Group 17
Shalika Neelaveni, Bryce Gunderman, Jillian Pote
Lab 1: Bungee - Activity 3
Activity 1: Rubber Band Characterization
Starting with activity 1, we created three variations of rubber band chains,
each of 4 inches (unit) length. The variations were: single link, double link,
and fishtail chains, as shown in order in figure 1. We then tested a 4 inch
chain of each variation in the MTS machine until they each deformed an
additional 150 mm or more from their original length. We graphed this data
and found a line of best fit through the linear portion of the Load vs.
Extension graphs, and thus calculated a value of k from the slope of these
lines. See the appendix to see our graphs, lines of best fit, and k-values for
each of the three chain variations.
Since graphing our initial MTS machine data with a linear line of best
fit, we have now added a nonlinear equation to our data. This equation (F)
will be used in:
𝑅(𝑥) = 𝑝𝐹 ( )
𝑥
𝑠
where x is the displacement of the rubber band, p is the number of 4 inch units in parallel and s is the
number of 4 inch units in series in our total chain. We will set the integral of 𝑅(𝑥) equal to the initial
potential energy of our system in order to solve for x, as in:
∆𝑥
𝑚𝑔ℎ = ∫ 𝑅(𝑥) 𝑑𝑥
0
where m is the mass we are given, h is the height we are given, and g = 9.81 m/s2.
Activity 2: Acceleration & Instrumentation
In order to measure acceleration, we used the phyphox app on an iPhone. We made sure to
measure the acceleration with g (gravitational acceleration) included. The phone was positioned
vertically in the plastic bag, and thus our direction of freefall acceleration was in the y direction and
other directions were ignored.
We dropped the phone accelerometer configuration,
weighing 250 grams, with our 3 different orientations of rubber
band links: fishtail, double links, and single links (see Figure 2
for a visualization of our experimental setup). Each chain was 8
cm in length (two 4 inch units in series) and was dropped 4
times from identical heights. We then graphed average
accelerations during the tests as shown below in Figures 3-5.
Since then, we have created 3 new chains, of the 3
orientations from before, but this time of 2ft in length (six 4 inch
units in series). We tested these configurations out by again
dropping our 250 gram phone accelerometer / jumper set up as seen under our Jumper section. We
dropped each configuration 4 times and measured accelerations as well as the total final length of the
configuration after maximum displacement occurred. We did this by having someone stand near eye
level of where the configuration would drop to its lowest point, and had them hold out their hand to
mark the lowest point which was achieved. We then lowered down a tape measurer and measured the
distance between the point dropped and the lowest point reached. See figures 6 - 8 to see our
acceleration data.
After examining our various acceleration data, we realized that we were reaching significant g’s
(4.8 - 6.7 g’s) with our stiffer chains (double link and fishtail) when only dropping to a distance of 5 ft or
less. Thus, we made the decision to use exclusively the least stiff chain, the single link chain. In this case,
it made the most sense to choose the chain with the lowest k value, since the k values could always be
increased by adding chains in parallel configurations.
Activity 2: Graphs of 8 inch Chain Accelerations
Figure 3: 8 inch Fishtail Chain Acceleration Data Figure 4: 8 inch Double Chain Acceleration Data
Figure 5: 8 inch Single Chain Acceleration Data
Activity 2 Graphs of 2ft Chain Accelerations
Figure 6: 2ft Single Link Acceleration Data Figure 7: 2ft Double Chain Acceleration Data
Figure 8: 2ft Fishtail Chain Acceleration Data
Jumper:
Figure 9: Jumper, front face Figure 10: Jumper angled view; masses were
placed inside the ziplock bag
The jumper includes a cell phone which we have secured inside a plastic bag. We attach this bag to a carabiner
which attaches to the end of our bungee. This was connected in testing with a card holder to add additional weight.
Our additional mass of 752 grams was added in the form of screws and bolts and placed inside the ziplock bag.
Chain Model Development (Code found in Appendix):
Assuming m = mass given, and h = height given, we started out by looping through all the different combinations of
p (number of bands in parallel) and s (number of bands in series), within reason. With each of these combinations,
we know the unstretched length (L) of each combination is simply the number of units in series multiplied by the
length of one unit. We calculated the expected change in length (∆𝑥) of each possibility by subtracting the total
height by the safety factor (1 meter in our case), and the unstretched length. Using this, we integrate to obtain the
expected energy at the bottom of the system and set it equal to the expected energy at the top:
∆𝑥
𝑚𝑔ℎ = ∫ 𝑅(𝑥) 𝑑𝑥,
0
where 𝑅(𝑥) = 𝑝𝐹 ( ) and 𝐹(𝑥)
𝑥
𝑠
−7 3 −5 2
= 2 × 10 𝑥 − 8 × 10 𝑥 + 0. 024𝑥 + 3. 0035
Realistically, our calculations will not result in these two expressions being exactly equal to one another. Therefore,
we extracted the combination of parallel and series units which produces the minimum difference in the energy at
the top and the energy at the bottom. We calculate the maximum acceleration using the force equation, and finally
check to make sure the acceleration of this combination falls within the given range. It then outputs this
combination of parallel and series, along with the total length, which yields the overall configuration of the rubber
bands.
Final Demo Analysis:
Link to drop video: https://youtube.com/shorts/cVqDolxlNA8?feature=share
Looking at our actual results, we achieved a final test acceleration of 4.68g. From our code, we obtained theoretical
length values of an unstretched length of 44 inches. This includes 11 links (1 link = 4 in of rubber band) attached in
series along with 5 links attached in parallel. We chose to use a safety factor of 1 meter (~39.37 in), and achieved a
drop distance of ~32 inches from ground, per video recording estimation. The data from our official drop was not
saved by the accelerometer, so our acceleration data from a test drop from the same location moments before the
official drop, the following acceleration was recorded:
We expected to see the distance to ground to be equal to our safety factor of 1 meter, while experimentally seeing a
distance to ground that was shorter at ~32 inches. Thus, there are sources of error present causing our safety factor
to be greater than needed. Additionally, the bungee was expected to stretch 195 inches (7.091 meters - 1 meter
safety factor - 44 inch initial length). We were able to see an experimental stretched value of 203.17 inches,
stemming from 7.091 meters - 32 inches from ground- 44 inch initial length. Foremost, this disparity comes from
the fact that the code is generated off the next closest link to the theoretical optimized bungee length. With each
length equating to 4 inches, if assuming the maximum error in distance, this would be able to account for over half
of the difference in values. Had we chosen to define each link as the length of a singular rubber band, thus
quartering the length of one link, this error margin would be able to be decreased by a similar order of magnitude.
Additionally, with the bungee top being held at a height different from that of the exact measured drop height, the
resulting distance from the ground at maximum stretch will lead to a resulting offset in the values calculated.
In analyzing the given time and y-direction acceleration data from the accelerometer, it can be observed that
acceleration reached its peak at the time at which the bungee is at its maximum stretch. This can be verified from
theory, since acceleration will reach a maximum when the upward force on the jumper is at its peak. Since force
increases as stretch increases, acceleration must peak at the maximum upward force.
Appendix:
Unit Length Chain Extension Graphs and k-values:
Figure 11: Line of best fit for single sink chain Figure 12: Line of best fit for double link chain yielding
yielding a k-value of 0.0429 N/mm. a k-value of 0.0991 N/mm.
Figure 13: Line of best fit for fishtail yielding a k-value of
0.0944 N/mm.
Matlab Code
Figure 14: Matlab code - inputting mass, height, and maximum acceleration and outputting rubberband configuration
(number in series and in parallel) and total length
