28 MATHEMATICAL MODELS FOR THE VEHICLE An increasingly competitive automotive market offers its products to increasingly demanding customers. Numerous standards and rules, primarily regarding safety and environmental impact, are issued by regulating bodies and governments, making today’s vehicles more and more complex. The specifications vehicles must comply with often contrast with each other. The time between the conception of a vehicle and its entering the market, the so called time to market, is an essential factor for its commercial success. The traditional approach, based on the construction of prototypes, subsequent ex- perimentation and modification, is no longer adequate. When design changes are introduced during the development of any ma- chine, they should be implemented as early as possible. The later such changes are made, the smaller the advantages, both in terms of economics but also more generally, while related costs are higher (Fig. 28.1). In the early stages of devel- opment, when the vehicle is merely an idea or the result of a preliminary study, major changes may be introduced at low cost, but when the design proceeds toward the construction of prototypes or the product launch, the freedom of designers is reduced and the costs linked to changes, not only to the design but also to prototypes and above all to production equipment, soar. This consideration alone can explain the increasing significance of activi- ties like the construction of mathematical models, virtual experimentation and simulation in the motor vehicle industry. The costs linked with the construction of prototypes and physical ex- perimentation are continuously growing, while the increase of computational power causes a decrease of computational times and thus of the costs linked with the construction of mathematical models and the ensuing numerical experimentation. G. Genta, L. Morello, The Automotive Chassis, Volume 2: System Design, 503 Mechanical Engineering Series, c  Springer Science+Business Media B.V. 2009 504 28. MATHEMATICAL MODELS FOR THE VEHICLE FIGURE 28.1. Qualitative trend of costs and advantages of design changes in the various stages of the design and construction of a machine. The importance of being able to predict the behavior of the system and its components before the construction of prototypes is therefore increasing in automotive technology. The goal, at present still far from being achieved, is to reduce the importance of physical testing to a simple verification of what virtual testing has ascertained. 28.1 MATHEMATICAL MODELS FOR DESIGN The computational predictions of the characteristics and the performance of a physical system are based on the construction of a mathematical model 1 ,con- structed from a number of equations, whose behavior is similar to that of the physical system it replaces. In the case of dynamic models, such as those used to predict the performance of motor vehicles, the model is usually built from a number of ordinary differential equations 2 (ODE). The complexity of the model depends on many factors, that represent the first choice the analyst has to make. The model must be complex enough to allow 1 Note that simulation is not always based on a mathematical model in a strict sense. In the case of analog computers, the model was made from an electric circuit whose behavior simulated that of the physical system. Simulation on digital computers is based on actual mathematical models. 2 A dynamic model, or a dynamic system, is a model expressed by one or more differential equations containing derivatives with respect to time. 28.1 Mathematical models for design 505 a realistic simulation of the system’s characteristics of interest, but no more. The more complex the model, the more data it requires, and the more complicated are the solution and interpretation of results. Today it is possible to built very complex models, but overly complex models yield results from which it is difficult to extract useful insights into the behavior of the system. Before building the model, the analyst must be certain what he wants to obtain from it. If the goal is a good physical understanding of the underlying phenomena, without the need for numerically precise results, simple models are best. Skilled analysts have simulated even complex phenomena with precision using models with a single degree of freedom. If, on the contrary, the aim is precise quantitative results, even at the price of more difficult interpretation, the use of complex models becomes unavoidable. Finally, it is important to take into account the available data at the stage reached by the project: Early in the definition phase, when most data are still not available, it can be useless to use complex models, into which more or less arbitrary estimates of the numerical values must be introduced. Simplified, or synthetic, models are the most suitable for a preliminary analysis. As the design is gradually defined, new features may be introduced into the model, reaching comprehensive and complex models for the final simulations. Such complex models, useful for simulating many characteristics of the vehi- cle, may be considered as true virtual prototypes. Virtual reality techniques allow these models to yield a large quantity of information, not only on performance and the dynamic behavior of the vehicle, but also on the space taken by the vari- ous components, the adequacy of details and the esthetic qualities of the vehicle, that is comparable to what was once obtainable only from physical prototypes. The models of a given vehicle often evolve initially toward a greater com- plexity, from synthetic models to virtual prototypes, to return later to simpler models. Models are useful not only to the designer in defining the vehicle and its components, but also to the test engineer in interpreting the results of testing and performing all adjustments. Simplified models can be used on the test track to allow the test engineer to understand the effect of adjustments and reduce the number of tests required, provided they are simple enough to give an immedi- ate idea of the effect of the relevant parameters. Here the final goal is to adjust the virtual prototype on the computer, transferring the results to the physical vehicle and hoping that at the end of this process only a few validation tests are required. Simplified models that can be integrated in real time on relatively low power hardware are also useful in control systems. A mathematical model of the con- trolled system (plant, in control jargon) may constitute an observer (always in the sense of the term used in control theory) and be a part of the control archi- tecture. The analyst has the duty not only of building, implementing and using the models correctly, but also of updating and maintaining them. The need to build a mathematical model of some