Mechanical Engineering (ME) 4201 Machine Design Laboratory

Lab Manual

PART-I & II

Compiled/Prepared by: Department of Mechanical Engineering

Louisiana State University

Fall - 2015

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Table of Contents Page Number REPORT WRITING GUIDE ............................................................................................ 3 FIRST CYCLE LABORATORY EXPERIMENTS

I. Gyroscope Experiment #1 ..................................................................... 10

II. Cam - Follower Experiment #2 .............................................................. 16

III. Journal Bearing Lubrication Experiment #3 ........................................ 22

IV. Static & Dynamic Balancing Experiment #4 ........................................ 31

V. Vibration Experiment #5…………………………………………………….35

VI. Strain Gage Experiment # 6(a) Cantilever Beam Experiment ...…….41 Thin-walled Lab Pressure Vessels# 6(b)…………………………………48

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REPORT WRITING GUIDE Introduction

The importance of good report writing and data presentation cannot be overemphasized. No matters how good an experiment, or how brilliant a discovery; it is worthless, unless the information is communicated to other people. As you complete each laboratory report, you will be gaining experience in writing technical reports and will be developing skills you will find invaluable throughout your career. There are a great many books available on report writing. However, the purpose of this document is to provide some basic pointers in the writing of laboratory reports. In particular, a major objective of this work is to provide the format for laboratory report writing, and to provide hints and tips to improve your reports. Each of these topics is discussed at length in the appropriate sections that follow. General Guidelines for Report Writing

All of the reports you submit during your career at Louisiana State University should follow one of two basic formats. There may be small differences due to individual preferences of the instructor, but for the most part, all of the components of the report will be identical. Style:

Third-person past tense is generally accepted as the most formal grammatical style for technical reports. However, in some isolated instances, it may be most effective to stress a point or to emphasize that a particular statement is primarily the opinion of the writer. An example of each of these styles is shown below.

Third person: Equation (6) is recommended for the final calculation procedure as a result of the limitations of the data as discussed above.

First-person: We recommend Eq. (6) for the final calculation procedure as a result of the limitations of the data as discussed above. As you can see, in the first-person statement, the writer is making the recommendation on a much more personal basis than in the third-person statement. The selection of the proper statement often depends on many factors, including the consideration of the people who will eventually read the report. In most cases, the third-person statement will be most preferable; however, if you have completed an engineering study for a particular individual, the first-person usage may be more appropriate. As most of the work that you will be completing will require a formal report, the third-person style is to be used. Format:

All reports must be typed or near-letter-quality (NLQ) printed. All work should be double- spaced, one side only. All pages must be trimmed or neatly folded to 8.5" - 11". Reports should be bound, such that the back of a page is on left and the front of the page is on the right (no binder, paper clips). Block format with separated side headings or an indented format with center of side heading may be used. This is an individual choice for the writer. Paragraphs may or may not be numbered, but all pages must be numbered.

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All pages should have identical margins. A reasonable set of margins is 1.0 inch at the top and the bottom, and 1.5 inches at the left, and 1.0 inch on the right. These margins allow for ease of binding, as well as clarity of reading. Rules for the Plotting of Data:

One of the more common tasks in completing engineering reports is plotting data. This section contains some rules to follow when making graphs. Automated graph-making programs such as Excel, Lotus, or SigmaPlot often take care of many of these details for you, but you are responsible for checking that these guidelines are followed.

All axes must be drawn in a location on the page, which will allow the numbering,

and labeling of the scales to occur without entering the margin. Each axis must be labeled and the proper units given. The lettering on the page must be oriented such that it is readable from either the bottom or the right-hand side of the page. This becomes important if the plot is drawn in landscape mode (i.e., sideways) on the page. The independent variable is always plotted along the abscissa (x-axis), and the dependent variable on the ordinate (y-axis).

Choose convenient scale factors for each of the scales, generally using multiples of 1, 2 or 5. An axis, with divisions of 2.5 or 3.33, is very troublesome when it is desired to read intermediate values from the curve, to say nothing of the difficulty you will experience in plotting the data. The plotted data points should be clearly marked by drawing a small circle about each point. These should be drawn with a template for neatness and usually are not more than 3/32 of an inch in diameter. Normally, this formality is automatically done when commercial computer graphing packages are used. If points for more than one curve are to be plotted on a single graph, or the data of different observers is to be indicated, a differentiation can be made by using small squares, triangles or other symbols in addition to the circles. On the other hand, the calculated points used in plotting a curve representing theoretical results should not be marked with such symbols. Only the curve itself (and not the points used to plot it) should be shown.

In most cases you will be required to draw a best-fit curve (with a French curve,

not freehand!!) through the data points. It is not necessary that the curves intersect all data points or even the first and last data points. It should, however, be a smooth curve which best represents (in some average sense) the data. The curve should not be allowed to pass through the circles or other symbols surrounding the points. The purpose of this is to leave the points visible so they may be checked at any time. Calibration curves and correction curves are drawn with small sections of straight line joining the points. This is done because the errors are mostly random and do not conform to a mathematical law.

Curves are drawn to as large a scale as is consistent with the precision with which the measurements are made. That is, the scale should not be so large that the curve can be read to a precision greater than that of the measured data, nor should the scale be so small that the curve cannot be read to as great a precision as that to which the measurements are made.

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Do not place too many curves on one sheet of paper, particularly if the curves cross one another. Typically, this leads to confusion and a cluttered appearance. Prepare all plots on ruled graph paper however, for computer-generated plots this is not necessary. Do not use quad paper or engineering paper. If more than one curve is present, label each curve as close to the curve as possible. Do not use colors to distinguish curves, as this distinction disappears upon reproduction. (Use full, dashed, or dotted lines instead). Each curve should have a figure number and a descriptive title such as “Frequency Response of a R-C circuit.” Non-descriptive titles such as “Voltage versus Frequency,” which are evident from axes labels, should be avoided. In calibration experiments, your name, the date on which the experiment was performed, the instruments manufacturer, model number, and serial or identification number should appear on the figure.

Full-page graphs in landscape mode (printed sideways on the page) must have the top of the graph at the left-hand side of the page. As in the text, any bibliographical references may be enclosed in brackets [xx]. Application of Probability and Statistics

Probability and statistics play an important role in experimental work, especially in industry. When presented without consideration of probability and statistical analysis, data can be misleading because measurement error prevents an engineer from determining the true value of measured quantities at any given time. Engineering measurements, repeatedly taken under seemingly identical conditions will normally show variations in measured values. The use of statistics in mechanical measurements provides a method of dealing with characteristics that have variability. Because the topic of probability and statistical analysis is presented in standard engineering texts, it is not necessary to present the full theory here. However, reports for this class may require some of the following analysis techniques. (1) If more than one measurement is taken for a given value, your report should

include calculation of mean and standard deviation. This information should be included whenever the data is reported. If data is reported in tabular form, one column each should be devoted to mean and standard deviation. If data is reported in a graph, error bars should be used to indicate the standard deviation for each mean value.

(2) If a curve-fit or regression is used, the correlation factor should be reported along

with the equation of the curve. In some cases, the two curves representing 95% confidence intervals (or other specified intervals) may also be required to demonstrate the goodness of fit.

(3) When specified in the manual, uncertainty analysis should be performed to

account for the effects of measurement uncertainties on calculated values.

For this specific class, always conduct analysis (1) and (2) whenever applicable. Conduct analysis (3) ONLY when specified in the individual experiment.

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CONTENT A good laboratory report answers the questions: What was done?, How was it

done?, and What were the results?. There are standard sections to each lab report which help answer these questions, and allow the reader the opportunity to move right to the section most appropriate to his or her interest. Basically, any formal lab report consists of the following sections: 1. Title Page & Abstract 2. Table of Contents 3. Introduction 4. Background and Theory 5. Equipment 6. Procedure 7. Results and Analysis 8. Conclusions and Recommendations 9. Bibliography 10. Appendix/Appendices.

