical in terms of computational and modelling complexity. It is not suitable for comprehensive vehicle simulation studies. However, this model was implemented in a quarter car vehicle simulation by Hall and Gill (1986) and it was established that typical linear and bi-linear models of dampers are inaccurate and lead to overestimated ride performances. Elementary models constructed with combinations of spring and ideal viscous damping elements fit well with certain measurements (Besinger et al., 1995) but in this case, simulations need numerical computation with differential equations, which can slow down parameter identification and simulation significantly. Moreover, as they are inherently non-tuneable they do not behave as a predictive tool for damper manufacturers. The same principle applies to non-parametric models, which do not depend on an a priori model of the structure. These black box methods make use of an elevated number of measurements correlating parameters without definite physical meaning. Such restoring force models characterise force as a 2D polynomial function of displacement and velocity but it must be taken into account that force-state maps are frequency dependent. Worden and Tomlinson (1992) limited the restoring force methods to unifrequent excitations, producing isofrequency maps which were made of amplitude varying sinusoidal motions with the same frequency. Differences between several isofrequency maps are resolved thanks to the so called ‘difference maps’. The aim of this paper is to create a ‘white-box’ explicit physical model to directly relate valve parameters to the resulting force of the studied shock absorber. This gives flexibility and simplicity to generate models for different shock absorber types; suitability for use in vehicle simulations and it also allows the model to behave as a fast predictive tool. Characterisation studies carried out by Duym and Reybrouck (2000) and Duym (1998) and in a similar manner by Simms and Crolla (2002) generate mechanistic or physical models more readily identifiable than those presented by Lang (1977) and Hall and Gill (1986). They consist of a set of first order non-linear differential equations, which determine internal pressures of the oil and the relations between pressure and oil flow. This results in an implicit function of pressure and flow. In order to avoid gross numerical solving, equations are developed partially into a Taylor series, which leads to explicit expressions. Otherwise, it would require huge iterative loops and thus make the model unsuitable for being implemented in real time vehicle simulations. These models have the advantage that parameters are identified only from dynamometer measurements. It is assumed that the user of the model has no access to open samples of the damper, although the damper manufacturer should provide some basic information regarding internal diameters and oil viscosity. Parameter identification is carried out so that they minimise the difference between the simulated force and the measured force, applying the least square method. Fluctuations of oil temperature modify damper dynamics, even though this variation is not generally taken into account for the sake of simplicity. Previously Reybrouck (1994) developed a model that derives also from others based upon more complex hydraulic theories, and gave a simpler explicit expression for the damper force as a function of displacement, velocity and acceleration. This model contains 14 valve parameters, seven for each displacement sense. Three of them are

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X. Carrera Akutain, J. Viñolas, J. Savall and J. Biera

simplified stiffness of leak, port (both permanent passages) and blow off valves. These last ones are opened by pressure. The blow off valve needs a preload parameter, another is needed to indicate friction of the damper and there are also two empirical coefficients that do not have such a physical meaning: a correlation coefficient of the acceleration with hysteretic effects and a factor connecting the effect of the leak valves to the effect of the blow-off valves. The model assures reliability between 0.5 Hz and 30 Hz. Yung and Cole (2001) researched damper behaviour at high frequencies (above 30 Hz) utilising wavelet analysis in order to develop a better mathematical prediction than that existing at the moment. The generation of high frequency harmonic forces appeared to depend strongly on the valve opening characteristics and on the behaviour at low velocities, where reversal of flow and friction are predominant. The simulation results were in good agreement with experimental data for low frequencies (under 4 Hz). Errors of more than 15% (and increasing with frequency) in the spectral density were found above 4 Hz. They later extended the frequency range, by replacing an existing Coulomb friction model with an improved friction model (Yung and Cole, 2003). Wavelet analysis showed significant improvement at frequencies between 50 Hz and 500 Hz. In this paper an explicit and parametric model for mono tube shock absorbers is shown, developed upon the same basis of the last mentioned ones. It has been tested not only in a dynamometer but also with real track data of a racing car. It is intended to achieve an accurate and manageable model in order to make it suitable for full vehicle simulations and for routines of a semi-active suspension. It will not result in complications due to excessive calculus and management and will allow simple parameter identification from the physical point of view. Software containing a user-friendly interface, called Damper-Testbench, has been developed in Matlab/ Simulink environment to make the model fit easier.

