1-s2.0-s0094114x0200037x-main

Synthesis and analysis of the ve-link rear suspensionsystem used in automobiles

P.A. Simionescu *, D. Beale

Department of Mechanical Engineering, Auburn University, 202 Ross Hall, Auburn, Al 36849, USA

Abstract

The paper deals with the optimum kinematic synthesis and analysis of the ve-link independent sus-pension system (also known as multilink suspension, mechanism commonly symbolized 5S5S). Thesynthesis goal is fullling a minimum variation of the wheel track, toe angle and camber angle duringjounce and rebound of the wheel. Two solutions obtained by synthesis are analyzed and compared to anexisting solution, and the displacement, velocity and acceleration of the wheel carrier relative to the carbody are determined, together with the variation of the momentary screw axis and the rear axle roll-centerheight. Both the kinematic synthesis and the analysis are performed in a simplied, easy to programmanner, using a ctitious mechanism that has all the links dismounted from their joints. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Rigid body guidance; Suspension linkage; Constraint equations; Optimization algorithm

1. Introduction

The ve-link suspension mechanism was rst introduced by Deimler-Benz on their W201 andW124 series under the name multilink suspension (Fig. 1a). Ever since has been successfullyimplemented both in independent suspension systems and in rear axle guiding mechanisms bymany automobile manufacturers. Due to the larger number of design parameters, it has the ca-pability of fullling better the complex kinematic and dynamic requirements imposed on sus-pension systems of todays automobiles. It is however much more dicult to synthesize than anyother suspension mechanism, due to its general spatial conguration. In case of multilink frontsuspension, the design problem is even more complex due to the fact that the kingpin is a virtual

*Corresponding author. Tel.: +1-334-844-5867; fax: +1-334-844-5865.

E-mail addresses: [email protected] (P.A. Simionescu), [email protected] (D. Beale).

0094-114X/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0094-114X(02)00037-X

Mechanism and Machine Theory 37 (2002) 815832www.elsevier.com/locate/mechmt

one [1,2] corresponding to the momentary screw axis of the wheel carrier performing the steeringmotion relative to the chassis.The rear independent wheel or axle guiding mechanism(s) are, in the sense of mechanism

theory, spatial motion generators (also known as rigid body guidance mechanisms). Research onmotion generators synthesis and analysis has been carried out on both abstract and appliedmechanisms by many researchers in the past. A general formulation of the mechanism synthesisproblem for path, function and rigid body guidance based on optimization techniques is proposedby Aviles et al. [3]. According to the authors, a global error function to be minimized is dened asa weighted-sum of some local error functions, previously minimized with respect to the Cartesiancoordinates of the basic points of the mechanism. The so-called basic points are the centers ofthe joints and the points of the links required to generate certain paths throughout the workingrange of the mechanism. Although the method is general in its formulation, the main disad-vantage lays in the large number of variables required to dene the objective function, as well as innot including the ground joint coordinates among the design parameters.An extension of the approach of Aviles et al. to the synthesis of spatial linkages is given by

Jimenez et al. [4]. However, the main drawback of an excessive number of design variables re-quired in formulating the synthesis problem was maintained. In the example presented of syn-thesizing a 5S5S suspension mechanism for only three prescribed positions of the wheel carrier,an objective function of 64 variables was dened.Suh [5] synthesized a double-wishbone suspension mechanism (an RSSR-SS spatial motion

generator), which can be considered a particular embodiment of the ve-link suspension [1], in amixed approach, with two nite and instantaneous exact positions, using displacement matricesand constraint equations.A combination of exact and approximate synthesis is performed by Sandor et al. [6] for the

same RSSR-SS motion generating mechanism. The authors considered part of the positionsimposed to the wheel carrier as exact positions and solved the corresponding set of equations. Thefree choices in these equations were further considered design variables in an objective function,penalized with the conditions of avoiding branching, achieving correct sequence of prescribedpositions and observing the shortest and longest links to be within prescribed limits in the re-maining imposed positions.

Fig. 1. Multilink independent suspension mechanism (a) and its kinematic diagram (b).