A short description of each of these topics follows. Title Page & Abstract:

This is one of the most important sections of your report, and will probably be the most carefully scrutinized section of any report you write. Basically, this section tells a short story about your lab, and very concisely answers the three questions posed above: What was done?, How was it done?, and What were the results?. The abstract is included as part of the title page, so that anyone reading the report can very quickly see who wrote the report, the subject of, and the procedure followed for the lab, and what results were obtained. This section should probably be the last section written, and will summarize all of the work done for the lab. A common mistake is to write this section first, and then force the rest of the report to fit with the abstract. Table of Contents:

Obviously, the table of contents is nothing more than a listing of headings with the

page numbers. However, the table of contents is often a good indicator of the organization of the report, and can be a good tool for the development of the lab report. When figures and tables are used in the report, an independent List of Figures and List of Tables is also required. Introduction:

An introduction sets the context of the experiment, and gives the relevant background to the experiment. This section is very important for the individual who wishes to read the entire report. The introduction should include a concise description of what you were trying to discover, as well as describe what is going to follow in the remainder of the report. It is not appropriate to include results in this section; rather, you are trying to set the stage for what follows in the rest of the report. Background and Theory:

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This section is used to lay the technical and theoretical groundwork for the process being studied. It is in this section that the technical basis for the work that you did is explained. In other words, why are the results that you obtained valid? Discuss the technical fundamentals, which lie behind the experiment. This includes describing (and laying the technical groundwork for) the analysis process for the data. This section does not have to be a master’s level thesis, but should demonstrate your understanding of the process and the sources of error in measuring it. A well-written background section greatly simplifies discussion of the results. Equipment:

This segment of the report, along with the section that immediately follows (the procedure section), allows the reader to duplicate your experiment should the need arise. Any equipment used in the experiment should be listed along with identifying marks (serial numbers, model numbers, etc...). Any particular settings of the instruments should be denoted in the procedure section. It is especially useful in this segment to include a sketch of the experimental setup. This sketch is especially helpful to avoid any ambiguities that might exist when the experiment is re-run, or when attempts are made to duplicate your results. Procedure:

A procedure section simply lists what was done in the lab, and the order in which it was done. As previously mentioned, it is this section, along with the equipment section, that allows the reader to reconstruct your experiment, if necessary. This section is most effective if it is written in the form of a list, following a chronological order as shown below.

1. A beaker was filled with 500 ml of water, and placed on a hot plate. 2. A thermocouple was placed in the beaker, and the temperature of the water

sampled once every 8 seconds. 3. The recording ...

Elements in the list are either all complete sentences, or all short phrases but not

both. Overhanging paragraph format is good for lists. The procedure section is also the appropriate place to discuss any deviations from the intended procedure. For example, if you originally intended to monitor the temperature of two beakers of water, you might note that only one thermometer was available, and that this was not possible. Results and Analysis:

This section may be the most crucial for your report. The results and the analysis to obtain these results should be presented here. This section is most effective if written in the past tense. “The data were taken ...”; “the curve was generated...” However, it is appropriate to say such things as ‘the data are well represented by a second order polynomial’ since this is a fact that extends into present. Additionally, estimate the error in measuring whatever your objective was to measure. Be particularly careful when referring to ‘human’ or ‘round-off’ errors that these errors are significant in terms of the discrepancies observed.

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Plots and figures tend to be the most effective ways to present data. It is extremely useful to include figures in the text at the point where they are being discussed (a helpful tool is to use the graphic import feature available with many word processors to import figures right into the text).

When graphs or tables will present the ideas clearly, use them, but also include a concise discussion of the graphs and tables focusing the reader’s attention on the salient features of data. Do not simply recite numbers or parameters that should be obvious upon simple inspection of the figures. Moreover, never forget to indicate units. The location of figures and tables should be included in the List of Figures/Tables in the Table of Contents section of the report.

Probability and statistical analysis should be included with your calculations in this section, if applicable. Please follow the requirements given above. Conclusions and Recommendations:

By the time the reader reaches this section of the report, most of the conclusions regarding the work should have already been presented. The object of the conclusion section is to gather all of the important results and interpretations in clear summary form. Recommend cost-effective feasible ways to improve the performance of the laboratory. Also remember, there will be many readers who focus only on the conclusion and abstract sections, so it is especially important that they be well written. Bibliography:

This section should include all references (including the lab manual), which were cited in the report. Citations should include all information that is not developed by the authors. A standard format should be used such as

“... Smith [1] discusses the effects of temperature...” which would refer to the following citation in the bibliography. 1. Smith, R., Turbulent Natural Convection for a Vertical Plane Surface,

Journal of Heat Transfer, Vol.76, p.234, 1979.

Your work will often require you to reference other contributions, not only in this particular course, but in others as well. Learn to do so. Appendices:

A laboratory report should be a complete, concise, self-contained document without appendices. These sections contain information not appropriate to any other section. For example, raw data, sample calculations, detailed derivations, etc. may be included in the appendices. This is the most ‘free’ space in the report. For example, you might include a sketch of an improved way to complete the experiment, or to present the data. An appendix can be a very valuable addition to the report. Hints to Writing Before You Begin:

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It is very difficult to write good lab reports if you don’t understand what you were doing when you carried out the experiment. Therefore, before coming into the laboratory, you are urged to carefully read both the experiment sheet and the partially completed laboratory report for that week’s experiment. Take advantage of opportunities to discuss with your instructor any questions you may have regarding the experiment. The experiment tends to run a little smoother if you designate one member of your group as the data taker. He or she is responsible for recording information pertinent to any device calibrated during the experiment. At the end of the lab, each member of the group should then copy this information directly onto their data sheets. It is this type of information, which should appear in your appendices. Writing the Report:

It is important to write your report as soon as possible after the experiment is completed. This will save you time since the experiment is still fresh in your mind. Remember, as always, that each section of the report should answer one of the three basic questions: What did you do? How did you do it?, or What did you find out?

In terms of these, think carefully about what you did in the experiment. Think about

what figures you want to include that help clarify the information. Decide the flow of information before actually sitting down to write. As you write, keep in mind the common rules of English grammar and punctuation. Proofread your report, and keep in mind that your report speaks for you. Does your report give the impression that you would want to make if you were in person?

Write a sketchy outline itemizing the basic sections of the report and listing the primary points to be made in each section. With the available text processors this should be relatively straightforward. The outline will help organize the report, help establish the flow of information, and can help indicate where figures and graphs are needed.

Once the content of the report is established with a multilevel of detail outline, it is much easier to begin writing. It should be much simpler to concentrate on rules of grammar and punctuation when not having to think about what to say also. After Getting Back Your Report:

After your report has been graded and returned to you, take the time to read and understand the comments that have been made. Consider these comments when writing your next lab report. Work with the instructor to improve the quality of your lab reports, and always keep in mind that the ability to write a good lab report is an excellent talent, that will be extremely useful throughout your educational and professional careers.

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GYROSCOPE EXPERIMENT #1 Introduction

The study of gyroscopic action is particularly important in the field of vehicle engineering. The gyroscopic couple produced by rotating components can often lead to undesirable effects, which affect the stability of vehicles. For example, when a road vehicle travels around a bend, the gyroscopic couple produced by turning the axis of the wheels tends to overturn the vehicle. In the case of an aircraft changing direction, the gyroscopic couple due to the rotating components of the engine causes the aircraft to pitch up or down. Gyroscopic effects can also be used to one’s advantage, as in the case of gyrostabilizers and gyroscopic instruments. If mounted in a suitable position, a gyroscope consisting of a rotating disc can be used to resist undesirable motion and so provide a means of stabilization. Gyroscopic action occurs whenever the axis of a spinning body rotates. The axis of rotation remains in the same direction so long as no external couple acts on the system. However, if a turning couple is applied to the axis, a torque reaction is produced which tends to turn the axis in a plane at right angles to the plane in which the applied couple acts. This torque reaction, or gyroscopic couple as it is called, results from attempting to alter the direction of angular momentum of the body.

If we have a stationary fly wheel, of moment of inertia I, on a shaft mounted in a trunnion frame such that it is supported but free to rotate about any axis, then the couple applied to the system will cause the shaft to move in the plane of application of the couple. Now consider Figure 1, where the flywheel disk is spinning with angular velocity ωr (or we may call ω, and the axis of spin is simultaneously rotating in the horizontal plane XOZ with angular velocity ωp. Then:

Active Gyroscopic Couple = I ωr x ωp -----------------------------Eq. 1. A torque is applied, to balance the active gyroscopic couple, by adding a mass (m)

to keep the axis of the disc from rotating in the horizontal plane. Therefore, the applied torque is a reactive gyroscopic couple or:

In order to investigate the validity of equation (1) it is necessary to determine the moment of inertia of the gyroscope rotor. In the experiment, this is done by suspending the rotor and disk on two wires as shown in Figure 2 and observing its torsional oscillation. This represents simple harmonic motion in which the periodic time τ (for

mgL = ωp x I ωr ------------------------------------------------------- Eq. 2

small oscillation angle θ such that sin θ is essentially equal to θ is given by:

dg M

l I 42 =

2rd

………………………… Eq. 3

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Where, Mrd is the mass of the rotor-disk (1100 gm.) combination, d is diameter of the disk (or the distance between the suspended strings), I is the mass moment of inertia about the centroidal axis of the rotor and disk, l is the length of the suspended string, and g is the acceleration constant due to gravity (9.81 m/sec2 or 386.4 in/sec2).

