Dynamic modelling and experimental validation of ...

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Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/nvsd20

Dynamic modelling and experimental validation of three wheeled tilting vehicles a

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Nicola Amati , Andrea Festini , Luigi Pelizza & Andrea Tonoli

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Mechanics Department, Mechatronics Laboratory, Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino, 10129, Italy b

Mechatronics Laboratory, Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino, 10129, Italy Version of record first published: 23 Feb 2011

To cite this article: Nicola Amati, Andrea Festini, Luigi Pelizza & Andrea Tonoli (2011): Dynamic modelling and experimental validation of three wheeled tilting vehicles, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 49:6, 889-914 To link to this article: http://dx.doi.org/10.1080/00423114.2010.503277

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Vehicle System Dynamics Vol. 49, No. 6, June 2011, 889–914

Dynamic modelling and experimental validation of three wheeled tilting vehicles

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Nicola Amatia *, Andrea Festinib , Luigi Pelizzab and Andrea Tonolia a Mechanics

Department, Mechatronics Laboratory, Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino 10129, Italy; b Mechatronics Laboratory, Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino 10129, Italy (Received 12 November 2008; final version received 17 June 2010; first published 23 February 2011 ) The present paper describes the study of the stability in the straight running of a three-wheeled tilting vehicle for urban and sub-urban mobility. The analysis was carried out by developing a multibody model in the Matlab/SimulinkSimMechanics environment. An Adams-Motorcycle model and an equivalent analytical model were developed for the cross-validation and for highlighting the similarities with the lateral dynamics of motorcycles. Field tests were carried out to validate the model and identify some critical parameters, such as the damping on the steering system. The stability analysis demonstrates that the lateral dynamic motions are characterised by vibration modes that are similar to that of a motorcycle. Additionally, it shows that the wobble mode is significantly affected by the castor trail, whereas it is only slightly affected by the dynamics of the front suspension. For the present case study, the frame compliance also has no influence on the weave and wobble. Keywords: tilting vehicles; narrow commuters; stability; multibody models; vehicle dynamics

1.

Introduction

Motorised tilting vehicles have been studied and developed since the beginning of the 1950s. Many other attempts have been made since the pioneering prototype proposed by Ernst Neumann [1] in 1945–1950. In the middle of the 1950s, the Ford Motor Company presented a gyroscopically stabilised two-wheeled lean vehicle (Gyron) with retractable wheel pods for parking. In the 1960s, the MIT proposed a tilting vehicle similar to a motorcycle but equipped with an active roll control. At the beginning of the 1970s, General Motors presented the ‘Lean Machine’ characterised by a fixed rear frame and a tilting body module which was controlled by the rider by using foot pedals to balance the tilting body [2]. These projects were confined to the field of research and therefore were far from the stage of mass production. Their development was restrained by the difficulties encountered in the control of the roll angle. The generation of a stabilising tilt torque by using a gyroscope (Gyron) was too much complex, ineffective at low speed and not safe. The option of delegating this *Corresponding author. Email: [email protected]

ISSN 0042-3114 print/ISSN 1744-5159 online © 2011 Taylor & Francis DOI: 10.1080/00423114.2010.503277 http://www.informaworld.com

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control to the driver (Lean Machine) turned out to be ineffective, as the effort required for protracted operation was unacceptable. Moreover, the learning process that was required for both configurations to manage an unconventional vehicle was not considered realistic. The MIT strategy was to develop an active tilt control, the only option in order to overcome the above-mentioned drawbacks. The results obtained during the steady-state cornering were promising but the delays in the transients were not satisfactory. This was due to the inadequate resolution of the sensors and to the moderate performance of the control system technology available at that time. Therefore, the interest in these kind of vehicles petered out up to the last decade, even if new configuration proposals continued to be made [1]. To this end, it is worth mentioning the Ariel 3 moped [3]. It is a three-wheeler with a two-wheel rear axle and a single front wheel. The front half is hinged to the rear frame and therefore is tiltable, whereas the rear remains flat on the ground. The vehicle was produced and commercialised in the UK by the BSA Company in 1971. The project was then acquired by Honda who produced the Honda Gyro model at the beginning of the 1980s. The problem of pollution, the energy costs, traffic congestion and the progress made in control system technology has more recently created new interest in narrow commuters for individual mobility. The F 300 Life-Jet [4], Clever [5], Carver One [6] and MP3 [7] are the projects that reached the highest level of development (the MP3 has been on the market since 2006 while the Carver One was commercially available from 2006 to mid-2009). The F 300 Life-Jet is a three-wheeled tilting vehicle developed by Mercedes-Benz and presented in 1997. It is characterised by a two-wheeled front axle and a single rear wheel, connected to the main body by a trailing arm suspension. The parallelogram mechanism of the front suspension allows the vehicle to lean up to 30◦ while maintaining the wheels almost parallel to the body. The Carver One (by Carver Engineering) and the Clever (by a Consortium including BMW and the Universities of Bath and Berlin) were presented respectively in 2002 and 2003. They both are characterised by a single front wheel that tilts with the main body and by a non-tilting two-wheel rear axle. The main body is connected to the rear frame by a suspension layout enabling it to lean like a motorcycle while the rear body remains in contact with the ground in the same way as a conventional automobile rear axle. The Carver One, Clever and F 300 Life-Jet (designed for two passengers, with a tandem seat structure) are driven with a small size car engine, have a conventional automotive steering line and are equipped with an active tilt control system. All the main vehicle features such as the chassis, the engine type, the style and the driving system highlight how these vehicles have been conceived as tall micro-cars with the added option of leaning the main vehicle body. They do not seem to be suited to manoeuvring through traffic. On the other hand, the Piaggio MP3 has been conceived as a scooter with an unconventional front body. The front suspension is characterised by a parallelogram in which the horizontal beams are hinged to the vehicle structure. This type of suspension allows the wheels to lean together with the main body, up to 40◦ . In the same way as a motorcycle, the lean angle is controlled by the rider. A brake locks the suspension to prevent falling at a low or zero speed. The configuration of the rear body and rear suspension, the limited front wheel track (640 mm), the moderate size of the tilting mechanism and the interface with the rider (a conventional scooter handlebar) allow the vehicle to be driven like a conventional scooter, moving in congested traffic efficiently. Unfortunately, the lack of any improvement in rider and passenger safety undermines the above-mentioned advantages and the high lateral stability described in [8]. The three tilting wheels (TTW) project, started at the Mechatronics Laboratory of the Politecnico di Torino in 2005, is aimed at developing a vehicle that is able to comply with the requirements of passenger safety and comfort, along with that of high manoeuvrability in

