Chapter 3
Simulink Model
3.1
Simulink Model
The principal aspects of the model we built using the software “Simulink” are presented in this chapter. In particular the next paragraphs are focused on the inevitable little differences existing between theoretical and real model.
3.1.1
Vehicle Subsystem
This subsystem is in turn made up of eleven subsystems (fig.3.1). We can observe on the left three subsystems in which the equations (2.1, 2.2, 2.3, 2.8, 2.9) are solved, and eight subsystems on the right in which the slip angles, the magnitudes of the wheel center speeds, and the signs of their x-components are calculated. In particular it is interesting to notice that the slip angles are not computed as in (2.12, 2.13, 2.14, 2.15), in fact we needed to improve these relations for two different reasons
1. Considering a turned wheel in particular conditions the slip angle calculated through the theoretical relation can be greater than 90 deg, and it is not correct. To un-derstand this fact we can think about two locked wheels, moving as represented in fig.3.2. The lateral forces must be the same, but without correcting the theoretical equation we work out two different slip angles, then different forces. For this reason actually we do not calculate (δ − α) (2.12 and 2.13) but (−α). The first two relations were obtained just through the analysis of the chassis motion (u, v, r), without taking into account the wheel position. On the contrary in our improved model we consider the rotation of each wheel reference frame, arriving at the following equations
Figure 3.1: Vehicle subsystem
tan (−α11) =
−(u − rt
2) sin δ + (v + ra) cos δ
(u − rt
2) cos δ + (v + ra) sin δ
(3.1)
tan (−α12) =
−(u + rt
2) sin δ + (v + ra) cos δ
(u + rt
2) cos δ + (v + ra) sin δ
(3.2) To take account of the position of each wheel reference frame is useful also in case we wanted to improve the tyre scheme in future. Indeed more complex models cannot
disregard the individual tyre position since the tyre motion can become too complex to analyze considering just the global vehicle reference frame (i, j, k).
2. As mentioned in the paragraph 2.2.2 a slip angle calculated by means of the the-oretical relations can represent correctly the direction of the side force only if the wheel reference frame changes depending on the x-speed direction. Nevertheless in our model we consider a “fixed” wheel reference frame, or better a reference frame rotating together with the wheel but in which the x axis keeps its initial positive di-rection independently of the x-speed didi-rection. Then we built a subsystem in which the slip angle computed through the theoretical formula is multiplied by the sign of the x-component of the wheel center speed (with respect to our “fixed” reference frame), as shown in figure 3.3.
Figure 3.3: Example of subsystem built to calculate the slip angles
In this way the direction of lateral forces is correctly represented even if the mentioned above speed changes its sign (fig.3.4).
3.1.2
Tyre Subsystem
The model is obviously provided with four of these subsystems, but since they are almost equal, we present just only one here.
This subsystem is made up of several subsystems in which the values obtained through (2.4, 2.19, 2.20 or 2.23, 2.21 or 2.24, 2.26, 2.27, 2.28) are calculated. In this block we can notice some differences in comparison with the theoretical equations.
Figure 3.4: Reference frame used to calculate the tyre forces
• The longitudinal slip k presents a problem similar to the slip angle α and, to solve it, to multiply the values worked out through the theoretical definition (2.11) by the x-speed sign would be necessary. However in our model we do not calculate k using the theoretical formula, but we compute
ˆ
k= −| Vx | −reΩ | Vx |
To obtain the desired value, we can multiply by the Vx sign just the |Vx| that is on
the numerator (fig.3.5).
• Since we were interested in building a model that could represent the vehicle during critical manoeuvre, we had, to calculate Fx0, to limit σ∗ at a value lower than 1.
In fact, observing (2.17) we can notice that within admissible limits for k (k ∈ [−1, +∞)), σx ∈ (−∞, 1]. However using (2.18) we can compute, for example if the
slip angle is very wide, σx values greater than 1. Then, applying the inverse formula
keq = σ
∗
1−σ∗ we can obtain keq < 0 even if σx > 0, and it obviously is not correct
Figure 3.5: First order filter subsystem
Figure 3.6: Limitation of σ∗
• To use the first order filter we preferred to multiply the equation (2.28) by |Vx|, such
a way to avoid to have a denominator that can be equal to zero (fig.3.5).
• To take into account load transfer in the “Magic Formula” created an algebraical loop which did not allow to solve the equations, and to solve this trouble a “Delay Block” was inserted in this subsystem. A “Delay Block” gives as output the value received in input a certain defined time before. We put one of them in each “Magic Formula Subsystem”, and its input (and output) is the stiffness factor (Bx,y, see par.2.2.3).
It is simple to understand the fact that this block needs a value to give as output during first instants of simulation, for a time equal to the delay time. Obviously we
fixed this value equal to the stiffness value calculated considering Fz0 (fig.3.7). In
this manner we managed to make the solver complete simulations.
Figure 3.7: Magic Formula subsystem
• The brake torque applied while the ESP is acting does not have a constant direction, but its sign depends on the sign of the wheel rotational speed. For example this fact does not occur as for the drive torque. Indeed a brake torque has every instant the direction opposite to the wheel rotational speed. To represent this phenomenon we multiply the magnitude of the brake torque, provided by the ESP subsystem, by a value obtained through the subsequent function (fig.3.8), where in abscissa we find the wheel rotational speed. This function is very similar to a “sign”. We used it in order to avoid that some problems arose during the simulations if a wheel had been close to lock. Indeed contrary to a “sign” it does not have any discontinuities and the solver can easily compute the solution.
Figure 3.8: Function used to determine the brake torque direction
3.1.3
Engine Model
In this block we calculate the torque provided by the engine. We have the wheel rotational speeds as inputs and, by means of a switch, we can choose to use the front wheel velocities (in case of front-wheel drive car) or the rear wheel speeds (in case of rear-wheel drive car) to work out the engine rotational velocity. Clearly to determine the crankshaft rotational speed we need to know the gear and the differential transmission ratios. Giving this value in input to the “Function Block” on the right we obtain the drive torque as output. This block is very flexible in fact, defining opportunely this function it is possible to keep constant torque, constant power, otherwise using an experimental curve we can simulate for example a constant engine load condition (fig. 3.9).
If the “Function Block” represents a real torque curve the output is the torque provided by the engine in full power conditions and, to simulate the choking of gas we multiply the output by a coefficient included between 0 and 1. We inserted in this subsystem also a controller which is able to make the engine provide a torque such to keep an almost constant vehicle speed (if the engine characteristic allows this), acting on a virtual throttle pedal.
Figure 3.9: Engine subsystem
3.1.4
Haldex Coupling Model
This subsystem (fig.3.10) realizes the equations (2.10) and to represent the coupling char-acteristic (fig.2.6) we used a “Look Up Table”. The output of the “Look up Table” is multiplied by a value (controlled by an external system described in Cap 5) representing the valve position . The result is the torque going towards the front axle (if the car is normally rear-wheel drive, or vice versa) before the differential (the torque that each wheel receives depends also on the differential transmission ratio). In any case the torque trans-mitted by the “Haldex” coupling is limited at a maximum value, usually included between 2000N m and 3000N m, because a greater value can damage the drive line components. In the real coupling this safety system is realized opening a pressure relief valve but in our model we just limited the maximum value by a “Minimum Block” without simulating the valve opening. Subtracting the transmitted torque from the total torque provided by the engine (after the gear-box) we calculate the torque going towards the driving axle. Also in this subsystem, as in the “Engine Subsystem”, to choose to simulate a front-wheel drive car or a rear-wheel drive car is possible simply controlling a shift.
