ACTIVE VIBRATION CONTROL OF A SELF-EXCITED VIBRATING SYSTEM UTILIZING PERIODIC EXTERNAL EXCITATION A Thesis Presented to the faculty of the Department of Mechanical Engineering California State University, Sacramento Submitted in partial satisfaction of the requirments for the degree of MASTER OF SCIENCE in Mechanical Engineering by Scott Donald Monroe FALL 2016 ©2016 Scott Donald Monroe ALL RIGHTS RESERVED ii ACTIVE VIBRATION CONTROL OF A SELF-EXCITED VIBRATING SYSTEM UTILIZING PERIODIC EXTERNAL EXCITATION A Thesis by Scott Donald Monroe Approved by: , Committee Chair Dr. Ilhan Tuzcu , Second Reader Dr. Estelle Eke Date iii Student: Scott Donald Monroe I certify that this student has met the requirements for format contained in the University format manual, and that this thesis is suitable for shelving in the Library and credit is to be awarded for the thesis. , Graduate Coordinator Akihiko Kumagai Date Department of Mechanical Engineering iv Abstract of ACTIVE VIBRATION CONTROL OF A SELF-EXCITED VIBRATING SYSTEM UTILIZING PERIODIC EXTERNAL EXCITATION by Scott Donald Monroe A common phenomenon observed in dynamic mechanical systems is the presence of self- excited vibrations under normal operating conditions. A classic example of this phenomenon can be seen on electrical power transmission lines that is commonly referred to as transmission line gallop. Transmission lines have been known to ”gallop”, oscillate up and down, with the oscillation increasing in magnitude until mechanical failure of the system occurs. The type of vibration that is occurring is classified as a self-excited vibration and is an example of a system that is dynamically unstable. In his classic text book, Mechanical Vibrations, J.P. Den Hartog presented this phenomena and described the dynamics of the phenomena in great detail. In addition, he proposed an apparatus that clearly and simply illustrates this phenomena to the reader. This apparatus is a simple mass- spring system that can be constructed and used to demonstrate the self-excited vibrations with the use of a small desk fan. This thesis will cover the theory and derivation of a proposed control method to settle the self-excited vibrating system. The approach for quenching the vibrations will utilize the theory of applying a controlled periodic force that mimics the forces created by a tuned mass damper. In addition to developing a control method, the thesis will also look at the development of the apparatus proposed by Den Hartog as well as implement the proposed control method and evaluate its effectiveness in canceling the self-excited vibration. And finally, in the development of this apparatus and control system, efforts will be made to utilize simple low cost micro-controller v (Arduino) and hobbyist hardware (servos and springs) with the intention that this system could be easily constructed by students and educators in the future. The goal of this thesis was to develop a control method and test the theory for effectiveness after constructing the aforementioned apparatus. The results of this research activity have the potential to be extended and adapted for many appli- cations where self-excited vibrations may occur. In addition, the platform that was developed to to test the theories in this thesis could be extended to further research and demonstration purposes. , Committee Chair Dr. Ilhan Tuzcu Date vi ACKNOWLEDGMENTS I would like to thank Dr. Ilhan Tuzcu for his guidance in selecting this topic as well as for his patience with me in getting this research completed. Most of all, I would like to thank my wonderful wife Kendall and my two little boys, Liam and Oliver, for their love and support during the many hours spent on this research. vii TABLE OF CONTENTS Page Acknowledgments . vii List of Tables . x List of Figures . xi Chapter 1 INTRODUCTION . 1 1.1 Self-Excited Vibrations . 1 1.2 Proposed Control of the System . 2 1.3 Previous Work on Self-Excited Vibration Control . 4 1.4 Problem Statement . 4 2 THEORY . 6 2.1 Passive Vibration Damping: The Tuned Mass-Damper . 6 2.2 Mathematical Derivation of System Response to an External Excitation . 8 2.3 Development of a Control Method to Minimize System Response . 15 2.3.1 Idealized Controller Function . 15 3 DESIGN AND IMPLEMENTATION . 17 3.1 Design and Fabrication of the Mass-Spring Apparatus . 17 3.1.1 Support Structure . 17 3.1.2 Mass . 17 3.1.3 Springs . 17 3.1.4 Sensing Element . 19 3.1.5 Micro-controller . 19 3.1.6 Actuation Device . 20 3.2 Development of the Program for Control of the System . 21 viii 3.2.1 Key Section of Arduino Code 1: Data Acquisition Loop . 21 3.2.2 Key Section of Arduino Code 2: Solenoid Force Calculation and Modulation 22 3.2.3 Key Section of Arduino Code 3: Serial Communication for Data Acquisi- tion in Matlab . 