Article

Parametric analysis of the stability of a bicycle taking into account geometrical, mass and compliance properties

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Abstract

Some studies of bicycle dynamics have applied the Whipple Carvallo bicycle model (WCBM) for the stability analysis. The WCBM is limited, since structural elements are assumed to be rigid bodies. In this paper, the WCBM is extended to include the front assembly lateral compliance, and analysis focuses on the study of the open loop stability of a benchmark bicycle. Experimental tests to identify fork and wheel properties are performed, this data is used in the stability analysis for ranking the influence of design parameters. Indexes from the eigenvalues analysis are applied in a full factorial approach. The results show that introducing front assembly compliance generates a wobble mode with little effect on self-stability. The forward displacement of the centre of mass of the rear frame and the increment in trail lead to large increments in the self-stability, whereas increments in front wheel radius and wheelbase reduce stability.

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... Lumped parameter models are described in [15]. This kind of modelling is also found in other sources, especially when a more complex non-rigid model is developed [16][17][18][19][20][21]. ...
... For this work, besides the fork bending compliance, two tyre parameters are of particular importance: the relaxation length and the cornering stiffness. Some research [3,16,17,20] has underlined that a tyre model with cornering stiffness is necessary in order to simulate the wobble mode. This also suggests an important consideration: the principal cause of the wobble is given by the tyre's response. ...
... The other important tyre parameter influencing wobble is the relaxation length [3,13,23]. Several works [3,16,17,20,24] investigated its effect on wobble damping, underlying how the relaxation length destabilises the wobble mode, since it generates a delay between the wheel steering and the tyre-lateral force generation. As a consequence, Sharp [3] (p. 323) clarifies that both tyre sideslip and tyre relaxation properties are fundamental for a proper representation of a motorcycle's dynamic properties. ...
Article
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The present work follows in the tracks of previous studies investigating the stability of motorcycles. Two principal oscillation modes of motorcycles are the well-known wobble and weave modes. The research in this field started about fifty years ago and showed how different motorcycle parameters influence the stability of the mentioned modes. However, there is sometimes a minor lack in the physical analysis of why a certain parameter influences the stability. The derived knowledge can be complemented by some mechanical momentum correlations. This work aims to ascertain, in depth, the physical phenomena that stand behind the influence of fork bending compliance on the wobble mode and behind the velocity dependence of the weave damping behaviour. After a summary of the relevant work in this field, this paper presents different rigid body simulation models with increasing complexity and discusses the related eigenvalue analysis and time behaviour. With these models, the mentioned modes are explained and the physical phenomena only partly covered by the literature are shown. Finally, the influence of the rider model on weave and wobble is presented.
... The development of a good model for a two-wheeled vehicle, as described previously, is not a trivial task indeed. In particular, considering that the predominant linearized equations from the literature are not based on a systematic linearization of full nonlinear differential equations, this task is even more challenging [52,53]. Thus far, systematic linearizations have not achieved analytical expressions for the linearized equation coefficients, until recently, when some authors, such as in [15,20,21], have currently achieved it by developing an automatic computer-aided linearization procedure, finding agreement with the well-known benchmark models [45]. ...
... In particular, for the proposed kinematic model, Equation (51) was found by means of symbolic manipulations and it was subsequently validated numerically through numerical experiments. In this way, it was possible to eliminate the second constraint equation of the surface parametrized model using Equations (53) and (54), thereby proving the independence between the nongeneralized parameter β and the angle φ f . To conclude, it is appropriate to mention that the final equation obtained to solve this issue is similar to that proposed by Cosalter [60]. ...
