Massimo Guiggiani

Università di Pisa | UNIPI · Department of Civil and Industrial Engineering

This paper derives the dynamic equations of a reduced-order race-car model using Lie-group methods. While these methods are familiar to computational dynamicists and roboticists, their adoption in the vehicle dynamics community is limited. We address this gap by demonstrating how this framework integrates smoothly with the Articulated-Body Algorith...

This paper presents an automatic procedure to enhance the accuracy of the numerical solution of an optimal control problem (OCP) discretized via direct collocation at Gauss-Legendre points. First, a numerical solution is obtained by solving a nonlinear program (NLP). Then, the method evaluates its accuracy and adaptively changes both the degree of...

In this paper we derive the dynamic equations of a race-car model via Lie-group methods. Lie-group methods are nowadays quite familiar to computational dynamicists and roboticists, but their diffusion within the vehicle dynamics community is still limited. We try to bridge this gap by showing that this framework merges gracefully with the Articulat...

This paper presents an automatic procedure to enhance the accuracy of the numerical solution of an optimal control problem (OCP) discretized via direct collocation at Gauss-Legendre points. First, a numerical solution is obtained by solving a nonlinear program (NLP). Then, the method evaluates its accuracy and adaptively changes both the degree of...

Real roads are far from flat. Even freshly paved highways have small imperfections that interact with the vehicle dynamics by exciting vehicle vertical vibrations.

The goal of this chapter is to understand how to stop a vehicle as soon as possible, avoiding wheel locking. This result can be achieved only if the vehicle has the right brake balance. Unfortunately, brake balance is affected by the value of the grip and by the position of the center of mass. This topic is addressed in detail, both analytically an...

Limited slip differential and wings are typical of race cars. Both greatly impact on the vehicle handling (otherwise would not be used). Therefore, the first part of this chapter is devoted to the formulation of a suitable vehicle model. The handling of Formula cars is addressed by means of the Map of Achievable Performance (MAP). With this new app...

The vehicle orientation is defined by means of the yaw-pitch-roll elemental rotations. Then, to define the vehicle position, a careful analysis of what happens when the vehicle rolls is performed. The key result is the definition of the Vehicle Invariant Point (VIP) as the best option for monitoring the vehicle position, and also for defining the l...

All road vehicles have wheels and almost all of them have wheels with pneumatic tires. Wheels have been around for many centuries, but only with the invention, and enhancement, of the pneumatic tire it has been possible to conceive fast and comfortable road vehicles [5].

The MAP approach provides a way to analyze the steady-state handling behavior of road/race cars. It is completely general, in the sense that is can be employed for any real car, and for any mathematical model as well. Two concepts play a central role in MAP: the achievable region, that is the totality of the achievable trim conditions for a given v...

Road cars are characterized by having an open differential and no significant aerodynamic downforces. These two aspects allow for some substantial simplifications of the vehicle model. With the additional assumption of equal gear ratio of the steering system for both front wheels, it is possible to formulate the single track model. Quite contrary t...

In this chapter a simple, yet significant, tire model is developed. It is basically a brush model, but with some noteworthy additions with respect to more common formulations. For instance, the model takes care of the transient phenomena that occur in the contact patch. A number of figures show the pattern of the local actions within the contact pa...

At the beginning of this chapter, the simplifying assumptions to formulate a simple, yet significant, vehicle model are listed. Then the kinematics of the vehicle as a whole is described in detail, followed by the kinematics of each wheel with tire. The next step is the formulation of the constitutive (tire) equations and of the global equilibrium...

Cars have to negotiate corners. But not all cars do that the same way. This is particularly evident in race cars, where the ability to negotiate a corner is a crucial aspect to minimize the lap time. In this chapter the kinematics of a vehicle while taking a corner is exploited. At first sight taking a corner looks quite a trivial task. But designi...

