Influence of Materials Physics on Seismometers

Transcription

1 Influence of Materials Physics on Seismometers Randall D. Peters, PhD 1,2 and Eric Daine 1 Professor Emeritus of Physics 2 Member of the Mercer University Seismological Association of Australia Macon, Georgia USA Copyright 31 August 2017 ABSTRACT The focus of the presently described experiment is on quality factor (Q). The functional dependence of Q on eigenfrequency (f0) has been determined from measurements of the displacement of a compound pendulum oscillating in free-decay. It is shown to be of the form Q f0 2. While experimenting with a zero-lengthspring seismograph many years ago, this same result was surprisingly noted by Gunar Streckeisen; who is famous in the seismology world as the builder of instruments such as the STS-2. Neither instrument can therefore be adequately described by the commonly assumed model of a simple harmonic oscillator influenced by viscous damping. BACKGROUND One of the most famous expressions in all of pedagogical physics is the following 2nd order, linear differential equation: d 2 x/dt 2 + o Q dx/dt + o 2 x = 0 where o = 2 fo (1) and the quality factor Q has been used to specify the damping coefficient, which multiplies the first derivative (velocity) term. We here use the same form of the equation as did the venerable Richard Feynman. Many textbooks work with what is called the damping 'constant' ; where the aforementioned 'coefficient' is replaced with 2. The factor of two is placed in the equation to begin with, so that the oscillator's free-decay involves e - t as a multiplicative factor acting on the harmonic component of the solution. This system oscillates exactly with the frequency fo only if = 0. Eq. (1) is beautifully elegant, but frequently impractical for the following reasons: First, whether stated explicitly or not; textbook literature assumes for the equation that is a constant. Second, the straightforward,

2 rigorous-to-the-math solution to the equation, predicts what is called the 'damping red-shift'. In other words, the (angular) eigenfrequency is necessarily a function of the damping, being given by = ( o 2-2 ). Numerous experiments by Peters over the last two decades have shown that neither of the features just stated is necessarily applicable to the description of mechanical oscillators of the real world [1]. Just because an oscillator exhibits exponential decay does not mean that it can be described by Eq.(1). Many real-world oscillators are influenced by 'damping anharmonicity' [2]; which derives from internal friction related to creep and hysteresis. Internal friction damping is associated with mesoscale phenomena that are not yet understood from first principles [3, 4]. INTRODUCTORY Surprise In Fig. (3) that follows, it is seen that the frequency dependence of Q for the pendulum of this study cannot be explained by means of Eq. (1). The figure shows clearly that Q f0 2 ; whereas it is readily proven that the equation yields a first-power rather than quadratic dependence on the eigenfrequency fo. INSTRUMENT OF THIS RESEARCH Shown in Fig. 1 is the pendulum that was used for collecting the data from which the graphical figures of this article were generated.

3 Figure 1. Photograph of the Pendulum. The pendulum pictured in this figure is the same instrument that was first used by Peters many years ago. A variety of other mechanical oscillators that he has studied have shown the same property mentioned in the abstract; i.e., that Q f0 2. This fact, coupled with the knowledge of other phenomena well-known to solidstate physicists, the claim was thus made that materials physics (structural) properties of an instrument are a greatly significant factor in determining the characteristics of a seismometer. It should be noted that the claim is not limited just to geo-science type instruments. Evidence is provided in bib-item [1] to believe in the universality of the claim; that it derives from features of creep and mechanical hysteresis due to defect properties of poly-crystalline metals, specifically dislocations.

4 Figure 2. Close-view of the Aluminum foil strip that was used as the moving electrode. As compared to similar experiments performed many years ago, data acquisition proved to be vastly more user-friendly for the work of the present study. Seen in the close-view of Fig. 2 is how a crudely constructed moving electrode was all that was needed to conduct the experiment. Adequacy of a strip of aluminum foil, squeezed to hold onto a wire connected to the pendulum; speaks to the advantages of latest technological advances. ELECTRONICS (State of the Art Digital type) The electronics used in this experiment was designed and built by Eric Daine; who also is the web-master of The electronics-heart of the sensor package (labeled sym cdc in Fig. 1) is the Analog Devices AD7745 integrated circuit. It is perfectly suited to function with the instrument via USB connection to the computer, receiving power to do so, directly from the computer. The package shown was specifically built by Eric for use with the TEL-Atomic Kater (user-friendly precision) pendulum that was designed by Peters [5]. SENSOR TYPE The sensor used by the TEL-Atomic box evolved from Peters' creation of the first fully-differential capacitive sensor, that was a rotary type [6]. The one in the box is of linear displacement type [7]. METHODOLOGY The pendulum pair of primary masses are nearly spherical in shape and fabricated from Lead. Near to the same size, each was drilled and tapped, so it can thus be adjusted vertically upward or downward relative to the pendulum axis. The axis was made of pieces taken from an old-style chemical analytic balance,

