Initial Phase Angle Research Articles

Nonlinear binary-mode interactions in a developing mixing layer

In this paper we present the formulation and results of two-wave interactions in a spatially developing shear layer, directed at understanding and interpreting the physical mechanisms that underlie the results of quantitative observation. Our study confirms the existence of Kelly's mechanism that augments the growth of a subharmonic disturbance by extracting energy from its fundamental or vice versa. This mechanism is shown to be strongest in the region where the fundamental begins to return energy to the mean flow and the two wave modes are of comparable energy levels. It is found that the initial conditions and especially the initial phase angle between the two disturbances play a very significant role in the modal development and that of the shear layer itself. A doubling of the shear-layer thickness is shown to take place; the two successive plateaux in its growth are attributed to the peaking in the energy production rates of the fundamental and subharmonic fluctuations.

Stability boundaries of an oscillator under high frequency multicomponent parametric excitation

The method of multiple scales is used to investigate certain of the stability boundaries of a damped single-degree-of-freedom system under a multifrequency parametric excitation having sinusoidal inputs with constant frequency spacing and with initial phase angles. The resonance case studied is that for which the difference between any two excitation frequencies is near to twice the natural frequency. Results obtained are compared with those obtained by the monodromy matrix method which is based on Floquet theory. The results compare closely when the excitation level is small, and agree qualitatively at larger excitation.

Optimal multiple-impulse time-fixed rendezvous between circular orbits

Minimum-fuel, multiple-impulse, time-fixed solutions are obtained for circle-to-circle rendezvous. The coplanar case and a restricted class of noncoplanar cases are analyzed. For several initial phase angles of the target relative to the vehicle, optimal solutions are obtained for a range of fixed transfer times and terminal radii. Primer vector theory is used to obtain the optimal number of impulses, their times and positions, and the presence of initial or final coasting arcs. For sufficiently large transfer times, the optimal solutions become the well-known time-open solutions, such as the Hohmann transfer for the coplanar case. The results obtained can be used to perform a time vs fuel trade-off for missions which have operational time constraints, such as space rescue operations and avoidance maneuvers.

Nonresonant interaction of heavy ions with electromagnetic ion cyclotron waves

The motion of a heavy ion in the presence of an intense ultralow‐frequency electromagnetic wave propagating along the dc magnetic field is analyzed. Starting from the basic equations of motion and from their associated two invariants, the heavy ion velocity‐space trajectories are drawn. It is shown that after a certain time, particles whose initial phase angles are randomly distributed tend to bunch together, provided that the wave intensity, b1 is sufficiently large. The maximum perpendicular velocity υa which the particle reaches during the bunching process can be a fraction of the Alfvén velocity. When the wave frequency f is smaller than the heavy ion gyrofrequency, F, υa is maximum for waves with frequencies slightly smaller than F. When the wave frequency is larger than the cutoff frequency fco, υa is a continuously decreasing function of f. In both cases, υa is proportional to b1/|F ‐ f| when b1 is small and/or |F ‐ f| is large, and it is proportional to (b1/f)1/3 when b1 is large and/or |F ‐ f| is small. The process is nonresonant in the sense that it is efficient however small is the heavy ion initial parallel velocity. A parameter study is done which gives the process efficiency for different wave or particle characteristics. Approximate analytical expressions are given, and they are compared with the results of numerical computations. The importance of these results for the interpretation of the recently observed acceleration of He+ ions in conjunction with the occurrence of large‐amplitude ion cyclotron waves in the equatorial magnetosphere is discussed.

Injection and Storage of Singly Charged Ions and Extension of the Principles of Ion Storage to Highly Charged Low Energy Recoil Ions

On-line experiments on low energy multiply charged ions can be performed with high sensitivity and resolution if the ions of interest are caught in flight and are subsequently accumulated in a finite observation volume. The various aspects of injecting singly and multiply charged ions into harmonic and nonharmonic ion trap devices are discussed. Injection of a continuous ion beam into an active trap was evaluated by numerical integration of the Mathieu equations to determine the finite confinement time of the injected ions. Three different modes of operation of the ion trap, namely, the radio-frequency (RF), the Penning, and the combined RF and Penning modes were considered. On the other hand the phase space method was used for a pulsed ion beam injected into the RF quadrupole ion trap for which the RF quadrupole field was switched on after the ion injection. In this way, a high trapping efficiency is reached for a suitable combination of the chosen initial RF voltage phase angle and the delay time after the ion injection. In addition, the trapped ions are confined indefinitely and energy- and mass-selective storage of the injected ions are also possible. The occurrence of ion motion induced side bands in the spectrum of stored ions and their importance for determining the ion temperature are discussed. Scaling relations are given which relate the results obtained for singly charged ions to the storage properties as they pertain to highly charged ions. Use is made of the unique environment of ion traps to speculate about novel collision and high resolution spectroscopy experiments on highly charged ions.

