WSJ Lab Manual

EXERCISES IN ENGINEERING EXPERIMENTATION

A Laboratory Manual

for

Mechanical Engineering 345

and

Aerospace Engineering 345

Prepared by

William S. Johnson

Mechanical and Aerospace Engineering Department

University of Tennessee

2009-2010 edition

INTRODUCTION

This manual contains laboratory instruction sheets and data sheets for all exercises to be done this semester. The student should read the lab instructions well in advance both to prepare for the experiment and to better understand material covered in the lecture. There are additional supplementary instructions that focus primarily on the equipment operation in the lab and on the course website. NOTE: bring a jump drive, a 3 ½ floppy disk or a writable CD to each lab.

Reporting Requirements:

The required report for experiments 1-8 should consist of the following, in the order listed:

1)Title page including experiment name and number, student name, lab partners, date performed, date submitted and name of lab instructor.

2)A brief written summary giving problem statement, brief general procedure and conclusions. Note that procedure should give the data collection procedure and should not include any discussion of equipment operation. The conclusions should be specific (numerical results, percent differences, etc) and supported by your results. Use past tense and avoid using first person.

3)Response to questions posed in the instructions. May be included in the summary or answered separately.

4)Results presented in graphical and/or tabular form.

A) Graphs: Number consecutively: Fig 1, Fig. 2, Fig. 3, each followed by a title

Several defaults found in Microsoft EXCEL are not acceptable for engineering reports and must be manually changed:

1. Titles must be below the figure. Sample: “Figure 1. Calibration of LVDT”
2. An X-Y plot should have vertical gridlines as well as horizontal.
3. Backgrounds should be white and not gray.
4. Omit the legend if there is only 1 curve.

B. Tables: Number consecutively; Table 1, Table 2, …each followed by a title.

1. The title for tables is placed at the top.

2. Everything listed in tabular form must have a title.

5)An appendix consisting of the following:

A)Original Data sheets

B)One Sample of ALL types of calculations

C)List/sketch and identification* of equipment

D)References: All material taken from another source must be referenced.

NOTE: Any material taken from other sources without documentation is academic dishonesty.

:

* for each instrument used, record range, resolution and serial number.

Experiment No. 1

INSTRUMENT SPECIFICATIONS

A weighing device utilizing a linear variable differential transformer* (LVDT) for output is to be evaluated and its specifications established. The device consists of a cantilevered beam which is deflected by the application of various weights. The beam deflection is measured by a LVDT whose voltage output is to be related directly to the applied weight.

The objective of this exercise is to calibrate the weighing device and determine instrument specifications based on your measurements. Determine the following: sensitivity, hysteresis, repeatability, linearity and zero error. Follow the procedure listed below.

Zero the output of the LVDT using the threaded adjustment. Once set do not readjust this setting.

A. Calibrate the LVDT

Run Program LVDT.VEE

Adjust the attached micrometer until the output is a small value, say 10 mv. Record deflection at this and four other deflections in increments of 0.050 in. up to a total deflection of 0.200 in. Plot mv vs inches of displacement and determine the best fit line. The slope of this line will be the calibration in mv / inch. Compare your result with the manufacturers specification below.

RETRACT THE MICROMETER BEFORE PROCEEDING TO THE NEXT PART.

1. Calibrate the weighing device

For weights corresponding to 0, 100, 200, 300, and 400 gm take 15 readings at each loading as follows:

1. 5 data points, Wi by increasing from a lower value (lift & release)

2. 5 data points, Wd by decreasing from a higher value, (depress & release)

3. 5 data points with neither increasing nor decreasing weight, W (simply remove and

replace the same weight 5 times)

At the end of the experiment, recheck the zero reading and record the value.

At each value of the applied mass determine the following:

Hysteresis = + .5*(ABS [Wd/5-Wi/5])

Repeatability = + (highest reading - lowest reading) / 2

Mean = Sum of all readings /15

Also find the zero error which is the difference between the initial and final millivolt reading with zero weight.

NOTE: The values found by any user/purchaser of this instrument at any applied mass should always fall within the instrument specification. Therefore the values of hysteresis, linearity and repeatability specified for the instrument should be the largest values found.

* Schaevitz Model 200 HR-DC; Mfg. Specs: range = + 0.2 in.; sensitivity = 15 v /in.; linearity = + 0.5% FSO; stability = + .125% FSO; temperature coefficient = .05% per deg. F; frequency response (-3db) = 20HZ.