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areas affected by heavy traffic. The description reported in Section 2 corresponds to a vehicle which combines the main features of the F 300 Life-Jet and the MP3. The front suspension, the optional active tilting control device and the main body layout are similar to the former. The steering system and the free tilting set-up are typical of the latter. The aim of the present paper is to study the lateral dynamics and the open loop stability of the TTW. Due to the complexity of the system, a multibody approach was considered appropriate. A simplified analytical model was used to analyse basic dynamic motions only. The SimMechanics multibody tool in the Matlab/Simulink environment was adopted to develop the model. This choice is related to the need to use the simulator in the future for the implementation of the active tilting control system and the hybrid powertrain. An Adams/Motorcyle model has also been developed to cross-validate the SimMechanics model. The literature on the lateral dynamics of tilting vehicles is limited to analytical models neglecting effects such as chassis compliance, suspensions kinematics, the motion of the steering system and appropriate model of the tyres taking into account their dynamics [9– 11]. These models have been developed to support the design of active tilt control strategies rather than to study the lateral vibration motions. As a matter of fact, the direct tilt control and the steer tilt control strategies described in [10,12–14] are based on basic inverted pendulum models. The only paper describing the stability of a tricycle is the article published by Sharp [15] in 1984. It describes the mathematical model of the vehicle making reference to the Sharp 1971 motorcycle model. This paper and the literature on motorcycle dynamics [16–28] has therefore been considered as a reference for the development of the TTW dynamic model used for the analysis of stability in the present article. To this end, evident similarities with motorcycle dynamics are presented and the effect of some parameters such as the castor trail (CT) as well as damping on the steering axis, chassis compliance and the layout of the front suspension are discussed. The sensitivity of most of these parameters was evident during the track tests that were carried out to validate the models experimentally. It is worth underlining that most of the considerations about the dynamic behaviour of the TTW can be extended to vehicles like F 300 Life-Jet, MP3 and to all the tricycles and quadricycles in which the suspension system enables the wheels to lean with the main body.

2.

Description of the TTW vehicle

The TTW is a tricycle with two wheels on the front axle with a track of 1.16 m (Figure 1). The size (L × W × H = 2.35 × 1.2 × 1.6 m, Table 1), the weight (m ∼ = 300 kg, driver included) and the leaning capacity up to 45◦ are typical of a narrow commuter. The vehicle is composed of two main parts (Figure 2) that are connected to each other by a bolted joint in 12. The rear part is composed of a lower beam structure (truss) which carries an upper frame and roll bar (17). The lower structure is configured for a double in-line seat configuration. It contains the gasoline engine (18 – Yamaha XT 660) which powers the rear wheel by a double-step chain transmission. The wheel (15) is supported by a motorcycle trailing arm (16) hinged to the main body. The upper frame is designed to support the bodywork and satisfy basic passive safety requirements. The front body of the vehicle is composed of the box beam (9) to which are connected the tilting lever, the steering system and the arms of the front suspension that supports the front wheels. The suspension is characterised by a double parallel wishbone configuration. The four arms (7–8 on the right and 5–6 on the left, Figures 1(c, d) and 2) are of the same length so that the front wheels are inclined as the main body. They are hinged to the frame (9) by revolute

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Figure 1. TTW vehicle. (a) The vehicle during track tests. (b) The layout of the steering system. (c) 3D view of the front suspension. (d) Front view of the suspension in the configuration in which the left wheel is moved up and consequently, the right one is gone down.

joints with longitudinal axis and to the uprights by the revolute joints within a fork-shaped body (1_5 in Figure 1). Two spring damper systems (10–11 in Figure 1) connect the upper wishbones to the tilting lever free end. The tilting lever is hinged to the frame (9) by a longitudinal axis revolute joint that is coaxial with the joints of the suspension arms. Using such a configuration, the tilting lever motion is coupled with the vehicle roll. If the lever is locked to the box beam (9) (e.g. when using a braking system), the suspension movement and therefore the vehicle roll is enabled only by the compliance and the damping of the shock absorbers (Figure 1(c)) as in conventional three-wheelers. By contrast, when the tilting lever is left free to rotate, the vehicle roll is free as in motorcycles. In this configuration, when one wheel moves up, the other goes down by the same amount without involving any deformation of the shock absorbers (Figure 1(d)). This also happens in dynamic conditions since the inertia of the tilting lever is negligibly small in comparison with the mass of the wheels and the uprights. The tilting lever can also be controlled by an active system (i.e. a rotative actuator between the front frame and the tilting lever). In this case, the controlled rotation of the tilting lever determines the lateral inclination of the wheels and therefore the roll of the body (Figure 1(d)).

Vehicle System Dynamics Table 1.

Main parameters of the TTW prototype.

Description

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Wheelbase (l) Front track (t1 ) Front wheels CT (e1 ) Vehicle mass + driver (m) Vehicle + driver moment of inertia about the roll axis (Jx ) Vehicle + driver moment of inertia about the pitch axis (Jy ) Vehicle + driver moment of inertia about jaw axis (Jz ) Steering line moment of inertia Front tyres Front tyres radial stiffness ktf Front tyres radial damping ctf Radius of the front tyre sidewall rtf Rear tyre Rear tyres radial stiffness ktr Rear tyre radial damping ctr Radius of the rear tyre sidewall rtr Mass of each front wheel Polar moment of inertia of each front wheel Mass of each the rear wheel Polar moment of inertia of each wheel Front axle – centre of gravity distance (vehicle + driver) (a) Height of the centre of gravity (vehicle + driver) (h) Powertrain Transmission Rear suspension Rear suspension damper Rear suspension spring Front suspension Front suspension damper Front suspension spring Tilting system Steering system

Unit

Value

m m mm kg kg m2

1.74 1.16 15, 29, 46 301 129

Kg m2

222

kg m2

105

kg m2 – N/m Ns/m M

0.5594 Dunlop 150/60 R17

– N/m Ns/m M kg kg m2

Dunlop 170/60 R17

12.5 0.47

kg m2

13 0.475

M

0.75

M

0.587

–

Internal combustion engine, Yamaha XT 660, Single cylinder, four valves, four strokes Chain Swing arm 1.85 · 104 9 · 104 Double parallel wishbone 2 × 1.85 · 104 2 × 9 · 104 Free Lever

– – Ns/m N/m – Ns/m N/m – –

The steering system (Figure 1(b) and 1 in Figure 2) is composed of a motorcycle handlebar connected to the steering column 1_2 with the lever 1_1 at the end. Two steering rods (1_3) are used to link the end of the lever to the longitudinal beams 1_4 of the uprights. For this reason, the rotation of the steering column is transformed into a lateral linear motion of the rods by the lever 1_1. The connection of the beams 1_4 to the end of the rods enables the motion of the rods to be converted into the steering motion of the wheels. Since the length of the beams 1_4 is approximately equal to that of the lever 1_1, the ratio between the rotation of the handlebar and the wheels is near to one. To allow decoupling of the tilting and steering motion, both the steering rod ends are equipped with spherical joints. Their ends, on the lever side, are aligned (one behind the other) to the longitudinal hinge axis of the upper wishbones. The wheels steer about the axis connecting the upper and the lower arm joints of the uprights. This axis has a lateral inclination (king pin angle) of 22.3◦ and a lateral offset (king pin offset)

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Figure 2. CAD assembly used to generate the multi-body model. (a) 3D view. (b) Lateral view evidencing the point in which the chassis compliance has been lumped.

inside the wheel track of 22.1 mm. The longitudinal offset (CT) can have three different values: 15, 29 and 46 mm. This is made possible as both the upper and the lower forks (1_5) can be moved longitudinally with respect to the uprights (1_6) in two different positions. Varying the CT, the longitudinal inclination of the steer axis (castor angle) remains almost constant and equal to 22◦ . Future models will be equipped with two optional subsystems: an active tilting system to control the vehicle leaning angle automatically and a hybrid powertrain characterised by two direct drive electric motors installed in the front wheel hubs.

3.

