Abstract
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 equations. A lot of work is devoted to the load transfers, which requires an in depth suspension analysis. This leads to the definition of the suspension and vehicle internal coordinates, of the no-roll centers and no-roll axis, for both independent and dependent suspensions. The case of three-axle vehicles is also considered. In the end, the vehicle model for handling and performance is formulated in a synthetic, yet precise way. A general description of the mechanics of differential mechanisms, either open or limited-slip is included.
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Change history
09 June 2023
The book was inadvertently published with some low resolution figures and wrong equations. The corresponding chapters have been corrected as follows:
In Chapter 3, Figures 3.70 to 3.75 have been replaced with higher resolution figures.
In Chapter 7, Figures 7.12 to 7.19 have been replaced with higher resolution figures.
In Chapter 10, Equations 10.13, 10.14, 10.15 and 10.18 have been corrected.
The correction chapters and the book have been updated with the changes.
Notes
- 1.
The reason is that \(\textrm{d}\! f = \cos \psi \,\textrm{d}x_0 + \sin \psi \,\textrm{d}y_0\) is not an exact differential since there does not exist a differentiable function \(f(x_0,y_0,\psi )\).
- 2.
In this book, lengths are usually indicated by a lower case letter. R and S are exceptions.
- 3.
Subscript v was chosen because the Italian translation of steering wheel is volante.
- 4.
The kinematic equations for roll steer, for both front and rear wheels, will be given in (3.210). Their presentation is delayed till the suspension analysis is completed.
- 5.
Pneumatic tires were invented about 70 years later.
- 6.
In a Formula 1 car we have \(J_z\simeq {900}{\,\textrm{kgm}^{2}}\), \(J_y\simeq {800}{\,\textrm{kgm}^{2}}\), \(J_x\simeq {100}{\,\textrm{kgm}^{2}}\), \(J_{zx}\simeq {3}{\,\textrm{kgm}^{2}}\), and \(J_w\simeq {0.8}{\,\textrm{kgm}^{2}}\), with \(|r|<{1}{\,\mathrm{rad/s}}\) and \(|\dot{r}|<{2}{\,\mathrm{rad/s}^{2}}\).
- 7.
In the first edition of this book it was the other way around.
- 8.
But not on the tire slips.
- 9.
At first it may look paradoxical, but it is not. Actually it is common practice in engineering. Just take the most classical cantilever beam, of length l with a concentrated load F at its end. Strictly speaking, the bending moment at the fixed end is not exactly equal to Fl, since the beam deflection takes the force a little closer to the wall. But this effect is usually neglected.
- 10.
A more precise definition of roll angle is given in Sect. 9.2.
- 11.
The symbol \(\hat{\phi }\) (instead of just \(\phi \)) is used to stress that this is not the roll angle under operating conditions, but the roll angle due to a pure rolling moment.
- 12.
The symbols \(\hat{z}_1\) and \(\hat{z}_2\) (instead of just \(z_1\) and \(z_2\)) are used to stress that these are not the vertical displacements under operating conditions.
- 13.
Let us state from the beginning that it is not always true that, when rounding a corner, the inner wheel rotates at a slower angular speed than the outer wheel.
- 14.
Here is an example of such wrong sentences: “A Torsen works on the principle that a spinning worm gear can rotate the wheel, but the rotating wheel cannot spin the worm gear”.
- 15.
Incidentally [10], we mention that a symbol like TBR could be interpreted as the product of three quantities if written using a mathematical font like in TBR.
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Guiggiani, M. (2023). Vehicle Model for Handling and Performance. In: The Science of Vehicle Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-031-06461-6_3
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DOI: https://doi.org/10.1007/978-3-031-06461-6_3
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