# Nyquist frequency

A pulse-repetition frequency of 15 shots/sec was sufficient for measurement and determination of eccentricity at the highest drawing speed. This was a reasonable value although it was not obtainable when digital output was based on averaging several shots. Another important question concerning measuring speed was the dynamic range in the measurements. Was every single measurement correct, when the wall thickness was changing from one point to the next A measuring fixture was made to control this (Figure 4). A piece of tube with correct diameter and wall thickness, and with a representative eccentricity was [c.897]

An interesting application of capillarity and drops in fields occurs in inkjet printing technology. In this process, illustrated in Fig. 11-19, ink resides in a small square chamber with a meniscus balanced at the exit orifice by the pressure in the reservoir and capillary forces. In the wall opposite the orifice is a thin film resistor that, upon heating at 10 °C/sec, causes rapid growth of a vapor bubble that ejects a drop of ink through the orifice (Fig. II-19h). The chamber refills and the process is repeated. The newest printers achieve a repetition frequency of 8000 Hz by carefully controlling the refilling process [100]. [c.32]

We can bring these qualitative remarks into focus by considering the amplitude, phase and frequency of a classical field interacting with an atomic transition dipole in a two-level atom. A detailed development of the following results is beyond the scope of the present article, but can be found elsewhere [12, 13]. The usual approach is semiclassical and consists in treating the atom as a two-level quantum system and the radiation as a classical electromagnetic field [14]. A full quantum approach can also be employed [15], but it will not be discussed here. Wlrat follows immediately is sometimes called the Doppler cooling model. It turns out that atoms with hyperfine stmcture in the ground state can be cooled below the Doppler limit predicted by this model and, to explain this unexpected sub-Doppler cooling, models involving interaction between a slowly moving atom and the polarization gradient of a standing wave have been invoked. We will sketch briefly in the next section the physics of these polarization gradient cooling mechanisms. [c.2458]

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship [c.293]

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). [c.288]

A simple mathematical manipulation of the dissociation energy of IFl, as determined from its absorption spectrum, yields the bond energy. Planck s law, A/f = hv. permits us to calculate the energy difference between the lowest level and the next higher level from its spectroscopic line at 2191 cm . This is the highest frequency line because the lowest is the largest of all energy spacings in Fig, 10-lb. We can also measure the second energy increment, which corresponds to the spectral peak of next lower frequency, the third, and so on. corresponding to the gradual diminution of energy spacing in Fig. 10-lb. The series approaches zero. The sum of all energies of transition is the dissociation energy. [c.302]

HyperChem integrates the equations of motion using very small time steps (At). At each step, the algorithm evaluates energy and forces of the molecular system. Use a time step of about 0.5 to 1.0 femtoseconds (fs) for an All Atom system or 1 to 2 fs for a United Atom system. Small time steps allow the simulation to adequately integrate the highest frequency motions of the system, usually bond stretching vibrations on the order of several picoseconds. Adjust At for each molecular system to obtain energy conservation (see the next section). [c.71]

Clearly, the pulsing electrode can be turned on and off at any frequency chosen, but there are some constraints on the frequencies actually used. At the fastest, there is little point in pulsing the electrode at such a rate that one lot of ions has not had time to travel the length of the flight tube before the next lot is on its way. With the physical dimensions of typical flight tubes and the magnitudes of accelerating voltages commonly used in mass spectrometers, an upper limit of about 30 kHz is found. The slowest rate of pulsing the electrode is almost anything At 30 KHz, the TOF instrument can measure one mass spectrum every 33 psec or, put another way, in one second the TOF instrument can accumulate and sum 30,000 spectra. Little wonder that the acquisition of a spectrum appears to be instantaneous on the human time scale. [c.197]

Force Fields, Molecular Dynamics, and Vibrational Spectroscopy. The details of the relationship between molecular mechanics force fields and spectroscopic vibrational force fields have been discussed (60). Other fundamental papers on molecular mechanics are available as well (59,62—63,65—67). The link between molecular mechanics and molecular dynamics comes about through the force field itself. In molecular mechanics, the main interest here is in computing the energy of molecules in the gas phase at room temperature in a single, discrete configuration and conformation time is not a variable in the equations. The goal is to know the stmcture of the molecule, bond lengths, angles, etc conformational energy relative to some reference stmcture (eg, gauche vs anti) the heat of formation and, perhaps, the vibrational frequencies. From molecular dynamics, the objectives are properties which represent a time-averaged ensemble of states, including, for example, conformationaHy excited states, rotational states, interconversion rates, or inversion barriers for amines or amides. From this ensemble of states, characterizing the existence of the excited states and their contribution to the total energy of the system (from a Boltzmann distribution) is next. From an understanding of the vibrational spectroscopic roots (64) of molecular mechanical force fields used in dynamics simulations it can easily be appreciated that molecular dynamics may be seen as an extension bridge between theory and experiment, linking "static" molecular mechanical representations of properties with "dynamical" experimental properties. [c.164]