816 P.A. Simionescu, D. Beale / Mechanism and Machine Theory 37 (2002) 815832

The method detailed in the present paper for kinematic synthesis of the 5S5S rigid bodyguidance mechanism considers the guided body (the wheel carrier) released from its joints andmoving in successive positions along the ideal trajectory. The synthesis problem thus becomes thatof nding the joint disposition for which the distances between the homologous released joints(the pair joints that in the real mechanism are connected by binary links) vary as little as possible.Essentially this is a numerical implementation of the nite-position spatial theory of kinematicsynthesis the object of which is determining those points which lie on special loci: spheres, cyl-inders, circles, lines, etc. [79].The same approach of considering the wheel carrier released from its joints was applied for

displacement analysis of the same mechanism. The interested reader can nd this procedure di-rectly applicable to solving the direct kinematic problem of a variety of parallel mechanisms of theGoughStewart type.

2. Synthesis problem formulation

The requirements upon the motion of the rear wheel that can be transposed into kinematicconditions when synthesizing the suspension mechanism are [10]:

minimum toe angle variation during compression and rebound; avoid excessive outward camber thrust on corners; avoid excessive sideways thrust and consequent rear end steering impulses on single wheelbump or rebounds;

supplementary, the suspension elements must ensure a minimum intrusion into the passengersand luggage accommodation, a condition that can be translated into constraints imposed to thepossible disposition of the ball joints on the chassis and on the wheel carrier.

The eect of the compliance of the rear wheel suspension upon the car ride behavior is im-portant and in the nal design must necessarily be considered by performing a dynamic analysisusing a multibody simulation software. However, in order to simplify both the kinematic synthesisand analysis procedures, it is common in the early stages of design to assume that the joints haveneither clearances nor elasticities, and the vehicle chassis and suspension elements are rigid. Whenequipped with compliant joints, it is to be expected that a rigid joint suspension that exhibit goodkinematic characteristics, will continue to perform satisfactory (provided that the stiness rates ofthe joints are properly selected).Taking the rst three above-mentioned conditions imposed to a suspension system, it can be

considered that the ideal wheel movement along its operation travel must be close to a verticaltranslation relative to the car body. This is in accordance with Raghavans ndings [11] that forstraight-line motion of the car, the motion of the wheel relative to the road should exhibit zero toeand camber change, and that track width should be maintained constant. The same authorconcluded that while turning, the wheels should remain at zero camber or should camber into theturn if possible, and the track should stay constant.In order to formulate the synthesis problem, all the ve links are removed from their joints (or

assumed of variable length), thus allowing the wheel carrier to be displaced in successive positions

P.A. Simionescu, D. Beale / Mechanism and Machine Theory 37 (2002) 815832 817

along any trajectory. If the distance between the homologous joints varies very little in thesesuccessive positions, the real mechanism with the ve links jointed back in place will guide thewheel very close to the imposed path.The above considerations are the basis for formulating the synthesis of the ve-link mechanism

as an optimization problem, i.e. of nding the minimum of the following objective function of 30variables (Fig. 1b):

F xAi; yAi; zAi; x0Bi; y 0Bi; z0Bii1...5 X5i1

Xnj1

lih

AiBiji2

1

with j 1 . . . n intermediate positions of the wheel carrier evenly spaced on the prescribed tra-jectory.In the followings this imposed trajectory will be a simple vertical translation of the wheel carrier

i.e. xN and yN are kept constant for zNj varying between a lower zN min and an upper zN max limit ofpoint N attached to the wheel carrier. One should not expect that the mechanism obtained bysynthesis to exactly generate this pure vertical motion. As will be seen later, the kinematic be-havior of the synthesized mechanism strongly depends on the values chosen for zN min and zN max(which should not necessarily be the upper and lower limits of wheel travel during jounce andrebound, nor even belong to the actual motion range of the wheel carrier).The reference lengths of the links noted li i 1; 5 in relation (1) are determined as the dis-

tances between the joints Ai and Bi for the wheel in its initial position, corresponding to the caraveragely loaded and in rest. The variable distances AiBij between the ve homologous joints Aiand Bi in a current position j of the wheel carrier is given by

