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A DOE Approach for Evaluating the Effect of Bicycle Properties on Stability
Abstract
The effect on stability of mass, geometric and stiffness parameters of a bicycle with compliant frame, fork and wheel is studied. Critical stiffnesses of the structural elements are identified by means of specific experimental tests based on modal analysis and static stiffness measurement. Numerical stability analysis is carried out by means of a MATLAB code and simulations are planned with the design of experiment (DOE) approach. Numerical results show that the rigid body properties that have the main influence on stability are front wheel radius, longitudinal position of the center of mass and trail. Compliance of structural elements strongly affects the weave mode, which becomes unstable at high speed, but have a small effect on self-stability.
... Parametric analyses have been previously presented in [5,10,11] using the hands-off model developed by Klinger et al. ...
... The parameters of the rider presented in [4] are used, while geometric, compliance and tire parameters are based on the nominal and typical range of variation of these parameters presented in [10,11]. These parameters are in a range that covers the geometry of most road racing bicycles. ...
Stability of bicycles has been studied by using models that assume both a rider with hands-off and hands-on the handlebars. Some authors have presented a general comparison between the behavior of the modes of vibration which affect stability in the two configurations; nonetheless, a systematic analysis of the influence of several parameters on each configuration has not been addressed. In addition, the concept of self-stability range has been used as a stability indicator especially when analyzing a rider’s hands-off situation. In contrast, in this paper, the stability area concept is used as a stability index given that it provides more information about the vibration modes in a broader speed range. Therefore, this paper has two major aims. First, the different influence on stability indexes of geometrical, compliance and tire parameters of road racing bicycles is analyzed using hands-off and hands-on models previously published in the literature. Numerical results show that some parameters have different effects on the stability of weave and wobble modes in hands-off and hands-on conditions. Second, a trade-off between the stability behavior of weave and wobble modes is found when using the hands-on model. Therefore, a multi-objective optimization problem for maximizing the stability of both modes is addressed. A Pareto front of the stability of weave and wobble modes is found that serves to analyze the stability trade-off.
In recent years the Whipple Carvallo Bicycle Model has been extended to analyse high speed stability of bicycles. Various researchers have developed models taking into account the effects of front frame compliance and tyre properties, nonetheless, a systematic analysis has not been yet carried out. This paper aims at analysing parametrically the influence of front frame compliance and tyre properties on the open loop stability of bicycles. Some indexes based on the eigenvalues of the dynamic system are defined to evaluate quantitatively bicycle stability. The parametric analysis is carried out with a factorial design approach to determine the most influential parameters. A commuting and a racing bicycle are considered and numerical results show different effects of the various parameters on each bicycle. In the commuting bicycle, the tyre properties have greater influence than front frame compliance, and the weave mode has the main effect on stability. Conversely, in the racing bicycle, the front frame compliance parameters have greater influence than tyre properties, and the wobble mode has the main effect on stability.
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.
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.
Copyright © 2015 by ASME Country-Specific Mortality and Growth Failure in Infancy and Yound Children and Association With Material Stature
Use interactive graphics and maps to view and sort country-specific infant and early dhildhood mortality and growth failure data and their association with maternal
This chapter provides a basic insight into the two-wheeled vehicle dynamics to be applied to vehicle modelling and control. The most relevant kinematic properties are discussed. In two-wheeled vehicles, the relationship between the handlebar and kinematic steering angles varies appreciably with the roll angle. The chapter discusses the peculiarities of motorcycle tyres. The performance of two-wheeled vehicles is largely influenced by the characteristics of their tyres. A fundamental concept when dealing with tyre behaviour is the coupling between longitudinal and lateral forces on the contact patch. The most popular suspension schemes are presented. The proper choice of front and rear suspension characteristics depends on many parameters like the road surface, the braking performance, and the engine power, among others. Different in-plane models of increasing complexity are presented to highlight the main vehicle dynamics involved. Finally, the chapter highlights the coupling between in-plane and out-of-plane dynamics.
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.
Mathematical models of a motorcycle and rider dependent on three alternative assumptions concerning the tyre behaviour are developed. Stability characteristics deduced from them are compared, and minimum requirements for the model greater than have been previously satisfied are established. Using the most sophisticated of the models, the effects of design changes are calculated, and the design implications are discussed.
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