Model description

The model was developed by taking into account the motion of the main body, the chassis compliance [22,24,29], the rotation of the steering line, the movement of the wheels due to the suspension linkages, and the steady state and dynamic forces developed by the tyres [16,24,25,30]. However, a set of assumptions was introduced to limit the model complexity without compromising its use. (1) The driver is not implemented. This means that the capsizing will remain in evidence as unstable during the stability analysis. (2) The driver is considered fixed to the

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Figure 3. Simulink model structure. In the block ‘Vehicle Structure’ is implemented the multibody model of the vehicle (Figure 4); in the block ‘Tyres’ is modelled the tyre-ground interaction.

seat. This is justified by the use of a car seat which has four point safety belts that limit any movement of the driver’s body outside of the symmetry plane. (3) The powertrain is not modelled. A constant longitudinal velocity V is achieved by applying the requested longitudinal force to the rear wheel tyre–ground contact point. (4) The static friction in the steering line and in the suspension joints is modelled by using viscous dampers. (5) The chassis compliance is assumed to be concentrated in the bolted joint (Figure 2(b)) which connects the front and rear bodies. (6) The interaction of the longitudinal and lateral tyre slip as well as the tyre side-slipcamber thrust coupling is ignored. (7) Deviations of the outer tyre surface from theoretical geometry (i.e. tyre conicity) are omitted. (8) The aerodynamic drag has been neglected. The model was developed in Matlab/Simulink/SimMechanics graphical programming language by using the structured layout reported in Figure 3. This structure is characterised by two main blocks: the ‘Vehicle Structure’ and the ‘Tyres’. The former implements the lateral vehicle dynamics, while the equations governing the forces generated by the tyres are included in the latter. The inputs to the vehicle model are the torque (Tsteer ) applied to the handlebar and the longitudinal force (Fext ) that is required to maintain a constant speed V . The ‘Vehicle Structure’ subsystem was modelled using a multibody approach. To this end, SimMechanics was adopted because it is a multibody tool for mechanical systems that can be interfaced to Matlab/Simulink blocks. This option is important to implement additional features such as active tilting control, a hybrid powertrain and electric power steering. The choice of using SimMechanics also has some drawbacks that are mainly related to the lack of prebuilt library models such as the driver controller and the tyre model. These subsystems are widely consolidated in codes like Adams-Motorcycle and MSC BikeSim. Currently the driver controller was not implemented while the effort of implementing the tyre model, as described in the subsection ‘tyre’, proved unavoidable. An equivalent Adams-Motorcycle multibody model was developed in parallel to compare simulations in different kind of manoeuvres from straight running to cornering. This model will be described in the second part of the present section. Additionally, a four degrees-of-freedom analytical model (derived from a motorcycle model) was implemented to evidence the similarities of free tilting tricycles and conventional

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Figure 4. Schematic representation of the multibody model.  = element with inertia tensor, ♦ = revolute joint, ◦ = spherical joint and d = compliant joint.

motorcycles sharing the same parameters. Such a model is presented in the third part of the present section. 3.1. Vehicle subsystem The CAD assembly of the ‘Vehicle Structure’subsystem is shown in Figure 2. The front (9) and rear (13) bodies are composed of rigid parts. They are highlighted in dark grey. The subsystems that are not rigidly connected to them (wheels and wheel hubs (2 and 3), front suspension wishbones (5, 6, 7, 8), rear suspension trailing arm (16), steering system (1) and driver body (14), are highlighted in light grey. Rigid revolute and universal joints are introduced to connect them to the frame, while a compliant joint is introduced between the front (9) and rear (13) bodies (element 12). The block diagram in Figure 4 highlights the connection of the different parts. The choice of modelling the torsional and lateral deformation of the frame with a two degrees-of-freedom compliant joint that is lumped at the connection of the front and rear body derives from the finite-elements analysis of the entire structure. This analysis [31] was also adopted to estimate the stiffnesses values, while the associated damping was tuned using experimental results and data reported in the literature [24]. The steering line was considered as a part of the front frame, all the parts of the system, including the steering column, being connected to the front frame. A revolute joint was adopted to connect the steering column to the front frame. This is a simplification of the prototype, where the steering column is connected to the front body with a single bearing at its lever end but it is also linked to the rear body by two bearings located close to the handlebar. The above description leads to a vehicle model with 16 degrees of freedom that are represented in Figure 5 and summarised in Table 2. The rear body (13) is characterised by six dofs

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Figure 5.

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Schematic representation of the degrees of freedom. Table 2. No. dof 1 2 3 4 5 6 7 8 9 10 11–13 14–16

Degrees of freedom of the model. Description Rear body vertical motion Rear body longitudinal motion Rear body yaw angle Rear body side-slip angle Rear body pitch angle Rear body roll angle Rotation of the front body with respect to the rear one Lateral displacement of the front body with respect to the rear one Rotation of the tilting lever Rotation of the steering column Vertical motion of the three wheels Rotation of the three wheels about their spin axis

Symbol zr V ψr βr θr φr φfr yfr φl δ z 1 , z 2 , z3 ω 1 , ω2 , ω3

(vertical zr , lateral βr and longitudinal linear motion V , roll φr , pitch θr and yaw ψr rotation). The compliant joint (12) allows the lateral displacement (yfr ) and torsional rotation (φfr ) of the front body with respect to the rear (two dofs). The front suspension allows the vertical movement of the front wheels (z1 , z2 ) relative to the front frame (two dofs). Similarly, the revolute joint connecting the trailing arm (16) to the rear body introduces an additional degree of freedom (z3 ). The spherical joints between the parts of the steering mechanism and the revolute joint linking the steering column to the front frame define the additional degree of freedom of the steering line (δ). The revolute joint connecting the tilting lever (4) to the front frame allows their respective rotation (φl − 1 dof). Finally, the revolution (ω1 , ω2 , ω3 ) of the three wheels with respect to their hubs is modelled by three revolute joints and leads to three additional dofs. 3.2. Tyres The generalised forces developed by each tyre are computed in the block ‘Tyres’ (Figure 3). They are estimated on the base of the following inputs: • rotation of the wheel body frame Oxh yh zh with respect to the reference frame OXY Z (Figure 5): ⎧ ⎫ ⎧ ⎫ ⎨X ⎬ ⎨xh ⎬ Y = Rw yh . (1) ⎩ ⎭ ⎩ ⎭ Z zh

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Figure 6. Reference frames adopted to obtain the tyre forces. (a) Tyre reference frame. (b) Frames evidencing the rotation θyh about axis yh to pass from the SimMechanics frame Oxh yh zh to the frame Oxt yh z . (c) Frames evidencing the rotation θxt about the axis xt to pass from the frame Oxt yh z to the tyre reference frame Oxt yt zt .

The frame OXY Z is defined with respect to the inertial frame HXYZ. The origin O is at the intersection between the equatorial plane and the ground. The axes X , Y , Z are parallel to X, Y, Z. Axis X lies on the ground and it is oriented towards the direction of motion, Z is normal to the ground and directed upwards. It follows that Y lies on the ground and is directed to the left. • Velocity (V) of the point O, with respect to the inertial frame HXYZ (Figure 6(a)). ˙ of the tyre, both contained in the wheel • Radial displacement (r) and relative velocity (r) equatorial plane. These data are used to compute the side-slip force (Fyt,α ), the aligning torque (Mzt ) and the camber thrust (Fyt,γ ) that are expressed in the tyre reference frame Oxt yt zt . It is worth to note that Oxt yt zt is just the conventional tyre reference frame (0 and 0) except for Oxt yt zt that is already rotated of the steering angle (δ). The generalised forces Fyt,α , Mzt , Fyt,γ are obtained for each tyre starting from the computation of camber angle (γ ), side-slip angle (α) ¯ and vertical load Fzt . The procedure adopted to compute α¯ and γ is described in Appendix 1, while the tyre vertical load is computed using

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the approach presented in [24] and summarised below. The wheel hub centre is linked to the origin O by the parallel connection in the equatorial plane of a spring with stiffness kt and a viscous damper with damping ct . Therefore, their resulting force is ˙ Fr = kt · r + ct · r.

