New Dynamic Analysis of Double Wishbone Front Suspension … · 2019. 1. 5. · Dynamic Analysis of Double Wishbone Front Suspension Systems on Sport Motorcycles 3 Table 1: GSX-R1000

Nonlinear Dynamics manuscript No.(will be inserted by the editor)

Dynamic Analysis of Double Wishbone Front Suspension Systems onSport Motorcycles

Ciro Moreno Ramırez · M. Tomas-Rodrıguez · Simos A. Evangelou

Received: date / Accepted: date

Abstract In this paper, two alternative front suspension sys-tems and their influence on motorcycle nonlinear dynamicsare investigated. Based on an existing high-fidelity motor-cycle mathematical model, the front end is modified to ac-commodate both Girder and Hossack suspension systems.Both of them have in common a double wishbone designthat varies the front end geometry on certain manoeuvringsand, consequently, the machine’s behaviour. The kinemat-ics of the two systems and their impact on the motorcycleperformance is analysed and compared to the well knowntelescopic fork suspension system. Stability study for bothsystems is carried out by combination of nonlinear dynami-cal simulation and root-loci analysis methods.

Keywords Motorcycle · Vehicle dynamics · Mathematicalmodelling · Stability analysis · Suspension systems

PACS PACS code1 · PACS code2 · more

Mathematics Subject Classification (2000) MSC code1 ·MSC code2 · more

Ciro Moreno RamırezDepartamento de Ingenierıa Industrial y Automocion - E.P.S.Universidad Antonio de NebrijaMadrid - [email protected]

M. Tomas-RodrıguezDepartment of Mechanical Engineering and AeronauticsCity, University of LondonUnited [email protected]

Simos A. EvangelouElectrical and Electronic Engineering DepartmentImperial College LondonUnited [email protected]

1 Introduction

The motorcycle’s front end links the front wheel to the mo-torcycle’s chassis and has two main functions: the front wheelsuspension and the vehicle steering. Up to this date, severalsuspension systems have been developed in order to achievethe best possible front end behaviour, being the telescopicfork the most common one.

(a) Fork (b) Girder (c) Hossack

Fig. 1: 3D models of Fork (a), Girder (b) and Hossack (c)suspension systems. Centres of masses and steering axis areshown with red dots and lines respectively.

A telescopic fork 3D model can be observed in Fig. 1a. Itconsists of a couple of outer tubes which contain the suspen-sion components (coil springs and dampers) internally andtwo inner tubes which slide into the outer ones allowing thesuspension travel. The outer tubes are attached to the framethrough two triple trees which connect the front end to themain frame through the steering bearings and allow the frontwheel to turn about the steering axis. This system keeps thefront wheel’s displacement in a straight line parallel to thesteering axis. However, there exist alternative suspension de-signs that allow different trajectories of the front wheel with

2 Ciro Moreno Ramırez et al.

the suspension travel. An extensive review of most of the ex-isting alternatives suspension systems can be found in [10],in which their advantages and drawbacks in relation to eachother are explained. This reference classifies front suspen-sion systems into two main groups depending on how theyare connected to the main frame: head stock mounted forkand alternatives systems. In the present research, the differ-ent suspension systems are classified in a different mannerconsidering kinematics criteria. Two different groups can bedefined depending on the steering axis’ location in the sus-pension assembly. The first group comprises those systemswhich present the steering axis located between the chassisand the suspension elements (pivoted arms or wishbones inmost of the cases). In the second group, this axis is placedbetween the suspension elements and the front wheel. Inboth cases, the system can be designed in order to provide adesired front wheel trajectory, however whilst the first groupkeeps the steering angle fixed in the chassis reference frame,the second one modifies it with the travel of the suspension.

A deep study of four alternative suspension systems be-longing to the second group of this later classification (twoHub-centre systems and two Dual-link systems) can be foundin [13]. The authors investigate their effects on motorcy-cle dynamics taking advantage of advanced multi-body soft-ware (namely Simpack) and performing several running sim-ulations close to real driving conditions. For the four differ-ent suspension systems, various aspects of the motorcycleresponse, such as the forces transmitted to the suspensionjoints, the resulting bending torque on the frame’s lateralaxis, the pitch angle achieved by the motorcycle or the max-imum roll velocity are studied as a means to understand theadvantages of each system under different design consider-ations.

In the present research, our main goal is to study the ef-fects of alternative front suspension systems of both groupsdefined above, not only in terms of motorcycle response butalso in terms of motorcycle dynamics and stability. That is,how the different modifications introduced by the new sus-pension systems, such as different mechanical designs orchanges in the front end geometry with new wheel trajec-tories, affect the suspension response and the motorcycle’soscillating normal modes.

Two double wishbone systems are considered in this re-search as representative of these two groups: the Girder sus-pension and the Hossack system respectively. In terms ofkinematics, these two types of designs cover most of the ex-isting double wishbone suspension systems. 3D models ofthese suspension systems are shown in Figure 1. The Girdersuspension system (Fig. 1b) consists of a pair of long up-rights where the front wheel is attached to. These uprightsare linked to the triple trees by an upper and a lower wish-bones which perform the suspension motion. Both triple treesrotate about the steering axis which is fixed to the motor-

cycle chassis. A spring-damper unit is usually attached be-tween the lower wishbone and the upper triple tree providingthe shock absorption function. On the other hand, the Hos-sack suspension system (Fig. 1c) consists of a double wish-bone structure directly attached to the chassis. The two wish-bones rotate both about axes perpendicular to the symmetryplane of the motorcycle, providing the suspension motion.An upright is linked to the front tips of the wishbones bytwo ball joints, which allow it to turn left and right as well asto move up and down. Therefore, the steering axis becomesdefined by the imaginary line passing through the geometriccentre of the ball joints. The control over the steering angleis applied by the rider to the handlebar which is connected tothe upright through the steering linkage. This is a system oftwo levers, connected by an axis, which can be compressedor elongated in order to reach the length between the han-dlebar and the upright. The front wheel is attached to thisupright and the suspension reaction is provided by a spring-damper unit attached between the lower wishbone and themotorcycle chassis.

The framework of this research is the mathematical mod-elling and numerical simulation. We took advantage of ahigh fidelity motorcycle model developed by [23] which hasbeen modified with the addition of the alternative suspen-sion system under study. The motorcycle model propertiesand the simulation tools used during this research are ex-plained in section 2. In order to obtain realistic descriptionsof the alternative front suspension systems in terms of dy-namical properties, we have designed and modelled thesesystems taking advantage of CAE tools. As a first step, thekinematics of the two systems are studied in comparisonwith the telescopic fork. Then, different kinematic config-urations are synthesized taking into consideration differentaspects of the front end geometry. This part of the researchis explained in section 3. The second step consists in 3Ddesign and compliance analysis of the suspension systems.This process allows to obtain realistic values for the dynami-cal properties (such as masses and moments of inertia) of thesuspension systems’ parts. Once the dynamical propertiesof all the suspension systems’ parts are obtained, the mo-torcycle model was modified to include the new suspensionsystems. This dynamical modelling process is explained insection 4. Several simulations were carried out with the newmotorcycle models in order to test the suspension responseunder different conditions (section 5). Full stability analyseswere performed to identify and reduce the eventual stabil-ity risks that the new suspension systems may introduce inthe motorcycle dynamics (section 6). Finally, conclusions ofthis research are presented in section 7.

Dynamic Analysis of Double Wishbone Front Suspension Systems on Sport Motorcycles 3

Table 1: GSX-R1000 geometrical model main points.

Point Description

P1 Aerodynamic reference point.P2 Twist body’s joint with rear frame.P3 Steer body’s centre of mass.P4 Front suspension body’s joint with steer body.P5 Front suspension body’s centre of mass.P6 Rear wheel’s centre of mass and attachment point.P7 Front wheel’s centre of mass and attachment point.P8 Main frame’s centre of mass.P9 Rider’s Upper Body attachment point on rear frameP10 Centre of mass of the rider’s upper body.P11 Swinging arm’s attachment point on main frame.P14 Swinging arm’s centre of mass.