Figurer 1. Fly Wheel Disk Spinning About Figure 2. Experiment Used to

About Disk Axis and Precessing determine Rotor-Disk Period About Y-Axis and Moment of Inertia

Active Gyroscopic

Couple

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Figure 3. Examples of the Balancing Couples needed for Different Rotor-Disk Rotation (ω or ωr) and Precession (ωp) Combinations Objective

The purpose of this experiment is to: Study the direction of gyroscopic couple, angular velocity of the rotor (ω), and the

precession velocity (ωp). Determine the mass moment of inertia of the rotor-disk experimentally by use of

Eq. (3), which is based on small oscillation theory. Investigate the validity of the gyroscopic couple relation, Eq. (1), by comparing the

I value based on Eqs. (1) and (3). Necessary Equipment and Materials Stop watch (determines precession speed) E64 electronic tachometer (determines rotor speed) Additional gyroscope rotor and armature assembly Fold-out bifilar suspension arm mount to base of gyroscope apparatus Gyroscope apparatus (see Figure- 4)

Balancing Couple

Balancing Couple

Balancing Couple

Balancing Couple

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- Small variable speed motor (B) carried in a gimbal frame (this frame not shown in Figure 4).

- Rotor disc (A) mounted on motor (B)’s shaft. - Second variable speed motor housed inside apparatus base (not

shown in Figure 4). - Torque arm (G) carrying a mass (D) at its end to balance the motor

and rotor disc. - Retaining plate (E) fitted over the torque arm to limit the angular

movement of the motor assembly. - Additional masses (F) attached to the end of the torque arm to

balance the gyroscopic couple (add mass as directed in Procedure). - Removable electrically interlocked transparent safety cover (or

dome) fitting completely over rotating assembly (cover not shown in Figure 4). (Note: Removing this cover automatically stops both motors.)

- Slip ring at the base of the gimbal frame supplying power to rotor motor (B).

- E66 Mains Transformer Unit. - TecQuipment E91 dual speed control units for both motors.

Figure 4. Gyroscope Apparatus Procedure 1. Make sure the following connections are completed before operating the

gyroscope apparatus. a. The 12V input terminals on the E91 Dual speed control unit to the 12V D.C.

power supply. b. One pair of the output terminals on the E91 unit to the rotor input terminals

on the apparatus, and the other pair to the precession input terminals. c. The E64 tachometer input to the output socket on the apparatus using the

signal lead provided.

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d. If all of the preceding connections are made, the apparatus is ready for operation. Switch on all the units.

2. Check that the gyroscope rotor assembly is adjusted so that with no weight added,

the line scribed on the end of the torque arm lies in the plane of the line scribed around the safety cover. If the arm does not lie in the plane, slacken the screws holding the rotor on the motor shaft, then adjust the rotor position until the rotor assembly balances. Replace the cover, check the alignment and when satisfactory, re-tighten the screws to clamp the rotor in position.

3. Check that the cover is correctly in position, then set the rotor and precession

motors running. Note the direction of rotation of the rotor, the direction of precession of the gyroscope and whether the torque arm rises of falls. By interchanging the motor input connection on the front panel, determine the direction of the gyroscope couple for each combination of rotor and precession directions.

4. Screw a 150 g mass onto the end of the torque arm and replace the safety dome.

Connect the rotor and precession motor’s supplies so that the gyroscopic couple will raise the torque arm.

5. Set the rotor speed to approximately 1000 RPM using the speed control unit. Vary

the precession velocity until the torque arm rises to a level at which the scribed line lies in the same plane as the line on the safety cover. This is the point of balance at which the gyroscopic couple is just equal to the moment produced by the mass on the torque arm.

6. At this point, measure the precession speed by timing a suitable number of

revolutions of the assembly using a stopwatch. The number of revolutions you will need to time depends on the test conditions. To obtain high accuracy always use a time period of at least 1 minute.

7. Decrease the rotor speed in order to get 4 more measurement points (for example,

1500 RPM, 2000 RPM, 2500 RPM.) and determine the precession speed at the balance point for each different rotor speed.

8. Add additional masses to the torque arm and obtain similar sets of results for each

value of mass as for the 150g mass. Use 150g, 200g, 250g, and 300g masses. (Please look in to the NOTE for the required weights and speeds to be carried out for the experiment)

9. Calculate the moment using T = mgL, where: L = Torque arm = 15 cm, m = mass

added, and g = 9.81 m/s2 10. Plot the reciprocal of the precession velocity (1/ωp) against the rotor velocity for

each mass. Obtain the linear regression and indicate its coefficient of correlation. Find the moment of inertia (I) using the slope of each graph.

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11. The second method to obtain the moment of inertia, I, is based on vibration theory

as described in the manual. Measure the vibration period, τ, for at least five times using various sampling time. Obtain the average τ and its standard deviation. Calculate the moment of inertia using Eq. (3).

12. Compare the moment of inertia obtained through steps #10 and 11. Discuss the

results. Note : Carry out the Experiments for the following weights and speed combination

Speed (rpm) 150 (gm) 200(gm) 250 (gm) 300 (gm)1000 1500 2000 2500

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CAM-FOLLOWER EXPERIMENT # 2 Introduction

A cam-follower system (sometimes referred to as a direct contact mechanism) is one of the simplest mechanisms used for control and conversion of one type of motion to another. However, the accuracy of output, which is generally the motion of follower, depends on the cam and follower being in contact at all times.

The forces acting on a follower may be shown to consist of: where,

W = Weight of the follower, retainer and applied weight (lb or N), β = Angle between the line of motion of the follower and the vertical

(direction of gravitational acceleration), ωc = Angular velocity of the cam (rad/s), Fs = Spring force behind the follower (lb or N), Z = Vertical displacement of the follower (in or mm), θ = Angular displacement of the cam (radians)

The spring force behind may be broken up into two components where:

where: Fso = Spring force behind the follower when the follower is at its

closest position the center of rotation of the cam (lb or N), k = Spring constant (lb/in or N/mm).

In Eq. (1), P must always be positive in order for the follower to remain in contact

with the cam. The combination of the first two terms is always positive, however a large negative d2Z/dθ2 term may cause P to be negative. For a given cam profile, the RPM of the camshaft that results in separation of the follower from the cam may be calculated from Eqs. (1) and (2) as:

where, d2Z/dθ2 is the maximum negative value. Note that in Equation 3, β = 0 since the follower and cam are in line.

2

22cs

d

Zdg

W + F + W = P cos ……………Equation 1

kZ + F = F sos F so = k[Initial (free length)-Final (Compressed length)] ………. Equation 2

2

2

cos

dZd

kZ + F + W-

2

60 = N

gW

so

1/2

RPM

……… Equation 3

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Obviously, the RPM at which the follower separates from the cam is a function of the weight of the follower, the spring force behind the follower, as well as the cam profile. Furthermore, if the cam-follower separation is going to take place, it will occur first at the point where d2Z/dθ2 has its maximum negative value during the cycle. Once the follower loses contact with the cam, in reestablishing contact a fairly large impact force is generated which may eventually result in fatigue failure of the surfaces of the cam as well as the follower. Objective

The objective of this experiment is to: Observe the follower behavior and determine the effect of weight of the follower

on the critical speed Ncr of the cam shaft (i.e., the speed at which the follower and cam temporarily lose contact),

Determine the effect of spring force on the critical speed of the camshaft. Necessary Equipment and Material Cams with different profiles A roller and flat follower Springs with different stiffness values Variable speed drive motor Necessary instrumentation and recording devices Experimental apparatus frame and assembly

(see Fig. 1)

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Equipment Specifications

See the following tables:

Cam-Follower Experiment Spring Dimensions Color

Dimensions

Spring Weight (lb)

Nominal Stiffness (lb/in)

Retainer Weight (lb)

Red

1.12" MD* x 1/8' Dia. x 2.99" Long

0.138 31.4 0.156

White

1.85" MD* x 1/8" Dia. x 3.02" Long

0.294 22.5 0.300

Black

1.37" MD* x 1/8" Dia. x 2.95" Long

0.156 19.7 0.144#

_______________________ #: Steel retainer

Cams

Quantity 2 1 1

Profile Convex Concave Tangent

Followers

Quantity 1 1

Type Roller Flat Face

* = Mean Diameter

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Miscellaneous

Weight of roller follower and attachment

3.78 lb (excluding spring, spring retainers, and additional weights).

Weight of flat follower and attachments

4.30 lb (excluding spring, spring retainers, and additional weights)

Diameter of roller follower 1-1/8" Dia. Diameter of paper recording drum 3.673" Dia.

Procedure 1. Select the direction of rotation of the cam by turning the switch provided for this

purpose. Do not change the direction of rotation while the motor is running. 2. Select a cam and obtain a trace of its profile on paper and then assemble it on the

camshaft. (The cam you select may depend upon the discretion of your instructor.) 3. Select one of the followers and assemble that also on the machine (note that if the

cam has any concave section the follower must be a roller follower). 4. Select one of the springs and place it in the machine with some preload (if desired). 5. Wrap a piece of Teledeltos paper around the recording drum and secure its ends

with scotch tape. Obtain, with the motor off, a trace of displacement of the follower and then remove the paper from the drum.