(2)

It can be expressed as a function of the vertical load Fzt and the lateral force Fyt as

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Fr = Fzt · cos(γ ) + (Fyt,α + Fyt,γ ) · sin(γ ).

(3)

Taking into account that the lateral force Fyt is a function of the vertical load Fzt and that Fr is computed from Equation (2), Equation (3) can be solved at each integration step to obtain the unknown vertical load Fzt . The tyre lateral force is computed using the tyre characteristics reported in Figure 7. These characteristics are obtained by a best fit of the experimental data supplied by the tyre manufacturer. They are implemented in the model as look-up tables. The effect of the tyre contact point migration is taken into account by computing the overturning moment Mxt about the longitudinal axis xt and the aligning torque Mzt about zt . They are obtained as a purely geometrical effect due to tyre round profile of radius rt that rolls on the ground plane: Mxt = Fzt · rt · tan(γ ), Mzt = Fxt · rt · tan(γ ).

(4)

The tyre generalised forces that are computed with respect to the reference frame Oxt yt zt are then expressed in the frame OXY Z (Figure 2) as ⎛ ⎞ ⎡ ⎞ ⎤ ⎛ FX , MX cos θZ − sin θZ 0 Fxt , Mxt ⎝ FY , MY ⎠ = ⎣ sin θZ cos θZ 0⎦ · ⎝Fyt , Myt ⎠ . (5) 0 0 1 FZ , MZ Fzt , Mzt where ϑZ is the angle between Oxt yt zt , and OXY Z . Angle ϑZ is computed as   nY · nxt θZ = arctan . nX · nxt Making reference to the frame OXY Z : ⎛ ⎞ ⎛ ⎞ 1 0 nX = ⎝0⎠ , nY = ⎝1⎠ , 0 0

nxt = Rw · Rϑyh

⎛ ⎞ 1 · ⎝0⎠ . 0

(6)

(7)

The rotation matrix Rϑyh is reported in explicit form in Appendix 1. 3.3. Adams-Motorcycle multibody model The present paragraph describes the Adams-Motorcycle model used to cross-validate the SimMechanics model presented above numerically. It was developed starting from a conventional motorcycle template in which the front subsystem (front body, front suspension, steering line, front hubs/uprights, front wheels) was rebuilt. The vehicle main frame is modelled as a rigid body. The trailing arm is hinged to the main frame by a rigid revolute joint and a spring–damper element modelling the shock absorber.

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Figure 7. Steady-state characteristics of the tyres. (a) Side force due to the side-slip angle α. (b) Aligning torque due to the side-slip angle α. (c) Camber thrust as a function of the camber angle γ .

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Figure 8. Layout of the front suspension system developed in Adams. The notation is consistent with that adopted in Figures 1 and 2. The circle and square marks indicate the cylindrical and spherical joints, respectively.

The rear wheel is hinged at the end of the trailing arm by a transversal axis revolute joint. The transmission system is modelled by a chain drive which connects the crown wheel that is fixed to the rear wheel to sprocket wheel that is connected to the gear box output shaft. The six-speed gear box is modelled, considering the transmission ratios and the equivalent inertia. The engine is modelled by introducing the torque map of the Yamaha 660 engine. The front body is connected to the rear by a lumped compliant joint enabling lateral motion along the y-axis and rotation about the longitudinal axis of the frame. Figure 8 shows the front subsystem as implemented in Adams. The different parts are labelled using the same notation used in Figures 1 and 2. The suspension arms are modelled as rigid elements connected to the front body 9 and to the uprights by rigid spherical joints. Similarly, the steering system is modelled by rigid bodies and torsionally rigid joints. The steering rods are connected to the uprights and to the steering lever by spherical and universal joints, respectively. The steering column is connected to the front body by a rigid revolute joint where the steering friction and damping are lumped. The tilting lever is modelled by a rigid beam hinged to the front frame by a rigid revolute joint. Spring–damper elements that connect the free end of the tilting lever to the upper wishbones are used to model the shock absorbers. The tyres are modelled using a standard tyre model (pac_mc_120_70R17.tir) included in the Adams library. The Paceika coefficients were tailored to fit properly the curves reported in Figure 7 and the relaxation model described by equations 18 and 19. The tyre sidewall profile was edited to take into account the tyre–ground contact point migration. The model described above has the same 16 degrees of freedom of the SimMechanics model. 3.4. Analytical equivalent model The analysis carried out with the multibody models shows that in straight running, the tricycle vehicle exhibits a dynamic behaviour very close to that of a motorcycle, with weave, wobble and capsize modes [32]. The question at this point is: how much of this behaviour is peculiar of the tricycle layout? To give an answer to this question, an analytical motorcycle model has been implemented with a set of parameters equivalent to that of the tricycle. The aim is to compare the dynamics of the tricycle and of the motorcycle to evidence if some fundamental differences exist. To this purpose, the model presented in [28] was adopted. It consists of two rigid frames joined

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Scheme of the equivalent motorcycle model. It is plotted on the vehicle layout.

at the steering axis with a steering degree of freedom of the front body with respect to the rear (Figure 9). The rear body is composed by the main vehicle body with the driver, the rear trailing arm and the rear wheel rigidly attached to it. The front body is composed by a single wheel, the hub and the front fork. The tyre forces are taken into account by considering only their linear behaviour. The stationary contribution of the camber thrust, side force and self aligning torque is then expressed as Fy = −Cγ · γ − Cα · α, Mz = CMz α,

(8)

while the moment about x axis due to the geometric effect of the tyre–ground contact migration is modelled as described in Equation (4). The vertical loads acting on the tyres are computed by considering the static equilibrium in the xz plane. The effect of the relaxation related to the tyre carcass distortion is modelled as described in Equation (A10). The aerodynamic forces are neglected. The schematic representation of the model is reported in Figure 9. The degrees of freedom are lateral displacement (y), yaw angle ( ), roll angle ( ) and steer angle (δ). The related mass, stiffness and damping matrices are reported in Appendix 2. The numerical values adopted for the model were extrapolated from those of the tricycle adopting the criteria described below. The inertia of the main body includes the contribution of all the parts connected to the rear frame (rear frame + rear suspension + rear wheel + driver). The computation of the front body parameters is based on the statement that the front wheels lean together with the vehicle body, maintaining the same inclination. The front wheels can therefore be considered collapsed in the vehicle symmetry plane xz. The steering axis lies on this plane. Its inclination angle η and the offset e1 represent the front wheel castor angles and the CTs. Due to the limitation of the model, the effects of the king pin offset and king pin angle are not taken into account. The mass and the inertia of the equivalent front body (m1 , J1x , J1z , J1xz , Jp1 ) are computed with reference to the equivalent steering axis. The contribution of all the parts subject to motion when the front wheels steer is taken into account. As a matter of fact, the front suspension arms and the tilt lever do not rotate as the vehicle rolls; they are only subject to lateral motion. Therefore, only their mass contribution is considered.

