It has frequently been observed that the front forks of bicycles often get bent backwards in a collision. It is reasonable to ask whether one can calculate the speed of the collision from the amount of deformation of the front fork. This became a subject for inquiry in the case of Johnson vs Derby. In JvD, Johnson, descending a hill on his bicycle, collided with a car ascending the hill that turned left in front of him. It was nighttime and the case concerned the reason why the cyclist was not using a headlamp, but the expert for the defense, James Green, attempted to show that Johnson was exceeding the speed limit, which was 35 mph at that location. Johnson's bicycle had little enough damage that the fork deformation could still be measured. Green set up an experimental track, along which he accelerated bicycles, loaded with iron weights to simulate Johnson's mass, into a steel plate. He measured the speed of impact and the amount of fork deformation, and concluded that Johnson was going faster than 41 mph at the point of impact. Green supported this calculation by another, supposedly based on basic physics, that showed that bicycles would, simply when coasting, accelerate on this grade to more than this speed.

Green reported his method of calculation from fork deformation in chapter 20 of the third edition of his book Bicycle Accident Reconstruction. The method of calculation is erroneous. Its largest error is assuming that in the collision the force required to bend the fork is the force that decelerates the cyclist and the bicycle. As a result of that error, Green calculates a force between the tip of the fork and the object collided against that is far too large. The bend strength of front forks runs from about 200 to 400 pounds (Green himself has measured this, to say nothing about many other engineers), but Green calculates this as 3,500 pounds and doesn't recognize the discrepancy. These erroneous calculations will be discussed in the next section.

The calculation that Green made of the bicycle's theoretical speed on this grade was the very elementary one of free fall in a vacuum. However, that is also erroneous, because it ignores all resistances, and air resistance is very significant. When a racing bicycle is being propelled at 25 mph, about 75% of the cyclist's power goes into overcoming air resistance. When that is included in a much more sophisticated algorithm, the calculated speed for Johnson's descent falls to the low 30s. An actual coasting test at this location, using a better bicycle than Johnson's, showed 33 mph at the point of impact.

However, it is possible that despite Green's errors in physics and mathematics, his experimental procedure could be reasonably accurate. Green reported this work in the fourth edition of his book, and the mathematical errors in this are discussed in the second section. One peculiarity about Green's calculations is that although he intended to calculate the speed from the deformation, he never went beyond the mathematics to calculate the deformation from the speed, which is of no use at all to the accident investigator. I have had to convert Green's mathematical expression into the required one in order to compare it against another that I prepared from his data.

Two entirely different issues need to be discussed. The first is the internal consistency of Green's procedure. Does his formula reasonably predict the results of his experiments? Green claims that his formula is so accurate that all of his data points are within 5% of the formula's prediction. Green's formula predicts deformation from speed, which is not the desired goal, but that is obviously the way that Green claims to have calculated. The attached calculation sheets show that in this calculation the averge error is 26%, with greatest errors of 112% and 87%. With a little understanding of the physical principles involved I produced a formula with average error of 14% and greatest error of 23% from the same data points. When the data are considered from the standpoint of predicting speed from deformation, Green's formula must be converted by mathematical methods. When this is done, his average error falls to 7%, with the two worst errors of 24% and 19%. Using a similar method to my previous one, I produced a formula that has average error of 6% with two greatest errors of 14% and 10%.

Finally, Green produces a formula that is simplified from that produced by the derivation process, a formula that directly predicts speed from deformation. "Numerous frontal impact tests in the laboratory ... reveal that fork deflection varies with velocity in the following relationship ..." giving a formula that simplifies down to:

speed = SQRT(1078 * deformation) / 1.47.

This is followed by the sentence "Laboratory data indicates that this method yields a deviation from actual impact speed of only three percent," This is surely an error, because when the deformations and impact velocities from Green's own data are compared against this formula, the formula has an average error of 86%. On average, the calculated speeds are only 16% of the actual measured speeds.

It is extremely difficult to square these numerous errors with either engineering competence or even with honesty.