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Description of shock absorbers utilised

Hydraulic mono tube shock absorbers were used as a first set, from now onwards called ‘set A’. These were of the telescopic mono-tube type, with 30 adjustable discrete positions and a gas tank. A shock-absorber manufacturer supplied a second set of shocks (set B), specifically designed prototypes based also on mono-tube variable technology. The same tube is used for extension and compression strokes and a floating piston produces physical separation between hydraulic and gaseous phases. Oil flows between rebound and compression chambers through a piston connected to the rod of the damper. The reserve chamber is separated from the compression chamber by a floating piston containing gas at around 20 bars. A selector with 17 different positions was implemented. This regulation is possible due to restrictions to oil passing areas: a rod and an attached valve rotate concentric to a drilled tiller, leaving oil passages partially open or closed. In Figure 1 an intermediate position with mentioned holes semi-overlapped is shown. These prototypes are double piston equipped, as they contain a primary piston and on the other hand, a conventional double effect piston, which gives the necessary load for rebound and compression strokes (see Figure 2). The amount of oil that each piston receives depends on the tiller’s hole levels. Positioning differences between the tiller and the rotating valve means not only restriction to oil flow but it determines also which piston

A parametric damper model validated on a track

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(or even both) is currently working. As they have different valve operating systems, the subsequent working modes of the damper can be achieved. Figure 1

Tiller holes. Intermediate position

Figure 2

Primary and conventional pistons

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The model

The model had to be efficient and meticulous but at the same time not so heavy as to embarrass simulation or process time; critical if the aim is to work on real time with an on board processor. An examination of the internal parameters of the damper must be avoided for the characterisation of the dynamic response of the shock absorber, as inner mechanics are considered unknown. The model itself consists of a total of 19 parameters. The force through the damper is a combination of hydraulic (Fdamping), gas (Fgas) and frictional forces (Ffriction). Fdamper = Fdamping + Fgas + Ffriction

(1)

Hydraulic force Fdamping is a function of area and blow off valves, viz. Fdamping =

Fleak × Fblow -off K tr

( Fleak ) K tr + ( Fblow -off ) K tr

Fleak = Kleak × υ0.25 × y 1.75

+ Fport

(2) (3)

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X. Carrera Akutain, J. Viñolas, J. Savall and J. Biera Fport = Kport × υ0.25 × y 1.75

(4)

Fblow-off = Fpreload + Kspring × y

(5)

The piston valve assembly is modelled as a combination of orifice type restrictions and an ideal blow-off valve. Parameters like Kport and Kleak relate to leak and port area valve systems and forces through this restriction. The leak restriction is smaller than the port one and it produces high resistance to oil flow with increasing velocity (Yung and Cole, 2001). Low velocity data (below 100 mm/s) are needed to adjust this parameter, as its influence range is very narrow, especially in mono tubes (zone 1 in Figure 3). Hence, noise and friction intervention must be taken into account. The port restriction presents negligible resistance at low damper velocity but it becomes the most important factor at high velocities. The individual valve pressure drop/flow rate relationships of the port and leak restrictions can be characterised by simple power law equations. Reybrouck (1994) proposed an exponent of 1.75 for turbulent flow (equations (3) and (4)). Figure 3

Force-velocity diagram zones

Port holes are predominant at high velocity (over 500 mm/s and not easy to get on every test) so, they become an important factor once the change of slope between zones 3 and 5 (zone 4 in Figure 3) is reached. As this point is not easy to visualise and the simplification is rather coarse, the best mode of fitting curves is a manual process utilising two or more high speed curves. Blow-off valves remain closed until pressure drop through parallel arranged (see Figure 4) leak valves exceed the preload of the spring of the blow off valve. For low flows, the oil passes through the leak. As flow paths are parallel, beyond that pressure drop, the blow off valves open and leak valves begin to lose importance. A switch set on 7 mm/s for transition zone 2 (1–3) is considered for the model. Below this switch a very high preload value is taken in order not to interfere leak values. Reybrouck (1994) considers pressure drop in blow-off valves to be proportional to a constant stiffness multiplied by the velocity of the piston equation (5). In spite of the rough simplification, this explicit expression seems to match quite well desired results. Medium velocity data is the most suitable one to adjust this parameter. As real dampers often do not show a sharp transition at the opening of the valves, an empirical parameter called Ktr is introduced. It joins leak and blow off pressure drops (forces). It makes the relation

