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Hindawi Publishing CorporationInternational Journal of Vehicular TechnologyVolume 2013, Article ID 562424, 16 pageshttp://dx.doi.org/10.1155/2013/562424

Research ArticleOptimisation of the Nonlinear Suspension Characteristics ofa Light Commercial Vehicle

Dinçer Özcan, Ümit Sönmez, and Levent Güvenç

Mekar Mechatronics Research Labs, Department of Mechanical Engineering, Istanbul Okan University, Akf𝚤rat-Tuzla,TR-34959 Istanbul, Turkey

Correspondence should be addressed to Levent Guvenc; [email protected]

Received 9 October 2012; Revised 16 December 2012; Accepted 30 December 2012

Academic Editor: Shinsuke Hara

Copyright © 2013 Dincer Ozcan et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The optimum functional characteristics of suspension components, namely, linear/nonlinear spring and nonlinear dampercharacteristic functions are determined using simple lumped parameter models. A quarter car model is used to represent the frontindependent suspension, and a half car model is used to represent the rear solid axle suspension of a light commercial vehicle.The functional shapes of the suspension characteristics used in the optimisation process are based on typical shapes supplied bya car manufacturer. The complexity of a nonlinear function optimisation problem is reduced by scaling it up or down from theaforementioned shape in the optimisation process.Thenonlinear optimised suspension characteristics are first obtained using lowercomplexity lumped parametermodels.Then, the performance of the optimised suspension units are verified using the higher fidelityand more realistic Carmaker model. An interactive software module is developed to ease the nonlinear suspension optimisationprocess using the Matlab Graphical User Interface tool.

1. Introduction

Vehicle suspension design and performance problems havebeen studied extensively using simple car models such as twodegrees-of-freedom (d.o.f.) quarter car, four or six d.o.f. halfcar, or seven d.o.f. full car models. Usually, the suspensiondesignmethodologies are based on analytical methods wherea linear vehicle model is investigated by solving linear ordi-nary differential equations. Laplace and Fourier transformsare valuable tools that are usedwhile investigating suspensionunits with linear characteristics. The performance functionsrepresented by transfer functions in Laplace and/or Fourierdomains are considered to be related to ride comfort, tireforces, and handling criteria versus road roughness inputto achieve an optimum design. On the other hand, theinvestigation of nonlinear suspension characteristics mustbe based more on numerical methods rather than analyticalmethods due to the more complicated nature of the problem.In this investigation, both linear and nonlinear spring anddamper characteristics of a light commercial vehicle areconsidered and used in an optimisation study.

Lumped parameter suspension models are used in thispaper. Mass, stiffness, and damping are distributed spatiallythroughout a mechanical system like a suspension. Mass,stiffness, and damping are therefore functions of spatialvariables (𝑥, 𝑦 and 𝑧 coordinates) in a mechanical system,resulting in what is called a distributed parameter systemsince the mass, stiffness, and damping parameter values aredistributed over the mechanical system. An easier approachis to lump the continuous mass, stiffness, and damping char-acteristics into ideal mass, stiffness, and damping elements.The result is a lumped parameter system, as the mass of themechanical system is concentrated at the ideal mass element,the stiffness of the mechanical system is concentrated inthe ideal stiffness element (the spring), and the damping ofthe mechanical system is concentrated in the ideal dampingelement (the damper).The lumped parametermodels used inthis paper are the quarter car and half car suspensionmodels.

The optimisation requirements of suspension systemsand the state-of-the-art of suspension research in the lastdecade are reviewed first. It should be noted, however, thatthe available literature is vast and only a small portion

2 International Journal of Vehicular Technology

of it can be presented here. This paper includes the well-known ride, handling trade-off optimisation, and geometricaloptimisation of light commercial vehicle suspension systems.Some heavy vehicle suspension optimisation papers arealso reviewed due to their conceptual contribution to thesubject.

Vehicle suspensions can be regarded as interconnectionsof rigid bodies with kinematic joints and compliance ele-ments such as springs, bushings, and stabilizers. Designof a suspension system requires detailed specification ofthe interconnection points (or so called hard points) andthe characteristics of compliance elements. Tak and Chung[1] proposed a systematic approach of achieving optimumgeometric design of suspension systems where design vari-ables are determined to meet some prescribed performancetargets expressed in terms of suspension design factors,such as toe, and camber, compliance steer. Koulocheris etal. [2] proposed to combine deterministic and stochasticoptimisation algorithms for determining optimum vehiclesuspension parameters. They showed that such combinationyields significantly faster and more reliable convergence tothe optimum. Their method combines the advantages ofboth categories of deterministic and stochastic optimisation.They used a half car model of suspension systems, subjectto various road profiles considering the improvement ofthe passenger ride comfort, leading to minimisation of themaximum acceleration of the sprung mass while payingattention to the geometrical constraints of the suspension aswell as the necessary traction of the vehicle.

Maximizing tractive effort is essential to competitiveperformance in the drag racing environment. Antisquat is atransient vehicle suspension phenomenonwhich can dramat-ically affect tractive effort available at the motorcycle drivetire. Wiers and Dhingra [3] addressed the design of a four-link rear suspension of a drag racing motorcycle to provideanti-squat. This design increases rear tire traction, therebyimproving vehicle acceleration performance. For the dragracing application considered, any increase in normal forcesat the tire patch helps improve race competitiveness. Mitchellet al. [4] used a genetic algorithm for the optimisation ofautomotive suspension geometries considering the descrip-tion of a suspension model and a scoring method. Theirapproach is to design with a unit-free measure of fitness foreach test and then to combine these with a weighing function.They showed that the genetic algorithm and the scoringmechanism worked effectively and significantly faster thanthemore common grid optimisation technique. Raghavan [5]presented an algorithm to determine the attachment pointlocations of the tie rod of an automotive suspension, in orderto achieve linear toe change characteristics with jounce andrebound of the wheel. This linear behaviour is advantageousfor achieving good ride and handling. Raghavan’s procedurecan be applied to all suspension mechanism types such asshort-long arm, McPherson struts, and five-link front andfive-link rear suspensions.

The design of suspension systems generally demands acompromise solution to the conflicting requirements of han-dling and ride comfort. The following examples demonstratethis compromise.

(i) For example, for better comfort a soft suspension andfor better handling a stiff suspension is needed.

