Abstract
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 to common belief, it is shown that the axle characteristics can take into account many vehicle features, like toe-in/toe-out, roll steering, camber angles and camber angle variations. The steady-state analysis is carried out first using the classical handling diagram. Then, the new global approach based on handling maps on achievable regions is introduced and discussed in detail. This new approach shows the overall vehicle behavior at a glance. Stability and control derivatives are introduced to study the vehicle transient behavior. Moreover, the relationship between data collected in steady-state tests and the vehicle transient behavior are thoroughly analyzed in a systematic framework. To prove the effectiveness of these results, a number of apparently different vehicles with exactly the same handling are generated.
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Ordinary road cars are by far the most common type of motor vehicle. Almost all of them share the following features relevant to handling:
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1.
four wheels (two axles);
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2.
two-wheel drive;
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3.
open differential;
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4.
no wings (and hence, no significant aerodynamic downforces);
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5.
no intervention by electronic active safety systems like ABS or ESP under ordinary operating conditions.
Moreover, in the mathematical models it is also typically assumed that the vehicle moves on a flat road at almost constant forward speed u, thus requiring small longitudinal forces by the tires.
The handling analysis of this kind of vehicles is somehow the simplest that can be envisaged.Footnote 1 That does not mean that it is simple at all.
The vehicle model developed in Chap. 3 is employed. However, owing to the above listed features of road cars, several additional simplifications can be made, which first lead to the double track model and eventually to the celebrated single track model. All the steps that lead to the single track model are thoroughly discussed to clarify when it is a suitable model for vehicle dynamics.
7.1 Additional Simplifying Assumptions for Road Car Modeling
The vehicle model introduced in Chap. 3, and whose equations were collected in Sect. 3.14, is simplified hereafter, taking into account the distinguishing features relevant to handling of road cars (Fig. 7.1).
7.1.1 Negligible Vertical Aerodynamic Loads
Aerodynamics of road cars is mostly concerned with attaining low drag, because of its impact on fuel consumption. Therefore, road cars normally do not have aerodynamic devices to generate significant vertical loads, that is \(Z_1^a \simeq 0\) and \(Z_2^a \simeq 0\). Basically, this means that the handling features of a road car are (almost) speed insensitive.
7.1.2 Open Differential
The main simplification is that the vehicle is equipped with an open differential. Since there is almost no friction inside an open differential mechanism, in (3.178) we have that its internal efficiency \(\eta _h \simeq 1\) , and hence \(M_l \simeq M_r\). In other words, both driving wheels receive always the same torque from the engine. Therefore, in the global equilibrium equations (3.94), the tire longitudinal forces \(F_{x_{ij}}\) (Fig. 7.1) are such that \(F_{x_{11}}=F_{x_{12}}\) and \(F_{x_{21}}=F_{x_{22}}\), and hence do not contribute to the yaw moment N applied to the vehicle. Summing up, in (3.87)
Basically, this means that the handling features of a road car are (almost) insensitive to the radius of curvature of its trajectory, provided the radius is not too small.
A look at Fig. 3.28 can be useful to better understand \(\varDelta \!X_1 = (X_{12} -X_{11})/2\).
7.1.3 Almost Constant Forward Speed
If the forward speed u is almost constant (\(\dot{u}\simeq 0\), and hence \(a_x\simeq 0\)), and the aerodynamic drag is not very high (like in ordinary cars, but not in a Formula 1 car, which, however, does not have an open differential),Footnote 2 the tire longitudinal forces are quite small (Fig. 7.1). That means that also the longitudinal slips are small and can be neglected. Therefore,
which means that all wheels are almost under longitudinal pure rolling conditions.
As a consequence, we have
where \(\omega _h\) is the angular velocity of the differential housing, and \(r_2\) is the rolling radius of the rear wheels.
Vehicle basic scheme (double track model)
7.2 Mathematical Model for Road Car Handling
The equations collected in Sect. 3.14 for the fairly general vehicle model described in Chap. 3 are now tailored to the case of road cars with open differential, no wings, and almost constant forward speed.
The general definitions (3.87) for the horizontal (in-plane) forces acting on the vehicle now become (Figs. 7.1 and 7.2)
7.2.1 Global Equilibrium
Since the forward speed u in (3.94) is given, the vehicle has basically only lateral and yaw dynamics (often simply called lateral dynamics), described by the following system of two differential equations (Fig. 7.2)
while
is now an algebraic equation, the unknown being the tire longitudinal force \(X_2\) (see (7.4)).
It looks like we are playing a dirty game. First we say \(X_2 \simeq 0\), and now we are computing it. This is indeed to check whether \(X_2\) is actually very small.
Global dynamics of the double track model
We recall that u is the vehicle longitudinal velocity, r is the vehicle yaw rate, v is the lateral velocity of G, \(a_x\) is the longitudinal acceleration of G, and \(a_y\) is the lateral acceleration of G. The vehicle has mass m and moment of inertia \(J_z\) with respect to a vertical axis located at G. It is worth noting that u and r are not affected by the position of G.
7.2.2 Approximate Lateral Forces
In all two-axle vehicles with an open differential, it is possible to solve (7.5) with respect to the front and rear lateral forces (cf. (3.100))
where, in the last terms, we took into account that \(|J_z \dot{r}| \ll |m a_y a_i|\), since in a car \(J_z < m a_1 a_2\) and \(|\dot{r}a_i| \ll |a_y|\). The other term \(\varDelta \! X_1 t_1\) becomes relevant if the wheel steer angle is at least 15 degrees. It is common practice to ignore this contribution. In most cases it is hardly mentioned, and almost always neglected, although it can be far from negligible. The main reason for this “ostracism” is that the analysis is much simpler if \(\varDelta \!X_1 t_1\) is set to zero. Well, not quite a reasonable reason...
Moreover, under ordinary operating conditions \(|\dot{v}|\ll |u r|\) (Fig. 3.7), and we can use
already defined in (3.28), instead of the full expression \(a_y =\dot{v}+ur\) of the lateral acceleration, to approximately evaluate the axle lateral forces (cf. (3.100))
Therefore, in a two-axle vehicle with open differential, the axle lateral forces are approximately linear functions of the lateral acceleration \(\tilde{a}_y\). This is a simple, yet fundamental result in vehicle dynamics of road cars, which greatly impacts on the whole vehicle model.
Equations (7.9) hold true only when \(\dot{v}=\dot{r}=0\), that is when the vehicle is in steady-state conditions. However, they are sufficiently accurate when employed to estimate lateral load transfers and roll angles, as will be shown. Actually, we should never forget that in the present analysis there is no roll dynamics (except in Chap. 9). Therefore, the roll angle is always assumed to be the angle at steady state.
7.2.3 Lateral Load Transfers and Vertical Loads
According to (3.151) and (3.155), both lateral load transfers \(\varDelta \!Z_1\) and \(\varDelta \!Z_2\) are linear functions of both lateral forces \(Y_1\) and \(Y_2\).
Inserting (7.9) into (3.151), we obtain the following simplified equations for the lateral load transfers in vehicles with open differential and linear springs
or, equivalently
where \(l = a_1+a_2\) is the wheelbase, \(q = (a_2 q_1+a_1 q_2)/l\), and
is the global roll stiffness.
The two quantities \(\eta _1\) and \(\eta _2\) , and hence the ratio \(\varDelta \! Z_1/\varDelta \!Z_2 = \eta _1/\eta _2\), depend in a peculiar way on the track widths \(t_i\), on the roll stiffnesses of the suspensions \(k_{\phi _i}^s\), on the roll stiffnesses of the tires \(k_{\phi _i}^p\), on the heights \(q_i\) of the no-roll centers \(Q_i\),Footnote 3 on the longitudinal position \((a_1,a_2)\) and height h of the center of gravity G (Fig. 7.3). The roll stiffnesses are defined in Sect. 3.10.6, and in particular in (3.148). The no-roll centers are defined in Sect. 3.10.9.
Geometric parameters that affect lateral load transfers
If the tires are supposed to be perfectly rigid, that is \(k_{\phi _i}^p\rightarrow \infty \) and \(k_{\phi _i}^s = k_{\phi _i}\), the expressions of the lateral load transfers (7.11) become much simpler
as in (3.158).
Taking (7.9) into account we also obtain that
The total vertical loads (3.109) on each tire can be further simplified because, in the present case, the longitudinal load transfer \(\varDelta \!Z\) is negligible. Moreover, cars with an open differential are not so sporty to have significant aerodynamic vertical loads. Therefore, combining (3.109) and (7.10), we obtain
which shows that the variations of vertical loads are (linear) functions of the lateral acceleration \(\tilde{a}_y= ur\).
Roll angle \(\phi _i^s\) due to suspension deflections only, roll angle \(\phi _i^p\) due to tire deformations only, and total vehicle roll angle \(\phi \) (front view)
7.2.4 Roll Angles
Also the (steady-state) roll angles due to suspension deflections (3.147) depend upon \(Y_1\) and \(Y_2\), and hence, according to (7.9), can be set as functions of the lateral acceleration only Footnote 4
The same applies to roll angles \(\phi _i^p\) due to tire deformations. According to (3.144) and (7.10) we obtain
If the tires are supposed to be rigid, we have \(\rho _1^p=\rho _2^p=0\), and \(\rho _1^s=\rho _2^s= (h-q)/k_\phi \).
The roll angles (Fig. 7.4) are important because they affect camber angles and steer angles of the wheels, as shown hereafter.
7.2.5 Camber Angle Variations
Let, \(\gamma _{i2}^0 = - \gamma _{i1}^0 = \gamma _i^0\) be the camber angles under static conditions (Fig. 7.5), and let \(\varDelta \!\gamma _{i1} = \varDelta \!\gamma _{i2} = \varDelta \gamma _i\) be the camber variations due to vehicle roll motion (Fig. 7.6). The camber angles of the two wheels of the same axle are thus given by
where the camber variation \(\varDelta \gamma _i\), according to (3.113), (7.15) and (7.16), depends on the roll angles, and hence on the lateral acceleration \(\tilde{a}_y\)
since the term \(\pm z_i^s/c_i\) is usually negligible in road cars on flat roads.
Positive static camber \(\gamma _i^0\) (front view)
Positive camber variations \(\varDelta \gamma _{i}\) due to roll motion (front view, left turn)
Three suspensions with the same \(t_i\), but with different values of \(c_i\), are shown in Fig. 7.7. We see that, as expected, the same amount of vehicle roll angle \(\phi _i^s\) yields different camber variations (tire roll angle \(\phi _i^p\) not considered). In all cases the roll angle is due to a left turn. Camber variations are negative in the first case.
Front view of three different suspensions (right), and their camber variations (left) due to the same positive vehicle roll angle \(\phi _i^s\) (tire roll angle \(\phi _i^p\) not considered)
7.2.6 Steer Angles
According to (3.210) and taking into account (7.15), we obtain the following (approximate, but very good) expressions for the steering angles of the two wheels of the same axle
which are, obviously, functions of the steering wheel rotation \(\delta _v\) imposed by the driver and, possibly, of the lateral acceleration \(\tilde{a}_y= ur\).
In (7.19), as discussed in Sect. 3.4, \(\delta _i^0\) is the static toe angle, \(\tau _i\) is the first-order gear ratio of the whole steering system, \(\varepsilon _i\) is the Ackermann coefficient for dynamic toe, \(\varUpsilon _i\) is the roll steer coefficient and \(\rho _i^s m \tilde{a}_y\) is the suspension roll angle \(\phi _i^s\). If the tires are supposed to be rigid, we have \(\rho _1^s=\rho _2^s= (h-q)/k_\phi \). The analysis is considerably simpler if \(\varUpsilon _i=0\), that is if there is no roll steer. Most cars have \(\tau _2=0\), that is no direct steering of the rear wheels.
7.2.7 Tire Slips
As already stated in Sect. 7.1.3, in the model under investigation all wheels are almost under longitudinal pure rolling conditions, that is \(\sigma _{x_{ij}}\simeq 0\). Therefore, according to (3.59)
where \(\omega _{ij}\) is the angular velocity of the corresponding rim and \(r_i\) is the wheel rolling radius, as defined in (2.38).
Under these assumed operating conditions, the tire lateral slips (3.60) become
where \(\delta _{ij}= \delta _{ij}(\delta _v, ur)\) as in (7.19).
Therefore, more compactly
It will turn useful to have these very same slips expressed in terms of \(\beta = v/u\) and \(\rho =r/u\)
and, more compactly
We see that the “main” dependence on u has disappeared.
7.2.8 Simplified Tire Slips
Equations (7.21) can be simplified without impairing their accuracy too much. More precisely, taking into account that \(u \gg |v|\), \(u \gg |r t_i|\), \(|\delta _{ij}|\ll 1\), and \(\omega _{ij}r_i \simeq u\), we obtain (see (3.55) and (3.57))
More explicitly, according to (7.19), we have
Most cars have \(\tau _2=0\), that is no direct steering of the rear wheels.
Equations (7.21)–(7.26) show how the lateral tire slips \(\sigma _{ij}\) are related to the global vehicle motion, to the kinematic steer angles, to the toe-in/out angles, and to the roll steer angle. None of these contributions can be neglected, in general.
Actual slip angles \(\alpha _{ij}\) in the double track model (see also Fig. 3.13)
We can also look at the actual tire slip angles \(\alpha _{ij}\) (Fig. 7.8), defined in (3.55) and (3.58). In this model (not to be taken as a general rule) we have
Just compare (3.58) with (7.25).
7.2.9 Tire Lateral Forces
The lateral force exerted by each tire on the vehicle depends on many quantities, as shown in the second equation in (2.82). For sure, there is a strong dependence on the vertical loads \(Z_{ij}\) and on the lateral slips \(\sigma _{y_{ij}}\), while, in this vehicle model, the longitudinal slips \(\sigma _{x_{ij}}\) are negligible. The camber angles \(\gamma _{ij}\) need to be considered as well, since they are quite influential, even if small. According to (3.212), the spin slips \(\varphi _{ij}\) are directly related to \(\gamma _{ij}\). Therefore, a suitable model for the lateral force of each wheel with tire is (Fig. 7.1)
Of course, extensive tire testing is required to make these functions available.
Needless to say, many other parameters affect the tire performance: road surface, temperature, inflation pressure, etc.
The lateral force \(Y_i\) for each axle of the vehicle is obtained by adding the lateral forces of the left wheel and of the right wheel (cf. (3.85) and (7.4), with \(F_{x_{ij}}\simeq 0\))
In general, the two wheels of the same axle undergo different vertical loads, different camber angles, and different lateral slips. Therefore, the two lateral forces are very different, as shown, e.g., in Fig. 3.28 and also in Fig 7.12. Equations (7.14) and (7.18), when inserted into (7.28), allow to take all these aspects into account. Therefore (Fig. 7.1)
It should be clearly understood that the functions in (7.30) are known algebraic functions.
A general comment on this vehicle model is in order here: some quantities depend (linearly) only on the lateral acceleration \(\tilde{a}_y=ur\). However, it must be remarked that this peculiarity needs an open differential, no aerodynamic forces, almost constant forward speed.
7.3 Double Track Model
7.3.1 Governing Equations of the Double Track Model
Summing up, the double track vehicle model for studying the handling of road cars is governed by the following three sets of equations:
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two equilibrium equations (lateral and yaw), as in (7.5)
$$\begin{aligned} \begin{aligned}&m(\dot{v}+ur) = Y_1 + Y_2 = Y \\[1ex]&J_z \dot{r}= Y_1 a_1 - Y_2 a_2 + \varDelta \!X_1 t_1 = N \end{aligned} \end{aligned}$$(7.31) -
three constitutive equations, as in (7.30), which are affected by several setup parameters and by the vertical loads
$$\begin{aligned} \begin{aligned} Y_1&= F_{y_1}(\sigma _{y_{11}},\sigma _{y_{12}}, \delta _v, ur) \\[1ex]Y_2&= F_{y_2}(\sigma _{y_{21}},\sigma _{y_{22}}, ur) \\[1ex]\varDelta \!X_1&= \varDelta \!X_1(\sigma _{y_{11}},\sigma _{y_{12}}, \delta _v, ur) \end{aligned} \end{aligned}$$(7.32) -
four congruence equations (tire lateral slips), as in (7.21), which take care, among other things, of the Ackermann coefficient
$$\begin{aligned} \begin{aligned} \sigma _{y_{11}}&=\sigma _{y_{11}}(v,r;u,\delta _{11}(\delta _v, ur)) \\[1ex]\sigma _{y_{12}}&=\sigma _{y_{12}}(v,r;u,\delta _{12}(\delta _v, ur)) \\[1ex]\sigma _{y_{21}}&=\sigma _{y_{21}}(v,r;u,\delta _{21}(\delta _v, ur)) \\[1ex]\sigma _{y_{22}}&=\sigma _{y_{22}}(v,r;u,\delta _{22}(\delta _v, ur)) \end{aligned} \end{aligned}$$(7.33)
We have simply \(\delta _{ij}= \delta _{ij}(\delta _v)\) if there is no roll steer.
This vehicle model for road vehicle handling is fairly general, and it is usually called double track model. A more classical formulation of the same model is obtained taking (7.27) into account. However, using \(\sigma _{ij}\) instead of \(\alpha _{ij}\) is conceptually clearer.
