lab manual 2013

ME/CE 96: Mechanical EngineeringLaboratory

Spring 2013

CONTENTS

1 Course Information 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Contact Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Class Webpage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Lectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Lab Notebooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 Prelab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.8 The Lab Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.9 Grading Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.10 Time Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.11 Deadlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.12 Lab Partners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.13 Collaboration Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.14 Signing Up for Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 61.15 Operating Guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.16 Starting an Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.17 Scheduling Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.18 What to Do if Something Breaks . . . . . . . . . . . . . . . . . . . . . . . 71.19 Lab Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.20 safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.21 Lab Safety Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Cantilever Beam 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Prelab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Optional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Turbomachinery 173.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

i

3.4 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6 Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.7 Operating the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.8 Precautions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.9 Prelab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Free and Forced Convection 294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5 Heat-Transfer Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . 344.6 Lab Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.8 Prelab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Turbulent Air Jet 375.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.2 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.4 Prelab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.5 Air Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.6 Appendix: Motion control . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Mechanical Properties of Metallic Materials - The Tensile Test 516.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.2 Test Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.3 Test Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526.4 Tests to Perform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.5 Fracture Surface Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 546.6 Lab Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.7 Pre-lab Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7 Dynamics of Coupled Mechanical Oscillators 577.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.3 Analysis of Experimental Model . . . . . . . . . . . . . . . . . . . . . . . 607.4 Lab Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607.5 Advanced Experiment (Optional) . . . . . . . . . . . . . . . . . . . . . . 617.6 Prelab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617.7 Appendix: Making Accurate Period and Phase Measurements . . . . . . . 61

A Keeping a Lab Notebook 63

B LabVIEW 65

ii

CHAPTER

ONE

Course Information

1.1 Overview

ME/CE 96 is a laboratory course with experiments in mechanics, dynamics, control, fluids,heat transfer, turbomachinery, combustion, and fuel cells. During the course of the term,you will do 4 experiments, selected from 9 that are available this year. Each experimentlasts two weeks.

A new aspect of the course is an emphasis on data acquisition methods. As part of thecourse, you will learn how to build “Virtual Instruments” using LabVIEW to acquire andanalyze experimental data and to control equipment.

1.2 Contact Information

Instructor: Monica Kohler [email protected] x4142 229 Thomas.

TAs: Juan Cardenas [email protected] Schwee [email protected]

1.3 Location

The experiments are located in the sub-basement of Thomas Laboratory, in rooms 0018 and0021. Please see Chris Silva in Thomas 208 to get a key to the lab rooms. Keys must bereturned at the end of the term to Chris Silva to recieve your grade.

1.4 Class Webpage

There is a webpage for the course, which may be found at http://sites.google.com/a/caltech.edu/me96spring/. This site contains the most up-to-date infor-mation about the course, handouts, class noted, useful programs, and an experiment sched-ule.

1

http://sites.google.com/a/caltech.edu/me96spring/

http://sites.google.com/a/caltech.edu/me96spring/

1.5 Lectures

A few lectures and demonstrations will be given early in the term. Each one will be 60 to90 minutes, and is designed to introduce you to useful methods and tools. For several of thelectures, there will be a homework assignment. The assignments will not be very difficultor time-consuming, but are designed to reinforce concepts from the lecture. They will bedue one week after the lecture. All of the homework assignments together will count for10% of your course grade.

Some of the planned lectures are:

1. Data acquisition with LabVIEW

2. Error analysis

The times and dates for these lectures/demos are to be determined. Consult the webpagefor more information.

1.6 Lab Notebooks

You will need two bound laboratory notebooks. The bookstore if still open, sells severalsuitable styles. All of your written work will be done in these notebooks and you willalternate from one to the other between labs, so that while one can be graded while theother is used for doing experiments. See Appendix A for more information on how to keepa lab notebook.

1.7 Prelab

Each lab has a set of questions, estimates to make and/or problems that must be solvedbefore beginning the experiment. These are designed to insure that you come to lab wellprepared. Write the answers to the prelab questions directly into your notebook. There isno need to copy the problem statements. When you meet the TA to go over the experiment,he or she will check that you and your lab partner have done the prelab appropriately.You will not be allowed to begin the experiment until you have completed the prelabsatisfactorily.

1.8 The Lab Report

The lab report is the primary means for you to communicate your results, and to demon-strate to the TA and the instructor your level of understanding and the quality of your pro-cedures and results. Consequently, it is expected that the reports will be written thoroughly,carefully, and thoughtfully.

The report should be written in your lab notebook. In most cases, your report will consistof entries written directly into the notebook with pen, along with computer-generated text

2 Chapter 1. Course Information

or plots. Quick plots made in the lab for a particular purpose (e.g. to check the linearityof a sensor) would typically be drawn by hand, but plots made after you have acquired thedata and have analyzed it (e.g. comparing measurements to theory) should be done usingthe computer for faithful reproduction of quantitative results.

We expect the report to be reasonably clear and legible, in addition to being well-organized.But we should emphasize that a lab report is different from the sort of reports you may havewritten for other classes. We don’t want a “finished” report like a term paper or a scientificpublication, but instead want to see the actual record of what you did in the lab, written inyour notebook.

A good lab report will contain crossed-out (but still readable) text where you made errors,will usually contain hand-drawn sketches of equipment, may contain hand-drawn quickplots done in the lab, and may have lots of things pasted into it, such as computer-generatedplots or photographs. Much of the report will be written by hand in pen (never pencil). It isperfectly fine to write the entire report by hand, with glued-in computer plots or printoutsas necessary. But if you are writing a lengthy discussion of the results, and would preferto use a computer, that is fine. Simply print out the text and glue it page by page into thenotebook.

There are special rules that govern making entries in lab notebooks, including lab reports.See Appendix A for more information.

The lab report should contain a complete1 record of the data you took in the lab, along withyour analysis and conclusions.

The report should begin with a brief statement of the objective of the experiment. Don’tsimply copy the objectives statement in the handout — state the objectives in your ownwords.

You should include a description of the apparatus, with a sketch with all significant partslabeled. Don’t simply cut and paste the sketch (if there is one) from the lab handout. It isrecommended that you draw the sketch by hand in your notebook — the act of drawing andlabeling each component makes you take notice of every part of the experiment, and givesyou a better physical “feel” for the experiment than can be obtained by simply passivelylooking at sketch drawn by someone else.

The report should include a description of all procedures followed, a record of all measure-ments made and a discussion and analysis section at the end, where the results are comparedto theory or other measurements, as appropriate, and conclusions are drawn. Your reportshould also contain an error analysis, in which the uncertainties in the measured values andin any functions derived from them are estimated. Any comparison of measured data totheory should always include error bars on the data points.

The analysis and discussion portions should be thorough with attention to detail. If youare asked to compare your measurements to theoretical or numerical predictions, and youfind discrepancies larger than experimental uncertainties, discuss why this is so. If youcan, attempt to resolve the problem — don’t be satisfied with unexplained discrepancies.If you conclude that an instrument was malfunctioning, or the measurements were doneincorrectly, you may want to go back into the lab to make more measurements or check thefirst ones (so don’t leave the report writing until the last minute!). Or it may be that the

1but see Appendix A for more on what constitutes “complete.”

1.8. The Lab Report 3

theoretical results are missing some important effects; can you account for them with bettertheory? It is easy to claim the problem is with the theory and not with the measurements.How do you know? What evidence supports your claim? Whatever you conclude, alwaysback up your statements with evidence.

As noted above, remember that a lab notebook is a lasting record of your experiment — ina research laboratory, they are often retained for years or decades, so that later researcherscan refer to the original data. (Millikan’s famed oil-drop experiments were recorded innotebooks that are in the Caltech Archives.) It is just good policy to get into the habit ofmaking top quality lab notebooks right from the start. In addition, the best of your labreports will be used to train next year’s TAs.

1.9 Grading Policy

The course grade will be based on the lab reports for the four experiments, plus a fewhomework assignments from the lectures early in the term. Grading will be done primarilyby the TAs and reviewed by the instructor. The grade for each experiment will be basedon a 100 point scale with the prelab counting for 20 points, the quality of the lab reportcounting for 40 points and lab performance counting for 40 points. The lab performancewill be based on a number of factors that may include meeting the deadlines describedbelow, resourcefulness in finding needed information, preparation before coming to the lab,safety practices, experimental technique and the quality of the resulting data. This portionof the grade attempts to assess the quality of what you do in the lab and how you do it. Inmost cases, it will be determined from your written lab report, although it may be based inpart on observations by the TA.

The lab report points will be based on the quality of the presentation, analysis, and discus-sion in your report. The emphasis is not on superficial appearance, except that the report isexpected to be legible and well-organized.

1.10 Time Management

You should expect to put in approximately 9 hours each week on this course, ideally ina few sessions of 2 or 3 hours each. Learning anything is most effective when you workon a project or assignment for short periods every day (3 hours maximum), instead of inone long session just before the assigment is due. When you get tired, or find you are notconcentrating well, it is time to rest or do something else for a while, and come back tothe project when you are refreshed. This is particularly important with lab work, sinceaccidents are much more likely if you are tired or not mentally alert and fresh.

Note that 9 hours per week is not the same as 18 hours every other week, or 90 hours in thelast week of the term! The experiments are designed to be done over a two-week period,and simply cannot be done in a marathon all-night session the night before the lab reportis due. It is fair neither to your lab partner nor to yourself to try to do the labs at the lastminute.

And remember, when doing experimental work, you have to expect the unexpected. Very


often things happen that require time to deal with. Maybe you can’t get the oscilloscope towork – is it broken, or do you simply not understand how to use it, or is there some buttonsomewhere to reset it, or . . . ? Or maybe you have a bug in your LabVIEW VI that you needto track down before you can make the measurements. Be sure to start the experiment earlyenough that if something unexpected occurs, you have time to resolve it.

To help insure that everyone keeps up with the schedule, several intermediate deadlineshave been established that must be met, as described in the next section.

1.11 Deadlines

The 8-week period beginning with the second week of the term is divided into four two-week blocks: Block A, Block B, Block C, and Block D. You will do one experiment in eachblock, and the lab report for each experiment will be due on the first day of the next Block.In addition, there are some important intermediate deadlines.

All of the deadlines are listed on the course calendar, which is accessible from the ME96class page. You can subscribe to this calendar from your own calendar program (Outlook,or Google Calendar, for example) so that you can set up reminders for yourself.

The deadlines are as listed below. Here “week n” means the nth week as measured fromthe start of a Block. So, for example, week 3 of Block B is the same as week 1 of Block C,etc.

1. Thursday of week 1. By this day, you must have completed the prelab and have metwith the TA to go over the experiment. You cannot begin the experiment until youhave met with the TA and he or she gives you the go-ahead to start the experiment,and he or she will not do so if your prelab is not commpleted.

2. Thursday of week 2. You must have completed the bulk of the measurements byThursday of week 2 at the latest, in order to allow sufficient time to analyze yourdata and finish writing the report. You can still go back into the lab and take moreor different measurements after this date, since you may find as you write the reportthat additional measurements are needed. When grading your reports, the TAs willnote the dates you enter in your lab notebook to determine whether or not you metthis deadline.

3. Monday of week 3. On the Monday following the end of the two-week block for theexperiment, the final lab report is due. Please turn your notebook in at my office (115Thomas) before 5 pm sharp. After this date, late lab reports will be accepted for oneweek only, with a 10 point reduction in the maximum possible points for each daylate. For example, if your report is 3 days late, and you would have received a scoreof 80 if it had been turned in on time, your grade would be 80 × 0.7 = 56. No labreports will be accepted more than one week late.

4. Monday of week 4. Your graded lab report will be available in front of my office oneweek after the deadline for turning in the lab reports.

5. Monday of week 5. If your lab report was submitted on time, you can correct andresubmit anything you missed in the lab report, for 80% of the original point value.

1.11. Deadlines 5

http://sites.google.com/a/caltech.edu/me96/Home


Any corrections must be submittted by 5 pm on the Monday of week 5, which is alsothe due date for the next lab report. Corrections will not be accepted after this date.Corrections cannot be submitted for lab reports that were turned in late.

Note that no lab reports will be accepted more than one week late. Since your grade isdetermined mostly by the four lab reports, it would be difficult at best to pass the coursewith an acceptable grade missing even one lab report. It is impossible to pass if you misstwo. If you don’t feel you can keep to this schedule, either due to Ditch Day, extracurricularactivities, a heavy course load, or any other reason, this is not the course for you. Youshould drop it now.

The only exceptions to this policy will be for genuine, unexpected medical or other seriousevents beyond your control. In such cases, please submit a note from the Dean or from adoctor explaining the situation.

Note that preparations for Ditch Day do not constitute a valid reason for missing thesedeadlines. If Ditch Day falls on a day with a deadline (a Monday or a Thursday), then thedeadline will be extended by one day.

1.12 Lab Partners

You will all work in teams of two. In most cases, you should arrange to have the samepartner over the whole term. You and your partner should both be present when taking data.All lab reports must be written individually.

1.13 Collaboration Policy

You may freely discuss the experiment and your interpretation of the results with yourpartner or other students. On the other hand, you are expected to do your own numericalsolutions, derivations, error analysis, etc.

