beam vibration lab

8/2/2019 Beam Vibration Lab

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Structures/Motion Lab20-263-571, Sections 001, 002, 003

STRUCTURAL BEAM TEST LAB

OBJECTIVE:

The objective of this test is to conduct a vibration test on a simple structure (free-freebeam) in order to determine the damped natural frequenciesr and mode shapesr ofthe beam. Dampingr (r = r + j r) will be present in the beam (material damping)but will be assumed to be small and constant for each mode. For the purposes ofprepar ing for the lab, assume that the the beam will be made of aluminum withdimensions 3.0 inches wide by 36.0 inches long by 0.75 inches thick.

BACKGROUND/THEORY:

The simplest way tofind the damped natural frequencies and mode shapes of a simplestr ucture is to recognize that the damped natural frequencies are approximately thefrequencies where the frequency response functions reach a relative maxima. This isdemonstrated by the following Figures.

0 100 200 300 400 500 6000

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Magnitude

Frequency Response Functions Magnitude

Frequency (Hz.)

Figure 1. Damped Natural Frequencies at Maxima of FRF Magnitude

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+ 20-263-571 Structures/Motion Lab UC-MINE +

220 230 240 250 260 270 280 290 300

0

10

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Magnitude

Frequency Response Functions Magnitude

Frequency (Hz.)

Damped NaturalFrequency

Figure 2. Damped Natural Frequencies at Maxima of FRF Magnitude

As long as the modal frequencies are not too close together, this simple approach willwork well. The mode shapes can be found at these modal frequencies byunderstanding the Expansion Theorem.

Expansion Theorem - Frequency Domain:

{ X(i) } =N

r=1 r {r }

The Expansion Theorem states that the response of the system is a linear combinationof the mode shapes. At a specific damped natural frequency, ther coefficient for thatmode shape will dominate. Thus, since r is a constant for each frequency, theresponse at each measurement point will be proportional to the mode shape. Since thefrequency response functions are normalized response functions, the values of thefrequency response functions, at the damped natural frequencies, will be approximatelyequal to the mode shapes. For a frequency response function, evaluated at the dampednatural frequencyr, the relationship is:

{ H(r) } pr

j 2 r Mr r{r }

This idea is demonstrated by the following Figures.



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0 100 200 300 400 500 60050

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Frequency (Hz.)

Frequency Response Functions Real Part

RealPart

Figure 3. Damped Natural Frequencies - FRF Real Part

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Frequency (Hz.)

Frequency Response Functions Real Part

RealPart

Damped

NaturalFrequency

Figure 4. Damped Natural Frequencies - FRF Real Part



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0 100 200 300 400 500 60080

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80Frequency Response Functions Imaginary Part

Frequency (Hz.)

ImaginaryPart

Figure 5. Damped Natural Frequencies - FRF Imaginary Part

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Frequency Response Functions Imaginary Part

Frequency (Hz.)

ImaginaryPart

Damped NaturalFrequency

Figure 6. Damped Natural Frequencies - FRF Imaginary Part



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This simple single degree of freedom (SDOF) approach assumes that the modalfrequencies are sufficiently separated in frequency so that the modal vector can beapproximated by the response vector with no contamination (error) from modes that arelower or higher in frequency. The following Figure illustrates how single degree offreedom contrabutions combine to create the realistic multiple degree of freedom(MDOF) frequency response function. Note that when the natural frequencies are close

together, the dashed line (SDOF) is not identical with the solid line (MDOF).

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Frequency, (Hertz)

LogMagnitude

,(dB)

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Frequency, (Hertz)

Phase,

(Degrees)

Figure 7. SDOF Contributions to Frequency Response Function

Fur ther theoretical details are included in Appendix B.



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PROCEDURE:

The testing procedure will utilize a single reference, impact test approach to estimatethe frequency response functions. This approach uses a singlefixed accelerometer asthe response (output) sensor and an instrumented hammer as the excitation (input).

The frequency response functions will be calculated by the Zonic Medalion SignalAnalyzer.

The load cell that is permanently mounted in the impact hammer is to beconnected to Channel 1 of the Zonic Medallion Signal Analyzer. Themeasurement sequence will be triggered on Channel 1 with a small pre-triggerdelay (1-3% of the total block). A force window (rectangular) should be applied tothe force signal to minimize noise problems. The impact hammer should have anylon or plastic tip appropriate for the frequency range of the modes of the teststr ucture.

The accelerometer is connected to Channel 2. A response window (exponential)

should be applied to the accelerometer signal to minimize the truncation errorknown as leakage. The accelerometer should be mounted with wax on the beam.

Typical views of the Analyzer Setup and a typical views of the time domainhistor ies (force and response) and frequency domain measurements (frequencyresponse function and coherence) are included at the end of this labdocumentation as Appendix B. Be sure to choose the frequency range(Bandwidth) large enough tofind the highest frequency mode of interest. Be sureto choose a Frame Size (number of time domain data points) of at least 1024.

