University of Puerto Rico
Mayagüez Campus
INME 4011 Machine Component Design 1, 2007-I
Prosthesis Device (Flexed Toe Part)
Jorge Corujo Sandín (802-02-1517)
Francisco Torres (802-03-8473)
Irvin de la Paz (802-03-1909)
Javier Cruz (802-00-1709)
Department of Mechanical Engineering
University of Puerto Rico at Mayagüez
May 7, 2007
Objective
The project outline is to design and analyses loads and stress
concentrations for the flexed toe part of the prosthesis of Tony Volpentest. At the
same time thinking in properties of materials, design and safety.
Description
The project consists of an existing design involving a prosthetic device
already in use by runners like Tony Volpentest. The goal of our team is to rethink
and if possible improve upon the current design. If no substantial new design can
be found we will concentrate our project on the existing design and how the
combine loads affect it as well as what material it should be in order to fulfill its
engineering and economical demands and specifications. The entire project (the
whole prosthesis) consist of various part but only one will be considered for the
full in depth design. As expressed before we will concentrate on the flexed toe
part. The focus of our design would be based on the standards already establish
to created a device that will be use under the diverse loads. Because we are
designing prosthesis for use in sport, it will be subjected to heavy a quick
changing loads, thus we need to choose a material which would be durable and
can resist fatigue. The other important detail when choosing a material for
designing sport prosthesis is the weigh, this needs to be strong and flexible but
as light as possible.
As mention, the flexed toe will be subjected to combine loads. Some of these,
which we are expecting to find during our analysis (but not limited to only these),
will be; Bending, sheer and normal compressive in the region where the socket
meets the flexed toe. Also there should be sheer and normal compressive forces
at the bottom of the toe (where the feet's toes should be). This would be cause
by the individual's weight and friction with the floor. We are currently not
expecting any mayor torsion force, since our
Preliminary analysis shows that for torsion to act it would require for part of the
flexed toe to somehow jam (get stuck) and for the individual to fall to his/her side.
Design Details
The design process first consisted in generating enough information, so
we could have a solid base of data to select the appropriate material. Several
consideration were taken into count, in our design the important aspects were
weight and the ability of the material to be flexible but still be strong enough to
resisted the alternating loads. The relationship of density with respect to tensile
stress was the major criteria. Using this has a departure point we were able to
come up with and equation to minimize mass by reducing this relation.
Where F is the maximum forces apply at the beam and L is the length.
This two are assumed constant for the purpose of this analysis. The value of the
force was 600 lb. and L was chosen to be 19 in. In the case of the other two
variables a different approach was taken, first it consists of presuming a
minimum value of tensile stress which was 29.00 psi. With the use of a graph of
strength-density, a range of materials were selected.
At the end we finish with four materials for which we after choose the
best that will fit our necessities. The four materials are a selecting of a
thermoplastic, titanium alloy, nickel alloy and a SRCR 6100. A thermoplastic is a
kind of material that is able to be deforming plastically, it becomes more brittle at
lower temperatures, but it has a great strength against fatigue. A titanium alloy
“Deutche Titan Tikrutan RT 18 Pd Low-Alloyed Titanium”, titanium is a material
that has a high YS value, but it has a higher density value than other material
with similar properties. Titanium is use in many aircraft applications and engines,
also for part that would be in constant use and expose to corrosion. A nickel alloy
N04400 it has a high value of YS, but it is a heavy material. The last material isn’t
that common and not that much information was obtained about it. A table with
the density and YS vale are shown next.
Table 1: Density, Yield and Mass of materials
Material Density(lb/in^3) "Yield" (psi) Mass (lb)
Thermoplastic 0.0495 36,000.00 0.5047
Titanium alloy 0.1630 46,400.00 1.2895
N04400 Nickel 0.3190 35,000.00 3.3457
SRCR 6100 0.0741 32,000.00 0.8500
With the respected values of density and YS mass was calculated for the
design part, results that any of the three materials would worked. Still Titanium
was selected because it has a low mass and the highest value of YS. This gives
us the advantages to design the prosthetic leg to resist higher loads. It has to be
mentioned that in our selecting of the material the cost of the material is not been
added to the parameters of the design.
