Figure 1 illustrates differences according to the nature of the nonlinearity.

  Figure 1. Spectral illustration of sum and difference artifact frequencies according to sensor nonlinear type.

The only 'real' signals of Fig. 1 are at frequencies f₁ and f₂. The number and type of other 'unreal' (artifact) signals depends on the type of nonlinearity. The sensor response for the left graph (quadratic) is of the form V = ax + bx² , whereas for the right graph V = ax+bx² + cx³ . The influence of terms other than V = ax (ideal, linear output voltage) were generated by (i) simulating the pair of harmonic signals, (ii) inputting these signals to each simulated sensor respectively, and (iii) performing an FFT on the output.

Although it is possible to mathematically understand the various artifacts using trigonometric identities, the phenomenon is much easier to demonstrate with the computer. For Fig. 1, all numerical operations were performed with code written by the author using Quickbasic. It was used to (i) simulate the harmonic signal that was written to a data file, after which it was (ii) read by the FFT algorithm based on the details supplied in Numerical Recipes [Press, 1986].

Frequency issues

The choice of a sensor depends largely on the frequencies to be measured. For higher frequencies an excellent instrument for data collection is a digital (storage) oscilloscope, where a microphone can often be directly connected to the instrument. At lower frequencies, a serial-port analog to digital converter (ADC) is generally adequate and user-friendly. Examples of each will be provided. The majority of examples considered in this document involve low frequencies, where the eigenmode is typically described not in terms of frequency but rather the period (reciprocal of frequency).

Data Acquisition

In the absence of sophisticated data collection and analysis tools, the true character of damping is not readily discovered. Proper characterization is important, since a crude estimate of the damping, based on a single parameter (such as the viscous linear model), may be inappropriate if the oscillator is driven at places (either frequency or amplitude) other than where the parameter was measured. Some of the examples from experiment which follow were selected to demonstrate the importance of nonlinearity. The probability that an oscillator, selected at random, might have a Q that varies in time is proving to be more significant than anticipated. Were it not for dramatic improvements in technology of numerical type, this improved understanding of damping would not have been possible.

As with computer technology in general over the last decade, ADC's have become much more powerful. The Dataq model 700, for example, is superior (at lower frequencies) to many of the 'plug-in' boards of previous generation type that were several times more expensive. The Dataq ADC operates through the USB port (Windows 98 and later), has 16 bit resolution; and the software support is excellent. Especially useful for present purposes are its ability to easily (i) do data compression with which to view long records, (ii) quickly compute an FFT according to different, useful options, and (iii) output files to a spreadsheet.

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, (iv) Basic, and he acquired a rudimentary knowledge of Pascal and C++. The drudgery of machine coding was a factor in his quest to better understand the Fourier transform [Peters 1992 and Peters 2003...speed..]. His philosophy with regard to numerical methods is similar to his view of hardware--choose the simplest package (lowest level of sophistication) consistent with 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.

Integration Technique

Too few have discovered the powerful integration scheme in which Alan Cromer 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 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, as she attempted to simulate planetary motion with the Euler method and accidently 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 degrees of freedom. A physics theorist at Texas Tech University, Prof. Thomas Gibson, now regularly uses the LPA as part of the graduate -level course which he teaches in numerical methods.

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 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.

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 is generally of temporal (rather than spatial) type, the techniqe is called the short time Fourier transform. For equivalent processing, where the independent variable has units of meter rather than second (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-of-freedom (SDOF) oscillator. For systems with multiple modes, whether they derive from eigenmodes recognized by most, or from mechanical noise recognized by 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. By this means, the free decay of a single component of the system can be readily extracted from the total system response. For present purposes, the individual intensities were copied by hand to paper and later typed to 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 Fig. 2. Unlike the methodology described above (operating on experimental data residing in a Dataq folder), the record of Fig. 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 in time. The triplet of components becomes obvious in the frequency domain (lower left), and when the intensities of each line is 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.

Figure 3. Illustration of decay character change in going from high to low energies of a pendulum.

The scales for the two graphs of Fig. 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.e., (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 Fig. 3. From hundreds of low level decays in different mechanical oscillators studied by the author, he 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. Unfortunately, the related phenomenon in mechanical systems (PLC effect) is hardly known among physicists.

The value of the STFT for the study of data such as that of Fig. 3 is illustrated in Fig. 4. Whereas Fig. 3 showed only the first and the last portions of a long record, the STFT was applied to the entire record, using 27 different FFT's.

Figure 4. Example of the use of STFT analysis applied to a data set, the first and last portions of which are shown in Fig. 3.

Shown in Box 1 are some representative sensors for damping measurements. The list is far from exhaustive; for a detailed description of each (plus discussion of other types), the reader is referred to Fraden, 1996. Of the transducer types indicated, position sensors are generally the most versatile; but the present document also provides (in the material that follows) examples of the use of (i) velocity, (ii) microphone, and (iii) photogate measurements.

