Project Report on “DYNAMIC CHARACTERIZATION OF

BONES”

Submitted by: SAHIL BAJAJ

Under the Guidance of:

DR. S. BHALLA

Department of Civil Engineering, Indian Institute of Technology Delhi

April 2007

2

CERTIFICATE

“I certify that this report explains the work carried out by me in the courses CED 310

(Minor Project), under the overall supervision of Dr. Suresh Bhalla. The contents of the

report including text, figures, tables etc. have not been reproduced from other sources

such as books, journals, reports, manuals, websites, etc. Wherever limited reproduction

from another source had been made the source has been duly acknowledged at that point

and also listed in the References section.”

Sahil Bajaj

Date: April 24th 2007

3

CERTIFICATE

“This is to certify that the report submitted by Mr. Sahil Bajaj describes the work carried

out by him in the course CED 310 – Mini Project under my overall supervision.”

Dr. Suresh Bhalla (Supervisor)

Date: April 24th 2007

4

ACKNOWLEDGEMENT

I would like to express my sincere thanks & gratitude to Dr. Suresh Bhalla for his

continuous and unfailing support, guidance and help, which have been invaluable during

the course of this project.

It has personally been a wonderful experience for me working under a professor as

sincere and committed as Dr. Bhalla. He has always been a source of inspiration for me

and his insistence on quality work has pushed me to put in my best.

I would also like to thank my friends Vivek and Aditya for their help and support during

the course of the project.

Sahil Bajaj

ABSTRACT

Structural Health Monitoring is an area of Civil Engineering, in which extensive research

has taken place in the last few years. Many processes, technologies and devices have

been developed to monitor the health of critical structures and warn engineers of any

impending failure/collapse.

The aim of this study is to expand on this pre-existing knowledge and expertise of Civil

Engineers and apply it to bio-mechanics. Dynamic characterization of bones has many

possible applications, some of which are discussed later in this report. The method discussed in this report uses a pair of PZT patches to find out the resonant

frequencies of a bone. These frequencies would obviously depend on physical

parameters like density and structural integrity of the bone. This fact has interesting

implications. It suggests that by finding the resonant frequency of a bone, and knowing

its normal base resonant frequency, we may be able to estimate its various physical

parameters.

6

TABLE OF CONTENTS PAGE

CERTIFICATES…..………………………………………………………..1 ACKNOWLEDGMENT……………………………………………………4

ABSTRACT………………………………………………………………...5

CHAPTER 1: THEORECTICAL BACKGROUND

1.1 The Peizoelectric effect………………………………………….8

1.1.1 Piezoelectricity ………………………………..…..........8

1.1.2 Piezoelectric Materials…………….……..…..……........9

1.1.3 Lead zirconate titanate (PZT)…………..……….. …….9

1.2 Principles of Operation ……..…………………………….. ……9

1.3 Mechanical Resonance ...………………..……………………...10

1.4 The Femur Bone………………..……………………………….11

CHAPTER 2: OBJECTIVES AND METHODOLOGY

2.1 Objectives…………...…..……………………………….……. 12

2.2 Experiment Methodology …………………... ………………...12

2.3 Voltage Amplification Circuit ……………………...….…........13

CHAPTER 3: EXPERIMENTS & RESULTS

3.1 Aluminum ruler experiment ……….……………………....... 14

3.1.1 Theoretical Estimate of Resonance Frequencies………14

3.1.2 Experiment and Results………….……...………...…....15

7

3.1.3 Analysis of the ‘ringing’ sound…….…….…………….15

3.1.4 Repeated Experiment……………………..…………….17

3.1.5 Analysis of Output Voltage vs. Input Frequency………17

3.2 Femur experiment ………………………………….…………18

3.2.1 Theoretical Estimate for Resonance Frequencies………18

3.2.2 Experiment and Results ……....……………...………...19

3.3 Fractured Femur experiment…………………………………….20

3.3.1 Crack Dimensions ………………………..…………….20

3.3.2 Experiment and Results …………………...…………...21

CHAPTER 4: CONCLUSIONS AND SUGGESTIONS

4.1 Inferences based on results…………………………………...…22

4.2 Suggestions and Future Work in Dynamic Characterization of

Bones ……………………………………………………………….23

REFERENCES //////////////////////.. 25

APPENDICES

Appendix A - Output Voltage vs. Frequency table for

Aluminum ruler////. ////////../////////..26

Appendix B - Output Voltage vs. Frequency table for

Femur/////////////////////.../////.27

Appendix C - Output Voltage vs. Frequency table for

Fractured Femur /.///////////////////// 29

8

CHAPTER 1 : THEORETICAL BACKGROU*D

1.1 The Piezoelectric Effect

1.1.1 Piezoelectricity

Piezoelectricity is the ability of crystals and certain ceramic materials to generate a

voltage in response to applied mechanical stress (See Fig 1.1). The piezoelectric effect is

reversible in that piezoelectric crystals, when subjected to an externally applied voltage,

can change shape by a small amount. (For instance, the deformation is about 0.1% of the

original dimension in PZT.)

