21.2 Data Processing

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21.2.1 Language Type

The author’s experience with software began with early computers and even included the loading of the

Fortran compiler of a PDP-11 using punch-tape. He has programmed computers (or hardware-specific

processors) with (i) machine code, (ii) assembly language, (iii) Fortran, and (iv) Basic, and he has

acquired a rudimentary knowledge of Pascal and Cþþ. The drudgery of machine coding was a factor

in his quest to better understand the Fourier transform (Peters, 1992, 2003a, 2003b, 2003c, 2003d).

His philosophy with regard to numerical methods is similar to his view of hardware: choose the simplest

package (lowest level of sophistication) consistent with the desired results for the problem at hand.

The reader may be surprised to learn that QuickBasic (which some have modernized to Visual Basic for

Windows) is his favorite language. Nearly all simulation results presented in this chapter were generated

with the DOS version of QuickBasic.

21.2.2 Integration Technique

Too few have discovered the powerful integration scheme in which Cromer (1981) modified the unstable

Euler algorithm. The difference between the two methods involves the sequencing (order) of updates to the

state vectors in the discrete approximation of the integrals. The method was called the “last point

Experimental Techniques in Damping 21-3

© 2005 by Taylor & Francis Group, LLC

approximation” (LPA) by Cromer, whereas the Euler technique would be called the “first point

approximation” according to this nomenclature. The LPA was discovered by a high school student working

for Cromer. She was attempting to simulate planetary motion with the Euler method and accidentally

coded the LPA. The author first used the LPA to do intercept analyses for the U.S. antisatellite program —

computing, among other things, orbital ephemerides. More recently he has used it in place of Runge Kutta

techniques to do all kinds of mechanical system simulations, including nonlinear types with several DoF. A

physics theorist at Texas Technical University, Professor Thomas Gibson, now regularly uses the LPA as

part of the graduate-level course which he teaches in numerical methods.

21.2.3 Fourier Transform

With the Cooley – Tukey improvement to make it fast, the Fourier transform has become a tool of major

software importance. Just as the integrated circuit dramatically changed hardware development, the FFT

has had a profound influence on the evolution of scientific code.

Whereas many recognize the value of the FFT for viewing “raw” spectral data, few have discovered

other powerful tools based in the FFT. For example, autocorrelation is unrivaled in its ability to uncover

low-frequency signals of fairly short duration that are corrupted by noise. The number of cycles is not

great enough (Heisenberg effect) for a well-defined line to be observed in the FFT by itself. Through the

Wiener – Khintchin theorem, the autocorrelation overcomes this limitation. It is computed by

multiplying the transform by its conjugate and then taking the inverse transform. The author has

used this technique to study free oscillations of the earth (Peters, 2004).

21.2.3.1 Short Time Fourier Transform

A powerful software tool is one in which the Fourier transform is not computed over the entire length of

a record. Instead, the record is subdivided (usually with some degree of overlap between adjacent

subsections), and the FFT is computed for each subsection. Because the data are generally of the temporal

(rather than spatial) type, the technique is called the short time Fourier transform (STFT). For equivalent

processing, where the independent variable has units of meters rather than seconds (as in optics

applications), the technique could just as well be called the short space Fourier transform.

The STFT is especially useful when waveforms are not pure harmonic, as from a single-degree-offreedom

(single-DoF) oscillator. For systems with multiple modes, whether they derive from eigenmodes

as recognized by most, or from mechanical noise as recognized by a few (generated as part of the internal

friction of load bearing members); the STFT is a powerful means for isolating and thus determining the

temporal history of individual spectral components.

The most common form of the STFT is the canned programs that are a part of software packages such

as LabVIEW. With the Dataq software it is easy to accomplish the same thing manually, since one can

readily step in time from place to place of a stored record, computing the FFT at any position. The

intensity at a given position is obtained by clicking on the displayed spectral line of interest, which

provides the value either in dB (Dataq version) or in volts. The amplitude history in dB of the line is thus

obtained (equally spaced-in-time values) with a simple click of the mouse. In this way, the free decay of a

single component of the system can be readily extracted from the total system response. For the

present purposes, the individual intensities were copied by hand to paper and later typed into a

spreadsheet for plotting. The process is not laborious, since the number of necessary points is typically

less than two dozen.

An example of a manually generated STFT is provided in Figure 21.2. Unlike the methodology

described above (operating on experimental data residing in a Dataq folder), the record of Figure 21.2

was generated by computer and written to an output file. The data correspond to three superposed,

exponentially damped sinusoids.

