Optimal Vehicle Suspension Characteristics for Increased Structural Fatigue Life

Available online at www.sciencedirect.com

Journal

www.elsevier.com/locate/jterra

Journal of Terramechanics 48 (2011) 397–408

ofTerramechanics

Optimal vehicle suspension characteristics for increasedstructural fatigue life

Braham Breytenbach, Pieter Schalk Els ⇑

Department of Mechanical and Aeronautical Engineering, University of Pretoria, 0002 Pretoria, South Africa

Received 8 December 2009; received in revised form 5 January 2011; accepted 26 September 2011Available online 20 October 2011

Abstract

Heavy off-road vehicle suspension systems face unique challenges. The ride comfort versus handling compromise in these vehicles hasbeen frequently investigated using mathematical optimisation. Further challenges exist due to the large variations in vehicle sprung mass.A passive suspension system can only provide optimal isolation at a single payload. The designer of such a suspension system must there-fore make a compromise between designing for a fully-laden or unladen payload state. This work deals with suspension optimisation forvehicle structural life. The paper mainly addresses two questions: (1) What are the suspension characteristics required to ensure optimalisolation of the vehicle structure from road loads? and (2) If such optimal suspension characteristics can be found, how sensitive are theyto changes in vehicle payload? The study aims to answer these questions by examining a Land Rover Defender 110 as test vehicle. Anexperimentally validated non-linear seven degree-of-freedom mathematical model of the test vehicle is constructed for the use in sensi-tivity studies. Mathematical optimisation is performed using the model in order to find the suspension characteristics for optimal struc-tural life for the vehicle under consideration. Sensitivity studies are conducted to determine the robustness of the optimal characteristicsand their sensitivity to vehicle payload variation. Recommendations are made for suspension characteristic selection for optimal struc-tural life.� 2011 ISTVS. Published by Elsevier Ltd. All rights reserved.

Keywords: Off-road vehicles; Structural life; Mathematical optimisation; Dynamic modelling

1. Introduction

Manufacturers in the commercial vehicle sector havegreat pressure on them to increase the payload capacityof the vehicle without exceeding the legal vehicle massrestrictions. This can only be achieved by reducing themass of the vehicle structure. This is to be accomplishedwithout compromising the reliability of the vehicle’s struc-tural and other systems.

The goal of reducing vehicle mass drives designers toexplore the use of exotic materials, novel constructiontechniques and innovative structural geometry. The reduc-tion of input loads to the vehicle structure must also be

0022-4898/$36.00 � 2011 ISTVS. Published by Elsevier Ltd. All rights reserve

doi:10.1016/j.jterra.2011.09.004

⇑ Corresponding author. Tel.: +27 12 420 2045; fax: +27 12 362 5087.E-mail addresses: [email protected] (B. Breytenbach), Schalk.

[email protected] (P.S. Els).

keenly considered. These loads may be due to payloadrequirements or road loads, passed to the vehicle structurethrough the tyres, wheels and suspension system. Thefocus here is on the selection of vehicle suspension charac-teristics as these parameters are under the control of thevehicle designer.

In order to address the problem some metric must beidentified to represent the quality of the suspension isola-tion in terms of damage to the vehicle structure. The chal-lenge in defining such a metric is that changes in load levelsare not a direct measure of changes in fatigue damage tothe structure. A simple parameter such as standard devia-tion of suspension forces will therefore not provide unam-biguous insight into the influence of suspensioncharacteristics on fatigue damage in the vehicle structure.The discussion of input loads to the vehicle structure natu-rally leads to the suggestion of cumulative fatigue damageas suitable metric.

d.

http://dx.doi.org/10.1016/j.jterra.2011.09.004

mailto:[email protected]



http://dx.doi.org/016/j.jterra.2011.09.004

398 B. Breytenbach, P.S. Els / Journal of Terramechanics 48 (2011) 397–408

Literature on the selection of suspension characteristicsfor ride comfort or handling can be found for virtually anytype of vehicle. The amount of literature available on theselection of suspension characteristics for fatigue damageto the vehicle structure stands in sharp contrast to this.The focus seems to have been on the prediction of roadloads on vehicle structures and subsequently fatigue lifeof these structures [1,2].

Zeiler and Barkey [3] did however conduct an analyticalstudy into the effect of suspension characteristics on fatiguedamage. They used an unvalidated linear pitch-bouncevehicle model combined with the Forman crack growthlaw to calculate the sensitivities of fatigue damage to springand damper characteristics. Their results indicate adecreasing fatigue life when spring and damper rates areincreased. No suggestion is made as to what the optimalcharacteristics may be.