1.14 Signing Up for Experiments

Contact the TA for the experiment you and you lab partner would like to perform. Each labgroup (of 2) should sign up for four experiments -one for each two week period. At most,experiments can accommodate three groups at once (working in the lab at different timesduring the week). To see what experiments in what experiment blocks are available, go tothe experiment schedule is listed on-line.

1.15 Operating Guides

In addition to this manual, each experiment there is an “Operating Guide” that describes indetail the practical aspects of how to run the experiment. These are PowerPoint documentswith photos of the equipment, and show, for example, where the power switch is, what


http://sites.google.com/a/caltech.edu/me96/Experiments/Experiment-Schedule

knobs do what function, etc. You should take a look at the operating guide before meetingwith the TA. They are all available on the ME96 class page.

1.16 Starting an Experiment

At the beginning of each two-week period you should make an appointment with the TAto meet in the lab at a mutually convenient time to have the TA go over the experiment.You must do the prelab before meeting with the TA, and cannot begin the experiment untilstarting the experiment. Grades will be reduced for those who are unprepared or for thosewho miss their appointments with the TAs (remember, the TAs are busy students just likeyou).

1.17 Scheduling Time

Each experiment will have a sign-up sheet where your group can reserve the experiment.Each group can reserve one block of time (up to four hours in duration) in advance. Onceyou have finished the lab time, you can sign up for an additional block of time. It is bestto space lab sessions by a day or two so that you can look over your results and check thatthings are looking reasonable before going back into the lab.

1.18 What to Do if Something Breaks

It is an unfortunate reality of working with equipment of any kind that sometimes thingsbreak, or simply fail to function as you expect or as the documentation describes. If some-thing is broken or not working, first see if you can diagnose what the problem is. Is theproblem in software? Is a connector loose? Is the problem constant or intermittent? Doesthe manufacturer’s web site have any relevant information? An important part of becominga resourceful engineer is learning how to troubleshoot and fix things.

But, having said that, don’t spend hours tracking down the problem. If you can’t figureout the problem after 30 minutes of work, contact the TA for the experiment. He or shemay know immediately what the problem is. Also, if any component is actually brokenand needs to be replaced, contact the TA or an instructor as soon as possible. We may havereplacement parts stored elsewhere, or may need to order a replacement part.

1.19 Lab Computers

Each experiment has a PC that is used to acquire data, or to control the experiment. All butone of the PCs run Windows XP. They should all have Microsoft Office, MATLAB R2006band LabVIEW 8.2, and are all on the Caltech network. In addition, other software tools thatallow you to transfer your data files to other machines are installed.

The PCs are all in Workgroup “ME96.” Each one has a folder entitled “Shared” that can beaccessed for reading and writing from any of the other PCs.

1.16. Starting an Experiment 7


Please keep your data files together in a folder . There is no way to protect your files fromaccidental deletion or corruption by others, so you should transfer or backup all data filesat the end of each lab session. A memory stick is convenient for this purpose.

Note that the folders belonging to other students should be regarded as their private property,and should not be perused without their permission. When you are completely finished withan experiment, please delete your folder containing your data (but retain a copy elsewhere,or burn a CD).

All of the PCs can access the B/W laser printer in 0018 attached to the airjet PC, the colorinkjet printer attached to the stress/strain PC, and the network printer in 0021.

Please do not install games or pirated software of any kind on these PCs. If you need anysoftware package installed that Caltech has a site license for, let the TA know and we willsee if we can install it.

1.20 safety

Please use caution when dealing with very hot or very cold entities. For example the flameburner in the fuel cell experiment or the liquid nitrogen in the boiling heat transfer exper-iment. Safety goggles and heavy gloves are available in the lab. Please use them whennecessary esp with these experiments. There is also a first aid kit for minor injuries and afire extinguisher. Hearing protection is available for the turbomachinery experiment.

1.21 Lab Safety Rules

To maintain a safe lab environment, there are a few rules that must be adhered to:

1. No working alone at any time. You can work in the lab whenever it suits your sched-ule best, but you must always have at least one other person present in the room withyou if you are doing anything involving lab equipment. This rule does not apply ifyou are only in the lab to use a computer. This should not present any problem, sincenormally you will work together with your lab partner.

2. No food or drink is allowed in the lab at any time.

3. Shoes must be worn in the lab at all times.

4. Do not wear dangling jewlery or clothing that could be caught when working onexperiments involving rapidly moving mechanical components (inverted pendulum,vibrations, turbomachines).

Also, please use common sense. It is not a good idea, for example, to work in any lab (orshop) when you are sleep-deprived, or are for any other reason less than 100 %.


CHAPTER

TWO

Cantilever Beam

2.1 Introduction

The experiment involves the bending and vibration of an aluminum beam. Measurementsare made of the deflection, strain rates, fundamental frequency, and damping constant. Thestudent is exposed to measurement techniques, data acquisition, and analysis. The experi-mental results are also compared with theory.

2.1.1 Analysis of beam strain and deflection

This section briefly reviews the material necessary to make calculations of beam deflectionand strain for an imposed load. The notation corresponds with the Figure 2.1.1 (a). Recallfor a linearly elastic beam that the strain in the x-direction, εx, is linearly related to theimposed normal stress in the x-direction, σx, by the following relation:

εx =σxE, (2.1)

where E is the modulus of elasticity.

In pure bending, the strain on a beam can be expressed in terms of the radius of curvatureof the beam R and the distance from the neutral axis of the beam y:

εx = − yR

(2.2)

Hence, the strain is zero along the neutral axis, and its magnitude increases with distancefrom the neutral axis. The strain is negative in the +y direction, which corresponds to anegative stress or compression in the +y part of the beam. Equations (1) and (2) can berearranged to give the stress at some position within the beam:

σx = −EyR. (2.3)

The moment M(x) acting on the beam can be calculated from the stress:

M(x) = −∫AyσxdA. (2.4)

9

Figure 2.1: Geometry for analyzing beam deflection.

Hence, a positive moment produces compression in the +y fibers of the beam. Recallingthe definition for the moment of inertia I:

I =

∫Ay2dA, (2.5)

the moment-curvature relation can be found for a homogeneous beam:

M =EI

R(2.6)

and the flexure formula follows as:

σx = −My

I. (2.7)

The radius of curvature of the beam can be related to the displacement v of the neutral axisof the beam due to bending. For small deflections of the beam compared to the length ofthe beam, the radius of curvature can be determined from the following equation:

1

R=

d2v/dx2

[1 + (dv/dx)2]3/2. (2.8)

If the slope dv/dx of the curve describing the loaded beam is at all points small relative tounity then one can set the denominator of the expression for curvature equal to one. If thisis the case, then this equation can be approximated as:

1

R= d2v/dx2. (2.9)

10 Chapter 2. Cantilever Beam

2.1.2 Analysis of beam vibration

Consider a beam of uniform cross section A and density ρ, as shown in Figure 1(b). Atsome position x within the beam, a balance of forces and moments can be drawn usingf(x) as some distributed load on the beam. The resulting equations for the balance of theforce and moments are as follows (assuming that dx approaches zero):

− f(x) =dV

dx(2.10)

andV =

dM

dx. (2.11)

These equations can be combined with the moment-deflection equation:

M = EI∂2v

∂x2. (2.12)

These equations can be combined to give:

EI∂4v

∂x4= −f(x) (2.13)

The distributed load is the inertial load due to vibration and can be represented as the prod-uct of the mass per unit length and the cross sectional area in the direction opposite of theacceleration:

f(x) = ρA∂2v

∂t2. (2.14)

Combining equations (2.13) and (2.14) results in the following fourth order partial differ-ential equation:

EI∂4v

∂x4+ ρA

∂2v

∂t2= 0. (2.15)

This differential equation can be solved subject to specified boundary and initial conditions.

The above analysis assumes that there is no damping of the vibrating beam, and hence thevibration of the beam would be of constant amplitude. However, in real systems dampingis usually always present due to friction. In a vibrating beam the friction is internal to themedium, and is associated with the energy being dissipated randomly in the crystal lattice.The energy dissipation manifests itself as an internal heating of the beam. The frictionaldamping reduces the amplitude of vibration over time.

For a vibrating system with a single degree of freedom (harmonic oscillator), the motion ofthe system is governed by the following equation:

y = a1 cos pt+ a2 sin pt, (2.16)

where p is the frequency of vibration and is related to the period of vibration by τ = 2π/p,and a1 and a2 are unknown constants. If the system is damped and the damping forceis proportional to velocity (as is the case for internal friction), the motion of the dampingsystem is governed by the following equation:

y = e−nt (a1 sin qt+ a2 cos qt) , (2.17)

2.1. Introduction 11

Figure 2.2: The cantilever beam experiment.

where n is the damping constant and vibratory motion now has the period:

τ =2π

q=

2π√(p2 − n2)

(2.18)

Note that if n is small relative to p, the period of vibration is close to the value obtainedwithout damping.

2.2 Experiment

2.2.1 Data Acquisition

In this experiment, you will measure the strain at five locations on a beam subjected tovarious static and dynamic loads. The measurements will be done using a strain indica-tor, which uses an electronic bridge circuit to accurately measure very small changes inresistance.

For static measurements, the strain may be read directly on the front panel. The strain isdisplayed in units of “microstrain.” For example, a reading of 5000 would correspond to astrain of 0.005, or 0.5%.

For dynamic measurements, the strain indicator unit provides an analog output on the back.Connect this to channel 1 of the digital oscilloscope. The oscilloscope can be used directly


to view the dynamic strain waveform, although it does not have the capability to store data.Therefore, a better approach is to control the oscilloscope from the computer.

A LabVIEW VI is provided that controls the digital scope, and displays the waveformvs. time. You will need to modify it to also display the frequency spectrum of the signal.

2.2.2 Measurements

1. Make a sketch in your notebook of the experimental setup, and give a short descrip-tion of the major components. Include any relevant dimensions you will need for theanalysis.

2. Apply different loads on the end of the beam and record the resulting strain and tipdeflection. Do the measurements for both axes of the beam and for all five straingauges for a single repetition (2 axes X 5 gauges X # of loads measurements). Plotyour results. Note that the strain indicator must be balanced separately for each straingauge. Therefore, it is simplest to take measurements for the full range of loads withone strain gauge, then select another strain gauge with the switch, re-balance thestrain indicator, and repeat the set of loads.

3. Investigate the repeatability of these measurements. For each axis, choose one straingauge (which one will give the biggest output?) apply a load multiple times to thebeam, each time beginning from an unloaded state. Record the strain (for the onegauge) and deflection readings each time. Do at least six repetitions. Repeat for atleast two loads. Repeat for the other beam axis (2 axes X 1 gauge X repetitions X #of loads).

4. Familiarize yourself with the oscilloscope and the LabVIEW VI that runs the oscil-loscope from the computer. Record the bandwidth, sampling rate, and resolution ofthe oscilloscope.

5. Using the oscilloscope (either directly, or from LabVIEW), select one strain gaugeand determine the fundamental vibrational frequency of the beam on each axis, bystriking the beam with the supplied hammer and examining the resulting dynamicstrain signal. Adjust the scope timebase and voltage scale so that you see several(ten or more) periods on the screen. You may need to adjust the trigger to get astable display. The easiest way to measure the fundamental frequency is to count thenumber of zero crossings in the voltage vs. time signal.

6. Now using LabVIEW to run the oscilloscope with the supplied VI, edit the VI toadd a graph of the power spectrum (you can find a convenient VI for this on the“Express” function menu). Now hit the beam with the hammer and examine thefrequency content of the signal. Some things to explore:

(a) What are the frequencies of the harmonics of the beam vibration?

(b) Do the amplitudes of the harmonics depend on where you strike the beam?

Make several measurements of the harmonics, so that you can estimate the repeata-bility. Do the same for the other axis.

2.2. Experiment 13

7. For one of the harmonics, for example the first overtone, examine the effect of sam-pling rate. Include at least one measurement with a sampling rate less than twice thefrequency of oscillation. Make a plot of the measured frequency vs. sampling rate.

2.3 Data Analysis

1. Compare your measured strain vs. load data to theoretical predictions for a linearlyelastic beam in pure bending. Show in a plot how the two compare, and include errorbars on the measured values. Discuss any discrepancies found, including possiblecauses.(Were there unaccounted-for systematic errors in the measurements? Doesthe theory make simplifying assumptions that aren’t valid? How do you know? Whatadditional measurements could you do to determine the problem?

2. Plot the tip deflection vs. load, and compare to the solution determined by integratingthe differential equation. As in (1), discuss any discrepancies, their possible causes,and the steps that could be taken to resolve the discrepancies.

3. If the mass of the beam were taken into account, how would it affect the deflectionvs. load data? (The density of aluminum is 2.77 g/cm3.)

4. Compare your measured fundamental frequencies, and the ratio of the harmonic fre-quencies to the fundamental frequency, to theoretical predictions. As in (1), discussdiscrepancies, etc.

5. Determine the damping constant for the fundamental frequency. This is most easilydone if you examine the signal with the scope timebase set to a long tim, so that youcan see on the screen the decaying-exponential “envelope” of the signal.

2.4 Prelab

Read the tutorial on measuring strain on the wiki.