Once the material is selected, the next step is to analyses the different
loads that are applied. For the purpose of this part only the bending moment was
taken to account because the part is more likely to fail by stress than shear. The
only effect the shear force will have is to create a force that would be counted as
the force of friction. This force is neglected because in comparison the magnitude
is smaller than the one of bending. The location of this force is applying parallel
to the curve beam. The force was assumed to be 200 lb
In tension and -600 lb in compression, the respect moments, stress and Von
Misses stress are shown below.
Table 2: Forces, moments and Von Misses stresses
Maximum Force
(lb)
Minimum Force
(lb)
Amplitude Force
(lb)
Mean Force
(lb)
200 -600 400 -200
Moment Amp.
(lb-in)
Moment Mean
(lb-in)
Stress of Amp.
(psi)
Stress Mean
(psi)
1000 -500 -11,120.79 5,560.39
Von Misses Amp.
(psi)
Von Misses Mean
(psi)
Length
(in.)
12,232.87 6,116.43 19
The negative value of the stress of amplitude reflects that the force is in
compression. With the values of the Von Misses stresses and the values of Se
and Sut the safety factor can be obtained with the Godman equation. The value
of Se is calculated first by determine what king of material is after a conversion
factor is applied to minimize it by half. After this process the new value of Se is
obtained, still this one is not correct the values of the different k’s were
calculated.
Table 3: K’s, Secorrected and Safety Factor
k (load) k (temp) k (size) k (surf)
1 1 0.8812 0.9881
k (reliability) Se (not corrected) Se (corrected) Safety factor
0.702 42,800.00 26,161.12 1.86
The value obtained is 1.86 which is a perfect value to design with. This
becomes clear after realizing the part was made to have and infinite live of more
than 1x10^6 cycles. By obtaining a safety factor for the project, the material
selected was a titanium alloy.
Static Analysis
Fig 1.0 Reactions of our Specimen
d
M=Fd
F
N
Ffr=μN
For our material
μN=FcosØ
We assumed: Fmax= -600lb, Ø = 75.1º and g=32.2ft/s2
This means that:
N= F sin 75.1= 579 lb
Ffr= Fcos 75.1 = -154lb
Now because we can assume this is one quarter circle thus we can assume the
greatest momentum and bending will occur when the plane of the circle is at 45º
degrees (see figure 2).
Fig 2.0 Internal forces
To calculate moment we have that M=Fd but because it is easier to convert N
and Fr in magnitude, magnitude is at 75.1
degrees. The distance D is 2.5 then moment is 2.5*600=1500.
Shear force is given by static equation:
Solving the previous two equations for V and Normal we have
V=-5.28lb
Normal=-813.54 lb
Now applying equation for curve we have
Because we have a rectangular cross section the shear simplify will be
This means shear has no noticeable effect on our static loading.
Now calculating principal stresses with matrix we have
We have that and
Shear max will be
R=
Figure 3.0 Mohr diagram calculated for force of -600lb
Figure 4.0 Plane Stress diagram for Static force of 600 pounds in compression.
Deflection of Curved Beams
For our particular material the challenge is develop a formula for a curved
beam. The application of the flexure formula for a straight beam results in error.
When all “fibers” of a member have the same center of curvature the concentric
or common type curved beam exists. Such a beam fortunately is defined by
Winkler – Bach theory. Also deflections can be found by use of Castigliano’s
theorem. In our case we used the moment area theory given by equations 1.2.
(1.2)
It is important for our material that the deflection not is to little or too much
because the materials have to absorb energy and release energy. We assumed
that our flex toe was a fourth of a circle for simplicity purposes. A program in
excel was made for faster calculations. The derivation of the formula is showed
below in fig 5.0
Fig.5.0 Flex Toe idealization
Discussion
With the changes on the materials that we assume, we hope to build more
long lasting and reliable prosthesis. This is without comparing with other
companies and without taking in considerations the cost. The finish line was to
build the best part for the prosthesis that we could design, just taking in
consideration the mechanical properties of the design and materials available.