A parametric damper model validated on a track

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showed in equation (2) closer to Fleak or to Fblow-off and also smoothens transition in the vicinity of the switch zone. Figure 4

Piston valving system scheme

As in zone 1 (Figure 3), the Fblow-off is forced by the model to take a very large value, the resulting force of both valves is almost equal to Fleak. Once the switch velocity is surpassed, equation (5) is applied to Fblow-off. The pressure drop over this combination of valves is added to the pressure drop over the port restriction equation (2). Friction force Fforce has the same sign as the damping hydraulic force. The friction of the damper is measured on a test-rig at very low speed, trying to avoid performance of shock’ hydraulics but it is not easy to fix a standard for this test or even to gauge it. Frictional terms have a direct relation with riding comfort. In this model, the same switch utilised for the blow-off valves is calibrated at 7 mm/s for friction. A dry Coulomb friction value equal to parameter Ffriction is selected beyond this velocity switch and a viscous type, below (Figure 5). Other shock absorbers may have a different switch value but generally mono-tubes behave approximately in this manner. Figure 5

Friction model

In the list of parameters of Table 1 parameters y(1) to y(19) are detailed. A data acquisition system records the displacement of the shock absorber, (y(1)). Afterwards, this is derived into velocity offline (y(2)). Parameters y(15) to y(19) are directly

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X. Carrera Akutain, J. Viñolas, J. Savall and J. Biera

measurable and specific for each shock absorber unit/type. The static force of the gas is measured on a test rig. The area of the damper, the static volume of the gas and the viscosity of the oil are data that must be supplied by the manufacturer. Blow-off and frictional switches and hysteretic constants are discovered from the visualisation of low velocity data and then applied in the code of the custom made software. They should not vary excessively for different mono-tube damper types. Table 1

List of parameters

Fixed parameters Code Name Description y(1) Disp. Damper Displacement (positive in compression) y(2) Velocity Damper Velocity (positive in rebound) y(15) Fgas_static Static Gas Force y(16) Vgas_static Static Gas Volume y(17) Arod Rod Section y(18) Kinematic Viscosity of Oil ν y(19) Travel Total Travel of the Damper Parameters to identify (rebound and compression) Reb. Comp Name Description

Units m m/s N m3 m2 mm2/s mm Units

y(3) y(4)

y(9) y(10)

K_leak F_preload

Leak restriction stiffness Preload of the blow-off spring

kg/(mm0.5 m0.75) N

y(5)

y(11)

K_spring

Spring stiffness

kg/s

y(6) y(7) y(8)

y(12) y(13) y(14)

K_tr K_port F_friction

Leak to blow-off transition parameter Port restriction stiffness Friction force

– kg/(mm0.5 m0.75) N

The parameters from y(3) to y(14) (six for compression and six for rebound) must be then discovered for a correct model fitting. Experience helps to provide a starting point for this purpose. Except for the ‘leak to blow off parameter’ Ktr, the remaining parameters own a direct physical meaning, therefore, this kind of analysis is favoured against pure mathematical approaches like least squares or simplex methods. A fast modelling software tool is developed and implemented in a Matlab/Simulink environment. Different simulations can be developed to analyse a track data history and to adjust a model to these data on a highly manageable graphical interface (Figure 6). This window enables the user to load data, select different formats, run simulations, change parameters, filter and visualise results and other features related to shock absorber analysis. Once data are loaded, it is possible to run a simulation. The user will be requested to use filters or not. If he does, a modifiable text box shows the cut off frequency of a butter worth, low pass third order filter. The user must take into account the selection of the mode. The filtered mode often achieves smoother results but at the expense of greater computational time. Simulink is forced to run an ode4 (Runge-Kutta) solver, with a fixed integration step size equal to the recorded data rate, with some limits, in order to avoid either rough or extremely reduced step sizes. If data is recorded under 100 Hz, the software imposes a minimum step size of 0.01 s. Analog applies to upper limit (max. 1000 Hz). Track data recorded at 100 Hz shows acceptable accuracy.

