# Mechanical Engineering Systems Laboratory

Mechanical Engineering Systems Laboratory Daniel S. Stutts Department of Mechanical and Aerospace Engineering Missouri University of Science and Technology March 5, 2015 2 Contents Preface 1 2 3 i Introduction 1.1 Prerequisites . . . . . . . . . . . . . . . . . . . . 1.2 Topical Schedule . . . . . . . . . . . . . . . . . 1.2.1 Lectures . . . . . . . . . . . . . . . . . . 1.2.2 Lab 1 . . . . . . . . . . . . . . . . . . . 1.2.3 Quiz 1 . . . . . . . . . . . . . . . . . . . 1.2.4 Lab 2 . . . . . . . . . . . . . . . . . . . 1.2.5 Lab 3 . . . . . . . . . . . . . . . . . . . 1.2.6 Quiz 3 . . . . . . . . . . . . . . . . . . . 1.2.7 Final Experiment Proposal . . . . . . . . 1.2.8 Lab 4 . . . . . . . . . . . . . . . . . . . 1.2.9 Final Report . . . . . . . . . . . . . . . 1.2.10 Final Symposium . . . . . . . . . . . . . 1.3 ME4842 Grading Policies . . . . . . . . . . . . . 1.4 Final Experiment Requirements . . . . . . . . . 1.4.1 Final Experiment Proposal Requirements 1.4.2 Budgetary Limitations . . . . . . . . . . 1.5 GTA Information . . . . . . . . . . . . . . . . . 1.6 ME4842 Schedule . . . . . . . . . . . . . . . . . Acoustics 2.1 General Objectives 2.2 Introduction . . . . 2.3 Experimental Setup 2.4 Data Acquisition . 2.4.1 Procedure . 2.5 Calculations . . . . 2.6 Results of Interest . 2.7 Data Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 4 5 6 6 . . . . . . . . 7 7 7 8 9 10 12 12 15 Wind Tunnel Experiment 17 3.1 General Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 4 CONTENTS 3.3 3.4 3.5 3.6 3.7 3.8 4 5 6 Theory . . . . . . . . . . Experimental Setup . . . Experimental Procedure . Calculations . . . . . . . Points of Interest . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convective Heat Transfer: Triangular Fin 4.1 General Objectives . . . . . . . . . . 4.2 Introduction . . . . . . . . . . . . . . 4.3 Theory . . . . . . . . . . . . . . . . . 4.4 Experimental Setup . . . . . . . . . . 4.5 Data Acquisition Procedures . . . . . 4.6 Data Reduction Procedure . . . . . . 4.7 Point of Interest . . . . . . . . . . . . 4.8 References . . . . . . . . . . . . . . . 4.9 Nomenclature . . . . . . . . . . . . . 4.10 Data Sheet . . . . . . . . . . . . . . . 4.11 Appendix of Figures . . . . . . . . . 4.12 Appendix A . . . . . . . . . . . . . . Dynamic Balancing Experiment 5.1 General Objectives . . . . . . . 5.2 Introduction . . . . . . . . . . . 5.3 Experimental Setup . . . . . . . 5.4 Procedure . . . . . . . . . . . . 5.5 Calculations . . . . . . . . . . . 5.6 Data Sheet . . . . . . . . . . . . 5.7 Results of Interest . . . . . . . . 5.8 Dynamic Balancing References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 21 21 23 23 23 . . . . . . . . . . . . 25 25 25 25 27 27 28 28 28 29 30 31 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 37 37 39 39 40 43 45 45 Brief Introduction to Vibrations 6.1 Dynamical System Modeling . . . . . . . . . . . . . . . . . . . 6.1.1 Discrete Versus Continuous System Models . . . . . . . 6.2 Equations of Motion for Discrete Systems . . . . . . . . . . . . 6.3 Response of a Single-DOF System . . . . . . . . . . . . . . . . 6.3.1 Response to Initial Excitation . . . . . . . . . . . . . . 6.3.2 The Logarithmic Decrement Method . . . . . . . . . . . 6.3.3 Steady-State Harmonic Response of One-DOF Systems 6.3.4 The Phasor Method . . . . . . . . . . . . . . . . . . . . 6.3.5 Frequency Response . . . . . . . . . . . . . . . . . . . 6.3.6 Frequency Response Metrics . . . . . . . . . . . . . . . 6.3.7 Response to Harmonic Base Excitation . . . . . . . . . 6.4 Displacement Transmissibility Under Base Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 47 47 48 49 50 52 54 54 55 56 59 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 7 8 9 Dynamic Vibration Absorbers 7.1 Basic Dynamic Absorber Theory . . . . 7.2 Vibration Absorber Numerical Example 7.3 Implementation . . . . . . . . . . . . . 7.4 Experimental Procedure . . . . . . . . . 7.5 Questions . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 63 66 68 69 73 Piezoelectric Beam Experiment 8.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Simplified (1-D) Theory of Piezoelectric Ceramic Elements . . . . . . . . . . . 8.3.1 Poling the Piezoceramic . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 The One-Dimensional Piezoelectric Constitutive Equations . . . . . . . 8.4 The Euler-Bernoulli Beam Model . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Free Vibration (Unforced) Solution . . . . . . . . . . . . . . . . . . . 8.4.2 The Effect of a Concentrated Mass on the Beam Natural Frequencies . 8.4.3 Solution of the Damped, Moment-Forced Cantilevered Beam . . . . . . 8.4.4 Modal Expansion: Orthogonality of the Natural Modes of Vibration . . 8.4.5 Application of the Initial Conditions to Solve the Initially Forced Beam 8.5 Experimental Determination of the Dimensionless Viscous Damping Parameter 8.6 Producing Voltage From Strain: The Direct Piezoelectric Effect . . . . . . . . . 8.6.1 Voltage Produced By Initial Beam Deflection . . . . . . . . . . . . . . 8.7 EXPERIMENTAL SETUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Data Acquisition Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Determining ζ1 and f1 . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Steady-State Amplitude and Voltage Measurements . . . . . . . . . . . 8.8.3 Direct Effect Transient Voltage Measurement . . . . . . . . . . . . . . 8.9 Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Discussion Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Piezoelectric Beam Experiment Data . . . . . . . . . . . . . . . . . . . . . . . 8.11.1 Beam data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11.2 Piezoceramic data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11.3 Measured data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 75 75 75 76 76 77 80 82 84 85 88 90 91 93 94 96 98 98 99 100 100 102 103 103 103 103 Pump Experiment 9.1 Introduction . . . . . . . . . . . . . . . . 9.2 Theory . . . . . . . . . . . . . . . . . . . 9.3 Experimental Setup . . . . . . . . . . . . 9.4 Data Acquisition Procedure . . . . . . . . 9.5 Uncertainty Analysis . . . . . . . . . . . 9.5.1 Random Versus Systematic Errors 9.5.2 Propagation of Error . . . . . . . 9.5.3 Design Error Analysis . . . . . . 9.6 Data Reduction and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 106 106 108 111 111 113 113 113 114 115 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 10 Uncertainties in Measurement 10.1 Experimental Errors . . . . . . . . . . . 10.1.1 Systematic errors . . . . . . . . 10.1.2 Random experimental error . . 10.2 Experimental Uncertainty Quantification 10.2.1 Normal distribution . . . . . . . 10.3 Combined Uncertainty . . . . . . . . . 10.4 Uncertainty Propagation . . . . . . . . 10.5 Conclusions . . . . . . . . . . . . . . . CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 119 120 121 121 123 123 125 127 11 Report Formats and Proposal Example 128 11.1 Short Memorandum Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 11.2 Long Journal Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 11.3 Final Lab Proposal Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Preface The purpose of this text is to consolidate all of the information pertaining to ME4842 – Mechanical Engineering System Laboratory – into a single source. Historically, this information, including experiment descriptions, policies, and schedules, was available piecemeal on the course website. This website-based approach served the course reasonably well for many years, but as enrollment has nearly doubled since this approach was adopted, necessitating twice as many lab sections, with a concomitant increase in the support effort required, less and less time was available to maintain the website-based content delivery approach. In addition, much of the content was in need of revision and updating to incorporate changes in instrumentation, equipment, and procedures. The primary motivation for this change, however, was student feedback. In the past few years, more and more student feedback, obtained in the final course review questionnaire, related to improved organization through consolidation of the course material. It is hoped that this document will serve to address both the content maintenance and organization issues. This text represents the work of a number of individuals over the course of over 25 years, and in one case, even specific authorship is no longer known. That said, current overall responsibility lies with the principal author, and specific subject responsibility is allocated to the responsible faculty and staff listed in Table 1. Table 1. Current list of primary authors and area experts. Area Acoustics Aerodynamics Convective Heat Transfer Dynamic Balancing Vibrations and Dynamic Absorbers Piezoelectric Actuation and Mechatronics Pumps and Fluid Mechanics Uncertainties in Measurment Current Expert Dr. Walter Eversman Mr. Yezad Anklesaria Dr. Kelly O. Homan Dr. Keith Nisbett Dr. Daniel S. Stutts Dr. Daniel S. Stutts Dr. James Drallmeier Dr. Xiaoping Du Primary Author same same Dr. Shravan Vudumu unknown same same same same Since schedules and graduate teaching assistant (GTA) assignments change nearly every semester, and new experiments are introduced from time to time, and old ones are retired, this text is intended to be a living document, and will be subject to frequent revision. After each revision, the i ii PREFACE new publication date will replace the previous one on the cover, an updated list of revisions will also be published. This text is intended to be web and mobile device friendly, and has been published in PDF format with numerous live links between internal content and embedded references, as well as external web-based resources, and was typeset using LATEX.1 1 See: http://web.mst.edu/˜stutts/LATEX/LaTeXVault.htm for more information on how to obtain and use LATEX. Chapter 1 Introduction 1.1 Prerequisites ME4842 is a “capstone” laboratory course because it requires knowledge of all of its prerequisites, which include: • ME240 • ME231 • ME221 • ME225 • ME213 Concurrent enrollment is allowed for only one of ME221, ME225, or ME213. A passing grade in ME240 is mandatory for enrollment in ME4842. As a capstone course, the primary goal of ME4842 is to enable you, the student, to integrate the knowledge and skills covered in the prerequisite courses, while, in at least a few cases, gaining new knowledge and skills. A secondary goal of ME4842 is to give you practice in communicating technical concepts in both written and verbal form. In the last experiment of the semester, you will be asked to conceive of, propose, conduct, and then report, in both verbal and written form, the results of an experiment of your own design. 1.2 Topical Schedule All lectures will be held in 199 Toomey Hall (TMH), and the experiments are located in TMH316 (the main ME4842 lab) and TMH314. 1.2.1 Lectures There will be lectures on Wednesdays and Fridays at 3:00–3:50PM during the first three weeks of the semester, and intermittently following the online calendar on Wednesdays after the first three weeks. 1 2 1.2.2 CHAPTER 1. INTRODUCTION Lab 1 In the first lab, you and your group will choose one of the standard experiments to conduct and then each group member will submit a report in the short memo format. Your GTA will grade this lab and return it for revision if necessary. If your report has been returned for revision, you can revise and resubmit it by the due date. You will receive the grade you earned on your revised report. 1.2.3 Quiz 1 You will take a short quiz over the first experiment you conducted at the beginning of the second lab. 1.2.4 Lab 2 In the second lab, you and your group will choose one of the remaining standard experiments to conduct and then you will report the results orally using powerpoint or equivalent. Each group member will be expected to present part of your results, and you should try to balance the time allotted to each. The maximum time will depend on the size of your section, but you should plan for six minutes each. 1.2.5 Lab 3 In the third lab, you and your group will choose one of the remaining standard experiments to conduct and then each group member will submit a report in the short memo format. There will be no revision cycle on this one, and you will receive the grade assigned by your GTA. However, if you have questions about format or content, you should feel free to show your GTA what you’ve written, and get his or her feedback before the report is due. 1.2.6 Quiz 3 You will take a short quiz over the third experiment you conducted at the beginning of the next lab session. 1.2.7 Final Experiment Proposal You and your group will author and submit a proposal for a final experiment of your own, in the format and satisfying the requirements detailed in Section 1.4.1. Your proposed experiment must also satisfy the requirements stipulated in Section 1.4. 1.2.8 Lab 4 In the fourth and final lab, each group will conduct the experiment described in their approved proposal. 1.3. ME4842 GRADING POLICIES 1.2.9 3 Final Report After completing your final experiment, each group will prepare and submit a final report in the same format as described in Section 11.3. 1.2.10 Final Symposium Each group will present the results of their final experiment in a two-hour poster symposium during the scheduled final exam time. You should dress professionally, suit and tie for men, and the equivalent business formal attire for women, and be prepared to describe your results to a general audience. You will have access to approximately three square feet of table space to display anything relevant to your experiment. If you require more space or an electrical outlet, you should notify your GTA to make the necessary arrangements. 1.3 ME4842 Grading Policies The points distribution for all components of ME4842 is shown in Table 1.1. Penalties for late submission of reports and final experiment proposal may be applied at the discretion of the GTA. You should come to each lab well prepared to conduct the experiment you’ve scheduled to do. The GTAs can help you, but cannot do the experiment for you. If your group is clearly not prepared, the GTA may, at his or her discretion, ask you to leave and reschedule a time to conduct your experiment when you are better prepared at the potential cost of a grading penalty. Table 1.1. ME4842 points distribution. 1.4 Final Experiment Requirements The following are the basic requirements for the final experiment: 1. It must contain an experimental component – i.e. measurement of a system property or properties. 2. It must be based on sound physical principles. 4 CHAPTER 1. INTRODUCTION There are two allowable types of experiment in ME4842: A. The experiment may statistically test a hypothesis pertaining to a measurable property after some sort of treatment, and must include at least 10 trial samples. For example, the effects of a certain heat treatment on the hardness and ultimate strength of annealed carbon steel may be quantified in this manner. The hypothesis that a certain heat treatment schedule would increase the hardness of the steel samples could be tested by first measuring the hardness of each sample before heat treatment, and then again after. The one-sided Student T test may then be applied to determine the probability that the difference in mean hardness between the before and after samples was due to random variation. This is known as the null hypothesis. A low probability of the means being due to random variations, say, 5%, allows the rejection of the null hypothesis, and the confirmation that the heat treatment increased the hardness of the samples. This kind of experiment does not seek to explain how or why the observed effect occurred – only that it was caused by the treatment applied. In other words, there is no predictive mathematical model involved. B. The experiment may validate a mathematical model. In this case, you will attempt to measure or estimate a parameter or parameters used in a known mathematical model derived from first principles. Such models exist in all fields and areas of the physical sciences, and are usually based on physical laws. Examples include Newton’s laws, Fourier’s law, or the various conservation laws, such as the conservation of energy. The convective fin experiment, where the convection coefficient, h, is estimated, is an example of this type of experiment. It is usually best if all experimental hypotheses can be reduced to mathematical models. This type of experiment is usually less time consuming than option A. 1.4.1 Final Experiment Proposal Requirements The requirements for the final experiment proposal include the following: 1. You should submit your proposal to your GTA by the published deadline to allow time to modify it. He or she will review it and return it to you promptly for revision if it is missing any requirements. It must be in PDF format because of (1) equation encoding issues between the Windows and Macintosh operating systems, and (2) because it will be digitally signed upon approval to authorize purchases. 2. Must include enough detail for a technically trained, non-specialist to determine feasibility and rationale for performing the experiment. 3. Must explain specific goals for the study – i.e. what you hope to determine. 4. Must include a literature review citing relevant background information. The S&T library has provided a webpage specifically for ME4842 with information directly relevant to kind of literature review you will conduct in preparation for your final experiment. This website may be accessed here: http://libguides.mst.edu/mechanical. 5. Must include at least one source from the primary literature – i.e. archival journals, or conference proceedings. These may be found in our library (see: 4. above) or ordered 1.4. FINAL EXPERIMENT REQUIREMENTS 5 via ILL (order early!). See: http://library.mst.edu/resources/databases/ dBList.html 6. Must include a detailed budget estimate. All purchases MUST be approved by Dr. Stutts. 1.4.2 Budgetary Limitations There are a number of constraints on what you can purchase for your final experiment, and how to make your purchases. These include: 1. The total budget for each group is $100.00. 2. The university does not pay tax, so you should use a university purchasing card with the tax exempt number on it. 3. The department will not reimburse for food. 4. No firearms or live ammunition are allowed on campus, nor will the university reimburse for ammunition purchases. 5. Publication and duplication costs will not be reimbursed in excess of $25.00. You may purchase 4800 by 3600 color posters from the Marketing and Communications department at a cost of $10.00.1 6. All purchases are made by the ME department purchasing agent, Ms. Tammy Vena: Phone(573) 341-4614 Address: 194J Toomey Hall Email: [email protected] 1 See: http://communications.mst.edu/templates/posters/ 6 1.5 CHAPTER 1. INTRODUCTION GTA Information Table 1.2. GTA Information Day & Time Monday 2:00 PM - 3:50 PM Tuesday 9:00 AM - 10:50 AM Tuesday 2:00 PM - 3:50 PM Wednesday 1:00 PM - 2:50 PM Thursday 9:00 AM - 10:50 AM Thursday 2:00 PM - 3:50 PM Friday 1:00 PM - 2:50 PM 1.6 Section Section A Section B Section C Section F Section D Section E Section G GTA Mahati Guntupalli Yiyu Shen Daniel Penn Abdulhakim Agll Yao Cheng Huixu Deng Gurjot Dhaliwal E-mail [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] [email protected] ME4842 Schedule All lectures, laboratory schedules, and due dates may be found in the Canvas calendar.2 3 2 Note: the lecture schedule is subject to change, but any additional lectures will occur on Wednesdays, and will be announced beforehand. 3 Canvas is a potential replacement for BlackBoard, and ME4842 has been included in the pilot study. For more information, please see: https://mst.instructure.com/. Chapter 2 Acoustics 2.1 General Objectives 1. To evaluate the absorption characteristics of materials used for noise control at various frequencies. 2. To gain insight into the effects of material thickness and air space behind the material on its normal incidence absorption coefficient. 2.2 Introduction Control of noise produced by mechanical devices almost always requires the use of acoustical materials, either individually or in combination. Two types of acoustical materials are available. One type “absorbs” incident sound waves by converting most of the acoustical energy into minute amounts of heat. A porous structure is required, so that incident sound waves will propagate into the material where viscous flow losses and friction dissipate some of the energy. The remaining acoustical energy is reflected from the structure or transmitted through it. Examples of absorbing materials are ceiling tile, glass fiber mats and boards of all types, and open cell foams. A second type of acoustical material serves as a barrier by attenuating incident sound waves from one side to the other. By nature, an acoustical absorber is a poor attenuator (or barrier) because too much of the acoustic energy propagates through its porous structure. On the other hand, the dense, nonporous characteristic of a barrier, make it a good reflector (and a poor absorber) of incident sound waves. Examples of barrier materials are metal panels,“loaded” vinyl sheet, masonry, wood gypsum board, and combinations thereof. All acoustical materials and structures perform better at some frequencies than others. For this reason, a frequency analysis of the noise to be controlled is almost always required. Once the “worst” frequencies (from an annoyance standpoint) have been identified, acoustical materials and structures can be selected or designed to provide maximum noise reduction at lowest cost. 7 8 CHAPTER 2. ACOUSTICS Measurement of the absorption characteristics of acoustical materials can be accomplished with normally incident, or randomly incident, sound waves. Although random incidence more nearly approximates conditions of actual use, measurements at normal incidence are much easier to obtain and are valuable for rank-ordering of acoustical absorbing materials and structures. The measurement of the acoustical absorption coefficient for normally incident sound, αn , requires a standing wave or impedance tube. The coefficient, αn , is the ratio of energy absorbed by a sample to the energy incident upon the sample, as a function of the frequency of the incident sound. The dependence on frequency is determined mainly by configuration and a spacing of pores, material thickness, thickness of air space behind the material, and the type of facing (cover, in front of material) employed. Because αn is frequency-selective, its value is usually measured over a range of frequencies (125 to 4000 Hz) that comprises the range of architectural interest. 2.3 Experimental Setup A standing wave tube is shown in Figure 2.1. Sound at a single preselected frequency is broadcast into the tube, where it produces a sound field consisting of incident and reflected sound waves. This field is explored by a probe tube, which is small enough to prevent interference with the field. The sound field itself is plane, providing that the diameter of the tube is less than λ /2, where λ is the wavelength of the test frequency. The movable microphone probe measures the sound intensity at Figure 2.1. Standing wave tube apparatus for Measuring αn . any desired location in the tube. The sound intensity level at that location in decibels can be read by the sound level meter. The sound intensity level is given by prms 2 prms L p = 10 log10 = 20 log10 dB (2.1) pre f pre f where, prms and pre f = 2 × 10−5 Pascals (Pa) denote the root mean squared value of pressure, and the ANSI reference value of pressure, respectively. 2.4. DATA ACQUISITION 9 A variety of acoustic samples of different thickness are provided along with a sample holder. The sample holder is also adjustable so that an airspace can be provided behind the sample. 