Synthesis and analysis of the two loop translational inputsteering mechanism

P.A. Simionescua, M.R. Smithb,*, I. Tempeaa

aDepartment of Mechanism and Robot Theory, Politehnica University of Bucharest, Spl. Independentei 313, Bucharest,

RomaniabDepartment of Mechanical, Materials and Manufacturing Engineering, University of Newcastle upon Tyne, NE1 7RU,

UK

Received 5 November 1997; accepted 30 August 1999

Abstract

A kinematic model of a rack-and-pinion type steering linkage is developed, on the basis of which thesynthesis of the mechanism is performed using the criteria of correct turning of the wheels and goodtransmissibility of the motion. The kinematic analysis of the mechanism is also carried out and outputdiagrams are presented for the case of a four-wheel-drive light tractor. 7 2000 Elsevier Science Ltd. Allrights reserved.

1. Introduction

The study of the steering and suspension mechanisms of vehicles has been the subject of

many publications in the past and consequently there exists a diversity of literature on this

subject [17]. However, the works dedicated to the motor vehicle engineer [8,9] are not

particularly explicit regarding the choice of main dimensions of these mechanisms or make use

of some simplifications which in certain cases are not sucient for a proper design.

On the other hand, in the literature oriented towards theoretical kinematic synthesis and

analysis of mechanisms [1013], the reference systems and parameters used for positioning the

Mechanism and Machine Theory 35 (2000) 927943

0094-114X/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.PII: S0094-114X(99)00056-7

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* Corresponding author. Tel.: +44-191-222-6206; fax: +44-191-222-8600.E-mail address: M.R.Smith@newcastle.ac.uk (M.R. Smith).

kinematic elements (which permit a simplification of the analytical calculations) are not directlyavailable to the designer from the blueprints or real model measurements.The present paper deals with the synthesis and analysis of a translational input, double loop

rack-and-pinion type mechanism (Fig. 1) employed in the steering of rigid-axle vehicles.Because the input element is the piston of a hydraulic actuator, which is fixed on the axle case,there is no coupling eect between the steering and the axle rotation, the control beingachieved through flexible hoses. This feature, together with the kinematic and force-actuatingsymmetry for turns to the left and to the right, is the cause of increasing use of this mechanismin the steering of four-wheel-drive tractors and the like.

2. The geometry of the mechanism

The geometric model of the mechanism is given in Fig. 2. The global reference frame Oxyzattached to the axle is chosen such that Ox is the front axle axis, Oy the vehicle longitudinalaxis and Oz such that the co-ordinate system is positive. The main parameters and notationused are as follows:

a is the kingpin inclination angle (see Appendix A1),b is the castor angle,yi and yo are the turning angles of the inner and outer wheels, respectively,j01 is the reference angle of the steering arm corresponding to the straight-ahead position ofthe steerable wheels (Fig. 4),

Fig. 1. Rigid axle with central hydraulic control piston steering mechanism (after Ref. [16]).

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943928

j1 is the angle of the steering arm while the vehicle is cornering measured from the straightahead position, such that j1 yi for the turn to the left and j1 yo for the turn to theright,OO1 is the half kingpin track length,O1A0 is the steering knuckle arm axial oset, measured along the kingpin axis,A0A is the eective steering knuckle arm length measured radially from ball-joint centre A tothe kingpin axis,AB is the tie-rod length,BB is the hydraulic actuator piston rod length,

Smax is the maximum stroke of the hydraulic piston (a characteristic of the hydraulic actuatorusually provided by the suppliers) for which the maximum turning angle of the inner wheelyi-max is assumed to be reached. This maximum turning angle is limited by the axle final driveperformance (the maximum achievable angle of the universal joints) or by possible interferencebetween the wheel and body.Since the mechanism is symmetrical about the vertical longitudinal plane yOz, a single loop

representation and analytical derivation are sucient for performing both the synthesis and theanalysis. In a general position (corresponding to j1 6 0), the co-ordinates of the centre A ofthe steering knuckle arm ball-joint, relative to the reference frame Oxyz, are given by:

Fig. 2. Geometric model of the left loop of the mechanism.

