Finite Elements in Analysis and Design 63 (2013) 98106Contents lists available at SciVerse ScienceDirectFinite Elements in Analysis and Designjournal homepage: www.elsevier.com/locate/nelAn analytical approach for design and performance evaluation of torsionbeam rear suspensionDongchan Lee a, Chulho Yang b,nabMetariver Technology Co. Ltd., B-801, Garden 5 Works, 52 Chungmin-ro, Songpa-gu, Seoul 138-961, South KoreaMechanical Engineering Technology, Oklahoma State University, 385 Cordell S., Stillwater, OK 74078, USAa r t i c l e i n f oArticle history:Received 1 November 2011Received in revised form13 August 2012Accepted 2 September 2012Keywords:Rear suspensionTorsion beamSection proleShear centerRoll steerRoll camberRoll center heightRoll stiffnessWarping constanta b s t r a c tTorsion beam rear suspension systems have been widely used in small passenger cars owing to theircompactness, light weight, and cost efciency. To study the roll behavior of torsion beam suspensionsystems, analytical equations to obtain roll center height, roll steer, and roll camber have beendeveloped in terms of geometry points through the kinematic consideration of the system on theassumption of rigid body motion. In most cases, commercial software for nite element methods orexperimental equations for individual parts are utilized for the design and evaluation of the suspensionsystems. This paper, however, proposes an analytical method to calculate the torsional stiffness of atorsion beam for various loading conditions based on the assumption that a torsion beam is in the rangeof linear torsional angles with a constant cross section. The proposed method is useful for exploringsuspension system design, especially in the early stage of vehicle system development in which detaildesign parameters such as layout, cross sections, and materials are not yet determined. It is shown thatthe torsional stiffness and roll stiffness predicted using the proposed method has sufcient precisionwith around four percent difference from the results of nite element analysis and bench tests. & 2012 Elsevier B.V. All rights reserved.1. Introduction Torsion beam rear suspension systems are commonly used forsmall passenger vehicles because of various advantages includinga reduced weight, lower cost, and compactness. These systemsconsist of two trailing arms interconnected by a exible beamwith the simple conguration shown in Fig. 1. This exible beamhas open or closed tubular cross sections that are rigidly con-nected to each trailing arm in order to provide the torsional andbending stiffness required for kinematic performance and com-pliance of the system. A V- or U-shaped prole is typically usedfor the open cross sections. The geometric prole of the torsionbeam consists of a constant section area, a transition area inwhich the section prole is changed along its distance, and ajunction area at each end that is welded to the trailing arms. Despite these relatively simple congurations, many designparameters need to be determined during the early stage ofsystem design considering the elastic deformation of the wholesystem. The torsion beam must allow large torsional displace-ment over its length without failure of the material and musthave enough stiffness to support lateral forces during cornering. Itmust also have enough exibility to allow each wheel to displacenCorresponding author. Tel.: 1 405 744 3033.E-mail address: chulho.yang@okstate.edu (C. Yang).independently over a bump. This suspension shows a non-linearrelationship between the wheel toe-in and the wheel verticaldisplacement when subjected to an anti-symmetrical roll loading.This non-linear relationship contributes to the overall handlingresponse of the vehicle. Therefore, engineers must consider abalance between the stiffness and the exibility of the compo-nents when they design a torsion beam suspension. In theconcept design stage it is necessary to predict changes inperformance that result from changes in the alignment andcompliance steer to see how the stiffness of each member hasan impact on the overall performance of the suspension. Mostresearches [15] have been conducted for in-house design toexamine deformation behavior and roll properties based on thekinematic motions. In addition, commercial nite element analy-sis (FEA) software or experimental equations for each member ismainly used for the evaluation of the system. However, in general,detail geometry (layout, cross section, and materials) of each partis not available for simulation or test in this stage. This research proposes design guidelines for a torsion beamrear suspension that can be used for determining design para-meters such as mounting positions, torsion beam cross section,and material through theoretical analysis of kinematic and elasticbehavior of the system. The proposed design approach is con-rmed through the feasibilities of simulation using the commer-cial FEA software Adams and bench tests of real automotivesuspension systems.0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nel.2012.09.002

