Amplitude of oscillation

The atomic harmonic oscillator follows the same frequency equation that the classical harmonic oscillator does. The difference is that the classical harmonic oscillator can have any amplitude of oscillation leading to a continuum of energy whereas the quantum harmonic oscillator can have only certain specific amplitudes of oscillation leading to a discrete set of allowed energy levels. 

The amplitudes of oscillations will depend upon weight, stiffness and configuration. The record of these oscillations is known as free vibration record. The rate of oscillations will determine the natural frequency of the object. Figure 14.20 shows one such free vibration record. 

Safety concerns about such thermohydrodynamic instabilities were raised after a rather unexpected occurrence of oscillations in the core of the LaSalle County Nuclear Station in 1988 (Phillips, 1990), following recirculation pump trips. Quite a large relative amplitude of oscillations was reached during that event, and the reactor was finally tripped from a high flux signal (Yadigaroglu, 1993). The safety... 

The amplitude of oscillation of cavity/bubble radius, which is reflected in the magnitude of the resultant pressure pulses of the cavity collapse... 

The curve shows the amplitude of oscillation of an object or system as the frequency of the input oscillation is steadily increased. Start by drawing a normal sine wave whose wavelength decreases as the input frequency increases. Demonstrate a particular frequency at which the amplitude rises to a peak. By no means does this have to occur at a high frequency it depends on what the natural frequency of the system is. Label the peak amplitude frequency as the resonant frequency. Make sure that, after the peak, the amplitude dies away again towards the baseline. 

An oscillated fuel flow was provided in the form of a central jet within the duct carrying the pilot stream. The dimensions of the tube carrying the oscillated flow implied that the mean velocity and equivalence ratio of the jet had to be larger than that of the pilot stream to enable the oscillation of at least 5% of the total fuel flow. An examination of the influence of the bulk mean velocities of the pilot stream and the central jet on the amplitude of oscillations in this flow arrangement showed that, for the present range of flow conditions, values of the bulk mean velocity of the pilot stream less than that of the annular flow had no effect on the amplitude of oscillations, although larger values led to a decrease in amplitude [20]. The amplitude was also insensitive to the bulk mean velocity of the oscillated jet for values up to 3.5 times... 

It has been shown recently [25] that concentrations of NOj, tend to reduce with increase in the amplitude of discrete-frequency oscillations. The mechanisms remain uncertain, but may be associated with the imposition of a near-sine wave on a skewed Gaussian distribution with consequent reduction in the residence time at the adiabatic flame temperature. Profiles of NO, concentrations in the exit plane of the burner are shown in Fig. 19.6 as a function of the amplitude of oscillations with active control used to regulate the amplitude of pressure oscillations. At an overall equivalence ratio of 0.7, the reduction in the antinodal RMS pressure fluctuation by 12 dB, from around 4 kPa to 1 kPa by the oscillation of fuel in the pilot stream, led to an increase of around 5% in the spatial mean value of NO, compared with a difference of the order of 20% with control by the oscillation of the pressure field in the experiments of [25]. The smaller net increase in NO, emissions in the present flow may be attributed to an increase in NOj due to the reduction in pressure fluctuations that is partly offset by a decrease in NOj, due to the oscillation of fuel on either side of stoichiometry at the centre of the duct. 

Figure 19.6 Influence of control by oscillation of fuel in inner duct on NOx emissions annular flow arrangement Um = Up = 7.5 m/s Re = 40,000 p = 1.1 4>j = 2.5 (a) effect of phase on amplitude of oscillation (6) exit profiles of NOx for three conditions of control 1 — RMS pressure 4.0 kPa, 2 — 1.0 kPa, dashed line corresponds to the case without control...

The flammability and stability limits of Fig. 19.7 were obtained using fuel-air mixtures with the same equivalence ratio in the radial and tangential inlets, and without an axial jet. The lean flammability limit decreased from 0.57 to 0.4 as the swirl number was increased from 0.6 to 3.75, and the region of high-heat release moved closer to the swirler which represented an acoustic pressure antinode for the naturally occurring oscillations associated with a quarter wave in the entire duct, with frequency close to 200 Hz. Thus, swirl led to an increase in the amplitude of oscillations and to an earlier transition from smooth to rough combustion with antinodal RMS pressures up to 10 kPa, and initiated at an equivalence ratio of 0.5 for a swirl number of 3.75... 

