Stiffness Coefficients - Energy and Damage
Copyright Ó George M. Bonnett, JD 2001 All Rights Reserved
Politics is not the only thing that makes for strange bedfellows, so do collisions. A wonderfully strange relationship exists between momentum, energy, force, damage and speed as uniquely illustrated in dealing with barrier collisions. Two of the critters in the menagerie that exemplify this relationship are Crush Energy Equivalent Speed (CEES) and Kinetic Energy Equivalent Speed1 (KEES), which are both used in computing stiffness coefficients.
Now before you think this matter is of little importance, remember that when dealing with a change of speed from damage, you come face to face with this relationship and the beasts therein.
In most instances when a testing organization crashes a vehicle, the vehicle is run into a non-deforming, immovable barrier of some type. These barriers run the gamut from a loading dock to the sophisticated devices used by the government. The barriers are usually for dealing with frontal impacts although there are some unusual contraptions designed for rear- and side-impact collisions.
In an effort to make this as mathematically easy as possible, speed and velocity (vector) will both be expressed in terms of units, which can be in any system you desire. As a result, energy will also be expressed in terms of units. While SI units are the choice for some, others prefer a different unit of measure. The purpose of this paper is not to quibble about which is more correct, but to convey an idea. The unit of speed or velocity is the correct unit for the energy formulae and for conversion to a unit of momentum. It is for the reader figure out what unit this should be considering the instant application.
When a vehicle is run into a non-deforming, immovable barrier like a loading dock, the speed of the vehicle can be measured and the vehicle can be weighed so that we know how much energy the vehicle possessed at impact. At the higher speeds, there is usually very little bounce (read restitution) and it is often ignored, as it does not have much influence on the measurements. Armed with the weight and speed and a tape measure for determining the measurements of the damage, we can determine the stiffness coefficients for the particular vehicle.
Sometimes the vehicle is not run into the barrier. Sometimes the barrier, or impactor, is run into the vehicle. Knowing the weight and speed of the impactor, the kinetic energy the impactor brings into collision can be determined.
The KEES computations are complicated by the necessity of determining the speed of both the impactor and the struck vehicle post-impact. This post-impact information is necessary in order to obtain the energy of these two bodies post-impact so that it may be subtracted from the energy of the impactor pre-impact in order to determine the energy that went into damage.
What happens when the post-impact speeds of the objects are not reported? It is out of this dilemma that KEES was spawned. The KEES, or similar, computation is necessary for rear and side impacts since the post-impact speed is not reported.
The KEES determination has at its core the following assumptions:
KEES is computed as follows:
KEES = Sqr (30 * (Energy before Impact – Energy after Impact)/Impactor Weight)
KEES = Sqr (30*0.5/32.2*(IW*(IS*5280/3600) 2- ((IW+VW)*(PIS*5280/3600) 2))/IW)
IW = Impactor Weight IS = Impact Speed VW = Vehicle Weight PIS = Post-impact Speed
Post-impact Speed (PIS) = (Impact Speed * Impactor Weight) / (Impactor Weight + Vehicle Weight)
KEES is that portion of the speed of the Impactor that actually goes into damaging the vehicle. This is then the closing speed necessary between the struck vehicle and a non-deforming, infinite mass barrier to duplicate the damage on the vehicle.
While this is one approach to the problem and might even be a solution, it is made more difficult than necessary, as we shall soon see.
We need to look at three related but different collisions:
1. Vehicle 1 hits an immovable barrier.
2. Vehicle 2 hits Vehicle 3 in a head on collision.
3. Vehicle 4 hits a stationary feather.
All of the vehicles are identical and are all traveling at the same speed of 30 units.
Which of these vehicles has the most damage? Which of them has the least damage? How many of them have identical damage? How does this relate to KEES?
A collision begins at first contact between the vehicles. The collision ends when the transfer of energy between the vehicles ceases. This may be at maximum engagement if there is no restitution, or if there is restitution, it is at separation of the vehicles. This time frame is the total duration of the collision impulse. Everything after this is post-impact and has nothing to do with damage or with the change of velocity (deltaV) of either vehicle due to the collision impulse. Post-impact information can help us compute the speeds of the vehicles at impact, but has absolutely nothing to do with damage.