D. Lee, C. Yang / Finite Elements in Analysis and Design 63 (2013) 9810699NomenclatureACLRLongitudinal distance from body mounting pivot toshear centerLateral span of body mounting bush pivotLongitudinal distance from body mounting pivot towheel centerArc radius of torsion beam proleTehphefyTread, lateral span between two wheel centersShear center of cross section of torsion beamHeight of body mounting bush pivotHeight from body mounting bush pivot to shearcenterRoll angle of torsion beam suspensionAngular displacement of torsion beam2. Design approach for torsion beam rear suspension For the conguration of a torsion beam rear suspension, manydesign parameters, such as position, cross section, materialproperties, and geometric dimensions of each component, needto be considered. The hierarchical design process provides betterperformance in achieving efciency and simple operability. Ifdesign engineers need to develop one of the components in detail,they must estimate the object of design for that component,which will be incorporated into the upper level of the entiresuspension system. In torsion beam rear suspension, it is impor-tant to select the design requirements that have a direct effect onthe performance of the vehicle to show the appropriate results,including weight, durability, roll center height, roll rate, andlateral stiffness. Design parameters should be considered inrelation to thickness and shape dimensions of each componentfor the given vehicle specication. In particular, the vehiclespecications for torsion beam rear suspension design are itstotal weight, pivot point and wheel center of trailing arm, shearcenter of torsion beam, and roll center height. Engineers shoulddetermine the fundamental requirements (roll steer, roll camber,roll stiffness) and layout of the torsion beam rear suspensionthrough the following considerations. For discussion of minimal variations of roll axis, the followingdenitions are described. In Fig. 2, the vectors from the bodymounting pivot point (point P) to shear center (point Q), wheelcenter (point W), and tire patch point (point G) can be given by! n !!PQ A u C=2 v o! The w ,oT1! n ! !PW L u TC=2 v !hw hp w,and2 o! n ! !! TPG L u TC=2 v hp w3 !!!where u , v , and w are the unit vectors in x, y, and z directions,respectively. A is the distance from the point P to shear center Q inx-direction, C is the span of the body mounting points in y-direction, L is the distance from the point P to wheel center Win x-direction, and T is the tread between the left and right wheelcenters. !The vector of rotation axis between point P and point Q, DaPQ , !can be calculated from its unit vector ( u PQ ) and magnitude

!

(DaPQ ):DaPQ2.1. Kinematic characteristics of torsion beam suspension The roll motion of torsion beam suspension occurs on the rollaxis of the rear suspension and the shear center of the torsionbeam. The shear center is very important in determining thekinematic characteristics of the suspension. If the shear center ofthe torsion beam is given or calculated, kinematic modeling canbe achieved. Under the roll motion, the shear center becomes therotation center of the beam section, and if the right- and left-handwheels are displaced differently over a bump, the shear centeraxis and the center plane of the vehicle make a point of instantcenter (point Q). The axis between the body mounting point(point P) of the trailing arm and point Q becomes the roll axis ofthe rear suspension. The design parameters, including the rollsteer and roll camber, can be estimated from the minimalvariations of roll axis [36].! !9DaPQ 9 n ! ! !Au 9DaPQ 9 u PQ ! 9 PQ 9 !!C=2 v he woT4 The roll angle (f) of the vehicle is equal to the portion of thevertical displacement (DZ W ) divided by the half tread (T=2), andthe total torsion angle (y) of the torsion beam is equal to two !times the y-direction component in DaPQ :f DZ W =T=2 !9DaPQ 9y ! C: 9 PQ 9 !9DaPQ 9 ! ATC CLT9 PQ 9and56 The displacement of wheel center (point W) can be given by ! !DaPQ PW and the displacement of wheel center in z-directionFig. 1. Descriptions of torsion beam axles (a) torsion beam rear suspension (b) suspension system for a typical front wheel drive vehicle.

