Search NASA⌕ Search

SEARCH · Search NASA

Results for “PERIODIC FUNCTION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1995

Exponentially accurate approximations to piece-wise smooth periodic functions

A family of simple, periodic basis functions with 'built-in' discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representations of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.

Greer, James↗
NASA NTRS1986

Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.

Sidi, A.↗
NASA NTRS1992

On the Gibbs phenomenon 1: Recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function

It is well known that the Fourier series of an analytic or periodic function, truncated after 2N+1 terms, converges exponentially with N, even in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. Here, we show that the first 2N+1 Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation (in the maximum norm) can be constructed.

Gottlieb, David↗
NASA NTRS2002

A critical period for functional vestibular development in zebrafish

We have determined a critical period for vestibular development in zebrafish by using a bioreactor designed by NASA to simulate microgravity for cells in culture. A critical period is defined as the briefest period of time during development when stimulus deprivation results in long lasting or permanent sensory deficits. Zebrafish eggs were collected within 3 hours of being laid and fertilized. In experiment 1, eggs were placed in the bioreactor at 3, 24, 30, 36, 48, or 72 hours postfertilization (hPF) and maintained in the bioreactor until 96 hPF. In experiment 2, eggs were placed in the bioreactor immediately after they were collected and maintained in the bioreactor until 24, 36, 48, 60, 66, 72, or 96 hPF. Beginning at 96 hPF, all larvae had their vestibulo-ocular reflexes (VOR) evaluated once each day for 5 days. Only larvae that hatched from eggs that were placed in the bioreactor before 30 hPF in experiment 1 or removed from the bioreactor later than 66 hPF in experiment 2 had VOR deficits that persisted for at least 5 days. These data suggest a critical period for vestibular development in the zebrafish that begins before 30 hPF and ends after 66 hPF. To confirm this, zebrafish eggs were placed in the bioreactor at 24 hPF and removed at 72 hPF. VORs were evaluated in these larvae once each day for 5 days beginning at 96 hPF. These larvae had VOR deficits that persisted for at least 5 days. In addition, larvae that had been maintained in the bioreactor from 24 to 66 hPF or from 30 to 72 hPF, had only temporary VOR deficits. In a final experiment, zebrafish eggs were placed in the bioreactor at 3 hPF and removed at 96 hPF but the bioreactor was turned off from 24 hPF to 72 hPF. These larvae had normal VORs when they were removed from the bioreactor at 96 hPF. Taken as a whole, these data support the idea that there is a critical period for functional maturation of the zebrafish vestibular system. The developmental period identified includes the timeframe during which the vestibular primary afferent neurons are born, innervate their central and peripheral targets, and remodel their central projections. Copyright 2002 Wiley-Liss, Inc.

Non-NASA Center↗
NASA NTRS2012

Optimal Force Control of Vibro-Impact Systems for Autonomous Drilling Applications

The need to maintain optimal energy efficiency is critical during the drilling operations performed on future and current planetary rover missions (see figure). Specifically, this innovation seeks to solve the following problem. Given a spring-loaded percussive drill driven by a voice-coil motor, one needs to determine the optimal input voltage waveform (periodic function) and the optimal hammering period that minimizes the dissipated energy, while ensuring that the hammer-to-rock impacts are made with sufficient (user-defined) impact velocity (or impact energy). To solve this problem, it was first observed that when voice-coil-actuated percussive drills are driven at high power, it is of paramount importance to ensure that the electrical current of the device remains in phase with the velocity of the hammer. Otherwise, negative work is performed and the drill experiences a loss of performance (i.e., reduced impact energy) and an increase in Joule heating (i.e., reduction in energy efficiency). This observation has motivated many drilling products to incorporate the standard bang-bang control approach for driving their percussive drills. However, the bang-bang control approach is significantly less efficient than the optimal energy-efficient control approach solved herein. To obtain this solution, the standard tools of classical optimal control theory were applied. It is worth noting that these tools inherently require the solution of a two-point boundary value problem (TPBVP), i.e., a system of differential equations where half the equations have unknown boundary conditions. Typically, the TPBVP is impossible to solve analytically for high-dimensional dynamic systems. However, for the case of the spring-loaded vibro-impactor, this approach yields the exact optimal control solution as the sum of four analytic functions whose coefficients are determined using a simple, easy-to-implement algorithm. Once the optimal control waveform is determined, it can be used optimally in the context of both open-loop and closed-loop control modes (using standard realtime control hardware).

Aldrich, Jack B.↗
NASA NTRS1972

A graphical test for checking the stability of a linear time-invariant feedback system.

A continuous-time scalar linear time-invariant feedback system is considered for the purpose of checking Willems' (1969, 1970) graphical test for a scalar linear time-invariant feedback system with constant feedback. Heavy reliance is placed on the theory of almost periodic functions. Following definition of the problem and layout of notation, attention is given to solution of the problem, considering only the almost periodic part of the open-loop transfer function.

Callier, F. M.↗
NASA NTRS1986

Periodicity of the earth's magnetic reversals

Results are presented from an attempt to perform a relatively comprehensive analysis of the evidence for a periodicity, with harmonics, of the observed regular reversals of the earth's magnetic field. The database considered covers 296 reversals over the past 165 Myr. Histograms with bins 1 Myr apart reveal only 30 Myr reversal patterns. The reversal dates are fitted to a linear periodic function and a spectrum is computed for the residuals at the adopted dates. The possible presence of multiple periodicities is evaluated and over various time intervals. The analysis shows that a recently observed 15 Myr periodicity is probably a harmonic of the 29.5-30.5 Myr period. The calculations do not confirm an inherent magnetic reversal property of the earth. The reversals may arise from tectonic events or from impacts from extraterrestrial objects.

