Handling and stability of Automobiles

Question 1

How do you find the differential of a rotating unit vector i with angular velocity ωk?

Answer 1

di/dt = ωk x i (analogous to v=ωr)

Question 2

What is the angular version of F=ma?

Answer 2

Torque = Moment of Inertia x Angular acceleration

Question 3

What are the 2 assumptions in the ‘Bicycle’/planar vehicle model?

Answer 3

-Small slip angles - Linear creep
-Neglect tyre realigning moments, longitudinal or spin creep
(also forward velocity is assumed constant)

Question 4

Why can a four wheeled vehicle be collapsed down to a bicycle in the bicycle model?

Answer 4

If the slip and steer angles are relatively small, both left and right tyres on each axle experience the same slip angle.

Question 5

What is in the data book is also referred to as the slip angle?

Answer 5

The lateral creep ratio alpha

Question 6

How to find force from slip angle alpha

Answer 6

multiply by the corresponding side force coefficient Cf or Cr

Question 7

How do you get the total cornering stiffness C?

Answer 7

Add the front and rear cornering stiffnesses, Cf and Cr

Question 8

How do you make the equations of motion for the bicycle model?

Answer 8

There are 2 equations of motion, one for Y force and one for moment N. They mare made using simple “F=ma” and into this use the equation for acceleration in moving coordinate frame on the datasheet for the COG, and the tyre forces from Cα, and moment of inertia needs to be considered for the moment equation. Note that αf and αr need to be substituted for as they relate to the forward velocity of the vehicle.

Question 9

How do you convert the equations of motion for the bicycle model to matrix form?

Answer 9

Use the top line for Y force and bottom line for N moment. Then use column vectors for the lateral acceleration, angular acceleration and one for lateral velocity and angular velocity, then fill in the 2x2 matrixes for the coefficients and put the constants on the RHS.

Question 10

What assumptions are used to find the stability conditions from the matrix form of the equation of motion?

Answer 10

Y=0, N=0, δ=0

no side force, moment or steering angle is applied to the vehicle

Question 11

What are the characteristic solutions that are used to find the stability conditions, and what does this allow you to do to the matrix form of the equation of motion?

Answer 11

v = vo * e^λt
Ω =Ωo * e^λt
It allows you to reduce it to one matrix/column vector of coefficients(v0 Ωo) = 0, called the characteristic equation.

Question 12

How do you find the coefficients for the Routh-Hurwitz equations?

Answer 12

you set the determinant of the 2x2 matrix in the characteristic equation (derived from the matrix form of the equations of motion by assuming characteristic solutions for v and Ω, and setting Y=N=δ=0) and the coefficients of λ^2, λ, constant give a2, a1, a0 respectively.

Question 13

What does the Routh-Hurwitz criterion reduce down to?

Answer 13

The vehicle will always be stable if:

bCr >= aCf

This means that if similar wheels (ie. Cr=Cf) are used, the COG position should FORWARD of the mid point of the wheel base.

Question 14

What is the equation for static margin?

Answer 14

static margin = -s / l

where: s = (aCf - bCr) / (Cf + Cr)
l = wheelbase

Question 15

What is the neutral steer point?

Answer 15

The point on the vehicle for which a side force produces no steady state yaw (the vehicle moves at an angle across the road, but doesn’t change direction), and occurs when x (distance forward of COG that the side force acts) equal the static margin.

Question 16

What happens if the side force is applied forward of the neutral steer point?

Answer 16

x > s and the front of the vehicle is pushed away

Question 17

What happens if the side force is applied behind the neutral steer point?

Answer 17

x < s and the rear of the vehicle is pushed away

Question 18

When does understeer occur in terms of the neutral steer point?

Answer 18

Understeer happens when s < 0, therefore the static margin (-s/l) is > 0, and thus the neutral steer point is behind the COG and the vehicle will understeer

Question 19

What is the static margin?

Answer 19

Static margin is given by the equation SM = -s/l, and is a measure of handling behaviour. It is a measure of how far the neutral steer point is BEHIND COG, normalised by the wheelbase l.

Question 20

When does oversteer occur in terms of the neutral steer point?

Answer 20

Oversteer happens when s > 0, therefore the static margin (-s/l) is < 0, and thus the neutral steer point is ahead of the COG and the vehicle will oversteer

Question 21

Basic equation for angular velocity

Answer 21

Ω = v/R

Question 22

Basic equation for angular acceleration

Answer 22

Ω^. = v^. (lateral acceleration=radial acceleration when travelling in a circle )
Note, these equal zero when the vehicle is traveling in a circle at steady state.

Question 23

How do you find the inverse of a 2x2 matrix?

Answer 23

Switch the values on the leaving diagonal, swith the signs of the values not on the leading diagonal and multiply by 1/ determinant.

Question 24

How to find if a car oversteers or understeers in terms of steering angle

Answer 24

Substitute the steady state cornering conditions: Ω = v/R, Ω^. = v^. = 0 into the matrix form of the equations of motion (rearranging slightly by taking out the 1/u factor and substituting β= Vss/u, 1/R = Ω/u, and setting Y=N=0) Then substituting into equations previously obtained for the general case of Ω/u (sorry!) from which an equation for steering angle δ can be obtained. Differentiate this wrt speed and this will tell you if its under or oversteering

Question 25

What influence can non-linear tyres and freedom to roll have on real cars in terms of under and oversteer?

Answer 25

It means that cars don't exhibit the same behaviour, either understeer or oversteer at all speeds, but can tend to understeer at low speeds and oversteer at high speeds.

Question 26

How do you form a handling diagram?

Answer 26

you take the original diagram of steer angle against speed, shift the x axis up by l/R (the steering angle at the neutral steer point), lot U^2/R (lateral acceleration) on the x axis instead of just U, and then rotate the whole plot by 90 degrees. This will create straight lines for the ideal linear vehicles, with -ve gradient for understeering and +ve gradient for oversteering vehicles.

Question 27

What is the first step when trying to find the equations of motion for a system?

Answer 27

Identify the number of degrees of freedom as this will give you the number of equations, and identify the variables/coordinates that define the system. Then set out to find force balance/moment balance equations for different directions/points in the system and these will act as the basis for the equations of motion.

Question 28

What does the fact that something is free to move about an axis tell you?

Answer 28

That there is no resultant moment about that point, so you can sum the moments about that point to 0.

Question 29

If one object is rotating while connected to another object which is rotating, what's the total rotational acceleration?

Answer 29

The sum of the two angular accelerations (make sure to get sign convention correct)

Question 30

When dealing with moments of inertia, what do you do when finding the moment at a particular point away from the spinning object?

Answer 30

You still transfer the moment of inertia to that point as it still has an effect (this is unlike a moment, which you would not translate to this different point)

Question 31

When dealing with sinusoidally varying inputs, what characteristic form of solutions should you use?

Answer 31

V = Vo e^iωt