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Figure 10. Picture of the vehicle during tests for the model validation. (a) Image of track tests with the driver that is hitting the handlebar. On the left there is the box that contains the acquisition device. Such a box is located in the rear of the vehicle. (b) Wire potentiometers to measure the steering angle δ. (c) Wire potentiometer to measure the tilting lever angle l .

The equivalent single front tyre develops a side force and an aligning torque twice that of the tricycle tyre for a given side-slip angle and camber angle. This is due to the fact that, on the equivalent front tyre, acts a vertical load that is the sum of the tricycle front tyre forces.

4.

Experimental tests and model validation

Experimental tests are considered essential to validate the models described before and to identify some basic parameters. The tests were carried out on a motor-racing test track. The vehicle was instrumented with two wire potentiometers (Vasahi – Rotative 971, range 0–110◦ ) that were installed on the front body. The former was used to measure the tilting lever position (φl ) with respect to the front body. It is worth to note that this angle is related to the roll angle by a fixed ratio (5/6) if the shock absorber system is not subject to deformation. This corresponds to what happens during the free tilting operation on flat surface, the inertia of the tilting lever being quite negligible. The latter was attacked by the front body too and was connected to the steering column to measure the steering angle δ. The zoomed images on the right of Figure 10 shows where the potentiometers were installed and how they were connected to the end of the tilting lever (Figure 10(c)) and to the steering column (Figure 10(b)). All sensor data were sampled at the rate of 1 kHz with a 12 bit converter. The image of Figure 10(d) shows the acquisition unit fixed at the back of the vehicle. The tests were performed as described below. The rider drives the vehicle in straight running. At the defined constant speed, the rider releases the handlebar and hits it with one hand to apply an impulsive torque to the steering column. This event is evidenced in the main picture of Figure 10. The tests were performed at different speeds in the speed range of 20–100 km/h for each of the three configurations of the CT (15, 29, 46 mm). The comparison of the experimental and numerical time histories of the steering column and tilting lever angles was used to: • tune the polar and transversal moment of inertia of the wheels, the tyre characteristics of Figure 7, the structural damping associated to the compliance of the frame; • identify the peak torque applied to the handle bar and the damping of the steering system;

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Figure 11. Numerical and experimental time history of the steering angle δ after torque input of the steering system. Thin full line: experimental data, dashed line: SimMechanics model results, thick full line: Adams model results. Thick dashdot line in (a): torque input. (a) CT = 15 mm, V = 56 km/h; (b) CT = 15 mm, V = 68 km/h; (c) CT = 29 mm, V = 65 km/h; (d) CT = 29 mm, V = 80 km/h; (e) CT = 46 mm, V = 60 km/h, (f) CT = 46 mm, V = 78 km/h.

• validate the multibody models by comparing the numerical and experimental data for different vehicle speeds and configurations of the CT. Figures 11 and 12 show the comparison of the numerical and experimental time histories of the steering and tilting lever angles. The identification and tuning process was carried out with the CT set at 29 mm and with a vehicle speed of 65 km/h (Figures 11(c) and 12(c)). The peak value of the input torque during steer input tests was identified by a best fit with the experimental steering angles restricted to the first peak. The steering damping was identified as best fit with the steering angle δ and tilting lever angle l experimental results in a time interval containing 2–3 periods. The value identified for the steering damping is equal to 17 Nms/rad. This is quite high if compared with the damping of the steering joint in conventional motorcycles, but it must be considered that it takes into account the friction losses inside all the suspension ball joints (eight in total) and bearings (three in total) that are present in the steering system. The feeling perceived by steering the handlebar of the TTW and that of a motorcycle when standing still confirm that there is an order of magnitude between the torque applied in the two cases. The validation process was carried out by considering all the parameters identified above frozen with the CT set at 29 mm and at a vehicle speed of 65 km/h (Figures 11(c) and 12(c)). The results are shown in Figures 11 and 12. The correlations are satisfactory with the trail set at 29 and 46 mm, while the comparisons with the 15 mm configuration show a fine matching in terms of frequency but a higher amplitude of the experimental data. This is ascribed to the excitations coming from the ground irregularities that become not negligible with such a CT. The interpretation fits with the feeling of the drivers. They perceived a scaring and dangerous vibration of the steering system without any excitation of the handlebar even at moderate speed. On the contrary, it was hard to excite a vibratory motion when hitting the handlebar at the maximum trail. The drivers confirmed that the intermediate trail is the best compromise in terms of stability and sensitivity. To this end, the choice of using the 29 mm trail for the identification and tuning process was appropriate.

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Figure 12. Numerical and experimental time history of the tilting lever angle l after torque input of the steering system. Thin full line: experimental data, dashed line: SimMechanics model results, thick full line: Adams model results. (a) CT = 15 mm, V = 56 km/h; (b) CT = 15 mm, V = 68 km/h; (c) CT = 29 mm, V = 65 km/h; (d) CT = 29 mm, V = 77 km/h; (e) CT = 46 mm, V = 60 km/h; (f) CT = 46 mm, V = 78 km/h.

The authors are confident that the results presented in Figures 11 and 12 prove the validity of the numerical models at least at null roll angle and in the considered speed range. No additional dynamic effects were evidenced in straight running while a non-expected instability of the front suspension raised up at large roll angles when the vehicle was tested on the steering pad. This phenomena is described in detail in [33].

5.

Stability of the TTW

The present section describes the vehicle stability and the vibration modes in straight running at constant speed. The analysis of the linearised model at null roll angle is based on the computation of the eigenvalues and eigenvectors at different vehicle speeds (5–120 km/h, step 5 km/h). The complete pole map, with the three possible configurations of the CT, is shown in Figure 13. The analysis of the results evidences that, among the solutions lying on the real axis, there is a pair (one positive and one negative) describing the unstable capsize. This is due to the lack of a driver control that was not introduced for the sake of simplicity, the capsize being uncoupled from the other vibration modes due to symmetry issues. The eigenvalues referred to the vertical motion of the rear and front wheels are indicated as Vr and Vf . The solutions of the vertical and pitching motion of the vehicle body are close to the origin (0/1 Hz, 0/ − 5 l/s) and therefore not evident in the map. Note that they are not of interest too, being uncoupled from the lateral vibration modes in straight running due to the vehicle symmetry. The eigenvector analysis shows that the modes 1/5 are related to the lateral dynamics of the vehicle. In detail, the poles 1 and 2 represent the typical motorcycle weave and wobble. As for motorcycles, the CT affects the wobble more than the weave. This is evident especially when the trail is reduced from 29 to 15 mm.

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Figure 13. Complete root loci as function of the longitudinal speed in straight running for the three CT configurations (round marks – 15 mm, cross marks – 29 mm, square marks – 46 mm). Speed range: 5–120 km/h, step 5 km/h. The first speed is evidenced with a ticker mark. The values in the speed range of 5–50 km/h are reported with larger marks while that in between 55 and 120 km/h are evidenced with a smaller mark. Labels of the curves: 1 – weave, 2 – wobble, 3 – lateral motion of the front body with respect to the rear, 4 – rotation of the front body about the longitudinal elastic axis, 5 – rear wobble, T – vertical out-of-phase motion of the front wheels, Vr – vertical motion of the rear wheel and Vf – vertical in-phase motion of the front wheels.

Table 3.

Chassis compliance.