A microcomputer-based visual stimulator

A microcomputer-based instrument is described that provides visual stimuli for a CRT display and collects and processes response data. The system produces a variety of visual patterns to facilitate the study of the properties of visual neurons of several different animals. The visual patterns include drifting sinusoidal or bar gratings, contrast reversal gratings, and diffuse light. By means of a keypad, the experimenter can select the visual pattern and its parameters, such as depth of modulation, spatial frequency, initial spatial phase, temporal frequency, and rotation angle.

Conditions under which a number of sinusoids may be instantaneously in phase (biological applications)

The literature on the phase relationships between frequency components of a Fourier analysis is reviewed, with examples and theories from acoustics and neurophysiology. Given n sinusoids of different frequencies and phase angles, it is shown that for n>or=2 the set of initial phase angles allowing the n sinusoids to be in phase at some time t0 consists of one or more planes of constant dimension 2 and that for n=2 such a time t0 always exists. The conditions under which the common phase of n sinusoids at one time t0 will be the same as the common phase at another time t0 are also investigated. The importance of incommensurately related frequency components is emphasized by proofs which do not depend on harmonic relationships. Proofs are formulated in a linear algebra format to demonstrate the versatility of the method for analysing long sequences of frequencies and phases.

Phase-bunching and other non-linear processes occurring in gyroresonant wave-particle interactions

We consider the movement of individual electrons in a magnetized plasma in which a monochromatic wave is propagating in the whistler mode. We derive simple expressions which give the displacement of the electrons as a function of time, the phase angle that their velocity vector makes with the magnetic component of the wave, their pitch angle and energy changes. A useful formula is obtained which gives the velocity range over which particles remain trapped inside the wave, as a function of the wave intensity and of the initial phase angle of the particle. It is shown that even strictly resonant particles can escape from the wave when their initial phase angle is very small. From the derived expressions, it is possible to compute the phase-bunching effect which occurs approximately at one trapping wavelength behind the leading edge of the interaction region. We deduce also the total amount of energy which is taken from (or given to) the wave by magnetospheric electrons in both cases of naturally existing or artificially injected particles. It is shown that these non-linear amplification processes can lead to very large VLF amplitude in the magnetosphere.

Two types of phase bunching in the whistler mode wave-particle interaction.

Importance of the “phase bunching” of resonant electrons in the whistler mode (WM) wave-particle interactions was pointed out by Brice. In this paper, we call attention on the point that there are two different types of phase bunching with two physically distinct phasing mechanisms in the case of the WM wave-particle interaction. One is a longitudinal phase bunching (the Brice type) which is caused by a longitudinal acceleration, while the other is a transverse phase bunching which is caused by a transverse acceleration. The both types are considered separately and examined mathematically and physically. In order to visualize the different characteristics of the two, numerical results on the phasing patterns and the resultant resonant current are shown by computing the nonlinear motion of electrons in a monochromatic WM wave with various initial velocities (36 different initial phase angles ×15 different speed |υ|).

Resonance Effects in Spiral Galaxies

view Abstract Citations (67) References (10) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Resonance Effects in Spiral Galaxies Contopoulos, G. Abstract The usual form of the distribution function in a spiral galaxy, j = Jo + Ji +. , where Jo is a function of the energy and angular momentum of the axisymmetric configuration, is not applicable, in general, near resonances. A resonance occurs whenever the ratio of the frequency of the radial oscillation to that of the rotational motion around the center of the galaxy (in a frame of reference rotating together with the spiral pattern) is rational. Near the Lindblad resonances, where this ratio is equal to 2, Jo has a secular term or a small divisor term that makes it large in comparison with Jo. In order to avoid secular terms in resonant cases, we use a new integral of the axisymmetric case, namely, the sine or cosine of the initial phase angle, which is then isolating. Thus we find a new form of Jo that has a quadruple symmetry in the phase angle 6' in the rotating frame. We consider various resonances expected to occur in our Galaxy. We study in some detail the forms of the orbits and invariant curves near the inner Lindblad resonance; we find two stable resonant periodic orbits and tube orbits around them. A comparison between the theoretical results and the numerical calculations is made. Finally, we see how the stars respond to a growing spiral wave. As the wave is established, the orbits near resonance change from their original epicyclic form to a resonant form near one or the other of the two resonant periodic orbits. The resulting density distribution has roughly a quadruple symmetry. Publication: The Astrophysical Journal Pub Date: April 1970 DOI: 10.1086/150411 Bibcode: 1970ApJ...160..113C full text sources ADS | Related Materials (2) Erratum: 1970ApJ...162.1057C Erratum: 1970ApJ...162.1057Y

