Stability, Ridability, Understeer &

Oversteer

Human Powered Vehicle: LSR-2000

Vehicle Dynamics

Technical Contribution

By: Jeremy Gramling

March 3, 2000

IntroductionThe purpose of this paper is a continuing view of understeer,

oversteer, stability and ridability. Specifically as understeer and oversteer

relate to the 2000 Human Powered Vehicle (HPV) tricycle and how stability

and ridability relate to the 1999 Land Speed Record (LSR) bicycle. In order

to look at what understeer & oversteer are and how they relate to a vehicle's

cornering we must first understand what’s affecting cornering. To do this we

will look at steady state cornering condition because it is easier to explore,

model, and understand. Assuming steady state, factors related to steering

will be examined first to gain a better understanding of what is affecting

cornering. Then the 1999 LSR will be examined for problems, it’s stability

calculated and what target speed is attainable. Steering torque for steering

design will be cover along with steering, and seat back & seat support

designs. The question of whether to heat-treat the LSR will be examined.

Factors Relating to Steering/ CorneringTo understand cornering, how a vehicle turns will have to be looked

at first. The key components for a vehicle to negotiate a corner are: the steer

angle (), Ackerman angle, slip angle (), cornering stiffness (C),

cornering force (Fy), centripetal acceleration, load on an axle (W),

wheelbase (L), radius of turn (R), and forward speed (V).

The steer angle (), in degrees, is the angle that a tire makes with

relation to the wheelbase axis. The steer axis is related to the Ackerman

angle in that it is when a line perpendicular to both the front and rear tires is

drawn that intersects at the point about which the turn is occurring. This is

the desired affect when turning because it reduces tire wear, allows for

proper centering torque, and yields an increasing steer torque with increasing

steer angle. From this point on it will be assumed that the front wheels of the

tricycle can be modeled as one because the difference of the outside and

inside tire’s slip angle at high speed (greater than 5-10 mph or parking lot

speed) will be minimal. Under this assumption the Ackerman angle is given

by:

= L/R

(1)

There are two forces that are being created during a turn, one is the

corning force (Fy), in lbf, and the other is the centrifugal force (or force

related to centripetal acceleration = mV2/R). This force is a lateral force that

acts perpendicular to the tire’s direction of heading in toward the center of

the turn. For steady state the sum of the cornering forces must equal the

centrifugal force. Associated with the lateral force is the slip angle (), in

degrees, and the cornering stiffness (C), in lbf/degree. The slip angle is the

angular displacement between the plane of rotation of the wheel (the

direction the rim is pointing or direction of heading) and the path that the tire

will follow on the road (direction of travel). The corning stiffness is a

proportionality constant and is related to the tires. The equation for

cornering force is given by:

Fy = C *

From the graph below it can be seen that at slip angles of less than 5 degrees

the relation between slip angle and later force is linear which can be seen by

the above equation.

Relating Newton’s Second Law to steady state cornering the equations for

cornering and the geometry of that cornering can be produced. Centripetal

acceleration times the mass must equal the sum of the forces in the lateral

direction (i.e. front and rear lateral forces).

Fy = Fyf +Fyr = m*V2/R (3)

Where m is the mass in lb., V is the forward velocity in ft/sec and R is the

radius of the turn in feet. Also, the sum of the moments from the lateral

forces must equal zero, which can be calculated about the center of gravity

(CG) using the distances c and b. This yields:

(Fy)r = (m*b/L)( V2/R) & (Fy)f = (m*c/L)( V2/R) (4)

In the above equation L is the wheelbase in feet. From the above equation,

the portion of the vehicle's mass the front axle carries, Wf/g, is [m*c/L] and

likewise the rear axle, Wr/G. The slip angle for the front and rear axis can

now be generated using the above equations.

Slip rear: r = Wr/G * V2/(R*Cr)

Slip front: f = Wf/G * V2/(R*Cf) (5)

This can all be related back to the steering geometry to find the steer angle

for both the front and rear wheel.

= (180/)(L/R) + f - r

With substitution:

= (180/)(L/R) + (Wf /Cf –Wr /Cr)*V2/(G*R) (6)

Having developed the factors related to steering and cornering; it is now

possible to look at how the vehicle steers. However, it is import to realize

that it is quiet possible for a vehicle to steer in all three of the following types

if steering without physically changing any of the above factors.