(ii) A high ground clearance is required on rough terrain,whereas a low centre of gravity height is desired forswift cornering and dynamic stability at high speeds.

(iii) It is advantageous to have low damping for low forcetransmission to the vehicle frame. On the contrary,high damping is desired for fast decay of oscillations.

Considering the aforementioned requirements, Deo andSuh [6] proposed a design for a customizable automotivesuspension system with independent control of stiffness,damping, and ride height. Their design enabled the achieve-ment of desired performance depending on user preference,road conditions, and maneuvering inputs while avoiding theperformance trade-offs.

Goncalves and Ambrosio [7] proposed a methodology inorder to investigate flexiblemultibodymodels for the ride andstability optimisations of vehicles. Their methodology allowsthe use of complex shaped deformable bodies, representedby finite elements. The ride optimisation is achieved byfinding the optimum of a ride index that is the outcomeof a metric that accounts for the acceleration measuredat several key points of the vehicle, weighed according totheir importance for occupant comfort. Duysinx et al. [8]developed a mechatronic approach to model, simulate andoptimize a passenger car (Audi A6) incorporating a con-trolled semiactive suspension. They paid particular attentionto the formulation of the mechatronic model of the carand compared two different modelling and optimisationapproaches. The first approach is carried out in the Matlab-Simulink environment and the derivation of the equationsis based on a symbolic multi-body model. The optimisationprocedure has also been investigated in Matlab. Their secondapproach relies on a multi-body model based on the finite-element method where the optimisation has been realizedwith an open-source industrial optimisation tool.

Eskandari et al. [9] optimized the handling behaviourof a midsized passenger car by altering its front suspensionparameters using Adams/Car software. They utilized anobjective function combination of eight criteria indicatinghandling characteristics of the car and reduced the amountof optimisation parameters, by implementing the design ofexperiments method capabilities. The amount of the param-eters was reduced from fifteen to ten by using a sensitivityanalysis. A similar Adams/Car-based study was conducted byBoyalı et al. [10]. More recently, He andMcPhee [11] reviewedthe state-of-the-art related to modelling approaches, con-sidering vehicle system models, design variable and perfor-mance criteria definitions, optimisation problem formulationmethods, optimisation search algorithms, sensitivity analysis,computational efficiency, and other related techniques. Theyapplied these techniques to the design synthesis of groundvehicle suspensions and proposed a methodology for auto-mated design synthesis of ground vehicle suspensions.

Li et al. [12] considered a five-link suspension opti-misation for improving the ride safety and comfort using

International Journal of Vehicular Technology 3

Adams/Insight. They investigated the relations among multi-link suspension structural parameter, wheel location param-eter, and wheel track. Uys et al. [13] reported an investigationto determine the spring and damper settings that will ensureoptimal ride comfort of an off-road vehicle, on differentroad profiles and at different speeds. Spring and dampersettings can be set either to the ride mode or the handlingmode, and therefore a compromise ride-handling suspensionis avoided.They found that optimizing for a combined driverplus rear passenger seat weighed root mean square verticalacceleration rather than using driver or passenger valuesonly returns the best results. Their results indicated thatoptimization of suspension settings using the same road andconstant speed will improve ride comfort on the same roadat different speeds and these settings will also improve ridecomfort for other roads at the optimisation speed and otherspeeds, although not as much as when optimisation has beendone for the particular road. In the present paper, one vehiclespeed and one road profile were used for optimization takingthis statement from [13] into account. They also concludedin [13] that for improved ride comfort, damping generallyhas to be lower than the standard (compromised) setting, therear spring as soft as possible, and the front spring rangingfrom as soft as possible to stiffer depending on road and speedconditions. Ride comfort is the most sensitive to a change inrear spring stiffness.

The roll steer of a front McPherson suspension systemis studied, and the design characteristics of the mechanismare optimized by Habibi et al. [14] using the genetic algo-rithm method. The roll steer affects handling, and dynamicstability of the vehicle due to variation of the angles of thewheel and the suspension links (i.e., camber, caster, andtoe). However, these changes cause other problems. In theirpaper, Habibi et al. [14] used a genetic algorithm methodto determine the optimum length and orientation of themechanism’s members to minimize the variations of thetoe, camber, and caster angles. They defined a performanceindex which expresses the overall variations of the mainparameters in the whole range of rolling of the body. Ageneral formulation for multi-body flexible systems, withlinear elastic deformations, is considered by Goncalves andAmbrosio [15] whose application involved a road vehiclewhere flexibility plays an important role in ride and handlingdynamic behaviour. Using finite elements to describe theflexibility of the body and the modal superposition methodhas the advantage of greatly reducing the dimensionality ofthe system. The results presented in [15] showed that theuse of the detailed vehicle model within the framework ofride optimisation leads to a measurable improvement of thecomfort conditions for different road profiles and drivingconditions.

An optimum concept to design “road-friendly” heavyvehicles with the recognition of pavement loads as a primaryobjective function of vehicle suspension design was investi-gated by Sun [16]. A walking-beam suspension system is usedas an illustrative example of the vehicle model to demonstratethe concept and process of optimisation. Dynamic responseof the walking-beam suspension system was obtained bymeans of stochastic process theory. Using the direct update

method, optimisation is carried out when tire load magni-tudes are taken as the objective function of suspension design.Their results showed that tires with high air pressure couldlead to more damage in pavement structures, and increasingsuspension damping and tire damping can reduce the tireloads and pavement damage.

This paper concentrates on optimisation of the nonlinearshape of the spring and damper of a light commercial vehicle.The work reported in the paper is motivated by the factthat the spring and damping characteristics in an actualroad vehicle are designed to be nonlinear on purpose bythe automanufacturer. Larger spring forces are used in therebound motion of the wheel in order to keep forcing anappropriate level of tire-road contact at all times, for example.When the automanufacturer starts working on a new model,a previous, successful suspension design is used as the basecharacteristic which is modified to fit the characteristics ofthe new vehicle model. The work presented here follows thesame approach as spring, and damper characteristics of anexisting base design are used as the starting point in theoptimisation. The optimisation procedure is embedded intoan interactiveMatlab Graphical User Interface to offer ease ofuse to suspension designers.

