Dynamic Systems and Controls Laboratory

Experiment 2:
Fundamental Frequency of a Beam

Due: October 9th, 2015
Submitted: October 9th, 2015

Table of Contents Abstract | 2 | List of Symbols and Units | 3 | Theory | 4 | Procedure and Experimental Setup | 8 | Sample Calculation | 9 | Error Analysis | 12 | Results | 13 | Discussion and Conclusion | 16 | References | 18 | Appendix | 19 |

Abstract The experimental analysis of rotating equipment and the fundamental frequency of an attached beam was achieved by attaching a motor to a simply supported beam with adjustable weights. The input from the motor provided a method of creating a frequency that could be correlated experimentally to the theoretical Dunkerley’s formula and Rayleigh’s energy method. Using the accumulated data, the experimental values obtained were Ke = 5285.4 lb/ft, me = .09773 slugs, and ωn = 232.56 rad/s; comparatively, the theoretical values were Ke = 5493.16 lb/ft, me = .06967 slugs, and ωn = 278.7 rad/s. The resultant margin of error for these values were 3.78%, 40.28%, and 16.56%, respectively.

List of Symbols and Units Symbols | Description | Units | ωns | Natural Frequency of System | radsec | ω1n | Natural Frequency of Beam Alone | radsec | ω2n | Natural Frequency of Motor and Added Mass | radsec | mb | Mass of Beam | slugs | me | Effective Mass of Beam | slugs | ke | Effective Stiffness of Beam | lbfft | M | Mass of Motor and Added Mass | slugs | E | Modulus of Elasticity | psi | I | Moment of Inertia | in4 | L | Length of Beam | in | h | Height of the Beam | in | b | Base of the Beam | in |

Theory
Rotating equipment may contain gears, disks, propellers, and other appurtences. These additions cause deflections in the expiermental shaft. Centrifugal forces, caused by the rotation of the machine and the deflection of the shaft, act cyclically on the shaft with a forcing frequency that is equal to the rotational speed. When the natural frequency of the rotating equipment and the shaft in deflection match, resonance occurs. In mechanical systems, resonance is very destructive and rapidly increased failure or reduce lifespan of equipment and materials. Thus arises the necessity to determine the natural frequency of the loaded shaft, to ensure its natural frequency is avoided.
The shaft has a distributed load and thus an infinate number of natural frequency components. The expiremental goal is to find the fundamental frequency, or lowest frequency, of the shaft. The natural frequency, ωn, in radians/second of a simple spring-mass system is defined as: Natural Frequency | ωn=km=kgW=gYs | [1a],[1b],[1c] |
Where:
* Ys = Static deflection of the spring * g = gravitational constant * W = weight

For estimating the fundamental natural frequency of a shaft or beam, equation (1) can be utilized in conjuction with the maximum static defection for Ys. This, however, assumes that the beam is massless carrying a concentrated load with Ys being the static deflection at the point of application of the load. Here the load is representative of the total weight. As this is a very crude approximation, it is better to find an equivalent system for a better estimation.

Rayleigh’s Energy Method is an analytical method used to determine the fundamental natural frequency of multi-degree of freedom systems. It is based on the use of the static deflection curve and on the energy consideration of the system. Rayleigh’s energy method estimates that the effective stiffness of the beam, e and the effective mass, e are given by: Effective Stiffness | ke=48EIL3 | [2] | Natural Frequency | me=0.4857mb | [3] |

Where: * E = Modulus of elasticity * I = Second moment of area of the beam section * b = Mass of the beam. If an additional concentrated mass is attached to the equivalent system as shown below, the natural frequency for this system can be written as

Natural Frequency of the system | ωns=keme+M | [4] |

Where: ωns = Natural frequency of the entire system (with additional concentrated mass).