We must remember that these are only the differences between the calculated speed and the measured speed under the conditions of Green's experimental procedure. They say nothing about the differences between Green's experimental procedure and real life.

The far more important point to consider is whether Green's experimental method is either generally useful, or even whether it reasonably simulated the facts of Johnson's accident. We know that it did not. It predicted a speed of greater than 41 mph on a slope that could only accelerate a better bicycle than Johnson's to about 33 mph. It is inconceivable that Johnson was pedaling down that slope at over 40 mph; he wasn't equipped, he wasn't experienced, and he wasn't in physical condition to do it, to say nothing of the fact that he was cycling home at midnight after a full shift of work.

Furthermore, the damage done to the bicycles in Green's experiments was far greater than the damage to Johnson's bicycle. In some, the handlebar and handlebar stem were torn off, in others they were bent, damage that did not appear at all on Johnson's bicycle. Quite obviously, the experimental bicycles suffered from much greater shock than did Johnson's. The greater damage can reasonably be attributed to the method of loading the iron weights that represented the mass of Johnson's body. While properly distributed initially, they obviously did not unload at the moment of collision in the same way that Johnson's body did, and applied greater force to the bicycle on the way off.

In addition, the collision was not a perpendicular collision, but at an angle and with the car moving laterally with respect to Johnson's movement. That is going to complicate the collision experiments. Green also stated that this kind of car, a Jeep, could not turn the corner faster than 8 mph, although I had turned equal corners in an identical jeep at 18 to 20 mph with no sign of mechanical distress. Green set the stationary steel plate that represented the car's side at an angle to represent the combined angular placement and lateral movement, but he didn't work out the movement vectors properly even for the assumptions that he used.

In summary, Green's method is not a valid predictor of the speed of bicycles in collisions, although it is possible that with better experimental procedures such a method could be improved to provide reasonable predictions.

Determination of the Velocity of a Cyclist from Deformation of the Cycle From a Frontal Impact

A Review of Chapter 20 of Bicycle Accident Reconstruction, by James M. Green

Saturday, 5 February, 1994

Derivation of the Velocity Formula

Green first tries to derive the equation for the average force that decelerates
a moving deformable body when that body hits a solid, immoveable, rigid object.
The equation assumes that the deformation occurs with constant force, or that
the force is so nearly constant that the deviations from it are unimportant to
the investigator. The derivation uses two velocities, the initial velocity Vi
and the final velocity, Vf. Green combines Vi and Vf incorrectly. Considering
that L is the contraction in length as the fork deforms and T is the time over
which this deformation occurs, and if the average velocity between the parts
coming together, then:

dT=d(L/Va) ........(d stands for DELTA in this and succeeding equations)

Green erroneously calculates the average velocity between the bodies as:

(Vi + Vf)/2

The correct calculation for the average velocity between the bodies is based not
on the sum of the initial and final velocities but the average of the difference
between their initial speeds and the difference between their combined speeds
after the collision, which by definition is zero. Therefore:

(Vi - Vf)/2

That this is easily seen to be the correct is shown by considering the impact
between a bicycle that hits a child and carries him with it. The average
velocity between them during the impact is less than if the bicycle hit a parked
car that doesn't move at all. Because Green assumes, for this example, that the
object stops the bicycle, the final velocity is zero. This allows Green to
escape, for this example, the consequences of his erroneous method of handling
the final velocity in his calculations.

The equation that Green derives is:

F = (M/g){(Vi^2 + Vf^2) / (2 * dL)}

Because of the error in handling the speed after impact, this should be:

F = (M/g){[Vi - Vf)^2] / (2 * dL)}

Green then applies his equation to the deformation of bicycle forks. He assumes
that the deformation force is 3,500 pounds (a value that he says has been
obtained by "numerous frontal impact tests in the laboratory"), the
front fork deforms 0.75 feet (9"), and the cyclist weighs 210 pounds. This
gives him the equation:

3500 = (210 * Vi^2) / (32.2 * 2 * 0.75)

This solves, says Green, to Vi = 28.4, units not disclosed. In his original
paper Green wrote miles per hour, but in the book he omits units altogether. The
correct unit is feet per second because all the values used are in feet and
seconds.