100D. Lee, C. Yang / Finite Elements in Analysis and Design 63 (2013) 98106Fig. 2. Kinematic model of torsion beam rear suspension (a) side view of suspension (b) top view of suspension (c) torsion angle of torsion beam (d) roll angle ofsuspension.can be derived from Eq. (2) and Eq. (4):DZ W _! _9DaPQ 9TC CL A !229 PQ 97The roll steer angle (bf ) and roll camber angle (gf ) of the wheelunder the roll angle of the rear suspension can be given bybf bThe f TCA CLand8gf gf TATCA CL9 The toe angle (b) and camber angle (g) are equal to thez-direction component and x-direction component in Eq. (4),respectively,Fig. 3. Typical section proles (a) thin-walled open section (b) thin-walled closedsection. DaPQb ! he

PQ

and10K fS 2K S DZ S 2f2 2K SAyS C=2xS 2r2 C 214 DaPQg ! A

PQ

11 The torsion stiffness of the beam is given by the twistingmoment per total twist angle using the shear modulus (G), torsionconstant (J), and length (LT ) of the beam as shown in Eq. (15a)[68],Mt2.2. Cross-sectional property of torsion beam with respect to torsion The total roll stiffness of a torsion beam rear suspension ismainly determined by the roll stiffness of the torsion beam (K f T )and spring (K fS ),K f K fT K fS12yJGJ ,LT15a n1X b t 3,3i1 i i15bandJ H4A @ds=ts j1k12 The roll stiffness due to the torsional stiffness of the torsionbeam, K fT , can be given by the lever ratio (r) of the twist angle ofthe torsion beam to the roll angle. The roll stiffness due to theleft- and right-hand side springs,K f S , can be given by thecondition that the stored potential energy due to the roll motion 2(1=2K fS f ) is equal to the stored potential energy of two springs(2 1=2K S DZ s 2 ):K fT KT ,r2r0jmax kmaxX X 12 ,jmax kmaxX X!!!r jD s k i ADsk =t n j1k115cJ in Eq. (15b) and (15c) represents the torsion constant along thecenter line of wall thickness of a thin-walled open section(Fig. 3(a)) and along the center line of wall thickness of a thin-walled closed section (Fig. 3(b)), respectively. r k denotes thedirection vector of thickness element (sk , sk 1 ). Dsk is thefy13

D. Lee, C. Yang / Finite Elements in Analysis and Design 63 (2013) 98106101 !circumferential vector of thickness element (sk , sk 1 ) and i is theunit vector in the longitudinal direction of the torsion beam. The vertical deformation of springs under the roll motion, DZ S ,

! !

is DaPQ PS ,IR 2tx2 dA"Z0f0 __2 #3l cos2 f0lcosf0 l Rsinf0 Rsinf tRdf 122220 If it is desired to bend a beam by transverse forces withouttwisting the beam, each transverse force should pass through theshear center of the cross section of application. When the sectionhas an axis of symmetry, the shear center lies on this axis, andonly loading in the plane perpendicular to this axis needs to beconsidered. The location of the shear center is independent of thedirection and magnitude of the transverse forces. For the symme-trical sections, the shear center lies on the axis of symmetry,while for a beam with two axes of symmetry, the shear centercoincides with their point of intersection. In general, it is notnecessary for the shear center to lie on a principal axis and it maybe located outside the cross section of the beam. For thin-walledsections, the shear stresses are taken to be uniformly distributedover the thickness of the wall and parallel to the boundary of thecross section. In Fig. 5, the moment M z of the shear stress due tothe shear ows about arbitrary point O is the external moment(eV) attributable to V about O. The distance between O and theshear center is given as Eq. (21). __iR h2Mz2 f0Re V V 0 qcosf cosf Rcosf qRsin f Rdf0 "# ___ _ R2 tR2f0 sin2f0E sinf0 Ef0 , 21 I cosf0 2cosf04whereE l =2R2 cosf0 Rl sinf0IR32DZ S A yS C=2xS f rC16 The roll center height of torsion beam suspension, HRCH , can begiven by the geometric behavior of the tire patch point in the rollmotion as shown in Fig. 4HRCH TA hp he LT DY G TCA CL2 DZ G17 ! !where the vector of the tire patch point is DaPQ PG ,DY G A hp he L, and DZ G TC=2 A CL=2.2.3. Cross-sectional property of torsion beam with respect toshear force Under the roll motion, the total longitudinal shear force istransmitted across the plane of cross section along the beam andis called the shear ow [810]. It is the resultant of the shear-stress distribution across the thickness. The relation for shear owin the presence of a shear force can be given byqVQ I18Q and I are the rst moment and the area moment of inertia ofthe shaded area about the Y axis in Fig. 5 and can be expressed asshown in Eqs. (19) and (20): "Z__#Zf0l cosf0QxdA tl19R2 sin fdf Rsinf0 2fand2223_f0 _ 3__2 sin2f0l cos2 f0lcosf0 l Rsinf0 241222.4. Cross-sectional property of the torsion beam with respect towarping Warping of the torsion beam cross section can seriously affectthe durability by increasing the maximum stress in the junctionarea between the torsion beam and trailing arm. It also affects thekinematic performance of the suspension system causing changesin toe angle. Therefore, warping must be carefully consideredwhen the suspension system is designed.Fig. 4. Roll center height of torsion beam rear suspension.Fig. 5. Shear ow and shear force of torsion beam section.Fig. 6. Warping shape of cross section of torsion beam.

