20 GENERAL CHARACTERISTICS 20.1 SYMMETRY CONSIDERATIONS Motor vehicles, like most machines, have a general bilateral symmetry. Only hypotheses can be advanced to explain why this occurs. Certainly to have a symmetry plane simplifies the study of the dynamic behavior of the system, for it can be modelled, within certain limits, using uncoupled equations. However, the reason is likely to be above all an aesthetic one: symmetry is considered an essential feature in most definitions of beauty. All complex animals that evolved on our planet, including humans, have a symmetry plane defined by a vertical axis and an axis running in the longitudinal direction; symmetry is, however, not complete since some internal organs are positioned in an unsymmetrical way and some small deviations from symmetry are always present even in exterior appearance. When such lack of symmetry is too evident, it is felt to be incompatible with the aesthetic canons developed by all human civilizations. A similar situation is encountered in all objects built by humans and, as in our interest here, in motor vehicles: a general outer symmetry and a certain lack of symmetry in the location of the internal components. Among the most common road vehicles, the only case where such a symmetry is not present is that of motorbikes with sidecar; these are, however, perceived to be made by a main unit, the motor bike, that has bilateral symmetry, plus a second unit, the sidecar, attached on a side, as its name suggests. G. Genta, L. Morello, The Automotive Chassis, Volume 2: System Design, 105 Mechanical Engineering Series, c Springer Science+Business Media B.V. 2009 106 20. GENERAL CHARACTERISTICS The sidecar often has its own symmetry plane, even if such characteristics are neither needed nor useful. This consideration may confirm the idea that symmetry has, in vehicles, purely an aesthetic justification. A few other vehicles, built for very specialized use, like mobile cranes and building yard vehicles, have a non-symmetrical shape when strong functional reasons dictate it, but these are vehicles in which aesthetic considerations are utterly unimportant. Many industrial vehicles may perhaps draw advantages from an asymmetri- cal architecture, for instance with the cab on one side and the loading surface on the other to use all the available length. Such a configuration seems, however, to be so unnatural as to be considered only if strictly needed. Imagination could at this point be set free to devise architectures that are not only without bilateral symmetry, but are even fully non-symmetrical, but this would likely be useless, since these configurations would seem to be unacceptable. If the vehicle were completely symmetrical, the center of mass would lie in the symmetry plane. Actually, as already said, the mechanical systems and the load distribution are often not exactly symmetrical, so that the mass center can be displaced from it. In practice, the distance of the mass center from the symmetry plane is small. 20.2 REFERENCE FRAMES The study of the motion of motor vehicles is usually performed with reference to some reference frames that are more or less standardized. They are (Fig. 20.1): • Earth-fixed axis system XY Z. This is a right-hand reference frame fixed on the road. In the following sections, it will always be regarded as an inertial frame, even if strictly speaking it is not such as it moves along with the Earth: The inertial effects due to its motion (rotation about its axis, orbiting about the Sun, the galactic center, ) are so small that they can be neglected in all phenomena studied in motor vehicle dynamics. Axes X and Y lie in a horizontal plane while axis Z is vertical, pointing upwards 1 . • Vehicle axis system xyz. This is a right-hand reference frame fixed to the vehicle’s center of mass and moving with it. As already stated, if the vehicle has a symmetry plane, the center of mass is assumed to lie on it. The x-axis lies in the symmetry plane of the vehicle in an almost horizontal direction 2 .Thez -axis lies in the symmetry plane, is perpendicular to the 1 Recommendation SAE J670 and ISO/TC 22/SC9 standard state that the Z-axis is vertical and points downwards. Note that in the present text the direction of the Y and Z axes is opposite to that suggested in the mentioned standard. 2 The mentioned standard states that the x-axis is contained in the plane of symmetry of the vehicle, is “substantially” horizontal and points forward. 