The organization of the rest of the paper is as follows.The vehicle models used in this study are introduced inSection 2 along with the scope of this paper. Some of theperformance criteria available in the literature are discussedin Section 3. The optimisation procedure used is the topic ofSection 4, while the optimisation results obtained using thisprocedure are treated in Section 5. The paper ends with theusual conclusions and recommendations being presented inSection 6.

2. Vehicle Models Used and Scope ofthe Investigation

In this section, the scope of the current investigation issummarized by presenting the vehicle models that are usedin this study. The use of a complex three-dimensional modelof the vehicle, with a detailed description of all suspensionsystems and road/tire interaction, is necessary to fully investi-gate the problem. However, such models are computationallyexpensive especially when used in an iterative optimisationdesign process. A good alternative which is used here isthe optimisation of a subsystem of a complex model. Thesuspension subsystem is very important in terms of vehicledynamics. Its spring and damper load deflection characteris-tics are treated as the basic design variables here.

The ride comfort optimisation is achieved by finding theoptimum of a ride comfort index which results from ametricthat accounts for the linear and the angular accelerations ofthemodel’s suspendedmass centre and properly combined ina cost function, considering their importance for the comfortof the occupant. Two lumped parameter models are built inMatlab considering the independent front suspension unit (aquarter car model) and the rear axle suspension unit (a halfcar model) of a light commercial vehicle. Vertical displace-ment 𝑧𝑠 and acceleration 𝑎𝑠 of the suspended mass and the


tire force 𝐹Tire of the quarter car model are considered asthe key variables in ride comfort and handling, respectively.Similarly, vertical and angular displacement 𝑧𝑠 and 𝜙𝑠 of themass centre and the tire forces at both of the rear wheels ofthe half car model are selected as the key variables for the rearsuspension half car model.

The dynamics of the quarter car and the half car rear axlesuspension models are governed by nonlinear differentialequations ofmotion.The road profile described inCebon [17]is used in the optimisation study and simulations to deter-mine suitable linear and/or nonlinear spring and nonlineardamper characteristics. Considering a nonlinear model, asuspension characteristic optimisation routine written as anm-file in Matlab gives more insight than using commercialvehicle simulation/analysis software. In some cases, thisoption (nonlinear suspension characteristics optimisation)is not readily available in commercial vehicle suspensionpackages. The main objective and the contribution of thisinvestigation are to determine the optimum nonlinear func-tions of the damper and the spring characteristics for theimprovement of the passengers’ ride comfort and vehiclehandling leading to minimisation of the chosen objectivefunction.

Basic shapes representing the spring and the dampercharacteristics (force versus deflection for the spring andforce versus velocity for the damper) are used accordingto automotive manufacturer’s specifications. Basic functionalshapes in each operating mode (extension or compressionregions of the spring and the damper) are predeterminedand the functional fits to these shapes are obtained. Thesefunctions and their linear combinations are then scaledsearching for the optimum characteristics. The emphasisof this investigation is placed on finding nonsymmetricoptimum nonlinear functions of the spring and the damperforce characteristics. Optimised functional relations are thenincorporated into a model built in a high fidelity, realisticcommercial vehicle dynamics software to evaluate the perfor-mance of the vehicle model with the optimised suspension.Carmaker software [18] is used for this purpose to study thehandling behaviour of the car in standard tests (double-lanechange, fishhook, etc.). The Carmaker vehicle model [18] isa highly realistic one that incorporates engine dynamics, tiredynamics, steering dynamics, suspension dynamics, vehiclesprung body dynamics, longitudinal and lateral dynamics,a driver model, and road and environment models. Theinvestigation also looks at a scenario where ride comfortand handling are simultaneously required. The corneringbehaviour of a road vehicle is an important performancemode often equated with handling. In order to analyze bothof ride and handling requirements, a double lane changemanoeuvre is performed after travelling over an irregularroad profile with a disturbance, and then the vehicle comes toa stop.The simulation results of optimised nonlinear damperand nonlinear spring characteristic functions are comparedwith those of the optimised linear ones in simulations.

2.1. Suspension Models Used. The mathematical models ofthe quarter car and the half car representing one of the

Road profile

Figure 1: Quarter car model.

front quarters and the rear axle suspension unit of a lightcommercial vehicle are presented in this section.

2.1.1. Quarter Car Model. The quarter car model subjectto road disturbances is shown in Figure 1. The equation ofmotion considering the vertical displacement of the vehiclebody with linear spring and nonlinear damper characteristicsmay be written as

𝑚𝑠 ..𝑧𝑠 + 𝐵𝐹 { .𝑧𝑠 − .𝑧𝑢} + 𝑘𝑠 (𝑧𝑠 − 𝑧𝑢) = 0. (1)

Similarly the equation of motion of the vehicle wheel maybe written as𝑚𝑢 + ..𝑧𝑢 + 𝐵𝐹 { .𝑧𝑢 − .𝑧𝑠} + 𝑘𝑠 (𝑧𝑢 − 𝑧𝑠) − 𝑏𝑢 ( .𝑧𝑢 − .𝑧𝑟)− 𝑘𝑢 (𝑧𝑟 − 𝑧𝑢) = 0, (2)

where𝐵𝐹{ .𝑧𝑢− .𝑧𝑠} represents the nonlinear functional relationof suspension damper force versus velocity characteristic.

2.1.2. Half Car Model. The rear solid axle suspension unit ofthe light commercial vehicle considered here is representedwith a half car model (see Figure 2) and subjected to roaddisturbances coming from both sides of the track (the left andthe right wheels).The torsional antiroll bar is also consideredin the half car model.

