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Parametric analysis of the stability of a bicycle taking into account geometrical, mass and compliance properties
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.
... 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. ...
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]. ...
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]. ...
It is known that the way a bicycle or motorcycle rider grasps the handlebar may have implications on the bike stability. Indeed, the rider closes a loop (with his/her hands) between the front assembly and the chassis of the bike, which in turn affects the open-loop stability of the vehicle. The aim of this paper is to experimentally quantify the interaction between the rider and the handlebar through the frequency response function (FRF), and to numerically simulate the effects of such interaction on the bicycle stability. In order to implement the effect of rider’s arms/hands, a mechanical model is employed to fit the experimental FRFs. The parameters of a number of different riders have been derived and used to assess the effect on bike dynamics.
In motorcycle dynamics, great importance is attributed to the study of the weave and wobble vibration modes and, in particular, to the effects of the flexibility of structural components on their stability. Therefore, appropriate motorcycle models for studying weave and wobble should include flexible elements for describing the flexural behavior of components such as the main frame, front assembly, and rear swingarm. Different approaches are possible for modeling flexibilities: the most common among them are the lumped stiffness and the flexible multibody approaches. While the latter certainly provides higher accuracy, the former has advantages in terms of computational load, but, above all, it makes it easier to understand in the design phase how technical parameters, such as torsional and bending stiffness of a given structural component, can influence the stability of weave and wobble. The accuracy of lumped stiffness models strongly depends on parameter identification. In this study, a general method is proposed to determine appropriate lumped stiffness parameters for any given motorcycle component. The proposed method is tested and validated by comparing the weave and wobble modal behavior with the results of flexible multibody analysis. The lumped stiffness model is then adopted to carry out a sensitivity analysis aimed at identifying the effects on the weave and wobble stability of the torsional and bending stiffness of specific structural components of the motorcycle to optimize their design.
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.
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.
Advanced simulation of the stability and handling properties of bicycles requires detailed road–tyre contact models. In order to develop these models, in this study, four bicycle tyres are tested by means of a rotating disc machine with the aim of measuring the components of tyre forces and torques that influence the safety and handling of bicycles. The effect of inflation pressure and tyre load is analysed. The measured properties of bicycle tyres are compared with those of motorcycle tyres.
We present canonical linearized equations of motion for the Whipple bicycle model consisting of four rigid laterally symmetri ideally hinged parts: two wheels, a frame and a front assembly. The wheels are also axisymmetric and make ideal knife-edg rolling point contact with the ground level. The mass distribution and geometry are otherwise arbitrary. This conservativ non-holonomic system has a seven-dimensional accessible configuration space and three velocity degrees of freedom parametrize by rates of frame lean, steer angle and rear wheel rotation. We construct the terms in the governing equations methodicall for easy implementation. The equations are suitable for e.g. the study of bicycle self-stability. We derived these equation by hand in two ways and also checked them against two nonlinear dynamics simulations. In the century-old literature, severa sets of equations fully agree with those here and several do not. Two benchmarks provide test cases for checking alternativ formulations of the equations of motion or alternative numerical solutions. Further, the results here can also serve as check for general purpose dynamic programs. For the benchmark bicycles, we accurately calculate the eigenvalues (the root of the characteristic equation) and the speeds at which bicycle lean and steer are self-stable, confirming the century-ol result that this conservative system can have asymptotic stability.
This paper addresses the influence of a passive rider on the lateral dynamics of a bicycle model and the controllability of the bicycle by steer or upper body sideway lean control. In the uncontrolled model proposed by Whipple in 1899, the rider is assumed to be rigidly connected to the rear frame of the bicycle and there are no hands on the handlebar. Contrarily, in normal bicycling the arms of a rider are connected to the handlebar and both steering and upper body rotations can be used for control. From observations, two distinct rider postures can be identified. In the first posture, the upper body leans forward with the arms stretched to the handlebar and the upper body twists while steering. In the second rider posture, the upper body is upright and stays fixed with respect to the rear frame and the arms, hinged at the shoulders and the elbows, exert the control force on the handlebar. Models can be made where neither posture adds any degrees of freedom to the original bicycle model. For both postures, the open loop, or uncontrolled, dynamics of the bicycle–rider system is investigated and compared with the dynamics of the rigid-rider model by examining the eigenvalues and eigenmotions in the forward speed range 0–10 m/s. The addition of the passive rider can dramatically change the eigenvalues and their structure. The controllability of the bicycles with passive rider models is investigated with either steer torque or upper body lean torque as a control input. Although some forward speeds exist for which the bicycle is uncontrollable, these are either considered stable modes or are at very low speeds. From a practical point of view, the bicycle is fully controllable either by steer torque or by upper body lean, where steer torque control seems much easier than upper body lean.