A correct methodology to evaluate the friction coefficient in lubricated gear pairs is paramount for both the estimation of energy losses and the prediction of wear. In the first part of the paper, a methodology for estimating the coefficient of friction with a semi-empirical formulation is presented, and its results are also employed to analyze me...

This paper addresses the problem of the link between the driving style of an ideal driver, modelled as an optimal controller, and fundamental set-up parameters of a vehicle in the GP2 motorsport class. The aim is to evaluate quantitatively how set-up parameters, like distribution of aerodynamic loads, weight and roll stiffness between front and rea...

A boundary element method (BEM) for the solution of lubrication problems on finite bearings is presented. The formulation requires the Reynolds equation to be transformed into a constant coefficient equation. Several film shapes that make the transformation possible are systematically obtained. Noticeably, they cover most practical cases. As an exa...

The capability to smooth down the road imperfections affects both the comfort and the road holding of the vehicle. Improving comfort means, basically, limiting the vertical acceleration fluctuations of the vehicle body and hence of passengers. Improving road holding means, among other things, limiting the fluctuations of the vertical force that eac...

Cars have to negotiate corners. But not all cars do that the same way. This is particularly evident in race cars, where the ability to negotiate a corner is a crucial aspect to minimize the lap time. In this Chapter the kinematics of a vehicle while taking a corner is exploited. At first sight taking a corner looks quite a trivial task. But designi...

The MAP approach provides a way to analyze the steady-state handling behavior of road/race cars. It is completely general, in the sense that is can be employed for any real car, and for any mathematical model as well. Two concepts play a central role in MAP: the achievable region, that is the totality of the achievable trim conditions for a given v...

The vehicle orientation is defined by means of the yaw-pitch-roll elemental rotations. Then, to define the vehicle position, a careful analysis of what happens when the vehicle rolls is performed. The key result is the definition of the Vehicle Invariant Point (VIP) as the best option for monitoring the vehicle position, and also for defining the l...

The goal of this chapter is to understand how to stop a vehicle as soon as possible, avoiding wheel locking. This result can be achieved only if the vehicle has the right brake balance. Unfortunately, brake balance is affected by the value of the grip and by the position of the center of mass. This topic is addressed in detail, both analytically an...

The first goal of this Chapter is to describe the kinematics of a wheel with tire, mainly under steady-state conditions. This leads to the definitions of slips as a measure of the extent to which the wheel with tire departs from pure rolling conditions. All aspects are discussed in detail and with a critical approach, showing that the use of the sl...

In this chapter a simple, yet significant, tire model is developed. It is basically a brush model, but with some noteworthy additions with respect to more common formulations. For instance, the model takes care of the transient phenomena that occur in the contact patch. A number of Figures show the pattern of the local actions within the contact pa...

Road cars are characterized by having an open differential and no significant aerodynamic downforces. These two aspects allow for some substantial simplifications of the vehicle model. With the additional assumption of equal gear ratio of the steering system for both front wheels, it is possible to formulate the single track model. Quite contrary t...

Limited slip differential and wings are typical of race cars. Both greatly impact on the vehicle handling (otherwise would not be used). Therefore, the first part of this Chapter is devoted to the formulation of a suitable vehicle model, which, in this case, cannot be single track. As a matter of fact, there is a strong interaction between lateral...

At the beginning of this chapter, the simplifying assumptions to formulate a simple, yet significant, vehicle model are listed. Then the kinematics of the vehicle as a whole is described in detail, followed by the kinematics of each wheel with tire. The next step is the formulation of the constitutive (tire) equations and of the global equilibrium...

This textbook covers handling and performance of both road and race cars. Mathematical models of vehicles are developed always paying attention to state the relevant assumptions and to provide explanations for each step. This innovative approach provides a deep, yet simple, analysis of the dynamics of vehicles. The reader will soon achieve a clear...

In the original version of the book, the Chapters 2, 3, 5, 6, 7, 8, 9 and 10 were revised.

