Technical Papers & Memos
Modeling and Simulation
Tech memos for several examples
This page contains a limited bibliography of technical papers related to the modling and simulation of motorcycle dynamics. Much of the work has been done in the UK, where Prof. Robin Sharp works with the same symbolic code generator (VehicleSim Lisp, formally known as AutoSim) used by Mechanical Simulation.
R. S. Sharp and Y. Watanabe. "Chatter vibrations of high-performance motorcycles," Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, Taylor and Francis, DOI:10.1080/00423114.2012.727440, October 2012. Show summary
Motorcycle racing teams occasionally experience speed-limiting vibrations of around 25 Hz frequency in mid-corner. The nature of the vibrations has not been closely defined yet and the mechanics are currently not properly understood. Conventional motorcycle-dynamics models are shown here to reveal the existence of a vibration mode that aligns with the experience being referred to, suggesting some explanations. Root loci for variations in speed or cornering vigour, demonstrating modal characteristics for small perturbations from trim states, are employed to indicate how the mode responds to changes in operation and design. Modal participation is examined for a lightly damped case. Influences on the natural frequency and damping of the mode are found and a way of stabilising the mode is suggested.
This paper describes modeling methods used in the commercial BikeSim® simulation package to represent alternative suspension design concepts. The modeling method used for automotive suspensions is applied to define generic suspensions for motorocycles. This method can represent multi-link suspension systems as well as traditional motorcycle suspensions with telescopic front forks and rear swing arms. Comparisons of two suspension types show a multi-link suspension can provide advantages over the traditional system for braking, acceleration (throttle), and cornering. Similar comparisons made with a chain-drive powertrain and a shaft-drive powertrain show less jacking with the chain-drive design. Although the math models include complex nonlinear motions, the computational efficiency supports fast operation; on a 2.8 GHz PC the simulation runs eight times faster than real time.
The main objectives of the work described are to devise an effective path-based motor- cycle simulation capability and to add to understanding of how riders control motor- cycles. Optimal linear preview control theory is applied to minimize a weighted sum of tracking errors, rider lean angle and control power. The time-invariant optimal control corresponding to a white noise disturbance and to an infinite optimization horizon is found for many situa- tions, involving variations in machine speed and performance priorities. Transformation of the problem from a global description, in which the optimal control is found, to a local description corresponding to the rider ís view, is described. It is concluded that a motorcycle rider model representing a useful combination of steering control capability and computational economy has been estab- lished. The model yields new insights into rider and motorcycle behavior.
R. S. Sharp, Optimal linear time-invariant preview steering control for motorcycles, The Dynamics of Vehicles on Roads and on Tracks (S. Bruni and G. Mastinu eds), supplement to VSD 44(1), Taylor and Francis (London), 2007.
R. S. Sharp, Motorcycle handling dynamics, Road and off-road Vehicle System Dynamics Handbook (eds Giampiero Mastinu and Manfred Plöchl), Taylor and Francis, London, 2007.
R. S. Sharp, Motorcycles and three-wheeled vehicles, Road and off-road Vehicle System Dynamics Handbook (eds Giampiero Mastinu and Manfred Plöchl), Taylor and Francis, London, 2007.
R. S. Sharp, Optimal stabilisation and path-following controls for a bicycle, Journal of Mechanical Engineering Science, Proc. I. Mech. E., Part C, 2007. Show summary
The article is about stabilising and path-tracking control of a bicycle by a rider. It builds on previously published work, in which it has been shown how a driver’s or rider’s preview of the roadway can be combined with the linear dynamics of an appropriate vehicle to yield a problem of discrete-time optimal-linear-control-theory form. In the previous work, it was shown how an optimal “driver” converts path preview sample values, modelled as deriving from a Gaussian white-noise process, into steering control inputs to cause the vehicle to follow the previewed path. The control compromises between precision and ease, to an extent that is controllable through choice of weights in the optimal control calculations.
Research into the dynamics of bicycles has yielded a benchmark model, with equations of motion firmly established by extensive cross-checking. Model predictions have been verified for modest speeds by experimental testing. The established optimal linear preview stabilising and tracking control theory is now brought together with the benchmark bicycle description to yield optimal controls for the bicycle for variations in speed and performance objectives. The resulting controls are installed in the bicycle, giving a virtual rider-controlled system, and frequency responses of the rider-controlled system are calculated to demonstrate tracking capability. Then path-tracking simulations are used to illustrate the behaviour of the controlled system. Tight and loose controls, representing different balances between tracking accuracy and control effort, are calculated and illustrated through the simulations.
