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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Automotive Series
- Sprache: Englisch
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Many books have been written on modelling, simulation and control of four-wheeled vehicles (cars, in particular). However, due to the very specific and different dynamics of two-wheeled vehicles, it is very difficult to reuse previous knowledge gained on cars for two-wheeled vehicles.
Modelling, Simulation and Control of Two-Wheeled Vehicles presents all of the unique features of two-wheeled vehicles, comprehensively covering the main methods, tools and approaches to address the modelling, simulation and control design issues. With contributions from leading researchers, this book also offers a perspective on the future trends in the field, outlining the challenges and the industrial and academic development scenarios. Extensive reference to real-world problems and experimental tests is also included throughout.
Key features:
- The first book to cover all aspects of two-wheeled vehicle dynamics and control
- Collates cutting-edge research from leading international researchers in the field
- Covers motorcycle control – a subject gaining more and more attention both from an academic and an industrial viewpoint
- Covers modelling, simulation and control, areas that are integrated in two-wheeled vehicles, and therefore must be considered together in order to gain an insight into this very specific field of research
- Presents analysis of experimental data and reports on the results obtained on instrumented vehicles.
Modelling, Simulation and Control of Two-Wheeled Vehicles is a comprehensive reference for those in academia who are interested in the state of the art of two-wheeled vehicles, and is also a useful source of information for industrial practitioners.
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Veröffentlichungsjahr: 2014
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Table of Contents
Automotive Series
Series Editor: Thomas Kurfess
Title Page
Copyright
About the Editors
Mara Tanelli
Matteo Corno
Sergio M. Savaresi
List of Contributors
Series Preface
Introduction
Organization of the book
Part One: Two-Wheeled Vehicles Modelling and Simulation
Chapter 1: Motorcycle Dynamics
1.1 Kinematics
1.2 Tyres
1.3 Suspensions
1.4 In-Plane Dynamics
1.5 Out-of-Plane Dynamics
1.6 In-Plane and Out-of-Plane Coupled Dynamics
References
Chapter 2: Dynamic Modelling of Riderless Motorcycles for Agile Manoeuvres
2.1 Introduction
2.2 Related Work
2.3 Motorcycle Dynamics
2.4 Tyre Dynamics Models
2.5 Conclusions
Nomenclature
Appendix A: Calculation of Ms
Appendix B: Calculation of Acceleration
Acknowledgements
References
Chapter 3: Identification and Analysis of Motorcycle Engine-to-Slip Dynamics
3.1 Introduction
3.2 Experimental Setup
3.3 Identification of Engine-to-Slip Dynamics
3.4 Engine-to-Slip Dynamics Analysis
3.5 Road Surface Sensitivity
3.6 Velocity Sensitivity
3.7 Conclusions
References
Chapter 4: Virtual Rider Design: Optimal Manoeuvre Definition and Tracking
4.1 Introduction
4.2 Principles of Minimum Time Trajectory Computation
4.3 Computing the Optimal Velocity Profile for a Point-Mass Motorcycle
4.4 The Virtual Rider
4.5 Dynamic Inversion: from Flatland to State-Input Trajectories
4.6 Closed-Loop Control: Executing the Planned Trajectory
4.7 Conclusions
4.8 Acknowledgements
References
Chapter 5: The Optimal Manoeuvre
5.1 The Optimal Manoeuvre Concept: Manoeuvrability and Handling
5.2 Optimal Manoeuvre as a Solution of an Optimal Control Problem
5.3 Applications of Optimal Manoeuvre to Motorcycle Dynamics
5.4 Conclusions
References
Chapter 6: Active Biomechanical Rider Model for Motorcycle Simulation
6.1 Human Biomechanics and Motor Control
6.2 The Model
6.3 Simulations and Results
6.4 Conclusions
References
Chapter 7: A Virtual-Reality Framework for the Hardware-in-the-Loop Motorcycle Simulation
7.1 Introduction
7.2 Architecture of the Motorcycle Simulator
7.3 Tuning and Validation
7.4 Application Examples
References
Part Two: Two-Wheeled Vehicles Control and Estimation Problems
Chapter 8: Traction Control Systems Design: A Systematic Approach
8.1 Introduction
8.2 Wheel Slip Dynamics
8.3 Traction Control System Design
8.4 Fine tuning and Experimental Validation
8.5 Conclusions
References
Chapter 9: Motorcycle Dynamic Modes and Passive Steering Compensation
9.1 Introduction
9.2 Motorcycle Main Oscillatory Modes and Dynamic Behaviour
9.3 Motorcycle Standard Model
9.4 Characteristics of the Standard Machine Oscillatory Modes and the Influence of Steering Damping
9.5 Compensator Frequency Response Design
9.6 Suppression of Burst Oscillations
9.7 Conclusions
References
Chapter 10: Semi-Active Steering Damper Control for Two-Wheeled Vehicles
10.1 Introduction and Motivation
10.2 Steering Dynamics Analysis
10.3 Control Strategies for Semi-Active Steering Dampers
10.4 Validation on Challenging Manoeuvres
10.5 Experimental Results
10.6 Conclusions
References
Chapter 11: Semi-Active Suspension Control in Two-Wheeled Vehicles: a Case Study
11.1 Introduction and Problem Statement
11.2 The Semi-Active Actuator
11.3 The Quarter-Car Model: a Description of a Semi-Active Suspension System
11.4 Evaluation Methods for Semi-Active Suspension Systems
11.5 Semi-Active Control Strategies
11.6 Experimental Set-up
11.7 Experimental Evaluation
11.8 Conclusions
References
Chapter 12: Autonomous Control of Riderless Motorcycles
12.1 Introduction
12.2 Trajectory Tracking Control Systems Design
12.3 Path-Following Control System Design
12.4 Conclusion
Acknowledgements
Appendix A: Calculation of the Lie Derivatives
References
Chapter 13: Estimation Problems in Two-Wheeled Vehicles
13.1 Introduction
13.2 Roll Angle Estimation
13.3 Vehicle Speed Estimation
13.4 Suspension Stroke Estimation
13.5 Conclusions
References
Index
Automotive Series
Series Editor: Thomas Kurfess
Modelling, Simulation and Control of Two-Wheeled Vehicles
Tanelli, Corno and Savaresi
February 2014
Modeling and Control of Engines and Drivelines
Eriksson and Nielsen
February 2014
Advanced Composite Materials for Automotive Applications: Structural Integrity and Crashworthiness
Elmarakbi
December 2013
Guide to Load Analysis for Durability in Vehicle Engineering
Johannesson and Speckert
November 2013
This edition first published 2014
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Library of Congress Cataloging-in-Publication Data
Tanelli, Mara.
Modelling, simulation and control of two-wheeled vehicles / Mara Tanelli, Sergio Savaresi and Matteo Corno.
pages cm
Includes bibliographical references and index.