2.4 Data Acquisition Let us first identify what data we should acquire to measure the normal incidence absorption coefficient, αn . Consider location x along the tube, and an incident plane sound wave with pressure, pi , given by, pi = A sin ω t (2.2) where A denotes the amplitude. The reflected wave at x is nothing more that an earlier incident wave that has had time to travel to the sample and back. Therefore, its amplitude may be reduced, and it will be shifted in phase relative to the incident wave. This phase shift results from the difference in distance traveled (2x), and a phase angle between incident and reflected waves introduced by the sample. The combined phase lag is given by φ = 2x(ω/c) + θ (2.3) where ω, c, and θ , denote the frequency of the sound wave (rad/s), the propagation velocity of sound in the tube (ft/s), and the phase shift measured at the sample (rad/s) respectively. And where ω = 2π f where f denotes the frequency in Hertz (Hz). The reflected wave, pr , is therefore, 2ω x pr = B sin ω t − +θ c (2.4) (2.5) The pressure at a point x from the termination of the pipe is the sum of the incident and reflected pressure waves: 2ω x pt = pi + pr = A sin ω t + B sin ω t − +θ (2.6) c The amplitude of this combined pressure wave is then: s 2ω x 2 2 |pt | = A + B + 2AB cos +θ (2.7) c As x is varied, pt will reach maximum and minimum values given by: p |pt |max = A2 + B2 + 2AB and |pt |min = p A2 + B2 − 2AB (2.8) (2.9) The particle velocity due to a sound wave is given by u= p ρc (2.10) 10 CHAPTER 2. ACOUSTICS and the sound intensity, by p2 (Watts/m2 ) (2.11) ρc where, again, c denotes the sound propagation velocity (nominally 1128 feet/s at 21◦ C), and where ρ denotes the mass density of the medium in which the sound is propagating – air, in this case.1 The product, ρ c, is referred to as the characteristic impedance of the medium. I = pu = If no attenuation of sound waves occurs within the standing wave tube, the intensity of the incident wave at the face of the sample is proportional to A2 ; the intensity of the reflected wave is proportional to B2 . Since no energy can escape (a rigid piston is located behind the sample), the energy absorbed is proportional to A2 − B2 . The normal absorption coefficient, αn , is defined as, I¯a average intensity absorbed αn = ¯ = average intensity incident Ii (2.12) where p2 1 T p2 (t) dt = rms I¯ = T 0 ρc ρc From Equations (2.8) through (2.13), the absorption coefficient may be written: 2 2 A B 2ρ c − 2ρ c 2 αn = Z A 2ρ c (2.13) (2.14) or B2 A2 Applying Equations (2.8) and (2.9), Equation (2.15) may be recast as: αn = αn = 1 − (2.15) 4 |pt |max |pt |min 2 |pt |max + |pt |min (2.16) Therefore, by measuring standing wave maxima and minima (pressures) at various frequencies, the performance of an acoustical material or structure (αn as a function of f ) can be determined. 2.4.1 Procedure 1. In order to qualify the standing wave tube, to begin with, use no sample or airspace, and mount the sample holder so that the tube is terminated with the rigid piston. 2. Set the sound level meter to the unweighted or FLAT (the display will indicate LP ), 1/3 octave, FAST, continuous measurement setting.2 If the readings are fluctuating too much, you can try the SLOW setting which will display the sound pressure levels averaged over one second. 1 An approximate formula for c as a function of temperature in degrees Celsius is given by Equation (2.17). information on the operation of the Rion-27 sound level meter can be down loaded from here: http:// web.mst.edu/˜stutts/ME242/LABMANUAL/AcousticsExpManuals/Rion_NA-27_Manual.pdf. 2 Detailed 2.4. DATA ACQUISITION 11 3. Use the probe to measure the sound pressure level (SPL) at a minimum of 10 approximately equidistant positions between a starting position with the probe positioned in the tube as far as it can go, and a position approximately one wavelength at the lowest frequency of interest out – without an input signal. The lowest frequency of interest will have the longest wavelength, so the wavelengths and corresponding peak and minimum SPLs will be traversed at the same time. After recording the sound levels, find the minimum value in each of the 1/3 octave bands of interest. These values represent the noise floor in each 1/3 octave bandwidth for the experiment. Note that these values can and will vary depending on ambient building noise levels, so you may need to repeat the qualification measurement if you notice any additional noise, perhaps due to another noisy experiment running in the next lab. This may be done during the experiment: after locating a minimum, simply turn the signal generator off, and record the SPL. This level should be less than or equal to the one recorded at the beginning – possibly lower if absorbing materials have been introduced. The reason that it is important to obtain the noise floor is that a high noise floor obscures the expected minimum sound pressure level described in the next step – thereby leading to erroneous absorption coefficient results. If the ambient noise contained no components anywhere near the excitation frequency, the ambient noise would not be a problem. Similarly, if ambient noise continuing components near, but not equal to the input signal frequency, these could be filtered out, thereby mitigating the noise floor issue. Unfortunately, the sound pressure meter, even when used on the one-third octave setting, may not be selective enough.3 Therefore, you should calculate the expected wavelength for each frequency beforehand. To facilitate this, the speed of sound in meters per second is related to the ambient temperature in degrees Celsius by4 p c = 20.0457 TC + 273.15 (2.17) For example, at 25◦ C, the speed of sound is approximately c = 346.13m/s (1135.5 feet/second), and corresponding wavelength at 1000 Hz, using Equation (2.18) below, is 1.135 ft, or 13.62 inches. 4. Set the frequency f (displayed on the electronic counter) to 125 Hz. 5. Traverse the receiver (microphone probe) to locate the first minimum pressure level (as shown by the sound level meter, set on the .Flat. scale with no filter). Record this value as L pmin and the x-location as xmin . 6. Continue traversing until the corresponding maximum pressure level is observed. Record this value as L pmax and the x-location as xmax . 7. For the next reading, double the frequency and repeat steps 3 and 4. Continue until the reading for 2000 Hz has been recorded. 3 See: http://www.engineeringtoolbox.com/octave-bands-frequency-limits-d_1602. html 4 See: http://en.wikipedia.org/wiki/Speed_of_sound 12 CHAPTER 2. ACOUSTICS 8. Select an acoustic sample from the materials provided. Mount it using the sample holder. To start with use no airspace behind the sample, i.e., the sample is flush with the surface of the piston. 9. Repeat steps 2 to 5 for this case of sample thickness. 10. Increase the airspace behind the material to first 100 and then 200 and repeat steps 2 to 5. 11. Now set the frequency at 1000 Hz. With no airspace behind the sample, but choosing at least five samples with different thicknesses, repeat steps 3 and 4. 2.5 Calculations Given the measured wavelength, λ , at a given frequency, the speed of sound in air may be computed as (2.18) c=λ f Defining the standing wave ratio as SWR = |pt |max |pt |min (2.19) From Equation (2.1), we have L pmax − L pmin = 20 log10 |pt |max dB |pt |min (2.20) or SWR = 10 L pmax −L pmin 20 (2.21) Hence, from Equations (2.16) and (2.19), the normal incidence absorption coefficient, αn , may be written as 4 SWR αn = (2.22) (SWR + 1)2 2.6 Results of Interest Normal incidence absorption coefficient αn varies with the frequency of the sound. To see the dependence of this variation with the airspace behind the sample, plot, and discuss: 1. αn versus f (in Hz) on a plot. Overlay the plots for airspace behind the sample of 0, 1, and 2 inches in thickness. 2. To study the dependence or material thickness, plot and discuss: αn vs h (sample thickness) for no airspace behind the sample and at a frequency f = 1000 Hz. 3. Find the percent differences between the experimental speeds of sound from set I and the speed of sound found by assuming that the air in the standing wave tube behaves as an ideal gas. What is the maximum percent difference? What is the significance of the difference between these ideal and experimental values? 2.6. RESULTS OF INTEREST 4. Could a frequency of 4000 Hz be used with this standing wave tube? Why or why not? 5. Could a frequency of 50 Hz be used with this standing wave tube? Why or why not? 13 14 CHAPTER 2. ACOUSTICS 2.7. DATA SHEETS 2.7 15 Data Sheets Data Sheet Date Section Group o Room Temperature, T C Standing Wave Tube, Diameter in. Standing Wave Tube, Length in. Tube Qualification Frequency (Hz) 125 250 500 1000 1800 SET II No Sample (Reflective Termination) and no input signal Xmin (in) LP,min (dB) Xmax (in) LP,max (in) Name of Sample used Thickness of Sample Frequency (Hz) 125 250 500 1000 1800 SET III in. Xmin (in) Airspace Behind Sample LP,min (dB) Xmax (in) 0 in. LP,max (in) Name of Sample used Thickness of Sample Frequency (Hz) 125 250 500 1000 1800 Xmin (in) in. Airspace Behind Sample LP,min (dB) Xmax (in) 1 in. LP,max (in) 16 CHAPTER 2. ACOUSTICS Data Sheet SET IV Name of Sample used Thickness of Sample Frequency (Hz) 125 250 500 1000 1800 SET V in. Xmin (in) Airspace Behind Sample LP,min (dB) Xmax (in) 2 in. LP,max (in) Name of Sample used Frequency Sample Thickneess (in) 1000 Hz. Xmin (in) Airspace Behind Sample LP,min (dB) Xmax (in) 0 in. LP,max (in) Chapter 3 Wind Tunnel Experiment 3.1 General Objectives 1. Gain experience with setting up and performing aerodynamic experiments 2. Learn concepts in aerodynamic force measurement. 3. Gain experience with flow visualization. 4. Gain hands-on experience with common instrumentation used in the aerodynamic experiments. 3.2 Introduction Mechanical engineering is a very a diverse degree field, and graduates find themselves working in a plethora of different job fields. Even for engineers not working in the aerospace industry, knowing basic aerodynamic properties is a useful skill to possess. Any object moving through air will have some aerodynamic forces associated with its motion. Therefore, it is valuable to have an understanding of some fundamental aerodynamic concepts. This experiment is designed to facilitate learning of these concepts and provide experience in performing an aerodynamic experiment and using common instrumentation used in aerodynamic force measurement experiments. 3.3 Theory Consider a two-dimensional flow over body as shown in Figure 3.1. The control volume is made of abcde f ghia The depth of control volume in the z direction is unity. Location one is inlet of the flow and location to is the outlet of the flow. Assume the contour abhi is far enough from the body such that the pressure is same everywhere and equal to free stream pressure. Assume the inlet velocity of flow u1 is uniform across ai and equal to free stream velocity. The outlet flow velocity u2 is not uniform across bh, because the airfoil has created a wake at outlet. But let us assume that both u1 and u2 are in the x direction; hence we can say, 17 18 CHAPTER 3. WIND TUNNEL EXPERIMENT Figure 3.1. Control Volume for obtaining drag on 2-D body u1 = constant u2 = f (y) (3.1) (3.2) Surface forces on the control volume occur from two sources: the pressure distribution over the surface abhi, and the viscous resistance of the body. The force due to the pressure distribution on the body is given by " Fabhi = − pdS (3.3) abhi where dS = dS n (3.4) and where n is the unit normal to the surface, dS. The surface force on de f created by the body is an equal and opposite reaction force to the shear stress and pressure distribution created by the flow over the surface of the body. To understand this further see Figure 3.2. The moving fluid exerts pressure and shear stress distribution over the body surface which creates resultant aerodynamic force per unit span R0 on body [1]. By Newton’s third law, the body exerts equal and opposite pressure and shear stress on the flow. Hence, the body exerts a force −R0 on the control surface as shown on right of Figure 3.2. Therefore total force on the entire control volume is given by Equation (3.5) below: " FS = − pdS − R0 (3.5) abhi 3.3. THEORY 19 Figure 3.2. Reaction Forces on Airfoil Now consider the integral momentum equation * * ∂ ρVdν + (ρV · dS)V = − pdS + ρfdν + Fviscous ∂t v s s (3.6) v The right hand side of Equation (3.6) is physically the force on the fluid moving through the control volume, and in this case, the force is represented by Equation (3.5). Assuming steady state flow ∂ /∂t terms go to zero. Therefore Equation 3.6 becomes " 0 R = (ρV · dS)V − pdS (3.7) s abhi Taking the x-component of Equation (3.7), and realizing the fact that x-component of R0 is the aerodynamic drag per unit span, denoted as D0 , we may write: " 0 D =− (ρ V · dS)u − (pdS)x (3.8) s abhi where u is the x-component of the velocity. Since the boundaries of the control volume are far enough from the body and the pressure is constant along the boundary, therefore " (pdS)x = 0 (3.9) abhi Thus we obtain, 20 CHAPTER 3. WIND TUNNEL EXPERIMENT 0 D =− (ρ V · dS)u (3.10) s Evaluating surface integral in Equation (3.10), we note the following: 1. The section ab, hi, and de f are streamlines of flow. By definition u i is parallel to stream lines and dS is perpendicular to the control surface, along these sections. Hence V · dS = 0. As a result there is no contribution by sections ab, hi, and de f to the integral. 2. The cuts cd and f g are adjacent to each other. The mass flux out of one is identically the mass flux into other, Hence, the contributions of sections cd and f g to the integral cancel each other. Thus the only contribution to integral in Equation (3.11) comes from sections ai and bh. These sections are oriented in the y-direction, and the control volume has unit depth, therefore, dS = dy(1), where the (1) term indicates unit depth in the z-direction. Thus Equation (3.10) simplifies to − (ρV · dS)u = − Za ρ1 u21 dy + i s Zb ρ2 u22 dy (3.11) h At this stage we step away from Equation (3.11) and consider the continuity Equation (3.12) and apply it to the control volume (ρV · dS) = − Za Zb ρ1 u1 dy + i s ρ2 u2 dy = 0 (3.12) h Thus, Za Zb ρ1 u1 dy = i ρ2 u2 dy (3.13) h Multiplying Equation (3.13) by u1 and substituting the result into Equation (3.11), yields: 0 Zb D = ρ2 u2 (u1 − u2 )dy (3.14) h Since the flow in this experiment is considered incompressible ρ2 = ρ. Thus, the result of interest, drag per unit span, is given by: 0 Zb D =ρ u2 (u1 − u2 )dy (3.15) h Recognizing the fact that u1 is the free stream velocity entering section one, u1 = u∞ . Thus we can take u2∞ out of the integral, and dividing by chord length c, we have: 3.4. EXPERIMENTAL SETUP 21 1 D0 = 2 ρu∞ c c Zb h u2 u2 (1 − )dy u∞ u∞ (3.16) The drag for a normal wing section is given by Equation 3.17 [1], 1 D = ρsu2∞ cd (3.17) 2 where s the wing area is given by s = bc where b is span of wing and c is the chord length. Since D0 is drag per unit length in the control volume, the drag on the wing wing is given by D = D0 b. Substituting Equation (3.17), along with definition of drag, and rearranging, we arrive at the final expression for the coefficient of drag: 2 cd = c Zb h 3.4 u2 u2 (1 − )dy u∞ u∞ (3.18) Experimental Setup Aerodynamics Experiment is set-up in 18 x18 inch low-speed wind tunnel section. The experimental setup is shown in Figure 3.3. In this experiment, a NACA 0015 airfoil wing section with a chord length of 16 inches, and span of 16 inches, is used (see Figure 3.4 for airfoil terminology). The maximum thickness of the airfoil is 2.24 inches. The wing is mounted on a support that connects to load cells which measure lift and drag. The setup also allows for change in the angle of attack, and flap and spoiler deflection which is used in the flow visualization process. The wind tunnel section is also equipped with a pitot tube and a flow processor to measure velocity. 3.5 Experimental Procedure 1. Record ambient temperature and pressure. 2. Set flow-meter for ambient conditions. 3. Activate labview software and Tare load cells. 4. Turn on the wind tunnel and set the steady state velocity in range of 3500 ft/min to 5000 ft/min and record the measured velocity. 5. Set the airfoil to angle of attack to 2◦ 6. Set the pitot tube at 600 behind the airfoil and at the top of the wind tunnel and record the velocity. 22 CHAPTER 3. WIND TUNNEL EXPERIMENT Figure 3.3. Experimental Setup Figure 3.4. Airfoil 3.6. CALCULATIONS 23 7. In steps of a quarter inch, move the pitot tube down the bottom of wind tunnel and record the velocity at each location. 8. Turn off the wind tunnel and restart to set the steady state velocity in range of 3000 ft/min to 4000 ft/min and record the measured velocity. 9. Change the angle of attack from −6◦ too 22◦ in steps of 2◦ ; at each step record lift and drag force measured by load cells. 10. For flow visualization turn off the wind tunnel and restart to set the steady state velocity in range of 1000 ft/min to 2000 ft/min and record the measured velocity. 3.6 Calculations 3.7 Points of Interest 1. Calculate the Reynolds number for the flow 2. Plot and calculate the area under the curve (Ui /U∞ )(1 − Ui /U∞ ) versus ((y1 − yn )/dc ) to calculate drag 3. Plot the velocity deficit (Ui /U∞ ) versus ((y1 − yn )/dc ) to show wake area of flow field 4. Plot coefficient of lift and coefficient of drag versus Angle of Attack. 5. Calculate the percent difference in drag measured using the load cell and calculated using drag equation. 3.8 Reference 1. Anderson, J.D., Fundamentals Of Aerodynamics, McGrawHill Publishers, Fourth edtion, pp. 40-50, 127-133, 197-210, 210-219. 24 CHAPTER 3. WIND TUNNEL EXPERIMENT Chapter 4 Convective Heat Transfer: Triangular Fin 4.1 General Objectives 1. To understand a one-dimensional experimental approximation. 2. To understand the art of experimental measurement; in particular, the judicious use of data. 3. To learn a practical method of measuring a convective heat transfer coefficient for a triangular fin using a statistical analysis. Particularly, the MATLAB API (application programmer’s interface) and the Levenberg-Marquardt algorithm to solve non-linear least squares problems. 4.2 Introduction While fins are used everyday, it is tempting to believe that the simplified analysis presented in basic textbooks is not a “real world” description or that it is an unrealistic approximation of what might be observed in the laboratory. This experiment is designed to demonstrate that the fin concept is straightforward and accurate. 4.3 Theory Consider a section of the one-dimensional triangular fin as in Figure 4.1. Note that the root is at a known temperature, TW , and that the ambient is at a known temperature, T∞ . Heat is lost from the fin by convection, the rate of which is proportional to the heat transfer coefficient, h, of the slanted surface. Presuming conduction with-in the fin to be primarily considered one-dimensional (i.e., the z dimension is effectively infinite and the perimeter can be approximated as 2z since l z), an energy balance for a differential slice of the fin takes the form 4.1 d dT ds kA(x) = hp(x)(T − T∞ ) (4.1) dx dx dx where T (x) is the temperature of the fin at a particular dimensional position x (refer to Heat Transfer from Extended Surfaces section in [1]). 25 26 CHAPTER 4. CONVECTIVE HEAT TRANSFER: TRIANGULAR FIN In order to simplify the expression, non-dimensionalize by introducing the following definitions: θ = T − T∞ and x∗ = x/L. Taking and to be constants, the expression simplifies to v" 2 # 2 u u d L dθ hl t 1+ l θ x∗ ∗ = ∗ dx dx k l L Further defining B = hl L 2 k l rh 1+ i , expression 4.2 simplifies to d ∗ dθ x = Bθ dx∗ dx∗ (4.2) l 2 L (4.3) or, on expanding dθ d2θ + ∗ − Bθ = 0 (4.4) ∗2 dx dx which may be recognized as a second-order ordinary differential equation with variable coefficients subject to the following boundary conditions: x∗ θ (x∗ = 0) = finite (4.5) θ (x∗ = 1) = θ0 (4.6) where θ0 = Tw − T∞ , and the solution is given by √ θ I0 (2 Bx∗ ) √ = θ0 I0 (2 B) (4.7) The function, I0 , on the right hand side of Equation (4.7) is the modified Bessel function of zeroth order. At this point, if all of the dimensional parameters (h, l, k, L, TW and T∞ ) were known, the axial temperature profile for the fin would be known as a function of x. Of the parameters listed, h, the convection heat transfer coefficient is the most difficult to determine, especially since its value depends entirely on the flow field characteristics. The present experiment seeks to estimate h from measured quantities. This method is based upon a statistical analysis of dimensionless position, x∗ , and dimensionless temperature, θ data using equation 4.7. The selected method is referred to as the Method of NonLinear Least Squares (NLLS) [4,5], a form of non-linear regression. Based upon the assumption that the errors in the experimental measurements follow a Gaussian distribution, the NLLS produces a unique value for the determined constants. That is, the magnitudes of the constants that are determined give the “most probable” form of the given equation that fits the data. The NLLS is based upon the differences between the independent values of collected experimental data and the expected values as provided by the theoretical result. For convenience, let θi = θ /θ0 , then from the theory at each position xi , 4.4. EXPERIMENTAL SETUP 27 θi = DI0 (2 p Bxi∗ ) (4.8) √ where D = 1/I0 (2 B) is a constant to be determined so as to fit that experimental data For simplicity, let Ii = I0 (2 p Bxi∗ ) (4.9) In this case, the residual for each data point is ρi = θ − DIi (4.10) Note that the dimensionless temperature θi , and the dimensionless position xi∗ (or the modified Bessel function at the ith specified position, Ii ) are the input data. Then the sum of the residuals squared is S = ∑ ρi2 (4.11) The value of h that would give the least value of residuals squared sum S is found by writing a MATLAB program using the Levenberg-Marquardt algorithm. Refer to Section 4.12 at the end for MATLAB API (application programmer’s interface) for more information on the LevenbergMarquardt algorithm. 4.4 Experimental Setup The configuration and schematic of the experimental setup (1-D triangular fin) is presented in Figures 4.1 and 4.2 respectively. The sides and the bottom are all well insulated with Thermal Ceramics Kaowool1 , which has a thermal conductivity of approximately 0.06W/m/◦ K at 100◦ C. This setup represents a physical approximation to the upper half of a symmetric, infinitely long fin. Figure 4.4 presents the actual experimental fin information. The temperature indicators are thermocouples embedded within the stainless steel fin as indicated in Figures 4.5 through 4.7. 4.5 Data Acquisition Procedures 1. Familiarize yourself completely with the experimental setup. Do NOT, under any circumstances, touch the stainless steel surfaces because the setup will be turned on at least an hour before class. Figure 4.3 is included so that you may view the setup without taking it apart and subjecting yourself to harm. 2. Confirm that the temperature profile is at steady state (not a function of time). Do not adjust the heater temperature setting. You should have data to prove that steady state exists. 1 http://www.morganthermalceramics.com/downloads/datasheet/ kaowool-board-1600-boards-shapes 28 CHAPTER 4. CONVECTIVE HEAT TRANSFER: TRIANGULAR FIN 3. Take the data as per the included data sheet, including T∞ . Repeat the data collection process to confirm the steadiness of the temperatures. 4.6 Data Reduction Procedure 1. First plot T (or θ ) versus x in order to check the data for the appropriate shape as well as to check its one-dimensional character. Note that Tw is the arithmetic average of the temperature readings at the fin root. 2. If the shape of the temperature variation is appropriate, enter the data sets (Ti and xi ) and T∞ into the MATLAB program to solve for h using Levenberg-Marquardt algorithm (non-linear least squares method). 3. Run the MATLAB program and read the convection coefficient, h. 4.7 Point of Interest In your technical memorandum you should include a discussion of the following 1. The physical significance of the boundary conditions listed by Equations 4.5 and 4.6. 2. Explain clearly how insulation on the two sides simulates a fin which is infinitely long in the z-direction. 3. Why is it important for the fin to be at steady state? 4. Plot the resulting model (Ti from using equation 4.8 with h value from MATLAB program and calculated values of D and B using h) and the measured steady state temperature versus. Discuss the accuracy of the model. 5. As an appendix, consider a differential element of the length dx of the experimental fin and obtain the governing equation for the fin. Apply the boundary conditions and hence obtain Equation 4.7. State the assumptions you made in the process of obtaining Equation 4.7. Are they all justified for the experimental setup used? Explain. (refer [1]) 4.8 References 1. Incropera, F.P., Fundamentals of Heat and Mass Transfer, John Wiley & Sons, Inc., New Jersey, 2007, pp. 137-162. 2. Chapman, A.J., Heat Transfer, Fourth ed., Macmillan Publishing Company, New York, Collier Macmillan Publishers, London, 1984, pp. 67-89. 3. Hildebrand, F.B., Advanced Calculus for Applications, second ed., Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976, pp. 141-153. 4.9. NOMENCLATURE 29 4. Bain, L.J., and M.Engelhardt, Introduction to Probability and Mathematical Statistics, PWS Publishers, 1987, pp. 300-307. 5. Lipson, C., and N.J. Sheth, Statistical Design of Engineering Experiments, McGraw-Hill Book Company, 1973, pp. 372-387. 6. Levenberg-Marquardt algorithm, http://en.wikipedia.org/wiki/Levenberg-Marquardt_algorithm 7. Thermal Ceramics. 25 June 1996. Web. 08 Dec. 2009, http://www.thermalceramics. com/. 