102D. Lee, C. Yang / Finite Elements in Analysis and Design 63 (2013) 98106 Assuming that the cross sections of the torsion beam (Fig. 6)rotate with respect to an axis passing through point A and parallelto the longitudinal axis while it is twisted, we nd that anylongitudinal position N in the middle surface of the wall becomesinclined to the axis of rotation by the angle y. [9,10] Thelongitudinal point N is dened by the distance s measured alongthe middle line of the cross section. The tangent to the middle lineat N remains perpendicular to the longitudinal layer, and thesmall angle between this tangent and the xy-plane is rycos a r yafter torsion. Having w denote the displacement of the middle line of thecross section in z direction and considering the twist moment tobe positive, the following expression can be obtained:@w r y@s243. Application to design and validation of a suspension system Fig. 7 shows the verication and validation ows through theentire process of designing a rear suspension from the conceptualdescription of the system to the nal design. The predictivecapability of models is limited to specic load ranges and hasbeen proven by showing the correlation through experiments. Table 1 shows the geometric parameters of a given rearsuspension. Table 2 shows the property of each section prolefor the torsion beam suspension using the numerical sectionproperty equations. Fig. 8 shows the typical nite element modelof the torsion beam rear suspension for the verication ofnumerical equations. The simulations and experiments are madefor four section proles of the torsion beam rear suspension.The suspension system is modeled as a single exible body withthe origin located at the vehicle center line, and hence alldeformation is described in this coordinate system. Large rotationof the suspension can occur at the origin of this coordinatesystem; therefore, the deformation is superimposed onto thelarge rotation at the vehicle center line. The toe and camber aregoverned mainly by linear elastic deformation ranges that can besuperimposed. In reality, the rotation at the left and right suspension bushescan be considered independently due to the exibility of thetorsion beam itself. The deformation needs to be describedseparately relative to each body mounting bush. A model iscreated by separating the suspension at the center of the torsionbeam and dening a exible body for each half. Each body isdescribed relative to its mounting bush and both bodies arejoined together using a xed joint. The linear deformation canbe superimposed onto the large rotation of each body of its ownmounting bush. Numerical validation has been achieved by comparing theresults obtained from commercial software and experiments.Fig. 9 shows the test set-ups for evaluating kinematic andcompliance performances of the torsion beam rear suspension.Table 3 shows roll steer, roll camber, roll center height, and rollstiffness compared with the quasi-static results obtained from thecommercial multi-body dynamic software, MSC/Adams, andexperiments. From the results of quasi-static analysis and experi-ments, it is known that the numerical equations can be veriedfor their feasibilities of estimations of kinematic characteristics ofa torsion beam rear suspension. Fig. 10 shows the average forceand vertical displacement measured at the left and right wheelcenter under the roll motions. Note that the estimated values ofkinematic performance are very close to the results of niteelement analysis and tests. When a new product is to be designed, various solutions areoften possible to be embodied in concept designs. A conceptdesign is an early representation of a product, incorporating onlya minimum of details in shape, material, and manufacturingprocess. While the concept denes the maximum achievableperformance of a product, the detail design determines the actualperformance. During detail design, each design parameter isrened through an optimization process. It is known that torsional stiffness of a torsion beam controls7085% of the roll stiffness of a vehicle and reinforcement about1530%. For the optimization of a beam cross section, rollstiffness, lateral stiffness, and weight are considered as they areprimarily affected by the cross section [11]. Durability of thebeam and junction area is additionally considered when thetopological optimization for the junction and reinforcementpanels is conducted. To obtain the best solution, it is necessary toselect major design parameters that affect the performance of thetorsion beam suspension system. The shape of the beam crosssection is an important design parameter related with roll stiffness, By integrating Eq. (24), we can obtain the warping displace-ment of the point from which s is measured in z direction: Z s rds25w w0 yo The average value w of the warping displacement can becalculated from Eq. (25) as follows: _ZZ _Z s1 my mwwds w0 rds ds26 m om oo Subtracting this value from the warping displacement given byEq. (25), the warping of the cross section can be found withrespect to the plane of average warping: _Z _Z sZ s y mdc ,27wrds dsyrds yws ws ws ws m odzoowhere the warping function ws represents the doubled sectionalarea corresponding to the arc s of the middle line of the crosssection, while ws represents the average value of ws . For the caseof non-uniform torsion, the constant angle of twist per unitlength, y, is replaced by the variable rate of change of the angleof twist dc=dz for the angle of rotation of any cross section, c. Since the rate of change in twist angle varies along the lengthof the torsion beam, adjacent cross sections will not be warpedequally and, as a result, there will be axial strain ez of thelongitudinal layer of the beam. We can obtain the axial strainand the normal stresses produced during non-uniform torsionfrom Hookes law:ez @wd c ws ws 2 ,@zdz2sz Eez Ews ws d cdz2228 The twist moment is applied only to the ends of the beam andits cross sections are free to warp. Under such conditions, warpingis the same for all cross sections and takes place without any axialstrain of the longitudinal direction. The case of non-uniformtorsion occurs if any cross sections are not free to warp or if thetwist moment varies along the length of the beam. The differ-ential equation for non-uniform torsion is given by dcd c C 2 3 ,Mt C 1 dzdz329where C 1 is the torsional rigidity of the beam, C 1 GJ, and C 2 isthe warping rigidity referred to as the portion of the twistmoment due to non-uniform torsion and non-uniform warpingof the cross section, C 2 EC w . C w is the warping constant given by RmC w 0 ws ws 2 tds and has units of length to the sixth power.The constraints of ange warping of the torsion beam canincrease the stress in the zones connected to the trailing arm, aswell as torsion stiffness of the beam.

