Department of Mechanical Engineering University of Puerto Rico at Mayagüez

INME 4011 Machine Component Design I, 2007-I Leg Prosthesis

Neysa Alicea Angélica Báez Marisa Bernal Beatriz Ramos Prof. Pablo Cáceres May 7, 2007. Objectives

Our objective is to analyze the stresses present in the socket of a prosthesis device. We will use the techniques learned during this semester to evaluate an object subjected to static and fluctuating loads. Our vision is to improve the design, making it more comfortable and thus improving the quality of life of people with disabilities.

Description

In the past years people like Dennis Oehler and Tony Volpentest established records using a leg prosthesis. Like them many people that loose a leg face a very difficult time. It is especially hard for an athlete or active person to resume their activities with this disability. A way to effectively replace that lost limb can bring new independence and mobility to the person. That's why the design of prostheses is so important. Our project consists of designing the socket of a leg prosthesis. Leg prostheses are composed of various parts including the liner, the socket, the knee and the foot. Each part has an important role in making the product a reliable and useful one. The purpose of the socket is to provide the base where the limb and the prosthesis join. The limb fits right into the socket where it is supported. The socket must be flexible allowing muscles, tendons and bones to function inside. In order to obtain a good performance the correct material should be chosen. The material which fits our characteristics is carbon fiber reinforced plastic which is ductile and strong to resist the stresses. The design considered many realistic facts such if the person has an accident and the socket is subjected to bigger pressures. Our goal is to design a durable, comfortable and reliable socket that will help people with disabilities to continue having an active life. Design Details

Material selection

The selection of the material for the prosthesis is very important since the trustworthiness of the product depends on its performance. The material we’ll use for the socket of the prosthesis device should provide the following characteristics:  Low density (lightweight to make it easier to move)  Strong ( to resist impact loads in a small area)  Ductile ( to help absorb the loads without breaking)  Wearable (comfortable for the user)  Easy to maintain and clean

We will focus on two main properties; that it is light and stiff so it will resist buckle. We will combine the equation of mass with the one of force while keeping the radius as a variable to obtain a material index. The equations used are: m  Lr 2 , 2 N EI  4 F  F  where I  r cr L2 4

1  4FL2  4 r     3   N E  1   2   4FL  2 By substituting for r in the equation of mass we obtain: m  L    1  N 3   E 2      Our material index is   . To obtain a light and strong material we should minimize  1   E 2   1  E 2 this value. Since the material index that is found in the graphs is   we will      maximize this value. By doing this we obtain certain materials which have the required properties. These materials are: wood, CFRP, composites and technical ceramics, foam. We choose to use carbon fiber reinforced plastic (CFRP). This material is light yet strong with a density of 1.8g/cc, an SUT of 3800MPa and an elastic modulus of 225GPa. Design Calculations for Static and Dynamic Loads

We propose a design of the socket of a prosthetic leg in which many assumptions are made in order to simplify the analysis of the design. In reality this part of the prosthesis has an irregular shape which adheres to the particular shape of a person’s limb. In our design we will approximate its shape as a beam, a cylinder or a cylinder with two different diameters depending on what we want to calculate.

Static Load Analysis

A socket is subjected to a distributed load due to impact as shown in figure 1. Since it would be difficult to calculate the effects of that distributed load in the real shape of the socket we represent it as a beam subjected to a distributed load as shown in figure 2. For the static analysis we find a resultant concentrated force of 800N and calculate the reactions at A and B which have a value of: 400N. We draw a shear and moment diagram in order to determine maximum shear force and moment shown in figure 3.

Figure 1

Figure 2

Figure 3

The values for Vmax and Mmax are 400N, 127.5N-m, respectively. Using the distributed load instead of a concentrated load the shear and moment diagram would be as in figure 4.

Figure 4 For the analysis of a static load we simply use the maximum moment and find a bending stress.

