Moderate Damping Research Articles

Advantageous Apodization Functions for Magnitude-Mode Fourier Transform Spectroscopy

A systematic examination of the efficacy of window functions for reducing the spectral skirt of magnitude-mode Fourier transform spectra is reported. The efficacy is examined for the general case of a damped time-domain signal, with specific cases ranging from undamped to essentially completely damped signals. The choice of the optimal window is dependent upon the required dynamic range and the amount of damping in the time-domain data. For a dynamic range of less than 100:1 and moderate damping, the Hamming window is the window of choice. For larger dynamic ranges or greater damping, the 3-term Blackman-Harris window and the Kaiser-Bessel window are the windows of choice. The 3-term Blackman-Harris window is preferred for a dynamic range of 1,000:1 and the Kaiser-Bessel window is preferred for a dynamic range of 10,000:1. The sensitivity (signal-to-noise ratio) reduction for windows is reported for a damping range from zero to essentially complete damping. All windows examined have the same sensitivity reduction within 25%.

The response of an incompressible, viscoelastic coating to pressure fluctuations in a turbulent boundary layer

The response of the interface between a compliant surface and a turbulent boundary-layer flow is examined theoretically. This response is forced by transient, convected, interfacial pressure pulses that represent the footprints of turbulent flow structures in the boundary layer. Calculations are presented for coatings with a wide range of damping and densities equal to the density of the flow. For coatings with moderate damping, three regimes of response are found. When the flow speed U∞ is less than about 1.2 (non-dimensionalized by the shear wave speed of the coating), the response is stable and primarily localized under the pressure pulse. For flow speeds from 1.2 to as high as 2.8, depending on the damping, the response is also stable, but it includes a wave pattern behind the pressure pulses. For flow speeds above 2.8, the response is unstable and eventually forms a two-dimensional wavetrain moving in the flow direction. At the highest stable flow velocity, the amplitude of the surface displacements reaches 4.0% of the boundary-layer displacement thickness δ* and the energy transfer from the pressure pulse reaches 5.0 × 10−4 U∞2 (δ*)2. For high damping, the coating response is again stable when the flow speed is below 2.8. However, there is no wavelike response regime; the path that is traversed by the pressure pulse is covered by a scar that heals according to the viscous relaxation properties of the material. The amplitude of the response is at most 0.01δ*.

Theory of activated rate processes: Exact solution of the Kramers problem

The Kramers theory for the escape rate of a Brownian particle from a potential well is extended to the full damping range. It is shown that the most adequate description of the underdamped Brownian motion in a deep potential well is provided by a Green function of the Fokker–Planck equation in the energy-position variables. The problem of lifetime of a particle in a single potential well is reduced to an integral equation in energy variable, with the Green function being the kernel of this equation. The straightforward solution by the Wiener–Hopf method yields an explicit expression for the lifetime, which describes the crossover from the extremely underdamped regime to that of a moderate damping. With the use of Kramer’s result for moderate-to-large damping an expression for the lifetime is presented, which holds at arbitrary damping. The problem of the rate of transitions between the two minima of a double-well potential is reduced to a system of two integral equations, which is also solved by the Wiener–Hopf method. An explicit expression for the relaxation time of nonequilibrium populations of the two minima is given.

Calculation of trim settings for a helicopter rotor by an optimized automatic controller

An automatic feedback system, based on continuous monitoring of hub loads, is used to find the control settings that are required to obtain a given flight condition for a helicopter rotor. Optimum values of gains and time constants are determined, and the limitations of the controller are examined for flap-pitch dynamics. It is found that the present method shows good convergence and is superior to other trim techniques for systems with moderate damping or with many degrees of freedom. B c c C-L CL CM CM [CM] CT CT d

Thermal activation in extremely underdamped Josephson-junction circuits

Noise-activated escape of a particle out of a metastable well was treated by Kramers, who derived results covering the three cases of heavy damping, moderate damping, and extreme underdamping. In the case of extreme underdamping, the escape occurs through diffusive motion along the energy coordinate. We extend Kramers's treatment of this extremely underdamped case to cover a wider range of damping constants, taking into account that at energies just above the barrier peak and within the initial well, the distribution is controlled both by uphill diffusion in energy, and by the flow out of the well. This new result is compared with computer simulations of escape events for the case of Josephson junctions under constant current bias. In addition to a conventional simulation which initially puts the particles at the bottom of the well, a method is developed which requires simulation only in an energy range close to the peak of the barrier.