2 Motorcycle model and simulation tools

The different mathematical models developed for this re-search are based on the model presented in [23]. This math-ematical model was built during several years of researchunderpinned by wide literature and experimental data. Thismodel has been extensively used in the past in several con-tributions such as [18], [7], [21], [6], [5] and [22]. Further-more, it has been widely tested and adopted by the industry.For instance, BikeSim software [14] is a motorcycle dynam-ics simulator which is based on this model and it is used bya large number of manufacturers to obtain high fidelity pre-diction on the dynamics of their machines.

Fig. 2: GSX-R1000 geometrical description. Blue circleswith diameter proportional to body mass are plotted aroundthe centre of mass of each body.

The model was developed using real dynamical parame-ters of an existing motorcycle. The Suzuki GSX-R1000 K1superbike, with 170 kg of mass, powered by an in-line fourcylinder and four stroke engine with 988 cc able to deliver160 hp, this machine is a good representative of contempo-rary commercial high performance motorcycles.

The motorcycle model consists of seven bodies: rear wheel,swinging arm, main frame (comprising rider’s lower body,engine and chassis), rider’s upper-body, steering frame, tele-scopic fork suspension and front wheel. It involves 13 de-grees of freedom: three rotational and three translational for

the main frame, two rotational for the wheels spin, one ro-tational for the swinging arm, one rotational for the rider’supper body, one rotational for the frame flexibility, one ro-tational for the steering body and one translational for thefront suspension fork. Figure 2 represents the main geomet-ric points and axes in the motorcycle’s geometry. The cen-tre of mass of each of the seven constituent bodies is rep-resented as a blue circle with a diameter proportional to itsmass. Table 1 contains the indexes of these points. For mod-elling purposes, a parent-child structure is used, as shown inFig. 3.

The tires are treated as wide, flexible in compression andthe migration of both contact points as the machine rolls,pitches and steers is tracked dynamically. The tyre’s forcesand moments are generated from the tyre’s camber anglerelative to the road, the normal load and the combined slipusing Magic Formulae models [16] and [15]. This model isapplicable to motorcycle tires operating at roll angles of upto 60◦.

The aerodynamic drag/lift forces and pitching momentare modelled as forces/moments applied to the aerodynamiccentre and they are proportional to the square of the mo-torcycle’s forward speed. In order to maintain steady-stateoperating conditions, the model contains a number of con-trol systems, which mimic the rider’s control action. Thesesystems control the throttle, the braking and braking distri-bution between the front and rear wheels, and the vehicle’ssteering.

Fig. 3: Parental structure of the GSX-R1000 model.

The forward speed is maintained by a driving torque ap-plied to the rear wheel and reacting on the main frame. Thistorque is derived from a proportional-integral (PI) controlleron the speed error with fixed gains.

For some manoeuvres, the motorcycle is not self-stable;in order to stabilise the machine in such situations a roll an-gle feedback controller is implemented. This allows to ob-tain different steady turning equilibrium states through sim-ple simulations, which will not be stable without the roll an-gle controller. The controller developed was a proportional-integral-derivative (PID) feedback of motorcycle lean angleerror to steering torque. The lean angle target is set by an


initial value and a constant change rate. Thus, the target leanangle is a ramp function of time which can be easily modi-fied. The PID gains are defined as speed adaptive in order toachieve an effective stabilisation of the motorcycle for thedifficult cases involving very low or very high speeds. Fi-nally, the steering control torque is applied to the steer bodyreacting on the rider’s upper body.

Some modification to the baseline model have been ap-plied for this research regarding the acceleration/braking sys-tems and road inputs. In order to study the response of thenew suspensions and their anti-dive properties, a brakingsystem capable of delivering a constant deceleration was im-plemented. In the new system, the braking torque is derivedfrom the PI controller on the speed error. On the other hand,the road input into the motorcycle tyres is also redefined.With the new approach, the longitudinal contact point mi-gration produced by a step bump is included in the dynamicdescription and its effects on the suspension response areconsidered.

The motorcycle model is implemented taking advantageof the VehicleSim multi-body software from Mechanical Sim-ulation Corporation [14]. This suite consists of two sepa-rated tools: VS Lisp and VS Browser. VS Lisp is the toolused to generate the equations of motions from a multi-bodydescription of any dynamical system. Making use of its owncomputer language (based on LISP) it is designed to auto-matically generate computationally efficient simulation pro-grams for those multi-body systems. It can be configured toreturn either the corresponding non-linear equations of mo-tion or the linearised equations of motion. Both non-linearand linearised equations of motion are symbolically describedas functions of all the parameters defining the model dynam-ics, such as suspensions or aerodynamics coefficients.

With the non-linear equations of motion, a solver can bebuilt with the same architecture and behaviour as those exist-ing in commercial packages such as BikeSim and fully com-patible with the VS Browser. VS Browser is the front end ofall the VehicleSim products. It provides a graphical contextwith a standard graphical user interface from which the non-linear simulations can be run and the different databases canbe managed. This includes the solvers created with VS Lisp,the external inputs and events and the data post processingand visualization.

On the other hand, the linearised equations of motion arereturned in a Matlab file with the state space description ofthe systems. The state space matrices obtained (A, B, C andD) depend on both the system parameters and the state vari-ables values. This feature become useful in the stability anal-ysis of complex non linear systems, which can be linearisedabout operating points corresponding to quasi-equilibriumstates. The frequency and damping associated to the sys-tem’s normal modes are found through the eigenvalues ofmatrix A. The previous version of this software was called

Autosim and was already used in the past in motorcyclesmulti-body modelling (see [19] and [4]).

In this research, following the approach of previous workssuch as [6] or [9], the system’s eigenvalues are plot in thecomplex plane for different values of the motorcycle speedand roll angle. In this way a general understanding on thestability properties of the different motorcycle models canbe obtained. To do so, the non-linear equations of motionsreturned by VS Lisp are integrated under a quasi-equilibriumvariation of the forward speed to obtain time histories ofthe state variables, for either straight running conditions orsteady turns. Then, the state space matrix A can be fed withthose values of the state variables for each time step, ob-taining an accurate linear description of the motorcycle sys-tem for different forward speeds and roll angles. Speed androll angle feedback controllers are used to reach the equi-librium states during the non-linear simulation. However, inthe model’s state space description these feedback controlsare disabled in order to study the open-loop system stability.In section 6 this technique is used to study the variation inthe root locus of the motorcycle model introduced by dou-ble wishbone suspension systems. That is, the effects on thestability of the motorcycle and its normal modes.

3 Front end kinematics

Fig. 4: Main motorcycle’s handling geometric parameters.The wheelbase (wb) is plotted in brown, the trail (t) is inmagenta, the normal trail (tn) is in green, the head angle (ε)is in black and the fork offset (o f s) is shown in blue.

Motorcycle handling properties are greatly influencedby some geometric parameters which are defined by the frontend design. Figure 4 presents the four most relevant geomet-ric parameters for motorcycle handling. These are the trail(t), the normal trail (tn), the head angle (ε) and the wheel-base (wb). The wheelbase is the distance between the frontand rear wheels contact points. The head angle is the anglebetween the steering axis and the vertical. The trail is thedistance between the front wheel contact point and the point


where the steering axis intersects with the ground. Finally,the normal trail is the projection of the trail distance into aplane perpendicular to the steering axis. This is the lever armof the front tyre forces appearing on its contact point, whichresult in a torque about the steering axis. The normal trail(tn) and the head angle (ε) are related to each other by thefollowing expression:

tn = r f w · sin(ε)−o f s (1)

where r f w is the tyre’s radius and ofs is the front wheel’sspindle offset from the steering axis. The wheelbase also de-pends on the rear frame construction including the swingingarm.

(a) Girder supension system (b) Hossack suspension system

Fig. 5: Design parameters on the four-bar linkage suspen-sion systems. a) Girder suspension system. b) Hossack sus-pension system.