The teeth of ordinary spur and helical gears are generated by a (virtual) rack provided with planar generating surfaces. The resulting tooth surface shapes are a circle-involute cylinder in the case of spur gears, and a circle-involute helicoid for helical gears. Advantages associated with involute geometry are well known. Beveloid gears are often...

These are all the errors and omissions for the book The Science of Vehicle Dynamics, published by Springer (2014), reported as of July 15, 2016. They are listed in two ways: • by page number (better for first reading); • in chronological order (better for updating).

The teeth of ordinary spur and helical gears are generated by a (virtual) rack provided with planar generating surfaces. The resulting tooth surface shapes are a circle-involute cylinder in the case of spur gears, and a circle-involute helicoid for helical gears. Advantages associated with involute geometry are well known: in particular, the motion...

Methods for accurate grinding of face-hobbed bevel gears have not been developed to date, the main obstacles being their epicycloidal lengthwise tooth curve and slot width taper. Grinding them while preserving their geometry would be desirable, as the epicycloidal tooth curve makes face-hobbed gears less sensitive to misalignments and deflections....

Book available at HALF PRICE (50% off), shipping included worldwide till MARCH 31, 2016 here http://www.springer.com/en/book/9789401785327 (Springer website) THE MORE YOU KNOW VEHICLE DYNAMICS, THE MORE YOU'LL BE SURPRISED Vehicle dynamics is often perceived as a quite intuitive subject. As a matter of fact, lots of people are able to drive a car....

The goal of this chapter is to understand how to stop a vehicle as soon as possible, avoiding wheel locking. This result can be achieved only if the vehicle has the right brake balance. Unfortunately, brake balance is affected by the value of the grip and by the position of the center of mass. This topic is addressed in detail, both analytically an...

Cars have to negotiate corners. But not all cars do that the same way. This is particularly evident in race cars, where the ability to negotiate a corner is a crucial aspect to minimize the lap time. In this chapter the kinematics of a vehicle while taking a corner is exploited. At first sight taking a corner looks quite a trivial task. But designi...

Road cars are characterized by having an open differential and no significant aerodynamic downforces. These two aspects allow for some substantial simplifications of the vehicle model. With the additional assumption of equal gear ratio of the steering system for both front wheels, it is possible to formulate the single track model. Quite contrary t...

The vehicle orientation is defined by means of the yaw-pitch-roll elemental rotations. Then, to define the vehicle position, a careful analysis of what happens when the vehicle rolls is performed. The key result is the definition of the Vehicle Invariant Point (VIP) as the best option for monitoring the vehicle position, and also for defining the l...

In this chapter a simple, yet significant, tire model is developed. It is basically a brush model, but with some noteworthy additions with respect to more common formulations. For instance, the model takes care of the transient phenomena that occur in the contact patch. A number of Figures show the pattern of the local actions within the contact pa...

The first goal of this chapter is to describe the kinematics of a wheel with tire, mainly under steady-state conditions. This leads to the definitions of slips as a measure of the extent to which the wheel with tire departs from pure rolling conditions. All aspects are discussed in detail and with a critical approach, showing that the use of the sl...

Limited slip differential and wings are typical of race cars. Both greatly impact on the vehicle handling (otherwise would not be used). Therefore, the first part of this chapter is devoted to the formulation of a suitable vehicle model, which, in this case, cannot be single track. As a matter of fact, there is a strong interaction between lateral...

The capability to smooth down the road imperfections affects both the comfort and the road holding of the vehicle. Improving comfort means, basically, limiting the vertical acceleration fluctuations of the vehicle body and hence of passengers. Improving road holding means, among other things, limiting the fluctuations of the vertical force that eac...

At the beginning of this chapter, the simplifying assumptions to formulate a simple, yet significant, vehicle model are listed. Then the kinematics of the vehicle as a whole is described in detail, followed by the kinematics of each wheel with tire. The next step is the formulation of the constitutive (tire) equations and of the global equilibrium...