The ideal vehicle suspension system would provide nearoptimal isolation at different speeds over varying road pro-files for widely varying payloads. A passive system cannotfunction optimally under these varying conditions and willinevitably result in a compromised solution. The largechanges in the sprung mass of heavy vehicles present evengreater challenges in selecting suitable suspension charac-teristics. A laden vs. unladen compromise thus exists,which is in many ways analogous to the ride comfort vs.handling compromise so often investigated in vehicledynamics research.

The promise of ideal behaviour over a wide range ofoperating conditions makes active suspension systemsattractive for solving these challenges. It is however well

Fig. 1. The 4S4 wo

known that the energy consumption of these systems maketheir use prohibitive. Semi-active or adaptive suspensionsystems offer many of the advantages of active systems ata fraction of the energy requirements and complexity [4].These systems have consequently been suggested to remedythe laden vs. unladen compromise in heavy vehicles [4,5].

1.1. Chosen test platform

The test vehicle chosen for the current study is a LandRover Defender 110 fitted with an experimental hydro-pneumatic suspension system, referred to as the 4 StateSemi-active Suspension System or 4S4. The 4S4 was devel-oped as a solution to the ride comfort vs. handling compro-mise for off-road vehicles [6]. The system has the ability toswitch semi-actively between four passive states. The fourstates are obtained by switching independently between ahigh and a low spring rate and a high and a low dampingrate. The working principle of a 4S4 suspension strut isillustrated in Fig. 1. The system also has load-levellingcapability.

The system, as currently implemented on the LandRover, only utilises two of the states. The two states (stiffspring, high damping and soft spring, low damping) arecurrently optimised for handling and ride comfort respec-tively. It is suggested that the system may have the poten-tial to solve the problems associated with high payloadvariations by optimising the stiff spring and high dampingcharacteristics for the fully laden vehicle while the softspring and low damping characteristics are optimised forthe unladen vehicle. Semi-active control then simply

rking principle.

Fig. 2. Comparison of vertical acceleration for three runs over symmet-rical trapezoidal bumps.

Fig. 3. Linear 4DOF pitch-bounce model: rear vertical acceleration.

B. Breytenbach, P.S. Els / Journal of Terramechanics 48 (2011) 397–408 399

switches between the “unladen” condition (soft spring andlow damping) and the “laden” condition (stiff spring andhigh damping) based on the vehicle payload.

2. Experimental work

The purpose of the experimental work was to gatherdata which could be used for the validation of a mathemat-ical model of the test vehicle as well as to determine therelationship between the suspension force and the strainin the suspension mounting bracket. The vehicle wasinstrumented to measure body vertical accelerations atthree locations, body pitch and roll velocities, all four sus-pension displacements, the pressure in each suspensionstrut and vehicle longitudinal velocity. Two of the acceler-ometers were placed below the rear passenger seats, whilethe third was attached to the top of the right front suspen-sion strut. The recorded suspension pressure data was usedto calculate an estimate of the force in the suspensionstruts.

A tension–compression load cell was fitted between theleft rear suspension strut and the rear axle. The load cellmeasured the vertical force in the suspension strut directly.Two 0�–45�–90� rosette type strain gauges were attached tothe left rear suspension mounting tower on the vehiclechassis. The strain gauges were used to measure the strainin the suspension mounting due to the vertical suspensionforces.

The vehicle dynamics was characterised over both dis-crete obstacles and random terrain as suggested in [7]. Atrapezoidal bump was used as discrete obstacle. The ran-dom terrain traversed was the Belgian Paving at the Gero-tek Test Facilities [8]. The use of this specific terrain isattractive as the terrain profile has been measured to highaccuracy using several methods [9]. The measured terrainprofiles were used as inputs to the vehicle model.

The Land Rover was driven at different speeds over eachterrain type. The speed was kept constant by drivingagainst the diesel engine’s governor in gear. The suspensionsetting was also manually varied between the ride comfortand handling modes. Each run was repeated at least threetimes to gain an estimate of the repeatability of the tests.The left rear body vertical accelerations for three differentruns over a symmetrical trapezoidal bump course are pre-sented in Fig. 2. The vehicle was travelling at 15 km/h withthe suspension set to ride comfort mode. The data from thethree runs is almost indistinguishable. The repeatability ofthe tests was found to be excellent. Similar repeatabilitywas observed with the other measured parameters for allthe test runs conducted.