In the case of a conventional telescopic fork, the steeringaxle is rigidly inserted into the chassis whilst the offset is aconstant value. Therefore, when the fork is compressed thewheelbase and the head angle decrease and, thus, the trailand the normal trail.

The vertical suspension travel (v.s.t.) is defined as thevertical travel of the front wheel centre when the suspensionsystem is compressed (v.s.t. > 0) or extended (v.s.t. < 0)considering the chassis fixed in the inertial frame. This def-inition is valid for all the different suspension systems andprovides a general magnitude that can be used to comparevarious behaviours.

Both Girder and Hossack systems consist in a four-barlinkage. The difference between them is the edge of the quadri-lateral to be considered as steering axis. Figure 5 shows thedesign parameters of the four-bar linkage for these two sys-tems: the lengths of the upper (l1) and lower (l2) wishbones,the distances between the attachment points of the wish-bones (h1 on the chassis side and h2 on the uprights side)and the angle between the upper wishbone and the horizon-tal at the nominal position (α). With these five parametersfull assembly kinematics are defined. The variation of oneof them will affect the overall behaviour of the handling

geometric parameters with the suspension travel. Differentconfigurations of these five parameters can be calculated toobtain different behaviours of the front suspension systems.

Head angle - deg. Normal trail - mm

Fig. 6: Effects of varying the design parameters on the headangle and the normal trail for the Girder suspension system.

The impact on the kinematic behaviour of varying eachdesign parameter value on the handling geometric parame-ters can be mapped. As an example, Fig. 6 shows the effectsof modifying l1, h1 and α parameters on the variation, withthe vertical suspension travel, of the head angle and the nor-mal trail for the Girder suspension system. Only the normaltrail will be taken into consideration as it is the actual leverarm of the front wheel force about the steering axis, whilstthe trail can be obtained as a simple function of the formeras:

t =tn

cos(ε)(2)

As it can be expected, as the steering axis is fixed to thechassis, the head angle variation behave similarly with dif-ferent values of any of the design parameters. However, thenormal trail variation is affected by these parameters values.The Girder suspension system can be designed to perform aprescribed behaviour of the wheelbase and the normal trailwhilst the head angle behaviour cannot be modified.

Similar results are shown in Fig. 7 for the Hossack sys-tem. In this case, a close relation between the head angle


Head angle - deg. Normal trail - mm

Fig. 7: Effects of varying the design parameters on the headangle and the normal trail for the Hossack suspension sys-tem.

and the normal trail behaviours can be observed due to thevariable steering axis and the constant offset. The head angleand the normal trail keep their nominal response for differ-ent values of α but are significantly modified when otherdesign parameters are varied. It is possible to obtain a de-sired head angle variation given certain vertical suspensiontravel with the Hossack suspension system, whilst the trailand the normal trail are closely related to it.

Table 2: Design parameters values obtained for the three dif-ferent configurations of the Girder suspension systems. l1,l2, h1 and h2 are expressed in mm and α is expressed indegrees.

Girder configurations l1 l2 h1 h2 α

Parallelogram 120 120 180 180 0.0Fork trajectory 107 135 171 172 0.0Constant tn 106 131 192 185 0.0

As it has been said, double wishbone suspension systemscan be design with different kinematic behaviours. In orderto study the effect of the front suspension systems’ kine-matic behaviour on the motorcycle dynamics, three mainkinematic configurations are considered for both the Girder

and Hossack suspension systems. The design parameters re-quired for each of these configurations were obtained bymeans of synthesis of mechanisms methodology. Several op-timization processes were developed for the different sus-pension systems taking advantage of Matlab’s optimizationtoolbox, which was proven to be an adequate framework forthis kind of problems. The following kinematic configura-tions were obtained:

- Parallelogram (prl): The suspension systems are designedwith l1, l2, h1 and h2 as two pairs of parallel sides andwith α = 0, being this the simplest configuration. No op-timization process is needed and the parallelogram’s sideslengths are chosen within the space restictions imposed bythe motorcycle’s design.

- Telescopic fork’s trajectory (tft): The suspension systemsare designed to obtain a front wheel trajectory similar tothat followed by the front wheel with a telescopic forksystem.

- Constant normal trail (cnt): The suspension systems aredesigned to obtain a constant normal trail along the fullsuspension travel.

The values of the design parameters obtained are givenin Table 2 and Table 3, where l1, l2, h1 and h2 are expressedin mm and α is in degrees.

Table 3: Design parameters values obtained for the three dif-ferent configurations of the Hossack suspension systems. l1,l2, h1 and h2 are expressed in mm and α is expressed indegrees.

Hossack configurations l1 l2 h1 h2 α

Parallelogram 170 170 120 120 5.7Fork trajectory 155 183 127 117 5.8Constant tn 173 190 102 123 6.0

Figure 8a shows the handling geometric parameters be-haviour of the Girder (green +) and Hossack (red ×) suspen-sion systems with parallelogram (prl) configuration com-pared to the telescopic fork (solid blue). It can be observedthat with both double wishbone suspension systems the headangle (ε) variation mostly follows that of the telescopic fork.Regarding to the normal trail (tn), its behaviour with theHossack system is similar to that with the telescopic fork,whilst the Girder suspension presents a more pronounceddecrease of this parameter with the vertical suspension travel(v.s.t.). Finally the wheelbase (wb) behaves similarly withboth Girder and Hossack suspension systems. This value isalways reduced relative to the nominal position although ina less relevant manner, compared to the telescopic fork.

Figure 8b shows the trajectories of the front wheel con-tact points along the full suspension travel obtained with par-allelogram configuration of the double wishbone suspension


(a) prl - geometric parameters (b) f.w. trajectory

Fig. 8: Kinematic behaviour of the Girder (green +) andHossack (red ×) suspension systems with prl configurationcompared with the telescopic fork (solid blue). The head an-gle (ε .), the normal trail (tn) and the wheelbase (wb) varia-tion with the vertical suspension travel (v.s.t.) are presentedin a). TThe trajectories of the front wheel’s contact pointalong the full suspension travel are plotted in b).

systems compared to that of the telescopic fork. Both Girderand Hossack systems produce curved trajectories. These tra-jectories are close to each other, have positive slopes be-tween -20 mm and +10 mm of v.s.t. and differ from thenominal trajectory returned by the telescopic fork, which hasconstant negative slope.

(a) tft - geometric parameters (b) f.w. trajectory

Fig. 9: Kinematic behaviour of the Girder (green +) andHossack (red ×) suspension systems with tft configurationcompared with the telescopic fork (solid blue). The head an-gle (ε .), the normal trail (tn) and the wheelbase (wb) varia-tion with the vertical suspension travel (v.s.t.) are presentedin a). The trajectories of the front wheel’s contact pointalong the full suspension travel are plotted in b).

Similar plots are obtained for the Girder and Hossacksuspension systems with telescopic fork’s trajectory config-uration (tft) in Fig. 9. The trajectories reached by both dou-ble wishbone suspension systems are almost identical to thatof the telescopic fork (Fig. 9b). In the case of the Girdersuspension system, the handling geometric parameters be-

have similarly to those with the telescopic fork suspension(Fig. 9a). This is the expected behaviour as the steering axisand the wheel trajectories are equal in both systems. How-ever, for the Hossack suspension system, the steering axisvaries with the suspension travel, which leads to differentbehaviours of the handling geometric parameters. The vari-ation in the head angle and the normal trail with the verti-cal suspension travel are significantly larger for the Hossacksystem than for the telescopic fork suspension. However, thewheelbase is modified in a similar way, as the front wheelfollows the same trajectory in both cases.

(a) cnt - geometric parameters (b) f.w. trajectory

Fig. 10: Kinematic behaviour of the Girder (green +) andHossack (red ×) suspension systems with cnt configura-tion compared with the telescopic fork (solid blue). Thehead angle (ε .), the normal trail (tn) and the wheelbase (wb)variation with the vertical suspension travel (v.s.t.) are pre-sented in a). The trajectories of the front wheel’s contactpoint along the full suspension travel are plotted in b).

