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