A variation on the transit time method is the frequency-difference or sing-around method. In this technique, pulses are transmitted between two pairs of diagonally mounted transducers. The receipt of a pulse is used to trigger the next pulse. Alternatively this can be done using one pair of transducers where each acts alternately as transmitter and receiver. The frequency of pulses in each loop is given by [c.67]

Operation. Small and medium sized coreless induction furnaces powered from high frequency power suppHes can be started with a charge of metal pieces at room temperature, usually scrap material of appropriate alloy. The charge material is selected to allow a reasonable power to be drawn from the power supply. As the metal charge begins to melt, a molten pool is estabUshed and the charge compacts, allowing additional charge to be added. AHoy additions and temperature adjustments complete the melting cycle (9). Higher operating efficiency is achieved if the next cycle is initiated prompdy after the charge is poured off so that the stored energy in the hearth refractory is not lost to the coil cooling water. Large coreless furnaces operating at line frequency are often started with a molten initial charge, although it is possible to start with a charge of soHd material. Typical operation of these furnaces involves dispensing 20 to 30% of the furnace capacity and immediately recharging dry or preheated material into the bath as power is appHed. These furnaces are usually held hiU during off shifts to maximize refractory life. An alternative is to empty the furnace and maintain the refractory continuously warm with supplemental heat. Furnaces with capacities of 4.5 to 13.6 t with input power ratings of 825 to 1100 kW/t produce Hquid iron at a consumption rate of 550 to 600 kW h/t. [c.130]

The analogue-to-digital converter (ADC) samples the fluctuating voltage produced in the coils of the probe at regular time intervals, storing each value as a binary encoded integer. The rate at which the ADC must sample the voltage is defined by the Nyquist theorem, which states that the sampling rate must be greater than or equal to twice the signal frequency. The maximum speed of the digitizer determines the maximum observable spectral width the number of bits used in storing each data point and the number of bits in each computer word determine the dynamic range of intensities that can be observed. [c.401]

Ideally a standard cell is constmcted simply and is characterized by a high constancy of emf, a low temperature coefficient of emf, and an emf close to one volt. The Weston cell, which uses a standard cadmium sulfate electrolyte and electrodes of cadmium amalgam and a paste of mercury and mercurous sulfate, essentially meets these conditions. The voltage of the cell is 1.0183 V at 20°C. The a-c Josephson effect, which relates the frequency of a superconducting oscillator to the potential difference between two superconducting components, is used by NIST to maintain the unit of emf. The definition of the volt, however, remains as the Q/A derivation described. [c.20]

Equipment. Screening devices can have stationary or moving screen decks. Moving screen decks can be designed as rotating cylinders (trommels) or vibrating surfaces. By far, the most popular screen deck design for sizing is the inclined vibrating screen (Fig. 3). Components of vibrating screens are the vibrating frame, deck support frame, screening deck, motor/drive assembly, and feed box/distributor. AuxiUary components include conveyor belts, feed chutes, and dust collection systems. Vibration is produced on inclined screen decks by circular motion in a vertical plane of 3—12.5-mm amphtude at frequencies of 700—1000 cycles per minute. Newer high speed screening designs operate at 3600 cycles per minute. The vibration lifts the material, producing stratification. Stratification places the larger particles on top, and the smaller particles on the bottom, next to the apertures. This also helps push near-size particles through the apertures, reducing blinding. When the screen deck is used on an incline, the material cascades down the slope, introducing the probabiUty that the particle either passes through the opening or over the screen surface. For horizontal screen decks, the motion must be capable of conveying the material without the assistance of gravity. Straight-line motion at an angle of approximately 45° to the horizontal produces a lifting component for stratification and a conveying component for probabiUty of separation as the material passes across the horizontal screen surface. Industrial screens typically range from 2—7.5 m in length. [c.435]

Some of the numerous factors influencing the passage of a particle through a screen opening are ratio between cross-section of particle and of opening, percentage of screen open area, angle of incidence of feed, efficiency of spread of feed over screen area, kinetic energy of particle approaching screen opening, moisture of feed, stickiness of particle and of aggregated particle, pressure of particles riding above those next to the screen deck, blinding of screen apertures, corrosion of deck material, electrostatic bunching, shape of particle, amount of near-size particles in the feed, rate of feed, thickness of the bed, tautness of deck, shape of screen apertures, trajectory of particle imparted by amphtude and frequency of vibration, bulk density of material, and particle integrity (6). Because of the multitude of factors acting individually and interactively, no fundamental procedure based on first principles has been developed for sizing commercial screens. For example, increases in the quantity of near-size particles, ie, particles where the sizes He between 0.5 and 1.5 times the aperture size, lead to more blinding, which in turn lead to a decrease in the percentage of open area, ie, the ratio of the aperture area to the screen deck area, which reduces the screening efficiency of the screen deck. Most manufacturers of vibrating inclined screens have developed variations on a design theme that involves estimating the design loading, defined as flow rate/screen area, and then dividing the design feed rate by the design loading to obtain the required deck area (7). [c.435]