Instinctively we “know” that Vehicle 4 has no damage. Vehicle 1 hits the barrier and comes to an immediate stop. Vehicles 2 and 3 hit each other at twice the closure speed between Vehicle 1 and the barrier. They also come to a complete stop at impact as the momentum of Vehicle 2 cancels the momentum of Vehicle 3 as the momenta are equal but opposite in direction. In collision 2 there is twice the energy at collision as in collision 1, but there are 2 vehicles that sustain damage. In actuality, the first three vehicles all absorb identical amounts of damage resulting in identical damage to each vehicle.
Why was there no damage in collision 3? The answer is “mass.” Not the mass of the vehicle, it is identical to the mass of all of the other vehicles. The mass, or lack thereof, of the feather is also critically important in the solution. While this may seem too simple to be discussed, it proves a very important principle. It illustrates, and convincingly so, the principle of “reduced mass.” It is the reduced mass of the objects in collision that determines the amount of damage sustained in the collision.
Reduced mass [M1*M2/(M1+M2)] allows the mass of each of the objects to be proportionally represented in the equation. It is necessary to know the impact speeds of both objects as well as the post-impact speeds of both objects in order to determine the energy going into damage if KEES is used. This amount of detail is unnecessary. If we can determine the closing velocity of the objects, assuming there is no rotation or separation velocity, we have all of the information necessary using the reduced mass principle to determine the energy that results in damage. The reduced mass must be applied to the closing velocity vector (resulting from subtracting the two impact velocity vectors) to determine the energy associated with the closing velocity of the two objects.
In central collisions, where the principal direction of force (PDOF) passes through the center of mass of both vehicles, it is the actual mass of the vehicles that is determinant. In non-central collisions, it is the effective (the gamma value - working through the lever arm) mass that is the determinant of the actual damage. The remainder of the damage energy goes into rotation. Effective mass deals with the PDOF acting on the center of mass through a lever arm requiring less force than acting directly on the center of mass.
How does this relate to KEES? Well, at least we now know that the damage in the collision is a related to the mass of the objects and the closing speeds. Do we really need the post-impact data?
The additional collisions consist of:
4. Vehicle 5 is traveling at 30 units and Vehicle 6 is stopped. Both of the vehicles in this rear-end collision are identical.
5. Vehicle 7 is traveling at 90 units and Vehicle 8 is traveling at 60 units. Both of the vehicles in this rear-end collision are identical to the vehicles in Collision 4.
How are these collisions similar and how are they different? Both collisions involve identical vehicles with identical closure rates of 30 units. If it were not for the post-impact information, we could not tell these collisions apart. Interesting!
Proof of Identical Damage
In Collision 4, using a mass of 1, we have a momentum value of 30 for Vehicle 5 and zero for Vehicle 6. With zero restitution, both vehicles will stick together after collision and each will have a momentum value of 15. Dividing the momentum by the mass, we get a speed of 15 units for both vehicles.
Using the kinetic energy formula (KE = 0.5 * Mass * Velocity 2) we have KE = 0.5 * 1 * 302 or KE = 450 units total into the collision. For each vehicle out of collision the KE = 0.5 * 1 * 152 or 112.5 units for a total out-of-collision energy of 225 units. This leaves 225 units of energy to be shared equally (identical vehicles) as damage.
In Collision 5, using a mass of 1, we have a momentum value of 90 for Vehicle 7 and 60 for Vehicle 8. With zero restitution, both vehicles will stick together after collision and each will have a momentum value of 75. Dividing the momentum by the mass, we get a speed of 75 units for both vehicles.
Using the kinetic energy formulae (KE = 0.5 * Mass * Velocity2) we have KE = 0.5 * 1 * 902 or KE = 4050 units for Vehicle 7 into the collision. For Vehicle 8, KE = 0.5 * 1 * 602 or KE = 1800 units into the collision. This gives a total energy into the collision of 5850 units. For each vehicle out of collision the KE = 0.5 * 1 * 752 or 2812.5 units for a total out of collision of 5625 units. This leaves 225 units of energy to be shared equally (identical vehicles) as damage, which is exactly what we have in Collision 4.
