ME 3456 Dynamics & Vibration Fall 2013Laboratory #3: Instructions and Worksheet

IntroductionLab #3 involves two vibration demonstration units: (A) a horizontal pivoted beam, and (B) a vertical-axis multi-disk torsional system. The schedule has been arranged to let every student participate in making measurements. ###Note, (B) is out of service until further notice.###

Figure 1 (A) Vibrating beam (B) Torsional system

During your 100-minute lab period, you will spend about 70 minutes with apparatus A as be part of a group effort to get response at many forcing frequencies, and about 20 minutes in smaller groups with apparatus B.

For apparatus A, you will be asked to write a short technical memorandum on the data you acquired. This memo will probably require about 4-6 pages. The technical memo should follow the instructions and template attached at the end of this lab document. Please pay attention to the sentences/paragraphs below with gray shading, which indicate the data you will need in order to complete your technical memo successfully. For apparatus B, you will complete a worksheet during the lab period, found in the final 3 pages of this document, and hand it in before you leave (like Lab 2).

1

APPARATUS A: INSTRUCTIONS Apparatus A is a slender horizontal bar, pivoted at one end, and supported against gravity by a vertical spring as shown in Figure 1A. The spring allows for oscillation about the equilibrium position, and an oil-filled dashpot removes energy from the vibrating system.

You will take measurements of Static Deflection, Free Vibration, and Forced Vibration due to rotating imbalance. You should attempt to make some of your observations two ways: crudely, with a stopwatch or tachometer; and electronically with an LVDT (linear variable displacement transducer) and oscilloscope. Both skills are worth knowing. (If you have a flash drive you can take recorded data from the oscilloscope.)

PREPARATORY DETAILS:1. Gently move the bar up and down to make sure that nothing is binding.

2. Connect the motor power cable to the power supply output; polarity doesn’t matter. You can switch it on and adjust the voltage just to make sure that the motor works.

Figure 2: Power supply, with cable to motor. Figure 3: Tachometer

3. Motor rpm is related to voltage. You will have two ways to determine rpm: the first is a contact tachometer (see figure 3). This is used by placing its tip at the center of the rotating disk and reading rpm. It can also be used as a rolling wheel, e.g. to determine the linear speed of a moving belt. The second way, when the motor is creating a steady vibration, is just to look at the vibration frequency with the oscilloscope. That is, measure the time for the zero crossings of several cycles.

4. The motion of the beam end is measured by an LVDT (linear variable differential transformer), as shown in figure 4. This device is a long cylinder containing a transformer coil, and a movable inductive core at the end of a slender wire. It has a range of + 2in, and the coupling core wire is delicate. The LVDT must be energized by a voltage source – the black power supply (fig. 4). The signal out, per inch of travel, is

2

proportional to the input voltage, so you need to maintain input voltage at a constant value – try to set it at 10.0 V, and be careful not to change it! The LVDT sensitivity is approximately 0.365 V out per inch of motion, per volt of excitation.

Figure 4: LVDT and its power supply

5. Connect a signal cable between the LVDT and channel 1 of the oscilloscope (figure 5). The oscilloscope power button is on the top of the instrument. Once switched on, it takes about 1 minute to perform startup routines and begin displaying. You may need to press the RUN/STOP button at the upper right corner if it is not displaying anything.

Figure 5: Oscilloscope with signal to channel 1 (yellow)

6. The main controls for simple use of the oscilloscope are the HORIZONTAL SCALE knob, found at the right; and the CHANNEL ONE (YELLOW BUTTON) VERTICAL SCALE (large) AND VERTICAL LEVEL (small) knobs, found at the left. The horizontal scale refers to how much time between major divisions (centimeters). Its value is shown at the bottom center of the screen. A value of 250 ms or 500 ms is good. The vertical scale refers to how many volts per major division (centimeter). Its value is shown at the lower left of the screen. 500 mV is a good value. Lastly, the vertical level is used to move the trace up

3

or down in the display. It is helpful to set the at-rest signal to zero (the midline) so amplitudes can be estimated.

7. Once the oscilloscope is set up you should be able to see a sinusoidal trace whenever the beam oscillates. To download a particular fixed trace, insert your flash drive and export data.