C.After the completion of Part B for all 5 values of mass to be considered, make a plot of the mean value of millivolts vs. weight (Newtons) and:

1.Determine sensitivity (the slope in mv/N of the straight line of best fit drawn through the mean value of each of the 5 different values of mass using least squares.) Determine the correlation coefficient. (A numerical indicator of how well the line fits the data -1.0 is perfect)

2.Determine linearity (use equation 1.9, text)

D. Theoretical Beam specifications

Measure the required dimensions of the beam setup (thickness should be carefully measured in several locations and averaged), assume the beam is aluminum with E = 10E6 psi and use the measured LVDT calibration (Part A) to estimate by calculation:

1. Theoretical maximum capacity in N (based on the range of: + 0.2 in. deflection)

2. Theoretical sensitivity in mv/N

Refer to the equation below for the relation between deflection, y of a cantilever beam at any point, x as a result of a load, P applied at x = a.

E.Compare calculated and measured sensitivity and discuss any reasons for differences found. Does the measured (non) linearity appear to be a result of data scatter about an actual linear function, or does the basic instrument appear non-linear?

F.List measured specifications for this weighing device as follows:

1. Range = 0 - ______N (max capacity from part D1 rounded off to one decimal place)

2. Resolution = 1 mv = ______N3. Meas. Sensitivity = ______mv / N

4. Linearity = + % FSO5. Calc. Sensitivity = ______mv / N

6. Repeatability = + % FSO7. Hysteresis = + % FSO

8. Zero error = ______% FSO9. Instrument error = ______% FSO

(equation 1.11 based on linearity, hysteresis, repeatability and zero error)

For a cantilevered beam:

where “l” is the distance from the base to the center of the weight and “x” is the distance from the base to the LVDT. Note: “b” is beam width and “h” is beam thickness.

Experiment No. 2

SIGNAL ANALYSIS

The objective of this experiment is to examine digital data collection and the analysis of complex signals utilizing the frequency domain.

A. Frequency domain measurements (FFTDEMO)*

Analyze the following periodic waves in both the time and frequency domains. For cases 1-5, measure amplitude and frequency in both domains, if possible. Compare results and discuss advantages and disadvantages of each domain of measurement. Explain the reason why some Cosine wave amplitudes and frequencies are accurate and some are not. Note that the input amplitude of each wave for cases 1 - 5 is 100 units.

For Case 5:

Record amplitudes of the first (fundamental) frequency and the first 4 harmonics. Compare these amplitudes to the first 5 terms in a Fourier Series expansion of a square wave of amplitude 100 and of frequency 16 Hz. (See Example 2.3)

B. Digital sampling rate (SCOPE)

Connect the signal generator to the A/D board and set a sampling rate of 1000 Hz. with a total of 100 samples. Measure output frequency using FFT to examine the following frequencies as provided by the signal generator: 50, 500, 600, 700, 800, 900, 1000, and 1500 Hz. Identify the real and the alias frequencies and compare all measured frequencies to input values and/or to the predictions of Figure 7.3 in the text.

1. Telephone Tones (SCOPE)

1. Analyze the frequencies of the sounds produced by the numbers 1 through 0 for a touch-tone telephone using FFT. For each case, first use a wide frequency range to be sure that all frequencies are displayed and then reduce the range as much as possible to minimize resolution. Record the results for each number on the phone and show the pattern for all numbers in a matrix.

D.Use of Accelerometer (SCOPE)

One technique to locate a break in a buried cable is to send a signal over the cable and measure the time for it to reflect from the “break”. Illustrate this principle by using the aluminum rod, accelerometer and hammer to measure the fundamental frequency of a sound wave in the rod produced by an axial hammer blow at the end opposite the accelerometer. Set a maximum frequency of 5000 Hz. MANUAL trigger may work better than automatic.

Calculate the length of the rod based on the measured fundamental frequency. Note that the speed of sound in a solid is (/)0.5 and for aluminum, , the bulk modulus is 10E6 lbf / in2 and the density is 165 lbm / ft3. Compare this calculated length to that measured in the lab.

• FFTDEMO generates the mathematical function indicated and analyzes the function by FFT. For example, in case 1, the function 100*COS(2*pi*205*t) is generated. It is then analyzed by the FFT software at frequencies of 0,1,2,3,…511 Hz. Since 205 is included, the full amplitude of 100 will be shown in the frequency domain. Note that true amplitude can be found in either the time or frequency domain.