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943 929

24 xAj1yAj1zAj1

35 M01 24A0A cosj01 j1A0A sinj01 j1O1A0

3524 xO1yO1zO1

35: 1The origin O1 of the kingpin reference frame relative to the global reference frame has beenchosen such that yO1 0 and zO1 0 while xO1 is equal to the half kingpin track.In the above relation the rotation matrix is [13]:

M01 24 1 0 00 cb sb0 sb cb

35 24 ca 0 0 sa 00 1 0

sa 0 0 ca 0

35 24 ca 0 0 sa 0sa 0 sb cb ca 0 sb

sa 0 cb sb ca 0 cb

35: 2where c stands for cos and s for sin.In the case of the piston rod ball-joint centre B, the co-ordinates xB, yB, zB relative to the

same global reference frame Oxyz are:24 xBSyBzB

35 24S 0:5 BB 0yTzT

35 3where S is the current piston displacement in the range Smax < S < Smax and yT and zT arethe co-ordinates defining the hydraulic trunion axis.Knowing the expressions of the co-ordinates of the centres A and B of the two ball- joints,

the equation of constraint governing the transmission law of the single-loop mechanism inFig. 2 can be written as:

xAj1 xBS2yAj1 yB2zAj1 zB2AB2 0: 4

The tie-rod length is calculated as the distance between joints A and B for the mechanism inthe reference position corresponding to j1 0:

AB xB0 xA0

2yB yA02zB zA02q : 5Eq. (4) may be solved either in a closed form or numerically for the unknown j1 (assuming thetranslation S of the piston to be the input motion) or for the unknown S (assuming therotation j1 of the steering knuckle to be the input).It is a straightforward matter to obtain the analytical expression S Sj1 of the

transmission law as follows.

S xAj1

AB2 yAj1 yB2zAj1 zB2q 0:5 BB 0: 6

For the reverse transmission law j1 j1S, used in the present paper only for the kinematicanalysis, analytical formulae are provided in Appendix A2.Important geometrical parameters that determine the eciency of motion transmissibility

through the linkage and avoid jamming are the pressure angles in the ball-joints FA and FB:These are defined as the angles between the velocity vector of the joint centres and the force

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943930

acting through the tie-rod (Fig. 3). In the case of ball-joint A, the velocity of its centre relativeto the global reference frame Oxyz has the following components:

vA 24 dxAdyA

dzA

35 1dt M01

24A0A sinj01 j1A0A cosj01 j10

35 dj1dt: 7

Similarly, the components of the velocity vector of joint centre B are:

vB 24 vxBvyBvzB

35 24 dS00

35 1dt: 8

where dS may be replaced by the unit vector (1, 0, 0) parallel to the piston rod axis. Finally,the pressure angles FA and FB in the respective joints are given by the relations:

FAj1 arccos

vA ABjvAj jABj

and FBj1 arccos

vB ABjvBj jABj

: 9

From the static equilibrium of the tie-rod (Fig. 3), the expression for the single loopmechanism transmission ratio dj1=dS FS=M1 may be obtained, which is useful in estimatingthe hydraulic force required to actuate the steering mechanism (see Appendix A3):

Fig. 3. Schematic for calculation of the pressure angles in the ball-joints and of the rotation angles of the kingpinreference frame.

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943 931

FSM1 dj1

dS cos FB

cos FA 1A0A

: 10

The expressions derived so far are sucient to perform both the synthesis and the analysis ofthe mechanism.

3. The synthesis problem

The present work solves the problem of optimally synthesising the steering linkage withregard to the condition of correct turning of the wheels and in the presence of a number ofconstraints using the penalty function approach.The components of the design variable vector X are chosen from those parameters defining

the mechanism geometry which are available for modification in the design process. Thekingpin angles a and b are given a priori, chosen from the condition of minimum wear of thetyre during vehicle riding and for providing a self-aligning eect of the wheels [8,9,14,15].Similarly, values of the kingpin track, as well as the maximum turning angle yi-max of thewheels and the maximum stroke of the hydraulic piston Smax are imposed. Values of thesteering knuckle arm length A0A and axial oset O1A0 remain available for the synthesis,together with the initial angle j01, and the co-ordinates yT and zT locating the hydraulictrunion axis, which can be varied within some minimum and maximum limits. In some types ofaxles this assembly must be associated with the cylindrical casing of the axle dierential, andpolar co-ordinates rT, lT are more suitable for defining the allowable region in which thehydraulic trunion is to be mounted (see Appendix A4).