4.9 Nomenclature h = surface convection coefficient for a triangular fin (W /m2C) k = thermal conductivity (W /mC) l = one half fin thickness at the root (m) L = fin length (m) s = slant length (m) T = fin temperature (C) Tw = fin root temperature (C) T∞ = ambient temperature (C) x = dimensional axial coordinate along fin length (m) x∗ = x/L,dimensionless axial coordinate y = coordinate along fin height z = coordinate along fin width A(x) = cross-sectional area parallel to the wall and enclosed by the perimeter (i.e. 2zy=2zxl/L) p(x) = perimeter of the cross sectional area of the triangular fin (i.e. 2(z+2y) = 2z, as z¿¿2y from a thermodynamic point of view) ds = ((dx)2 + (dy)2 )1/2 θ = T − T∞ θ0 = Tw − T∞ 30 4.10 CHAPTER 4. CONVECTIVE HEAT TRANSFER: TRIANGULAR FIN Data Sheet Table 4.1. Data for triangular fin convection experiment. Thermocouple Depth from fin Bottom(in) Distance from root (in) Distance from tip (in) 1 0 1 5 2 0.1875 2.125 3.875 3 0.875 3.5 2.5 4 0.0625 5.375 0.625 5 0.5 1 5 6 0.3125 2.125 3.875 7 0.3125 3.5 2.5 8 0 4.625 1.375 9 0.625 1 5 10 0.625 3.5 2.5 11 1.0 1 5 12 0 2.125 3.875 13 0.375 4.625 1.375 Table 4.2. Measured thermocouple temperatures. Time 5 min 10 min 15 min 20 min 25 min 1 2 3 4 5 6 7 8 9 10 11 12 13 4.11. APPENDIX OF FIGURES 4.11 Appendix of Figures Figure 4.1. Configuration of the Triangular Fin Figure 4.2. Simple One-Dimensional Fin Setup 31 32 CHAPTER 4. CONVECTIVE HEAT TRANSFER: TRIANGULAR FIN Figure 4.3. Components of Experiment 4.12 Appendix A MATLAB API (application programmers interface) for the Levenberg-Marquardt algorithm to solve non-linear least squares problems. Example:Rosenbrock function - in mathematical optimization, the Rosenbrock function is a non-convex function used as a test problem for optimization algorithms. The global minimum is inside a long, narrow, parabolic shaped flat valley. To find the valley is trivial, however to converge to the global minimum is difficult. The Rosenbrock function is defined as f (x, y) = (1 − x)2 + 100(y − x2 )2 (4.12) To test if Levenberg-Marquardt algorithm is able to find the global minimum (1,1): create a new file in MATLAB with the following commands (make sure the new MATLAB file and LMFsolve.m, LevenbergMarquardt algorithm are in the same working directory). In this case m=[x;y], m(1)=x and m(2)=y. rosenbrock= @(m) [1-m(1); 10*(m(2)-m(1)ˆ2)]; [m,residualssquaressum,iterations] = ... 4.12. APPENDIX A 33 Figure 4.4. Fin draft and dimensions in inches. Figure 4.5. Bottom View of Fin with Thermocouple Holes LMFsolve(rosenbrock,[-1.2,3],’Display’,1,’MaxIter’,15000) “1-m(1)” is the first function and “10 ∗ (m(2) − m(1)2 )” is the second function 34 CHAPTER 4. CONVECTIVE HEAT TRANSFER: TRIANGULAR FIN Figure 4.6. General Side View of Thermocouple Depths Figure 4.7. Dimensions of Thermocouple Placements and Depths with Levenberg-Marquardt algorithm we are trying to find the minimum of the squares-sum of these two functions. [-1.2,3] is the initial guess value for m “’Display’,1,’ ” is for displaying iteration information after every iteration “ ’MaxIter’,15000 ” is to set the maximum number of iterations to 15000 Running the program will return “m = [1; 1]” i.e, m(1) = x = 1 and m(2) = y = 1 (in approximately 4000 iterations) which is the global minimum for the Rosenbrock function. 4.12. APPENDIX A 35 Fin Experiment:Develop a new MATLAB program for this experiment with the help of the example as shown above. Hint: The important commands are residuals= @(h) [ra;rb;rc;rd;re]; [h,residualssquaressum,iterations]= ... LMFsolve(residuals,[100],’Display’,1,’MaxIter’,15000) ra, rb, rc, rd, and re are the residual errors (ρ in Equation 4.10) at different x values as a function of h and 100 is the initial guess value for h. ————————— besseli(nu,Z); nu is the order of the system Z is the input argument. This can be complex. 36 CHAPTER 4. CONVECTIVE HEAT TRANSFER: TRIANGULAR FIN Chapter 5 Dynamic Balancing Experiment 5.1 General Objectives 1. To gain insight into the causes of undesirable vibration of rotors and to understand static and dynamic unbalance conditions of rotors. 2. To learn a simple, practical method of dynamic balancing that can be used in the field and appreciate its advantages and limitations. 5.2 Introduction Production tolerances used in the manufacture of rotors are adjusted as closely as possible without running up the cost of manufacturing prohibitively. In general, it is more economical to produce parts, which are not quite true, and then to subject them to a balancing procedure than to produce such perfect parts that no correction is needed. Typical examples of such machinery are crankshafts, electric armatures, turbo-machinery, printing rollers, centrifuges, flywheels, and gear wheels. Some common causes of irregularity during production are machining error, cumulative assembly tolerances, distortions due to heat treatment, blow holes or inclusions in castings, and material non-homogeneity. Because of these irregularities the actual axis of rotation does not coincide with one of the principal axes of inertia of the body, and variable disturbing forces are produced which result in vibrations. In order to remove these vibrations and establish proper operation, balancing becomes necessary. The forces generated due to an unbalance are proportional to the rotating speed of the rotor squared. Therefore, the balancing of high-speed equipment is especially important. Frequently, a machine already in operation will need re-balancing or a new machine when assembled at its permanent location will need balancing. In some cases, the cost of disassembly, shipping to a balancing machine, and delay, are prohibitive and the machine must be rebalanced in the field in its bearings. The system of balancing discussed in this experiment was developed to satisfy the need to perform field balancing of equipment easily and accurately. Although there are many possible causes of vibration in rotating equipment, this technique will deal only with that component of vibration, which occurs at running speed (frequency), and is caused by a mass unbalance in the rotor. Note: This component is the only component of vibration, 37 38 CHAPTER 5. DYNAMIC BALANCING EXPERIMENT which can be eliminated by the addition or removal of weight from the rotor. The condition of unbalance of a rotating body may be classified as static or dynamic unbalance. In the case of static unbalance, the unbalance appears in a single axial plane. In the case of dynamic unbalance, the unbalance can be in different axial planes. As a result, while in rotation, the two unbalanced forces form a couple, which rocks the axis of rotation and causes undesirable vibration of the rotor, mounted in its bearings. Let us now consider a single rigid rotating mass mounted in two supporting bearings and assume that the axis of rotation is horizontal.It can be shown that for the correct balance of such a rotor, two weights placed in different radial planes of the rotor are necessary and sufficient to balance the rotor. The vibratory motion of either bearing may be represented by three components,the horizontal and vertical radial components and the axial component. The purpose of balancing at running speed of the rotor is to reduce the greatest of these three components to a practical minimum. The other two components will be reduced to negligible amounts from their original magnitudes by this technique. Assume in this example that the radial component is the greatest. Therefore, only this component will be measured and analyzed in this technique. It follows that if the vertical components of vibration of two points, one chosen on each bearing, are reduced to zero (or near zero) the purpose of balancing has been accomplished and no vibration will be transmitted to the support structure. There are four variables to be dealt with when balancing any rigid rotor. They are the amount and position of the two correction weights required to balance the rotor. Each correction weight is located in one of the arbitrary chosen radial reference planes on the rotor. These reference planes are usually placed near the support bearings. In general, the farther apart the radial reference planes are located, the smaller the required correction weight. This technique deals with these four variables simultaneously as the amount and position of the correction weight in the other reference plane. The data necessary to determine the magnitudes and position (angle) of the two correction weights are obtained by test runs, all at the same speed by measuring the vibration amplitude and phase angle at each bearing. Some commercial equipment allows measurement of the vibration amplitude and phase relative to a geometric trigger reference point on the rotor. Lacking the instrumentation to measure the phase angle, this technique will obtain data to allow calculation of the phase angle. An important assumption made by this technique is that the system follows linear relationships i.e., the vibration amplitude is proportional to the force producing the vibration. This assumption is reasonably valid. Most simple rotors/systems can be balanced by applying this technique iteratively. 5.3. EXPERIMENTAL SETUP 5.3 39 Experimental Setup The apparatus provided to you is shown in Figure 5.1. The long, rigid rotor is supported in its bearings and is driven by a DC motor whose speed can be controlled. Note that the cradle on which these are mounted has been isolated from the supporting structure. Vibration amplitudes (not the phase angles) at bearings A and B are measured by accelerometers attached to them. The signal from the accelerometers is converted to an electrical signal, which is then integrated and displayed on the Labview program. Motor speed is also measured using the Labview program. Labview system Figure 5.1. Schematic of Dynamic Balancing Experiment 5.4 Procedure Bearing in mind that our experimental apparatus measures only vibration amplitudes, we will measure values at both bearings with the trial mass at 0◦ , 90◦ , 180◦ , and 270◦ , to determine the phase angle of vibration at both bearings. The procedure to acquire the required data and balance the rotor is as follows: 1. Verify Labview program has been started 2. Set the sampling frequency to maximum 3. Set the filter to value of 2 4. Set the rotor speed to approximately 10 Hz (600 RPM) 1 1 It is important to balance a viscoelastically supported rotating system well below any of its resonance frequencies due to the associated phase shift between the unbalance forcing and the system response near and above the resonance frequency. Please refer to Section 6.3.6 for more information on the phase lag between forcing and steady-state harmonic response. 40 CHAPTER 5. DYNAMIC BALANCING EXPERIMENT 5. With only the original mass unbalances of the rotor, record vibration amplitudes at both bearings at the indicated rotor frequency. 6. Arbitrarily choose a trial mass, mtrial and mount it on plane A of the rotor at an angle of 0◦ . Record readings at both bearings. Repeat this step with the trial mass at 90◦ , 180◦ , and 270◦ . 7. Repeat step 7 with the difference that the trial mass is now mounted on plane B of the rotor. 8. Mount the trial mass at 45◦ on plane A of the rotor and record reading at both bearings. This measurement is not needed for balancing the rotor, but will be used later as a check on calculations. 9. Using the equations described in section 5.5 (calculations) determine the 2 masses required at both planes A & B of the rotor to dynamically balance the rotor. These calculations will be performed manually. 5.5 Calculations Let A = vibration amplitude at Bearing A due to original unbalance of rotor. ΨA = vibration phase angle at Bearing A due to original unbalance of rotor. ∆A1 = additional vibration amplitude at Bearing A due to the trial mass mounted at any angle on plane A – refer to Figure 5.2. Figure 5.2. Balancing vector diagram. From these quantities we can derive the location and size of the necessary balancing mass. NOTE: You should prove this to yourself before you run the experiment. 5.5. CALCULATIONS 41 2 2 2 (A90 1 ) − (∆A1 ) − A ΨA = arctan (A01 )2 − (∆A1 )2 − A2 s s 0 2 180 2 2 (A1 ) + (A1 ) − 2A (A1 90 )2 + (A1 270 )2 − 2A2 ∆A1 = = 2 2 (5.1) (5.2) ∆A1 = additional vibration amplitude at Bearing A due to trial mass mounted at any angle on plane B. s s 0 2 180 2 2 (A2 ) + (A2 ) − 2A (A2 90 )2 + (A2 270 )2 − 2A2 = (5.3) ∆A2 = 2 2 Defining B, ΨB , ∆B1 , and ∆ B2 similar to the quantities above (except that they are measured at Bearing B), we can show that, ! 2 (B1 90 ) − (∆B1 )2 − B2 ΨB = arctan (5.4) 2 (B1 0 ) − (∆B1 )2 − B2 s s 0 2 180 2 2 (B1 ) + (B1 ) − 2B (B1 90 )2 + (B1 270 )2 − 2B2 ∆B1 = = (5.5) 2 2 s s 0 2 180 2 2 (B2 ) + (B2 ) − 2B (B2 90 )2 + (B2 270 )2 − 2B2 = (5.6) ∆B2 = 2 2 The purpose of calculating these ∆A and ∆b values is ultimately to determine the masses and their locations on planes A and B that will eliminate the vibration at both bearings. Let these correction masses be identified as A M¯ A and M¯ B for planes A and B respectively. In this experiment there are only a limited number of locations where M¯ A and M¯ B can be attached. This problem is easily solved by calculating an x and y component of both M¯ A and M¯ B . Define vectors R¯ A and R¯ B as correction factors to be applied to the trial masses on planes A and B. The following equations express the relationships between the trial masses and the final correction masses. M¯ A = R¯ A Mtrial (5.7) M¯ B = R¯ B Mtrial (5.8) MAx = RAx Mtrial (5.9) MAy = RAy Mtrial (5.10) MBx = RBx Mtrial (5.11) MBy = RBy Mtrial (5.12) or in component form 42 CHAPTER 5. DYNAMIC BALANCING EXPERIMENT Recall that the vibration at bearing A was affected by the trial mass attached to plane A, ∆A1 , as well as by the trial mass attached to plane B, ∆A2 . It is desired to determine correction factors R¯ A and R¯ B which, when applied to the trial masses on planes A and B respectively, will cause the total change in vibration at bearing A to offset the original unbalance at bearing A. This can be expressed as: R¯ A ∆A1 + R¯ B ∆A2 = −A¯ (5.13) It is also necessary for the same correction factors to simultaneously balance bearing B. Thus, R¯ A ∆B1 + R¯ B ∆B2 = −B¯ (5.14) Equations 5.13 and 5.14 state mathematically that the proper location of MA and MB on planes A and B will produce vectors equal to, but opposite in direction to the existing vibration (A and B) thus eliminating the vibration. Resolving equations 5.13 and 5.14 into their x component equations gives: RAx ∆A1 + RBx ∆A2 = −A cos ΨA (5.15) RAx ∆B1 + RBx ∆B2 = −B cos ΨB (5.16) and Equations 5.17 and 5.16 can be readily solved for RAx and RBx since all of the other quantities are known: RAx = B∆A2 cos ΨB − A∆B2 cos ΨA ∆A1 ∆B2 − ∆A2 ∆B1 (5.17) −A cos ΨA − RAx ∆A1 (5.18) ∆A2 Similarly, Eqns. 5.13 and 5.14 can be resolved into their corresponding y component equations: RBx = RAy ∆A1 + RBy ∆A2 = −A sin ΨA (5.19) RAy ∆B1 + RBy ∆B2 = −B sin ΨB (5.20) and Again, equations 5.19 and 5.20 can be solved for since all other quantities are known: RAy = B∆A2 sin ΨB − A∆B2 sin ΨA ∆A1 ∆B2 − ∆A2 ∆B1 (5.21) −A sin ΨA − RAy ∆A1 (5.22) ∆A2 The 4 individual required masses can now be determined from Eqns. 5.9, 5.11, 5.10 and 5.12. To place these masses at the correct locations, keep the following in mind: RBy = x component placed at 0◦ position if positive, placed at 180◦ position if negative y component placed at 90◦ position if positive, placed at 270◦ position if negative. 5.6. DATA SHEET 5.6 43 Data Sheet Table 5.1. Measured vibration amplitudes. Motor Speed (Hz) Motor Speed: Plane A Hz Plane B RPM Plane A: Plane B: Trial mass mtrial (grams): Table 5.2. Vibration Amplitudes with trial mass on Plane A. Trial Mass Postion Bearing A Bearing B 0◦ A01 B01 90◦ A90 1 B90 1 180◦ A180 1 B180 1 270◦ A270 1 B270 1 44 CHAPTER 5. DYNAMIC BALANCING EXPERIMENT Table 5.3. Vibration Amplitudes with trial mass on Plane B. Trial Mass Postion Bearing A Bearing B 0◦ A02 B02 90◦ A90 2 B90 2 180◦ A180 2 B180 2 270◦ A270 2 B270 2 Table 5.4. Vibration Amplitudes with trial mass on Plane A. Trial Mass Postion 45◦ Bearing A A45 1 Bearing B B45 1 Table 5.5. Corrected Mass Values and Location. Trial Mass Postion Bearing A MAx MAy MBx MBy Vibration Amplitudes with corrected masses added Plane A: Plane B: Bearing B 5.7. RESULTS OF INTEREST 5.7 45 Results of Interest 1. Determine the percent reduction at each bearing. How can these results be improved further? 2. Using your calculated values, verify Equations (13) and (14) graphically. 3. Neatly sketch the original screen showing the original unbalance and the screen after the correction mass has been added. Explain all important features. 4. Plot vibration amplitudes of the unbalance rotor versus the motor speed. Explain this behavior. 5. Construct a scaled figure similar to Fig. 6.2 using the following procedure: (a) Draw vector using your measured magnitude and calculated direction. (b) From the tip of A, draw vertical and horizontal construction lines. 180 270 (c) Draw vectors A01 , A90 1 , A1 and A1 using your measured magnitudes. The directions should be determined by placing the tips of the vectors on the horizontal or vertical construction lanes. (d) Measure from your figure the four distances corresponding to ∆A1 in Figure to your calculated ∆A1 . (e) Using the calculated values of ∆A1 and ∆B1 , draw vector diagrams to predict the vibration at bearings A and B due to the trial mass mounted at 45◦ on plane A. (You may use the above vector diagram for bearing A). Compare these values to the vibration amplitudes actually obtained when this trial mass is mounted. 5.8 Dynamic Balancing References 1. Shigley, Joseph E. and Vicker, John J., Jr., “Theory of Machines and Mechanisms”, McGraw Hill inc., 1980. (pp 478-500 on Balancing) 2. Thearle, E. L. and Scheuectady, N. Y., “Dynamic Balancing of Rotating Machinery in the Field”, ASME Transactions, Paper AMP 56-19, Vol. 56, 1934, pp 745-753. 3. Marks’ Standard Handbook of Mechanical Engineers, edited by Baumeister, T., et. al, McGraw Hill Book Company, eighth edition, 1988, pp 3-17 to 3-18, 5-70 to 5-72, 5-75 to 5-76. 46 CHAPTER 5. DYNAMIC BALANCING EXPERIMENT Chapter 6 Brief Introduction to Vibrations This chapter is a brief, and necessarily incomplete, introduction to the field of vibrations analysis and mitigation. Here we will focus on the development of discrete dynamical system models, and the subsequent solution of these models to determine system steady-state response to various forms of harmonic excitation. We will then present a collection of definitions and techniques to characterize the behavior of a vibrating structure based on simple measurements. 6.1 Dynamical System Modeling A system’s governing equations of motion are composed of at least two distinct sets of physical properties: (1) state variables (described earlier), and (2) constitutive parameters. The constitutive parameters, such as mass and stiffness, relate the system’s state to forces both internal and externally applied to the system. In addition, specified externally applied forces, torques, or pressures, if present, constitute a third physical property to be accounted for in the system equations of motion. In general, every macroscopically identifiable particle of mass in a system may be considered to exhibit degrees of freedom – i.e. directions in which it may translate or rotate. We limit the size of the particles to “macroscopic” size in order to avoid consideration of quantum effects. Thus, in general, every single particle in a system could exhibit as many as six degrees of freedom. These degrees of freedom correspond to the three perpendicular translations and rotations, each particle having its own translational and rotational state. Since it is difficult or impossible to account for so many degrees of freedom, a different modeling approach is most often used. 6.1.1 Discrete Versus Continuous System Models Although the physical world on a macro scale appears continuous in nature, it is often convenient to model dynamical systems using discrete states and lumped parameters. A parameter is said to be lumped when it is considered to act at one or two points in a system. An example of this discretizing approach is shown in Figure 6.1. Here, due to the greater mass of the floor slabs, relative to the structural walls, all of the building’s mass is “lumped” into the three above ground floor slabs. Likewise, the elastic stiffness of the structural steel beams which support the floor slabs is lumped into discrete linear elastic stiffnesses. In this way, a building, where every single particle of mass 47 48 CHAPTER 6. BRIEF INTRODUCTION TO VIBRATIONS could be considered to have as many as six degrees of freedom, may be modeled using only three! Figure 6.1. Discrete or lumped parameter building model. Discrete models are common in physics. For example, the resistance in a circuit wire or element is often considered to be lumped or concentrated at a single point, denoted as a resistor, in the circuit. Similarly, because the capacitance and inductance of a circuit board trace are relatively small compared to that of discrete capacitors or inductors, they are usually ignored in circuit models. However, when the structures of interest have one or two dimensions much thinner than the third, such as is the case for strings, wires, rods, beams, plates, and shells of rotation, there may be no simple way to discretize or lump the relevant constitutive parameters without resorting to numerical approaches such as the finite element method. Such systems are governed by partial differential equations having solutions in the form of continuous functions.1 We will study both discrete and continuous models in this course. 6.2 Equations of Motion for Discrete Systems Occam’s razor2 : “entia non sunt multiplicanda praeter necessitatem,” translated as “entities must not be multiplied beyond necessity,” and actually attributed to the theologian, John Punch3 , translates to “entities must not be multiplied beyond necessity,” and essentially implies that the best answer is the simplest one that describes what is observed. In other words, in the context of mechanics, it is unnecessary to include mathematical modeling details beyond those which are required to describe the salient physics of the system being modeled. Therefore, when possible, the vibrations analyst is compelled to develop discrete or “lumped-parameter” models using the fewest degrees of freedom required to predict the dynamics of interest for the system under study. As illustrated in Figure 6.1, a building might be treated as a three-degree-of-freedom (3DOF) structure of masses representing the concrete floor decks, and springs representing the interconnected steel frame. Such a model might provide sufficient prediction of the building’s first three natural frequencies and modes of 1 http://en.wikipedia.org/wiki/Partial_differential_equations 2 http://en.wikipedia.org/wiki/Occam’s_razor 3 http://en.wikipedia.org/wiki/John_Punch_(theologian) 6.3. RESPONSE OF A SINGLE-DOF SYSTEM 49 vibration. 6.3 Response of a Single-DOF System Figure 6.2. Prototypical forced mass-spring-damper system. Before we investigate the two-DOF discrete model necessary to describe the simplest application of a dynamic absorber, we will consider the well-known single-DOF model shown in Figure 6.2. Application of Newton’s 2nd law based on the free-body diagram (FBD) shown in Figure 6.3 yields Figure 6.3. Free-body diagram of a linear single-DOF system. the equation of motion: M x¨ = F(t) −Cx˙ − Kx (6.1) M x¨ +Cx˙ + Kx = F(t) (6.2) or where M, C, and K denote the mass, linear damping, and linear stiffness coefficients, respectively. In actuality, the artistic rendering of the figure, complete with wheels, belies the fact that the model it represents may only be considered a particle or purely translating object of mass, M. Since no dimensions are specified, we must assume that no rotation is allowed, and the decorative wheels contribute negligible inertia. In addition, Equation (6.2) is often parameterized for convenience 50 CHAPTER 6. BRIEF INTRODUCTION TO VIBRATIONS by dividing by M and recasting the result into what we shall refer to as the canonical form for single-degree of freedom vibration models: x¨ + 2ζ ωn x˙ + ωn2 x = 1 F(t) M (6.3) where r ωn = 2π fn = K M (6.4) is the system natural frequency and 2ζ ωn = C M (6.5) so that C ζ= √ 2 KM (6.6) defines the dimensionless damping constant – also known as the dimensionless damping ratio. Hence, Equation (6.3) may also be expressed as x¨ + 2ζ ωn x˙ + ωn2 x = 6.3.1 ωn2 F(t) K (6.7) Response to Initial Excitation Next, we assume that no external force is applied (F(t) = 0) but that the mass is given some initial excitation in the form of an initial displacement, x(0) = x0 , and velocity, x(0) ˙ = v0 . Equation (6.7) thus assumes its homogeneous form: x¨ + 2ζ ωn x˙ + ωn2 x = 0 (6.8) Since Equation (6.8) is a linear, homogeneous, ordinary differential equation with constant coefficients, the general form of its solution is given by x(t) = Aeλ t where A is an arbitrary constant. Substitution of (6.9) into (6.8) yields λ 2 + 2ζ ωn λ + ωn2 Aeλ t = 0 (6.9) (6.10) which implies, for nontrivial constant, A, λ 2 + 2ζ ωn λ + ωn2 = 0 (6.11) Equation (6.11) is the so-called characteristic equation for the single-degree of freedom linear oscillator equation; its solutions are given by q λ1 = −ζ ωn − ωn ζ 2 − 1 (6.12) 6.3. RESPONSE OF A SINGLE-DOF SYSTEM 51 and q λ2 = −ζ ωn + ωn ζ 2 − 1 (6.13) From the above, it is clear that the character of the homogeneous, or transient, response is governed by the value of the dimensionless damping constant, ζ . As you may recall, there are four distinct cases: 1. Undamped =⇒ ζ = 0 2. Over damped =⇒ ζ > 1 3. Critically damped =⇒ ζ = 1 and 4. Underdamped =⇒ 0 < ζ < 1 The most commonly encountered case, and therefore, the one we will focus on here, is the underdamped √ case, where the quantity under the square root becomes negative. Therefore, factoring out j = −1, we may write q λ1,2 = −ζ ωn ± jωn 1−ζ2 (6.14) Since there are two eigenvalues (solutions to the characteristic equation) there must be two free constants to account for the initial conditions, so we have x(t) = A1 eλ1 t + A2 eλ2 t (6.15) Alternatively, substitution of (6.14) into (6.15) and using Euler’s relationship: e± jθ = cos θ ± j sin θ (6.16) x(t) = e−ζ ωn t (B1 cos ωnd t + B2 sin ωnd t) (6.17) equation (6.15) may be recast as where B1 and B2 are real constants to be defined by the system initial conditions, and were q ωnd = ωn 1 − ζ 2 (6.18) is known as the damped natural frequency. Application of the initial conditions in this example yields ζ ωn x0 + v0 −ζ ωn t sin ωnd t x(t) = e x0 cos ωnd t + (6.19) ωnd For an initially displaced system with x(0) ˙ = v0 = 0, Equation (6.19) may be written: ζ x(t) = x0 e−ζ ωn t cos ωnd t + p sin ωnd t 1−ζ2 ! (6.20) 52 CHAPTER 6. BRIEF INTRODUCTION TO VIBRATIONS or4 x0 x(t) = p e−ζ ωn t cos (ωnd t − θ ) 1−ζ2 (6.21) where θ = arctan ζ p ! 