ISBN 978-1-119-95018-9 (cloth)
1. Motorcycles— Dynamics. I. Savaresi, Sergio M. II. Corno, Mauro, 1970- III. Title.
TL243.T36 2014
629.2′31— dc23
2013036260
A catalogue record for this book is available from the British Library.
ISBN 9781119950189
About the Editors
Mara Tanelli
Mara Tanelli was born in Lodi, Italy, in 1978. She is an Assistant Professor of Automatic Control at the Dipartimento di Elettronica, Informazione e Bioingegneria of the Politecnico di Milano, Italy, where she obtained the Laurea degree in Computer Engineering in 2003 and the PhD in Information Engineering in 2007. She also holds an MSc in Computer Science from the University of Illinois at Chicago.
Her main research interests focus on control systems design for vehicles, energy management of electric vehicles, control for energy-aware IT systems and sliding mode control.
She is co-author of more than 100 peer-reviewed scientific publications and seven patents in the above research areas. She is also co-author of the monograph Active braking control systems design for vehicles, published in 2010 by Springer. She is a Senior Member of the IEEE and a member of the Conference Editorial Board of the IEEE Control Systems Society.
In the past few years, she has gained considerable experience in industrial projects carried out in collaboration with leading manufacturers of four- and two-wheeled vehicles, that involved – besides her scientific research activities – prototyping, implementation and experimental testing.
Matteo Corno
Matteo Corno was born in Italy in 1980. He received his MSc degree in Computer and Electrical Engineering (University of Illinois) and his PhD cum laude degree with a thesis on active stability control of two-wheeled vehicles (Politecnico di Milano) in 2005 and 2009, respectively.
He is currently an Assistant Professor at the Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italy.
In 2011, his paper “On Optimal Motorcycle Braking” was awarded the best-paper prize for Control Engineering Practice, published in the period 2008–2010.
In 2012 and 2013, he co-founded two highly innovative start-ups: E-Novia and Zehus.
His current research interests include dynamics and control of vehicles; lithium-ion battery modelling; estimation and control; and modelling and control of human-powered electric vehicles. He has held research positions at Thales Alenia Space, University of Illinois, Harley Davidson, University of Minnesota, Johannes Kepler University in Linz, and TU Delft. He is author or co-author of more than 50 peer-reviewed scientific publications and of six patents. Most of his publications are the result of strong industrial collaboration with leading companies in the automotive and motorcycle industries.
Sergio M. Savaresi
Born in Manerbio, Italy, in 1968, Sergio Savaresi holds an MSc in Electrical Engineering and a PhD in Systems and Control Engineering, both from the Politecnico di Milano, and an MSc in Applied Mathematics from Università Cattolica, Brescia. After receiving his PhD, he became a consultant for McKinsey & Co., Milan Office. He has been a Full Professor in Automatic Control since 2006, and has been visiting scholar at Lund University, Sweden; University of Twente, the Netherlands; Canberra National University, Australia; Minnesota University at Minneapolis, USA and Johannes Kepler University, Linz, Austria.
He is an Associate Editor of several international journals and he has been on the international program committees of many international conferences.
His main research interests are in the areas of vehicles control, automotive systems, data analysis and modelling, nonlinear control and industrial control applications. He is co-author of the monographs Active Braking Control Systems Design for Vehicles, and Semi-Active Suspension Control For Vehicles. He is also author or co-author of more than 300 peer-reviewed scientific publications and of 28 patents.
He is the Chair of the Systems and Control Section of Politecnico di Milano, and Head of the MOVE research team (http://move.dei.polimi.it). He is a Lecturer on a Masters Course in “Automation and Control in Vehicles”, and has been Principal Investigator in more than 100 research cooperation projects between Politecnico di Milano and private companies, mostly in the fields of automotive and motorcycle dynamics and control.
List of Contributors
Series Preface
The motorcycle is the most prevalent form of mechanized transportation on the planet. In its human-powered form, the bicycle, it is one of the first pragmatic and useful vehicles that most people encounter. The dynamics of two-wheeled vehicles have been studied for many years, and provide the foundation for most vehicle dynamic analyses. Not only are these dynamics fundamental to the transportation sector, but they are quite elegant in nature, linking various aspects of kinematics, dynamics and physics. In fact, the dynamics of the motorcycle and bicycle are inherently linked to their functionality; one cannot easily balance these vehicles unless they are in motion!
Modelling, Simulation and Control of Two-Wheeled Vehicles is a comprehensive text of the dynamics, modelling and control of motorcycles. It provides a broad and in-depth perspective of all the necessary information required to fully understand, design and utilize the motorcycle. Topics covered in this text range from basic two-wheeled dynamics that are used as the foundation for most vehicle dynamic analyses to advanced control and estimation theory applied to fully developed complex systems models. This text is part of the Automotive Series whose primary goal is to publish practical and topical books for researchers and practitioners in industry, and for postgraduate or advanced undergraduates in automotive engineering. The series addresses new and emerging technologies in automotive engineering, supporting the development of the next generation transportation systems. The series covers a wide range of topics, including design, modelling and manufacturing, and it provides a source of relevant information that will be of interest and benefit to people working in the field of automotive engineering.
Modelling, Simulation and Control of Two-Wheeled Vehicles presents a number of different design and analysis considerations related to motorcycle transportation systems including integration dynamics, agile manoeuvring systems integration, rider biomechanical models, passive and active steering control and autonomous control of riderless motorcycles. The theory and supporting applications are second to none, as are the authors of this wonderful book. The text provides a strong foundational basis for motorcycle design and development, and is a welcome addition to the Automotive Series.
Thomas KurfessAugust 2013
Introduction
Mara Tanelli, Matteo Corno, and Sergio M. Savaresi
Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Italy
Plenty of books have been written on modeling, simulation and control on four-wheeled vehicles (cars, in short). As such, one would be tempted to ask: is a “two-wheeled-specific” book really needed or missing? The editors and authors of this book strongly believe that the answer is yes.
A thorough technical motivation for this answer will be implicitly given throughout the book. A simpler, somehow naive, but effective answer, however, is: we drive a car, but we ride a two-wheeled vehicle: this crucial difference highlights that they just cannot be “similar” vehicles. In the field of vehicle modeling and automatic control, the majority of scientists and practitioners have been working on automotive (car)-related modeling and control problems. As such, one may think that moving from four to two-wheeled vehicles just requires a small re-casting (re-modeling, re-design of the controllers, re-tuning, etc.) effort. This sort of prejudice typically vanishes when dealing with real problems, on real two-wheeled vehicles. Most of the authors of this book have been through this enlightening process. Discovering that two-wheeled vehicles are not just a “subset of cars” is both challenging and fascinating.
This book helps the reader discover all the peculiar features of modeling and control of this very special class of vehicles.