3.1. Minimising the steering error

The classical approach is to express the objective function as the error between the specifiedmotion of the steering wheels and the real motion produced by the mechanism [11]. Thedesired motion of the wheels, which is the condition of correct turning of the vehicle, must besuch that the steerable wheel axes and the rear wheel axes intersect at the same point C, calledthe turning centre. The condition that points C1 and C2 in Fig. 4 coincide (known as theAckermann law) can be analytically expressed as:

1=tg yo 1=tg yi Wt=Wb 11where Wb is the wheel base and Wt the front wheel track of the vehicle. It is usual toapproximate the wheel track to the kingpin track which is 2OO1. The departure between thedesired transmission law and the actual transmission law of the mechanism is called thesteering error dy and must be suciently small to allow the wheels not to work against eachother producing an increased wear of the tyres.A very convenient way of defining the objective function, which brings about a significant

amount of analytical simplification [10,12] is to consider a variable length of the piston rodwhich allows the two wheels to be exactly positioned relative to each other according to thesteering law (11). For a number j j 1, . . . ,n of correlated positions of the wheels within their

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943932

working range, the corresponding objective function to be minimised can be considered asfollows [10]:

f1X "Xn

j1

dBB 0j

2#1=2or f1X maxjdBB 0j j with dBB 0j BB 0j BB: 12

The former objective function is dierentiable and a gradient-based optimisation subroutinecan be employed, while the latter (which is non-monotonic) assures the closest coincidencebetween the actual and desired transmission laws of the mechanism after minimisation. In aparticular design position j the length variation dBB 0j is given by the following expression,involving S() as defined in Eq. (6):

dBB 0j Sj1j S j2Thj1j 13where j2Th is the theoretical turning angle of the right wheel, obtained from the Ackermannsteering law (11), i.e.:

j2Thj1j arctg1

ctg j1j 1=Wt=Wb14

corresponding to the whole range of j1j which in turn is:

j1j yi-max j Dy with Dy yi-max yo-max=n 15and where the corresponding maximum turning angle of the outer wheel is that given by thetheoretical transmission law, i.e. yo-maxj2Thyi-max:

Fig. 4. Schematic diagram of a four-wheel vehicle during turning.

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The objective function must be calculated together with penalty functions describing thefollowing constraints:

. g1X: the condition that the maximum turning angle yi-max is achieved for the piston strokeS = Smax (equality constraint);

. g2X: the design variables must be within the prescribed limits;

. g3X: the pressure angles in the ball-joints must be less than a limiting value Fadm;

. g4X: the mechanism loop must close for any position of the steering knuckle arm within itsworking limits which is the condition that the expression under the square root in relation(6) is positive.

The pressure angles, FAj and FBj, are evaluated for every position j1j given by relation (15). Inthe real mechanism the minimum value of these angles can be below the one given by thetheoretical law (because the outer wheel maximum turning angle might exceed the estimatedyo-max), but this uncertainty can be overcome by considering a slightly lower admissible valuefor checking in the penalty function for instance 598 instead of 608.Extra complications are encountered when synthesising a leading mechanism, where the

steering knuckle arms are divergent and the possible optimum angle j01 together with thelength A0A are constrained by the feasible location of the corresponding ball-joint A in thespace between the wheel rim and the wheel hub. In this case it is useful to define an extrapenalty function g5X, which becomes eective in the case when the centre of the ball-joint Ais situated outside the available space i.e. the cylindrical envelope of diameters d1 and d2 inFig. A2 (Appendix A5).An improvement of the objective function f1X is that of correcting the length variation

dBB 0j so as to have an approximation to the steering error of the mechanism (the actualkinematic characteristic we wish to minimise). By associating in relation (10), dBB 0j with dS weobtain:

dj2j dBB 0j cos FA j2Thj1

cos FB j2Thj1 A0A16

which is a close estimate to the steering error of the mechanism dy j2 j2Th: Because thepressure angles FA and FB have already been calculated in the penalty function g3, the increaseof the CPU time required to evaluate the objective function once is insignificant. The errorintroduced decreases during the optimisation and the closer to the minimum (where dBB 0j issmall), the better the approximation. For the numerical example presented below, in the caseof the optimised mechanism with rounded-value parameters, the dierence between the realand approximate maximum steering errors is less than 20 (0.67918 and 0.67998, respectively).Consequently, the objective functions in relation (12) can be replaced by the expression:

f 1X "Xn

j1

dj2j

2#1=2or f 1X maxjdj2jj: 17

Finally, merging one of the above functions with the penalty functions yields:

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943934

F1X f 1X X

giX: 18

3.2. Maximising the eciency of motion transmissibility

In many cases the steering error is not the main criterion of evaluation of the steeringmechanism performance. There are cases when the motion and force transmissibility are moreimportant and cause special concern during the design process.According to Ref. [13] the optimisation problem may be formulated to minimise the pressure

angles while the steering error is constrained to remain less than a maximum permissible value.In this case the objective function can be defined as the maximum norm of the pressure anglesoccurring in the ball-joints A and B, for the whole working range of the mechanism i.e.:

f2X maxFAj1j, FBjj1j: 19The values considered for the pressure angles FA and FB were those given by relations (9) forthe same range of j1j as in Eq. (15). The final objective function to be minimised will have thesame form as F1X in relation (18):

F2X f2X X

giX 20

with the dierence that the penalty function g3 corresponding to the pressure angles has beenreplaced by one checking that the maximum steering error as calculated approximately usingrelation (16) is less than an imposed value dyadm:

3.3. Numerical example

The case of synthesising the steering mechanism of a light tractor has been considered. Inthe optimisation process two variants have been examined, that of steering error minimisation(objective function F1X and that of increasing the eciency of motion transmissibility byminimising the maximum pressure angle (objective function F2X).In both cases a maximum norm-based objective function has been defined with a number n

= 40 of design points. The authors of the present paper have used the Simplex optimisationmethod due to Nealder and Mead which has the advantage of not requiring the calculation ofthe derivatives and copes successfully with the non-monotonicity in both of the objectivefunctions. Another advantage is that it can deal with dependent components of the designvariable vector XO1A0, A0A, BB 0, xT, yT, j1: It can be shown that for mechanisms withdierent disposition of the steering knuckle arm along the kingpin axis, by a proper correlationof the co-ordinates yT and zT and of the length BB , identical kinematical behaviour of themechanism can be obtained. Obviously some of these parameters can be maintained fixed, andthe minimisation performed only with regard to j1 and A0A, both of which strongly aect thetransmission law of the mechanism, adding eventually a third variable O1A0 which aectsmainly the pressure angle in the ball-joints.The strict equality constraint of Smax 95 mm, which is actually of no practical use, was

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943 935

replaced with the inequality constraint 95 0:25RSmaxR95 0:25 mm, that facilitates theconvergence in the optimisation subroutine.The corresponding input numerical data are as follows: a 108, b 88, Wb 1400 mm,

yi-max 458, OO1 437 mm, j1 2 2508; 2908 (which means a leading steering linkageconfiguration is being sought), A0A 2 100; 250 mm, O1A0 2 50; 50 mm, BB 0 600 mm,yT 126 mm and zT 28 mm, Smax 95 mm (the position and characteristics of thehydraulic actuator were kept unchanged from a previous design). For expressing the constraintupon the position of the ball-joint centre A, the values of the constructive parameters in Fig.A2 were d1 322 mm, d2 476 mm, O1O 01 72 (measured along the kingpin axis relative tothe origin O1) and the permitted maximum distance b 114 mm in Fig. A2 (measured alongthe wheel axis O 01x

01).

In the case of the objective function F2, the maximum allowable steering error has beentaken as dyadm 58:The exact and rounded design parameters obtained after optimisation are presented in

Table 1 together with the resulting length of the tie-rod, which in most designs is adjustable viasome threaded parts, permitting variation of the toe-angle of the front wheels.

4. Analysis of the mechanism and final conclusions

The purpose of analysing the kinematic behaviour of the mechanisms obtained afteroptimisation is to check whether the requirements imposed are fulfilled, since approximationsto the steering error and transmission angles have been used and also because the exactoptimum design variables have been rounded to practical values.By inspecting the input/output law j1 j1S, it may be seen that one of the main

requirements is fulfilled, i.e. for the maximum permissive stroke of the piston Smax 95 mm),the inner maximum turning angle achieved by the wheels is 44.968 for the first design(minimisation of the objective function F1) and 44.668 for the second design (minimisation ofthe objective function F2).Fig. 5 is a graphical representation of the steering error dy for the whole range of j1 for the

cases of minimising the steering error and for minimising the maximum transmission angle. Atthis point, since the departure between the desired and the actual transmission law of themechanism is the main parameter of interest in a steering linkage, it is useful to adopt someconventions, on the basis of which comparison between dierent steering mechanisms can be