Lateral stiffness Lateral equivalent damping Torsional stiffness Torsional equivalent damping

N/m Ns/m Nm/rad Nms/rad

5.64 · 105 97.67 1.98·105 15.81

Eigenvalues 3 and 4 are related to the lateral and torsional motion of the front body with respect to the rear. This is due to the frame compliance. The relative frequencies are almost three and five times higher than the frequency of the wobble mode. Differently to what is evidenced in several studies presented in the literature [8,22,29], these modes slightly affect both the wobble and the weave. This is due to the high stiffness of the frame (about twice that of race motorcycles, Table 3, [24]) and to the reduced vertical distance of the longitudinal elastic axis of the frame from the ground. Eigenvalue 5 is the rear wobble, a mode in which the roll and yaw angles are almost in phase and both 180◦ out of phase with respect to the steering angle. This eigenvalue moves down and to the left in the root loci plot at increasing speed and results to be non-oscillatory above 50 km/h. Eigenvalue T is related to the front wheels 180◦ out-of-phase vertical motion. This means that when one wheel moves up, the other moves down of the same quantity. This is due to the quadrilateral layout of the front suspension arms and to the linkage between the shock absorbers by means of the tilting lever. Being the inertia of the tilting lever negligibly small in comparison with the mass of the wheels and the uprights, this movement does not involve the deformation of the shock absorbers. Therefore, the mode results to be governed by the inertia of the front unsprung masses and the radial front tyre compliance and damping. The

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Figure 14. Pole map restricted to lower frequency and low day rate. The circle marks are obtained by the SimMechanics model (speed range: 5–120 km/h, step 5 km/h). The first speed is evidenced with a ticker mark. The values in the speed range of 5–50 km/h are reported with larger marks while that in between 55 and 120 km/h are evidenced with smaller marks. The cross and asterisk marks are referred to the experimental data obtained by using the decaying exponential described by Equation (9). The solutions evidenced with triangles in graph b refer to the equivalent analytical model. Labels of the curves: 1 – weave, 2 – wobble, 5 – rear wobble, T – vertical out-of-phase motion of the front wheels, Vr – vertical motion of the rear wheel and Vf – vertical in-phase motion of the front wheels.

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Figure 15. Comparison of the experimental steering angle time history with the decaying exponential described by Equation (9). CT = 29 mm, vehicle speed = 77 km/h.

root loci evidences how this mode is slightly damped and weakly dependent on the speed during straight running. It seems to be decoupled from the lateral motion. The plots in Figure 14 represent the pole maps with reference to the three CTs in a limited frequency range of 14 Hz to evidence the modes of interest. Figure 14(b) shows also the results of the equivalent motorcycle model. The comparison with the numerical results of the tricycle models confirms that the three oscillatory modes depending on the vehicle speed can be ascribed to the weave, wobble and rear wobble. Additionally, the comparative analysis evidences how the weave mode is well described also by an equivalent two wheels model. Differently, the discrepancies on the wobble and rear wobble are evident. They are mainly ascribed to the layout of the TTW steering mechanism similar to that of an automobile. It must be noted that the front wheels of the TTW rotate about a steering axis having inclinations and ground offsets both in the longitudinal and transversal plane. The cross and asterisk marks in the plots of Figure 14 derive from the fitting process of the steering angle experimental data with the decaying exponential function δ=

2 

δ0k eωnk (−ζk +j

√

1−ζk2 )t

,

(9)

k=1

 where δ0k , ωnk · ζk and ωnk 1 − ζk2 represent the amplitudes, the decay rates and the vibration frequencies of the two harmonics that have been accounted for. Figure 15 shows the best fit of Equation (9) with the experimental results at a vehicle speed of 77 km/h with the CT set at 29 mm. The correlation between the two curves is considered satisfactory in frequency, while the discrepancy in the amplitude is ascribed to the damping model (a pure viscous damping) that cannot take properly into account all the actual dissipations in the steering system. Nevertheless, the comparison in Figure 14 between the values identified for ωnk · ζk

and ωnk 1 − ζk2 with the corresponding numerical solutions is fine. This proves that the lateral dynamics of the TTW tricycle is mainly related to the contribution of the weave and wobble modes.

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Figure 16.

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Decay rate of the main modes as function of the vehicle speed.

To evidence the vehicle stability at a low speed, in Figure 16 the decay rate of the of the most significant solutions described above is plotted as a function of the vehicle speed. The plot highlights two positive non-oscillatory eigenvalues up to 16 km/h. From 16 to 20 km/h, they converge into two complex conjugate solutions with positive real part and an oscillatory contribution having a frequency that is between 1 and 2 rad/s. In the speed range of 20– 40 km/h, all the solutions have negative real part. Above that speed, there is always a nonoscillatory solution with positive real part. Such a decay rate plot is similar to that presented in the literature about motorcycles [28].

6.

Conclusions

The paper describes the model implementing the lateral dynamics of the TTW vehicle and the stability analysis performed in straight running. The complexity of the layout and the goal of taking into account the main dynamic effects outlined in the recent literature on motorcycle dynamics suggested adopting a multi-body approach. To this end, the implementation was done in Matlab/Simulink/SimMechanics environment while an Adams-Motorcycle model was adopted for cross-validation. An equivalent motorcycle four degrees-of-freedom model was used to evidence the effects that can be predicted with such a simplified analytical model and to highlight the similarities in terms of stability between a motorcycle and a tricycle like the TTW. Tests performed on a test track were carried out to get sensitivity on the stability of the vehicle, tune some model parameters, identify the damping on the steering axis and prove the validity of the model. The stability analysis leads to the following conclusions: (1) The vibration modes and the issues on the stability of the TTW are similar to that of a motorcycle. (2) As in motorcycles, the vertical and lateral dynamics are uncoupled in straight running.

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(3) The effect of the lateral and torsional chassis compliance on the modes involving the motion of the front body is not dominant, as in motorcycles being the frame stiff enough and the elastic axis of rotation close to the ground. (4) The decay rate of the wobble mode is very sensitive to the CT so that a fine analysis of its value must be followed when designing the vehicle to guarantee stability. (5) The out-of-phase motion of the front wheels seems to be weakly coupled with the weave and wobble modes in straight running. This consideration will be verified with further analysis. If it will be confirmed, it will be possible to conclude that the track of the wheels does not influence the lateral stability of the vehicle in straight running. Excluding the remark 3 that is mainly related to the particular configuration of the TTW, the results obtained in the present paper and those summarised above can be extended to all the free tilting tricycles and quadricycles [1,4,7] in which the suspension kinematics enables the vehicle body to roll in parallel with the wheels. Acknowledgements The authors acknowledge the administration of the Guida Sicura test track in Susa for granting free the use of the test and the people of the TTW srl Company for their support.

