Stability, Ridability, Understeer & Oversteer
Human Powered Vehicle: LSR-2000
Vehicle Dynamics
Technical Contribution
By: Jeremy Gramling
March 3, 2000
Introduction
The 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/ Cornering
To 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.
Understeer
Understeer 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.
Oversteer
Oversteer 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 Steer
Neutral 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 Problems
When 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 Recommendations
Recommendations 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 Stability
The 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.
Steering Design
The 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:
Seat and Back Support Design
The 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 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 Speed
A 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 Recommendations
The 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-Treatment
As 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 7075Yield Strength (psi)Tensile Strength (psi)Rockwell HardnessElongation in 2”
O:15000330006017%
T6:730008300012011%
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.