1. What is the Gauge Factor (GF) of a strain gauge?

2. Suppose a strain gauge with GF = 2.5 experiences a strain of 0.1%. What is thepercent change in resistance?

3. Describe how bridge circuits can be used to accurately measure the very smallchanges in resistance that result when a strain gauge is subjected to strain.

4. Determine the fundamental frequencies, ωn of vibration in terms of E, I , ρ, and Aby solving equation 2.15. Assume that the beam is undamped.

• Use separation of variables to obtain the solution v(x, t) = X(x)T (t).

• Note that the solution of T −ω2T = 0 is T (t) = a1 cos(ωt)+a2 sin(ωt) wherethe a’s are constants determined from the initial conditions.


• Note that the solution of XIV + λ4X = 0 is X(x) = b1 sinλx + b2 cosλx +b3 sinhλx + b4 coshλx where the b’s are unknown constants determined fromthe boundary conditions. This solution can be rewritten as

c1(cosλx+coshλx)+c2(cosλx−coshλx)+c3(sinλx+sinhλx)+c4(sinλx−sinhλx),(2.19)

• The zeros of the transcendental equation cosλL coshλL = −1 are λL =(1.875, 4.694, 7.855, 10.996, . . . ).

• λ relates the frequencies to the beam properties and the initial conditions.

5. Modify the spectralCant.vi to perform a frequency analysis on a signal generatedusing the function generator.

• Connect the function generator to channel 1 of the DAQ.

• Generate a sinusoidal signal at roughly 2000 Hz.

• Modify the VI to perform a spectral analysis on the signal and plot the output.Include a printout of the modified front-panel and block-diagram (screenshot).

• Verify that the spectral analysis gives reasonable results (since we know theinput frequency and amplitude).

• Experiment with the sampling frequency and number of samples to see effectthese have on the measured frequency. Include at least one sampling rate that isbelow 4000 Hz. Plot this data.

2.5 Optional

1. Time permitting, explore other effects of your own choosing. Some interesting ques-tions: What is the effect of combined static and dynamic loading? Does it change thevibration frequency? What would the dynamic response look like if you were to hanga mass from a spring or a pendulum attached to the end? How would you analyzesuch a problem theoretically?

2. Measure the damping constants of the harmonic frequencies. How do they compareto that of the fundamental?

3. To appreciate how much care must be taken with strain gauges (or for that matterwith many types of delicate instruments), take one of the uninstrumented beams andapply strain gauges to it. Each student should apply at least one strain gauge. Repeatselected measurements of the static and dynamic response and compare the results.Do your results agree to within previously-determined uncertainties? If not, can youdiagnose what went wrong with your strain gauge application? What evidence canyou give that the discrepancy is due to one cause, and not some other?

2.5. Optional 15

CHAPTER

THREE

Turbomachinery

3.1 Introduction

By turbomachine we mean a fan, pump, compressor, or turbine that changes the energy con-tent of a flowing fluid by means of momentum exchange. Sometimes the term rotordynamicmachine is used for this class of machines to distinguish them from positive displacementdevices. Turbomachines are second only to electric motors in their number and are wide-spread in practically all industries, ranging in power levels from a few watts to more than100 MW. Because of the requirements of aircraft jet propulsion, new power plants (bothsteam and gas turbine) and rocket propulsion research, development continues actively inthis field today.

The present experimental turbomachine is an axial-flow fan powered by a three phase in-duction motor. The fan is used on the MD-11 aircraft to cool electronic equipment in thecockpit. The fan consists of a rotor which receives axial non-swirling air from the inlettube; the rotor imparts an angular velocity Vθ to this oncoming flow, which is subsequentlystraightened out to be purely axial in a stationary row of vanes termed a stator. These statorvanes also serve as heat-transfer surfaces needed to cool the drive motor that is mountedinternally inside the hub of the stator. The rotor consists of a cylindrical hub structure onwhich equally spaced airfoil-shaped blades are mounted. The centroids of each radial bladesection are aligned on a radial line (this is the stack line) and are set at various angles to thetangential direction to provide the requisite pressure rise (or enthalpy rise) of each radialsection of the rotor.

The present facility includes a motor controller that provides a variable voltage and fre-quency (up to 400 Hz) for the compressor motor, permitting operation up to a synchronousspeed of 12,000 rpm. The motor is a four pole, 3 phase, Y wound induction motor. Thesynchronous frequency in revolutions per minute (rpm) of the rotating field is given by

rpm =

(f(cycles/s)

(P/2)

)60s

min(3.1)

where f is the frequency in Hertz of the supply (read from the LED readout on the wall-mounted motor controller) and P is the number of poles of the motor. Directions for oper-ating the supply are given in the Operation section 3.7. An induction motor must operate ata speed slightly less than the synchronous speed; the difference or slip is about 3% in thepresent case. For the purpose of the experiment we will assume the fan operates at 97%of the synchronous speed since we have no independent measurement of the actual rotor

17

Figure 3.1: The MD-11 aircraft. The turbomachine in this experiment is an axial flow fanused to cool electronic equipment in the cockpit. Of course, the engines are also turboma-chines — only much larger and more powerful!

speed.

A schematic of the apparatus is shown in the Appendix. It consists of a semi-elliptical inletnozzle, upstream and downstream static pressure ports, the fan, and a manually adjustablethrottle plate. The inlet nozzle provides a smooth inlet flow, the pressure ports enabledetermination of inlet flow and fan compression, and the throttle plate provides a meansto adjust the load on tthe fan. This test apparatus is built to the AMCA 210-85 AmericanNational Standard Laboratory Methods of Testing Fans for Rating.?

3.2 Background

The energy level change brought about by a turbomachine is expressed as the total enthalpyrise (for a fan, compressor or pump) per unit mass of the fluid. In most turbomachines, thiswork is accomplished by momentum exchange with the rotor and the total enthalpy rise isgiven by a famous formula, the Euler Turbine Equation

∆ht = ∆(UVθ) (3.2)

where ∆ht is the enthalpy increase in J/kg, U is the rotative speed of the impeller tip, i.e. rωat a particular radius r (ω is the angular velocity), and Vθ is the component of the absolutevelocity in the direction of U . For an isentropic, incompressible flow,

∆ht =∆ptρ

(3.3)

where ∆pt is the total pressure increase.

Closely associated with the Euler equation is the notion of geometric flow similarity; thevelocity vector triangle formed by the tip speed U , the absolute velocity V , and relativevelocity W (see ?, fig. 12.18 for example) remains similar when flow rate and speed are

18 Chapter 3. Turbomachinery

Figure 3.2: The turbomachinery experiment in 0021 Thomas.

changed. Then it follows that

Vθ ∼ U

Q ∼ UAm

where Am is the meridional cross section of the discharge flow. What follows, then are

∆pt ∼ ρD2ω2

Q ∼ D3ω

Power ∼ ρD5ω3

where D is a reference length dimension such as the diameter of the pipe containing thecompressor flow. By means of these relations, the flow rate, pressure rise, and power can bescaled to a standard reference speed and density. The scaling of these relations implies thatthe effect of Reynolds number is weak; this is approximately so if the Reynolds number(Re = WL/ν) based on blade chord length L, relative flow velocity W , and kinematicviscosity of air ν is about 105.

3.3 Experiment

There are several aims of this experiment: (1) to gain some knowledge of the typical behav-ior of a heavily loaded axial fan as a function of flow and operating speed and to relate thisbehavior to basic fluid mechanics; (2) to learn how to measure fan performance; (3) to learnbasic similarity laws of fan performance; and (4) to make some qualitative dynamic mea-surements of flow phenomena in stall — a subject of much current turbomachine research.

3.3. Experiment 19

3.4 Instrumentation

This experiment is equiped with a water-equivalent manometer, two differential pressuretransducers, and a linear gauge. The manometer is used to calibrate the pressure transducersthat are in turn used to measure the pressure drop across the inlet nozzle and the pressurerise across the fan. The pressure transducers are also capable of resolving time-varyingchanges in the static pressures. The linear gauge is used to measure the throttle opening.

3.4.1 Experimental Variables

The following are the variables which the operator of this system is free to adjust:

• Throttle setting — The throttle plate is adjusted by rotating the plate on threaded leadscrew. Throttle displacements can be measured using the provided linear gauge, toadjust the operating point of the fan. Always start the fan with the throttle plate atleast two inches open. The fan should not be operated at shut-off conditions (i.e.throttle closed) for more than five seconds maximum since flow through the fan isused to cool it; operation at shut-off can lead to thermal overload and damage to thefan.

• Fan speed — The maximum fan speed is about 12,000 rpm (400 Hz drive frequency).For the purpose of this experiment the maximum speed has been limited to 10,500rpm (350 Hz drive frequency). Operation at frequencies above about 200 Hz is verynoisy and the use of provided ear protection is mandatory. The minimum frequency islimited by the ability to measure the pressure differences with the provided pressuretransducers.

to adjust the operating point of the fan. Always start the fan with the throttle plate atleast two inches open. The fan should not be operated at shut-off conditions for morethan five seconds maximum.

• Fan speed — The maximum fan speed is about 12,000 rpm; this results in a verynoisy operation. For the purpose of this experiment the maximum frequency shouldnot exceed 250 Hz. The minimum frequency is limited by the ability to measurethe pressure differences with the provided pressure transducers. The directions forstarting the fan and adjusting the speed are given in the Appendix under “Operatingthe System.”

3.4.2 Inlet Flow

The flow rate entering the compressor is determined by measuring the pressure drop acrossthe inlet nozzle. From Bernoulli’s equation:

ρ(V 22 − V 2

1 )

2= p1 − p2, (3.4)

where V represents the flow velocity, p represents the static pressure, and ρ representsthe density. The subscripts 1 and 2 represent the conditions before and after the nozzle,


respectively. Since condition 1 is the ambient air of the laboratory, V1 is zero; therefore,the velocity downstream of the nozzle (upstream of the fan) is determined by the pressuredifference across the nozzle:

V2 =

√2(p1 − p2)

ρ. (3.5)

So, the theoretical volumetric flow rate Qideal, assuming uniform flow across the pipe, is:

Qideal = AV2 = A

√2(p1 − p2)

ρ, (3.6)

where A is the cross-sectional area of the pipe. However, the real rate is actually less thanthis theoretical value due to boundary-layer effects in the nozzle and other fluid losses. Toaccommodate the difference between the actual and the ideal flow rates, it is customary todefine a discharge coefficient

cd =Qactual

Qideal. (3.7)

The value of the discharge coefficient is determined by making measurements of the actualvelocity profile across the pipe just downstream of the nozzle. This was done before fora very similar facility with the same inlet nozzle and 0.98 was the value found for thedischarge coefficient. Therefore, the actual flow rate for this facility is approximately:

Qactual = cdA

√2(p1 − p2)

ρ, (3.8)

So, at each throttle position, the flow rate through the compressor can be determined fromthe measurement of the pressure drop across the inlet nozzle.

3.4.3 Fan Performance Curve

Incompressible turbomachines, fans and pumps, are tested at constant speed and fluid den-sity; the performance curve is a plot of total pressure rise and input power vs. flow rate allat constant rotative speed and density. We are not able to measure the power in the presentexperiment. In American industrial practice the pressure rise is given in inches of water andthe flow rate in cubic feet/minute and the pressure rise and power are corrected to a standarddensity for air (0.075 lbm/ft3 or 1.205 kg/m3) corresponding to 0.1013 MPa pressure and293 K temperature. In this experiment we will use SI units: pressure rise in Pascals andflow rate in cubic meters/second.

The performance curve for the present experiment is to be measured for three different rota-tive speeds. At each speed, make measurements at a sufficient number of flow rates to mapthe performance curve; start at a wide-open throttle setting and closing the throttle towardthe shut off then increase the throttle opening until the fan returns to normal operation. Usethe inlet nozzle pressure difference to measure the flow rate. As the throttle plate is closed,the fan will eventually reach a point at which the head rise abruptly and drastically falls.This is a result of rotating stall, an instability in compressors. The stall point should bedetermined for decreasing flow rates, and the bottom of the stall region should be found aswell as shut-off point. As the throttle plate is reopened, the compressor will recover fromstall (stall-recovery point), signaled by a rapid pressure rise. It is useful to keep track of thethrottle opening to enable careful mapping of the stall point and the stall-recovery point.

3.4. Instrumentation 21

3.4.4 Unsteady Behavior

At some operating conditions the performance of the fan becomes unsteady and the mea-sured pressures will oscillate. In particular, at the peak of the performance curve (justbefore stall) and in deep stall (the bottom). These pressure oscillations are indicative of aphenomenon known as rotating stall.

The onset of this condition is best visualized in the blade-fixed frame of reference. As thethrottle opening is reduced, the flow rate through the compressor is decreased, so the axialvelocity v is reduced. However, the tip speed of the compressor blades remains constant,since the compressor is running at constant speed. Therefore, the angle of incidence of theair flow with respect to the blades is reduced translating to an increase in the angle of attackof the blades. At larger angles of attack the blades will stall (flow will separate from theupper side of the airfoil shape) just as an aircraft wing will stall at large angles of attack.When a blade stalls, the flow that normally passes through the now-stalled region is divertedto the neighboring blades. The angle of attack of the preceding blade then falls and the angleof attack of the following blade rises. This decreases the tendency of the preceding blade tostall while increasing the tendency of the following blade to stall. Therefore, in the blade-fixed frame of reference, the stalled region of the blade row (stall cell) appears to movein the opposite direction of the compressor rotation. In the lab-fixed frame of referencethe stall cell rotates in the same direction as the compressor, but at a lower speed. As thestall cell rotates around the inlet tube it creates an oscillation in the pressures measured atthe pressure ports. From a measurement of the oscillation frequency of the pressure, it ispossible to determine the speed of the rotating stall cell.