6. Increase the speed of the motor slowly until you detect the tapping noise, indicating

the impact between cam and follower. Decrease and then increase the speed several times to make sure that you are detecting the tapping noise the very moment that it starts. Read the speed of the camshaft on the tachometer and record it. This is the experimental value of the speed, Ncr, at which P in Eq. (1) becomes zero. Because the collection of data is dependent on each individual’s hearing sensitivity, each person needs to obtain his/her own data and record them without getting influence from others.

7. Repeat step 6 five times, each time adding a weight of 400 grams to the follower. 8. Change the spring force Fs by either a) replacing the spring or b) increasing the

initial compression. Then repeat steps 6 and 7 for one other level of spring force. 9. Calculate the critical speed (Ncr) analytically by differentiating twice, the Z-θ curve

and plotting dZ/dθ and d2Z/dθ2 verses θ, obtaining the maximum negative value of d2Z/dθ2 from the plot and substituting this maximum negative value into Eq. (3).

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Because a curve-fit is required to approximate the cam surface in this step, 95% confidence intervals should also be shown on the Z-θ plot.

10. Plot the variation of Ncr versus the weight of the follower as well as the spring force

(either Fso or k if the spring was replaced). Be sure to indicate the mean and standard deviation of each data point on graphs containing measured values.

11. Compare the theoretical Ncr with the experimental Ncr values obtained and

comment on the results.

Cam-Follower Experiment

Name:

Section:

Group:

Date:

Cam Profile:

Spring(s) Used:

Critical Speeds (RPM) Data Sheet (Preload #1)

Extra Weight (grams)

Person#1 Person # 2

Person #3

0

400

800

1200

1600

2000

Critical Speeds (RPM) Data Sheet (Preload #2)

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Extra Weight (grams)

Person#1 Person # 2

Person #3

0

400

800

1200

1600

2000

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JOURNAL BEARING LUBRICATION EXPERIMENT #3 Introduction

The major objective of lubrication of journal bearings is to induce and maintain a film of lubricant between the journal and the bearing. The purpose of this film of lubricant is to keep the two surfaces separate at all times and thus prevent metal to metal or dry contact which otherwise will create bearing failure.

Hydrodynamic lubrication is the most common method of lubrication of journal bearings. In this method, as the shaft rotates it will, due to the load applied to it (as well as its own weight), take a slightly eccentric position relative to the bearing. The eccentric rotation of the shaft in the bearing, as shown in Fig. 1, acts some-what like a rotary pump and generates a relatively high hydrodynamic pressure in the con-verging zone. The hydrodynamic pressure for a properly designed bearing is responsible for supporting the shaft without allowing it to come in contact with the bearing.

It can be shown, analytically, that the hydrodynamic pressure distribution around the bearing is related to other parameters by:

and that

where,

) + (1 ) + (2

) + (2

c

Nr12- = )P - (P 222

2

0

cos

sincos……….. Eq. 1

)cos + (1

sin)cos + (2 K)(- = 2

1

11max

)P - (P 0 ……….. Eq. 2

23

The remaining terms in the above equations are defined as,

r = Journal radius, (in) c = Radial clearance, (in) μ = Oil viscosity at operating temperature, (reyn) ε = Eccentricity ratio (= e/c) θ1 = Location of maximum film pressure N = Journal speed (RPS)1 P0 = Ambient or oil supply pressure (psi) P = Hydrodynamic pressure at position θ

A dimensionless number called the Sommerfeld Number or Characteristic

Number defined by relates the bearing performance to the design parameters:

and

where, P = Load per projected area of the bearing (psi), L = Bearing length (in), W = Load carried by the bearing (lb).

Other relationships that can be obtained analytically are as follows:

1 RPS denotes revolutions per second or rev/sec

) + (2 c

Nr12 = K

22

2

……………….. Eq. 3

2rL

W = P ……………… Eq. 5

P

N

c

r = S

2

……………… Eq. 4

24

and

Where, cosθ1 is the location of maximum pressure relative to the line of centers and h0 is the minimum oil film thickness. The h0/c ratio and the friction coefficient are plotted vs. the “Sommerfeld No”. in Figs. 2 and 3. Objective

The purpose of this experiment is to: Measure hydrodynamic pressure variation in a journal bearing at different speeds, Calculate load carrying capacity of the journal bearing and compare it with theory; Measure the location of maximum film pressure, Measure the friction loss in the bearing and compare it with theory. Necessary Equipment and Material Journal bearing with adjustable speed journal Instrumentation to measure the pressure around the bearings

(manometer tube equipment) Instrumentation to measure journal RPM (stroboscope) Dead weights to adjust the load on the bearing Equipment Specifications Bearing diameter = 2.166 in Journal diameter = 1.984 in Effective bearing length = 2.766 in Bearing weight with attachments = 1.43 lb Weight of each movable load = 0.22 lb Lubricant = SAE 15 W 50 Lubricant’s density = 0.0282 lb/in3 Procedure

21 + 2

3- = cos …………… Eq. 6

- 1 = ch0

……………. Eq. 7

25

1. Select the direction of rotation of the motor by turning the switch provided for this purpose. Do not change the direction of rotation while the motor is running.

2. Start the motor and let it run for about half an hour for the temperature and the oil

viscosity to reach the steady state condition. 3. Apply about a one-pound load on the bearing, using the dead weights. 4. Increase the shaft speed gradually until you observe instability and vibration of the

bearing. Measure the speed of rotation ω of the shaft. Observe and comment about the pressure distribution around the bearing.

5. Using the strobe light set the journal speed to 1200 RPM. (Make sure to

periodically check this speed as it may increase during the course of the experiment.) Allow enough time for the oil to level in the barometer tubes to stabilize (about 3 min.)

6. Read the pressures for locations 1 through 16 and convert the readings to psi. 7. Change the shaft speed to 1400 and 1600 and repeat steps 5 through 7 for the

new RPM’s. For Each RPM Setting 8. Plot the variation of pressure along the bearing axis (pressure taps 1 through 5)

and obtain the average pressure along the axis as well as the ratio (R) of average pressure to the maximum pressure along the axis (pressure tap No. 3).

9. Multiply the reading of pressure tap 3 by the ratio R obtained in Step 8 to obtain

the axially averaged pressure. Repeat the same multiplication procedure for the readings of pressure taps 6 through 16. Plot the axially averaged pressure vs. theta in Cartesian as well as polar coordinates.

10. Find 2 points A and B on the experimental pressure curve that are 180o apart but

having equal pressure. Note that for any pressure curve there will be only one such pair of points possible. These two points, A and B, form the axis P - P0 = 0 for the “Sommerfeld” curve.

11. Of these two points choose as the origin the point with a larger thickness of oil film

and take the axis θ = 0 to pass through this point. 12. From your pressure distribution graph determine the location (θ1) of maximum

pressure and then from Eq. (6) find the eccentricity ratio. 13. Using Eq. (2) to calculate the constant K, and then plot the Sommerfeld curve

(theoretical pressure distribution), using Eq. (1). A typical curve is shown in Fig. 4.

26

14. Calculate the load carrying capacity of the bearing from:

Where, (P-P0)i is the pressure at the midpoint between any two consecutive pressure tap points and βi is the angle between that mid-point and the line of action of load (pressure tap point no. 3). Compare this load with the actual load on the bearing.

15. Calculate the Sommerfeld number from:

and then the friction coefficient from Fig. 3.

16. Calculate and plot theoretical HP lost in the bearing based on the friction coefficient

obtained in the previous step. 17. Discuss your results and comment on them.

i0 i

12

=1i

P - P12

Ld = W cos

P12

+ 2K =

P

N

c

r = S

22

27

Figure 2. Variation of ho/c with Sommerfeld No.

Figure 3. Variation of (r/c)f with Sommerfeld No.

28

Figure 4. Comparison of Theoretical & Experimental Pressure Curves.

29

Figure 5. Polar Pressure Diagram.

30

Journal Bearing Lubrication Experiment Name:

Date:

Section:

Group:

h0:

Rotation Speed (ω) of Shaft when Oil Whirl Instability Occurs:

RPM

Tap Pressure1 (lb/in2) Tap Number

RPM RPM RPM

RPM

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

1 Computation of the tap pressure in psi: P = ρ(h - h0)

31

STATIC AND DYNAMIC BALANCING EXPERIMENT #4 Objective The purpose of this experiment is to understand the principles of static and dynamic balancing of rotating machinery. Necessary Equipmentand V4

3 relative rms velocity of channel 4 with respect to channel 3. For V2

1 set “1=accel 1” as reference and “2 = accel2” as channel in signal processing window. Similarly for V4

3 change reference to “3 = accel 3” and “4 = accel4” as channel.

1. To measure the rms (root mean square) velocity at a given frequency (Hz), use the markers 1 or 2 to reach the set frequency of the rotor. Use only one Marker i.e., M1=15 Hz. By looking at the window of M1 or M2 at the given frequency (Hz), the velocity is noted. This gives the base data of vibration without any unbalance.