1−ζ2 (6.22) The homogeneous solution given by Equations (6.20) and (6.21) is usually referred to in the field of vibrations as the transient solution because it becomes vanishingly small at an exponential rate governed by the product, ζ ωn . 6.3.2 The Logarithmic Decrement Method Determining the viscous loss parameter, C, is actually quite difficult in most cases; fortunately, the dimensionless damping constant, ζ , is quite easy to calculate from a direct measurement of the decaying response of an initially excited single-DOF system. The damping constant, C, if needed, may be calculated from ζ using Equations (6.5) or (6.6). Consider the following example: A typical free response of an under damped (ζ < 1) second-order system is shown in Figure 8.9. In this hypothetical single degree of freedom system, whose total free response is given by x(t) = e−0.2t cos(9.998t), we note that there are seven peaks (labeled on the graph) before t = 4 seconds. This corresponds to seven complete cycles or periods of motion. As you can see from the figure, each successive peak is lower than the previous one. This decay in amplitude is governed by the indicated exponential envelope, e−0.2t . From the p previous development, we know that Figure 8.9 represents a system where ζ ωn = 0.2, and ωd = ωn 1 − ζ 2 = 9.998. Hence, our hypothetical system has an undamped natural frequency of ωn = 10 radians per second, and a dimensionless damping parameter of ζ = 0.02, or 2% of critical (ζ = 1) damping. In general, the amplitude of a second-order system at the kth peak at time tk is given by x(tk ) = xk = x1 e−ζ ωtk (6.23) where x1 is the initial amplitude. Thus, at the first peak, tk = t1 = 0, and x(0) = x1 . Similarly, at the third peak, x(t3 ) = x3 = x1 e−ζ ωt3 = x1 e−ζ ω2Td (6.24) since there are two damped natural periods (Td = 2π/ωd ) between the first and third peaks. Hence, the ratio of the first peak to the kth peak is given by x1 = eζ ω(k−1)Td xk 4 See: http://web.mst.edu/˜stutts/SupplementalNotes/TrigComplex.pdf. (6.25) 6.3. RESPONSE OF A SINGLE-DOF SYSTEM 53 Figure 6.4. Typical free response of an under damped system. but Td = 2π 2π p = ωd ωn 1 − ζ 2 (6.26) Hence, 2(k−1)π √ ζ x1 ζ ω(k−1)Td =e = e 1−ζ 2 xk Taking the natural log of Equation (8.98), we obtain x1 2(k − 1)π ln = p ζ = δk xk 1−ζ2 (6.27) (6.28) where δk is called the logarithmic decrement over k − 1 periods. It should be noted that the log ratio xi ln xi+1 is constant. In most vibrations texts, the logarithmic decrement is defined between any two adjacent peaks. We use the more general formula given in Equation (8.100) for this application, because the difference in amplitude between any two peaks is usually too small to accurately measure. Hence, we measure over k peaks. Solving equation (8.99) for ζ yields ζ= q δk (6.29) δk2 + 4(k − 1)2 π 2 Thus, by measuring the amplitude of any two sequential peaks, and computing the logarithmic decrement, we may use Equation (8.100) to obtain the system dimensionless damping parameter. 54 CHAPTER 6. BRIEF INTRODUCTION TO VIBRATIONS Since the damped period, Td , and ζ are known, the undamped natural frequency may be found to be ωd 2π ωn = p = p 1 − ζ 2 Td 1 − ζ 2 (6.30) For most lightly damped systems, ωn ≈ ωd 6.3.3 (6.31) Steady-State Harmonic Response of One-DOF Systems The most commonly encountered, and therefore the most important system response case is the steady-state response due to harmonic forcing. Therefore, in the following development, we will assume F(t) to be sinusoidal (another term for harmonic) in form. 6.3.4 The Phasor Method If only the steady-state solution of a harmonically-forced system is required, the phasor method provides a very efficient means to obtain it. The phasor method is presented in more detail, and in comparison to the method of undetermined coefficients in Dr. Stutts’ “The Phasor Analysis Method For Harmonically Forced Linear Systems”5 . In the present development, the underdamped case, which is the most commonly encountered, is assumed. Assuming F(t) = F0 sin ω t, substitution of x(t) = XIm e jωt into Equation (6.7), yields 2 F0 F0 ωn2 jωt ωn − ω 2 + j2ζ ωn ω XIm e jωt = Im e jωt = Im e (6.32) M K Here, Im e jωt denotes the imaginary part of the complex exponential, often referred to as the phasor, defined by e jωt .6 Canceling the phasors on either side of Equation (6.32), dividing by ωn2 , and solving for X, yields X= F0 = K [1 − r2 + j2ζ r] F0 e− jφ q 2 K (1 − r2 ) + (2ζ r)2 (6.33) ω ωn (6.34) where, r= 5 http://web.mst.edu/ ˜stutts/SupplementalNotes/PhasorMethod.pdf this context, the Im{} notation may be thought of as a place holder to remind you which component (sin or cos) of the complex exponential was actually present in the forcing term. Otherwise, all operations, such as differentiation, are applied on e jω t . 6 In 6.3. RESPONSE OF A SINGLE-DOF SYSTEM is defined as the forcing frequency ratio, and φ is given by tan−1 2ζ r2 for r ≤ 1 1−r φ= 2ζ r −1 π − tan for r > 1 r2 −1 55 (6.35) Thus, the steady-state solution is given by x(t) = o n F0 e− jφ F0 q Im e jωt = q Im e j(ωt−φ ) 2 2 K (1 − r2 ) + (2ζ r)2 K (1 − r2 ) + (2ζ r)2 (6.36) or, x(t) = F0 q sin (ωt − φ ) = |X| sin (ωt − φ ) 2 K (1 − r2 ) + (2ζ r)2 (6.37) The approach is identical for f (t) = F0 cos ωt, except that a solution of the form x(t) = Re e jωt is assumed instead.7 Figure 6.5. Force vector relationships for a harmonically forced spring-mass-damper system. The phase relationship between the individual force components of the system is shown graphically in Figure 6.5, where it may be seen that the damper force and inertial force are 90◦ and 180◦ out of phase with the spring force, respectively. 6.3.5 Frequency Response The term frequency response refers to how a periodically-forced system responds as a function of frequency, and is probably the most important concept in the study of vibrations. There are a 7 Another reference which provides details regarding complex exponential and its use may be found here: http: //web.mst.edu/˜stutts/SupplementalNotes/TrigComplex.pdf. 56 CHAPTER 6. BRIEF INTRODUCTION TO VIBRATIONS number of metrics used to characterize a system’s frequency response, which we will discuss in detail in the next section, but perhaps the best known is the “dynamic amplification” or resonance curve. From Equation (6.33), we define the dimensionless amplitude frequency response, or dynamic amplification factor, as |X|K 1 M(r, ζ ) = (6.38) = p 2 F0 (1 − r ) + 4ζ 2 r2 Another term for M(r, ζ ) is the displacement transmissibility due to applied harmonic forcing. It may be thought of as a dimensionless transfer function relating the displacement response to the harmonic input force. Graphs of M(r, ζ ) and the corresponding phase, φ (r, ζ ), are shown in Figure 6.6 assuming relatively light damping (ζ = 0.05) such that the peak response occurs when ω ≈ ωn , and thus, at r ≈ 1. As can be seen, the frequency response amplitude is considerably larger near the resonance frequency, and then decreases to zero in the limit as ω → ∞. It is this dynamic amplification that is usually responsible for vibration-induced metal fatigue and the collapse of large civil structures such as the Tacoma Narrows Bridge.8 Figure 6.6. Dynamic amplification resonance curve. 6.3.6 Frequency Response Metrics There are four metrics used to describe the frequency response of a system: 1. Percent of critical damping, ζ – otherwise known as the dimensionless damping constant 8 See: http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridges. 6.3. RESPONSE OF A SINGLE-DOF SYSTEM 57 2. Bandwidth 3. Quality factor, or simply, Q 4. Phase As we have seen, the system response phase with respect to the harmonic forcing, given by Equation (6.35), depends on ζ . The system quality factor also depends on ζ . Therefore, we now discuss another means to determine ζ . There are two commonly-used methods to determine the value of ζ or its complement, the system quality factor, Q; we have already discussed the logarithmic decrement method, which is applicable to initially excited systems. The technique which is applicable in the case of harmonically forced systems, as described here, is the so-called half-power method. This method requires the system under test to be excited over a range of frequencies encompassing its resonance frequency. From Equation (6.38), and recognizing that for ζ << 1, Q is defined as the maximum value of M(r, ζ ), which is given by 1 Q = Mmax ≈ M(1, ζ ) = (6.39) 2ζ The maximum value of M(r, ζ ) actually occurs at r= q 1 − 2ζ 2 (6.40) but for values of ζ < 0.1 or so, there is little error in assuming that the peak amplitude occurs at r = 1. Figure 6.7. Displacement transmissibility of an underdamped 1DOF system showing key points for the half-power method. 58 CHAPTER 6. BRIEF INTRODUCTION TO VIBRATIONS √ The system bandwidth is related to Q as follows: setting M(r, ζ ) = Q/ 2, yields r4 − 2 1 − 2ζ 2 r2 + 1 − 8ζ 2 = 0 (6.41) Solving for the two roots in r2 yields 2 r1,2 q = 1 − 2ζ ∓ 2ζ 1 + ζ 2 2 (6.42) Hence, for small ζ , we have r12 ≈ 1 − 2ζ (6.43) r22 ≈ 1 + 2ζ (6.44) f22 − f12 = r22 − r12 fn2 = ( f2 − f1 ) ( f2 + f1 ) ≈ ( f2 − f1 ) 2 fn (6.45) and Thus, where use has been made of the fact that for small ζ , f1 + f2 ≈ 2 fn . Substitution of Equations (6.43) and (6.44) into (6.45) yields 4ζ fn2 = ( f2 − f1 ) 2 fn (6.46) so that we may define the bandwidth in Hertz as ∆ f = BW = f2 − f1 = 2ζ fn (6.47) Hence, we see that f2 − f1 ω2 − ω1 = 2 fn 2 ωn (6.48) fn ωn 1 = = 2ζ f2 − f1 ω2 − ω1 (6.49) ζ= and Q= where fn , f1 , and f2 are the resonance frequency, and first and second half-power points as illustrated in Figure 6.7. Another important characteristic of the steady-state harmonic response is its phase, as defined in Equation (6.35), with respect to the forcing. Examining Equation (6.35) at limiting values of forcing frequency ratio, r, reveal that the phase is zero at very low frequencies, 90◦ at resonance, and lim φ (r, ζ ) = π r→∞ (6.50) or 180◦ , as may also be observed in Figure 6.6. The implication of this result is that the system response lags the input by a phase approaching 180◦ at forcing frequencies well above the resonant frequency. It is extremely important to understand that All harmonically excited systems with viscoelastic support or coupling exhibit a response which lags the excitation by a phase which is a function of the forcing frequency and the system mass, stiffness, and damping parameters! 6.4. DISPLACEMENT TRANSMISSIBILITY UNDER BASE EXCITATION 59 Figure 6.8. Single-DOF base excitation model. 6.3.7 Response to Harmonic Base Excitation The previous development is valid for systems under the direct action of harmonic forcing. However, the situation is significantly different when the harmonic forcing is due to base or support excitation such as is the case for buildings during earthquakes, and automobiles on periodically bumpy roads. The prototypical single-DOF model for base excitation and the corresponding free-body diagram are shown in Figure 6.8. Application of Newton’s 2nd law allows us to write the equation of motion as: M x¨ = K (y(t) − x) +C (y(t) ˙ − x) ˙ (6.51) or M x¨ +Cx˙ + Kx = Cy˙ + Ky 6.4 (6.52) Displacement Transmissibility Under Base Excitation As before, we’ll divide by M, and recast the equation of motion using the canonical parameters, ωn and ζ . x¨ + 2ζ ωn x˙ + ωn2 x = 2ζ ωn y(t) ˙ + ωn2 y(t) (6.53) Assuming harmonic displacement excitation y(t) = Y0 e jωt (6.54) the steady-state response may also be assumed to be harmonic: x(t) = X e jωt Substitution of Equation (6.55) into (6.53) yields ωn2 − ω 2 + j2ζ ωn ω X = Y0 j2ζ ωn ω + ωn2 (6.55) (6.56) 60 CHAPTER 6. BRIEF INTRODUCTION TO VIBRATIONS Recasting the complex terms from Cartesian to polar form yields q q 2 (ωn2 − ω 2 ) + 4ζ 2 ωn2 ω 2 e jφ X = Y0 ωn4 + 4ζ 2 ωn2 ω 2 e jψ (6.57) Thus, p Y0 ωn4 + 4ζ 2 ωn2 ω 2 e( jψ−φ ) X= q 2 (ωn2 − ω 2 ) + 4ζ 2 ωn2 ω 2 (6.58) Dividing the numerator and denominator of the right-hand side of Equation (6.58) by ωn2 , yields p Y0 1 + 4ζ 2 r2 e( jψ−φ ) X= q (6.59) 2 2 2 2 2 (1 − r ) + 4ζ r ω where, again, the forcing frequency ratio, r, is defined in Equation (6.34), and where φ is given by Equation (6.35), and ψ = arctan(2ζ r) (6.60) Equation (6.59) describes the steady-state frequency response of the base-excited system in terms of complex number in polar form, which facilitates the separation of the magnitude and phase behavior as a function of r and ζ . Dividing the magnitude of X by the amplitude of the base excitation, Y0 , we define the displacement transmissibility for a single-degree of freedom, base excited system: p |X| 1 + 4ζ 2 r2 Mdbe (r, ζ ) = = q Y0 2 (1 − r2 ) + 4ζ 2 r2 ω 2 (6.61) The magnitude and phase frequency response of this system is shown for various values of damping ratio over the range of r = 0 to r = 3 in Figure 6.9. There are several important characteristics which may be observed: 1. As expected, larger ζ values attenuate the magnitude of the response. √ √ 2. The magnitude curves for all values of ζ pass through the point r = 2, Mdbe ( 2, ζ ) = 1. 3. limr→∞ Mdbe (r, ζ ) = 0. 4. Larger values of ζ have the effect of increasing the phase lag below resonance (r = 1) and decreasing it above resonance. The phase behavior at high frequencies (r >> 1) is not apparent in figure 6.9, but may be easily derived from Equations (6.34) and (6.60): lim ψ(r, ζ ) − φ (r, ζ ) = r→∞ or 90◦ . π 2 (6.62) 6.4. DISPLACEMENT TRANSMISSIBILITY UNDER BASE EXCITATION 61 Figure 6.9. Mdbe (r, ζ ) and phase, ψ − φ , for various values of ζ . The peak magnitude of Mdbe (r, ζ ) may be found to be p 4 8ζ2 +1 Qdbe = s 2 √ p 1− 8 ζ 2 +1 −1+ 8ζ2 +1 1+ 4ζ 2 (6.63) and may be shown to occur at qp 8ζ2 +1−1 1 rmax = 2 ζ (6.64) rmax ≈ 1 − ζ 2 ≈ 1 (6.65) or for small values of ζ : which implies that the peak amplitude occurs approximately at resonance – i.e. fmax = rmax ωn = rmax fn = fn 2π (6.66) The corresponding orders of approximation for Qdbe are Qdbe ≈ 1 2+5ζ2 1 ≈ 4 ζ 2ζ (6.67) The final approximation in Equation (6.67) is identical to that derived for single DOF systems excited by applied harmonic forces given by Equation (6.49). However, the corresponding approximate r 62 CHAPTER 6. BRIEF INTRODUCTION TO VIBRATIONS values for the half-power points, which may be derived from the truncated Taylor series expansion of the very complicated exact expressions, are given by r1 ≈ 1 − ζ (6.68) r2 ≈ 1 + ζ (6.69) and Thus, the corresponding frequencies are given by f1 = r1 fn = (1 − ζ ) fn (6.70) f2 = r2 fn = (1 + ζ ) fn (6.71) and Hence, as before for the directly forced case, the dimensionless damping constant is given by ζ= f2 − f1 2 fn (6.72) Chapter 7 Dynamic Vibration Absorbers Dynamic absorbers, also called tuned mass dampers, are widely applied to attenuate vibration at a specific frequency[1].1 The terms “tuned mass damper” and “dynamic absorber” may be considered synonymous for the most part, but the former denotes the explicit addition of a loss mechanism – usually a viscous or friction damper. Most often, dynamic absorbers are employed when the frequency where vibration reduction is desired lies at or near a system resonance or natural frequency. Dynamic absorbers have have a long history; Den Hartog published an exact solution of system combining both Coulomb (friction) and viscous damping in 1931 [2]. Most of the literature on dynamic absorbers describes their application in buildings and other civil structures such as bridges [3]. One of the most common applications of dynamic absorbers is the attenuation of wind-induced galloping in electrical power lines; this type of dynamic absorber is known as a Stockbridge damper. Sauter and Hagadorn characterized the attenuation of a Stockbridge absorber by using hysteric Coulomb damping elements in numerically solved nonlinear model [4]. The model described here assumes simple velocity-proportional viscous damping. When used in buildings, tuned mass dampers (TMD) often take the form of a pendulum whose fundamental period of motion is tuned to match that of the first vibration mode of the building. The first mode of vibration of a skyscraper is the most likely to be excited by an earthquake or vortex shedding in high winds.2 When a tall building sways back and forth, the motion primarily corresponds to the first building mode and natural frequency. Probably the largest example of of a pendular TMD may be found in Taipei 101.3 7.1 Basic Dynamic Absorber Theory Consider a base-excited, single-degree of freedom system such as that shown in Figure 6.8; perhaps it represents an instrument attached to a manufacturing process which undergoes constant harmonic vibration during operation. In addition, assume that the vibration occurs at or near the resonant frequency of the instrument and its viscoelastic support structure, and that altering the frequency 1 See: http://en.wikipedia.org/wiki/Tuned_mass_damper. https://www.youtube.com/watch?v=HB2jgJJG2is. 3 See:https://www.youtube.com/watch?v=NYSgd1XSZXc and https://www.youtube.com/ watch?v=uybEXOkkrsw. 2 See: 63 64 CHAPTER 7. DYNAMIC VIBRATION ABSORBERS of operation or changing the instrument support are not options. In such a case, if the vibration amplitude that the instrument undergoes too high, one of the simplest and least expensive methods to reduce the instrument vibration is by attaching a dynamic absorber or tuned-mass damper to the instrument. Doing so changes the single-degree of freedom system into a two-degree of freedom system as shown in Figure 7.1. The equations of motion for each mass may be determined using Newton’s 2nd law: Figure 7.1. Dynamic absorber model with damping; also known as a tuned-mass damper. M1 x¨1 = F(t) +C2 (x˙2 − x˙1 ) + K2 (x2 − x1 ) − K1 x1 −C1 x˙1 or M1 x¨1 + (C1 +C2 )x˙1 + (K1 + K2 )x1 −C2 x˙2 − K2 x2 = F(t) (7.1) and M2 x¨2 = −C2 (x˙2 − x˙1 ) − K2 (x2 − x1 ) or M2 x¨2 −C2 x˙1 − K2 x1 +C2 x˙2 + K2 x2 = 0 Assuming F(t) = F0 sin Ωt = F0 ℑ e jΩt and thus assuming x(t) = Xℑ e jΩt , where, n o jΩt ℑ e = sin Ωt denotes the Imaginary part of e jΩt , we have K1 + K2 − M1 Ω2 + j(C1 +C2 )Ω − jC2 Ω − K2 X1 F0 = X2 0 − jC2 Ω − K2 K2 − M2 Ω2 + jC2 Ω By Cramer’s rule (7.2) (7.3) (7.4) 7.1. BASIC DYNAMIC ABSORBER THEORY det X1 = det F0 − jC2 Ω − K2 0 K2 − M2 Ω2 + jC2 Ω 65 K1 + K2 + j(C1 +C2 )Ω − M1 Ω2 −( jC2 Ω + K2 ) −( jC2 Ω + K2 ) jC2 Ω − M2 Ω2 = (K2 − M2 Ω2 + jC2 Ω)F0 4 (7.5) (7.6) where 4 = M1 M2 Ω4 − [C1C2 + (K1 + K2 )M2 + K2 M1 ]Ω2 + K1 K2 + j[(K1C2 + K2C1 )Ω − (M1C2 + (C1 +C2 )M2 )Ω3 ] (7.7) 4 = M1 M2 Ω4 − [C1C2 + (K1 + K2 )M2 + K2 M1 ]Ω2 + K1 K2 + j[(K1C2 + K2C1 )Ω − (M1C2 + (C1 +C2 )M2 )Ω3 ] = ℜ {4} + jℑ {4} (7.8) where n o jΩt ℜ e = cos Ωt (7.9) is the Real part of e jΩt . Hence, the magnitude of X1 is given by s |X1 | = F0 (K2 − M2 Ω2 )2 +C22 Ω2 ℜ2 {4} + ℑ2 {4} (7.10) where ℜ {4} = M1 M2 Ω4 − [C1C2 + (K1 + K2 )M2 + K2 M1 ]Ω2 + K1 K2 (7.11) ℑ {4} = (K1C2 + K2C1 )Ω − (M1C2 + (C1 +C2 )M2 )Ω3 (7.12) and Based on Equation (7.10), amplitude of x1 may be significantly reduced by choosing the ratio of K2 and M2 such that the system operates at its antiresonance frequency. In other words, setting K2 = Ω2 M2 (7.13) While the above analysis assumes harmonic forcing directly applied to the original mass, q the result is applicable for base-excited systems as well, except that F0 would be replaced by Y0 where Y0 denotes the amplitude of the harmonic base excitation. C12 Ω2 + K1 , 66 7.2 CHAPTER 7. DYNAMIC VIBRATION ABSORBERS Vibration Absorber Numerical Example The following is an example of the amount of vibration attenuation possible through the use of a dynamic absorber. A numerical simulation was performed via Maple 16 using the following parameters: M1 = 10 kg, K1 = 3, 947, 842 N/m, C2 = C1 = 3, 948 N-s/m M2 = 2 kg, K2 = 7,895,681 N/m, and F0 = 3, 948 N. The natural frequency before the addition of the dynamic absorber was fn = 100 Hz. The operating frequency was 100 Hz. Resonance frequencies after the addition of the absorber were: f1 = 80 Hz, and f2 = 124 Hz. A remarkable 99% reduction at 100 Hz was predicted by the simulation. Figure 7.2. Amplitude of x1 ( f ) before and after addition of vibration absorber. Based on Equation (7.10), it would seem that the amplitude of x1 (t) is mostly a function of the ratio of K2 to M2 and C2 . If that were the case, one could pick a tiny absorber to put on a large machine or even a building as long as the correct absorber stiffness to mass ratio were correct, and fully expect it to absorb all of the vibrational energy of the original structure! Alas, life is not so simple... The absorber mass and stiffness must be reasonably large in order to separate the resulting two modes sufficiently, which implies that they must be chosen large enough to absorb an appreciable amount of the kinetic and potential energy of the original structure. Qualitatively, the dynamic absorber can split the original resonance mode of vibration into two new resonance modes, one above and the other below the original resonance, thereby leaving the system forced at an antiresonance frequency. Furthermore, since the input energy remains the same, each of the new modes must vibrate at a lower amplitude than the original single resonance in order to obey the law of conservation of energy. The resulting amplitude and phase of the original mass, X1 , and tuned mass damper, X2 , in the 7.2. VIBRATION ABSORBER NUMERICAL EXAMPLE 67 Figure 7.3. Amplitude frequency response of the original mass, X1 , and tuned mass damper, X2 . combined structure are shown in Figures 7.3 and 7.4 respectively. Note that while the tuned mass damper is in phase with the forcing at 100 Hz, the original mass is about 90 degrees out of phase. Figure 7.4. Phase frequency response of the original mass, X1 , and tuned mass damper, X2 . 68 7.3 CHAPTER 7. DYNAMIC VIBRATION ABSORBERS Implementation Figure 7.5. Schematic of dynamic absorber experimental apparatus. A schematic of the dynamic absorber apparatus is shown in Figure 7.5. The dynamic absorber will be implemented using a cantilevered beam with a mass attached to its end, consisting of two steel plates connected together and to the beam via set screws. The cantilevered beam may be positioned in a clamp such that the end mass is either closer or further away from the clamp to increase or decrease the spring stiffness, respectively, thereby tuning the absorber. A wing nut is provided to facilitate the tuning process. The spring stiffness of the attached dynamic absorber varies approximately according to: 3E I Kb ≈ 3 (7.14) L where E, I, and L, are the Young’s modulus, second moment of area of the beam cross section, and distance between the spring attachment and the mass, respectively. If the attached end mass is much heavier than the beam, the beam mass may be neglected, and the dynamic absorber system treated exactly as described in Section 7.1, with K2 = Kb , and M2 denoting the mass of the attached steel plates. Therefore, from Equations (7.13) and (7.14), the required location of the dynamic absorber 7.4. EXPERIMENTAL PROCEDURE 69 end-mass with respect to the mounting clamp is given by4 r L= 7.4 3 3EI M2 Ω2 (7.15) Experimental Procedure To determine the effect of absorber mass, the system will be tested using two different masses: 116.48±0.008660 grams, and 206.7±0.08660 grams. The basic theory, as detailed in section 7.1 does not predict any difference in attenuation as long as the stiffness to mass ratio is held constant. However, energy considerations predict greater attenuation with increasing mass, and this is the hypothesis that will be tested in this study. The resulting resonances and before-and-after vibration amplitude will be recorded, and the corresponding percent attenuation calculated. In order to accomplish this comparison, the following steps must be taken: 1. Measure the natural frequency of the single-degree of freedom system. This may be done as follows: (a) First attach an accelerometers to the mass (M1 ) and the base. Make sure the other end of each accelerometer is attached to signal conditioning hardware or an appropriate data acquisition module. If an oscilloscope is used, the accelerometer conditioning hardware must be in the signal path between the accelerometer and the oscilloscope to obtain a signal.5 (b) Drive the shaker starting at a low frequency, say 10 Hz, and then increasing the drive frequency in increments of 1 Hz until a peak amplitude is observed. Try to refine the frequency increments near the peak to make sure it’s really the maximum response. Systems with low damping have sharp peaks, so being a little off in the drive frequency can lead to significant errors in measuring the maximum amplitude. Record the signal measured at the peak amplitude of the mass, Xmax , the corresponding base amplitude signal, and drive frequency, fmax . The accelerometer signals measured will be linearly proportional to the accelerations, so when the ratio of the M1 peak accelerometer signal max to the base accelerometer signal, V Vacc base , is calculated, it will correspond to the 6 maximum displacement transmissibility: Tdmax = max κΩ2max X1max X1max Vacc = = Vbase κΩ2maxY0 Y0 (7.16) 4 Note that care must be taken to accurately measure the distance L from clamp to clamp – i.e. between where the beam is clamped and to where the end-mass is clamped. Also, make sure that the beam is tightly clamped, and that the vise and table experience as little as possible vibration during the test. Therefore, do not lean on or bump the table, and ask anyone who might be running the dynamic balancing experiment to turn it off while you perform this part of your experiment. 