20.2 Reference frames 107 FIGURE 20.1. Reference frames, forces and moments used for the dynamic study of motor vehicles. FIGURE 20.2. Projection of the axes of the vehiocle fixed frame in XY plane of the inertial frame. x-axis, and points upwards. The y-axis is perpendicular to the other two 3 . The y -axis,then, points to the left of the driver. In the case of vehicles without a symmetry plane, the xz plane is identified by the direction of motion when the wheels are not steered and by a direction perpendicular to the road in the reference position of the vehicle. The projections of the vehicle-fixed axes xyz and of the velocity on the plane XY of the inertial frame are shown in Fig. 20.2. The angle between the projection of the x-axis and the X-axis is the yaw angle ψ. The projection of the velocity of the center of mass on the XY -plane is here conflated with the 3 Here again there is a deviation from the mentioned standard. 108 20. GENERAL CHARACTERISTICS absolute velocity, since the component of the latter in a direction perpendicular to the road is usually very small. The velocity of the air with respect to the ground, or ambient wind velocity v a , is defined as the horizontal component of the air velocity relative to the earth-fixed axis system in the vicinity of the vehicle in the inertial frame XY Z. The resultant air velocity (i.e. the velocity of the air with respect to the vehicle) V r is the difference between the ambient wind velocity and the projection of the absolute velocity V of the vehicle. The velocity of the vehicle with respect to air is −V r and coincides with the absolute velocity V when the air is still, i.e. when the ambient wind velocity v a is zero. The angle between the projection on the XY plane of the x-axis and that of the velocity vector V is the sideslip angle β of the whole vehicle; sometimes referred to as the attitude angle. As it was the case for the sideslip angle of the tire α, it is positive when vector V points to the left of the driver (in forward motion). In a similar way it is possible to define an aerodynamic angle of sideslip β a as the angle between the projections on the XY plane of the x-axis and the relative velocity −V r . Angles β and β a usually refer to the velocity of the centre of mass, but can be referred to the velocity of any specific point of the vehicle. 20.3 POSITION OF THE CENTER OF MASS The position of the center of mass is very important in determining the behavior of the vehicle and must be computed, or at least assessed, at the design stage and then experimentally determined. If the various components of the vehicle are designed using CAD techniques, it is generally possible to have their mass and the positions of their centers of mass among the outputs of the code. The position of the center of mass of the vehicle can then be assessed from the results obtained for the components, but this evaluation is generally approximate and at any rate is a long computation. Then what is of interest is not the position of the center of mass of the empty vehicle, but that of the vehicle in the various operating conditions (vehicle with all liquids, the driver, a variable number of passengers and possibly luggage). The longitudinal position of the center of mass can be obtained by simply weighing the vehicle on a level road at the front and rear axles (Fig. 20.3a). If F z 1 and F z 2 are the vertical forces measured at the two axles, the equilibrium equations for translations in the z direction and for rotations about the y-axis can be written as: F z 1 + F z 2 = mg lF z 1 = bmg , (20.1) 20.3 Position of the center of mass 109 mg mg G h b z a x 1 α G a 1 b a) b) F z 1 F z 2 +ΔF z F z 1 −ΔF z R 1 2 h G F z 2 R 1 2 h G FIGURE 20.3. Sketch of the experimental determination of the longitudinal and vertical position of the center of mass. and then: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ a = l F z 2 F z 1 + F z 2 b = l F z 1 F z 1 + F z 2 . (20.2) If the center of mass does not lie in the symmetry plane, it is possible to compute the transversal position of the center of mass by measuring the forces under the right and left wheels. It is more difficult to determine the height of the center of mass. Once the longitudinal position is known (once a and b have been determined) it is possible to measure again the forces on the ground after the front (or the rear) axle has been raised from the ground (Fig. 20.3b). Let the front axle be set on a platform with height h with respect to the platform on which