Mu l t i - L i n k F i v e - P o i n t

8/3/2019 Mu l t i - L i n k F i v e - P o i n t

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K S M E Interna tional Journal, VoL 17 No. 8. pp. 113 3-- 1139, 2003 1133

Ki n e m at ic A n a l ys is o f th e M u l t i -L i n k F i ve - P o i n t S u s p e n s ionSystem in Point Coordinates

H a z e m A l i A t t i a *D e p a r t m e n t o f M a t h e m a ti c s, C o ll eg e o f S c ie n ce , K i n g S a u d U n i ve r si ty ( A l - Q a s s e e m B r a n c h ) ,P .O . Boa" 237, Bu ra id ah 81999 , K S A

In this paper, a numerical algorithm for the kinematic analysis of a multi link five pointsuspensi on system is presented. The kinemati c analysis is carri ed out in terms of the rectangularCartesian coordi nates of some defined points in the links and at the joints. Ge ometr ic constraintsare introduced to fix the relative positions between the points belonging to the same rigid body.Position, velocity and acceleration analyses are carried out. The presented formulation in termsof this system of coordi nates is simple and involves only elementary mathematics. The results ofthe kinematic analysis are presented and discussed.Key Words:Kinematic Analysis, Suspension Systems, Five Point Suspension,

Point Coordinates

I . I n t r o d u c t i o n

In recent years various methods for the kine-matic analysis of spatial mechanisms have beendeveloped. Th e different methods can be classifiedaccording to the type of coordinates chosen todetermine their configuration and specify theirconstraints. Some tbrmulations use a large set ofabsolute coordinates (Wehage and Haug, 1982,Nikravesh, 1988). The position and orientation ofthe rigid links in the mechanism are describedwith respect to the global reference coordinatesystem. The algebraic equ ations of constraints areintroduced to represent the kinematic joints thatconnect the rigid bodies. Although in this type offormulation the constraint equations are easy toconstruct, it has the disadvantage of the largenumber of defined coordinates. Other lbrmul a-tions use sets of relative coordi nates (Denavit andHartenberg, 1955: Paul and Krajcinovic, 1970).

* E-mail : ahl113 @yahoo.comTEL : +966-6-3800319" FAX : +966-6-380091 IDepartment of Mathematics, College of Science, KingSaud University (AI-Qasseem Branch), P.O. Box 237,Buraidah 81999, KSA. (Manuscript Re c e ive d October30, 2002:Re v i se d May 6, 2003)

The pos itio n of each link is defined with respectto the previous link by means of relative joi ntcoordin ates that depend on the type of the joi ntused. This type of formulation yields the cons-traints as a minimal set of algebraic equations.The constraint equations are derived based onloop closure equations and the resulting con-straint equations are highly nonlinear and containcomplex circular functions.

Another tbrmulation which is based on pointcoordinates is discussed in (Garcia de Jalon et al.,1981, 1982; Vilallong et al., 1984; Akhras andAngeles, 1990 : Attia, 1993). T he conf igurat ion o fthe system is described in terms of the rectangularCartesian co ordinat es of some defined points inthe links and at the joints. The system constraintequations are then written to fix the relativepositions of the points in each rigid link and alsothe relative positions between the different linksdetermined by the type of joint s connect ing them.

In this paper the kinematic analysis of themulti-link five-point suspension system is carriedout in terms of point coordinates. The position,velocity, and acceleration analyses are carried outto determine the positions, velocities, and acce-lerations for the unknown points and links in themechanism. The velocities and accelerations of


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1134 H az e m A l i A t t i a

o t h e r p o i n t s o f i n t e re s t c a n a l s o b e c a l c u l a t e d . T h ea n g u l a r v e l o c it y a n d a c c e l e ra t i o n o f a n y l i n k i nt h e m e c h a n i s m a r e e v a lu a t e d i n te r m s o f th e C a r -t e s i a n c o o r d i n a t e s , v e l o c i ti e s , a n d a c c e l e r a t i o n s o ft h e a s s i g n e d p o i n t s .