The half car model can represent the bounce (𝑧𝑠, 𝑧𝑢)and roll motions (𝜙𝑠, 𝜙𝑢) of the car body and solid rearaxle. Therefore, it has four d.o.f. The equations of motionsof the sprung (car body) and unsprung (rear axle) massesconsidering the bounce and roll motions may be written as

𝑚𝑠 ..𝑧𝑠 + 𝐹𝑠left + 𝐹𝑠right = 0 (3)

for the bounce motion of the sprung mass and

𝑚𝑢 ..𝑧𝑢 − 𝐹𝑠left − 𝐹𝑠right + 𝐹𝑢left + 𝐹𝑢right = 0 (4)

for the bounce motion of the rear axle. The roll motion of thesprung inertia is given by

𝐼𝑥𝑥 ..𝜙𝑠 + 𝐹𝑠left 𝐿2 − 𝐹𝑠right 𝐿2 = 0 (5)


𝑉𝑧𝑠

𝑧u

𝑚𝑠

𝐾𝐹𝑅

𝐾𝑢𝐿

𝐵𝐹𝑅

𝑍𝑅𝐿 𝑍𝑅𝑅

𝑚𝑢

𝐿

𝐾𝐹𝐿 𝐵𝐹𝐿

𝜙𝑠

𝜙𝑢

𝐾𝜙𝑢

Antiroll bar𝐾𝑢𝑅

Figure 2: Half car model.

and the roll motion of the rear axle (considering rotationalunsprung inertia 𝐼𝑢𝑥𝑥) is represented by

𝐼𝑢𝑥𝑥 ..𝜙𝑢+ 𝐾𝜙𝑢𝜙𝑢 − (𝐹𝑠left + 𝐹𝑢right) 𝐿2+ (𝐹𝑠right + 𝐹𝑢left) 𝐿2 = 0,

(6)

where the forces 𝐹𝑠left , 𝐹𝑠right, 𝐹𝑢left , and 𝐹𝑢right acting on thesprung mass are given by

𝐹𝑠left = 𝐵𝐹𝐿 {( .𝑧𝑠 + .𝜙𝑠 𝐿2) − ( .𝑧𝑢 +.𝜙𝑢 𝐿2)}

+ 𝐾𝐹𝐿 {(𝑧𝑠 + 𝜙𝑠 𝐿2) − (𝑧𝑢 + 𝜙𝑢 𝐿2)} ,𝐹𝑠right = 𝐵𝐹𝑅 {( .𝑧𝑠 − .𝜙𝑠 𝐿2) − ( .𝑧𝑢 −

.𝜙𝑢 𝐿2)}+ 𝐾𝐹𝑅 {(𝑧𝑠 − 𝜙𝑠 𝐿2) − (𝑧𝑢 − 𝜙𝑢 𝐿2)} ,𝐹𝑢left = 𝑘𝑢𝐿 (𝑧𝑢 + 𝜙𝑢 𝐿2 − 𝑧𝑅𝐿) ,𝐹𝑢right = 𝑘𝑢𝑅 (𝑧𝑢 − 𝜙𝑢 𝐿2 − 𝑧𝑅𝑅) ,

(7)

where 𝐿 stands for the track width, and 𝐼𝑥𝑥 and 𝐼𝑢𝑥𝑥represent themoments of inertia of the sprungmass and axle,respectively.𝐾𝜙𝑢 is the torsional spring stiffness of the antirollbar which is calculated as

𝐾𝜙𝑢 = 𝐺𝐽𝐿𝑏 , (8)

where 𝐺 is the shear modulus, 𝐽 is the polar moment ofinertia, and 𝐿𝑏 is the length of the stabilizer bar which is

subjected to torsional stress. In the simulations and in theoptimisation process, spring and damper characteristics ofthe left and right sides are assumed to be identical.

3. Performance Criteria Available inthe Literature

3.1. Some Performance Indices Available in the Literature. Thechoice of subobjective functions and their weights in thecombined (main objective) function plays a very critical rolein the optimisation process. In this section, the literaturereview of the vehicle suspension objective functions and theperformance indices are summarized, and our approach tothe objective function formulation is presented. The nonlin-ear stiffness and damping characteristics are optimized byKoulocheris et al. [2] considering a half car model subject todifferent types of road irregularities. As an objective function,the maximum value of vertical acceleration of the vehiclebody at the passenger seat is minimized from the view pointof ride comfort. The objective function is formed accordingto the quadratic penalty given by

𝑓(𝑧) = max ( ..𝑧𝑠) + 𝑀∑𝑐2𝑖 (𝑧) , (9)

where ..𝑧𝑠 is the vertical acceleration of the vehicle mass,𝑀 isthe penalty parameter, and 𝑐𝑖 are the constraint functions forparameter vector 𝑧.

Geometrical parameters of the suspension were consid-ered in Mitchell et al. [4] when determining the fitness ofa given suspension design. Since these parameters are notall at the same magnitude or even have the same units,coming up with a single fitness value is difficult. Their basicapproach was to carry out the design with a unitless measureof fitness for each test and then to combine these results witha weighing function. Several functions were analyzed andcompared while evaluating the speed and the accuracy ofthe method using the genetic optimisation algorithm. A first-order normal distribution was chosen due to its convergencespeed. The score equals to 100 for the ideal score and 28.3 atthe bound. Each metric score (score𝑖) is combined by way ofa weighing function (𝑊𝑖). Then, the scoring metric and totalscore are normalised using

Total Score = ∑𝑖𝑊𝑖Score𝑖∑𝑖𝑊𝑖 . (10)

The coordinates of the front and rear suspension hard points,the stiffness and damping properties of the front and rearsuspension springs and damper, sprung mass, gear ratio, theinertia of steering wheel and so forth. were selected by Li etal. [19] as the design variables, considering vehicle handling.The objective evaluation index was adapted to evaluate theperformance of vehicle handling. The index included coursefollowing indices, driver burden indices, indices for the risksof roll over, index for driver’s road feeling, and index forlateral slip. The double lane change maneuver was selectedfor a virtual test, and the objective evaluation index wascalculated.


The objective function in Eskandari et al. [9] was selectedto represent several aspects of the handling behaviour of avehicle and is of the form

𝐹 = ∑𝑖

𝑊𝑖𝑋𝑖, (11)

where 𝑋𝑖 represent yaw velocity overshoot, yaw velocity risetime, lateral acceleration overshoot, lateral acceleration risetime, roll angle steady-state response, RMS of the under-steering coefficient, RMS of the steering torque, and RMS ofthe steering sensitivity. Determination of theweighing factors𝑊𝑖 was made based on the importance of each quantity andis adjustable.