Equation [4] can be written as: Natural Frequency of the system | 1ωns2=meke+Mke | [5] |

Now with 1ω1n2=keme representing the natural frequency of the beam and 1ω2n2=keM representing the natural frequency of the mass M, attached to the massless beam, Equation [5] can be redescribed as: Natural Frequency of the system | 1ωns2=1ω1s2+1ω2s2 | [6] |

This Equation [6], is known as Dunkerley’s equation. In Equation [4], b and b are constants, with the variable being ωns and ω2n If the mass, M, changes, then the systems natural frequency, ωns changes accordingly.
Plotting 1ωns2 versus M would produce a straight line with the following characteristics:

Figure 1: System Natural Frequency vs Mass

Note: * The slope is equal to 1ke, and the beam stiffness, ke can be found. * The intercept at the vertical axis (M=0), occurs at 1ωns2, and the experimental ωn can be found. * The horizontal axis intercept, (1ωns2=0) occurs at (me=-M)

The theoretical value of the natural frequency of a simply supported beam with a uniform cross section can be found using the following: Theoretical natural frequency of a simply supported beam | ωn=π2EImbL3 | [7] |

Procedure and Experimental Setup 1. Ensure the motor assembly on the beam and LabVIEW software is set-up properly, starting with no added weight. 2. Press record in the LabVIEW software and turn on the motor. 3. With the motor running, gradually increase the speed using the knob just left of the meter display (outlined in red in the image below) until the maximum beam vibration is reached. This will be when the beam is in resonance and can be observed by extreme beam vibration and excessive noise.

Image 1: Motor Speed Unit 4. While at resonance, take a screenshot of the LabVIEW software. 5. Save screenshot and label it properly. 6. While at resonance, use the tachometer to record the motor RPM. To do this, aim the tachometer at the spinning motor. There is an area to aim at the bottom of the circular disk (red dot).

Image 2: Tachometer Image 3: Motor (front) 7. Record and label the RPM properly. 8. Stop LabVIEW software. 9. Gradually reduce motor speed via the same knob used in step 3, image 1, then turn it off. 10. Attach weight to the motor assembly in increments of four pounds. Each weight weighs one pound, so a total of four weights. 11. Repeat steps one through 10 until you reach a maximum of sixteen pounds of added weight.

Sample Calculations
From Dunkerley’s equation [5]; | 1ωns2=meke+Mke | [S.1] |

Using values from Excel the equation can be rewritten as; | 1wns2=P1M+P2 | [S.2] |
Values from the best fit line in Excel;
P1= 0.0001892 ; P2= 1.849e-05
Solving for ke, | P1=1ke | [S.3] |

ke=10.0001892 ke= 5285.4 lb/ft2 Solving for me, 0=0.0001892me+ 0.00001849 -M=me me=0.09773 Slugs

Solving for wn, | ωn=keme | [S.4] | ωn=5285.40.09773 ωn=232.555 rad/sec

Summarized experimental values (Tachometer): * Ke = 5285.4 lb/ft2 * me = .09773 slugs * Wn = 232.56 rad/s
Summarized experimental values (Labview): * Ke = 5055.9 lb/ft2 * me = .07093 slugs * Wn = 266.98 rad/s

Theoretical Values (From Procedure) mb=13.813-9.195=4.618 lb≈0.1434 slugs me=0.4857mb=2.243 lb≈0.06967 slugs

I=bh312=1(0.5)312=0.01042 ft4

Ke=48EIl3=48EIl3=457.91121=5493.16lbft

wn=EImbl3=278.7rads

Summarized theoretical values: * Ke = 5493.16 lb/ft2 * Me = .06967 slugs * Wn = 278.7 rad/s

Error Analysis

% error=theoretical value-experimental valuetheoretical value×100

% error=theoretical[ke, me,wn]-experimental[ke, me,wn]theoretical[ke, me,wn]×100

Summarized margins of error (Tachometer): * Ke = 3.78% * Me = 40.28% * Wn = 16.56%
Summarized margins of error (LabVIEW): * Ke = 7.96% * Me = 1.81% * Wn = 4.20%

Results | Added weight (lbs) | M (slugs) | v (Hz) | wn | 1/wn2 | 0 | 0.318982919 | 17.82 | 111.9454182 | 7.97971E-05 | 4 | 0.443206522 | 15.85 | 99.58848712 | 1.00828E-04 | 8 | 0.567430124 | 14.14 | 88.85471222 | 1.2666E-04 | 12 | 0.691653727 | 13.07 | 82.10028801 | 1.48358E-04 | 16 | 0.815877329 | 12.08 | 75.91135049 | 1.73535E-04 |
Table 1: Tachometer Results