D. Lee, C. Yang / Finite Elements in Analysis and Design 63 (2013) 98106103Fig. 7. Product verication plans (a) Modeling and simulation phases for product design and (b) design ows for torsion beam rear suspension.Table 1Geometric parameters of torsion beam suspension.Parameterhp (mm)he (mm)A (mm)C (mm)L (mm)T (mm)Value 263.6 13 188.71147.6 423.561410Roll Stiffness Z Lower limitW/C Lateral Stiffness Z Lower limit Note that the upper and lower limits are not described with detailvalues because they are proprietary values dened by each auto-motive maker for the performance of its own vehicles. iSIGHT,developed by DASSAULT Systems, is used for the optimization basedon an SQP (Sequential Quadratic Programming) optimizer interoper-ating with Hypermesh and NX.NASTRAN. The shape of the torsionbeam is calculated and determined with respect to each designparameter using morphing functions given in Hypermesh. Linearanalysis of stress, roll stiffness, and lateral stiffness are performedusing NX.NASTRAN, and the stiffness of each part is calculated usingnumerical equations while roll behavior of a vehicle is analyzed. In the design phase focusing on the kinematic roll performances,roll strength under the roll motions of a 4.5 degree angle can bedetermined. Fig. 11 shows the design parameters of the cross sectionprole and the geometric parameters of the transition zone in thetorsion beam. Fig. 12 shows the design plans for the roll strength inthe roll durability. Fig. 13 shows the stress contours for the initialand optimum torsion beam suspension under roll motion. Changesin system characteristics and performances for the initial andoptimum designs are compared in Table 4.lateral stiffness, durability, and weight. Therefore, thickness andshape of the beam cross section are optimized to satisfy variousdesign requirements for the torsion beam suspension system asfollows:Objective: Minimize maximum stress (smax)Design Variables: Thickness (t) and shape of the beam cross section (a, R, R2)Dimensions in transition zone (Tr1, Tr2, Tr3)Constraints: Roll Center Height r Upper limitWeight r Upper limit (or initial value)