7.3.2 Dynamical Equations of the Double Track Model
The dynamical equations for road vehicle handling are now promptly obtained. As a final step, it suffices to insert (7.32) and (7.33) into (7.31)
This is a dynamical system with two state variables, namely, but not necessarily, v(t) and r(t), as discussed in Sect. 7.3.3. The driver controls the steering wheel angle \(\delta _v(t)\) and the forward speed u.
The double track model can be used to simulate and investigate the vehicle handling behavior under steady-state or transient conditions (i.e., nonconstant \(\delta _v(t)\)).
Unfortunately, the double track model is not as popular as the single track model (often and mistakenly also named “bicycle model”). The effort required to build a computer program and to run simulations with the double track model is comparable to the effort required by the less accurate single track model (introduced and discussed in Sect. 7.5).
7.3.3 Alternative State Variables (\(\beta \) and \(\rho \))
The use of v(t) and r(t) as state variables is not mandatory, and other options can be envisaged. Other state variables may provide a better insight into vehicle handling, if properly handled.
The state variables \(\beta (t)\) and \(\rho (t)\) have been already introduced in (3.16) and (3.17). They are repeated here for ease of reading
and
They are just v and r normalized with respect to u.
The corresponding three sets of equations of the double track model become:
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equilibrium equations (cf. (3.23), (3.27) and (7.31))
$$\begin{aligned} \begin{aligned} m(\dot{\beta }u +\beta \dot{u}+u^2 \rho )&= Y =Y_1 + Y_2 \\[1ex]J_z (\dot{\rho }u + \rho \dot{u})&= N =Y_1 a_1 - Y_2 a_2 +\varDelta \!X_1 t_1 \\[1ex]\end{aligned} \end{aligned}$$(7.35) -
constitutive equations (as in (7.32), with \(\tilde{a}_y= u r = u^2\rho \))
$$\begin{aligned} \begin{aligned} Y_1&= F_{y_1}(\sigma _{y_{11}},\sigma _{y_{12}},\delta _v, u^2\rho ) \\[1ex]Y_2&= F_{y_2}(\sigma _{y_{21}},\sigma _{y_{22}}, u^2\rho ) \\[1ex]\varDelta \!X_1&= \varDelta \!X_1(\sigma _{y_{11}},\sigma _{y_{12}},\delta _v, u^2\rho ) \end{aligned} \end{aligned}$$(7.36) -
congruence equations (cf. (7.33), with \(\tilde{a}_y= u r = u^2\rho \))
$$\begin{aligned} \begin{aligned} \sigma _{y_{11}}&=\sigma _{y_{11}}(\beta ,\rho ;\delta _{11}(\delta _v, u^2\rho )) \\[1ex]\sigma _{y_{12}}&=\sigma _{y_{12}}(\beta ,\rho ;\delta _{12}(\delta _v, u^2\rho )) \\[1ex]\sigma _{y_{21}}&=\sigma _{y_{21}}(\beta ,\rho ;\delta _{21}(\delta _v, u^2\rho )) \\[1ex]\sigma _{y_{22}}&=\sigma _{y_{22}}(\beta ,\rho ;\delta _{22}(\delta _v, u^2\rho )) \end{aligned} \end{aligned}$$(7.37)
Therefore, in this case, the two dynamical equations (7.34) of the double track model become
where \(|\dot{u}| \simeq 0\) and can be discarded. The dependence of Y and N on the lateral acceleration \(u^2\rho \), and hence on the forward speed u, disappears if there is no roll steer. This is the main advantage in using \(\beta \) and \(\rho \) as state variables in the double track model for road cars.
Quite remarkably, we will see in (7.75) that in the single track model, when \(\beta \) and \(\rho \) are used as state variables, there is no dependence of Y and N on u even if roll steer is taken into account.
7.4 Vehicle in Steady-State Conditions
An essential step in understanding the behavior of a dynamical system, and therefore of a motor vehicle, is the determination of the steady-state (equilibrium) configurations \((v_p, r_p)\). In physical terms, a vehicle is in steady-state conditions when, with fixed position \(\delta _v\) of the steering wheel and at constant forward speed u, it goes around with circular trajectories of all of its points.
After having set \(\dot{\delta }_v=0\) and \(\dot{u}=0\), the mathematical conditions for the system being in steady state is to have \(\dot{v}= 0\) and \(\dot{r}= 0\) in (7.34). Accordingly, the lateral acceleration drops the \(\dot{v}\) term and becomes at steady state
This equation was already introduced in (3.28).
Finding the equilibrium points \((v_p,r_p)\), that is how the vehicle moves under given and constant \(\delta _v\) and u, amounts to solving the system of two algebraic equations
or, equivalently and more formally
to get \((v_p,r_p)\) such that
Because of the nonlinearity of the tire behavior, the number of possible solutions \((v_p,r_p)\), for given \((u,\delta _v)\), is not known a priori. Typically, if more than one solution exists, at most only one is stable.
Equations (7.42) define implicitly the two maps
that is, the totality of steady-state (equilibrium) conditions as functions of the forward speed u and of the steering wheel angle \(\delta _v\). Given and kept constant the forward speed u and the steering wheel angle \(\delta _v\), after a while (a few seconds at most) the vehicle reaches the corresponding steady-state condition, characterized by a constant lateral speed \(v_p\) and a constant yaw rate \(r_p\).
For a more “geometric”, and hence more intuitive, analysis of the handling of vehicles, it is convenient to employ \(\beta =v/u\) and \(\rho =r/u\) instead of v and r, as done in Sect. 7.3.3. Therefore, (7.43) can be replaced by
The steady-state handling behavior is completely characterized by these handling maps of \(\beta \) and \(\rho \), both as functions of two variables, namely, but not necessarily, u and \(\delta _v\)
Indeed, it is common practice to employ \((\delta _v,\tilde{a}_y)\), instead of \((u,\delta _v)\), as parameters to characterize a steady-state condition. This is possible because
Therefore, (7.43) becomes
and, accordingly, (7.44) becomes
At first it may look a bit odd to employ \((\delta _v, \tilde{a}_y)\) instead of \((u,\delta _v)\), but it is not, since it happens that in most road cars a few steady-state quantities are functions of \(\tilde{a}_y\) only. This is quite a remarkable fact, but it should not be taken as a general rule.Footnote 5
Similarly, we could use \((u,\tilde{a}_y)\) as parameters to characterize a steady-state condition.
Equations (7.44) or (7.48) provide a fairly general point of view that led to the new global approach that we called Map of Achievable Performance (MAP) in Chap. 6. Additional, relevant information are provided in Sects. 7.7 and 7.8. These MAPs can be obtained experimentally or through simulations. Therefore, they are not limited to mathematical models. Actually, as will be discussed in the next chapter, they exist also for race cars, including Formula cars with very high aerodynamic downforces.
Now we address more classical topics, like the single track model and the associated handling diagram.
7.5 Single Track Model
The goal of this Section is to present a comprehensive analysis of the single track model [2, 5, 6, 8, 12, 17, 19], thus showing also its limitations. In many courses or books on vehicle dynamics (e.g., [8, p. 199]) the single track model, shown in Fig. 7.9, is proposed without explaining in detail why, despite its awful appearance, it can provide in some cases useful insights into vehicle handling, particularly for educational purposes. Vehicle engineers should be well aware of the steps taken to simplify the model, and hence realize that in some cases the single track model may miss some crucial phenomena, and the double track model should be used instead.
Equivalent schematic representations of the single track model (with \(\delta _2=0\))
7.5.1 From Double Track to Single Track
The double track model is shown in Figs. 7.1 and 7.2. Of course, it has four wheels. In the corresponding schematic representations of the single track model shown in Fig. 7.9 it looks like there are only two wheels. It is not so. It is not a bicycle. In the single track model we look at each axle as a whole. Therefore, better names would be “four wheel” model and “two axle” model, respectively.
To go from the double track model to the single track model (Fig. 7.9) we need to further simplify (7.26): the Ackermann corrections have to be set equal to zero, that is
which is consistent with small steering angles. Indeed the Ackermann correction is a second order contribution in (7.26) and, important as it can be, it cannot be included in the single track model. You see that we are missing something.
This (not necessarily true) hypothesis (7.49) on the Ackermann coefficients, if combined with the simplified expressions (7.26), leads to the following (first-order) expressions for the lateral slips of the four wheels
where we can still take into account the toe-in/toe-out terms \(\delta _i^0\), and also the roll steer contributions.
In (7.50) it is convenient to define what may be called the apparent slip angles \(\alpha _1\) and \(\alpha _2\) of the front and rear axles, respectively (Fig. 7.20)
Combining (7.50) and (7.51), we obtain that both front lateral slips \(\sigma _{y_{11}}\) and \(\sigma _{y_{12}}\) are known functions of the same two variables \(\alpha _1\) and \(\tilde{a}_y\). Similarly, both rear lateral slips \(\sigma _{y_{21}}\) and \(\sigma _{y_{22}}\) are known functions of the same two variables \(\alpha _2\) and \(\tilde{a}_y\)
The two wheels of the same axle undergo the same apparent slip angle, but not necessarily the same lateral slip. The key point for the model to be single track is that the difference between left and right lateral slips must be a function only of \(\tilde{a}_y=ur\). This is the peculiar feature of the single track model (cf. (7.26)). It is the fundamental brick for the next step.
But before doing that, it is worth noting the crucial difference between the actual slip angles \(\alpha _{ij}\) of each wheel, defined in (3.58) (and also in (7.26)), and the apparent slip angles \(\alpha _i\) of each axle, defined in (7.51).
In general, the two apparent slip angles \(\alpha _i\) can be defined only in the single track model (Fig. 7.20). In real vehicles there are four actual slip angles \(\alpha _{ij}\).
It is very common in traditional (oversimplified) vehicle dynamics not to take into account toe-in/toe-out and roll steering, thus having
7.5.2 “Forcing” the Lateral Forces
Owing to (7.52), the first two equations in (7.30) become
while the third equation in (7.30) is set to zero because of the assumed very small steer angles
It is really crucial for a vehicle engineer to understand and keep in mind the differences between (7.30) and (7.55). In the final expressions of \(Y_i\) in (7.55) there appear only variables associated to the corresponding axle, not anymore to the single wheel.
As already obtained in (7.9) at the beginning of this chapter, we have that the lateral forces are basically linear functions of \(\tilde{a}_y\) (open differential)
Therefore, \(F_{y_1}(\alpha _1, \tilde{a}_y)\) and \(F_{y_2}(\alpha _2, \tilde{a}_y)\) must be such that
which can be solved with respect to the lateral acceleration, to obtainFootnote 6
These relationships are affected by many setup parameters, like camber angles, roll steer, toe-in/toe-out, etc., as discussed in detail in Sect. 7.5.3.
The final, crucial, step is inserting (i.e., “forcing”) these results back into (7.55), thus obtaining the axle characteristics of the single track model
that is, two functions, one per axle, that give the axle lateral force as a function of only the corresponding apparent slip angle. In other words, each axle behaves formally as an equivalent single wheel with tire.
Forcing the lateral forces to be as in (7.60) is an approximation when the vehicle is in transient conditions. Moreover, to go from (7.30) to (7.55) we made assumptions (7.49) about the steer kinematics (parallel steering) and small steer angles.
The double track model provides more accurate results when running simulations. On the other hand, the single track model is a useful tool for educational purposes and for investigating steady-state conditions. It is less accurate, but more intuitive.
Equation (7.59) implies that in the single track model there is a link between \(\alpha _1\) and \(\alpha _2\)
and that this link is not affected by u or \(\delta _v\). In a real vehicle this is not necessarily true. Vehicle engineers should be aware that the single track model is somehow an inconsistent model, albeit very appealing.
7.5.3 Axle Characteristics
As done in (7.60), by axle characteristics we mean two algebraic functions (one per axle) of the form
which provide the total lateral force as a function of the apparent slip angle only, with the effects, e.g., of the lateral load transfers already accounted for. They were obtained in (7.60), but the topic is so relevant to deserve an in-depth discussion.
Tire tested under symmetric vertical loads with respect to the static load \(Z_0/2\)
7.5.3.1 The Basics
The basic procedure to obtain the axle characteristics is described here. The goal is to provide an intuitive and physical approach to the construction of the axle characteristics in the single track model. “Basic” means that only the effects of the lateral load transfers \(\varDelta \!Z_i\) are taken into account. Of course, lateral load transfers cannot be omitted. They must necessarily be included in the analysis (unless linear tire behavior is assumed, Fig. 7.62).
The first step is to test the tire under symmetric vertical loads with respect to the reference value \(Z_i^0/2\), as shown in Fig. 7.10. “Symmetric” means that tests have to be carried out in pairs, that is with \(F_z = Z_i^0/2\pm \varDelta \! Z_i\). In Fig. 7.10 two such pairs are shown.
Basic graphic construction of the axle characteristic and influence of changing the roll stiffness
The second step is to add the two tire curves obtained with symmetric vertical loads, as shown in Fig. 7.11(top), thus getting a sort of axle curve for each value of the lateral load transfer. To legitimate this second step it is mandatory that the inner wheel and the outer wheel of the same axle undergo the same apparent slip angle \(\alpha _i\). As expected, the higher the lateral load transfer \(\varDelta \!Z_i\), the lower the corresponding axle curve.
The third step is to draw a straight line according to (7.13), to linearly relate the lateral load transfer \(\varDelta \!Z_i\) to the axle lateral force \(F_{y_i}\).
The fourth and final step is to pick the unique point on each axle curve that corresponds to a real operating (steady-state) condition for the vehicle, as shown in Fig. 7.11(top). As a matter of fact, each axle curve was obtained testing the tire with given and constant \(\pm \varDelta \!Z_i\), but this amount of lateral load transfer requires a definite value of the lateral force \(F_{y_i}\) in the vehicle, and hence a definite value of \(\alpha _i\).
The sought axle characteristic \(Y_i(\alpha _i)\) is just the curve connecting all these points, as schematically shown in Fig. 7.11.
Changing the value of \(\eta _i\) in (7.13) results in a different straight line and hence in different axle characteristics, as shown in Fig. 7.11(bottom). The axle curves are not affected by \(\eta _i\), but the points corresponding to real operating conditions are.
7.5.3.2 The General Case
Now we are ready to address the construction of the axle characteristics with greater generality. It means that we will use the lateral acceleration \(\tilde{a}_y\) as a parameter.
According to (7.30), (7.52), (7.55) and (7.57), the general framework, at steady state, for a given vehicle is that:
-
1.
each axle lateral force \(Y_i\) is determined solely by the lateral acceleration \(\tilde{a}_y\), see (7.9) (open differential);
-
2.
there is a one-to-one correspondence between the lateral acceleration \(\tilde{a}_y\) and the following quantities:
-
3.
both left and right tire lateral forces are known functions of the lateral acceleration \(\tilde{a}_y\) and of the same apparent slip angle \(\alpha _i\), see (7.55).
Therefore, as discussed in Sect. 7.5.2, for any given value of \(\tilde{a}_y\) we can obtain the corresponding load transfers, camber angles and roll steer angles. Consequently, we can plot (measure) the lateral forces \(F_{y_{ij}}(\alpha _i)\) of each wheel as functions only of the apparent slip angle \(\alpha _i\)
This way we can single out the contribution of each wheel. Typical curves for the inner and the outer tires of the same axle are shown in Fig. 7.12. As expected, the outer wheel (green plot) provides a lateral force larger than the inner wheel (red plot).
Each axle characteristic is given by (Fig. 7.12, blue plot)
Of course, any calculation of this type assumes the availability of tire data.
As shown in Fig. 7.12, the maximum lateral forces of the two wheels are not attained, in general, for the same apparent slip angle. This is not desirable, if we are interested in maximising the lateral acceleration. To mitigate this phenomenon we can resort on tuning some setup parameters. We have to understand the effects on the axle characteristics of changing these setup parameters. This extremely relevant practical topic is discussed hereafter, taking into account the effects of changing the:
-
1.
lateral load transfer;
-
2.
static camber angles;
-
3.
roll camber;
-
4.
toe-in/toe-out;
-
5.
roll steer.
Lateral forces exerted by the inner wheel (red), by the outer wheel (green) of the same axle, and resulting axle characteristic (blue). Maximum lateral force is 5 kN at 10.34 deg
All plots in this section are for a car making a left turn (\(\tilde{a}_y>0\)). In all plots in this section, the apparent slip angles are in degrees and the lateral forces are in kN.
As in Fig. 7.12, but with lower load transfer (left: max force 5.15 kN at 8.20 deg) or higher load transfer (right: max force 4.97 kN at 10.95 deg)
7.5.3.3 Lateral Load Transfer \(\varDelta \!Z_i\)
Two additional basic examples are shown in Fig. 7.13. They are basic in the sense that it is assumed that the lateral acceleration \(\tilde{a}_y\) affects only the lateral load transfer \(\varDelta \!Z_i\). More precisely, it is assumed that \(\gamma _{ij}=\delta _i^0=\varUpsilon _i=0\).
The two cases in Fig. 7.13 have different values of \(\eta _i\), and hence different load transfers for the same lateral acceleration, with respect to the case in Fig. 7.12. One has lower \(\eta _i\) (left: max force 5.15 kN at 8.20 deg) and one has higher \(\eta _i\) (right: max force 4.97 kN at 10.95 deg).