Being the normal trail a crucial parameter in the motor-cycle handling, it could be an interesting feature for a sus-pension system to maintain this value constant at any po-sition of the suspension travel. Kinematic behaviours of theGirder and Hossack suspension systems with a constant nor-mal trail configuration (cnt) are represented in Fig. 10a. Al-most constant normal trails are achieved by both Girder andHossack suspension systems, with small deviations from theirnominal values. As expected, in the case of the Girder sus-pension, the head angle maintains its nominal behaviour withthe vertical suspension travel whilst the wheelbase is re-duced. However, the Hossack suspension system with cntconfiguration returns an almost constant head angle and, op-positely to the Girder and telescopic fork systems, the wheel-base is increased in compression and decreased in extension.

Regarding to the front wheel’s contact point in Fig. 10b,it can be observed that with the Girder system, the trajectoryis mostly a straight line at an angle with the vertical whichis greater than that of the telescopic fork’s trajectory. In thecase of the Hossack system, the front wheel follows a curvedtrajectory. The angle with the vertical becomes negative in


this case, reducing its value under compression and increas-ing it under extension. These trajectory angles will directlyaffect the suspension systems’ anti-dive capabilities. Con-trary to the behaviour of the telescopic fork system, bothdouble wishbone suspension systems show a wide range ofpossible kinematic configurations. Either Hossack or Girdersystems could be a good choice depending on the motorcy-cle’s kinematics requirements.

4 Dynamical modelling

In order to study the dynamic properties of Girder and Hos-sack suspension systems, two mathematical models have beenbuilt using VehicleSim. Each of these models geometry hasbeen modified with the design parameters values obtained inthe previous section for the three kinematic configurations:parallelogram (prl), telescopic fork’s trajectory (tft) and con-stant normal trail (cnt). Therefore, three different configura-tions of each of the Girder and Hossack suspension systemsare obtained.

The mathematical models here presented are developedas modifications of the Suzuki GSX-R1000 nominal model,derived in [23], which was built considering the actual phys-ical properties of the original motorcycle’s parts. The masses,the moments of inertia and the centres of masses were di-rectly measured for each part. However, real GSX-R1000motorcycles are not fitted with either Girder or Hossack sus-pensions. Therefore, the physical properties of these partscannot be measured and included in the mathematical model.In order to obtain accurate values of these properties, a 3Dcomputer-aided design (CAD) has been developed for eachsuspension system. The software used for this task was Solid-Works [3], which also allowed to perform the different com-pliance analyses (FEA) through its SolidWorks Simulationtool, needed to determine the designs consistency and relia-bility.

4.1 3D design and compliance analysis

It is important to note that this part of the research is notintended to obtain high performance commercial suspen-sion systems, but to provide a good approximation of themechanical parts of each suspension system under study.Therefore, the masses, the centres of mass, the inertia mo-ments, etc. represent close values to those of a possible realsuspension system implementation.

Both Girder and Hossack suspension systems are de-signed in order to keep the same front end assembly’s massas that of the original telescopic fork of the GSX-R1000model. Each part’s mass tends to be equal to the equiva-lent part in the telescopic fork case. However, due to thestructural differences between the three suspension systems,

(a) (b) (c)

(d) (e) (f)

Fig. 11: Factor of safety computed by FEA for the Girder(top row) and Hossack (bottom row) models under maxi-mum lateral load (a and d), maximum longitudinal load forextended suspension (b and e) and maximum longitudinalload for compressed suspension (c and f).

this is not always possible. For instance, in the case of theHossack suspension, the steering assembly consists only ofa triple tree. This makes the Hossack systems much lighterthan the telescopic fork system. The Girder suspension isonly a few grams lighter than the telescopic fork. Neverthe-less, in both Girder and Hossack cases the mass differenceis added to the chassis body as a mass placed at the samecoordinates as those of the steering body’s centre of mass.

A construction material is associated to the different partsof the suspension systems, so that the dynamic properties ofeach part can be calculated. The material chosen for bothGirder and Hossack suspension systems was 7075 aluminiumalloy, commonly used in automotive industry due to its lightweight and strength. The suspension systems have been de-signed in order to support the maximum loads that appearduring extreme running conditions, such as extreme brakingor high speed and high lean angle manoeuvres. Various fi-nite element analyses were carried out for each suspensionsystem in order to ensure their integrity and reliable perfor-mance under those heavy loads. A factor of safety greaterthan two is found for all the simulations, even though theassemblies masses are smaller than the conventional tele-scopic fork and the loads applied were increased in simula-tion an extra 50% over the theoretical maximum loads calcu-lated. Figure 11 shows the FEA results for maximum lateralload, maximum longitudinal load for extended suspensionand maximum longitudinal load for compressed suspension.Although more detailed studies would be needed, these re-sults suggest that even lighter designs could be obtained forthese kinds of suspension. Despite that is not the objectiveof this research, these results allow to be confident in the


Table 4: Masses for the bodies of the different suspensionsystems models.

mass (kg) STR SUS UWB LWB Total

Fork 9.990 7.250 —— —— 17.240Girder 7.863 7.930 0.666 0.764 17.223Hossack 1.1692 7.1489 0.976 0.924 10.218

dynamic properties computed for the parts of the differentassemblies, being close to those properties of eventual real-istic designs of these suspension systems.

In order to obtain a coherent comparison between thethree different suspension systems, they have been dividedin four subsystems each of them containing different parts.The parts belonging to each subsystem depend on whichsuspension system is considered.

STR: It is the body that allows the steering action. Itcomprises the triple trees and eventually other parts depend-ing on the model under consideration. In the case of the tele-scopic fork it also includes the upper tubes. In the case ofthe Girder, the mass of the upper part of the spring damperunit is also included. In the case of the Hossack system, itincludes the upper lever of the steering linkage.

SUS: It represents the body holding the front wheel. Inthe case of the telescopic fork it comprises the lower tubesof the fork. In the case of the Girder and Hossack suspensionsystems, this body corresponds to the uprights, the lowerpart of the spring-damper unit and, only for the Hossack sys-tem, the lower lever of the steering linkage.

UWB: This part is exclusively defined for the Girder andHossack systems. It only consists of the upper wishbone.

LWB: This part is exclusively defined for the Girder andHossack systems. It only consists of the lower wishbone.

The dynamic properties of each subsystem can be ob-tained from the 3D model in SolidWorks. Table 4 showsthe masses of each part on the different suspension systemscompared to the original telescopic fork parts masses. In thecase of the Hossack system, the remaining mass needed toequal that of the original telescopic fork is 7.022 kg. How-ever, as it has been said, an equivalent mass is added to thechassis body in the same coordinates than those of the steer-ing body’s centre of mass maintaining the overall motorcy-cle mass for both suspension systems.

4.2 Mathematical modelling

The GSX-R1000 model presented in section 2 has been mod-ified to include multi-body representations for both the Girderand the Hossack suspension systems. In both cases, and sim-ilarly to the original nominal model, a massless body is in-cluded (the twist body) to represent the frame’s flexibility.The flexibility is defined as a rotational degree of freedom

between the motorcycle chassis (rear frame) and the frontsuspension (front frame) about the twist axis. This is an axisperpendicular to the steering one which lies within the mo-torcycle’s symmetry plane and passes through the attach-ment point of the twist body. This point is defined in bothsuspension systems as the middle point between the upperand the lower wishbones joint coordinates (points q1 and q2in Fig. 5). For each of the suspension models, a differentparent-child relation between the different bodies is imple-mented.