If we disregard the post-impact information, the collisions are identical as far as the damage is concerned. Unless we need to know an exact impact speed, we can disregard the post-impact data, as the collisions are identical as far as the energy for damage is concerned. Damage does not involve anything post-impact. If identical vehicles are involved, it is the closing speed of the vehicles that is determinant in the damage resulting from the collision. Interesting, most Interesting!
A Second Look
In Collision 1, Vehicle 1 strikes the barrier at 30 units. However, what if the barrier was traveling at 30 units when it struck Vehicle 1, which was stationary?
As we saw in the first three collisions, it is the reduced mass of the two objects in collision that determines the amount of damage if the objects are non-deforming. In collisions 4 and 5 we discovered that it is not the instantaneous speed of the individual vehicles that determines the amount of damage but the closing speed.
If both of these principles apply, then it does not matter if the vehicle strikes the barrier or the barrier strikes the vehicle. It is the closing speed (Vc) and the reduced mass of the two objects in the collision that determines the amount of damage.
The energy absorbed by each object is directly related to the stiffness coefficients of the objects and the volume of the damage.
Since the only variables that go into determining the damage in a central (or nearly central) collision without restitution are the closing speeds of the objects and the reduced mass of the objects involved, why do we need the post-impact speed of anything in order to determine the energy that goes into damage? It is as important as the color of the vehicle in helping to determine the damage.
Frame of Reference
Are there still skeptics? Let us take two more looks at Collision 1:
1. In this look, we will be in the same reference frame as the barrier. We see the vehicle approaching the barrier at 30 units until collision. After collision, all motion ceases between the barrier and the vehicle.
2. In the next look, we will be in the same reference frame as the car. We now see the barrier approaching the car at 30 units until collision. After collision, all motion ceases between the barrier and the vehicle.
In these collisions, which was the moving object? In actuality, both were in motion in space. The point is it really does not matter which object is the moving object. Again, it is the closing speed and the reduced mass of the objects that determines the amount of damage in a collision.
We must remember that a definition of mass is the resistance of the object to a change in velocity. When the two objects in collision reach a common velocity, the vehicles have reached maximum engagement and there can be no further damage from the collision.
Maximum Damage Energy
The maximum energy available for damage is equal to 0.5 times the reduced mass times the closing velocity squared. Collisions 7 and 8 have a closing velocity of 30 units regardless of the reference frame. With this formula, it is not necessary to compute the total collision energy and subtract the energy after collision to determine the damage.
Determination of reduced mass involves dividing the product of the masses by the sum of the masses. This fraction is then used as the reduced mass. In Collision 2, 4, and 5 the result is 1 * 1 divided by 1 + 1 or 1/2.
The maximum energy available for damage using this method for Collisions 4 and 5 would be 0.5 times 1/2 times the closing velocity squared. With a closing velocity of 30, the maximum energy available for damage is EMax = 0.5 * (1/2) * 302 or 225 units. This is the same energy figure computed previously.
For Collision 2 the energy of each vehicle coming into collision is KE = .5 * 1 * 302 or 450 units. Since both vehicles stop at impact, there is no post-impact energy. This means both vehicles share a total of 900 units of energy that is available for damage.
With a closing velocity of 30, using the reduced mass principle, the maximum energy available for damage is EMax = 0.5 * (1/2) * 602 or 900 units. This is the same energy figure computed using the other method.
What happens in Collision 1 with the infinite mass, non-deforming barrier? In order to make the problem a little more manageable, lets assign a mass of 1000 to the barrier since it is difficult to add and multiply using Infinity as one of the numbers.
If the frame of reference is the car and the barrier is moving at 30 units, the total KE into collision is then KE = .5 * 1000 * 302 or 450,000 units of energy. We then have to fuss with all of the post-impact data and KEES formula to arrive at the Kinetic Energy Equivalent Speed and then convert back to energy so that figure can be subtracted from the total energy into the collision to get the energy that goes into damage.
If the frame of reference is the barrier and the car is moving at 30 units, the total KE into collision is then KE = .5 * 1 * 302 or 450 units of energy. Since in this frame of reference the car will come to a stop, there is no post-impact information to deal with and there are a total of 450 units of energy going into damage. With a closing velocity of 30, using the Reduced Mass formula, the maximum energy available for damage is EMax = 0.5 * (1000/1001) * 302 = 450,000/1001 or 449.5504 units. This is not the same energy figure computed using the other method.