2. Add weight of say 10 g on the any one of the holes at the outer radius (r =

69.91mm). Note down the values of V21 (rms velocity of 2 w.r.t. 1) and V4

3 (rms velocity of 4 w.r.t. 3). By looking at the data one can make out that there is an unbalance in the rotor.

3. To nullify the effect of the unbalance we need to add a counter weight of 10 g at

the same radius and at an angle of 1800 apart. Now note down V21 and V4

3. Remove both the weights.

4. To see the effect of radius, add weight of say 10 g on the any one of the holes at

inner radius (r = 57.21mm). Note down the values of V21 and V4

3. Again add the counter weight of same mass on same radius at an angle 1800 apart to balance the rotor. Note down the values of V2

1 and V43.

5. Repeat the steps 5-10 for speeds 30 Hz and 45 Hz.

Part 2 1. Set the motor speed to 25 Hz. Remove all the weights from the system. 2. Run the motor and note down the values of V2

1 and V43. This would be the base

data. 3. Add a weight of 10 g at 300 from reference and 10 g at 1300 from the reference.

Run the motor and note down the values of V21 and V4

3. This would give the results for the unbalance.

4. To balance the rotor, we need to add counter weight, where mass and position has to be calculated. The procedure is given at the end of the section.

5. Add the calculated mass at the position determined in step 4. 6. Run the motor and note down the values of V2

1 and V43, this would give the results

for balanced rotor. 7. Repeat steps 3-6 with weights of 9.37 g @ 2100 and 10 g @ 1700

32

The magnitude and direction of the balancing mass may be obtained analytically and graphically as discussed below. Analytical method

1. Find out the centrifugal force (Product of the mass and its radius of rotation) x 2 (where is thew angular velocity) exerted by each mass on the rotating shaft, Fc = mr 2.

2. Resolve the centrifugal forces horizontally and vertically and find their sums, i.e. Σ H and Σ V.

Σ H = [m1 r1 cosθ1 + m2 r2 cosθ2 + ….] 2 Σ V = [m1 r1 sinθ1 + m2 r2 sinθ2 + ….] 2

3. Magnitude of the resultant centrifugal force, FC = 22 VH

4. If θ is the angle, the resultant force makes with the horizontal or the reference, then,

Tan θ = Σ V / Σ H. (From Tan θ calculation we could have two angles, θ and θ+1800). 5. The balancing force is then equal to the resultant force but in opposite direction,

hence add 180 to the θ obtained. 6. Now find the magnitude of the balancing mass, such that

FC = m r 2

Where m = Balancing mass, r = radius of rotation (57.21 mm or 69.91 mm, depending on the location), is the angular velocity.

7. Knowing FC, r, determine m, the magnitude of the balancing mass.

8. Verify the above results using Graphical method.

Tabular column Part 1

R = 69.91 mm R = 57.21 mm

33

Base Data

Unbalance Balanced Unbalance Balanced

15 Hz V2

1 in/s

V43 in/s

30 Hz V2

1 in/s

V43 in/s

45 Hz V2

1 in/s

V43 in/s

Part 2 Speed = 25 Hz Case 1, m1 = 10 g , θ1 = 300 , m2 = 10 g , θ2 = 1300 , m = ……….., θ = ………… Case 2, m1 = 9.37 g , θ1 = 2100 , m2 = 10 g , θ2 = 1700 , m = …… ., θ = …………

Base Data Case 1 Case 2

Unbalance Balanced Unbalance Balanced

V21 in/s

V43 in/s

34

VIBRATION EXPERIMENT# 5 Objective The purpose of this experiment is to use the measured oscillations from a coupled oscillator to mathematically derive the values of the physical components used. Necessary Equipment

Model 210a Rectilinear Dynamic System. High stiffness, Medium stiffness and Low stiffness springs. Brass masses. Allen wrenches. Computer with data acquisition board for recording and plot generation of the

Rectilinear Dynamic System. Flash drive.

Procedure for Dynamic Parameter Identification

1. With computer powered up, enter the Driving Function box via the Set-Up menu and select Force (Torque), then select Setup Driving Function, then Ok (from within the Force (Torque) dialog box), then Enable Driving Function, and finally Ok again to return the background screen.

2. Enter the Command menu, go to Input Shape and select Step Input. Input a Step

size of 0 (zero), duration of 3000 ms and 1 repetition. Exit to background screen by consecutively selecting Ok. This puts the controller board in a mode for acquiring 6 seconds of data on command but without actually introducing drive force (via the drive motor).

3. Go to Data Menu - Setup Data Acquisition in the Data menu and select Encoder

# 1 and Encoder # 2 as data to acquire and specify data sampling every 2 (two) servo cycles, i.e., every 2 Ts ’s. Select Ok to exit.

CASE1

4. Clamp the carriage # 2 using a ¼ in. threaded nut between the stop tab and the

stop bumper so as not to engage the limit switch. See to that the centerline mark of the carriage #1 coincides with the 0 of the scale provided along the carriage #1. Move and fix the limit switches for the carriage # 1 at its extreme positions. Fix the medium stiffness spring between the carriage # 1 and carriage # 2.

5. Secure four 500g masses on the carriage # 1.

6. In the Utility menu select the Zero position to zero set the initial readings.

7. Select Execute from the Command menu. Prepare to manually displace the

carriage # 1 approximately 2.5 cm. Exercise caution in displacing the carriage so

35

as not to engage the limit switch. With the first mass held at 2.5 cm from the initial position, select Run from the Execute box and then release the mass approximately 1 second later. The mass will oscillate and attenuate while encoder data is collected to record this response. Select Ok after data is uploaded.

Note: If at any time during this procedure a limit switch is engaged, you must return to the Driving Function box and Enable Driving Function before proceeding.

8. Select Set-up Plot from the Plotting menu and choose Encoder # 1 position.

Then select Plot data from the Plotting menu. You will see the first mass time response.

9. In the Data menu, select import raw data and save the data in a .txt format in the

a:\ drive.

Fig: Typical Step Response

10. Plot the above saved data for encoder position #1 vs. time (2266 counts = 1 cm). Divide the number of cycles by the time taken to complete them being sure to take beginning and end times from the same phase (e.g. the local amplitude peak) of the respective cycles. Convert the resulting frequency in Hz to radian/sec. This damped frequency ωd approximates the natural frequency ωn according to:

1111 112

111

m

m m

m

dn d

n

(1)

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2

Time (s)

En

cod

er P

osi

tio

n (

cm)

X n

Steady StateError

t 0

X 0

t n

36

(For small 1m )

Where the “ nm11” subscript denotes mass #1, trial #1.

11. Calculate 11mn , i.e., natural frequency of mass 1 and trial 1.

CASE 2

1. Remove mass from the first carriage. 2. In the Utility menu, zero set the carriage position.

3. Repeat steps 7, 8 and 9 of the previous case.

4. Plot the saved data.

5. Measure the initial cycle amplitude X0 and the last cycle amplitude Xn for the “n”

cycles measured. Using relationships associated with the logarithmic decrement:

12

12 0 012

12

1 1ln ln

2 21

m

mm

n nm

for small

X X

n X n X

(2)

Find the damping ratio ζm12 and show that for this small value the approximations

of the previous equation are valid. 6. Calculate ωnm12, i.e., natural frequency of mass 1 and trial 2.

CASE 3

1. Unclamp the carriage # 2 and clamp the carriage # 1 using a ¼ in. threaded nut

between the stop tab and the stop bumper so as not to engage the limit switch. See to that the centerline mark of the carriage #2 coincides with the 0 of the scale provided along the carriage #2. Move and fix the limit switches for the carriage # 2 at its extreme positions.

2. Disconnect the dashpot from carriage # 2.

3. Retain the medium stiffness spring between carriage # 1 and carriage # 2.

4. Secure four 500g masses on carriage # 2.

5. In the Utility menu select Zero position to zero set the initial readings.

37

6. Select Execute from the Command menu. Prepare to manually displace the carriage # 2 approximately 2.5 cm. Exercise caution in displacing the carriage so as not to engage the limit switch. With the first mass held at 2.5 cm from the initial position, select Run from the Execute box and then release the mass approximately 1 second later. The mass will oscillate and attenuate while encoder data is collected to record this response. Select Ok after data is uploaded.

7. Select Set-up Plot from the Plotting menu and remove Encoder # 1 position and

add Encoder # 2 position to the left axis and then select Plot data from the Plotting menu. You will see the second mass time response.

8. In the Data menu, select upload raw data and save the data in a .txt format in the

a:\ drive.

9. Repeat step # 10 and 11 of Case 1 to obtain ωnm21. CASE 4

1. Remove the mass from carriage # 2. 2. Repeat steps 5 to 8 from the previous case 3, to get ωnm22, i.e., natural frequency

of mass 2 and trial 2.

