Stability Analysis of a Two-wheeler during Curve Negotiation under Braking

Transcription

1 NIT, Durgapur, India, Decemer 17-18, 009 NaCoMM-009-DVAMMG4 Staility Analysis of a Two-wheeler during Cure Negotiation under Braking M Ghosh 1, S Mukhopadhyay * 1 Dept. of Mech. Engg., Bengal College of Engg. and Technology, Durgapur, India Dept. of Mech. Engg., NIT, Durgapur, India * Corresponding author ( msumitnit@yahoo.co.in) Astract The two-wheelers show interesting dynamic characteristics. They are statically unstale. But the roll instaility disappears as the forward speed increases. Here a simplified model is considered to analyze the effects of forward speed and raking force on the roll instaility during cornering of a two-wheeler. Variations of ike parameters are also studied in the releance of roll staility. The present work helps to understand some important concepts aout a two-wheeler negotiating a turn under applied raking force. eywords: Cornering, Braking, Roll-staility, Tire properties. 1 Introduction Steering a two-wheeler inoles a complex interaction etween centrifugal force, graitational force and the torque applied to the handle ar together with the geometry of the two-wheeler and rider. Leaning the two-wheeler into the turn and maintaining an appropriate forward speed allows graitational forces to alance the centrifugal forces, leading to a controlled and stale turn. One method of estalishing the proper lean is counter-steering in which handlear is turned counter to the desired turn and thus deeloping a centrifugal torque that leans the two-wheeler appropriately. Alternately the required lean can e generated y throwing the rider s hips in the direction counter to the turn. The situation ecomes een more complicated when a sudden raking force is applied. During raking, the raking forces, the load distriution, changes in the cornering stiffness and the camer stiffness and lateral adherence mainly affects the lateral staility of the twowheeler. The major prolem in applying rake during cure negotiation is the tendency of tire skidding. The staility analysis of the two-wheeler proides important information aout its handling capailities and riding safety. A numer of important studies hae een reported on ehicle dynamics and control to optimize the performance of ground ehicle with regard to staility and road handling. Simple models of icycle dynamics are gien in the text ook [1].. J. Åstrom, R. E. lein and A. Lennartsson hae treated the icycles from the perspectie of control []. Models of different complexity are presented and models that capture essential ehaior such as self stailization and demonstrate the difficulties with rear wheel steering are presented. R. S. Sharp and D. J. N. Limeeer discussed aout the confirmation and extension of motorcycle model using automated model uilding platform, AutoSim [3]. V. Cossalter R. Lot and F. Maggio hae presented a study on the raking style and its implication on motorcycle staility [5]. The ideal raking strategy and raking maneuers are analyzed using a detailed multi-ody model. D. J. N. Limeeer, R. S. Sharp discussed aout different icycle and motorcycle models for staility and control [6]. Daid J. N. Limeeer and Amrit Sharma studied the dynamics of the accelerating icycle under straightrunning and cornering conditions [8]. Two-wheeler model and Equations.1 Two-wheeler model for cure negotiation without raking For the purpose of the analysis a simple twowheeler is considered as shown in fig.1. The coordinates used to analyze the two-wheeler is shown in fig. (a). The inertial system is with the axes uw and origin O. The system uw is fixed to inertial frame and the system xyz has its origin at contact point of rear wheel with the xy plane. The x-axis is in the direction of the contact line of the rear wheel with xy plane and z- axis is ertical and y-axis is perpendicular to x and positie on the left side of the two-wheeler. The roll angle Ф of rear frame is positie when leaning to the right and the steer angle δ is positie for steering left. The angle etween axes x and u, is ψ indicating the orientation of rear wheel plane. 304

2 NIT, Durgapur, India, Decemer 17-18, 009 NaCoMM-009-DVAMMG4 amg N f (3) Fig. 1: Parameters defining a simple two-wheeler. Due to tilt in steer axis y an angle, the effectie front angle is [] δf δ sin (1) Fig. 3: (a) Tire model, () Reaction forces on front wheel. Effectie centrifugal force on front wheel am Fcf r am Fcf δ f am sin Fcf δ (4) Lateral force resisting sideslip [5] F C θ + C (5) s f Fig. (a): Coordinate system. The static torque alance equation T [ F + ( C θ + C )] c sin 0 cf Sustituting equations (1) - (5) in equation (6) 1 C C δ T c sin f θ (6) (7) Angular momentum alance for the two-wheeler frame [] D m J & sin ( mgh & δ + h acg) sin δ (8) Sustituting δ from equation (7) in main frame equation (8) Fig. (): Tyre forces and sideslip angle. The effectie front fork roll angle is gien y f δ cos () Normal reaction force on front wheel excluding dynamic and centrifugal effect J && + D c Where, DC sin & + [ T& + c T [ C C sin mgh] sin ]θ (9) 305