complexity is often felt at a certain stage of the design process, but the model is then used much less than necessary, and 506 28. MATHEMATICAL MODELS FOR THE VEHICLE above all is not updated with subsequent design changes, with the result that it becomes completely useless or needs updating when the need for it again arises. There are usually two different approaches to mathematical modelling: mod- els made by equations describing the physics of the relevant phenomena, − these may be defined as analytical models − and empirical models, often called black box models. In analytical models the equations approximating the behavior of the various parts of the system, along with the required approximations and simplifications, are written. Even if no real world spring behaves exactly like the linear spring, producing a force proportional to the relative displacement of its ends through a constant called stiffness, and even if no device dissipating energy is a true linear damper, the dynamics of a mass-spring-damper system can be described, often to a very good approximation, by the usual ordinary differential equation (ODE) m¨x + c ˙x + kx = f(t) . (28.1) The behavior of a tire, on the other hand, is so complex that writing equa- tions to describe it beginning with the physical and geometrical characteristics of its structure is forbiddingly difficult. The magic formula is a typical example of the empirical, black box, model. The behavior of the tire is studied exper- imentally after which a mathematical expression able to describe it is sought, identifying the various parameters from the experimental data. While each of the parameters m, c and k included in the equation of motion of the mass-spring- damper system refers to one of the parts of the system and has a true physical meaning, the many coefficients a i , b i and c i appearing in the expressions for A, B, C, etc. in the magic formula have no physical meaning, and refer to the system as a whole. Among the many ways to build black box models, that based on neural networks must be mentioned 3 . Such networks can simulate complex and highly nonlinear systems, adapting their parameters (the weights of the network) to produce an output with a relationship to the input that simulates the input- output relationship of the actual system. Actually, the difference between analytical and black-box models is not as clear cut as it may seem. The complexity of the system is often such that it is difficult to write equations precisely describing the behavior of its parts, while the values of the parameters cannot always be known with the required precision. In such cases the model is built by writing equations approximating the general pattern of the response of the system, with the parameters identified to make the response of the model as close as possible to that of the actual system. In this case, the identified parameters lose a good deal of their physical meaning related to the various parts of the system they are conceptually linked to and become global parameters of the system. 3 Strictly speaking, neural networks are not sets of equations and thus do not belong to the mathematical models here described. However, at present neural networks are usually simulated on digital computers, in which case their model is created by a set of equations. 28.2 Continuous and discretized models 507 In the following, primarily analytical models will be described and an at- tempt will be made to link the various parameters to the components of the system. 28.2 CONTINUOUS AND DISCRETIZED MODELS The objects constituting our real world are all more or less compliant and are quite well modelled as continuous systems. A compliant body is usually mod- elled as a continuum or, if its behavior can be considered linear and damping is neglected, as a linear elastic continuum. It is clear that the elastic continuum is only a model, because no actual body is such at an atomic scale, but for most objects studied by structural dynamics the continuum model is more than adequate. An elastic body may then be thought of as consisting of an infinity of points. The configuration at any time t can be obtained from the initial configura- tion once a vector function expressing the displacements of all points is known (Fig. 28.2). The displacement of a point is a vector, with a number of components equal to the number of dimensions of the reference frame. The components of this vector are usually taken as the degrees of freedom of each point, and thus the number of degrees of freedom of a deformable body is infinite. The correspond- ing generalized coordinates can be manipulated as functions of space and time coordinates, usually continuous and differentiable up to a suitable order, while the characteristics of the material are defined by functions of the coordinates in the whole part of space occupied by the continuum. In general, these functions need not be continuous. FIGURE 28.2. Deformation of an elastic continuum; reference frame and displacement vector. 508 28. MATHEMATICAL MODELS FOR THE VEHICLE The theory of continuous functions is the natural tool for dealing with deformable continua. Function −→ u (x, y, z, t) describing the displacement of the points of the body is differentiable with respect to time at least twice; the first derivative gives the displacement velocity and the second the acceleration. Usu- ally, however, higher-order derivatives also exist. Assuming that the forces acting on the body are expressed by function  f(x, y, z, t), the equation of motion can generally be written as D[u(x, y,z, ˙x, ˙y, ˙z, t)] =  f(x, y, z, t) , (28.2) where the differential operator D completely describes the behavior of the body. Eq. (28.2) is a partial derivatives differential equation, in general nonlinear, con- taining derivatives with respect to both time and space coordinates and veloci- ties. If the system is linear, Eq. (28.2) is also linear and does not contain the velocities ˙x, ˙y, ˙z if the system is conservative. To such an equation other equations expressing the boundary conditions and the initial conditions must be added. The actual form of the differential operator may be obtained by resorting directly to the dynamic equilibrium equations or by writing the kinetic and po- tential energies and using Lagrange equations. The boundary conditions usually follow from geometrical considerations. Equation (28.2) may be solved in closed form only in a small number of cases, owing to the difficulties deriving from the differential equation and especially from the boundary conditions. For complex systems the only feasible approach is the discretization of the continuum and then the application of the methods used for discrete systems. The replacement of a continuous system, characterized by an infinite number of degrees of freedom, with a discrete system, sometimes