Table 1

Main parameters of the optimised mechanisms

F1X 0:668 (max. steering error) F2X 56:48 (max. pressure angle)

j01 (deg) 301.7 (302) 298.4 (298)O1A0 (mm) 45.0 (45) 44.9 (45)A0A (mm) 194.4 (194.5) 186.3 (186)

AB (mm) (resultant) 250.2 (250.9) 234.6 (233.2)

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943936

made. For instance, Ardayfio and Qiao [1] studied the actual trajectory of the centre ofintersection C between the front wheel axes and the desired straight-line trajectory coincidentwith the rear wheel axis, while Felzien and Cronin [3] considered the steering error as thedistance between the points of intersection C1 and C2 of the same axes and the rear wheel axes(see Fig. 4). Visa and Alexandru [5] analysed the steering linkage as a function-generatingmechanism, and considered the steering error as the angular departure between the actual andthe desired transmission law evaluated at the inner wheel. Because the inner wheel has animposed definite maximum pivoting angle, yi-max, the authors of the present paper suggest thatit is better to assess the steering error at the the outer wheel and to consider the inner wheel asthe input. In Fig. 5 the steering error is shown for both the inner and outer wheel, for thewhole range of j1:Fig. 6 shows that the pressure angle variation is in compliance with the requirements that

these angles must be less than 608. As expected, for the case of minimising the transmissionangles, the maximum values are smaller than for the case of minimisation of the steering error.Fig. 7 shows the variation of individual input/output transmission ratios of Fs/M1 and Fs/

M2. It is known that the wheel pivoting moments M1 and M2 depend strongly on the nature ofthe surface on which the vehicle stands and on the load upon the front axle. They also varywith the wheel turning angle j1 and j2 due to the eect of the load upon the front axle and ofthe castor oset and of kingpin axis inclination. Fig. 8 shows the sum of these individualtransmission ratios which is the overall transmission ratio Fs/M. In this case, a constantresisting turning moment of a single wheel, M, must be assumed to permit the calculation ofthe approximate hydraulic force Fs required to actuate the steering mechanism. This graphshows the actuating symmetry of the mechanism with minimum control force occurring at thestraight-ahead position of the vehicle.The final graph in Fig. 9 was obtained by considering a variable tie-rod length and a fixed

piston in its median position S 0 in the equation of constraint (4): j1 j1AB dAB:The adjusting range of the tie-rod has been considered as dAB 2 1:5; 1:5 mm, for which one

Fig. 5. Steering error of the optimised mechanisms.

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943 937

can observe that there is no significant dierence between the two variants. This graph is veryuseful for the designer in choosing the toe-out angle of the wheels. Since the tie-rod is the maindeformable element in the steering mechanism (due to both elasticities and clearances occurringin the bushings), the toe-out angle can be estimated with ease, j1dAB being almostrectilinear.The final choice of one or other of the variants is up to the designer. It might be

useful to consider a greater steering error and further improve the force and motiontransmissibility of the mechanism. The consequent outer wheel oversteer occurring in the

Fig. 6. Variation of ball-joint pressure angles.

Fig. 7. Input/output transmission ratios of left and right loops of the mechanism.

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943938

second variant is also useful according to Reimpell and Stoll [9], permitting a diminishingof the minimum turning radius of the vehicle which is of some importance in everydaypractice.In the case analysed, the steering mechanism is located on the vehicle axle, and there is no

coupling eect between the axle oscillation and the steering linkage kinematics. However, theabove considerations can also be used in the design of an automobile steering mechanism withindependent wheel suspension. The kinematics of the suspension must first be assessed and asupplementary constraint must be added to the objective functions F1X or F2X which is that

Fig. 8. Overall transmission ratio of the steering mechanism.

Fig. 9. Left wheel toe angle vs. tie-rod length variation.

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943 939

the point B is located at the approximate centre of curvature of the trajectory of point A whilethe wheel oscillates within its range. The reader is referred to Felzien and Cronin [3] for a moredetailed solution to this problem.

Acknowledgements

The research work was supported by the Romanian Ministry of Education grant number5843/1996 while the first author was a guest member of sta at Newcastle University. Thanksare also extended to Mr. L. Nicoara and Mr. V. Marosi from Tractor Project S.A. Brasov,Romania for providing the numerical data of the example presented in the paper.