3.5 Measurements

These are the measurements that are to be made on the apparatus:

• Air density — For the purpose of this experiment we may assume the air to be aperfect dry gas with a gas constant of 287 J/kgK (in practice we measure the abso-lute humidity and so determine the effect of water vapor on the density). The localbarometric pressure and temperature are measured on the wall-mounted barometer tocalculate the density.

The pressure rise across the fan is small compared to the absolute pressure (less than2 percent) so that the flow may be considered as effectively incompressible. The“best” effective density is the average of the inlet and discharge densities. For thepurpose of this experiment, the inlet stagnation density may be used as the referencedensity.

• Calibrate the Differential Pressure Transducers — Both pressure transducers shouldbe calibrated using the water-equivalent differential manometer. A lab View VI (Vir-tual Instrument) is provided on the data acquisition system to facilitate this process.The throttle plate is set to a small opening (about 1 cm), the number of desired pointsfor the calibration is entered into the appropriate field on the VI control panel, andthe color code of the transducer to be calibrated is selected. Upon running the VI,the user will be prompted to enter the pressure reading from the manometer into the


appropriate field and click the continue button. The compressor speed will be rampedunder computer control and the user will be prompted for pressure readings until thedesired number of points is acquired. The VI will determine the best fit line to the dataand prompt the user for a pressure that encompasses your expected measurements.

• Performance Curves — Measure performance curves of the compressor at three op-erating speeds. Be sure to carefully resolve the stall onset, stall bottom, and stall-recovery points. A Lab View program may be available to aid you in this process.

• Repeatability — For one of the compressor speeds used above, repeat the perfor-mance curve measurements two more times to assess the repeatability of the mea-surements. Base this repeatability on how well you can reposition the throttle plate.It is not necessary to repeat every point on the original performance curve, chooseabout 5 points that are distributed throughout various portions of the performancecurve.

• Unsteady Behavior — During the measurement of one of the performance curves,record time series of oscillatory pressure near stall onset and in the bottom of stall todetermine the frequency content of these signals. A Lab View VI may be available toaid you in this measurement. If not, temporary use of the oscilloscope to determinethe frequency content is recommended. Look at both the frequency and amplitude.

• Speed Curves (Optional) — If time and group interest permits, modify a copy ofthe LabView VI used for calibration to make measurements of performance along aspeed curve. At a fixed throttle setting (you can repeat for several throttle settingsif you like), step through a range of compressor speeds and measure the differentialpressure. Be sure to wait long enough at each speed setting before measuring thepressures. Take note of any apparent stall phenomena and the throttle opening(s)used . . . it may be necessary to take longer time averages of pressure under oscillatingconditions to get a measure of mean performance. This exercise is intended to giveyou an opportunity to explore the use of LabView in a data acquisition and controlsituation. If you are unable to achieve measured results in a reasonable time, don’tfret, view this as an opportunity to learn more about LabView. Remember this isoptional and it is up to you whether you include any of this in your lab report. If youdo include speed curves in your report, print out your LabView VI for the appendix.

3.6 Report

Your results should include:

1. For each of the three speeds, plot ∆p vs. Q; indicate by error bars the uncertainty ofthe measurements. Use estimates of measurement uncertainties of the instrumenta-tion along with results from your repeatability measurements.

2. Using the scaling laws outlined in Section 2, non-dimensionalize the performancecurves plotted above to test the similarity of the compressor performance.

3. Carefully identify the stall, stall-bottom, and stall-recovery points on your perfor-mance curves.

3.6. Report 23

4. Speed Curves, if you decide to measure and include them. Identify points where stallor rotating stall were observed.

Your discussion should include:

1. Describe the performance curves noting any features of special interest.

2. Discuss the results of the scaling calculations.

3. The design flow rate for the fan is 0.566 m3/s (1200 ft3/min) when the exciting fre-quency is 400 Hz. Predict the performance of this compressor, based on a similaritydiscussion, using your acquired data.

4. If you decided to measure and include speed curves, describe these curves and noteany features of special interest.


Figure 3.3: Test Set-up Schematic

3.7 Operating the System

There are several details concerning the actual operation of the fan rig which require atten-tion before actually attempting to operate the system.

The axial fan, an Able 29680, is driven by an electric motor. It is designed to run continu-ously at 11700 rpm, to draw 2700 Watts of power, and 10 Amps of current. Its commercialuse is in the cooling of the avionics (the control circuitry) of an MD-11 Aircraft.

In the ME 96 Experiment, you will be running the fan at speeds slower than this. (Whenrunning full speed, the fan produces a great deal of noise). The power supply, since itis variable frequency, can run the fan at most any speed. This is done by changing thefrequency of the input power, which (since the fan uses an induction motor) will change therotor speed. In fact, the fan impeller will run at a speed half the speed of the input powerfrequency. Standard airplane power frequency is 400 Hz, which corresponds to a fan speedof 12000 rpm (nominal) which is 200 Hz. The power supply outputs power up to 400 Hz.

To actually operate the system, follow these steps:

1. Turn on the LARGE wall switch to allow current to flow to the power supply (motorcontroller). Then turn on the power toggle switch on the tethered remote control boxmounted to the same table as the axial fan.

2. Use the switch on the side of the remote control box to select manual or BNC (re-mote) control of the fan speed. Under manual (knob) control turn the potentiometer

3.7. Operating the System 25

knob until the desired frequency appears on the LED readout on the power supply(motor controller). Under BNC control, a voltage (0 – 8 Vdc) is supplied to the BNCconnector (usually from a D/A channel of the data acquisition system) to set the drivefrequency in the range of 0 to 350 Hz.

3. Turn off the power toggle switch on the tethered remote control to shut down the fan.This will signal the power supply to automatically ramp down the drive frequency.

4. Be sure to turn the LARGE wall switch off only AFTER the fan has come to a com-plete stop.

Note that the fan speed may be changed while the fan is running by simply following steps2 and 3. The power supply will automatically ramp the voltage in the appropriate way.Please see the precautions section for cautionary information.

3.8 Precautions

The fan rig uses a significant amount of power and includes some very high speed rotatingmachinery. Therefore it is very important to handle the entire system with care to avoiddamaging the setup and, more importantly, to ensure operator safety. There are severalprecautions to take whenever using the setup which are described in the following list.

1. Do not let anything come close to the inlet of the nozzle. Loose items could be suckedinto the impeller, damaging it. A flow straightener is located just inside the nozzleto prevent large items from entering, but these large items may become lodged in theinlet. Items small enough to fit through an 8 mm square hole could pass through thestraightener and damage the impeller.

2. Do not turn off the LARGE switch on the wall that provides power to the powersupply (motor controller) before shutting the fan off from the tethered remote control.Turning off the power switch on the tethered remote control causes the power supplyto ramp down and safely shut off the fan.

3. There are several wires and hoses which run from the facility to the power supply,monometer, and computer. Use caution when moving around the rig to avoid disturb-ing these connections.

4. Do not run the fan with the throttle closed for any significant amount of time. The fanrequires flow through it to expel the generated heat. Running with the throttle closedcan cause the fan to overheat.

5. Avoid running the fan at too low a speed (say lower than 66 Hz drive frequency or2000 rpm). This may not produce enough flow for the fan to cool properly.

6. Do not start the fan in a throttled condition. Starting up with high losses in the system(e.g. a closed throttle) can cause undue stress.

7. Avoid leaning on, sitting on, or manipulating the fan. This may cause misalignmentof the assembly.


8. When adjusting the throttle plate, be sure to avoid positioning your head (especiallyyour eyes) in the flow. The flow velocity is high and the potential for injury exists.

9. Wear hearing protection while operating the experiment. Headsets are providedin the room.

3.9 Prelab

It may be helpfuul to refer to a fluid mechanics textbook that has a chapter on turbomachin-ery, such as ?. (There are some in 0018).

1. Derive the Euler Turbine Equation.

2. Discuss what is meant by geometric flow similarity, and draw a sketch showing thevelocity vectors.

3.9. Prelab 27

CHAPTER

FOUR

Free and Forced Convection

4.1 Introduction

The term forced convection refers to heat transport that results from fluid motion caused byexternal means such as a pump, a fan, or atmospheric winds. The flow velocity depends onthe fluid mechanical properties of the system and not (ideally) on the heat transfer processesthat occur in the system. Typical examples of forced convection include forced-air heatingof homes, liquid filled active solar heating systems, and nuclear reactor cooling (duringnormal operation).

For free or natural convection, the flow velocity depends on both the fluid mechanicalproperties of the system and the heat transfer processes that occur. Free convection isdriven by buoyant forces that result from density differences in the convecting fluid. Inmost situations, the density gradients are caused by temperature variations in the fluid.Examples of free convection include a single-phase closed loop thermosyphone (roughlythe condition in a nuclear reactor immediately after pump failure), a Trombe wall (a passivesolar heating device), and the shimmering visible above a paved highway on a hot summerday.

The essential difference between free and forced convection manifests itself in the govern-ing equations of the two convection modes. The boundary layer equations for laminar freeconvection over a heated vertical surface are (neglecting viscous dissipation and pressuregradients):

Energy: u∂T

∂x+ v

∂T

∂y= α

∂2T

∂y2

Momentum: u∂u

∂x+ v

∂u

∂y= ν

∂2u

∂y2+ gβ(T − T∞)

(4.1)

where β is the coefficient of thermal expansion, β = −1/ρ(∂ρ/∂T )p. The importantpoint to be noted about these equations is that they are coupled. For forced convection themomentum equation does not include the buoyancy term, and hence does not depend on thetemperature gradients within the flow.

29

Figure 4.1: The free and forced convection experiment.

Figure 4.2: Experimental apparatus.

30 Chapter 4. Free and Forced Convection

Figure 4.3: Plate and cylinder thermocouple and heater layout.


4.2 Experimental Apparatus

This experiment is designed to study both free and forced convection heat transfer using aheated plate and a heated cylinder. A schematic of the experimental apparatus is shown inFig. 1. The plate and the cylinder are shown in Fig. 2. Either one can be mounted above thefan and the honeycomb flow straightener. The plate can be oriented at different angles tothe oncoming flow. The plexiglass side walls are to minimize disturbances from the room.

The heat input is regulated by separate control of three nominally 1 in. × 5 in. kapton-insulated flexible heaters. The power dissipated through each heater is controlled using aten-turn potentiometer mounted in the control panel. The power can be calculated from thevoltage input as displayed on the control panel, and the heater resistance (60Ω± 1%). Themaximum input voltage is 21 volts. The central heater serves as the experimental heater,and the two side heaters act as guard heaters. The power input to the guard heaters shouldbe adjusted so that there is no lateral temperature gradient across the central heater, andhence no heat loss from the central heater to the sides and the metal supports. Typically,the voltage input to the side heaters should be 3 to 7 volts higher than to the central heater.The temperature gradient across the central heater can be monitored using the three alignedthermocouples.

Chromel-alumel (“type K”) thermocouples are attached to the heated surfaces as shown inFig. 2. There is an additional thermocouple to measure the ambient temperature. The ther-mocouples are connected to a USB data acquisition unit designed specifically for acquiringthermal data from thermocouples and other temperature sensors. Thermocouples 1 through6 are connected, in numerical order, to channels 0 through 5 of this unit. A thermocoupleto measure the ambient air temperature is connected on channel 6.

In addition to the thermocoupls, a heat-flux gauge is mounted on one surface of the plate.The calibration for the heat flux gauge is in the lab folder. The output for the heat flux gaugeis read on the analog microvolt meter. In the range of 0–3 mV, the meter has an accuracy of±5%.

The fan is controlled by a DC power supply. The flow velocity is measured using a hot-wireanemometer. The anemometer can be inserted through the collar mounted on the plexiglasswall. Be careful not to break the wire, which is very fragile!

4.3 Data Acquisition

The acquisition of the thermocouple data for this experiment is to be done using LabVIEW.In addition to the primary objetive of investigating convective heat transfer phenomena, asecondary objective of this experiment is to teach you something about modern, computer-ized data acquisition methods.

You must have worked through the LabVIEW tutorial or completed the LabVIEW home-work assignment before beginning this lab. If you have not yet done so, do this beforemeeting with the TA to begin this experiment.

You are provided with a small USB data acquisition unit that is designed specifically foracquiring thermal data from thermocouples, thermistors, or other types of temperature sen-sors.


But you should configure it in software. Run the “InstaCal” program (under “MeasurementComputing” on the Windows “Start” menu) on the lab PC to configure, calibrate, and testthis unit. Make sure the channels you will use are configured for Type K thermocouples.Within InstaCal, you can access from the Help menu the manual for this unit (model USB-TEMP). Once you are confident the unit is operating as expected, you can start writing aLabVIEW VI to use it to make the desired measurements for this experiment.

