Appendix A Rear-Wheel Drive Car Simulations II

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Appendix A

Rear-Wheel Drive Car Simulations II

In this appendix we illustrate some other simulations, exhibiting the potential of our sys-tem. In particular we want to show the effectiveness of our ESP in case it has to correct understeer, and the effects of the tyres which size is 205/55-R15.

A.1 External Disturbance (Understeer, 195/65-R15)

In this simulation the vehicle is supplied with tyres 195/65-R15 and the car is subjected to a negative lateral force which makes it widen the corner. The external force is applied for 1.5 s, 0.3 m ahead the center of gravity and its magnitude is 2000 N. The initial speed

is 35 m

s and the steer angle follows a step function with final value equal to 1 deg.

310 320 330 340 350 360 60 80 100 120 140 160 distance (m) distance (m) NO control ESP Undisturbed

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0 2 4 6 8 10 12 14 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 time (s) beta (deg) NO control ESP

Figure A.2: Body slip angle

9.5 10 10.5 11 11.5 12 12.5 13 13.5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 time (s) alpha 21 (rad) NO control ESP

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9.5 10 10.5 11 11.5 12 12.5 13 13.5 0 100 200 300 400 500 600 time (s) Mb 21 (N*m) ESP

Figure A.4: Brake torque applied to the rear-inner wheel

Since the vehicle represented in this simulation is a rear wheel drive car, and it is subjected to an external disturbance which makes it understeering, obviously we did not study the effects of the engine torque redistribution. Anyhow we can appreciate the quality of our ESP set-up since the ESP intervention improves really much the vehicle motion.

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A.2 External Disturbance (195/65-R15)

The results here presented refer to a rear wheel vehicle liable to a loss of grip in the rear

outer wheel. The tyre size is 195/65-R15. The initial speed is 35 m

s and the steer angle is represented by a step function and it reaches the value of 1 deg.

9 10 11 12 13 600 700 800 900 1000 1100 time (s) a22 10 11 12 13 600 700 800 900 1000 1100 1200 time (s) b22 a 22 b22

Figure A.5: Friction coefficients

10 11 12 13 −5 −4 −3 −2 −1 0 1 2 3 time (s)

Yaw accel. (rad/s

2) 10 11 12 13 −0.5 0 0.5 1 1.5 2 2.5 time (s)

Yaw accel. (rad/s

2)

NO control Redistribution

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10 11 12 13 −1 −0.5 0 0.5 1 time (s)

Yaw accel. (rad/s

2) 10 11 12 13 −2 −1.5 −1 −0.5 0 0.5 1 1.5 time (s)

Yaw accel. (rad/s

2)

ESP Redistribution & ESP

Figure A.7: Yaw acceleration

10 10.5 11 11.5 12 12.5 13 −80 −60 −40 −20 0 20 time (s) beta (deg) NO control Redistribution Redistribution & ESP ESP

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10 11 12 13 0 100 200 300 400 500 600 700 800 time (s) Mb 12 (N*m) 10 11 12 13 0 100 200 300 400 500 600 700 800 time (s) Mb 12 (N*m)

Redistribution & ESP

ESP

Figure A.9: Brake torque applied to the front-outer wheel

10 11 12 13 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)

valve (1 closed; 0 open)

10 11 12 13 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)

Redistribution Redistribution & ESP

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We want to point out the fact that the “Redistribution of Torque” system is not enough powerful to avoid that the car spins around, but on the contrary the ESP can keep the vehicle stability. Nevertheless to supply the car with both the Active Yaw Controls is not useless since it improves the comfort (fig.A.6 and fig.A.7) and the performances (fig.A.9)

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A.3 High Speed (205/55-R15)

This vehicle is provided with tyres which size is 205/55-R15. The initial speed is 60 m

s and

the steer angle function is a step with final value equal to 0,5 deg.

380 400 420 440 460 480 500 520 5 10 15 20 25 30 35 40 45 50 distance (m) distance (m) NO control Redistribution ESP

Figure A.11: Driven paths

5 5.5 6 6.5 7 7.5 8 8.5 9 −70 −60 −50 −40 −30 −20 −10 0 time (s) beta (deg) NO control Redistribution ESP

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4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 0 100 200 300 400 500 600 time (s) corner radius (m) NO control Redistribution Reference ESP

Figure A.13: “R” distance

4 6 8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 time (s)

yaw accel. (rad/s

2) 5 6 7 8 9 −0.1 0 0.1 0.2 0.3 0.4 0.5 5 6 7 8 9 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

NO control Redistribution ESP

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5 5.5 6 6.5 7 7.5 8 8.5 9 52 53 54 55 56 57 58 59 time (s) speed (m/s) _{Redistribution} ESP

Figure A.15: Center of gravity speed

4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)

valve (0 open; 1 closed)

valve

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4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 0 100 200 300 400 500 600 700 800 time (s) Mb 12 (N*m) ESP

Figure A.17: Brake torque applied to the front-outer wheel

5 5.5 6 6.5 7 7.5 8 8.5 9 0 100 200 300 400 500 600 700 800 time (s) Mb 21 (N*m) ESP

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5 5.5 6 6.5 7 7.5 8 8.5 9 0 2 4 6 8 10 time (s) α11 (deg) NO control Redistribution ESP

Figure A.19: Front-inner tyre slip angle

5 5.5 6 6.5 7 7.5 8 8.5 9 −2 0 2 4 6 8 10 time (s) α21 (deg) NO control Redistribution ESP

Figure A.20: Front-outer tyre slip angle

This simulation provided very good results (“R” distance, longitudinal acceleration, body slip angle, cornering speed) although the Active Yaw Control set-up was developed and defined testing a vehicle equipped with a different type of tyre.