The parent-child structure of the Girder suspension isshown in Fig. 12. The steer body is attached to the twistbody, allowing the rotation about its z axis. The twist’s bodyreference frame shares its y axis with the main frame’s yaxis. The twist body’s reference frame is rotated about the yaxis making coincident its x axis with the twist axis in themain frame. All the bodies after the twist body have a sim-ilar reference frame orientation. Therefore, the z axes of thetwist and the steer bodies are parallel with the steering axisin the main body reference frame. The mass, the momentsof inertia and the inertia products of this body correspondto those of the Girder’s STR subsystem stated in the pre-vious section. The rider’s steering moment and the steeringdamper moment are applied to the steer body about its z axisand react on the rider’s upper body and on the main body re-spectively. The upper wishbone and lower wishbone bodiesare children of the steer body and both rotate about the yaxis. Their masses and their moments and products of in-ertia are obtained from the CAD designs and correspond tothe Girder’s UWB and LWB subsystems respectively. Thesuspension body is a child of the upper wishbone body. Italso rotates about the y axis (only) and its mass, momentsand products of inertia correspond to those of the Girder’sSUS subsystem. The closure of the kinematic loop is donebetween the lower wishbone and the suspension body. Thegeometrical point of the lower wishbone’s tip is defined ascoincident with the wishbone attachment point in the sus-pension body (point q3 in Fig. 5a) by position constraintsin x, y and z directions. The rotational degrees of freedomabout axes x and z are also removed. Therefore, the onlymotion allowed between the lower wishbone and the sus-pension body is the rotation about their common y axis. Fi-nally, the suspension body is the parent of the front wheelbody, which has same properties and kinematics as the orig-inal GSX-R1000 nominal model, rotating about its y axis.

Following the different mechanical configuration of theHossack suspension system, in which the steering axis isplaced on the four bar linkage’s opposite side, a differentparent-child structure must now be considered. This is shownin Fig. 13. In this case, the two wishbones are connecteddirectly to the twist body and rotate about their y axis cor-responding to that of the twist body. Their mass and iner-tia properties were found previously as those of the Hos-


Fig. 12: Parent-child structure of the motorcycle modelmodified with the Girder suspension system.

sack’s UWB and LWB subsystems. The suspension body isa child of the upper wishbone body. Its mass, inertia mo-ments and inertia products correspond to the Hossack SUSsubsystem. In the Hossack model, the suspension body alsoperforms the system’s steering action. Thus, in its joint withthe upper wishbone, the suspension body is allowed to rotateabout the common y axis and its own z axis. The kinematicloop closure in this case, as in the Girder suspension, is donebetween the lower wishbone and the suspension body. Thegeometrical point of the lower wishbone’s tip is defined ascoincident with the lower wishbone attachment point in thesuspension body (point q3 in Fig. 5b) by position constraintsin x, y and z directions. However, in this case, in addition tothe common y axis rotation allowed between the lower wish-bone and suspension body in the Girder suspension case,rotation about the suspension body z axis is also allowed be-tween the lower wishbone and suspension body. Finally, thefront wheel body is connected to the suspension body androtates about its y axis. It has a similar definition to that inthe telescopic fork and the Girder suspension models. Con-sidering that the inertia moment and products obtained forthe Hossack STR subsystem are negligible and that it doesnot play a significant role on the front end kinematics, itsmass is directly lumped into the main body’s mass, whosecentre of masses is modified according to the relative posi-tion of this subsystem. In the Hossack suspension system,the rider’s steering and the steering damper moments are di-rectly applied to the suspension body about its z axis. Thefirst reacts on the rider’s upper body whilst the second doesso on the main body.

4.3 Suspension tuning

The suspension forces in both Girder and Hossack systemsare modelled as two moments applied to the lower wish-bones and reacting on the steer body and the twist body re-spectively. These suspension moments depend on the lower

Fig. 13: Parent-child structure of the motorcycle modelmodified with the Hossack suspension system.

wishbones angular displacements and speeds, providing thereactive and the dissipative suspension actions. This workis focused on comparing the two alternative suspension sys-tems performance to that of the telescopic fork system. Thus,in order to introduce the minimum systems variations, it issought a stiffness/damping tuning, for both Girder and Hos-sack systems, equivalent to that of the telescopic fork sus-pension. Following the approach in [2], the equivalent sus-pension moments to the linear suspension force of the tele-scopic fork can be calculated considering an energy conser-vation condition: the sum of the energy stored and dissipatedby the torsional spring and damper respectively in the dou-ble wishbone system is equivalent to that of the linear springand damper in the telescopic fork, for the same vertical dis-placement of the front wheel attachment point and in thesame time. Finally, the stiffness moment and the dampingcoefficient are described through third order polynomial fitson the angle rotated by the lower wishbone. The dampingmoment results from multiply the damping coefficient andthe angular rate. The maximum error achieved by this ap-proximation is less than 2 % along the full suspension travel.

5 Suspension response

The responses of the different suspensions systems are testedand compared to the conventional telescopic fork responseunder two different running situation. The first considers themotorcycle passing straight through a step bump input. Inthe second one, the motorcycle performs a hard front brak-ing manoeuvre. Several simulations are carried out in bothcases for different motorcycle forward speeds. The resultsare found to be qualitatively similar for the different for-ward speeds under study. In here, the simulations at 40m/sare taken as example to illustrate these results.


5.1 Road bump input

The road bump input simulation is performed with the mo-torcycle running in straight line. After a few meters, a stepinput of height hb = 50 mm is applied. This step bump iscomputed taking into account both vertical and horizontalforces appearing on the tyres when they reach the bump cor-ner.

Figure 14 compares the front end responses to a bumpinput of the motorcycle fitted with a telescopic fork and withGirder and a Hossack suspension systems designed with aparallelogram (prl) configuration. In Fig. 14a the verticaldisplacement of the motorcycle handle bar is presented whilstFig. 14b presents the vertical displacement of the front wheelcentre. Although the overshoot is slightly higher for bothGirder and Hossack suspension systems, it can be appreci-ated that the behaviours of the three models are very similar.

When both suspension systems are designed with a fork’strajectory (tft) configuration, the Girder system shows a re-sponse almost identical to the telescopic fork suspensioncase. It is the Hossack suspension system which introducessmall behaviour differences. These results, shown in Fig. 15,are coherent with the fact that the steering axle in the Girderand the fork suspension systems is fixed to the chassis, andin both cases, the front wheel follows the same trajectory.Consequently, the motions of the masses in both fork andGirder systems are close.

(a) Handlebar height (b) Front wheel height

Fig. 14: Motorcycle front end response after a 50 mm roadbump input with a forward speed of v = 40 m/s for the par-allelogram configuration (prl) of both Girder and Hossacksuspension systems.

Figure 16 shows the results for the road bump input sim-ulation for the two alternative suspension systems designedin order to introduce a minimal normal trail variation (cntconfiguration). Whilst the Hossack system response slightlydiffers, the Girder suspension response is closer to that ofthe telescopic fork.

Both alternative suspension systems have been designedto obtain similar stiffness and damping properties to the tele-scopic fork. However, the different mechanical configura-

(a) handlebar height (b) front wheel height

Fig. 15: Motorcycle front end response after a 50 mm roadbump input with a forward speed of v = 40 m/s for the tele-scopic fork’s trajectory configuration (tft) of both Girderand Hossack suspension systems.

tions and mass distribution of these systems introduce smallvariation in the motorcycle front end response. Although amore precise tuning of each alternative suspension systemwould result in more efficient response of the front end, itcan be said that different kinematics configurations(prl, tftand cnt) do not introduce significant variation in the suspen-sion performance.

(a) Handlebar height (b) Front wheel height

Fig. 16: Motorcycle front end response after a 50 mm roadbump input with a forward speed of v = 40 m/s for the con-stant normal trail configuration (cnt) of both Girder andHossack suspension systems.

5.2 Front wheel braking

Front wheel braking manoeuvres are simulated for a con-stant deceleration of a = −0.5 G. As it has been explainedin section 2, a PD controller implemented in the model cal-culates the braking moment to be applied to the front wheelin order to obtain the desired deceleration. In order to fo-cus on the pure braking effects only, the aerodynamic forceshave not been taken into account by setting the drag, liftand pitch aerodynamic coefficients to zero during this sim-ulations. The diving of the motorcycle’s front end and thevariation in normal trail are compared between the double


wishbone suspension systems and the telescopic fork sus-pension.

