Mechanical Engineering Series Giancarlo Genta Lorenzo Morello The Automotive Chassis Volume 2: System Design Second Edition Tai ngay!!! Ban co the xoa dong chu nay!!! Mechanical Engineering Series Series Editor Francis A Kulacki, Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA The Mechanical Engineering Series presents advanced level treatment of topics on the cutting edge of mechanical engineering Designed for use by students, researchers and practicing engineers, the series presents modern developments in mechanical engineering and its innovative applications in applied mechanics, bioengineering, dynamic systems and control, energy, energy conversion and energy systems, fluid mechanics and fluid machinery, heat and mass transfer, manufacturing science and technology, mechanical design, mechanics of materials, micro- and nano-science technology, thermal physics, tribology, and vibration and acoustics The series features graduate-level texts, professional books, and research monographs in key engineering science concentrations More information about this series at http://www.springer.com/series/1161 Giancarlo Genta Lorenzo Morello • The Automotive Chassis Volume 2: System Design Second Edition 123 Giancarlo Genta Politecnico di Torino Turin, Italy Lorenzo Morello Politecnico di Torino Turin, Italy ISSN 0941-5122 ISSN 2192-063X (electronic) Mechanical Engineering Series ISBN 978-3-030-35708-5 ISBN 978-3-030-35709-2 (eBook) https://doi.org/10.1007/978-3-030-35709-2 1st edition: © Springer Science+Business Media B.V 2009 2nd edition: © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Foreword Each book—even one that at first glance might seem like a “cold” university engineering text—tells a fascinating story of experience and knowledge When I was invited to write an introduction to this volume, based on the experiences of a great professor and a very experienced industrial manager, I felt the pleasant feeling of a puzzle just completed: the text has, in fact, achieved the important goal of helping readers understand what it really means to move from the concept to the creation of a new car, from design to assembly To describe the impact of the text, it is sufficient to highlight how the authors’ passion for creating this manual—so much so that it has become an international point of reference in the design of chassis—coincides substantially with the enthusiasm of the thousands of people who work every day at Fiat Chrysler Automobiles with the aim of conceiving and creating cars that are increasingly innovative Before they even start studying on these pages, I would like students to be aware that creating a new car—or even just contributing to its birth—is a fascinating job, made up of successive joints of creative and technical skills, which requires an extraordinary commitment The same commitment is needed to lay the foundations for new engineers to grow and make their contribution to creating ever more cutting-edge cars Flipping through the pages of the book and investigating the various design steps, it is clear that the authors have achieved an important objective: to give due importance to the fact that a university text must not only tell the theory on how to build new cars, but also describe in a coherent and comprehensive way the content of a car starting from the manufacturer’s point of view, sometimes very different from the theory, too often the only subject in university texts And in telling this small story of effective integration between the world of study and the world of work, I rediscover a bit of the experiences I lived first by studying Engineering at the University and then working in Fiat Chrysler Automobiles: I find out how much effort and dedication both paths have taken But, above all, how much satisfaction can they bring v vi Foreword The approach used by the authors shows the indissoluble link between the academic system and the professional