Vehicle dynamics can be set as a truly scientific subject, it actually needs to be set as such to achieve a deep comprehension of what is going on when, e.g., a race car negotiates a bend. To formulate vehicle dynamics on sound concepts we must rely on clear definitions and model formulations, and then on a rigorous mathematical analysis. We must,...

Tooth surface modifications are small, micron-level intentional deviations from perfect involute geometries of spur and helical gears. Such modifications are aimed at improving contact pressure distribution, while minimizing the motion transmission error to reduce noise excitations. In actual practice, optimal modification requirements vary with th...

Micro-geometry optimization has become an important phase of gear design that can remarkably enhance gear performance. For spiral bevel and hypoid gears, micro-geometry is typically represented by ease-off topography. The optimal ease-off shape can be defined as the outcome of a process where generally conflicting objective functions are simultaneo...

In this paper we set out to investigate the performances of some of the algorithms proposed in the gear literature for identifying the machine-settings required to obtain predesigned gear tooth surface topographies, or needed to compensate for flank form deviations of real teeth. For the ease of comparison, the problem is formulated as a nonlinear...

This paper presents an automatic procedure to optimize the loaded tooth contact pattern of face-milled hypoid gears with misalignments varying within prescribed ranges. A two-step approach is proposed to solve the problem: in the first step, the pinion tooth micro-topography is automatically modified to bring the perturbed contact patterns (as the...

This paper presents a novel approach to estimate the contact pattern for gear drives. The proposed method is based on the geometric properties of the generated surfaces of the pinion and the gear and it neglects the mechanical characteristics of the mating members. The key feature of the method is the superimposition of a virtual marking compound o...

Systematic optimization of the tooth contact pattern under load is an open problem in the design of spiral bevel and hypoid gears. In order to enhance its shape and position, gear engineers have been assisted by numerical tools based on trial-and-error approaches, and/or they have been relying on the expertise of skilled operators. The present pape...

This paper presents a new systematic method for identifying the values of the machine-tool settings required to obtain flank form modifications in hypoid gears. The problem is given a nonlinear least-squares formulation, and it is solved by the Levenberg-Marquardt method with a trust-region strategy. To test the method, the same ease-off topography...

The analysis of the steady-state cornering behaviour of a vehicle is generally based on the classical single track model. As a consequence, some results, such as the handling diagram and the understeer gradient, are not general and should be used only for the category of vehicles that they represent. A general approach, which is not dependent on th...

This paper deals with the analysis of the handling behaviour of a novel three-wheeled motorcycle. This vehicle has two front steering wheels and a single rear wheel and can be driven much like a common two wheeler. In order to analyse the handling behaviour of such vehicle and to compare it to an ordinary two wheeler, an experimental campaign was c...

In this work the modal analysis of a three-wheeled tilting motorcycle is presented. This new kind of vehicle has two front wheels and a single rear wheel, but is driven like a common motorcycle. In order to study the stability of the system in straight running, two models have been developed: a simplified motorcycle model, with locked suspensions a...

Quite frequently mathematical formulæ in reports, theses or even articles are not written correctly. Indeed, the basic rules for composing the formulæ are almost never explicitly stated. This article provides some suggestions to ll this gap.

The problem of describing the understeer–oversteer behaviour of a general vehicle, such as one with locked differential or tandem rear axle, is addressed taking a new perspective. The well-known handling diagram and the associated classical understeer gradient may be inadequate, mainly because they are no longer unique. The new concept of handling...

In this paper a new general algorithm is presented for the direct evaluation of all singular double integrals arising in the 2D Galerkin BEM, including those with hypersingular kernels. A distinguishing feature of the proposed method is that double singular integrals are treated as a whole, that is not as inner integrals followed by outer ones. The...