On the attracting set for Duffing's equation: II. A geometrical model for moderate force and damping

After a brief review of some earlier work on Duffing's equation in the small force and damping regions, we use the results of numerical integrations to construct a geometrically defined Poincaré map which captures the qualitative features of the attracting set of larger force and damping levels. This map has a (small) constant Jacobian determinant and can be regarded as a perturbation of a non-invertible one-dimensional map. We give a partial analysis of the map and pose some important open questions regarding perturbations of one-dimensional maps and the creation of “strange attractors” during bifurcation to horseshoes.

Unsteady aerodynamics and vortex induced aeroelastic instability of a structural angle section

Wake-body interactions for a two-dimensional structural angle member during stationary and vortex induced oscillatory conditions are studied using a conventional low turbulence wind tunnel. The response of an angle section with combined plunging and torsion indicates that the oscillations occur essentially in one of the two degrees of freedom. The measurements of frequency and phase substantiated this observation. The plunging resonance exhibits the familiar vortex capture phenomenon where the shedding frequency is controlled by the cylinder motion over a finite wind speed range. On the other hand, the torsional vibration shows a vortex control phenomenon where the vortex shedding governs the frequency of oscillation. The vortex induced torsional resonance was found to be severe even at moderate damping levels. The results should prove useful in structural designs such as high voltage transmission towers, antenna masts, bridges, etc. where angle sections are often used as secondary members.

A NUMERICAL METHOD IN SOLVING A COUPLED THERMOELASTICITY EQUATION AND SOME RESULTS

Abstract This paper discusses a numerical method for solving a coupled dynamical ther-moelasticity problem in a long hollow cylinder. The inner boundary conditions are the traction and the fluid temperature imposed on the inner surface of the cylinder, whereas the outer boundary is subjected to the ambient fluid temperature and is free of traction. This problem is solved by a finite element method in which the spatial and the time variables are discretized by several schemes. The study shows that the central explicit scheme for space discretization combined with a time marching scheme, which incorporates moderate damping, is suitable for this particular problem. Numerical results of the temperature and stress responses due to a modified step or single pulse are presented and discussed. One interesting observation is that, under high rate of stress loading, the coupling in the energy equation could generate temperature variations of significant magnitude.

A mesoscale semi‐implicit numerical model

AbstractA three‐dimensional semi‐implicit mesoscale numerical model based on the Navier‐Stokes equations is described and tested. Numerical techniques for achieving numerical stability with long time steps are discussed. In simulating flow over mountains the model accurately reproduces the results of an explicit model. Low vertical resolution causes moderate damping of the orographically forced responses, but adjustment rates are not seriously affected. the model is stable with time steps six times those of a similar explicit model and its computational efficiency is about three times greater.A cross‐section version is used to examine the response characteristics. With four layers in the vertical, the phase speeds of external gravity waves with wavelengths 8 Δx to 30 Δx are slowed to 50% to 85% of the analytic phase speeds, respectively. Even so, simulations with eight and sixteen layers show that the semi‐implicit model reproduces the features of simulations with the explicit model and has comparable mass and energy conservation characteristics.Adjustment rates of the mass and momentum fields towards a steady state are studied for mountain wave and jet stream circulations. Despite the reduced gravity wave phase speeds, root‐mean‐square error calculations show very little difference between the semi‐implicit and explicit models. This is attributed to the fact that most of the adjustment is effected by relatively slow internal modes which are not greatly slowed in the semi‐implicit model. the verification that the semi‐implicit model produces adjustments between mass and momentum fields about as well as an explicit model is an important step in ensuring that the semi‐implicit technique can be successfully applied to mesoscale forecasting.

Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Attractors

The forced vibrations of a buckled beam show nonperiodic, chaotic behavior for forced deterministic excitations. Using magnetic forces to buckle the beam, two and three stable equilibrium positions for the postbuckling state of the beam are found. The deflection of the beam under nonlinear magnetic forces behaves statically as a butterfly catastrophe and dynamically as a strange attractor. The forced nonperiodic vibrations about these multiple equilibrium positions are studied experimentally using Poincare plots in the phase plane. The apparent chaotic motions are shown to possess an intricate but well-defined structure in the Poincare plane for moderate damping. The structure of the strange attractor is unravelled experimentally by looking at different Poincare projections around the toroidal product space of the phase plane and phase angle of the forcing function. An experimental criterion on the forcing amplitude and frequency for strange attractor motions is obtained and compared with the Holmes-Melnikov criterion and a heuristic formula developed by the author.

Some aspects of the control and measurement of the dynamic properties of fiber reinforced plastics

The damping of carbon fiber reinforced plastics (CFRP) is generally low at fiber volume fractions used for practical structures, such as aircraft components. If resonant behavior is induced in service, there is obvious merit in increasing material damping. Experimental and theoretical studies have been carried out to examine methods for increasing the damping of CFRP by use of short aligned fibers and combinations of fiber types; some configurations yielding moderate damping increases have been found. Stiffness to weight ratio is also a significant parameter for high performance structures, therefore both the loss factor and dynamic modulus of elasticity of specimens must be measured if experimental studies are carried out. Techniques for measuring these dynamic characteristics of material specimens have been developed. Attention has been given to the development of simple, rapid methods for measuring the properties, such as structural loss factor, of beams through a frequency range for comparative purposes. A technique for the more fundamental determination of material loss factor is also described. Results from tests on various types of CFRP are given. [Work supported by the Ministry of Defence.]

Radial motion of unichiral magnetic bubbles with in-plane field

The radial motion of unichiral magnetic bubbles in an in-plane field has been investigated using transient photography. It is found that when a 20-Oe in-plane field is applied to a unichiral bubble, the wall section whose center spin is opposed to the in-plane field is initially immobile during a bias-field step pulse. This immobility lasts for about 25 nsec, depending on the pulse field. The normal wall section, whose center spin is along the in-plane field, moves with a drive-dependent velocity on the order of 20 m/sec. The velocity difference between the two walls occurs in films of all types, but is particularly prominent in the film with moderate damping (α=0.14) that is used here. Numerical calculations show that the in-plane field produces a different static-spin structure in each of these two wall sections. In the normal wall section, the in-plane field merely reduces the total twist due to the surface stray field. However, in the wall whose center spin is opposed to the in-plane field, a statically stable 2π horizontal Bloch line is nucleated when the in-plane field is above a threshold, which is 12 Oe for the film used here. The velocity of the normal wall is compared with that predicted by a one-dimensional model, and they are found to be in agreement for pulse fields less than 8 Oe. Numerical calculations of the velocity of the wall with the 2π HBL are made using Hubert’s finite-difference method. These calculations confirm that the wall section is essentially immobile until the HBL approaches the film surface. When the pulse field is greater than 8.6 Oe, punch-through of the 2π HBL occurs, leaving two opposite-winding VBL’s. An implanted layer suppresses punch-through.

Surface plasma waves in the noble metals

The dispersoon curves for surface plasma waves (SPW) in Ag have been determined from calculated reflectivity minima as exhibited by attenuated total reflection (ATR spectra) for the prism-metal-air (PMA) configuration and from the direct calculation of the dispersion relation for the same configuration. Comparison is made with published results. The dispersion curves for Au and Cu have been determined for the same configuration from calculated ATR spectra and verified experimentally. Comparison is made with the direct calculation of the dispersion relation. The direct calculation yields two types of modes; namely the surface or Brewster mode and the virtual mode. Both modes are discussed. The marked effect of small electronic damping as exhibited by Ag and moderate electronic damping as exhibited by Au and Cu upon the PMA dispersion curves from ATR spectra is examined. Finally the direct calculation of the dielectric shift (perturbing effect of the prism-metal surface upon the active metal-air surface dispersion curve) is presented for the three metals as a function of film thickness and photon energy.