A global performance index is considered in Tak andChung [1] as the linear combination of each individualperformance index. Through kinematic analysis, toe andcamber curves were obtained, and target values of the toe andcamber curves were set up. The squared value of the reactionforce at each tie rod was also included in the performanceindex. The global performance index was determined as theweighed linear combination of the wheel angles and reactionforces at the joints as

𝐼 = 𝑊1𝐼toe +𝑊2𝐼camber + ⋅ ⋅ ⋅ + 𝑊𝑗𝐼roll rate+ ⋅ ⋅ ⋅ + 𝑊𝑘𝐼reaction forces . (12)

The performance of an active suspension system was evalu-ated by Jonasson and Roos [20], covering comfort and roadholding capabilities as well as the energy consumption ofthe system. The formulation of three different performanceindices was considered: two of them are based on the RMSnorm described by

|𝑧|RMS = √1𝜏 ∫𝜏

0𝑧2 𝑑𝑡. (13)

Comfort is strongly related to the vertical accelerations of thevehicle body. Therefore, the performance index for comfortis formulated considering vertical accelerations. The comfortindex for a vehicle with an active suspension system isweighed with respect to the acceleration of the body ina conventional system and is described as a ratio 𝐼1 =| ..𝑧active|RMS/| ..𝑧passive|RMS. A value of more than one meansthat the current design is inferior as compared to the passivesuspension. Road holding capability is directly related tothe variation in vertical tire force, a constant tire forceis ideal, and therefore the second index is described by𝐼2 = |𝐹𝑇,active|RMS/|𝐹𝑇,passive|RMS. The third index 𝐼3 of theobjective function considers the energy used by the activesystem as represented with 𝐼3 = |𝑃loss,active|/|𝑃loss,passive|.Then, these indices are combined to form an overall objectivefunction for the optimisation algorithm

𝐼 = 𝑊1𝐼1 +𝑊2𝐼2 +𝑊3𝐼3. (14)

3.2. Use of Frequency Weighing Based on the ISO2631 Stan-dard. In order to optimize ride characteristics, human sensi-tivity to vibrations needs to be considered. For that purpose,

Frequency (Hz)

Fre

qu

ency

wei

ghin

gs (

dB

)

Frequency weighing according to ISO 2631-1

−50

−100

−150 10

0

−1 100 101 102

Figure 3: Frequency weights as specified in ISO 2631-1 standard.

the vertical motion is weighed according to the ISO 2631[21] standard. The different characteristics of the excitation,including magnitude, frequency, axis, and duration based onthe human tolerance for vibrations should be considered. Assuggested by the ISO 2631 standard, the complete accelerationtime histories for each of the target points are measured.Then, each Cartesian component of the acceleration historyis decomposed into a Fourier series. After that, a frequencyweight given in ISO 2631 standards is multiplied by each termof the Fourier series. The single objective function value isdetermined as the sum of the weighed terms of the Fourierseries previously obtained in the decomposition process.Frequency weights of acceleration as specified in ISO 2631-1standards are shown in Figure 3.

Figure 4 shows the body acceleration response of a quar-ter car model to a chirp (swept sine) signal, its power spectraldensity (PSD), and their weighed counterparts obtained afterusing the ISO 2631 standard. The original and the weighedsignals are presented in time domain (top plot) for comparingtheir magnitudes and in the frequency domain (bottom plot)for comparing their power spectral densities (see Figure 4).

The target accelerations of the vehiclemodels are weighedaccording to the ISO 2631 standards in this investigation[21, 22]. The cases belonging to each vehicle model (total oftwo cases) are investigated. The performance indices of thesecases are presented in the simulation section (Section 6).

4. The Optimisation Procedure

In this section, the optimisation procedure is explainedin detail. In the current investigation, the complexity ofthe optimisation problem is reduced by deciding on thebasic shape/behaviour of the force versus displacement andforce versus velocity characteristics and then scaling themup or down during optimisation. This is motivated by thesuspension design procedure used in the automotive industrywhere a previous, successful design is used as the basedesign, modified for the vehicle model being developed. Theprocedure used here has two steps.


0 2 4 6 8 10

Time (s)

Acceleration

Weighed acceleration

−5−10

1050

Bo

dy

acce

lera

tio

n(m

/s2 )

Weighed versus unweighed body acceleration

(a)

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 1500

5000

10000

Frequency (Hz)

Mag

nit

ud

e(m

/s2 ) Weighed versus unweighed acceleration

Acceleration

Weighed acceleration

(b)

Figure 4: Body acceleration signal is weighed according to ISO2631-1 standards.

Velocity (mm/s)

Fo

rce

(N)

Fitted

Real

−0.25−0.5−2000 −1000

Front damper real versus fitted damping curves

0

1

0.75

0.5

0.25

0

1000 2000

Figure 5: Normalized damper characteristic and its functional fit(front).

Relative velocity (mm/s)

No

rmal

ized

dam

pin

g fo

rce

Fitted

Real

−0.5−2000 −1000

Rear damper real versus fitted damping curves

0

1

0.5

0

1000 2000

Figure 6: Normalized damper characteristic and its functional fit(rear).

(1) A function like a polynomial, rational, or an exponen-tial function that can fit the basic initial data of forceversus displacement/velocity profile is chosen first.

(2) Then, a scaling factor 𝐶opt is used for the function,such that

𝐹opt (𝑥) = 𝐶opt𝐹 (𝑥) . (15)

The optimisation procedure requires computations of theobjective function at each iteration step. The performanceindex used here includes both time and frequency domainanalyses. The performance index evaluation can be summa-rized with the following steps.

(1) First, the simulations of the quarter and the half carmodels subjected to a road excitation are carriedout in the time domain. Note that while runningthe optimisation routines of the vehicle suspensionmodels, two aspects could significantly affect resultsand might cause errors in the optimisation process asfollows:

(a) it is preferable to consider the steady stateresponse of the vehicle run. Since a constantvehicle speed is assumed, it takes some timefor the vehicle to reach the steady state con-ditions. Therefore, the beginning part of thetime domain simulation containing the tran-sient response is omitted,

(b) attention should be paid to the static deflec-tions (due to weights) and the initial conditionsconsidering the static equilibrium points for thesprings.

(2) Then, the target point’s accelerations are weighedaccording to ISO-2631 standards in the frequencydomain and used as part of the objective function.

(3) A suitable global objective function is establishedaccording to the needs of automotive manufacturers.This is the most subjective step of the methodology.Since the choice of the objective function and weigh-ing of the particular objectives will result in different


0 50 100 150 2000

0.25

0.5

0.75

1

Deflection (mm)

Leaf spring characteristic curve

No

rmal

ized

sp

rin

g fo

rce

Figure 7: Normalized spring characteristic (rear).

