Forces frequency, oscillation

X = Spring displacement at time, t Zst = Static spring deflection under constant load, Fo Cl) = Forced frequency ci)o = Natural frequency of the oscillation t = Time... 

FolK = Deflection of the spring under load, (also called static deflection, Zst) a> = Forced frequency ft)jj = Natural frequency of the oscillation = Frequency ratio... 

The outputs of the sensors were used in two closed-loop control strategies developed for combustor performance optimization [7]. The objective of the first strategy, based on an adaptive least-mean squares (LMS) algorithm, was to maximize the magnitude and coherence of temperature oscillations at the forcing frequency /o in the measured region. The LMS algorithm was used to determine... 

Note 5 Under a forced flexural oscillation the resonance frequency... 

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

A monograph of Shchelkin Troshin (Ref 11) is based on studies at the Institute of Chemical Physics, Academy of Science, Moscow during the period 1952-1962, of theoretical analyses of detonation, deflagration, flame acceleration, nonsteady-state double discontinuities, and high-frequency oscillations in forced-combustion chambers. The book has been translated into English (See Ref 11a)... 

Dynamic melt viscosity studies on the star blocks and a similar triblock were carried out using a Rheometric Mechanical Spectrometer (RMS) (Rheometrics 800). Circular molded samples with -1.5 mm thickness and 2 cm diameter were subjected to forced sinusoidal oscillations (2% strain) between two parallel plates. The experiment was set in the frequency sweep mode. Data were collected at 180 and 210 °C. 

The dynamic melt viscosity measurements of select star blocks and a similar triblock were carried out on a rheometric mechanical spectrometer, RMS. Circular molded samples of 2 cm diameter and -1.5 mm thickness were subjected to forced sinusoidal oscillations. Dynamic viscosities were recorded in the frequency range of 0.01-100 rad/s at 180 °C. Figure 10 shows the complex viscosities of two select star blocks and a similar linear triblock. The plots showed characteristic behavior of thermoplastic elastomers, i.e., absence of Newtonian behavior even in the low frequency region. The complex viscosity of the star block... 

If vibration is applied both through the cantilever tip (at frequency heterodyne detection. The AFM tip detects the oscillating force at the difference frequency cot — cos, very much like a heterodyne radio receiver. This technique is known as heterodyne force microscopy (HFM Cuberes et al. 2000). Once again, the tip-surface force non-linearity plays a critical role. The low-frequency beating oscillation carries information on the phase of the original high-frequency oscillations. 

Fig. 13.9. The forced Takoudis-Schmidt-Aris model with a forcing frequency twice that of the natural oscillation (a) zero-amplitude forcing (autonomous oscillation and limit cycle) (b) r, = 0.002 (c) r, = 0.003 (d) r, = 0.004 (e) r, = 0.005 (f) rr = 0.006 (g) rf = 0.007 (h) rf = 0.01. Traces show the time series 0p(t) over 10 natural periods (or 20 forcing periods) and the associated limit cycle in the 0 -6, plane.

In the previous subsection, the forcing frequency was exactly twice the natural oscillatory frequency. Thus the motion around one oscillation gives exactly two circuits of the forcing cycle for one revolution of the natural limit cycle. The full oscillation of the forced system has the same period as the autonomous cycle and twice the forcing period. The concentrations 0p and 6r return to exactly the same point at the top of the cycle, and subsequent oscillatory cycles follow the same close path across the toroidal surface. This is known as phase locking or resonance. We can expect such locking, with a closed loop on the torus, whenever the ratio of the natural and forcing... 

A very rich scenario was, on the other hand, observed with Pt(110) if perturbed under conditions for which regular autonomous oscillations with well-defined frequencies existed (91-93). Typically, these autonomous oscillations were established at fixed external parameters and then one of the partial pressures was periodically modulated by use of a feedback-regulated gas inlet system with frequencies up to 0.5 s l and relative amplitudes around 1% (31, 33). Following the pioneering mathematical treatment of forced nonlinear oscillations by Kai and Tomita (SO), the results can be rationalized in terms of a dynamic phase diagram characterizing the response of the system as a function of the amplitude A and of the period of the pressure modulation Tcx with respect to that of the... 

Reaction rate oscillations may be accompanied by temperature oscillations [temperature fluctuations of up to 500 K have been reported (24)] or they may be isothermal. Isothermality occurs either because the catalyst can conduct heat away much faster than the rate at which it is produced by the reaction, as is the case in UHV studies, or because isothermal conditions are forced on the system by anemometry, as described in the work of Luss and co-workers (757). Oscillation frequencies can range from more than 10 Hz (24) up to periods of several hours (217,219). Often there is evidence for several time scales in a single oscillating stem. Relatively regular high-frequency oscillations may be superimposed over relaxation oscillations (93,98), with the two types of oscillations caused by different changes on the catalyst surface. 

Cantilevers in AFM function as force transducers converting unknown force to measurable deflection. The value of the unknown force can then be expressed by Hookean mechanics following spring constant calibrations. In addition to static point loads, cantilevers can also be vibrated, e.g., by an oscillation piezo to which the fixed end of the beam is attached (or by other approaches). Excitation frequency, oscillation amplitude, and phase relationships are variables that govern dynamic tapping (intermittent contact) imaging. This problem will be discussed in the next section. 

Fig. 1.16 Average tip—sample force and oscillation frequency as a function of reduced amplitude (setpoint). The dashed resonance corresponds to a damped driven oscillator without sample—tip interactions. Reprinted from [16], copyright American Physical Society...

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