Articlets

Package "e-Factor" Profiling for Improved Equivalent Drop Height Measurement and Package Design
Rodney J. Lambert
Lead Engineer / R & D Group Leader
Instrumented Sensor Technology
Okemos MI 48864 

Abstract 
The Coefficient of Restitution 
Application of the Coefficient of Restitution 
A New Approach 
What is "Package Profiling" 
A More Detailed Description 
Verifying the Procedure 
The Coefficient of Restitution as a Function of Impact Energy 
Evaluating a Package’s Ability to Absorb Energy 
Conclusion

Abstract

Portable acceleration data recorders have been used increasingly for measuring the drop height environments of packages in distribution. A common difficulty in drop height calculation is determining "equivalent" drop heights, when no actual "free fall" has occurred. In this case one relies on an assumed coefficient of restitution, i.e. "e-factor", to estimate package impact velocity, and subsequently drop height. This paper introduces a new technique for profiling the "e-factor" of a package, and is useful in both field measurement and laboratory package design. In this approach the "e-factor" is measured as a function of impact direction, creating a complete functional form that describes the package under test. The functional form of the "e-factor" can then be used to visually evaluate the package design, allowing the package to be modified to meet minimum drop height survivability from impacts of any orientation. This mathematical description of the "e-factor" can also be used to accurately measure equivalent drop heights in the distribution environment. Previous techniques have relied on values entered by the user and were discrete in nature. Package profiling is based on physical measurements of the system in question and provides a continuum of values resulting in more accurate equivalent drop height calculation and a graphical view of the package's ability to absorb energy.

The Coefficient of Restitution

The coefficient of restitution or e-factor, as it is also known, is the ratio of the rebound velocity to the impact velocity. In a perfectly elastic collision no energy is dissipated and the rebound velocity is equal to the impact velocity. This type of impact would have a coefficient of restitution equal to 1.0. The other extreme would be an impact where all energy is dissipated and the rebound velocity is zero. Impacts of this type have a coefficient of restitution of 0.0. Historically the coefficient of restitution has been treated as a scalar quantity. This paper introduces a method of generating a vector field for the coefficient of restitution, thus allowing the magnitude to be a function of the direction of impact. 

Application of the Coefficient of Restitution

The coefficient of restitution can be extremely helpful in reproducing impacts in the laboratory. First, the distribution environment is recorded. Then the recorded events are transformed into drop heights of equivalent impact energy. If the event was the result of a free fall then the equivalent height is just the height of the free fall, providing that the impact cushioning is roughly the same. Things become considerably more difficult when the recorded event was not the result of a free fall. Several different methods have been used to provide the missing information needed to perform this more difficult calculation. Most, if not all, require the investigator to enter a limited amount of information about the coefficient of restitution of the package under test. There are several problems with this approach:

real packages have complicated packing materials
corners, edges, and surfaces have different properties
the coefficient is often a function of the energy
an investigative approach should provide the user with information about the package not the other way around


A New Approach

Why should a packaging engineer have to tell the analysis software what values to use for the coefficient of restitution? Can’t the coefficient of restitution be measured? And if we are directly measuring the coefficient can’t we do it for all directions and energies? Then we could use this data to determine the equivalent drop height in a purely experimental manner. We would also have a valuable mapping of the package’s ability to absorb energy in all directions. Instrumented Sensor Technology has named this new technique "Package Profiling". 

What is "Package Profiling"

In short form, "Package Profiling" consists of placing a data recorder in the package under test and then dropping the package at random orientations. Instrumented Sensor Technology’s DynaMax software scans the events for free falls and then calculates the coefficient of restitution at each impact that was the result of a free fall. DynaMax will then fit a three dimensional surface to the data. The resulting surface represents the coefficient of restitution of the package in all directions. Once a surface has been calculated from the drop data it can be used to determine equivalent drop heights for any impact that the package experiences. The surface can also be viewed interactively to provide a level of insight into the package design that has never been seen before. Figures 1 and 2 represent a typical profile of a package that employs two types of cushioning material. 



Figure 1: Profiler wire frame model showing the discrete impacts and resulting computed surface

Figure 2: Profile rendered in solid model. Each color corresponds to a particular coefficient of restitution.

A More Detailed Description of Using "Package Profiling" to Reproduce the Distribution Environment

Let’s say your company is experiencing damage problems with one of its products while in route to your customers. How could you use "Package Profiling" to determine the cause of the damage and more importantly resolve the issue. The first step would be to mount a data recorder in the package that your product is normally shipped in. You want the total mass in the package to be equivalent to the product. This may require the use of a fixture to replace your product or the mounting of the recorder directly to an example of the product. The recorder will need to be programmed to record drop events. Drop the package at all orientations. You will want several impacts per face, edge and corner for each energy range of interest. Download the data to a UDF file and open the file in DynaMax. You will need to process the data with the "Calculate Drop Heights" option turned on. Now open the Package Profile View. This view will allow you to determine how well the surface fits your data.