AiBij xAij xBij2 yAij yBij2 zAij zBij2

q2

where the coordinates x, y and z must be specied relative to the same reference frame, preferablethe xed reference frame Oxyz. Because the disposition of the ball-joint centers Bi is given in thereference frame attached to the wheel carrier Nx0y 0z0, the following transformations must be ap-plied in order to make use of Eq. (2):

xBijyBijzBij

24

35Oxyz

x0Biy 0Biz0Bi

24

35Nx0y0z0

xNjyNjzNj

24

35Oxyz

3

In the initial position, the reference frame Oxyz attached to the chassis and the frame Nx0y0z0

attached to the wheel carrier are considered parallel. Knowing the coordinates (xN0, yN0, zN0) ofthe origin of Nx0y 0z0 frame relative to the chassis reference frame, the coordinates of the same pointN relative to Oxyz reference frame will be (xN0, yN0, zNj) for a current prescribed position j, wherezNj zN min DzNj with DzNj jzN max zN min=n.The limitations upon the possible locations of the ball joints on the chassis and wheel carrier

can be prescribed as side constraints of the form:


xAimin6 xAi6 xAimaxyAimin6 yAi6 yAimaxzAimin6 zAi6 zAimax

i 1 . . . 5 4

and

x0Bimin6 x0Bi6 x0Bimaxy0Bimin6 y0Bi6 y 0Bimaxz0Bimin6 z0Bi6 z0Bimax

i 1 . . . 5 5

and must necessarily be imposed in order to avoid convergence to unpractical solutions with linksexcessively long.The objective function (1) together with the constraints (4) and (5) can be minimized using a

proper optimization subroutine. Of the maximum number of design variables (30 in totalirrespective of the number of intermediate positions n of the wheel carrier), some of the ball-jointcenters can be imposed xed values and the number of design variables further reduced.In theory it is possible to prescribe a trajectory to the wheel carrier that can be exactly generated

by a real mechanism (case in which the global minima of the objective function F will be zero). Inpractice however, there will always be a departure between the prescribed motion and the actualmotion of the real mechanism. Therefore a kinematic analysis is required in order to determine theactual behavior of the suspension mechanism obtained by synthesis.

3. Kinematic analysis of the ve-link suspension mechanism

The analysis of the ve-link suspension mechanism has been tackled by a number of researchersin the past. Lee et al. [1] derived the velocity equations of the wheel carrier and applied a step-wiselinearization to solve the position problem. Mohamed and Attia [12] used the constrainedequations obtained from the condition that the ve connecting rods and the wheel carrier are rigidbodies. Knapzyk and Dzierzec [13] considered a modied mechanism with two of the guidinglinks disassembled and solved an optimization problem describing the condition that the distancesbetween the homologous released joints remain equal to the lengths of the disconnected members.Following [6], Unkoo and Byeongeui [2] used 4 4 displacement and dierential-displacement

matrices and constraint equations to solve the position and velocity problem of ve-link and strut-type multilink suspensions. The referred authors also determined the imaginary kingpin axis ofthese suspensions systems using screw-axis theory and compared the results with those obtainedby nite-center analysis.The same approach of considering all the ve connecting rods removed will be further con-

sidered. For successive values of the input parameter zN, the position of the point N relative to thehorizontal axis and the orientation angles of the wheel carrier will be tuned in a searching process,until the distances between the released joints Ai and Bi become equal (within some error limits) tothe lengths of the respective links AiBi.


3.1. Position problem

The ve-link suspension mechanism has six degrees-of-freedom, of which ve are trivial ro-tations of the connecting links around their own axes. Correspondingly, the position of the wheelcarrier can be specied using only one independent parameter viz the coordinate zN of the originof the Nx0y0z0 reference frame relative to the central reference frame Oxyz. The remaining veparameters: coordinates xN , yN and angles a, b and c that dene the position and orientation of thewheel carrier can be determined by solving the following equations of constraint:

xAi xBi2 yAi yBi2 zAi zBi2 l2i i 1 . . . 5 6describing the condition of the distance between joints Ai and Bi to remain constant during theworking range of the mechanism. In the above Eq. (6), the coordinates xBi, yBi and zBi are de-termined by applying the following transformation to the Nx0y0z0 reference frame:

xBiyBizBi

24

35Oxyz

Rbacx0Biy 0Biz0Bi

24

35Nx0y0z0

xNyNzN

24

35Oxyz

7

where Rbac is the transformation matrix that express the successive rotation of the wheel carrierrelative to Oxyz by the pitch angle b, yaw angle a and roll angle c [14]:

Rbac Rc;xRa;zRb;y ca cb sa ca sb

sa cb cc sb sc ca cc sa sb cc cb scsa cb sc sb cc ca sc sa sb sc cb cc

24

35 8

In the above equation Ra;z, Rb;y and Rc;x are the basic rotation matrices while ca cos a,sa sin a and so forth.For a given value of the independent parameter zN, the system of Eq. (6) in the unknowns a, b,

c, xN and yN can be very conveniently solved by minimizing the following objective function:

F0a; b; c; xN; yN X5i1

xAi xBi2 yAi yBi2 zAi zBi2 9

In order to facilitate convergence, the starting point when minimizing F0 can be taken the positionof the wheel carrier (the same xN, yN and orientation angles a, b, c) imposed during synthesis forthe same zNj. Once the displacement problem of the wheel carrier is solved, the diagram of thewheel track, recessional wheel motion, camber and toe angle alteration can be generated.

3.2. Linear velocity and acceleration analysis

The velocities of points Bi on the wheel carrier can be determined by dierentiating once withrespect to time the equations of constraint (6). The number of unknowns thus emerging is 15, andtherefore 10 more equations must be added, like the time derivatives of following equations:

xBj xBk2 yBj yBk2 zBj zBk2 const j 1 . . . 4 and k j 1 . . . 5 10


Table 1

The coecients of the linear system of equations used to determine the linear velocity of points Bi (i 1 . . . 5)_xxB1 _yyB1 _zzB1 _xxB2 _yyB2 _zzB2 _xxB3 _yyB3 _zzB3 _xxB4 _yyB4 _zzB4 _xxB5 _yyB5 _zzB5 _xxN _yyN

xA1 xB1 yA1 yB1 zA1 zB1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 xA2

xB2yA2yB2

zA2zB2

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 xA3xB3

yA3yB3

zA3zB3

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 xA4xB4

yA4yB4

zA4zB4

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 xA5xB5

yA5yB5

zA5zB5

0 0 0

xB1 xB2 yB1 yB2 zB1 zB2 xB2xB1

yB2yB1

zB2zB1

0 0 0 0 0 0 0 0 0 0 0 0

xB1 xB3 yB1 yB3 zB1 zB3 0 0 0 xB3xB1

yB3yB1

zB3zB1

0 0 0 0 0 0 0 0 0

xB1 xB4 yB1 yB4 zB1 zB4 0 0 0 0 0 0 xB4xB1

yB4yB1

zB4zB1

0 0 0 0 0 0

xB1 xB5 yB1 yB5 zB1 zB5 0 0 0 0 0 0 0 0 0 xB5xB1

yB5yB1

zB5zB1

0 0 0

0 0 0 xB2xB3

yB2yB3

zB2zB3

xB3xB2

yB3yB2

zB3zB2

0 0 0 0 0 0 0 0 0

0 0 0 xB2xB4

yB2yB4

zB2zB4

0 0 0 xB4xB2

yB4yB2

zB4zB2

0 0 0 0 0 0

0 0 0 xB2xB5

yB2yB5

zB2zB5

0 0 0 0 0 0 xB5xB2

yB5yB2

zB5zB2

0 0 0

xB1 xN yB1 yN zB1 zN 0 0 0 0 0 0 0 0 0 0 0 0 xNxB1

yNyB1

zB1zN_zzN

0 0 0 xB2xN

yB2yN

zB2zN

0 0 0 0 0 0 0 0 0 xNxB2

yNyB2

zB2zN_zzN

0 0 0 0 0 xB3xN

yB3yN

zB3zN

0 0 0 0 0 0 xNxB3

yNyB3

zB3zN_zzN

0 0 0 0 0 0 0 0 0 xB4xN

yB4yN

zB4zN

0 0 0 xNxB4

yNyB4

zB4zN_zzN

0 0 0 0 0 0 0 0 0 0 0 0 xB5xN

yB5yN

zB5zN

xNxB5

yNyB5

zB5zN_zzN

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Table 2

The coecients of the linear system of equations used to determine the linear accelerations of points Bi (i 1 . . . 5)axxB1 yyB1 zzB1 xxB2 yyB2 zzB2 xxB3 yyB3 zzB3 xxB4 yyB4 zzB4 xxB5 yyB5 zzB5 _xxN _yyN

* * * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 _xx2B1 _yy2B1 _zz2B10 0 0 * * * 0 0 0 0 0 0 0 0 0 0 0 _xx2B2 _yy2B2 _zz2B20 0 0 0 0 0 * * * 0 0 0 0 0 0 0 0 _xx2B3 _yy2B3 _zz2B30 0 0 0 0 0 0 0 0 * * * 0 0 0 0 0 _xx2B4 _yy2B4 _zz2B40 0 0 0 0 0 0 0 0 0 0 0 * * * 0 0 _xx2B5 _yy2B5 _zz2B5* * * * * * 0 0 0 0 0 0 0 0 0 0 0 _xxB1 _xxB22 _yyB1 _yyB22 _zzB1 _zzB22* * * 0 0 0 * * * 0 0 0 0 0 0 0 0 _xxB1 _xxB32 _yyB1 _yyB32 _zzB1 _zzB32* * * 0 0 0 0 0 0 * * * 0 0 0 0 0 _xxB1 _xxB42 _yyB1 _yyB42 _zzB1 _zzB42* * * 0 0 0 0 0 0 0 0 0 * * * 0 0 _xxB1 _xxB52 _yyB1 _yyB52 _zzB1 _zzB520 0 0 * * * * * * 0 0 0 0 0 0 0 0 _xxB2 _xxB32 _yyB2 _yyB32 _zzB2 _zzB320 0 0 * * * 0 0 0 * * * 0 0 0 0 0 _xxB2 _xxB42 _yyB2 _yyB42 _zzB2 _zzB420 0 0 * * * 0 0 0 0 0 0 * * * 0 0 _xxB2 _xxB52 _yyB2 _yyB52 _zzB2 _zzB52* * * 0 0 0 0 0 0 0 0 0 0 0 0 * * zzNzB1 zN _xxB1 _xxN2 _yyB1 _yyN2 _zzB1 _zzN20 0 0 * * * 0 0 0 0 0 0 0 0 0 * * zzNzB2 zN _xxB2 _xxN2 _yyB2 _yyN2 _zzB2 _zzN20 0 0 0 0 0 * * * 0 0 0 0 0 0 * * zzNzB3 zN _xxB3 _xxN2 _yyB3 _yyN2 _zzB3 _zzN20 0 0 0 0 0 0 0 0 * * * 0 0 0 * * zzNzB4 zN _xxB4 _xxN2 _yyB4 _yyN2 _zzB4 _zzN20 0 0 0 0 0 0 0 0 0 0 0 * * * * * zzNzB5 zN _xxB5 _xxN2 _yyB5 _yyN2 _zzB5 _zzN2a The stars in the table designate coecients identical to the corresponding ones in Table 1.

822

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and

xBi xN2 yBi yN2 zBi zN2 const i 1 . . . 5 11describing the condition that the wheel carrier is a rigid body. By dierentiation these equationsonce with respect to time, a new independent parameter _zzN will emerge, which, the same as zNmust be specied as input during the numerical analysis. The coecients of the system of linearequations in the 17 unknowns _xxBi, _yyBi, _zzBi (i 1 . . . 5), _xxN and _yyN used for velocity analysis aresummarized in Table 1.By dierentiating with respect to time the equations used to solve the velocity problem, a

second system of linear equations in the unknowns xxBi, yyBi, zzBi (i 1 . . . 5), xxN and yyN will befurther obtained, the coecients of which are given in Table 2. In this case zN, _zzN and zzN will bethe independent parameters that must be specied as inputs.

3.3. Angular velocity and acceleration analysis

The components of the angular-velocity vector (xx, xy , xz) relative to the xed reference frameOxyz can be determined using the following equation known from the rigid body kinematics:

_xxBi_yyBi_zzBi

24

35

_xxN_yyN_zzN

24

35

xxxyxz

24

35

xBi xNyBi yNzBi zN

24

35 12

written for any two dierent points of the wheel carrier for which the linear velocity are known.The expressions of xx, xy and xz as derived from relation (12) are given in Eq.(A.1) in AppendixA.The components of the angular-acceleration vector (ex, ey, ez) can be determined writing the

following equation, the same for two dierent points of the wheel carrier:

xxBiyyBizzBi

24

35

xxNyyNzzN

24

35

exexex

24

35

xBi xNyBi yNzBi zN

24

35

xxxxxx

24

35

xxxxxx

24

35

0@

xBi xNyBi yNzBi zN

24

351A 13

The expressions of ex, ey and ez derived through analytical manipulations of relation (13) aresummarized in Eq.(A.2) in Appendix A. Alternatively, the components of the angular accelerationcan be determined by dierentiating once with respect to time the components of the angularvelocity:

ex _xxx; ey _xxy and ez _xxz 14The results of velocity and acceleration analysis will be further used in determining the location ofthe instantaneous screw axis of the wheel carrier moving relative to the chassis. Position, velocityand acceleration problems are also stages required in solving the dynamic problem of the sus-pension system. According to Hiller [15], of the total CPU time needed to simulate the response ofa ve-link suspension to a road input, almost 70% was required for solving the kinematics of thesystem. The method described above for solving the position problem it is likely to reduce thisamount of time, since requires solving a system of only ve nonlinear equations.