5 comprising a hard steel knife-edge that rests on a sapphire flat to support the full weight of the pendulum. Specifics of pendulum size, weight, etc are presently irrelevant; since previous experiments have demonstrated that the only thing important for present purposes is to establish a free-decay; and then measure the motion so that the period can be determined along with its associated quality factor. It should be intuitively obvious to the reader that raising the upper mass (increasing its distance from the axis of rotation) gives rise to a lengthening of the period when everything else remains the same. (Of course, as the period begins to increase dramatically with altitude increase, a critical point is quickly reached where the pendulum is no longer stable). Contrariwise, increasing the distance of the lower mass from the axis shortens the period. These two degrees of freedom were used to establish five different free-decays, for purpose of plotting (i) Q versus eigenfrequency for one graph, and (ii) Q versus square of eigenfrequency for the other case. These results are shown in Fig. 1. Figure 3. Graphics showing that the pendulum's damping is not consistent with the simple harmonic oscillator model assuming viscous damping. Observe from the left-side plot of Fig. 1, that when Q is plotted versus frequency with the linear trendline forced through the origin, the fit is poor, having a regression coefficient (R 2 ) of But when plotted versus the square of the frequency as in the right-side plot; excellent fit to the linear regression line is realized with the near-perfect value of DETAILS of calculating Q and Eigenfrequency To obtain the eigenfrequency for each of the five free-decay cases, use was made of the Fast Fourier Transform (FFT, called 'Fourier Analysis' by Excel, part of the 'Tool Pak'). Since the Excel maximum number of values that can be treated is 4096; that was the number used; which must be equal to \2 n with n being an integer (for the Cooley-Tukey (fast) algorithm to work). (One interested in a better understanding of the FFT is referred to [8]). Though precision suffers somewhat, as compared to a discrete FT of an entire set, the technique is adequate for present purposes.

6 Shown in Fig. 4 are the graphics for the highest Q case of 480. The Q for every case was determined by trial and error fit of an exponential to the turning points of the motion, shown by the red curve. Two parameters were involved in the fit: (i) amplitude estimate (best guess) at the start of the record, and (ii) the guess value of Q. (The 'amplitudes' are in terms of CDC counts, standing for Capacitive to Digital converter Counts) Figure 4. Graphics pertaining to the highest Q case of the five cases considered in this experiment. Notice that the density spectrum has been plotted for this case in terms of a linear plot of period rather

7 than frequency. The 'shape' of the spectrum would show mirror-symmetry equivalance between a frequency plot and a period plot, but only if using a logarithmic-scaled abscissa. The use of a linear scale for period (although log-scale for 'intensity') was for purpose of showing clearly the 2nd and 3rd harmonic distortions, smaller than the signal 'line intensity' by a factor of about 1000 (meaning down by 30 db from the fundamental at Hz, euivalent to a period of 4.10 s. Because of the high-q, the record persisted long enough that the decay plot has too many cycles for individual ones to be discerned. To show 'wave-form' of the record, the first 20 seconds are shown to the right. Shown in Fig. 5 are the graphics for the lowest Q case where the value was 20. The motion dies away quickly enough in this case to see all cycles of the recorded motion.

8 Figure 5. Graphics pertaining to the lowest Q case with a value of 20. The log-log plot was selected for this case (with abscissa being frequency), to highlight the noise features of the electronics. It is seen to be about 60 db below the signal of interest. CONCLUSIONS This article has emphasized two facts; i.e., (i) that there is no such thing as a truly 'simple' harmonic oscillator [9], and (ii) greater need in scientific circles ought to be paid to improving science education in the 21 st century [10]. Bibliography [1] R. Peters, "Damping Theory" (Ch. 2) and Ëxperimental Techniques in Damping" (Ch. 3), Vibration damping, control, and design, ed. C.. de Silva ISBN , Taylor and Francis (2007) (the reader is also referred to the following half-a-dozen other closely related publications): R. Peters, "Model of internal friction damping in solids", online at R. Peters, "Nonlinear damping of the linear pendulum, online at R. Peters``Oscillator damping with more than one mechanism of internal friction dissipation'', online at R. Peters, ``Toward a universal damping model-modified Coulomb friction'', online at R. Peters, "Beyond the linear damping model for mechanical harmonic oscillators" R. Peters, "Creep and mechanical oscillator damping", online at [2] R. Peters, Änharmonic oscillator", McGraw Hill Encyclopedia of Science and Technology 10th edition online at [3] R. Peters, "Friction at the Mesoscale", Contemporary Physics Vol. 45, No. 6 Taylor and Francis, (2004). [4] R. Peters, "Compton Energy Scale of Friction Quantization", online at [5] Kater pendulum electronics, online at

9 [6] Linear rotary differential capacitance transducer, Patent US 5,028,875 (1991) [7] "Symmetric differential capacitance transducer employing cross coupled conductive plates to form equipotential pairs US A (1995) [8] R. Peters, "Graphical Explanation for the speed of the Fast Fourier Transform" (2003), online at [9] R. Peters and T. Pritchett, "The not-so-simple harmonic oscillator", American Journal of Physics (1997). [10] R. Peters, "Building on old foundations with new technologies", Ch. 1 of Science Education in the 21st Century, Nova Science Publishers (2008). File translated from T EX by T TH, version On 31 Aug 2017, 09:06.