In physics, the piezoelectric effect can be described as the link between electrostatics and

mechanics.

Fig 1.1 – How Piezoelectricity Works

1.1.2 Piezoelectric materials

A piezoelectric sensor is a device that uses the piezoelectric effect to measure

mechanical signals like pressure, acceleration, strain or force by converting them to an

electrical signal. An actuator does the exact opposite.

Two main groups of materials are used for piezoelectric sensors and actuators:

piezoelectric ceramics and single crystal materials. The ceramic materials (such as

PZT ceramic) have a piezoelectric constant / sensitivity that is roughly two orders of

magnitude higher than those of single crystal materials and can be produced by

inexpensive sintering processes. The piezoeffect in piezoceramics is "trained", so

unfortunately their high sensitivity degrades over time. The degradation is highly

correlated with temperature. The less sensitive crystal materials (gallium phosphate,

9

quartz, tourmaline) have a much higher – when carefully handled, almost infinite – long

term stability (Wikipedia, 2007).

One disadvantage of piezoelectric sensors is that they cannot be used very effectively

for true static measurements. A static force will result in a fixed amount of charges on

the piezoelectric material. Working with conventional readout electronics, not perfect

insulating materials, and reduction in internal sensor resistance will result in a constant

loss of electrons, yielding a decreasing signal. However, for dynamic measurements,

they are very effective.

1.1.3 Lead zirconate titanate (PZT)

Lead zirconate titanate (Pb[ZrxTi1-x]O3 0<x<1) is a ceramic perovskite material that

shows a marked piezoelectric effect. It is also known as lead zirconium titanate or PZT,

an abbreviation of the chemical formula.

Being piezoelectric, it develops a voltage difference across two of its faces when

compressed (useful for sensor applications), or physically changes shape when an

external electric field is applied (useful for actuators and the like).

The material features an extremely large dielectric constant at the morphotropic phase

boundary (MPB). These properties make PZT-based compounds one of the most

prominent and useful electroceramics. Commercially, it is usually not used in its pure

form, rather it is doped with either acceptor dopants, which create oxygen (anion)

vacancies, or donor dopants, which create metal (cation) vacancies and facilitate domain

wall motion in the material. In general, acceptor doping creates hard PZT while donor

doping creates soft PZT (Wikipedia, 2007).

1.2 Principles of Operation

Depending on how a piezoelectric material is cut, three main modes of operation can be

distinguished: transverse, longitudinal, and shear.

1. Transverse effect

A force is applied along a neutral axis (y) and the charges are generated along the

(x) direction, perpendicular to the line of force. The amount of charge depends on

the geometrical dimensions of the respective piezoelectric element. When

dimensions a, b, c apply,

Cx = dxyFyb / a,

where a is the dimension in line with the neutral axis, b is in line with the charge

generating axis and d is the corresponding piezoelectric coefficient.

10

2. Longitudinal effect

The amount of charge produced is strictly proportional to the applied force and is

independent of size and shape of the piezoelectric element. Using several

elements that are mechanically in series and electrically in parallel is the only way

to increase the charge output. The resulting charge is

Cx = dxxFxn,

where dxx is the piezoelectric coefficient for a charge in x-direction released by

forces applied along x-direction (in pC/N). Fx is the applied Force in x-direction

[N] and n corresponds to the number of stacked elements .

3. Shear effect

Again, the charges produced are strictly proportional to the applied forces and are

independent of the element’s size and shape. For n elements mechanically in

series and electrically in parallel the charge is

Cx = 2dxxFxn. (Wikipedia, 2007).

1.3 Mechanical Resonance

Mechanical resonance is the tendency of a mechanical system to absorb more energy

when the frequency of its oscillations matches the system's natural frequency of

vibration (its resonant frequency) than it does at other frequencies.