In addition to the need for linearity (discussed in sec. 1 above), the ideal sensor will be 'noninvasive'. In actuality, there is no possibility of performing a measurement that does not at some level perturb the system under study. The least perturbative types of direct measurement are (i) optical, and (ii) electrical--capacitive, followed by inductive.

Some of the assets/liabilities of the devices indicated in Box 1 are as follows:

The inductive linear variable differential transformer (LVDT) is a sensor that is common to engineering application. Thus, it has been a natural choice for many position sensing purposes; but it is both more invasive and noisier than capacitive sensors. Wielandt [2001] notes the following, concerning the advantage of capacitive over inductive sensors: "Their sensitivity is .... typically a hundred times better than that of the inductive type."

Optical encoders are also readily available and have been used extensively. Because of their digital nature, based in a finite number of elements; their low-level resolution is poor compared to capacitive devices.

Optical sensing by shadow means is easy to employ--for example, using a solar cell of the type discussed later. The method is afflicted, however, by (i) an offset voltage, and (ii) degrading influence of background light.

Potentiometers are very easy to use, but compared to other position sensors they are extremely invasive, because of friction in the bearings that support the slider.

The most important velocity sensor is that which functions on the basis of Faraday's law. Using a magnet and a coil; an emf is generated in the wire of the coil when it experiences a changing magnetic flux. Prevalent in seismometers before the advent of broadband (feedback) instruments, its primary shortcoming is poor sensitivity at low frequencies.

Photogates have become the primary means for kinematic studies in introductory physics laboratories. Combined with compact, user-friendly timers, it is possible to measure both (i) period and (ii) velocity. As illustrated later, they can be easily used to measure damping in slowly oscillating systems, but only in a limited amplitude range.

In the cases of (i) pressure, (ii) acceleration, and (iii) force/stress sensing, the measurement is an indirect one. Consider, for example, the Ruchhardt experiment to measure the ratio of heat capacities of a gas (discussed in part I of this document). The oscillation of the piston could be measured several different ways. Direct position sensing could be accomplished by attaching a small electrode to the piston and allowing it to move between stationary capacitor plates. Alternatively, a 'flag' on the piston could be used to interrupt the light beam of a photogate. Depending on constraints, however, the easiest method might be an indirect measurement in which a pressure sensor monitors the gas through a catheter communicating with the cylinder of the apparatus.

Accelerometers can sometimes be connected directly to an oscillator, but only if the mass of the instrument is very small compared to the system being studied. As with the measurement of velocity, their sensitivity at low frequency is very poor (the second derivative of position yielding a response that is proportional to frequency-squared).

Strain gauges are easy to employ but also lack sensitivity (compared to position measurement), since they communicate with a very small portion of the oscillating sample (if non-invasive).

Figure 5. Free-decay record of a vibrating bar.

The voltage versus time of the microphone output was saved to memory in the oscilloscope, from which the digital record was output to a floppy disk, using the CSV format. Data from the disk was read into columns A and B of an Excel spreadsheet using 'Open file'. An envelope fit was then performed on the turning points by placement of trial and error data into column 'C', using 'autofill'. A separate graph was generated for each value of the constant 'b' in the expression '=0.04*exp(-b*A1)' typed into 'C1'. (The lower turning points were obtained by typing '=-b1' into 'D1' and using autofill. (Additional details concerning the use of Excel in this manner will be provided in the discussion of seismometer damping that follows.)

Although optimizing algorithms could be generated to do such a fit (with a probable slight increase in accuracy), this visual technique is presently preferred for reason of (i) understandability, (ii) user-friendliness, and (iii) proven performance. The total time required in Excel to generate Fig. 5 using a 2K record (2048 points) is typically only a few minutes with a modern Pentium computer.

A disclaimer is in order at this point. Present comments by the author should not be interpreted as an endorsement of Microsoft products in general. Although QuickBasic and Excel have both proven unusually beneficial to the work described in this document, they are the only software packages marketed by the company to have received a strong endorsement from the author.

Once a satisfactory fit was obtained (b = 1.6 in Fig. 5), the Q was estimated using

(1)

There is a fair amount of electronic noise in Fig. 5 because the microphone was connected to the oscilloscope directly. The smallest bandwidth of the oscilloscope, at 20 MHz, causes a large amount of Johnson (white) noise. Narrowing the bandpass by means of a preamplifier would improve the quality of the data dramatically. Such is typically true (SNR improvement) by tailoring the electronics to the need.

To illustrate another sensing technique, the system shown in Fig. 6 was used.

Figure 6. Setup for measurement of vibrating reed free-decay.