Figure 2: Tachometer Frequency vs. Mass

| Added weight (lbs) | M (slugs) | v (Hz) | wn | 1/wn2 | 0 | 0.318982919 | 18.03 | 113.2858311 | 7.792E-05 | 4 | 0.443206522 | 15.95 | 100.2168056 | 9.95678E-05 | 8 | 0.567430124 | 14.05 | 88.27875357 | 1.28318E-04 | 12 | 0.691653727 | 13 | 81.68140899 | 1.49883E-04 | 16 | 0.815877329 | 12.01 | 75.46105554 | 1.75612E-04 |

Table 2: LabView Results

Figure 3: LabVIEW Frequency vs. Mass

Figure 4: LabVIEW and Tachometer Frequency vs. Mass

Discussion and Conclusion This experiment explored finding the fundamental frequency of a beam, the effective stiffness of the beam and the effective mass, using a motor as the frequency generation and changes in mass via removable disks. Data was collected by two different methods. The first method being the data acquisition through LabVIEW. In this method, an accelerometer was attached to a motor (rotating machine) weighing 10.27 lbs, and the motor was attached to a simply supported 1” x 5” x 32” steel beam. LabVIEW plotted the acquired data into peak frequency curves of the system. (Appendix A). The peak frequencies for added weights of [0, 4, 8, 12, 16] lbs were [18.03, 15.95, 14.05, 13, 12.01] hz, or [113.29, 100.22, 88.28, 81.68, 75.46] rad/s, respectively. A graph of the inverse of the system frequency vs. the mass, shown in Figure 3, results in a least squares curve of y = 0.0001977883x + .0000140292. Using this curve, the effective stiffness and mass, as well as the fundamental frequency were determined to be 5055.9 lb/ft2, .07093 slugs, and 266.98 rad/s, respectively.
For the second method, data was obtained using a tachometer to read the motor RPM at observed resonance with the given weight intervals. The frequencies calculated for the same added weights of [0, 4, 8, 12, 16] lbs were [17.82, 15.85, 14.14, 13.07, 12.08] hz, or [111.95, 99.59, 88.85, 82.10, 75.91] rad/s, respectively. A graph of the inverse of the system frequency vs. the mass, shown in Figure 2, results in a least squares curve of y = 0.0001891791x + .0000184897. Using this curve, the effective stiffness and mass, as well as the fundamental frequency were determined to be 5285.4 lb/ft2, .09773 slugs, and 232.56 rad/s, respectively.
Comparing these against the theoretical values obtain from the given procedural beam and motor weight, gave error margins that would be close to acceptable depending on the level of work the data was needed for. From the data obtain from both experimental data sets, the [ke, me, wn] error percentages for the LabVIEW results were [7.96%, 1.81%, 4.20%] and for the tachometer results, [3.78%, 40.28%, 16.56%]. The closest result for the fundamental frequency after error analysis was from LabView at 266.98 rad/s.
For more accurate and trustworthy results, the LabVIEW software would be preferred over the tachometer. The attached accelerometer and the software resolution gave more accurate results, as shown in the resulting error analysis. A higher caliber tachometer that was mounted in a more optimal position, as well as having a dedicated clean reflective surface for the tachometer to read the RPM from would likely help get closer to theoretical results. However under the current circumstance, with the tachometer being handheld and the motor vibrating, these are the leading cause for the higher error margins within the experiment.

This experiment carries forward one of the biggest or most important lessons in mechanical and civil engineering. Understanding the resonant frequencies of materials and how to obtain them is of extreme importance in order to ensure that designs are not made within this destructive frequency band or near it. Visual observation of the beam at its resonant frequency give immediate understanding of the exponential amplitude relationship that is met under these “optimal” conditions.

References 1. https://en.wikipedia.org/wiki/Resonance 2. Beer, Ferdinand P. and E. R. Johnson, Jr., Mechanics of Materials, McGraw-Hill Book Company, New York

Appendix

Figure A1: LabVIEW Plot; Weight Added: 0 lbs

Figure A2: LabVIEW Plot; Weight Added: 4 lbs

Figure A3: LabVIEW Plot; Weight Added: 8 lbs

Figure A4: LabVIEW Plot; Weight Added: 12 lbs

Figure A5: LabVIEW Plot; Weight Added: 16 lbs