Multi-link suspension analysis: An elastokinematic ... - ArchiMeDes

Introduction 

29 th ISATA - Florence, Italy, 3 rd - 6 th June 1996 - 96VR031 

Multi-link suspension analysis: 

An elastokinematic model 

Danilo Cambiaghi, Marco Gadola, David Vetturi, Luca Manzo 

Mechanical Engineering Dept. - University of Brescia 

Via Branze 38, 25123 Brescia - Italy 

tel. ++39.30.3715426 fax ++39.30.3702448 

e-mail: gadola@bsing.ing.unibs.it 

96VR031 

Multi-link suspensions are the latest development in road car chassis design. They are conceived 

to provide high stiffness in the lateral direction and a reasonable compliance in the fore/aft 

(longitudinal) direction. This is due to the need of ensuring a good ride quality without affecting 

handling and safety. 

Ellis defines compliance as “The change in suspension characteristics due to external forces” ( 1 ). 

The elastokinematic characteristics are usually tuned up to minimise camber and track changes and 

correctly tune toe variations. At the rear axle of a rear-drive car the central axis of elasticity can be 

positioned properly to result in a toe-in tendency under throttle-off condition, thus improving 

stability. The front axle of a front-drive car can be designed with a "virtual steering axis" to 

optimise the steering geometry. The paper presents the application of a simple algorithm for the 

force-displacement analysis of a five-rod suspension which allows for rubber bushing compliance. 

Multi-link geometry: advantages 

The tyre/road interface is a source of vibration which can heavily affect the ride comfort in a 

passenger car. Moreover, the radial-type tire is structurally stiffer than its cross-ply predecessor 

hence more prone to generate noise and harshness. As Bastow states ( 2 ) “it has long been realised 

that road noise finds its way into the car through the suspension system and that rubber in each of 

the paths from wheel to car structure could provide a barrier (...)”. Providing some compliance with 

the use of rubber bushings (the so-called silent-blocks) at the chassis end of the suspension arms 

can be an effective way to improve the comfort level but care has to be taken to avoid a flexibility 

excess. 

Fig. 1 - five-link scheme 

Generally speaking a multi-link suspension is usually a five-rod mechanism which can be seen as 

a double-wishbone system whose arms are split at the outer apices (Fig. 1). The geometry and the


rubber bush elasticity can be properly tuned to minimise camber and track changes and obtain the 

desired toe variations. At the rear axle of a RWD car the central axis of elasticity can be positioned 

properly to result in a toe-in tendency under throttle-off condition, thus improving stability and a 

slight toe-out tendency under power to improve the handling characterictics. The front axle can be 

designed with a "virtual steering axis" to optimise the steering geometry and minimise castor 

variation under traction in a front-drive car. 

The multi-link system offers total flexibility of wheel/body movement control, lift, squat and dive 

compensation, and mounting point placement with low unsprung mass figures. This can be 

achieved provided that an advanced computer model is available to the designer, since a multi-link 

can not be studied with the use of traditional drafting methods. A closed-form solution model is not 

very handy either since it requires the solution of a 20 th degree polynomial equation in a single 

unknown. 

Quite a few papers have been written on the subject (see ( 3 ), ( 4 ), ( 5 )). This paper deals with the 

application of a simple numerical method for the force-displacement analysis of the five-rod 

suspension which allows for rubber bushing compliance. 

Previous papers have presented the computation of a double-wishbone, ball joint-mounted racing 

car suspension ( 6 ), ( 7 ), ( 8 ). In that case suspension kinematics and component compliance were 

analysed in a three-step process : geometry solution, force equilibrium, and compliance analysis, 

where “compliance” means that not only the spring but also the suspension arms were considered 

elastic i.e compliant. This sort of process can be used to study a five-link suspension as well ; in this 

case “compliance” stays for rubber bushing as well as spring elastic deflection because the 

suspension arm deflection is negligible. 

Geometry solution 

Inputs to the first step are the suspension node co-ordinate array and a spring deflection value s, 

either in bump or in droop. The node co-ordinate array fully describes the static suspension 

geometry i.e. the so-called “design configuration”. The chassis is held still while the spring is 

deflected by the given amount s and the wheel is displaced accordingly provided the movement is 

within the mechanical limits. The procedure leads to determination of all moving nodes new coordinates 

and outputs the altered node array as a function of s. In the double-wishbone case basic, 

closed-form stereometry formulas were employed (Fig. 2). 