A parametric damper model validated on a track Figure 6

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Analysis software

At this very moment of the analysis, it is possible to fit model outputs to the imaginary line that could be drawn within measured force signal curves, since no hysteretic effects have been yet introduced as can be seen in Figure 7a. The model considers hysteresis due to two factors: •

Mono-tubes contain a gas chamber to compensate volume interchange between rebound and compression chambers (Figure 8). It produces a static force even when the damper is not excited. When it is, hysteresis loops appear. Gas pressure can be predicted from an isentropic process equation, which needs instantaneous values of dynamic volume. Flow entering compression chamber: Atube × ( y − z )

(6)

Volume gain (see Figure 8): (Atube – Arod) × y

(7)

then, z =

Arod × y Atube

(8)

Vdyn = Vstatic − Atube × z = Vstatic − Arod × y Fgas, dyn =

•

(9)

Fgas, static × V

1.4 gas, static

(Vgas, static − Arod × y )1.4

(10)

In addition, the filtered mode of the analysis tool makes use of the acceleration signal (displacement derivative twice) multiplied by a constant. This procedure tries to simulate the effects of oil compressibility through the leak valve passages. The non-filtered mode does not make use of this hysteresis factor because of the hazard inherent in non-filtered signal derivatives. It is not difficult to implement one accelerometer on each shock absorber in order to avoid derivatives, but the minimal benefit is not worth the extra work. As it requires lower CPU time, the non-filtered mode becomes very suitable for long time or high data rate fast simulations.

10 Figure 7

X. Carrera Akutain, J. Viñolas, J. Savall and J. Biera Force-velocity diagrams (a) without and (b) with hysteresis. Set B

(a)

(b)

A parametric damper model validated on a track Figure 8

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Damper chambers

Experimental tools

Input data were collected from dynamometers and track shakedowns. The testing vehicle was a Car-Cross single seater, a kind of sports car with a double wishbone suspension scheme, which allows large suspension displacements (Figure 9). Regarding measurements on a track, deflection of the suspension was gauged with four potentiometric displacement transducers. A connecting wire between the outers of the damper, coils a pulley acting on a linear potentiometer. The achieved sensibility is 4.2 mV/V/mm, and linearity almost 100% (Alonso Marichalar, 2000). A CAD package, using Pro-Engineer, of the front of the car, including detail of the displacement sensor, can be seen in Figure 10.

12 Figure 9

X. Carrera Akutain, J. Viñolas, J. Savall and J. Biera Melmac car cross on a track. Driver: Ander Vilariño

Figure 10 CAD model of the vehicle. Detail: displacement sensor

Model outputs of the model with displacement signals of a track were compared with measurements from four on board set force transducers. The front and rear suspensions are equipped with differently designed parts but all based on strain gauges (Full Wheastone bridge) (Carrera Akutain, 2001). The front right force transducer, part of the displacement sensor and a suspension containing the shock absorber are shown in Figure 11. The force due to spring compression, calculated as the product of measured displacement and real spring stiffness needs to be subtracted from the force transducers’ signals, in order to get the net force through the damper. A two post rig was also used to perform futher testing, as can be seen in Figure 12.

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Figure 11 Front right scheme

Figure 12 Test-rig

(a)

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(b)

Results and discussion

The software interface contains an editable box to change preload of the spring or to compensate possible calibration offset deviations. In this manner the model output and the measured force can be set at the same level. A correlation coefficient between model output and transducer signal is showed in the force-displacement plot. At the top left border of the interface (Figure 6), a results list is created as long simulations are carried on. Run number, parameters, data and the filtering mode are listed. In Figures 13 and 14 it can be seen that the model behaved well for the first set of dampers on a dynamometer and on the two-post test rig, respectively. A force-velocity diagram of the second set of shock absorbers on medium hard position can be seen in Figure 7(b), this time including the aforementioned hysteresis. The entrapped gas

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X. Carrera Akutain, J. Viñolas, J. Savall and J. Biera

produces a closed loop all trough the curve and the effect of acceleration makes it faster in the low velocity zone. Figure 13 Force-displacement plot. Dyno: Sine input (v = 300 mm/s; f = 1.5 Hz). Set A

Figure 14 Force-displacement plot. Rig: Step input (50 mm). Set A

In the force-time diagrams of Figures 15 and 16, results of the same simulation of free driving without and with filtering can be seen, respectively. The filtered mode shows a ‘cleaner’ and a more damped shape, although they do not differ in excess. The correlation coefficient is always over 0.90 for track data and the non filtered mode runs,

A parametric damper model validated on a track

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of course, much faster. In the next two plots we can observe a skid-pad with the damper in a soft position (Figure 17) and response to full throttle acceleration and heavy braking in a hard position (Figure 18). Figure 15 Free driving. Hard position. Non-filtered. Set B