Well, what happened? It looks like there is a problem with the Reduced Mass formula. Well, not really. Remember we assigned a mass of 1000 to the barrier. The higher the number we assign the closer this number gets to 450. With a mass, less than Infinity, the barrier and the car have a different reduced mass. They will remain together but both reach a common velocity that is no longer zero and the total energy coming into collision goes into both damage and translational motion of the car/barrier object. In other words, the barrier moves ever so slightly and that takes some energy away from the total going into collision with the remainder (449.5504) going into damage.
Reduced Mass Formula
The closing velocity (VC) is the magnitude of the resultant of the vector subtraction of the velocity vectors of the objects. It (VC) is the third side of the triangle, computed using the Law of Cosines.
VC = |V1 - V2|
V1 * M1 + V2 * M2 = V3 * M1 + V4 * M2
The maximum possible deformation is reached when both vehicles achieve a common velocity (Vcommon).
V3 = V4 = V
Substitute into the momentum equation:
V1 * M1 + V2 * M2 = V * (M1 + M2)
Solve for V':
V = (V1 * M1 + V2 * M2) / (M1 + M2)
An energy equation similar to the momentum equation is:
Ke1 + Ke2 = ED + Ke3 + Ke4
Ke1 and Ke2 are the kinetic energies at first contact for the two vehicles, respectively, Ke3 and Ke4 are the post-impact energies for the two vehicles and ED is the total energy due to deformation to both vehicles.
Kinetic energy formula:
Ke = .5 * M * V2
Therefore: .5 * M1 * V12 + .5 * M2 * V22 = ED + .5 * M3 * V32 + .5 * M4 * V42
Multiply by 2:
M1 * V12 + M2 * V22 = 2 * ED + M3 * V32 + M4 * V42
Substitute V for V3 and V4:
M1 * V12 + M2 * V22 = 2 * ED + M3 * V2 + M4 * V2
M1 * V12 + M2 * V22 = 2 * ED + V2 * (M3 + M4)
Substitute for V:
M1 * V12 + M2 * V22 = 2 * ED + ((V1 * M1 + V2 * M2) / (M1 + M2))2 * (M3 + M4)
M1 * V12 + M2 * V22 = 2 * ED + (V1 * M1 + V2 * M2)2 / (M1 + M2)
Multiply by (M1 + M2) and cancel:
(M1 + M2) * (M1 * V12 + M2 * V22) = 2 * ED + (V1 * M1 + V2 * M2)2
M12 * V12 + M1 M2 * V12 + M1 M2 * V22 + M22 * V22 =
2 * ED * (M1 + M2) + M12 * V12 + 2 * M1 * M2 * V1* V2 + M22 * V22
Simplify by canceling like terms:
M1 M2 * V12 + M1 M2 * V22 = 2 * ED * (M1 + M2) + 2 * M1 * M2 * V1* V2
Subtract 2 * M1 * M2 * V1* V2 from both sides:
M1 M2 * V12 - 2 * M1 * M2 * V1* V2 + M1 M2 * V22 = 2 * ED * (M1 + M2)
Factor the left side of the equation:
M1 M2 * (V12 - 2 * V1* V2 + V22) = 2 * ED * (M1 + M2)
M1 M2 * (V1 - V2) 2 = 2 * ED * (M1 + M2)
Substitute VC for (V1 - V2):
M1 M2 * VC 2 = 2 * ED * (M1 + M2)
Solve for ED:
ED = .5 * M1 M2 * VC 2 / (M1 + M2)
This reduced mass formula can be used to find the maximum damage energy for a collision. By substituting separation velocity (VS) for closing velocity (VC) the formula can be used to solve for the energy restored to the system in the form of translational motion.
EMaximumDamage = .5 * M1 M2 * VC 2 / (M1 + M2)
ESeparationTranslationalMotion = .5 * M1 M2 * VS 2 / (M1 + M2)
Reduced Mass Formulae
The practical value of the formula for the Energy for Maximum Damage and the formula for Energy of Separation Translational Motion is a different matter entirely. Either, or both, may play a significant role in the reconstruction.