The potential interest for a book specifically dealing with two-wheeled vehicles is amplified by the current and future mobility trends: traffic congestion in urban and metropolitan areas and the need to reduce energy consumption and pollutant emission are pushing towards a strong downsizing of vehicles used for urban mobility. The number of E-Bikes, scooters, motorcycles, narrow-track vehicles (tilting or non-tilting) is expected to grow exponentially in the next decades, especially around large metropolitan areas. Along this trend, two-wheeled vehicles can play a key role: they have the appealing features of being light and having a very small energy footprint. Thus, there are good chances that the two-wheeled vehicles market will soon compete (in volume, and, possibly, in technology) with the today larger and more advanced automotive market.
Such an expansion, then, will see an increasing interest in finding innovative andoriginal solutions for solving many challenging problems that deal with the dynamic analysis and control of such vehicles, and this book can be one of the first comprehensive answer to such needs.
In this respect, this book lies in the class of edited-books, namely books that are a collection of chapters written by different authors. The price to pay is a limited homogeneity of notation and presentation style, but their main advantage is that a single book embeds the perspective of an (almost) entire scientific community, rather than that of a single research group.
This book has been carefully conceived in order to provide at the same time a broad perspective and a rigorous structure. Its contents have been clearly divided into two parts: in the first part, the modeling and simulation issues are considered, while in the second one the problem of controlling (mostly by feedback electronic control systems) the vehicle is analyzed. In many cases, there are pairs of chapters written by the same authors: one in the first, one in the second part, stressing the fact that modeling and control are just two sides of the same coin. The valuable coin is the dynamic behavior of a two-wheeled vehicle: weird, exhilarating, challenging. We want to understand it. We want to control it.
Organization of the book
The two parts of the book are organized as follows.
Part one:
Chapter 1 (by Vittore Cossalter, Roberto Lot and Matteo Massaro—University of Padova) is a comprehensive and introductory chapter that describes all the aspects of the kinematic and dynamic behavior of a two-wheeled vehicle.
Chapter 2 (by Yizhai Zhang and Jingang Yi—Rutgers University—and by Dezhen Song—Texas A&M University) further develops the modeling topic, with a special focus on a reduced-order model suited for modeling fast-dynamics maneuvers.
Chapter 3 (by Matteo Corno and Sergio M. Savaresi—Politecnico di Milano) explores the field of black-box control-oriented modeling, by presenting a case study of direct identification from experiments of the engine-to-slip dynamics, ancillary to traction-control design. The design-of-experiment in this context represents a major issue and is described in detail.
Chapter 4 (by Alessandro Saccon—TU Eindhoven—John Hauser—University of Colorado Boulder—and Alessandro Beghi—University of Padova) andChapter 5 (by Francesco Biral, Enrico Bertolazzi and Mauro Da Lio—University of Trento) present, with different approaches, the problem of simulating the motorcycle dynamics in a time-optimal maneuver. This problem is a combination of dynamics modeling, optimization and optimal control issues. This topic is highly relevant not only for the purpose of automatic (electronic) feedback control, but mostly for better understanding the sensitivity of the performance of a motorcycle, with respect to different parameter configurations.
Chapter 6 (by Valentin Keppler—University of Tubingen) deeply explores the issue of rider modeling and simulation. This issue is a key element of two-wheeled vehicles simulation, since the rider is so deeply linked with the vehicle dynamics that the two elements can hardly be simulated separately. In this chapter, the rider simulation is dealt with a sophisticated bio-mechanics approach.
Chapter 7 (by Vittore Cossalter and Roberto Lot—University of Padova) ends part one by presenting a research work that can be considered in-between simulation and control: the development of a virtual-reality system for the hardware-in-the-loop simulation of vehicle dynamics. The system described in this chapter might have multi-faceted applications: it can be used as a design and testing tool for advanced electronic control, as a rider-training system and…even as a sophisticated and fun-to-use
toy
).
Part two:
Chapter 8 (by Matteo Corno and Giulio Panzani—Politecnico di Milano) presents a complete (from model-based design to experimental validation) design procedure for a traction control system of a high-performance motorcycle. Traction control is the most used electronic control system in high-end motorcycles today, and has a key role both on the safety and the performance of a sport motorcycle. This Chapter is somehow the continuation of Chapter 3.
Chapter 9 (by Simos A. Evangelou—Imperial College—and Maria Tomas-Rodriguez— City University of London) focuses on the key issue of steer dynamics, with an approach that aims to improve the dynamic behavior by passive mechanical elements.
Chapter 10 (by Pierpaolo De Filippi, Mara Tanelli and Matteo Corno—Politecnico di Milano) addresses again the problem of steer dynamics, proposing a solution that employs closed-loop electronic control systems and relies on a semi-active damping technology.
Chapter 11 (by Diego del Vecchio—Politecnico di Milano, and Cristiano Spelta—University of Bergamo) focuses on vertical and pitch dynamics, by presenting a complete case-study of semi-active suspension control design. Semi-active suspensions have been first presented in motorcycle applications at the end of 2012, and they constitute—today—one of the fastest-growing electronic-control technology in motorcycles).
Chapter 12 (by Yizhai Zhang and Jingang Yi—Rutgers University—and by Dezhen Song—Texas A&M University) is the natural continuation of Chapter 2, and the problem of designing an electronic-rider for the autonomous control of a 2-wheeled vehicle is analyzed.
Chapter 13 (Ivo Boniolo—University of Bergamo, Giulio Panzani, Diego del Vecchio, Matteo Corno and Mara Tanelli—Politecnico di Milano, Cristiano Spelta—University of Bergamo—and Sergio M. Savaresi—Politecnico di Milano) is a sort of appendix chapter, where three important problems of variable estimation (or software sensing) are considered: roll-angle estimation, vehicle-speed estimation, and suspension stroke estimation. Variable estimation from indirect measurements is, today, a key element for the optimization of sensors layout, both for reducing the cost and for improving the safety of the control systems.
This overview of chapters shows that the book provides a broad perspective on all the main modeling, simulation and control issues of modern two-wheeled vehicles. Moreover, the style and content of the chapters (with a good balancing between theory and experimental results) make this book potentially useful for both practitioners and researchers. From a technological and industrial point of view, the content of the book is up-to-date: it contains the latest technologies both in terms of electronic control systems (traction control, suspension control, steer-damping control), vehicle-dynamics optimization, rider-modeling and virtual-reality hardware-in-the-loop frameworks.
A last comment about the authorship: Italy is largely represented and this reflects the fact that the motorcycle (and bicycle) Italian industry has been, and still is, one of the most vital and technologically advanced worldwide, with a vast number of large, medium and small bike and motorbike companies and prestigious brands. UK, USA and Germany are also represented, consistently with the location of the main motorcycle industries. The most evident missing contribution is from Japan, that has expressed, in the last 30 years, an enormous industrial power and potential, but, somehow, this potential has not been equally represented in the academic research activities (which, in this field, do exist but are quite fragmented).