Figure 16 Free driving. hard position. filtered. Set B

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X. Carrera Akutain, J. Viñolas, J. Savall and J. Biera

Figure 17 Skid pad driving. soft position. Non-filtered. Set B

Figure 18 Acceleration and braking. hard position. Non-filtered. Set B

A parametric damper model validated on a track

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Employed simulation times are showed shown in Table 2 for a couple of simulations. Configuration utilised: Pentium 4 CPU 2.00 GHz; 512MB SDRAM; W2K OS. As it can be seen, real time simulations run near to 900 Hz. The simulation time can be slightly reduced by employing a dynamic link library (dll) for the equations of the model instead of a Matlab function call (up to 6%). The results of Table 2 are those obtained using Matlab. Table 2

Simulation times

Filter mode

Step size(s)

Non-filtered Non-filtered Filtered Filtered

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Sim time(s)

Run time(s)

0.01

100

10.10

0.001

1.33

1.46

0.01

100

36.11

0.001

1.33

4.99

Conclusions and further research

The proposed model has proved to be suitable for shock absorber characterisation. Thanks to its simplicity, physical meaning and ease of use of the model, a user with some background regarding dampers can adjust a shock absorber in a short period of time. Fitting the same damper for another selection lies almost direct (only Kleak and Kport must be varied). The model matches imposed criteria relative to simplicity, accuracy and speed and it has shown enough agreement with real track data collected with experimental tools to evolve towards a full model subroutine. As the model runs fast on a standard PC, it lets us have a good feeling about further projects regarding vehicle multibody simulations and semi active suspensions currently being developed.

Acknowledgments The author wishes to thank Mr. Juan Alberdi Urbieta, who started this labour and made a great contribution to it. He currently works for AP Amortiguadores , S.A., the firm that helps in the support of this research.

References Alonso Marichalar, A. (2000) Master Thesis, Cálculo, Construcción Y Ensayo De Un Sensor De Desplazamiento, Mechanical Department, University of Navarra TECNUN, San Sebastian. http://www.tecnun.es/automocion/proyectos/sensor_desplazamiento/inicio.htm. Besinger, F.H., Cebon, D. and Cole, D.J. (1995) ‘Damper models for heavy vehicle ride dynamics’, Vehicle System Dynamics, Vol. 24, pp.35–64. Carrera Akutain, X. (2001) Determinación Experimental De Las Cargas Existentes Sobre La Estructura De Un Vehículo Automóvil, Mechanical Department, University of Navarra TECNUN, Master Thesis, San Sebastian. http://www.tecnun.es/automocion/proyectos/sensor_fuerza/inicio.htm. Duym, S. (2000) ‘Simulation tools, modelling and identification, for an automotive shock absorber in the context of vehicle dynamic’, Vehicle System Dynamics, Vol. 33, pp.261–285.

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Duym, S. and Reybrouck, K. (1998) ‘Physical characterization of nonlinear shock absorber dynamics’, European Journal Mech. Eng., Vol. 43, No. 4, pp.181–188. Hall, B.B. and Gill, K.F. (1986) ‘Performance of a telescopic dual-tube automotive damper and the implications for vehicle ride prediction’, IMechE 200, D2, pp.115–123. Lang, H.H. (1977) A Study of the Characteristics of Automotive Hydraulic Dampers at High Stroking Frequencies, Mechanical Engineering, The University of Michigan, Ann Arbor, p.288. Reybrouck, K. (1994) ‘A non linear parametric model of an automotive shock absorber’, Vehicle Suspension System Advancements SAE SP-1031, Warrendale, pp.79–86. Simms, A.J. and Crolla, D.A. (2002) The Influence of Damper Properties on Vehicle Ride Behaviour, SAE Paper 2002-01-0319, pp.69–80. Worden, K. and Tomlinson, G.R. (1992) ‘Parametric and nonparametric identification of automotive shock absorbers’, Proc. of the 10th. International Modal Analysis Conference (IMAC), San Diego, CA USA. Yung, V.Y.B. and Cole. D.J. (2001) ‘Analysis of high frequency forces generated by hydraulic automotive dampers’, 17th IAVSD Symposium, Copenhaguen, Denmark. Yung, V.Y.B. and Cole. D.J. (2003) ‘Mechanisms of high-frequency force generation in hydraulic automotive dampers’, 18th IAVSD Symposium, Kanagawa, Japan.