In this paper an innovative joint for the transmission of motion between parallel and incident axes is presented. It is made up of two frontal pin-wheels with cylindrical pins. The kinematics in case of parallel axes is discussed in detail. It is shown that, quite surprisingly, it may behave in many different ways, depending on the value of the cen...

This paper outlines a systematic methodology for finding the machine setting corrections required to obtain a predesigned ease-off surface in spiral bevel and hypoid gear teeth. The problem is given a nonlinear least squares formulation which, however, is highly prone to numerical instabilities. The Levenberg–Marquardt algorithm with a trust region...

The steady-state cornering behaviour of rear-wheel drive vehicles fitted with locked differential is critically analysed by means of simple, albeit carefully formulated, vehicle models, which allow for a rigorous theoretical analysis. Results obtained for some classical manoeuvres, with either constant forward speed, steer angle or turning radius,...

In some recent papers a new vector approach to the theory of gearing has been proposed by the present authors. However the formulation there presented was limited to fixed or, at most, translating axes. The present paper presents the non-trivial extension to the case of gear generation with supplemental spatial motions (helical motion, tilt motion,...

In this work, three motorcycle models of increased accuracy are presented. They were developed as a tool for investigating how the level of accuracy, for example, in the front kinematics, affects the handling judgment, as obtained on the basis of some handling indexes taken from the technical literature, in some typical manoeuvres. A simple driver...

In the literature, some methods for curvature analysis of gears can be found, apparently very different from one another. This article presents a comparison of three approaches to stress their similarities or differences and field of application: the classic one by Litvin, its development by Chen and another one proposed by Wu and Luo and revisited...

In the first part, the alternative formulation of the theory of gearing presented in [F. Di Puccio, M. Gabiccini, M. Gui-ggiani, Alternative formulation of the theory of gearing, Mechanism and Machine Theory 40 (5) (2005) 613–637] is gen-eralized to cope with the case of modified roll and translating axes of the mating surfaces, that is with a sett...

In this paper the theory of gearing is presented by means of a new approach that does not need reference systems. All equations are written in terms of vectors and therefore are valid regardless of the reference system actually employed. The overall formulation becomes more compact and clearer. As an example, the proposed approach is applied to the...

The alternative formulation of the theory of gearing presented in [1], which does not need an early estabilishment of reference systems like in [2], is applied to model the grinding process of a real spiral bevel gear drive for aerospace applications. All the information collected about the geometry is employed to generate a finite element model ca...

In this paper the theory of gearing is presented by means of a purely geometric approach without any recourse to reference systems. All vector equations are therefore valid regardless of the reference system actually being employed to perform calculations. The overall theory becomes more compact and, hopefully, clearer.

In this paper a new general algorithm is developed for the direct evaluation of all singular double integrals arising in the 2D Galerkin BEM, including those with hypersingular kernels. A distinguishing feature of the proposed method is that double singular integrals are treated as a whole, that is, not as inner integrals followed by outer ones. Th...

Based on a recently developed geometric approach to the theory of gearing that does not make use of any reference systems [1], this paper presents some useful relations between the geometric properties of the enveloping surface and those of its envelope. Treating vectors as such, that is without expressing their components in any reference systems,...

In this paper two basic issues concerning hypersingular boundary integral equations (HBIE's) for three-dimensional prob-lems are addressed. Firstly, a new general method for the evaluation of all free-term coefficients is presented. Secondly, the so-called Tricomi-Mikhlin compatibility conditions at non-smooth bounday points are proved. In both cas...

Boundary integral equations with extremely singular (i.e., more than hypersingular) kernels would be useful to obtain second and third order derivatives of the primary variable on the boundary. In this paper it is shown how to obtain these boundary integral equations with still unnamed singularities and, moreover, how to efficiently and reliably co...

Boundary integral equations with extremely singular (i.e., more than hypersingular) kernels would be useful in several ÿelds of applied mechanics, particularly when second-and third-order derivatives of the primary variable are required. However, their deÿnition and numerical treatment pose several problems. In this paper, it is shown how to obtain...