R. S. Sharp, Dynamics of Motorcycles: Stability and Control, in Dynamical Analysis of Vehicle Systems: Theoretical Foundations and Advanced Applications (ed. W Schiehlen), Springer, Wien/ New York, 2007, pp. 47, in press.
R. S. Sharp, Application of optimal preview control to speed tracking of road vehicles, Multibody Dynamics, Milan, 25-28 June, 2007. Show summary
In previous work, high-fidelity motorcycle-rider dynamics models have been developed. Among other things, such models have been linearized for small perturbations from equilibrium turning ``trim'' states and used to establish the existence of resonant conditions when road undulations are regular and are tuned to a system natural frequency. The resonances have been linked with loss-of-rider-control accidents, in which anecdotal accounts have not infrequently referred to road undulations. The prior research is now extended to include nonlinear behaviour, allowing larger amplitude motions to be considered. Based on an appreciation of the potential vibration responses of complex smooth systems to sustained sinusoidal forcing, challenging running conditions are determined via linear theory. Subharmonic, superharmonic, internal and combination resonances could potentially occur. The interesting circumstances are simulated, and the time-history results for steer angle and yaw rate are time-frequency analysed usingshort-period Fourier-Transform techniques. Practical implications of the observed behaviour are discussed.
Evangelou, S., Limebeer, D. J. N., Sharp, R. S. and Smith, M. C., Mechanical steering compensators for high-performance motorcycles, ASME Journal of Applied Mechanics, 73 (5), 2006. Show summary
This paper introduces the idea of using mechanical steering compensators to improve the dynamic behavior of high-performance motorcycles. These compensators are seen as possible replacements for a conventional steering damper, and comprise networks of springs, dampers and a less familiar component called the inerter. The inerter was recently introduced to allow the synthesis of arbitrary passive mechanical impedances, and finds a new potential application in the present work. The design and synthesis of these compensation systems make use of the analogy between passive electrical and mechanical networks. This analogy is reviewed alongside the links between passivity, positive reality and network synthesis. Compensator design methods that are based on classical Bode-Nyquist frequency-response ideas are presented. Initial designs are subsequently optimized using a sequential quadratic programming algorithm. This optimization process ensures improved performance over the machine's entire operating regime. The investigation is developed from an analysis of specific mechanical networks to the class of all bi-quadratic positive real functions. This aspect of the research is directed to answering the question: what is the best possible system performance achievable using any simple passive mechanical network compensator?'' The study makes use of computer simulations, which exploit a state-of-the-art motorcycle model whose parameter set is based on a Suzuki GSX-R1000 sports machine. The results show that, compared with a conventional steering damper, it is possible to obtain significant improvements in the dynamic properties of the primary oscillatory modes, known as `wobble' and `weave'.
D. J. N. Limebeer and R. S. Sharp, Bicycles, motorcycles and models, IEEE Control Systems Magazine, Special issue on Advances in Motorcycle Design and Control, October 2006, 26(5), 34-61.
S. Evangelou, D. J. N. Limebeer, R. S. Sharp and M. C. Smith, Steering compensation for high-performance motorcycles: Passive mechanical compensators incorporating inerters, IEEE Control Systems Magazine, Special issue on Advances in Motorcycle Design and Control, October 2006, 26(5), 78-88.
S. Evangelou, D. J. N. Limebeer, R. S. Sharp, and M. C. Smith, An H-inf loop-shaping approach to steering control for high-performance motorcycles , in Control of Uncertain Systems: Modelling, Approximation, and Design (Bruce A. Francis, Malcolm C. Smith, Jan. C. Willems eds), Springer-Verlag, Berlin Heidelberg, 2006, 257-275. Show summary
A fixed-parameter active steering compensation scheme that is designed to improve the dynamic behavior of high-performance motorcycles is introduced. The design methodology is based on the Glover-McFarlane H-infinity loop-shaping procedure. The active steering compensator designed is seen as a possible replacement for a passive steering damper, or as an alternative to the more recently introduced passive mechanical compensation networks. In comparison with these networks, active compensation has several potential advantages including: (i) the positive-reality of the compensator is no longer a requirement; (ii) it is no longer necessary for the device to be low-order; (iii) in a software implementation, it is easy to adjust the compensator parameters and (iv) an adaptive, or parameter varying version of this scheme is a routine extension. The study makes use of computer simulations that exploit a state-of-the-art motorcycle model whose parameter set is based on a Suzuki GSX-R1000 sports machine. The results extend further the significant improvements achieved in the dynamic properties of the primary oscillatory modes (‘wobble’ and ‘weave’) obtained previously by replacing the conventional steering damper with passive mechanical steering compensation schemes.