My   I

M= Mmax = 63.83N-m

101.8mm y=  50.9mm  0.0509m 2

bh3 (0.319m)(0.1018m)3 I =   2.80 105 m4 12 12 (127.5N  m)(0.0509m) KN    115.8 2.80 105 m4 m2

We can also calculate a shear force using the maximum shear force shown in the shear force diagram. 3V   2A

V = Vmax = 400N A = (0.319m)(.1018m)=0.0324m2

3(400N ) N KN    18518.5  18.5 2(0.0324m2 ) m2 m2

Since the beam is symmetrical we know that the maximum deflection will appear in the middle of the beam. This conclusion can be verified by executing the following procedure: 4 EIw(x)  , where w(x) is the distributed load x 4

3 EIM (x)  , M(x) is the moment x3  2 EI (x)  , ө(x) is the angle of deflection x 2  EI(x)  , ν(x) is the deflection. x The resulting equation for deflection is: qx3 C x2  1  (x)   1  C x  C  2 3    6 2  EI  C1=0, C2= -127.4, C3=0 627.4x3  1  (x)   127.4x   6  EI  (x)  2.68 x 10-6m= 2.6 x 10-3mm

Calculation of the safety factor

We found σ1=118.6KN. To calculate the safety factor we use Modified Mohr’s Theory

SUT ns  1 3800MPa n   32.8 103 s 115.8KPa

Obviously this factor doesn’t make any sense. It means that for this material and load the product is way over-designed. To find a sensible value using CFRP we would need a larger stress to compare against the ultimate tensile strength. We opted for leaving all the data as it was since changing it would mean changing all the calculations for static and dynamic load analysis. Dynamic loads are much larger and a sensible safety factor can be found with our data. Dynamic Load Analysis

For a dynamic load analysis the force that acts on the socket is an impact force that acts normal to its surface. To calculate this impact force we first calculate the maximum Mv 2 L elongation using the following equation:   , where : max EA M = mass , v = velocity of the impact load, L = length, E = elasticity modulus, A = area

The values we used for each are: M = 81.61kg, v = 4.57m/s, L= 0.3048 m, E =225 GPa, A = 0.03242m2.

Mv 2 L (81.61kg)(4.57m / s)2 (0.3048m)   =  0.000266962m max EA (225GPa)(0.03242m2 ) E Once we know the maximum elongation we use   max to calculate the maximum max L stress.

E (225GPa)(0.0002669)   max =  197.06MPa max L (0.3048m)

To find the maximum force that is applied we simply multiply the maximum stress by the area. 2 Pmax  197.06MPa  0.03242m  6390.77KN

With this force we calculate amplitude and a mean forces which we use to find the corresponding stresses. The values obtained are:

 Pa=3195.38KN = Pmean

 σa= 128.69MPa= σmean

Since the design of the socket includes stress concentrators we must calculate a stress concentrator factor (Kt) and dynamic stress concentration factor (Kf). To do this we now approximate the shape of the socket to that of a cylinder with two different diameters, and a certain radius at the fillet as shown in figure 5.

Figure 5

To calculate Kt :

D 203mm r 25.24mm   1.14   .14 d 177mm d 177mm

Using the graph for tension we found Kt =1.5

For dynamic loads we need to calculate a Kf,bending . We used the equation

K f  1  q(Kt 1) . Since we don’t have the necessary information to find the sensitivity factor q, we approximated it to 1. We can justify this assumption since as the value of SUT of the material increases the value of a decreases. The value of SUT of CFRP is 551ksi and it is much larger than any of the values for aluminum. We approximated the value of

a to 0.0001593 and we obtain a sensitivity factor of 1. We obtain a value of Kf equal to 1.5.

Once Kf is found and assuming that the stress acts in the fillet which is a point with stress concentration we can use it to correct the value of the amplitude and mean stresses. The values obtained are:

 σa,sc= 192.95MPa

 σmean,sc = 192.95MPa We have stresses but no shear stresses so using DET fracture theory our σvm,a will be the same as σa,sc, and our σvm.mean will be the same as σmean,sc.