3 NIT, Durgapur, India, Decemer 17-18, 009 NaCoMM-009-DVAMMG4 am sin C m( acg) D mah cos J mh The aoe equation reeals that torque at the handle ar has a great influence on roll angle as well as staility of the two-wheeler.. Two-wheeler model for cure negotiation with raking For staility analysis of the two-wheeler during cornering with raking the model considered is shown in fig.3. The key purpose of the model is to discuss the alancing prolem for two-wheeler during cornering under raking. The distriution of forces during raking is shown in fig.4. Where, m sin 1 Fi f m Da sin h acg Equation (11) is deried considering deceleration due to raking and therefore, elocity as a ariale with time. Breaking force largely affects the staility of the two-wheeler which can e controlled suitaly applying torque on the handle ar. Thus equations (9) and (11) proide qualitatie explanation for stailization. 3 Simulation Simulation is carried out for a motor cycle using equations (9), (10) and (11) to study the response of roll angle as the two-wheeler negotiates a cure, oth at constant speed as well as under uniform raking using a program written in C language. The parameters of the two-wheeler used for this simulation are shown in tale- 1. The flow chart for the programming is as follows: Fig. 4: Forces acting on the two-wheeler during raking. Vertical reaction force due to raking on front and rear wheel N N f r amg mfh + ( a) mg mfh (10) Sustituting δ in main frame equation (8) from equation (7) J && + D c DC sin & + [ C mgh ] 1 T& + T [ 1C c sin 1 ] θ (11) Fig. 5: Flow chart for simulation 306

4 NIT, Durgapur, India, Decemer 17-18, 009 NaCoMM-009-DVAMMG4 Tale-1: Two-wheeler parameters Two-wheeler Parameters: Value Unit Mass (m) 00 kg Length of wheel ase () 1.54 m Position of C.G in plane 0.6 m containing rear wheel (h) Position of C.G along x m axis from P 1 (a) Head angle () 70 Deg. Trail (c) m Sideslip angle (θ) Deg. rider. It is also seen that such instaility of the twowheeler gradually decreases with the increase in speed. The two-wheeler shows unstale roll ehaiour unless the rider applies a suitale torque at the handle-ar. Rider is assumed to ehae like a PID controller y applying a steer torque proportional to icycle lean [9]. The handle ar torque T p ( d ) + & + d t i 0 Where, p, d and i are controller gains. ( d )dt Tire parameters C and C Ф changes with ertical load and raking force [5]. The response of roll angle is studied during cornering with a desired lean ( d ) of 0.5 radian (approx ) at different running speed of the two-wheeler first, without raking and then with seeral raking forces. The effect of raking is plotted (fig.6) at a running speed of 0 km/hr. Fig. 7: Root locus at arious running speeds under different raking conditions. The response of the roll angle for a desired lean under arious raking conditions are shown for running speed of 40 km/hr (fig.8) and 50km/hr (fig.9). At a running speed of 40 km/hr the two-wheeler shows good staility ehaiour during a steady cornering without the application of any raking force (fig.8). Fig. 6: Roll response of the two-wheeler as it negotiates a cure with and without raking forces. 4 Results and Discussion Staility analysis for the two-wheeler is carried out under arious running speeds without any raking force as well as applying different magnitudes of raking forces. The left hand side of the system equations (9) and (11) and the root locus plot for the corresponding characteristic equation (fig.7) reeals that the system is unstale with the characteristic roots falling at the right hand side of the complex plane. The inherently unstale two-wheeler can only e made stale with an appropriate steering torque proided at the handle ar y the Fig. 8: Roll response of the two-wheeler as it negotiates a turn with and without raking forces. The staility of the two-wheeler is seriously affected when the raking forces are introduced. It is also osered that peak amplitude of roll decreases to some extent with the application of increased raking forces. Moreoer the roll oscillation seems to settle faster with an increase in raking force and the application of appropriate torque on the handle ar. Similar oserations are made for a higher running speed of 50 km/hr (fig.9). These oserations are important for deciding the safety of the rider ecause an increase in the peak amplitude 307