Green then simplifies the equation that he says has been verified by numerous
laboratory tests, producing a form that would have been useful had it been
accurate:

Vi = SQRT(1078 * dL) / 1.47

Green claims that laboratory tests demonstrate that this equation predicts the
impact velocity within 3%. Considering that this equation has been incorrectly
applied, with the very large error factor of the ratio of the weight of the
cyclist (given as 210 lbs) to the weight of the bicycle (about 30 lbs), this
claim must be a lie. Any laboratory test must have demonstrated that his
equation was wrong by about one order of magnitude. In fact, when this formula
is run against Green's own data, the calculated speeds that it produces are, on
average, only 16% of the speeds that Green measured.

Derivation of the Instantaneous Maximum Force Formula

Green then uses the same formula to calculate what he believes to be the maximum
force produced by the fork during its crush sequence. He describes his method as
"The fork is driven into a pad that can record the amount of force over
time. When one examines the printout of the force applied over the deceleration
period, an instantaneous maximum force can be extrapolated." Measuring
force versus distance as an object is either stretched or crushed is a standard
laboratory practice for which tensile test machines are easily available. Their
output is a plot of distance versus force for each increment. The instantaneous
force is easily read as the force at any one point in time. Green reports his
own use of this procedure in chapter 16. This present section shows that Green
thinks that inertial effects in rapid deceleration will greatly increase the
forces over those produced in the slow procedure used by tensile test machines.

Green is correct in a sophomoric way. If the fork tips were infinitely rigid and
impacted against an infinitely rigid body, the instantaneous force between them
would be infinitely large for an infinitely short time. However, materials
aren't infinitely rigid, and the first part of the bicycle to contact the other
body is the tire of the front wheel, which is quite compressible. After the tire
gets compressed, once the force increases to that which bends the front fork,
this bending insulates the fork tips from the inertia of the rest of the
bicycle, leaving only the inertia of the fork blades, which cannot have a
significant effect because they weigh only a pound or so.

Green uses the same formula to compute instantaneous force as he used for the
average force calculation, except that now he specifies the initial and final
speeds and calculates the force. Green writes: "Obviously, a much higher
value of this decelerated dynamic force will be recorded at this instantaneous
moment than the average force values described earlier." Given that he uses
the same formula, he must obtain the same results. In other words, a given force
equals a given speed, whether the force is calculated from the speed or the
speed from the force. That is the way that arithmetic works; it is a
mathematical certainty. Green's idea that working in one direction computes
average force and in the other direction computes a much higher instantaneous
force is bogus.

Green also errs in his handling of the final velocity after the impact, as
described above. He uses the same equation as before:

F = (M/g){[Vi^2 + Vf^2] / (2 * dL)}

Green gives no definition of Vf or any method of measuring it. He then provides
the following example:

Impact speed = 51.5 feet per second

Residual speed = 7.4 feet per second

Fork deflection = 0.75 feet

Rider mass = 210 pounds

Using the equation that Green gives, this solves to 11,770 pounds. However,
Green says that the answer is 11,289 pounds. The difference between the two
results is caused by two errors by Green. He subtracts the squares of the
velocities instead of adding them, as is required by his formula, and he uses
7.5 feet per second instead of the 7.4 fps that he specified. I am not guessing;
both errors appear in the printed numerical calculations.

Had Green handled the residual velocity correctly his answer would have been
8,417 pounds, substantially less than either of the above.

Green's claim that a pair of fork blades, being bent backward, will require
11,289 pounds of force at some point as they deform backwards is utterly
ridiculous. The typical value for the deformation force is in the range of 200
to 400 pounds, as he himself has measured. Generally, in deformation of this
type, where the initial empty space nowhere closes up but the items are merely
bent more and more, the bending force remains fairly constant during the bend.
An increase of 37 times at one point is impossible. The idea that the two fork
blades, if they were held horizontally by the fork crown, could support almost 6
tons of weight is utterly absurd.