3.4. Instantaneous screw axis

Considering the instantaneous motion of the ve-link suspension, the wheel carrier motionrelative to the car body is a screw motion of the circle-point-surface xed to the wheel carrier withrespect to the center-point-surface xed to the car body [16]. The common tangent of these twosurfaces is the instantaneous screw axis of the spatial motion (see Fig. 2), and corresponds to thepoints of minimum velocity of the wheel carrier relative to the car body. Therefore, the parameterspositioning the momentary screw axis can very well be determined by formulating a minimizationproblem.A dierent approach is to solve the system of equations expressing the condition the linear

velocity ( _xx, _yy, _zz) of a point (x; y; z) attached to the wheel carrier is parallel to the angular-velocityvector (xx, xy , xz):

_xx=xx _yy=xy _zz=xz 15

Fig. 2. Center-point-surface and circle-point-surface of a ve-link independent suspension (solution 2 in paragraph 4)

in perspective view (a) and top view (b).


Based on Eq. (12), the above equalities becomes:

_xxN xyz zN xzy yNxx

_yyN xzx xN xxz zNxy

_xxN xyz zN xzy yNxx

_zzN xzx xN xxz zNxz

16

The resulting expressions of the parametric equation of the momentary screw axis xy and zyare given in Eq.(A.6) in Appendix A.Determining of the screw axis due to steering input is of much signicant importance in the case

of the ve-link suspension mechanism used for guiding the front wheels, which has 2 DOF cor-responding to the steering input. When only the steering input is active, the resulting momentaryscrew axis (which can be calculated following a similar approach) will be the virtual kingpin of thewheel during the steering motion [2].

3.5. Suspension roll center

Each suspension has a roll center dened as the point in the vertical plane through the wheelcenters at which lateral forces may be applied to the sprung mass without producing suspensionroll [17,18]. According to Reimpell and Stoll [19] there is a direct correlation between the wheeltrack variation and the roll-center height hR. According to the same authors, this correlation isalso conicting, in that a high roll center (which is desirable for a favorable car body attitudeduring cornering) implies a larger track alteration. The suspension roll center can be approxi-mately determine by nite-center analysis as the intersection between the normal to the trajectoryof the path center point S projected on the vertical plane Oxz and the cars longitudinal plane Oyz(Fig. 3). The following formula:

Fig. 3. Schematic for calculating the roll-center height of the rear axle.


hRzNj 0:5x2SzNj x2SzNj1 z2SzNj z2SjzNj1

zSzNj zSjzNj 17

has been derived for calculating the roll-center height relative to the chassis reference frame. Theheight of the roll-center measured from the ground will be

hRzNj h0RzNj x2SzNj 18In the above equations zNj and zNj1 are two successive positions of the wheel center, sucientlyclose one to the other to allow a tangent-chord approximation along the trajectory of the pathcenter point.

4. Numerical results

Based on the procedure described above, the synthesis of a ve-link rear wheel independentsuspension system was performed. The numerical data corresponding to the Mercedes-190 mul-tilink suspension available [13] was used in dening the allowable positions of the ball-jointcenters (Table 3).The origin of the Nx0y0z0 coordinate system in the reference position was xN0 705 mm, yN0 0

and zN0 302 mm, while the wheel radius was R 314 mm.Two intervals of the wheel carrier vertical travel have been considered in the objective function

F. The rst numerical solution recorded (Table 4) was obtained for the reference frame Nx0y0z0

translating verticaly between zN min 50 mm and zN max 100 mm. The second solution (TableTable 3

Searching domains of the design variables xAi , yAi , zAi , xBi, yBi, zBi (i 1 . . . 5)1906 xA16 220 876 yA16 117 2166 zA16 2464816 xA26 511 3366 yA26 306 2366 zA26 2663896 xA36 419 2246 yA36 194 2816 zA36 3114226 xA46 452 2246 yA46 194 3876 zA46 4173416 xA56 371 106 yA56 20 4016 zA56 431