2 . M o d e l li n g o f t h e M u l t i - L i n kF i v e - P o i n t S u s p e n s i o n S y s t e m

T h e m u l t i - l i n k f i v e - p o i n t s u s p e n s i o n s y st emi s u s u a l l y u s e d f o r r e a r d r i v e n a x l e s o f c u r r e n tp r o d u c t i o n s o f M e r c e d e s - B e n z c a rs , M a z d a 9 2 9,s o m e B M W a n d T o y o t a S u p r a c ar s. T h e m e c h a n i -c a l s y s te m c o n s i s t s o f th e m a i n c h a s s i s , a m u l t i -l i n k f i v e - p o i n t s u s p e n s i o n m e c h a n i s m , a n d t h ew h e e l a s s h o w n i n F i g . I . T h e s y s t em h a s t h r e ed e g r e e s o f f r e e d o m ( D O F ) . T h e c h a s s is , s i n c e it i sc o n s t r a i n e d t o m o v e v e r t ic a l ly u p w a r d o r d o w n -w a r d , o n l y o n e D O F o u t o f i ts s ix D O F r e m a i n s .T h e w h e e l ha s o n e - D O F c o r r e s p o n d i n g t o t her o l l i n g m o t i o n . S i n c e t h e s u s p e n s i o n m e c h a n i s mc o n n e c t s t h e d r i v e n w h e e l t o t h e c h a s s is ( s p e c-i f ic a l ly t o th e a x l e c a r r i e r ) b y r u b b e r m o u n t i n g s ,it c a n b e s i m u l a t e d a s f iv e b i n a r y l i n k s c o n n e c t i n gt h e c h a s si s a n d t h e w h e e l k n u c k l e t h r o u g h s p h e -r i c al j o i n t s a t b o t h e n d s o f e a c h l i n k . T h u s , t h es u s p e n s i o n m e c h a n i s m c o n s i st s o f fi ve l i n k s a n dt e n s p h e r ic a l j o i n t s , a n d h a s o n l y o n e - D O F ( se eF i g . I ) .

2 .1 D i s p l a c e m e n t a n a l y s i sT h e c o n f i g u r a t i o n o f th e m e c h a n i s m c a n b e

s p e c i fi e d b y d e f i n i n g a s e t o f p o i n t s o n t h e l i n k sa n d a t t h e j o i n t s . F i g u r e 2 p r e s e n t s th e m e c h a n i s m

Fig. i

fI CH ASSIS "

/KNUCKLE

WHEELl

IT h e m u l t i l i n k f i v e - p o i n t s u s p e n s i o n s ys te m

w i t h th e a s s ig n e d p o i n t s . E a c h b i n a r y l i n k isr e p l a c e d b y t w o p o i n t s l o c a t e d a t t h e c e n t r e o f t h es p h e r i c a l j o i n t s a t b o t h e n d s , w h i l e t h e a d j a c e n tl i n k s a r e s h a r i n g c o m m o n p o i n t s. T h e w h o l em e c h a n i s m is t h e n r e p l a c e d b y t e n p o i n t s . P o i n t s1 , ' - - , 5 t h a t a r e l o c a t e d a t t h e c h a s s is , a r e k n o w np o i n t s. T h e C a r t e s i a n c o o r d i n a t e s o f th e u n -k n o w n p o i n t s 6. " " , 1 0 l o c a t e d o n t h e k n u c k l ed e f i n e t h e m o t i o n v a r i a b l e s . T h e r e f o r e , 1 5 c o n -s t r a i n t e q u a t i o n s a r e n e e d e d t o s o l v e f o r t h e 15u n k n o w n C a r t e s i a n c o o r d i n a t e s . T h e i n i t ia l p o s i -t i o n s , v e lo c i t ie s , a n d a c c e l e r a t i o n s o f p o i n t s I , .. . ,5 a r e k n o w n f r o m t h e d r i v e r d a t a .

T h e c o n s t r a i n t s a r e e i t h e r g e o m e t r i c o r k i n e -m a t i c c o n s t r a i n t s . G e o m e t r i c c o n s t r a i n t s a r e d i s -t a n c e c o n s t r a i n t s t h a t f i x t h e r e l a t i v e p o s i t i o n s o ft h e p o i n t s o n a r i g id l i n k o f t h e m e c h a n i s m . T h eg e o m e t r i c c o n s t r a i n t e q u a t i o n s a r e e x p r e s s ed i nt h e C a r t e s i a n c o o r d i n a t e s o f th e p o i n t s a s lb l l o w s ,( x 6 - x l ) 2 +( x T - x 2 ) 2 +( x ~ - x 3) 2+

( Y6-- Y l ) 2+ (Z~-- z i) 2- - d~ i = 0(y7--Y2) 2 + ( Z T - Z 2 ) 2 - d Z 2 = 0(ya- - y3) 2+ ( z s - z 3) E - d~.3 = 0