For all the detectors with spatially distinct signal recording, the numeric pixel size (such as scaiming pixel size for photographic fihn and IP, or chip-element size for ssCCD) must be distinguished from the actual obtainable resolution. This resolution can be affected by the primary scattering process of electrons in the detecting medium, or by the scattering of a produced light signal or a scaiming light spot in the detecting medium. Therefore, a point signal is delocalized, mathematically described by the PSF. The Fourier transfonn of the PSF is called the modulation transfer fiinction (MTF), describing the spatial frequency response of the detector. Wliereas the ideal detector has a MTF = 1 over the complete spatial frequency range, real detectors exhibit a moderate to strong fall-off of the MTF at the Nyquist frequency, i.e. their maximal detectable spatial resolution. In addition to spatial resolution, another important quantity characterizing a detector is the detection quantum efficiency (DQE). It is a measure of the detector noise and gives an assessment for the detection of single electrons. [c.1632]

Based on experience with the measurement of thin layers and related deconvolution techniques [5], [6] air-borne ultrasonics and a new deconvolution algorithm have been investigated [7]. Focussed and optimized composite probes have been used and were excited with a square wave pulser in pulse-echo mode. The signals have been acquired, digitized and after preprocessing (filtering) the difference of Time of Flight (TOF) of two overlapping reflection pulses have been deconvolved. The time resolution of the deconvolution is independant of the sampling and can be much better than the actual sampling rate. Of course the Nyquist theorem has to be fullfilled. This is no restriction since it just says that the samplerate has to be at least twice the maximum frequency (bandlimit, filter) present in the signal itself. [c.843]

The third region is one for which the Q values are of the order of chemical bond energies the r values become quite large, indicating that desorption may be slow, and F as computed by Eq. XVII-3 becomes preposterously large. Such values are evidently meaningless, and the difficulty lies in the assumption embodied in Eq. XVII-3 that the collision frequency gives the number of molecules hitting and sticking to the surface. As monolayer coverage is approached, it is to be expected that more and more impinging molecules will hit occupied areas and rebound without experiencing the full Q value. One way of correcting for this effect is taken up in the next section, which deals with the Langmuir adsorption equation. [c.603]

The previous calculations, while not altogether trivial, are among the simplest uses one can make of kinetic theory arguments. Next we turn to a somewhat more sophisticated calculation, that for the mean free path of a particle between collisions witii other particles in the gas. We will use the general fonn of the distribution fiinction at first, before restricting ourselves to the equilibrium case, so as to set the stage for discussions m later sections where we describe die fomial kinetic theory. Our approach will be first to compute the average frequency with which a particle collides with other particles. The inverse of this frequency is the mean time between collisions. If we then multiply the mean time between collisions by the mean speed, given by equation (A3.1.8), we will obtain the desired result for the mean free path between collisions. It is important to point out that one might choose to define the mean free path somewhat differently, by using die root mean square velocity instead of v, for example. The only change will be in a mimerical coefficient. The important issue will be to obtain the dependence of the mean free path upon the density and temperature of the gas and on the size of the particles. The mimerical factors are not that unportant. [c.669]

The next two temis (Lorentzians) arise from the mechanical part of the density fluctuations, the pressure fluctuations at constant entropy. These are the adiabatic sound modes (l/y)exp[-FA t ]cos[co(A) t ] with (D(k) = ck, and lead to the two spectral lines (Lorentzians) which are shifted in frequency by -ck (Stokes line) and +ck (anti-Stokes line). These are known as the Brillouin-Mandehtarn, doublet. The half-width at [c.724]

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RE is generated at two frequencies one is fixed at the free nnclear frequency appropriate to die sort of nnclei under scrutiny and the second is swept. These two frequencies are mnltiplied to obtain the sum and the difference frequencies, which are nsed to irradiate the sample. The experiment can be understood [c.1571]

Introducing the complex notation enables the impedance relationships to be presented as Argand diagrams in both Cartesian and polar co-ordinates (r,rp). The fomier leads to the Nyquist impedance spectrum, where the real impedance is plotted against the imaginary and the latter to the Bode spectrum, where both the modulus of impedance, r, and the phase angle are plotted as a fiinction of the frequency. In AC impedance tire cell is essentially replaced by a suitable model system in which the properties of the interface and the electrolyte are represented by appropriate electrical analogues and the impedance of the cell is then measured over a wide [c.1944]