Figure 17 shows the vertical suspension travel and thenormal trail variation of the three different motorcycle mod-els fitted with the telescopic fork, the Girder and the Hossacksuspension systems with a parallelogram configuration (prl).The anti-dive effect is shown to increase in Fig. 17a for bothdouble wishbone suspension systems. This is produced bytheir front wheels contact points trajectories which can beobserved in Fig. 8. However, regarding to the normal trailvariation, the Girder and Hossack systems behave oppositeto each other. Whilst the Girder suspension reaches smallernormal trail values than the telescopic fork, the Hossack sys-tem presents larger normal trail values than the fork for allthe suspension travel.

(a) Vertical suspension travel (b) Normal trail

Fig. 17: Vertical suspension travel (v.s.t.) and normal trail(tn) for the telescopic fork suspension compared to theGirder and Hossack systems with a parallelogram configu-ration (prl). A straight line front wheel braking manoeuvreat an initial forward speed of v = 40 m/s with a constantdeceleration of a =−4.9 m/s2 is performed.

When both Girder and Hossack suspension systems aredesigned with a fork’s trajectory configuration (tft), theirdiving properties become similar to those of the telescopicfork, as it is shown in Fig. 18. The vertical suspension travelreached under the braking manoeuvre is similar for the threesystems. In the Girder suspension and telescopic fork cases,the common steering axles and front wheel contact pointstrajectories, produce similar kinematics in both systems, whichresults in similar normal trail behaviour.

In the Hossack suspension system case, the normal trailis highly reduced from tn = 88 mm up to tn = 68 mm, whichexceeds significantly the reduction of this parameter reachedby the telescopic fork and Girder suspension systems. TheHossack system’s geometry magnifies the normal trail re-duction. In order to obtain a front wheel contact point tra-jectory similar to that of the telescopic fork, the steering an-gle is necessarily reduced with the suspension travel. Thisleads to a greater normal trail reduction compared to otherconfigurations.


Fig. 18: Vertical suspension travel (v.s.t.) and normal trail(tn) for the telescopic fork suspension compared to theGirder and Hossack systems with a telescopic fork’s trajec-tory configuration (tft). A straight line front wheel brakingmanoeuvre at an initial forward speed of v = 40 m/s with aconstant deceleration of a =−4.9 m/s2 is performed.

Figure 19 shows the front end behaviour of the Girderand the Hossack systems configured to present a minimalnormal trail variation under a front wheel braking manoeu-vre. The front wheel contact point trajectory becomes highlyrelevant on the suspension’s anti-dive effect. Looking at Fig. 10bit can be observed a trajectory of the contact point with alarger angle with the vertical that the Girder suspension pro-duces, which makes this configuration of the Girder suspen-sion system more prone to dive than the telescopic fork. Thisresults in an increase of the vertical suspension travel about10 mm with respect to the telescopic fork system.


Fig. 19: Vertical suspension travel (v.s.t.) and normal trail(tn) for the telescopic fork suspension compared to theGirder and Hossack systems with a constant normal trailconfiguration (cnt). A straight line front wheel braking ma-noeuvre at an initial forward speed of v = 40 m/s with aconstant deceleration of a =−4.9 m/s2 is performed.

Conversely, the Hossack suspension system with this con-figuration shows a negative angle of its front wheel contactpoint trajectory with the vertical. This results in an oppo-site behaviour of the front end, which rises from its nom-inal position about 15 mm. Regarding to the normal trail,both Girder and Hossack suspension systems experience a


reduction (limited by the geometrical configuration) of thisvalue, as shown in Fig. 19b. Even though they were both de-signed to keep this value constant, this can only be achievedby considering the static suspension compression. Depend-ing on the different accelerations on the motorcycle, otherelastic parts different than those of the front suspension sys-tem will be compressed or extended: these are the tyres car-casses and the swinging arm assembly. This geometry vari-ation changes the kinematics design and produces a normaltrail reduction, however this reduction is not as large as forthe telescopic forks suspension.

Fig. 20: Root loci of the nominal motorcycle model fittedwith a telescopic fork suspension. The speed is increasedfrom 10 m/s (�) up to 80 m/s (∗) at different roll angles: 0◦

(blue ×), 15◦ (green ◦), 30◦ (red +) and 45◦ (black 3).

Whilst the kinematics of the front suspension systems donot affect in a significant manner the suspension response toa bump, it does affect the anti-dive effect and the motorcy-cle geometry variation under braking manoeuvres. Depend-ing on the specifications sought, different systems may beadopted. For instance, if higher values of anti-dive and areduction of normal trail under braking are required, thena Girder suspension systems would provide a good kine-matic solution. On the other hand, for a reduced variation ofthe normal trail avoiding the diving of the motorcycle whilebraking, a Hossack system can be used.

6 Stability analysis

In order to understand how the double wishbone suspen-sion systems may affect the motorcycle oscillatory dynam-ics, stability analyses are performed taking advantage of rootloci of the different linearised motorcycle models. The sta-bility of both double wishbone suspension systems are stud-ied for various parameter variations such as geometry con-figuration, front frame compliance and steering damper co-efficients. Following the approach stated in section 2, thestate space models derived from the linearised equations ofmotion are fed with the quasi-equilibrium states, integrated

from the nonlinear equations. These states have been ob-tained for each model, from four motorcycle simulationswith four different roll angles (0◦ , 15◦ , 30◦ and 45◦). Ineach simulation, the forward speed is increased from 10 m/sup to 80 m/s with an acceleration of a = 0.001 m/s2.

Figure 20 shows these root loci of the nominal motorcy-cle model fitted with a telescopic front fork. The rider lean,weave and wobble oscillating modes are shown. Also thepitch mode appears in the area of interest, but only for thecase of a 45◦ roll angle. The rest of the normal modes arehighly damped and are not visible in this area.

(a) Girder - tft config.

(b) Girder - cnt config.

Fig. 21: Root loci for the Girder suspension with telescopicfork’s trajectory (tft) and constant normal trail (cnt) configu-rations. The speed is increased from 10 m/s (�) up to 80 m/s(∗) at different roll angles: 0◦ (blue ×), 15◦ (green ◦), 30◦

(red +) and 45◦ (black 3).

Under straight running conditions, the rider lean, weaveand wobble are out-of-plane modes whilst the pitch modeis an in-plane one. When cornering, the in-plane and out-of plane variables become coupled. The pitch mode consistsin the pitching of the motorcycle through the front and rearsuspension compression and extension in an almost phaseopposition motion. The rider lean appears in the root locuswhen the rider upper-body degree of freedom is included inthe mathematical model. It represents the oscillation of therider’s upper-body. The weave mode involves roll, yaw andsteering angle oscillations combined in a fishtailing motion.


The wobble mode is characterized by a shaking of the frontframe about the steering axis whilst the rear frame is slightlyaffected. The in-plane and the out-of-plane degrees of free-dom become coupled for roll angles different to zero, whenthe motorcycle symmetry plane is no longer vertical. Weaveand wobble oscillating modes have been widely studied inthe literature (e.g. [20], [1], [17], [11], [12], [6] just to citea few) and they are of main relevance in this research dueto their lightly-damped nature and the possibility to becomeunstable under some running conditions.

6.1 Geometry variation

Fitting a different suspension system may affect the motor-cycle stability. However, for each suspension system understudy, its geometrical configuration (prl, tft and cnt) does notmodify in a significant manner the system’s roots positionsof the motorcycle fitted with that suspension.

(a) Hossack - prl config.

(b) Hossack - cnt config.

Fig. 22: Root loci for the Hossack suspension system withparallelogram (prl) and constant normal trail (cnt) configu-rations. The speed is increased from 10 m/s (�) up to 80 m/s(∗) at different roll angles: 0◦ (blue ×), 15◦ (green ◦), 30◦

(red +) and 45◦ (black 3).