A very relevant fact in vehicle dynamics, as stated in Sects 2.10.2 and 2.12, is that the lateral force exerted by a single tire grows less than proportionally with respect to the vertical load. This is clearly shown in Fig. 7.13, where the higher the lateral load transfer, the lower the resulting curve of \(Y_i\).
7.5.3.4 Static Camber \(\gamma _i^0\)
The definition of static camber is given in Fig. 7.5 and in (7.17). The effects of negative and positive static camber angles, i.e. \(\gamma _i^0 \ne 0\), are shown in Fig. 7.14, left and right, respectively. If the top of the wheel is farther out than the bottom (that is, away from the axle), it is called positive static camber. If the bottom of the wheel is farther out than the top, it is called negative camber. We see in Fig. 7.14 that there are lateral forces on each wheel (camber thrust) when the car is going straight. We also see that the peak (max) value of the axle lateral force is higher with negative static camber. The main reason is that the inner and outer wheel reach their peak values for apparent slip angles that are less far away from each other.
As in Fig. 7.12, but with negative static camber (left: max 5.03 kN at 10.23 deg) or positive static camber (right: max 4.97 kN at 10.47 deg)
As in Fig. 7.12, but with negative roll camber (left: max 5.02 kN at 10.29 deg) or positive roll camber (right: max 4.98 kN at 10.40 deg)
7.5.3.5 Roll Camber \(\varDelta \!\gamma _i\)
As shown in Fig. 7.15 and in (7.18), roll camber \(\varDelta \!\gamma _i\) is an anti-symmetric setup modification. Therefore, it also affects the slope in the origin of the axle characteristic. Negative and positive camber variations due to roll motion are shown in Fig. 7.6. Also useful may be Fig. 7.7, which shows how the suspension architecture strongly affects roll camber.
As in Fig. 7.12, but with toe-in (left: max 5.03 kN at 9.40 deg) or toe-out (right: max 4.97 kN at 11.42 deg)
7.5.3.6 Toe-in/Toe-out \(\delta _i^0\)
The definition of toe-in/toe-out is given in Fig. 3.15 and in (7.19). The effects of toe-in (\(\delta _i^0 > 0\)) and toe-out (\(\delta _i^0 < 0\)), are shown in Fig. 7.16, left and right, respectively. We see that also in this case there are lateral forces on the wheels when the car is going straight.
The beneficial cumulative effects of negative static camber, negative roll camber and toe-in are shown in Fig. 7.17. We see that the peak value from 5 kN reached 5.09 kN. This result was achieved because the inner and outer wheel reach their peak values for apparent slip angles that are now closer to each other.
As in Fig. 7.12, but with the cumulative effects of negative static camber, negative roll camber, and toe-in (max 5.09 kN at 9.24 deg)
7.5.3.7 Roll Steer \(\varUpsilon _i \rho _i^s m\tilde{a}_y\)
Also interesting is the case of roll steer, i.e. \(\varUpsilon _i\ne 0\), shown in Fig. 7.18. While most effects are symmetric with respect to the vehicle axis, and hence the contributions of the two wheels cancel each other at low lateral acceleration, the roll steer is anti-symmetric, and hence it affects the axle characteristic also at low lateral accelerations. However, contrary to all other setup parameters here considered, it does not affect the peak value of the axle lateral force.
As in Fig. 7.12, but with positive roll steer (left: max 5 kN at 9.12 deg) or negative roll steer (right: max 5 kN at 11.61 deg)
7.5.3.8 General (Mixed-up) Case
As already shown in Fig. 7.17, in general all these effects may very well coexist in a real car. The axle characteristics are what most characterize vehicle dynamics. They may differ in the initial slope (slip stiffness) and in the maximum value. Both aspects have a big influence on vehicle handling.
We remark that the axle characteristics, under an apparent simplicity, contain a lot of information about the vehicle features and setup (see also [12, Chap. 6]).
7.5.3.9 Look at the Peak Positions
For just a moment, let us assume the grip coefficient being not dependent on the vertical load. More precisely, set \(a_1 = 0\) in (2.99). Even in this case, the amount of lateral load transfer does affect the peak value of the axle lateral force, as shown in Fig. 7.19. Therefore, we should do our best to keep the apparent slip angles of the inner and outer peaks as close together as possible. This issue is rarely discussed, maybe because it is unusual to draw Figures like 7.12, 7.13, 7.14, 7.15, 7.16, 7.17, 7.18 and 7.19.
Higher load transfers lower the peak value of the axle characteristic even if assuming (erroneously) load independent grip
7.5.4 Governing Equations of the Single Track Model
Summing up, the single track model is governed by the following three sets of fairly simple equations:
-
two equilibrium equations (lateral and yaw), as in (7.5)
$$\begin{aligned} \begin{aligned} m(\dot{v}+ur)&= Y = Y_1 + Y_2 \\[1ex]J_z \dot{r}&= N = Y_1 a_1 - Y_2 a_2 \end{aligned} \end{aligned}$$(7.65) -
two constitutive equations (axle characteristics, which include the effects of several setup parameters), as in (7.60)
$$\begin{aligned} \begin{aligned} Y_1&= Y_1(\alpha _1) \\[1ex]Y_2&= Y_2(\alpha _2) \end{aligned} \end{aligned}$$(7.66) -
two congruence equations (apparent slip angles), as in (7.51)
$$\begin{aligned} \begin{aligned} \alpha _1&= \tau _1 \delta _v -\frac{v+r a_1}{u} \\[1ex]\alpha _2&= \tau _2 \delta _v -\frac{v-r a_2}{u} \end{aligned} \end{aligned}$$(7.67)
Single track model
A comparison with the governing equations of the double track model (Sect. 7.3.1) shows that in the single track model:
-
the term \(\varDelta \!X_1 t_1\) has disappeared from the equilibrium equations;
-
there are two, instead of four, constitutive equations;
-
there are two, instead of four, congruence equations.
A pictorial version of the single track model is shown in Fig. 7.20, where
with
Reasonably, but also arbitrarily, we call \(\delta _1\) the steer angle of the front axle. A similar thing can be done for the rear axle.
The angle \(\delta \) is called net steer angle of the (single track model of the) vehicle. Usually, \(\kappa =0\) and hence \(\delta \) is just the steering angle \(\delta _1\) of the front axle. However, \(\kappa \ne 0\) leaves room for rear steering \(\delta _2\) as well, without affecting \(\delta \).
In this single track model there is a one-to-one relationship between \(\delta \) and \(\delta _v\), that is we have a rigid steering system. A vehicle model with compliant steering system is developed in Sect. 7.16.
Indeed, the equations governing the dynamical system of Fig. 7.20 are precisely (7.65), (7.66) and (7.67). Therefore, Fig. 7.20 can be used as a shortcut to quickly obtain the simplified equations of a vehicle. However, the vehicle model still has four wheels, lateral load transfers, camber and camber variations, roll steer, as discussed in Sect. 7.5.3 on the axle characteristics.
The main feature of this model is that the two wheels of the same axle undergo the same apparent slip angle \(\alpha _i\), and hence can be replaced by a sort of equivalent wheel, like in Fig. 7.20. However, that does not imply that the real slip angles of the two wheels of the same axle are the same. Neither are the camber angles, the roll steer angles, the vertical loads. Therefore, the single track model is not really single track! It retains many of the features of the double track model.
It is not necessary to assume that the center of mass G of the vehicle is at road level [11, p. 170], neither that the lateral forces of the left and right tires to be equal to each other [1, p. 53]. Actually, both assumptions would be strikingly false in any car.
Assuming the total mass to be concentrated at G, as if the vehicle were like a point mass [18, p. 223], is another unrealistic, and unnecessary, assumption.
7.5.5 Dynamical Equations of the Single Track Model
Among the governing equations, only the two equilibrium equations are differential equations, and both are first-order. The other four algebraic equations must be inserted into the equilibrium equations to ultimately obtain the two dynamical equations of the single track model
or, more compactly
Therefore, the single track model is a dynamical system with two state variables, namely, but not necessarily, v(t) and r(t), as discussed in Sect. 7.5.6. The driver controls the steering wheel angle \(\delta _v(t)\) and the forward speed u.
7.5.6 Alternative State Variables (\(\beta \) and \(\rho \))
As already done in Sect. 7.3.3, instead of v(t) and r(t), we can use \(\beta (t) =v/u\) and \(\rho (t)=r/u\) to describe the handling of a vehicle.
The corresponding governing equations of the single track model become:
-
equilibrium equations (cf. (7.65))
$$\begin{aligned} \begin{aligned} m(\dot{\beta }u +\beta \dot{u}+u^2 \rho )&= Y = Y_1 + Y_2 \\[1ex]J_z (\dot{\rho }u + \rho \dot{u})&= N = Y_1 a_1 - Y_2 a_2 \end{aligned} \end{aligned}$$(7.72) -
constitutive equations (cf. (7.66))
$$\begin{aligned} \begin{aligned} Y_1&= Y_1(\alpha _1) \\[1ex]Y_2&= Y_2(\alpha _2) \end{aligned} \end{aligned}$$(7.73) -
congruence equations (cf. (7.67))
$$\begin{aligned} \begin{aligned} \alpha _1&= \delta _v \tau _1 - \beta - \rho a_1 \\[1ex]\alpha _2&= \delta _v \tau _2 - \beta + \rho a_2 \end{aligned} \end{aligned}$$(7.74)
Combining these three sets of equations, we obtain the dynamical equations, that is the counterpart of (7.70)
where \(|\dot{u}|\simeq 0\).
It is worth noting that, differently from (7.38) of the double track model, the axle lateral forces \(Y_1\) and \(Y_2\), and hence also the total lateral force Y and the yaw moment N, do not depend explicitly on the forward speed u, even if roll steer is taken into account. All the effects of the lateral acceleration \(\tilde{a}_y=u r = u^2\rho \) on Y and N are already included in the axle characteristics. Moreover, the expressions of Y and N in (7.75) are even simpler than those in (7.71).
7.5.7 Inverse Congruence Equations
The state variables v and r appear in both congruence equations (7.67). However, it is possible to invert these equations to obtain two other equivalent equations, with \(\rho =r/u\) appearing only in the first equation and \(\beta =v/u\) only in the second equation
where the more compact notation \(\delta _1 =\delta _v \tau _1\) and \(\delta _2 = \delta _v \tau _2\) has been used.
It is important to realize that all these inverse congruence equations are not limited to steady-state conditions, although they are mostly used for the evaluation of some steady-state features.
Another very common way to rewrite the first equation in (7.76) is as follows
where \(R =u/r\). Should \(\alpha _1=\alpha _2=0\) (very low speed), then \(\delta = l/R\), which is often called Ackermann angle (not to be confused with Ackermann steering geometry, discussed in Sect. 3.4).
7.5.8 \(\beta _1\) and \(\beta _2\) as State Variables
Another useful set of state variables may be the vehicle slip angles at each axle midpoint (Fig. 7.20)
The inverse equations are
The corresponding governing equations of the single track model become:
-
equilibrium equations
$$\begin{aligned} \begin{aligned} \dot{\beta }_1 u +\beta _1 \dot{u}+(\beta _1-\beta _2) \frac{u^2}{l}&= \frac{Y}{m} + \frac{N }{J_z}\,a_1 \\[1ex]\dot{\beta }_2 u +\beta _2 \dot{u}+ (\beta _1-\beta _2) \frac{u^2}{l}&= \frac{Y}{m} - \frac{N }{J_z}\,a_2 \end{aligned} \end{aligned}$$(7.80) -
constitutive equations (from the axle characteristics)
$$\begin{aligned} \begin{aligned} Y_1&= Y_1(\alpha _1) \\[1ex]Y_2&= Y_2(\alpha _2) \end{aligned} \end{aligned}$$(7.81) -
congruence equations
$$\begin{aligned} \begin{aligned} \alpha _1&= \delta _v \tau _1 - \beta _1= \delta _1 - \beta _1 \\[1ex]\alpha _2&= \delta _v \tau _2 - \beta _2= \delta _2 - \beta _2 \end{aligned} \end{aligned}$$(7.82)
The two first-order differential equations (7.70) or (7.75), governing the dynamical system, become
where, again, the terms on the r.h.s. do not depend on u.
These equations highlight an interesting feature. The terms \((J_z - m a_1 a_2)\), which appear in both equations, are often very small in road cars, and could even be purposely set equal to zero. Therefore, the coupling between the two equations is fairly weak.
We observe that (7.77) becomes
and we also have
7.5.9 Driving Force
At the beginning of this chapter, and precisely in (7.2), we made the assumption of small longitudinal forces. But small does not mean zero. Indeed, a small amount of power is necessary even for keeping a vehicle in steady-state conditions. To make this statement quantitative, let us consider a rear-wheel-drive single track model (Fig. 7.20, with \(F_{x_1}=0\)). The power balance
provides the following driving force \(F_{x_2}\)
This force \(F_{x_2}\) has to counteract the aerodynamic drag (obvious) and also the drag due to tire slips (maybe not so obvious at first). Indeed, lateral axle forces \(F_{y_i}\) are not orthogonal to the corresponding velocity and hence absorb mechanical power.
Graphical evaluation of the driving force \(F_{x_2}\) at steady state
That tire slips induce drag can be better appreciated from Fig. 7.21 (where, for simplicity, the aerodynamic drag is not considered). Points C and A do not coincide because of the apparent slip angles \(\alpha _1\) and \(\alpha _2\). Therefore, a longitudinal driving force \(F_{x_2}\) is required to achieve the dynamic equilibrium (cf. [12, p. 67]).
Of course, Fig. 7.21 is just a scheme. In real cases, slip angles are much smaller. The last line in (7.87), which is valid for about \(|\alpha _i| < 0.05 \text { rad}\), clearly shows that \(F_{x_2}\) is much smaller than \(F_{y_i}\), as assumed in (7.2).
7.5.10 The Role of the Steady-State Lateral Acceleration
As already stated in Sect. 7.4, it is common practice to employ \((\delta _v, \tilde{a}_y)\), instead of \((u,\delta _v)\), as parameters to characterize a steady-state condition. In the single track model some steady-state quantities are functions of \(\tilde{a}_y\) only.
The reason for such a fortunate coincidence in the case under examination is promptly explained. Just look at the equilibrium equations at steady state, with the inclusion of the constitutive equations (axle characteristics), that is for the single track model
They yield this result (already obtained in (7.9) and (7.57))
which can be more conveniently rewritten as
where \(Z_1^0\) and \(Z_2^0\) are the static vertical loads on each axle.
Therefore, if we take the monotone part of each axle characteristic, there is a one-to-one correspondence between \(\tilde{a}_y\) and the apparent slip angles at steady state (Fig. 7.24)
This is the key fact for using \(\tilde{a}_y\) as a parameter.
Both apparent slip angles \(\alpha _1\) and \(\alpha _2\) only “feel” the lateral acceleration, no matter if the vehicle has small u and large \(\delta _v\) or, vice versa, large u and small \(\delta _v\). In other words, the radius of the circular trajectory of the vehicle does not matter at all (in this model). Only \(\tilde{a}_y\) matters to the lateral forces and hence to the apparent slip angles. Actually, this very same property has been already used to build the axle characteristics. Equations (7.91) are just the inverse functions of (7.59).
We remark that (7.91) must not be taken as a general rule, but rather as a fortunate coincidence (it applies only to vehicles with two axles, open differential, no wings and parallel steering).
Another very important result comes directly from (7.90)
that is, at steady state, the lateral forces are always proportional to the corresponding static vertical loads. Therefore, the normalized axle characteristics
are what really matters in the vehicle dynamics of the single track model. The normalized axle characteristics are non-dimensional. Their maximum value is equal to the grip available in the lateral direction and is, therefore, a very relevant piece of information.
7.5.11 Slopes of the Axle Characteristics
It turns out that vehicle handling is pretty much affected by the slopes (derivatives) of the axle characteristics
Obviously, \(\varPhi _i>0\) in the monotone increasing part of the axle characteristics.
According to (7.91), in the single track model we have that the slopes \(\varPhi _i\) of the axle characteristics are functions of the lateral acceleration only
From (7.89)
and hence
7.6 Double Track, or Single Track?
Equations for the double track model and equations for the single track model are quite similar. Therefore, the effort for building a numerical model and running simulations is pretty much the same. Of course the double track model is more general. It does not require the assumptions of parallel steering and open differential. Moreover, it can deal with cases in which \(u \gg |r t_i|\) does not hold.
The single track model is less realistic, but simpler, and hence more predictable for a human being. Almost all the complexity boils down to the axle characteristics. As already mentioned, the single track model can provide in many cases useful insights into vehicle handling, particularly for educational purposes. But “many cases” does not mean “all cases”.
7.7 Steady-State Maps
We have already stated that the two functions (7.48) define all steady-state conditions of the double track model. However, the topic is so relevant to deserve additional attention and discussion.
From (7.46), (7.47), (7.68), (7.76) and (7.91) we have, at steady state, the following maps
A vehicle-road system has unique functions \(\rho _p(\delta _v,\tilde{a}_y)\) and \(\beta _p(\delta _v,\tilde{a}_y)\). As will be shown, they tell us a lot about the global vehicle steady-state behavior. In other words, these two maps fully characterize any steady-state condition of the vehicle. Of course, the r.h.s. part of (7.98) is strictly related to the single track model, and it is useful to the vehicle engineer to understand how to modify the vehicle behavior.