The formula for the Energy for Maximum Damage, EMD = .5 M1M2VC2/(M1+M2), eliminates the need to subtract the zero restitution kinetic energy of the vehicles departing the collision from the total kinetic energy of the vehicles entering a collision in order to arrive at the maximum possible energy going into deformation. In addition to the mass of each of the vehicles involved, only the closing velocity is required for the computation.
Substituting the separation velocity for the closing velocity, the formula now computes the energy restored to the system in the form of translational motion, ER(STM) = .5 M1M2VS2/(M1+M2).
The formulae can also work together. If .5 M1M2VS2/(M1+M2) is divided by .5 M1M2VC2/(M1+M2), the expression will factor into VS / VC, which is the formula for restitution. If instead of dividing, we subtract the components .5 M1M2VC2/(M1+M2) - .5 M1M2VS2/(M1+M2) the result is the actual energy of the collision going into damage. This damage may consist of deformation, heat, light, sound, rotation, and/or the (viscoelastic) waves that are generated within the object.
Tsunami and Other Waves
One of the contenders for the energy contained in the pool is the viscoelastic wave that is generated within the object itself. This wave is similar in many respects to the Tsunami or "harbor wave" generated by an earthquake or resulting from landslides under the surface of a body of water. The tsunami, almost invisible until it reaches shallow water, causes damage that is spatially far removed from the origin of the wave. As the viscoelastic wave reaches structurally weaker sections, it may cause stress within the object. The oscillation created by the wave causes the object to damage itself. This is commonly referred to as induced damage and may be spatially removed from the contact damage of the collision. If no damage is caused this energy is radiated by the object just as the object radiates sound wave energy.
Reduced Mass Applications
The Reduced Mass set of formulae can be used to determine the actual energy going into damage, which can be of great importance to the Reconstructionist. It is an easy method of determining the total energy in the damage pool, which must be apportioned out to the different contenders.
Energy: Kinetic versus Potential
The kinetic energy formula (KE = 0.5 * Mass * Velocity2) and its variant (E = Weight * friction coefficient * Displacement) differ from the potential energy formula (PE = 0.5 * k * x2) in one very important aspect - Mass. This formula for Potential Energy does not deal with mass.
Crush energy uses the CRASH3 model developed by McHenry. This system is based on the spring model using Hooke's Law (FS = - kx), or Forcespring = - spring constant * displacement. Newton's Third Law states F1 = F2. If this spring model is used then we must also accept that the energy resulting in damage is treated as potential energy. Potential energy equals 0.5 * spring constant * displacement2. E1 = 0.5 * k1 * x12 and E2 = 0.5 * k2 * x22. Therefore, E1 / E2 = x1 / x2.
Newton's Third Law of Motion tells us that the forces must be equal and opposite and must act on different bodies. Since the forces are equal and opposite (F1 = F2), and a "spring model" is used to model the collision damage, it follows that in the collision, the energy related to damage is potential energy and not kinetic energy.
E1 = 0.5 * k1 * x12 and E2 = 0.5 * k2 * x22. Therefore, E1 / E2 = x1 / x2.
In order to illustrate the reduced mass principle, two more collisions will be considered. The first involves two identical 18-wheelers loaded to 80,000 pounds each. Attached to the front of each is a giant spring. Both springs are identical. These springs can absorb all of the energy that would normally go into damage. When the vehicles collide, both springs compress absorbing all of the energy. The springs compress equally, but this is to be expected, as both the vehicles and the springs are identical.
In the next collision, the second vehicle is a 1000-pound dune buggy. It has a spring attached to the front that is identical to that of the 18-wheeler. The closing speed is identical to the collision between the two 18-wheelers. How much damage do we expect in this collision?
It is obvious that if each of the vehicles is traveling at 30 units at impact, the post-impact motion will be in the same direction as the pre-impact direction of the 18-wheeler. This is the only realistic way that momentum can be conserved in this collision, as momentum is a result of the inertial qualities of the vehicles. Mass is really a measure of the inertia of an object. If the dune buggy has very little relative mass, it must accelerate faster than the 18-wheeler when equal forces are applied to both. If it takes less force to accelerate it to the same speed in the same time frame, the springs will not compress as much as they did in the collision involving the two 18-wheelers. To belabor a point, for those non-believers, instead of a dune buggy, imagine again the feather. How much spring compression will the feather generate?