3. Using equation 2 determine the damping ratio ζm22. CASE 5

1. Connect dashpot to carriage # 2. 2. The damping adjustment knob is set to fully closed position, i.e., when it first begins

to resist tightening. “Do not over tighten”. At this position make a reference mark on the knob.

3. Open the damping adjustment knob 2 turns from its fully closed position.

4. Add four 500 g mass to carriage # 2.

5. Repeat steps 5 to 8 of case 3.

6. You should subtract the ‘steady state error’ from each measured amplitude if it is

greater than 0.05 cm to calculate ζd. 7. Using equation 1 determine the damping ratio ζd.

CASE 6

1. Remove the dashpot and replace medium stiffness spring by high stiffness spring between carriage #1 and carriage #2.

38

2. Repeat CASE 4, i.e., without mass. Determine ωnm23 for spring of high stiffness.

CASE 7

1. Replace high stiffness spring by low stiffness spring. 2. Repeat CASE 4, i.e., without mass. Determine ωnm24 for spring of low stiffness.

Calculations and Tabulation m :- 4* 500 = 2000 g mc1 :- Mass of carriage # 1 mc2 :- Mass of carriage # 2 kh :- Stiffness of high stiffness spring km :- Stiffness of medium stiffness spring kl :- Stiffness of low stiffness spring cm1 :- Damping coefficient of carriage 1 cm2 :- Damping coefficient of carriage 2 cd :- Damping co efficient of dash pot For Medium Stiffness Spring and mass 1 (measure and calculate from time response of case 1 and 2): ωnm11 :- ……………….. rad/s ωnm12 :- ……………….. rad/s

39

ζm12 :- ………….. For Medium Stiffness Spring and mass 2 (measure and calculate from time response of case 3 and 4): ωnm21 :- ……………….. rad/s ωnm22 :- ……………….. rad/s ζm22 :- ………….. For Medium Stiffness Spring and mass 2 attached to dashpot (measure and calculate from time response of case 5): ζd :- ………….. ωd :- ……………….. rad/s (measured from time response) ωn :- ………………… rad/s (calculated from eqn # 1) The spring stiffness (kl, km, kh), mass of carriages (mc1 and mc2) and the damping coefficients (cm1, cm2 and cd) are found by equating the coefficients of same order in the following general equation:

2 2 22 n ns s s c m s k m (5)

i.e., 2

n

k

m (6)

2 n

c

m (7)

Note: Equations (5), (6) & (7) are general equations. For finding above said parameters, their respective natural frequency, mass, and damping ratio have to be considered. e.g., For solving the unloaded carriage mass mc1 and spring constant k, the following two equations are solved:

40

2

122

mnw c

k

m m

2

222

mnc

k

m

And for finding, the damping coefficient, the following equation is used:

2 22 22 22m m nm cc m

And, similarly for other masses and damping coefficients.

STRAIN GAGE EXPERIMENT # 6(a)

INTRODUCTION

41

The design of the systems that involve components with complicated geometry, complicated loading and perhaps points of stress concentration, are often verified by actual measurement of strains, from which the actual stresses can be calculated. A common technique used in practice is the application of strain gages. The utilization of strain gage is comparatively easier and less expensive than photo elasticity. It measures the strain only at the surface of the object. In order to find the three unknowns εx, εy, and γxy at any point on a plane, it is necessary to use three gages oriented in three arbitrary directions. However, if the direction of principal stresses is known, only two gages would be sufficient in that, they can be installed parallel to the principal axes along which the shearing stress and strain are zero. These principal stress directions can be easily determined by using the stress coat application method. The stress coat is a brittle lacquer that is applied on the surface in liquid form (normally by spraying). When dried, in about 3-4 hours, it will become brittle. The coating will crack perpendicular to the direction of tensile principal stresses if the member is subjected to stress and if the strain on the surface is above the coating’s threshold strain. Observing the cracks, one can determine the direction of principal stresses. Even though the stress coat method will not be investigated in this experiment, it is still good to have a familiarity of its existence and usage. Principle of Strain Gage:

The strain gage principle is based on change of electrical resistance of a conductor. Consider the wire, shown below, cemented to the surface of a specimen.

Any strain applied to the specimen is directly transmitted to the wire. The electrical

resistance of the wire before applying any strain is given by

where ρ is resistivity, L is length and A is the cross sectional area of the wire. When a strain is applied to the surface, the length of the wire will change as well as its cross sectional area (due to Poison’s effect). This change can be written as

or

A

L = R

A

AL -

A

L = R

2

L

L f =

A

A -

L

L =

R

R

42

Here ∆L/L is the strain on the surface. This equation leads to

where f is the gage factor provided by the manufacturer. Normally to obtain a measurable amount of resistance change, the length of conductor is to be on the order of several inches. However, the strain may not really represent the strain at a point. For this reason, the conductor is instead formed or stamped out of a thin metal foil in the configuration shown below.

Figure 2. Typical Strain Gages Configurations

The semi-circular loops formed will cause the gage to read, also, some strain in directions other than that of the gage axis. This can be corrected by a factor called “cross sensitivity” given by the gage manufacturer. The foil gages have relatively large cross sectional area at the loops, therefore their cross sensitivity will be very small and the error is negligible if no correction is made for cross sensitivity. Commercial gages are available in different lengths L and widths W. Depending on the application, sometimes gages are needed to be smaller than 1/64 of an inch in length and width.

A strain gage may contain only one gage, which is called single element gage, or

it may contain two or three independent gages set at certain angle relative to one another. The three-element gage is called delta rosette or rectangular rosette depending on the angle of gages.

f

R / R = =

L

L

43

A Wheatstone bridge as showed in Fig. 3 measures the change of electrical resistance in the gage due to strain. The instrument that does this is called strain indicator. Once the gage is mounted on a surface, it is connected to the bridge and the circuit is balanced (zero potential between B and D) before the member is loaded. This is accomplished by changing the resistance of other legs until the indicator reads zero. When the strain is applied the bridge will have to be rebalanced. The rebalancing is done automatically or manually, depending on the strain indicator. The gage factor f is set on the instrument and the reading of instrument is directly in terms of micro inches per inch of strain (i.e., μin/in).

In many situations more than one gage will be involved. In such cases to save time, a switch and balance unit is used to which as many as ten gages can be connected and balanced simultaneously. The switch and balance unit will connect one gage at a time to the strain indicator.

Determining Experimental & Theoretical Stress in a Round Beam

The stress distribution in a round beam subjected to a bending moment and torsion can be determined using three strain gages mounted on top of the beam as shown below:

44

where P is the weight of the load, M is the moment along the beam axis due to P considered at the end of the beam, and T is the torque is the product of the load times the arm length.

The strains obtained from a strain indicator can be used to compute the strains in the longitudinal and tangential directions of the round beam. Equation (5) gives the strain in any direction on a plane tangent to the beam as a function of the longitudinal, tangential, and shear strains,

Substituting the strain readings from the three strain gauges, we will obtain three equations and three unknowns (εl, εt, γlt). The principal strains (εl and ε2) can now be calculated using Eq (6),

The principal stresses, σ1 and σ2, and their direction, α, can be found from the following

equations:

)(2 2

+ )(2 2

- +

2

+ = lttltl sincos

2

+ -

2

+ =

2lttl

2tl

1,2

2121 + -1

E =

45

Where, E is the modulus of elasticity, and is Poisson’s ratio.

The theoretical principal stresses are calculated from the bending moment acting

along the beam’s axis and the torsional stress acting perpendicular to the beam:

and,

where, I is the area moment of inertia of the section, and J is the polar moment of inertia of the beam section, y is the vertical distance from the neutral axis to the point of where the strain is measured, and r is the radius of the beam. Objective

The purpose of this experiment is to: Learn how to mount the strain gage and how to use the associated instruments to

measure the strain. Measure the strain on a simple specimen, calculate the stress and compare the

results with theory. Necessary Equipment and Tools Aluminum bar specimen Strain gages Switch and balance unit Strain indicator Ohmmeter (i.e., Multimeter)

tl

lt

- = )(2 tan

1222 + -1

E =

+ 2

-

2

+ = 2

xyyx

2yx

1,2

yx

xy

-

2 = )(2 tan

J

Tr = and ,0 = ,

I

My = xyyx

46

Means to apply load to the specimen Specimen Load (weights) Procedure 1. Select the aluminum bar specimen. Carefully examine the construction of the

gauge as well as the manner in which each gauge is mounted to the specimen. Please handle the specimen with care so that no gage is damaged!

2. Make sure there is a good connection between the wire leads and the strain gauge by measuring the resistance across each gage with the ohmmeter.

3. Connect the gages to the strain indicator through a switch and balance unit and balance them. Please follow carefully the instructions provided inside the strain indicator.

4. Apply load to the specimen in ten steps from 2 lbs to 20 lbs. 5. Record the load and the strains at each step. 6. Repeat steps 4 and 5 for obtaining two additional sets of data. 7. Provide a linear regression for the data load vs. strain for θ = +45°, 0°, and -45°.