As an example, Fig. 21a and Fig. 21b show the root locifor four different simulations at various motorcycle’s leanangles for the Girder suspension with the telescopic fork’s

trajectory (tft) and the constant normal trail (cnt) configura-tions.

Compared to the root loci for the telescopic fork suspen-sion (Fig. 20), three things can be observed: first, the desta-bilization of the weave mode at zero roll angle for speedsabove 70 m/s. At higher roll angles (15◦, 30◦ and 45◦), thismode is less damped than in the telescopic fork suspen-sion case but does not cross the stability limit. Secondly,the wobble mode is unstable for speeds lower than 16 m/sat 45◦. However, it becomes more damped for higher speedsand smaller roll angles. Finally, the third effect of fitting themotorcycle with such a Girder suspension system is an ap-preciable increase of the wobble frequency. The rest of themodes remain mostly unaffected by the inclusion of this sus-pension system on the motorcycle model.

(a) Girder - roll = 0◦ (Twist moment variation)

(b) Girder - roll = 45◦ (Twist moment variation)

Fig. 23: Root loci for the Girder suspension system withconstant normal trail (cnt) configuration. The twist mo-ment coefficients are varied as 60%(blue ×), 80%(green ◦),100%(magenta ·), 120% (red +) and 140% (black 3) of thenominal value. The speed is increased from 10 m/s (�) upto 80 m/s (∗) at different roll angles.

On the other hand, the geometrical configuration of theHossack system does not introduce relevant differences onthe system’s roots. As an example Fig. 22a and Fig. 22bshow the root loci for four lean angles simulations of theHossack suspension system with the parallelogram (prl) andthe constant normal trail (cnt) configurations respectively.


Compared to the root loci of the telescopic fork suspen-sion case, the Hossack suspension system’s wobble modebecomes more damped at higher forward speeds for all rollangles whilst it is less damped at lower speeds. In the caseof 45◦ roll angle, this mode is unstable below 20 m/s. Theremaining normal modes are not substantially affected bythe inclusion of this suspension system in the motorcyclemodel.

6.2 Front frame compliance

The design of a front suspension system will determine itscompliance and hence, the stiffness at the front end. It isinteresting to study how this compliance can affect the sta-bility of a motorcycle assembly. In the GSX-R1000 nomi-nal mathematical model, the compliance is introduced as areacting moment applied between the chassis and the frontend about the twist axis which is an axis perpendicular tothe steering one and belonging to the motorcycle symmetryplane.

(a) Hossack - roll = 0◦ (Twist moment variation)

(b) Hossack - roll = 45◦ (Twist moment variation)

Fig. 24: Root loci for the Hossack suspension system withconstant normal trail (cnt) configuration. The twist mo-ment coefficients are varied as 60%(blue ×), 80%(green ◦),100%(magenta ·), 120% (red +) and 140% (black 3) of thenominal value. The speed is increased from 10 m/s (�) upto 80 m/s (∗) at different roll angles.

The twist moment was defined as a torsional spring anddamper combination whose stiffness and damping param-eter nominal value was set to kt0 = 100 kNm and ct0 =

100 Nms. In order to study the variation on the rigidity ofboth front suspension systems, these stiffness and dampingcoefficients are modified proportionally from 60% of theirnominal values up to the 140%. These maximum values maybe difficult to be achieved in a real mechanical implementa-tion, but become useful to be considered in order to highlightthe trends of the system’s behaviour.

As it has been shown, since the variation in the geomet-rical configuration of both Girder and Hossack suspensionsystems do not affect their stability properties significantly,Figs. 23 and 24 show the root-loci of the Girder and Hos-sack systems for the different values of the twist momentsonly for the constant trail geometrical configuration (cnt).On the other hand, considering that the more relevant stabil-ity issues appear for roll angles of 0◦ and 45◦, the root locifor 15◦ and 30◦ are not presented in order to obtain clearerresults. Therefore, the root loci are plot for an unique roll an-gle, with different stiffness and damping coefficients valuesand with the speed varied from 10 m/s up to 80 m/s.

(a) Fork - roll = 0◦ (Twist moment variation)

(b) Fork - roll = 45◦ (Twist moment variation)

Fig. 25: Root loci for the telescopic fork suspension system.The twist moment coefficients are varied as 60%(blue ×),80%(green ◦), 100%(magenta ·), 120% (red +) and 140%(black 3) of the nominal value. The speed is increased from10 m/s (�) up to 80 m/s (∗) at different roll angles.


For the Girder suspension system with nominal valuesof twist moment coefficients, the weave mode is unstablefor speeds above 70 m/s at a zero roll angle, whilst the wob-ble mode instability happens for a 45◦ roll angle and for-ward speed values below 16 m/s. When the twist rigidity isincreased, the wobble mode becomes more unstable whilstthe weave mode’s stability increases, narrowing in this wayits unstable forward speed range at zero roll angle. A re-duction of the twist rigidity results in the opposite effect.Consequently, weave and wobble modes’ stability cannot besatisfied simultaneously by modifying the front suspensionsystem’s compliance.

(a) Fork - roll = 0◦ (Steering damper variation)

(b) Fork - roll = 45◦ (Steering damper variation)

Fig. 26: Root loci for the telescopic fork suspension system.The steering damper coefficient is varied as 60%(blue ×),80%(green ◦), 100%(magenta ·), 120% (red +) and 140%(black 3) of the nominal value. The speed is increased from10 m/s (�) up to 80 m/s (∗) at different roll angles.

Considering the Hossack suspension system, the frontend’s compliance variation has an impact on the motorcy-cle’s stability behaviour similar to that in the case of theGirder suspension. In this case, the wobble mode at 45◦ rollangle is stable for speed values larger than 20 m/s whilst theweave mode is stable for practically all the speed range atany roll angle for the nominal value of the twist moment co-efficient. However, if the stabilization of the wobble modefor a 45◦ roll angle at the lower speed range is sought bydecreasing the front frame’s rigidity, the weave mode will

become unstable for the straight line case at its higher speedrange.

If theses results are compared to a similar study for thetelescopic fork (see Fig. 25), it can be observed that theeffect of modifying the front end’s compliance on the mo-torcycle general stability is similar for the three suspensionsystems: A positive increment of the twist bending momentincreases the weave mode’s stability and decreases the wob-ble mode’s stability. On the other hand, by reducing the twistbending moment the weave mode become less stable whilstthe wobble mode stability is increased.

6.3 Steering damper coefficient

The steering damper is a passive device which links the steer-ing body and the chassis, whose mission is to attenuate wob-ble oscillations. Nowadays, a steering damper is fitted inmost of the commercial sport motorcycles. The nominal GSX-R1000 model has a steering damper which is mathemat-ically modelled as a linear reacting moment between thesteering body and the motorcycle’s main body (chassis).

(a) Girder - roll = 0◦ (Steering damper variation)

(b) Girder - roll = 45◦ (Steering damper variation)

Fig. 27: Root loci for the Girder suspension system withconstant normal trail (cnt) configuration. The steeringdamper coefficient is varied as 60%(blue ×), 80%(green ◦),100%(magenta ·), 120% (red +) and 140% (black 3) of thenominal value. The speed is increased from 10 m/s (�) upto 80 m/s (∗) at different roll angles.


Figure 26, Fig. 27 and Fig. 28 show the root loci ofthe motorcycle system fitted with telescopic fork, Girderand Hossack suspension systems for several simulations inwhich the steering damper coefficient was varied from the60 % of its nominal value (csd0 = 6.94 Nms) up to the 140 %.For the three suspension systems the results are similar. Itis well known ([7], [8]) that in the case of telescopic forksuspension systems, by increasing the steering damper co-efficient the wobble mode becomes more stable, however,the weave mode stability at high forward speeds is compro-mised by the action of the steering damper. On the otherhand, when the steering damping coefficient is reduced theeffect on the motorcycle stability is inverted. Girder and Hos-sack systems follow the same trend in terms of stability whenthe damping coefficient is varied, which is to be expected.