A final comment for the reader: the books has been conceived for being readable both end-to-end or by cherry-picking some chapters. Each chapter is almostcompletely self-consistent, with the (partial) exception of the twin-chapters (one in part one, one in part two) written by the same authors.
Part One
Two-Wheeled Vehicles Modelling and Simulation
Chapter 1
Motorcycle Dynamics
Vittore Cossalter, Roberto Lot, and Matteo Massaro
University of Padova, Italy
This chapter aims at giving 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 Section 1.1, the peculiarities of motorcycle tyres are reported in Section 1.2, the most popular suspension schemes are presented in Section 1.3, while Sections 1.4 and 1.5 are devoted to the analysis of the vehicle in-plane and out-of-plane vibration modes. Finally, Section 1.6 highlights the coupling between in-plane and out-of-plane dynamics.
1.1 Kinematics
From the kinematic point of view, every mechanical system consists of a number of rigid bodies connected to each other by a number of joints. Each body has six degrees of freedom (DOF) since its position and orientation in the space are fully defined by six parameters, such as the three coordinates of a point and three angles (yaw, roll, pitch). When a joint is included, the number of DOFs reduces according to the type of joint: the revolute joint (e.g., the one defining the motorcycle steering axis) inhibits five DOFs, the prismatic joint (e.g., the one defining the telescopic fork sliding axis) inhibits five DOFs, the wheel–road contact joint inhibits three DOFs when pure rolling is assumed (only three rotations about the contact point are allowed while no sliding is permitted), or one DOF when longitudinal and lateral slippage is allowed (the only constraint being in the vertical direction, where the compenetration between the wheel and the road is avoided).
1.1.1 Basics of Motorcycle Kinematics
Two-wheeled vehicles can be considered spatial mechanisms composed of six bodies:
the rear wheel;
the swingarm;
the chassis (including saddle, tank, drivetrain, etc.);
the handlebar (including rear view mirrors, headlamp, the upper part of the front suspension, etc.);
the front usprung mass (i.e., the lower part of the front suspension, front brake calliper, etc.);
the front wheel.
These bodies are connected each other and with the road surface by seven joints:
a contact joint between the rear wheel and the road surface;
a revolute joint between the rear wheel and the swingarm, to give the rear wheel spin axis;
a revolute joint between the swingarm and the chassis, to give the swingarm pivot on the chassis;
a revolute joint between the chassis and the handlebar, to give the steering axis;
a prismatic joint between the handlebar and the front unsprung, to give the sliding axis of the telescopic fork;
a revolute joint between the front unsprung and the front wheel, to give the front wheel spin axis;
a contact joint between the front wheel and the road plane.
Therefore, the two-wheeled vehicle has nine DOFs, given the 20 DOFs inhibited by the four revolute joints, five DOFs inhibited by the prismatic joint and the two DOFs inhibited by the two contact joints (tyre slippage allowed), subtracted from the 36 DOFs related to the six rigid bodies. It is also common to include the rear and front tyre deformation due to the tyre compliance, and consequently the number of DOFs rises to 11.
Among the many different sets of 11 parameters that can be selected to define the vehicle configuration, it is common (e.g. Cossalter et al. 2011b, 2011c) to use the ones depicted in Figure 1.1: position and orientation of the chassis, steering angle, front suspension travel, swingarm rotation and wheel spin rotations.
Figure 1.1 Degrees of freedom of a two-wheeled vehicle
Finally, it is worth mentioning that these DOFs are related to the gross motion of the vehicle, while additional DOFs are necessary whenever some kind of vehicle structural flexibility is considered, e.g. Cossalter et al. (2007b).
Some geometric parameters such as the wheelbase , normal trail and caster angle , are very important when it comes to the vehicle stability, manoeuvrability and handling. In more detail, the wheelbase is the distance between the contact points on the road and usually ranges between 1.2 and 1.6 m, the normal trail is the distance between the front contact point and the steering axis (usually 80–120 mm) and the caster angle is the angle between the vertical axis and the steering axis (usually 19–35).
In general, an increase in the wheelbase, assuming that the other parameters remain constant, leads to an unfavourable increase in the flexional and torsional deformability of the frame (this may reduce vehicle manoeuvrability), an unfavourable increase in the minimum curvature radius, a favourable decrease in the load transfer during accelerating and braking (this makes wheelie and stoppie more difficult) and a favourable increase in the directional stability of the motorcycle.
The trail and the caster angle are especially important inasmuch as they define the geometric characteristics of the steering head. The definition of the properties of manoeuvrability and directional stability of two-wheeled vehicles depend on these two parameters, among others. Small values of trail and caster characterize sport vehicles, while higher values are typical of touring and cruiser vehicles. The trail and caster are related to each other by the following relationship:
1.1
where is the front tyre radius and is the fork offset; see Figure 1.2.
Figure 1.2 Wheelbase, caster angle and trail
Finally, it is worth noting that all these parameters are usually given for the nominal (standstill) trim configuration, while they change as the vehicle speed, longitudinal and lateral accelerations change.
1.1.2 Handlebar Steering Angle and Kinematic Steering Angle
While the driver operates the handlebar steering angle, the vehicle cornering behaviour is determined by the projection on the road surface of the angle between the rear and front wheel planes, the so-called kinematic steering angle. In two-wheeled vehicles, the relationship between the handlebar and kinematic steering angles varies appreciably with the roll angle. In particular, the steering mechanism is attenuated (i.e. the kinematic angle is lower than the handlebar angle) up to a certain value of the roll angle (close to the value of the caster angle), then it is amplified (i.e. the kinematic angle is higher than the handlebar angle); see Figure 1.3 for example.
Figure 1.3 Kinematic steering angle as a function of the handlebar steering angle δ for different values of the roll angle
The following simplified expression can be used to estimate the kinematic steering angle from the handlebar steering angle , the caster angle and the roll angle :
1.2
The local curvature of the vehicle trajectory (or the turning radius ) can be estimated from the kinematic angle and the wheelbase using the following expression:
1.3
Note that Equation 1.3 does not include the effect of tyre slippage, whose contribution will be described in Sections 1.2 and 1.5.2.
1.2 Tyres
The performance of two-wheeled vehicles is largely influenced by the characteristics of their tyres. Indeed, the control of the vehicle's equilibrium and motion occurs through the generation of longitudinal and lateral forces resulting from the rider's actions on the steering mechanism, throttle and braking system. The peculiarity of motorcycle tyres is that they work with camber angles up to 50 and even more, while car tyres rarely reach 10.
1.2.1 Contact Forces and Torques
From a macroscopic viewpoint, the interaction of the tyre with the road can be represented by a system composed of three forces and three torques, as in Figure 1.4:
a longitudinal force (positive if driving and negative if braking);
a lateral force ;
a force normal to the road surface;
an overturning moment ;
a rolling resistance moment ;
a yawing moment .