Green has made a very serious conceptual error that shows how little he
understands the physics and engineering of the situation that he is analyzing.
He believes that the force exerted by the front forks is the force that
decelerates the cyclist. In actual fact, the cyclist is connected to the bicycle
by only the most fragile of links. His feet, when not using foot retainers, are
not connected at all. His upper body is connected only by his hands, which
cannot transmit an unexpected, suddenly applied strong force. His lower body is
connected to the saddle only by small frictional forces. The effect of this is
that when the bicycle hits a solid object the cyclist's body continues forward
until it hits either the same object or travels over it and hits something else.
Green considers that the mass of the cyclist's body is the object being
decelerated by the deforming force of the front forks, rather than considering
that only the mass of the bicycle is being decelerated by that force.

Because of this basic misunderstanding, Green's calculations won't provide
credible values for speed unless he provides a value for force that is far too
large. Green has made three crash tests in which bicycles loaded with cast-iron
weights have been driven against barriers made of steel plate. He did so as part
of his participation in a bicycle accident case in which a cyclist going
downhill collided with a motor vehicle that had been coming up the hill and
turned left in front of him. Green was trying to prove that the cyclist had been
exceeding the speed limit, which there was 35 mph. The weights totaled 180
pounds to match the weight of the victim and the value of 210 pounds that Green
uses in his examples is the 180 pound cyclist and the 30 pound bicycle. The
weights were mounted on forward-facing bars so that they would slide off when
the bicycle decelerated during impact. Whether they slid off in the same manner
that a cyclist would is doubtful, and the deformation of the bicycles was very
unlike that of the accident bicycle. As near as I can figure, Green inserted the
data on speed and fork deformation from the three crash tests into his formula
to calculate the force. The answers probably averaged to approximately 3,500
pounds. If the bicycle had weighed 210 pounds the answers would be approximately
correct, but since it weighed only 30 pounds the answers were about 7 times too
great. Dividing 3,500 by 7 gives 500 pounds, which is more nearly the
appropriate value, as shown by tests of typical front forks.

The fact that Green's concept is incorrect is demonstrated by the accident
itself. The bicycle was stopped in its tracks by the collision with the motor
vehicle, but the cyclist continued right through the superstructure of the
vehicle, smashing fiberglass and methacrylate as he went, to hit the ground some
feet on the far side of the vehicle. The cyclist was not stopped by the effect
of the bicycle hitting the vehicle; he was first slowed by tearing out the
fiberglass and methacrylate and then stopped by friction with the roadway. Any
slowing by, for example, his legs hitting the handlebars as he passed over them,
was minor. The judge in the case ruled that Green's methods did not meet
scientific standards and excluded testimony about them from the trial.

Review of Chapter 23 of Bicycle Accident Reconstruction, 4th ed., by James M. Green

From crash tests of three bicycles, Green developed a formula relating speed versus fork deformation. He then tested 11 more bicycles, and states that in these tests "The speed and fork deformation relationships were within five percent of the values produced with the algorithm (meaning his formula)." That formula is:

Deformation = 28.376 * log(speed) - 37.208 ...........Green

where speed is in miles per hour and deformation is in inches. Green provides a table giving the speed and deformation of each test run, from which it is easy to run one's own calculations. This formula is not the one that is necessary, because it enables one to calculate only the deformation from the speed, not the speed from the deformation. It is apparent that Green was not properly considering the use to which he would put the data that he was gathering in his experiments, and was not using the formula to make the calculations that he said that he was making.

Using what I thought would be a form of formula that better matched physical realities, I produced a formula for the same purpose as Green's:

Deformation = 0.003s^2 + 0.118 s -1.739.........Forester

The error in Green's formula, computed against his own data, was 26%, while the error in Forester's formula was only 14%.

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