A riderless bicycle can automatically steer itself so as to recover from falls. The common view is that this self-steering
is caused by gyroscopic precession of the front wheel, or by the wheel contact trailing like a caster behind the steer axis.
We show that neither effect is necessary for self-stability. Using linearized stability calculations as a guide, we built
a bicycle with extra counter-rotating wheels (canceling the wheel spin angular momentum) and with its front-wheel ground-contact
forward of the steer axis (making the trailing distance negative). When laterally disturbed from rolling straight, this bicycle
automatically recovers to upright travel. Our results show that various design variables, like the front mass location and
the steer axis tilt, contribute to stability in complex interacting ways.
The definitive book on tire mechanics by the acknowledged world expert. © 2012 Hans Pacejka Published by Elsevier Ltd All rights reserved.
In this paper the effect of front fork compliance on uncontrolled bicycle stability is analyzed. First the benchmark model of a bicycle is improved to take into account either torsion compliance or bending compliance of front fork, a lumped element approach is adopted introducing additional joints restrained by rotational springs and dampers. Two models having three degrees of freedom are developed and implemented in MATLAB codes to perform stability analysis. Then series of experimental tests are carried out on an advanced carbon fork and a standard steel fork, the modal analysis approach is adopted. Experimental methods and results are presented and discussed. A specific method is developed for identifying the stiffness and damping properties from the bending and torsion modes of the forks. Results obtained with the proposed method agree with data presented in literature. Finally, the identified stiffness and damping parameters are implemented in the simulation codes and some numerical simulations are carried out. Results presented in the paper show a small influence of torsion compliance on stability and a large influence of bending compliance on high speed stability.
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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.
The passive response of the rider's body to bicycle oscillations is experimentally studied by means of laboratory tests. Lumped element models of the rider's body are developed and the relevant stiffness and damping parameters are identified from experimental results. The biomechanical model of the rider is coupled with the benchmark model of the bicycle and open-loop stability analysis is carried out. Results show that the stiffness and damping parameters of the waist do not strongly affect bicycle stability. Uncontrolled arm stiffness has a very detrimental effect on stability and destroys the self-stabilization mechanism. Arm damping has a more complex effect and reduces the self-stability region.
So far fundamental papers on the understanding of the wobble mode at motorcycles have been published, but in contrast, little research has been published on the wobble mode at bicycles. Wobble denotes a characteristic unstable oscillatory mode dominated by oscillations of the front wheel about the steering axis. The wobble mode of a trekking bicycle at low speeds has already been analysed, where no influence of the rider's hands on the steering system is taken into account. The wobble mode of a racing bicycle at higher speeds has not been addressed in more detail so far. The paper points out the difference between a trekking bicycle and a racing bicycle in particular with respect to the wobble mode. Different geometry, mass and stiffness properties of both types of bicycles and different characteristic positions of the rider are considered. As the wobble at racing bicycles often occurs at high speeds, when riding down a grade with hands on a dropped handlebar, a passive rider model, that takes into account the movement of the rider's arms, is presented.
This paper first presents original data records of the linear accelerations, along with angular velocities and forward velocity, for a high-speed hands-on bicycle shimmy. These describe the oscillation in terms of duration, magnitude, correlation with forward speed, and related danger for the rider. Then additional experimental data are reported which support the conjecture that the shimmy while riding occurs essentially at the same frequency as the lateral resonance of the head tube when the saddle of a stationary bicycle is laterally constrained.
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.
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.
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
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