the rear axle is located. If the height of the center of mass h G is greater than the radius under load of the wheels, the force F z 1 measured at the front axle will be smaller than that measured on level road. It is then possible to write: F z 1 = F z 1 − ΔF z . (20.3) If the force at the rear axle is measured, it would be: F z 2 = F z 2 +ΔF z . (20.4) The equilibrium equation for rotations about the center of the front axle is: mg [a cos (α)+(h G − R l 1 )sin(α)] = (20.5) =(F z 2 +ΔF z )[l cos (α)+(R l 2 − R l 1 )sin(α)] . The height of the center of mass is then: h G = F z 2 +ΔF z mg l tan (α) + R l 2 − R l 1 − a tan (α) + R l 1 . (20.6) 110 20. GENERAL CHARACTERISTICS Remembering that: F z 2 mg = a l , it follows that h G = a l [R l 2 − R l 1 ]+R l 1 + ΔF z mg l tan (α) + R l 2 − R l 1 . (20.7) If the loaded radius of all wheels is the same, as it is usually the case (at least approximately), Eq. (20.7) can be simplified as: h G = R l + ΔF z mg l tan (α) , (20.8) that is: h G = R l + ΔF z mg l √ l 2 − h 2 h . (20.9) To measure accurately the height of the center of mass, the wheels must be completely free (brakes released and gear in neutral), and the vehicle must be restrained from rolling by chocks at one of the axles. Moreover, the suspensions must be locked at a height corresponding to the load distribution on level road, and the tires must be equally compressed. This last condition can be obtained in an approximate way by increasing the inflation pressure. It is then possible to take several measurements of ΔF z at different values of tan (α) and to plot ΔF z versus tan (α). If the radii of the wheels are all equal, such a curve is a straight line, whose slope ΔF z tan (α) = mg l (h G − R l ) (20.10) allows one to compute the height of the center of mass. 20.4 MASS DISTRIBUTION AMONG THE VARIOUS BODIES In terms of dynamic behavior, vehicles are often modeled as a number of rigid bodies with different inertial properties. The simplest way to model a vehicle is to consider the body as a rigid body (sprung mass), to which a further rigid body is added to model each rigid axle, and two rigid bodies for each axle with independent suspension. This approach is approximate, since many vehicle el- ements (suspension linkages, springs, shock absorbers) do not belong to any of these bodies. Half of the mass of the elements located between two rigid bodies may be attributed to each one of them, but this is an approximation, although usually an acceptable one. 20.5 Moments of inertia 111 There is no alternative either to computing or experimentally determining the mass of the various components separately to evaluate the mass of the various subsystems. For a first approximation evaluation, the center of mass of a rigid axle can be assumed to lie on the line connecting the centers of the wheels in the symmetry planes. The center of mass of each independent suspension can be located at a distance from the symmetry plane equal to half of the track of the relevant axle. Better estimations can come from a detailed analysis of the drawings of the suspension. Once the positions of the centers of mass of the whole vehicle and of the suspensions (unsprung masses) are known, it is straightforward to locate the center of mass of the sprung mass. More detailed models can be obtained from computer codes based on multi- body modelling. Not only do they allow us to take into account the various parts constituting the vehicle in much greater detail (each element of the suspension can be introduced separately, with its mass, moments of inertia and exact kine- matics), but their compliance can also be introduced using the finite element method (FEM). 20.5 MOMENTS OF INERTIA The inertia tensor of a rigid body is: J = ⎡ ⎣ J x −J xy −J xz −J xy J y −J yz −J xz −J yz J z ⎤ ⎦ (20.11) where the moments of inertia are: J x = V y 2 + z 2 dm J y = V x 2 + z 2 dm J z = V x 2 + y 2 dm (20.12) and the products of inertia 4 are: J xy = V xydm J xz = V xzdm J yz = V yzdm. (20.13) If the vehicle has a plane of symmetry that coincides with the xz-plane, the products of inertia J xy and J yz vanish and the inertia tensor of the vehicle reduces to: J = ⎡ ⎣ J x 0 −J xz 0 J y 0 −J xz 0 J z ⎤ ⎦ . (20.14) 4 Often products of inertia are defined as J xy = − V xydm,etc.andsigns(−)arenot included in Eq. (20.11). 