(a) Hossack - roll = 0◦ (Steering damper variation)

(b) Hossack - roll = 45◦ (Steering damper variation)

Fig. 28: Root loci for the Hossack suspension systemwith constant normal trail (cnt) configuration. The steeringdamper coefficient is varied as 60%(blue ×), 80%(green ◦),100%(magenta ·), 120% (red +) and 140% (black 3) of thenominal value. The speed is increased from 10 m/s (�) upto 80 m/s (∗) at different roll angles:

In the case of the Girder suspension, the weave modestability is always compromised at high speeds ranges evenfor the smallest value of the steering damper coefficient. Inthis case, the wobble mode is unstable for half of the speedrange at 45◦ roll angle. Stability of both modes cannot beachieved simultaneously with the steering damper. However,

as a compromise solution, it can be used to stabilize the wob-ble mode at lower speed and high roll angles by sacrificingthe weave stability at higher forward speed values.

(a) Girder - Roll = 0◦ (CM variation)

(b) Girder - Roll = 45◦ (CM variation)

Fig. 29: Root loci for the Girder suspension system in whichits centre of mass is displaced towards the steering axis.68mm forward (blue ×), 34mm forward (green ◦), 0mm(magenta ·) -34mm backwards (red +) and -68mm back-wards (black 3)

It was found that for the GSX-R1000 model fitted withGirder suspension system with a constant normal trail con-figuration, the wobble mode becomes fully stable for a valueof the steering damper coefficient of csd = 7.85 Ns, whichis a 13 % higher than the nominal value. The maximum for-ward speed at which the weave mode still remains stableis v = 70 m/s, which for non racing conditions is a con-siderable speed (252 km/h) above the legal speed limits.On the other hand, with a Hossack suspension system, theweave mode is better damped. A steering damper coeffi-cient value which keeps both modes stable for almost allthe studied running conditions can be found for the GSX-R1000 model fitted with this suspension system. This valueis csd = 7.92 Ns, which is a 14 % higher than the nominalvalue.


6.4 Location of the front frame’s centre of mass

Due to the different designs of the three suspension systemstheir centres of mass are differently located, as it is shownin Fig. 1. For the telescopic fork and the Hossack suspen-sion their centres of mass are next to the steering axis. How-ever, in the case of the girder suspension its centre of massis placed about 68mm ahead its steering axis.

(a) Hossack - Roll = 0◦ (CM variation)

(b) Hossack - Roll = 45◦ (CM variation)

Fig. 30: Root loci for the Hossack suspension system inwhich its centre of mass is displaced towards the steeringaxis. 68mm forward (blue ×), 34mm forward (green ◦),0mm (magenta ·) -34mm backwards (red +) and -68mmbackwards (black 3)

This study was made by modifying the coordinates ofthe centres of mass of the SUS and STR parts in the differentsuspension systems in order to obtain the desired assembly’scentre of mass. Figure 29 presents the the impact on the rootloci of displacing the centre of mass of the Girder suspen-sion system. The distance of this point to the steering axistakes the following values: 68mm forward (blue ×), 34mmforward (green ◦), laying on the steering axis (magenta ·),-34mm backwards (red +) and -68mm backwards (black 3).Similar studies are performed for the Hossack and the tele-scopic fork suspension systems. Figure 30 and Fig 31 showthese results.

For the three suspension systems it is found that, by dis-placing the centre of mass towards the steering axis from its

(a) Fork - Roll = 0◦ (CM variation)

(b) Fork - Roll = 45◦ (CM variation)

Fig. 31: Root loci for the telescopic fork suspension systemin which its centre of mass is displaced towards the steer-ing axis. 68mm forward (blue ×), 34mm forward (green◦), 0mm (magenta ·) -34mm backwards (red +) and -68mmbackwards (black 3)

nominal position, the stability of weave and wobble modesis significantly increased. The stability of the weave modeis always improved by displacing backwards the centre ofmass. The wobble mode becomes more stable when the cen-tre of mass is close to the steering axis. However, if it isdisplaced backwards over a certain limit the wobble modestarts to approach instability.

In both Girder and Hossack suspension system, when thecentre of mass is laying on the steering axis, the weave andwobble normal modes become stable even with the nomi-nal value of the steering damper. These results suggest thatwhilst a Hossack suspension system can be designed as afully stable system just by tuning the steering damper, aGirder suspension system will present stability issues dueto its geometry and mass distribution.

7 Conclusions

The main goal of this research is to understand how the dif-ferent geometrical configurations of double wishbone sus-pension systems may affect the motorcycle dynamics andstability. The dynamic response and the stability propertiesof an existing motorcycle model modified with different ge-


ometrical configurations of two representative double wish-bone suspension systems (Girder and Hossack) have beeninvestigated and compared to those of the baseline modelfitted with a telescopic fork suspension.

Several simulations were performed to test the dynam-ical response of the double wishbone suspension systems.For front wheel braking manoeuvres, the designs of the dif-ferent suspension systems become crucial for their responses.The anti-dive capabilities of double wishbone suspensionsystems depend directly on the front wheel trajectory thatthey can perform. For the nominal parallelogram configu-ration (prl), both Girder and Hossack suspension systemshave similar responses providing some anti-dive comparedto the telescopic fork. However, when both Girder and Hos-sack suspension systems are designed with the telescopicfork trajectory configuration (tft) their behaviour is similarto that of the telescopic fork and do not provide extra anti-dive. Finally, when constant normal trail configuration (cnt)is implemented, the kinematic differences in the two doublewishbones suspensions designs imply different behaviours.Whilst Girder suspension dives further than the telescopicfork, the Hossack system does not dive but rises.

In terms of shock absorption capabilities, the responseof the three suspension systems is similar for road bump in-puts regardless of their geometrical configuration, althoughsome small differences can be observed between differentsystems. It has to be noted that Girder and Hossack suspen-sions were tuned to obtain the same equivalent stiffness anddamping coefficient than those of the telescopic fork, de-spite these equivalent coefficients are not optimal values forthe double wishbone suspension systems due to their par-ticular designs. More precise responses would be achievedwith individualized tuning of each suspension.

The results of stability analyses showed that differentkinematic designs of a double wishbone suspension systemsdo not modify the motorcycle’s stability. For each of the sus-pension systems (Girder and Hossack), the normal modesproperties remain unaffected with the three geometrical con-figurations. However, these normal modes differ from onesuspension system to the other. It was seen that normal modesbehaviour depends on dynamic parameters, such as the frontframe flexibility, the steering damping coefficient and thecentre of mass location, in a similar manner for the Girder,Hossack and telescopic fork suspension systems. Althougheach suspension system presents its particular behaviour, gen-eral properties could be obtained for the three systems understudy.

On one hand, variations on front frame compliance af-fect the motorcycle’s stability in an opposite manner thanthe variations on the steering damper coefficient. Reduc-ing twist rigidity results in decreased stability of the weavemode whilst the general wobble mode stability is improved.

If the twist rigidity is increased, the wobble mode becomeless stable and the weave mode’s stability is improved.

On the other hand, the centre of mass location with re-spect to the steering axis of the front frame has a remarkableimpact on the motorcycle stability. For the three suspensionsystems under study, the weave and wobble modes’ stabil-ity is improved if the centre of mass is displaced backwards.The wobble mode is observed to be more stable when thecentre of mass is close to the steering axis, while displacingbackwards the centre of mass over a certain limit reduces thestability of this mode.

In conclusion, both Girder and Hossack systems presentinteresting properties in terms of anti-dive and kinematic de-sign that can not be exploited with conventional telescopicfork suspensions. The different geometric configurations thatcan be obtained with these systems do not affect neither theirshock absorption capabilities nor the stability of the motor-cycle. However, the effect on the stability of dynamic prop-erties such as centre of mass location or assembly’s compli-ance have to be carefully considered in the design of thesesuspension systems.

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