The two functions \(\rho _p(\delta _v,\tilde{a}_y)\) and \(\beta _p(\delta _v,\tilde{a}_y)\) can also be obtained experimentally [4], once a prototype vehicle is available, by performing some rather standard tests on a flat proving ground. With the vehicle driven at almost constant speed u and a slowly increasing steering wheel angle \(\delta _v\) (Slow Ramp Steer, often performed at 80 km/h), it suffices to measure the following quantities: \(r_p\), \(v_p\), u, \(\tilde{a}_y\) and \(\delta _v\). It is worth noting that none of these quantities does require to know whether the vehicle has two axles or more, or how long the wheelbase is. In other words, they are all well defined in any vehicle, including race cars.
A key feature, confirmed by tests on real road cars (with open differential and no wings), is that the \(\delta _v\)-dependence and the \(\tilde{a}_y\)-dependence are clearly separated.Footnote 7
As shown in Figs. 7.22 and 7.23, both maps in (7.98) are (in the single track model) linear with respect to the steering wheel angle \(\delta _v\), whereas they are nonlinear with respect to the steady-state lateral acceleration \(\tilde{a}_y\). The linear parts are totally under control, in the sense that both of them are simple functions of the steer gear ratios and of \(a_1\) and \(a_2\). The nonlinear parts are more challenging, coming directly from the interplay of the axle characteristics.
Figures 7.22 and 7.23, where \(\delta =\tau \delta _v\), anticipate the Map of Achievable Performance (MAP) approach, discussed in Sect. 7.8.
Curves at constant \(\rho \) (1/m) in the plane \((\delta ,\tilde{a}_y)\), for an understeer vehicle
Curves at constant \(\beta \) (deg) in the plane \((\delta ,\tilde{a}_y)\), for an understeer vehicle
7.7.1 Steady-State Gradients
It is informative, and hence quite useful, to define and compute/measure the gradients of the two maps \(\beta _p(\delta _v,\tilde{a}_y)\) and \(\rho _p(\delta _v,\tilde{a}_y)\), defined in (7.98)
As well known, gradients are vectors orthogonal to the level curves.
For the single track model, the explicit expressions of the components of the gradients \({{\,\textrm{grad}\,}}\rho _p\) and \({{\,\textrm{grad}\,}}\beta _p\) are as follows
where, to compute \(\beta _y\) and \(\rho _y\), we took into account (7.97).
It is worth noting that, for a given single track model of a vehicle, the two gradient components \(\beta _\delta \) and \(\rho _\delta \) are constant, whereas the other two gradient components \(\beta _y\) and \(\rho _y\) are functions of \(\tilde{a}_y\) only.
As will be discussed shortly, only one out of four gradient components is usually employed in classical vehicle dynamics,Footnote 8 thus missing a lot of information. But this is not the only case in which classical vehicle dynamics turns out to be far from systematic and rigorous. This lack of generality of classical vehicle dynamics is the motivation for some of the next sections.
7.7.2 Alternative Steady-State Gradients
Although not commonly done, we evaluate the gradients of the front and rear slip angles \(\beta _1(\delta _v,\tilde{a}_y)\) and \(\beta _2(\delta _v,\tilde{a}_y)\), which were defined in (7.78)
A fairly obvious result, but that can turn out to be useful in some cases.
7.7.3 Understeer and Oversteer
For further developments, it is convenient to rewrite (7.98) in a more compact form
where, in the single track model
The two known functions \(f_{\rho }(\tilde{a}_y)\) and \(f_{\beta }(\tilde{a}_y)\) are nonlinear functions, peculiar to a given road vehicle. They are called here slip functions.
Let us discuss this topic by means of a few examples.
First, let us consider the normalized axle characteristics (7.93) (multiplied by g) shown in Fig. 7.24(left). In this example, it has been assumed that both axles have the same lateral grip equal to 1. Moreover, to keep, for the moment, the analysis as simple as possible, we also assume that \(\hat{Y}_1(x) = \hat{Y}_2(kx)\), with \(k>0\). When inverted, they provide the apparent slip angles \(\alpha _1(\tilde{a}_y)\) and \(\alpha _2(\tilde{a}_y)\) shown in Fig. 7.24(right). Assuming a wheelbase \(l = 2.5\,{\text {m}}\), \(a_1 =1.125\,{\text {m}}\), and \(a_2 =1.375\,{\text {m}}\), we get from (7.103) the two slip functions \(f_{\rho }\) and \(f_{\beta }\) of Fig. 7.25.
In all figures, angles are in degree, accelerations in m/s\(^2\), and \(a_y\) should be read as \(\tilde{a}_y\).
Normalized axle characteristics (multiplied by g) of an understeer vehicle (left) and corresponding apparent slip angles (right)
Slip functions of an understeer vehicle
A vehicle with a monotone increasing slip function \(f_{\rho }(\tilde{a}_y)\), as in Fig. 7.25, is said to be an understeer vehicle. A more precise definition is given in (7.108).
As a second example, let us consider the normalized axle characteristics (multiplied by g) shown in Fig. 7.26(left). They are like in Fig. 7.24, but interchanged. When inverted, they provide the two functions \(\alpha _1(\tilde{a}_y)\) and \(\alpha _2(\tilde{a}_y)\) shown in Fig. 7.26(right). In this case the two slip functions \(f_{\rho }\) and \(f_{\beta }\) are as in Fig. 7.27.
Normalized axle characteristics (multiplied by g) of an oversteer vehicle (left) and corresponding apparent slip angles (right)
Slip functions of an oversteer vehicle
A vehicle with a monotone decreasing function \(f_{\rho }(\tilde{a}_y)\), as in Fig. 7.27, is said to be an oversteer vehicle. A more precise definition is given in (7.110).
7.7.4 Handling Diagram
Usually, only the function \(f_{\rho }(\tilde{a}_y)\) is considered in classical vehicle dynamics, while \(f_{\beta }(\tilde{a}_y)\) is neglected.
Since, at steady state, \(\rho _p=\tilde{a}_y/u^2 = r/u = 1/R\), the first equation in (7.102) becomes
which, for given u and \(\delta _v\), is an equation for the unknown \(\tilde{a}_y=\tilde{a}_y(u,\delta _v)\). See also (7.46).
Another, most classical, way to recast (7.104) is
where
is the net steer angle, already defined in (7.69).
It is customary [13,14,15] to rewrite (7.104) as a system of two equations
Solving this system amounts to obtaining the values of \((\tilde{a}_y, f_{\rho })\) attained under the imposed operating conditions \((u,\delta _v)\). Geometrically, that can be seen as the intersection between a straight line (i.e., the first equation in (7.107)) and the so-called handling curve \(y= f_{\rho }(\tilde{a}_y)\) (i.e., the second equation in (7.107)).
Handling diagram of an understeer vehicle (y in deg/m)
Handling diagram of an oversteer vehicle (y in deg/m)
Together, the handling curve and the straight lines form the celebrated handling diagram [13,14,15]. Examples are shown in Figs. 7.28 and 7.29 (where \(a_y\) is indeed \(\tilde{a}_y\) and y is in deg/m).
The handling curve \(y = f_{\rho }(\tilde{a}_y)\) is peculiar to each vehicle-road system (in the single track model it depends on the normalized axle characteristics only). Therefore, for a given vehicle-road system it has to be drawn once and for all.
On the other hand, the straight line depends on the selected operating conditions \((u,\delta _v)\). For instance, in Fig. 7.28 the two intersecting lines correspond to two operating conditions with the same value of \(\delta _v\), while the two parallel lines share the same value of u.
Perhaps, the best way to understand the handling diagram (Figs. 7.28 and 7.29) is by assuming that the steering wheel angle \(\delta _v\) is kept constant, while the forward speed u is (slowly) increased.
In Fig. 7.28, an increasing u, with constant \(\delta _v\), results also in an increasing y. Therefore, from (7.105) with constant \(\delta _v\) (and hence constant \(\delta \)), the higher the forward speed u, the larger the radius R of the trajectory of the vehicle. This is called understeer behavior. More precisely, we have understeer whenever
where, in the single track model
On the contrary, if the handling curve is, e.g., like in Fig. 7.29, the higher the forward speed u, with constant \(\delta _v\), the smaller the radius R. This is called oversteer behavior. More precisely, we have oversteer whenever
Actually, when the straight line becomes tangent to the handling curve, as shown in Fig. 7.29, the vehicle becomes unstable. It means that the vehicle has reached the critical speed associated to that value of \(\delta _v\). The concept of critical speed will be discussed in Sect. 7.13 in a more general framework.
A less classical, but maybe more interesting formula is (see (7.84))
The relevance of this result is that it combines a very weak term \(\delta - (\alpha _1-\alpha _2)\) with a very robust term \(\beta _1 - \beta _2\). The first one is not well defined in real vehicles, whereas the last one is. Understeer/oversteer should be defined and evaluated using the robust term.
Vehicles with aerodynamic devices and/or limited-slip differential do not exhibit a handling curve [13, p. 172], but a handling surface [7], instead. More precisely, (7.105) still holds true, but with \(f_{\rho }(\tilde{a}_y,1/R)\). Therefore, definitions (7.108) and (7.110) of understeer/ovesteer become meaningless. This topic is addressed in Sect. 8.5.2.
Classical vehicle dynamics stops about here. In the next section a fresh, more comprehensive, global approach is developed. It brings new insights into the global steady-state behavior of real vehicles, along with some new hints about the transient behavior.
7.8 Map of Achievable Performance (MAP)
The handling diagram [13,14,15], although noteworthy, does not provide a complete picture of the handling behavior. Just consider that the use of \(\tilde{a}_y\) as input variable, that is one variable instead of two, hides some features of the vehicle handling behavior.
It would be better to have a more general approach, able to unveil at a glance the overall steady-state features of the vehicle under investigation, thus making it easier to distinguish between a “good” vehicle and a “not-so-good” one.
As already stated in (7.44), the steady-state handling behavior is completely described by the handling maps
where the last terms are peculiar to the single track model.
In the single track model, it is convenient to define the net steer angle \(\delta \), as already done in (7.69) and in (7.106)
Usually, \(\kappa =0\) and hence \(\delta =\delta _1\) is just the steering angle of the front wheel. However, \(\kappa \ne 0\) leaves room for direct rear steering as well. In general,
With this notation, the handling maps (7.112) become
where, for the sake of compactness, we dropped the subscript p.
These two maps fully characterize the steady-state behavior of the vehicle. This is a fairly general point of view that leads to the global approach presented in Chap. 6, that we called Map of Achievable Performance (MAP).
Actually, under the acronym MAP we will present several types of possible graphical representations of the handling maps, each one on the corresponding achievable region. This is another key concept.
Figures in this section are for road cars with the following features: mass \(m = 2000\,{\text {kg}}\), wheelbase \(l =2.5\,{\text {m}}\), \(a_1 =1.125\,{\text {m}}\), \(a_2 =1.375\,{\text {m}}\), grip coefficient \(\mu =1\), maximum speed \(u_\text {max}= 40\,{\text {m}}\big /{\text {s}}\), maximum steer angle of the front wheels \(\delta _\text {max}= 15^{\circ }\). The understeer version has normalized axle characteristics as in Fig. 7.24. The oversteer version has normalized axle characteristics as in Fig. 7.26. In all figures, angles are in degree, accelerations in m/s\(^2\), and \(\rho \) in m\(^{-1}\).
Achievable input region (left) and achievable output region (right) for an understeer vehicle
7.8.1 MAP Fundamentals
The main idea behind the MAP approach is simple: the driver controls \((u,\delta )\), the vehicle reacts with \((\rho ,\beta )\). That is
The input values \((u,\delta )\) that a given vehicle can really achieve are subject to three limitations:
-
maximum steer angle \(\delta _\text {max}\);
-
maximum speed \(u_\text {max}\), or critical speed \(u_\text {cr}\), if \(u_\text {cr}<u_\text {max}\);
-
maximum lateral acceleration (grip limited).
This is shown in Fig. 6.1 (left). Each achievable point \((\delta ,u)\) results in the vehicle performing with precise values \((\rho ,\beta )\). Therefore, the achievable input region of Fig. 6.1 (left) is mapped onto the achievable output region shown in Fig. 6.1 (right).
Lines at constant lateral acceleration \(\tilde{a}_y\) for an understeer vehicle
Quite interesting are the MAPs (Maps of Achievable Performance) that can be drawn inside these achievable regions. For instance, curves at constant \(\tilde{a}_y\) are drawn on both regions in Fig. 7.31. In an understeer vehicle without significant aerodynamic vertical loads, the grip-limited bound is just the curve at constant \(\tilde{a}_y=\mu g\).
While the yaw rate \(r_p\) has typically the same sign as \(\delta \), the same does not apply to the lateral speed \(v_p\). As shown in Fig. 7.32, in a left turn the vehicle slip angle \(\beta =v_p/u\) can either be positive or negative. As a rule of thumb, at low forward speed the vehicle goes around “nose-out” (\(\beta > 0\)), whereas at high speed the vehicle goes around “nose-in” (\(\beta < 0\)). This statement can be made quantitative by drawing the curves at constant \(\beta \) on the achievable input region, as shown in Fig. 7.33. The almost horizontal line \(\beta =0\) clearly splits the region into a lower part with \(\beta >0\), and an upper part with \(\beta <0\).
Steady-state behavior: a nose-out (low speed), b nose-in (high speed)
u-\(\delta \) MAP with curves at constant vehicle slip angle \(\beta \) for an understeer vehicle
Drawing curves at constant curvature \(\rho \) also highlights the overall understeer/oversteer behavior of a vehicle. For instance, it is quite obvious that the pattern of Fig. 7.34 is typical of an understeer vehicle: the faster you go, the more you have to steer to keep \(\rho \) constant. Moreover, it is worth comparing Fig. 7.22, which is the contour plot of \(\rho (\delta ,\tilde{a}_y)\), and Fig. 7.34, which is the contour plot of \(\rho (u,\delta )\). For instance, the first MAP is linear with respect to \(\delta \), whereas the second one is not. The reason is that the other independent variable is different: linear behavior with respect to \(\delta \) requires constant lateral acceleration \(\tilde{a}_y\), not constant forward speed u.
u-\(\delta \) MAP with curves at constant curvature \(\rho \) for an understeer vehicle
\(\rho \)-\(\beta \) MAP with curves at constant u and lines at constant \(\delta \) for an understeer vehicle
Curves at constant speed u, and also lines at constant steer angle \(\delta \), are shown in Fig. 7.35 for an understeer vehicle. As expected, moving top to bottom along each line at constant steer angle, that is with increasing speed, brings smaller values of the curvature \(\rho \). Also interesting is to observe that at low speed the slip angle \(\beta \) grows with \(\delta \), whereas at high speed it is the other way around. The same phenomena can be observed more clearly in Fig. 7.33.
These MAPs can be obtained experimentally or through simulations. Therefore, they are not limited to the single track model. Actually, as will be discussed in the next chapter, they exist also for race cars, including cars with very high aerodynamic downforces.
The effects of rear steering (in addition to front steering, of course) are shown in Fig. 7.36. The picture on the left is for the case of rear wheels turning opposite of the front wheels with \(\delta _2=-0.1 \delta _1\), whereas the picture on the right is for rear wheels turning like the front wheels, with \(\delta _2= 0.1 \delta _1\). The vehicle slip angle \(\beta \) is affected pretty much. Basically, a positive \(\chi = \delta _2/\delta _1\) moves the achievable region upwards, and vice versa. On the other hand, rear steering does not impinge on the achievable region in the plane \((\delta ,\rho )\), as will be discussed in Sect. 7.8.2.
Effects of rear steering on the achievable region: rear wheels turning opposite of the front wheels (left), rear wheels turning like the front wheels (right)
\(\rho \)-\(\beta \) MAP for a vehicle with rear wheels turning opposite of the front wheels at low speed and like the front wheels at high speed
Example of the effects of rear steering on \(\beta \): front steering only (top); front and rear steering (bottom). All cases have the same \(\tilde{a}_y\), and hence the same \(\alpha _1\) and \(\alpha _2\)
Vehicles behave in a better way if the vehicle slip angle \(\beta \) spans a small range. To have a narrower achievable output region in the plane \((\rho ,\beta )\) we have to move down the upper part and move up the lower part. This is indeed the effect of a steering system with rear wheels turning opposite of the front wheels at low speed, and turning like the front wheels at high speed. That is a steering system (7.113) with, e.g., \(\kappa (u) = -\kappa _0 \cos (\uppi u/u_{\text {max}})\). The net result can be appreciated by comparing Fig. 7.37 with Fig. 7.35. The MAP approach provides a better insight into rear steering effects than by looking at, e.g., Fig. 7.38.
Oversteer vehicle: u-\(\delta \) MAPs with curves at constant \(\rho \) (both), constant \(\tilde{a}_y\) (left) and constant \(\beta \) (right)
Oversteer vehicle: \(\rho \)-\(\beta \) MAP with curves at constant speed u and lines at constant steer angle \(\delta \)
Vehicle with too much understeer: \(\rho \)-\(\beta \) MAP with lines at constant u, \(\tilde{a}_y\) and \(\delta \)
The achievable region in case of an oversteer vehicle is limited by the critical speeds, not by grip. A typical achievable input region, with noteworthy lines, is shown in Fig. 7.39.