(X 9--X4) 2-] (Y9--Y4) 2-'~ (Zg--Z 4) 2--d 24 = 0( X l o - X s ) 2+ , 2 2 2 _(Y10-3.5) + ( z l o - z s ) - d l o , s - - O( X T - - X 6 ) 2 - ' } (yT--Y6) 2+ (.ZT--Z6)2--d72,6 ~--0( x s - - x6 ) 2+ ( Y s - - y6 ) 2+ ( z s - - z ~ ) 2 - - d2 ,6 = 0( X 9- -X 6) 2__ ( Y g- -Y 6) 2+ ( 2 9 - - 6 ) 2 - - d92 . 6 : 0(XIo -- X6) 2+ (Ylo - 3"6) 2+ ( 21o - - 6,6) 2 __ d12o,6= 0( X s - .' :7 ) 2 + ( Y 8 - y 7 ) 2 + ( z ~ - zT ) 2 _ d82.7= 0(X9--X7) 2+ [Yg--YT)2+ (2 ,9--27) z d~7 = 0( X l o - X 7) 2+ (Ylo--YT) z+ (Z lo- - Z7) 2__ d20 ,7 :0(m-xs) 2+ (_Vg-ys)2+ (Zg-Z~)Z-d~.s=O

( I . 1 )( 1 . 2 )(1.3)( 1 . 4 )(1 .5)( i .6 )( I . 7 )( i .8 )(I .9)l.lO)I . I I )1.12),.13)

IO

Fig. 2 The m ul t i - l ink f ive-point suspens ion wi th theass igned points


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Kinematic Analysis o f the Mu lti -Li nk Five Point Suspension System in Point Coordinates 1135(xx0 -- xs) 2q_ (Yl0 -- Y8) z+ ( z~0 -- zs) 2_ d20,8 0 (1.14)where di.~ is the distance between points i andj belonging to the same rigid link, and xi, Yi,and zi are the Cartesian coordinates of point i.Kinematic constraints result from the conditionsimposed by the kinematic join ts on the relativemotion between the bodies they comprise. Pointslocated at the centre of a spherical joint or at theaxis of a revolute joi nt automatica lly eliminate allthe kinematic constraints due to these joints.Moreover, driving constraints are added to theabove constraints as functions of the inpu t drivingang ula r positio n 0 (see Fig. 2) in the form,

( z ~ - z , ) - & , , c o s ( P ) = 0 ( I. 15)Equation (I) expresses the required 15 indepen-dent constraint equations in terms of the Car-tesian coordinates of the assigned points. Giventhe set of the known coordinates of points 1, --.,5 and the driving variable * at each instant, thenon lin ear Eq. (1) can be solved by any iterativenumerical method (Molian, 1968) to determinethe 15 unknown Cartesian coordinates of points6,-", 10.

It should be noted that in this formulation, thekinematic constraints due to some common typesof kinematic join ts (e.g. revolute or sphericaljoints) can be automatically eliminated by prop-erly locating the assigned points. The remainingkinematic constraints along with the geometricconstraints are, in general, either linear or quad-ratic in the Cartesian coordinates of the particles.Theretbre, the coefficients of their Jacobian ma-trix are constants or linear in the rectangularCartesian coordinates. Where as in the formula-tion based on the relative coordinates, the con-straint equations are derived based on loopclosure equations which have the disadvantagethat they do not directly determine the positionsof the links and points of interest which makes theestablishment of the dynamic problem more diffi-cult. Also, the resulting constraint equations arehighly nonlinear and contain complex circularfunctions. The absence of these circular functionsin the point coor dinate formu lation leads to fasterconvergence and better accuracy. Furthermore,preprocessing the mechanism by the topological

graph theory is not necessary as it would be thecase with loop constraints.

Also, in comparison with the absolute coor-dinates formulat ion, the manua l work of the localaxes attachment and local coordinates evaluationas well as the use of the rotational variables andthe rotation matrices in the absolute coordinateformulation are not required in the point coordi-nate formulation. This leads to fully computerizedanalysis and accounts for a reduction in com-putational time and memory storage. In additionto that, the constraint equations take much sim-pler forms as compared with the absolute coor-dinates.

