CLARKSON UNIVERSITY Effects of an In-Plane Axisymmetric Electromagnetic Field on the Vibration of a Thin, Spinning Disk A Thesis by Arthur J. Michalek Department of Mechanical and Aeronautical Engineering Submitted in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Accepted by the Graduate School Date Dean 1
The undersigned have examined the thesis entitled Effects of an In-Plane Axisymmetric Electromagnetic Field on the Vibration of a Thin, Spinning Disk presented by Arthur J. Michalek a candidate for the degree of Master of Science, and hereby certify that it is worthy of acceptance. Date Advisor Examining Committee: Cetin Cetinkaya Piergiovanni Marzocca John Moosbrugger 2
Abstract Thin, spinning disks are present in a wide range of mechanical devices, ranging from data storage drives to lumber processing equipment. Transverse vibration presents a significant design challenge to these devices. A particular problem limiting the performance of many of these applications is the occurrence of elastic instability at or above a certain speed where the natural frequency of the disk diminishes to zero. This work proposes a passive method to increase the maximum effective operating speed of a spinning disk. This method utilizes an axisymmetric in-plane electromagnetic field to stiffen the disk, and thus raise its critical speed. The equations governing the magnetoelastic loading resulting from a current carrier concentric with the disk are developed here for the first time. A review of previously employed techniques to solve the equations of linear free vibration of a spinning disk and an appropriate method to solve the proposed case are also presented. A parametric study of the solutions is provided in order to determine the practical limitations of the passive control method. Also presented here is a program of experimental investigation aimed at validating the theoretical model. Findings from the analytical investigation imply that the elastic critical speed of a spinning disk may be significantly increased within practical ranges of disk geometry, material, and magnetic field strength. The effect of the magnetic field has been found to be inversely proportional to both thickness and density. In addition to a change in critical speed, a change in the lowest critical mode is also predicted for a disk spinning in the presence of and axisymmetric in-plane magnetic field. The experimental portion of the work failed to validate the analytical predictions. It is believed that the discrepancy 3
between analytical and experimental results is the result of an assumption of perfectly conductive disk material made in the theoretical model. Possible avenues of future research are also presented. 4
Acknowledgements I wish to thank my colleagues Dr. Piergiovanni Marzocca and Dr. Davresh Hasanyan for their extensive help in the development of the equations governing magnetoelastic loading. The experimental portion of this work was made possible by the following people s contributions; Dr. David Morrison and Dr. John Moosbrugger for helping find laboratory space, Will Meenan at Kaman Aerospace for generously lending his time and equipment, Sean Fahey at CSA Engineering for assisting with the design of the experiment, and Ted Ritzko and Leroy Willard at the Clarkson machine shop for their help with the construction of the experimental apparatus. Lastly, I would like to thank Miss Mary Anderson, without who's great patience and proofreading ability, this work would not be possible. 5
Table of Contents Title Signatures Abstract Acknowledgements Table of Contents List of Figures List of Tables Nomenclature Chapter 1: Introduction 1.1: Physical Phenomenon and Motivation 1.2: State of the Art 1.3: Objectives Chapter 2: Development of Governing Equations of Motion 2.1: Physical System 2.2: Modeling Assumptions 2.3: Magnetoelastic Loading 2.4: Linear Transverse Vibration Chapter 3: Solution of Governing Equations 3.1: Overview of Solution Techniques 3.1.1: Early Analytical Approximation 3.1.2: Continuous Galerkin Approach 3.1.3: Finite Element Method 3.1.4: Extended Models 3.2: Single Mode Approximation 3.3: Galerkin Expansion with Frobenius Polynomial 3.4: Verification 3.5: Analytical Results and Discussion 3.6: Conclusions Chapter 4: Experimental Investigation 4.1: Introduction 4.2: Equipment 4.3: Procedure 4.4: Results Chapter 5: Conclusion 5.1: Overview of Results 5.2: Avenues of Future Research References Appendix A: Details of the Mathematical Model A.1: Definition of the Polar Biharmonic Operator A.2: Development of Magnetoelastic Loading A.3: Computational Solution of Governing Equations of Motion Appendix B: Dynamic Analysis of Test Stand B.1: Introduction B.2: Analysis Parameters i ii iii v vi viii ix x 1 2 7 11 11 12 13 17 17 17 18 18 19 19 21 22 24 32 32 34 34 44 45 48 51 51 58 62 62 vi
B.3: Results Appendix C: Performance of Electromagnetic Field Generation C.1: Introduction C.2: Development of Magnetometer Correlation Model C.3: Results vii