WHAT'S NEW -- Research of Randall D. Peters
Debunking the Linear Damping Model for Mechanical Harmonic Oscillators
In a recent experiment, a folded pendulum was driven to steady state at frequencies below resonance. The results cannot be explained by linear damping models.
Flex-Pendulum--basis for an Improved Timepiece
By extending the model that was used to describe damping of the long-period pendulum, new properties of the compound pendulum have been discovered. The potential function converts to a Duffing type at long period for a flexible upper structure. The dependence of frequency on amplitude for the pendulum has a trend that is opposite that of the rigid pendulum. Thus compensation is possible for a carefully tailored instrument.
Even when sine of the angle can be approximated by the angle, the pendulum is not a linear oscillator. This paper demonstrates the importance of nonlinear damping of hysteretic ('structural') type. It also shows, when only viscous damping is important--that it is not the simple(minded) behavior suggested by physics textbooks.
A model for harmonic oscillator damping due to the internal friction of solids has been developed, based on considerations of a long period pendulum. The assumption of a complex elastic modulus to describe stress-strain hysteresis in the support structure of the pendulum yields an expression for the figure of merit Q that agrees with many experiments involving material damping. As such, the approximations of this linear model stand in contrast with common theory.
A modification of Coulomb’s law of friction uses a variable coefficient of friction that depends on a power law in the energy of mechanical oscillation. Through the use of three different exponents--0, ½, and 1; all commonly encountered non-viscous forms of damping are accommodated. The nonlinear model appears to yield good agreement with experiment in cases of surface, internal, and amplitude dependent damping.
When identifying instruments that have had great influence on the history of physics, none comes to mind more quickly than the pendulum. Though `birthed' by Galileo in the 16th century, and in some respects nearly `dead' by the middle of the 20th century; the pendulum experienced `rebirth' by becoming an archetype of chaos. With the resulting acclaim for its surprising behavior at large amplitudes, one might expect that there would already be widespread interest in another of its significant nonlinearities. Such is not the case, however, and the complex motions of small amplitude physical pendula are barely known. The present paper shows that a simply-constructed metallic rod pendulum is capable of demonstrating rich physics in a largely unstudied area.
The heat engine that comes closest to the ideal cycle of Sadi Carnot is probably that of Rev. Robert Stirling, D.D., minister in 19th century Scotland. Although theoretical treatments of the Stirling cycle seem to have adequately modeled the thermodynamics of the engine, they appear to have ignored the Newtonian mechanics of the flywheel ,as it is influenced by the temperature gradient between hot and cold reservoirs. The following paper blends flywheel mechanics with system thermo--hence the word, 'thermo-mechanics':
"The Stirling-Engine Refrigerator--Rich Pedagogy from Applied Physics"
"Creep and Mechanical Oscillator Damping", an archived Los Alamos paper--was featured in Complexity Digest:
http://www.comdig.de/ComDig01-40/#2
It assumes secondary creep as the basis for a modified Coulomb type of internal friction (structural or hysteretic) damping.
Viscous damping for the simple harmonic oscillator (SHO) has been a near universal assumption among physicists for many years. Even efforts at modelling the mirror supports of LIGO appear to diligently try and avoid what is probably inevitable--that the equation of motion for the damped SHO must be nonlinear. One would think that various similar failures before the new science of chaos would have resulted in a greater willingness to depart from the futility of linear models. Those who have followed the history of the slow acceptance of chaos and complexity are not surprised to find a 'herding' instinct , even among physicists.
MESODYNAMICS AND SEISMIC INSTRUMENTATION
Dr. Peters has revived some work that was first done a decade ago--using long-period mechanical oscillators (pendula) to study the earth. It was discovered by accident (1990) that a novel tiltmeter that he designed was an excellent earth-shape monitor--and thus responsive to free earth vibrations that are triggered by rapid internal relaxations of the earth as it is stressed by the tidal force. A combination instrument package of vertical seismometer and tiltmeter have been placed in a quiet location from which they are being monitored.
The Not-So-Simple Harmonic Oscillator
For many years the physics community has assumed that the free decay of a mechanical oscillator could be adequately described by a viscous (first power in the velocity) damping term. Recent studies of long-period pendula have shown that this is not a reasonable assumption. A paper having the same title as the page heading and which documents this claim, was published in the American Journal of Physics (65, 11, 1067, 1997) An alternate model, which agrees much better with experiment, is described in the paper. It is referred to as the generalized flip-flop model of damping.
The viscous damping model is especially poor in the mesodynamic realm, where changes in amplitude of the motion are not always continuous. It is even possible to place the pendulum's support structure in a far-from-equilibrium condition which results in negative damping.
SENSORS
The experiments which demonstrated the inadequacy of conventional wisdom concerning pendulum damping; and others which point toward a new frontier--mesoanelastic complexity, were made possible by a new sensor technology. These sensors are referred to as Symmetric Differential Capacitive (SDC).
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