3. Mathematical modelling

A mathematical model of the vehicle provides a power-ful tool to the vehicle designer. A model must be as simpleas possible to ensure low computational expense. It musthowever contain enough detail to capture the dynamics

which it was intended to predict with sufficient accuracy.A vehicle designer cannot use a vehicle model in designstudies with any confidence if the limitations of the modelare not known through comparison with experimentalresults.

The requirement for simplicity of the vehicle model inthis study was met by starting with a simple model. Themodel was then incrementally improved by adding com-plexity as dictated by the correlation with experimentaldata or indeed lack thereof.

The first attempt at modelling the dynamics took theform of a linear four degree of freedom pitch-bouncemodel. The model consisted of three masses, representingvehicle sprung and unsprung masses, connected by linearsprings and dampers.

Figs. 3 and 4 show the rear vertical acceleration and leftrear suspension force from the simulation, superimposedon the measured results. The data is for a run at 15 km/h

Fig. 4. Linear 4DOF pitch-bounce model: rear suspension force. Fig. 5. Non-linear 4DOF pitch-bounce model: rear vertical acceleration.

Fig. 6. Non-linear 4DOF pitch-bounce model rear suspension force.


over symmetrical trapezoidal bumps with the 4S4 set to ridecomfort mode. The results from the model show someresemblance to the measured data over discrete obstacles,but were not considered acceptable for the current purpose.

The model was subsequently improved by introducingnon-linear spring and damper characteristics as well asthe effects the rubber bushings in the suspension trailingarms and rubber bump stops. Analysis of the suspensionforce measured with the load cell, compared to the suspen-sion force calculated from the measured pressure revealedthe presence of significant friction in the hydro-pneumaticsuspension system, which was experimentally characterisedand included in the model. A static friction model wasemployed. This greatly improved the model’s correlationover discrete obstacles, as indicated in Figs. 5 and 6. Cor-relation over the random terrain was however still unsatis-factory. This was attributed to the absence of the rolldegree of freedom, which becomes important on the rough,unsymmetrical Belgian paving terrain.

The model was extended to a seven degree of freedom,non-linear, full vehicle model as indicated in Fig. 7. Thefront and rear axles are modeled with vertical and rolldegree of freedom each. The vehicle body has pitch, rolland vertical degrees of freedom. This results in a set ofseven non-linear second order differential equations thatare solved by numerical integration. The complete mathe-matical derivation of the equations of motion, as well asthe non-linear spring and damper force characteristicsand friction models are documented in [10]. The modelnow captured the roll degrees of freedom which wereabsent in the pitch-bounce model. The correlation over dis-crete obstacles was similar to the pitch-bounce model. Thequality of the correlation, by inspection, is excellent.

The correlation over the Belgian paving in terms of theRMS values of selected quantities is presented in Tables 1–4. The data is presented for four runs at different speedswith suspension setting manually alternated between ridecomfort and handling modes. The model correlation over

the Belgian paving was much improved at high speed(55 km/h), by the introduction of the additional degreesof freedom. The correlation at low speed (15 km/h) stillleaves room for improvement. The model can thereforenot be used to make inferences about events at low speeds.

Some improvement in the model correlation may begained by constructing a high fidelity multi-body-systemsmodel of the vehicle, which can include the non-linearitiesin suspension geometry. The linear approximation to thesuspension kinematics currently employed is howeverthought to be reasonable assumption, when consideringthe simple suspension geometry on the Land Rover. Thetyre model is thought to harbour the most potential forimprovement.

Currently a linear point follower tyre model is employed,with experimentally determined stiffness and damping rate.On the rough terrain the point follower tends to “drop” intoruts which are filtered by the tyre’s geometry on the realvehicle. Attempts to mimic this filtering by a rigid wheel

Fig. 7. The 7DOF model schematic.

Table 1Belgian paving correlation: 15 km/h, 4S4 soft.

Quantity Measured Simulation Error (%)

Rear accel. RMS (m/s2) 1.401 1.993 42Left rear force RMS (N) 752 1105 47Damage 0.55 � 10�6 1.88 � 10�6 239

Table 2Belgian paving correlation: 55 km/h, 4S4 soft.


Rear accel. RMS (m/s2) 3.149 3.276 �4.0Left rear force RMS (N) 1715 1687 �1.6Damage 2.41 � 10�6 2.28 � 10�6 �5.1

Table 3Belgian paving correlation: 15 km/h, 4S4 hard.