Figure 1.4 Tyre forces and torques
Experimental observations show that the force and torque generation is mainly related to the following input quantities:
tyre longitudinal slip ;
tyre lateral slip ;
tyre camber angle ;
tyre radial deflection ;
tyre spin rate .
Therefore we can write:
1.4
with the longitudinal force mainly related to longitudinal slip , lateral force mainly related to the lateral slip and the camber angle , overturning moment mainly related to the camber angle , rolling resistance mainly related to the wheel spin rate and yawing moment mainly related to the lateral slip and camber angle .
The longitudinal slip (positive when driving and negative when braking) is defined as:
1.5
where is the tyre longitudinal velocity, is the tyre spin rate and is the tyre effective rolling radius. In particular, the effective rolling radius can be computed from the freely rolling tyre as
1.6
Note that the effective rolling radius does not coincide with either the tyre loaded radius or the the tyre unloaded radius ; see Figure 1.5. This should not be surprising since the tyre is not a rigid body. Experimental observations show that . However, a common assumption is .
Figure 1.5 Tyre radii
Sometimes a slightly different formulation of longitudinal slip is adopted:
1.7
It can easily be shown that
1.8
and the relative difference between the two is:
1.9
which is typically lower than 5% in normal conditions (i.e. no skidding).
The lateral slip is defined as:
1.10
where is the lateral velocity of the tyre and is the longitudinal velocity. The sign is chosen to give positive force for positive slip.
Sometimes another input quantity is considered, the turn slip :
1.11
where is the curvature of the tyre contact point path and is the yaw rate. This quantity is important only at very low speed and therefore is not considered in the following sections.
1.2.2 Steady-State Behaviour
A widely used model for computing the steady-state tyre forces and moment is based on the so-called Magic Formula (Pacejka 2006). The general form is:
1.12
where passes through the origin , reaches a maximum and subsequently tends to a horizontal asymptote; see Figure 1.6. For given values of the coefficients , the curve shows an anti-symmetric shape with respect to the origin. To allow the curve to have an offset with respect to the origin (e.g. because of ply-steer and conicity of the tyre), two shifts and can be introduced:
1.13
Coefficient represents the peak value of the curve, while the product corresponds to the slope of the curve at the origin (e.g. the lateral slip stiffness when the lateral force is reported in the vertical axis and the lateral slip is reported in the horizontal axis). The shape factor determines the shape of the resulting curve. The factor is used to determine the slope at the origin and is called the stiffness factor. The factor is introduced to control the curvature at the peak and at the same time the horizontal position of the peak. The various factors depend on the tyre normal load (or tyre radial deflection).
Figure 1.6 Main parameters of the tyre Magic Formula
In particular, the slope of the lateral force is especially sensitive to load variation, and is usually modelled as follows (Figure 1.7):
1.14
The sideslip stiffness attains a maximum at a normal load .
Figure 1.7 Tyre cornering stiffness as a function of normal load
Another widely used tyre formula is the Burckhardt model (Kiencke and Nielsen 2001):
1.15
Again, the curve typically passes through the origin , reaches a maximum and subsequently decreases. An offset can be added, following the same approach used above.
Typical tyre curves are depicted in Figure 1.8.
Figure 1.8 Typical tyre curves
A fundamental concept when dealing with tyre behaviour is the coupling between longitudinal and lateral forces on the contact patch. In practice, the tyre gives the maximum longitudinal (lateral) force when in pure longitudinal (lateral) slip condition. Indeed, the theoretical analysis on physical models (Pacejka 2006) shows that the tyre longitudinal and lateral force generation depends on the following theoretical slip quantities:
1.16
rather than on the practical slip quantities and , and that there exists a total slip:
1.17
which defines the maximum friction force available from the tyre. The corresponding total force can be split between the longitudinal and lateral directions, according to the slip and . Also, the effect of camber can be included into the sideslip as follows:
1.18
and the formulas for forces read:
1.19
where and are the longitudinal and lateral forces in pure slip condition.
There is also a newer empirical approach to modelling force coupling (Pacejka 2006). To describe the effect of combined slip on the lateral force and longitudinal force characteristics, the following hill-shaped function is employed:
1.20
where is either the longitudinal slip or the lateral slip (or ). The coefficient is the peak value, determines the height of the hill's base and influences the sharpness of the hill, which is the main factor responsible for the shape of the function. The formulas in combined slip conditions read
1.21
1.2.3 Dynamic Behaviour
The relationships between the tyre inputs (slips, camber, load/deflection and spin) and the tyre outputs (forces and torques) described in the previous section hold in steady-state conditions. However, the tyre forces do not arise instantaneously: to appear the tyre needs to travel a certain distance, which depends on the tyre characteristics. The physical reason is the tyre flexibility, and the related behaviour can be explained as follows.
We consider a tyre whose contact point has longitudinal velocity and lateral velocity , where and are the velocities of the contact point when neglecting tyre deformation, while and are the deflection velocities; see Figure 1.9.
Figure 1.9 Tyre contact area with deflections
The observed longitudinal slip (e.g. with sensors on the rim) is
1.22
with the rim spin rate and the effective rolling radius, while the actual (or instantaneous) longitudinal slip experienced by the contact point is
1.23
and therefore
1.24
Similarly, the observed lateral slip is
1.25
while the actual (or instantaneous) lateral slip is
1.26
Under small angle assumption it is
1.27
At the tyre–road contact point, the slip-induced longitudinal and lateral forces balance the deflection-induced forces. Under small slips assumption the following relationships hold:
1.28
where and are the lateral slip stiffness and longitudinal slip stiffness respectively, and are the lateral and longitudinal structural stiffness and is the tyre normal load. When introducing Equations 1.24 and 1.27 into 1.28 one obtains:
1.29
1.30
which yields, after the elimination of carcass deflections and
1.31
1.32
where and are the actual tyre forces (i.e. computed with the instantaneous slip and ), while and are the tyre forces computed with the practical slips and and:
1.33
In practice, there is a deformation-induced lag between and . The resulting first-order differential equations are called relaxation equations, with the relaxation lengths. Equation 1.33 shows that the longitudinal (lateral) relaxation length increases with the longitudinal (lateral) slip stiffness and with the normal load, while it reduces with the longitudinal (lateral) structural stiffness. The relaxation length represents the space that the wheel has to cover in order for the force to be 63% of the steady-state force. Typical values of relaxation length are in the range 0.10–0.4 m, the higher values corresponding to higher tyre normal load and higher speeds.