The achievable region in the plane \((\rho ,\beta )\) for an oversteer vehicle is shown in Fig. 7.40, along with curves at constant speed u and lines at constant steer angle \(\delta \). As expected, moving top to bottom along the lines at constant steer angles, that is with increasing speed, entails larger values of the curvature \(\rho \).
Very instructive is the comparison between Figs. 7.35 and 7.40, that is between an understeer and an oversteer vehicle. The two achievable regions have different shapes also because an oversteer vehicle becomes unstable for certain combinations of speed and steer angle. These critical combinations form a sort of stability boundary which collects all points where the u-curves and \(\delta \)-lines are tangent to each other, as shown in Fig. 7.40.
On the opposite side, a vehicle with too much understeer has an achievable region like in Fig. 7.41 (see also Fig. 7.45 for a more intuitive MAP).
7.8.2 MAP Curvature \(\rho \) Versus Steer Angle \(\delta \)
A central issue in vehicle dynamics is how a vehicle responds to the driver input commands (namely, the steering wheel angle \(\delta _v\) and the forward speed u). Well, let us map it. The plane \((\delta ,\rho )\) suits the purpose in a fairly intuitive and quantitative way.
Constant speed lines on the \(\rho \)-\(\delta \) MAP for an understeer vehicle
Constant lateral acceleration lines on the \(\rho \)-\(\delta \) MAP for an understeer vehicle
\(\rho \)-\(\delta \) MAP for an understeer vehicle
Let us consider again a vehicle with the front and rear normalized axle characteristics (multiplied by g) shown in Fig. 7.24.Footnote 9 We recall that it is an understeer vehicle and that the corresponding slip functions and handling diagram are shown in Fig. 7.25 and Fig. 7.28, respectively.
If we draw the lines at constant speed u in the plane \((\delta ,\rho )\), we get the plot shown in Fig. 7.42, if \(\rho \ge 0\). In the same achievable region, we can draw the lines at constant lateral acceleration \(\tilde{a}_y\), as shown in Fig. 7.43. According to (7.98), they are parallel straight lines. In Fig. 7.44, both lines at constant u and constant \(\tilde{a}_y\) are drawn on the whole achievable region.
The achievable region is bounded by:
-
1.
maximum speed (dashed line in Fig. 7.43);
-
2.
maximum lateral acceleration (dashed line in Fig. 7.42);
-
3.
zero lateral acceleration;
-
4.
maximum steer angle.
We see that the driver must act on both u and \(\delta \) to control the vehicle, that is to drive it on a curve with curvature \(\rho \) and lateral acceleration \(\tilde{a}_y\). But, the key feature is that it can be done fairly easily because the lines at constant speed are “well shaped”, that is quite far apart from each other and neither too flat, nor too steep (Fig. 7.42).
In Fig. 7.44, all lines at constant speed intersect all lines at constant lateral acceleration. This is typical of all vehicles without significant aerodynamic vertical loads. This is another piece of information that is provided by this kind of maps on the achievable region.
\(\rho \)-\(\delta \) MAP for a vehicle with too much understeer
An example of a not-so-nice achievable region is shown in Fig. 7.45. A vehicle with a map like in Fig. 7.45 shows too much understeer: the lines at high speed are too flat, showing that the driver can increase \(\delta \) without getting a significant increase in \(\rho \). Not a desirable behavior.
Another example of undesirable behavior, but for opposite reasons, is shown in Fig. 7.46. This is a vehicle with too little understeer. It has a very narrow achievable region, which means that the driver has a very heavy task in controlling the vehicle: the lines at zero and maximum lateral acceleration are very close together.
An oversteer vehicle (whose corresponding slip functions and handling diagram are shown in Fig. 7.27 and Fig. 7.29, respectively) has an achievable region as in Fig. 7.47. The lines at constant \(\tilde{a}_y\), shown in Fig. 7.47, are quite far apart like in Fig. 7.43, but the lines at constant speed u are very badly shaped. At high speed they are too steep, meaning that a small variation of \(\delta \) drastically changes \(\rho \) and \(\tilde{a}_y\).
Moreover, the vehicle becomes unstable when the u-lines have vertical slope. Accordingly, the truly achievable region becomes smaller, as shown in Fig. 7.48, where the truly achievable region is bounded by the stability boundary (long-dashed line).
All these examples show how the map curvature versus steer angle provides a very clear and global picture of the vehicle handling behavior. It makes clear why a well tuned vehicle must be moderately understeer. Too much or too little understeer are not desirable because the vehicle becomes much more difficult to drive (for opposite reasons).
Constant lateral acceleration lines on the \(\rho \)-\(\delta \) MAP for a vehicle with too little understeer
Apparent achievable region on the \(\rho \)-\(\delta \) MAP for an oversteer vehicle
Constant speed lines and truly achievable region on the \(\rho \)-\(\delta \) MAP for an oversteer vehicle
The difference between understeer and oversteer is laid bare (Figs. 7.44 and 7.47). Both have far apart \(\tilde{a}_y\)-lines, but covering achievable regions on opposite sides. In fact, the u-lines are totally different.
The more one observes these handling MAPs on the corresponding achievable regions, the more the global handling behavior becomes clear.
7.8.3 Other Possible MAPs
So far we have discussed the fundamental MAPs \((\delta ,u)\) and \((\rho ,\beta )\), and also the fairly intuitive, and very useful, MAP \((\rho ,\delta )\).
Of course, several other MAPs are possible. For instance, in Fig. 7.49, curves at constant \(\delta \) are drawn in the planes \((\rho ,u)\) and \((\beta ,u)\) for an understeer vehicle. The same kind of MAPs, but for an oversteer vehicle, are shown in Fig. 7.50. The onset of instability is clearly indicated, e.g., by the vertical tangent of the curves at constant \(\delta \) in the plane \((\beta ,u)\).
Moreover, the MAP \((\delta ,\tilde{a}_y)\) was introduced in Sect. 7.7 and is extensively employed in Sect. 7.10.
Constant steer curves for an understeer vehicle
Constant steer curves for an oversteer vehicle
7.9 Weak Concepts in Classical Vehicle Dynamics
Some “fundamental” concepts in vehicle dynamics are indeed very weak if addressed with open mind. They are either not well defined, particularly when we look at real vehicles, or they are commonly defined in an unsatisfactory way. This is a serious practical drawback that can lead to wrong results and conclusions.
7.9.1 The Understeer Gradient
According to the SAE J266 Standard, Steady-State Directional Control Test Procedures For Passenger Cars and Light Trucks
understeer/oversteer gradient K is defined as the difference between steer angle gradient and Ackermann steer angle gradient.
This definition of K is equivalent to the following formula
which comes directly from (7.105). See also (7.109).
Therefore, to compute/measure K we need both the net steer angle \(\delta \) and the Ackermann steer angle l/R. Unfortunately, none of them is clearly defined in a real vehicle. In fact, they are well defined only in the single track model, as it is done, e.g., in Figure A1 in the SAE J266 Standard.
In a real vehicle, the two front wheels have typically different steer angles (Fig. 7.51). Therefore, the net steer angle \(\delta \) is not precisely defined.
The Ackermann steer angle l/R also gets in trouble whenever a vehicle has three or more axles, as the wheelbase l is no longer a clear concept (Fig. 7.51). One may object that almost all cars have two axles. Nonetheless, we cannot ground a theory on such a weak concept.
Case not covered by the classical theory
The understeer gradient K has been an important performance metric in analyzing the handling behavior of vehicles. Unfortunately, it should not have been. It will be demonstrated in Sect. 7.14.1 that it is not a good parameter to measure the handling behavior of a vehicle. Nor even of a single track model. A much better parameter is \(\rho _y\), discussed in Sect. 7.13.
7.9.2 Popular Definitions of Understeer/Oversteer
Perhaps, the most astonishing case of use of unclear concepts is the popular way to “define” understeer and oversteer:
Oversteer is what occurs when a car steers by more than the amount commanded by the driver. Conversely, understeer is what occurs when a car steers less than the amount commanded by the driver.Footnote 10
Understeer: a tendency of an automobile to turn less sharply than the driver intends (or would expect).
The term understeer means that you have to give your car more steering input than the corner should require to get it to go around.
What is the “amount commanded by the driver”? What is the scientific, quantitative meaning of what “the driver intends”? What does “than the corner should require” mean?
Figure 7.52 exemplifies this paradoxical situation. Three different curves, three identical trajectories, only one is fine in each case. What about the understeer/oversteer behavior of the car? What did the driver intend to do?
What did the driver intend to do?
7.10 Double Track Model in Transient Conditions
Steady-state analysis cannot be the whole story. Indeed, a vehicle is quite often in transient conditions, that is with time-varying quantities (forces, speeds, yaw rate, etc.). Addressing the transient behavior is, of course, more difficult than “simply” analyzing the steady state. More precisely, the steady-state conditions (also called trim conditions) are just the equilibrium points from which a transient behavior can start or can end.
The general way to study the transient behavior of any dynamical system is through in-time simulations. However, this approach has some drawbacks. Even after a large number of simulations it is quite hard to predict beforehand what the outcome of the next simulation will be.
One way to simplify the analysis of a non-linear dynamical system is to consider only small perturbations (oscillations) about steady-state (trim) conditions. This idea leads to the approach based on stability derivatives and control derivatives (as they are called in aerospace engineering [12, p. 151]).
The nonlinear equations of motion of the double track model of the vehicle are (cf. (7.38))
We prefer to use \((\rho ,\beta )\) as state variables, instead of (v, r), because they provide a more “geometric” description of the vehicle motion. Since \(\beta =v/u\) and \(\rho =r/u\), it is pretty much like having normalized with respect to the forward speed u.
7.10.1 Equilibrium Points
At steady state we have, by definition, \(\dot{v}=\dot{r}=0\), that is \(\dot{\beta }=\dot{\rho }=0\). The driver has direct control on u and \(\delta _v\), which are kept constant and whose trim values are named \(u_a\) and \(\delta _{va}\). The subscript a is introduced here to distinguish clearly between the generic and the trim values (i.e., assigned values).
The equations of motion (7.118) become
which can be solved to get the steady-state maps (exactly like in (7.44) or (7.112))
These maps have been thoroughly discussed in Sect. 7.8, where the new concept of MAP (Map of Achievable Performance) was introduced.
Actually, when applying the MAP approach to the vehicle transient behavior it is more convenient to do like in (7.98), that is to use \(\tilde{a}_y= u_a r_p(u_a,\delta _{va})\), which provides (exactly like in (7.46))
and hence
An example of achievable region in \((\delta ,\tilde{a}_y)\) is shown in Fig. 7.53 for an understeer vehicle, along with lines at constant \(\beta \) (left) and constant \(\rho \) (right). In a real vehicle, these maps can be obtained by means of classical steady-state tests. Therefore, they do not require departing from the traditional way of vehicle testing.
MAPs in the plane \((\delta ,\tilde{a}_y)\) with curves at constant \(\beta \) (left) and constant \(\rho \) (right) for an understeer vehicle
7.10.2 Free Oscillations (No Driver Action)
The basic idea is to linearize around an equilibrium point to obtain information in its neighborhood about the dynamical behavior. It is a standard approach for almost any kind of nonlinear dynamical systems.
Assuming that the driver takes no action (i.e., both \(u=u_a\) and \(\delta _v=\delta _{va}\) are constant in time), the first-order Taylor series expansion of the equations of motion (7.118) around the equilibrium point (7.120) are as follows
where
The stability derivatives \(Y_\beta \), \(Y_\rho \), \(N_\beta \) and \(N_\rho \) are simply the partial derivatives
all evaluated at the equilibrium (trim) conditions \((\beta _p,\rho _p; u_a,\delta _{va})\). Like Y and N, each stability derivative depends on the whole set of chosen coordinates. When evaluated at an equilibrium point, they depend ultimately on the two input coordinates.
According to Maxwell’s Reciprocal Theorem
and hence there are only three independent stability derivatives. See Sect. 7.14 for an example.
It is convenient to introduce the shifted coordinates
into the linearized system of Eqs. (7.123), thus getting
where \(\dot{\beta }= \dot{\beta _t}\) and \(\dot{\rho }= \dot{\rho _t}\). The shifted coordinates are just the distance of the current values from the selected trim values.
The same system of two first-order linear differential equations with constant coefficients can be rewritten in matrix notation as
where the matrix \(\textbf{A}\) is not time dependent.
As a possible further analytical step, we can reformulate the problem as two identical second order linear differential equations, with constant coefficients, one in \(\rho _t(t)\) and the other in \(\beta _t(t)\) (see Sect. 7.18.6 for details)
where
The solutions of (7.129) depend on two initial conditions, i.e. \(\beta _t(0)\) and \(\rho _t(0)\). From the system of equations (7.128) we get \(\dot{\beta }(0)\) and \(\dot{\rho }(0)\), which are the two additional initial conditions needed in (7.130). Therefore, the two state variables have identical dynamic behavior (i.e., same \(\zeta \) and \(\omega _n\)) and are not independent from each other.
From (7.130) and (7.131) we see that the vehicle behaves as a mechanical vibrating system with
The derivatives \(Y_\beta \) and \(N_\rho \) are always negative and act as viscous dampers. Apparently, the quantity \(Y_\beta N_\rho - Y_\rho N_\beta \) is always positive (see (7.194)). The derivatives \(Y_\rho = N_\beta \) can be positive or negative. Understeer vehicles have \(N_\beta > 0\), oversteer vehicles have \(N_\beta < 0\). It is important to understand the physical significance of each stability derivative [12, p. 151].
The matrix \(\textbf{A}\) in (7.129) has two eigenvalues
From (7.131) we can obtain the damping ratio \(\zeta \)
If \(\zeta <1\), the two eigenvalues are complex conjugate
and the system has a damped oscillation with natural angular frequency \(\omega _s\)
It is kind of interesting to observe that all these relevant dynamic parameters \(\zeta \), \(\omega _n\) and \(\omega _s\) depend on the following four quantities
Real and imaginary parts of the two eigenvalues (7.135), for \(\delta =0\)
Real and imaginary parts of the two eigenvalues (7.135), for \(\delta = 5^{\circ }\)
Of course, the eigenvalues depend on \((u_a,\delta _{va})\), as shown in Figs. 7.54 and 7.55 for (the single track model of) an understeer vehicle. In these figures, the real part (gray lines) and the imaginary parts (black lines) are plotted as functions of the forward speed \(u_a\). In Fig. 7.54 the car is going straight, that is with \(\delta =0\). In Fig. 7.55 the car has a net steer angle \(\delta =5^{\circ }\) (defined in (7.68)). In both cases, the eigenvalues are complex conjugate for speeds higher than about \(4\,{\text {m}}\big /{\text {s}}\).
Interestingly enough, when the car goes straight (Fig. 7.54), the real part \(-\zeta \omega _n\) and the imaginary part \(\omega _s\) are almost constant for \(u_a > 25\,{\text {m}}\big /{\text {s}}\), that is for about \(u_a >90\,{\text {km}}\big /{\text {h}}\). Indeed, it is at \(u_a \simeq 100\,{\text {km}}\big /{\text {h}}\) that car makers typically perform the steering harmonic sweep test, in which the steer input is a harmonic function but with a slowly increasing frequency.
As expected, Fig. 7.55 is almost like Fig. 7.54 for low speeds, say \(u_a < 10\,{\text {m}}\big /{\text {s}}\). For higher speeds, the two figures are very different. The maximum speed is limited by grip when a vehicle is making a turn.
A clearer picture of the global dynamical features of the vehicle is provided by the MAP approach (7.122) when applied to the damping ratio \(\zeta (\delta ,\tilde{a}_y)\) and to the damped natural frequency \(\omega _s(\delta ,\tilde{a}_y)\), as in Fig. 7.56. It immediately arises that the closer the vehicle is to the grip limit (maximum lateral acceleration), the lower both \(\zeta \) and \(\omega _s\). Therefore, the dynamical behavior of the vehicle changes significantly. Perhaps, an expert driver may take advantage of these phenomena to “feel” how close the vehicle is to the grip limit.
MAP in the plane \((\delta ,\tilde{a}_y)\) with curves at constant damping ratio \(\zeta \) (dashed lines) and constant damped natural angular frequency \(\omega _s\) (solid lines)
Summing up, we have seen that the dynamical features of the vehicle in the neighborhood of an equilibrium point depend on the four stability derivatives (7.125), besides m, \(J_z\) and \(u_a\). Actually, we know that \(Y_\rho =N_\beta \), and hence there are only three independent stability derivatives. See also Sect. 7.14.
The characterization of the vehicle requires knowledge of these stability derivatives.
7.10.3 Stability of the Equilibrium
An equilibrium point can be either stable or unstable. The typical way to assess whether there is stability or not is by looking at the eigenvalues (7.133). As well known
that is, both eigenvalues must have a negative real part. A convenient way to check this condition without computing the two eigenvalues is
Typically, vehicles may become unstable because one of the two real eigenvalues becomes positive. From (7.131), the mathematical condition is
As already mentioned, instability may occur only if \(N_\beta < 0\).