Rear accel. RMS (m/s2) 3.173 5.061 55Left rear force RMS (N) 2007 2868 43Damage 6.0 � 10�6 20.6 � 10�6 243

Table 4Belgian paving correlation: 55 km/h, 4S4 hard.


Rear accel. RMS (m/s2) 6.106 6.616 �8.4Left rear force RMS (N) 3776 3676 �2.6Damage 12.7 � 10�6 16.1 � 10�6 27


travelling over the road profile have delivered limited suc-cess. Footprint or flexible ring tyre models should be inves-tigated to improve the vehicle dynamics correlation infuture work.

3.1. Stress and damage prediction

It has been shown thus far that the correlation of thedeveloped model is of sufficient accuracy to predict roadinput loads to the vehicle structure at 55 km/h based onthe comparison of vehicle body acceleration and suspen-sion force. The prediction of structural damage requiresthat these loads are translated to strains or stresses in thevehicle structure.

A quasi-static stress analysis approach was used to con-vert the predicted loads to strains and subsequently stressesin the vehicle suspension mounting. This approach can beapplied in cases were a structure is operating in its linear(elastic) region, and where the frequency content of theapplied load is well below the structures natural frequencies[11].

Fig. 9. Overview of fatigue damage calculation procedure.


Comparison of the strains (measured with the two tri-axial strain gauges on the suspension mounting) to theapplied vertical force (measured with the load cell on thesuspension strut) over various terrains at different speedsrevealed that quasi-static analysis was indeed feasible.The measured strain which most accurately representedthe strain due to suspension vertical load was identified.The identified strain channel combined with the measuredvertical suspension force was then used to calculate a con-stant of proportionality or “stiffness” for the structure asindicated in Fig. 8. The constant could then be used to cal-culate the strains in the structure due to the vertical suspen-sion force. The strain calculated by this method is within5% of the measured strain, given the measured suspensionforce from runs over the Belgian paving at various speeds.

The strains were converted to stress using the uni-axialform of Hooke’s law of elasticity. Range–pair–range cyclecounting, as suggested in [12], was performed on the calcu-lated stress histories to extract individual loading cycles.Miner’s rule of damage summation was used to estimatethe damage due to each cycle extracted. The effect of meanstresses was neglected. Miners rule of damage summationwas used to estimate the damage from the simulatedstress–strain histories. The S–N curve determined for aSANS class E weld [13] was used to represent the fatiguebehaviour of the material. This choice was largely arbi-trary, as the fatigue behaviour of the material wasunknown and the prediction of the exact fatigue life ofthe structure was not of primary concern. The model cor-relation in terms of fatigue damage is also given in Tables1–4. An overview of this fatigue damage predictionmethodology is given in Fig. 9.

The friction model used in the 4S4 suspension strutsshould be further investigated to improve the fatigue dam-age correlation of the model. The static friction character-istic currently employed causes sharp suspension forcechanges as the suspension displacement reverses direction.This leads to damage over-prediction. Dynamic friction

Fig. 8. Relationship between suspension force and strain on suspensionbracket.

models may improve the damage correlation and shouldbe investigated in future work.

4. Mathematical optimisation

The gradient-based Dynamic-Q mathematical optimisa-tion algorithm developed by Snyman and Hay [14], waschosen for the mathematical optimisation due to its provenefficiency and robustness when applied to vehicle suspen-sion optimisation problems. The algorithm makes use ofsuccessive spherical quadratic approximations of the costfunction, which are then solved by the LFOPC dynamictrajectory method. Dynamic-Q has been shown to behighly efficient at solving problems with severe inherentnoise content, which is common when simulations are usedfor cost function evaluations or gradient approximations.The Dynamic-Q algorithm is described in detail in [15].Great success has been achieved in applying the algorithmto the problems of suspension optimisation for ride com-fort and handling [16–18]. Thoresson [16] employed centralfinite differences for gradient approximation instead of theless expensive forward finite differences usually employed.He found that the improved convergence due to morerobust and less noisy gradient approximation was worththe added computational expense.

4.1. Problem formulation

The aim of the mathematical optimisation is to minimisethe damage to the vehicle structure due to road input loads,by the selection of appropriate suspension spring and dam-per characteristics. The problem can be formulated as astandard optimisation problem:

minw:r:t:�x

f ð�xÞ;�x ¼ ½x1; x2; . . . ; xn�T 2 Rn;

subject to the constraints:


gjð�xÞ 6 0; j ¼ 1; 2

In the equations given above, f ð�xÞ represents the objectivefunction (or cost function) and �x represents the vector ofdesign variables.