UndersteerUndersteer is considered to be essentially a stable condition. When

understeering is in effect the vehicle follows a greater radius circle than that

of the steering angle or the front wheels. Another way to look at it is the car

turns wider than the driver inputs or intends. Therefore, since the slip angle

of the front tires is greater than the slip angle of the rear tires the driver will

have to increase the steer angle as speed increases to maintain a constant

radius circle. Relating this to equation (6), it can be seen that for understeer

the load on the front axle must be larger then the load on the rear axle,

Wf /Cf >Wr /Cr.

During understeer conditions a driver often comments on how the

vehicle feels tight, which makes sense due to the reduced rear slip angle but

this is still an understeer condition. For a driver aware of the understeer

condition they can compensate by adjusting their corner entry speed and

steering angle, which will head the vehicle in the intended corner regardless

of where the front wheels are pointing. In addition, if there is room to play

in the corner they can reduce speed to recover if the radius of the turn is

tighter than expected or poor judgement was made. It is important to avoid

excessive understeer because it causes a large front tire scrub and may

require slower cornering speeds.

To attain an understeer condition there are a variety of modifications

that can be made. When making changes be sure to make one change at a

time and then record the results before proceeding to another change. Start

by lowering the tire pressure in the front wheels and raising the pressure in

the rear wheels. For more understeer move more weight to the front of the

vehicle. Another change is to decrease the width of the front tire and

increase the width of the rear. These changes assist in allowing the

understeer of the car to be controlled.

OversteerOversteer is a condition that can be very unstable. A vehicle

experiencing oversteer is trying to spin, the spin must be stopped before

directional control can be of concern and often by this point the vehicle has

left the track. Oversteer in a constant radius turn is when the slip angle of

the rear tire becomes greater than the slip angle of the front tire. Another

way to look at this is that the vehicle is turning too far into the apex of the

turn for a lesser steering angle. To compensate for this the driver must steer

less and often in the direction that the rear of the vehicle is moving to

maintain a constant radius turn and/or keep the vehicle from spinning.

Relating this to equation (6) it can be seen that for oversteer the load on the

rear axle must be larger then the load on the front axle, Wf /Cf <Wr /Cr.

For the oversteer condition the driver will likely say that the vehicle

feels loose. This is because he/she is wrestling with the vehicle to get it

through the corners. If the vehicle becomes to loose or the driver is unable to

wrestle the vehicle through a corner the vehicle will spin, putting the driver

at possible grave danger. However, oversteer is often desired when coming

out of a corner because the vehicle straightens into the straight faster. This

yields a longer straightaway distance.

To attain an oversteer condition there, are a number of variables that

can be adjusted. The front tire pressure can be raised and the rear tire

pressure can be dropped, the width of the front tires can be increased and the

rear can be decreased, and the load should be moved back to the rear if

oversteer is desired.

Neutral SteerNeutral steer is the condition existing between oversteer and

understeer. It’s something of a continental divide by the fact that you are

either on one side or the other, but the highest point is still the continental

divide. Neutral steer is where the slip angle of the front wheels and the rear

wheels are equal. This mean that the load between them is equal (assuming

the same cornering stiffness), Wf /Cf = Wr /Cr. Therefore, on a constant

radius turn as speed is increased there is no steering adjustment required.

Under this condition the driver should feel that the vehicle is

responding precisely to his inputs. The driver should not need to make

adjustments while cornering at any speed.

To attain neutral steer the adjustments for oversteer need to be moved

toward understeer and/or the adjustment for understeer need to be moved

toward oversteer.

Analysis of LSR ProblemsWhen examining the 1999 LSR there were many problems

discovered. First, from our experience riding the LSR it was found to be

unstable and rather unridable for a majority of riders. Then there is the

problem of the rear triangle to main tubing weld cracking which was do to

never being heat-treated. Another problem was that the handle bar welds

were also crack. The front wheel of the LSR did not “flop” (which will be

detailed later). The steering design was a major contributing factor to the

bikes unstability and ridability. Therefore, a new design had to be produced

along with a back support and seat design. The LSR had no operational

brakes or shifting. The headset was found to be broken and will have to be

replaced. Also, the front tire was not that of the proper race setup and will be

replaced.