104D. Lee, C. Yang / Finite Elements in Analysis and Design 63 (2013) 98106Table 2Section properties and proles for torsion beam suspension.Section1t (mm)2.6Area (mm2)806.11I1 (mm4)574,321I2 (mm4)274,941J (mm4)33,979Stiffness (N m/rad)28.7Section prole22.6805.90630,351259,31234,24627.932.6806.39639,279259,01324,60224.642.6806.42692,674244,36923,69223.4Fig. 8. Typical tubular torsion beam suspension (a) CAD model (b) nite element model.Fig. 10. Force-displacement curves for roll motion of torsion beam rearsuspension.Fig. 9. Test set-ups for torsion beam rear suspension.4. Conclusion A new analytical method for designing a torsion beam rearsuspension system is proposed in this paper. This method utilizestheoretical descriptions for kinematic and elastic behavior of atorsion beam suspension. The analytical investigations can giveengineers an efcient and easy way to design a torsion beam rearsuspension by enabling design engineers to quickly perform basicdesign and property evaluations in the suspension planning stage.Quasi-static nite element analyses and experiments are included inthe paper to validate the feasibility of the proposed approachTable 3Kinematic performances of torsion beam suspension.PerformancesRollRollRollRollsteer (%)camber (%)center height (mm)stiffness (N m/rad)Numerical equation 0.026 0.31109 12.17Adams 0.029 0.42112 12.8Test 0.031 0.45117 12.65

D. Lee, C. Yang / Finite Elements in Analysis and Design 63 (2013) 98106105Fig. 11. Design parameters for the durability strength (a) design parameters of cross section prole (b) design parameters in transition zone mounted to trailing arm (c)shape prole of design parameter Tr1.Fig. 12. Design plans for the roll durability strength.

106D. Lee, C. Yang / Finite Elements in Analysis and Design 63 (2013) 98106Fig. 13. Roll motion strength of torsion beam suspension (a) roll motion strength of initial model (b) roll motion strength of optimized model.Table 4Comparison of the values of objective function constraints between initial andoptimum designs.PerformancesObjective function (MPa)Roll center height (mm)Weight (kg)Roll stiffness (N m/rad)W/C lateral stiffness (N m)Lower limit12.00390Initial 478 1091020 12.17 345Optimum 370 1161085 13.5 412Upper limit1301130under different loading conditions, such as braking and cornering, tocomplete the evaluation of the torsion beam suspension system.References [1] H. Horntrich, Rear suspension design with front wheel drive vehicles, SAE 810421 (1981). [2] H. Shimatani, S. Murata, K. Watanabe, T. Kaneko, H. Sakai, Development of torsion beam rear suspension system with toe control links, SAE (1999) 1999-01-0045. [3] D.C. Lee, J.H. Byun, A study on the structural characteristics and roll behavior of suspension for the section prole of torsion beam, Trans. KSAE 7 (9) (1999) 195202. [4] T.L. Satchell, The design trailing twist axles, SAE 810420 (1981). [5] J.M. Lee, J.R. Yun, J.S. Kang, S.W. Bae, A study on the steady-state cornering of a vehicle considering roll motion, Trans. KSAE 5 (6) (1997) 89102. [6] J. Kang, Kinematic analysis of torsion beam rear suspension, Trans. KSAE 12 (5) (2004) 146153. [7] S.H. Crandall, N.C. Dahl, T.J. Lardner, An Introduction to the Mechanics of Solids, second ed., McGraw-Hill, New York, USA, 1978. [8] H. Sugiura, Y. Mizutani, H. Nishigaki, First-order analysis for automotive suspension design, R&D Rev. Toyota CRDL 37 (1) (2002) 2530. [9] A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, third ed., Prentice-Hall, New Jersey, USA, 1995.[10] S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, second ed., McGraw- Hill, New York, USA, 1963.[11] D.C. Lee, C.S. Han, A frequency response function-based updating technique for the nite element model of automotive structures, Proc. Inst. Mech. Eng. Part D J. Automob. Eng. 222 (10) (2008) 17811791.with condence in predicting the behavior of the exible body. Thesimulation and experiments show that the numerical calculations ofsuspension properties are sufciently precise to allow engineers tojudge the acceptability of design plans in the development stage.Several processes necessary for analysis of the suspension areintegrated and automated, thus the analysis and design works fora suspension system are efciently performed. Design optimizationconsidering kinematic performances, compliance, and strength ofthe torsion beam axle suspension is also successfully performed byapplying a deterministic optimization procedure and discrete designsolution. The characteristics of the exible body are also affected bythe lateral stiffness of the bushings. Therefore, further studies need tobe carried out to predict the behavior of non-linear elastic bushes