536 x0B16 33 336 y0B16 53 1046 z0B16 84836 x0B26 63 546 y0B26 34 1496 z0B26 129496 x0B36 29 1516 y0B36 131 436 z0B36 23536 x0B46 33 886 y0B46 68 876 z0B46 105836 x0B56 63 56 y0B56 15 1156 z0B56 135

Table 4

Numerical solution obtained for 50 mm6 zN6 100 mm in the objective function F (variant 1)i 1 i 2 i 3 i 4 i 5

xAi 190.436 482.605 401.068 422.000 344.310yAi 87.591 317.292 210.635 198.545 3.447zAi 238.816 236.036 289.298 410.077 430.258x0Bi 33.737 63.000 31.195 46.577 67.136y0Bi 43.949 36.344 135.524 78.843 3.353z0Bi 90.997 129.034 43.000 87.000 115.002li 483.584 329.115 284.509 265.835 293.853


5) was obtained for zN min 3000 mm and zN max 3100 mm. This unusual domain of zN facili-tated obtaining a kinematic solution that ensures a higher location of the suspension roll centerrelative to the ground. According to [19], both rear and front suspension roll centers should be ashigh as possible and at approximately the same height. However, limitations imposed to the wheeltrack alteration restrict choosing a rear suspension that ensures a roll center located to high.The two solutions obtained by synthesis noted 1 and 2, were compared with an existing solution

noted 0 also available in [13]. The wheel track variation:

DSzN xSzN0 xSzN 19is visibly improved in case of solution 1. Also improved is the camber angle alteration Dd ascompared to the existing solution 0 (Fig. 4).In Fig. 5 is given the diagram of the recessional motion of the wheel as the variation of yS

coordinate of the center path S.

Table 5

Numerical solution obtained for 3000 mm6 zN6 3100 mm in the objective function F (variant 2)i 1 i 2 i 3 i 4 i 5

xAi 203.760 493.038 390.597 422.066 348.449yAi 111.186 330.546 197.256 211.100 20.000zAi 243.559 261.996 308.938 392.424 426.300x0Bi 33.000 69.814 29.608 35.783 72.934y0Bi 37.798 34.000 133.477 82.386 15.000z0Bi 100.101 148.828 30.041 87.057 115.226li 475.783 346.393 294.182 278.679 283.807

Fig. 4. Wheel track alteration (a) and recessional wheel motion (b) during jounce and rebound for an initial solution 0,

and the two solutions obtained by synthesis, 1 and 2.


DY zN ySzN0 ySzN 20This parameter describes the fore and aft motion of the wheel during jounce and rebound.However, since it occurs along the direction of car travel, is has a smaller eect upon the cardynamics than the wheel track alteration.The camber angle variation Dd was determined as the projection of the angle between the axes

Oz and Nz0 on the vertical transverse plane (Fig. 6a).Similarly, the toe angle alteration Du shown in Fig. 7b was determined as the angle between the

axes Ox and Nx0 projected on the horizontal plane. In this case, for 150 mm6DzN6 150 mm thetoe angle of solution 1 is slightly larger than that of the existing solution 0, being howevercompensated by the understeer eect of track widening during jounce.For illustrative purposes, the diagrams of the magnitude of the angular velocity x and angular

acceleration e of the wheel carrier have been plotted (Fig. 6) for _zzN 1:0 m/s and zzN 0 usingEqs.(A.1) and (A.2) in Appendix A.The results of the kinematic analysis have been also used in the 3D visualization of the motion

of the mechanism. Fig. 7 shows superimposed positions of the suspension mechanism solution 1,corresponding to zN0 and zN0 150 mm, viewed from the front (a) and from above (b).The circle-point-surface and the center-point-surface in Fig. 3 were produced for solution 2.

They have been generated as ruled surfaces of the momentary screw axis relative to the chassis

Fig. 5. Camber alteration DdDzN (a) and toe angle alteration DuDzN (b) of the wheel relative to the chassis duringjounce and rebound, for the same numerical variants in Fig. 4.