The equations above describe the effect of flexibilities on the longitudinal force due to longitudinal slip, and lateral force due to lateral slip. Actually, in two-wheeled vehicles there is a significant component of the lateral force related to the camber angle . Therefore, under small assumption, Equation 1.28 becomes
1.34
where is the camber stiffness, and after substitution Equation 1.32 gives
1.35
Finally, it is worth mentioning the gyroscopic couple that arises as a result of the time rate of change of the tyre camber distortion, the wheel spin rate and the belt inertia. This effect is visible for certain types of tyre at high speeds, and leads to an increase of the observed relaxation length (De Vries and Pacejka 1998).
1.3 Suspensions
Suspensions serve several purposes such as contributing to the vehicle's road-holding/handling, keeping the rider comfortable and reasonably well isolated from road noise. These goals are generally at odds. In addition, the suspensions affect the vehicle's trim while accelerating, braking, turning and so on. The proper choice of front and rear suspension characteristics depends on many parameters: the weight of the rider and the vehicle, the position of the centre of gravity, the characteristics of stiffness and vertical damping of the tyres, the geometry of the motorcycle, the conditions of use, the road surface, the braking performance, the engine power and the driving technique, among others.
1.3.1 Suspension Forces
The total force exerted by the spring–damper group is the sum of the following different actions:
1.36
where is the elastic force exerted by the coil spring and/or air spring (or different elastic components), is the damping force exerted by the shock absorber, is the friction force and is the end-stroke pad force.
Preloading is commonly used to adjust the initial position of the suspension with the weight of the vehicle and rider acting on it. It consists of a precompression of the spring: as a consequence, if the spring is stressed with forces that are lower than or equal to the preload, it is not compressed. In practice, this adjustment shifts the curve of the elastic force as a function of travel ; see Figure 1.10.
Figure 1.10 Elastic force as a function of preload
1.3.2 Suspensions Layout
Several suspension layouts have been used over the years and the following sections present a brief overview.
1.3.2.1 Front Suspension Types
The most widespread front suspension is the telescopic fork (Figure 1.11a). It is made up of two telescopic sliders which run along the interior of two fork tubes and form a prismatic joint between the unsprung mass of the front wheel and the sprung mass of the chassis. The telescopic fork is characterized by limited inertia around the axis of the steering head.
Figure 1.11 Example of front suspensions
Two limitations of the telescopic fork are the impossibility of attaining progressive force displacement and the rather high values of the unsprung mass that is an integral part of the wheel. To overcome the typical defects of the telescopic fork, alternative solutions have been proposed: push arm (Figure 1.11b), trail arm (Figure 1.11c) and four-bar linkage (like the BMW Duolever).
The front arm suspension and four-bar linkage suspensions can be designed so as to present total or partial anti-dive behaviour in braking conditions. Further, the absence of a prismatic joint eliminates the typical dry friction problems of telescopic forks.
1.3.2.2 Rear Suspension Types
The classic rear suspension is composed of a swingarm (a rocker made up of two oscillating arms) with two spring–damper elements, one on each side (Figure 1.12a). The main advantages are the simplicity of construction and the modest reactive forces transmitted to the chassis. Among its disadvantages are a poorly progressive force–displacement characteristic and the possibility that the two spring–damper units generate different forces and therefore torsional stress on the swingarm.
Figure 1.12 Example of rear suspensions
An alternative is represented by the cantilever mono-shock system, characterized by only one spring–damper unit. However, this suspension does not enable a progressive force–displacement characteristic and the positioning of the spring–shock absorber unit close to the engine can cause problems with the absorber's heat dissipation.
The introduction of a four-bar linkage in the rear suspension makes it easier to obtain the desired stiffness curves. Different attachment points of the spring–damper elements can be chosen: for example, in the Kawasaki Uni-Trak the suspension element is between the rocker and the chassis (Figure 1.12b), in the Suzuki Full-Floater it is between the rocker and the swingarm and in the Honda Pro-Link the element is between the connecting rod and the swingarm. Modest unsprung masses are obtained, as well as large wheel amplitude, but langer reactive forces are exchanged between the various parts of the four-bar linkage.
The four-bar linkage (Figure 1.12c) is also the basis of a suspension used especially on the final shaft transmission with universal joints (e.g. the BMW Paralever). The wheel is attached to the connecting rod of the four-bar linkage. The suspension acts as if it were composed of a very long fork fastened to the chassis at the centre of rotation (the point of intersection of the axes of the two rockers). An additional small four-bar linkage can be added to provide a suitable attachment point for the spring–shock element and thus a proper suspension behaviour.
1.3.3 Equivalent Stiffness and Damping
From a dynamics point of view, the vehicle can be considered as a main sprung body (chassis and rider) connected to two unsprung bodies (wheels) with two elastic systems (front and rear suspension). Also, rather than the characteristics of the spring–damper units, it is important to consider the characteristics of the suspensions in terms of wheel vertical displacement as a function of the vertical force applied. Therefore it is useful to reduce the real suspensions to equivalent, simpler, suspensions represented by two vertical spring–damper elements that connect the unsprung masses to the sprung mass. The parameters defining the equivalent suspension are: reduced stiffness, reduced damping, dependence of the reduced stiffness on the vertical displacement (progressive/regressive suspension), maximum travel and preloading.
To derive the equivalent (or reduced) stiffness, we consider the expression of the variation of the spring force as a function of the travel:
1.37
where is the spring force at the initial suspension travel , is the stiffness at and is the travel after variation. The power balance between the actual spring force and its equivalent vertical force at the wheel centre is
1.38
Therefore
1.39
where is the velocity ratio, between the suspension travel velocity and the wheel vertical velocity. The equivalent stiffness is
1.40
When assuming a constant velocity ratio the expression simplifies to
1.41
The derivation of the equivalent (reduced) damping is carried out using the same approach, and therefore Equation 1.40 can also be used for the damping by replacing , with , .
The preload of the equivalent suspension can be computed with the following expression (again from force power balance):
1.42
Finally, the dependence of the reduced stiffness on the vertical displacement can be affected either by changing the characteristic of the velocity ratio or by changing the characteristics of the spring element .
1.3.3.1 Front
The equivalent stiffness and damping can be easily computed for the widely spread telescopic fork (Figure 1.13). The velocity ratio is derived by considering the geometric relationship between the fork travel and the wheel vertical displacement:
1.43
where is the caster angle. Under the assumption of constant stiffness and damping coefficients, and constant velocity ratio , the reduced values are
1.44
In more complex linkages, the velocity ratio is computed numerically from a kinematic analysis of the mechanism. When it is not constant, or the spring/damper coefficients are not constant, the full Equation 1.40 should be used.
Figure 1.13 Reduced stiffness for the telescopic fork
Usually the suspension stiffness is in the range 13–25 kN/m, and the equivalent stiffness in the range 15–37 kN/m, while the damping coefficient is in the range 500–2000 Ns/m, and the equivalent in the range 550–2200 Ns/m, with the velocity ratio in the range 1.05–1.25 and the caster angle in the range 19–35.