Some resonant objects have more than one resonant frequency, particularly at harmonics

of the strongest resonance. It will vibrate easily at those frequencies, and less so at other

frequencies. It will "pick out" its resonant frequency from a complex excitation, such as

an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies

other than its resonance.

A swing set is a simple example of a resonant system that most people have practical

experience with. It is a form of pendulum. If the system is excited (pushed) with a period

between pushes equal to the inverse of the pendulum's natural frequency, the swing will

swing higher and higher, but if excited it at a different frequency, it will be difficult to

move (Wikipedia, 2007).

This study focuses on dynamic characterization of a femur bone (see below) by

determining its resonant frequencies, and finding out its relationship to the structural

integrity of the bone.

11

1.4 The Femur (thigh) Bone

The femur or thigh bone is the longest, most voluminous, and strongest bone of the

mammalian bodies. It forms part of the hip and part of the knee (See Fig 1.2).

Fig 1.2 – The Femur Bone

Femur bone fractures, on occasion, are liable to cause permanent disability because the

thigh muscles pull the fragments so they overlap, and the fragments re-unite incorrectly.

With modern medical procedures, such as the insertion of rods and screws by way of

surgery (known as Antegrade [through the hip] or Retrograde [through the knee], those

suffering from femur fractures can now generally expect to make a full recovery.

However, the recovery process is very sensitive and there is often a need to regularly

monitor the healing process through repeated X-Ray tests. (Orthopedics - About.com)

Using the method employed in this study, it may be possible, in the future, to develop

techniques to monitor the healing of such fractured bones in real time. This has been

discussed later in the report.

*ote : For the purpose of this study, the femur bone of a chicken was used.

12

CHAPTER 2 : OBJECTIVES &

METHODOLOGY

2.1 Objectives of the study

1. First, to determine the resonant frequencies of an aluminum ruler (under free-

free condition) using a pair of actuator and sensor PZT patches and confirm that

the method and the setup are working.

2. Second, once objective 1 is achieved, then to determine the resonant

frequencies of a femur bone (under free-free condition) using the same setup.

3. To determine the effect of a bone fracture on the resonant frequencies.

2.2 Experiment Methodology

1. 2 PZT patches are bonded to the bone (or ruler), one near each end (See Fig 2.1).

2. The free-free condition is (approximately) emulated by keeping the bone on an

elastic, sponge-like cuboid. This allows the bone to have sufficient good freedom

in vibrations.

3. The actuator patch is connected to a function generator (through soldered wires).

The function generator provides an AC voltage at a controllable frequency.

4. The alternating electrical signal is converted to mechanical vibrations by the

actuator.

5. The sensor converts the vibrations back to electrical signals. These are

recorded through the multimeter which is connected to the sensor patch.

6. Charts of input frequency vs. output AC voltage are prepared.

7. Resonant frequencies are now identified by peaks in the output voltage.

13

Fig 2.1 – Experimental Setup

2.2 Voltage Amplification Circuit

Based on the literature of previous experiments involving the use of PZT actuators, it was

predicted that there could be a significant difference in the order of the impedance of

the multimeter (normally order of MΩ) and PZT patch (order of GΩ). The multimeter,

therefore, may not be sensitive enough to record the voltage variations across the sensor.

(Sirohi And Chopra ,2000)

Hence a voltage amplification circuit was procured. It was decided to use it for the

amplification of the output voltage (before it was fed into the multimeter).

However, as it turned out, the need for the circuit did not arise as the multimeter was

sensitive enough to record the voltage variations even in the case of the femur experiment

where the output voltage was of the order of mV.

A PZT patch is bonded here. This is connected to a function generator

Another PZT patch is bonded here. This is connected to a multimeter

14

CHAPTER 3 : EXPERIEME*TS & RESULTS

3.1 Aluminum Ruler Experiment

In order to confirm the validity of this method, it was decided that the test would first

be carried under more controlled conditions with predictable output.

Therefore the experiment was setup with an aluminum ruler instead of a bone (See

Fig 3.1)

3.1.1 Theoretical Estimate of Resonance Frequencies

The approximate expected resonant frequency (primary mode) was calculated using

the following formula:

f1 = (n/2l)*√√√√ (E/ρρρρ)

n= mode no.=1 for primary mode

l= 2 * (distance from the actuator patch to the sensor end of the bone) =

2*0.16=0.32 m

E= stiffness modulus= 70 GPa and,

ρ = density of aluminum = 2700 kg/m^3 (International Union of Pure & Applied

Chemistry, iupac.org)

Putting in the above values, we get f1 = 7.96 KHz

Fig 3.1 – The Aluminum Ruler

PZT patch

15

3.1.2 Experiment and Results

The experiment was setup and the output frequency of the function generator was varied

at a constant AC voltage of 5 V.

