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Two-Axle Vehicle with Driving Profile

Overview

This demonstration shows a two-axle electric vehicle model with front-wheel drive and highlights the PLECS mechanical domain.

Note that this model contains model initializations in the InitFcn model callback, which can be accessed from the File menu, in the Callbacks tab of the Model Properties window.

System model

The electric motor is connected to the front-wheel axle of the two-axle vehicle model. An ideal torque source is used to apply the motor torque on the rotor, modeled here as a rotating body of fixed inertia. The rotor shaft is then connected to the wheel axle through a gearbox. The shaft stiffness and damping are modeled using a torsion spring and a rotational damper, respectively.

The total traction force generated by both the front and rear wheels is then applied to the vehicle, represented here as a lumped mass. Rotational friction blocks are used to model the front and rear wheel brakes. The vehicle model converts the applied torque on the wheel axle into a traction force applied on the vehicle using a slip-based wheel model. In addition, the vehicle is connected to a road load through the translational mechanical terminal.

Vehicle model

The weight on the front and rear axle is distributed according to the location of the center of gravity. In this example, the center of gravity is assumed to be closer to the front wheel axle to model a front-engine vehicle. The wheel model converts the torque applied on the wheel axle into a traction force. The total traction force generated by both the front and rear wheels is then applied to the vehicle represented here as a lumped mass. The vehicle model also incorporates the effect of acceleration/deceleration on the weight distribution of the vehicle. The “Pitch Change Delay” is used to model the stiffness of the suspension system of the vehicle.

Wheel model

The torque applied to the wheel axle acts on a wheel, which is modeled in a lumped fashion as a rotational inertia block. The rotational speed of the wheel as well as the translational speed of the vehicle (vs) are both measured. The rotational speed of the wheel is scaled by the radius of the wheel to determine the translational speed of the tire (vw) near the point of contact with the road. The normal force acting on the wheel axle and the measured speeds (vw and vs) are used to calculate the traction force (Ft) applied on the vehicle by the wheels. The applied traction force will then move the total vehicular system.

There are two implementations for the calculation of the tire slip. The "Basic Tire Model" ignores the transient behavior of the tires while the "Restricted Fully Nonlinear Model" incorporates the transient behavior. For both implementations, the calculated slip is then translated into a traction force using Pacejka's Magic Formula:

FTraction = Dx*sin(Cx*arctan(Bx*lambda)),

where lambda is the tire slip and Bx, Cx, and Dx are Magic Formula coefficients that are determined through experimental characterization of the vehicular system.

Simulation

Driving profile

The figure above shows the torque applied by the electric motor on the vehicular system over the length of the simulation. A ramped accelerating torque demand is applied on the motor for the first 100 seconds. At t=100 seconds, the accelerator is slammed down leading to a step change in the torque applied by the electric machine. At t=200 seconds, the driver uses the mechanical brakes to slow the vehicle down, which leads to zero torque being applied by the motor. At t=300 seconds, a negative torque is applied by the motor to reverse the vehicle. This torque is maintained over the next 150 seconds when the mechanical brakes are employed again to bring the vehicle to a standstill. Additionally, the power required by the motor is also plotted. As can be seen, the electric machine is only operated in motoring mode in this drive cycle.

Velocity

The vehicle is initially at rest. The vehicle accelerates slowly from rest to 36 km/h over the first 100 seconds. At t=100 seconds, a step change in torque causes a faster acceleration of the vehicle. The vehicle continues to accelerate until it reaches its final steady-state speed of 74 km/h at around t=150 seconds. This steady-state speed is maintained until t=200 seconds and then the brakes are applied, bringing the vehicle to a standstill in 4.7 seconds. At t=300 seconds, the vehicle is reversed, reaching a final velocity of -34.9 m/s at t= 380 seconds. This speed is maintained until t=450 seconds when the brakes are applied to bring the vehicle to standstill in 3.2 seconds.

Weight distribution and accelerating traction force

The figure above shows the effect of the simulated drive cycle on the weight distribution of this vehicle. The accelerating force applied on the vehicle is shown in an additional plot. As can be seen, the ramped increase in forward acceleration over the first 100 seconds causes the weight of the vehicle to be slowly shifted to the rear of the vehicle. A step change in driving torque contributes to a sudden shift in the weight distribution of the vehicle to the rear. As the vehicle reaches steady-state speed, a constant weight distribution is maintained. As the vehicle is brought to a quick standstill, the weight is shifted to the front of the vehicle, as can be seen by the green spike at t=200 seconds. As the vehicle is reversed, the weight of the vehicle is shifted towards the front of the vehicle due to acceleration in the negative direction. As the reversing vehicle is brought to a stop, the weight of the vehicle is shifted to the back of the vehicle, as can be seen by the red spike at t=450 seconds. The waveform between t=205 seconds and 300 seconds, and t=450 seconds and 500 seconds, shows the weight distribution of the vehicle at standstill.

The Traction Force plot shows the accelerating/decelerating force acting on the vehicle over the entire drive cycle. It can be seen that as the vehicle reaches steady-state speed, between t=150 seconds and 200 seconds, the vehicle accelerating traction force is zero. During this period, the torque generated by the electric motor is used to overcome the force due to the road load.

Regenerative braking

An alternative to decelerating the vehicle using the mechanical brakes is to use the generation capabilities of the electric machine to recapture the kinetic energy, hence slowing down the vehicle. The figure above shows a second driving profile for the electric vehicle. The negative torque between t=200 seconds and 205 seconds is used to slow down the vehicle. As can be seen in the power graph, this leads to the machine being operated in generator mode, allowing the kinetic energy to be recaptured. Similarly, the positive torque applied between t=450 seconds and 453 seconds is used to slow down the reversing vehicle. Here the machine is again operated in generator mode allowing the kinetic energy to be recaptured.

Restricted non-linear model vs basic model

The simulations may also be repeated using the "Restricted Fully Nonlinear Model". This implementation models the transient behavior of the tire/wheel, which results in a wind-up oscillation due to a step change in torque applied to the wheel. The figure above shows the accelerating/decelerating force applied on the vehicle due to instantaneous mechanical braking of the vehicle. These results correlate with the observations in Chapter 8 of the textbook "Tire and Vehicle Dynamics".

References

Pacejka, H.B.: Tire and Vehicle Dynamics, 3rd Edition, Butterworth-Heinemann, Oxford, 2012, Chapters: 1, 4, 7, and 8.