Start LabVIEW 8.2, create a new, blank VI, and view the block diagram window. Onthe “Functions” palette, select “User Functions,” where you will find the MeasurementComputing drivers. Under “Analog Input,” you will find one called “Tin.” This functionwill acquire a temperature reading from the USB-TEMP unit. This should be all you needfrom this palette to build your VI.

Your LabVIEW VI should do the following:

1. Continuously monitor the temperature for all thermocouples, and alert you in someway when the temperatures have stabilized sufficiently to begin measurements. (Per-haps it could generate a sound, or send you an e-mail message, or...)

2. Provide a convenient display of the three horizontally-spaced thermocouples to watchwhile adjusting the guard heater voltages to minimize the ∆T in this direction.

Optionally, you can acquire data from the heat flux sensor also using LabVIEW.

4.4 Experiments

4.4.1 Free Convection

Begin by orienting the plate in the vertical position, and do not use the fan. Set the powerto each of the heaters and monitor the surface temperature until it reaches a steady state.Use power settings close to the maximum value. This warm up process may take about 40minutes.

After the heater reaches a steady state, record the surface temperatures and the ambienttemperature. Record the voltage input to the heaters, and the output from the heat fluxsensor. From the measurements of the heat input one can calculate the surface heat flux, q′′

in W/m2 by dividing the input power by the surface area. Remember that the heater has twosides. Check to see if the two methods for determining q′′ agree. If the difference is morethan 10% and the measurement from the heat flux gauge is lower, reduce the heat input tothe central heater.

Without changing the input power, change the plate orientation. Again allow the surface toequilibrate. Record the temperatures and the voltage output from the heat flux gauge. Tryat least two different orientations.

Natural convection experiments should be performed in quiescent environments. Roomcurrents will affect the results.

4.4. Experiments 33

4.4.2 Forced Convection

Keep the input power at the same value. With the plate in the vertical position, turn onthe fan to a low speed. Use the anemometer to record the incoming flow velocity over thecentral heater. Allow the plate to equilibrate, and record the temperatures and the heat flux.Record the conditions for at least three different fan speeds. The warm-up time should befaster than for the free-convection conditions.

Also try at least one experiment with the plate at a different inclination.

4.4.3 Flow Over a Cylinder

Disconnect the plate and connect the cylinder. Orient the cylinder so that there is a thermo-couple directly at the bottom and at the top. Repeat the measurements for natural convectionand for forced convection using several different flow speeds.

There is no heat flux sensor in the cylinder. Start with the same heat input settings used forthe forced convection experiment; however, the settings may need to be adjusted.

4.5 Heat-Transfer Predictions

The Nusselt number, Nu is defined as follows

Nu =h`ck

=q′′`c

(Tw − Ta) · k(4.2)

where h is the heat transfer coefficient, `c is a characteristic length, Tw and Ta are the walland ambient temperatures, and k is the thermal conductivity of the fluid. The Nusselt num-ber can be either a local value, Nux where `c is the distance from the beginning of the plateto the location of interest (`c = x) and Tw is the temperature at that location Tw = T (x), orthe average value, Nua, using the total plate length (`c = L) and the average temperature.The average temperature is often difficult to define, and so the midpoint temperature (Tw atx = L/2) is often used.

For forced convection, the Nusselt number is presented as a function of Reynolds number,Re = u∞`c/ν, where u∞ is the approach velocity, and ν is the kinematic viscosity. For freeconvection, the appropriate number is the Rayleigh number, Ra = gβ(Tw−Ta)`3c/να. Thecoefficient of thermal expansion for an ideal gas is equal to the inverse of the absolute fluidtemperature. All properties should be evaluated at the film temperature, Tf = (Ta+Tw)/2.

For laminar forced convection (ReL < 5 × 105), over a constant heat-flux surface, thetheoretical local Nusselt number is

Nux = 0.453Re1/2x Pr1/3 (4.3)

and the average value isNua = 0.679Re

1/2L Pr1/3 (4.4)

Pr is the fluid Prandtl number, Pr = 0.7 for gases.


For laminar free convection of air over a vertical plate (0 < RaL < 109), the average valueis

Nua = 0.68 + 0.52Ra1/4L (4.5)

For flow over a heated cylinder, the characteristic length is the cylinder diameter. TheNusselt number based on the cylinder diameter for air flow is (10−5 < RaD < 1012)

Nua =[0.6 + 0.32Ra

1/6D

]2(4.6)

For forced convection, the Nusselt number for air flow is (ReDPr > 2)

Nua = 0.3 + 0.54Re1/2D Pr1/3

[1 +

(ReD

28200

)5/8]4/5

(4.7)

Remember that the heated surfaces lose heat by conduction, convection and radiation. Theeffects of conduction are minimized using the guard heaters. The radiation loss, however,should be accounted for. The radiation loss from a surface can be estimated from the fol-lowing equation:

q′′ = εσ(T 4w − T 4

a ) (4.8)

where ε is the surface emissivity, and σ is the Stefan-Boltzmann constant.

4.6 Lab Report

1. For the natural convection results, plot the average Nusselt number as a function ofinclination angle. For each inclination you can obtain two Nusselt numbers corre-sponding with positive and negative angles from the vertical. Also show on the graphthe theoretical value for a vertical plate. Why are there differences?

2. For the forced convection results, plot the average Nusselt number as a function ofReynolds number. Show theoretical values. Also indicate results from the differentorientations.

3. For the cylinder, plot the Nusselt number as a function of Reynolds number. Show thetheoretical values. Determine the Nusselt number for natural convection and compareit to the theoretical value.

4.7 Discussion

In the discussion section of the lab report, the following issues should be addressed.

1. How large is the radiation contribution for each of the flows?

2. How well do the theoretical and experimental values compare? What are possiblereasons for the differences?

3. How does the angle of attack of the plate affect the heat transfer?

4.6. Lab Report 35

4.8 Prelab

Before beginning the lab, you need to have worked through the LabVIEW tutorial on thewiki, and have read the thermocouple tutorial, also on the wiki.

Answer the following questions in your lab notebook before the first lab session.

1. What is the Prandtl number of air at 300 K? What is it for water?

2. Estimate the Rayleigh number for air at 1 atm pressure, 300 K, for a length scale of1 cm, and a temperature difference of 10 C. Repeat the calculation for water.

3. Using the parameters in the last question for air, calculate the heat loss by free con-vection from a 1 cm radius heated cylinder. How large an air speed transverse to thecylinder would be required to achieve the same heat flux by forced convection?

4. Describe how a hot-wire anemometer measures air speed.

5. How do you think the heat flux gauge works?

It may be helpful to refer to a heat transfer textbook, such as ? or ?.


CHAPTER

FIVE

Turbulent Air Jet

5.1 Introduction

If you have taken a course in fluid mechanics, you probably learned all about laminar flows –laminar boundary layers, laminar flow in tubes, etc. Unfortunately, in most real engineeringapplications this is not at all the sort of flow that is observed. Instead, the flow is turbulent,consisting of a cascade of eddies, resulting in a chaotic, unsteady flow.

Turbulent flow is much more difficult to characterize than laminar flow, and although agreat deal has been learned about many of the properties of turbulent flows, we still cannotpredict from theory some features of turbulence that are easy to observe. For example,the spreading angle of a turbulent jet (see Figure 5.1) is easy to measure experimentally,but cannot be predicted from theory, even though we know the governing equations ofthe jet (the Navier-Stokes equations)! One problem is that a direct simulation of the flow,resolving even the smallest eddies, is still too large a calculation even for the fastest parallelcomputers today, at least for Reynolds numbers of practical interest. So much effort hasgone into developing approximate, statistical theories of turbulence, but so far there is noapproximation that is good for all turbulent flows.

Much of what we know about turbulence comes from careful study of model flow problems.One of the classic model problems is that if a round jet, issuing into still air. This is the flowthat you will investigate in this lab.

Figure 5.1: A turbulent jet.

37

Figure 5.2: Notation for the jet experiment.

5.1.1 Background

For any turbulent flow, the instantaneous velocity fluctuates in time at every point in theflow. But for steady turbulent flows, the time-averaged mean velocity is independent oftime:

u(x, r) =1

T

∫ T

0u(x, r, t′)dt′. (5.1)

Here the averaging period T is chosen to be long enough to capture the motions of verymany eddies. For a truly steady turbulent flow, u is independent of T , as long as it is“sufficiently large.”

We define the fluctuating or turbulent velocity by

u′(x, r, t) = u(x, r, t)− u(x, r). (5.2)

Of course, by this definition ∫ T

0u′(x, r, t′)dt′ = 0. (5.3)

We will use the notation shown in Fig. 5.1.1. All mean flow quantities are assumed to beaxisymmetric, so that all mean quantities depend only on the axial distance downstream ofthe nozzle x as the radial distance r from the jet centerline. We will use δ(x) to denote thejet width at a given axial distance. While this could be defined in many ways, it is usuallytaken as the radial distance from the jet centerline to the point where the mean axial velocityhas dropped to half its value on the centerline (half-velocity point).

For a narrow jet issuing into still air, we can make the following assumptions:

1. The flow is nearly parallel and the inclination of the (mean) streamlines is small.This assumption requires v u, or, equivalently, a negligible transverse pressuregradient.

2. The jet is issuing into infinite space, and therefore there is a negligible axial pressuregradient. (If, however, we were to place an object into the jet, the pressure would rise

38 Chapter 5. Turbulent Air Jet

as the stagnation point on the leading edge is approached, in order to decelerate theflow and bring it to rest there.)

These assumptions clearly define a uniform pressure field. Further, we assume the jet mo-mentum flux remains constant at all x:

J = 2π

∫ ∞0

ρu2rdr = constant = J0 (5.4)

where ρ is the density.

We want to determine the jet growth law – that is, how δ and u(x, 0) depend on x. Wecan determine the functional form, just from dimensional analysis, without even solvingthe governing equations (which are too hard to solve anyway).

An important non-dimensional parameter in any flow is the Reynolds number. By consid-ering the momentum flux at the nozzle exit, it is easy to show that the Reynolds number fora round jet can be written as

Re =J

ρν2(5.5)

(to within a factor of π/4, which for the present analysis is close enough to 1 to neglect).

The Reynolds number is the only non-dimensional parameter that we can form from molec-ular parameters (ρ, ν) and conserved quantities (J).

Now consider how to non-dimensionalize δ. The only available length scale to use as areference is x itself (we assume we are far enough downstream of the nozzle that the nozzlewidth is not an important parameter), so we conclude that

δ

x= fn(Re). (5.6)

(Here “fn” means “a function of”).

Dimensional analysis tells us that δ/x can be written as some function of J/ρν2, but it can’ttell us what the form of that function is. For a turbulent flow, we are particularly interestedin the limiting behavior for J/ρν2 1. There are three possibilities – δ/x might divergeas J/ρν2 →∞, or it might go to zero, or it might asymptotically approach a constant.

We can rule out the second possibility right away, since we know the jet spreads, it doesnot contract. We can also rule out a diverging function, since in that case very small valuesof viscosity would have a large effect on δ/x. In reality, most aspects of a turbulent flowbecome independent of ν, since the churning eddy motion of turbulence is a much more ef-fective way to disperse fluid momentum than is molecular viscosity. So the only possibilitythat is consistent with observations is that the function approaches a constant. Therefore, ina turbulent flow,

δ

x= C, (5.7)

where C must be determined from experiment.

Now to non-dimensionalize u(x, 0), we need to form a quantity with units of m/s from theavailable independent variables and parameters. We could choose ν/x, which would beappropriate for a laminar flow, but this would be a poor choice for a turbulent flow, sincewe know ν is not an important parameter for most aspects of a turbulent flow. We can form


a quantity with velocity units that is independent of viscosity using the Reynolds number:(ν/x)

√(J/ρν2) =

√J/ρ/x.

Therefore, we construct the non-dimensional velocity

u(0)x√J/ρ

, (5.8)

and require that it too, like δ/x be a function only of J/ρν2. And, following the samearguments as above, we require that this approach a constant in the limit of high Reynoldsnumber, with the result

u(0)x√J/ρ

= B, or u(0) =B

x

√J

ρ. (5.9)

The constant B must also be determined from experiment.

Equations (5.7) and (5.9) show that the jet width increases linearly with x, and the centerlinevelocity decreases with x−1. The constants of proportionality, C and B, can be determinedexperimentally.

The same arguments we used for u(x, 0) could have been applied to the mean velocityat any other radial location u(x, r), with the only difference that we would have had tointroduce another non-dimensional parameter x/r.

As a result, we can write

u(x, r)

u(x, 0)= f1

( rx

)or u∗ = f1(r

∗) (5.10)

which defines u∗ and r∗.

The concept of self-preservation in turbulent flow requires the turbulent time scales be muchsmaller than the mean flow time scales. Again we normalize with the local conditions togive:

u′RMS

um= f2

( rx

)(5.11)

where u′ is the RMS value of the fluctuating velocity component, defined as

u′RMS =

[1

T

∫ T

0(u′)2

]1/2(5.12)

and um is the centerline velocity.

Note that since we neglected the nozzle diameter in this analysis, the jet velocity profileclose to the nozzle will not be similar to profiles far downstream, because near the nozzlethe diameter is an important parameter. The flow does not become self-preserving until wereach the “far field” of the jet.