8. You should calibrate the LVDT to make sure you have a realistic relation between oscilloscope voltage and beam angle (radians). Do this by setting the vertical sensitivity to 200 mV/division, and adjusting the trace vertical position to a line near the top of the screen. Take note of the mm position of the beam tip. Then place a 500 g weight on top of the existing weight near the right end of the beam. The beam will move lower, resulting in both a lower oscilloscope signal (if the signal goes higher you could reverse the voltage polarity to the LVDT) and a lower mm reading at the tip. For calibration purposes, record the change in LVDT signal (should exceed 1 volt) _______ and displacement at the tip of the beam (mm) ______. From this you can compute mm/Volt _______. Lastly, measure the distance (mm) from the beam pivot to the tip scale_____. Tip displacement divided by this distance is angular displacement, radians. Please compute radians/volt _______. With this information, you can interpret oscilloscope readings in terms of beam angle.

9. Once you have completed your use of Apparatus A, please switch off all instruments and disconnect the cables.

Apparatus A Analysis GuidelinesTo model and analyze this rotational system. You have to think of ‘equivalent rotational quantities’. For example the LVDT output should be expressed in terms of radians per volt, the equivalent mass is a moment of inertia, the equivalent spring is a torsional spring, etc. (See page 7 for related comments.)

(1) Static Test: Determining Keq In the LVDT calibration routine, you used a 500 g mass to move the beam lower. From the tip displacement and beam length, you can determine the angle change of the beam, due to the weight. Now measure the location (distance from beam pivot) where the weight was applied. Determine the weight in Newtons of the mass, and multiply by the distance to the pivot to get moment (N-m). The ratio of applied moment to angle change is Keq, the torsional stiffness, defined as N-m/radian of rotation about the pivot.

Weight: _______ Moment arm: ______ Keq:__________

(2) Free Vibration: Determining ωd and ζ Displace the end of the beam an inch or so and allow it to oscillate freely. You will be looking for the damped natural frequency ωd, and the

4

damping ratio ζ. From these quantities you will calculate the undamped natural frequency ωn, which in combination with Keq will let you determine the mass moment of inertia J0. You will then have all the information you need to calculate Ceq. In other words, you will be able to find all three coefficients J0, Ceq and Keq of the unforced equation of motion.

Here is where the oscilloscope shines: If you set the vertical scale to 500 mV/division, and center the signal, then you can get the signal to oscillate the full height of the screen. Once a decaying signal is visible on the screen, push RUN/STOP to freeze it for measurement.

The way to determine ζ is via the logarithmic decrement δ: count the number of cycles for the amplitude to reduce by about 50%. Try to measure the precise amplitude reduction for that many cycles. δ = (1/N) ln(θ1/θN+1) _______________ For your report, download the oscilloscope record to a flash drive, and import it into Excel.

Estimate the frequency ωd by determining the time for 10-20 cycles. By using many cycles, your estimate of the frequency will be more accurate than trying to ‘time’ a single cycle. Cycles are best measured by their zero crossings.

Time ______ # cycles ______ Hz_____ ωd ________

Please also estimate the frequency by timing ten cycles with a stopwatch – your result should match! Hz_________

(3) Vibration Forced by Rotational Imbalance: Determining Frequency Response

Mounted on the beam is a small DC gearmotor with an unbalanced flywheel. The motor voltage controls the rotational speed, which excites the steady vibration of the beam. Note that when the motor is turned on, or the speed is changed, it takes a little while for the beam to settle into steady motion. This delay is simply the time for a free vibration to die away. To clearly see the approach to steady state, once the motor is rotating, use your hand to further deflect the beam. After that oscillation visibly dies away, what remains is the forced vibration.

To model the unbalanced forcing, you will need the ‘missing mass’ of the hole in the disk. Therefore, measure the hole diameter and depth, and use the density of steel to calculate missing mass. You will also need the eccentricity of the hole (distance from motor shaft to hole center), and the moment arm of motor shaft relative to beam pivot (i.e., distance from pivot to motor shaft. Hole diameter _______ Hole depth ______ Hole center from motor shaft ______ Motor shaft from beam pivot _______

The idea is to measure steady vibration amplitude as a function of motor rotational speed, and see how it compares to Equation 3.81 which is plotted in Figure 3.17. Measure at least 15 points, including the voltages suggested below (some below resonance and some above).

5

In order for the LVDT signal to represent angle, you need the calibration information found above. That is, for the given power supply voltage to the LVDT, what is the ratio of beam angle to LVDT voltage.