First we take in consideration just one critical point because the second critical
point is depreciable in comparison. This let us change the material and look for
other properties for a better design. Changing the material to titanium alloy we
can make a stronger prosthesis and more reliable significantly increasing the
safety factor. The properties of the titanium make the prosthesis a long lasting
one since it was design with infinite life in mind. So far the only trade off has
been in cost. With this we can assure the client that his prosthesis isn’t going to
brake while he is using it. One of the disadvantage against this design is that it
might weight a slightly more than the commercially available, composite made
prosthesis. It also would cost significantly more because titanium material isn’t
cheap, this can be seen when we compare the titanium cost with the composite
cost.
Some of the merits that have our design are:
Resistible to corrosion
Light weight
Simple and very user friendly for users
Multipurpose
Wide range of customer applications
Some suggestions to make better our design are find cheaper materials that
are known but we don’t have the properties. Also with these materials we can
make the prosthesis more light weight for the runner. Main problems of our
design are the cost and the viability of the prosthesis because is hard to fabricate
and to buy.
Conclusions
Thru this whole process we where expose to a series of new concepts we
where not accustomed to use. In particular the mechanics of working with
curved beams, this is due since the whole part in essence is simply a curved
beam. Once the whole mechanics where understood we where able to dive into
the whole process of material selection. This process proved to be a challenging
one, since it required a material with exceptional high strength yet a density low
enough so that the runner didn’t feel he was running with cement block attached
to his legs. This led us to appreciate how far material sciences have advanced in
the last 30 years.
Material selection was without a doubt the toughest part as a whole,
followed closely by the dynamic analysis; one is directly affected by the other.
This was emphasized by the fact that the materials we where working with such
a titanium, carbon fiber, and other strong ultra light materials we aren’t used to
dealing with. Thru trial an error we where able to establish a list of finalists out of
which we finally made our decision, an alloy of titanium discovered thru our
search in the www.matweb.com database. Although in reality and practicality
this alloy is very likely to expensive for the design, a lack of reliable and complete
information found about newer and stronger polymers and thermoplastics made
us choose said alloy.
Finally we obtain a design capable of being utilized by an up to 200 pound
person, with a safety factor of n = 1.86 yet still enjoying a remarkably light design.
Appendix
Governing Equations
sin][ inout RRy
dRRds inout ][
Radio Interno (in.) Radio Externo (in.) Ancho (in.) Altura (in.)7.49 7.95 2.5 0.46
Area Seccional (in.^2) Distancia (in.) Angulo (grados) Largo (in.)1.15 2.5 75 19
r (barra) (in.) R r (in.) S (ut) (psi)7.720 7.718 7.950 85600
Fuerza Maxima (lb) Fuerza Minima (lb) Fuerza Amplitud (lb) Fuerza Mean (lb)200 -600 400 -200
Momento Amp. (lb-in) Momento Mean (lb-in) Esfuerzo de Amp. (psi) Esfuerzo Mean (psi)1000 -500 -11,120.79 5,560.39
Von Mises Amp. (psi) Von Mises Mean (psi) Largo (in.) Kf12,232.87 6,116.43 19 1.1
k (load) k (temp) k (size) k (surf)1 1 0.8812 0.9881
k (reliability) Se (no corregido) Se (corregido) Factor de Seguridad0.702 42,800.00 26,161.12 1.86
Material Densidad (lb/in^3) "Yield" (psi) Masa (lb)Termoplastico 0.0495 36,000.00 0.5047Titanium alloy 0.1630 46,400.00 1.2895N04400 Nickel 0.3190 35,000.00 3.3457
SRCR 6100 0.0741 32,000.00 0.8500
Links used in the project:
1985 The Seattle Foot http://www.washington.edu/research/pathbreakers/1985a.html
Bending of Cantilever Beam http://documents.wolfram.com/applications/structural/BendingofCantileverBeams.html
Comparison of the Seattle Lite Foot and Genesis II Prosthetic Foot during walking and running http://www.oandp.org/jpo/library/2000_01_009.asp
Delrin Material http://en.wikipedia.org/wiki/Delrin
Is the Use of Advanced Materials in Sports Equipment Unethical?http://www.tms.org/pubs/journals/JOM/9702/Froes-9702.html
Tony Volpentest http://encarta.msn.com/media_461550907_761562123_-1_1/Tony_Volpentest.html
Transtibial prosthesishttp://en.wikipedia.org/wiki/Transtibial_prosthesis
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