A simple method for maintaining the resonance condition in a spin echo nuclear magnetic resonance spectrometer

A device is described which will correct for drifts in the resonance condition of a spin echo spectrometer. It operates by measuring the voltage at the output of the spectrometer's phase sensitive detector a definite time after the nuclei in the sample under study have received a 90° rf pulse. The phase detector is used with an initial phase angle difference of 90°. The measured voltage is used to correct the magnetic field towards its resonance value. The particular unit described in detail accomplishes both field-frequency correction and nuclear resonance signal measurement with the same phase detector.

100 Mc/s gated clock oscillator

A simple gated clock oscillator is analyzed, where gating pulse rise and fall times are comparable to the period of oscillations. Resonator response to ramp forcing function assuming a linear damping device is found and curves for calculating the amplitude and initial phase angle of the oscillation are shown. Condition for compensation of resonator losses is found and interval of damping of oscillations is analyzed. Also the circuit diagram and the waveforms at the 100 Mc/s clock frequency are shown.

Transient Vibration by Vector Methods

SummaryIn control theory, stability is investigated by established techniques such as the root-locus method. The same techniques may be applied to the control equivalent of a mechanical system and will show the transient state of the system. Vector methods such as are used to solve vibration problems may also be extended to give transients.This note gives a vector representation of the transient vibration of a one degree of freedom system possessing inertia, elasticity and viscous damping. For the transient which is set up by a change in the steady vibration of the system, the initial amplitude and phase angle are readily found by vector methods. This information is sufficient to define the transient.

Integral pulse frequency modulated control systems

The paper describes a new class of discrete data control systems in which a portion of the system makes use of pulse frequency modulation. This concept arose as an abstraction from the study of neural communication links in physiological control systems. An integral pulse frequency modulator is defined as one which emits a standard pulse whenever the integral of the input variable reaches a threshold value. The frequency of the output pulse train varies in a linear manner with the input magnitude. Modulators producing pulse trains of one or two signs are introduced and approximate frequency characteristics are obtained; they have discontinuous and discrete features, and depend upon signal amplitude, frequency, and initial phase angle. Feedback control systems making use of single-signed modulators in two parallel feedforward paths are described, and some of the unique stability considerations are studied. Continuous systems that approximate these stable discrete-data systems are constructed, and their transient properties considered.

Cross-Compound Turbine-Generator Turning-Gear Synchronization Study II-Field Test Investigation

1. The internal phase angle between rotors at the instant of initial field excitation was found equally as important as the magnitude and rate of application of excitation voltage in affecting the turning-gear synchronization process for this particular machine. As both elements have identical generators and turning speeds, and nearly equal inertias, the optimum initial phase angle was found near 0 degrees. For cross-compound machines having widely dissimilar elements and different nominal turning speeds, however, the same conclusions may not apply. The worst initial phase angle for synchronizing this machine was found to be near 180 degrees. Although initial phase angle is normally a purely random variable, monitoring devices such as those used in these tests may be warranted in some cases for remotely indicating phase difference angle for use during either manual or automatic start-up operation. 2. The optimum excitation voltage level for smoothest synchronization was found to be near 60-65 volts for each machine, or about 60% of rated opencircuit excitation. With excitation reduced to 38 volts (37% open-circuit) or increased to 80 volts (75% open-circuit) synchronization failure resulted. Best results were obtained by presetting the exciter voltage to the desired level followed by simultaneous closure of both field breakers. Sequential operation of field breakers gave less satisfactory results, and, in some cases, resulted in synchronization failure. 3. Data from the tests illustrated in Figs.