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# Vehicle Dynamics – It’s all about the Calculus… PowerPoint PPT Presentation

Vehicle Dynamics – It’s all about the Calculus…. J. Christian Gerdes Associate Professor Mechanical Engineering Department Stanford University. Future Vehicles…. Clean Multi-Combustion-Mode Engines Control of HCCI with VVA Electric Vehicle Design. Safe By-wire Vehicle Diagnostics

Vehicle Dynamics – It’s all about the Calculus…

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## Vehicle Dynamics – It’s all about the Calculus…

J. Christian Gerdes

Associate Professor

Mechanical Engineering Department

Stanford University

### Future Vehicles…

Clean

Multi-Combustion-Mode Engines

Control of HCCI with VVA

Electric Vehicle Design

Safe

By-wire Vehicle Diagnostics

Lanekeeping Assistance

Rollover Avoidance

Fun

Handling Customization

Variable Force Feedback

Control at Handling Limits

### Electric Vehicle Design

• How do we calculate the 0-60 time?

### Basic Dynamics

• Newton’s Second Law

• With Calculus

• If we know forces, we can figure out velocity

### What are the Forces?

• Forces from:

• Engine

• Aerodynamic Drag

• Tire Rolling Resistance

### Some numbers for the Tesla Roadster

• From Tesla’s web site:

• m = mass = 1238 kg

• Rgear = final drive gear ratio = 8.28

• A = Frontal area = Height*width

• Overall height is 1.13m

• Overall width is 1.85m

• This gives A = 2.1m2 but the car is not a box. Taking into account the overall shape, I think A = 1.8 m2 is a better value to use.

• CD = drag coefficient = 0.365

• This comes from the message board but seems reasonable

### More numbers for the roadster

• From other sources

• rwheel = wheel radius = 0.33m (a reasonable value)

• Frr = rolling resistance = 0.01*m*g

• For reference, see:

http://www.greenseal.org/resources/reports/CGR_tire_rollingresistance.pdf

• r = air density = 1.2 kg/m3

• Density of dry air at 20 degrees C and 1 atm

• To keep in mind:

• Engine speed w is in radians/sec

• The Tesla data is in RPM

• 1 rad/s = .1047 RPM

• (or 0.1 for back of the envelope calculations)

• 1mph = 0.44704 m/s

### Motor issues

• The website lists a motor peak torque of 375 Nm up to 4500RPM. This doesn’t match the graph.

• They made changes to the motor when they chose to go with a single speed transmission. I think the specs are from the new motor and the graph from the old one.

• Here is something that works well with the new specs:

### Results of my simulation

• Pretty cool – it gives a 0-60 time of about 3.8s

• Tesla says “under 4 seconds”

• Top speed is 128 mph (they electronically limit to 125)

### P1 Steer-by-wire Vehicle

• “P1” Steer-by-wire vehicle

• Independent front steering

• Independent rear drive

• Manual brakes

• Entirely built by students

• 5 students, 15 months from start to first driving tests

steering motors

handwheel

### Future Systems

• Change your handling… … in software

• Customize real cars like those in a video game

• Use GPS/vision to assist the driver with lanekeeping

• Nudge the vehicle back to the lane center

handwheel

handwheel angle sensor

handwheel feedback motor

shaft angle sensor

steering actuator

power steering unit

pinion

steering rack

### Steer-by-Wire Systems

• Like fly-by-wire aircraft

• Motor for steering wheel

• Like throttle and brakes

• Diagnosis

• Look at aircraft

a

b

b

ar

d

V

af

r

### Bicycle Model

• Basic variables

• Speed V (constant)

• Yaw rate r – angular velocity of the car

• Sideslip angle b – Angle between velocity and heading

• Steering angle d – our input

• Model

• Get slip angles, then tire forces, then derivatives

### Vehicle Model

• Get forces from slip angles (we already did this)

• Vehicle Dynamics

• This is a pair of first order differential equations

• Calculate slip angles from V, r, d and b

• Calculate front and rear forces from slip angles

• Calculate changes in r and b

a

b

b

ar

d

V

af

r

ar

d+ af

### Lateral Force Behavior

• ms=1.0 and mp=1.0

• Fiala model

### When Do Cars Spin Out?

• Can we figure out when the car will spin and avoid it?

loss of control

linear

nonlinear

### Lanekeeping with Potential Fields

• Interpret lane boundaries as a potential field

• Add this force to existing dynamics to assist

• System redefines dynamics of driving but driver controls

### Lanekeeping Assistance

• Energy predictions work!

• Comfortable, guaranteed lanekeeping

• Another example with more drama…

### Handling Limits

• What happens when tire forces saturate?

• Front tire

• Reduces “spring” force

• Loss of control input

• Rear tire

• Vehicle will tend to spin

• Loss of stability

handling limits

linear region

Is the lanekeeping system safe at the limits?

### Countersteering

• Simple lanekeeping algorithm will countersteer

• Large heading error will change direction of steering

• Lanekeeping system also turns out of a skid

Lateral

error

Projected

error

Example: Loss of rear tire traction

### Yaw Stability from Lanekeeping

Lanekeeping Active

Lanekeeping Deactivated

Controller countersteers to prevent spinout

### A Closer Look

Controller response to heading error prevents the vehicle from spinning

### Conclusions

• Engineers really can change the world

• In our case, change how cars work