7.10.4 Forced Oscillations (Driver Action)
Linearized systems can also be used to study the effect of small driver actions on the forward speed and/or on the steering wheel angle to control the vehicle. More precisely, we have \(u = u_a + u_t\) and \(\delta _v=\delta _{va}+ \delta _{vt}\).
The linearized inertial terms in (7.118) are
where \(m u_a^2 \rho _p = Y_0\), according to (7.119).
The linearized system becomes
where, in addition to the four stability derivatives (7.125), there are also four control derivatives
evaluated, like the others, at the equilibrium point \((\beta _p,\rho _p; u_a,\delta _{va})\). A better way to write (7.142) is
which generalizes (7.128).
The most intuitive case is the driver acting only on the steering wheel, which is described by the simplified set of equations
since \(u_t=\dot{u}=0\). Moreover, \(\dot{u}=0\) is consistent with the assumptions made at the beginning of this chapter.
In matrix notation, (7.144) become
or, in an even more compact notation
where the entries of matrix \(\textbf{A}\) are, exactly as in (7.129)
and the components of vector \(\textbf{b}\) are
Like in (7.130), we can recast the problem (7.144) as two second-order linear differential equations, only apparently independent from each other
where
Again, if the driver acts only on the steering wheel, like in (7.145), all these expressions become much simpler. More precisely
and hence
The two differential equations (7.150) have identical values of \(\zeta \) and \(\omega _n\), but different forcing terms \(F_\beta \) and \(F_\rho \). However, in (7.153) we still find the four quantities listed in (7.137).
The fundamental result of this analysis is that the transient dynamics of a vehicle in the neighborhood of an equilibrium point is fully characterized by a finite number of normalized stability derivatives and control derivatives:
-
normalized stability derivatives \(Y_\beta /m\), \(Y_\rho /m\), \(N_\beta /J_z\), and \(N_\rho /J_z\);
-
normalized control derivatives \(Y_u/m\), \(Y_\delta /m\), \(N_u/J_z\), and \(N_\delta /J_z\).
It will be discussed shortly that in most cases \(Y_u=N_u=0\), thus leaving six derivatives. It is worth noting that the equality \(Y_\rho = N_\beta \) does not reduce the number of relevant derivatives to five. Indeed, we still have \(Y_\rho /m \ne N_\beta /J_z\).
The key point is how to measure (identify) all the stability derivatives and all the control derivatives. Their knowledge would be very relevant practical information. The next section presents indeed a novel method to extract these data from the results of steady-state tests. This approach appears to be simpler and more reliable than direct measurements.
7.11 Relationship Between Steady-State Data and Transient Behavior
Most classical vehicle dynamics deals with steady-state data. Understeer and oversteer are steady-state concepts. Or they are not? This is a crucial question. What does a professional driver mean when he/she complains about his/her car being understeer or oversteer? Does it have anything to do with the classical definition of understeer/oversteer as discussed in Sect. 7.7?
Two aspects should be carefully taken into account. While the concepts of velocity, acceleration, mass, stability etc. arise in any branch of mechanics, why do the concepts of understeer and oversteer only belong to vehicle dynamics? This is rather surprising. Why are vehicles so special dynamical systems that they need concepts conceived uniquely for them?
The other aspect is somehow more practical. Why should steady-state tests tell us anything about the transient behavior of a vehicle? In more technical terms, why should steady-state data be related to stability derivatives? Are they or not? If they are related, what is the relationship? Indeed, in [3] it is admitted “Transient responses are related to understeer to some extent, but there is no one-to-one relationship between steady-state and transient response”.
This section is devoted to the investigation of the link between the universe of steady-state data and the universe of the dynamical, hence transient, behavior of a vehicle. It will be shown that a link does indeed exist, but it is not direct, not to mention obvious.
It is worth noting that this section is not strictly related to the single track model. The theory developed here is applicable to real road vehicles.
7.11.1 Stability Derivatives from Steady-State Gradients
The starting point is a sort of mathematical trick. At steady state, the lateral force Y and the yawing moment N have very simple values
Nevertheless, by combining (7.119), (7.121), and (7.122), they can be given, as functions, the following expressions
Now, the key idea is to take the partial derivatives of the just defined function \(Y_0(\delta _{va},\tilde{a}_y)\), thus obtaining
The same steps can be taken for the yawing moment \(N_0(\delta _{va},\tilde{a}_y)\), getting
In a road vehicle, that is without significant aerodynamic vertical loads, it is reasonable to assume
if we take \(\beta \) and \(\rho \) as state variables to describe the vehicle motion.Footnote 11 In other words, Y and N do not change if we modify only u, keeping constant \(\beta \), \(\rho \) and \(\delta _v\), that is keeping constant \(\alpha _1\) and \(\alpha _2\) (cf. (7.51)). It would not be so in Formula cars, that is in cars with aerodynamic devices.
The two equations in (7.156), with \(Y_u=N_u=0\), yield the system of linear equations
and, similarly, from (7.157)
These two systems of equations
have the same matrix, whose coefficients are the four components of the gradients defined in (7.99)
of the two steady-state maps (7.122). After having performed the standard steady-state tests, all these gradient components (already introduced in Sect. 7.7.1) are known functions.
The four stability derivatives are the solution of the two systems of equations (7.161)
Therefore, they are known functions of the gradient components and of the control derivatives \(Y_\delta \) and \(N_\delta \). This is a fundamental original result, as it shows why steady-state data can indeed provide information about the transient behavior, although not in an obvious way.
Moreover, from (7.126) (i.e., \(Y_\rho = N_\beta \)) and (7.162) we have that
The transient behavior of the vehicle is characterized by the stability derivatives. This is well known. What is new is that the stability derivatives are strictly related to the gradients of steady-state maps. This result opens up new perspectives in the objective evaluation of the handling of vehicles (cf. [10]).
7.11.2 Equations of Motion
Now, we can go back to the linearized equations of motion (7.145). The stability derivatives can be replaced by the expressions in (7.162), thus obtaining
where \(\beta _t\) and \(\rho _t\) are the shifted coordinates defined in (7.127).
In some cases it is convenient to define and use the generalized control derivatives
thus obtaining
This is quite a remarkable (and original) result. It shows how the equations of motion can be given in terms of data collected in steady-state tests. It is the link between the realm of steady-state gradients and the realm of transient behavior.
7.11.3 Estimation of the Control Derivatives
The control derivatives \(\hat{Y}_\delta \) and \(\hat{N}_\delta \) can be estimated by means of standard dynamic tests. For instance, let us consider a generalized step steering input, that is a sudden increase \(\delta _{vt}\) of the steering wheel angle \(\delta _v\) applied to a vehicle in a steady-state (equilibrium) configuration. We say “generalized” since it should and can be done from any steady-state configuration, not necessarily from a straight-line trajectory. Since, by definition \(\beta _t(0)=0\) and \(\rho _t(0)=0\), from (7.166) we obtain
Combining this result with (7.163), we also get that in a step steering input
7.11.4 Objective Evaluation of Car Handling
The two coefficients \(2\zeta \omega _n = -(\lambda _1+\lambda _2)\) and \(\omega _n^2=\lambda _1\lambda _2\) of the differential equations (7.130), can now be expressed as combinations of steady-state gradient components and control derivatives
or, equivalently
Exactly like in (7.132), we have the physical interpretation as a vibrating system
Once again, the dynamic features of the vehicle are strictly related to the gradients of data obtained in steady-state tests.
From (7.169), we see that the vehicle becomes unstable (\(\det (\textbf{A})<0\)) if
which requires \(\rho _y >0\) (oversteer). This condition is completely equivalent to (7.140) (see also Sect. 7.13). From (7.162), we obtain that \(N_\beta \) and \(\rho _y\) have opposite signs
since \(N_\delta > 0\) and \((\beta _y \rho _\delta - \beta _\delta \rho _y) < 0\).
Following the same path of reasoning, the two forcing terms \(F_\beta \) and \(F_\rho \) in (7.153) can be rewritten as
and
Typical patterns are shown in the MAP in Fig. 7.57 for \(F_\beta \), and in the MAP of Fig. 7.58 for \(F_\rho \).
MAP in the plane \((\delta ,\tilde{a}_y)\) for \(F_\beta \), with curves at constant \(n_3\) (left) and constant \(n_4\) (right)
MAP in the plane \((\delta ,\tilde{a}_y)\) for \(F_\rho \), with curves at constant \(n_5\) (left) and constant \(n_6\) (right)
7.11.4.1 Vehicle “DNA”
Equations (7.169), (7.174) and (7.175) show that the dynamical behavior of a road vehicle in the neighborhood of any equilibrium point is fully described by six maps \(n_i(\delta _{va},\tilde{a}_y)\). These maps (functions) can be seen as a sort of “DNA” of the vehicle, in the sense that they determine the vehicle transient behavior. To help the reader, these six maps are listed below:
However, all these quantities are, ultimately, combinations of the following six fundamental “handling bricks”:
all of them, in general, functions of two variables like, e.g., \(\tilde{a}_y\) and \(\delta _v\).
Two vehicles with the same \(s_i\), and hence with the same \(n_i\), have identical transient handling behavior, notwithstanding their size, weight, etc. In other words, the two vehicles react in exactly the same way to given driver input. Therefore, there is indeed a strong relationship between data collected in steady-state tests and the transient dynamical behavior of a vehicle.
Objective measures of car handling should be based on the quantities defined in (7.176).
On the practical side, we see that the components of the gradients (7.99) of the steady-state maps \(\beta _p(\delta _v,\tilde{a}_y)\) and \(\rho _p(\delta _v,\tilde{a}_y)\) provide four out of six “handling bricks”, the other two being the generalized control derivatives. Basically, we have found a more feasible way, based on the gradient components of the steady-state MAPs, to measure the six stability and control derivatives listed on Sect. 7.11.
7.12 Stability (Again)
According to (7.139), an equilibrium point is stable if and only if \({\text {tr}}(\textbf{A})<0\) and \(\det (\textbf{A})>0\). These two conditions, after (7.169), can be expressed in terms of the six fundamental handling bricks (7.177) and the forward speed. However, provided \(\varPhi _1 > 0\) and \(\varPhi _2 > 0\), the onset of instability is given by (7.172).
7.13 New Understeer Gradient
Let us discuss in detail the component \(\rho _y\) of the new understeer gradient introduced in (7.99). In general it is a function of two variables
except in some special cases, like the single track model with open differential, where, according to (7.102), \(\rho _y=\rho _y(\tilde{a}_y) = -\textrm{d}f_\rho /\textrm{d}\tilde{a}_y\).
More explicitly,
This is similar to the definition (7.117) of the classical understeer gradient K, but with a few fundamental differences.
The definition of \(\rho _y\) does not involve any weak concept, like the wheelbase l or the Ackermann steer angle, as discussed in Sect. 7.9. Therefore, it is much more general. This new understeer gradient is defined for any vehicle.
Moreover, it is the correct measure of understeer/oversteer, while K is not. This may look surprising, but that is the way it is, as will be shown in Sect. 7.14.1 (see in particular Table 7.1).
Of course, the partial derivative in (7.179) requires the steer angle to be kept constant, according to (7.178).Footnote 12
But there are other reasons that support \(\rho _y\) as a good handling parameter. Let us consider a constant steering wheel test and monitor the yaw rate \(r_p=r_p(u_a;\delta _{va})\) as a function of the forward speed \(u_a\), keeping constant the steering wheel angle \(\delta _{va}\). For brevity, let \(r_p' = \textrm{d}r_p/\textrm{d}u_a\). Equation (7.179) can be rewritten as
This general equation provides a way to obtain the critical speed and the characteristic speed.
If \(\rho _y < 0\) (understeer), the characteristic speed \(u_\text {ch}\) is, by definition [12, pp. 181–185], the speed at which \(r_p' = 0\), that is the yaw velocity gain \(r_p(u_a;\delta _{va})\) is maximum. By letting \(r_p' \rightarrow 0\) in (7.180), we obtain that the characteristic speed must satisfy the following equation
Similarly, if \(\rho _y > 0\) (oversteer), the critical speed \(u_\text {cr}\) is, by definition [12, p. 177], the speed at which \(r_p' \rightarrow \infty \), which means
Summing up:
-
\(\rho _y\) has been defined without any recourse to weak concepts, like a reference vehicle having Ackermann steering [16];
-
\(\rho _y\) can be easily measured in constant steering wheel tests;
-
the critical speed and the characteristic speed come out naturally as special cases.Footnote 13
A similar treatment applies to the other gradient component \(\beta _y\). In this case \(v_p=v_p(u_a;\delta _{va})\), thus obtaining
In general
except in cases like the single track model with open differential, where, according to (7.102), \(\beta _y = \beta _y(\tilde{a}_y) = -\textrm{d}f_\beta /\textrm{d}\tilde{a}_y\).
7.14 The Nonlinear Single Track Model Revisited
The general approach presented in Sect. 7.11, which explains why steady-state data are also relevant for the transient behavior, is applied here to the single track model. The goal is to clarify the matter by a significant worked-out example.
For simplicity, we assume \(u=u_a\) and \(\dot{u}=0\) and hence start with the linearized equations of motion (7.145).
In the single track model (with open differential), the stability derivatives (7.125) can be obtained directly (cf. (7.88)), taking into account the congruence equations (7.51) and the axle characteristics (7.73)
and
where
are the slopes of the axle characteristics at the equilibrium point, defined in (7.94). Obviously, \(\varPhi _i>0\) in the monotone increasing part of the axle characteristics. These slopes are simple to be defined, but not so simple to be measured directly.
It is also worth recalling that
To proceed further, as already done in (7.68), let
thus linking the rear steer angle \(\delta _2\) to the front steer angle \(\delta _1\) in such a way to keep constant the net steer angle \(\tau \delta _v = \delta _1-\delta _2= \delta \). To have front steering only it suffices to set \(\kappa =0\).
We can now obtain also the explicit expressions of the control derivatives
In this vehicle model, all stability derivatives and all control derivatives are functions of \(\tilde{a}_y\) only, that is \(Y_\beta = Y_\beta (\tilde{a}_y)\), and so on.
The linearized equations of motions (7.145) become
Similarly, (7.131) becomes, in this case
and
Quite remarkable is that in (7.131)
Therefore this quantity is always positive, provided the vehicle operates with positive \(\varPhi _1\) and \(\varPhi _2\).
The damping ratio (7.134) has the following expression
and the natural angular frequency (7.136) becomes
or, equivalently
These parameters characterize the handling behavior in the neighborhood of an equilibrium point. More explicitly, like in (7.132), we have that the vehicle behaves as a mechanical vibrating system with
It is a kind of interesting comparing (7.132), (7.171), and (7.198). They provide the same information with very different tools.
In the single track model, the explicit expressions of the two forcing functions (7.153) can also be obtained
with obvious simplifications if \(\kappa =0\) (front steering only).
All the equations obtained in this section show that for a single track model there are seven design parameters
in addition to the control parameters u and \(\delta _{v}(t)\), with constant \(u = u_a\).
Now, we can relate these design parameters to the six fundamental “handling bricks” of (7.177).
The components of the gradients \({{\,\textrm{grad}\,}}\beta _p\) and \({{\,\textrm{grad}\,}}\rho _p\), defined in (7.99), have been obtained for the single track model in (7.100)
As already stated, all these components can be measured experimentally from standard steady-state tests, and without having to bother about Ackermann steer angle and the like.
Also interesting is that
In case of no rear steering (i.e., \(\kappa = 0\)) it becomes
The generalized control derivatives \(\hat{Y}_\delta = Y_\delta /m\) and \(\hat{N}_\delta = N_\delta /J_z\) are immediately obtained from (7.190).
Summing up, for the single track model the six “handling bricks” \(s_i(\delta _{va},\tilde{a}_y)\) in (7.177) are
Therefore, we have six “handling bricks” depending on seven design parameters. This means that there exist infinitely many different vehicles sharing the same handling transient behavior. This observation opens up many new paths of reasoning.
One of these paths of reasoning is worked out in the next section. The results are quite surprising.
7.14.1 Very Different Vehicles with Identical Handling
As a test of the new theory presented in Sect. 7.14, we are going to compare the transient handling behavior of, say, three linear single track models. These vehicles will be very different, and identical at the same time. How is it possible?
These three vehicles will share exactly the same values of all the six handling bricks listed in (7.203). Therefore, they will have the same handling behavior. However, they need not to be exactly alike, since we can play with seven design parameter to fulfill the six handling requirements.
A good test is to define a first vehicle with front steer only, a second vehicle with also negative rear steer, and a third one with also positive rear steer. This can be easily done by means of parameter \(\kappa \), introduced in (7.189)
Parameter \(\kappa \) controls the amount of rear steer with respect to front steer, while keeping constant the net steer angle \(\delta = \tau \delta _v = \delta _1-\delta _2\). The rear wheels turn opposite to the front wheels if \(\kappa <0\), while both front and rear wheels turn alike if \(\kappa >0\). Typically, \(|\kappa |<0.1\), that is the rear wheels cannot turn as far as the front wheels.