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A.4 External Disturbance (205/55-R15)

The vehicle is running at the speed of 30 m

s, the steer angle follows a step function (final value equal to 0.8 deg), and the rear-outer wheel is liable to a loss of grip. The tyre size is 205/55-R15. 0 5 10 15 600 700 800 900 1000 1100 1200 1300 time (s) a2 1 0 5 10 15 600 700 800 900 1000 1100 1200 1300 time (s) b2 1 a2₁ b21

Figure A.21: Friction coefficients

300 310 320 330 340 350 60 80 100 120 140 160 distance (m) NO control Redistribution ESP

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10 11 12 13 14 15 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 time (s) beta (deg) NO control Redistribution ESP

Figure A.23: Body slip angle

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 −300 −200 −100 0 100 200 300 time (s) corner radius (m) NO control Redistribution ESP

Redistribution & ESP Reference

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10 11 12 13 14 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)

valve position (1 closed; 0 open)

10 11 12 13 14 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)

valve position (1 closed; 0 open)

Figure A.25: Valve position

10 11 12 13 14 15 0 100 200 300 400 500 time (s) Mb 12 (N*m) 10 11 12 13 14 15 0 100 200 300 400 500 600 700 time (s) Mb 12 (N*m)

Redistribution & ESP ESP

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10 11 12 13 14 15 0 100 200 300 400 500 600 time (s) Mb 21 (N*m) 10 11 12 13 14 0 50 100 150 200 250 300 350 400 450 500 time (s) Mb 21 (N*m)

Figure A.27: Brake torque applied to the rear-inner wheel

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 −90 −80 −70 −60 −50 −40 −30 −20 −10 0 10 time (s) α11

(deg) NO controlRedistribution

ESP

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10 11 12 13 14 15 −4 −3 −2 −1 0 1 2 3 4 time (s)

yaw accel. (rad/s

2) 10 12 14 16 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 time (s)

yaw accel. (rad/s

2)

NO control Redistribution

10 11 12 13 14 −1 −0.5 0 0.5 1 time (s)

yaw accel. (rad/s

2) 10 11 12 13 14 15 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 time (s)

yaw accel. (rad/s

2)

ESP _{Redistribution & ESP}

Also in this case the results provided by the simulations are quite good, in concordance with the results obtained in the previous simulations. Just the performance of the rear brake is not perfect (fig.A.27), since it is activated too many times, and in the car equipped

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A.5 Avoidance Manoeuvre (205/55-R15)

In this simulation we tested the vehicle in conditions of continuously variable steer input. In fact the steer function is the same as that imposed in the manoeuvre illustrated in the

main report (“Avoidance Manoeuvre” fig.6.27). The initial speed is 60 m

s, and the vehicle is equipped with tyres which size is 205/55-R15.

0 100 200 300 400 500 600 −15 −10 −5 0 5 10 15 20 distance (m) distance (m) NO control Redistribution ESP

Figure A.31: Driven paths

5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 −25 −20 −15 −10 −5 0 5 time (s)

beta (deg) Redistribution

NO controller ESP

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6 7 8 9 0 100 200 300 400 500 600 700 800 900 time (s) Mb 11 (N*m) 5.5 6 6.5 7 7.5 8 0 200 400 600 800 1000 time (s) Mb 11 (N*m)

Figure A.33: Brake torque applied to the front-left wheel

6 7 8 9 10 0 100 200 300 400 500 600 700 800 900 time (s) Mb 12 (N*m) 5 6 7 8 9 10 0 100 200 300 400 500 600 700 800 900 time (s) Mb 12 (N*m)

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6 7 8 9 10 0 100 200 300 400 500 600 700 800 900 time (s) Mb 21 (N*m) 5 6 7 8 9 0 100 200 300 400 500 600 700 time (s) Mb 21 (N*m)

Figure A.35: Brake torque applied to the rear-left wheel

5 6 7 8 9 0 100 200 300 400 500 600 700 time (s) Mb 22 (N*m) 5 6 7 8 0 100 200 300 400 500 600 700 time (s)

Mb

22

(N*m)

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5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 time (s)

5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 time (s)

Redistribution Redistribution & ESP

Figure A.37: Valve position

We want to remark a notable handling improvement supplying the car with the ESP or with both the Active Yaw Controls, in fact we manage to make the trajectory much narrower (fig.A.31) than the one driven along by the uncontrolled vehicle. Furthermore in fig.A.32 we can realize that just the redistribution of torque cannot correct the vehicle behavior such a way to make the car motion acceptable.