As can be seen from the collisions above, it is the reduced mass of the vehicles in collision that accounts for the difference in the spring compression. If a "spring model" is used for damage, then it follows that it is the reduced mass that is the "effective mass" in any collinear collision with 100 percent overlap or any central collision where all action passes through the centers of mass of both vehicles.
However, by using a spring model for damage, it is not necessary to consider mass in order to determine how the energy is distributed between the vehicles. The mass is in effect factored out of the distribution of the energy. It is important to get rid of mass in the damage distribution model for a collision in order to eliminate all of the qualifiers at the end of the last sentence in the preceding paragraph.
Once the collision is non-central (action does not pass through the center of mass), the mass of the object or vehicle is not what is being displaced. It is the end of the moment arm that the force is acting on, not the mass of the object. It is rotational inertia that the force is acting against, not the inertial mass of the object.
The proceeding section demonstrated how energy is shared using springs to illustrate a point. Just as ice cream comes in different flavors, springs come in different sizes. If we put a spring capable of resisting the weight of the 18-wheeler by only compressing a fraction of an inch and place it in front of the 18-wheeler while using a spring that will compress 100 times as much in front of the dune buggy, and then re-stage the collision, the results will be much different. The springs are equivalent to the A and B stiffness values. They represent the resistance to deformation of the object. Since the non-deforming, infinite mass barrier cannot be deformed, all of the energy available for damage goes into damaging the vehicle. Energy is only shared equally between identical vehicles. If the resistance to deformation is not identical, the energy is no longer shared equally for the same volume of damage. The force is still equal and opposite between the vehicles, but energy is not force.
When the vehicle is struck on the side by a vehicle with less mass than the struck vehicle, it is analogous to being struck by the feather. It will not cause the same amount of damage as would have occurred had the vehicle been struck by an infinite mass, non-deforming barrier. The damage is based on the spring model and the reduced mass principle. Remember also, the distinction of struck and being struck is simply one involving the frame of reference.
Since Kinetic Energy Equivalent Speed computations result in the same energy values obtained from using the reduced mass computations, and the closing speed of the barrier is known, then KEES computations are really nothing but an involved mathematical exercise. In order to distinguish between the KEES computations and the reduced mass computations, the term Crush Energy Equivalent Speed (CEES) will be applied to the latter.
It is important to emphasize the crush reference in the term CEES. This term (CEES) refers to the process of converting the potential energy of actual crush damage into a speed for a vehicle of identical weight (mass) as the damaged vehicle using the kinetic energy formulae.
A Fundamental Error
An analysis of two more collisions is necessary. The striking vehicle in both collisions will have a speed of 30 units and a mass of 1.
6. In Collision 6, the vehicle strikes a non-deforming infinite mass barrier.
7. In Collision 7, the vehicle strikes a non-deforming object with a mass of 1.
These collisions are not identical.
An examination of both collisions using the conservation of momentum reveals that the striking vehicle has 30 units of momentum. Both of the non-deforming struck objects are stationary and therefore generate a momentum value of zero. As both objects are non-deforming, they will absorb no damage energy as a result of the collision. So far, both collisions are identical.
A closer examination, again using momentum, reveals a slightly different story. After collision, according to the law of Conservation of Momentum, we must have the same total momentum as before the collision.
In Collision 6, using the reduced mass formula, all of the energy brought into collision goes into damaging the striking vehicle, as an infinite mass object, by the very definition of mass, cannot move. If the non-deforming infinite mass barrier does not move, there is no momentum for that object in the post collision phase. If zero restitution is a given, then both objects must reach a common velocity of zero.
In Collision 7, using the reduced mass formula, only a portion of the energy brought into collision goes into damaging the striking vehicle. Some of the energy goes into accelerating the non-deforming object. If the non-deforming object moves, the energy used to accelerate this object must be subtracted from the total energy pool available for damage. Conservation of Momentum tells us that the two objects will reach a common speed. Again, with zero restitution, there is no force to separate the vehicles so they will remain at this common velocity. With both objects having a mass of 1, there is a total mass of 2 with a momentum value of 30 that gives a common velocity of 15 units. With both units traveling at 15 units, the energy (which must also be conserved) can be computed.
