Forcing frequency

Bellows can vibrate, both from internal fluid flow and externally imposed mechanical vibrations. Internal flow liner sleeves prevent flow-induced resonance, which produces bellows fatigue failure in minutes at high flow velocities. Mechanically induced resonant vibration is avoided by a bellows with a natural frequency far away from the forcing frequency, if known. Multiple-ply bellows are less susceptible to vibration failure because of the damping effect of interply friction. 

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... 

In a damped forced vibration system such as the one shown in Figure 43.14, the motion of the mass M has two parts (1) the damped free vibration at the damped natural frequency and (2) the steady-state harmonic motions at the forcing frequency. The damped natural frequency component decays quickly, but the steady state harmonic associated with the external force remains as long as the energy force is present. 

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

Fatigue of muscles is found post-exercise and in some patients with disorders of limb or respiratory muscles. Peripheral muscle fatigue is generally characterized by the changes in force frequency relationships that occur. The process is traditionally divided into a failure of force production at either low or high frequencies of electrical stimulation. 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

Dispersion Forces, Frequency Shifts on Condensation, and the VPIE... 

Dynamic mechanical testers apply a small sinusoidal stress or strain to a small sample of the polymer to be examined and measure resonant frequency and damping versus temperature and forced frequency. Instrument software computes dynamic storage modulus (G ), dynamic loss modulus (G") and tan delta or damping factor. Measurements over a wide range of frequency and temperature provide a fingerprint of the polymer with sensitivity highly superior to DSC. 

Figure 7.4 Time averaged axial distribution of 15 /Lim particles when injected continuously and in and out of phase with the forcing frequency of 1380 Hz. 1 — continuous, 2 — in phase, and 3 — out of phase...

The air flow rate was set at 0.3 g/s, and the fuel flow rate was set at 0.9 g/s (above the rich flammability limit). The forcing frequency was chosen to be 592 Hz. This frequency was chosen to coincide with the natural frequency of the air chamber since the resonance would amplify the forcing signal. The amplitude of the forcing is 20% of the mean exit velocity at the nozzle. 

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... 

It increase force, frequency and duration of uterine contractions. It is used to control and prevent postpartum haemorrhage. It is also used to prevent uterine atony after cesarean or instrumental delivery. 

Here, r0 is the mean value of the dimensionless partial pressure, rf is the amplitude of the forcing, so r varies between r0 + r, and a> is the dimensionless forcing frequency (the forcing peiod if is given by In/co). 

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... 

Because of the enormous amount of integration required, computation costs on conventional computers would have been prohibitive and thus, all of our computations were performed on the Cray 1 and Cray 2 supercomputers at the University of Minnesota. The results are shown in figure 2 where the forcing frequency w has been scaled by the system s natural frequency co0... 

In the limiting case of very large-amplitude forcing, the system is completely dominated by the forcing and the only response possible is a stable limit cycle of period 1 (same period as the forcing). The amplitude at which this becomes the case varies greatly and seems to increase with the forcing frequency. 

The attractive force frequency shift, Ava is assumed to depend linearly on density solvent density (23)... 

Stimulus trains of varying lengths (0.5-1.2 s) and frequencies of 10-200 pps can provide additional information. As stimulus frequency is increased, there is greater tetanic fusion and higher peak forces are produced. A force-frequency curve (F-F) can also be generated to evaluate factors that affect muscle contractile speed (time-to-peak, half-relaxation). If, for example, disease processes lead to slower contraction/relaxation time, more tetanization occurs at lower frequencies and the shape of the F-F curve will be shifted compared to normal muscle. 

Georgakopoulos, D. and Kass, D. A. (2004). Minimal force-frequency modulation of inotropy and relaxation of in situ murine heart. J. Physiol. 15, 535-545. 

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