112 20. GENERAL CHARACTERISTICS The computation of the moments of inertia of the vehicle and of its parts is a complex operation, but if the design of the various components has been performed using CAD, the approximate values of the moments of inertia are among the outputs of the code. Usually, the moments of inertia of interest are: • the baricentric roll moment of inertia (about the x-axis) of the sprung mass J S x , • the baricentric pitch moment of inertia (about the y-axis) of the sprung mass J S y and • the baricentric yaw moment of inertia (about the z-axis) of the whole vehicle J z . The first two are obviously referred to the center of mass of the sprung mass, while the latter is referred to the center of mass of the whole vehicle. Some empirical formulae for their computation from the mass of the vehi- cle are often used. These usually yield the radius of gyration ρ instead of the corresponding moment of inertia: J i = mρ 2 i ,i= x, y, z. (20.15) A first rough approximation for the yaw radius of gyration is: ρ z = √ ab. (20.16) First approximation values of the radii of gyration (in meters) that may be used for a medium size car are: 5 Load condition ρ S x ρ S y ρ z Empty 0,65 1,21 1,20 2 passengers 0,64 1,13 1,15 4 passengers 0,60 1,10 1,14 4 passengers + luggage 0,56 1,13 1,18 Note also that the values for the sprung mass are referred to the total mass of the vehicle. As a first approximation, inertia products may be neglected. Remark 20.1 The moments and products of inertia must be known with preci- sion, since they are important in assessing the performance of the vehicle, both for its handling and its comfort. It is then important to measure them accurately once prototypes are built. The simplest way to measure the moments of inertia of any object is to suspend it in such a way that it can rotate as a pendulum about an axis that 5 J. Reimpell, H. Stoll, The Automotive Chassis, SAE, Warrendale, 1996. 20.5 Moments of inertia 113 FIGURE 20.4. Measurement of the moments of inertia. (a): Oscillation on a knife located on an axis parallel to the baricentric axis. (b): Quadrifilar pendulum to measure the inertia tensor of a motor vehicle. is parallel to the baricentric axis about which the moment of inertia must be measured (Fig. 20.4a). If d is the distance between the center of mass and the suspension axis, the moment of inertia about the latter is: J = J G + md 2 . (20.17) The period of oscillation of the pendulum made of the body suspended on the knife is T =2π J G + md 2 mgd . (20.18) By measuring the period of the small oscillations it is then possible to obtain the baricentric moment of inertia J G = m gd 4π 2 T 2 − d 2 . (20.19) By repeating the measurement three times with the object suspended about three axes parallel to the x-, y-andz-axes, the three moments of inertia can be measured. To measure the products of inertia, the measurements can be repeated mea- suring the moments of inertia about three axes different from the previous ones. Since the inertia tensor about the latter is J = R T JR, (20.20) where R is a known rotation matrix, it is possible to compute the values of the elements of the unknown inertia tensor J from the three values of J computed 114 20. GENERAL CHARACTERISTICS in the new test. The measurements of the moments of inertia are theoretically simple, but it is not easy to actually perform them, particularly if precision is required 6 . It is possible to use test rigs based, for instance, on the multifilar pendulum layout (Fig. 20.4b 7 ). The vehicle is suspended from it and the three moments of inertia can be computed in a single test. The pendulum has three degrees of free- dom and, once it starts oscillating, the three moments of inertia can be obtained by analyzing the time history of the oscillations. 6 G. Genta, C. Delprete, Some Considerations on the Experimental Determination of the Moments of Inertia, Meccanica, 29, pp.125-141, 1994. 7 G. Mastinu, M. Gobbi, C.M. Miano, The Influence of the Body Inertia Tensor on the Active Safety and Ride Comfort of Road Vehicles, SAE Paper 2002-01-2058, SAE, Warrendale, 2002. . 20 GENERAL CHARACTERISTICS 20.1 SYMMETRY CONSIDERATIONS Motor vehicles, like most machines, have a general bilateral symmetry. Only hypotheses can. Series, c Springer Science+Business Media B.V. 2009 106 20. GENERAL CHARACTERISTICS The sidecar often has its own symmetry plane, even if such characteristics are neither needed nor useful. This consideration. here conflated with the 3 Here again there is a deviation from the mentioned standard. 108 20. GENERAL CHARACTERISTICS absolute velocity, since the component of the latter in a direction perpendicular to