paths that a car company like FCA is able to offer, making it well understood that “knowing how to do” is very different from “doing” but that the two voices together are a winning combination, an essential method to always keep up with the times and make a difference in a context in which knowledge remains the fundamental competitive advantage Torino, Italy June 2019 Daniele Chiari Head of Product Planning & Institutional Relations, FCA, Emea Foreword to the Second Edition It is a great pleasure and honor for me to write the foreword to the second edition of The Automotive Chassis First of all, I want to express the gratitude that I have for the authors, who have been great masters of my education: Professor Genta as my Tutor at Politecnico di Torino and Professor Morello as Head of Engineering at Fiat Auto Their innovative, methodical, rational approach, and their effort to promote and develop the technical competence have helped to form my core values and beliefs The two books of The Automotive Chassis represent an exceptional masterpiece that has been useful in these years to the engineering students and also to the automotive engineers of my generation Thanks to this work, we have further developed our knowledge of the most complex and fascinating area of the vehicle where the real technical competence of the engineer is tested In this second edition, the first book maintains its robust and structured approach to “Components Design” and the second book on “Systems Design” has been further enriched and updated according to the rapid growth of our car industry toward NEVs and Autonomy With these actions, The Automotive Chassis will continue its role of spreading the chassis engineering culture in our fascinating automotive world Shangai, China April 2019 Giorgio Cornacchia Head of APAC Product Development at FCA vii Preface This book is the result of a double decades-long experience: from one side a teaching experience of courses such as Vehicle Mechanics, Vehicle System Design, Chassis Design, and more to students of Engineering, from the other side from the design praxis of vehicle and chassis components in a large automotive company This book is primarily addressed to students of Automotive engineering and secondarily to all technicians and designers working in this field It also addressed to all people enthusiast of cars that are looking for a technical guide The tradition and the diversity of disciplines involved in road vehicle design lead us to divide the vehicle into three main subsystems: the engine, the body, and the chassis The chassis isn’t today a visible subsystem anymore, tangible as a result of a certain part of the fabrication process, while engine and body are; chassis components are assembled, as a matter of fact, directly on the body For this reason, the function of the chassis cannot be assessed separately from the rest of the car As we will see better, reading the chapters dedicated in the first and in the second part of this book, to the historical evolution, the situation was completely different in the past; in the first cars the chassis was defined as a real self-moving subassembly that included the following: • a structure, usually a ladder framework, able to carry on all the remaining components of the vehicle; • the suspensions for the mechanical linkage of wheels with the framework; • the wheels completed with tires; • the steering system to change wheel angles accordingly to the vehicle path; • the brake system to reduce the speed or to stop the vehicle; and • the transmission to apply the engine torque to the driving wheels This group of components, after the engine assembly, was able to move autonomously; this happened at least in many experimental tests, where the body was simulated with a ballast and during the fabrication process, to move the chassis from the shop of the carmaker to that of the body maker ix x Preface Customers