5.2 Instrumentation

In the experiments, a pitot tube and a hot wire anemometer will be used to measure thevelocity. The pitot tube is robust and inexpensive, and the anemometer is fragile and ex-pensive. The benefit of the anemometer is that the transient characteristics of the flow can


Figure 5.3: Airjet experiment.

be measured, which cannot be obtained with the pitot tube. In the experiments take greatcare in handling the anemometer. If the wire is broken, it must be sent back to the vendorfor repair, which is expensive and time consuming.

5.2.1 The Pitot Tube

The pitot tube measures the total or dynamic pressure of the flow. At a point in the flow,the relation between the total pressure, Pt, the static pressure, P , and the velocity, u, can bedetermined from Bernoulli’s equation:

Pt = P +1

2ρu2 (5.13)

where ρ is the density of the fluid. The static pressure is the pressure measured for no-flowconditions. The total pressure is measured using a manometer.

5.2.2 The Hot-Wire Anemometer

The hot wire anemometer is based on principles of heat transfer. An energy balance on theelectrically-heated wire yields,

V 2

R= h(u)A(Tw − Ta) (5.14)

where V is the voltage, R is the wire resistance, h is the heat transfer coefficient (whichdepends on the flow speed u), A is the wire surface area, and Tw and Ta are the wire andair temperatures.

For a constant-temperature anemometer (CTA), the CTA electronics use closed-loop con-trol to maintain the wire resistance at a fixed value (and thereby maintain the wire tem-perature fixed) by continually adjusting the electrical current to the wire is continuously

5.2. Instrumentation 41

adjusted to maintain the resistance of the wire at a fixed value. Therefore, if the flow isisothermal, then the temperature difference between the wire and the flow is a constant.

The CTA voltage is then given by

V 2 = Rh(u)A∆T (5.15)

The dependence of the heat transfer coefficient on flow speed can usually be represented by

h = c1 + c2un (5.16)

where n ≈ 1/2. In this case, the speed is related to the measured voltage by

u = A1

(V 2 − V 2

0

)1/n (5.17)

where A1 is a constant, and V0 is the voltage reading in still air. If n = 1/2, then this rela-tionship is a fourth-order polynomial. In practice, a CTA should be calibrated by measuringthe voltage for a few different known speeds (measued, for example, with a pitot probe),and fit to a polynomial.

5.2.3 Data Acquisition and Stage Control

LabView will be used to acquire the data from the hot-wire anemometer and from the Pitotprobe. The probes are mounted on two orthogonal motorized translation stages, allowingthem to be positioned vertically and horizontally under computer control. The data acqui-sitioning and stage control are accomplished with the same VI (provided).

The main parameters of the DAQ that can be changed are the sampling frequency and thenumber of samples. Note that two channels are connected, so the number of samples foreach channel is different.

The VI controls the horizontal and vertical stages on which the probes are mounted. Thestages can be controlled in 3 modes:

1. Set the horizontal spacing, vertical spacing, and dwell time, and let the VI control thesystem taking measurements at each location automatically

2. Control the stages manually using buttons

3. Control the stages manually using command-line inputs

The speed of the stages can be set, with an available range of 0.025-20 rev/s. This canbe controlled either from the System Configuration tab, or from the command line on theCommand tab. The set of available commands for command line control is given in table.Note that the encoders measure relative to some home location. Thus, to command thestages to a specific location, the stages must be commanded to the ”Home” location.

5.3 Experiment

In this experiment, you will use a hot-wire anemometer and pitot tube to measure velocity offlow in a turbulent jet. The hot-wire anemometer will be used to measure the both transient


and mean velocity profile, while the pitot tube is used to measure the mean velocity. Thepitot tube and anemometer need to be calibrated. The anemometer is used to measure theshedding frequency from a bluff body immersed in the flow. The flow velocity profile (meanand fluctuating velocities) are measured at different distances from the nozzle.

5.3.1 The Pitot Tube Calibration

The pitot tube is connected to a pressure transducer which converts the pressure to a volt-age. This transducer is in turn connected to the DAQ board. The conversion factor fromtransducer voltage output to flow velocity, u(V ), needs to be measured. To do this, manu-ally control the stages to position the pitot tube in the center of the jet just above the nozzle.Take care that the probe do not hit the nozzle! Take total pressure measurements, Pt, atdifferent flow speeds (controlled by throttling the air intake). One measurement should betaken with the fan off (to get the static pressure, P ). The total pressure should be read offthe manometer (the manual is the the lab-folder). At the same time, the voltage should berecorded. The flow velocity can be calculated as:

u =√

2(Pt− P )/ρ (5.18)

1. Record data.

2. Using Excel, MATLAB, or other means of your choice, generate an appropriate poly-nomial curve fit for u(V ).

3. Include a plot of this fit and your velocity vs. voltage data in your lab report. Makesure the curve is a good fit to the data, and is monotonic.

5.3.2 The Anemometer Calibration

The objective is now to use the calibrated pitot tube to calibrate the anemometer. Carefullyposition the anemometer just above the nozzle by traversing the stages vertically and hor-izontally. Measure the time-average voltage from the anemometer. Move the pitot tube tothe same position and measure the flow velocity. The Pitot tube has a slow time response,since the tube is small. Make sure you wait long enough that the pressure signal has stabi-lized. Make sure to take measurements at several different intake valve throttle positions,ranging from wide-open, to nearly closed. Also, take one CTA voltage measurement withthe pump off.

1. Record data.

2. Using Excel, MATLAB, or other means of your choice, generate an appropriate poly-nomial curve fit for u(V ).

3. Include a plot of this fit and your velocity vs. voltage data in your lab report. Makesure the curve is a good fit to the data, and is monotonic. If you find you need moredata points to determine the polynomial, take more readings until you are satisfiedwith the accuracy of your curve fit.

4. Enter the polynomial coefficients for your fit into the appropriate place in the VI(System Configuration tab).

5.3. Experiment 43

5.3.3 Vortex Shedding

Vortex shedding from a bluff body immersed in a cross flow leads to a periodic disturbanceat the vortex shedding frequency f . The Strouhal Number Sr is defined as

St =fD

U(5.19)

where f is the vortex shedding frequency,D is a characteristic length, andU the flow speed.For a long cylinder, with D equal to the cylinder diameter, the Strouhal number is given bythe empirical formula

St = 0.198(1− 19.7/Re) (5.20)

At some height in the flow, place a stretched-out paperclip or other wire that can be modeledas a cylinder across the flow. Two mounted wires are provided for this purpose. Positionthe anemometer probe approximately 5 cylinder diameters behind the cylinder using thecommand-line control of the probe. With the anemometer you should be able to measurethe turbulent fluctuations caused by the cylinder, and by examining the power spectrum youcan determine the shedding frequencies. Theoretically, the frequency should correspond toa Strouhal number of approximately 0.2 where S = fD/u and f is the shedding frequency,D is the cylinder diameter, and u is the approach velocity of the air stream.

The provided VI has a frequency analysis tab to do this analysis. .

1. Experiment with the averaging in the frequency analysis to see the effect on the plot.

2. Record the list of command-line inputs to move from the home location to the appro-priate location.

3. Compare the power spectrum with the cylinder in the flow to that obtained withoutthe cylinder.

4. What is the experimental Strouhal number?

5. Repeat frequency measurement for different flow velocities and plot the measuredshedding frequency vs flow velocity.

5.3.4 Mean and Fluctuating Velocity Profiles

Adjust the pump inlet to an intermediate value and leave it at that position for the remainderof the experiment. Using LabVIEW, traverse the probe across the jet at a fixed height andmeasure profiles of the mean and fluctuating velocity. Repeat the measurements for severaldifferent heights, from the near-field just above the nozzle, out into the far field (or as highas you can go with the translations stage).

Make the measurements at a sufficient number of heights so that you can generate a 2Dmap of mean velocity and fluctuating velocity. The scans do not need to be equally spacedin height – you will probably need more closely spaced ones near the nozzle.

1. Record mean velocity vs position data for both the anemometer and the pitot tube.


2. Record fluctuating velocity vs position data for the anemometer.

Compare your results to expectations based on theory discussed above. Your discussionshould include, but not necessarily be limited to, the following points:

1. Test whether self-similar mean velocity profiles are obtained: make a plot ofu(x, r)/u(x, 0) as a function of r/x, where u(x, 0) is the centerline velocity at heightx. Include the results for multiple heights on the same plot. If the profiles are self-similar, they will collapse to a single curve.

2. Do a similar analysis of the fluctuating component u′. Does the fluctuating velocitybecome self-similar at the same distance from the nozzle as does the mean velocity?

3. Is the jet momentum actually constant, as assumed in the theory?

4. Does the far-field jet width grow with distance as expected?

5. How far downstream (measured in number of nozzle diameters) do you need to go toreach the far-field?

6. What isC? What isB? Compare to values from Ref. 1: C ∼= 0.084 (Fig. 1),B ∼= 6.1(Fig. 3). To find B, use J0 = ρu20πr

20 where r0 is the jet radius and u0 is a constant

velocity across the diameter.

5.4 Prelab

1. Describe how a constant-temperature hot-wire anemometer works. Why is the wireso fine?

2. Calculate the Reynolds number based on jet diameter of an air jet with nozzle diam-eter = 2 cm and velocity = 40 m/s.

3. Calculate the expected vortex shedding frequency for a 1 mm diameter cylinder im-mersed in a 30 m/s flow.

4. Modify the spectralTurbAirJet.vi to perform a frequency analysis on a signal gener-ated using the function generator.

• Connect the function generator to channel 1 of the NI Scope DAQ on the can-talever beam experiment.

• Generate a sinusoidal signal at roughly 2000 Hz.

• Modify the VI to perform a spectral analysis on the signal and plot the output.Include a printout of the modified front-panel and block-diagram (screenshot).

• Verify that the spectral analysis gives reasonable results (since we know theinput frequency and amplitude).

• Experiment with the sampling frequency and number of samples to see effectthese have on the measured frequency. Include at least one sampling rate that isbelow 4000 Hz. Plot this data.

5.4. Prelab 45

5.5 Air Properties

T (C) Density (kg/m3) Viscosity (Pa-s)0 1.294 1.72× 10−5

50 1.093 1.95× 10−5

100 0.947 2.17× 10−5

References

1. Wyganski, I. and Fielder, H. (1969) “Some Measurements in the Self-Preserving Jet,”JFM, Vol. 38, Part 3, pp. 577–612.

2. Sabersky, R. H., Acosta, A. J., and Hauptmann, E. G. (1989) Fluid Flow, 3rd edition,MacMillan Publishing Co.

3. Dantec literature on hot-wire anemometry.

4. Schlichting, H. (1979) Boundary-Layer Theory, 7th edition, McGraw-Hill, pp. 21–32.

5. White, F. M. (1991) Viscous Flud Flow, 2nd edition, McGraw-Hill, pp. 470–476.


5.6 Appendix: Motion control

A goal of this lab is to measure the air velocity as a function of position above the nozzlei.e. the velocity field. To start, we can assume that the jet is axially symmetric so that weonly need to measure in single plane above the edge of the nozzle. As you would expect,we will define the r direction as perpendicular to the symmetry axis of the nozzle (the flowdirection) and passing through the center of the nozzle. We will define the x direction asparallel to the symmetry axis and also passing through the center of the nozzle. These axiswill form the r-x plane in which we make our measurement.

We will measure the air velocity using two instruments. The first probe is a hot-wireanomometer, which is a direct measure of air speed, and the second probe is the Pitottube, which is measure of the local air pressure and is an indirect method of determiningthe local air velocity. The Pitot tube is connected in parallel to an electronic pressure trans-ducer and an old-fashioned water manometer. Don’t be fooled. The latter pressure gage ismuch more sensitive than the electronic gage! Be aware that the dynamic response of thePitot is much less than than that of the hot-wire anomomter. Both the hot-wire anonometerand the electronic pressure transducer are fed to the USB DAQ card on channels 1 and 0,respectively.

Since both the hot-wire anonometer and the Pito tube measurements are point-wise i.e. thevelocity in the immediate vicinity of each the probe, we will need to sample the r-x plane tofully determine how the velocity field and how it changes with the addition of disturbances.Sampling of the velocity field could of course be done in an arbitrary point by point manner.However, it is interesting to measure a velocity profile of the jet. That is to say, the change inthe velocity in the r direction as the position of x is held constant. An example of a velocityprofile is shown in Figure ??. This velocity profile was taken close to the nozzle, i.e. smallx, and the flow region is well defined and uniform. Note that in this case we measuredthe air velocity with time as the stage moved in the r direction; the velocity profiles youmeasure will of course need to be measured with respect to position. As you will see later,velocity profiles are a powerful tool for characterizing the air jet. By combining velocityprofiles for a range of x, we can construct the entire velocity field of the jet.

In past ME96 labs, velocity profiles were measured by attaching the probes to a translationstage that moved in the r direction. The stage was driven by a hand crank and the positionwas read from a mechanical indicator. The x position could only be set much less pre-cisely using a coarse rail guide. This year we have replaced both the manual r-x translationmechanisms with computer controlled stepper motor linear translation stages. These willallow you to make very accurate measurements of the field, and because they are computercontrolled the entire measurement process can be automated using LabView.