5 PCB Piezotronics accelerometers are used in this lab; these have built-in charge amplifiers which require power from the signal conditioning hardware, or from data acquisition units designed to drive PCB accelerometers. 6 This assumes that the base amplitude remains constant during the sweep; this may not be the case unless the base signal is monitored and adjusted as necessary. 70 CHAPTER 7. DYNAMIC VIBRATION ABSORBERS where κ is the constant of proportionality. 2. Estimate the dimensionless damping constant, ζ , using the half-power method described in Sections Sections 6.3.6 and 6.4. To do this, you will need to search for the half-power frequency. Sweep the drive frequency from well below the resonance peak until the resonance frequency recorded in the previous step. Refine your search until you’ve located the halfpower point √ frequency, f1 . This frequency will correspond to an accelerometer signal equal max / 2 ≈ 0.707V max . to Vacc acc 3. Repeat the previous step, but from fmax up until the second half-power point frequency, f2 , is located. 4. Use Equation (6.72) to calculate ζ . 5. Estimate Young’s modulus for the dynamic absorber beam, which may be done by first measuring the natural frequency of the cantilevered beam. (a) This may be done by clamping the beam without the attached end mass in the vise provided with a known cantilever length – say approximately 170 mm. The chosen cantilevered length is not important, but an accurate measurement of it using the provided calipers is. (b) Attach one of the accelerometers to the beam relatively close to the vise clamp. The closer the better in order to minimize the mass contribution of the accelerometer to the beam. The tradeoff, however, is that the accelerometer must experience sufficient motion to generate a large enough signal. Make sure the other end of the accelerometer is attached to signal conditioning hardware or an appropriate data acquisition module. If an oscilloscope is used, the accelerometer conditioning hardware must be in the signal path between the accelerometer and the oscilloscope to obtain a signal. (c) Deflect the end of the beam approximately 3 mm and abruptly release it. This is best done using one’s fingernail. (d) Record the decaying signal long enough to obtain the so-called damped natural period of oscillation for the first mode of the cantilevered beam, Td1 , by determining the time between two successive zero crossings. A more accurate reading may be obtained by measuring the time between 11 successive zero crossings (and therefore five periods) and dividing this time by 10. The frequency of vibration of the first mode will be given approximately (assuming light damping) by q q 1 ωnd ωn 2 fd1 = = = 1 − ζ = f1 1 − ζ12 ≈ f1 (7.17) Td1 2π 2π The difference between f1 and fd1 must be accounted for if the damping is significant. To determine the degree of approximation in the above assumption, utilize the logarithmic decrement method, discussed in Section 6.3.2, to estimate ζ1 . (e) Once f1 is known, the value of value of the bending rigidity, EI, may be calculated from the theoretical natural frequency given by s (βn L)2 EI ωn = (7.18) L2 ρ bh 7.4. EXPERIMENTAL PROCEDURE 71 where βn denotes the nth eigenvalue for a cantilevered beam, and ρ denotes the mass per unit volume of the beam, b, the width of the beam, and h, the thickness of the beam in the direction of displacement. Hence, the rigidity is given by EI = ρbhL4 2 4ρbhL4 π 2 2 ω = f (βn L)4 n (βn L)4 n (7.19) 4ρbhL4 π 2 2 f (β1 L)4 1 (7.20) or for the first (n = 1) mode: EI = or, recalling that the area moment of inertia with respect to the bending axis of a rectangular beam is given by Z I= r2 dA = b Z h/2 y2 dy = −h/2 A 1 3 bh 12 (7.21) Young’s modulus is given by E= 48ρL4 π 2 2 f h2 (β1 L)4 1 (7.22) The first eigenvalue, beam-length product is given (to seven significant figures) by β1 L = 1.875104 ± 0.5E-7. The width and thickness of the beam, based on 11 equally spaced measurements along the length of the beam, are given in Table 7.1, along with the measured density and all corresponding uncertainties.7 The uncertainty in the first mode frequency measurement, f1 , is a function of the sample rate of the data acquisition system as well as the estimated period of vibration, T1 : 1 T1 The time increment between any two consecutive samples is given by f1 = ∆t = 1 fs (7.23) (7.24) where fs denotes the sampling frequency. The location of a peak is determined by the relative amplitude of three sequential data points, hence its uncertainty is ±∆t. Consider the time between two peaks separated by n periods, Tm , the uncertainty in the determination of Tm is at best, ±2∆t. Hence, we have: Tm = nT1 ± 2∆t 7 The (7.25) uncertainties were determined using the expanded uncertainty (Uk=2 ) calculations as described in ASME report: ASME B89.7.3.2-2007, which provide a 95% probability that the measurand lies within ±Uk=2 hence,√ they represent twice the standard (one standard deviation) uncertainty, u, assuming a Gaussian distribution, and 3u, assuming a uniform distribution. A Gaussian distribution of uncertainty is often assumed for all uncertainties; at worst, this leads to a slight overestimation of the combined uncertainty, and is acceptable for this experiment. Please refer to Chapter 10. 72 CHAPTER 7. DYNAMIC VIBRATION ABSORBERS Thus, the fundamental period is given by T1 = Tm 2∆t ± n n (7.26) uncertainty in the f1 estimate may be approximated as: ∂f 1 uf = uT = − 2 ∂T T1 2∆t n (7.27) where n denotes the number of periods between any two peaks. Hence, the magnitude of the fundamental frequency measurement is given by: 1 uf = 2 T1 2∆t n = 2 f12 n fs (7.28) Therefore, the fundamental beam natural frequency must be stated as f1 ± u f where u f is given by Equation (7.28). Clearly, the larger the number of periods between the two peaks, the lower the uncertainty. Table 7.1. Original system mass and dynamic absorber beam parameters and uncertainties. M1 kg 0.2651 ± 0.0000866 b (mm) 25.07 ± 0.04899 h (mm) 1.571 ± 0.01028 ρ (kg/m3 ) 7565 ± 52.05 The caliper used to measure the beam has a resolution of 0.001 ± 0.0008660 millimeters.8 6. With the value of Young’s modulus calculated in 5e and the resonant frequency, calculate the required cantilevered length of the dynamic absorber using Equation (7.15) assuming M2 is the smaller of the two masses provided. 7. Repeat Step 6 assuming M2 is the larger mass of the two masses. 8. Set up the dynamic absorber using the smaller mass, and drive it at the original single-degree of freedom resonance frequency. Record the base and M1 signal amplitudes, and calculate the displacement transmissibility, X1 /Y0 . Adjust the location of the mass up and down by a few millimeters to see if the transmissibility is better or worse. 9. Repeat Step 8 using the larger mass. 8 Resolution uncertainties are assumed to occur according to a uniform distribution, therefore the extended uncertainty √ given a resolution of 0.001 is ± 3(0.001/2) ≈ ±0.0008660. 7.5. QUESTIONS 7.5 73 Questions 1. Did the design values of L calculated in Steps 8 and 9 using Equation (7.15) yield the greatest reduction in transmissibility or did the position of M2 require adjustment to obtain maximum reduction? 2. Did L need to be increased or decreased for the dynamic absorber to perform best? What can you infer from this result? 3. Calculate the percentage reduction in displacement transmissibility between the original system running at its resonance frequency, fn , and the system with the tuned mass damper (dynamic absorber) attached running at the same frequency. In other words, compare the best transmissibility reduction achieved in Steps 8 and 9 with the maximum transmissibility obtained in Step 1b. The transmissibility as defined in Equation (6.61) is the ratio of the amplitude of the original mass (M1 ) to that of the base. Hence we can define the transmissibility before and after addition of the dynamic absorber as Tb = X1b Y0 (7.29) and X1a Y0 respectively. Thus, we can define the transmissibility reduction ratio as Ta = RT = Tb Ta (7.30) (7.31) It is not uncommon to achieve RT values well over 1000. 4. Did the larger dynamic absorber mass yield a greater reduction in the displacement transmissibility than the smaller one? If so, by what percentage? 5. How did the estimated value of Young’s modulus compare to the nominal published value of 200 GPa? Does the value of E that you estimated lie within the extended combined uncertainties for this calculation? To make this determination, you’ll need to calculate the combined standard uncertainties as is done in the Pump Experiment, and then scale this value to the Extended value assuming random error by doubling it. The interpretation of the extended uncertainty is that the E value calculated has a 95% probability of occurring between E − Uk=2 and E + Uk=2 . For this calculation, you can assume the quantity (β1 L) provided to be accurate to 0.5E − 7, so practically exact compared to the other uncertainties. 6. Use the value of fn obtained in Step 1b and the mass of M1 given in Table 7.1 to estimate the stiffness of the original system, K1 . You may find Equation (6.4) useful here. 7. Using the value of ζ obtained in Step 4, and the values of M1 and K2 (obtained in Question 6) to estimate the value of C1 . What are the units of C1 ? Bibliography [1] W. Contributors. (2013, January) Tuned mass damper. On line. Wikipedia. [Online]. Available: http://en.wikipedia.org/wiki/Tuned mass damper [2] J. P. Den Hartog, “Forced vibrations with constrained coulomb damping and viscous friction,” Transactions of the ASME Advance Papers, p. 107115, 1931. [3] R. Ibrahim, “Recent advances in nonlinear passive vibration isolators,” Journal of Sound and Vibration, vol. 314, no. 3-5, pp. 371 – 452, 2008, euler spring isolators;Nonlinear vibration isolation;Structural installations;. [Online]. Available: http: //dx.doi.org/10.1016/j.jsv.2008.01.014 [4] D. Sauter and P. Hagedorn, “On the hysteresis of wire cables in stockbridge dampers,” International Journal of Non-Linear Mechanics, vol. 37, no. 8, pp. 1453 – 1459, 2002, cable hysteresis;Dynamic flexural deformations;. [Online]. Available: http://dx.doi.org/10.1016/S0020-7462(02)00028-8 74 Chapter 8 Piezoelectric Beam Experiment 8.1 Objectives 1. To provide students with a basic introduction to the concepts of electromechanical actuation and transduction, known as mechatronics. 2. To determine natural frequencies for a cantilevered beam system. 3. To estimate the damping in the cantilevered beam system 4. To determine the effect of an additional attached mass on the natural frequencies of the beam. 5. To verify the direct piezoelectric effect for application in strain sensing or energy harvesting. Note that due to time limitations, it is acceptable to chose either 4 or 5 for this lab. 8.2 Background The use of piezoelectric actuators in ultrasonic motors, active damping of smart structures, ultrasonic welding and cutting, etc., has become widespread [1, 2, 3, 4, 5]. All of these applications combine the area of Mechatronics, with the theory of vibrations. Therefore, it is important to understand the basic principles in each area. Mechatronics may be defined as the combination and integration of mechanics, electronics, materials science, and electrical and control engineering. Computer science also comes in to play when higher levels of system control behavior are desired. Hence, mechatronics is a very broad and interdisciplinary field. This laboratory will touch on the materials and electrical aspects of mechatronics, and focus mostly on the mechanical aspects by developing the mathematical model of a piezoelectrically driven cantilevered beam. No development of vibration control strategies will be attempted, but may be readily undertaken as an extension if desired. 8.2.1 Disclaimer The author freely admits that this manual contains WAY too much material for the average student to absorb in the time alloted for ME242! Don’t worry! You will only be tested on the material in the 75 76 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT experimental procedure and the discussion questions. The object of including so much information in this document was to provide fertile ground for the development of extensions, and to provide the interested person with enough information to get him or her started in the field of mechatronics requiring piezoelectric actuation. 8.3 Simplified (1-D) Theory of Piezoelectric Ceramic Elements In this section, we examine the basic theory behind the thin and narrow PZT elements used to drive the cantilevered beam in this experiment. The PZT plates bonded to the aluminum beam are thin enough (14 mils or 0.356 mm) that the additional mass and stiffness contributed to the beam is negligible. In addition, since they are long and relatively narrow, we will neglect any induced strain in the y-direction perpendicular to the long axis of the beam. Hence, we need only consider the one-dimensional piezoelectric constitutive equations which pertain to the case at hand, and therefore avoid the much more complicated general case. Before going further, we must first define the piezoelectric effect [6]: Definition 8.1 The direct piezoelectric effect refers the charge produced when a piezoelectric substance is subjected to a stress or strain. Definition 8.2 The converse piezoelectric effect refers to the stress or strain produced when an electric field is applied to a piezoelectric substance in its poled direction. Hence, the direct piezoelectric effect is useful in sensors such as some microphones, accelerometers, sonar, and ultrasound transducers. The converse effect is useful in actuators such as ultrasonic welders, ultrasonic motors, and the beam in this experiment. Interestingly, both effects are employed simultaneously in sonar and ultrasound. That is, the same element which produces the sonic signal also registers the return signal. 8.3.1 Poling the Piezoceramic Before PZT can be used in actuation and sensing, it must be electroded and poled. The term poling refers to the application of a strong electric DC field across the electrodes of the PZT. This applied coercive field causes most of the randomly aligned polar domains in the initially inert PZT to align with the applied field. When the field is removed, most of these realigned polar domains remain aligned in the direction of the coercive or poling field – a condition known as remnant polarization. The PZT is now said to be poled in the direction of the poling field, and when another electric field is applied in the same direction (+ poling direction), the PZT will expand in that direction and contract in the transverse direction. Conversely, when a field is applied in the opposite direction, the PZT will contract in the poled direction and expand in the transverse direction. This is why the piezoelectric coefficients in the transverse direction are given negative signs (see Section 8.11.2), and the effect is to render the transverse forcing 180 degrees out of phase with the electrical field. The situation is shown in Figure 8.1 where the initial (before the field is applied) size of the PZT element is indicated by dashed lines. The relevant geometry of poling is shown in Figure 8.2. The double arrow passing through the PZT plate indicates the positive or down poling direction which is 8.3. SIMPLIFIED (1-D) THEORY OF PIEZOELECTRIC CERAMIC ELEMENTS 77 in the oposite direction as the poling potential. Mathematically, the electric field is related to the potential by Φ E = −∇Φ (8.1) where ∇ is the gradient of the applied field. By convention, the Cartesian directions x, y, and z are indicated by the numbers 1, 2, and 3 respectively. This is because in the general case, the constitutive equations are described using tensors and matrices. Also by convention, The 3-direction is always chosen to align with the positive direction of poling. The field induced in the PZT in this case is simply the applied voltage divided by the thickness of the PZT E3 = − Φ hPZT (8.2) . Figure 8.1. Strain as a function of applied field with respect to the + poling direction. There are numerous formulations of PZT and other piezoceramics, and each has an optimal poling schedule. Most manufacturers of finished piezoceramics treat the poling temperature and schedule for their ceramics as proprietary information. Figure 8.2. Schematic of relevant poling geometry. 8.3.2 The One-Dimensional Piezoelectric Constitutive Equations Whereas the generalized Hook’s law relating stress and strain is sufficient to describe the behavior of elastic materials, such as the aluminum beam in this experiment, piezoelectric materials are more complicated. In Piezoelectric materials there is coupling between the electrical and mechanical constitutive relationships. It is, of course, this coupling which makes piezoelectric ceramics like 78 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT PZT so useful. For an approximately one-dimensional piezoelectric element such as the PZT used in this experiment, the constitutive equations are given by σ1 = YPZT ε1 − e31 E3 , (8.3) D3 = e31 ε1 + ε3 E3 , (8.4) and where the parameters are defined below in Table 8.1. Manufacturers commonly provide the d31 coefficient which relates the strain in the 1-direction produced in the PZT due to an applied field in the 3-direction. The corresponding Stress relationship is given by the e31 = d31YPZT coefficient. The first subscript in this notation refers to the poled direction, and the second to the direction in which strain or stress is measured, so d31 is the piezoelectric coefficient relating electric field in the 3-direction to strain in the 1-direction. Equation (8.3) applies to the converse piezoelectric effect, and is most applicable to actuation, and Equation (8.4) is a statement of the direct piezoelectric effect and is most applicable to sensors.1 These two equations will be applied in both actuation and sensing later on. Table 8.1. Piezoelectric constitutive variables and constants. Symbol σ1 YPZT ε1 E3 d31 e31 = YPZT d31 D3 ε3 Φ Variable Name Stress in the 1-direction Young’s modulus of PZT in the 1-direction Strain in the 1-direction Electric Field in the 3-direction Strain-Form Piezoelectric constant Stress-Form Piezoelectric constant Electric Displacement in the 3-direction Permittivity in the 3-direction Voltage potential in the 3-direction SI Units Newton/meter2 Newton/meter2 Dimensionless volt/meter Coulomb/Newton or meter/volt Coulomb/meter2 or Newton/volt-meter Coulomb/meter2 Farad/meter Volts Piezoelectric behavior is further complicated by the fact that the material constants are highly dependant upon the state of the material. Important variables include stress, strain, electric field, and electric displacement2 , D. This dependence may be characterized by another important piezoelectric metric: the electromechanical coupling coefficient. There are several frequently used definitions for the coupling coefficient, but in its basic form it is defined in one of two ways [7, 6]: k2 = 1 The Electrical energy stored Mechanical energy input (8.5) permittivity of the ceramic must usually be calculated from the manufacturer’s published value of the dielectric constant, K3 = εε03 , where ε0 = 8.85 × 10−12 Farads/m is the permittivity of free space (vacuum). 2 D is also called the dielectric displacement and the electric flux density in the literature. 8.3. SIMPLIFIED (1-D) THEORY OF PIEZOELECTRIC CERAMIC ELEMENTS 79 or Mechanical energy stored . (8.6) Electrical energy input It turns out that k is the same in either definition for a given piezoelectric. As an example, Equations (8.3) and (8.4) are more properly written k2 = E σ1 = YPZT ε1 − e31 E3 , (8.7) D3 = e31 ε1 + ε3ε E3 , (8.8) and to indicate that Young’s modulus is measured at constant (actually, zero) electrical field, E, and that the dielectric constant is measured at constant (zero) strain, ε. Zero field is achieved by shorting the electrodes, and zero strain is achieved by blocking or preventing the PZT from expanding while the field is applied. In all cases where it matters, a superscript indicates that the measurement was taken with that variable held constant, and usually zero. For example, in the open-circuit condition, D = 0, and we have Y E = (1 − k2 )Y D . (8.9) Equation (8.9) indicates that Young’s modulus measured at constant field (short circuit) is lower than when measured at constant D (open circuit). When used as a sensor, PZT elements may be treated in the open-circuit state since the input impedance of most voltage measuring circuits is usually high. The PZT used in this experiment has coupling coefficient of k31 = 0.53, where the subscripts refer to the fact that the strain is in the 1-direction, and the induced field is in the 3-direction, so Young’s modulus under the short condition is about 28% less than it is in the open circuit condition. When electrically driving the PZT in actuation, the condition is neither short nor open circuit, but there is an electric field effect as you will see during your experiments. Since the PZT is relatively thin compared to the aluminum beam, it contributes negligible stiffness, and minimal mass. The principal effect due to the variable Young’s modulus is the change it causes in the e31 coefficient during actuation [7]. When D = 0, the case when PZT is used to sense strain, the strain-induced field is given by E3 = − D d ε e31 ε1 −YPZT 31 1 = . ε ε ε3 ε3 (8.10) To obtain the voltage at a point in the PZT, the field must be integrated over the thickness of the PZT which is usually constant (hPZT ). Hence, from Equation (8.1) we have Φ= D d ε −hPZT YPZT 31 1 ε3ε (8.11) However, to obtain the total voltage potential of the PZT element, the pointwise field must be integrated over the entire electroded area of the PZT element, so Equation (8.11) becomes D d −hPZT YPZT 31 ε1 dAPZT APZT Φ = , ε3ε R (8.12) 80 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT so the total voltage induced in the PZT due to applied strain is D d −hPZT YPZT 31 Φd = APZT ε3ε Z x2 x1 ε1 dAPZT , (8.13) or, since dAPZT = b dx, D d −b hPZT YPZT 31 Φd = ε APZT ε3 Z x2 x1 D d −hPZT YPZT 31 ε1 d x = ε (x2 − x1 ) ε3 Z x2 x1 ε1 d x (8.14) where APZT = b (x2 − x1 ) is the electroded area of the PZT. The subscript d on Φd is there to denote that this voltage potential is due to the direct piezoelectric effect – i.e. due to the strain induced in the sensing PZT layer. 