A 90 degree twist was given to a hacksaw blade after heating with a torch and quenching. One end was clamped to the vertical post shown and a piece of cardboard was taped to the other end. Above the cardboard, which vibrates horizontally, is placed an incandescent lamp; and below it the solar panel used as sensor. The solar panel in this case is a commerical unit that comes with a cigarette lighter plug for charging automobile batteries. The output from the panel goes to the Tektronix digital scope also in the picture.

Operationally, unlike other sensing schemes described in this document, the solar panel output is not bipolar, but rather has a constant voltage offset corresponding to the equilibrium position of the reed.

The frequency of oscillation is too low to operate the oscilloscope with a.c. coupling. Thus, it is important to make the d.c. offset as small as possible. This was accomplished by shielding non-active parts of the solar cell from the lamp. Although the d.c. offset could be removed with a 'voltage bucking' battery, this was found to introduce unacceptable noise spikes. With the offset, the gain of the electronics was limited by the amount of vertical position shift allowed by the scope.

The results of this study are illustrated in Fig. 7.

Fig. 7. Vibrating reed decay with amplitude-dependent damping.

The non-zero value of b (0.018) in Fig. 7 indicates the presence of amplitude dependent damping [refer to Eq. (61), PART I]. The nonlinear damping in this case probably derives from the air, rather than internal friction of the hacksaw blade. Its presence causes the Q of the system to increase with time.

The Q of the system is calculated from the expression

(2)

Case 3. Seismometer

Since the detectability of tremors by a seismometer is proportional to the square of the period of the instrument, they require a good low-frequency sensor. The most common sensor for the latest generation commercial instruments is a half-bridge (differential) capacitive type. Because of the greater sensitivity and linearity of the full-bridge SDC sensor mentioned in PART I, it is well suited to these applications, being easy to employ. A significant advantage of the SDC symmetry (equivalent electrically to the inductive LVDT) is its relative insensitivity to construction imperfections, such as roughness of surface and non-parallelism of electrodes. Thus construction can be done crudely without serious degradation of performance. For example, electrodes of the first prototype of the SDC sensor were fabricated from sheet copper that was cut with shears and subsequently flattened by hammer on a 'hard' plane surface. In this regard, the sensors stand in stark contrast to the optically polished surfaces that were necessary for the author's PhD work, involving ultrasonic harmonic generation in single crystalline solids.

The Sprengnether vertical seismometer discussed several places in PART I of this chapter was studied in a configuration for which Q = 4.9, as determined by an external 990 ohm resistor to provide induced-current damping. The resistor was connected across the coil that is part of the original equipment and which moves (with the mass of the instrument) in the field of a stationary magnet; i.e., a Faraday's law (velocity) detector. Excitation to initiate the free-decay study was accomplished by applying an alternating (square wave) current to the coil, reversing the direction of the current at each turning point of the motion of the mass. The fundamental (Fourier series) of a square-drive generated this way is shifted 90 degrees from the mass motion, corresponding therefore to resonance. After cessation of the drive, data for the record of Fig. 8 was collected with a Dataq DI-700 analog to digital converter (16-bit).

Figure 8. Free-decay record of the Sprengnether vertical seismomter, period 17 s.

' = A_n+1/(sample rate)' was typed, to increment the time. Then the lower right hand corner 'small solid square' of the box containing this time was 'grabbed and held' with the left button of the mouse to 'autofill' all the way to the last time point of the data. The computer generated exponentials which correspond to the turning points were obtained by generating two additional columns. These were obtained by placing the 'cursor' at a row corresponding to the time A_n (in column C) and then typing

Note that in Table I a new excel worksheet was employed (column A no longer the time as in the discussion above). The voltages corrresponding to the turning points (maximum and minimum) were each read by placing the cursor at an extremum and manually recording the value displayed in-turn by the Dataq software. These values were then typed into excel, as opposed to the 'file open' method for importing large data sets; i.e., as used to generate Fig. 7.

One of the simplest ways to measure damping at larger levels is to use a photogate of the type common to general education physics laboratories. The infrared beam of the photogate is tripped by a 'flag' attached to the oscillator. The rod pendulum pictured in Fig. 9 was studied by this means. Shown in Fig. 10 is a closeup of the flag which is attached to the top of the pendulum and which passes through the photogate during oscillation.

As seen in the figures, various parts of the pendulum are clamped to a vertical steel rod. Both upper and lower masses are of lead, drilled each with a pair of holes--one to pass the brass rod of the pendulum and the other (after tapping) to hold a thumbscrew for securing the mass at different vertical positions on the rod.