From the upper graph (time record), it is not clear how the individual components are changing with

time. The triplet of components becomes obvious in the frequency domain (lower left), and when the

21-4 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

intensities of each line are plotted versus time (lower right) the exponential character of each becomes

visible. To test the viability of this method, the individual damping parameters were evaluated from a

given STFT graph and found to be in excellent agreement with the values supplied to the simulation.

21.2.3.2 Example Use of the STFT

The majority of the examples given in this chapter avoid low-energy oscillations as space does not allow

mesoanelastic regimes to be treated at length. The following case was chosen to illustrate (i) the

importance of the Portevin – LeChatelier (PLC) effect on damping (Portevin and Le Chatelier, 1923), and

(ii) the power of the STFT in eliminating the influence of clutter. The STFT benefit was expected, since

the difference between a noisy record and a multimode case like the previous example is in the number of

modes. The PLC effect is significant for high-energy internal friction dissipation even though the jumps

for which the effect is known are not obvious at these energies. Evidently, this is a consequence of the

large number of events, for which the average effect is a fairly smooth decay.

The scales for the two graphs of Figure 21.3 are different by nearly three orders of magnitude, with the

level of the sensor output for each case being indicated (mid-range values). Two features are evident from

a direct visual inspection: (i) the change from a smooth to a jerky decay in going from high to low levels,

and (ii) a 4% increase in the frequency of oscillation at the lower energy. The latter is recognizable from

the vertical lines that have been added (every fifth peak). One of the more interesting (and surprising to

most) features of the lower graph is that phase of the oscillation is not significantly altered as the result of

mean position jumps.

The following information is provided for those skeptical of the comments concerning the lower trace

of Figure 21.3. From the study of hundreds of low-level decays in different mechanical oscillators, the

author has become confident that the jumps shown are not sensor (or other electronic) artifacts.

Prejudice against this conclusion has been considerable over the last 14 years, in spite of the fact that a

similar phenomenon was noted (and accepted) in magnetic materials many years ago, i.e., the

Barkhausen effect (Barkhausen, 1919). Unfortunately, the related phenomenon in mechanical systems

(PLC effect) is hardly known among physicists.

FIGURE 21.2 Example of separating the time decay of superposed components using the STFT.

Experimental Techniques in Damping 21-5

© 2005 by Taylor & Francis Group, LLC

The value of the STFT for the study of data such as that of Figure 21.3 is illustrated in Figure 21.4.

Whereas Figure 21.3 showed only the first and the last portions of a long record, here the STFT was

applied to the entire record using 27 different FFTs.

As seen in the upper left curve, there is a sharp decrease in the damping in STFT No. 5. Thus, the

intensity was replotted in each of the intervals from 0 to 5 and 5 to 27. Both intervals show a near perfect

linear trendline fit, indicating exponential decay. Thus the Q was found to quickly change from 78 to 340

at an energy level in the region of 10210 J. Although air damping was a factor in the early part of the

record, it is not thought to be capable of causing the rapid change in Q that was observed. A similar sharp

Decay history, Rod-Pendulum in air

STFT no. (Δn = 1 ↔ 125 s)

STFT no. (Δn = 1 ↔ 125 s)

STFT no. (Δn = 1 ↔ 125 s)

0

0

0

0 0.5 1 1.5 2 2.5

0 5 10 15 20

y = –1.6512x-24.445

y = –7.3197x-7.8533

25 30

Q=340

Q=78

R2 = 0.9995

R2 = 0.9993

0 5 10 15 20 25

–20

–10

–20

–30

–40

–50

–60

–70

–80

–40

–5

–10

–15

–20

–25

–60

–80

dB

(Slope change at 1 e-10 J)

FIGURE 21.4 Example of the use of STFT analysis applied to a dataset, the first and last portions of which are

shown in Figure 21.3.

FIGURE 21.3 Illustration of the character changes in decay in going from high to low energies of a pendulum.

21-6 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

change in the damping of this same pendulum was seen at roughly 10211 J with the pendulum swinging

in a high vacuum. The slope change was equally rapid for the vacuum case, but the change in Q was from

120 to 210. It is not known why the damping at low-level energy in a vacuum would have a lower Q than

in air. Perhaps the difference derives from a different placement of the knife-edges on the silicon flats.

Some of the damping of this pendulum is the result of the knife-edges being fabricated from brass rather

than a harder metal such as carbon steel.

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