with a large but finite number of degrees of freedom, is usually referred to as discretization. This step is of primary importance in the solution of practical problems, because the accuracy of the results depends largely on the adequacy of the discrete model to represent the actual system. In recent centuries many discretization techniques were developed with the aim of substituting the partial derivatives differential equation of motion (with derivatives with respect to time and space coordinates) with a set of ordinary differential equations containing only derivatives with respect to time. The set of equations so obtained is often made by a number of second-order equations equal to the number of degrees of freedom of the discretized system. When the model is made by equations that are not all of the second-order (and often when they are) it is expedient to reduce it to a set of equations of the first order, by resorting to a number of auxiliary variables. If the model has n degrees of freedom (defined by n generalized coordinates) and is made of a set of n second-order equations, n auxiliary variables (generally the generalized veloc- ities) are needed, and the resulting model is made by 2n first-order equations. The 2n variables (n generalized coordinates and n generalized velocities) are the 28.3 Analytical and numerical models 509 state variables of the system. The vector containing the state variables is called the state vector and is usually indicated by z. If the set of 2n first-order equations is solved in the derivatives of the state variables, or it is solved in monic form, it has the form ˙z = f(z,t) . (28.3) The simplest way to discretize a model is by concentrating its inertial char- acteristics in a certain number of rigid bodies, or even material points, with its elastic and damping properties in massless springs and dampers. The models seen in the previous chapters to analyze the dynamic behavior of motor vehi- cles belong mostly to this type, as for instance the spring-mass-damper model expressed by Eq. (28.1). Because a point has 3 degrees of freedom in three-dimensional space, the most obvious choice is to use as generalized coordinates the 3 coordinates of the point referred to an inertial frame. A rigid body has, in a three-dimensional space, 6 degrees of freedom. Thus a reasonable choice for the generalized coordinates is to use the three components of the displacement of one of its points, usually the center of mass, and 3 rotations. While the displacement is a vector, so that there is no difficulty in choosing the 3 coordinates related to displacement, things are much more complicated for the coordinates linked to rotations and, as will be seen later, different choices are possible. The choice of the generalized coordinates and the equations of motion of a rigid body will be described in detail in Appendix A. If it becomes impossible to neglect the compliance of some parts of the sys- tem, it is possible to resort to the Finite Element Method (FEM). The body is subdivided into a number of regions, the finite elements, so-called to specify their difference from the infinitesimal regions of space used to write the equations of motion of continuous systems. The inflected shape of each region is approximated by the linear combination of a number of functions of space coordinates through parameters that are taken as the generalized coordinates of the element. Usually such functions of space coordinates (the shape functions) are very simple and the generalized coordinates have a physical meaning, such as generalized displace- ments of some points of the element, called nodes. The analysis then proceeds by writing a set of differential equations of the type typical of discrete systems. 28.3 ANALYTICAL AND NUMERICAL MODELS Once the model has been discretized and the equations of motion written, there is no difficulty in studying the response to any input, assuming the initial con- ditions are stated. A general approach is to numerically integrate the ordinary differential equation constituting the model, using any of the many available nu- merical integration algorithms. In this way, the time history of the generalized coordinates (or of the state variables) is obtained from any given time history of the inputs (or of the forcing functions). 510 28. MATHEMATICAL MODELS FOR THE VEHICLE This approach, usually referred to as simulation or numerical experimenta- tion, is equivalent to physical experimentation, where the system is subjected to given conditions and its response measured. This method is broadly applicable, because it • may be used on models of any type and complexity, and • allows the response to any type of input to be computed. Its limitations are also clear: • it doesn’t allow the general behavior of the system to be known, but only its response to given experimental conditions, • it may require long computation time (and thus high costs) if the model is complex, or has characteristics that make performing the numerical in- tegration difficult, and • it allows the effects of changes of the values of the parameters to be pre- dicted only at the cost of a number of different simulations. If the model can be reduced to a set of linear differential equations with constant coefficients, it is possible to obtain a general solution of the equations of motion. The free behavior of the system can be studied independently from its forced behavior, and it is possible to use mathematical instruments such as Laplace or Fourier transforms to obtain solutions in the frequency domain or in the Laplace domain. These solutions are often much more expedient than solutions in the time domain that, as stated above, are in general the only type of solution available for nonlinear systems. The possibility of obtaining general results makes it convenient to start the study by writing a linear model through suitable linearization techniques. Only after a good insight into the behavior of the linearized models is obtained will the study of the nonlinear model be undertaken. When dealing with nonlinear systems it is often expedient to begin with simplified methods, such as harmonic balance, or to look for series solutions before starting to integrate the equations numerically. . ascertained. 28.1 MATHEMATICAL MODELS FOR DESIGN The computational predictions of the characteristics and the performance of a physical system are based on the construction of a mathematical model 1 ,con- structed. 28.2. Deformation of an elastic continuum; reference frame and displacement vector. 508 28. MATHEMATICAL MODELS FOR THE VEHICLE The theory of continuous functions is the natural tool for dealing. stage of the design process, but the model is then used much less than necessary, and 506 28. MATHEMATICAL MODELS FOR THE VEHICLE above all is not updated with subsequent design changes, with the