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Mechanical Impedance Techniques Applied to Measuring the Complex Modulus of Bone Joel Wallace Gerdeen Iowa State University

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Mechanical Impedance Techniques Applied to Measuring the Complex Modulus of Bone Joel Wallace Gerdeen Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1975 Mechanical impedance techniques applied to measuring the complex modulus of bone Joel Wallace Gerdeen Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Applied Mechanics Commons Recommended Citation Gerdeen, Joel Wallace, "Mechanical impedance techniques applied to measuring the complex modulus of bone " (1975). Retrospective Theses and Dissertations. 5373. https://lib.dr.iastate.edu/rtd/5373 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. 1. The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a larc^ round black mark, it is an indication that the photographer suspectai that the copy may have moved during exposure and thus cause a blurred ima^. You will find a good ima^ of the page in the adjacent frame. 3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left har.d corner cf s large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complets. 4. The majority of users indicate that the textual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation. Silver prints of "photographs" n?ay be ordered at additiona! charge by writing the Order Department, giving the catalog number, title, author and specific pages you wish reproduced. 5. PLEASE NOTE: Some pages may have indistinct print. Filmed as received. Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 'I J 75-17,391 GERDEEN, Joel Wallace. 1948- MECHANICAL IMPEDANCE TECHNIQUES APPLIED TO MEASURING THE COMPLEX MODULUS OF BONE. Iowa State University, Ph.D., 1975 Engineering Mechanics XorOX UnîVGPSÎty Microfilms, Ann Arbor, Michigan 48106 THIS OiSSERTATÎÛM HAS BEEN miCROFiLiviED EXACTLY AS nECEiVED. Mechanical impedance techniques applied to measuring the complex modulus of bone bj Joel Wallace Gerdeen A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Departments: Engineering Science and Mechanics Biomedical Engineering Co-majors: Engineering Mechanics Biomedical Engineering approved: Signature was redacted for privacy. In Charge of Maj^r Work Signature was redacted for privacy. For the Major Departments Signature was redacted for privacy. For the Graduate College Iowa State University Ames, Iowa 1975 ii TABLE OF CONTENTS Page CHAPTER I. ISTSGDUCTXON 1 CHiPTEB 11= LITERATURE REVIEW % History of Mechanical Impedance 4 Application to Biomechanics 6 complex Modulus investigations 5 CHAPTER III. CONCEPT OF MECHANICAL IMPEDANCE 10 Definitions and Variations 10 Elementary Examples 12 Impedance Graph 15 Mechanical Systems Analysis 18 Kirchhoff s laws 19 Principle of superposition 19 Reciprocity theorem 19 Thevenin's theorem 20 Norton's theorem 20 Building block approach 21 Impedance Measurement 22 Stiffness considerations 24 Mass considerations 33 CHAPTER IV. DEVELOPMENT OF IMPEDANCE ANALYSIS SYSTEM 36 Instrumentation 35 PDP-8/E Laboratory Computer 38 Programming Techniques hO Dynamic Analysis Program 44 Input monitoring 45 Switch register input 47 Frequency determination 48 Sample and store 53 Magnitude determination 55 Phase angle determination 57 Impedance calculations 54 Impedance output 64 CHAPTER V. THEORETICAL ANALYSIS 66 Complex Modulus 66 Cylindrical Rod Analysis 69 Impedance to Complex Modulus 79 lii CHAPTER VI. TSSTIKG HETHODS 87 Horizontal Suspension 87 Vertical Suspension 89 Specimen Preparation 90 Specimen Attachaent 91 Environmental Control 94 Testing Procedure 95 CHAPTER VII. DATA AHALTSIS 96 Confidence of Measurement 96 Computer Analysis 10b Bakelite Variability 108 Bone Variability 112 CHAPTER VIII. COHCLOSIONS AND RECOHHEHDATIONS 121 BIBLIOGRAPHY 124 ACKNOMLEDGEBEliTS 123 APPENDIX A: PROGRAMHING INSTBOCTIONS 129 PAL Instructions 130 FOCAL Commands 132 FOCAL Functions 133 FOCAL Function Changes 134 APrBnDXX oS DinASIC ASALzSXS PaCGSAn 135 Flowcharts 136 Program Listing 144a APPENDIX C: CYLINDRICAL BOD IMPEDANCE 171 Derivation of Impedance 172 FOCAL Program for plotting Impedance 177a FOCAL Program to Calculate Complex Modulus 180a APPENDIX D: LEAST SQÏÏA83S CURVE PITTING 183 Statistical Methods 184 FOCAL Curve Fitting Program 188a APPENDIX E: DATA PLOTTING PROGRAM 192 1 CHAPTER I. INTRODUCTION From studies made at constant and varying strain rates, the elastic modulas of bone has been shown to be strain rate dependent. Bone also exhibits the phenomena of creep and re­ laxation, necessitating its characterization as a viscoelas- tic material. Investigations have been made to determine the parameters of assumed riscoslastic models of bone. Restric­ tions were placed on such tests due to the constitutive equations of the mathematical models. Then the specimens were perturbed in some manner and the model parameters deter­ mined from their response. The first dynamic tests applied to bone have been at constant strain rates. Through testing at different strain rates, bone's dependence upon strain rate is determined. In order to reach higher strain rates, impact tests are also made. & common method cf testing response to high strain rates is to use sinusoidal motion. Viscoelastic properties are then determined from amplitude ratio and phase angle be­ tween input and output signals. These properties can be a function of the frequency of excitation. Different methods may be used to determine the frequency dependence of the complex modulus of bone. Wave propagation in a uniaxial test can be used. This method is limited to higher frequencies due to the specimen size. At least one wavelength of vibration per length of the specimen is needed 2 which sets a lower limit on the frequency due to the limited length of a bone specimen. At lower frequencies, sinusoidal tests consist of measurements of complex impedance, transmis- sibility and magnification factors. But such tests are usu­ ally impaired, to some extent, by extraneous damping or com­ plex impedance of the apparatus. These extraneous effects include damping caused by the specimen's motion in the sur­ rounding air as well as that introduced by attachments to the testing device. I noted that there is a need for more coafplex modulus data of bone and a difficulty in accounting for sources of damping and extraneous impedances, so I chose mechanical im­ pedance techniques for use in developing a test method at lower frequencies. Hechanical impedance techniques are the mechanical equivalents of electrical circuit analysis. When used correctly, the individual elements of a system can be described and the overall systes response detersinei* Cor­ rections for complex impedance errors can also be introduced. Assurance that the test procedure meets the requirements of the bone's mathematical model is then possible. This thesis presents the concept of mechanical impedance in the development of a test for the complex modulus of bone. The history of mechanical impedance, its previous use in biomechanics, aud related investigations of bone are reviewed. Mechanical impedance concepts and theoretical analysis 3 Of tests are developed. The development of an impedance mea­ suring system using a PDF 8/E digital computer on line for control and analysis of vibration test data is also present­ ed. The application and limitations of a test method for the complex modulus of bone is shown. This research is a limited effort in bone property determination, and attempts only to combine the expanding technique of mechanical impedance in an application for bone analysis. 4 CHAPTER II. LITERATURE REVIEW History of Hechanical Impedance The early part of the nineteenth century saw the begin­ ning of the transition of engineering froa an art to a sci­ ence. flaxwell, in studying tfee new science of electrical en­ gineering ^ proposed that electrical systems were dynaaic. "This led to the use of oechanical systems and of mechanical theory in dynamics for the representation of the electrical theory of that time, and for many years mechanical models of electrical systems were used for purposes of study and demon­ stration "(Sutherland, 1950). Thus, the
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