These optimal characteristics are to be determined forthe vehicle in the unladen condition and for a fully ladencondition. The unladen condition sees the vehicle weighingin at 2190 kg. The fully laden condition was chosen to berepresentative of the unladen to laden load variationobserved in heavy trucks. Cole [19] states that the ratioof unsprung to sprung mass can be as low as 0.1 in fullyladen vehicles of this type. This implies vehicle sprung massof 4500 kg for the Land Rover. This load condition is sim-ilar to the fully laden mass of armoured versions of theLand Rover Defender.

The objective function, f ð�xÞ, is defined as the cumulativedamage incurred by the vehicle when driving over a roadconsisting of 100 m of an ISO 8608 class A road [20] and100 m of the Gerotek Belgian paving at a constant speedof 55 km/h.

The design variables used in the optimisation are thenon-linear spring and damper characteristics of the vehicle.The force displacement characteristic of the hydro-pneumatic spring is fully defined by the spring static gasvolume [6]. The baseline static volume is 0.3 ‘ and the staticvolume was allowed to vary between 0.05 and 0.85 ‘ in theoptimisation.

The damper curve is not so easily defined by a singleparameter. A set of four quadratic polynomials was fittedto the baseline force displacement characteristic [16]. Theshapes and magnitudes of the polynomials are determinedby six damper scale factors. The bump and rebound direc-tions of the damper characteristic are each governed bythree of the six damper scale factors. The damper scale fac-tors were allowed to vary between 0.1 and 4 in theoptimisation.

The optimisation process is also bounded by certainconstraints. The suspension working space on the vehicleis fixed by the vehicle design and limited by rubber bumpstops. It is undesirable that the suspension encounters itsbump stops from both a fatigue damage and ride comfortperspective. A constraint was constructed to ensure thatthe optimal suspension characteristics do not allow the sus-pension to encounter the bump stops.

The optimal suspension characteristics should also besuch that wheel hop is limited. Lack of damping in the sus-pension may cause large variations in vertical wheel load,which is detrimental to the longitudinal and lateral forcegeneration capability of the tyre and therefore a safety con-cern [4,21]. High dynamic wheel load fluctuations are also acontributing factor for pavement fatigue failures [22].Thoresson [16] employed a wheel hop constraint whichonly allowed the wheels to loose ground contact for lessthan 10% of the total simulation time in his optimisationfor ride comfort. Thoresson’s constraint was included inthe current study’s optimisation for fatigue damage.

In a heavy vehicle application, road damage is also animportant parameter to consider. Attempts to include aroad damage constraint based on Sweatman’s observationsof the tyre Dynamic Load Coefficient (DLC) [23] have asyet proved unsuccessful. The DLC is defined as the ratiobetween the RMS dynamic tyre loads to the static tyreload. The DLC was chosen to represent the road damagein this study due to the fact that road damage is usuallymodelled as fourth power of the DLC. The reader isreferred to [23] for a detailed discussion on pavement dam-age estimation.

The cost function, design variables and constraint func-tions were all scaled in an attempt to promote sphericalityof the design space and aid in the rapid convergence of theDynamic-Q algorithm. A gradient sensitivity study wasperformed to find an appropriate variable perturbation sizefor the gradient approximations. The gradient approxima-tions were found to be well behaved for variable perturba-tions sizes ranging from 10�4 almost down to machineprecision. A move limit of one fifth the radius of the regionof interest was found to give the best convergence over arange of different starting points.

4.2. Two-variable optimisation

The optimisation methodology was developed andrefined using a two-variable formulation described here.The spring static gas volume is defined as the first designvariable, while the six damper scale factors are combinedinto a single damper scale factor representing the seconddesign variable. The front and rear static gas volume as wellas the front and rear damper characteristics are thereforethe same in this approach. The two-variable formulationof the cost function allows one to visualise the cost functionin three dimensional space. A surface plot of the cost func-tion, for both the unladen and fully laden conditions, isgiven in Figs. 10 and 11. The cost function values in the plothave been scaled as percentages of the baseline damage. Thebaseline is defined as the damage in the unladen case withthe baseline spring and damper characteristics, i.e. staticgas volume = 0.3 ‘ and all damper scale factors = 1.

The cost functions in both cases indicate a largely uni-modal design space. Minima can be found in the lowdamping, soft spring region of the design space. The costfunctions exhibit a large region of near minimum values.