The main kinematical properties of the suspen-sion are described by the coordinates of the wheelcentre point and the kingpin angle a' and camberangle ~ (Adler, U.). The wheel centre point(po int 11, see Fig. 3) is defined as the point atwhich the wheel spin axis intersects the wheelplane. Points 8 and 11 define the wheel spin axis.The coordinates of the wheel centre point caneasily be determined by specifying its positionrelative to three other points located on the knuc-kle. Ki ngp in angle determines the steering ali-gning torque in conjunction with steering offsetand wheel caster. The kingpin angle ct is definedas the inc lin ati on angle of a fixed line on theknuckle (connecting points 6 and 8) relative tothe vertical longitudinal plane, measured in thetransverse plane of the vehicle (Adler, U.) andtherefore from Fig. 3;

a, =t an _ ~ Ys--Y~

z\ , t 8

F i g . 3 Kingpin and camber angles


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1 1 3 6 H a z e m A f t A t t i a

A p o s i t i v e a n g l e a s i g n i f ie s a d i s p l a c e m e n t o fp o i n t 6 i n t h e n e g a t i v e y d i r e c t i o n . T h e c a m b e ra n g l e , 8 i s t h e i n c l i n a t i o n o f th e w h e e l p l a n er e l a t i v e t o t h e l o n g i t u d i n a l v e h i c l e p la n e , m e a -s u r e d i n t h e t r a n s v e r s e p l a n e o f t h e v e h i c l e a n dt h e r e f o r e ;

- ( z s - z . )f l = t a n - 1 ( Y s - - Y H )P o s i t i v e c a m b e r m e a n s t h a t t h e w h e e l s a r e t i l t e do u t a t th e t o p r a t h e r t h a n a t t h e b o t t o m .

2 .2 Ve loc i ty and acce l era t ion ana ly sesT h e v e l o c i ty e q u a t i o n s a r e d e r iv e d b y d i ff e r e n -

t i a t i n g E q . ( 1) w i t h r e s p e c t t o ti m e a n d t a k e t h ef o r m

[ c ~ ] q = o ( 2 )S i n c e t h e v e l o c i t y e q u a t i o n s a r e l i n e a r , t h e v e c t o ro f v e l o c i t i e s ; ~ 1 = [ 0 1 , 0 2 , " ' , q 3 o , O] r c a n b ep a r t i t i o n e d a s ; e l = [ t ~ r , z ~ r ] r , a n d t h e v e l o c i t ye q u a t i o n i s w r i t t e n i n t h e p a r t i t i o n e d m a t r i xf o r m ,

[c~]a=[c~]w ( 3 )w h e r e t i = [ 2 ~ . 9 6 , 2 6. ' " , 2 ~ o] r a n d g r = [ 2 ~ . _ g t , 21 ."" , 25, O ] r a r e t h e u n k n o w n a n d k n o w n v e c t o rs o fv e l o c i t i e s , r e s p e c t i v e l y a n d a r e g i v e n b y .

[ C ~ ] =

2xsa 2y ~ . , 2z~,1 0 0 00 0 0 2x 7 , 2 2y 7 , 2 2 z7 ,20 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

2 x s a 2y 6 . 7 2 z c a 2x 7 . 6 2y T , s 2Z7,62X 6,s 2ys,s 2Z 6.s 0 0 02X6,9 2y6,9 2& ,9 0 0 02x~ ,lo 2Y6,19 2z~ ,io 0 0 0

0 0 00 0 0

2X s,s 2 28,3 22.8,30 0 00 0 00 0 0

2Xs,s 2ys,s 2Zs,60 0 00 0 0

0 0 2 x T , s 2 y T , s 2 zT,S 2X s , 7 2 y s , 7 2 ~ , 70 0 2 x 7 ,9 2 y 7 .9 2 z7,9 0 0 00 0 2X 7, to2y7,1o 2Zr,m 0 0 00 00 00 0

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

2x9.4 2y9,4 2z9,4 0 0 00 0 0 2X lo ,s 2ylo ,5 2Zlo,50 0 0 0 0 00 0 0 0 0 0

2x9,6 2y g,s 219,6 0 0 00 0 0 2X ,o ,s 2ym ,6 2Zlo ,60 0 0 0 0 0

22"9,7 2y9 ,7 2Z9,7 0 0 00 0 0 2Xio.7 2yxo,72Zlo,7

0 0 02Xxo.s 2ylo,s 2Z lo.s

0 0 0

0 0 0 2 X s ,9 2 y s ,9 2 Z s ,9 2 Xg , s 2 y g , s 2 zg, s0 0 0 2X s, ,o 2ys , lo 2Zsao 0 0 00 0 0 0 0 0 0 0 0

( 4 )

a n d

[ C . ] =

2X~,1 2y6,1 2Z6,10 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 0

0 0 0 0 0 0 0 0 02XTa 2yT a 2z7,2 0 0 0 0 0 0

0 0 0 2 x s , s 2ys .s 2zs,s 0 0 00 0 . 0 0 0 0 23{9,4 2y9,4 2~ ,40 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

0 0 0 00 0 0 00 0 0 00 0 0 0

2X lo,s 2yxo,s 2 2 : 1 o , 8 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 00 0 0 - & , l s i n ( 0 )