5.6.1 Motion Control

Stepper Motor Linear Translation Stage

A stepper motor linear translation stages is shown in Figure ??. Conceptually, the stageconsists of a carriage (i.e. where the probes are mounted) attached to a pair of rails vialinear bearings. The linear bearings allow the carriage to slide along the rails with minimalfriction. (A very important feature of linear bearing is that there is little side-to-side “play”

5.6. Appendix: Motion control 47

or “slop” so that accuracy is guaranteed.) The motion of the plate is controlled by ball screwthat passes through a threaded mount that is attached to underside of the carriage. The screwin turn is driven by a stepper motor. As the motor turns, the plate moves precisely along therails.

As the name implies, a stepper motor rotates in discrete steps i.e. angular increments. Thisis unlike a simple DC motor where rotation occurs in a continuous fashion. The discretesteps are controlled by current pulses sent to multiple electromagnets inside the motor froman external controller. Each step is some small fraction of a revolution, and the amount ofrotation will depend on parameters set by the controller for individual motor. The pitch ofthe lead screw i.e. the threads per inch, attached to the stepper motor determines how farthe stage will move with each revolution. Using this and the angular rotation per step, youcan determine how far the stage is translated per step. For our stages the screw pitch is 2mm. The controller can be set between 2000 and 50,800 steps per revolution. However theabsolute resolution of the stage is limited mostly by play in the ball screw to about 50 µm.

Stepper Motor Controller

A special controller is needed to activate and manage all the necessary hardware settingsand activation required to control each stage. The stepper controller we are using, theSi3540, can itself be controlled using commands sent from the computer using the RS232serial protocol. Si3540 controller, Figure ??, can be thought of as the mediator between thecomputer and the stepper motor stage both receiving commands as well as responding toqueries. The controller comes with two canned software packages. The Si Programmer andthe SCL program. The Si Programmer package is best used for setting motor parameterssuch as the current and the phase. The motion program is included with this package isbizarre and should be avoided if possible. The second software package is SCL. This issimply a command terminal for sending and receiving ASCII commands and responses.We have written an equivalent LabView VI that you can use for your experiments.

Motion Commands

The Si3540 controller can receive commands as well as respond to queries from the com-puter. For example the “set position” command SP can be used to arbitrarily set the positionof the stage in steps. By sending SP0. The step count is set to 0 at that stage position. How-ever, by sending just SP, the controller will read and send the current position from thestage to the computer. As mentioned above, commands are sent as ASCII characters. thereturn character \r. The communication uses the RS-232 protocal i.e. serial lines. There isa separate Si3540 controller for each translation stage, and in our setup COM4 and COM5are used for the x and r axis, respectively.

There are two main modes for operating the motors. Steps can be taken in discrete amounts,which is in the Si3540 command language is referred to as “feed length” or steps can befed continuously until, which is universally referred to as “jogging”. To move the stagea discrete number of steps you will need two commands, the distance index DI and feedlength FL. The FL command moves the stepper motor a number of steps and directionas defined by the DI command. For example by sending DI-1000. The stage will moveto the left by 1000 steps when the FL command is issued. By sending DI1000 the stage


will move the right by 1000 steps. Finally, by sending DI to the controller, the controllerwill send back the current “index distance” which is this case is 1000. Jogging and moreimportantly stopping the jogging requires CJ “ commence jog” to start jogging and then SJ“stop jogging” to stop the jogging. It is important to note that the direction of the joggingwill be determined by the last DI command. If DI=-1000, the stage will jog to the right, andif DI= 1000 the stage will jog to the right. The feed to position “FP” will jog to an absoluteposition. For example if you define your starting point for your measurements as 0 (SP0)you can move back to this point by commanding FP0 PROVIDED that the DI is set in theproper direction. Shown in Table 5.1 are the basic commands you will need to commandthe Si3540 controller.

It is important to note that the stages are equipped with hardware limit switches sothat if the stage moves to end of either travel range, the motors will automatically stop.This is very important to prevent damage to the stage! The switches have a red diodethat lights when limit has been reached.

Computer control of the stage

Although almost any programming language could be used to communicate with the Si3540controller, we are going to use LabView because it will allow us to easily integrate both themotion and the measurements into one VI. The VI you will uses is a stand alone executablei.e. you cannot edit it. It is located in the Windows “Run” menu. This VI allows you tocreate a full 2-D scan of the velocity field automatically, record data to text file, configure theDAQ, and change the stepper motor setting. It also allows you to enter motion commandsto move the stage manually. Your TA or the instructor will cover the program with you. Ascreen shot of this Vi shown in Figure ??.

5.6.2 Troubleshooting

Stage does not move after limit stop This can happen if motion of the stage is stoppedby the limit switch and the stage sits idle for more than a few minutes. The fix is tocycle the power on the Si3540. NOTE–The plug must pulled until the red power lightextinguishes.

Constant voltage from the hot-wire anomometer If the voltage reads a constant outputof a few volts and does not change values when you unplug the BNC connection.The hot-wire is broken. You can verify using a 60X objective microscope.

5.6. Appendix: Motion control 49

Cmmnd Descr. Param. W Only R Only Units Range DefDI distance or position dstnc steps ±16,000,000 20,000FL feed to length xFP feed to position xVE velocity setting speed rev/sec 0.025-50 1MR microstep resolution 3-16 8SJ stop jogging xSP set abs position ±16,000,000ST stop x x ±16,000,000

Table 5.1: Brief table of Si3540 commands. Additional commands can be found in thebound lab folder

MR Code Steps/rev MR Code Steps/rev MR Code Steps/rev3 2000 8 20000 13 360004 5000 9 21600 14 500005 10000 10 25000 15 508006 12800 11 254007 18000 12 25600

Table 5.2: MR code and the actual steps/rev represented


CHAPTER

SIX

Mechanical Properties of MetallicMaterials - The Tensile Test

6.1 Objectives

The tensile test is one of the most important methods for characterizing the mechanicalproperties of a metallic material. Several important engineering characteristics of a mate-rial can be deduced from this test. Examples include: Elastic modulus, yield and tensilestrengths, energy absorbed to failure, ductility, etc. These mechanical properties allow theengineer to predict how a part will react when placed under various working loads.

The major objectives of this laboratory exercise are:

1. Understand the basic process of deformation due to tensile loading

2. Characterize the mechanical properties of various metals from their stress-straincurves

3. Determine how annealing a metal affects its mechanical behavior

4. Examine the fractured surfaces using a Scanning Electron Microscope (SEM) anddetermine the mechanism of failure.

In this lab you will use the available apparatus (this lab manual will provide you only withgeneral guidelines on how to operate the equipment. There is also a user manual providedby the manufacturer of the equipment in the lab that is very useful, along with some tutorialCDs).

6.2 Test Equipment

The available axial loading system consists of a PASCO AP-8216 Stress Strain Appa-ratus, a PASCO CI-6746 Force Sensor and a PASCO CI-6538 Rotary Motion Sensor(www.pasco.com). The combined system gathers the necessary data to analyze stress strainbehavior of material samples using the accompanying software.

The furnace used for annealing is a Lindberg Tube Furnace, which produces heated environ-ments up to 1150 C. It is next the experiment. The tube is made of quartz. It has stainless

51

Figure 6.1: The stress/strain analysis equipment and test samples.

steel end caps with Swagelok fittings for the nitrogen purge. The exit of the purge is passedthrough water to prevent backflow. The furnace is controlled by a Eurotherm temperaturecontroller, and instructions for a temperature ramp are on the class web page.

The materials to be tested are: steel, brass and aluminum

6.3 Test Procedures

1. Get Started.

Plug in the apparatus (there is no on/off switch) and turn on the accompanying com-puter. Double-click the DataStudio icon, and select create experiment.

2. Set Start/Stop Parameters.

It’s important that the forces applied to the PASCO Force sensor not exceed the limitof the components within the sensor. Click the Setup button near the top of thescreen, and within the experiment setup window click options. In the sampling op-tions window, go to the delayed start tab. Select Data measurement with Force,push Positive as the limiting parameter. Set the value so that recording begins whenthe force is above 1 N.

Next go to the Automatic Stop tab in the sampling options window. Again selectData measurement with Force, push positive as the limiting parameter. Set thelimit at 45 N. Take care to watch the software during any testing so that you can stopcranking immediately after the software stops taking data (when the force registeredreaches your set limit of 45 N).

3. Familiarize Yourself with the Program and Apparatus.

52 Chapter 6. Mechanical Properties of Metallic Materials - The Tensile Test

• Learn how to load and operate the testing apparatus (the user manual is a goodplace to start). You can also use some of the plastic samples for practice.

• Determine what measurements the program can extract from the testing appa-ratus.

• Explore the various ways to manipulate data provided by the program (creatinggraphs, determining slopes, calculating functions of measured properties).

Remember:

• The extension of the samples can be determined from the angular rotation of the knobby manually measuring the relationship between them.

• As you test samples, the tensile force will cause the samples to stretch. However, theapplied force also causes the apparatus platform and the force sensor to bend. Thedisplacement being calculated by the software will be the combination of the samplestretching and the bending of the rest of the apparatus. Regardless of how the samplestretches, the bending within the apparatus is constant for a given force. You canmeasure this deformation directly by using the calibration bar, which does not stretchsignificantly.

• The force measured by the force sensor is applied by a moment arm. The force actingon the sample being pulled is not equal to the force measured by the sensor, but ratherrelated according to the physics of how levers work and the distance from the pivotto the places where the forces act.

6.4 Tests to Perform

You will measure stress-strain curves for three different metals – steel, brass, and aluminum.

6.4.1 Steels

Obtain 10 tensile samples from the TA. Divide them into two batches ( 5 samples/batch).Conduct the tensile tests on the first batch as described above, ,and plot the stress-straincurve. From these curves, determine:

1. Young’s Modulus

2. Yield Strength

3. Tensile Strength

4. 0.2% Yield Stress

5. Fracture Stress

6. Toughness

7. Ductility.

6.4. Tests to Perform 53

Report each one of the above properties as a mean value, along with the standard deviation,and the Coefficient of Variation (100* SD/ mean).

Place the second batch of 5 samples in the tube furnace and follow the heating procedure.The samples with ramp up to the desired maxium, hold for a preset period of time, andthen down to room temperature. The entire cycle takes plance under a nitrogen purge. Seethe class webpage for details on the oven and purge. Test these samples as a batch, anddetermine the same 7 parameters that you tested on the first batch of 5.

6.4.2 Brass and Aluminum

Obtain 5 samples of brass, and 5 samples of aluminum from the TA and conduct the exper-iments as in the case of the first batch for steel.

6.5 Fracture Surface Imaging

Once your test samples have fractured, remove them carefully from the test apparatus with-out touching the fracture surface. Tape them directly into your lab notebook (it doesn’tmatter which partner’s notebook) to save them for later SEM imaging. Give each samplean ID number or name, and record any relevant notes about how it was processed or testedthat you might want to know later.

We will arrange for all lab groups in each block to go together to the scanning electronmicroscope (SEM) in the Geology Division. Selected samples from each group will beimaged, focusing in particular on the fracture surface. Include the images of your samplesin your lab report.

6.6 Lab Report

You lab report should include a complete description of all measurements made, includingstress strain curves measured for all samples. It should also contain a table of the parametersyou determined from the stress-strain curves, and SEM images of the fracture surfaces.

In your discussion section, compare the results for the steel samples that you heat-treatedwith the results for those you did not. Also compare the results for steel, brass, and copper.Compare a few of the parameters you measured to literature values for these materials(Young’s modulus, yield strength, and possibly others if you find them). What factorsmight explain any differences between your results and those in the literature?

For the heat-treated steels, include a TTT plot with the time-temperature history of yoursamples sketched in. Discuss in terms of this plot and the iron-carbon phase diagram thetransformation in microstructure produced by the heat-treatingn process.

From the images of the fracture surfaces, discuss what you can conclude about the failuremode. Was the failure brittle or ductile?

54 Chapter 6. Mechanical Properties of Metallic Materials - The Tensile Test

6.7 Pre-lab Questions

1. Give a brief physical description of stress, and strain, including relevant equations.

2. Draw a typical stress strain curve for a metal specimen that underwent tensile loadinguntil fracture. Label and describe the different stages of deformation.

3. Define each of the following terms. Also describe how you can determine each ofthem from a stress-strain graph: (include diagrams if useful)

(a) Young’s Modulus

(b) .2

(c) Ultimate Tensile Stress

(d) Fracture Stress

6.7. Pre-lab Questions 55

CHAPTER

SEVEN

Dynamics of Coupled MechanicalOscillators

7.1 Introduction

In many engineering fields, the input signal to a dynamical system is unknown, but theoutput signal can be observed and measured. An example of such a system is a seismo-graph used to measure the strength of an earthquake. Often the characteristics of the inputsignal can be determined by constructing an analytical model of the dynamical system andexamining the response of the system to different input signals. In addition, the dynamiccharacteristics of various mechanical systems such as an automobile suspension are evalu-ated by observing the output response to various input test signals.

This experiment is designed to illustrate the dynamic response of a mass-spring-dampersystem. The input signal is sinusoidal with variable frequency. The movement of twomasses is measured with linear variable-displacement transformers (LVDT).