1.3.3.2 Rear
The velocity ratio of a rear suspension featuring a linkage depends on many parameters and sometimes cannot be expressed analytically. In any case, the ratio can be easily computed numerically from a kinematic analysis of the mechanism. Typical values of for a swingarm with a four-bar linkage are in the range 0.3–0.6. Usually the suspension stiffness is in the range 100–150 kN/m, and the equivalent stiffness in the range 10–55 kN/m, while the damping coefficient is in the range 5–15 kNs/m and the equivalent in the range 450–5400 Ns/m.
1.4 In-Plane Dynamics
Vehicle dynamics can be divided between in-plane and out-of-plane dynamics. The former involve the motion of the vehicle in its symmetry plane (e.g. pitch, bounce, suspension travel) and mostly affect the riding comfort and road-holding, while the latter involve the lateral motion of the vehicle (e.g. yaw, roll, steer) and strongly affect the stability and safety. In straight running, two-wheeled vehicles are substantially symmetric and in-plane and out-of-plane motions are decoupled (therefore they can be examined separately), whereas while cornering strong interactions occur.
In this section, different in-plane models of increasing complexity are presented to highlight the main vehicle dynamics involved.
In practice, these dynamics are excited by road undulations and/or by the inertial forces generated while accelerating/braking. Suppose that the vehicle travels with constant velocity on a road with equidistant irregularities (e.g. the bays of a viaduct), Figure 1.14. The time required to cover the distance between two irregularities (length of the bay) is equal to
1.45
and represents the period of the external excitation. A resonance condition occurs whenever the excitation frequency is equal to the natural frequency of one of the in-plane vibration modes of the vehicle. As an example, with and m/s, it is , so the excitation has a frequency of 2 Hz. In general, several frequency components are present at the same time, depending on the road characteristics.
Figure 1.14 Road undulation
1.4.1 Pitch, Bounce and Hops Modes
Among the 11 DOFs necessary to fully define the vehicle trim (see Section 1.1), only seven are involved in the in-plane dynamics: longitudinal and vertical motion of the chassis, pitch of the chassis, suspension travel, wheels rotation. If we assume that the vehicle is traveling at constant speed (which is a common assumption when dealing with comfort analysis), the DOFs further reduce to four: vertical motion of the chassis, pitch angle and suspension travel. This DOFs are related to four physical vibration modes: bounce, pitch, wheel front hop and rear hop. Bounce is mainly related to the vertical motion of the chassis, pitch is mainly related to the pitch of the chassis and hops are mainly related to the wheels' vertical motion. In practice, every mode involves some contribution from all four DOFs. Figure 1.15 depicts an example of in-plane vibration modes, with typical values of natural frequency and damping ratio.
Figure 1.15 In-plane vibration modes
In order to model in-plane dynamics, several simple models are commonly used in addition to complex multibody models. The most popular are reported in the following sections.
1.4.1.1 Half-Vehicle Models
The half-vehicle model is a simple yet widespread model used to analyse the suspension–tyre dynamics. The name is due to the fact that only one tyre and one suspension are considered. Two versions are used: one DOF (or SDF) and two DOF.
In the simplest version (Figure 1.16) the model features a mass suspended by a spring–damper element. The mass may represent either the sprung mass (whose share is computed from the whole vehicle mass by considering the tyre loads distribution) or the unsprung mass (wheel rim, brake calliper, etc.). In the former case the spring–damper element represents the suspension, while the tyre compliance is neglected (Figure 1.16a), in the latter case the spring–damper element represents the tyre compliance while the suspension dynamics are neglected (Figure 1.16b). The undamped natural frequency , damped frequency and damping ratio are
1.46
where is the stiffness, the damping coefficient and the mass. As an example, we consider a vehicle with a mass of 200 kg, including two wheels of mass 15 kg each, a rider with a mass of 80 kg and the whole centre of mass exactly in the middle. We aim at estimating the natural frequency of the front wheel–suspension system, given the front suspension reduced vertical stiffness (see Section 1.3) and the tyre radial stiffness of . If we want the model to capture the suspension mode, we use and , obtaining a natural frequency . Another option is to use a combination of the suspension spring and tyre spring:
1.47
Figure 1.16 Half-vehicle model with (a) one DOF for sprung mass and (b) for unsprung mass
In this case, the stiffness reduces to and the natural frequency to . Otherwise, if we want the model to capture the wheel hop mode, we use and , obtaining a natural frequency of 17 Hz.
In the version with two DOFs (Figure 1.17), the model features two masses, representing the sprung mass and unsprung mass , and two spring–damper elements, representing the suspension characteristic and the tyre radial compliance. Therefore two modes influencing each other are captured. The expressions for the natural undamped frequencies are:
1.48
with:
1.49
Using the vehicle parameters defined above, gives (suspension mode) and (wheel hop mode).
Figure 1.17 Half-vehicle model with two DOFs
1.4.1.2 Full-Vehicle Models
The full-vehicle model has four DOFs and is depicted in Figure 1.18. Since there are no analytic and compact expressions for the system natural frequencies, the equations of motion are reported as:
1.50
with:
1.51
where and are the front and rear unsprung masses respectively, is the sprung mass, and are the front and rear suspension reduced stiffness, and are the front and rear suspension reduced damping, and are the front and rear tyre radial stiffness and and are the front and rear tyre radial damping.
Figure 1.18 Full-vehicle model with four DOFs
Note that the undamped radian frequencies can be numerically derived from
1.52
while for the full modal analysis the system must be reduced to a standard first-order formulation before computing the eigenvalues.
1.4.2 Powertrain
Powertrain dynamics involve the fluctuation of the vehicle's longitudinal velocity and wheel rotations, the three DOFs not considered in the models presented in the previous section. Figure 1.19 depicts a common motorcycle powertrain layout, which includes the crankshaft (where the engine propulsive or braking torque is generated), the primary and secondary shafts (whose velocity ratio is set by the rider operating the gearbox lever), the chain final transmission and the rear wheel.
Figure 1.19 Motorcycle powertrain
It is worth noting that the geometry of the final transmission strongly affects the vehicle trim while varying the propulsive force. In particular, it can be shown (Cossalter 2006) that the variation of the trim of the vehicle rear end (with respect to the standstill configuration) mainly depends on a parameter called the squat ratio :
1.53
where is the angle of the load transfer line and is the angle of the squat line; see Figure 1.20, where the lines are depicted for the swingarm with chain final transmission (Figure 1.20a) and for the four-bar linkage with shaft final transmission (Figure 1.20b).
Figure 1.20 Squat and load transfer lines
In more detail, the transfer load line represents the direction of the transfer tyre force acting on the rear contact point. The load transfer is originated by the vehicle acceleration and/or aerodynamic forces. Therefore two transfer lines can be computed: in the case of significant acceleration, the transfer line is the load line, whereas for mild acceleration, the transfer line is the aerodynamic line:
1.54
where is the height of the vehicle centre of mass, is the height of the aerodynamic centre and is the wheelbase.