22 4 EXPERIMENTATION AND RESULTS . 23 4.1 Baseline System Response to No Control . 23 4.2 System Response to Control: Scenario 1 . 24 4.3 System Response to Control: Scenario 2 . 25 4.4 System Response to Control: Scenario 3 . 27 5 CONCLUSION . 29 Appendix A Photos of Apparatus . 31 Appendix B Bill of Materials . 34 Appendix C Arduino Code . 35 Appendix D Matlab Code for System Model . 37 Appendix E Matlab Code for Data Acquisition . 39 Bibliography . 41 ix LIST OF TABLES Tables Page 3.1 Table of force and displacement values and calculated spring constant k . 19 3.2 Table of force and voltage measurements of the solenoid . 21 B.1 Bill of materials including costs and source of component . 34 x LIST OF FIGURES Figures Page 1.1 Schematic of mass-spring system defined by J.P. Den Hartog (Den Hartog [1]) . 3 2.1 Schematic of 2 DOF system consisting of a main mass m1 and a tuned mass damper m2 ........................................... 6 2.2 Schematic of mass-spring system, simplified with equivalent springs and dampers . 9 2.3 Transient response xtr of the mass-spring system with a negative damping constant c 12 2.4 steady-state response xss of the mass-spring system to a periodic mass driving force 13 2.5 System response together with the transient and steady-state response of the system 14 2.6 Function block diagram representation of the proposed control system . 15 4.1 Scenario 1 system position response . 24 4.2 System position response during Baseline experiment . 25 4.3 Overlay of Scenario 1 system response on Baseline system response . 26 4.4 System position response during Scenario 2 experiment . 27 4.5 System position response during Scenario 3 experiment . 28 A.1 Iso-view of mass-spring apparatus . 31 A.2 Front view of mass-spring apparatus . 32 A.3 Bottom view of mass-spring apparatus . 32 A.4 Top and Side views of mass-spring apparatus . 33 A.5 Solenoid detail view of mass-spring apparatus . 33 xi 1 1. INTRODUCTION 1.1. Self-Excited Vibrations Self-excited vibrations can found in dynamic mechanical systems across many different fields and applications. It can typically be defined as a vibration in which the alternating force that sustains the motion is controlled and influenced by the motion itself (Den Hartog [1]). An airfoil through a fluid medium such as a fluttering airplane’s wing can exhibit self excited vibrations under certain conditions. In addition, torsional vibrations of rotating shafts in high speed machinery is another example (Jansen and Steen [2]). A well known example of this phenomena can be seen on electrical transmission lines known as transmission line gallop (Pansini [3]). In cold weather conditions, power transmission lines have been known to ”gallop,” or oscillate at an exponentially increasing magnitude. This is due to the formation of ice on the lines that creates an airfoil. The wind blowing over the airfoil created by the ice causes the galloping motion of the system. The vibrations are typically excited by a very small stimulus in comparison to the final magnitude. These vibrations can escalate to the point where mechanical damage will occur in the transmission lines and associated hardware. This phenomena is classified as a self excited-vibration and is an example of a system that demonstrates dynamic instability. Den Hartog ((Den Hartog [1]) presented this phenomena and described the physics of the prob- lem in great detail. In addition to describing and deriving the problem, he proposed an apparatus that easily illustrates this phenomena to the reader. This apparatus is a simple mass-spring system that can be constructed to demonstrate the self excited vibrations with the use of a small desk fan. The apparatus proposed not only gives insight to the phenomena witnessed on high voltage transmission lines, but also gives the reader a visual example of a system with dynamic instability. The system is described as a mass-spring system where the mass is a rod with a semi-circular cross-section. The rod is suspended from both ends by springs, see Figure 1.1. Once assembled, if care is taken to select the spring stifness and mass, the system can be placed in front of a small desk fan and the 2 self-excited vibration will occur. The operation may seem somewhat mysterious, but it has a fairly simple explanation. When the air from the fan blows over the flat face of the rod, small aerodynamic forces act on the mass and proceed to excite vibrations that gradually increase to the physical limit of the system. The aerodynamic forces are the result of an alternating pressure differential between the top and bottom surfaces of the mass. The alternating pressure differential is caused by the vertical oscillations of the mass. The oscillations increas as long as the motion exists.