8.4 The Euler-Bernoulli Beam Model Figure 8.3. Schematic of piezoelectrically-forced cantilevered beam with accelerometer showing relevant geometry. A schematic of the beam used in this experiment, showing the relevant geometry is shown in Figure 8.3. The cantilevered beam is modeled by first considering a differential element of the length dx as shown in Figure 8.4. This element has properties including Young’s Modulus, Yb , area moment of inertia, I, and linear density, ρ, or mass per unit length. Assuming the presence of a distributed force per unit length, f (x,t), summing forces in the y-direction (8.15) and summing the moments about the element (8.17) yields Equations (8.16) and (8.18) respectively. The higher order terms (dx2 ) can be neglected in the limit as dx approaches zero. Numerous texts on the basic theory of vibrations are available which detail various aspects of beam modeling. Kelly [8], is one such example. ∂V ∑ Fz = ρ dx u¨ = f (x,t) dx +V −V − ∂ x dx ρ u¨ + ∂V = f (x,t) ∂x (8.15) (8.16) 8.4. THE EULER-BERNOULLI BEAM MODEL 81 Figure 8.4. Differential beam element. dx ∂V dx ∂M dx − M −V − (V + dx) = 0 (8.17) ∂x 2 ∂x 2 Equation (8.17) is valid under the Euler-Bernoulli assumptions of negligible mass moment of inertia and shear deformation. ∑M = M + ∂M =V ∂x (8.18) It can be shown that for small slope ( ∂∂ ux << 1), that M = Yb I ∂ 2u , ∂ x2 (8.19) where Yb is the Young’s modulus of the beam. Substituting Equation (8.19) into Equation (8.18) and then into Equation (8.16) yields, ∂2 ∂ 2u (Y I(x) ) = f (x,t). (8.20) b ∂ x2 ∂ x2 Equation (8.20) neglects any losses which could be modeled as velocity proportional damping. In order to solve this system we will need four boundary conditions. The boundary conditions are given by Equations (8.21) through (8.24). The boundary conditions represent zero displacement at the fixed end, zero slope at the fixed end, zero moment at the free end, and zero shear force at the free end, respectively. ρ u¨ + u(0,t) = 0 (8.21) ∂u (0,t) = 0 ∂x (8.22) ∂ 2u (Lb ,t) = 0 ∂ x2 (8.23) ∂ 3u (Lb ,t) = 0 ∂ x3 (8.24) 82 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT 8.4.1 Free Vibration (Unforced) Solution Using the technique of separation of variables3 (SOV), a general solution with f (x,t) = 0 (Free Vibration) may be written as u(x,t) = U(x) T (t). (8.25) Substitution into Equation (8.20) yields, d2 T d2 d2 U ρ U 2 = − 2 (Yb I 2 T ) (8.26) dt dx dx Separating the spatial and temporal components and assuming that bending stiffness, Yb I, is constant, yields 1 d2 T Yb I d 4 U =− = −ω 2 , (8.27) 2 4 T dt ρ U dx where ω 2 is a constant, and will be shown to be equal to the natural frequencies of the beam in radians per second. That for the left side of Equation (8.27) to be equal to the right hand side requires that both sides are constant stands to reason. The left side is a function of time only, and the right side is purely a function of the spatial variable, x. Hence, for Equation (8.27) to be valid for all values of t and x, requires that each side is equal to a constant. Hence, two equations result: and d2 T + ω 2 T = 0, 2 dt (8.28) d4 U − β 4U = 0, dx4 (8.29) where ρ ω2 . Yb I Equation (8.29) has an exponential solution of the form β4 = U(x) = A eλ x . (8.30) (8.31) Substituting Equation (8.31) into Equation (8.29) yields, (λ 4 − β 4 ) A eλ x = 0, (8.32) (λ 2 + β 2 )(λ 2 − β 2 ) = 0. (8.33) or Solving for the eigenvalues (λ ), yields λ = β , λ = −β , λ = jβ , and λ = − jβ . Hence, the general form of the spatial solution is a sum of all of the possible eigen functions, and may be written U(x) = A1 e j β x + A2 e− j β x + A3 eβ x + A4 e−β x . 3 See (8.34) my presentation on the mathematical model of a tight string: http://web.mst.edu/˜stutts/ PRESENTATIONS/ModalAnalysisofaTightString.pdf. 8.4. THE EULER-BERNOULLI BEAM MODEL 83 The general solution may be rewritten in terms of trigonometric functions as U(x) = B1 cosh β x + B2 sinh β x + B3 cos β x + B4 sin β x. (8.35) Applying the boundary conditions,we obtain U(0) = B1 + B3 = 0, (8.36) U 0 (0) = β (B2 + B4 ) = 0, (8.37) U 00 (Lb ) = β 2 [B1 (cosh β Lb + cos β Lb ) + B2 (sinh β Lb + sin β Lb )] = 0, (8.38) U 000 (Lb ) = β 3 [B1 (sinh β Lb − sin β Lb ) + B2 (cosh β Lb + cos β Lb )] = 0. (8.39) and Combining the above yields the following equations for B1 and B2 , (cosh β Lb + cos β Lb )B1 + (sinh β Lb + sin β Lb )B2 = 0, (8.40) (sinh β Lb − sin β Lb )B1 + (cosh β Lb + cos β Lb )B2 = 0, (8.41) B1 cosh β Lb + cos β Lb sinh β Lb + sin β Lb = 0. sinh β Lb − sin β Lb cosh β Lb + cos β Lb B2 (8.42) and or in matrix form The determinate of the coefficient matrix in Equation (8.42) must vanish for nontrivial B1 and B2 . Hence, we obtain after simplification cosh β Lb cos β Lb + 1 = 0. (8.43) Equation (8.43) must be solved numerically for βn Lb where the subscript, n, denotes the fact that Equation (8.43) has a countable infinity of discrete roots. Table 8.2 gives the first four βn Lb values, where Lb is the length of the beam. Table 8.2. First four eigenvalues multiplied by the beam length. n βn Lb 1 1.875104 2 4.694091 3 7.854757 4 10.995541 Hence, the eigenvalues, βn , may be calculated from the βn Lb values tabulated in Table 8.2 as: βn = (βn Lb ) . Lb (8.44) 84 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT Solving Equation (8.40) for the ratio B2 B1 B2 B1 at a given root, βn , we have = sin βn Lb − sinh βn Lb . cos βn Lb + cosh βn Lb n (8.45) It should be noted that either (8.40) or (8.41) may be solved for this ratio, and that the resulting eigen function merely describes the shape of the individual modes, but not the amplitude of vibration. The situation here is analogous to that in finite dimensional vector spaces studied in linear algebra except that instead of discrete vectors, we have continuous functions. From Equation (8.30), and recognizing the discrete nature of the eigenvalues, βn , the natural frequencies are given by (βn Lb )2 ωn = Lb2 s Yb I . ρ (8.46) where ρ = ρ Ab , where ρ is the mass density of the aluminum beam in kg/m3 , and Ab is the cross sectional area of the beam. The frequency in Hertz may be obtained by dividing ωn by 2π. Thus, (βn Lb )2 fn = 2πLb2 s Yb I . ρ (8.47) In view of Equations (8.36), through (8.39), and Equation (8.45), Equation (8.35) may be written B2 Un (x) = cosh βn x − cos βn x + B1 (sinh βn x − sin βn x) . (8.48) n where the βn values are given by Equation (8.44). Equation (8.48) represents the nth spatial solution, or mode shape. It is important to note at this point that the mode shape, Un (x) is dimensionless! The corresponding temporal solution is given by Tn (t) = D1 cos ωn t + D2 sin ωn t, (8.49) where D1 and D2 must be determined from the initial displacement and velocity of the beam. The total solution is thus the infinite sum of all of the modal solutions, and may be written ∞ u(x,t) = ∑ Un(x)Tn(t). (8.50) n=1 8.4.2 The Effect of a Concentrated Mass on the Beam Natural Frequencies The addition of a concentrated mass located at a point, x = x∗ , on the beam has the effect of lowering some of the natural frequencies. An approximate expression for the perturbed (including the additional mass) natural frequencies may be obtained via the so-called receptance method 8.4. THE EULER-BERNOULLI BEAM MODEL 85 as described in Soedel’s text [9]. For the cantilevered beam, the approximate expression for the perturbed natural frequencies in Hertz is fn ≈ ω q n 2 ∗ , MU (x ) 2π 1 + Mn b (8.51) where Un2 (x∗ ) is the nth mode squared and evaluated at the location of the point-mass, Mb is the total mass of the beam (without the added mass), and M is the point-wise added mass. It can be seen that the attached point-mass will lower frequencies for all modes accept at those where the point of attachment corresponds to a node (U(x∗ ) = 0) which will be unchanged. For M = 0, we recover the original, unperturbed, natural frequency. A more accurate version of equation (8.51) will be given in Section 4.4. 8.4.3 Solution of the Damped, Moment-Forced Cantilevered Beam Now let us consider the forced solution, where the external forcing function is not zero. In this experiment, we are forcing the beam to vibrate through the use of piezoelectric material bonded to it. The PZT forces the beam through distributed moment forcing [9]. As shown in Figure 8.6, the effective moment arm of the PZT with respect to the neutral axis of the beam is given by rPZT = h2b + hPZT 2 . In the present case, PZT elements are bonded symmetrically to both sides of the beam, so the resulting forcing can be doubled if both are driven. The double arrows in Figure 8.6 indicate the positive poling direction as mounted, so for pure bending to occur when both PZT elements are driven, two driving phases must be used with one 180◦ out of phase with respect to the other. We will neglect any external forcing perpendicular to the beam, so in this case f (x,t) = 0 as well, but now there is forcing due to the PZT. Again, we take the sum of the forces in the z-direction and the sum of the moments: Figure 8.5. Differential element of a beam with PZT attached. ∂V ∑ Fz = ρdxu¨ = − ∂ x dx − γ u,˙ (8.52) where γ represents the distributed damping parameter, and has units of N-s/m2 , and ∑M = ∂M dx −V dx = 0. ∂x (8.53) 86 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT Figure 8.6. Schematic of PZT attachment to beam. Combining Equations (8.52) and (8.53) yields, ∂ 2M = 0. (8.54) ∂ x2 The moment, M(x,t), stems from two sources: mechanical stress from bending, and the converse piezoelectric effect from the PZT [10]. Neglecting the axial component of stress in the beam, i.e. assuming pure bending, the moment, M, can be defined as ρ u¨ + γ u˙ + n Z zk+1 M=b∑ k=1 zk zσ1k dz (8.55) where b is the width of the beam and PZT layer, z is the distance from the neutral axis, n is the total number of layers, and σ1k is the total stress in the layer. From Equation (8.3), σ1k can be defined as, σ1k = Y1k ε1k − ek31 E3k (8.56) Where Y1k is Young’s Modulus in the 1-direction, denoted as Yb earlier, ek31 is the converse piezoelectric constant, E3k is the transverse electric field, and ε1k is the strain in the kth layer of material. Of course, not all of the possible layers are piezoelectric, and we will assume a negligible stiffness contribution from the PZT layer since it is so thin, and only located on discrete areas of the beam. The mechanical strain in the kth layer is defined as ε1k = z ∂ 2u ∂ x2 (8.57) where, again, z is the distance from the neutral axis of the beam. Ashton and Whitney authored a nice text on the modeling of composite structures [11]. Equation (8.57) may be substituted into Equation (8.13) with z = rPZT to predict the voltage produced by a sensing PZT element. Substituting Equation (8.57) into (8.56) and then (8.56) into Equation (8.55) yields upon integration, z 3 2 1 k k 2 k+1 kz ∂ u M = b ∑ Y1 − e E z , 3 ∂ x2 2 31 3 zk k=1 n or (8.58) n 2 bY1k 3 1 3 ∂ u M= ∑ (zk+1 − zk ) 2 − b ek31 E3k (z2k+1 − z2k ). ∂x 2 k=1 3 (8.59) 8.4. THE EULER-BERNOULLI BEAM MODEL 87 Figure 8.7. Schematic of lamination geometry (after Soedel [9]). The relevant geometry is shown in Figure 8.7, and where in this case hb + hPZT ) 2 hb = − 2 hb = 2 hb + hPZT = 2 z1 = −( (8.60) z2 (8.61) z3 z4 (8.62) (8.63) The resulting moment may be expressed as a sum mechanical and electrical moments M(x,t) = M m (x,t) + M e (x,t). (8.64) If the stiffness contribution of the PZT is neglected, mechanical and electrical moments may be shown to be 1 3 ∂ 2u b h Yb 12 b ∂ x2 (8.65) M e (x,t) = −κ p brPZT d31 YPZT Φ(x,t), (8.66) M m (x,t) = where we recognize the quantity bh3 b/12 to be the area moment of inertia for a beam of rectangular cross section, usually denoted as I, and Yb is the Youngs modulus of the beam in the 1 or x-direction. In addition, the symbol, κ p , denotes the number of driven PZT elements. If one PZT element is driven, and the opposite used as a sensor, then κ p = 1. The spatially distributed voltage for a single PZT element beginning at x = x1 , and ending at x = x2 may be expressed in terms of Heaviside step functions as Φ(x,t) = Φ0 [H(x − x1 ) − H(x − x2 )] sin ωt (8.67) where Φ0 is the amplitude of the applied voltage. The resulting voltage distribution is shown in Figure 8.8. 88 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT Figure 8.8. Voltage distribution. Substituting the moment, M, into Equation (8.54) yields, ∂ 4u = κ p b rPZT d31 YPZT Φ0 δ 0 (x − x1 ) − δ 0 (x − x2 ) sin ωt, (8.68) 4 ∂x where the symbol, δ 0 , represents the first derivative with respect to x of the Dirac delta function, which is in turn, the derivative of the Heaviside step function.4 ρ u¨ + γ u˙ +Yb I 8.4.4 Modal Expansion: Orthogonality of the Natural Modes of Vibration The concept of the orthogonality of modes represents a generalization of Fourier’s series, and, having been developed by Daniel Bernoulli, actually predates Fourier’s contribution [9]. In essence, the concept is analogous to the concept of orthogonal vectors in finite dimensional vector space wherein the dot (or inner) product of orthogonal vectors is zero. In function space, the inner product between two functions is defined as I = Z f g dD, (8.69) D where D denotes the domain over which the inner product of the functions, f , and g is defined. In general, f , and g may be functions of any number of variables. The functions f , and g are said to be orthogonal over the domain, D, if and only if Z f g dD = 0. (8.70) D The natural modes of a cantilevered beam are orthogonal, so we have Z Lb 0 for m , n Un (x)Um (x) dx = Nn for m = n 0 (8.71) where Um (x), and Un (x) are any two different modes of the beam corresponding to the mth , and nth eigenvalues respectively, and Nn is called the normalization factor. The power of the orthogonality of modes will be illustrated in the following solution of Equation (8.68). 4 Please refer to my notes on Laplace transformation which may be found at: http://web.mst.edu/ ˜stutts/SupplementalNotes/LaplaceT.PDF. 8.4. THE EULER-BERNOULLI BEAM MODEL 89 The more accurate version of Equation (8.51) to account for the change in frequency due to an attached point mass, given here now that we know about the normalization factor, is fn = ω q n 2 ∗ . MUn (x ) 2π 1 + ρN n (8.72) As in the case of the free vibration solution, a separable solution will be assumed, but in this case, we already know the spatial solution. We will use the natural modes found in the free vibration solution. This is the so-called method of modal expansion. The solution may be written ∞ u(x,t) = ∑ Un(x) ηn(t) (8.73) n=1 and is similar in form to that of the free-vibration solution given in Equation (8.50), except that the spatial solution, Un (x), is known, and the temporal solution, also known as the modal participation factor, ηn (t), must be determined. Substitution of Equation (8.73) into Equation (8.68) yields 0 ∂ 4Un (x) 0 ¨ ˙ η(t) = κ b r d Y Φ δ (x − x ) − δ (x − x ) sin ωt n (x) + γ η(t)U n (x) +Yb I p PZT PZT 31 0 1 2 ∑ ρ η(t)U ∂ x4 n=1 (8.74) Recognizing from Equation (8.35) that ∞ ∂ 4Un (x) = βn4Un (x), ∂ x4 (8.75) we have ∞ ∑ ¨ + γ η(t) ˙ +Yb Iβn4 η(t) Un (x) = −κ p b rPZT d31 YPZT Φ0 δ 0 (x − x1 ) − δ 0 (x − x2 ) sin ωt. ρ η(t) n=1 (8.76) Equation (8.76) may be rewritten in canonical form as ∞ ∑ η¨ n + 2ζn ωn η˙ n + ωn2 ηn Un (x) = F0 δ 0 (x − x1 ) − δ 0 (x − x2 ) sin ωt, (8.77) n=1 where, F0 = κ p b rPZT d31 YPZT Φ0 , ρ and 2ζn ωn = γ . ρ (8.78) (8.79) or ζn = γLb2 γ = √ 2ρωn 2 (βn Lb )2 Yb Iρ (8.80) 90 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT Equation (8.81) would be impossible to solve in its present form because of the infinite series on the left-hand side, so lets apply orthogonality and Equation (8.71) to eliminate the summation! Multiplying both sides of Equation (8.81) by Um (x), and integrating over the length of the beam, we have Lb ∞ Z 2 ¨ ˙ η + 2ζ ω η + ω η ∑ n n n n n n 0 n=1 Un (x)Um (x)dx = F0 Z Lb δ 0 (x − x1 ) − δ 0 (x − x2 ) Um (x)dx sin ωt. 0 (8.81) However, all of the terms in the summation vanish according to (8.71) except for when m = n leaving: η¨ n + 2ζn ωn η˙ n + ωn2 ηn = Fn (t), (8.82) where Fn (t) = κ p b rPZT d31YPZT Φ0 [Un0 (x2 ) −Un0 (x1 )] sin ωt. ρNn (8.83) Note that the differentiation of the Dirac delta functions has been transferred to the modes, Un , by integration by parts. Equation (8.82) is an ordinary differential equation with time as an independent variable, and may be easily solved given the initial conditions of the beam. However, we are most often only interested in the steady-state solution due to the harmonic input given by Equation (8.83) which may be shown to be ηn (t) = Λn sin(ωn q 1 − ζn2t − φn ), (8.84) where, Fn∗ p Λn = , ωn2 (1 − rn2 )2 + 4ζn2 rn2 (8.85) o n arctan 2ζn ω2n for rn ≤ 1 1−r nn o φn = , π + arctan 2ζn ω2n for r > 1 n 1−r (8.86) n Fn∗ = κ p b rPZT d31YPZT V0 [Un0 (x2 ) −Un0 (x1 )] , ρNn (8.87) and rn = ωωn . Note also that due the fact that d31 will carry a negative sign, modal participation factor, ηn (t), will lag the forcing, Fn (t), by 180◦ . 8.4.5 Application of the Initial Conditions to Solve the Initially Forced Beam The beam may be given an initial shape due to deflection, and even an initial velocity. In this case, the beam is said to be initially forced. For the present, we will assume the beam has been initially deformed due to a specified end deflection, u(Lb , 0) = uL at x = Lb , but is otherwise stationary 8.5. EXPERIMENTAL DETERMINATION OF THE DIMENSIONLESS VISCOUS DAMPING PARAMETER9 until released from its initial deflection – i.e. u(x, ˙ 0) = 0. The static shape assumed by the beam according to the Euler-Bernoulli model is given by uL x3 2 u(x, 0) = 2 3x − (8.88) Lb 2Lb Substitution of Equation (8.73) into Equation (8.88) yields uL x3 2 u(x, 0) = ∑ Un (x)ηn (0) = 2 3x − Lb 2Lb n=1 ∞ (8.89) We now need to identify ηn (0) in terms of u(x, 0). This is easily accomplished by invoking the orthogonality of the modes of vibration: as demonstrated in Section 8.4.4, first, we multiply Equation (8.89) by Um (x), where m , n in general, then we integrate over the domain [0, Lb ]. The orthogonality property collapses the infinite series down to a single non-zero term for the case when m = n, yielding R Lb 2 x 3 uL 0 3x − Lb Un (x)d x ηn (0) = (8.90) R 2Lb2 0Lb Un2 d x Application of the velocity initial condition can be shown to reveal that η˙ n (0) = 0. From Equation (8.82) with Fn (t) = 0, and assuming underdamped (0 < ζ < 1) behavior, we find that q ηn (0) −ζn ωnt 2 ηn (t) = p e cos ωn 1 − ζn t − ψn 1 − ζn2 (8.91) where ωn is given by Equation (8.46), and ψn = arctan ζn ! p 1 − ζn2 (8.92) The response of the initially forced beam due to tip deflection uL is given by Equation (8.73) but with ηn (t) defined by Equation (8.91). 8.5 Experimental Determination of the Dimensionless Viscous Damping Parameter Equations (8.84) through (8.86) depend on the determination of the dimensionless damping constant, ζn . Fortunately, there are several ways to make this determination experimentally, and one of the most straightforward is the logarithmic decrement method. In this case, damping will be determined by initially deflecting the cantilevered beam about 1cm, and releasing it to allow it to freely vibrate. Hence, we need only examine the homogeneous (unforced) case of Equation (8.82) η¨ n + 2 ζn ωn η˙ n + ωn2 ηn = 0. (8.93) 92 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT In this experiment, the beam will vibrate primarily in its first mode, so the particular damping parameter we will measure is ζ1 . A typical free response of an under damped (ζ < 1) second-order system is shown in Figure 8.9. In this hypothetical single degree of freedom system, whose total free response is given by x(t) = e−0.2t cos(9.998t), we note that there are seven peaks (labeled on the graph) before t = 4 seconds. This corresponds to seven complete cycles or periods of motion. As you can see from the figure, each successive peak is lower than the previous one. This decay in amplitude is governed by the indicated exponential envelope, e−0.2t . From our knowledge p of linear systems, we know that Figure 8.9 represents a system where ζ ωn = 0.2, and ωd = ωn 1 − ζ 2 = 9.998, where ωd is the so-called damped natural frequency. Hence, our hypothetical system has an undamped natural frequency of ωn = 10 radians per second, and a dimensionless damping parameter of ζ = 0.02, or 2% of critical (ζ = 1) damping. In general, the amplitude of a second-order system at the kth peak at time tk is given by x(tk ) = xk = x1 e−ζ ωtk , (8.94) where x1 is the initial amplitude. Thus, at the first peak, tk = t1 = 0, and x(0) = x1 . Similarly, at the third peak, x(t3 ) = x3 = x1 e−ζ ωt3 = x1 e−ζ ω2Td , (8.95) since there are two damped natural periods (Td = 2π/ωd ) between the first and third peaks. Hence, the ratio of the first peak to the kth peak is given by Figure 8.9. Typical free response of an under damped system. x1 = eζ ω(k−1)Td , xk but (8.96) 8.6. PRODUCING VOLTAGE FROM STRAIN: THE DIRECT PIEZOELECTRIC EFFECT 93 Td = 2π 2π p . = ωd ωn 1 − ζ 2 (8.97) Hence, 2(k−1)π √ ζ x1 = eζ ω(k−1)Td = e 1−ζ 2 . xk Taking the natural log of Equation (8.98), we obtain x1 2(k − 1)π ζ = δk , ln = p xk 1−ζ2 (8.98) (8.99) where δk is called the logarithmic decrement over k − 1 periods. It should be noted that the log ratio xi ln xi+1 is constant. In most vibrations texts, the logarithmic decrement is defined between any two adjacent peaks. We use the more general formula given in Equation (8.100) for this application, because the difference in amplitude between any two peaks is usually too small to accurately measure. Hence, we measure over k peaks. Solving equation (8.99) for ζ yields δk ζ= q . δk2 + 4(k − 1)2 π 2 (8.100) Thus, by measuring the amplitude of any two sequential peaks, and computing the logarithmic decrement, we may use Equation (8.100) to obtain the system dimensionless damping parameter. Since the damped period, Td , and ζ are known, the undamped natural frequency may be found to be ωd 2π ωn = p = p . 1 − ζ 2 Td 1 − ζ 2 (8.101) For most lightly damped systems, ωn ≈ ωd . 8.6 (8.102) Producing Voltage From Strain: The Direct Piezoelectric Effect From Equation (8.14), repeated below for convenience, D d −hPZT YPZT 31 Φd = ε (x2 − x1 ) ε3 Z x2 x1 ε1 d x (8.103) where, from Equation (8.57) under pure bending, the strain in the x-direction at the middle of the PZT layer is given by ∂ 2u ε1 = rPZT 2 (8.104) ∂x 94 CHAPTER 8. PIEZOELECTRIC BEAM EXPERIMENT Hence, substitution of (8.104) into (8.103) yields D d −hPZT YPZT 31 Φd = rPZT ε (x2 − x1 ) ε3 ∂u ∂u (x2 ,t) − (x1 ,t) ∂x ∂x (8.105) Substitution of Equation (8.73) into Equation (8.105) yields ∞ D d d Un dUn −hPZT YPZT 31 Φd = rPZT ∑ d x (x2) − d x (x1) ηn(t) (x2 − x1 ) ε3ε n=1 (8.106) If the beam is vibrating primarily in its first mode, we can neglect the higher modes, and Equation (8.106) becomes D d hPZT YPZT 31 Φd = rPZT (x2 − x1 ) ε3ε d U1 dU1 (x1 ) − (x2 ) η1 (t) dx dx (8.107) Substitution of Equation (8.84) into Equation (8.106) or with n = 1 into Equation (8.107) yields the steady-state voltage produced by the sensing PZT element due to forcing by the driven element. The amplitude of the voltage produced by steady-state vibration in the first mode may be calculated from Equations (8.85) and (8.87), with n = κ = 1, as D d dU dU h Y PZT 31 1 1 PZT Φd = rPZT (x ) − (x ) Λ 1 2 1 (x2 − x1 ) ε3ε dx dx 8.6.1 (8.108) Voltage Produced By Initial Beam Deflection The voltage produced by the vibrating beam due to an initial tip deflection, uL , may be determined using Equation (8.106) with ηn (t) defined by Equation (8.91). Bibliography [1] A. K UMADA U. S. Patent No. 4,868,446, Sep. 19, 1989. Piezoelectric revolving resonator and ultrasonic motor. 10 Claims, 17 Drawing Sheets. [2] A. K UMADA , T. I OCHI AND M. O KADA U. S. Patent No. 5,008,581, Apr. 16, 1991. Piezoelectric revolving resonator and single-phase ultrasonic motor. 6 Claims, 6 Drawing Sheets. [3] S. U EHA AND Y. T OMIKAWA 1993 Ultrasonic Motors—Theory and Applications. Clarendon Press, Oxford. 297. [4] T. S ASHIDA U. S. Patent No. 4,562,374, Dec. 31, 1985. Motor device utilizing ultrasonic oscillation. 29 Claims, 22 Drawings. [5] T. S ASHIDA AND T. K ENJO 1993 An Introduction to Ultrasonic Motors. Clarendon Press, Oxford. 242. [6] B. JAFFE , W. C OOK AND H. JAFFE 1971 Piezoelectric Ceramics. Academic Press Limited, New York. 317. [7] TAKURO I KEDA 1990 Acoustic Fields and Waves in Solids. Oxford University Press, New York. 263. [8] S. G RAHAM K ELLY 2000 Fundamentals of Mechanical Vibrations. McGraw-Hill. 629. [9] W. S OEDEL 1993 Vibrations of Shells and Plates. Second edition. Marcel Dekker, New York. 470. [10] H. S. T ZOU 1993 Piezoelectric Shells—Distributed Sensing and Control of Continua. Vol. 19, Solid Mechanics and Its Applications. Kluwer, Boston. 470. [11] J. E. A SHTON AND J. M. W HITNEY 1970 Theory of Laminated Plates. Technomic, Stamford, CT. 153. [12] E RNEST O. D OEBELIN 1990 Measurement Systems Applications and Design. Fourth Edition McGraw-Hill. 960. 95 96 BIBLIOGRAPHY Figure 8.10. Schematic of experimental layout. 8.7 EXPERIMENTAL SETUP A schematic of the experimental layout is shown in Figure 8.10, and a photograph of the connection box on top of the amplifier is shown in Figure 8.12. The equipment consists of the following: 1. Two-channel signal generator – The signal generator is used to generate the sinusoidal drive signals to the power amplifier. The GTAs are familiar with the signal generator’s operation and will explain the relevant controls on the front panel. The two channels may be adjusted independently. the adjustments include: (a) Frequency (b) Amplitude – the minimum amplitude is 0.05 volts, and the co-axial cables between the signal generator and the power amplifier provide a voltage limit to about 200 volts peak to peak. (c) Phase – the phase difference between the two channels may be adjusted from zero to 180 degrees. 2. Power amplifier – The power amplifier provides voltage amplification by a factor of 100. 3. Accelerometer – The accelerometer provides a voltage proportional to acceleration. 