Show the error bar of each load based on max/min measurements. 8. Using load = 15 lbs and the curve-fitted data from step 7, first calculate εl, εt, and

γlt, then ε1 and ε2, and finally σ1 and σ2. Also calculate the principal direction α using Eq. (9).

9. For load = 15 lbs, calculate σ1 and σ2 using Eq. (10). Also calculate the principal direction α using Eq. (11).

10. Compare the results between the two methods shown in Steps 8 and 9. 11. Provide additional discussion including the maximum load Pmax that the Aluminum

bar can safely support. Strain Gage Experiment

Name:

Section:

Group:

Date:

Specimen Used:

47

Specimen dimensions: Gage factor: 2.0 Strain gage resistance:

Micro-Strain Data Sheet

Load P

(lb)

Strain (μin/in)

Channel # ____ Strain (μin/in)

Channel # ____ Strain (μin/in)

Channel # ____

0

2

4

6

8

10

12

14

16

18

20

Running ANSYS 15.0 on LSU Virtual Lab (VLab) 1. Open vlab2.lsu.edu in web browser, and install VMware View Client. 2. Open VMware View Client on your desktop or perform a search.

3. Use address: vlab2.lsu.edu and click Connect.

4. Type in your myLSU ID and password and click Login. (Domain: LSU)

5. Select Engineering Desktop and click Connect.

6. After VLab session opens, go to Start > All Programs > ANSYS 15.0 > Mechanical APDL

Product Launcher.

7. Enter Working Directory and Job Name, and click Run to start ANSYS.

ME4201 VI(b)

48

VI(b). Thin-Walled Pressure Vessel Experiment

Introduction When a design involves stress concentrations or is comprised of a complex

geometry, theoretical calculations alone may not be sufficient for engineering analysis. The time necessary to complete hand calculations may be impractical, especially when multiple points of interest exist. In these instances strain measurement tools and mathematical modeling can be used as alternatives to determine the stresses present under various loading conditions. Common techniques include: Strain gages, Photoelastics, and FEA modeling.

The application of strain gages is often easier and less expensive than implementing photo elastics. However, photoelastics can provide measurable data over a large region simultaneously, while strain gages only provide readings directly at the application site. Similarly, FEA modeling can provide data at any point on the model and at unlimited loading conditions, but building and meshing an accurate model with proper boundary conditions can be quite difficult. This lab aims to explore strain measurement using all three of these techniques to highlight the pros and cons of each method.

The sample used in this experiment is a commonly found thin walled pressure vessel with uniform internal pressure. For a symmetrical cylindrical thin walled pressure vessel the hoop stress (X-axis) and longitudinal stress (Y-axis) can be easily calculated at the center of the cylindrical body. However, near the non-symmetrical end caps this does not hold true. When exploring these regions the aforementioned techniques are much more effective.

Principle of Pressure Vessels

A pressure vessel is any container designed to hold a liquid or fluid at a pressure considerably larger than the ambient temperature. In order for a pressure vessel to be thin-walled it must have a radius of at least 10 times the thickness. There are two main stresses induced on thin-walled pressure vessels, the first being the hoop stress ( ) or stress in the circumferential direction. And the second being the longitudinal stress ( ) or stress in the longitudinal direction. The figures below show give a visual representation of how the stresses act.

ME4201 VI(b)

49

For a thin walled pressure vessel with hemispherical end caps the hoop and

longitudinal stresses can be calculated using the following equations, given that is the internal pressure, is the radius of the inner portion, and is the wall thickness.

1

2 2

Note: that these equations give you the stress on the inner portion of the pressure vessel; it must be assumed that the stress acts uniformly across the thickness in order to calculate the theoretical stresses at locations of interest for this experiment. Also the pressure vessel does not have hemispherical however these equations are still valid at the center of the pressure vessel.

The hoop stress can also be written as the first principle stress ( and likewise the longitudinal as the second . There cannot be a negative stress induced because the system is only in tension therefore the third stress ( ) is 0. Knowing this the maximum shear stress can be calculated from the following equation.

2 3

Figure1:HoopStress Figure2:LongitudinalStress

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Principle of Strain Gages

Basic Concept The principal of strain gages is an important concept that can be used in a

variety of different engineering applications. If you recall from instrumentations strain gages were used on a cantilever beam in order to measure the system response and the oscillation of the beam was modeled as a second order system. In this experiment you will use strain gages in different configurations in order to calculate principal stresses and direction in thin-walled pressure vessels.

The theory is based on the change in electrical resistance of a conductor. The electrical resistance, , of the wire can be found before any strain is applied by the following equation:

4

is the resistivity of the conductor, is its length, and is the cross sectional area

of the wire. After a strain is applied, the length of the wire changes as well as the cross sectional area, due to Poisson’s effect. The change can be found by either of the following two equations:

Δ Δ

Δ

Δ

Δ

Δ

Δ 5&6

is known as the gage factor of the strain gage, the manufacturer provides this.

It can also be found by manipulating the following equation:

Δ

Δ ⁄ 7

is the strain on the surface of the wire. The length of the conductor needs to

be several inches in order to obtain a measureable resistance change, Figure 3 below shows the common configuration for a strain gage

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The Wheatstone bridge is device used to measure the resistance in the gage due to the strain. The configuration for the Wheatstone bridge is shown in Figure 4 below. The strain DAQ for this experiment is set up in the quarter bridge configuration. This configuration is only capable of measuring tension. The strains obtained from the DAQ system can be used to compute the longitudinal, tangential, and shear strains for the tee and rectangular rosette configurations.

Tee Rosette The Tee rosette configuration is set up with two mutually perpendicular

grids; this can be seen in the figure below.

Figure3:TypicalStrainGageConfiguration

Figure4:Wheatstonebridge Configuration

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Figure 5: Tee Rosette Configuration

This configuration should only be used when the principal strain directions are know, for example in this experiment we know the principal directions are known at the center of the pressure vessel, this is why there were tee gages there. If there is uncertainty about the principal directions, a three rosette system in preferred.

Knowing the principal directions for the configuration in can be concluded that at the tee rosette the strains measured are the principal strains. Therefore

8 9

The principal stresses ( , ) and direction must then be calculated using Hooke’s Law. The stresses can be determined using the following equations where is the modulus of elasticity and is Poisson’s ratio.

1

10

1

11

Rectangular Rosette The rectangular rosette configuration consists of three grids, with the second and third grids angularly displaced from the first grid by 45° and 90°respectively. The figure below shows the rectangular setup.

Figure6:RectangularRosetteConfiguration

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Once the strains have been measured using the DAQ system the tangential longitudinal and shear strains must then be calculated. The equation below is the general form of strain at any given direction on the plane tangent to the bottle based on the tangential ( ), longitudinal , and shear strains .

2

2cos 2

2sin 2 12

The equations below have modified form the general form and been

adapted for rectangular rosettes in this experiment. The strains and directions at each of the three locations for each rosette strain gage is known; thus this information can then be used to calculate the tangential, longitudinal, and shear strains given that there are 3 equations and 3 unknowns.

13 14 15

Knowing the given directions (0°, 45°, 90°) the equations can be simplified into much plainer terms. The three equations below are the reduced forms of the original equations for the rectangular rosette configuration.

16 17

2 18 Once the tangential, longitudinal and shear strains are known the principal strains can then be calculated the principal strains must then be calculated by substituting them into the equation below.

, 2

2

19

The principal stresses ( , ) and direction can then be calculated knowing the principal strains using Hooke’s Law, using the same equations from the Tee Rosette section (# and #). The principal direction can be calculated using the equation below.

tan 2 20

Principles of Photoelasticity Photoelasticity is another technique that can be very useful for visually

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determining stress concentrations on an object. It is the method of coating an object using photoelastic materials in order to take advantage of their unique material properties to analyze stresses. A photoelastic material is one that exhibits the property of Birefringence or contains dual-refraction indices. These materials may be applied in the form of an adhesive and liquid coating. This process leads to the surface of the object showing a high number or degree of contours. In the presence of a polarized light these materials will produce a colorful or black and white fringe of light, the magnitude of which is directly proportional to the stress at that point. In areas of stress concentrations, a polariscope can then be used in order to measure these fringes for the magnitude and direction of stresses.

Finite Element Analysis Finite Element Analysis (FEA) is an invaluable tool for engineers to

simulate real world physical effects on an object including heat transfer, stress analysis, fluid flow, and vibration. It utilizes the Finite Element Method, which is a numerical tool to approximate solutions to boundary value problems. FEA software has the potential to be extremely accurate but is only as good as the engineer operating the software. Objects can be extremely difficult to mesh and any mistakes or oversight can lead to inaccurate results. It often takes a seasoned professional to produce an accurate simulation and it can take a very long time to correctly design your simulation. In this experiment the FEA software used will be ANSYS.