often bought from the carmaker a chassis to be completed later on by a body maker, according to their desire and specification On contemporary vehicles, this particular architecture and function is only provided for industrial vehicles, with the exception of buses where the structure, even if built by some body maker, participates with the chassis framework to the total stiffness, such as a kind of unitized body On almost every car, the chassis structure cannot be separated from the body as being part of its floor (platform); sometimes some auxiliary framework is also added to interface suspensions or power train to the body and to enable their pre-assembly on the side of the main assembly line Nevertheless, tradition and some particular technical aspect of these components have justified the development of a particular discipline within vehicle engineering; as a consequence, almost all car manufacturers have a technical organization addressed to the chassis, separated from those addressed to the body or to the engine A new reason has been added in recent times to justify a different discipline and a specific organization and is the setting up of the so-called technological platforms: the modern trend of the market calls for an unprecedented product diversification, never reached in the past; sometimes marketing expert calls this phenomenon fragmentation This high diversification couldn’t be sustained with acceptable production cost without a strong cross standardization of non-visible or of non-specific part of a certain model This situation has been very well known since years to all industrial vehicle manufacturers The term platform implying the underbody and the front side members, with the addition of the adjective technological, describes a set of components substantially equal to the former chassis; the particular technical and scientific issues, the different development cycle, and the longer economic life have reinforced the specificity of engineers that are dedicated to this car subsystem The contents of this book are divided into five parts, organized into two volumes The first volume describes main chassis subsystems in two parts The first part describes the main components of the chassis from the tire to the chassis structure, including wheels, suspension, steering, and braking systems, not forgetting the control systems that show an increasing importance, due to the diffusion of active and automatic systems The second part is addressed to the transmission and to the related components; the complexity of this topic justifies a separated presentation It should be noticed that, by many car manufacturers, the engineering and production organization dedicated to this subsystem are integrated into the power train organization, instead of the chassis organization This has obviously no influence on the technical contents of this book and can be justified by the standardization issues and by the life cycle of this component, in certain aspects more similar to the engine than to the chassis 32.2 Linearized Rigid Body Model 751 ⎡ cos(ψ) − sin(ψ) ⎢ sin(ψ) cos(ψ) A=⎢ ⎣ 0 0 ⎤ 0⎥ ⎥ 0⎦ (32.53) Because in this case A is a rotation matrix, the inverse transformation is q˙ = Bw = Aw The vector defining the position of the center of the sprung mass GS with respect to point H is, in the body-fixed frame, T r1 = h 0 (32.54) In the inertial frame the position of the same point is (GS −O’) = (H − O’) + Rr1 (32.55) Because r1 is constant, the velocity of point GS is i.e T ˙ 1, VGS = X˙ Y˙ + Rr (32.56) ˙ 1, VGS = R1 V + Rr (32.57) and then the translational kinetic energy of the sprung mass is Tt = T ˙ T Rr ˙ +2VT R1T Rr ˙ m V V + r1 T R (32.58) Because plane x z is a symmetry plane for the sprung mass, its inertia tensor is ⎡ ⎤ Jx −Jx z J = ⎣ Jy ⎦ −Jx z Jz (32.59) The rotational kinetic energy of the sprung mass is then Tr = T J (32.60) By performing the relevant computations, expressing