Stothers, R. B.↗
NASA NTRS1974

Aeroelastic stability of periodic systems with application to rotor blade flutter

The dynamics of a helicopter blade in forward flight are described by a system of linear differential equations with periodic coefficients. The stability of this periodic aeroelastic system is determined, using multivariable Floquet-Liapunov theory. The transition matrix at the end of the period is evaluated by: (1) direct numerical integration, and (2) a new, approximate method, which consists in approximating a periodic function by a series of step functions. The numerical accuracy and efficiency of the methods is compared, and the second method is shown to be superior by far. Results illustrating the effect of the periodic coefficients and various blade parameters are presented.

Friedmann, P.↗
NASA NTRS1988

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, and free surface flows. The use of the quadrature methods is demonstrated with numerical examples.

Sidi, Avram↗
NASA NTRS1999

"Dust Devils": Gardening Agents on the Surface of Mars, and Hidden Hazards to Human Exploration?

Dust devils are familiar sites in the and regions of the world: they can produce quite spectacular displays of dust lofting when the vortices scavenge very loose dust from a dry lake bed or from recently disturbed agricultural fields. If one were to arrive at the center of an arid region, take one photograph, or even a series of photographs over a period of several days, then return the images for laboratory analysis, it would be most likely concluded that the region was inactive from an aeolian perspective. No images of general dust movement were obtained, nor were any dust devils "caught on camera" owing to their ephemeral and unpredictable appearance, and the fact that there was deceptively little residue of their actions. If, however, a camera were to take a 360 degree continuous recording over a period of a year, and the film were then to be shown at high speed over a period a several minutes, the impression might be that of a region ravaged by air vorticity and dust movement. Extrapolate this over geological time, and it is possible to visualize dust devils as prime aeolian agents, rather than insignificant vagaries of nature, On Mars, the thin atmosphere permits the surface of the planet to be heated but it does not itself retain heat with the capacity of the earth's atmosphere. This gives rise to greater thermal instability near the surface of Mars as "warm" air pockets diapiritically inject themselves into higher atmospheric layers. Resulting boundary-layer vorticity on Mars might therefore be expected to produce dust devils in abundance, if only seasonally. The spectacular images of dust devils obtained by Pathfinder within its brief functional period on the planet testify to the probability of highly frequent surface vorticity in light of the above reasoning about observational probability. Notably, the Pathfinder devils appeared to be at least a kilometer in height. There are several consequences for the geology of Mars, and for human exploration, if dust devils are to be expected in reasonable abundance. First, from a geological perspective, the vortices will act as "gardening" agents for the top few centimeters of entrainable material. Over time (hundreds of millions, or billions of years being available), they will cover the surface with scouring paths, and the grain sizes that can be lofted by a vortex probably extends over the whole sand to dust range. The depositional paths are, of course, much larger, so that vortex-induced deposition is more widespread than vortex-induced erosion, and will without doubt, affect the whole region in which the dust devils occur (this might explain why rocks at the Viking site seemed oddly capped with dust in a region apparently subject to general aeolian scouring). On Mars, the lift forces in dust devils might be less than on earth owing to the much thinner atmosphere, but this may be counterbalanced by lower gravity and greater vortex velocities. Certainly, when active, other aeolian phenomena on Mars --sand motion and dust storms, seem no less energetic and no less capable of lofting sediments than equivalent terrestrial aeolian phenomena. Every several years, within the current climatic regime, the surface of Mars is subject to light dust fall from global dust storms. Over time, this should develop a very uniform surface layer, with commensurate uniformity in grain size, mineralogy, albedo, color, and general spectroscopic properties. Dust devils will disturb this situation by continually mixing the surface dust with underlying layers, perhaps composed of silt and sand. This size mixing will also involve compositional mixing. After some years, the thin layer of dust that may be difficult to entrain alone, becomes progressively mixed with coarser materials that could reduce the general aeolian threshold of the soil. Certainly the continual disturbance by vorticity will prevent surface stabilization that may bind or indurate grains (caused by slow cementation or ice welding at grain boundaries). If dust devils continually loft dust to kilometer heights, and the dust is sprayed into many cubic kilometers of atmosphere each time, could the devils produce a continual background of atmospheric dust that might be mistaken for the fallout of a distant large-scale dust storm? From a human exploration perspective, dust devils are unlikely to pose any, life- threatening situation for an astronaut unfortunate enough to encounter a momentary swirling cloud of loose soil. However, it is noted that pervasive dust is probably one of the greatest long-term hazards for a human encampment. The fineness and penetration capabilities of the dust, its electrostatic adhesive properties, and its complete ubiquity, render the material a persistent nuisance at best, but at worst, over a period of many months it is possible that space suits, machinery, habitat interiors, air filters, and so forth, could become jeopardized. Owing to dust penetration, the space suits used in the Apollo landings were rendered unusable after a few EVA activities. There will be a definite attempt to situate a human colony on Mars in an area that is far removed from the regions of the planet known for being the centers of major dust storms. At the heart of these storm systems, the dust lofting mechanics are unknown, but they are energetic and perhaps potentially life-threatening for an astronaut. Locating a colony in a region that appears from space to be meteorologically benign may lead to colony placement in a region prone to dust devils, but dust devils are not (or have not been) detectable from orbital observations: the region surveyed for placement will appear like the apparently inactive and area referred to earlier. The region may be spared from highly energetic weather systems, but it may not be necessarily immune from continual dust disturbance. Additional information is contained in the original.

Marshall, J.↗