• However no voltage peak was observed in the multimeter output.

• Even on disconnecting the multimeter, the voltage output didn’t change

indicating that the output observed was only noise.

The failure to observe resonance through the multimeter output may have been

possibly due to some defect in the PZT patches and/or improper soldering.

Therefore, for the time being it was decided to establish the resonance frequencies

through an analysis of the ringing sound produced by the metal at resonance.

The sound was therefore recorded and analyzed using a sound editing software

(Audacity 1.2) to verify the resonant frequency.

3.1.3 Analysis of ‘ringing’ sound

The amplitude profile (with respect to time) that was obtained is given in Fig 3.2

Fig 3.2 – Amplitude Analysis of the ‘ringing’ sound

• The sharp jabs seen at regular intervals represent the clicking sound that was

made at intervals of 1 KHz as the frequency was being varied in the function

generator.

• The first jab (at about t=3 sec) represents a frequency of 3 KHz.

• As can be seen, the first mode occurred at frequency of about 5 KHz.

resonance

16

For a more detailed analysis, a frequency spectrum analysis of the recorded sound

was performed. Before this a noise reduction algorithm was applied to the sound

file to get more meaningful results. The result is given in Fig 3.3.

Fig 3.3 – Frequency Spectrum Analysis

As can be seen from the figure, resonances were observed at the following frequencies (KHz) : 5.0 10.5 13.0 21.0

This compares with the theoretical primary mode frequency of 7.96 KHz.

The amplitudes at other frequencies around 5 KHz may be attributed to either

1. noise that was not removed during the noise reduction process.

and/or

2. the fact that structure may vibrate at certain frequencies at all times irrespective of

the frequencies to which it is excited.

Primary mode at 5

KHz

Secondary mode at

10.5 KHz

17

3.1.4 Repeat Experiment

After the analysis was complete the experiment was again setup with new PZT patches

bonded to the ruler. This time, resonances were indicated by the multimeter output.

The output voltage was recorded at continuously varying frequencies from 1.6 to 22 KHz

(see Appendix A).

3.1.4 Analysis of output voltage vs. input frequency

A chart of output voltage vs. input frequency was prepared using Microsoft Excel. It is

given below (Fig 3.4)

0.15

0.2

0.25

0.3

0.35

0.4

0.45

1.6

2.2

3.4

7

6.6

3

7.0

6

8.0

8

9.4

3

10.6

11.8 13

14.8

15.2

18.1

18.3

20.7

21.8

22.2

Input Frequency (KHz)

Output Voltage (V)

Fig 3.4 – Output voltage vs. Input Frequency Chart (Aluminum Ruler)

Major peaks in amplitude are observed at the following frequencies (KHz): 7.03 9.46 15.14 18.35 21.84 This compares with the theoretical primary mode frequency of 7.96 KHz.

18

3.1 Femur Experiment

The experiment was next performed on the femur bone. (See Fig. 3.5)

Fig 3.5 – Setup for the Femur Experiment

3.1.1 Theoretical Estimate of Resonance Frequencies

The same formula is used to get a theoretical estimate of the primary mode frequency of

the bone.

f1 = (n/2l)*√√√√ (E/ρρρρ)

n= mode no.=1 for primary mode

l= 2 * (distance from the actuator patch to the sensor end of the bone) =

2*0.10=0.20 m

E= stiffness modulus= 20 GPa and,

ρ = density of bone = 2000 kg/m^3 (Erickson and Catanese , 2002)

Putting in the above values, we get f1 = 7.91 KHz

Sponge on which the

femur was placed

Femur with bonded

PZT patches

19

3.1.2 Experiment and Results

• The experiment was setup and the output frequency of the function generator

was varied at a constant AC voltage of 5 V.

• The output voltage was recorded at continuously varying frequencies from 0.5

to 23 KHz (see Appendix B)

A chart of output voltage vs. input frequency was prepared using Microsoft Excel. It is

given below (Fig 3.6)

0

5

10

15

20

25

0.5

1.7

2

1.8

9

3.2

5

4.1

8

4.5

4.6

7

4.8

1

5.0

9

5.6

7

6.3

7

7.6

7

8.8

4

10

.3

11

.1

13

.2

14

.3

16

.7

18

.2

22

.1

Frequency (KHz)

Ou

tpu

t (m

V)

Fig 3.6 – Output voltage vs. Input Frequency Chart (Femur)

Major peaks in amplitude are observed at the following frequencies (KHz): 4.95 15.11 This compares with the theoretical primary mode frequency of 7.91 KHz.