Sharp, R. S., Evangelou, S. and Limebeer, D. J. N., Multibody aspects of motorcycle modelling, Advances in Computational Multibody Systems, J. Ambrósio (Ed.), Springer-Verlag, Dordrecht, The Netherlands, 2005, 45-68. Show summary
Modelling of the ride and handling dynamics of motorcycles using the symbolic mechanical multibody system package Autosim has been carried out since 1995. Motorcycles are principally of tree structure but their geometry is complex in relation to the tyre to road contact and tyre force and moment descriptions and to the chain drive system. They may contain closed kinematic loops, according to common suspension and steering design variations. Various aspects of the modelling problem are discussed and some implications, from a multibody standpoint, of choosing different options are exposed. Simulation results illustrate the “anti-squat” behaviour of a chain drive transmission and the “anti-dive” behaviour of a Telelever front suspension system.
Sharp, R. S., Limebeer, D. J. N. and Shaeri, A., Nonlinear vibrations of motorcycles forced by regular road roughness, Proc. 5 th Euromech Nonlinear Dynamics Conference, August 7-12, 2005, Eindhoven.
R. S. Sharp and D. J. N. Limebeer, On steering wobble oscillations of motorcycles,Proc. I. Mech. E., Part C, Journal of Mechanical Engineering Science, 218, 2004, 1449-1456. Show summary
The paper is aimed at an improved understanding of steering wobble oscillations of motorcycles through simulation. The background to the problem is discussed first. Then, an existing mathematical model of a manoeuvring motorcycle and rider is extended to include a yaw freedom for the upper body of the rider. The rider upper body and arm structural parameters are chosen in the light of newly published results from the testing of human subjects in a driving simulator, with forced motion of the steering wheel by means of an electric motor. Results show that steering wobble oscillations grow more vigorously as their amplitude increases beyond a few degrees of steering and that the stabilising influence of the rider tensing his/her muscles in response to a growing wobble problem is small. The work supports the idea that any machine which has a very lightly damped wobble mode at some operating condition may be made unstable by an unusual set of initial conditions and that the natural response of the rider to the problem will be largely ineffective. This idea is closely aligned with anecdotal accounts from general motorcycle usage.
Sharp, R. S., Evangelou, S. and Limebeer, D. J. N., Advances in the modelling of motorcycle dynamics, Multibody System Dynamics, 12(3), 2004, 251-283. Show summary
Starting from an existing advanced motorcycle dynamics model, which allows simulation of reasonably general motions and stability, modal and response computations for small perturbations from any trim condition, improvements are described. These concern (a) tyre / road contact geometry; (b) tyre shear force and moment descriptions, as functions of load, slip and camber; (c) tyre relaxation properties; (d) a new analytic treatment of the monoshock rear suspension mechanism with sample results, (e) parameter values describing a contemporary high performance machine and rider, (f) steady-state equilibrium and power checking and (g) steering control. In particular, the “Magic Formula” motorcycle tyre model is utilised and complete sets of parameter values for contemporary tyres are derived by identification methods. The new model is used for steady turning, stability, design parameter sensitivity and response to road forcing calculations. The results show the predictions of the model to be in general agreement with observations of motorcycle behaviour from the field and they suggest that frame flexibility remains an important design and analysis area, despite improvements in frame designs over recent years. Motorcycle rider parameters have significant influences on the behaviour, with results consistent with a commonly held view, that lightweight riders are more likely to suffer oscillation problems than heavyweight ones.
S. Evangelou, D. J. N. Limebeer, R. S. Sharp, and M. C. Smith, Steering compensation for high-performance motorcycles, Proc. 43 rd IEEE Conference on Decision and Control, Bahamas, Dec. 14-17, 2004, 749-754. Show summary
This paper introduces the idea of using a mechanical steering compensator to influence the dynamic behaviour of a high-performance motorcycle. The compensator is seen as a possible replacement for a conventional steering damper, and comprises a network of a spring, a damper and a less familiar component called the inerter. The inerter was recently introduced to allow the synthesis of arbitrary passive mechanical impedances, and finds a new potential application in the present work. The approach taken here to design the compensator is based on classical Bode-Nyquist frequency response ideas. The vehicle study involves computer simulations, which make use of a state-of-the-art motorcycle model whose parameter set is based on a Suzuki GSX-R1000 sports machine. The study shows that it is possible to obtain significant improvements in the dynamic properties of the primary oscillatory modes, known as “wobble” and “weave”, over a full range of lean angles, as compared with the standard machine fitted with a conventional steering damper.