Using the Kinetic energy formula (Ke = 0.5 * M * V^2), the energy figure for each vehicle is 112.5 units for a total energy out of collision of 225 units.
The energy brought by the vehicle into each of the collisions using this same formula was 450 units.
In Collision 6, all of the energy (450 units) went into damage. In Collision 7, only 225 units of energy were available for damage.
Clearly, the vehicle in Collision 6 will sustain twice the damage as the vehicle in Collision 7. If this is true, should both vehicles have identical stiffness coefficients?
Assume that both vehicles have identical resistance to deformation. With this as a given, both vehicles will have identical deformation if they both strike an infinite-mass, non-deforming barrier and both vehicles will have identical deformation if they both strike a non-deforming impactor with a mass of 1. So where is the problem?
Wait, how can they have identical resistance to damage when one has twice the damage as the other and they were both closing at the same speed?
A Note for the Purists
There is no such thing as an infinite mass barrier here on our little planet. It is a theoretical object, an object of our fantasy and a figment of our imagination. In reality, momentum is truly conserved, as the earth moves when our infinite mass object is struck, and this is where the momentum goes. It also requires energy to move the earth, but not as much as you might suspect, as it is proportional not to the speed, but to the square of the speed.
Where the Rubber Meets the Road
A problem arises when we try to compute the stiffness values from the information garnered from the test collision. The formula calls for the following variables:
1. The (average for average stiffness) crush depth,
2. The maximum speed without damage,
3. The width of the damage,
4. The weight of the vehicle, and
5. The speed of the vehicle.
This appears to be fairly straightforward data that is required. So where is the problem?
The fundamental flaw is very carefully hiding. The flaw, like the devil, is in the details. The average crush depth is relatively easy to compute even though many improperly perform this computation. The maximum speed without damage seems simple enough, as do the width of the damage and the weight of the vehicle. This only leaves the speed of the vehicle, and that can be measured very precisely. So, again, where is the Big Problem, the Fundamental Flaw?
Is the culprit the term speed? Do we want a speed, or is it a velocity that is required? It must have a quantity, but it also has a direction. Is it a vector, or is it a scalar? That depends on how you look at it. Not unlike light that can be either a particle or a wave, depending upon which is being sought. But this is not in and of itself the answer.
Does the term speed mean closing speed and if so, between which objects? Here is the real problem! Speed does not necessarily mean closing speed, as it refers to the impact speed with a non-deforming infinite mass barrier and needs to be adjusted accordingly. The term speed refers to that speed computed using either KEES or CEES. Using the term CEES has an advantage over KEES in that it helps to denote that the speed obtained from the energy computation is equivalent to the closing speed between the vehicle and the non-deforming infinite mass barrier required to produce identical damage.
CEES, KEES and BES all generate a value for speed. Their computed values are identical, as we have seen. They have a distant cousin – absolute speed. It is only in a collision with an infinite mass (immovable) non-deforming barrier that absolute speed has the same value as the others. The absolute speed should never be used to determine stiffness coefficients in a test collision. With an immovable barrier, they are all identical in value, but even in this situation, absolute speed is not the correct input into the stiffness computations.
Using the absolute speed of the impactor, which is not an infinite mass object, introduces the fundamental flaw into the calculations. Energy is transformed into a change of speed for the impactor. This energy is no longer available for damage and therefore must be taken out of the pool of energy used to deform the test vehicle. Only the speed computed as a result of the energy of actual damage can be used in determining the stiffness coefficients of the test vehicle. Using the absolute speed of the impactor, especially one with a mass nearly identical to the test vehicle, results in abnormally high stiffness coefficients. When these stiffness coefficients are used in computations that employ the CRASH3 algorithms, abnormally high speeds are then generated.
Suppose a test is preformed with an impactor at 30 miles per hour that has a mass equal to that of the test vehicle. If the absolute speed (30 M/H) of the impactor is used to compute the stiffness coefficients, the resulting CRASH3 speed changes are almost 50% higher than those computed using the correct stiffness values.
How much of an effect these incorrect stiffness values and the correspondingly erroneous speed change computations have had is a matter of speculation. Certainly, the flawed computations have been used as evidence of either fault or negligence within the civil and criminal divisions of our legal system.
Better the Devil You Know?