Figure 19.7 also shows that the amplitude of oscillations decreased with equivalence ratios greater than around 0.8 for swirl numbers up to 1.35, and at smaller values of equivalence ratio for larger swirl numbers. This is in contrast with results for ducted flames behind steps and bluff bodies, where the amplitude is nearly always a maximum near stoichiometry. This appears to be due to a shift in the location of flame stabilization by up to 50 mm, from close to the exit of the swirler to the end of the expansion section, since the amplitude of oscillations depends strongly on the intensity of heat release near the acoustic pressure antinode. This shift in flame location may have been related to the movement of the flame attachment with pressure oscillations. 

The needle valve oscillated up to 12% of the total fuel, and flow rates larger than around 3% of the total implied that the bulk mean velocity in the axial jet was greater than the bulk flow in the swirler and the mean equivalence ratio was greater than unity. It was found that the amplitude of oscillations was unaffected by values of bulk mean velocity of the axial jet greater than 2.5 times that in the swirler for a main flow swirl number of 1.35, and 4 times that value for a swirl number of 0.6. Larger values of axial jet velocity led to a decrease in amplitude due to the penetration of the swirl-induced recirculation region by the jet and the consequences for the distribution of heat release. 

Control was implemented by the oscillation of fuel in the axial jet. Tests with a wide range of flow conditions showed that the oscillation of more than around 6% of the total fuel flow did not result in improved attenuation of the oscillations, except for small overall equivalence ratios around 0.65 for which the larger flow rates of fuel in the axial jet led to an increase in amplitude of oscillations and the oscillation of that flow resulted in an increase in attenuation. 

The results of Fig. 19.8 for a swirl number of 1.35 show that the attenuation increased to 10 dB with the velocity of the axial jet up to 42 m/s, and further increase to 47 m/s caused the amplitude to fall from around 6 kPa to less than 1.5 kPa and the attenuation to decrease from 10 dB to almost zero. Similar results were observed with the swirl number of 0.6 the attenuation improved with axial jet velocity up to 60 m/s, after which the amplitude and attenuation decreased. The decline in the amplitude of oscillation and its attenuation by active control was due to the interaction between the axial jet with a large velocity and the central recirculation zone, which caused the flame to move further downstream of the swirler and heat release to occur further from the pressure antinode. The consequent increase in the distance between the point of entry of the oscillated fuel and the active burning zone reduced the effectiveness of the oscillated input due to increased fluid dynamic damping and development of a large difference in phase between different parts of the oscillated flow, especially with swirl surrounding the oscillated axial jet. 

Figure 19.11 shows that, for an overall equivalence ratio of 0.73 and a swirl number of 0.6, the amplitude of oscillation increased with the proportion of oscillated fuel due to the unpremixedness caused by the higher value of mean fuel concentration in the oscillated flow. The attenuation also increased with the fuel flow to around 5.5 dB with oscillation of around 10% of the fuel, compared with around 7 dB by the oscillation of 7% of the fuel in the axial jet and the same overall equivalence ratio and swirl number. As expected, control was less effective with higher swirl numbers due to the greater dissipation of the oscillated input. 

The tendency of premixed flames to detach from the flame holder to stabilize further downstream has also been reported close to the flammability limit in a two-dimensional sudden expansion flow [27]. The change in flame position in the present annular flow arrangement was a consequence of flow oscillations associated with rough combustion, and the flame can be particularly susceptible to detachment and possible extinction, especially at values of equivalence ratio close to the lean flammability limit. Measurements of extinction in opposed jet flames subject to pressure oscillations [28] show that a number of cycles of local flame extinction and relight were required before the flame finally blew off. The number of cycles over which the extinction process occurred depended on the frequency and amplitude of the oscillated input and the equivalence ratios in the opposed jets. Thus the onset of large amplitudes of oscillations in the lean combustor is not likely to lead to instantaneous blow-off, and the availability of a control mechanism to respond to the naturally occurring oscillations at their onset can slow down the progress towards total extinction and restore a stable flame. 