An interesting observation which can be made is that inboth cases the problem seems to be largely insensitive tothe static gas volume. The cost function in the fully ladencase does become more sensitive to spring stiffness asdamping is decreased. The stiffness of the hydro-pneumaticspring is a hyperbolic function of static gas volume [6]. Thisis at least the partial cause of the cost function’s low sensi-tivity to static gas volume at large static gas volumes.

The convergence histories for two optimisation runsfrom different starting points are plotted for both cost func-tions (Figs. 10 and 11.). The circular markers indicatepoints were function and gradient values were evaluated.

Fig. 10. Cost function plot for the unladen vehicle.

Fig. 11. Cost function plot for the fully laden vehicle.


The arrows indicate the direction of convergence. The con-vergence is generally well behaved, with the algorithm fol-lowing a direct path to the minima. Deviations from thispath are due to noisy gradient information or the hin-drance by constraint functions. The reliability of the con-vergence deteriorated as the cost function entered theshallow dish-like region near the optima. The numericalnoise is thought to dominate the gradient approximationin these regions of low gradient value.

In the unladen payload case the optimisation convergedto a static gas volume ranging between 0.3–0.7 ‘, while thedamper scale factor ranged between 0.2 and 0.4. The costfunction values for these minima ranged between 24%and 28% of the baseline structural damage. A similar result

was found for the fully laden case. The optimisation con-verged to static gas volumes ranging from 0.40 to 0.85 ‘,with damper scale factors ranging between 0.35 and 0.7.The cost function value at the minima varied between65% and 76% of the baseline damage.

The widely distributed near optimum points indicatevery robust optima. In order to confirm this, a MonteCarlo simulation, using 500 cost function evaluations,was performed across the region to which the optimisationconverged. The results of the simulation are presented inFigs. 12 and 13. The feasible function evaluations are indi-cated as circles while the crosses indicate infeasible points.The infeasible region in both plots is due to the violation ofthe suspension workspace constraint.

Fig. 12. Contour plot of the cost function for the unladen vehicle.

Fig. 13. Contour plot of the cost function for the fully laden vehicle.


The selection of a robust minimum requires that thedesign variables lie far enough from the constraint functionboundary. The robust optima were chosen such that a 10%variation in the design variable values would not cause aconstraint violation.

The robust optimum for the unladen vehicle is found ata damper scale factor of 0.4 and a static gas volume ofbetween 0.4 and 0.85 ‘. The mean cost function value inthis region is 29% with a standard deviation of 2.9% ofthe function value. The robust optimum for the fully laden

vehicle is at static gas volumes between 0.5 and 0.85 ‘ and adamper scale factor of 0.75. The corresponding mean func-tion value in the region is 86%, while the coefficient of var-iation is 7.9% of the baseline damage.

The results for the unladen and fully laden cases aresummarised in Table 5. The ride comfort and handlingoptima for the unladen case, as determined by Thoresson[16], have been included for comparison.

The fatigue damage optima and ride comfort optima forthe unladen case are very similar. This result is not entirely

Table 5Optimal suspension characteristics for the two-variable optimisation.

Static gas volume (‘) Damper scale factor

Unladen vehicle

Fatigue damage 0.3–0.85 0.4Ride comfort 0.5 0.3Handling 0.1 3

Fully laden vehicle

Fatigue damage 0.5–0.85 0.75

Table 6Optimal suspension characteristics for the three-variable optimisation.

Static gasvolume (‘)

Damper scalefactor (bump)

Damper scale factor(rebound)

Unladen vehicle

Fatiguedamage

0.3–0.85 0.49 0.17

Fully laden vehicle

Fatiguedamage

0.5–0.85 0.6 0.10


unexpected. The fatigue damage optimum for the fullyladen case requires a slightly larger static gas volume thanfor the unladen case. The large overlap of the static gas vol-umes required for the unladen and laden cases are due tothe load-levelling operation of the hydro-pneumatic sus-pension system. The result should therefore not be extrap-olated to non-levelling suspension systems. The damperscale factor required for the fully laden vehicle is almostdouble that required for the unladen case.

4.3. Multi-variable optimisation

The optimal shape of the fatigue damage optimaldamper characteristic is also of interest. A seven-variableformulation of the optimisation problem was defined using

Fig. 14. Suspension strut force r

the static gas volume as well as the six independent damperscale factors. This formulation proved too complex andclear trends could not be distinguished from the results,i.e. there may be many diverse local minima. It is thoughtthat the uni-modality of the optimisation problem is lostwhen multiple variables are considered.