Free-free boundary conditions are to be approximated by resting the beam on afoam pad. Damping will be small enough to ignore. The stiffness of the foam pad

may alter the free-free modal frequencies estimates by a ver y small amount. The location of the reference accelerometer and the location of the impact points

is at the discretion of each lab group.

Use 3-5 averages for each measured frequency response function. Be sure totur n on Manual Rejection for each average so that you can reject any average thathas a double impact on the force signal or has an overloaded signal on either theforce or response signal.

Utilize the coherence function (measure of linearity between the measured inputand output) to determine the quality of the measured frequency responsefunctions.

Save the measured frequency response functions to disc and move the data toMATLAB for plotting and/or processing.

All damped natural frequencies and mode shapes should be identified beforeleaving the lab.



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RESULTS:

PRIOR TO COMING TO LAB, the damped natural frequencies for thefirst six (6)defor mation modes should be estimated analytically. Sev eral references areavailable that provide the analytical solution for a simple free-free beam. For

example, see Reference 1 or 2. Note that thefirst six defor mation modes mayinclude bending or torsion modes involving any orientation of the rectangularbeam.

Exper imentally deter mine the damped natural frequencies and associated modeshapes for thefirst six defor mation modes of the beam. Compare these values toyour predictions. Be sure to note that your exper imental data has a finitefrequency resolution.

Plots of the mode shapes should be included in the report along with a verbaldescr iption of each mode. The measured frequency response functions should besaved and moved to MATLAB to simplify the plotting of the mode shapes.

Be sure to measure and weigh the beam to ver ify your analytical estimation.

DISCUSSION

The discussion should include the following issues:

Summar ize the procedure used to identify the damped natural frequencies andmode shapes.

Comment on any difficulties encountered in identifying the mode shapes.

Compare/contrast damped natural frequencies with theoretical values estimatedfrom a reference. Discuss the reasons for any differences (what are the actualboundar y conditions?). Which answer is correct?

REFERENCES

1. Blevins, Rober t D., Formulas for Natural Frequency and Mode Shapes, VanNostrand Reinhold Company, 1979, 492 pp. (Call Number TA654.B56)

2. Volterra, Enrico, Zachmanoglou, E.C., Dynamics of Vibration, Char les E. Merril

Books, Inc. 1965, 622 pp. (pp. 293-366)3. HP-35660 Dynamic Signal Analyz er, Getting Started Guide , The Hewlett-

Packard Company, Manual Number 35660-90005, 1988.



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Figure 9. First Bending Mode at First Damped Natural Frequency (1)

Figure 10. Second Bending Mode at Second Damped Natural Frequency (2)

In order to understand why this infor mation can be determined from the imaginary par tof the frequency response functions, SDOF theory must be reviewed and extendedslightly, primar ily from a notational point of view.

The general mathematical representation of a single degree of freedom system isexpressed using Newtons second law in Equation 1:

M x(t) + C x(t) + K x(t) = f(t) (1)

For the general case with a forcing function that can be represented as a summation ofsin and cosine terms, the forcing function can be represented as:

f(t) =

=0 F() e

jt+ F*() ejt (2)



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Assuming that the system is underdamped and that enough time has passed that anytransient response of the system due to initial condition or startup of the excitation hasdecayed to zero, the response of the system can be represented as:

x(t) =

=0 X() e

jt+ X*() ejt (3)

Note that, while x(t) and f(t) are real valued functions, X() and F() are complexvalued. Wor king with any arbitrar y frequency term in Equations 2 and 3, Equation 2,Equation 3 and the derivatives of Equation 3 can substituted into Equation 1 yielding thefollowing frequency response function (FRF) relationship for a SDOF system:

H() =X()

F()=

1

M2 + C j + K(4)

Note the character istic of the above frequency response function when it is evaluated(measured) at the undamped natural frequency. At the undamped natural frequency,

the mass and stiffness terms cancel each other and the FRF is purely imaginary valued.

Thefirst extension that is necessary provides a description for the case wherex(t) andf(t) are not located at the same point. On a single degree-of-freedom system, thiswould provide redundant infor mation (no new infor mation) but it becomes important asthe extension to multple degrees-of-freedom occurs. For example, assume that thepar ticular point (and direction) on the mass where the force is applied is referred to asDOF p and the particular point (and direction) on the mass where the response ismeasured is referred to as DOFq. Equation 4 now can be written as follows to note thisinfor mation.

Hqp() =Xq()Fp()

=1

M2 + C j + K(5)

The system is still a SDOF system so Hqp = Hpp = Hqq = Hqs = . . . but the input andoutput location can now be descr ibed. This clearly demonstrates that the number ofmodes (one in this case) is unrelated to the number of input and output sensors that areused to measure the system.