Wheel 

displacement 's' 

Node co-ordinates 

STEREOMETRY 

new configuration= f(s): 

Displaced node co-ordinates 

Fig. 2 - geometry solution, double wishbone case


A multi-link system can be seen as an in-parallel five-rod mechanism which can be studied with 

the same methods which are commonly applied to platform robots ( 9 ), ( 10 ). The chassis can be seen 

as the ground (where the global co-ordinate system is placed) while the wheel-upright assembly can 

be seen as the moving part of the system (where a local co-ordinate system is placed). In the 

simplest case the upright is connected to the chassis by five fixed-length rods plus one variablelength 

rod (the spring-damper unit). 

If well-designed this sort of mechanism can give a continuous solution in the working range hence 

the analysis can be based on a simple application of Taylor’s formula around the known static 

position. A rigid body in a 3-D space has 6 d.o.f. therefore the local reference system can be defined 

with the use of 6 parameters. The typical wheel angles α (castor), β (toe) and γ (camber) can be 

used to define the axis rotations while the wheel center co-ordinates X, Y, Z can define the relative 

origin position. 

T 

Now if the 6x1 parameter vector is S 0 = [ α β γ X Y Z] = [ s1 K s6] 

, a 4x4 

rototranslation matrix M0 = M(S0) can be written in the initial “design configuration” to compute 

the global co-ordinates (x0, y0, z0) starting from the local co-ordinates (x’0, y’0, z’0) for any point: 

⎡x 

0 ⎤ ⎡x' 

0 ⎤ 

⎡ Cβ⋅Cα −Sβ −Cβ⋅Sα X⎤ 

⎢ 

y 

⎥ ⎢ 

0 y' 

⎥ 

⎢ 

⎢ ⎥ 

0 

Cγ ⋅Sβ⋅Cα −Sγ ⋅Sα Cγ ⋅Cβ −Cγ ⋅Sβ⋅Sα −Cα⋅Sγ Y 

⎥ 

= M ⋅⎢ 

⎥ where M = ⎢ 

⎥ 

⎢z 

0 ⎥ ⎢z' 

0 ⎥ 

⎢Sγ 

⋅Sβ⋅ Cα + Cγ ⋅Sα Sγ ⋅Cβ −Sγ ⋅Sβ⋅ Sα + Cα⋅Cγ Z⎥ 

⎢ ⎥ ⎢ ⎥ 

⎢ 

⎥ 

⎣ 1 ⎦ ⎣ 1 ⎦ 

⎣ 0 0 0 1⎦ 

“Sα” stays for “sinα”, “Cα” stays for “cosα”, and the last line is added to define both translation 

and rotation with the same matrix. Also the suspension joints on the chassis (global co-ordinates) 

and on the upright (local co-ordinates) can be represented in matrix notation: 

A i 

⎡ X i ⎤ 

⎢ 

Y 

⎥ 

i 

= ⎢ ⎥ 

⎢ Z ⎥ 

⎢ 

⎣ 1 

i 

⎥ 

⎦i 

= 1... 6 

(chassis joints, global co-ords) B i 

⎡x' 

i ⎤ 

⎢ 

y' 

⎥ 

i 

= ⎢ ⎥ 

⎢z' 

⎥ 

⎢ 

⎣ 1 

i 

⎥ 

⎦i 

= 1... 6 

(upright joints, local co-ords) 

and, applying the rototranslation matrix: Ci = M⋅ Bi 

for i=1...6 (upright joints, global co-ords). 

Now the link length vector can be written as follows: Q [ ] 

T 

= d1 d2 d3 d4 d5 d6 

where the 

distances between joints can be computed as a 2 nd order norm : 

di = M⋅Bi − Ai = Ci −A 

i for i=1...6 

In the five-link suspension case d1, d2, d3, d4 and d5 are constant values while the spring length d6 

is a variable. In the design configuration S0, d6 is known hence Q0 = F(S0) is a known function. 

When a small deflection is applied to d6, Taylor's series can be used to write : 

Q≅ ( S ) + ⋅( S−S ) 

S F 

∂ F 

∂ F 

0 0 or ( Q−Q0) ≅ ⋅( S−S0) ∂ 

∂ S 

(Q-Q0) represents the spring length variation, (S-S0) is the unknown term and represents the 

motion between the design configuration and the new position, and the 6x6 matrix ∂F/∂S is the socalled 

Jacobian matrix J, whose terms look like 

∂ d d ( s s ds s ) d ( s s s 

i i 1... j + j... 6 − i 1... j... 