Wavenumber (cycle/m)

PSD

Robson road profile used in optimisation

3

50

800

3000Profile

10−5

10−110−210−10 100 101 102

100

Figure 8: Road profile PSD versus wavenumber.

0 2 4 6 8 10

0

0.05

0.1

Time (s)

Ro

ad i

rreg

ula

rity

(m

)

Robson road profile used in optimisation

−0.05

−0.1

Figure 9: Generated road profile versus time.

5 10 15 20 25 30 3590

100

110

120

130Objective function values through optimisation process

Number of runs

Val

ues

Figure 10: Objective function’s run history.

36

1

11

2134

7

38 40 42 44 46 48 50

57

56.5

55.5

55

56

57.5

58

58.5Change of indices through optimisation

Body acceleration index

Tir

e fo

rce

chan

ge i

nd

ex

Figure 11: Change of indices throughout optimisation process.

optimum outcomes. Our choice of objective func-tions (performance index) for the current paper isexplained in the previous section and in the followingsection on optimisation results.

(4) As the final step, the optimisation type and algorithmare selected, and the optimisation step is performedin Matlab. The optimisation toolbox SQP algorithmwith Quasi-Newton line search is implemented. TheSQP algorithm like Simplex, Complex, and Hook-Jeeves belongs to the family of local search algorithms.The local search algorithms converge to the nearestoptimum, since they depend upon the starting valuesof the design variables. Examples in the followingsection illustrate the simulation result for quarter carand half car vehicle models used here. Numericalsimulation results show that the SQP algorithm canefficiently and reliably find the optimum in the neigh-bourhood of the initial point.

Finally, the optimum spring and damper characteristicsobtained should be checked to see if they aremanufacturable.


5 10 15 20 25 30 350.67

0.83

1

Val

ues

Number of runs

(a)

5 10 15 20 25 30 350

1

2Damping curve optimisation coefficient

Number of runs

Val

ues

(b)

Figure 12: Optimum normalised linear spring stiffness and nonlinear damper characteristic scaling coefficients.

5 10 15 20 25 30 3520

40

60

Body acceleration RMS values

Val

ues

Number of runs

(a)

5 10 15 20 25 30 35

60

80

100Tire force change

Number of runs

Val

ues

(b)

Figure 13: Performance indices throughout the optimisation process.

0 1000 2000

0

0.5

1


Dam

pin

g fo

rce

(N)

(before and after optimisation)

Default After optimisation

−0.5−2000 −1000

Front damper comparison—normalised

Figure 14: Default and optimized damper characteristic curves.

The nonlinear damper characteristic of the front inde-pendent suspension and rear solid axle of a light commercialvehicle are presented in normalized form in Figures 5 and6, respectively. Since the shape of the curve is essentialfor manufacturing, an appropriate functional representationshould be used in the optimisation process. A function withthe following form is suitable for the whole range of thedamper data

𝐹(𝑣) = 𝐴𝑒−𝑘𝑣 + 𝐵𝑒𝑞𝑣. (16)

Equation (16) is used as an empirical curve fit to real dampercharacteristics. The parameters 𝐴, 𝐵, 𝑘 and 𝑞 in (16) are,therefore, determined empirically. 𝑣 in (16) represents the

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1

Deflection (mm)

Spri

ng

forc

e (N

)



Coil spring comparison—normalised

Figure 15: Default and optimised spring stiffness.

independent variable, which is suspension velocity here.The real experimentally determined suspension dampingcharacteristics are shown as solid lines in Figures 5 and6. The empirical data fit obtained using (16) is shown asdashed lines in Figures 5 and 6. The close fit between the realand fit (using (16)) damping characteristics shows that (16)provides a highly accurate representation of the real dampingcharacteristics considered in this paper.

For the nonlinear modelling of the spring of the rear axle,a lookup table which represents the nonlinear characteristicof the spring is used (see Figure 7). The spring characteristicof the front independent suspension is linear in all optimisa-tion processes.


0.51

1 1.5

0.670.83

50

100

150

200

Damping curve coefficient

Objective function

Spring stiffness

Ob

ject

ive

fun

ctio

n

Figure 16: Values of the objective throughout two-dimensionalsearch domain.

0 2 4 6 8 10

0

0.05

0.1

Time (s)

Ro

ad i

rreg

ula

rity

(m

)

Track 1 and track 2

−0.05

−0.1

Figure 17: Left and right side of the road generated with RobsonMethod.

0 10 20 30 40 5090

100

110

120

130

140Objective function values through optimisation process

Number of runs

Val

ues

Figure 18: Total objective function progress.

5. Optimisation Simulation Results

The optimisations results for the front and rear suspensionunits of the quarter and the half car models are presentedin this section. The performance indices used in the opti-misation process are presented in the following Sections (5.1and 5.2). Finally, Carmaker software is used to check and to

452

50

48

46 44

1

2141

47

31

46

48

4.5

5

5.5

Body acceleration indexTire force change index

Ro

ll a

ngl

e in

dex

Change of indices through optimisation

Figure 19: Change of indices throughout optimisation process.

Table 1: Different road classes.

Road class C/10−8 m0.5 cycle1.5

Motorway 3–50Principal road 3–800Minor road 50–3000

confirm the optimised results using a high fidelity, full vehiclemodel.

5.1. Quarter Car Optimisation Results. Optimisation runsare performed using the quarter car model in this part ofthe investigation considering the vehicle with a nonlineardamper and linear spring unit subjected to a generated road.It is required to have road surface input profiles for therealistic response of the vehicle dynamic simulations. Theseinput profiles may come directly from the measurementsmade by a test vehicle. It is also possible to artificially generatethe random road profiles like Robson’s Method, presentedin Kamash and Robson [23]. In Robson’s Method, the roadprofile spectra can be given by

𝐺 (𝑛) = 𝐶𝑛−𝑤, (17)

where 𝑤 is a constant which is equal to 2.5, 𝑛 is thewavenumber in cycle/m as described in ISO-8608, and 𝐺(𝑛)is the displacement spectral density in m3/cycle. For differentclasses of roads 𝐶 takes the values listed in Table 1 [24].