5 NIT, Durgapur, India, Decemer 17-18, 009 NaCoMM-009-DVAMMG4 may cause the wheel to skid if sufficient support is not otained from road and tire interaction. Fig. 9: Roll response of the two-wheeler as it takes a turn with and without raking forces. Taking the fact that the rider acts as a PID controller, the roll oscillation characteristics can e much improed with a sustantial (nearly six times) increase in the deriatie gain (fig.10). Fig. 11: Effect of parameter changes on the roll ehaiour at a speed of 40 km/h (a) change of mass, () change of trail. 5 Conclusions Fig. 10: Roll response with an increased deriatie gain for handle ar torque. It is osered that some of the parameters, e.g., the mass and the trail at the front wheel of the two-wheeler contriute significantly towards its staility. Tire characteristics together with road conditions are also important in this regard. Effects of change in mass and change in trail towards roll staility ehaiour is shown in fig.11 for a running speed of 40 km/h under the condition of a raking force of 500 N. As a comparison the aoe cases are also shown without the application of any raking force. The present paper considers a simplified model of a two-wheeler assuming only its roll freedom. The analysis reeals important facts aout running the twowheeler in a cure especially when a raking force is applied on it. To alance the centrifugal force and the graity force a two-wheeler in a turn must lean towards the centre of turn. This is essential for ensuring the staility of the two-wheeler. The time responses show that roll ehaiour worsen with the raking force applied during a turn. The condition ecomes een more worse if the mass of the two-wheeler increases (with a pillion rider) or if the front wheel trail is reduced. But as the peak amplitude is less with harder raking force, staility of the two-wheeler in a cure under such condition of hard raking may e etter if proper road condition is met. References [1] S. Timoshenko and D. H. Young. Adanced Dynamics. McGraw Hill, New York, 1948, pp. 39. [] Åström,.J., R. E. lein and A. Lennartsson (005) Bicycle Dynamics and Control. IEEE CSM. August, [3] R. S. Sharp and D. J. N. Limeeer, A motorcycle model for staility and control analysis, Multiody System Dynamics, ol. 6, no., pp ,

6 NIT, Durgapur, India, Decemer 17-18, 009 NaCoMM-009-DVAMMG4 [4] V. Cossalter R. Lot and F. Maggio: The Modal Analysis of a Motorcycle in Straight Running and on a Cure, Meccanica, luwered, (003) [5] Vittore Cossalter, Roerto Lot and Faiano Maggio: On the Braking Behaior of Motorcycles, Copyright 004 SAE International and Copyright 004 Society of Automotie Engineers of Japan, Inc / [6] D. J. N. Limeeer, R. S. Sharp, Bicycles, Motorcycles, and Models IEEE Control System Magazine, pp , Octoer 006. [7] D. J. N. Limeeer, R. S. Sharp, and S. Eangelou, The staility of motorcycles under acceleration and raking, Proc. I. Mech. E., Part C, Journal of Mechanical Engineering Science, ol. 15, pp , 001. [8] Daid J. N. Limeeer and Amrit Sharma The Dynamics of the Accelerating Bicycle Control and Signal Processing, 008. ISCCSP, 008. [9] D. Bortoluzzi, R. Lot, N. Ruffo, Motorcycle steady turning: the significance of geometry and inertia Nomenclature N f Normal force at front wheel, N N r Normal force at rear wheel, N F cf Effectie centrifugal force on front wheel, N F s Lateral force resisting side slip, N F i Inertia force due to raking, N S f Front raking force, N S r Rear raking force, N C Cornering stiffness, N/rad. C Φ Camer stiffness, N/rad. m Mass, kg g Graitational acceleration, m/s Head angle, rad δ Steer angle with zero trail, rad δ f Effectie steer angle, rad Φ Roll angle, rad Φ f Front fork roll angle, rad h Position of C.G in plane containing rear wheel, m a Position of C.G along X-axis, m Length of wheel ase, m c Trail, m ψ Orientation of rear wheel plane, rad T Torque applied to handle ar, Nm θ Sideslip angle, rad Forward elocity, m/s f Deceleration during raking, m/s p, d, i Controller gains 309