(the circle-point-surface) and to the wheel carrier (the center-point-surface). The inclined positionof the screw axis relative to cars longitudinal axis it is due to the wheel carrier rotation around itsown axis, which for solution 2 corresponds to a maximum angle c of 16.2 occurring forzN0 150 mm.Finally, the plot in Fig. 8 of the alteration of the roll-center height with DzN have been pro-

duced. As compared to the existing solution, both variant 1 and 2 have a favorable smaller dropof the roll center under load. According to [19], in case of the real vehicle with compliant sus-pension, the roll center will be higher than for the simplied mechanism with rigid joints.

Fig. 6. Variation of the angular velocity x (a) and angular acceleration e (b) for _zzN 1:0 m/s and zzN 0 for the samevariants in Fig. 4.

Fig. 7. Superimposed positions of the suspension mechanism solution 1, corresponding to DzN 0 and DzN 150mm, viewed from the rear (a) and from above (b).


5. Conclusions

A classic rigid body guidance mechanism synthesis problem was presented, that of designing ave-link independent rear suspension system under the condition of ensuring a proper motion ofthe wheel carrier. Also given were complete kinematic analysis equations that allow determiningthe wheel recession, wheel track, toe angle, camber angle and roll-center height variation togetherwith the linear and angular velocities and accelerations of the wheel carrier of a given ve-linksuspension system. Two variants obtained by synthesis were analyzed and compared to an ex-isting solution of a Mercedes-190 suspension system. Though the characteristics of the samemechanisms equipped with compliant joints will dier, the good behavior of the rigid jointmechanisms obtain by synthesis are likely to be preserved.Both the synthesis and the analysis procedures advanced in the paper can be extended to de-

signing and simulating other suspension systems. For example the RSSR-SS double-wishbonesuspension can be synthesized in the same manner. Multilink suspensions used for front wheels ofpassenger cars (that have a second DOF needed for wheel steering) can also be synthesized fol-lowing a similar approach, as well as the 5S5S mechanisms used in guiding rigid axles. In thiscase however, the eect of joint elasticities must necessary be assessed using a multibody simu-lation software, since they have an essential contribution to the combined translationrotationmotion of the real axle.

Acknowledgements

The authors would like to thank Dr. Madhu Raghavan as well as to the anonymous reviewersfor their comments and suggestions.

Fig. 8. Variation of the suspension roll-center height, measured relative to the car reference frame (a) and relative to the

ground (b). In the reference position (DzN 0), hR 138:6 mm for variant 0, hR 73:2 mm for variant 1 andhR 150:4 mm for variant 2.


Appendix A

Considering two points Bj and Bk (j 6 k), the components of the angular velocity of the wheelcarrier are

xx P1 Dxj Dxk P2 Dyj Dxk P3 Dxj DzjDxj Dzj Dyk Dyj Dzj Dxkxy Dyk xx P3=Dxkxz Dzj xy P1=Dyj

A:1

while the components of the angular acceleration are

ex Q1 Dxj Dxk Q2 Dyj Dxk Q3 Dxj DzjDxj Dzj Dyk Dyj Dzj Dxkey Dyk ex Q3=Dxkez Dzj ey Q1=Dyj

A:2

with

Dxj xBj xN; Dxk xBk xNDyj yBj yN; Dyk yBk yNDzj zBj zN; Dzk zBk zN

A:3

P1 _xxBj _xxN; P2 _yyBj _yyN; P3 _zzBk _zzN; A:4Q1 x2y

x2z

Dxj xxxy Dyj xz Dzj xxBj xxN

Q2 x2x x2z

Dyj xyxx Dxj xz Dzj yyBj yyNQ3 x2x

x2y

Dzk xzxx Dxk xy Dyk zzBk zzN

A:5

The parametric equations of the screw axis of the wheel carrier are

xy T1 xy xzh

T2x2x x2yi.

xx xy x2

zy T1 xy T2 xzxy x2

A:6

where

T1 xy _xxN xx _yyN xx xz xN xy xzyN yC zNx2x x2yT2 xx _zzN xz _xxN xy xz zN xx xy xN yC yNx2x x2z

A:7

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Synthesis and analysis of the five-link rear suspension system used in automobilesIntroductionSynthesis problem formulationKinematic analysis of the five-link suspension mechanismPosition problemLinear velocity and acceleration analysisAngular velocity and acceleration analysisInstantaneous screw axisSuspension roll center

Numerical resultsConclusionsAcknowledgementsAppendix AReferences

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