Observing the Center of Mass and Structural Analysis of Irregularly Shaped Objects in Static Equilibrium

Observing the Center of Mass and Structural Analysis of
Irregularly Shaped Objects in Static Equilibrium
Charles Maloneya, Jake Hamborb, John Schererc
a Mechanical and Aerospace Engineering Department
b Mechanical and Aerospace Engineering Department
c Department of Industrial Engineering and Management Systems
ABSTRACT
One of the main challenges in statics, and in designing structures in general, is stabilizing
the structure so that it will be in a static equilibrium. Unless the structure is specifically
meant to accelerate, this is needed in all structures, from large buildings to small toys.
Making a structure remain in equilibrium, especially when under external forces, is the main
challenge when designing these structures. One common way to achieve this is through the
application of more external forces, through the use of additional supports, so that both the
resultant force and resultant moment on the object is zero. However, if the object is balanced
along its center of mass, it no longer needs as many additional supports to stay stable. In
addition to requiring less additional supports, thus making the structure cheaper to construct,
designs that are stable on their own make structures safer, because, if a support were to fail,
the structure could continue to be stable on its own, or at least until evacuations took place.
For our IDEAS project, we will be studying the method of achieving equilibrium through
use of rebalancing the center of mass of a structure. In particular, we will be examining how
irregular objects can be balanced by a single normal force placed along a point on the vertical
axis which runs through the center of mass. We researched this topic by creating an
irregularly shaped object that was easily able to balance on a single point, located slightly
above the center of mass. We found that the structure was able to achieve a static equilibrium,
having no resultant force or moment, and that it was even able to resist some external forces
in various directions, and then return back to its original position.
1 INTRODUCTION
It’s safe to say that the human race has gotten general structures down. Apart from having
an entire area of research dedicated to itself, it also is required to have a strong idea of how
structures stand on their own in any area of engineering, from architects, to computer scientists.
However, as structures get bigger and bigger they require more and more supports to make
sure they are, and stay, stable throughout their lifetimes. However, even with additional
internal supports, buildings can still become unstable if an unforeseen event, such as an
earthquake or tsunami puts additional strain on the base of the structure. Some buildings now
integrate an internal counterweight to help the building balance, even if some of its supports
are compromised. In our paper, we will be looking at a specific structure, specifically a
balancing bird, and seeing if its ability to balance can be applied to larger structures.
2 BACKGROUND RESEARCH
In order for an object to remain in equilibrium, the net force on the object must be zero.
Normally, reaction forces are used to keep the object in balance. To minimize the number of
reaction forces needed, an object can be balanced along its center of mass. By applying a single
force along this axis, all net forces and moment of the object are canceled out, allowing it to
remain in equilibrium.
a
Email address: charlesmaloney412@gmail.com
b
Email address: jakehambor2@gmail.com
c
Email address: Case.Scherer@gmail.com

2.1 Application: Pagodas
Interestingly enough, the same balancing properties used by the balancing bird are also
used in traditional Japanese architecture, specifically the pagoda. Although earthquakes in the
past years have ravaged many modern buildings in Japan, these ancient structures seem to
always go untouched. Upon further inspection, we see that many of these structures were
designed specifically with structural security in mind.
Apart from the wood and the use of lashings and fittings instead of nails, which help the
structure bend rather than break when under intense strain, the layers of the building themselves
are designed to allow the center of gravity to stay low, and towards the center. The building
traditionally has five stories, with the largest on the bottom and the smallest on the top. This
keeps the majority of the structure’s weight at the bottom, which in turn keeps the center of
mass low.
Each layer of the building also has its weight distributed almost evenly among each floor.
In the case of an earthquake, if a layer was beginning to fall over, the weight of each floor
would pull the center of balance back to the center.
The pagoda also has a large support running through the middle of the structure, which acts
as both a support, but also as a sort of reverse pendulum. When the structure begins to sway,
the support acts as both a way to make sure that the layers of the building stay together, but
also allows the building to sway a bit without breaking, giving the structure time for its internal
mechanisms to stabilize the structure without it breaking.
Figure 1: Cross section of a traditional
Pagoda [1]

3 METHODS AND PROCEDRES
3.1 Goals and Outline
Overall, our goal for this project is to analyze the distribution of weight on the balancing
bird, and compare our observation to the result of our equations. Then, we will construct two
additional models of the bird, the first made out of a heavier paper, specifically card stock, and
the final bird out of aluminum. We will then compare the balancing ability of these birds to the
original, smaller model, and see if the same balancing ability translates onto the larger models.
We will use this information to make an assumption of whether or not a much larger model of
a balancing bird could be build, with the same stability features that the smaller model has.
3.2 Materials Used
 0.066 by 0.156 m (2.61 by 6.14 in) Dollar Bill
 0.130 by 0.305 m (5.10 by 12.0 in) piece of cardstock
 0.305 by 0.716 m (12.0 by 28.2 inch) piece of aluminum foil
 Triton T3 digital scale
3.3 Procedures
3.3.1 Experimental Center of Mass of Dollar bill, Cardstock, and Aluminum Foil d.
In order to find the experimental center of mass of each of the birds, we used a singular
point, and searched for a point on the point where the bird could balance on its own. We
Figure 3: Grid used and calculations for center of mass
Figure 2: Bird used in experimentation and
calculations