( 5 )


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Kinematic Analysis of the Multi-Link Five-Point Suspension System in Point Coordinates 1137

S i m i l a r l y , t h e a c c e l e r a t i o n e q u a t i o n i s d e r i v e d b yd i f f e r e n t i a t i n g th e v e l o c i t y E q . ( 2 ) w i t h r e s p e c t tot i m e a s f o l l o w s ,

[ C q ] q + ( [ C q ] ~ 1) q d l = O ( 6 )P a r t i t i o n i n g t h e v e c t o r o f a c c e l e r a t i o n s i l i n t o[ f i r W r ] w h e r e f f a n d ",# a r e t h e v e c t o r s o fu n k n o w n a n d k n o w n a c c e l e r a t io n s re s p ec t iv e l y .t h e a c c e l e r a t i o n e q u a t i o n i s e x p r e s se d a s.

[C , ]u= [cw]w- ( [G ]q )~q=o (7 )w h e r e t h e tw o s u b m a t r i c e s [Cu] a n d [ C w ] a r ed e f i n e d b y E q s . ( 4 ) a n d ( 5) r e s p e c t i v e l y a n d t h es q u a r e v e l o c i t y t e r m ( [ C q ] l ) o l S l i s e x p r e s s e d a sf o l l o w s ,

2d '~ . , + 25 ,g ,t + 2 ~ . ,2372.2+ 2 Q27.2+ 2~7.22X Z ,s + 29z,3 + 22~8,s2:eL + 2_9L + 2&,

2,~ ~ o.s+ 2920 .5 + 22tZo,5

2./'6Z.s+ 2_9~,s+ 2 ~ , s[ C ' q ] q = 2 2 ~ , 9 - 233~ ,9q - 2 J2'~6,9 (8 )

2 3 .2 -2 x s . t o + - Y 6 a o + 2 ~ a o2 2 2 .8 + 2 9 2 s + 2 ~ , s2.~ ,9 + 23",72.9+ 2 ~ . 9

2 ~ , o + 2 y-L o + 2 & , 0222 , 9+ 2 9~ ,9+ 2k~8,9

2 v-2 - - 2 "2 - - 9" 2. t 8 .10 ~- V8 , t0 ~- ~8 ,10& ,, COS (0 ) t~~

R e g a r d l e ss o f th e o r d e r o f n o n l i n e a r i t y o f th ec o n s t r a i n t E q . ( 1 ) , th e v e lo c i t y a n d a c c e l e r a t i o ne q u a t i o n s a r e l i n e a r i n t e r m s o f d l a n d i~ . , r e s p e c -t iv e l y . I f t h e p o s i t i o n a n a l y s i s h a s b e e n f o r-m u l a t e d c o r r e c t ly , t h e n t h e m a t r i x [ C, ~ ] i s o f as u f fi c ie n t r a n k a n d b e c o m e s n o n s i n g u l a r . T h e r e -f o re , th e v e l o c i t i e s a n d a c c e l e r a t i o n s o f t h e u n -k n o w n p o i n t s c a n b e ea s il y d e t e r m i n e d b y s o l v i n gb o t h t h e l i n e a r E q s . ( 3) a n d ( 7) u s i n g a n y n u -m e r i c a l m e t h o d . T h e v e l o c i t i e s a n d a c c e l e r a t i o n so f o t h e r p o i n t s o f i n t e r e s t c a n a l s o b e c a l c u l a t e d i ft h e i r p o s i t i o n s a r e s p e c if i e d . T h e a n g u l a r v e l o c i tya n d a c c e l e r at i o n o f a n y l i n k i n th e m e c h a n i s m ,c a n b e e v a l u a t e d f r o m t h e C a r t e s i a n c o o r d i n a t e s ,v e l o c i ti e s , a n d a c c e l e r a t i o n s o f a n y t h r e e d e f i n e dp o i n t s o n t h e l i n k a n d a r e r e s p e c t i v e l y g i v e n a s ,

(1 3 = V j r i X V k - ' i (9 )v . i , i ' r k , i

^ ^a d , i x a k , ia = x ( 1 0)aj, i'rk,iw h e r e ~ t ~ , i = a ~ , i - - wx ( w x r g .

