## What is Restitution? (w/r to Vehicle Collisions)

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### What is Restitution? (w/r to Vehicle Collisions)

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Q: What is the Coefficient of Restitution?
A: In a central collision (i.e., where the collision force acts directly through the centers of mass of the colliding bodies), the speed at which the centers of mass are moving toward each other at the instant of initial contact is called the speed of approach.
Subsequent to the initial contact, the centers of mass of the colliding bodies continue to approach each other, as deformations occur at the contact regions, until the relative motion has been stopped by the deceleration-acceleration action of the collision force between the two bodies. Thus, the speed of approach is reduced to zero during the collision.
If the collision force does not immediately vanish at the end of the approach period, the continued deceleration-acceleration of the collision partners will produce a separation speed.
In the extreme case of a perfectly elastic recovery of the deformed contact regions, a separation speed equal and opposite to the approach speed will be produced.
At the other extreme, the case of a perfectly inelastic, or “plastic” behavior of the deformed contact regions, no separation speed will be generated and the colliding bodies will remain in contact until acted upon by external forces (i.e., forces external to the two-body system, such as tire-terrain forces in the case of an automobile collision).
The ratio of the speed of separation to the speed of approach is referred to as the coefficient of restitution.
coeficient of restitution.jpg
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### Re: What is Restitution? (w/r to Vehicle Collisions)

This topic came up on the NAPARS facebook page so i added a couple other sources for the definition:
• Oxford Reference
• The coefficient of restitution always satisfies 0<=e<=1. When e=0, the balls remain in contact after the collision. When e=1, the collision is elastic: there is no loss of kinetic energy.
• Study.com Netwon's defintion
• "it never comes out as a negative number. The coefficient of restitution is always positive...coefficient of restitution. This coefficient is equal to the relative speed between the two objects after the collision, divided by the relative speed between the two objects before the collision"
• ScienceABC defintion
• the coefficient of restitution is a measure of how much kinetic energy remains after the collision of two bodies. Its value ranges from 0 to 1.
• Wilkapedia:
• "The coefficient of restitution (COR), also denoted by (e), is the ratio of the final to initial relative velocity between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision."
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### Re: What is Restitution? (w/r to Vehicle Collisions)

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### Re: What is Restitution? (w/r to Vehicle Collisions)

As part of the NAPARS facebook page presentation on restitution they included NHTSA Crash test from test #10156 which was a 2018 Chevrolet Traverse 4D SUV 35 MPH barrier crash.
They calculated the coefficient of restitution for the test at 18%!!
• As Posted on the Facebook NAPARS Page:
• "NCAP report, page 2-20 shows that based on the accelerometers, the velocity change was 66.56 km/hr (41.35mph). But the impact speed was only 56.39 km/hr (35.03 mph).
With these numbers, we can calculate the coefficient of restitution, commonly written as the Greek Epsilon, or a lower case e.
Restitution is defined as the separation speed divided by the approach speed, both being absolute values (i.e. always positive).
In this test that looks like this :
• e= ABS[0-(-10.17)] / ABS[56.39-0] = 0.18. That's 18% rebound.
That's a lot, right? Not quite superball-levels, but it is 2 or 3 times what we commonly expect in the analysis of big crashes. I think this is because we usually deal with car-to-car crashes, where both "impact partners" can plastically deform.
Restitution for us ranges from 0 and 1, with zero being like silly putty going "splat" on the floor and stopping right there, and a "one" when the separation velocity equals the approach velocity - nothing getting lost in the exchange. But there should always be something lost, even if just a little, in the form of noise, heat, deformation, and hysteresis. As you approach zero speed change (very very small crashes) you get more and more of the impact speed back, so restitution goes up, approaching one as you approach zero change in speed.
It [normally] drops pretty quickly to the 20-30% range for even light impacts, and down to the 5% range for highway events."
That seems high for a 35 MPH barrier crash. I tried to examine the vehicle accelerometer data from the NHTSA Test Report 10156 however it did not include any data from the 8 vehicle accelerometers.
On careful examination of the video it appears that the tires may have participated in the bounceback from the barrier.

The following plot shows generally at 35 MPH (~56 kph) the coefficient of restitution would be expected to be lower than 0.18.
However some do approach that value.
So question is: Are the higher values related to a smaller front overhang and additional rebound produced by the tires being loaded/unloaded on the barrier? And if so, would we see that extra amount of restitution in car to car crashes?
Might need to do some research on that!
restitution over a range.png (85.63 KiB) Viewed 566 times
Here is another chart of measured restitution at lower speeds
plot of restitution.png (64.94 KiB) Viewed 544 times
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### Re: What is Restitution? (w/r to Vehicle Collisions)

• Monson and Germane reported on a bunch of full-barrier crash tests at 30 and 35mph, and found a higher restitution value for the higher speeds, more or less across the board.
This is contrary to our expectation that e should go down with higher speeds, and hints at a more complex system than a simple spring. They concluded that restitution should be above 10 percent (0.10) "until impact velocities exceed 70km/h (43.5mph)"
Apart from the weird 30/35mph-epsilon speed relationship, I was most impressed by the fact that pickups are basically in the same neighborhood, despite having rigid front bumpers.
The extra "bounce" afforded by egg-crate bumpers or piston isolators, which is so prominent at very low speeds (under DV=5mph) is negligible once you start really bending sheetmetal and structural components.
• The paper is SAE 1999-01-0097, Determination and Mechanisms of Motor Vehicle Structural Restitution from Crash Test Data; Kenneth Monson and Geoff Germane.
Rose, et al., reported 3 equations for estimating crash-specific restitution values from barrier crash tests results, with slightly mixed success, but in the four car-to-car crash tests they evaluated, the measured restitution was 8.9% to 15%. Oddly, the four analyses showed that either calculated speeds matched well (within 2%) or calculated restitution matched well (within 5%) but not both at the same time.
The paper is SAE 2006-01-0908, Restitution Modeling for Crush Analysis: Theory and Validation; Nathan Rose, Stephen Fenton and Gray Beauchamp
The following are some interesting tables and figures from the 1999 paper, after the holidaze and once we get time to review we will post up more information.
See our post and video above for our slow motion video of the tires influence in a test. We can't help but wonder if the smaller front overhang and larger wheels contribute to the higher than expected restitution values in frontal impacts. However, what about the side and rear impacts?

Frontal Collision Data:
full frontal comparison.jpg (94.28 KiB) Viewed 375 times
Side Collision Data:
Side Collision Data.jpg (85.57 KiB) Viewed 375 times
Rear v Side Data:
Rear v Side.jpg (156.42 KiB) Viewed 375 times
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