Surface plasma waves in silver and gold

The dispersion curves for surface plasma waves (SPW) in Ag have been determined from calculated reflectivity minima as exhibited by attenuated total reflection (ATR spectra) for the prism-air-metal (PAM) configuration and from the direct calculation of the dispersion relation for the same configuration. The dispersion curves for Au have been determined by measuring the ATR spectra for the prism-dielectric-metal (PDM) configuration, by calculating the ATR spectra from published optical constants and from the direct calculation of the despersion relation for the PDM configuration. We have found two general types of solutions from the direct calculation of the dispersion relation for both configurations. The two solutions are the surface or Brewster modes and the virtual modes. The characteristics of both modes are discussed. The effect of electronic damping upon the dispersion curves for Ag which exhibits low electronic damping and Au which exhibits moderate electronic damping is demonstrated. Finally the perturbing effect of the dielectric (referred herein as dielectric shift) upon the displacement of the dispersion curves to higher wave number for the PAM configuration for Ag and for the PDM configuration for Au is shown.

An adjustable spring rate suspension system

NE of the most challenging tasks in aerospace design occurs in the design of suspension systems subjected to both shock and vibration. The usual isolation method for decreasing the transmission of structural borne energy from one system to another is to interpose a relatively flexible member between the two structures. In the linear spring domain, the isolation of steady-state vibration is accomplished by means of a soft suspension having low damping. The isolation of transient shocks is accomplished by means of hard springs and moderate damping. Unfortunately, linear design solutions that are good for shock isolation are poor for vibration isolation and vice versa. Thus the design process encompasses a tradeoff for a best compromise in performance. It has been shown1 that nonlinear elastic suspensions that stiffen as they deflect can incorporate both the advantages of soft and hard systems to achieve both transient and steady-state isolation. Unfortunately, the variety of simple, nonlinear suspension devices is somewhat limited. In a previous Note2 a special suspension spring known as the elastica suspension was presented. Because of its simple construction and its symmetrical, nonlinear behavior, this device was found to have much promise for aerospace and transportation applications. It is the purpose of this Note to add to the versatility of the elastica suspension by demonstrating how the device can be adjusted to provide an enhanced selection of nonlinear spring rates.

The Brownian motion theory of chemical transition rates

The complete Fokker-Planck equation of the Brownian motion in a potential consisting of harmonic parts is studied as a method of providing chemical transition rates. General, analytic solutions are obtained for the situations of moderate or strong damping. These results generalize the treatment of Kramers 1), primarily based upon the simpler Smoluchowski equation. For a sharp potential maximum, there is an appreciable deviation from the earlier results. However, for a potential maximum, around which, there is a smooth, harmonic character over the entire region where the deviation from equilibrium is appreciable, it is found that a simple result of Kramers is valid in a wide range of the parameters involved. There is no restriction upon the energy barrier in the method, which would be applicable for calculating transformation or dissociation rates of macromolecular weak bonds.

Surface-plasmon-polariton dispersion of semiconductors with depletion layers

For a sample with plasma frequency varying exponentially from ω ps at the surface to a larger value ω pb in the bulk, and moderate damping, we find, contrary to other calculations, that the long wavelength surface plasmon dispersion has a single branch that is reentrant at ω = ω ps .

Optical properties of aggregated metal systems: Real metals

In a previous paper, the simple Maxwell-Garnett theory of aggregated metal systems was analyzed in the free-electron regime for low damping (ωpτ>1). Expressions were derived for the magnitude and resonance frequencies of plasma and conduction resonances as functions of particle constants and the aggregation parameter. In this paper the effect of moderate and high damping (ωpτ≲1) and the effect of a dielectric interparticle matrix with n > 1 is examined, and the results are compared with experiment.

Rational adjustment of gyro-theodolite observations

Formulae for the reduction of gyro-theodolite observations using the reversal point method are reviewed and relative precisions are compared. The significance of theoretical and practical comparison models is discussed and the flexibility and stability of different reduction proposals are considered. More stable formulae than are currently available are derived for five, seven and nine observations for moderate damping. The optimum number of observations to be taken for any spin-up of the gyroscope is also examined.

Response of Laminates with Dissipative Skins

Laminates made with plastic or reinforced plastic skins were sometimes found to have different damping characteristics than expected on the basis of conventional laminate theory. Investigation of the magnitude of damping of the plastic skins indicated that they had moderate internal damping. Laminates made with such skin materials can be considered as two-phase damping structures. The core contributes shear damping, and the skins add their own characteristic internal damping. Mathematical analyses of the effects of dissipative skins on the dynamic response of laminates are presented. Experimental verification is shown, and some practical design considerations are discussed.