A three-variable formulation of the optimisation prob-lem was subsequently constructed in an attempt to revealmore distinct trends in the results. The three-variable for-mulation kept the static gas volume as design variable,but redefined the damper characteristic using two damperscale factors. The first scale factor governs the negativevelocity (bump) region, while the second scales the positivevelocity (rebound) region. The optimisation was conducted

esponse to sinusoidal input.


for both payload cases using the three-variable formula-tion. The resulting optima are listed in Table 6.

The observation can be made that the damper scale fac-tor in the rebound direction is significantly reduced fromthat required by the two-variable optimisation optimum.Significant reductions in cost function value were alsoachieved. The cost function value for the unladen casewas reduced to 25.3%, while the cost function value forthe fully laden case was 29% of baseline. As with thetwo-variable formulation the problem was relatively insen-sitive to changes in static gas volume and the dampingrequired increased with vehicle payload.

The observations concerning the reduction in rebounddamping and the associated improvement in fatigue dam-age can be explained by examining the suspension strutforce responses to sinusoidal input. The strut responseforces for both symmetric and asymmetric damper scalefactors shown in Fig. 14. It can be seen that the forceranges for the asymmetric damper characteristic are signif-icantly smaller than those for the symmetric damper char-acteristic. The reduced force ranges are thought to be thecause for the reduction in fatigue damage, when the asym-metric damper characteristic is employed.

It is concluded that an asymmetric damper characteristiccan significantly improve fatigue performance. This standsin contrast to the damper characteristics required for ridecomfort and handling. In his optimisation for ride comfortand handling Thoresson [16] found that the optimal dampercharacteristics were required to be symmetric. Assumingthat a ride comfort optimal damper characteristic is alsofatigue damage optimal is therefore not necessarily correct.

5. Conclusions

The seven degree of freedom non-linear vehicle modelgave excellent correlation over discrete obstacles and goodcorrelation at high speed over rough, asymmetrical randomterrain. The model’s poor correlation over the random ter-rain at low speed is attributed to the point follower tyremodel employed. Footprint or flexible ring tyre modelsare recommended for investigation in future work toimprove the vehicle dynamics correlation. The static modelfor the friction in the 4S4 suspension struts should berefined to include dynamic effects if the fatigue damage cor-relation is to be improved.

The model was used in the mathematical minimisationof the fatigue damage to the vehicle structure for both afully laden and unladen payload case. The hydro-pneu-matic spring static gas volume and a damper scale factorwhere defined as design variables in a two variable formu-lation of the optimisation problem. Visualisations of thecost functions revealed that the optimisation problemsare uni-modal, with minima in the soft spring, low damp-ing region of the design space. The cost functions werefound to be insensitive to variation of the static gas volumefor both load cases. The insensitivity is attributed to the

fact that the stiffness of the hydro-pneumatic spring is ahyperbolic function of the static gas volume.

The optimisation converged to a large region of feasibleand near optimal values, for both payload conditions. Thechosen optima were shown to be highly robust in terms ofvariation of both spring and damper characteristics.

The optimal suspension characteristics for fatigue dam-age in the unladen case are similar to the characteristicsrequired for ride comfort. The spring characteristicrequired for the fully laden condition is a subset of the opti-mum for the unladen case. The optimal damper scale factorfor the fully laden case is however almost double thatrequired for the unladen vehicle. The problem is thus moresensitive to damper characteristic with respect to changingpayloads. The load-levelling operation of the hydro-pneu-matic suspension system contributes to the insensitivity ofthe static gas volume.

A three-variable formulation of the optimisation prob-lem was used in order to determine the optimal shape forthe damping curve. The investigation showed that damageto the vehicle structure could be reduced if the rebounddamping could be minimised.

Due to the problem’s general lack of sensitivity tochanges in static gas volume, the implementation of a sys-tem such as the 4S4 cannot be recommended based struc-tural fatigue life considerations alone. The mostadvantage would be gained from the use of a suspensionsystem with ride height control and a two-state semi-activedamper. The situation might still favour the 4S4 if handlingcriteria are taken into consideration.

References

[1] Medepalli S, Rao R. Prediction of road loads for fatigue design – asensitivity study. Int J Veh Des 2000;23(1/2):161–75.

[2] Conle FA, Mousseau CW. Using vehicle dynamics simulations andfinite-element results to generate fatigue life contours for chassiscomponents. Int J Fatigue 1991;13(3):195–205.