2 . 3 R e s u l t s o f t h e s i m u l a t io nT h e c h a s s i s is a s s u m e d t o b e s t a t i o n a r y a n d

t h e r e f o r e t h e v a l u e s o f th e v e l o c i t i e s a n d a c c -e l e r a t io n s o f a l l k n o w n p o i n t s ( p o i n t s I , " ' , 5 )f i xe d o n i t a r e i d e n t i c a l l y z e ro . T h e C a r t e s i a nc o o r d i n a t e s o f t h e k n o w n p o i n t s a r e l i st ed i nT a b l e 1. T h e d r i v i n g v a r i a b l e 0 is t a k e n a s f u n c -t i o n o f ti m e i n th e f o r m , 0 ( t ) = 0 .7 + 0 .5 +0 .1 2 5 z. T h e n o n l i n e a r e q u a t i o n s o f c o n s t r a i n t s(1 ) a r e s o lv e d b y N e w t o n - R a p h s o n ' s m e t h o d o fs u c ce s si v e a p p r o x i m a t i o n t o d e t e r m i n e t he C a r -t e s ia n c o o r d i n a t e s o f th e u n k n o w n p o i n t s f o rd i f f e r en t t i m e s te p s . A l s o , t h e C a r t e s i a n c o o r -d i n a t e s o f t h e w h e e l c e n t r e ( p o i n t 1 1) a r e e s t i -m a t e d . S i n c e th e p o s i t i o n a n a l y s i s is a n o n l i n e a rp r o b l e m w h i c h i s s o l v e d b y a n it e r a t iv e n u m e r i c a lm e t h o d , i t i s e x p e c t e d t h a t t h e p r o b l e m o f m u l t i -p l e s o l u t i o n s a r r i se s . I n o r d e r t o a v o i d s u c hp r o b l e m , k n o w i n g t h e in p u t d r i v i n g v a r ia b l e s ,m e a s u r e m e n t s c a n b e u s e d i n i t i a l ly to o b t a i n ag o o d i n i t i a l g u e s s a t t h e s t a r t i n g p o i n t o f th ep o s i t i o n a n a l y s i s . I n t h e s u b s e q u e n t i t e r a t io n s , t h ep r o b l e m o f m u l t i p l e s o l u t i o n c a n b e o v e r c o m et a k i n g a s a g o o d i n i t i a l g u e s s th e p r e v i o u s c o n -f i g u r a t i o n o f t h e s y st e m . T h e v e l o c i t y a n d a c c e l-e , a t i o n e q u a t i o n s a r e s o l v e d u s i n g th e L U fa c to -r i z a t io n w i th p i v o t i n g m e t h o d . T a b l e 2 p r e s e n tss o m e r e s u lt s o f t h e k i n e m a t i c a n a l y s i s f o r tw os e c o n d s o f s i m u l a t i o n a s a r e su l t o f c h a n g i n g t h ed r iv e l" a n g l e w i t h t i m e . T a b l e 2 i n d i c a t e s a l s o t h ei n i ti a l g u e ss f o r th e c o o r d i n a t e s o f th e u n k n o w np o i n t s . T a b l e 3 p r e s e n t s t h e a n g u l a r v e l o c i t y a n da n g u l a r a c c e l e r a ti o n o f t h e k n u c k l e a t v a r i o u sT a b l e I C a r t e si z n c o o r d i n a t e s ( m ) o f t h e k n o w n

p o i n t sP o i n t 1 P o i n t 2

X - 1 . 3 9 6 - -1 .3 0 1Y - -0 .02 5 0 .1168Z - - 0 . 082 0 .106

P o i n t 3 P o i n t 4 P o i n t 5- 1 .0941 - -0 .983 8 - 1 .0950.1987 0.2609 0.183

0.090 0 .0501 - -0 .01 58


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1138 H a z e m ,4 l i A t t i a

Table 2 Cartesian coordinates (m) of the unkn own pointsTime 0.00(sec.l

Po in t 0 40.1X --1.196

6 Y --0.181Z 0.259X --1.114

7 Y --0. 013Z 0.299X --1.107

8 Y --0. 010Z 0.221X -- 1.205

9 Y 0.183Z 0.161X -- 1.028

10 Y 0.094Z 0.276

T a b l e 3

0.25 0.50 0.75 1.00 1.25 1.50 1.7547.72 56.22 65.62 75.92 87.11 99.19 112.17

--1.231 --1.261 --1.292 --I.315 --1.324 --1.316 --1.2840.261 0.320 0.367 0.400 0.415 0.408 0.3720.218 0.166 0.102 0.027 --0.0 59 --0. 153 --0. 250

--I.158 --1.212 --1.262 --1.302 --1.327 --1.336 --1.3190.126 0245 0.330 0.383 0.410 0.409 0.3770.368 0.361 0.311 0.240 0.155 0.060 --0.039

--1.144 --1.193 --1.238 --1.270 --1.287 --1.285 --1.255--0. 084 --0.183 0.263 0.315 0.345 0.350 0.332