This mechanical oscillator is a compound pendulum; the period can be made long as the knife edge (also clamped to the rod) approaches the center of mass. At long period, the instrument is not very responsive to external accelerations of the supporting frame; but it is sensitive to internal structural changes. Low-frequency instability is encountered as the upper parts of the instrument experiences creep, particularly in the materials just above the knife-edge. Having similarities to an inverted pendulum, except rigidly connected to the 'lower pendulum, the 'upper' pendulum causes the oscillator to eventually exhibit double-well (Duffing) characteristics. This happens at larger amplitudes as the period is varied toward really long times. The tendency toward mesoanelastic complexity depends on the dimensions of the rod. As expected because of the well known engineering properties of rods and tubes, a large diameter thin-wall tube will behave differently than a solid rod made from the same amount of material (same total mass). This will be true if the tube does not experience localized (sharp) deformation prone to creasing.

Damping measurements with a photogate require that the time required for the flag to pass through the beam be fairly small--thus meaning larger amplitudes of the motion than with other sensors. Of course really large motion would result in period increase, consistent with long-understood pendulum dynamics. For amplitudes within the acceptable range (which in practice is not overly restrictive), the velocity of the pendulum as it passes through the equilibrium position is inversely proportional to the time interval between interrupts of the photogate beam by the two vertical arms of the flag. If the period does not change with amplitude, then there is also then an inverse relationship between the gate time and the amplitude. A plot of the inverse of these times versus cycle number is, for the constraints indicated, a reasonable approximation of the turning points of the free-decay.

The instrument presently used to make these 'single gate' time interval measurements is the Pasco 'Smart Timer'. It is a user friendly instrument that also permits the flag (by using two different lengths of the flag arms) to accurately measure the period of the pendulum. For experiments of this type, it may prove more convenient to measure the period with a stopwatch (infrequently as compared to the velocity). The sequentially increasing time intervals are read manually from the smart timer and written on paper with a pen, once per cycle. Of course, to do so requires that the period be long enough to permit these operations. The recorded values are conveniently analyzed by typing to a spreadsheet, which is then used to graph damping curves such as shown in Fig. 11.

A pure exponential fit is not appropriate to the decay of Fig. 11, in which the upper mass had been removed and a business card taped to the bottom of the pendulum to cause turbulent air damping ( period near 1 s ). The fit shown, however, involving both linear and quadratic damping, is seen to be quite reasonable. As in the case of the vibrating reed discussed earlier, this system is adequately described by the nonlinear damping Eq. (61), PART I. For the data of Fig. 11, Q = 25 initially and increases to 70 at the end of the record.

Without the business card, and with the upper mass in place; the decay of this pendulum was found with the photogate measurement technique to be exponential, as expected for viscous damping with periods of about 5 s. At periods in excess of about 10 s, however, internal friction of the rod becomes more important than air damping. Although the decay is then still exponential at larger levels, the frequency dependence is not the same as required by linear air damping.

The data of Fig. 11 were collected with the knife-edges resting on hard ceramic alumina flats. When supported by other materials, the damping of a rod pendulum can be influenced by anelastic flexure other than that of the rod. Hardness of the material does not guarantee low damping, as will be seen in the following examples. The data that follows was collected with a different pendulum, pictured in Fig. 12.

Figure 12. Long period rod pendulum used to study the influence of different material under the knife edge.

The sensor in this case was an SDC unit, connected to the computer through the Dataq DI-700 A/D converter. The upper and lower masses are each approximately 1 kg and their separation distance on the aluminum hunting arrow from which the pendulum was fabricated was about 70 cm.

Figure 13. Illustration of damping difference according to specimen type under the knife edge.

The samples used to collect the data in Fig. 13 were identical pairs, except that one pair had been irradiated with a huge dose of gamma rays. The resulting changes to the structure of the crystal are responsible not only for color centers as noted in the photograph of Fig. 14, but also a dramatic change to the internal friction. It is clear from Fig. 13 that internal friction in the LiF is the dominant source of damping of the rod pendulum that was used ("Oscillator damping with more than one mechanism of internal friction dissipation", http://arxiv.org/html/physics/0302003/).

Figure 14. Photograph of the LiF singles crystals used to obtain the data of Fig. 13.

Lithium fluoride is used in thermoluminescent film badges (radiation monitors). When exposed to

energetic radiation, atoms are `knocked' from their crystal lattice sites into metastable states corresponding to interstitial positions of the lattice. Upon ramping the temperature of the sample in an oven fitted with a photomultiplier tube, jumps from the metastable state are accompanied by the release of photons. The amount of light so generated is a measure of the dose that was received by the crystal. Because light flashes are observed with rather small changes in the temperature, it is reasonable to expect that mechanical strains might also cause significant change to the defect state of such crystals. This postulate is confirmed in the data of Fig. 13 by the dramatic difference in the decay character of the pure (clear) crystals (bottom figure) and those which were extensively damaged by gammas (top figure).

In both decays of Fig. 13 there is significant nonlinear damping, as evidenced in the early portions of each of the two records. The top case is nearly pure Coulombic, and the bottom case is partially amplitude-dependent. This is revealed from estimates of the Q, shown in Fig. 15.