References [1] Tilting-Three Whelers reference web site. Available at http://www.maxmatic.com/ttw_moto.htm [2] R. Hibbard and D. Karnopp, Twenty first century transportation system solutions – a new type of small, relatively tall and narrow active tilting commuter vehicle, Veh. Syst. Dyn. 25 (1996), pp. 321–347. [3] Ariel 3 web page. Available at http://www.3wheelers.com/ariel.html [4] Tilting vehicles images web site. Available at http://www.seriouswheels.com/cars/top-1997-Mercedes-BenzF-300-Life-Jet-Concept.htm [5] Clever Official Site. Available at http://www.clever-project.net/ [6] Carver Engineering, official site. Available at http://www.carver-engineering.com/ [7] Piaggio&C. s.p.a., official MP3 web site. Available at http://www.MP3.piaggio.com/ [8] N. Amati, A. Festini, P. Macchi, P. Massai, and A. Tonoli, Analisi di Stabilità di Veicoli Inclinabili a Tre Ruote: Confronto con Motoveicoli Convenzionali, Proceedings of the XXXVI AIAS Conference, AIAS 2007, Ischia, Napoli, 2007. [9] J. Gohl, R. Rajamani, P. Starr, and L. Alexander, Development of a novel tilt-controlled narrow commuter vehicle, Intelligent Transportation Systems Institute, CTS06-05, 2006. Available at http://www.cts.umn.edu/pdf/CTS06-05.pdf. [10] J. Gohl, R. Rajamani, L. Alexander, and P. Starr, Active roll mode control implementation on a narrow tilting vehicle, Veh. Syst. Dyn. 42 (2004), pp. 347–372. [11] S. Kidane, L. Alexander, R. Rajamani, P. Starr, and M. Donath, A fundamental investigation of tilt control systems for narrow commuter vehicles, Veh. Syst. Dyn. 46 (2008), pp. 295–322. [12] A. Snell, An active roll moment control strategy for narrow tilting commuter vehicles, Veh. Syst. Dyn. 29 (1998), pp. 277–307. [13] D. Karnopp and S. So, Active dual mode tilting control for narrow ground vehicles, Veh. Syst. Dyn. 27 (1997), pp. 19–36. [14] D. Karnopp, Vehicle Stability, Marcel Dekker, New York, 2004. [15] R.S. Sharp, The stability and control of pivot framed tricycles, in Proceedings of the 8th International Association for Vehicle System Dynamics Symposium on the Dynamics of Vehicles on Roads and Tracks, J. Karl Hedrick, ed., Swets and Zeitlinger, Amsterdam, 1984. [16] V. Cossalter, A. Doria, R. Lot, N. Ruffo, and M. Salvador, Dynamic properties of motorcycle and scooter tires: Measurement and comparison, Veh. Syst. Dyn. 39 (2003), pp. 329–352. [17] V. Cossalter, A. Doria, and R. Lot, Steady turning of two-wheeled vehicles, Veh. Syst. Dyn. 31 (1999), pp. 157–181. [18] V. Cossalter, A. Doria, M. Da Lio, R. Lot, and L. Fabbri, A general method for the evaluation of vehicle manoeuvrability with special emphasis on motorcycles, Veh. Syst. Dyn. 31 (1999), pp. 113–135. [19] V. Cossalter and R. Lot, A motorcycle multi-body model for real time simulations based on the natural coordinates approach, Veh. Syst. Dyn. 37 (2002), pp. 423–447. [20] R. Sharp, Wobble and weave of motorcycles with reference to police usage, Autom. Eng. 17 (1992), pp. 25–27. [21] R. Sharp, Stability, control and steering responses of motorcycles, Veh. Syst. Dyn. 35 (1999), pp. 291–318.

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[22] R. Sharp and C. Alstead, The influence of structural flexibilities on the straight-running stability of motorcycles, Veh. Syst. Dyn. 9 (1980), pp. 327–357. [23] R. Sharp, S. Evangelou, and J. Limebeer, Multibody aspects of motorcycle modelling with special reference to Autosim, Advances in Computational Multibody Systems, Springer, Dordrecht, The Netherlands, 2005, pp. 45–68. [24] R. Sharp, S. Evangelou, and J. Limebeer, Advances in the modelling of motorcycle dynamics, Multibody Syst. Dyn. 12 (2004), pp. 251–283. [25] R. Sharp and J. Limebeer, A motorcycle model for stability and control analysis, Multibody Syst. Dyn. 6 (2001), pp 123–142. [26] R. Sharp and J. Limebeer, On steering wobble oscillations of motorcycles, J. Mech. Eng. Sci. (2004), pp. 1449–1456. [27] V. Cossalter, Cinematica e dinamica della motocicletta, Edizioni Progetto, Padova, Italy, 2001. [28] R.S. Sharp, The stability and control of motorcycles, J. Mech. Eng. Sci. 13 (1971), pp. 316–329. [29] P.T.J. Spierings, The effects of lateral front fork flexibility on the vibrational modes of straight running single track vehicles, Veh. Syst. Dyn. 10 (1981), pp. 37–38. [30] H.B. Pacejka, Tire and Vehicle Dynamics, Society of Automotive, Engineers/Butterworth-Heinemann, Oxford, 2002. [31] A. Renna, Analisi e progetto di telaio per veicolo leggero, Master Degree, Politecnico di Torino, 2006. [32] G. Genta, Motor Vehicle Dynamics, 2nd ed., World Scientific, Singapore, 2003. [33] N. Amati, A. Festini, M. Porrati, A. Tonoli, Accoppiamento tra dinamica laterale e verticale nei veicoli inclinabili a tre ruote, Proceedings of the XXXVIII AIAS Conference, Torino, 2009.

Appendix 1. Computation of the tyre camber angle (γ) and side-slip angle (α) The present section describes the procedure implemented in the block ‘Tyre’to obtain the tyre reference frame Oxt yt zt (Figure 6), the tyre camber angle (γ ) and the tyre side-slip angle (α). The variables that are available from the block ‘Vehicle Structure’ are: • the inclination of each wheel equatorial plane (matrix Rw ) with respect to the wheel reference frame OXY Z (Figure 6, Equation (1)). • The velocity (V) of the point O (Figure 6(a)) in the inertial frame HXYZ. The adopted procedure is described here below. Step 1: Rotation of the reference frame Oxh yh zh about axis yh of an angle ϑyh . This leads to superimposing of the xh axis to xt lying on the ground plane. The new reference frame is then Oxt yh z . The relation between Oxh yh zh and Oxt yh z is defined as ⎧ ⎫ ⎧ ⎫ ⎤ ⎡ ⎨xh ⎬ ⎨ xt ⎬ cos ϑyh 0 − sin ϑyh ⎦. y 1 0 = Rϑyh yh where Rϑyh = ⎣ 0 (A1) ⎩zh ⎭ ⎩ z ⎭ sin ϑyh 0 cos ϑyh h Angle ϑyh is determined by letting nZ  · nxt · = 0. nZ  and nxt are the unit vectors of axis Z and xt . Making reference to the frame OXY Z ⎛ ⎞ ⎛ ⎞ 1 0 nZ = ⎝0⎠ , nxt = Rw · Rϑyh ⎝0⎠ . 0 1 Angle ϑyh is then found applying the condition (A2): 

0

0



1 · Rw · Rϑyh

⎛ ⎞ 1 · ⎝0⎠ = 0. 0

(A2)

(A3)

(A4)

Step 2: Once ϑyh is determined, the reference system Oxt yh z is rotated about the axis xt of angle ϑxt such as to superimpose z axis to Z . The relation between the frame Oxt yh z and Oxt yt zt is ⎧ ⎫ ⎧ ⎫ ⎤ ⎡ ⎨ xt ⎬ ⎨xt ⎬ 1 0 0 y sin ϑxt ⎦ . cos ϑxt = Rϑxt yt where Rϑxt = ⎣0 (A5) ⎩ zh ⎭ ⎩z ⎭ 0 − sin ϑxt cos ϑxt t Angle ϑxt is determined by letting nZ  · nzt · = 1, or nZ  · nyt · = 0, or nZ  · nxt · = 0.