20

3.1 Fractured Femur Experiment

The experiment was next performed after fracturing the femur bone. (See Fig. below)

Fig 3.7 – Fractured Femur

3.1.2 Crack Dimensions

The experiment was again performed on the femur bone after giving it a crack fracture (a

fracture that does no run through the entire depth of bone i.e. the bone is not broken into

2 parts).

This was done using a saw. The approximate crack dimensions were as follows:

Length = 8 mm

Width =1 mm

Depth =3 mm

crack

21

3.1.2 Experiment and Results

• The experiment was setup and the output frequency of the function generator

was varied at a constant AC voltage of 5 V.

• The output voltage was recorded at continuously varying frequencies from 0.5

to 23 KHz (see Appendix C)

A chart of output voltage vs. input frequency was prepared using Microsoft Excel. It is

given below (Fig 3.6)

0

5

10

15

20

25

30

35

40

45

0.5 1.72.3

24.6

85.7

66.8

17.6

79.6

311 .34

11 .7212 .58

13 .4814 .5

17 .5920 .01

22 .83

Frequencies (KHz)

Outp

ut (m

V)

Fig 3.8 – Output voltage vs. Input Frequency Chart (Cracked Femur)

Major peaks in amplitude are observed at the following frequencies (KHz): 5.76 11.34 16.25 This compares with the theoretical primary mode frequency of 7.91 KHz.

22

CHAPTER 4 : CO*CLUSIO*S A*D

SUGGESTIO*S

3.1 Inferences based on the results

1. As expected, the output voltage signals in case of the bone are much lower

compared to those in the aluminum ruler experiment (There is a difference of 1

order). This can be explained with the higher damping that is expected in the

case of a bone – as a result the energy loss is greater in the bone.

If damping is very high and the multimeter used is not sensitive enough to pick up

changes in the output voltage, then use of a voltage amplification circuit may be

necessary (although, it wasn’t in this case).

2. In the case of the fractured bone one can observe the following :

a. There is a slight increase (about 1 KHz) in the resonance frequencies

when compared with the unfractured bone.

b. There is a sharp increase (1.5-2 times) in the output voltage peaks when

compared to unfractured bone.

This indicates that for detection of small crack factures, it may be better to focus

on detecting amplitude changes at resonant frequencies instead of changes in

the frequency itself.

However this inference is based only on the experiments conducted under this

study. More detailed research and a much larger sample is needed to confirm this

result and verify if it can be generalized.

3. In all the above experiments, the difference between theoretical and experimental

values can be considered to be within limits of acceptability indicating the

soundness of the theoretical model (for calculating resonant frequencies) in

providing a good estimate for objects of varying shapes.

23

3.2 Suggestions & Future Work in Dynamic Characterization of Bones

Following is a list that I have prepared for some possible applications of doing

dynamic characterization of human bones.

1. A study, such as this, for dynamic characterization of bones could be of aid in

research on road accident related damages to the bone structure.

For example, it may be possible, in future, to identify specific frequencies ranges

at which the bone structure of our body resonates and thus design vehicles to

prevent such vibrations from occurring in case of an impact/accident so that

serious injuries are avoided.

2. It may also be possible, to develop new methods for real time monitoring of the

health of a bone after a surgery, and even possibly quantify the healing process.

Similarly, the bone health in osteoporotic women may be studied in real time to

chart out the reduction in bone density with time. Such a study may be of great

use to medical researchers to better understand conditions (such as osteoporosis)

where the bone health progressively declines.

This is how it may be done –

In order to monitor the health of a bone, the surgeon will bond a PZT patches to

the bone and may possibly use a chip to transfer voltage signals to a receiver

outside.

Now, when parameters such as bone density need to be measured, the patient

would stand on a vibrating platform and the voltage signals of the PZT patch

will be recorded by the external receiver. This will enable us to detect resonant

frequencies of the bone and relate them to its density.

3. Research indicates that bones themselves have piezoelectric properties. It has

suggested that if a particular bone is made to vibrate at certain frequencies, the

current thus induced stimulates bone growth and recovery.