LSR 1999 RecommendationsRecommendations for the 1999 LSR are as follows. First, a stability

criterion of between 1 and 3 must be achieved. Second, a steering design

that allows for proper balancing, counter force application, smooth steering

and minor steering adjustments for small directional changes. Next, an

adjustable back angle backrest must be designed that will also be adjustable

for differing height riders. The seat design must also follow those same

lines.

LSR StabilityThe LSR is unstable for a number of reasons. The initial problem is

that the friction created by the broken headset is negating the steering torque

and it’s ability to steer smoothly. The current steering design does not allow

a person to balance himself or herself well and does not allow for good

counter steering forces. This is seen in the current design where the hands

are positioned too close together. The stability is also severely hampered by

lack adjustability where a rider’s positioning is involved.

A key factor when analyzing stability is that of the stability criterion

(U) or “flop”. When looking at the Jones Stability criterion chart (seen

below), the LSR can be plotted to help determine its stability. The stability

criterion (U) determines whether a bike is stable of not. If U is positive the

bike is unstable as compared to a negative value of U which yields a stable

bike. To do this the relative frontal projection must be found by dividing the

head tube to hub offset by the diameter of the front wheel. This is then

plotted against the head angle, H. The following plot shows where the LSR

falls in relation to other common bikes of today.

Steering torque is another key factor in bicycle stability. In relation

to the LSR, the steering torque was negligible do to the high friction created

by a broken head set. A steering torque is created when the vertical force

(Fv) and the head tube angle are not in the same plane. This means that when

you lean a bike a steering torque is created about the head tube that causes

the bike to track the arc of a circle. If a steering torque is not created a rider

would fall when they leaned. The steering torque is calculated using the

following equation:

TH = C * Fv * sin H (7)

In equation (7) the trail (C) is multiplied by the vertical force (Fv) and the

head tube angle. As the trail increases do does the stability of the bike.

However, if the trail becomes too large the bike becomes to stable and is

difficult to steer.

C

Steering DesignThe steering was designed to improve upon hand positioning. This

was accomplished by having a wider cross bar and a better hand grip angle.

The handgrips are positioned so that the force exerted on the handlebar acts

tangentially to the arc about the head tube the handlebars makes. The wider

crossbar will allow a rider to apply countering forces increasing stability

through smoother steering. Also the wider and more accurate steering design

will allow minor adjustments in steering to be made again increasing

stability. The following is a sample calculation to show how the handgrip

angle was found.

This CAD drawing is the design for LSR’s new steering mechanism is:

L

W/2

R

Cross Bar

Handgrip

Seat and Back Support DesignThe back support was designed with adjustability in mind for two

major reasons. First, to allow individual riders to make adjustments in their

back angle to maximize their balance. Second, to allow for different height

riders to be able to achieve the same level of stability. To do this a back

plateform will have four threaded rods attached to it. Two threaded rods at

the top and two at the bottom, these rods will run threw a drill pipe that is

weld horizontally at the top and bottom of the rear triangle. Using different

sized spacer at the top and bottom will allow you to move the back support

forward and backward as well as change the back angle by varying the top

and bottom spacer size. This can be seen in this CAD drawing.

The seat design was chosen to be a regular bike seat for several

reasons. First, a bicycle seat will apply the same pressure points whether in

an upright or recumbent position, however the amount of pressure will vary

slightly. This will help keep a riders muscles from tiring prematurely and a

rider should not find the seat causing any pain as is typically true of exercise

Back platform

Threaded Rods

Horizontal Pipe

equipment that has been trained on. Second, ease of adjustibility. The seat

can be slid on it’s rails for different sized riders. And, individual riders can

use seats of there own preference. This is because some riders like more

buttocks support as compared to those who like a thin narrow seat to

straddle. With all of these put together the LSR should be very adjustable

for any rider which will increase the stability.

Target SpeedA target speed for a stable and ridable LSR was deemed to be

necessary. Therefore, using the power equation

P * = (v * Cr * m * g) + (.5 * * CD * A * v3) (8)

Using measured data and tabulated data from “Scientific America” for a

recumbent bike identical to ours (CD =.77, Cr =.005, =95%, A =3.8). From

this a plot was constructed of the power generated, in watts, versus the speed

of the vehicle in miles per hour. That graph is seen below and from it, it can

be seen that at a maximum power output of 1100 watts the 1999 LSR will

achieve speeds of 40 mph.