7.1.1 Description of the Experiment

Figure 7.1.1 shows a simplified dynamic model of the experiment. The system is composedof two large masses of mass m (approximately 700 grams) on sliding tracks. The massesare separated by three linear springs with spring constant k and two air dashpots with vis-cous friction coefficients b1 and b2. The system is driven by a variable speed motor withan eccentric shaft to provide a sinusoidally oscillating input. There are three LVDT’s tomeasure the displacement of the mass from the stationary reference frame: one on each ofthe two masses and one on the driver plate. The output from the sensors is sent through aLabJack to the PC via USB. The LabVIEW VI ’Coupled Oscillators’ can be used to monitorthe signals.

Simply navigate to the “Data Acquisition” tab and click “Enable Stream” to start data ac-quisition from the LVDT’s. Please refer to the “Documentation” tab for further instructionson how to use the calibration and data analysis functions.

57

Figure 7.1: The coupled oscillator experiment.

7.1.2 System Model

By doing a force balance on each of the masses, two coupled linear ordinary differentialequations are determined that we use to model the system. Let X = [x1 x1 x2 x2]

T bethe state vector of this system, where x1 and x2 are the respective centers of mass. Thedynamic model has the form X = AX +Bu, where:

A =

0 1 0 0

−2k/m −b1/m k/m 00 0 0 1

k/m 0 −2k/m −b2/m

B =

000

k/m

(7.1)

The transfer function can be derived as: C(sI − A)−1B where A and B are as above, andC is the “output matrix.” In this case, if one is interested in measuring the response of bothx1 and x2 then C takes the form:

C =

[1 0 0 00 0 1 0

](7.2)

Figure 7.2: Simplified dynamic model of experiment. (Note b1 is incorrectly labeled as band b2 is not shown.

58 Chapter 7. Dynamics of Coupled Mechanical Oscillators

The characteristic polynomial (e.g., D(s) = det(sI −A)) of this system has the form:

D(s) = s4 + ζ1s3 + ζ2s

3 + 4ω2ns

2 + ζ1ζ2s2 + 2ζ1ω

2ns+ 2ζ2ω

2ns+ 3ω4

n (7.3)

where ζ1 = b1/m, ζ2 = b2/m, and ω2n = k/m. Then the transfer functions from u(s) to

x1(s) and x2(s) are of the form:[x1(s)u(s)x2(s)u(s)

]=

[ω2n(s

2+ζ2s+2ω2n)

D(s)ω4n

D(s)

]. (7.4)

An alternate approach to finding the transfer function is to use Laplace transforms. Thetransfer function is defined as the ratio of the Laplace transform of the output function tothe Laplace transform of the input or driving function. Using this approach the transferfunction is the same as given in equation (7.4).

7.2 Experiments

7.2.1 Calibration of LVDT transducers

Calibrate the LVDT transducer by determining the displacement versus transducer voltageoutput curve. The calibration can be done by simply moving the mass a known distance.Check the linearity of the device and be sure to take measurements over the entire range ofmotion of the masses.

7.2.2 Response of System to Step Input

Disconnect the two masses from each other. For the single-mass, single-spring system,record the response of the damped and undamped system to a step input. This can beobtained by moving the uncoupled masses with your hand and recording the output. Thisinformation will be used in determining k, b1, and b2.

7.2.3 Frequency Response (Bode Plot) of the Undamped System

Using the results of Sections 7.2.1 and 7.2.2 you should construct plots of the theoretical fre-quency response. Using these plots as a guide, you should measure the frequency responseof the undamped system. The term frequency response refers to the steady-state responseto a sinusoidal input. The frequency response information is presented in two separate fig-ures, one giving the magnitude of the ratio of the output to input signals as a function offrequency, and the other presenting the phase angle between the input and output signalsversus frequency. The figures are usually presented on a log-log scale for the magnitude anda log-linear scale for the phase. This representation of the frequency response is referred toas a Bode plot.

Disconnect and remove the damper from both masses. To find the frequency response,you want to examine several different input frequencies and record the displacement of theinput signal and the displacement of mass 1 and 2. Start with a low frequency and covera frequency range that includes the first two resonant frequencies. Be sure to take enoughdata to characterize each of the resonant frequencies.

7.2. Experiments 59

7.2.4 Frequency Response of the Damped System

Reconnect the dashpot to both masses. Set the damper to a relatively low damping coef-ficient and do not readjust the knob during the course of your experiments. As done inSection 7.2.3, record the displacement of the input signal and the response of mass 1 and 2.

(Note: be sure to take the response of the damped system with the same damping settingsas you had for the step response. The damper has a small knob which controls the amountof damping. If this knob has been moved, you will have to take another step response atthis time to compute the correct damping factor.)

7.3 Analysis of Experimental Model

To match the experimental data to the analytical model, we make a number of simplifyingassumptions:

1. Assume that the friction of the LVDT transducer is very small and neglect stictionin the sliding bearings. Thus the friction will be dominated by the air dash pots.These dashpots can nominally be modeled as viscous friction devices, with viscousdamping coefficients b1 and b2.

2. Neglect the mass of the structure connected to the eccentric input shaft.

3. Assume that both larges masses have uniform mass, m.

4. Assume that all 3 springs are linear and have uniform spring constant k.

7.4 Lab Report

1. Sketch and briefly discuss the experimental setup.

2. Record the calibration data, and estimate the uncertainty in the amplitude measure-ment. Can the transducer be modeled as linear?

3. Measure the step response and the undamped and damped frequency response, asdescribed in the text. Record all data.

4. From your measurements, estimate as accurately as you can the model parameters(spring constants, damping coefficients). There are two ways you might do this. Youcould determine them from the vibration period and decay rate of the step-responsetime series data. Alternatively, you could work in the frequency domain, by adjustingthe parameters in the theoretical amplitude and/or phase frequency response func-tions (using the program you wrote) to achieve the best fit to the measured frequencyresponse data. However you do it, make sure your conclusions are self-consistent -i.e., that both the measured step response and frequency response are consistent withthe same model parameters.


5. Plot the theoretical phase response vs. frequency for both masses. At what frequencyor frequencies do the two masses move in phase? Out of phase? Verify these predic-tions experimentally.

6. Note that there is a frequency between the two resonances where mass 2 is predictedto remain motionless, even though mass 1 and the input are both moving. Verify thisexperimentally, and check that the frequency where this occurs is where it is predictedto occur. (This frequency location might be useful in part 4 too)

7. Discuss any problems, discrepancies with theory, possible causes, etc.

8. Time permitting, explore other topics of your own choosing. Can you replace a springby a nonlinear one, for example? What new phenomena, if any, result from nonlin-earity?

7.5 Advanced Experiment (Optional)

Find or make a nonlinear spring(s). Insert this spring(s) in place of an existing spring,and try to observe the non-linear jump resonance phenomenon which occurs as you slowlyincrease the frequency of excitation. Alternatively, the springs currently installed in thesystem are slightly nonlinear. With the proper initial conditions and frequency of excitation,they will exhibit a steady-state nonlinear response. Have the TA show you the conditionswhich lead to this phenomena.

7.6 Prelab

1. Derive Equation (7.1).

2. Explain what a transfer function is, and derive Equation (7.4).

3. Write a program or spreadsheet to make Bode plots of the amplitude and phase fre-quency response from Equation (7.4).

4. Using the program, explore how damping affects the frequency response functions.Print out and put into your notebook an amplitude Bode plot showing the frequencyresponse for a few damping coefficients ranging from very small to large.

7.7 Appendix: Making Accurate Period and Phase Measure-ments

1. You should measure period and phase information by using zero crossing times in-stead of peaks in data. The data near the peak is very noisy while zero crossing datais much cleaner (due to slope crossing zero).

2. To measure zero crossing times you must calculate the mean of the data from the minand max of the waveform and not the mean reported by the data capture program.

7.5. Advanced Experiment (Optional) 61

The reason for this is that the data capture program calculates the mean for all of thecaptured samples which usually don’t correspond to an integral number of cycles.

3. Getting error information for stiffness and damping is easy using different cycles ofthe data and also using multiple runs.


APPENDIX

A

Keeping a Lab Notebook

Learning how to keep a lab notebook is critical to success as an experimental scientist orengineer. The lab notebook is the primary record of what was done in an experiment, andshould be written in a manner so that someone reading it years later (perhaps you) couldunderstand exactly what was done, and if necessary replicate the experiments.

Lab notebooks are legal documents, and are often called upon in legal disputes over patentsor scientific priority. Don’t let your patent rights for your brilliant idea, or your Nobel prizefor your brilliant discovery, slip away due to sloppy lab notebook practices!

There is no single way to keep a lab notebook, but there are some generally-agreed-uponpoints. These are summarized below.

1. Use a lab notebook with pages bound securely, not a spiral-bound notebook or onewith easily-removed pages. The pages should be numbered; if they are not, thennumber them yourself.

2. Never remove a page from the notebook.

3. Write in ink, preferably blue or black. Never make any entry in pencil.

4. Do not make notes or write down measured values on scrap paper, to later be enteredinto the lab notebook. Write your observations and measurements directly into thelab notebook.

5. Write legibly. It is not necessary to have a beautiful notebook, but it should at leastbe possible to read what you write.

6. Never erase anything, or use white-out, or cover anything with scribbled lines. Ifsomething is incorrect, simply draw a single, horizontal line through it, so that thetext can still be read. You may decide later that the values were correct after all, so itis important that they be still readable. If a large section of material is in error, simplydraw an “X” covering it.

7. Whenever you begin making entries into the notebook, first date the entry. It shouldbe possible years later to determine the exact date on which each entry was recorded.When you start a new page, put the date at the top.

8. Never leave a blank space to be filled in later. All entries must be in chronologicalorder.

63

9. When you need to include an image or a graph, sometimes a sketch or quick plot byhand is sufficient. For computer-generated images or plots, print it out and fasten itinto the notebook. Glue is best for this.

10. Write down how the measurements are made as you make them; if you wait untillater, you will forget the details. Was the length of a beam measured from its end, orfrom its support post? Was the voltmeter zeroed before making the reading?

11. In many experiments, it is not practical to include all data generated by the exper-iment in the notebook. This is particularly true if your primary measurements aremade using a computer. In this case, there is usually little point in fastening into thenotebook many pages of printouts listing numbers. Instead, save the data in a file onthe computer, and write in the notebook the name of the file(s) containing the data.

Summarize the salient features of the data in the notebook, possibly with a sketch ora graph. What is worth noting about the data? Why did you acquire it? What does itshow? The interested reader (perhaps you years later, or the patent examiner, or theNobel committee) should be able to first read the notebook to see the record of whatyou did and were thinking at the time, with the raw data location clearly identified sothat he/she can access all of it (and possibly re-analyze it) if necessary.

12. Make sure your data files are secure; you may want to transfer them to a memorystick at the end of each lab session. When you are finished with the experiment (orwith the course), burn a CD containing all raw data files, and attach it to your labnotebook.

13. If you are acquiring data by computer, save all data files. Just like erasing entries ina lab notebook, if you delete data files it is all too tempting to delete the ones thatdon’t show the phenomena you expect to see. If there is a good reason to think thatsome data files are in error (for example, due to equipment that was not functioningproperly), then note this in the notebook, but retain the data files anyway.

14. Never falsify data or dates in a lab notebook. Either one is serious scientific miscon-duct. Intentionally entering an incorrect date may affect claims of scientific priorityor patent claims. And recording measurments that were never made can lead (andhas led many times in the past) to fraudulent claims of new physical phenomena.Inevitably the fraud is discovered when no one else can repeat the purported exper-iments, and the result is always the sudden end of a scientific career in disgrace. Inthis course, false entries in a lab notebook will be taken as seriously as any other formof academic Honor Code violation.

64 Appendix A. Keeping a Lab Notebook

APPENDIX

B

LabVIEW

A new feature of ME 96 is the emphasis on teaching modern data acquisition methods, andin particular, data acquisition using LabVIEW. LabVIEW is a product of National Instru-ments, Inc. and is the leading software package by far for measurement and test systems.Like many other tools we use routinely (Mathematica, MATLAB, Excel, . . . ), LabVIEW iscommercial software, and is not free. But an inexpensive student editition is available thatcosts no more than a textbook (¡ $100). See the wiki for more information.

Most of the experiments use LabVIEW to acquire data and/or to control the equipment. Inseveral of the experiments, you are asked to write your own VI, or modify an existing oneto add new functionality. These experiments are:

1. Boiling Heat Transfer. Write a VI to acquire thermocouple data and plot temperaturevs. time and the boiling curve (heat flux vs. ∆T ).

2. Convection. Write a VI to monitor thermocouples and notify you when steady-stateconditions are achieved.

3. Cantilever Beam. Modify a supplied VI to plot the frequency spectrum of a time-varying signal.

4. Flame Fuel Cell. Write a VI to display the temperature read by the thermocoupleprobe and the voltage across the fuel cell.

5. Turbulent Air Jet. (To be determined.)

In addition, the Coupled Oscillators and Turbomachine experiments use LabVIEW to ac-quire data. Currently, these run using supplied VIs that do not require modification.

65

BIBLIOGRAPHY

67

lab manual 2013

Documents

lab rooms

lab report546

lab report354

lab report607

lab questions

lab computers

lab notebooks

lab partners