Shaeri, A., Limebeer, D. J. N. and Sharp, R. S., Nonlinear steering oscillations of motorcycles, Proc. 43 rd IEEE Conference on Decision and Control, Bahamas, Dec. 14-17, 2004, 773-778. Show summary
Extensive prior modelling of the three-dimensional motions of motorcycles, which has depended heavily on linearization for small perturbations from equilibrium “trim” states, is extended to cover large amplitude, non-linear operating regimes. For a cornering machine, road undulation displacement forcing is shown to be capable of exciting subharmonic and superharmonic responses. A procedure for identifying particular operating conditions that may be expected to yield internal or combination resonances for a baseline modern machine/rider combination is devised. Interesting cases are examined by simulation and the results analysed by short time Fourier transform processing of the output signals. Internal and combination resonances are shown to occur under specially chosen circumstances. A procedure for choosing these special circumstances is described. Some practical implications are also considered.
Sharp, R. S., Evangelou, S. and Limebeer, D. J. N., Improved modelling of motorcycle dynamics, ECCOMAS Thematic Conference, Multibody Dynamics 2003, Lisbon, July 1-4, MB2003-029 (CD-ROM). Show summary
Starting from an existing advanced motorcycle dynamics model, which allows simulation of reasonably general motions and stability and modal computations for small perturbations from any trim condition, improvements are described. These concern (a) tyre / road contact geometry; (b) tyre shear force and moment descriptions, as functions of load, slip and camber; (c) tyre relaxation properties; (d) a new analytic treatment of the monoshock rear suspension mechanism with sample results and (e) parameter values describing a contemporary high performance machine and rider. The new model is used for limited stability and design parameter sensitivity calculations. The results show the predictions of the model to be in general agreement with observations of motorcycle behaviour from the field and they suggest that frame flexibility remains an important design and analysis area, despite improvements in frame designs over recent years. They also suggest that the choice of rider parameters, for which there is no effective conventional wisdom, is less critical.
D. J. N. Limebeer, R. S. Sharp and S. Evangelou, “Motorcycle steering oscillations due to road profiling”, Proc. ASME, Journal of Applied Mechanics, 69 (6), 2002, 724-739. Show summary
A study of the effects of regular road undulations on the dynamics of a cornering motorcycle is presented. This work makes use of a modified version of a motorcycle model that was first described in . We make use of root-locus and frequency response plots that were derived from a linearized version of the model; the linearization is for small perturbations from a general steady cornering equilibrium state. The root-locus plots provide information about the damping and resonant frequencies of the key motorcycle modes at different machine speeds, while the frequency response plots are used to study the propagation of road forcing signals to the motorcycle steering system. Our results are based on the assumption that there is road forcing associated with both wheels and that there is a time delay between the front and rear wheel forcing signals – this is sometimes referred to as wheelbase filtering. The time delay is simply the ratio of the machine’s wheelbase to the machine speed. As has been explained before , control systems are used in the simulation model to maintain both the machine speed and the machine roll angle at preset values for flat road running.
The results show that at various critical cornering conditions, regular road undulations of a particular wavelength can cause severe steering oscillations. At low speeds the machine is susceptible to road forcing signals that excite the lightly damped wobble and front suspension pitch modes. At higher speeds it is the weave mode and front wheel hop modes that become vulnerable. We believe that the results and theory presented here explain many of the stability related accidents that have been reported in the popular literature and are therefore of practical import.
The models used in this research make use of the multibody modelling package AUTOSIM . The motorcycle and tyre parameters can be found at the end of the code.
R. S. Sharp and D. J. N. Limebeer, "A motorcycle model for stability and control analysis", Multibody System Dynamics, 6(2), 2001, 123-142. Show summary
The observed dynamic behaviour of motorcycles suggests that interesting and significant motions occur that are not currently understood. The most elaborate modelling exercise completed so far has produced results that need confirmation and extension. The construction of these models necessitates the use of automated methods and one such modelling methodology is described. The automated model building platform that was used here is AutoSim. This code was used to generate a variety of linear and nonlinear models in symbolic form. The relatively complex geometry of the steering system and the front tyre force system is discussed in detail and a new method of checking the self-consistency of the model is described and exploited. Sample results in the form of root-locus plots for small perturbations from straight running and cornering equilibrium states are presented. These are used to reproduce important findings from the literature. Conclusions are drawn on the basis of the results presented.