HPV 2000 Tricycle RecommendationsThe following graph shows the effect of each of the three steering

scenarios as speed increased.

From the graph it can be seen that as speed increases there becomes a critical

point where vehicle control is lost with oversteer and the inability to make a

corner at such high speeds for understeer. With all of these things in mind, it

appears that the neutral steer condition is the optimal condition for steering

and cornering. This would especially be true for more inexperienced drivers

and without prior known corner geometry. However, while driving straight

the best situation for the vehicle is to understeer lightly. This is so that

driver’s inputs are minimized on the vehicle when there are sudden lateral

forces such as bumps, wind, road camber changes or aerodynamic

disturbances.

Therefore, for the 2000 HPV it is my recommendation that we

attempt to attain as neutral steering a vehicle as possible erring toward

understeer if erring at all.

Heat-TreatmentAs mentioned earlier the LSR’s welds were never heat-treated and

consequently failed during the competition. Therefore it is our view that the

LSR must be heat-treated. The tubing supplied by Easton is 7075-T6

aluminum. The T6 extension means that the metal has already been heat-

treated. In the case of T6 Al. The metal will have an improved yield strength

of 5 times, tensile strength will increase by a factor of three, and the hardness

will more than double all from the “O” state or fully annealed state.

Aluminum 7075 Yield Strength (psi) Tensile Strength (psi) Rockwell Hardness

Elongation in 2”

O: 15000 33000 60

17%

T6: 73000 83000 120

11%

Since the aluminum welds on the LSR are in the “O” state and the

frame itself is in the “T6” the decision was made to return the entire vehicle

to the fully annealed condition and bring it all back to the “T6” state. The

following is the necessary procedure with times and temperatures.

I. Return to the fully annealed state

Heat oven to 415o C and hold for 2-3 hours

Cool uncontrolled until a temperature of 205o C is

reached

Reheat to a temperature of 230o C and hold for 4

hours

** This will yield a fully annealed or “O” state.

II. To achieve the “T6” conditions

Heat oven to 120o C and hold for 3 hours

Increase temperature to 175o C for an additional 3

hours

Then quench

With the entire LSR in a T6 state there will no longer be a concern of

weld failer do to yield strength.

Measurements & TestsCornering stiffness plays a key roll in calculating the lateral forces,

which can be seen in equation (2) above. A question of how to

experimentally measure this on the 1999 HPV and on the 2000 HPV is

needed in the continued attempt to optimize the HPV for speed and handling.

To determine what type of steering the vehicle has you need only to

steer through a given corner at moderate speed and see if you need to apply

more steering input (understeer), less steering input (oversteer) or no steering

input (neutral steer).

The major difficulty is a method by which to find the corning

stiffness. One possibility, for a fixed radius circle and fixed speed, is to

attach a protractor type device to the axle right next to the wheel. Then a

needle with ink on the end could be attached to the wheel. From this you

should be able to find what the steer angle of the vehicle is and using

equation (6) find the cornering stiffness. This would only work at low speed

for a perfect Ackerman angle or neutral steer scenario.

This is something that future HPV teams should consider trying to

measure in order to assure increasing performance in our HPV’s handling

and stability characteristics. Also when looking into the yield strength of

their HPV’s I recommend doing sample welds and performing Rockwell

hardness tests on those sample to determine what stresses the vehicle can

handle.

BIBLIOGRAPHY

Jenkinson, Denis The Racing Driver, 1st ed., (MA): (Publisher), 1958.

Smith, Carol Tunee To Win, 1st ed., (MA): (Publisher), 1958.

Jenkinson, Denis The Racing Driver, 1st ed., (MA): (Publisher), 1958.

Whitt, Frank R. and Wilson, David G. Bicycling Science.

2nd ed., MIT Press, 1982.

Smith, Chris. Theory and the Art of Communications Design. State of the University Press, 1997.

Gillespie, Thomas D. Fundamentals of Vehicle Dynamics. SAE, 1992.

ASM Handbook. 10th ed., ASM International, 1991