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Overview

In this section we will delve deeper into the vehicle dynamics, examining stability and control parameters.

How will a vehicle behave in the middle of a turning manoeuvre? What determines whether the corner can be taken faster, or must be taken at reduced speed? What are the parameters that govern the vehicle in these situations and how do these parameters interact? These are questions that we aim to answer in this discussion.

Objective

Continue to examine vehicle stability and control characteristics.

Study time: 4 hours

Topic 1 - Stability parameters

In an effort to develop and verify the relationships affecting the handling of a high-performance vehicle, the Millikens designed, constructed and tested a unique vehicle with near infinitely variable suspension geometry. This vehicle was highly instrumental in aiding them to develop the relationships which went into their iconic text Race Car Vehicle Dynamics.

racing cars

(Milliken and Milliken 2006:498)

A number of relationships which the Millikens developed were Stability Factor, Understeer Factor, Stability Index, Static Margin, and Cornering Compliance are summarized in this table. Each of the parameters is related, as shown, to the ride frequency (ω_n) which has been previously discussed.

Under/oversteer factor	Natural frequency, $ω_{n}^{2} =$
Stability factor (K)	$\frac{C_{F} C_{R} ℓ^{2}}{m^{2} k^{2} V^{2}} (1 + {KV}^{2})$
Understeer factor $(\overline{K})$	$\frac{C_{F} C_{R} ℓ}{m^{2} k^{2} V^{2} g} (ℓ g + \overline{K} V^{2})$
Stability index (SI)	$\frac{(C_{F} + C_{R}) ℓ}{{mk}^{2}} (SI)$
Static margin (SM)	$\frac{(C_{F} + C_{R}) ℓ}{{mk}^{2}} (\frac{C_{0} ℓ g}{V^{2}} - SM)$
Cornering compliance (D)	$\frac{C_{F} C_{R} ℓ}{m^{2} k^{2} V^{2} g} (ℓ g + (D_{F} - D_{R}) V^{2})$

Where

$K = - (\frac{SM}{C_{0} ℓg}) {, 1/(ft./sec.)}^{2} (classical stability factor)$
$\overline{K} = - (\frac{SM}{C_{0}}), nondimensional$
$\frac{\overline{K}}{g} = K ℓ, 1/g$
$UG = 57.3 Kℓg = D_{F} - D_{R}$
$D_{F} - D_{R} = 57.3 \frac{\overline{K}}{g} = 57.3 Kℓ$
$k = radius of gyration in yaw$

(Milliken and Milliken 1995:242)

Term	Symbol	Units	Sign
CG location	a, b	ft.	always +
Wheelbase	ℓ	ft.	always +
Mass of vehicle (W/g)	m	slugs	always +
Weight of vehicle	W	lb.	always +
Gravitational acceleration	g	ft./sec.²	always +
Yawing moment of inertia (k²)M	l_z	slug-ft.²	always +
Radius of gyration in yaw	k	ft	always +
Yawing moment	N	lb.-ft.	+ for clockwise
Lateral force (also F_y)	Y, Y_F, Y_R	lb.	+ to right
Lateral acceleration	a_y	ft./sec.²	+ for RH turn
Lateral acceleration (a_y/g)	A_Y	g	+ for RH turn
Path radius	R	ft.	+ for RH turn
Path curvature	1/R	1/ft.	+ for RH turn
Vehicle velocity $(\sqrt{u^{2} + v^{2}}; V \approx u)$	V	ft./sec.	+ for forward
Yawing velocity	r	radian/sec.	+ for clockwise
Lateral velocity	v, v_F, v_R	ft./sec.	+ to right
Longitudinal velocity	u	ft./sec.	+ for forward
Steer angle front wheels	$δ$	radian	+ for clockwise
Slip angles	α_F, α_R	radian	+ for slip to right
Vehicle slip angle at CG	β	radian	+ for slip to right
Cornering stiffness (two tyres)	C_F, C_R	lb./radian	always -
Total cornering stiffness (C_F + C_R)	C	lb./radian	always -
Ackermann angle	ℓ/R	radians	+ for RH turn
Stability factor	K	$\frac{1}{{(ft./sec.)}^{2}}$	+ for US
Centrifugal force	CF	lb.	- for RH turn
Overall steering ratio	G	deg./deg.	always +

(Milliken and Milliken 1995:145)

We recognize that higher values of ω_n are usually desirable for performance vehicles because the higher ω_n corresponds to a lighter car with a suspension which is stiffer, and these are generally characteristics which improve performance.

