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Page Nomenclature 
g = gravity, 32.2 ft/secsec

H = height of c.g. above
datum, in.

µ = mu = coefficient of
friction

SF = rollover safety factor,
%

SUV = sport utility vehicle 
T = wheel track, in. 
Rollover safety factor
It is now useful to further discuss the concept of
safety factor, SF. Safety factor, in
engineering, refers to the number of times better the specifications are to be
assigned to a part, than those specifications that part needs to minimally do the job
assigned. A bolt, for example may have to
carry a load of 1000 pounds, but one may be fitted strong enough to handle
3000 pounds. The bolt is then said to have a safety factor of 3. Safety
factor therefore makes a part more reliable.
I wish I could say it was possible to have a rollover
safety factor of 3, but it isn't. The physics of the problem tell us that,
given the dimensional restraint of 80 inch vehicle width and practical minimum
ground clearance, about all that can be done with cars is a SF less than or equal to
75%. It will be seen by calculation that SF is far less, possibly inadequate, for most SUVs.
You may skip this paragraph, if desired. Explaining
safety factor is best done not by description, but by vector analysis.
Achieving 75% is not easy and I don't know if any Big Three cars can approach
it. To give 75% some meaning, this really translates into 1.75 x µ. And that
turns out to equal 1.3 g. If a car could somehow be able to boost its
grip on the road, that is, increase µ from 0.75 to 1.3, it would be able to overspeed
through a curve fast enough to generate 1.3 g centrifugal force and not tip.
But since µ is fixed at a maximum of about 0.75, the car will surely slide,
not tip. Even though the car itself could take more centrifugal force, the
tiretoroad, friction gives up and the car slides. This is what we want.
Mathematically speaking, V sub t is greater than or equal to V sub s, where V
= velocity, t is tipover and s is sliding. The reason for this explanation is
to satisfy those who need to be assured that the formulae are rational and
that they have merit. SF is not an imaginary concept; it has real world value
because it accurately predicts whether a vehicle will tip or remain upright
when it is mishandled.
The safety factor equations will yield anything from a
negative SF, 0, to a positive SF. Units of SF are expressed in %.
A table will follow below the formulae to help evaluate results of
calculations.
To help with the calculations, actual examples will be
used, with values substituted. The first example will be in detail. The
others, abbreviated. All are important to show the effect of changing the
magnitude of critical dimensions, even by relatively small values.
Lets start the math!
Obtaining dimensions of T and H
Only two measurements plugged into a simple
algebraic equation are needed to determine whether or not any vehicle will tip
over or slide in a situation which exceeds the physical limits of land
vehicles. The same equation will yield a
"safety factor" value which then can be compared with other vehicles
under consideration as well as judge whether or not the antiupsetting
properties of the vehicle is sufficient to be considered safe for passenger
car use.
Note: trucks used for cargo are in a different category; professional
truck drivers realize the inherent limitations of necessarily topheavy
vehicles to suit a commercial purpose; truck drivers are expected to drive
much more conservatively than drivers of regular passenger cars. The rule that a vehicle be very
tipover resistant applies only to those cars, SUV's, and pickup trucks that are
intended for and sold for passenger car use.
You need to obtain two dimensions for the
vehicle under investigation. See Fig. 5 above. The first measurement is designated: Wheel
track, T and the second is loaded vehicle center of gravity height, H.
Heretofore, no consideration was given to c.g. with respect to whether or not
the vehicle was at curb weight or maximum gross weight. From hereon, all
calculations involving SF must use H as calculated at maximum gross weight.
The reason for this will be obvious later as it will be shown that H increases
with load.
Getting T is easy. It will be found in the
general specifications of the owner's manual, in the shop service manual for
the car or on an internet file. It is also simple to just take a steel tape
measure and measure from tiretotire at the outside of the tread and subtract
the width of one tread to establish the centertocenter distance. The other
dimension, H is going to be more difficult to get, but it can be
done. Amazingly, H is not in any public source that I can determine,
despite its crucial importance. It is not in the manuals. Manuals
sometimes show a c.g. longitudinally for placing a car on a hoist safely, but
H is not dimensioned and often placed at some ridiculous,
arbitrary place on the drawings. The dealer will not have the information
either. Its my understanding that the Federal Agency, D.O.T. had to take their
own c.g measurements. Measuring H is difficult and requires very elaborate
equipment. If you are interested in a General Motors car, try going thru the
chain of command starting at the dealer,
progressing to customer service and then to engineering archives. Do not
accept the answer that the manufacturer doesn't know. General Motors, for
instance, uses two giant pendulums to swing the car under study. From the intervals of the
oscillations, complex calculations will yield H to within 1%. Remember, if you
do get H from the factory it represents vehicle empty (curb) weight H, not the
vehicle loaded weight H needed. It will be shown later how to convert from H sub curb weight
to H sub maximum gross weight, the value needed to substitute into the
equations.
Safety factor formulae
The two
formulae are presented as follows:
Evaluation of a compact station wagon
Let's now look at a practical
example. I will start by evaluating a compact station wagon.
Case
1
Compact Station Wagon (Front Wheel Drive)
Weight, curb 2600 lbs.
Load, max 1050 lbs.
Wheel track, T = 57.6 in.
Empty c.g. height, H = 25.0 in.
The first step is to recalculate the
c.g. to determine its new height after the vehicle is loaded to its
maximum weight as specified by the manufacturer. It should be noted that
many, if not all proving tests done by testing agencies fail to load the
test vehicle. If Consumer's Reports tester had loaded the controversial
Suzuki, there would have been no question at all as to its propensity to
tip over in hard "S" turns onto the fitted outrigger safety
device. Calculating loaded vehicle H, is not
difficult if the curb weight and empty vehicle H is known. Click_here
to see the method and calculations. Don't worry, you will be able to
return to this spot.
You now know the new H is 28 in. to be
used in the equations.
Substituting the know values into
equation (1):
SF% = (67 x 57.6/28) 
100, which reduces to
137.83  100 = 37.83,
rounded off to
38%
The second equation (2)
works as well. Any small difference is some rounding off taken in the
derivation of (1) above. This formula is exact. I suggest you use a
scientific calculator which allow you to insert all the parentheses. Be
sure to follow the exact format as in the illustration. Substituting
into (2), the key strokes are: 100
x ((57.6  (1.5 x 28) = / (1.5 x 28)) = 37
(percent) Please
note that the / symbol above refers to the "division" key! 
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