But, let us do some numerical examples. Let us consider a vehicle with front steering only (\(\kappa =0\)), with the following features:
-
\(\tau = 1/20\);
-
\(m = 1300\,{\text {kg}}\);
-
\(J_z = 2000\,{\text {kgm}}^{2}\);
-
\(a_1 = 1\,{\text {m}}\);
-
\(a_2 = 1.60\,{\text {m}}\);
-
\(\varPhi _1=\varPhi _1(0) = 70000\,{\text {N}}/{\text {rad}}\);
-
\(\varPhi _2=\varPhi _2(0) =90000\,{\text {N}}/{\text {rad}}\).
From (7.203) we can compute all six handling bricks \(s_i\) for this vehicle, and then use them for the other two vehicles. This way, it is possible to create vehicles that look very different, but which ultimately have exactly the same handling behavior.
The vehicle features for \(\kappa \pm 0.1\), that is two very high amounts of rear steer, are shown in Table 7.1. The three vehicles there reported are strikingly different (Fig. 7.71), yet they have the same handling behavior, and not limited to steady state. For the driver, they behave exactly the same way under any transient conditions.
For instance, starting from a straight trajectory, let us impose a step steering input \(\delta _v = 60^{\circ }\), the forward velocity being \(u=30\,{\text {m}}\big /{\text {s}}\). Figures 7.59 and 7.60 show the lateral velocity v(t) and the yaw rate r(t), respectively. They are identical for the three vehicles, thus confirming the theoretical claims.
Of course, the slip angles are not identical, as shown in Fig. 7.61. The three vehicles are indeed different. It is left to the reader to figure out which curve is for \(\kappa =0.1\), etc.
Lateral velocity v(t) of any of the three vehicles after a step steering input
Yaw rate r(t) of any of the three vehicles after a step steering input
Front and rear slip angles of the three vehicles after a step steering input
Just out of curiosity, the most extreme vehicles that can be obtained with this algorithm are shown in Table 7.2. Of course, we are not suggesting that they are feasible vehicles. They are reported here because they provide some rigorous evidence that rear steer must be kept small to have good handling behavior, as intuitively everybody knows.
But perhaps the most astonishing result obtained in this section is that all these vehicles of Tables 7.1 and 7.2, although with identical handling behavior, do not have the same classical understeer gradient K. Just have a look at the next to last column in Table 7.1. In other words, they would have been classified as very different if evaluated in terms of their classical understeer gradient K [16].
The conclusion is that the classical understeer gradient K is not a good parameter and should be abandoned. It should be replaced by the gradient components proposed in (7.99) and discussed in Sect. 7.13, which have proven to really provide a measure of the dynamic features of a vehicle. In particular, the gradient component \(\rho _y\), shown in the last column in Tables 7.1 and 7.2, is the real measure of understeer/oversteer.
7.15 Linear Single Track Model
The simplest dynamical systems are those governed by linear ordinary differential equations with constant coefficients. The single track model of Fig. 7.20 is governed by the nonlinear ordinary differential equations (7.162), unless the axle characteristics are replaced by linear functions
where
The axle lateral slip stiffness \(C_i\) is usually equal to twice the tire lateral slip stiffness, firstly introduced in (2.88). It is affected by the static vertical load (Fig. 2.23), but not by the load transfer, neither by the amount of grip. The influence of roll steer is quite peculiar (Fig. 7.18).
However, as shown in Fig. 7.62, this linear approximation is acceptable only if \(|\alpha _i|< 2^{\circ }\), that is for very low values of \(\tilde{a}_y\).
Linear approximation of the axle characteristics
The main advantage of the linear single track model lies in its simplicity, the main disadvantage is that it does not model the vehicle behavior at all, unless the lateral acceleration is really small (typically, \(\tilde{a}_y< {0.2} g\) on dry asphalt). In some sense, it is a “dangerous” model because one may be tempted to use it outside its range of validity. Indeed, too often it is the only handling model that is presented and discussed in detail.
However, in some cases it is useful to have a model where everything can be obtained analytically. For this reason, the linear single track model is included in this book as well, albeit not in a prominent position.
7.15.1 Governing Equations
The linear single track model differs from the more general nonlinear model only in its constitutive equations. However, we list here all relevant equations, that is equilibrium equations (7.5)
congruence equations (7.67) (with \(|\chi |\ll 1\), and often equal to zero)
and the just defined linear constitutive equations (7.204) [12, Chap. 5]
Combining congruence and constitutive equations we get
which are linear in v and r, but not in u.
Inserting these equations into the equilibrium equations, we obtain the governing equations, that is two linear differential equations
In matrix notation, (7.210) becomes
where \(\textbf{w}(t) = \bigl (v(t),r(t)\bigr )\) is the vector of state variables, the r.h.s. known vector is
and
is the coefficient matrix. It is important to note that \(\textbf{A}\) depends on the forward speed u, but not on the steer angle \(\delta _v\), which multiplies the known vector \(\textbf{b}\).
7.15.2 Solution for Constant Forward Speed
As well known, the general solution \(\textbf{w}(t)\) of (7.211) is given by the solution \(\textbf{w}_o\) of the homogeneous equation plus a particular solution \(\textbf{w}_p\)
Unfortunately, analytical solutions are not available if \(u(t) \ne \text {const}\).
If u is constant (\(\dot{u}=0\)), the system (7.211) has constant coefficients and the homogeneous solution must fulfill
with a constant matrix \(\textbf{A}\). Assuming constant u is therefore a very relevant assumption. We look for a solution among the exponential functions
which implies \(\dot{\textbf{w}}_o(t) = \lambda \textbf{x}\textrm{e}^{\lambda t}\), and consequently yields an eigenvalue problem for the matrix \(\textbf{A}\)
The eigenvalues are the solutions of the characteristic equation
which, for a \((2 \times 2)\) matrix, becomes
The two eigenvalues \(\lambda _1\) and \(\lambda _2\) are
If the discriminant is negative, that is if \(\zeta <1\), the dynamical system is underdamped and the eigenvalues are complex conjugates.
From (7.213) we get the trace
and the determinant
These two quantities are very important because they provide handy information about the two eigenvalues \(\lambda _1\) and \(\lambda _2\) of \(\textbf{A}\), since
These two relationships can be obtained easily writing the characteristic equation as \((\lambda - \lambda _1)(\lambda - \lambda _2) = 0\).
Once the two eigenvalues have been obtained, we can compute the two eigenvectors \(\textbf{x}_1\) and \(\textbf{x}_2\).
Therefore, the solution of the homogeneous system is
where \(\gamma _1\) and \(\gamma _2\) are constants still to be determined. In components we have
where \(\textbf{x}_1 = (x_{11}, x_{21})\) and \(\textbf{x}_2 = (x_{12}, x_{22})\).
The particular integral \(\textbf{w}_p(t) = (v_p(t), r_p(t))\) depends on the known vector \(\textbf{b}\) and on the steering wheel angle \(\delta _v(t)\). The simplest case is for constant \(\delta _v\), but analytical solutions are available also when \(\delta _v(t)\) is a polynomial or trigonometric function.
Summing up, the general solution of the system (7.211) is
in which the two constants \(\gamma _1\) and \(\gamma _2\) are to be determined from the initial conditions \(\textbf{w}(0) = (v(0), r(0))\), that is solving the system
where \(\textbf{y}= (\gamma _1, \gamma _2)\) and \(\textbf{S}\) is the matrix whose columns are the two eigenvectors of \(\textbf{A}\).
7.15.3 Critical Speed
The two parts \(\textbf{w}_o\) and \(\textbf{w}_p\) of the general solution have distinct physical meanings. The particular integral is what the vehicle does asymptotically, that is basically at steady-state. The solution of the homogeneous system shows how the vehicle behaves before reaching the steady-state condition, if the vehicle is stable.
As already discussed in Sect. 7.10.3, the stability of the vehicle is completely determined by the two eigenvalues \(\lambda _1\) and \(\lambda _2\), or better, by the sign of their real parts \({{\,\textrm{Re}\,}}(\lambda _1)\) and \({{\,\textrm{Re}\,}}(\lambda _2)\). The rule is very simple: the system is asymptotically stable if and only if both eigenvalues have negative real parts
If just one eigenvalue has a positive real part, the corresponding exponential solution grows without bound in time, and the system is unstable.
Fortunately, we can check the stability without computing the two eigenvalues explicitly, but simply looking at (7.223) and (7.224). To have an asymptotically stable vehicle it suffices to check that
From (7.221) we see immediately that \({\text {tr}}(\textbf{A}) < 0\) is always fulfilled. Stability is therefore completely due to the second condition in (7.230). Setting \(\det (\textbf{A})=0\) in (7.193) yields an equation in the unknown forward speed u, whose solution, if it exists, is the critical speed \(u_\text {cr}\)
Beyond the critical speed the vehicle becomes unstable. It is worth noting that \(u_\text {cr}\) does not depend on \(J_z\).
In the linear single track model, the critical speed exists if and only if
that is, if the vehicle is oversteer. In this vehicle model (which, we recall, has a very limited range of applicability), the critical speed is not affected by the steer angle.
7.15.4 Transient Vehicle Behavior
It may be of some interest to know how the eigenvalues evolve as speed changes. To this end, it is useful to plot \({\text {tr}}(\textbf{A})\) vs \(\det (\textbf{A})\), which, according to (7.221) and (7.222), can be compactly expressed asFootnote 14
where \(\alpha \) and \(\gamma \) are always positive, while \(\beta = (C_2 a_2 - C_1 a_1)/J_z\) can be either positive or negative, depending on the vehicle being understeer or oversteer, respectively.
Both functions are monotone in u (if \(u>0\)). They can be combined to get
Moreover, it is easy to show that
Therefore, as u grows, we draw parabolas, as shown in Fig. 7.63, up to their vertex in \((0, \beta )\).
Evolution of \(\det (A)\) and \({\text {tr}}(A)\) when u grows
Also plotted in Fig. 7.63 is the parabola \(\det = {\text {tr}}^2/4\). According to (7.220), it corresponds to the points where \(\lambda _1 = \lambda _2\). Below this parabola, i.e. \(u < u_t\), the two eigenvalues are real, whereas above it they are complex conjugates.
Evolution of the real part and of the imaginary part of \(\lambda _1\) and \(\lambda _2\) as functions of the forward speed u, for an understeer vehicle
It can be shown that
where \(J_z = m k^2\). Since it attains its maximum value 1/4 when \(C_1 a_1 = C_2 a_2\) (neutral vehicle) and \(J_z = m a_1 a_2\), we see that all vehicles at sufficiently low speed have real negative eigenvalues.
As the speed increases, the following evolutions are possible. An oversteer vehicle (actually, an oversteer linear single track model) has always two real eigenvalues. When the parabola in Fig. 7.63 crosses the horizontal axis (\(\det = 0\)), one eigenvalue becomes positive and the vehicle becomes unstable. That happens for \(u = u_\text {cr}\).
An understeer vehicle has two negative real eigenvalues at low speed. For speeds higher than \(u = u_t\), they become complex conjugate with negative real parts (Fig. 7.63): \(\lambda _1 = -\zeta \omega _n + \textrm{i}\omega _n\sqrt{1-\zeta ^2}\), \(\lambda _2 = -\zeta \omega _n - \textrm{i}\omega _n\sqrt{1-\zeta ^2}\). Therefore, at sufficiently high speed, the transient motion is a damped oscillation (very damped, indeed). The speed \(u_t\) is given by
From Fig. 7.64, we see that the imaginary part of the eigenvalues, that is the angular frequency \(\omega _s=\omega _n\sqrt{1-\zeta ^2}\), is almost constant up to relatively high speeds. This is typical and makes the classical sine sweep test quite insensitive to the selected speed.
The general solution is given by (7.227). However, when the eigenvalues are complex conjugates, also the eigenvectors \(\textbf{x}_1\) and \(\textbf{x}_2\) and the constants \(\gamma _1\) and \(\gamma _2\) are complex conjugates. Having to deal with so many complex numbers to eventually get a real function \(\textbf{w}(t)\) is not very convenient. Fortunately, we can rearrange it in a way that it involves only real numbers. As well known, \(\textrm{e}^{(\zeta + \textrm{i}\omega )t} = \textrm{e}^{\zeta t}[\cos (\omega t) + \textrm{i}\sin (\omega t)]\), and the general solution can be written as
where \(\omega _s = \omega _n \sqrt{1-\zeta ^2}\).
To obtain \(\textbf{z}_1\) and \(\textbf{z}_2\) we can proceed as follows. Vector \(\textbf{z}_1\) is simply obtained setting \(t=0\) in the last expression in (7.238)
where \(\textbf{w}(0)\) is the vector of the initial conditions. To obtain the other vector, just consider that
and hence
7.15.5 Steady-State Behavior: Steering Pad
As already stated, the particular integral \(\textbf{w}_p(t) = (v_p(t), r_p(t))\) is determined, in this linear model, by the known vector \(\textbf{b}\), and hence by the function \(\delta _v(t)\). The simplest case is when \(\delta _v = \text {const.}\)
Keeping the steering wheel in a fixed position and driving at constant speed makes the vehicle go round in a circle. This is called steering pad. To obtain the steady-state solution, we have to solve the system
thus getting
Once we have obtained \(v_p\) and \(r_p\), we can easily compute all other relevant quantities, like the vehicle slip angle \(\beta _p\) and the curvature \(\rho _p\)
According to (7.207), we can compute the steady-state front and rear slip angles
A non-zero lateral speed \(v_p\) at steady state may look a bit strange, at first sight. It simply means that the trajectory of G is not tangent to the vehicle longitudinal axis, as shown in Fig. 7.32.
The speed \(u_\beta \) that makes \(\beta _p=v_p=0\) is given by (7.243) and is equal to (if \(\chi =0\))
It is called tangent speed [12, p. 174].
7.15.6 Lateral Wind Gust
It is of some practical interest to study the behavior of a vehicle (albeit a very linear one) when suddenly subjected to a lateral force, like the force due to a lateral wind gust hitting the car when, e.g., exiting a tunnel. As shown in Sect. 7.15.7, the same mathematical problem also covers the case of a car going straight along a banked road.
We have only to modify the equilibrium equations (7.206) by adding a lateral force \(\mathbf {F\!}_l = -F_l \,\textbf{j}\), applied at a distance x from G
where \(x>0\) if \(\mathbf {F\!}_l\) is applied between G and the front axle. The other equations are not affected directly by \( \mathbf {F\!}_l\).
The equations of motion are like in (7.211), with the only difference that the term
must be added to the known vector.
Lateral force applied at the neutral point \(N_p\) (i.e., \(x=e\))
If we assume \(\delta _v=0\), the steady-state conditions \(\textbf{w}_p\) are obtained, as usual, by solving the system of equations \(-\textbf{A}\textbf{w}_p = \textbf{b}_F\), with \(\textbf{A}\) as given in (7.213). Accordingly, we have the following quantities at steady-state
where
Should the steer angle be non-zero, it suffices to superimpose the effects. This is legitimate because of the linearity of the equations.
This quantity e in (7.250) is often called static margin. The yaw rate is zero, that is \(r_p=0\), if and only if the lateral force is applied at a distance e from G. This is the distance that makes the vehicle translate diagonally under the action of a lateral force, as shown in Fig. 7.65. The point \(N_p\) on the axis of the vehicle at a distance e from G is called neutral steer point.
Obviously, the condition \(r_p=0\) with \(\delta _v=0\) is equivalent to \(\alpha _{1p}=\alpha _{2p}=\alpha _p\). Inserting this condition into (7.247) we get
which provide another way to obtain e.
An oversteer vehicle has \(e > 0\), whereas \(e < 0\) in an understeer vehicle.
If \(\delta _v=0\), the steady-state distance \(R_p\) is
The numerator is always positive if \(u < u_\text {cr}\). Therefore, \(R_p >0\) if \(x<e\), and vice versa.
Lateral force applied at a point ahead of the neutral point (\(x>e\))
Lateral force applied at a point behind the neutral point (\(x<e\))
If the point of application of the lateral force is located ahead of the neutral point \(N_p\), the vehicle behaves like in Fig. 7.66, turning in the same direction as the lateral force. This is commonly considered good behavior.
If the point of application of the lateral force is behind the neutral point \(N_p\), the vehicle behaves like in Fig. 7.67. This is commonly considered bad behavior.
Of course, since an oversteer vehicle has the neutral point \(N_p\) ahead of G, the likelihood that a wind gust applies a force behind the neutral point is higher, much higher, than in an understeer vehicle.
Lateral force applied by means of a rocket (General Motors Corporation, circa 1960)
To understand why the first case is considered good, while the second is considered bad, we have to look at the lateral forces that the tires have to exert. In the first case, the inertial effects counteract the wind gust, thus alleviating the tire job. In the second case, the inertial effects add to the lateral force, making the tire job harder.
Figures 7.66 and 7.67 show a lateral force constantly perpendicular to the vehicle axis, pretty much like if a rocket were strapped on the side of the car. Indeed, in some cases a rocket has been really employed as shown in Fig. 7.68, taken from a presentation by Tom Bundorf at the SAE Automotive Dynamics and Stability Conference (2000).
7.15.7 Banked Road
A car going straight on a banked road is subject to a lateral force due to its own weight. Therefore, it is a situation somehow similar to a lateral wind gust, but not equal. The main difference is that the lateral force is now applied at G.