(A6)

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Again, making reference to the frame OXY Z , ⎛ ⎞ 0 = ⎝0⎠ , 1

nZ

nzt = Rw · Rϑyh · Rϑxt

⎛ ⎞ 0 ⎝0⎠ . 1

(A7)

Angle ϑxt is then found by imposing the first condition in Equation (A6) that in explicit form is expressed as

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 0

0



1 · Rw · Rϑyh · Rϑxt

⎛ ⎞ 0 · ⎝0⎠ = 1. 1

(A8)

The two rotations about axes yh and xt lead to determine the tyre reference frame Oxt yt zt (always defined with respect to the frame OXY Z ) and the camber angle γ that is actually the angle ϑxt . Step 3: The knowledge of the frame Oxt yt zt and of the velocity vector V (both with respect to OXY Z ) leads to computing the side-slip angle α at steady state: u = V · xt , v = V · yt , α = arctan

v u

(A9) .

Being the force generated by the side-slip angle α related to the distortion of the tyre carcass (A3), the time or rolled distance delay is taken into account by using the following first-order filter [21–23]: α¯ =

1 α, σ¯ s + 1

(A10)

where s is the Laplace variable, and σ¯ is given by the following expression [23] depending on the tyre velocity (V) and on the tyre cornering stiffness Kyα :  σ¯ = Kyα

9.694 · 10−6 + 1.333 · 10−8 + 1.898 · 10−9 · V V

 .

(A11)

The numerical values reported in Equation (A11) are taken from [23]. They are referred to a tyre 180/55 that is quite similar to that adopted for the TTW model in terms of aspect ratio.

Appendix 2. Matrices describing the linearised equivalent motorcycle model The mass, stiffness and damping matrices of the equivalent motorcycle model are represented in the present section. They are referred to the generalised displacement vector x = (y · · · δ)T . Matrix M: ⎡

1 m − 2 (C1α σ¯ 1 + C2α σ¯ 2 ) ⎢ V ⎢ 1 ⎢− ⎢ V 2 (σ¯ 1 (C1α a + C1Mz ) + σ¯ 2 (C2Mz − C2α b)) ⎢ −mh M=⎢ ⎢ σ¯ 1 ⎢ m e − (C 1 1α e1 + C1Mz cos(η)) ⎢ V2 ⎣

−mh ×

Jxz Jx −m1 eh1 − J1z sin(η) + J1xz cos(η)

1 (C1α σ¯ 1 a − C2α σ¯ 2 b) V2 1 2 Jz − 2 (σ¯ 1 (C1α a + C1Mz a) + σ¯ 2 (C2α b2 − C2Mz b)) V Jxz −

m1 ec1 + J1z cos(η) + J1xz sin(η) σ¯ 1 a − 2 (C1α e1 + C1Mz cos(η)) V

⎤ e1 C1α σ¯ 1 V2 ⎥ σ¯ 1 e1 ⎥ m1 ec1 + J1z cos(η) + J1xz sin(η) + 2 (C1α a + C1Mz )⎥ ⎥ (A12) V ⎥ −m1 eh1 − J1z sin(η) + J1xz cos(η) ⎦ σ¯ 1 e1 J1x + 2 (C1α e1 + C1Mz cos(η)) V m1 e +

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Matrix C: ⎡

1 − (C1α + C2α ) V ⎢ ⎢ 1 ⎢− (C1α a − C2α b + C1Mz + C2Mz ) ⎢ V C=⎢ ⎢ 0 ⎢ ⎢ ⎣ 1 − (C1α e1 + C1Mz cos(η)) V



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V

0 Jp2 Jp1 + R01 R02



0 V cos(η)

Jp1 R01

1 (C1α a − C2α b) V 1 2 − (C1α a + C2α b2 + C1Mz a − C2Mz b) V   Jp1 Jp2 −V mh + + R02   R01 Jp1 a V m1 e + sin(η) − (C1α e1 + C1Mz cos(η)) R01 V mV −

⎤ C1α (e1 + σ¯ 1 cos(η)) V ⎥ ⎥ Jp1 1 sin(η) + (σ¯ 1 cos(η) + e1 )(C1α a + C1Mz )⎥ −V ⎥ R01 V ⎥ ⎥ Jp1 ⎥ −V cos(η) ⎥ R01 ⎦ 1 + (C1α e1 + C1Mz cos(η))(σ¯ 1 cos(η) + e1 ) + cδ V +

(A13)

Matrix K: ⎡

0 ⎢0 ⎢ K = ⎢0 ⎣0

0 0 0 0

−C1γ − C2γ −aC1γ + bC2γ −mg(h − rt ) + Fz1 rt1 + Fz2 rt2 m1 ge + e1 C1γ − Fz1 (rt1 sin(η) + e1 )

⎤ C1α cos(η) + C1γ sin(η) (aC1α + C1Mz ) cos(η) + aC1γ sin(η) ⎥ ⎥ m1 ge − Fz1 rt1 sin(η) ⎥ −(m1 ge + e1 C1γ ) sin(η) + Fz1 (rt1 sin(η) + e1 ) sin(η)⎦ +(C1α e1 + C1Mz cos(η)) cos(η) (A14)

All the terms reported in the matrices are listed in Table 4. Table 4.

Parameters adopted in the motorcycle equivalent model.

Parameter Total mass (vehicle + driver (75 kg) Mass of body 1 Moment of inertia of body 1 about x1 Moment of inertia of body 1 about y1 Moment of inertia of body 1 about z1 Mixed moment of inertia of body 1 in x1 z1 plane Moment of inertia of the vehicle about x axis (roll) Moment of inertia of the vehicle about y axis (pitch) Moment of inertia of the vehicle about yaw axis (yaw) Mixed moment of inertia of the vehicle in xz plane Distance G – front axle in x Distance G – rear axle in x Distance G–G1 in x Distance G–G2 in x Distance G1 – steering axis CT, x1 direction Caster angle Height of G, z direction

Symbol

Unit

Value

m

kg

301

m1 J1x

kg kg m2

48.5 11.02

J1y

kg m2

3.19

J1z

kg m2

0.5594

J1xz

kg m2

0

Jx

kg m2

128.9840

Jy

kg m2

222.6500

Jz

kg m2

104.86

Jxz

kg m2

−19.62

a b c1 c2 e e1 η h

m m m m m m rad m

0.713 1.03 0.693 0.133 0.0115 0.0287 0.1442 0.587 (continued)

914

N. Amati et al. Table 4.

Continued

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Parameter Height of G1 , z direction Damping coefficient of the steering line Rolling radius of the front tyre Rolling radius of the rear tyre Polar moment of inertia of the front wheel Polar moment of inertia of the rear wheel Front tyre side-slip stiffness Rear tyre side-slip stiffness Camber thrust coefficient of the front wheel Camber thrust coefficient of the rear wheel Front wheel self-aligning torque coefficient Rear wheel self-aligning torque coefficient Relaxation length front and rear tyre Radius of the tyre sidewall Front axle load vertical load Rear axle load

Symbol

Unit

Value

h1 cδ

m Nms/rad

0.378 17

R01 R02 Jp1

m m kg m2

0.308 0.319 2 · 0.475

Jp2

kg m2

0.475

(C1α )/Fz (C2α )/Fz (C1γ )/Fz

N/rad/N N/rad/N N/rad/N

7.579 7.579 1.02

(C2γ )/Fz

N/rad/N

1.02

C1Mz /Fz

Nm/rad/N

0.206

C2Mz /Fz

Nm/rad/N

0.206

σ¯ 12

m

0.0259

rt Fz1 Fz2

m N N

0.084 1744.2 1210.2