This would mean that, if such frequencies are known, then bonded PZT patches

could be used to stimulate bone vibrations of the desired frequencies. As in the

previous case, these patches would again be bonded at the time of surgery, and

could aid in recovery from major fractures.

The electric field required to excite the PZT inside the patient’s body may be

indirectly produced through an external varying magnetic field.

24

However, for implementing any of the three suggestions listed above, a lot of future

research needs to be carried out in this area. Over the long term, researchers may focus

on developing a comprehensive database of dynamic characteristics of individual bones

as well as entire skeletal structures.

Also, detailed empirical relationships between resonant frequencies and parameters

such as a bone’s density and structural integrity need to be derived before any useful

applications, such as the ones given above, can be realized.

25

REFERE*CES

• Sirohi and Chopra (2000), Fundamental Understanding of Piezoelectric Strain

Sensors.

• Erickson and Catanese (2002), Evolution of the Biomechanical Material

Properties of the Femur.

Online resources:

o Wikipedia (2007) , http://www.wikipedia.org

o International Union of Pure & Applied Chemistry, http://iupac.org

o About.com – Orthopedics,

http://orthopedics.about.com/od/brokenbones/a/femur.htm

26

Appendix A - Output Voltage vs. Frequency table for Aluminum ruler

Frequency(Hz) Output Voltage (V)

1.6 0.253

2 0.249

2.08 0.259

2.1 0.259

2.2 0.256

3.33 0.274

3.37 0.25

3.38 0.24

3.47 0.245

3.63 0.25

5.03 0.235

5.78 0.262

6.63 0.265

6.79 0.271

6.95 0.291

7.03 0.32

7.06 0.297

7.08 0.248

7.09 0.23

7.13 0.213

8.08 0.249

9.19 0.23

9.35 0.213

9.4 0.27

9.43 0.29

9.46 0.3

9.53 0.29

9.73 0.27

10.64 0.255

10.82 0.261

10.94 0.266

11.22 0.263

11.81 0.27

11.96 0.28

12.08 0.25

12.12 0.24

13.03 0.26

13.22 0.244

13.48 0.25

14.27 0.26

14.83 0.267

27

15.02 0.282

15.08 0.27

15.14 0.3

15.22 0.226

15.68 0.24

15.99 0.257

17.56 0.239

18.1 0.201

18.19 0.197

18.26 0.238

18.29 0.288

18.32 0.33

18.35 0.362

18.57 0.324

18.77 0.3

20.67 0.28

21.25 0.289

21.56 0.332

21.77 0.4

21.84 0.416

21.9 0.37

21.96 0.302

22.07 0.202

22.16 0.178

22.35 0.201

Appendix B - Output Voltage vs. Frequency table for Femur Frequency (KHz) Output (mV)

0.5 1.94

1 2.06

1.5 2.434

1.72 4

1.78 5

1.84 4

1.89 3

2.05 1.93

2.32 1.78

3.25 1.9

3.73 2.33

4.09 3.58

4.18 4

4.35 5

4.44 6

28

4.5 7

4.55 8

4.59 9

4.67 11.16

4.69 11.97

4.74 14

4.81 17.32

4.87 19.94

4.95 22.32

5.09 19.95

5.18 17.1

5.53 11

5.67 10

6.07 9.04

6.28 6

6.37 5.65

6.88 6.65

7.42 7.85

7.67 6.65

8.29 10.07

8.38 10.38

8.84 7.11

9.25 4.04

9.41 3.71

10.32 7

10.58 8.59

10.92 6.5

11.13 5

11.7 9.9

12.35 13

13.21 7.08

13.44 6.3

13.71 9.1

14.3 18.8

15.11 10.03

15.63 7.04

16.68 6.72

16.96 10.08

17.57 13.1

18.23 9.03

20.04 6.53

20.39 9.82

22.13 3.4

22.96 4.19

29

Appendix C - Output Voltage vs. Frequency table for

Fractured Femur

0.5 1.87

1.38 1.96

1.7 2.24

1.99 3.86

2.32 1.86

3.78 1.84

4.68 3.04

5.39 10.2

5.76 29.3

6.31 9.96

6.81 7.03

7.04 4

7.67 3.68

8.26 5.9

9.63 11.22

10.36 24.93

11.34 31

12.21 20.16

11.72 14.66

12.19 20.35

12.58 13.5

12.92 15.4

13.48 9.6

14.03 31.8

14.5 24.5

16.25 41.6

17.59 13.6

18.76 22.4

20.01 3.4

20.59 7

22.83 2.7

30