0.303 0.316 0.281 0.218 0.140 0.051 --0.042--I.232 --1.256 --I.278 --I.292 --1.290 --1.268 --1.215

0.206 0.240 0.279 0.309 0.325 0.34 0.3021.136 0.108 0.060 --0.0 05 --0.0 84 --0. 172 --0.261

--I.067 --1.109 --I.153 --1.188 --I.212 --1.223 --I.2190.202 0.290 0.364 0.418 0.453 0.467 0.4580.294 0.277 0.232 0.167 0.089 0.001 I --0.0 94

i

I

Angular velocity

1.95123.20

-- 1.2230.305

--0.326-- 1.267

0.313--0.116--1.190

0.299--0.114--1.130

0.271--0.329-- 1.209

0.424--0.178

(rad /s) and acceleration (ra d/s 2) of the knuckle

Initialguess

-- 1.3090.412

--0.086-- 1.225

0.4050.111

- - 1.1640.4350.070

- 1.2310.532

--0.145--1.137

0.489--0.058

Tim e (sec.) 0.00~x -- 3.47O)y --0.36w z 0.43a x 18.7a y 1.41Cez --7.74

0.25 0.50 0.75 1.00 1.25 1.5--1.83 --1.02 --0.51 --0.27 --0.17 --0.15--0.43 --0.47 --0.38 --0.32 --0.30 --0.30--0.17 0.01 0.22 0.34 0.48 0.74

3.55 2.78 1.37 0.62 0.24 --0 .11!--0.61 0.27 0.35 0.12 0.01 0.02--0.21 1.09 0.58 0.48 0.74 1.43

0 .5

E l l

0 .4 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 . 3

0 . 2

0 . 1

7 \

/ i - ........

0 0 . 5 1 1 . 5 2T i m e

Fig. 4 Time (s) variation of zu (m)2 . 5

1.75 ] 1.95--0.27 --1.14--0.27 0.33

1.35 4.46-- 1.0 7 --20.8

0.36 20.64.19 77.6

, [8 0 [ . . . . . . . . .6 O4 02 0

0- 2 0- 4 0- 6 0- 8 0

F i g . 5

i fO 0 . 5 1 1 . 5 2 2. 5

T i m ek i n g p i n a n g l e ~ - - c a m b e r a n g le

Time (s) variation of the kingpin and cambe rangles (Deg)


7/7

Kinematic Analysis of the Multi-Link Five-Point Suspension System in Point Coordinates 1139t i m e l e v el s . F i g u r e 4 p r e s e n t s t h e t im e v a r i a t i o n o ft h e z - c o o r d i n a t e o f t h e w h e e l c e n tr e . F i g u r e 5s h o w s t h e v a r i a t i o n o f t h e k i n g p i n a n g l e * a n dt h e c a m b e r a n g l e * w i t h ti m e . It s h o u l d b e n o t e dt h a t t h e r e s u l t s o f t h e s i m u l a t i o n a r e t e st e d a n dc o m p a r e d w i t h t h e s i m u l a t i o n r e s u l t s b a s e d o n t h ea b s o l u t e c o o r d i n a te s . T h i s c o m p a r i s o n p r o v e s th ev a l i d i ty o f th e p r o p o s e d m e t h o d .

3 . C o n c l u s i o n sI n t h i s p a p e r , a n e f f i c i e n t a l g o r i t h m f o r t h e

n u m e r i c a l k i n e m a t i c a n a l y s is o f th e s p a t i a l m o t o r -v e h ic l e m u l t i - l i n k f i v e - p o i n t s u s p e n s io n m e c -h a n i s m i s p r e s e n t e d . T h e k i n e m a t i c a n a l y s i s i sc a r r i e d o u t i n t er m s o f th e r e c t a n g u l a r C a r t e s i a nc o o r d i n a t e s o f s o m e d e f i n e d p o i n t s i n t h e l i n k sa n d a t th e k i n e m a t i c j o i n t s . T h e s u g g e s t e d a l g o -r i t h m e l i m i n a t e s th e n e e d t o w r i t e r e d u n d a n tc o n s t r a i n t s a n d a l l o w s s o l v i n g a r e d u c e d s y s t e mo f e q u a t i o n s . T h e a l g o r i t h m c a n b e u s e d t o s o l v et h e i n i t i a l p o s i t i o n a s w e l l a s t h e f i n i t e d i s p l a c e -m e n t p r o b l e m s . T h e r e s u lt s o f t h e a n a l y s i s i n -d i c a t e t h e s i m p l i c i t y a n d g e n e r a l i t y o f t h e p r o -p o s e d a l g o r i t h m .