The second extension that is necessary provides a way to indicate that the modalcharacter istics (modal coefficients) of both the input and output are represented in thefrequency response function model. The modal frequency is already represented bynoting that the denominator is related to the character istic equation. A form of modalscaling is already represented by noting the the mass term in the denominator scalesthe equation. Modal coefficient infor mation, which is relative not absolute infor mation,can be added by changing the numerator to reflect this.

Hqp() =Xq()

Fp()=

q p

M2 + C j + K(6)



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Note that, since the system is still a SDOF system, the relative motion at each DOFwould be normalized to 1 such thatp =q =s = 1 which shows that Equation 5 and 6still represent the same infor mation. Note as before, if the FRF is evaluated (measured)at the undamped natural frequency), the FRF is once again imaginary valued and is afunction of the modal coefficients and damping. Assuming the damping is unknown butconstant means that the product of the modal coefficients is proportional to the

imaginar y par t of the FRF.

Finally, the third extension that is necessary provides for the change from SDOF toMDOF. Note that for a linear system, linear superposition can be used in the frequencydomain to add the infor mation associated with each mode together to represent thefrequency response function of a MDOF system. To desr ibe this, every ter m inEquation 6 will need a subcript (r) to indicate which mode the infor mation is associatedwith. Thefinal for m of the frequency response function is:

Hqp() =Xq()

Fp()=

r=1

qr pr

Mr 2 + Cr j + Kr

(7)

Equation 7 is one common representation of the FRF of a MDOF system. Note that theMr, Cr and Kr ter ms in the denominator are the modal or generalized mass, dampingand stifness parameters, not the physical mass, damping and stiffness parameters. Themodal or generalized parameters can be found analytically from the physical mass,damping and stiffness parameters or exper imentally using more complicated parameterestimation algorithms.

Note that, as long as the modes are well separated in frequency, the infor mation in theneighborhood of the undamped natural frequency for a given mode can be found from:

Hqp() =Xq()

Fp()

qr pr

Mr 2 + Cr j + Kr

(8)

This character istic is observable in Figure 4. Note that the SDOF contribution to theFRF (dotted line) is nearly the same as the MDOF contribution to the FRF (solid line) inthe neighborhood of the undamped natural frequency, as long as the naturalfrequencies are well separated.

Note that, if the output DOF (point and location) is held fixed while the input DOF ismoved, the only infor mation that changes in Equation 8 as different FRFs are measured

is the infor mation relative to the modal coefficient for the particular mode of interest.Note that, since mechanical systems obey Maxwells Reciprocity Theorem (Hpq = Hqp),

either sensor (input or output) can be heldfixed with the other sensor allowed to rove ormap the DOFs that define the modal vector.

{H()}p {r}pr

Mr 2 + Cr j + Kr

(9)

If Equation 8 is evaluated (measured) near the undamped natural frequency, this means



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that the imaginary par t of the FRF will be proportional to the modal coefficient. Thepropor tionality constantr) is:

r pr

Mr 2 + Cr j + Kr

(10)

Since mode shapes are relative patter ns, not absolute motions, the value of theconstant is not important unless damping or modal scaling is required.

Therefore, modal vectors can be estimated from the imaginary par t of the frequencyresponse functions at the damped natural frequencies (or from the magnitude andphase infor mation of the frequency response functions at the damped naturalfrequencies). This will be reasonably accurate as long as the undamped naturalfrequencies are well separated and the damping is small (undamped and dampednatural frequencies nearly equal).

This result is consistent with the expansion theorem concept (the response of the

system at any instant in time or at any frequency is a linear combination of the modalvectors):

Expansion Theorem - Time Domain:

{ x(ti) } =N

r=1 r {r }

Expansion Theorem - Frequency Domain:

{ X(i) }=

N

r=1 r {r }

Using the frequency domain for m of the expansion theorem, if the response is evaluatedat the undamped natural frequency of mode r, the expansion coefficient r willdominate and be approximately equal to alpha defined in Equation 9.

Damping Issues

Note that the exper imental analysis assumes that damping is samll and constant foreach mode, less than 10 percent of critical damping ( less than 0.10) and constant forall measurements. This is a practical assumption for many realistic, simple systems.Even with damping at 10 percent of critical damping, the difference between thedamped natural frequency and the undamped natural frequency is less than onepercent.



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Appendix B: Impact Testing



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Time Domain Histories

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Figure 11. Typical Force Signal (Impact) - Time Domain

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Time Domain Histories

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Time (Sec.)

Figure 14. Typical Responce Signal (Exponential Decay) - Time Domain



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Frequency Domain Measurements

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Phase(Degrees)

Figure 15. Typical Frequency Response Function (Phase) - Frequency Domain

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Magnitude(Volt/Volt)

Frequency (Hz)

Figure 16. Typical Frequency Response Function (Magnitude) - Frequency Domain

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Coherence

Figure 17. Typical Coherence Function - Frequency Domain



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beam vibration lab

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