expected this to be located at the tip of the bird’s beak, since the bird is created specifically to
be able to balance on its beak.
3.3.2 Theoretical Center of Mass for Cardstock Bird
To look at why the bird is able to balance on its beak, we decided to calculate where the
center of mass should be for the cardstock bird. We decided to calculate this for the cardstock
bird, because it was the sturdiest of all the birds, and was a reasonable size for us to easily
measure.
First, we divided the bird into six segments, each fitting into a 0.0508 m (2.000 in) square.
Since these cutouts were very close to simple shapes, we assumed uniform density for each of
the parts, and found their center of masses, and found them with respect to the beak (Which we
placed at (0,0)). After marking these points on the paper, we took each individual piece off,
and measured them separately on our scale. We then used each pieces weight and coordinates
to find the birds total moment with respect to the X axis and Y axis in g * in. We then divided
by the bird’s total mass to find the point at which the bird would have its center of mass.
4 RESULTS
4.1 Comparison of Theoretical Center of Mass to Actual Center of Mass for Cardstock Bird
In all cases, the bird’s center of mass was located at the beak, and it was easily able to
balance on the tip of a finger. Our calculated center of mass was very close to the actual center
of mass for the cardstock bird, only being off by about 0.00526 m off our experimental value.
This can be overlooked, as it was probably due to human error in measurement.
We can conclude that the reason these birds balance so well is due to the fact that they are
designed to have a summation of weights that allows the center of mass to be located towards
the beak, so when a singular normal force, such as one from a table or a finger, is exerted onto
the bird’s beak, it is able to balance with ease.
4.2 Comparison of Balancing Abilities of Varying Sizes of Bird Models.
We also made several different birds of various sizes, and out of different materials. We
found was that the heavier and larger aluminum bird was less stiff than the other two, and
tended to droop more. However, this caused its center of mass to be lower than the other two
birds, which actually made it more stable. The cardstock bird we made was slightly larger than
the traditional bird, which was folded out of a dollar. Due to the fact that it was made out of a
stiffer material, it was much more rigid. This caused the center of mass to be higher, which
made it more difficult to balance. The standard dollar bird was the smallest bird, and exhibited
average stability compared to the other birds.
4.3 Possible Areas of Human Error
Although we tried to keep our errors to a minimum, we still must keep possible errors in
mind when looking at these results. The most likely error is probably in the uniformity in the
birds. Although we tried to keep them uniform across the different materials, since they were
hand folded it is very likely they all have slight variations from the original design.
Another possible error comes in the form of our actual measurements. The scale we used
only recorded values to the nearest hundredth of a gram, and if we had access to a more specific
measurement system we could have used more specific values in our final calculations.
Finally, although we used only 6 sections for our final calculations, we could have certainly
used more and gotten a more accurate result. However, we felt like using more than this would

overcomplicate our calculations, and also give more opportunities for human measurement
error.
5 CONCLUSION
Through our experimentation, we found that the balancing bird is designed specifically to
focus its center of mass on the tip of its beak, which gives it it’s seemingly gravity defying
properties. We also found that this property remains active even if the model is scaled up, or
made out of a different material.
We also found that the birds that balanced best were the birds that were made out of
materials that weren’t very stiff, such as the aluminum foil bird, because the wings drooping
down allow for the center of mass to be below the normal force, stabilizing it further.
5.1 Use in Future Structures
When testing these birds, we found that we only needed a singular force to keep them
upright, and the rest of the bird was held up by the moments around its center. This was due to
the shape and weight distribution of the bird, but any structure with its center of mass around
its primary support should have a balancing property similar to the birds. Several sculptures
and toys already use this property, but this property is not just limited to Knick knacks and
artistic pieces. If a larger structure, such as a skyscraper, was designed to have its center of
mass near its primary support, it could balance like the bird, without many other exterior
supporting mechanisms.
5.2 Use in Prevention of Earthquake Damage
Another interesting property seen with the birds were their uncanny ability to remain stable,
even if its support is shaken or moved. This is due to the bird’s center of mass staying still
relative to the support, which helps it move back to a state where it is stable. As discussed with
the Pagoda in the introduction, buildings that are designed to have a low, centered center of
mass are less vulnerable to earthquakes, and other disrupting forces. In fact, some buildings in
japan are being designed with a large pendulum like counterweight in their centers, which helps
to keep the top of the building balanced if the base is shaken.
5.3 Areas of possible further research
A property we would like to study further is the bird’s ability to reorient itself if put under
an exterior force, such as a gust of air or a tap of a finger. We believe this is due to the center
of mass shifting to the left if the right side is lowered, and vice versa, but knowing for certain
would take additional testing. Also, we would like to go even bigger with our bird design, and
see if a larger statue could be made practically by following the diagram, although this would
obviously require both time and money, both of which we have very little of.
6 REFERENCES
[1] Atsushi, U. (2005, June 15). Five-story Pagodas: Why Can't Earthquakes Knock Them
Down? Retrieved from Web Japan: http://web-japan.org/nipponia/nipponia33/en/topic/
[2] Shafer, J. (1999). Balancing $ Eagle. Retrieved from
http://www.barf.cc/jeremy/origami/PDF_diags/Designs/Eagle.pdf
[3] Yirka, B. (2013, August 2nd). Japanese companies develop quake damping pendulums
for tall buildings. Retrieved from Phys.org: News and Articles on Science and
Technology: http://phys.org/news/2013-08-japanese-companies-quake-damping-
pendulums.html

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