Pressure oscillations in the first arrangement depended on the equivalence ratio of the flow in the annulus and decreased with velocities in the pilot stream greater than that in the main flow due to decrease in size of the recirculation zone behind the annular ring and its deflection towards the wall. Increase in swirl number of the second arrangement caused the lean flammability limit to decrease, and the pressure oscillations to increase at smaller values of equivalence ratio. Unpremixedness associated with large fuel concentrations at the centre of the duct increased the pressure oscillations. Pressure oscillations caused the position of flame attachment to move downstream in both flows with a decrease in amplitude of oscillations. 

At Re = 130, a weak long-period oscillation appears in the tip of the wake (T2). Its amplitude increases with Re, but the flow behind the attached wake remains laminar to Re above 200. The amplitude of oscillation at the tip reaches 10% of the sphere diameter at Re = 270 (GIO). At about this Re, large vortices, associated with pulsations of the fluid circulating in the wake, periodically form and move downstream (S6). Vortex shedding appears to result from flow instability, originating in the free surface layer and moving downstream to affect the position of the wake tip (Rll, R12, S6). 

The upper bound of the region of stable steady motion is shown in Fig. 6.7 as a function of (CdRcx ) and /. For large /, secondary motion starts at Rcy = 100, i.e., (Cd Rcj ) = 23.4. At lower /, steady motion persists to higher Rcy the boundary shows a maximum at Rcy = 172, (CdRcj ) = 32.6 for/ = 8 X 10 Three kinds of secondary motion have been observed (S8), although the distinctions between them are not sharp. Immediately above the transition to unsteady motion, a disk shows regular oscillations about a diameter the amplitude of oscillation and of the associated horizontal motion increases with... 

In general, oscillations may be oblate-prolate (H8, S5), oblate-spherical, or oblate-less oblate (E2, FI, H8, R3, R4, S5). Correlations of the amplitude of fluctuation have been given (R3, S5), but these are at best approximate since the amplitude varies erratically as noted above. For low M systems, secondary motion may become marked, leading to what has been described as random wobbling (E2, S4, Wl). There appears to have been little systematic work on oscillations of liquid drops in gases. Such oscillations have been observed (FI, M4) and undoubtedly influence drag as noted earlier in this chapter. Measurements (Y3) for 3-6 mm water drops in air show that the amplitude of oscillation increases with while the frequency is initially close to the Lamb value (Eq. 7-30) but decays with distance of fall. 

It was shown by [51] that the film energy is a function of the particle diameter for a fixed value of volume fraction. The amplitude of oscillation in the... 

Tumbling regime At very low shear rates, the birefringence axis (or the director) of the nematic solution tumbles continuously up to a reduced shear rate T < 9.5. While the time for complete rotation stays approximately equal to that calculated from Eq. (85), the scalar order parameter S,dy) oscillates around its equilibrium value S. Maximum positive departures of S(dy) from S occur at 0 n/4 and — 3n/4, and maximum negative departures at 0 x — k/4 and — 5it/4, while the amplitude of oscillation increases with increasing T. 

Fig. 4. NOx reduction behavior on Pd/alumina catalyst as a function of oscillation periods and amplitudes in an engine test. Engine 2L, l,600rpm and —440 Torr catalyst Pd 0.05g/L. A/F amplitude of oscillation 0.4(A), 0.7(0) and 1.0(D).

Figure 10.2 — Mechanical interpretation of the interaction between a light wave and a polar bond. Upon interaction of a bond with light of the same frequency, the amplitude of oscillation is changed, not the mechanical frequency.

A role of other parameters of the model is investigated by Kuzovkov [26], It is demonstrated that an increase of the ratio a//3 for a fixed lj0 = (cr/3)1/2 and the control parameter k acts to accelerate a change of the focal regime for chaotic. Simultaneously, the amplitudes of oscillations in concentration for particles of different kinds are no longer close. A study of the stochastic Lotka-Volterra model performed here shows that irregular concentration motion observed experimentally in the Belousov-Zhabotinsky systems [8] indeed could take place in a system with mono- and bimolecular stages and two intermediate products only. 

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