Article
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In this investigation, a closed-chain kinematic model for two-wheeled vehicles is devised. The kinematic model developed in this work is general and, therefore, it is suitable for describing the complex geometry of the motion of both bicycles and motorcycles. Since the proposed kinematic model is systematically developed in the paper by employing a sound multibody system approach, which is grounded on the use of a straightforward closed-chain kinematic description, it allows for readily evaluating the effectiveness of two alternative methods to formulate the wheel-road contact constraints. The methods employed for this purpose are a technique based on the geometry of the vector cross-product and a strategy based on a simple surface parameterization of the front wheel. To this end, considering a kinematically driven vehicle system, a comparative analysis is performed to analyze the geometry of the contact between the front wheel of the vehicle and the ground, which represents a fundamental problem in the study of the motion of two-wheeled vehicles in general. Subsequently, an exhaustive and extensive numerical analysis, based on the systematic multibody approach mentioned before, is carried out in this work to study the system kinematics in detail. Furthermore, the orientation of the front assembly, which includes the frontal fork, the handlebars, and the front wheel in a seamless subsystem, is implicitly formulated through the definition of three successive rotations, and this approach is used to propose an explicit formulation of its inherent set of Euler angles. In general, the numerical results developed in the present work compare favorably with those found in the literature about vehicle kinematics and contact geometry.
... The structural components of a bicycle (frame and fork) are not as stiff as the structural components of other vehicles and sometimes the compliance of the structural elements is enough to modify the stability properties of a bicycle [10]. ...
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The basic Whipple-Carvallo bicycle model for the study of stability takes into account only geometric and mass properties. Analytical bicycle models of increasing complexity are now available, they consider frame compliance, tire properties, and rider posture. From the point of view of the designer, it is important to know if geometric and mass properties affect the stability of an actual bicycle as they affect the stability of a simple bicycle model. This paper addresses this problem in a numeric way by evaluating stability indices from the real parts of the eigenvalues of the bicycle's modes (i.e., weave, capsize, wobble) in a range of forward speeds typical of city bicycles. The sensitivity indices and correlation coefficients between the main geometric and mass properties of the bicycle and the stability indices are calculated by means of bicycle models of increasing complexity. Results show that the simpler models correctly predict the effect of most of geometric and mass properties on the stability of the single modes of the bicycle. Nevertheless, when the global stability indices of the bicycle are considered, often the simpler models fail their prediction. This phenomenon takes place because with the basic model some design parameters have opposite effects on the stability of weave and capsize, but, when tire sliding is included, the capsize mode is always stable and low speed stability is chiefly determined by weave stability.
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To work towards an advanced model of the bicycle-rider-environment system, an open-loop bicycle-rider model was developed in the commercial multibody dynamics software ADAMS. The main contribution of this article to bicycle dynamics is the analysis of tyre and rider properties that influence bicycle stability. A system identification method is used to extract linear stability properties from time domain analysis. The weave and capsize eigenmodes of the bicycle-rider system are analysed. The effect of tyre properties is studied using the tyre's forces and torques that have been measured in several operating conditions. The main result is that extending simplified models with a realistic tyre model leads to a notable decrease in the weave stability and a stabilization of the capsize mode. This effect is mainly caused by the twisting torque. Different tyres and tyre inflation pressures have little effect on the bicycle's stability, in the case of riding straight at a constant forward speed. On the other hand, the tyre load does have a large effect on bicycle stability. The sensitivity study of rider properties shows that body stiffness and damping have a small effect on the weave and capsize mode, whereas arm stiffness destabilizes the capsize mode and arm damping destabilizes the weave mode.
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The definitive book on tire mechanics by the acknowledged world expert. © 2012 Hans Pacejka Published by Elsevier Ltd All rights reserved.
Conference Paper
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The static and dynamic properties of the frame and the front fork of a single track vehicle play a critical role from the point of view of vehicle stability. A turning point in the study of motorcycle stability was established by the introduction of lumped stiffness elements to characterize the critical compliances of the motorcycle elements, this approach being still in use with advanced multibody codes. Nonetheless, up to now very few scientific studies have been carried out to identify the parameters that account for the stiffness and damping properties of motorcycle front forks and frames. This work addresses the problem of identifying the parameters needed for developing lumped element models of motorcycles from experimental results. Two motorcycle frames are studied performing static, dynamic, and modal tests by means of a specific testing equipment. The frames have been tested in two different conditions: fixing them at the steering head or at the swing-arm pivot. In the first section of the paper a general definition of the twist axis, based on the concept of “Mozzi” or instantaneous screw axis, is presented. The twist axis is used for characterizing the deformation patterns of the tested frames. The static twist axis is identified loading the frames at low rate by means of a servo-hydraulic actuator and measuring the deformation of a reference plate by means of three laser sensors; the dynamic twist axis is identified exerting an impulsive excitation and measuring the vibration of a reference plate by means of three accelerometers. In the last section of the paper, experimental results obtained on motorcycle frames are shown. A method to identify the stiffness properties of the frames from the measured twist axes is presented. Results obtained with the proposed method are in good agreement with the ones presented in literature.