Objective Investigate various types of strain gages Preliminary exposure to ANSYS Introduction to thin-walled pressure vessel theory Measure stress concentrations in thin-walled pressure vessels using strain

gages, photoelastic coatings with polariscope, and ANSYS Compare the results obtained from the various methods

Necessary Equipment and Instrumentation NI cDAQ-9174, CompactDAQ chassis NI 9235, 120ohm, 8-Ch, 24-Bit, 1/4 Bridge Input Module NI 9269 4ch voltage output, ±10V, ch-ch ISO Laptop with LabVIew and ANSYS Software Pressure source capable of 50 psi Tee and Rectangular Rosette strain gages Omega General Purpose Electropnumatic Transducer Alumi-Tek 16oz Bottles Photoelastic coating Polariscope

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Procedure

Remote Operation: TeamViewer 1. Run TeamViewer (Click “Sign Up” to create a free account if you have not

done so yet) and sign in within the TeamViewer program. 2. Note the values next to “Your ID” and “Password” and provide them to

whomever you wish to operate this computer remotely. 3. On remote computer make sure you have TeamViewer downloaded

(Available here: http://download.teamviewer.com/download/TeamViewerQS_en.exe) and enter the information from the host computer to take over.

4. Open “teamviewer” and record your id and password

System Preparation: 1. Safety glasses must be worn when running this experiment. 2. Ensure the air compressor is connected via the quick disconnect 3. Pressurize the main reservoir to around 100 psi (check that reservoir drain on

back is closed) 4. Open the line pressure till it reaches 50 psi. To do so turn the red knob on the

front clockwise slowly. Ensure that the pressure remains constant throughout all of testing (PSV will engage at around 58 psi)

5. Run LabVIEW program below. 6. Release the line pressure by turning the red knob on the front of the compressor

counter clockwise all the way 7. Release air in the main reservoir via the valve on the back of the compressor

LabVIEW: 1. Open NI Max from the NI Launcher on the taskbar. 2. On the left sidebar expand “Devices and Interfaces” and then “NI cDAQ-

9174 ‘cDaq1’” 3. Click “1: NI 9235 ‘cDAQMod1’” and then on the top option toolbar select

“Create Task” 4. Expand “Acquire Signals” then “Analog Input” and select “Strain” 5. Highlight all channels (ai0 through ai7) and click next. 6. Enter “GROUP#_StrainTask” and select “finish” 7. For each channel enter the gage factor as listed in the table below:

Strain_0 2.06Strain_1 2.02Strain_2 2.04Strain_3 2.05Strain_4 2.04Strain_5 2.04Strain_6 2.05

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Strain_7 2.04

8. Hold “Ctrl” and click each channel in order making sure all 8 channels are highlighted and then click the “devices” tab under “Strain Setup” and click “Strain Calibration”

9. Make sure “Enable Offset Nulling” and “Enable Shunt Calibration” are ticked and enter 50k under the Shunt Resistor Value then click “next”

10. Once the measurements are collected click “Calibrate” and then “Finish” 11. Under “Timing Settings” make sure the Acquisition Mode is “N Samples”

with Samples to read and Rate set to 5k. 12. On the left sidebar select ‘NI 9269 “cDaq1Mod2’” and select “Test Panels”

from the top menu. 13. Under “Test Panels” input .5 for the “Output Value (V) and select “Update.”

Close out and return to the Task you created under “Ni-DAQmx Tasks” 14. Record the pressure and under the charter area select “Table” for Display

type. 15. Select Run at the top of the table and record your measurements. 16. Repeat steps 13-15 for output values of .5-2 V in increments of .5 V. (DO

NOT EXCEED 2 VOLTS WILL DAMAGE REGULATOR)

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ANSYS Procedure: Step 1: Open the PV-ME4201-1 file on the Desktop. To view the model in Element form, click on PlotElements

Figure 1: Nodal model

Figure 2: Element model Applying Loads on Model Step 2: Click on Preprocessor LoadsDefine LoadsApplyStructuralPressureOn Areas

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Figure 3

Step 3: Apply Pressure on the bottle by selecting the Pick All button

Figure 4

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Step 4: Type in the values of pressure obtained in the lab in the box that is circled red and then click OK. (Ex: if given 10 psi in lab, type in -10 in ANSYS) (Make sure that it is set to Constant value and Load key is set to 1)

Figure 5

Step 5: Click SolutionSolveCurrent LS

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Figure 6

Step 6: Click OK to begin Solution of Current Load Step. (If there are any Warnings that pop up in the VERIFY box, click OK to continue the Solution process)

Figure 7

Step 7: Once the model is solved, click Close in the window as shown in Figure 8 and X out of the /STATUS Command window

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Figure 8

Step 8: Click on General PostprocResult Viewer

Figure 9

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Step 9: Click on the scroll down arrow where Choose a result item is located. Click on Nodal SolutionStress1st Principal stress (Hoop Stress) as shown in Figure 10 below.

Figure 10

Step 10: Click on the Plot Results icon circled in red shown in Figure 11 below. The model will now be defined with various colors signifying the various stress concentration areas on the bottle.

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Figure 11

Step 11: Click on the Front View Icon as shown in Figure 12 below.

Figure 12

Step 12: Click on the Query Results icon as shown in Figure 13 below.

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Figure 13

Find Stresses at 3 identified locations on the model Step 12: To find the stresses at the top, middle and bottom of the model, left click, hold and drag cursor to designated nodes. Top: node 1076 Middle: node 1593 Bottom: node 15739

Figure 14

Because of curvatures at the top and bottom of the model, multiple elements may be found. For top node 4 elements may be found (1078, 1076, 1074, 1072). These numbers will be shown in the Query Subgrid Results box shown in Figure 14. Once you reach one of these 4 element (Node Nos.) click next in the

Top

Middle

Bottom Rosette

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Multiple_Entities window shown below in Figure 15, UNTIL designated nodes mentioned in Step 12 are reached. For middle node 4 elements may be found (1593, 15233, 48192, 15234) For bottom node 4 elements may be found (15740, 1639, 15739, 48326)

Figure 15

Step 13: Record the 1st Principal Stresses for the Top, Middle, and Bottom locations on the model Step 14: Click on the scroll down arrow where Choose a result item is located. Click on Nodal SolutionStress2st Principal stress (Longitudinal Stress) as shown in Figure 16 below.

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Figure 16

Step 15: Repeat Steps 10-13 for the 2nd Principal Stresses (Longitudinal Stresses) Step 16: Repeat steps 1-15 for 3 other pressures given in the Lab manual (10, 23.5, 35, 47.5 PSI) Step 17: Repeat steps 1-15 for 200 psi which is the rupture pressure.

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Stress Concentration Experiment

Names: ____________________________________________ ____________________________________________ ____________________________________________ ____________________________________________

Section: ____________________________________________ Group: _____________________________________________ Date: _______________________________________________

Pressure Vessel #: __________________________________

Given: 3104 H19 Aluminum properties 10000ksi 0.34

0.0075in 2.6in

Theoretical Hoop and Longitudinal Stresses (using Eq. 1 & 2)

Pressure (psi)

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Experimental Data and Results

Run #

Strain Gage Tee Rosette

Top Rosette Bottom Rosette

Channel Strain_0 Strain_1 Strain_

2 Strain_

3 Strain_

4 Strain_

5 Strain_

6 Strain_

7 Excitation (V)

Pressure (psi)

° ° ° ° ° °

1

2

3

Average of 3 runs

Strain Gage Tee Rosette

Top Rosette Bottom Rosette

Channel Strain_0 Strain_1 Strain_2 Strain_3 Strain_4 Strain_5 Strain_6 Strain_7

Excitation (V)

Pressure (psi)

° ° ° ° ° °

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Tee rosette

Pressure (psi)

Top rosette

Pressure (psi)

Bottom rosette

Pressure (psi)

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ANSYS Results

Report Requirements Obtain strain gage readings from the aluminum can experiment Calculate theoretical hoop and longitudinal stresses at the Tee-gage using

Eq. 1 & 2, i.e. assuming thin-walled cylindrical vessel with hemispherical end-caps

Calculate principal stresses from strain gages (experimental) Analyze ANSYS model for different pressures and obtain principal stresses

at the three strain gage locations Compare results as graphs

o Hoop stress at Tee-gage v pressure (theoretical , experimental, ANSYS)

o Longitudinal stress at Tee-gage v pressure (theoretical, experimental, ANSYS)

o Max. stress (S1) at top rosette v pressure (experimental, ANSYS) o Max. stress (S1) at bottom rosette v pressure (experimental, ANSYS)

Note: You may notice discrepancies between experimental and theoretical/ANSYS stress results due to incorrect orientation of the Tee-rosette

ANSYS

Pressure (PSI) Hoop Longitudinal σ1 σ2 σ1 σ2

T (stress psi) Top Rosette (stress psi)Bottom Rosette (stress ps