These relationships, which the Millikens developed, allow us to make observations about a car’s transient response. Let us look at how we might do that.

Let us take the Stability Factor equation for example.

Rearranging it slightly, we have

$ω_{n}^{2} = \frac{C_{F} C_{R} l^{2}}{m^{2} k^{2}} [(1 + {KV}^{2}) / V^{2})]$

Consider the C_FC_R/m²portion of this term

For ω_n to be large, we would want the product of the product of the cornering stiffnesses (C_FC_R) to be high and mass, m, to be low, because these would both drive this portion higher, which would result in a higher ω_n.

stifness graph

Further, if we assume that a maximum cornering stiffness is available from a given tyre and suspension, then as the figure shows, the product C_FC_Rwill be highest when the two terms are most nearly equal. From this, we would conclude that best handling response, a lightweight car with equal front and rear cornering stiffness would be desirable.

Does this term C_FC_R/m²have any physical meaning? Well, we could think of it as the ratio of cornering ability to weight - sort of like a power-to-weight ratio for cornering.

How does the conclusion compare to reality? As the table below indicates, the higher the performance of the car, the larger this parametric term.

Type of vehicle	Parametric term
Mini Van	15500
Sedan	20500
Small sports car	44500
Large sports car	97500
Formula One car	1500000

Next look at the term l²/k² where l is the length of the car and k is the radius of gyration. So this term tells us how the weight of the car is distributed along its length.

In other words, for best handling, l²/k² should be large, meaning the weight should be near the centre of the vehicle, making a mid-engine car good. This is probably why most race cars strive to put the bulk of the weight near the centre of the car.

Does this make sense? Of course, it does - a car with a small moment of inertia should turn better because it will take less turning moment to get the car to rotate about its centre.

Does this make sense in terms of real cars? Yes, it does, as the table below indicates, the higher the performance of the car, the larger this parametric term.

Type of vehicle	Parametric term
Mini Van	3
Small sports car	5
Formula One car	8

Continuing, the velocity squared term in the denominator says the car is more stable at lower velocities, which makes sense.

Topic 2 - Understeer versus oversteer

Let us consider two handling characteristics, referred to as understeer, and oversteer. Either oversteer or understeer occurs when one end of the car is reaching the limit of its lateral force capability.

Understeer

Understeer occurs when the limit is first reached at the front, changes in slip angle no longer increase the destabilizing moment from the front. The stabilizing moment from the rear takes over, and the vehicle path tends to straighten, the vehicle aligns itself to the path and thus tends to drive off the outside of the corner. About all the driver can do is slow down, or break traction on the rear tyres to reduce their capability

understeer

Understeer

Text version

Understeer diagram showing steering input and motion of the vehicle

Oversteer

Oversteer occurs when the limit is reached first at the rear, increase in slip angle results in no more stabilizing moment from the rear and the destabilizing moment at the front takes over. The car tends to aim towards the inside of the corner, resulting in a tendency for the car to spin, with the tail of the car trying to pass the nose of the car. Steering control remains with the driver and proper corrective steering action may avert a spin.

oversteer

Oversteer

Text version

Oversteer diagram showing the steering input and motion of the vehicle.

Relative to the Stability term, K, the car is more stable in understeer (K positive) than oversteer (K negative). This makes sense because a car is indeed more likely to spin due to oversteer.

And clearly a positive K would generate a higher ω_n term than would a negative K.

Once a parametric relationship has been established, it is possible for us to use that relationship to endeavour to determine the effect of changing the independent parameters within the relationship, as we have just demonstrated.

Summary

Choosing the independent parameters affecting vehicle handling is an open-ended problem because there are so many parameters, many of which are interrelated. For this reason, a number of dependent parameters have been developed which can aid in the study of the interconnectedness of the various terms.

Reference and bibliography

Gillespie, T. (2000) Fundamentals of Vehicle Dynamics. Detroit USA: SAE International.

Milliken, W. & Milliken, D. (1995) Race Car Vehicle Dynamics. Detroit, USA: SAE International.

Milliken, W. (2006) Equations of Motion. Cambridge, USA: Bentley.

Seward, D. (2014) Race Car Design. London: MacMillan.

Parametric analysis of a dynamic system 3

Overview

Objective

Topic 1 - Stability parameters

Topic 2 - Understeer versus oversteer

Understeer

Oversteer

Summary

Reference and bibliography