The criterion that single molecules are being observed is often taken to be the appearance in a fluorescence excitation frequency scan of well separated peaks that have about the same intensity and width, separated by regions of flat background. However, one cannot be certain that a given peak does not arise from two molecules with accidentally degenerate resonant frequencies. Stronger evidence is provided by observing that when spontaneous or photoinduced spectral jumps or photobleaching occur, the fluorescence excitation feature jumps to a new frequency or disappears entirely from the scanned region in a quantized, all-or-nothing manner. Probably the best evidence for single-molecule observation comes from the statistics of photon emission on short time scales. Since a single molecule must experience a nonzero time interval between successive photon emissions (after emitting a photon, it must absorb one prior to its next emission), the probability of emitting two photons with zero time delay goes to zero for a single molecule, the phenomenon known as photon antibunching. [c.2486]

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if [c.3063]

The method presented in the next section is an attempt to overcome the barrier due to the highest frequencies whatever their origin. Although it has been implemented and tested for unconstrained dynamics only, there is no fundamental reason why it cannot be applied to overcome the less restrictive time step barrier arising in constrained dynamics. [c.325]

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. [c.216]

Force constants can be calculated if a spectral line can be associated with a specific mode of motion. For example, if we take the C—H stretching frequency to be at 2900 cm in the infra red spectrum of hydrocarbons, we have k = 457 N m for the force constant of the C—H bond. The actual value of k taken from the MM3 force field differs somewhat from this value, being 4.74 mdyn/A = 474 N m . The frequency enters into this calculation as the square, making k very sensitive to v. Infrared C—H stretching frequencies are not identical from one molecule to the next, giving a range of values of v from which to calculate k. A choice of 2960 cm (Barrow, 1999) leads to A = 480 N m . General-purpose force fields like MM3 are parameterized to reproduce not only spectral vibrational frequencies but molecular geometry and energy as well. Selection of the stretching parameters is guided by peaks in the vibrational spectr um, but the final choice is intended to give the best results for all calculated values. In the absence of a methodically precise method of parameter generation, some degree of trial and error enters into the process. [c.114]

More difficult to study are the nitrogen chemical shifts the resonance of in natural abundance has been determined for thiazole, among a lot of other compounds, by Warren and Roberts (259), who propose an empirical rule for the evaluation of these chemical shifts. Their results agree with the experimental determinations of resonance frequencies done by Whitanovsky et al. (128) and later by Noth et al, (260), as well as with the measurements made by Nagata et al. (261) as part of a study, using the INDOR method, of the chemical shifts of five- and six-membered nitrogen containing heterocycles as a function of the nature (proton accepting or donating) of the solvent. [c.77]

To predict the properties of a population on the basis of a sample, it is necessary to know something about the population s expected distribution around its central value. The distribution of a population can be represented by plotting the frequency of occurrence of individual values as a function of the values themselves. Such plots are called prohahility distrihutions. Unfortunately, we are rarely able to calculate the exact probability distribution for a chemical system. In fact, the probability distribution can take any shape, depending on the nature of the chemical system being investigated. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed next. [c.71]

According to the Nyquist theorem, to determine a periodic signal s true frequency, we must sample the signal at a rate that is at least twice its frequency (Figure 7.3b) that is, the signal must be sampled at least twice during a single cycle or period. When samples are collected at an interval of At, the highest frequency that can be accurately monitored has a frequency of (2 At) k For example, if samples are collected every hour, the highest frequency that we can monitor is 0.5 h k or a periodic cycle lasting 2 h. A signal with a cycling period of less than 2 h (a frequency of more than 0.5 h k cannot be monitored. Ideally, the sampling frequency should be at least three to four times that of the highest frequency signal of interest. Thus, if an hourly periodic cycle is of interest, samples should be collected at least every 15-20 min. [c.184]

The next step in the Fourier imaging procedure is to encode some property of the spins as to the location in the selected plane. Spins could easily be encoded as to their x position by applying a gradient G after the r-f pulse and during the acquisition of the FID. The difficulty is in encoding the spins with information as to theirjy location. This is accompHshed by encoding the phase of the precessing spin packets withjy position (see Fig. 3). Phase encoding is accompHshed by turning on a gradient in thejy direction immediately after the sHce selection gradient is turned off and before the frequency-encoding gradient is turned on. The spins in the excited plane now precess at a frequency dependent on thejy position. After a period of time T the gradient is turned off and the spins have acquired a phase 7T equal to [c.55]

See pages that mention the term

**Nyquist frequency**:

**[c.1167] [c.292] [c.216] [c.908] [c.620] [c.226] [c.1027] [c.1209] [c.1295] [c.1509] [c.3028] [c.499] [c.124] [c.65] [c.92] [c.195] [c.53]**

Advanced control engineering (2001) -- [ c.200 ]