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Analytical and Experimental Analysis of the Large Deflection

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Analytical and Experimental Analysis of the Large Deflection © 2007 JOHN C. VISNER ALL RIGHTS RESERVED ANALYTICAL AND EXPERIMENTAL ANALYSIS OF THE LARGE DEFLECTION OF A CANTILEVER BEAM SUBJECTED TO A CONSTANT, CONCENTRATED FORCE, WITH A CONSTANT ANGLE, APPLIED AT THE FREE END A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science John C. Visner December, 2007 ANALYTICAL AND EXPERIMENTAL ANALYSIS OF THE LARGE DEFLECTION OF A CANTILEVER BEAM SUBJECTED TO A CONSTANT, CONCENTRATED FORCE, WITH A CONSTANT ANGLE, APPLIED AT THE FREE END John C. Visner Thesis Approved: Accepted: ___________________________ ___________________________ Advisor Dean of the College Dr. Paul C. K. Lam Dr. George K. Haritos ___________________________ ___________________________ Co-Advisor Dean of the Graduate School Dr. Daniel C. Deckler Dr. George R. Newkome ___________________________ ___________________________ Co-Advisor Date Dr. Jiang Zhe ___________________________ Department Chair Dr. Celal Batur ii ABSTRACT Large deflection of a cantilever beam subjected to a constant force is modeled. The motivation for this work is derived from an excellent example of large cantilever beam deflection, the archery limb. With the development of a program that models the deflection of an archery limb comes the possibility to improve upon existing designs, which in turn could have large impacts on a rapidly growing multi-million dollar market. This study investigates a long, slender cantilever beam of constant cross section with homogeneous and isotropic material properties. The beam modeled is subjected to a concentrated force applied at the free end. This force has constant components in two orthogonal directions. For this model, the weight of the beam is assumed to be negligible. It is also assumed that the beam is non-extensible and therefore the strains are negligible. Considering these assumptions, a second order nonlinear differential deflection curve equation is obtained by means of a static analysis. Because an exact analytical solution does not exist, a FORTRAN Program using Euler’s numerical method is created to solve this equation. The first of two boundary conditions, the curvature at the free end, is known to be zero. However, the boundary condition at the fixed end is unknown. A shooting method is employed within the program to obtain the correct curvature at the fixed end to yield the deflected beam shape. Experiments are then performed to verify the numerical results. Comparisons with published numerical results show excellent agreement, and excellent agreement is also obtained between the numerical results and experimental data. iii ACKNOWLEDGEMENTS To my advisor and friend, Daniel C. Deckler, Ph. D, P.E., for his motivation and support throughout my undergraduate and graduate education. iv TABLE OF CONTENTS Page LIST OF TABLES ...........................................................................................vi LIST OF FIGURES ........................................................................................vii CHAPTER I. INTRODUCTION ................................................................................... 1 II. BACKGROUND SURVEY.......................................................................... 3 III. THEORETICAL ANALYSIS....................................................................... 9 3.1 Deflection Curve Equation Development........................................ 10 3.2 Analytical Solution....................................................................... 15 3.3 Numerical Solution ...................................................................... 18 3.4 Program Description.................................................................... 20 IV. RESULTS............................................................................................ 23 4.1 Theoretical Results...................................................................... 23 4.2 Experimental Procedure and Results ............................................. 27 V. CONCLUSION ..................................................................................... 35 5.1 Future Work ............................................................................... 36 REFERENCES.............................................................................................. 37 APPENDICES .............................................................................................. 39 A. FORTRAN PROGRAM CODE.......................................................... 40 B. FORTRAN PROGRAM OUTPUT OF EXPERIMENT 1.......................... 44 C. FORTRAN PROGRAM OUTPUT OF EXPERIMENT 2.......................... 51 v LIST OF TABLES Table Page 4.1 Changes in Tip Deflection for Different Stepsizes, Δs.............................. 25 vi LIST OF FIGURES Figure Page 3.1 Cantilever Beam.................................................................................. 10 3.2 Beam Free Body Diagram .................................................................... 12 3.3 Free Body Diagram of Cut Beam .......................................................... 13 3.4 Infinitesimally Small Section of Beam.................................................... 15 3.5 FORTRAN Program Flowchart .............................................................. 22 4.1 Comparison of Belendez and FORTRAN Program Theoretical Curves........ 25 4.2 FORTRAN Program Results with Varying Force ...................................... 26 4.3 FORTRAN Program Results with Constant Force of 3.92N ....................... 26 4.4 FORTRAN Program Results with Constant Force of 5.92N ....................... 27 4.5 Experimental Beam with 3.92N Applied Vertically Downward .................. 29 4.6 Experimental Beam Measurement......................................................... 30 4.7 Comparison of FORTRAN Program Curve and Experimental Curve........... 31 4.8 Experimental Beam with 3.92N Applied at Angle of 53 degrees............... 33 4.9 FORTRAN Program Theoretical Curves vs. Experimental Curves .............. 34 vii CHAPTER I INTRODUCTION While beams receive very little recognition, they play a very important role in our everyday life. From bridges to cranes, decks to any roofed structure, beams are everywhere and we most likely use them every day and never realize it. Many types of beams exist today, however this study examined only one type, the cantilever beam. By definition, a cantilever beam is a beam that is fixed at one end, while the other end is suspended and unsupported, much like a diving board. The inspiration for this study was derived from perhaps one of the best examples of a cantilever beam, an archery limb. Made from highly elastic material and capable of projecting an arrow at extremely high speeds, archery limbs represent an excellent example of a cantilever beam made of linear elastic material that is capable of sustaining large deflections. The traditional long bow, which is a curved stick with a string attached to each end that is drawn and released while the bow is oriented vertically, has served archers of many types for thousands of years. However, increasing popularity in archery hunting and competitive target archery has presented a need for increased performance in archery equipment. This need prompted the invention of the compound bow, which uses a cam pinned to the end of the limb along with a series of cables to provide a mechanical advantage allowing the bow to store more energy while requiring less force from the archer to draw the bow. The result is higher potential energy with less work. 1 Today, the archery industry has blossomed into a multi-million dollar industry with dozens of manufacturers all competing to design and manufacture the fastest, lightest, quietest and most cost-competitive compound bow. Today’s archery manufacturers are utilizing the best available technology to date to remain competitive and increase their product’s performance. Because the majority of the performance of a compound bow lies in the limb/cam combination, a model that describes the relationship between the limb deflection and the action of the cam could provide insight into the inner workings of this system as well as reveal areas within the system that could be improved upon. While the relationship between the limb and cam is one of extreme complexity due to several unknown variables, the development of such a model could take compound bow performance to the next level and revolutionize the archery industry. This study takes the first step to design the aforementioned model by addressing large deflections of cantilever beams of linear elastic material subjected to a constant force applied at a constant angle to the free end. The goal of this study is two-fold: to develop a program that will solve a second order, non-linear differential equation governing the behavior of a deflected beam and then perform a series of experiments that will verify the results of the program to build confidence in the program’s accuracy. 2 CHAPTER II BACKGROUND SURVEY Deflection of cantilever beams has been the subject of numerous analyses to date. An excellent example of a cantilever beam subjected to a vertical concentrated force at the free end can be found in Mechanics of Materials [1], as well as many other textbooks on physics and mechanics. In this case, the small angle assumption is valid and an equation that describes the deflection of the free end, showing proportionality between the deflection and the externally applied force that is
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