In case 2 the same function is analyzed at 0,2,4,6,…1022 Hz. Since analysis is done at 204 Hz and 206 Hz, but not at 205 Hz, the true amplitude of 100 will not be displayed. It can only be measured in the time domain.

In case 4, the function 100*COS(2*pi*50*t) + 100*COS(2*p1*205*t) + 100*COS(2*pi*605*t) is analyzed. Note that almost nothing can be found from the time domain and that only the amplitude at 50 Hz is accurately displayed in the frequency domain.

Experiment No. 3

DYNAMICS OF INSTRUMENTS -I

(FIRST ORDER SYSTEMS)

The objective of this exercise is to determine the dynamic characteristics of first order systems including low and high pass RC filters and a thermocouple.

A.RC filters

Filters are used to eliminate unwanted parts of a signal so that a specific frequency range may be analyzed. This exercise will involve 2 different filters as follows:

Filter no. 1. Low-Pass, R= 10 K-ohm ± 1%, C = 1.0 microfarad ± 0.1%

2. Low-Pass, R = 5 K-ohm  1%, C = 1.0 microfarad  0.1%

Filter no. 3. High-Pass, (Same as 1)

1.Determination of time constant (SCOPE)

Set the oscillator for square wave output with a frequency of 5 HZ. Adjust the oscillator amplitude for about 80% of full scale deflection of the filter output. For each filter do the following (sample @ 10 kHz & 1000 samples):

1. Record one complete half cycle and measure the time constant.

1. Place the first cursor at the start of the step and the second at the final value.
2. Read the magnitude of the step and multiply by 0.632
3. Move the second cursor to the value calculated in #2 and read the time difference.

b.Calculate the time constant (T = RC) and its maximum and minimum theoretical values using the individual uncertainties given for R and C.

c.Compare measured and theoretical values. Are the measured values within the theoretical range of uncertainty?

2.Determination of the magnitude ratio of the three filters as a function of frequency

(SCOPE)

The magnitude ratio, M() = (filter output amplitude)/(filter input amplitude)

For the low pass filters, input amplitude should be the amplitude at 5.0 Hz and for the high pass filter, use the amplitude at 200 Hz. (In each case M()  1.0)

Set the oscillator to the sine wave mode and adjust the initial oscillator output with M() = 1.0(Use 5 Hz. to set low-pass and 200 Hz. to set high-pass) to produce about 80% of full-scale deflection on the computer screen.

a.For each filter, obtain amplitude data at frequencies of 5, 10, 20, 40, 70, 100, 150 and 200 Hz.

1. Plot amplitude ratio (db) vs. LOG10 (frequency) for each filter. (Eqn. 3.11 )

c.For filter no. 1, determine 1) cutoff frequency (frequency where the amplitude ratio equals -3db) and 2) the rolloff (slope in db/decade where one decade represents a ten fold frequency increase, from 10 to 100 or 20 to 200, for example: See figure 6.29)

.

B.Time constant of a thermal system (SCOPE)

1.Measure the time constant of the thermocouple under two conditions. In both cases the data should be copied to a floppy disk.

a)Take a thermocouple that is in equilibrium with room air and suddenly plunge it into an ice bath. Find the time constant by two methods:

1. Present a plot of temperature vs. time and identify the time when the process is 63.2% complete as the time constant.
2. Linearize the data using equation 3.6 (example 3.3) and fit a straight line to the data. Determine the time constant from the slope of the line and compare the result with Part 1.

b)Remove the thermocouple from the ice bath, dry the junction quickly with a cloth or towel and allow it to warm up in room air. To find the time constant for this case, calculate the 63.2% temperature based on ice bath and previously measured room temperature. Plot the data and identify the time constant. Note that several minutes are required for the thermocouple to reach room temperature. Consequently, it is not necessary to plot the data all the way to the end of the process.

2.Explain why the two time constant values measured above in parts 1a and 1b are so different.

Experiment No. 4

DYNAMICS OF INSTRUMENTS II

(SECOND ORDER SYSTEMS)

The purpose of this experiment is to determine the dynamic characteristics of a cantilever beam using displacement, velocity and acceleration measurement.