R e f e r e n c e sA d l e r , U .. A u t o m o t i v e H a n d b o o k , B o s ch ,

F D I - V e r l a g , A p p r o v e d E d i t io n U n d e r L i ce n ceS A E , I S P N 0 8 9 2 8 3 - 5 1 8 - 6 .

A k h r a s , R . a n d A n g e l e s , J ., 1 99 0, " U n c o n s -t r a in e d N o n l i n e a r L e a s t - S q u a r e O p t i m i z a t i o n o fP l a n a r L i n k a g e s fo r R i g i d - B o d y G u i d a n c e , " Mec-hanism and Machine Theory, V o l . 2 5 , p p . 9 7 - -118.

A t t i a , H . A . , 1 9 9 3 , A c o m p u t e r - O r i e n t e d D y n a -m i c a l F o r m u l a t i o n w i th A p p l i c a t i o n s t o M u l t i -b o d y S y s t em s , P h . D . D i s s e r t a ti o n , D e p a r t m e n t o fE n g i n e e r i n g M a t h e m a t i c s a n d P h y s ic s , F a c u l t y o f

E n g i n e e r i n g , C a i r o U n i v e r s i t y .D e n a v i t , J . a n d H a r t e n b e r g , R . S . , 1 95 5, " A

K i n e m a t i c N o t a t i o n f o r L o w e r - P a i r M e c h an i sm sB a s e d o n M a t r i c e s , " ASME Journal of AppliedMechanics, p p . 2 1 5 - - 2 2 1 .G a r c i a d e J a l o n , J . , S e r n a , M . A . a n d A v i l e s ,R . . 1 98 1, " C o m p u t e r M e t h o d f o r K i n e m a t i c A n -a l y si s o f L o w e r - P a i r M e c h a n i s m s - I V e l o ci ti e sa n d A c c e l e r a t i o n s , " Mechanism and MachineTheory, V o l . 1 6, p p . 5 4 3 ~ 5 5 6 .

G a r c i a d e J a l o n , J ., S e r n a , M . A . a n d A v i l e s ,R . , 1 9 8 1 , " C o m p u t e r M e t h o d f o r K i n e m a t i cA n a l y s is o f L o w e r - P a i r M e c h a n i s m s - l l P o s i ti o nP r o b l e m s , " Mechanism and Machine Theory,V o l . 1 6, p p . 5 5 7 - - 5 6 6 .

G a r c i a d e J a l o n , J . e t a l . , 1 9 8 2 , " S i m p l e N u -m e r i c a l M e t h o d f or th e K i n e m a t i c A n a l y s i s o fS p a t i a l M e c h a n i s m s , " ASME Journal on Mec-hanical Design, V o l . 1 0 4 , p p . 7 8 - - 8 2 .

M o l i a n , S ., 1 9 68 , " S o l u t i o n o f K i n e m a t i c sE q u a t i o n s b y N e w t o n ' s M e t h o d , " Journal ofMechanical Engineering Science, V ol . 10 . No . 4 ,p p . 3 6 0 ~ 3 62 .

N i k r a v e s h , P . E . , 1 98 8, Computer-Aided An-alysis of Mechanical Systems, P r e n t i c e - H a l l .

P a u l , B . a n d K r a j c i n o v i c , D ., 1 97 0, " C o m p u t e rA n a l y s i s o f M a c h i n e s w i t h P l a n a r M o t i o n - l .K i n e m a t i c s , 2 . D y n a m i c s , " ASME Journal ofApplied Mechanics, V o l . 3 7 , p p . 6 9 7 - - 7 1 2 .

V i l a l l o n g a , G . . U n d a , J . a n d G a r c i a d e J a l o n ,J ., 1 98 4, " N u m e r i c a l K i n e m a t i c A n a l y s i s o fT h r e e - D i m e n s i o n a l M e c h a n i s m s U s i n g a " N a t u -r a l " S y s t e m o f L a g r a n g i a n C o o r d i n a t e s , " ASMEPaper N o . 8 4 - D E T - 1 9 9 .

W e h a g e , R . A . a n d H a u g , E . J ., 1 9 8 2, " 'G e n e r a -l iz e d C o o r d i n a t e P a r t i ti o n i n g o f D i m e n s i o n R e -d u c t i o n , in A n a l y s i s o f C o n s t r a i n e d D y n a m i cS y s t e m s , " A SM E Journal of Applied MechanicalDesign, V o l . 1 04 , p p . 2 4 7 - - 2 5 5 .

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