Figure 15. Temporal dependence of the Q, LiF crystal experiments.

The Q values of Fig. 15 were computed from successive triplet-values of the turning points of the motion, read directly from the decay pattern displayed on the monitor by the Dataq software. The equation used is:

, n = 0, 1, 2, ... (3)

It is commonly (and mistakenly) thought that hard materials must necessarily also have low damping. It is seen in the two examples that follow that this is not necessarily so. Even though cast iron is very hard, it is also quite dissipative, which makes it an ideal material for engine blocks. Shown in Fig. 16 is a decay curve for the steel knife edges of the pendulum resting on cast iron samples.

Figure 16. Data collected using cast iron samples.

At the start of the record the damping with cast iron is nearly twice as great as that of steel-on-sapphire or steel-on-silicon where the Q was found to be of the order of 80. This measured large damping is consistent with the known excellent properties of cast iron for use in engine blocks; although the frequencies for such applications are much higher.

Shown in Fig. 17 is another very hard material which has large damping-- the ceramic piezoelectric wafer formed from lead, zirconium, and titanium (PZT); which by means of a mechanical impulse is commonly used to generate an electric spark to ignite a gas grill. The secular decline of Q based on the short temporal record indicates Coulomb damping. It is consistent with the nearly straight line turning points for the early part of the long term record, also shown. The long-term record is labeled `anomalous' because it does not appear to be consistent with several simultaneously acting dissipation mechanisms. Rather, the strong early Coulomb damping appears to later disappear, once the amplitude has dropped below some level. This suggests activation processes of a quantal type. It would be interesting to study the PZT wafers in a different pendulum configuration, and not operating `open- circuit' as the present case; but rather with different resistors connected between the top and bottom of the wafers.

Figure 17. Data from an experiment involving PZT ceramic wafers.

With polaroid material (H sheet) placed under the knife edges, it was found that the damping depends on the direction of the long-chain polymeric molecules. The direction of the molecules in a sample is readily determined by looking through the polaroid at reflected light from a polished floor. When the reflection occurs close to the Brewster angle, only the horizontal component of the electric field is significant in the reflected light, for unpolarized incident light. The direction of the molecules is thus determined by rotating the sample until a minimum of light is passed; and when this happens the molecular chains are then situated horizontally.

It was reasoned that the molecular properties of polaroid might result in mechanical as well as optical anisotropies. This postulate proved to be true, as shown in Fig. 18.

Figure 18. Free-decay curves showing anisotropy of the internal friction in polaroid material.

When oscillating on silicon at a period of 10 s, the instrument had been found from previous studies to decay consistently with a Q of 80 (uncertainty 3%). For the present studies, half a dozen free-decay records were obtained for each of (i) edges parallel to the chains, and (ii) edges perpendicular to the chains. The average Q of oscillation was estimated at 50 for the parallel case and 43 for the perpendicular case. Reproducibility proved slightly better for the parallel case (4%) as compared to the perpendicular case (5%). For more details one may refer to http://arxiv.org/html/physics/0302055

(4)

where f is the frequency in Hz and the STFT slope is specified in dB per s.

MUL apparatus.

Some of the techniques applicable to driven systems are illustrated by the multipurpose undergraduate laboratory (MUL) apparatus shown in Fig. 19, that has been used by students in the physics department at Mercer University.

Figure 19. Apparatus for studying resonance and the Lorenz force law.

For purpose of measuring the Lorenz force (basis for defining the current unit, ampere) a constant current is supplied through the posts to the pivoted-on-points brass wire on which a weight W is shown hanging on one of the horizontal arms of the wire. Current enters the wire through one post, via the banana plug inserted into a drilled hole. It thereafter travels through the lower (invisible) shorter straight segment of the wire located between the poles of the drive magnet; and it finally exits through the banana plug on the opposite post. When carrying a current, the force on the wire from the part inside the magnet causes vertical deflection, the direction up or down being determined by the direction of the current. This results in a rotation about the pivot points (indented tops of the posts), and the position is measured by the capacitive sensor S (one of several variants of the SDC patent).

The sensitivity of this current balance depends on the center of mass location of the oscillatory wire, which is determined in part by the position of the rare earth magnet M which hangs from a steel nut on the threaded part of the heavier brass rod having a 90 degree bend. The upper end of this threaded rod is held by a plexiglass member which also holds the ends of the oscillatory wire.

The MUL becomes a driven harmonic oscillator when the excitation current is a.c. rather than the d.c. used for the Lorenz force study. The damping is determined primarily by eddy currents in the aluminum ring R that lies on the wooden base underneath and in close proximity to magnet M.

Figure 20. Screens of LabVIEW program used with the MUL to study both transient and resonance phenomena.