Understeer and oversteer vehicles behave differently, as shown in Fig. 7.69. Both axles must exert lateral forces directed uphill to counteract the weight force \(m g \sin \varepsilon \). Therefore, both must work with positive slip angles \(\alpha _1\) and \(\alpha _2\), if the banking is like in Fig. 7.69. However, due to the different locations of the neutral point \(N_p\) with respect to G, the two front axles cannot have the same slip angle. To go straight, we must steer the front wheels uphill in an understeer vehicle and (apparently) downhill in an oversteer vehicle, as shown in Fig. 7.69. More precisely, in both cases \(\alpha _1-\delta _1 = \alpha _2\), where \(\delta _1 > 0\) if the vehicle is understeer, while \(\delta _1 <0\) if the vehicle is oversteer.
7.16 Compliant Steering System
Many modern cars use rack and pinion steering mechanisms. The steering wheel turns the pinion gear, which moves the rack, thus converting rotational motion into linear motion. This motion applies steering torque to the front wheels via tie rods and a short lever arm called the steering arm.
So far we have assumed the steering system to be perfectly rigid, as stated on Sects. 3.1 and 7.5.4. More precisely, equations (3.210) have been used to relate the steer angles \(\delta _{ij}\) of each wheel to the angle \(\delta _v\) of the steering wheel.
In the single track model (Fig. 7.20) we have taken a further step, assuming that the left and right gear ratios of the steering system are almost equal, that is
thus getting (7.68)
Now, in the framework of the linear single track model, we relax the assumption of rigid steering system. This means to make a few changes in the congruence equations (7.207), since \(\delta _1\) and \(\tau _1 \delta _v \) are no longer equal to each other.
Understeer and oversteer vehicles going straight on a banked road
Single track model with compliant steering system
7.16.1 Governing Equations
As shown in Fig. 7.70, the steering system now has a finite angular stiffness \(k_{s_1}\) with respect to the axis about which the front wheel steers. In a turn, the lateral force \(Y_1\) exerts a vertical moment with respect to the steering axis A because of the pneumatic trail \(t_{c_1}\) and also of the trail \(t_{s_1}\) due to the suspension layout (see Fig. 3.1). The effect of this vertical moment \(Y_1(t_{c_1}+t_{s_1})\) on a compliant steering system is to make the front wheel steer less than \(\tau _1 \delta _v\). More precisely, we have that (Fig. 7.70)
The computation of the pneumatic trail \(t_{c_1}\) is discussed on page xx.
Accordingly, the congruence equations (7.207) of the linear single track model become
with the additional equation (7.253).
On the other hand, the equilibrium equations
and the constitutive equations
do not change at all.
7.16.2 Effects of Steer Compliance
Equation (7.253) can be rewritten taking the first equation in (7.208) into account
where
The first congruence equation becomes
which leads naturally to define a fictitious slip angle
and, consequently, a fictitious slip stiffness
Summing up, the linear single track model with compliant steering system is governed by the set of equations
which is formally identical to the set governing the single track model with rigid steering system. Therefore, the analysis developed in Sect. 7.15 applies entirely, provided we take into account that \(C_1 \rightarrow \tilde{C}_1\) and \(\alpha _1 \rightarrow \tilde{\alpha }_1\).
Since \(\tilde{C}_1<C_1\), a compliant steering system makes the vehicle behavior more understeer.
7.16.3 There Is Something Unsafe
Apparently, the critical speed of the linear single track model is not affected by the steer angle, as shown in Sect. 7.15.3. However, according to (7.243), if \(\det (\textbf{A})=0\) the model predicts unlimited values of lateral velocity \(v_p\) and yaw rate \(r_p\), unless \(\delta _v = 0\). These unrealistic results suggest that in the analysis something relevant is missing.
Indeed, instead of imposing the forward speed u, it would more realistic to impose the longitudinal force \(F_{x_2}\), thus having also u as a state variable. Of course, the steady-state results (7.243) do not change. But what about stability?
This more general single track model, with three state variables (u, v, r), is governed by the following differential equations
Even with linear tires, the governing equations are no longer linear.
The counterpart of matrix \(\textbf{A}\) in Sect. 7.10.2 is this matrix \(\textbf{B}\)
which contains \(\textbf{A}\) as a submatrix. It is worth noting that the new entries (first raw and column) depend on \(v_p\), \(r_p\), and \(\delta _1\).
Stability requires the real part of all eigenvalues of \(\textbf{B}\) to be negative.
In case of straight running \(\delta _1=0\) and, obviously, \(v_p=r_p=0\). Two eigenvalues are exactly the same of \(\textbf{A}\). The third eigenvalue \(\lambda _3 = -\rho S C_x u_p\) is always negative. Therefore, the classical critical speed \(u_\text {cr}\) in (7.231) is confirmed.
What happens when \(\delta _1 \ne 0\)? The characteristic equation is something like
According to Routh criterion, all eigenvalues have negative real part if and only if \(b_1 > 0\), \(b_3 > 0\), and \(b_1 b_2 - b_0 b_3 >0\).
Since analytic expressions are very complex, we prefer to perform just a numerical test. For instance, let \(m = {1000}~\text {kg}\), \(a_1 = 1.4~\text {m}\), \(a_2 = 1.2~\text {m}\), \(J_z = {1680}~\text {kg m}^2\), \(C_1 = C_2 = {100000}~\text {N/rad}\). Moreover, \(\rho = 1.3~\text {kg/m}^3\), \(S_a = 1.8~\text {m}^2\), \(C_x = 0.35\).
This vehicle has, in straight running, a critical speed \(u_\text {cr}=\) 58.14 m/s = 209 km/h. However, if, e.g., \(\delta _1 = 6\) deg, the quantity \(b_1 b_2 - b_0 b_3\) becomes negative at the speed \(\tilde{u}_{\text {cr}}=53.21~\text {m/s} = 191~\text {km/h}\), which is quite lower than \(u_\text {cr}\). Therefore, imposing u may not be a safe assumption!
7.17 Road Vehicles with Locked or Limited Slip Differential
The handling of cars equipped with either a locked or a limited-slip differential is addressed in Chap. 8, that is in the chapter devoted to the handling behavior of race cars. This has been done because the limited-slip differential is a peculiarity of almost all race cars, whereas very few road cars have it.
7.18 Exercises
7.18.1 Camber Variations
As shown in (7.18) and in Fig. 7.7, camber variations due to vehicle roll motion are determined by some suspension parameters. Given the track length \(t_i\), find the values of \(c_i\) to have:
-
1.
\(\varDelta \!\gamma _i/\phi _i^s = -1\);
-
2.
\(\varDelta \!\gamma _i/\phi _i^s = 0\);
-
3.
\(\varDelta \!\gamma _i/\phi _i^s = 1\).
Solution
It is a simple calculation to obtain
-
1.
\(c_i = t_i/4\);
-
2.
\(c_i = t_i/2\);
-
3.
\(c_i = +\infty \).
Quite a big difference.
7.18.2 Ackermann Coefficient
According to (7.19), and assuming \(\delta _1^0=0\), \(l=2.6\,{\text {m}}\), \(t_1 = 1.6\,{\text {m}}\), and \(\varepsilon _1 = 1\) (Ackermann steering), compute \(\delta _{11}\) and \(\delta _{12}\) when \(\tau _1 \delta _v\) is equal to \(5\,{\text {deg}}\), \(10\,{\text {deg}}\), and \(15\,{\text {deg}}\).
Solution
It is a simple calculation to obtain
-
1.
\(\delta _{11} = 5.13\,{\text {deg}}\), \(\delta _{12} = 4.87\,{\text {deg}}\);
-
2.
\(\delta _{11} = 10.54\,{\text {deg}}\), \(\delta _{12} = 9.46\,{\text {deg}}\);
-
3.
\(\delta _{11} = 16.21\,{\text {deg}}\), \(\delta _{12} = 13.79\,{\text {deg}}\).
We see that the Ackermann correction is relevant, with respect to parallel steering, only for not so small steer angles.
7.18.3 Toe-In
Repeat the previous calculations now with \(1\,{\text {deg}}\) of toe-in.
Solution
-
1.
\(\delta _{11} = 4.13\,{\text {deg}}\), \(\delta _{12} = 5.87\,{\text {deg}}\);
-
2.
\(\delta _{11} = 9.54\,{\text {deg}}\), \(\delta _{12} = 10.46\,{\text {deg}}\);
-
3.
\(\delta _{11} = 15.21\,{\text {deg}}\), \(\delta _{12} = 14.79\,{\text {deg}}\).
Quite influential.
7.18.4 Steering Angles
With reference to (7.68), obtain the relationship between \(\chi \) and \(\kappa \) for any steering system.
Solution
From the following system of equations
we obtain
7.18.5 Axle Characteristics
Axle characteristics are very important in vehicle dynamics. In Sect. 7.5.3, the effects of the following setup parameters were discussed (not in this order):
- 1.:
-
roll stiffness;
- 2.:
-
static camber angles;
- 3.:
-
roll camber;
- 4.:
-
roll steer;
- 5.:
-
toe-in/toe-out.
Some of these parameters have similar effects on the axle characteristics. Before going back to Sect. 7.5.3, think about the physics of each parameter and try to figure out the similarities.
Solution
Have a look at Sect. 7.5.3.
7.18.6 Playing with Linear Differential Equations
Find out how to go from (7.128) to (7.130), that is, from a system of two first-order linear differential equations with constant coefficients to two second-order equations.
Solution
Like in (7.128), the starting point is
where \(a_{ij}\) are the entries of matrix \(\textbf{A}\), as in (7.148).
We can see (7.266) as a system of two algebraic equations and solve it with respect to \(\dot{\beta }_t\) and \(\beta _t\), thus getting
Differentiating the first equation in (7.267) and setting it equal to the second equation in (7.267) provides the sought second-order linear differential equation
exactly like in (7.130).
7.18.7 Static Margin
Compute the static margin for the single track model defined on Sect. 7.14.1.
Solution
To compute the static margin we have to use (7.250). The result is \(e = -0.46\,{\text {m}}\). A negative value is typical of understeer vehicles.
7.18.8 Banked Road
The same vehicle is travelling on a straight road with \(6\,{\text {deg}}\) of banking. Compute the steering wheel angle required to have a trajectory parallel to the road (that is to go straight ahead).
Solution
With the aid of Fig. 7.69, we see that the rear axle has to counteract a lateral force \(Y_2= m g \sin (6\,{\text {deg}}) a_1/l = 452.8\,{\text {N}}\). That means that the rear axle operates with a slip angle \(\alpha _2 = Y_2/\varPhi _2(0) = 0.29\,{\text {deg}}\).
Similarly, the front axle has to balance a force \(Y_1= m g \sin (6\,{\text {deg}}) a_2/l = 724.4\,{\text {N}}\), which needs a slip angle \(\alpha _1= Y_1/\varPhi _1(0) = 0.59\,{\text {deg}}\).
Therefore, the front steer angle has to be \(\delta _1 = 0.59-0.29 = 0.3\,{\text {deg}}\). The steering wheel angle is \(\delta _v=20 \times 0.3 = 6\,{\text {deg}}\).
Of course, the vehicle slip angle is \(\beta = -\alpha _2 = -0.29\,{\text {deg}}\).
7.18.9 Rear Steer
Repeat the calculations of the banked road for the two vehicles with rear steer whose features are listed in Table 7.1.
Solution
First we consider the vehicle with \(\kappa =-0.1\). We have \(Y_2=331.8\,{\text {N}}\) and hence \(\alpha _2 = 0.26\,{\text {deg}}\). Similarly, \(Y_1=845.4\,{\text {N}}\), and \(\alpha _1 = 0.56\,{\text {deg}}\).
To obtain the net steer angle \(\delta \) we have to solve the equation
with \(\kappa =-0.1\), which provides \(\delta = 0.3\,{\text {deg}}\), and hence a steering wheel angle \(\delta _v = 0.3\times 20/0.99 = 6.0\,{\text {deg}}\).
The vehicle slip angle is \(\beta =-(0.26+0.1\times 0.3)=-0.29\,{\text {deg}}\).
Then we consider the vehicle with \(\kappa =0.1\). We have \(Y_2=622.2\,{\text {N}}\) and hence \(\alpha _2 = 0.32\,{\text {deg}}\). Similarly \(Y_1=555.0\,{\text {N}}\), and \(\alpha _1 = 0.65\,{\text {deg}}\).
To obtain the net steer angle \(\delta \) we have to solve (7.269), with \(\kappa =0.1\), which provides \(\delta = 0.33\,{\text {deg}}\), and hence a steering wheel angle \(\delta _v = 0.3\times 20/0.99 = 6.0\,{\text {deg}}\).
The vehicle slip angle is \(\beta =-(0.32-0.1\times 0.33)=-0.29\,{\text {deg}}\).
As expected, for the driver the three vehicles behave exactly the same way: same steer wheel angle \(\delta _v\), same vehicle slip angle \(\beta \). The three vehicles also have the same static margin \(e=-0.46\,{\text {m}}\).
Comparison of the three vehicles of Table 7.1 under a lateral wind gust
7.18.10 Wind Gust
Are the three vehicles of Table 7.1 fully equivalent with respect to a lateral wind gust?
Solution
These three vehicles are compared in Fig. 7.71. The point of application of a lateral force \(F_l\) due to a wind gust depends on the shape of the vehicle. However, we can reasonably assume \(F_l\) be applied like in Fig. 7.71. Should this be the case, the three vehicles would behave very differently.
Vehicle (a), which has \(\kappa =0\), would do like in Fig. 7.65. Vehicle (b), which has \(\kappa =-1\), would do like in Fig. 7.67. Vehicle (c), which has \(\kappa =1\), would do like in Fig. 7.66.
Therefore, the three vehicles are not equivalent with respect to a lateral wind gust. Actually, their behaviors can be completely different.
7.19 Summary
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 ratios of the steering system for both front wheels, we have been able to formulate the single track model.
Quite contrary to common belief, we have shown that the axle characteristics can take into account many vehicle features, like toe in/out, roll steering, camber angles and camber angle variations.
The steady-state analysis has been carried out first using the classical handling diagram. Then, the new global approach MAP (Map of Achievable Performance), based on handling maps on achievable regions has been introduced and discussed in detail. This new approach shows the overall vehicle behavior at a glance.
Stability and control derivatives have been introduced to study the vehicle transient behavior. Moreover, the relationship between data collected in steady-state tests and vehicle transient behavior has been thoroughly analyzed in a systematic framework. To prove the effectiveness of these results, a number of apparently different vehicles with almost the same handling characteristics have been generated.
7.20 List of Some Relevant Concepts
Section 7.1.1—road cars are normally equipped with an open differential;
Section 7.4—some steady-state quantities are functions of the lateral acceleration only because of the open differential and no significant downforces;
Section 7.5—to go from the double track to the single track model we need the following additional assumption: the left and right gear ratios of the steering system are almost equal;
Section 7.5.4—the main feature of the single track model is that the two wheels of the same axle undergo the same apparent slip angle;
Section 7.9—some “fundamental” concepts in classical vehicle dynamics are indeed very weak if addressed with open mind;
Section 7.12—the classical understeer gradient is not a good parameter and should be dismissed.
7.21 Key Symbols
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.
Some sports cars and all race cars have a limited-slip differential. Several race cars also have wings that provide fairly high aerodynamic downforces at high speed. The handling of these vehicles is somehow more involved than that of ordinary road cars and will be addressed in Chap. 8.
- 2.
The left and right wheels of the same axle are normally equipped with the same kind of brake. Therefore, the braking torque is pretty much the same under ordinary operating conditions, and, again, (7.1) holds true. However, there are important exceptions. The left and right braking forces can be different if: (a) the grip is different and at least one wheel is locked; (b) the friction coefficients inside the two brakes is different (for instance, because of different temperatures, which is often the case in racing cars); (c) some electronic stability system, like ESP or ABS, has been activated.
- 3.
We call no-roll center what is commonly called roll center.
- 4.
In this model the roll inertial effects are totally disregarded.
- 5.
For instance, vehicles equipped with a locked differential and/or with relevant aerodynamic downforces always need (at least) two parameters.
- 6.
This step would not be possible with \(F_{y_i}\) as in (7.30).
- 7.
We remark that this is no longer true in vehicles with limited-slip differential and/or aerodynamic vertical loads.
- 8.
- 9.
To keep, for the moment, the analysis as simple as possible, we also assume that \(\hat{Y}_1(x) = \hat{Y}_2(kx)\), with \(k>0\).
- 10.
From Wikipedia.
- 11.
- 12.
Tests with constant steer angle are the most general: they can be performed on any kind of vehicle.
- 13.
Actually, the real critical speed can be lower than the value predicted by (7.182), as shown in Sect. 7.16.3 [9, pp. 216–219]. Basically, (7.182) may not predict the right value because in real vehicles we control the longitudinal force, not directly the forward speed. Therefore, a real vehicle is a system with three state variables, not just two. This additional degree of freedom does affect the critical speed, unless the vehicle is going straight.
- 14.
Here \(\alpha \), \(\beta \) and \(\gamma \) are just constants. They have no connection with slip and camber angles.
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Guiggiani, M. (2023). Handling of Road Cars. In: The Science of Vehicle Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-031-06461-6_7
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