Now that we have our package profiled we can ship the instrumented package through the normal distribution channels and capture the possible damaging events. For the actual shipment we will only need to program the recorder to capture the impacts. Once the recorder is retrieved and downloaded you will want to process the data with the "Calculate Drop Heights" on and choose "Profile Surface" from the "Source of Coefficient of Restitution" field. You will also need to pick the file that contains the profile of your package. After processing you can view the equivalent drop heights. 

Verifying the Procedure

The drop height can be determined with a high degree of accuracy by the time of flight. This technique can generally yield results that are within 5% of the actual free fall distance. The time of flight can be used to verify a profile of a package when both impacts and their associated free falls are recorded. Normally, when we have a profile for the package the recorder would subsequently only be programmed to record impacts but for the verification we will program the recorder to capture both the free fall and the impact. Then we will use the profile to generate equivalent drop heights and compare them to the recorded free falls. 
 

Figure 3: Solid 3D Rendering of the coefficient of restitution for the package under test.


FIGURE 4: This is an event from the set used to create the profile discussed below. The cursors mark the free fall portion of the data.

 

Event Number	Time of Event	FF Drop Height	I Drop Height
		m	m	%Diff
1:01	5/31/99 15:06	0.646	0.664	-2.79%
1:02	5/31/99 15:06	0.587	0.599	-2.04%
1:03	5/31/99 15:07	0.533	0.527	1.13%
1:05	5/31/99 15:07	0.593	0.597	-0.67%
1:06	5/31/99 15:07	0.668	0.698	-4.49%
1:07	5/31/99 15:07	0.648	0.646	0.31%
1:09	5/31/99 15:07	0.664	0.659	0.75%
1:10	5/31/99 15:07	0.592	0.605	-2.20%
1:12	5/31/99 15:07	0.513	0.448	12.67%
1:14	5/31/99 15:07	0.453	0.517	-14.13%
1:15	5/31/99 15:07	0.491	0.52	-5.91%
1:17	5/31/99 15:07	0.525	0.505	3.81%
1:19	5/31/99 15:08	0.556	0.627	-12.77%
1:20	5/31/99 15:08	0.542	0.596	-9.96%
1:22	5/31/99 15:08	0.526	0.456	13.31%
1:23	5/31/99 15:08	0.592	0.538	9.12%
1:24	5/31/99 15:08	0.549	0.595	-8.38%
1:25	5/31/99 15:08	0.568	0.534	5.99%
1:27	5/31/99 15:08	0.502	0.486	3.19%
1:28	5/31/99 15:08	0.459	0.509	-10.89%
1:29	5/31/99 15:08	0.541	0.461	14.79%
1:30	5/31/99 15:08	0.504	0.449	10.91%
1:31	5/31/99 15:08	0.457	0.483	-5.69%
1:33	5/31/99 15:09	0.545	0.621	-13.94%
1:34	5/31/99 15:09	0.448	0.495	-10.49%
1:35	5/31/99 15:09	0.483	0.461	4.55%
1:36	5/31/99 15:09	0.545	0.467	14.31%
1:37	5/31/99 15:09	0.433	0.398	8.08%
1:38	5/31/99 15:09	0.525	0.456	13.14%
1:39	5/31/99 15:09	0.521	0.57	-9.40%

TABLE 1

"FF Drop Height" is the measured free fall distance and "I Drop Height" is the calculated equivalent drop height. The Average percent difference is 0.08% and the standard deviation of the percent differences is 9%.

The package used for this part of the discussion contained two types of cushioning foam. A high coefficient of restitution foam on the bottom and middle layers and a very low coefficient of restitution foam on the top. The recorder was located in the middle of the package. A profile was created by randomly dropping the package by hand (dropping by hand can add additional error in the calculation due to the gradual transition to free-fall). The recorder was reset and a second set of random drops was recorded. Table 1 contains the measured free falls from the second set of data as well as their corresponding equivalent drop heights calculated with the profile. The "%Diff" column represents the error in each calculation. On average the profile doesn’t over or under estimate the coefficient of restitution as illustrated by an average error of 0.08%. In this example the standard deviation of the error is 9%. That means that the majority of the equivalent drops are within 9% of the actual free fall. At first that might not seem like much of a feat but when you consider the complexity of the package this accuracy would be nearly impossible with other techniques. You could expect even better results if the profile drops were created with a drop table or some other means of generating clean drops. 

The Coefficient of Restitution as a Function of Impact Energy

The above example uses a single surface profile, which means that all of the drops are included in the creation of the profile. This makes the discussion more simplistic but does lead to more error in the calculation of the equivalent drop heights. An additional advantage of "Package Profiling" is the ability to calculate the coefficient of restitution as a function of the impact energy. 


Figure 5A                                                                                                    Figure 5B