An example Lorentzian (resonance response) is pictured in Fig. 20. (For additional information concerning the original student experiment, one may consult the following URL: http://physics.mercer.edu/labs/sho.htm)

Figure 21. Mechanical oscillator with multiple nonlinearities

The instrument is a modified extensometer that was sold by TEL-Atomic and which was designed around the SDC sensor to measure Young's modulus and thermal expansion coefficients. The wire sample normally used with the instrument (along with hollow power resistor that fits in the black clamp) has been removed, and two rare earth magnets have been employed. One magnet was superglued to the bottom of the 'pan' where weights are normally placed, as shown in Fig. 22; and the other magnet is attached to the bottom of the inductor that is sitting on the top of the oscillator (cased instrument, Pasco) used for drive. The pair of magnets are positioned in close proximity so as to repel each other; thus supporting the mass of the moveable arm of the extensometer.

Figure 22. Closeup picture of the oscillator of Fig. 21 (non-operational configuration) , showing placement of the rare earth magnets.

The study of nonlinear systems requires a linear sensor; i.e., any nonlinear contributions from the detector must be negligible. In Fig. 23 are calibration results for the instrument, showing its linear response for the range of amplitudes used in the study.

Figure 23. Calibration data for the sensor used with the oscillator having multiple nonlinearities.

The potential energy of this oscillator was assumed to have the following form

(5)

and the parameters were estimated by measuring x as small masses were placed on the pan. A linear regression fit to a log-log plot (using the sensor calibration constant of 550 V/m) yielded b = 1.02 x 10^-5, n = 1.526, and c = 0.304 (system international units). Anharmonicity of the potential is readily apparent in the plot shown in Fig. 24; with the force of restoration being greater in compression (x decreasing) than it is in extension. This feature is reminiscent of interatomic potentials, with anharmonicity being responsible for thermal expansion.

Figure 24. Plot of the potential energy function of the oscillator.

Because of the elastic nonlinearity, the mean position depends on amplitude of oscillation, as is evident in the free-decay curve of Fig. 25.

Figure 25. Free-decay curve showing mean position shift as a function of oscillator amplitude. (Decreasing sensor voltage corresponds to increasing x). The frequency of oscillation is 6.01 Hz.

Figure 26. Free-decay character as determined using the short time Fourier transform.

The oscillator exhibits hysteresis when driven at larger amplitudes, as shown in Fig. 27; where it is seen that the place of an amplitude jump depends on which way the oscillator is adjusted, either up or down in frequency. Such jumps (well known with oscillators having nonlinear elasticity) stand in stark contrast with the behavior of a linear oscillator, as can be seen by comparing Fig. 27 with the screen picture of Fig. 20.

Figure 27. 'Resonance' response (steady-state) of the driven oscillator.

A surprise of this study involves the frequency of oscillation. In general, the oscillator does not entrain to the drive. Moreover, the preferred frequencies are not necessarily the same as the free-decay frequency of 6.01 Hz. Some of the frequencies (measured with power spectra) are indicated in Fig. 27. When a frequency change occurred, it was always rapid, and the magnitude was in every case approximately 3%.

Fig. 27 demonstrates why nonlinear damping measurements should be done in free-decay. Fig. 25 demonstrates how exponential fits make no sense for some oscillators. Fortunately, the STFT can be used to determine the amplitude dependence of the Q.

For the picture, the apparatus was disassembled and the plate allowed to rest on the top of the bottom electrode set. Operationally, the plate was positioned midway between the upper and lower static electrode sets (separation distance of 4 mm); and there was no mechanical contact during oscillation. The top side of the circuit board containing the upper electrode set contains more than a dozen electronic components that can be seen; these are of the surface mount technology type. The detector is of the SDC type and this particular embodiment is manufactured in Poland for TEL-Atomic Inc., for use in the Computerized Cavendish Balance (references provided elsewhere in this chapter).

As can be seen in the picture, the wire was rather kinky instead of straight, which is expected to be a signficant source of nonlinearity. For this reason, not to mention that it is very difficult to make larger diameter tungsten wires reasonably straight; no serious attempt was undertaken to remove the kinks.

An example decay record generated with this apparatus is illustrated in Fig. 29.

Where the rows are blank for the end of record case, the values were insignificantly small.

The decibel values in the table are referenced to the bit-size (16 corresponding to 65536) of the ADC. In terms of the sensor output voltage, V, it is defined by Dataq as:

(6)

where FS is the full-scale voltage as determined by the gain setting.

In other places of this document, the decibel is calculated with a different reference. For example, an FFT spectral line having real and imaginary components R and I respectively (voltage based); the 'intensity' in dB is calculated using

(7)

where n is the number of points in the FFT. This is convenient for determining noise levels.. For example, from later graphs showing electronics noise, the 'floor' of the SDC sensor is found to be of the order of -120 dB, corresponding to a microvolt. The position resolution defined by this noise level is about 500 nm; i.e., the wavelength of visible light.