the components of the angular velocity as functions of the derivatives of the coordinates and neglecting the terms containing powers of small quantities higher than the second, it follows that 752 32 Models for Tilting Body Vehicles T = 21 m vx2 + v 2y + 21 Jx∗ φ˙ + 21 Jy∗ sin2 (φ) + Jz cos2 (φ) ψ˙ −Jx z cos (φ) ψ˙ φ˙ + mvx h ψ˙ sin (φ) − mv y h φ˙ cos (φ) , where (32.61) Jx∗ = mh + Jx , Jy∗ = mh + Jy The height of the center of mass of the sprung mass on the ground is Z G = h cos (φ) , (32.62) and then the gravitational potential energy of the vehicle is Ug = mgh cos (φ) (32.63) The potential energy reduces to its gravitational components in the case of a twowheeled vehicle In vehicles with three or more wheels with suspensions, the elastic potential energy due to the springs must also be accounted for In the following study the elastic potential energy will be assumed to depend only on the roll angle; however, it is not a simple quadratic function as in the case of linearized models, because the roll angle may be large In general, it is possible to state that Us = Us (φ) (32.64) If the vehicle has suspensions for the roll motion and the latter are provided with dampers, a dissipative function may be defined, F = F φ, φ˙ (32.65) It must be expressly stated that the equations above were obtained without resorting to the assumption that all variables of motion, with the exception of the roll angle φ, are small quantities Moreover, these equations are more general and hold even if the roll axis does not lie on the ground or is exactly horizontal, provided that the angle between the roll axis and the ground plane (referred to as θ0 in the previous chapters) is a small angle and that h is the distance between the center of mass and the roll axis instead of its height on the ground 32.2.2 Rotation of the Wheels Because it has been assumed that, as in the case of vehicles with two wheels (see Appendix B), the rotation axis of the wheels is perpendicular to the symmetry plane, the absolute angular velocity of the ith wheel expressed in the reference frame of the sprung mass is 32.2 Linearized Rigid Body Model 753 ⎧ ⎨ ⎫ x ⎬ i = y + χ˙ i , ⎩ ⎭ z (32.66) where χi is the rotation angle of the wheel If the wheel steers, the reference frame of the ith wheel will be rotated by a steering angle δi about an axis, the kingpin axis, that in general is not perpendicular to the ground If ek is the unit vector of the kingpin axis (its components will be indicated as xk , yk and z k ),3 the rotation matrix Rki to rotate the reference frame fixed to the sprung mass in such a way that its z axis coincides with the kingpin axis of the ith wheel is ⎡ ⎤ z k −xk yk xk xk2 + z k2 ⎢ ⎥ ⎢ ⎥ (32.67) Rki = ⎢ xk2 + z k2 yk xk2 + z k2 ⎥ ⎦ 2 ⎣ xk + z k −xk −z k yk z k xk2 + z k2 The caster and the inclination angles of the kingpin are usually small in suspensions for two-wheeled axles and, as seen in the previous sections, rotation matrix Rki reduces to ⎡ ⎤ xk Rki ≈ ⎣ yk ⎦ , (32.68) −xk −yk where xk and yk are the caster and the inclination angles (the latter changed in sign) of the kingpin axis For symmetry reasons xk D = xk S , yk D = −yk S (32.69) In motorcycles yk is zero, while the caster angle xk may be large In the following parts of this section this possibility will not be considered A further rotation matrix ⎡ ⎤ cos(δi ) − sin(δi ) R4i = ⎣ sin(δi ) cos(δi ) ⎦ (32.70) 0 can be defined for the rotation of the wheel about the kingpin axis The angular velocity of the wheel in the reference frame of the sprung mass is then T e2 (32.71) wi = +δ˙i Rki e3 + χ˙ i Rki R4i Rki T T ) to obtain the angular Equation (32.71) must be premultiplied by (Rki R4i Rki velocity of the wheel in its own reference frame Remembering that R4i e3 = e3 , it Obviously xk2 + yk2 + z k2 = 754 32 Models for Tilting Body Vehicles follows that wi = χ˙ i e2 + δ˙i + , (32.72) = Rki e3 , (32.73) where T T = Rki R4i Rki Because the