In the conventional spectral analysis the process is simplysquaring the spectral coefficients which are determined byusing discrete Fourier transform. It is required to haveuniformly distributed random phases between 0 and 2𝜋. Theinverse Fourier transform is given by:

𝑧𝑟1 = 𝑁−1∑𝑘=0

√𝐺𝑘𝑒𝑖(𝜃𝑘+(2𝜋𝑘𝑟/𝑁)), 𝑟 = 0, 1, 2, . . . (𝑁 − 1) , (18)

where 𝐺𝑘 is described by

𝐺𝑘 = ( 2𝜋𝑁Δ)𝐺11 (𝛾𝑘) (19)


0 10 20 30 40 500.5

1

1.5Spring coefficient

Val

ues

Number of runs

(a)

0 10 20 30 40 500

2

4Damper coefficient

Number of runs

Val

ues

(b)

Figure 20: Spring and damper characteristics scaling coefficient progress.

0 10 20 30 40 50

406080

Body acceleration RMS values

Val

ues

Number of runs

(a)

0 10 20 30 40 5040

60

80Tire force change values

Val

ues

Number of runs

(b)

0 10 20 30 40 50

456

Roll angle RMS values

Number of runs

Val

ues

(c)

Figure 21: Objective function components’ progress.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

Deflection (mm)

Spri

ng

forc

e (N

)



Coil spring comparison—normalised

Figure 22: Optimized and the default spring characteristics.

and 𝛾𝑘 is given by

𝛾𝑘 = 2𝜋𝑘𝑁Δ , (20)

where 𝛾𝑘 is the wave-number in rad/m, 𝐺11(𝛾𝑘) is the targetspectral density, Δ is the distance interval between two spotheights, and 𝜃𝑘 is a set of independent random phase anglesuniformly distributed between 0 and 2𝜋.The road profile wasgenerated via Robson’s Method as is given in Kamash and

Robson [23] and also in Cebon [17]. The road profile used inthe optimisation has 𝐶 = 250 (see Figures 8 and 9).

The objective function has two components which arethe weighed body acceleration RMS value and the penaltyfunction of tire force difference. The objective function isdefined as

𝐼 = 𝑤1 ..𝑍𝑠 + 𝑤2Δ𝐹tire, (21)

where..𝑍𝑠 is the RMS value of the weighed body acceleration,

and Δ𝐹tire is the difference between the maximum andminimum tire force picked from the tire force history. Thetotal objective function changes as shown in Figure 10 duringthe optimisation run, and an optimum point is reachedafter 35 iterations. The minimisation of both indices inthe objective function as the process goes on is illustratedin Figure 11 via representative steps. Figure 12 shows thenormalised values of the design variables: the spring stiffnessand damper characteristic scaling coefficients for the quartercar model. The spring stiffness increases, and the dampercharacteristic scaling coefficient decreases until an optimumpoint is reached. How the body acceleration RMS value..𝑍𝑠 and the tire force change Δ𝐹tire change during theoptimisation run are displayed in Figure 13. The initial andoptimised damper and spring characteristics are shown inFigures 14 and 15, respectively. Using a brute force approach,the objective function is evaluated on a grid of spring stiffnessand damping curve scaling coefficient values. The resultingplot in Figure 16 demonstrates that the optimisation resultsshown in Figure 13 are accurate.


0 1000 2000

0

0.5

1


Dam

pin

g fo

rce

(N)



−0.5−2000 −1000

Rear damper comparison—normalised

Figure 23: Optimised and default damper characteristics.

0.51

1.5 0.51

1.580

100

120

140

160

Spring curve coefficient

Objective function

Damper curve coefficient

Ob

ject

ive

fun

ctio

n

Figure 24: Values of the objective throughout two-dimensionalsearch domain.

Figure 25: Snapshot of Carmaker animation.

5.2. Half CarModel. Thehalf carmodel embodying a nonlin-ear rear suspension unit of the light commercial vehicle incor-porating nonlinear dampers and nonlinear springs whosebasic characteristic curve shapes are given in the previoussections is considered in this subsection. The performanceindex used in the previous case (the quarter car) is also con-sidered here with the addition of a function including bodyrollmotion.The objective function (performance index) used

24 26 28 30 32

0

2

4Roll angle comparison

Time (s)

Ro

ll a

ngl

e (d

eg)

Default

After optimisation

−2

−4

Figure 26: Roll angle comparison double lane change.

24 26 28 30 32

0

5

10

15Roll rate comparison

Time (s)

Ro

ll v

elo

city

(d

eg/s

)

Default

After optimisation

−5−10−15

Figure 27: Roll rate comparison double-lane change.

24 26 28 30 32

0

10

20Yaw rate comparison

Time (s)

Yaw

vel

oci

ty (

deg

/s)

Default

After optimisation

−10

−20

Figure 28: Yaw rate change during the manoeuvre.


0

50

Frequency (Hz)


−50−100−150−200−250 100 101 102

Mag

nit

ud

e (d

B)

FL body

(a)

0

50


−50−100−150−200−250 101100 102

Mag

nit

ud

e (d

B)

Fr

FR body

equency (Hz)

(b)

0

50


−50−100−150−200−250 101100 102

Mag

nit

ud

e (d

B)

Frequency (Hz)

RL body

(c)

0

50


−50−100−150−200−250 101100 102

Mag

nit

ud

e (d

B)

Frequency (Hz)

RR body

(d)

Figure 29: Vertical vehicle body acceleration PSD plots at four corners with high order model.

in optimisation ismade up from three functions, as describedbelow:

𝐼 = 𝑤1 ..𝑍𝑠 + 𝑤2Δ𝐹tire + 𝑤3Φ𝑠, (22)

where Φ𝑠 represents the RMS value of the roll angle of thevehicle. The difference in the performance index for this caselies on the existence of the roll angle.

The left and the right sides of the road being followedare shown in Figure 17, and they are generated independentlyusing the Robson Method. Since both sides of the roadare independent of each other, they have zero correlationbetween them. Arbitrary body roll motions will be inducedas a result of this road profile in the simulation runs.The rearaxle nonlinear suspension unit characteristics are optimizedconsidering body roll angle.

The optimisation results for this case are presented inFigures 18–21. The change of the performance index duringthe course of the optimisation run that is given in Figures18 and 19 emphasizes the minimisation of all three indices

step by step throughout the process. How the nonlinearspring and nonlinear damper scaling coefficients changeduring the optimisation is shown in Figure 20.The changes ofcomponents of the performance index are shown separatelyin Figure 21. Figures 22 and 23 show the initial and opti-mised damper and spring characteristic curves, respectively.Figure 24 demonstrates the search domain of the objectivefunction utilized according to two design parameters chang-ing in a predetermined interval. Based on these figures, it isseen that after the optimisation the spring characteristic tendsto be softer, while the damper characteristic tends to be moredamped.