D. J. N. Limebeer, R. S. Sharp and S. Evangelou, “The stability of motorcycles under acceleration and braking” Proc. I. Mech. E., Journal of Mechanical Engineering Science, 215(C9), 2001, 1095 - 1109. Show summary
A comprehensive study of the effects of acceleration and braking on motorcycle stability is presented. This work is based on a modified version of the dynamic model that was first presented in , and is thought to be the most comprehensive motorcycle dynamic model in the public domain. Extensive use is made of both nonlinear and linearised models. The models are written in LISP and make use of the multibody modelling package AUTOSIM . There is novelty in the way in which control systems have been used to control the motorcycle drive and brake systems in order that the machine maintains desired rates of acceleration and deceleration.
The results show that the wobble mode of a motorcycle is significantly destabilized when the machine is descending an incline, or braking on a level surface. Conversely, the damping of the wobble mode is substantially increased when the machine is ascending an incline at constant speed, or accelerating on a level surface. This probably accounts for the pleasingly stable “feel” of the machine under firm acceleration. Except at very low speeds, inclines, acceleration, and deceleration appear to have little effect on the damping or frequency of the weave mode. Nonlinear simulations have quantified the known difficulties to do with rear tyre adhesion in heavy braking situations that are dominated by rear wheel braking.
Sharp, R. S., Stability, control and steering responses of motorcycles, Vehicle System Dynamics, 35(4-5), 2001, 291-318. Show summary
Following on from the author's reviews of the stability and control of motorcycles in 1978 and 1985, the paper treats the earlier material in tutorial fashion and adds more recent information. Fixed and free control properties are compared and it is concluded essential to the operation of a motorcycle that the steering system is free. Motorcycles typically possess lightly damped oscillatory modes and the properties of these modes are discussed. Small perturbations from straight line motion and from cornering equilibrium states are treated. Steering control by handlebar torque and by rider upper body lean torque are compared. The behaviour of the rider as an extension of the structure of the machine and as a controller is also discussed. Theoretical analysis, experimental measurements and general experience are linked as far as possible.
More recent years have seen a strong movement towards the use of structurally efficient frames for large motorcycles, in particular, and multibody dynamics software for handling analysis. Theoretical predictions can be extended in ways hardly possible before automated multibody analysis became available. The state of the art is illustrated with a description of a contemporary motorcycle / rider model and some results which are derived from it. Directions for future work are indicated.
Robin S. Sharp, “Variable geometry active rear suspension for motorcycles”, Proceedings of AVEC 2000, Ann Arbor, MI, August 2000, 389-395. Show summary
Active rear suspension by variable geometry is associated with a powerful motorcycle. The variable geometry principle here involves the variation of the leverage ratio between conventional spring damper unit and road wheel. An actuator is imagined to act as a displacement controller to vary the geometry. Levelling and cornering weave mode stabilisation controls are devised and tested by modelling and simulation methods.
The model used is a development of a published one. The original is described briefly and the developments are discussed. Computational methods employed are outlined. Results are included to illustrate steady turning equilibrium states, root locus plots with speed as the variable parameter for unstabilised and stabilised machines and motion time histories. It is concluded that the variable geometry system is effective in levelling and cornering weave mode stabilisation. It also appears to be practically feasible.
Robin S. Sharp, David J. N. Limebeer and Mahbub R. Gani, "A motorcycle model for stability and control analysis", Advances in Computational and Multibody Dynamics (Jorge A. C. Ambrosio and Werner O. Schiehlen Eds), IDMEC/IST Euromech Colloquium 404, 1999, 287-312.
M. Gani, D. J. N. Limebeer and R. S. Sharp, "The analysis of motorcycle dynamics and control", International Workshop on Modelling and Control of Mechanical Systems, Imperial College, London, June 17-20, 1997.
M. Gani, D. Limebeer and R.S. Sharp, "Multibody simulation software in the analysis of motorcycle dynamics", 8th IFAC/IFIP/IFORS Symposium on Transportation Systems '97, June 16-18, 1997, Chania, Greece.
M. Gani, RS Sharp and D.J.N. Limebeer, "Multibody simulation software in the study of two-wheeled road vehicles", 35th IEEE Conf. Decision and Control, Kobe, Japan, Dec. 11-13, 1996.