Initial rpm scan: To get an initial idea of the behavior due to rotating imbalance, first just try approximate motor voltages 30, 25, 20, 15, 10, 8.2, 5, and 3.5, and observe the LVDT signal without measuring. Above 10V and below 5V, you will probably want to use the 200 mV/division vertical scale (with the signal centered), and 250 ms/div horizontal scale. However near the natural frequency (resonance) the angular vibration amplitude will be too great. Then you will need the 500 mV/division vertical scale (and will have to re-center the trace). What you see: At the higher frequencies, the amplitude is essentially constant. That is because the center of mass of the beam-motor system hardly moves – the heavy part of the flywheel moves up, and the beam moves down in compensation to leave the CM undisturbed. Near resonance, the motor base exerts a cyclic force on the beam that tends to increase its vibration amplitude. Below resonance, the imbalance force decreases rapidly, and near zero the beam amplitude is proportional to ω2. This corresponds to figure 3.17 in the text.

Data to be taken by the group: set motor voltage to values near 4, 5, 6, 7, 7.5, 8, 8.25, 8.5, 9, 10, 12, 15, 20 and at least three or four additional values that you choose. For each voltage, the lab team will determine motor speed (rad/s) by two methods, and determine the steady oscillation amplitude (radians). Please also try to find the motor frequency (or rpm) giving the very highest vibration amplitude.

Exact Voltage

Motor rpm, tachometer

Motor Hz oscilloscope

Motor rad/s calculated*

Displacement amplitude, volts

Angle amplitude, radians, calculated.

6

*Motor speed is a very smooth function of voltage. Therefore if you plot motor radians per second, determined by both methods, against motor voltage, you may see that one method (rpm or oscilloscope) is more accurate than the other.

TO HAND IN FOR (A): You will hand in approximately 3-5 pages, i.e. a short technical memo; guidelines and template are attached at the end of this hand-out. Be sure to:

1. Present your static stiffness and free vibration data in a small table, then briefly show your calculation of Keq, Ceq, J0, and an oscilloscope record of the decaying vibration.

2. Likewise, present your forced vibration results in a table like the one above. Then provide an Excel plot of these data points along with a theoretical curve from eq. 3.81 based on the ωn and ζ values determined in the free vibration experiments.

Note that equation 3.78, on which 3.81 depends, can be connected to rotational quantities, as follows: first Keq, Ceq, and J0 are used as coefficients of a differential equation for θ

J0 θ̈+Ceq θ̇+Keqθ=Rmeω2sinω t

where R is the distance R from the pivot to motor shaft, m is the missing mass of the hole in the disk, and e is the radius to the hole center. You can measure e with a ruler, and estimate m from the hole volume and the density of steel.

3. Please also take note of equation 3.83, which gives the resonant amplification in terms of ζ. How close is the theoretical value to maximum amplitude you observe with voltages near 8.2?

7

APPENDIX: REPORT FORMAT

ME 3456 TECHNICAL MEMORANDUM REPORT FORMAT

TO: LAB T.A. NAME

FROM: STUDENT NAME, CLASS SECTION

LAB TITLE: VIBRATION DUE TO ROTATIONAL UNBALANCE

LAB DATE: 3/18/2013

DATE SUBMITTED: 4/1/2013

INTRODUCTION

What was the purpose of the experiment? What were the primary results you were attempting to get? What does the reader need to know to understand your results and analysis? This part can be brief – a paragraph or two.

EQUIPMENT DESCRIPTION

What equipment and sensors did you use? Provide a simple labeled sketch of the main hardware and instruments, and key hardware dimensions.

PROCEDURE

Give an overview of what experiments you did and what you measured.

RESULTS

Here is where to present your requested data, graphs, equations, computed results, and anything else that presents what you got. Explain what the reader is seeing.

DISCUSSION

What went wrong? What errors did you encounter? Did you get any unexpected results? Did you run out of time or have any equipment failure? What steps did you take to rectify the situation? What troubleshooting was necessary to make the experiment work?

8

APPENDICES

No appendices are expected for these labs. They would normally be a good place for a detailed description of the experiment, or to record a mass of data.

GENERAL NOTES ABOUT WRITING MEMOS

Write in a concise, direct manner. First person is acceptable in memo format.

Use complete sentences, proper spelling and grammar. Proofread your document.

Use headings to guide your reader. Make sure your figures are readable, and that they have captions, and

figure numbers if there is more than one. Plots must have labeled axes and units.

These are individual lab reports. This means that although you can discuss concepts with your lab group, the work presented in the lab report must represent your individual understanding of the material.

All procedures and mathematical manipulations that have been applied to the raw data to obtain new tables need to be explained.

Equations that are used in the report should be defined and referenced as shown in equation 1 below:

F⃗=m a⃗ (Eq. 1)

Here, F⃗ is the force, m is the mass, and a⃗ is the acceleration. (All variables should be clearly explained.)

9