4. Accelerometer integration unit – Integrates the accelerometer signal twice to yield a signal proportional to position – see Figure 8.13. The output from the accelerometer integrator/amplifier must be connected to the A13 input on the interconnection box! 5. Data acquisition lap computer – Provides two National Instruments LabviewTM VIs (virtual instruments) to acquire the damping factor, and the steady-state amplitude. The data may be saved as tab-delimited text to a file and later imported into Excel or another spreadsheet. 8.7. EXPERIMENTAL SETUP 97 6. Aluminum beam with two PZT elements bonded to it – The beam should be securely clamped to a solid surface before beginning the experiment. All connections to the PZT elements are independent, and may be switched around for different driving and sensing configurations. Care should be taken not to deflect the end of the beam any more than 0.5 cm to avoid damaging the PZT elements. 7. Interconnection box – The various inputs and outputs to and from the beam may be switched using the interconnection box. The interconnection box schematic is shown in Figure 8.11, and a photograph of the connection box and the amplifier is shown in Figure 8.12. 8. If the NI-Labview-based data acquisition system malfunctions, a Tektronics digital storage oscilloscope is available to capture the PZT voltage waveform or the output from the accelerometer ICP power and signal integration box. The scope should be set in single trigger mode with sample rate slow enough to capture a couple of seconds of data without filling the scope buffer. The captured data can then be transferred from the scope to a USB flash drive. Figure 8.11. Interconnection box diagram. 98 BIBLIOGRAPHY 8.8 Data Acquisition Procedure Before beginning any of the following experiments, please do the following: 1. Familiarize yourself with the equipment, and read ALL of the procedures pertaining to the experimental measurements you are about to make. 2. Measure the dimensions of the aluminum beam. Its length should be measured from the point where it is clamped to the support. Measure the distance from the base of the beam to the location of the accelerometer (xacc ). This will also be the location of the attached masses (x∗ ). If you are planning on doing any measurements pertaining to either the direct or indirect piezoelectric effects – i.e. measuring the voltage produces by one of the PZT elements due to tip deflection or drive voltage amplitude applied to the opposite PZT element, or the steady-state displacement produced by a given drive voltage amplitude – make sure to measure the x1 , and x2 locations of the PZT boundaries relative to the clamped end of the beam. A data worksheet is provided for this purpose. 8.8.1 Determining ζ1 and f1 In this experiment, you will determine both the dimensionless damping constant and the damped natural frequency for the first mode of the cantilevered beam. Because the system is relatively lightly damped, you should find that there is very little difference between the damped and undamped natural frequency. 1. Familiarize yourself with the equipment, and read ALL of the following instructions. 2. Measure the dimensions of the aluminum beam. Its length should be measured from the point where it is clamped to the support. Measure the distance from the base of the beam to the location of the accelerometer (xacc ). This will also be the location of the attached masses (x∗ ). 3. Make sure that the signal generator output and the power amplifier are off, and start the damping measurement VI on the computer. 4. Make sure that the integrator circuit is turned on, and the accelerometer is affixed to the end of the beam as indicated in Figure 8.10, then deflect the end of the beam approximately 0.5 cm, and release it. Press the acquire data button on the VI. 5. Position the yellow vertical line on a peak, and then the red vertical line on a subsequent peak (about 10 peaks away is best). Set the peak number including the first peak, and record the calculated value of ζn . 6. Determine the damped natural period, Td , by averaging over 10 periods – i.e. the time between 11 peaks. 7. Using the sticky wax, attach a 2 gram mass to the beam just opposite to the accelerometer. Deflect the beam tip (0.5 cm), and repeat the damping measurement and period measurement. 8. Repeat the last step for 5 and 10 gram masses. Remove the attached mass. 9. Terminate the damping VI. 8.8. DATA ACQUISITION PROCEDURE 8.8.2 99 Steady-State Amplitude and Voltage Measurements 1. Execute the steady-state measurement VI. 2. With the signal generator and power amplifier outputs off, measure the indicated peak amplitude of the accelerometer. This value represents any baseline noise or system bias. Take care not to vibrate the beam at all during this measurement – make sure nobody is leaning on or touching the table on which the experiment is set up. 3. Use the value of fn determined from Section 8.8.1 with no extra masses attached to set the initial frequency of the signal generator. 4. Set the amplitude of the signal generator to 0.5 Volts. Also make sure that the connection box is configured correctly to drive one PZT element, and measure the voltage produced in the other. Do not turn on the amplifier yet! After you’ve set the drive voltage, be sure to turn the channel output off before turning the amplifier on. 5. Turn on the power amplifier and the signal generator outputs, and record the indicated amplitude after allowing the beam to reach steady-state. Select Hold Data and record the maximum amplitude indicated by the accelerometer. Record the peak value of the amplified voltage as well. Use the forms provided. 6. Turn off all of the equipment. Figure 8.12. Picture of connection box and amplifier. 100 8.8.3 BIBLIOGRAPHY Direct Effect Transient Voltage Measurement In this experiment, you will measure the transient voltage produced by initially deflecting the end of the beam by 2.5 mm and releasing it. To do this, you must deflect the beam with the PZT elements until one edge of its end is even with the gage mark in the middle of the adjacent gage beam before releasing it. Deflecting the beam too much may damage the PZT elements. 1. Turn on the steady-state Figure 8.13. Accelerometer signal integration unit. 8.9 Experimental Analysis The following procedures will be used to determine the full-scale and proportional errors found in the first set of amplitude versus voltage measurements done at low voltage (0 to 0.5 volts). This analysis is by no means a complete characterization of the experimental errors or the quantization errors inherent in the analog to digital conversion taking place in the data acquisition board. These errors are certainly important, but in the interest of time, we will only investigate two error metrics – full-scale and proportional error [12]. A least squares line through a hypothetical set of data is shown in Figure 8.14, where Uacc is the displacement of the accelerometer in microns (1 × 10−6 meters), Um = Uacc (Φm ) is the value of 8.9. EXPERIMENTAL ANALYSIS 101 Figure 8.14. Least squares fit of data showing error bars. accelerometer displacement which deviates most from the least squares line, Φ is the voltage amplitude, and Φm is the voltage at Um . Also shown are the full scale error lines, and the proportional error lines which will be defined below. Definition 8.3 Full scale error is defined as the magnitude of maximum deviation from the least squares line, em , and is assumed to be of plus or minus constant value over the full scale. Hence, the upper and lower dotted lines parallel to the least squares line. The full scale error bounding curves may be computed as E f s = aΦ + b ± em , (8.109) where a, and b are the least squares coefficients to be defined below. Definition 8.4 Proportional Error is defined as the maximum percentage error of the measurement at the point of maximum error, and is assumed to apply to all other measurements. All other errors are assumed to occur in the same proportion to the measurement as that at the point of maximum error. Hence, the upper and lower proportional error bounding lines diverge from the least squares line, and are equal to the full scale error at the point of maximum error. The proportional error bounding line equations are given by E p = (a ± α)Φ + b, where α= em . Φm (8.110) (8.111) Usually, both error metrics are reported, and one can assume the worst of either as a lower bound of the measurement system accuracy. 102 BIBLIOGRAPHY Definition 8.5 Sensitivity, a, is defined as the slope of the least squares line. The least squares coefficient, a is a measure of the system measurement sensitivity. Definition 8.6 Bias, b, is defined as the intercept of the least squares line. The least squares parameter, b represents the system measurement bias. Perform the following analyses: 1. Plot the accelerometer amplitude versus the drive (amplified) voltage amplitude, and perform a least squares fit through the data to determine measurement sensitivity and bias for the low voltage range measurements (0 to 0.5 volts). 2. Determine em , Φm , and compute α, then plot the full-scale and proportional error bounding lines on the same plot as the least squares line and original data. 3. Plot the high voltage data indicating the direction (increasing or decreasing drive voltage). 4. Use Equation (8.51) to determine the theoretical first mode natural frequencies for each mass case – note that the accelerometer weighs 0.78 gram. 5. Compute the % error in the between the theoretical and measured natural frequencies for each mass case. 6. Compute the difference between the measured first mode natural frquencies of the beam with only the accelerometer in place as compared to that with the 10 gram mass in place. 8.10 Discussion Questions 1. Compare the measured natural frequency with the theoretical value, f1 . Why do you think these values are different if they are? 2. Using any means at your disposal, (Maple, Mathematica, MATLAB, etc.), show (at least numerically) that equation (8.71) is valid for modes one and two – i.e. when n = 1, and m = 2, the inner product is effectively (numerically within round off error) zero. 3. Show by integration by parts and the filtering property that Z c f (x)δ 0 (x − b)dx = − f 0 (b) for a < b < c, (8.112) a where δ 0 (x − b) is the first derivative of the Dirac delta function evaluated at x − b. 4. How might you use Equation (8.14) in a practical application? 5. How was the damping constant, ζ , effected by the attached masses? Why do you suppose this is? Hint: You might examine Equation (8.80). 8.11. PIEZOELECTRIC BEAM EXPERIMENT DATA 8.11 Piezoelectric Beam Experiment Data 8.11.1 Beam data Yb = Ys := 69 × 109 Pa (Young’s modulus of the aluminum beam substrate) ρb = 2, 700 kg/m3 (Density of the aluminum beam) hb = please measure (Thickness of the beam in the bending (z) direction) b = please measure (Width of the beam) Ab = hb b (nominal cross-sectional area of beam) Macc = 0.78 grams (mass of the attached accelerometer. 8.11.2 Piezoceramic data hPZT = 0.00053 m (0.53 mm) rPZT = hb +h2 PZT m YPZT = 71 × 109 Pa (Young’s modulus) ρPZT = 7, 700 kg/m3 ε3 = 1.6 × 10− 8 Farad/m d31 = −190 × 10−12 m/Volt or Coulombs/Newton 8.11.3 Measured data Lb = (measured from clamp) x1 = (measured from clamp) x2 = (measured from clamp) hb = b= x∗ = xacc (measured from clamp) f1 = (theoretical) f1 = (measured) 103 104 BIBLIOGRAPHY f1 2 gram = (measured) (Equation (8.51)) f1 5 gram = (measured) (Equation (8.51)) f1 10 gram = (measured) (Equation (8.51)) Table 8.3. Low range data. ΦPA (volts) 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.0 Φ (volts) Uacc(µm) 8.11. PIEZOELECTRIC BEAM EXPERIMENT DATA 105 Table 8.4. High range data. ΦPA (volts) 0.0 0.05 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.05 0.0 Φ (volts) Uacc(µm) Chapter 9 Pump Experiment 9.1 Introduction OBJECTIVES 1. To determine the typical operating characteristics of an external gear pump and a rotary screw pump under various load and speed conditions. 2. To understand instrument accuracy and design error analysis. In general, pumps may be divided into two major categories: 1. Dynamic, in which energy is continuously added to the fluid by means of fast moving blades or vanes. 2. Positive displacement, in which energy is periodically added by application of force to one or more fluid volumes, resulting in an increase in pressure. Dynamic pumps usually require some type of priming system to begin the pump’s operation. An example of a dynamic pump is a centrifugal type, shown below, which depends on centrifugal force for the pumping action. Fluid enters the pump through this eye of the impeller and travels outward between the vanes of the impeller to its edge. At this point fluid enters the casing of the pump and is discharged through the exit port. An increase in fluid pressure is obtained by the rotary action of the impeller. In the same way that fluid pressure increases toward the edge of a rotating tank due to the radial acceleration of the fluid (as studied in your undergraduate fluid mechanics course), the fluid pressure rises in the casing. Centrifugal pumps are typically used where high volumetric flow rates at low pressure are needed. Positive displacement pumps are in general self-priming and may require some type of mechanical bypass valve to protect the system from excessive pressures. This class of pumps is further divided into reciprocating or piston type and rotary type. Both the gear pump and the rotary screw 106 9.1. INTRODUCTION 107 Figure 9.1. Cross-sectional schematic of a centrifugal pump. pump used in this experiment falls into the class of positive-displacement rotary pumps. Rotary pumps are typically used for high pressure, low flow rate, applications. Excessive leakage is in many cases a problem since no satisfactory method of sealing the moving surfaces to compensate for wear has been developed. Consequently, rotary pumps are best suited for pumping oils and other liquids having lubricating value and sufficient viscosity to prevent excessive leakage. The gear pump you will test in this experiment is an external gear type. The pump consists of two spur gears inside an enclosure and in mesh with each other, as shown in Figure 9.2. The liquid Figure 9.2. Cross-sectional schematic of a gear pump. 108 CHAPTER 9. PUMP EXPERIMENT is carried in the pockets formed between the gear teeth and the housing as the two gears rotate. The volume flow per revolution is approximately equal to the total volume of the liquid pockets. The place where the two gears mesh serves as a seal in addition to the toothcasing interface, to prevent back flow from the high pressure discharge to the low pressure intake. The rotary screw or progressive cavity pump is a positive displacement pump designed to handle solid particles in suspension such as abrasive slurries, water base paints, dirty oils, etc. The operating principle can be understood in reference to Figure 9.3. The rotor has an eccentric axial pitch while the elastic stator has an axial pitch one half that of the rotor. The rotor and stator form a constant sealing line creating a series of sealed cavities. When the shaft rotates, these sealed cavities progress axially down the stator to the discharge end. Meanwhile the opened cavities at the suction end pull in new material. Both the gear pump and the rotary screw pump suffer from pump leakage at high-back pressures due to imperfect sealing at the sealing interfaces. Figure 9.3. Schematic of a rotary screw pump. 9.2 Theory Consider the energy transfer of a pump by applying the steady state form of the energy equation: 2 2 V V ˙ Q + W˙ x = m˙ out h + +gz − m˙ in h + +gz (9.1) 2 2 out in By definition, the enthalpy is given by P ρ (9.2) m˙ in = m˙ out (9.3) h = u+ This implies that on a per unit mass basis, P V2 P V2 q + ws = u + + +gz − u+ + +gz ρ 2 ρ 2 out in (9.4) 9.2. THEORY 109 Since the flow is considered incompressible, and assuming inlet and outlet areas are approximately equal, Equation (9.3) implies that the inlet and outlet velocities are also approximately equal. In addition, negligible change in potential energy occurs across the pump, resulting in P P Ws uout − uin − q + (9.5) − = ρ g out ρ g in g g The last group of terms represent frictional losses and are grouped into a frictional head loss term, h f . Similarly, expressing the shaft work put into the pump as a shaft work equivalent head term, hs , yields 1 (Pout − Pin ) = hs − h f (9.6) H= ρg H is the “net” or “pump” head and is the primary output parameter for a pump. Note that this is the difference between the shaft work and the frictional head. The energy ultimately delivered to the fluid results from the difference between the input energy and the energy lost to friction, both mechanical and viscous, and losses due to pump leakage. The power delivered to the fluid is by tradition termed the water horsepower, Pw = ρ g Q H (9.7) where Q is the volumetric flow rate of the fluid. In contrast, the power required to drive the pump is termed the shaft or brake horsepower, Ps = 2π N T (9.8) where T is the measured shaft torque and N is the shaft speed in revolutions per minute. The overall efficiency of the pump is defined as the ratio between the power delivered to the fluid and the input power, Pw ρ g Q H η= (9.9) = Ps 2π N T The overall efficiency given by Equation (9.9) is a function of both mechanical and fluid mechanical losses. Contributing factors to the overall inefficiency are viscous frictional effects and mechanical frictional effects in the bearings, packing and other contact points in the pump. For both the gear pump and the rotary screw pump we will also consider the volumetric efficiency, which compares the actual pumped fluid flow rate to the loss of flow, QL , due to leakage of the fluid around the gear and casing. The volumetric efficiency is defined as, ηv = Q Q + QL (9.10) The leakage flow rate is defined as the difference between the flow rate at a given head and speed and the flow rate at that speed if no head were developed, Q0 (i.e. zero back pressure). You will be collecting data on the head developed by the pump for various back pressures for a fixed speed. 110 CHAPTER 9. PUMP EXPERIMENT Figure 9.4. Pump head versus flowrate. Hence, to find Q0 , a plot of flow rate versus head need simply be extrapolated to zero head as illustrated in Figure 9.4. For lower head valves, the data will be fairly linear. Thus a method of linear least squares can be used to accurately extrapolate the plot to zero head. The general nature of the approach will be discussed here and applied in the discussion section. If we assume the flow rate to be linearly proportional to the head (this should be true for most of your data at lower head values), then we can write: Q = k H + Q0 (9.11) where k is some constant. The method of linear least squares is based on finding the equation of a line (i.e. k and Q0 ) which best fits the data by minimizing the sum of the square of the residual. The ith residual is given by ri = k Hi + (Q0 − Qi ) (9.12) where Hi and Qi are the ith measured values of head and flow rate. Hence, we wish to minimize the sum of the squared residuals given by n n n n i=1 i=1 i=1 i=1 S = ∑ ri2 = k2 ∑ Hi2 + 2k ∑ Hi (Q0 − Qi ) + ∑ (Q0 − Qi )2 (9.13) where n is the number of data points. Because S is a convex function1 , its extremum in k and Q0 is a minimum. Hence, taking the derivative of S with respect to each yields two equations in k and Q0 : and or n n ∂S = 2k ∑ Hi + 2nQ0 − 2 ∑ Qi = 0 ∂ Q0 i=1 i=1 (9.14) n n n ∂S = 2k ∑ Hi2 + 2Q0 ∑ Hi − 2 ∑ Hi Qi = 0 ∂k i=1 i=1 i=1 (9.15) n n k ∑ Hi + nQ0 = ∑ Qi i=1 1 See: i=1 http://en.wikipedia.org/wiki/Convex_function. (9.16) 9.3. EXPERIMENTAL SETUP and 111 n n n k ∑ Hi2 + Q0 ∑ Hi = ∑ Hi Qi i=1 i=1 (9.17) i=1 It is important to note that Equations (9.16) and (9.17) are valid only at low head where head as a function of flow rate, H(Q), appears linear. Obviously, the more data points, (Qi , Hi ), obtained at low head, the more accurate the resulting linear extrapolation will become. Recasting the above into matrix form yields: n n k ∑i=1 Hi Qi ∑i=1 Hi2 ∑ni=1 Hi = (9.18) Q0 n ∑ni=1 Qi ∑ni=1 Hi The unknown flow rate at zero head, Q0 , and constant, k, may be determined using Cramer’s rule, yielding:2 n ∑ni=1 Hi Qi − (∑ni=1 Hi ) (∑ni=1 Qi ) k= (9.19) n ∑ni=1 Hi2 − (∑ni=1 Hi )2 and Q0 = 9.3 ∑ni=1 Hi2 (∑ni=1 Qi ) − (∑ni=1 Hi ) (∑ni=1 Hi Qi ) n ∑ni=1 Hi2 − (∑ni=1 Hi )2 (9.20) Experimental Setup A schematic of the apparatus used is given in Figure 9.5. The gear pump you will test is driven by a variable speed motor, through a coupling that gives a read out of torque and rotational speed. The torque is sensed by measuring torsional strain, using electrical resistance strain gages, and transmitting the signal through slip rings. The liquid flowing through the pump is water. There are pressure gages to measure static pressure at the inlet and outlet of the pump. There is a valve on the outlet side of the pump to control the discharge pressure. Turning the valve handle clockwise closes the valve and increases the discharge pressure. The flow rate is measured by passing the water into a tank mounted on a scale. A stopwatch will be used to determine the time for the scale reading to change by a noted amount (50 lbm is suggested.) 9.4 Data Acquisition Procedure A. Gear Pump (1) Familiarize yourself with the pressure gage scales, balance and stopwatch operation. (2) Check and confirm that the Outlet Pressure control valve is fully open (counterclockwise). Switch the variable speed motor ON. 2 See Dr. Stutts’ Linear Algebra Primer: http://web.mst.edu/˜stutts/SupplementalNotes/ LinearAlgebraPrimer4.pdf. 112 CHAPTER 9. PUMP EXPERIMENT Figure 9.5. Schematic of pump experiment. (3) Adjust the speed of the motor to 900 rpm. (4) Adjust the outlet pressure to 10 psig. (5) Set the initial balance on the scale such that the scale is balanced with the drain valve open. (6) Record the motor speed, inlet and outlet pressures, input torque and the time taken to collect a given amount of water ( suggest 50 lbm). (7) Increase the outlet pressure by closing the control valve and repeat step (6). Continue until values have been recorded for outlet pressures shown on the data sheets.3 (8) Increase the pump speed and repeat steps (3) through (7) for 1300 rpm. B. Rotary Screw Pump (1) Remove the gear-pump and replace it with the rotary screw pump. (Be sure that the quick disconnects are fully locked!). (2) Run the pump at 1500 RPM for one minute to insure stable operation. (3) We will conduct this test similar to the gear pump tests by fixing the speed and varying the outlet pressure. However, you will find that this pump cannot withstand as high back pressures as the gear pump before cavitation begins in the pump. Set the initial speed to 900 RPM. (4) Set the balance on the scale such that the scale is balanced with the drain valve open. (5) Record the motor speed, inlet and outlet pressures, input torque and the time taken to collect a given amount of water (use 50 lbs). 3 Caution: when the outlet pressure is increased, check to be sure that the motor speed is still at the set point. 9.5. UNCERTAINTY ANALYSIS 113 (6) Increase the outlet pressure by closing the control valve and repeat step (5). Continue until values have been recorded for outlet pressures as shown on the data sheets. (7) Increase the pump speed and repeat steps (4) through (6) for 1300 rpm. 9.5 Uncertainty Analysis For the pump experiment we would like to pay special attention to experimental errors and how we account for them in the final analysis. In this light, the next few sections are devoted to understanding sources of error and how to properly account for these. 9.5.1 Random Versus Systematic Errors Experimental uncertainty or error can generally be divided into two categories; random and systematic. Random errors are associated with the random fluctuation of repeated measurements. True random errors are subject to statistical analysis and will typically follow a normal distribution. Common examples of random error sources are insufficient equipment sensitivity and environmental effects such as temperature fluctuations. The second category is systematic errors. Systematic errors are due to effects that influence the result but with repeated measurements are not subject to statistical analysis. These are also called bias errors in that often their value remains roughly constant during a series of measurements under fixed operating conditions. Examples here include calibration errors, improper meter reading techniques or instrument resolution. Often systematic errors are established based on the instrument’s manufacturer specifications and by the experience and intuition of the experimenter. In an error analysis, random and systematic errors are treated differently. However, it is sometimes difficult to clearly establish whether an error is random or systematic. The best rule of thumb is to treat an uncertainty as a random error, if it can be treated statistically, otherwise treat it as a systematic error. 