5.3. Handling Tests Using Carmaker Software. Finally, the ridehandling and comfort performances of the optimized suspen-sion parameters are confirmed by employing the high fidelityCarmaker model of the considered vehicle (see Figure 25). Astandard double lane change maneuver is utilized to observethe body roll improvement which mostly indicates better


0

50

100

Frequency (Hz)


−50−100−150

100 101 102

Mag

nit

ud

e (d

B)

FL tire

(a)

0

50

100

Frequency (Hz)


−50−100−150

100 101 102

Mag

nit

ud

e (d

B)

FR tire

(b)

0

50

100

Frequency (Hz)


−50−100−150

100 101 102

Mag

nit

ud

e (d

B)

RL tire

(c)

0

50

100

Frequency (Hz)


−50−100−150

100 101 102

Mag

nit

ud

e (d

B)

RR tire

(d)

Figure 30: Vertical wheel acceleration PSD plots at four corners with high order model.

handling.The simulation results for roll angle and roll rate inFigures 26 and 27 show that the nonlinear optimisation hasimproved roll handling performance. Figure 28 shows thatthe yaw rate handling has not changed significantly. A testtrackwith road irregularities which contains various frequen-cies is employed for the confirmation of comfort performanceof the optimized suspension settings. The simulation resultsin Figure 29 show the ride comfort improvement in the formof lower vehicle body vertical acceleration PSD values atthe four corners of the vehicle. The simulation results inFigure 30 show that the tire accelerations have not changedsignificantly, illustrating the maintenance of the previousroad holding characteristic.

6. Conclusions

Light commercial vehicle front and rear suspension unitsincorporating both linear or nonlinear springs and nonlinear

dampers were optimized to improve vehicle ride and han-dling. The nonlinear equations of motions of the quarter carand the half car representing the front and rear suspensionmodels were presented and simulated in theMatlab/Simulinkenvironment. Several aspects of performance criteria wereconsidered for ride comfort and handling such as RMSvalues of weighed body acceleration, the range of tireforces, and RMS values of body roll angle. For each aspectof performance, time-domain performance measures wereevaluated after the optimisation run. A simple optimisationmethodology of nonlinear suspension unit was presented,incorporating typical data provided by car manufacturers forthe initial characteristics. The methodology was based onkeeping the shape of the damper and spring properties andcurve fitting a proper function to these data and then scalingit throughout the optimisation process. Finally, the advantageof the nonlinear optimised suspension unit compared to adefault suspension unit was demonstrated using a double lanechange maneuver with a high fidelity full vehicle model in


Carmaker. In order to generalise the nonlinear suspensionunit optimisation problem, an interactiveMatlab toolbox wasconstructed and used in obtaining the results presented here.

The fact that the improvement between original perfor-mances and optimized ones is not too big in the resultspresented in the paper is due to the fact that we started withan already optimized suspension which had been designedearlier. We decided against creating a nonideal startingvalue for the suspension parameters and showing a largeimprovement in performance. We were indeed able to showthat the suspension design provided to us was very closeto its optimum configuration, and only small performanceachievementswere possible.This is also a very useful outcomeas the suspension designer can analytically evaluate hisdesign against the optimal one. The method presented in thepaper is fully automated in the form of a Matlab programand its interactive graphical user interface. The results arealso obtained much faster as compared to the conventionalmethod of suspension design including a lot of trial and error.

Abbreviations

DOF: Degree of freedom𝐼𝑖: Performance index, 𝑖 = 1, 2, 3 . . .𝐼: Total indexRMS: Root mean squarePSD: Power spectral density.

List of Symbols

Quarter Car Model

𝑚𝑠: Sprung mass, kg𝑚𝑢: Unsprung mass, kg𝑧𝑠: Sprung mass vertical displacement, m𝑧𝑢: Unsprung mass vertical displacement, m𝑧𝑟: Road irregularity, m𝑏𝑢: Linear tire damping coefficient𝑘𝑠: Linear spring stiffness𝑘𝑢: Linear tire stiffness𝐵𝐹: Nonlinear damper characteristic function.

Half Car Model

𝑚𝑠: Sprung mass, kg𝑚𝑢: Unsprung mass, kg𝑧𝑠: Sprung mass vertical displacement, m𝑧𝑢: Unsprung mass vertical displacement, m𝑧𝑅𝐿, 𝑧𝑅𝑅: Left and right road irregularities, m𝑘𝑢𝐿, 𝑘𝑢𝑅 : Left and right linear tire stiffness,𝑁/m𝐵𝐹𝐿, 𝐵𝐹𝑅: Left and right nonlinear dampercharacteristic functions, Ns/m𝐾𝐹𝐿, 𝐾𝐹𝑅: Left and right nonlinear dampercharacteristic functions, N/m𝐾Θ: Antiroll bar torsional stiffness, Nm/rad𝐿: Track width, m𝜙𝑠: Sprung mass roll angle, deg/sec

𝜙𝑢: Unsprung mass roll angle, deg/sec𝐼𝑥𝑥: Sprung mass moment of inertia about 𝑥-axis𝐼𝑢𝑥𝑥: Unsprungmassmoment of inertia about𝑥-axis.

Optimisation Process

𝑤𝑖: Weighing, 𝑖 = 1, 2, 3 . . .Δ𝐹tire: Tire force change, NΔ𝐹tire opt: Optimised value of tire force change, N..𝑍𝑠: RMS of weighed sprung mass acceleration..𝑍spot: Optimised value of the RMS of weighed

sprung mass accelerationΦ𝑠: RMS of sprung mass roll angle.

Acknowledgments

The authors thank Yıldıray Koray for his help in the roadprofile used.The authors thank Server Ersolmaz, Erhan Eyol,Mustafa Sinal. and Selcuk Kervancıoglu of Ford Otosan forhelpful discussions on suspension design. The authors alsothank Ford Otosan and the EU FP6 project AUTOCOMINCO-16426 for their support. The authors dedicate thispaper to the dear memory of their second coauthor ProfessorU. Sonmez who passed away so suddenly and unexpectedly.

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