1. Calibrate the LVDT for displacement using the micrometer. Plot output vs displacement and determine the sensitivity as _____ mv = 1.0 mm of displacement using least squares. (CALBEAM)

NOTES: 1. For each part below, use the digital filter as needed to eliminate noise and drift from the signal. The bandpass filter will eliminate all but the frequency of interest. Measure amplitudes and frequencies from the time domain, not the frequency domain. Take an average of several cycles.

DO NOT OPERATE THE BEAM AT 9 Hz. – excessive vibration will occur!

B. Measure Damping and Natural Frequency. Pluck the beam to get LVDT amplitude as a function of time. (Set 512 Hz. & 1024 samples – SCOPE; Pluck the beam, wait about 1 second and then hit “enter”.)

1. Measure the damped natural frequency averaged over several cycles.

1. Measure the amplitude of about 8 consecutive positive peaks on the screen. (BE SURE TO SUBTRACT THE DC COMPONENT) Determine damping ratio by two methods:Use the log decrement method. Determine 7 different values using peaks 1&2, 2&3, 3&4, etc. Report the mean damping coefficient for the 7 calculations.

1. Plot Ln of the amplitude vs. time, recognizing that this plot should be a straight line of slope -n (see eqn. 3.15a). Note that this requires the solution of two simultaneous equations since  affects the measured value of nd.

1. Determine the undamped natural frequency.

C. Determine Magnification Ratio as a Function of Frequency. Excite the beam over the available range of frequencies and measure peak amplitude from the LVDT output as a function of frequency. Set the amplitude at the indicated mark. (Set 512 Hz. & 512 samples – SCOPE) Plot measured and calculated (Eqn. 3.21) amplitude ratio vs. frequency ratio using the damping ratio from B.2. To get input amplitude, measure the output at a low frequency and note that the amplitude ratio should be 1.0.

NOTE: Move quickly from 8 Hz. To 10 Hz.

1. Calibrate the Velocity and Acceleration Transducers. Excite the beam at 10 Hz and successively read displacement, velocity and acceleration output as a function of time. Measure frequency and the amplitude (mv) of each. (512 Hz., 512 samples)

1. Use the calibration from Part A to convert the maximum displacement amplitude from mv. to mm and express displacement as y(mm) = A* SIN(2**f*t), where A is the peak amplitude in mm, f is frequency in Hz and t is time.
2. Analytically differentiate the equation from Part D.1 to get velocity amplitude in mm/sec. By equating the peak amplitude obtained through differentiation (mm/s) to the direct peak velocity measurement in mv, determine the calibration of the velocity sensor as _____ mv = 1.0 mm/sec.
3. Differentiate a second time to get acceleration. Compare calculated peak acceleration (mm/sec2) to the accelerometer peak output in mv to obtain a calibration of the accelerometer as _____ mv = 1.0 mm/sec2.

Experiment No. 5

ANALYSIS OF EXPERIMENTAL DATA

The objective of this experiment is to utilize a pitot tube, pressure transducer and A/D board to determine volumetric flow rate and its 95% uncertainty when the velocity is fluctuating. Uncertainties to be considered include: velocity fluctuation, Pitot Tube, pressure transducer, data acquisition board and ambient temperature and pressure.

1. DATA COLLECTION (FAN-PITOT.VEE)

Determine the volumetric flow rate produced by a small axial fan. Use the Pitot Tube to measure mean velocity, V at eight radial locations in the fan exhaust. Assuming no circumferential variations, integrate 2*pi*r*V over the radius to estimate the true mean volumetric flow rate and its uncertainty.

1.Take 200 samples at the rate of 10 per second at each one of the eight marked locations and record mean velocity pressure, VP and the standard deviation, Sx at each location. (all in inches of H2O), Note that mean velocity, V (ft./sec.) is also calculated.

2. Draw a histogram of the velocity pressure data at location #4. On the graph, identify the mean, median and most probable value. Note that these are all the same in a true "normal" distribution. NOTE: Total range of data considered is Xmean  3Sx. Consequently, the range for each one of the 9 individual bins is 2*(3*Sx)/9.

1. UNCERTAINTY EVALUATION (UNCERTAINTY.VEE)

1. Evaluate the overall instrument error of the pressure transducer, e1 based on uncertainty in sensitivity, hysteresis, repeatability and linearity. Use Equation 1.12.

2.Estimate the overall instrument error for the data acquisition board, e2 and the pitot tube, e3.(both in inches of H2O)

3. Estimate the overall instrument uncertainty in determining the velocity pressures, u1 . Note that u1 is based on FSO and is the same at all locations.

(IN. H2O)