wheel is a gyroscopic body (two of its principal moments of inertia are equal) with a principal axis of inertia coinciding with its rotation axis, its inertia matrix is diagonal and has the form Jwi = diag Jti J pi Jti , (32.74) where J pi is the polar moment of inertia and Jti is the transversal moment of inertia of the ith wheel The rotational kinetic energy of the ith wheel is Twri = 21 T 2T Jwi + 21 χ˙ i2 e2T Jwi e2 + 21 δ˙i2 1T Jwi + +χ˙ i δ˙i e2T Jwi + χ˙ i e2T Jwi + δ˙i 1T Jwi (32.75) By performing the relevant computations and assuming that all variables of motion, except for φ and χi , are small, it follows that Twri = 21 Jti φ˙ + 21 J pi sin2 (φ) + Jti cos2 (φ) ψ˙ + 21 J pi χ˙ i2 + (32.76) + 21 δ˙i2 Jti − J pi δi φ˙ χ˙ i + J pi yki χ˙ i δ˙i + J pi sin (φ) ψ˙ χ˙ i + Jti cos (φ) ψ˙ δ˙i The first two terms express the rotational kinetic energy of the wheel due to angular velocity of the vehicle and thus have already been included in the expression of the kinetic energy of the vehicle, if the moments of inertia of the wheels have been taken into account when computing the total inertia 32.2.3 Lagrangian Function The Lagrangian function of the vehicle is then L = 21 m vx2 + v 2y + 21 Jx∗ φ˙ + 21 Jy∗ sin2 (φ) + Jz cos2 (φ) ψ˙ + −Jx z cos (φ) ψ˙ φ˙ + mvx h ψ˙ sin (φ) − mv y h φ˙ cos (φ) + ! + ∀i 21 J pi χ˙ i2 + 21 δ˙i2 Jti − J piδi φ˙ χ˙ i + J pi yki χ˙ i δ˙i + +J pi sin (φ) ψ˙ χ˙ i + Jti cos (φ) ψ˙ δ˙i − mgh cos (φ) − Us (φ) (32.77) If the longitudinal slip of the wheels is neglected, their angular velocity is χ˙ i = V Rei (32.78) 32.2 Linearized Rigid Body Model 755 In a way similar to our treatment of the four-wheeled vehicle, the kinetic energy linked with the steering velocity δ˙ may be neglected in the locked control motion The Lagrangian reduces to L = 21 m at V + 21 mv 2y + 21 Jx∗ φ˙ + 21 Jy∗ sin2 (φ) + Jz cos2 (φ) ψ˙ + −Jx z cos (φ) ψ˙ φ˙ + V Js ψ˙ sin (φ) − mv y h φ˙ cos (φ) + ! J −V ∀i Rpie δi φ˙ − mgh cos (φ) − Us (φ) , (32.79) i where m at = m + " J pi " J pi , Js = m h + , Rei Rei ∀i ∀i Jx∗ = mh + Jx , Jy∗ = mh + Jy The derivatives of the Lagrangian function are then ∂L = m at V + Js ψ˙ sin (φ) , ∂V (32.80) ∂L = mv y − mh φ˙ cos (φ) , ∂v y (32.81) " J pi ∂L δi , = Jx∗ φ˙ − Jx z cos (φ) ψ˙ − mv y h cos (φ) − V Rei ∂ φ˙ ∀i (32.82) ∂L ∗ = Jy sin (φ) + Jz cos2 (φ) ψ˙ − Jx z cos (φ) φ˙ + V Js h sin (φ) ∂ ψ˙ (32.83) The derivative with respect to time of the derivatives with respect to the generalized velocities contains products that are themselves the products of two or more small quantities, and thus must be neglected in the linearization process Also V˙ may be considered as a small quantity, and then terms containing, for instance, product V˙ δ may be neglected It then follows that d dt d dt d dt ∂L ∂ φ˙ ∂L ∂V = m at V˙ + Js ă sin () , (32.84) = m v y mh ă cos () , (32.85) = Jx ă Jx z cos () ă m v y h cos (φ) , (32.86) ∂L ∂v y 756 32 Models for Tilting Body Vehicles d dt ∂L ă = Jy sin2 () + Jz cos2 () ă Jx z cos () φ+ (32.87) +Js V˙ sin (φ) + Js V cos (φ) φ˙ , ∂L ∂L ∂L =0, = = ∂x ∗ ∂ y∗ ∂ψ (32.88) ∂L ∂Us (φ) = Js V ψ˙ cos (φ) + mgh sin (φ) − ∂φ ∂φ (32.89) 32.2.4 Kinematic Equations Matrix A is what we have already seen for the model with 10◦ of freedom, except that the last six rows and columns are not present here The equation of motion in the configuration space is ∂ ∂t ∂L ∂w ∂L ∂L ∂F T +B −B + = BT Q ∂w ∂q ∂w T (32.90) The column matrix BT Q containing the four components of the generalized forces vector will be computed later, when the virtual work of the forces acting on the system is described In the following its elements will be written as Q x , Q y , Q φ , Q ψ As usual, the most difficult part is writing matrix BT By performing somewhat complex computations, following the procedure outlined in Appendix A, it follows that ⎤ ⎤ ⎡⎡ −ψ˙ ⎥ ⎢ ⎢ ψ˙ ⎥ ⎥ 04×2 ⎥ ⎢ BT = ⎢ ⎦ ⎣⎣ 0 ⎦ −v y vx By introducing the values of the derivatives and linearizing, it follows