[3] Zeiler TA, Barkey ME. Design sensitivities of fatigue performanceand structural dynamic responses in an automotive application.Struct. Multidisc. Optim. 2001;21:309–15.

[4] von Glasner E-C, Goehring E, Povel R, Schuetzner P. Analysis ofintelligent suspension systems for commercial vehicle. SAE TechnicalPaper No. 933008; 1993. p. 35–43.

[5] Bode O, von Glasner E-C, Holnkaus H, Pflug H-C, Povel R. Einflussvon Fahrzustand und Fahrwerk auf die Vertikaldynamik vonNutzfahrzeugen. Automobiltech. Zeitsch. 1997;99(7/8):396–402.

[6] Els PS. The ride comfort vs. handling compromise for off-road vehicles.Unpublished PhD thesis. South Africa: University of Pretoria; 2006.

[7] Letherwood MD, Gunter DD. Ground vehicle modelling andsimulation of military vehicles using high performance computing.Parallel Comput 2001;27:109–40.

[8] http://www.armscorbusiness.com/Subsites/Gero-tek/Gerotek01_landing.asp [accessed 18.09.09].

[9] Els PS, Becker CM. Profiling of rough terrain. In: Proceedings: 11thEuropean regional conference of the International Society forTerrain-Vehicle Systems, Bremen, Germany, 5–8 October 2009.

[10] Breytenbach HGA., Optimal vehicle suspension characteristics forincreased structural fatigue life. Unpublished M.Eng dissertation.South Africa: University of Pretoria; 2010. <http://upetd.up.ac.za/thesis/available/etd-09172010-190101/> [accessed 28.12.10].

http://www.armscorbusiness.com/Subsites/Gero-tek/Gerotek01_landing.asp

http://www.armscorbusiness.com/Subsites/Gero-tek/Gerotek01_landing.asp

http://www.upetd.up.ac.za/thesis/available/etd-09172010-190101/



[11] Haiba M, Barton DC, Brooks PC, Levesley MC. Review of lifeassessment techniques applied to dynamically loaded automotivecomponents. Comput Struct 2002;80:481–94.

[12] Dowling NE. Mechanical behaviour of materials: engineering meth-ods for deformation, fracture, and fatigue. 3rd ed. Pearson – PrenticeHall; 2007.

[13] Standards South Africa. South African National Standard SANS10162: the structural use of steel. SANS 10162-1:2005, 2nd ed.; 2005.

[14] Snyman JA, Hay AM. The Dynamic-Q optimization method: analternative to SQP? Comput Math Appl 2001;44:1589–98.

[15] Snyman JA. Practical mathematical optimisation – an introduction tobasic optimisation theory and classical and new gradient-basedalgorithms. Springer; 2005. 119-125.

[16] Thoresson MJ. Efficient gradient-based optimisation of suspensioncharacteristics for an off-road vehicle. Unpublished PhD thesis. SouthAfrica: University of Pretoria; 2007. <http://upetd.up.ac.za/thesis/available/etd-08042008-093103/> [accessed 28.12.10].

[17] Thoresson MJ, Uys PE, Els PS, Snyman JA. Efficient optimisation ofa vehicle suspension system, using a gradient-based approximation

method. Part 1. Mathematical modelling. Math. Comput. Model.2009;50:1421–36.

[18] Thoresson MJ, Uys PE, Els PS, Snyman JA. Efficient optimisation ofa vehicle suspension system, using a gradient-based approximationmethod. Part 2. Optimisation results. Math. Comput. Model.2009;50:1437–47.

[19] Cole DJ. Fundamental issues in suspension design for heavy roadvehicles. Vehicle Syst. Dynam. 2001;35(4-5):319–60.

[20] International Organization of Standardization. International Stan-dard ISO 8606: mechanical vibration – road surface profiles –reporting of measured data. ISO 8608:1995(E); 1995.

[21] Uys PE, Els PS, Thoresson M. Suspension settings for optimal ridecomfort of off-road vehicles travelling on roads with differentroughness and speeds. J Terrramech 2006;44:163–75.

[22] Cebon D. Vehicle-generated road damage: a review. Vehicle Syst.Dynam. 1989;89:107–50.

[23] Sweatman PF. A study of dynamic wheel forces in Axle Groupsuspensions of heavy vehicles. Special Report No. 27. AustralianRoad Research Board; 1995.



Documents

Optimal Vehicle Suspension Characteristics for Increased Structural Fatigue Life