6) 

Jij 

= ≅ 

∂ s 

ds 

j 

j


J can be computed by writing the analytical equations or approximated by giving very small 

increments to each parameter sj of S [ ] 

T 

= α β γ X Y Z then computing the length 

variation di for each link. This method is usually quicker. 

Now an iterative process can be started. A linear system 

∂ F 

( Q− QK) 

= ⋅∆ 

S K with QK=F(SK) 

∂ S 

must be solved at each step until Q− F( SK) 

≤ε 

. It should be noted that small wheel displacement 

steps make a quick and precise computation easier. 

Force equilibrium 

Rigid body 

load set 'F' 


EQUILIBRIUM EQUATIONS 

Restraint reactions 

Fig. 5 - force equilibrium 

The suspension is considered locked in its actual position e.g. the design configuration. As stated 

before the wheel/upright assembly has 6 d.o.f. in a 3-D space. In the multi-link case it is restrained 

by six pin-ended links : each link is equivalent to one restraint, thus reducing the structure from 

redundant (indeterminate) to isostatic (determinate). 


Wheel load set 'F' 

iteration 

FORCE EQUILIBRIUM 

Axial loads 

Spring and bushing 

properties 

Fig. 6 - compliance module 

Spring and rubber bushing 

deflections 

N 

GEOMETRY SOLUTION 

New node co-ordinates 

convergence? 

Y 

Main geometry parameters


Orientation hence direction cosines are known for each link. Any load set can be applied at the 

wheel/road surface interface (accelerating, braking, turning, crossing a bump) and six equilibrium 

equations in six unknowns i.e. axial forces are obtained. This set of simultaneous equations is 

solved with the Gauss-Jordan method (Fig. 5). 

Compliance analysis 

Now both the kinematic behaviour and the force distribution are known. But when it comes to 

determine spring and rubber bushing deflection when the wheel is laden, a deterministic solution is 

impossible due to non-linearities like the suspension ratio. It is possible to overcome the nonlinearity 

by linking the previously seen modules through an iterative process. When a load case is 

given (braking, turning etc. as before) the load on the coil spring as well as on each silent-block can 

be determined by running the force equilibrium module. The deformed geometry can be computed 

by deflecting each elastic component. If coil spring deflection will probably be the most important 

contribution to wheel displacement also elastic joint effects (compliance) will upset wheel position 

by a certain amount. This leads to a new geometry hence a new load distribution through the 

system. The process must then be restarted to compute the consequent load distribution and altered 

geometry until the iteration converges (Fig. 6). 

Conclusion 

A method for solving the multi-link suspension problem is presented. It is fairly straightforward 

and enables the designer to fully understand the suspension behaviour under road loads with the use 

of a simple PC program. A comprehensive, windows-based software and a case study will be 

presented in the near future. 

References 

1 

Ellis J.R. : “Road Vehicle Dynamics”. published by John R. Ellis, Akron (Ohio, USA), 1989. 

2 

Bastow D., Howard G. : “Car Suspension and Handling - Third Edition”. Pentech Press - SAE, 

1993 

3 

Hiller M., Woernle C. : “Elasto-Kinematical analysis of a five-point wheel suspension”. Revue de 

la SIA, Ingenieurs de l’Automobile, Paris 1985, s. 77-80. 

4 Knapczyk J., Dzierzek S. : “Displacement and force analysis of five-rod suspension considering 

silent-block compliances”. Trans. ASME, Jnl. of Mech. Design. 

5 Knapczyk J., Dzierzek S. : “Displacement analysis of five-rod wheel-guiding mechanism by using 

vector method”. Proc. of the 25 th ISATA, Florence 1992, v. Mechatronics, s. 687-695. 

6 

Gadola M., Cambiaghi D. : “Racecar suspension kinematics design : a report on the tool 

developed at the University of Brescia”. Internal paper, University of Brescia, Italy, January 1994. 

7 

Gadola M., Cambiaghi D. : “MMGB : a computer-based approach to racing car suspension 

design”. ATA - Ingegneria Automotoristica, 6/7 - June/july 1994. 

8 

Gadola M., Cambiaghi D. : “Computer-aided racing car suspension design and development at the 

University of Brescia, Italy”. SAE paper n. 942507. Proc. of the Motorsports Engineering 

Conference and Exposition, Dearborn (USA), December 1994. 

9 

Legnani G. : “Appunti di meccanica dei robot”. Edizioni Città Studi, Milano 1992. 

10 

Chonggao Liang, Lin Han, Fuan Wen : “Forward displacement of the 5-6 Stewart platforms”.

Multi-link suspension analysis: An elastokinematic ... - ArchiMeDes

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