As regards the squat line, it passes through the rear tyre contact point and the point , which is the intersection of the swingarm axis with the chain axis, in the case of a final transmission with chain and swingarm (Figure 1.20a), the swingarm pivot on the chassis in the case of a final transmission with shaft and swingarm, and the intersection of the two rockers in the case of a final transmission with shaft and four-bar linkage (Figure 1.20b).
When (often a design target) there is no variation of the trim of the vehicle rear end while changing the tyre thrust force . In practice, there may be a small variation, due to the theoretical assumptions. When increasing the longitudinal force with the rear suspension extends, while in the case of the rear suspension compresses.
1.4.3 Engine-to-Slip Dynamics
The engine torque generated at the crankshaft is transferred through the powertrain to the rear tyre, which generates a longitudinal force as a function of the longitudinal slip. These dynamics are especially important when it comes to the design of traction control systems (Massaro et al. 2011a, 2011b, Corno and Savaresi 2010): three simple models are described below to highlight the physical characteristics of the system.
First we consider a very simple model of the powertrain which does not account for either the tyre or the sprocket absorber flexibilities; see Figure 1.21(a). Note that the sprocket absorber is a device usually placed between the rear wheel sprocket and the rim, with the aim of damping the torsional vibration of the transmission system. This simple model is presented because it is widespread.
Figure 1.21 Half-vehicle model
The engine torque is applied to the crankshaft, it is transmitted through the gearbox (according to the selected gear ratio) to the output shaft (whose spin rate is in Figure 1.21a), it passes through the chain to the rear wheel rim (whose angular velocity is ) and then to the contact point. The external forces acting on the model are the tyre longitudinal force and the tyre normal load , but rolling resistance is neglected for simplicity. The equation of motion reads:
1.55
where is the rear wheel spin inertia, is the transmission inertia reduced to the rear wheel, is the spin rate of the wheel rim, is the whole transmission ratio, the engine torque at the crankshaft, the longitudinal force arm (assumed equal to the rolling radius) and the longitudinal force.
In more detail, the transmission inertia reduced to the rear wheel is computed from the engine spin inertia (plus clutch, starter, etc.), the gearbox primary shaft spin inertia and the gearbox output shaft spin inertia , given the primary ratio (between the crankshaft and the primary shaft of the gearbox), the gear ratio (between the primary and the output shaft of the gearbox thus depending on the selected gear) and the final ratio (between the output shaft and the rear wheel):
1.56
Moreover the product of the primary ratio, the gearbox ratio and the final ratio is defined as the whole transmission ratio:
1.57
and represents the ratio between the engine spin rate and the rear wheel spin rate.
For computation of the longitudinal road–tyre force, the full non linear formula is linearized about the steady-state condition (i.e. steady-state longitudinal slip and steady-state longitudinal force ), thus giving the following relationship between the actual force and the slip :
1.58
where is the slope of the longitudinal slip curve at the linearization point (also called the normalized longitudinal slip stiffness at the linearization point) and is the tyre normal load. The longitudinal slip is defined as:
1.59
The model equation can be written in state space formulation:
1.60
and the transfer function between engine torque and longitudinal slip can be expressed as:
1.61
where is the identity matrix, the Laplace variable and:
1.62
Therefore the system has one pole at:
1.63
Since all the parameters of Equation 1.63 are always positive but , the plant stability (i.e., the sign of the pole) is bound to the sign of the normalized longitudinal slip stiffness . In particular, the system is unstable when the slip stiffness is negative, and this usually happens only for high values of slip (skidding condition), after the peak of the force–slip curve, which usually occurs for slip values in the range 0.1–0.2. Finally, it is worth highlighting that the plant dynamic is very fast at low speeds, , and very slow at high speeds, .
As a second step, the tyre circumferential compliance is also considered; see Figure 1.21(b). With respect to the previous model, now the rim angular velocity differs from the tyre circumferential angular velocity, because of the deflection . As a consequence, the longitudinal slip expression changes to
1.64
In practice, when it comes to road tests the tyre circumferential deflection is not considered and the slip is computed according to Equation 1.59. For this reason, it is common to refer to Equation 1.59 as practical slip, since this is the slip which is measured in practice, and to Equation 1.64 as instantaneous slip (Lot 2004), since this is the actual slip at the contact point, which is the physical reason of the tyre longitudinal force. In other words, the instantaneous slip generates the longitudinal force, while the practical slip is what is observed. Therefore, the instantaneous slip is used to compute the road–tyre force while the practical slip is observed when computing the engine-to-slip transfer function Equation 1.61 and when comparing numerical results with road tests. The physical effect of the flexibility is to generate a phase lag between the practical slip and the actual force. Indeed, the actual force is in phase with the instantaneous slip.
A spring–damper element is used to take into account the flexibility of the tyre, and in particular the following expression relates the tyre circumferential deflection and deflection rate to the tyre longitudinal force :
1.65
At the contact point, there is force equilibrium between the force due to the elastic deflection and the force due to the slippage :
1.66
The system now has two equations (1.56 and 1.66) and two state variables ( and ). The following state space matrices are found:
1.67
When inspecting the engine-to-slip transfer function (Equation 1.61), it turns out that at null longitudinal speed the system is vibrating with undamped natural frequency and damping ratio :
1.68
As the speed increases, the frequency reduces and the damping increases up to a critical velocity (usually in the range 30–60 m/s):
1.69
Above this the system is no longer vibrating (the two poles turn from complex conjugate pairs to real poles).
It is worth noting that there is an alternative approach to account for this tyre force lag. Instead of considering the tyre flexibility, it is possible to add a first-order differential equation (relaxation equation, see Section 1.2.3) which replaces Equation 1.66 :
1.70
where is the relaxation length, is the longitudinal velocity, the actual longitudinal force, the longitudinal force computed with the practical slip of Equation 1.59. The two approaches give similar results when
1.71
As a third step, a flexible sprocket absorber is introduced between the rear wheel chain sprocket and the rear wheel rim, in addition to the compliant tyre; see Figure 1.21(c). The following expression is used to compute the absorber torque as a function of its deflection :
1.72
where is the absorber stiffness, the damping coefficient and the absorber deflection rate. The state space matrices now read:
1.73
No compact expressions for poles are available, but they can be easily computed numerically from Equation 1.73. The four complex poles of the system are associated with two torsional vibrating modes, which may be either identified in the tyre circumferential and sprocketabsorber deflection, or in the wheel and transmission spin.
Finally, it should be noted that when the engine-to-slip dynamics are of interest for frequencies above 30 Hz, the tyre belt dynamics should also be added to the model (thus increasing the number of state variables above four).