that BT ∂L ∂w = ⎧ ⎪ ⎪ ⎪ ⎨ ⎫ ⎪ ⎪ ⎪ ⎬ m at V ψ˙ ⎪ # $ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V −mh φ˙ cos (φ) − v y ! ⎭ ∀k J pr R (32.91) e Finally B T ∂L ∂q = ∂L ∂q (32.92) 32.2 Linearized Rigid Body Model 757 32.2.5 Equations of Motion First Equation: Longitudinal Translation m at V + Js ă sin () = Q x (32.93) Second Equation: Lateral Translation m v˙ y + m at V mh ă cos (φ) = Q y (32.94) Third Equation: Roll Rotation Jx ă Jx z cos () ă m v˙ y h cos (φ) − Js V ψ˙ cos (φ) + −mgh sin (φ) + ∂Us (φ) ∂φ + ∂F (φ,φ˙ ) ∂ φ˙ (32.95) = Qφ Fourth Equation: Yaw Rotation ă Jy sin2 () + Jz cos2 () ă Jx z cos () + +Js V˙ sin (φ) + V cos (φ) φ˙ ! J pi ∀i Rei − V vy ! J pi ∀k Rei2 (32.96) = Qψ 32.2.6 Sideslip Angles of the Wheels The sideslip angles of the wheels may be computed from the components of the velocities of the centers of the contact areas of the wheels in the x ∗ y ∗ z frame If the roll axis lies on the ground, some simplifications may be introduced: The roll angle and the roll velocity not appear in the expression of the velocity of the wheel-ground contact points, if the track variations due to roll are neglected The expression of the sideslip angle coincides with that seen for the rigid vehicle, except for the term containing the steering angle Assuming that the sideslip angle is small, it follows that αk = vy x Pk + ψ˙ − δk cos (φ) − δk (φ) cos (φ) , V V (32.97) where subscript k refers to the axle, because the two wheels of the same axle have the same sideslip angle The term cos (φ) multiplying the steering angle is linked to the circumstance that the steering loses its effectiveness with increasing roll angle, and was computed assuming that the kingpin axis is, when the roll angle vanishes, essentially perpendicular to the ground If it is not, the caster and inclination angles had to be taken 758 32 Models for Tilting Body Vehicles into account, together with their variation with the roll angle The term δk (φ) is roll steer that, in case of large roll angles, may be too large to be linearized 32.2.7 Generalized Forces The generalized forces Q k to be introduced into the equations of motion include the forces due to the tires, the aerodynamic forces and possible forces applied on the vehicle by external agents The virtual displacement of the center of the contact area of the left (right) wheel of the kth axle is ⎧ ∗ ⎫ ⎨ δx − δψ y Pk ⎬ (32.98) {δs Pk L(R) }x ∗ y ∗ z = δ y ∗ + δψx Pk , ⎩ ⎭ where x Pk and y Pk are the coordinates of the center of the contact area in the reference frame x ∗ y ∗ z ∗ By writing as Fx∗ and Fy∗ the forces exerted by the tire in the direction of the x ∗ and y ∗ axes, assuming that the longitudinal forces acting on the wheels of the same axle are equal, the expression of the virtual work is δLk = δx ∗ Fx∗ + δ y ∗ Fy∗ + δψ Fy∗ x Pk + Mz (32.99) Because of the small steering angle, forces Fx∗ and Fy∗ will be confused in the following sections with the forces expressed in the reference frame of the wheel In a similar way, the virtual displacement of the center of mass for the computation of the aerodynamic forces is, in the x ∗ y ∗ z ∗ frame, {δsG S }x ∗ y ∗ z ∗ ⎧ ∗ ⎫ ⎨ δx + h sin (φ) δψ ⎬ = δ y ∗ − h cos (φ) δφ ⎩ ⎭ −h sin (φ) δφ (32.100) The aerodynamic forces and moments are referred to the x yz frame and not to the x ∗ y ∗ z ∗ frame Force Fz a , for example, lies in the symmetry plane of the vehicle and is not perpendicular to the road In this way it may be assumed that aerodynamic forces not depend on the roll angle φ A rotation of the reference frame is then needed: ⎫ ⎧ ∗ ⎫ ⎧ Fxa ⎬ ⎨ Fxa ⎬ ⎨ ∗ Fya = Fya cos (φ) − Fza sin (φ) , (32.101) ⎭ ⎩ ∗ ⎭ ⎩ Fya sin (φ) + Fza cos (φ) Fza ⎧ ∗ ⎫ ⎧ ⎫ Mxa ⎨ Mxa ⎬ ⎨ ⎬ ∗ M ya = M ya cos (φ) − Mza sin (φ) ⎩ ∗ ⎭ ⎩ ⎭ M ya sin (φ) + Mza cos (φ) Mza (32.102) 32.2 Linearized Rigid Body Model 759 The virtual work of the aerodynamic forces and moments is then δLa = Fxa δx ∗ + Fya cos (φ) − Fza sin (φ) δ y ∗ +