The typical Boundary Element Method (BEM) for fourth-order problems, like bending of thin elastic plates, is based on two coupled boundary integral equations, one strongly singular and the other hypersingular. In this paper all singular integrals are evaluated directly, extending a general method formerly proposed for second-order problems. Actuall...

This paper deals with a direct approach for the evaluation of singular element integrals arising from the discretization of hypersingular boundary integral equations (BIEs) in symmetric Galerkin form, for two-dimensional problems.

An integral formulation for heat conduction problems in non-homogeneous media has recently been proposed by Kassab and Divo [Engineering Analysis with Boundary Elements 1996;18:273]. The goal of this communication is to revisit and clarify two key features of the formulation of Kassab and Divo. First, the contention that the integral equation formu...

This paper investigates the evaluation of the sensitivity, with respect to tangential perturbations of the singular point, of boundary integrals having either weak or strong singularity. Both scalar potential and elastic problems are considered. A proper definition of the derivative of a strongly singular integral with respect to singular point per...

Boundary integral equations with strongly singular and hypersingular kernels can be very useful in many fields of applied mechanics. In this paper two basic aspects are addressed. First, a method for the formulation of boundary integral equations with kernels of any order of singularity is outlined. The method never employs arbitrary interpretation...

The subject is the sensitivity analysis of approximate boundary element solutions with respect to the positions of the collocation points. Since only the collocation points are perturbed, the shape of the body and the corresponding discretization remain unaltered. Sensitivities are shown to be related to the residual of hypersingular integral equat...

The subject of this paper is the sensitivity analysis of approximate boundary element solutions with respect to the positions of the collocation points. The direct differentiation approach is considered here and the analysis is performed analytically. Since only the collocation points are perturbed, the shape of the body and the corresponding discr...

In this paper it is shown that hypersingular boundary integral equations may have an additional free term which has been erroneously omitted in former analyses.

In this paper it is shown that hypersingular boundary integral equations may have an additional free term which has been erroneously omitted in former analyses.

This paper aims to show that the limits involved in boundary integral equations with singular kernels do not produce unbounded terms even if the density function and/or its derivatives are not continuous at the singular point. Therefore, the common requirement for the density funtion of being Hölder continuous of some degree is not strictly necessa...

Instead of using shape function derivatives and Hooke's law, the full stress tensor is evaluated at boundary points by direct application of boundary integral identities for the displacement derivatives. It is first shown that integral equations with singular or hypersingular kernels do not give rise to unbounded terms, even when the source point i...

A self-adaptive boundary element method as implemented in the new computer program Sherpa is presented. Error indicators are obtained by comparing two BEM solutions obtained from exactly the same discretization but with (partly) different sets of collocation points. The code Sherpa has some advanced features, including fully automatic mesh refineme...

The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analy...

Current methods to deal with Cauchy principal-value (CPV) integrals in advanced boundary-element implementations have been almost entirely based on indirect approaches (such as the rigid-body motion in elastostatics). The present paper illustrates an alternative direct approach for the rigorous treatment and numerical evaluation of general CPV inte...

A general direct method for the evaluation of hypersingular integrals in the BEM is presented with reference to two-dimensional problems. It is first shown that there are no special problems in the derivation of hypersingular boundary integral equations (HBIE’s), that is, in taking the singular point on the boundary. No unbounded quantities ultimat...

In this paper several methods of dealing with Cauchy Principal Value integrals in advanced boundary element methods are discussed and compared. An attempt is made to present a comprehensive description of these methods in a unified, systematic manner. It is shown that the methods can be grouped into two basic approaches, the (more classical) indire...

In the first part, hypersingular boundary integral equations are obtained through proper consideration of the limiting process. It is proved that no divergent terms actually arise, and that interpretations of the integrals are not required. In the second part, a general algorithm for the direct numerical treatment of hypersingular integrals in the...