Vehicle Dynamics – It’s all about the Calculus…

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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

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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
• 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
Calculate Slip Angles

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?
Comparing our Model to Reality

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