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Modelling and assembly of the full vehicle 355 10000 3000 Axle load (N) 8000 6000 4000 2000 Front axle load Rear axle load 0 0 0.2 0.4 0.6 0.8 1 Ideal rear axle brake force (N) Ideal 2000 Typical 1000 0 0 2000 4000 6000 8000 10000 Deceleration (g) Front axle brake force (N) Fig. 6.30 Force distribution for ideal and typical braking events the vehicle, such as drive or transmission clutches. Typical values of the convection constant hAc are around 150 W K 1 for a front disc brake installation, around 80 W K 1 for a rear brake installation and as low as 20 W K 1 for a rear drum brake. A further key factor in modelling brake performance is the distribution of brake torques around the vehicle. While decelerating, the vertical loads on the axles change as described in section 4.8.1 due to the fact that the mass centre of the vehicle is above the ground. It may be presumed that for ideal braking, the longitudinal forces should be distributed according to the vertical forces. Using the above expressions, the graphs in Figure 6.30 can be calculated for vertical axle load versus deceleration. Knowing the total force necessary to decelerate the vehicle it is possible to calculate the horizontal forces for ‘ideal’ (i.e. matched to vertical load distribution) deceleration. Plotting rear force against front force leads to the characteristic curve shown in Figure 6.30. However, in general it is not possible to arrange for such a distribution of force and so the typical installed force distribution is something like that shown by the dashed line in the figure. Note that the ideal distribution of braking force varies with loading condition and so many vehicles have a brake force distribution that varies with vehicle loading condition. For more detailed information on brake system performance and design, Limpert (1999) gives a detailed breakdown of performance characteristics and behaviour, all of which may be incorporated within a multibody system model of the vehicle using an approach similar to that shown in Table 6.1 if desired. Described in some detail in Limpert’s work is the function of a vehicle ABS system. The key ingredient of such a system is the ability to control brake pressure in one of three modes, often described as ‘hold, dump and pump’. Hold is fairly self-explanatory, the wheel cylinder pressure is maintained regardless of further demanded increases in pressure from the driver’s pedal. ‘Dump’ is a controlled reduction in pressure, usually at a predetermined rate and ‘pump’ is a controlled increase in pressure, again usually at a predetermined rate. The main variable is the brake pressure p. In the work by Ozdalyan (1998) a slip control model was initially developed as a precursor to the implementation of an ABS model. This is illustrated in Figure 6.31 where it can be seen that on initial application of the brakes the brake force rises approximately

356 Multibody Systems Approach to Vehicle Dynamics Braking force Fx (N) ON ON ON ON Braking force versus slip ratio OFF OFF OFF OFF Slip angle 0 Camber angle 0 Fz 8 kN Fz 6 kN Fz 4 kN Fz 2 kN Fig. 6.31 0.0 Slip ratio 1.0 Principle of a brake slip control model linearly with slip ratio depending on the wheel load. If the braking is severe the slip ratio increases past the point where the optimum brake force is generated. To prevent the slip ratio increasing further to the point where the wheel is locked an ABS system will then cycle the brake pressure on and off maintaining peak braking performance and a rolling wheel to assist manoeuvres during the braking event. In this model the brake pressure is found by integrating the rate of change of brake pressure, this having set values for any initial brake application or subsequent application during the ABS cycle phase. Implementation of these changing dump, pump and hold states requires care to ensure no discontinuities in the brake pressure formulation. The modelling in MBS of more realistic ABS algorithms (van der Jagt et al., 1989) is more challenging as the forward velocity and hence slip ratio is not directly available for implementation in the model. The implementation of such a model allows the angular velocity of the wheel to be factored with the rolling radius to produce an output commonly referred to as wheel speed by practitioners in this area. A plot of wheel speed is compared with vehicle speed in Figure 6.32 where the typical oscillatory nature of the predicted wheel speed reflects the cycling of the brake pressure during the activation of the ABS model in this vehicle simulation. 6.10 Modelling traction For some simulations it is necessary to maintain the vehicle at a constant velocity. Without some form of driving torque the vehicle will ‘drift’ through the manoeuvre using the momentum available from the velocity defined with the initial conditions for the analysis. Ignoring rolling resistance and aerodynamic drag will reduce losses but the vehicle will still lose momentum during the manoeuvre due to the ‘drag’ components of tyre cornering forces generated during the manoeuvre. An example is provided

356 Multibody Systems Approach to Vehicle Dynamics<br />

Braking<br />

force<br />

Fx (N)<br />

ON<br />

ON<br />

ON<br />

ON<br />

Braking force versus slip ratio<br />

OFF<br />

OFF<br />

OFF<br />

OFF<br />

Slip angle 0<br />

Camber angle 0<br />

Fz 8 kN<br />

Fz 6 kN<br />

Fz 4 kN<br />

Fz 2 kN<br />

Fig. 6.31<br />

0.0 Slip ratio<br />

1.0<br />

Principle of a brake slip control model<br />

linearly with slip ratio depending on the wheel load. If the braking is severe<br />

the slip ratio increases past the point where the optimum brake force is generated.<br />

To prevent the slip ratio increasing further to the point where the<br />

wheel is locked an ABS system will then cycle the brake pressure on and<br />

off maintaining peak braking performance and a rolling wheel to assist manoeuvres<br />

during the braking event.<br />

In this model the brake pressure is found by integrating the rate of change<br />

of brake pressure, this having set values for any initial brake application<br />

or subsequent application during the ABS cycle phase. Implementation<br />

of these changing dump, pump and hold states requires care to ensure no<br />

discontinuities in the brake pressure formulation.<br />

The modelling in MBS of more realistic ABS algorithms (van der Jagt<br />

et al., 1989) is more challenging as the forward velocity and hence slip ratio<br />

is not directly available for implementation in the model. The implementation<br />

of such a model allows the angular velocity of the wheel to be factored<br />

with the rolling radius to produce an output commonly referred to as wheel<br />

speed by practitioners in this area. A plot of wheel speed is compared with<br />

vehicle speed in Figure 6.32 where the typical oscillatory nature of the predicted<br />

wheel speed reflects the cycling of the brake pressure during the<br />

activation of the ABS model in this vehicle simulation.<br />

6.10 Modelling traction<br />

For some simulations it is necessary to maintain the vehicle at a constant<br />

velocity. Without some form of driving torque the vehicle will ‘drift’<br />

through the manoeuvre using the momentum available from the velocity<br />

defined with the initial conditions for the analysis. Ignoring rolling resistance<br />

and aerodynamic drag will reduce losses but the vehicle will still lose<br />

momentum during the manoeuvre due to the ‘drag’ components of tyre<br />

cornering forces generated during the manoeuvre. An example is provided

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