Appendix A

1. In Fig. 3 the following three relations hold:

tan a 0 P2P3O1P2

, cos b O1P1O1P2

and tan a P1P4O1P1

P2P3O1P1

A1

which, by elimination, gives:

tan a 0 cos b tan a A22. Substituting into the equation of constraint (5), the co-ordinates of the ball-joint centre A

derived from relations (1)(3) yields the transmission law in the following form:

j1 2arctanQ12

Q21 Q23 Q22

qQ3 Q2 A3

where

Q1 2a2cb a3sb A0A

Q2 2a1ca 0 a2sa 0 sb a3sa 0 cb

A0AQ3 a21 a22 a23 AB2 A0A2 A4

and

a1 OO1 xB sa 0 O1A0

a2 ca 0 sb O1A0 yB

a3 ca 0 cb O1A0 zB: A5

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943940

The sign in front of the square root must be chosen in accordance with the actualconfiguration of the mechanism (leading or trailing steering linkage). Numerical solution ofthe equation of constraint does not give rise to this problem of choosing between twopossible configurations, since in the optimised mechanism the theoretical output angle j2Thwill be a very good initial estimate for the actual angle being sought.

3. The condition of static equilibrium of the tie-rod in Fig. 3 is written as:

FAcos FA

FScos FB

where FA M1A0A

A6

which permits determination of the inputoutput ratio Fs/M1. Since it is also equal todj1=dS, an equivalent expression can be deduced by dierentiating the equation ofconstraint (5) resulting in:

dj1dS xA xBdxA yA yBdyA zA zdzA

xA xB : A7

4. In Fig. A1, the admissible position of the hydraulic trunion axis has been defined in polarco-ordinates relative to the yOz system plane of the main reference frame Oxyz. The polarradius rT is considered to remain in the domain rminRrTRrmax while lT must be withinlminRlTRlmax, limits given by the external edges of the casing and by the dierential inputshaft position, and also by the type of linkage leading or trailing link. The Cartesian co-ordinates xT and yT appearing in the text, must be replaced in accordance with the knownrelations that hold between the polar and Cartesian co-ordinates.

5. The condition that the centre A of the steering knuckle arm ball-joint must be within thepermissive cylindrical envelope in Fig. A2, delimited by the wheel rim and the wheel hub, isanalytically expressed as:

Fig. A1. Polar co-ordinate positioning of the hydraulic trunion relative to the front axle.

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943 941

x 0ARb and d1=2Ry 0A2z 0A2q Rd2=2 A8

the latter expression being the condition that the distance between the point A and the wheelaxis lies within the allowable range [d1/2d2/2]. Both inequalities are easier to checkemploying the co-ordinates of the ball-joint centre x 0A, y

0A and z

0A relative to a co-ordinate

system obtained from O 01x01y01z01 attached to the wheel. This is obtained from the kingpin co-

ordinate system by applying a translation O1O01 along the z1-axis followed by a rotation g

about the y-axis. Numerical values of O1O01, g and b are directly available from the

blueprints (the angle g can be considered approximately equal to the sum of the wheelcamber angle and the kingpin inclination angle). The described co-ordinate systemtransformation can be expressed in matrix form as follows:24 x 0Ay 0A

z 0A

35 24 cos g 0 sin g0 1 0sin g 0 cos g

35 24A0A cos j01A0A sin j01O1A0 O1O 01

35 A9

References

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Fig. A2. Schematic for calculating the permissible situation of the steering knuckle arm ball-joint.

P.A. Simionescu et al. / Mechanism and Machine Theory 35 (2000) 927943942

[3] M.L. Felzien, D.L. Cronin, Mechanism and Machine Theory 20 (1985) 17.[4] P.A. Simionescu, I. Tempea, M.R. Smith, in: Proceedings of the Seventh International Symposium

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[12] C.H. Suh, A.W. Mecklenburg, Mechanism and Machine Theory 8 (1973) 479.[13] C.H. Suh, C.W. Radclie, Kinematics and Mechanisms Design, Wiley, New York, 1978.[14] R. Ivanov, S. Liubenov, Proceedings of the Institution of Mechanical Engineers 209 (D) (1995) 111.[15] P.A. Taylor, R. Birtwistle, Proceedings of the Institution of Mechanical Engineers 181 (A2) (1967) 133.

[16] Carraro SpA, Axle Catalogue, Via Olmo 37, Campodarsego, Italy, 1992.

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