Of the two primary modes of this kinked-wire case study, the higher frequency (2.19 Hz) is the twisting mode, and the lower frequency (1.19 Hz) is the 'swinging' mode. The swinging mode is a little higher frequency than that which would result if the wire were completely flexible, yielding a near simple pendulum (1.02 Hz for 24 cm length). The swinging mode is two-dimensional (pendulum equivalent called 'conical'), but the sensor only responds (first-order) to motion perpendicular to the long axis of the electrodes. It should also be noted that this motion is attenuated, relative to the twisting response, because of the mechanical common-mode rejection feature discussed in PART I.

The manual STFT was used on the data that generated Fig. 29 to estimate the decay history of three different modes--each of the primary ones (twist and swing), and also the mode whose frequency is the difference of the frequencies of the primaries; i.e., 1 Hz. Shown in Fig. 31 are the results, in which a Hanning window was used, and the total number of points in the record permitted five equally spaced in time FFT's, when working with a 1024 point transform.

At least two signals in the spectra are the result of nonlinearity; i.e., the lines corresponding to the sum and difference of the frequencies of the primary pair--at 3.38 Hz and 1.00 Hz respectively. If the system of oscillator and detector were completely linear, then no such sum and difference cases are possible. It is also to be noted that these mixtures are not the result of sensor nonlinearity; which as noted previously one must be careful to avoid.

It was expected that the amplitude of a mix signal should approximately obey the following relation

(8)

To test this premise, the STFT was used to estimate the amplitudes of each of the three components indicated in equation (8). The amplitudes were all normalized, relative to the starting value for each case, and the results used to generate the graphs of Fig. 32.

The amplitude of oscillation for a given mode, at the time of the transform, is found by using the peak value in dB of the intensity of the spectral line for that mode, according to

(9)

where the factor of 20 is used since the spectral intensities were calculated in terms of voltages. Although calibration constants (in V/m and V/rad) could be used to express the amplitude in meters or in radians, corresponding to the mode, nothing is gained for present purposes by doing so.

The mixing index for these cases is defined by the expression

(10)

which is similar to expressions encountered in optics. It can be seen that the sum and difference frequencies are approximated reasonably well by theoretical expectation.

The claim is made in PART I of this chapter that internal friction is responsible not only for damping but also for significant mechanical noise of 1/f type. The following figures are provided in support of that claim.

Figure 33. Power spectrum and associated temporal record showing mechanical 1/f noise (right pair). For reference, electronics noise is also provided (left pair).

The pendulum in these experiments (lead spheres near ends of an aluminum tube, pair of steel-points for axis) was operated in high vacuum; to eliminate air influence. What is labeled 'maximum electronics noise' was obtained by removing the top mass and measuring the motion after oscillation had come to a minimum (frequency approximately 1 Hz); some pendulous mode remains because of pump noise transmitted through the vacuum hose. Similarly, the pump vibrations excite a vibratory mode in the long-period pendulum (sharp spectral line at 3.5 Hz, right spectrum) In this vibratory mode the lead masses move in the same direction relative to the stationary axis (similar to the bending mode of the carbon dioxide molecule). It is interesting to note that there is no coherent oscillation to be seen above noise corresponding to the period of 5.7 s. The mechanical noise is seen to include bistability, which is not uncommon for this type of system before 'hardening', following a significant force disturbance. The data of Fig. 33 were collected after replacing the upper mass, which had been removed to measure the electronics noise, and after pumping to the operating pressure (below 5 microns Hg).

The mechanical noise is seen to be 1/f for f < 1.5 Hz, which is where electronics begins to contribute noticeably. Everywhere below 1 Hz, the electronics noise is an order of magnitude smaller than the mechanical noise.

After the pendulum had stabilized overnight and been allowed to oscillate through a number of free-decays (initialization by tilting the chamber), the data of Fig. 34 was collected.

Figure 34. Same as Fig. 33, but after the pendulum had 'stabilized'.

It is seen that the noise of mechanical type has mostly 'settled out', leaving the remanent electronics noise.

Recent experiments have shown important subtleties of viscous damping [Peters, 2003, "Nonlinear ...]. It is true that the dissipation at a specified frequency can be adequately modelled by simply multiplying the velocity term in the differential equation by a coefficient. It is not proper, however, to call this coefficient a 'constant'; since the damping coefficient is frequency dependent and also involves the density as well as the viscosity of the fluid in which oscillation takes place.

Some engineers have known about the 'history term' which is most simply treated in the case of a sphere executing simple harmonic motion. The friction force acting on the sphere in this case can be reasonably approximated by

(11)

Using Eq. 11 in the equation of motion for a pendulum yields for the Q

(12)

Reasonable experimental validation of the estimate for the Q was provided, as demonstrated in Fig. 35.