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Wheel shimmy and wobble are well-known dynamic phenomena at automobiles, aeroplanes and motorcycles. In particular, wobble at the motorcycle is an (unstable) eigenmode with oscillations of the wheel about the steering axis, and it is no surprise that unstable bicycle wobble is perceived unpleasant or may be dangerous, if not controlled by the rider in time. Basic research on wobble at motorcycles within the last decades has revealed a better understanding of the sudden onset of wobble, and the complex relations between parameters affecting wobble have been identified. These fundamental findings have been transferred to bicycles. As mass distribution and inertial properties, rider influence and lateral compliances of tyre and frame differ at bicycle and motorcycle, models to represent wobble at motorcycles have to prove themselves, when applied to bicycles. For that purpose numerical results are compared with measurements from test runs, and parametric influences on the stability of the wobble mode at bicycles have been evolved. All numerical analysis and measurements are based on a specific test bicycle equipped with steering angle sensor, wheel-speed sensor, global positioning system (GPS) 3-axis accelerometer, and 3-axis angular velocity gyroscopic sensor.
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A mathematical model of a unicycle and rider, with a uniquely realistic tyre force and moment representation, is set up with the aid of multibody modelling software. The rider's upper body is joined to the lower body through a spherical joint, so that wheel, yaw, pitch and roll torques are available for control. The rider's bandwidth is restricted by low-pass filters. The linear equations describing small perturbations from a straight-running state are shown, which equations derive from a parallel derivation yielding the same eigenvalues as obtained from the first method. A nonlinear simulation model and the linear model for small perturbations from a general trim (or dynamic equilibrium) state are constructed. The linear model is used to reveal the stability properties for the uncontrolled machine and rider near to straight running, and for the derivation of optimal controls. These controls minimize a cost function made up of tracking errors and control efforts. Optimal controls for near-straight-running conditions, with left/right symmetry, and more complex ones for cornering trims are included. Frequency responses of some closed-loop systems, from the former class, demonstrate excellent path-tracking qualities within bandwidth and amplitude limits. Controls are installed for path-following trials. Lane-change and clothoid manoeuvres are simulated, demonstrating good-quality tracking of longitudinal and lateral demands. Pitch torque control is little used by the rider, while yaw and roll torques are complementary, with the former being more useful in transients, while the latter has value also in steady states. Wheel torque is influential on lateral control in turning. Adaptive control by gain switching is used to enable clothoid tracking up to lateral accelerations greater than 1 m s⁻². General control of the motions of a virtual or robotic unicycle will be possible through the addition of more comprehensive adaptation to the control scheme described.
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The development of bicycles and motorcycles since the first patented running machine, or draisine, in 1817 is described. Bicycle modeling and control were also discussed. These models include: derivatives or simplifications of Whipple's bicycle dynamics model in which the lateral motion constraints at the road contact are nonholonomic, requiring special techniques to form correct equations of motion; and the Timoshenko-Young model in which the steer angle and speed completely determine the lateral motion of the base point of an inverted pendulum that represents the vehicle's roll dynamics
Bicycle dynamics and control
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Åström, K., J., Klein, R., E., Lennartsson, A., (2005) 'Bicycle dynamics and control', IEEE Control System Magazine, August 2005, 25(4), pp. 26-47.
Design and Analysis of Experiments
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Montgomery, D.C., (2008) Design and Analysis of Experiments, 3rd edn., John Wiley & Sons, Hoboken NJ.
Influence of frame stiffness and rider position on bike dynamics: analytical study', Proceedings of ASME 2015 International Mechanical Engineering Congress and Exposition
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Williams T., Kaul S., Dhingra A. (2016) 'Influence of frame stiffness and rider position on bike dynamics: analytical study', Proceedings of ASME 2015 International Mechanical Engineering Congress and Exposition, IMECE 2015;