9.5.2 Propagation of Error The measurement of each parameter in the lab (e.g. pump speed, pressure, etc.) has one or more errors associated with it. For example, in using a pressure transducer for making a pressure measurement, we may have a systematic error due to the resolution of the device in addition to random errors associated with fluctuation environmental conditions. To then estimate the total uncertainty associated with the pressure measurement we must have some scheme for combining these errors. The technique we will use is termed the root-sum-squares (RSS) method. The uncertainty, ux , for the measurement of variable x is given by root-sum-square of all of the errors associated with that measurement, q ux = ± e21 + e22 (9.21) where for the previous example e1 would be the systematic error and e2 would be the random error due to fluctuating environmental conditions. Often in experimental work, measurements of several variables (e.g. torque, speed, pressure, etc.) are combined using some functional relationship to give a final answer (e.g. water horsepower). We will again use the RSS method to combine all of 114 CHAPTER 9. PUMP EXPERIMENT the errors associated with the different measurements. Let us assume we wish to calculate y and its associated uncertainty uy based on measurements of x1 , x2 and x3 : y = f (x1 , x2 , x3 ) (9.22) The best value for y would be based on the mean values for x1 , x2 , and x3 . The uncertainty in y, uy , is calculated by s 2 2 2 ∂f ∂f ∂f ux + ux + ux (9.23) uy = ± ∂ x1 1 ∂ x2 2 ∂ x3 3 The partial derivatives are often called sensitivities since they represent the sensitivity of the final result, y, to the measured parameter. As a final note, one must be very careful to use consistent units in this calculation. 9.5.3 Design Error Analysis For this experiment we will only consider errors due to the measurement system’s resolution. These instrumentation errors will be treated as systematic errors since we are unable to treat them statistically. Random errors will not be considered simply because we do not have the lab time to obtain statistically significant sample sizes for the parameters of interest. This brief discussion on design error analysis is focused on understanding how the limitations of an existing system, impacts the accuracy of the results. However, the strategy employed here could also be used to evaluate the accuracy of a system before it is built (design stage). We will consider two types of errors that will contribute to the uncertainty of a parameter measured with a particular instrument: interpolation errors, uo , and instrument errors, uc . An interpolation error is due to the finite ability to resolve the information provided by the instrument. The resolution of the information is typically dependent on the scale provided on the instrument. The rule of thumb for interpolation error is to use one-half of the resolution. For example, if a pressure gage has increments of 1 psi, the interpolation error, uo , is ±1/2 psi. Instrument errors are based on the manufacturer’s statement as to the accuracy of the instrument. Here we are depending on the experience of the manufacturer to establish the uncertainty. Often the uncertainty given in the specifications for the instrument will be in several parts (e.g. linearity, hysteresis, repeatability, etc.). What the manufacturer is saying is that all of these uncertainties will contribute to the total uncertainty in the reading. All of these instrument errors must then be combined using RSS (using the same units) to give the total instrument error, uc . Once we have established the interpolation and instrument errors, the design stage uncertainty for that particular measurement, ud , is given by q ud = u20 + u2c (9.24) All of the design stage uncertainties from all of the measured parameters should then be combined as outlined in Section 9.5.2 to give the final uncertainty in the final answer. 9.6. DATA REDUCTION AND DISCUSSION 115 Just as a final note, if a large enough number of samples of, say, pressure data were taken to account for random errors, these would simply be combined with the design stage uncertainty using the RSS method. Typically the uncertainty associated with the random errors would be based on the 95% confidence interval. 9.6 Data Reduction and Discussion A. To illustrate the accuracy of your results, develop: (1) A table (see Table 9.1) showing inlet pressure, outlet pressure, flow rates, torques, heads, water horsepower, shaft horsepower for each motor speed and pump. Note: Be sure to include with each measured value the design stage uncertainty based on the instrument specs given in Table 9.2. Also, for each calculated value, include the uncertainty based on the methods outlined in Section 9.5 Note: The error for the flow rate value should be estimated at 5% of the flow rate value since we do not have enough data for a statistical analysis. (2) A handwritten appendix of one sample calculation of each variable (including uncertainty calculations). (3) Let us assume that a design stage uncertainty of less than 15% for each measured value is acceptable. Based on this, which, if any, instruments are NOT acceptable for this experiment? B. To describe the operating characteristics of each pump, plot: (1) Leakage flow rate versus total head. Recall the leakage flow rate, QL (ft3 /min), is defined as the difference between the flow rate at a given total head and speed and the flow rate at that speed if there were no head developed (i.e. 0 back pressure). The flow rate at zero head can be determined by using your plot of head versus flow rate and extrapolating to 0 head using the method of least squares outline in Section 9.2. Discuss why leakage flow increases with increasing head. (2) Pump volumetric efficiency versus volumetric flow rate for both pumps and speeds on the same plot. (3) Pump overall efficiency versus volumetric flow rate for both pumps at both speeds on the same plot. Under what conditions (speed, head and flowrate) is each pump best suited? 116 CHAPTER 9. PUMP EXPERIMENT Table 9.1. Sample data table. Date: Motor Speed (RPM) Inlet Pressure (in Hg) Outlet Pressure (psig) Torque (in-oz) Water mass flow rate (lbs/s) Pump Head (ft) Water Power (HP) Shaft Power (HP) 900±1/2 -2±0.5 10±1.6 95±3 0.6±0.03 25±3.7 0.03±0.004 0.09±0.003 Table 9.2. Instrument specifications. Resolution: Accuracy: Nonlinearity: Motor Speed (RPM) 1 Inlet Pressure (in-Hg) 0.5 1.5% full-scale Outlet Pressure (psig) 1 1.5% full-scale Torque (oz-in) 1 0.05% full-scale 0.1% of rated capacity Hysteresis: Repeatability: 0.05% of rated capacity Zero balance: 0.03% of rated capacity Full Scale or rated capacity: 30 100 1000 9.6. DATA REDUCTION AND DISCUSSION 117 Table 9.3. Gear pump data. Date: Run No. Motor Speed (RPM) 1 900 Inlet Pressure (in Hg) Outlet Pressure (psig) 5 2 10 3 20 4 30 5 40 6 50 7 60 8 70 9 1300 5 10 10 11 20 12 30 13 40 14 50 15 60 16 70 Torque (in-oz) Water mass (lbm) 50 Time (sec) 118 CHAPTER 9. PUMP EXPERIMENT Table 9.4. Screw pump data. Run No. Motor Speed (RPM) 1 900 Inlet Pressure (in Hg) Outlet Pressure (psig) 5 2 10 3 20 4 30 5 40 6 1300 5 7 10 8 20 9 30 10 40 11 50 Torque (in-oz) Water mass (lbm) 50 Time (sec) Chapter 10 Uncertainties in Measurement Experiments require taking measurements of physical quantities, such as velocity, time, and voltage. We generally assume that the true values of the quantities to be measured exist if we had a perfect measuring apparatus and followed a perfect procedure. The measurements, however, are always subject to unavoidable uncertainty due to the limitations of the measuring apparatus, random environment, and even fluctuations in the value of the quantity being measured. As uncertainty is an unavoidable part of the measurement process, we should at first identify its sources and effects, and then quantify and report it. We should also seek to reduce measurement uncertainty whenever possible. The result of any measurement has two components as shown in the following expression for a measured temperature. T = 20◦ C ± 1◦ C (10.1) The first term on the right-hand side is a numerical value that gives the best estimate of the quantity measured, and the second term indicates the degree of uncertainty associated with the estimated value. This result tells that the temperature measured is most likely to be 20◦ C, but it could be between 19◦ C and 21◦ C. In this chapter we study how to get the expression as the one in Equation (10.1) for a quantity measured. The other task of this chapter is uncertainty propagation. If the measured quality, for example, the temperature T in Equation (10.1), is used as an input variable for an analysis, the analysis result for the output variable Y will also be naturally reported in the same form as Equation (10.1), having the best estimate of Y and the associated uncertainty term. The second term is the result of the uncertainty in T propagated to Y . The task of uncertainty propagation is to find both of the two terms of Y . 10.1 Experimental Errors Experimental error is the difference between the true value of the parameter being measured and the measured value. The error of a measurement is never exact because the true value is never exactly known. Measurement errors could be either positive or negative. 119 120 CHAPTER 10. UNCERTAINTIES IN MEASUREMENT A measurement error can be assessed by the accuracy and precision of the measurement. Accuracy measures how close a measured value is to the true value. As discussed above, the true value may never be exactly known, and it is difficult or even impossible to determine the accuracy of a measurement. Precision measures how closely two or more repeated measurements agree with each other. Better repeatability means higher precision. The distinction between accuracy and precision is illustrated in Figure 10.1. Figure 10.1. Accuracy and precision. Generally speaking, the accuracy and precision can be increased by decreasing the systematic and random errors, respectively. These two errors constitute the experimental error. Next we discuss the two types of error. 10.1.1 Systematic errors Systematic errors are those that affect the accuracy of a measurement. Systematic errors are not determined by chance but are introduced by an inaccuracy inherent in a measuring instrument or measuring process. In other words, systematic errors may occur because of something wrong with the instrument or its data handling system or because of the wrong use of the instrument. In the absence of other types of errors, systematic errors yield results systematically in repeated measurements, either greater than or less than the true value. In this sense, systematic errors are “one-sided” errors. For example, you use a cloth tape measure to measure the length of a table. The tape measure has been stretched out from a number years of use. As a result, your length measurements will always 10.2. EXPERIMENTAL UNCERTAINTY QUANTIFICATION 121 be shorter than the actual length. If a systematic error is known to be present in the measurement, you should either to correct it or report it in your uncertainty statement. It is, however, hard to detect or reduce systematic errors. Below are some general guidelines. • Calibrate the measuring instrument if the systematic error comes from poor calibration. • Compare experimental results from your instrument with those from a more accurate instrument so that you have a good idea about how large systematic error of your instrument is. • Change the environment, which interferes with the measurement process, so that the accuracy of the measuring instrument is highest. 10.1.2 Random experimental error Random errors are errors affecting the precision of a measurement. Random errors can be easily detected by different observations from repeated measurements. Random errors are commonly form unpredictable variations in the experimental conditions under which the experiment is performed. For example, random errors can come from electric fluctuations within components used in a measuring instrument or variations in temperature change in a lengthy experiment. In the absence of other types of errors, repeated measurements yield results fluctuating above and below the true value or the average of the measurements. This indicates that random errors are “two-sided” errors. 10.2 Experimental Uncertainty Quantification As shown in the expression in Equation (10.1), when reporting the experimental result, we have the best estimate term (20◦ C) and the uncertainty term (±1◦ C). In this section, we focus on using statistical techniques to find both of the terms. Uncertainty herein is a quantification of the double about the measurement result. Uncertainty quantification provides us with an estimate of the limits to which we can expect an error to go as shown in Equation (10.1). Suppose a quantity to be measured is X, and its measurements are x1 , x2 , . . ., xn , where n is the number of repeated measurements. With the n measurements, the obvious question we may ask is: “What is the best estimate of X?” If the only error source is from random fluctuations, given that the random error is a “two-sided” error, a reasonable answer is to use the average of the measurements. Averaging the measurements makes the fluctuations on both sides cancel out to some degree. A key point to understand here is that the average that we calculate is based on a, necessarily, finite number of samples, so it is only an estimate of the mean value. Another important point is that if we could obtain an infinite number of measurements, the mean value of these measurements would be the exact value of the quantity we are measuring – assuming purely random errors in measurement. 122 CHAPTER 10. UNCERTAINTIES IN MEASUREMENT Thus, the estimated average or mean is calculated by x1 + x2 + . . . + xn 1 n = ∑ xi X¯ = n n i=1 (10.2) After obtaining the best estimate of X, we now look at the uncertainty of our estimate. The uncertainty in the set of the measurements x1 , x2 , . . ., xn can be quantified by the degree of scatter of the measurements around the mean. The most commonly used measure of scatter is the unbiased sample standard deviation, s, defined by1 s s s= (x1 − x) ¯ 2 + (x2 − x) ¯ 2 + . . . + (xn − x) ¯2 = n−1 ¯2 ∑ni=1 (xi − x) n−1 (10.3) Figure 10.2. Slider-bar mechanism. Example 10.1 A slider mechanism is shown in Figure 10.2. The motion input, which is the angular velocity ωAB rad/s of link AB, is measured. The ten measurements are given by (xi )i=1,10 = (3.96, 3.99, 4.02, 4.01, 3.98, 4.02, 3.97, 3.99, 3.98, 4.0) rad/s. Determine the average and standard deviation of the measurements. The average is given by 3.96 + 3.99 + 4.02 + 4.01 + 3.98 + 4.02 + 3.97 + 3.99 + 3.98 + 4.0 1 n X¯ = ∑ xi = = 3.994 = 3.99 rad/s n i=1 10 The average is considered the best estimate the angular velocity. 1 See: https://www.khanacademy.org/math/probability/descriptive-statistics/variance_std_deviation/v/review-and-intuition-why-we-divide-by-n-1-for-the-unbiased-sample-variance 10.3. COMBINED UNCERTAINTY 123 The standard deviation is computed by s s= ¯2 ∑ni=1 (xi − x) n−1 s = 2 ∑10 i=1 (xi − 3.994) = 0.0204 = 0.02 rad/s 10 − 1 After we have done the statistical analysis, we could state that the best estimate of ωAB is 3.99 rad/s. Note that the number of significant digits used in the final result of the average (3.99 rad/s) is the same as the number of significant digits in the measurements. It does not make sense to use the calculated one (3.994 rad/s) because its last digit is beyond the precision of the measuring instrument. Of course, there is some degree of uncertainty because of the non-zero standard deviation. We should also report the associated uncertainty at a certainty confidence level. This requires us to know something about probability distributions. Next we discuss some basics about the normal distribution, which is the most commonly used distribution. 10.2.1 Normal distribution A normal distribution for random variable X is determined by the mean and standard deviation of X ¯ s2 ). The probability density function (PDF) of X, as shown in Figure and is denoted by X ∼ N(X, 10.3, tells us everything about X, especially the likelihood of the occurrence of certain possible values of X. It is easy to see that the values around the mean, X¯ have the highest chance to occur. Fig. 3 also indicates that the range defined by X¯ ± 2 s covers about 95% possible values of X. In other words, the probability that the actual values of X fall into the interval [X¯ − 2 s, X¯ + 2 s] is about 95%. We can then define the, so-called, extended uncertainty as U = 2s (10.4) Example 10.2 The angular velocity ωAB of link AB of the mechanism shown in Figure 10.3 is measured, and the ten measurements are given in Example 1. Report the measurement result in a standard form. In Example 10.1, we have obtained the average X¯ = 3.99 rad/s, and the standard deviation s = 0.02 rad/s. The uncertainty term is then U = 2s = 0.04 rad/s. The result is that we expect that the chance of the true angular velocity ωAB being within [3.95, 4.03] is 95%. 10.3 Combined Uncertainty In Example 10.1, the uncertainty came from only one source. If uncertainty arises from multiple independent or independent sources, their effects may be accounted for by the root-sum-of-squares method.2 2 Independent in this context implies that an arbitrary change in one variable has no effect whatsoever on any of the others. Moreover, uncorrelated variables subject to the same probability distribution may be assumed to contribute orthogonal uncertainties to the total uncertainty space. 124 CHAPTER 10. UNCERTAINTIES IN MEASUREMENT Figure 10.3. Normal probability distribution of a random variable. Assume that random variables X1 and X2 are uncorrelated and that their standard deviation are s1 and s2 , respectively. The standard deviation of X1 + X2 is then given by q (10.5) s = s21 + s22 Thus, the combined extended uncertainty term is q q 2 2 U = (2s1 ) + (2s2 ) = 2 s21 + s22 or U= q U12 +U22 (10.6) (10.7) In general, given n random variables, X1 , X2 , . . . , Xn , the corresponding extended uncertainties may be combined as s n U= ∑ Ui2 (10.8) i=1 Example 10.3 The angular velocity ωAB of link AB of the mechanism shown in Figure 10.2 is measured, and the ten measurements are given in Example 10.1. The measuring equipment manufacturer claims an accuracy of ±0.03 rad/s on the equipment readout. This accuracy is assumed at 95% confidence. Estimate the overall measurement uncertainty and report the measurement result in the standard notation. There are two sources of uncertainty. We have found the uncertainty term from random fluctuations U1 = 0.04 rad/s in Example 10.2. The other source of error is from the measuring instrument itself with U2 = 0.03 RPM. According to Equation (10.8), the combined overall uncertainty term is 10.4. UNCERTAINTY PROPAGATION U= 125 q q U12 +U22 = (0.04)2 + (0.03)2 = 0.05 rad/s The measurement is then expressed as ωAB = 3.99 ± 0.05 rad/s. 10.4 Uncertainty Propagation Measured quantities may be used for an analysis. Let the measured quantities be X = (X1 , X2 , X3 , . . . , Xn ) and the output of the analysis be Y . Also assume Y = f (X). From experiments, we have Xi = X¯i ±Ui (for i = 1, 2, 3, . . . , n). Uncertainties in X will be propagated to Y through the function, f . Our task is to find Y¯ ±UY . We start our discussions with a linear function: n Y = f (X) = c0 + c1 X1 + c2 X2 + . . . + cn Xn = c0 + ∑ ci Xi (10.9) i=1 where the ci are constants. If the measured variables, Xi , are independent, we have n Y¯ = c0 + ∑ ci X¯i (10.10) i=1 where the X¯i of the means of each independent measured variable. Therefore, the standard deviation of Y is given by s n sY = ∑ c2i s2i (10.11) i=1 where the si are the standard deviations of the corresponding measurements, Xi ; the corresponding extended uncertainties are Ui = 2si . Therefore, the uncertainty associated with Y is given by s n UY = ∑ c2i Ui2 (10.12) i=1 We now look at the general case where Y = f (X) is a nonlinear function. To use the results we have obtained for a linear function, we linearize f (X) at the mean values of the measured variables, X¯i . Denoting the ensemble of the means as ¯ = (X¯1 , X¯2 , . . . , X¯n ) X we define the linear approximation of the mean of the function Y as n ∂f (Xi − X¯i ) ¯ +∑ ¯ =f X Y¯ ≈ fˆ (X) ∂ X i X¯i i=1 where fˆ denotes the linear approximation of the nonlinear function, f . Hence, we have ¯ Y¯ ≈ fˆ X (10.13) (10.14) (10.15) 126 CHAPTER 10. UNCERTAINTIES IN MEASUREMENT It follows that s sY ≈ n ∑ c2i s2i (10.16) i=1 where the ci = ∂f ∂ Xi X¯ , and the extended uncertainty is given by Equation (10.12). Example 10.4 Ten measurements of the angular velocity ωAB of link AB of the mechanism shown in Figure 10.2 were given in Example 10.1. The measuring equipment manufacturer claims an accuracy of ±0.03 rad/s on the equipment readout. This accuracy is assumed at 95% confidence. The measured value of the length of link AB is 100 ± 0.1 mm. Determine the velocity of the slider, vC , and state the result in the standard notation. Letting X1 = ωAB , and X2 = LAB , we thus have X1 = 3.99 ± 0.03 rad/s, with extended uncertainty U1 = 0.05 rad/s, as determined in Example 10.3, and X2 = 100 mm, with extended uncertainty U2 = 0.1 mm. Performing the kinematic analysis to determine vC , we have LAB ωAB cos 45◦ Letting Y = vC , X1 = LAB , and X2 = ωAB , we define the nonlinear relationship between the vC = kinematic variables in terms of the previously defined statistical variables, X1 , X2 , and Y : X1 X2 cos 45◦ Substitution of the numerical values of ωAB and LAB yields: Y¯ = 3.99(100.0)/ cos 45◦ = 564.3 Y= mm/s. We now calculate the uncertainties: c1 = ∂f 1 1 = X¯2 = (100.0) = 141.4214 ◦ ∂ X1 cos 45 cos 45◦ and c2 = ∂f 1 1 = X¯1 = (3.99) = 5.6427 ◦ ∂ X2 cos 45 cos 45◦ Hence, UY = q c1U12 + c2U22 q = (141.4214(0.05))2 + (5.6427(0.1))2 = 7.09 mm/s The velocity of the slider is then reported as vC = 564.3 ± 7.1 mm/s 10.5. CONCLUSIONS 10.5 127 Conclusions The measurement error is the difference between the quantity being measured and its true value. The measurement error consists of systematic error and random error. The measurement error can be characterized by uncertainty analysis, and the measurement results is commonly stated in the form of X¯ ±U, where X¯ is the best estimate (usually the average of repeated measurements), and U is the uncertainty term with a stated confidence level (usually 95%). When a measured quantity is used in an analysis, the effect of the uncertainty in the measurement quantity on the analysis result can be quantified through uncertainty propagation, which is often based on the first order Taylor series expansion.3 3 See: http://en.wikipedia.org/wiki/Taylor_series. Chapter 11 Report Formats and Proposal Example 11.1 Short Memorandum Guidelines The following is meant to be a general, guide to the SHORT report format in ME242. Your audience may be assumed to be very familiar with the subject that you are reporting on. Therefore, much less background information is needed as compared to a journal paper, or general technical report. In essence, your audience may be assumed to be students, like yourselves, or GTAs who have done the same study before. You should follow the guidelines established in the Journal Paper Format document for equations, figures, tables, and references. Title Section, Group Designation, Group Members Names GTA: GTA’s Name Prepared by Your Name Date Abstract The Abstract should be approximately 100-150 words in length. This may be a very strict limit in some cases. For ME 242 the limit will be somewhat flexible. It should contain the bare essentials of what was studied, why it was done, where the work fits in with current studies elsewhere, and essential results. The abstract should contain predominantly fact-based results. Procedure Here, you will explain in brief what steps you followed in conducting the experiment. Results Here, you will explain the data you took, and any calculations that you made. You need NOT repeat the published experimental protocols in the lab manual sections you need only refer to the manual when necessary to clarify a procedure. Of course, you must properly cite any references, and list them in the bibliography or references section. You most show sample calculations in appendix. This section should be the longest in the report. 128 11.2. LONG JOURNAL FORMAT 129 Answers to Lab Manual Questions In this section, simply answer each question in order. Conclusion Summarize the experiment in one paragraph . Recommend suggestions to improve experimental data collection. References List any references that you used including figures, verbal communications, internet sources, etc. Table 11.1. Grading Scheme for Short Memo Component Abstract Procedure Results Answer to lab question Conclusion Refrence Graphs & Table Formatting Language 11.2 Percent of Grade 10% 5% 20% 25% 10% 5% 15% 5% 5% Long Journal Format A document containing guidelines for writing journal articles as well as a humorous example may be found at the following link: http: //web.mst.edu/˜stutts/ME242/LABMANUAL/Journal_Paper_Guide.pdf 11.3 Final Lab Proposal Example Please see: http://web.mst.edu/˜stutts/ME242/Example4thLabProposals/ TunedMassDamperExperimentProposal.pdf.

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