Figure 35. Comparison of theory and experiment for a pendulum damped by water.

The instrument in this case was a compound pendulum in which a mass was located above the axis of rotation, as well as the usual situation of mass below the axis. The water damping was provided through a small sphere at the bottom of the pendulum, immersed in water held by a rectangular container.

To ignore the history term of Eq. 12 can result in huge errors. For example, in the case of water damping, this can cause one to underestimate the damping by 1000 to 3000 % , as shown in Fig. 36.

Figure 36. Illustration of how huge errors can result in damping estimates if one ignores the history term.

At low frequencies, it is also important to correct for the influence of hysteretic damping of the pendulum. Figure 37 shows the large errors which result when one fails to do so.

Figure 37. Illustration of significant low-frequency errors that result from a failure to recognize the hysteretic damping component of the pendulum.

For some cases, buoyancy and added mass of the fluid are also quite significant to the frequency of oscillation, as shown in Figure 38.

Figure 38. Example of how fluid properties influence the frequency as well as damping of an oscillator.

As seen from Fig. 37, low frequency motions are likely to be influenced more by internal friction than by any fluids that interact with the oscillator. The most important fluid is of course air, and a true delineation beween external and internal effects requires that the oscillator be studied in a high vacuum. It is not enough to just remove 'most' of the air, since the viscosity of gases is suprisingly constant until the mean free path between collisions becomes a significant fraction of chamber dimensions.

Theoretically, it is possible to roughly estimate air influence only in the simplest of geometries, such as a sphere. For that case, Eq. 11 could be used (with account for the history term, using appropriate values for the viscosity and density). It is also possible in some cases to estimate air influence experimentally, as in the example that follows.

Rod Pendula of Brass and Solder

Because of its malleability, the internal friction of solder (lead-tin alloy) is large, compared to that of much harder brass. A pendulum of each material was studied, both having a length of about 50 cm and a diameter of about 3 mm. The technique used was the photogate method described in sec. 4 (case 4 above). Unlike the previous study, no lead masses were clamped on the rod--but it used the same adjustable knife-edge.

Figure 39 shows clearly that the internal friction for the solder pendulum is much greater than that of the brass pendulum.

Figure 39. Free-decay curves for brass and solder pendula at two different frequencies, showing the larger internal friction of solder. The velocity is that of the peak value (amplitude) at the top of the pendulum, approx. 22 cm above the axis.

A nonlinear fit was generated for each decay curve, from which the history of the quality factor was graphed as a function of velocity amplitude, as shown in Fig. 40.

Figure 40. Illustration of amplitude-dependent damping in a rod pendulum made of (i) brass, and (ii) solder. The two different matched periods of oscillation are indicated in s.

Consider the pair of brass curves of Fig. 40. The large difference in Q at 10 cm/s (387 compared to 266) stands in stark contrast with their near equality at 50 cm/s. This is primarily a consequence of air drag that is quadratic in the velocity at the larger amplitude. It is more important to brass than to solder because of the small internal friction of the brass.

From the large difference in internal friction of the two materials, a first order correction for air influence on the solder pendulum is to simply subtract 1/Q of the brass from 1/Q (raw data) of the solder, to yield the reciprocal Q (corrected) due to internal friction of the solder. This has been done in Fig. 41.

Figure 41. Amplitude dependence of the estimated quality factor due only to internal friction in the Solder pendulum. Dashed lines show the Q before correction for air damping.

It is seen from Fig. 41 that the internal friction damping is not simply hysteretic (constant Q); rather it is a function of amplitude. It is also seen from the close proximity of the solid and dashed curves, that the air influence on the solder pendulum is much less than that of the internal friction. By contrast, air influence is of comparable magnitude to the internal friction in the case of the brass pendulum (or even larger, at large amplitude).

A minimum of two frequencies were considered for the study, since frequency variation of the damping is different for external and internal friction. (Note: Although the period is a function of amplitude, the amount of non-isochronism is small compared to the damping changes and is presently ignored.) The periods were matched for the two pendula at each of 2.03 s and 2.51 s. For hysteretic-only (internal friction) damping, the Q at the shorter period should by theory be 1.53 times that of the longer period, for both brass and solder. If the damping were viscous only, the factor should be 1.24. In the case of solder at 10 cm/s (corrected), the ratio is 1.66 = 131/71, and for brass it is 1.46 = 387/266. Although the ratio for solder is greater than the expected 1.53, the difference is within experimental uncertainty for individual Q values; which from other, more detailed experiments were in the neighborhood of 5 to 10%.

The ratio for brass (1.46) is between 1.24 and 1.53, as expected, because of the comparable influence of air and internal friction.

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