This identification of a serious problem poses another one that is potentially more serious. Now, where do we go from here? All of the stiffness computations made using absolute speeds involving impactors with a mass that was not several orders of magnitude greater than the mass of the vehicle being tested have this fundamental flaw, unless they were computed using either CEES, KEES or BEV as the Impact Speed in the computation.
A Cautionary Note
CEES and KEES do not deal with Rotational Energy! Usually minimal rotation is involved in the tests, but any energy that does go into rotation must be subtracted from the Damage energy pool before calculating any equivalent speed. This work does not deal with rotational energy, as its inclusion would only detract from our central focus. This cautionary note is to serve only as a reminder that rotational energy must also be removed from the pool of energy before calculating a barrier equivalent speed that represents the crush damage to the vehicle.
The procedure for computing stiffness coefficients requires using the speed obtained from energy computations as the impact speed. The energy equivalent speed and the absolute speed are identical in value only in a collision with an infinite mass barrier.
Either by oversight or carelessness, in test collisions with a movable barrier instead of an infinite mass barrier, the absolute speed of the test vehicle has been used in the stiffness computations instead of the crush energy equivalent speed. This work illustrates the correct procedures for determining the stiffness coefficients of vehicles in collision as well as the relationship between momentum, energy and force and their relationship to speed and damage.
The underlying problem of allotting all of the pre-impact energy of the vehicle (or impactor) to damage is not confined to the calculation of stiffness coefficients. Americans seem to be obsessed with the CRASH3 methodology for determining deltaV and accident severity. The European community seems to prefer a comparison between the damage severity of the accident vehicle and of exemplar photographs of damage where the speed of the vehicle was known or could be accurately determined.
The inherent problem still exists with this methodology and has the potential of infecting any resulting calculations. It is a wolf hiding in sheep’s clothing. If using the comparison for establishing absolute speed, it is just as flawed as the stiffness calculations, and for exactly same reason. The deformation of a vehicle is not related to absolute speed – EVER! It does not matter if the deformation occurred during a staged test crash, or a real world collision, the deformation does not result from absolute speed.
Deformation results from a change of velocity or deltaV. In dealing with collisions involving vehicles, it is the change of velocity resulting from the collision impulse. A more precise statement would be that deformation results from the force of acceleration, or deltaV divided by deltaT where deltaT is the time interval for deltaV. It is not usually the deltaV of the fall with a large deltaT that causes injury; it is the deltaV of the stop with the small deltaT of the impact that can be fatal.
A comparison of photographs, where the first involves a collision with a bridge abutment and the second involves an identical vehicle striking a small stationary vehicle, will yield remarkably different deformation characteristics if the bullet vehicle was a large sedan traveling at identical speeds in both collisions. The deltaV in the first instance will be extremely close to the absolute speed of the vehicle, as the speed of the rotation of the earth will not be markedly changed. While close in value, or even identical in value, the absolute speed is not the deltaV. The deltaV in the second instance will be much less, as the smaller vehicle will be accelerated to a common velocity with the larger vehicle and they will both continue in motion after maximum engagement.
This is the reasoning behind the correct approach for stiffness coefficient determination and why photographs of damage should not be used to determine absolute speed.
An understanding of the basic principles involved will prevent errors that in the world of the accident reconstructionist can have serious civil and criminal consequences.
1. The first use of the term Kinetic Energy Equivalent Speed was, to the best of my knowledge, by Daniel W. Vomhof III to identify a speed (mph) equivalent to the energy used in damaging a vehicle, specifically in the NHTSA side- and rear-crash tests. The problem he found with using the more commonly referred to term "Barrier Equivalent Speed" was that in most vehicle vs. vehicle accidents, no "barrier" is involved. This term, he believes, more correctly describes what is being computed when making speed computations from "crush" and has been in the AutoStats® program since originally offered for sale in 1991. Dan and I agree in principle and on methodology; we disagree on terminology, understandable in light of the fact that he lives in California and I live in Florida.
2. Glenn A. Burdick Ph.D. introduced me to the reduced mass principle. Glenn received his Ph.D. in physics from Massachusetts Institute of Technology and is a Dean Emeritus of Engineering and a Distinguished Professor of Engineering at the University of South Florida. He is also a Full Professor of both Transportation and Electrical Engineering.