To calculate the safety factor we need to find the fatigue strength (Sf) of the material. To do this we approximate the material as aluminum in order to use the data from the slides. We can justify this approximation since aluminum is light and ductile and we need a material with those properties. ’ S f = 0.4 SUT ’ S f =1520MPa We need to correct this factor and to do so we need to find Ksize, Kload, Ksurface, Ktemperature, Kreliability. To determine Ksize we used a non-rotating cylinder, using the smaller diameter to find the equivalent diameter. Kload was calculated for axial load, while room temperature was assumed. Also we assumed that the surface of the part was ground and we used a reliability of 99.9999% since only one prosthesis will be fabricated and it must be of the best quality.

Ksize = 0.792 Kload =0.85 Ksurface =0.7841 Ktemperature =1 Kreliability=0.620 Sf = (0.792)(0.85)(0.7841)(1)(0.62)(1520MPa) = 497.5MPa

Ksize  0.792

Kload  0.85

Ksurface  0.7841

Ktermperature  1

Kreliability  0.620

S f  (0.792)(0.85)(0.7841)(1)(0.620)

S f  497.5MPa

Calculation of the safety factor

Using Modified Goodman theory to calculate the safety factor:

1 1 n    2.28   192.95MPa 192.95MPa vm,a  vm,m    497.5MPa 3800MPa S f SUT Calculation of the component life Data: σmax =197.06MPa Sf = 495.5MPa SUT = 3800MPa

Sm = 0.75SUT = 2850MPa 3 Sm @ 10 cycles 8 Sf @ 5 x 10 cycles log Sn = log a + b log(N) log(2850) = log a + b log(103) log (495.5) = log a + b log(5 x 105)

0.7598 = -b (2.69) b= -0.2815 log(2850) = log a + -0.2815 log(103) a = 19922.54

With a and b we substitute and find:

Log(197.06) = log(19922.54) – 0.2815 log(N)

N = 1.3 x 107 cycles

S m

S m

S f

1.3 x 107 5 x 108 Discussion

Our design has been made including many assumptions. This represents one of its main disadvantages. By acquiring the necessary specific data we could improve our design by making it more accurate. Another disadvantage is the cost of the prosthesis. We selected carbon fiber reinforced plastic which is an expensive material. The selection of this material can be justified since one of the most important parts of the design is the selection of a material that will perform as intended. The selection of CFRP will guarantee the best performance. An additional disadvantage of this design is that it cannot be mass produced since its dimensions are for a specific person. On the other hand this can also be considered an advantage for the costumer because it fits all their needs and complaints. The user will not feel uncomfortable with this device because it was customized for him. Another advantage of this design was that real situations like accidents that may occur to a person were considered. By adding stress concentrators on purpose at strategic places between the socket and the knee, we localize a place where the piece would intentionally fail in a way that would protect the user. In case of failure instead of failing at the socket it would fail below and the “knee” would not become jammed on the patient’s limb. The prosthesis we have designed is specifically made for athletes. Conclusion

This project has given us the opportunity to apply what we learned in class directly to a real situation. As is our case with our knowledge we are designing something that will drastically change another human being’s life. By completing the analysis of the socket we realize that it is much more complex than we thought. There are many details to be considered and some of them are beyond our knowledge. Nevertheless we have come up with the calculations that were required of us. This socket is subjected to static and fluctuating loads which cause stresses. The stresses acting on the sockets are bending, shear and axial stress. We created stress concentrators in the socket to prevent users from getting hurt when the prosthesis is subjected to bigger loads than the ones expected. Using the Modified Goodman Theory we calculated the safety factor which was found to be 2.28. For disabled people to continue achieving world records like Oehler and Volpentest the redesign and continuous improvement of prosthetic devices is necessary. In this project we improve the quality of life of the person with a better design.

References: http://www.matweb.com/search/SpecificMaterial.asp?bassnum=ESGL17