Force power spectrum

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

It is useful to define reduced frequencies <3 = a>/a>o and di = tojato. Sceats et aL have evaluated the mean force power spectrum of this poten-... 

Consider first the force power spectrum. At modest bath densities this is to be derived for particle pairs initially at R moving inward. The strong repulsive potential for R < Rp leads to a power sjjectrum 8/iBMk7BBM(o>) of a form discussed in Section III C (with di = 0 because the trajectories begin at Rp, not Rj, and do not exjjerience the acceleration effect). The delay in time between initiation at Rp and recoil is so small that collisional effects are... 

To measure the strength of the forces exerted on particles, various analytical techniques have been developed [6, 7]. Unfortunately, since most of these techniques are based on hydrodynamics, assumption of the potential profiles is required and the viscosities of the fiuid and the particle sizes must be precisely determined in separate experiments, for example, using the viscous flow technique [8,9] and power spectrum analysis of position fluctuation [10]. Furthermore, these methods provide information on ensemble averages for a mass of many particles. The sizes, shapes, and physical and chemical properties of individual particles may be different from each other, which will result in a variety of force strengths. Thus, single-particle... 

Dipole and force. In a study of nuclear electric dipole relaxation, Purcell pointed out that even at low gas densities, the force pulses experienced by a molecule cannot be treated as uncorrelated [328]. Instead, the correlation is such that the power spectrum of the net intermolecular force on the... 

Rohde, F., Normand, M.D. and Peleg, M. (1993). Characterisation of the power spectrum of force-deformation relationships of crunchy foods. J. Texture Studies 24,45-62. 

In the adiabatic regime, the solvent relaxation time rc reaction coordinate. This limit corresponds to (t) = 5(t), so the power spectrum (Eq. (11.87)) is equal to , that is, to white noise . The GLE is reduced to the simple Langevin equation with a time-local friction force — x. Xr is found from Eq. (11.85) ... 

Briefly returning to the Coulomb force theme of Section II, although not presented in Ref. 57, subsequent (unpublished) results have shown that Coulombic forces are dominant in the OH stretch relaxation. Figure 4 displays the results for the bend power spectrum for two different frequency ranges, including the contribution of Coulomb and non-Coulomb forces. From Fig. 4a, Coulomb forces clearly dominate over the full range of interest a similar behavior has also been found for the stretchings spectra (not shown).1 ... 

Ohmine has observed that the dissipation of the excess ethylene energy occurs considerably faster in water than it does in liquid Ar. He has shown that this rate of energy loss is consistent with the overlap between the velocity power spectrum of the ethylene motions and the power spectrum of the forces that the solvent exerts on the ethylene molecule. The water solvent is able to exert forces that have a much greater range of frequencies than do the forces from the Ar solvent, and is therefore able to dissipate energy more efficiently from the higher frequency motions of the ethylene molecule. (A similar effect can be seen in simulations of the vibrational relaxation of diatomic molecules in rare gas solution by Chesnoy and Weis o and by Whitnell, Wilson, and Hynes in water.13 135) Ohmine also found when the depth of the Ar-Ar attractive well was increased by a factor of 50, that the rate of dissipation of energy into the solvent increased markedly, as did the power spearum of the forces that the solvent exerts at all frequencies. 

In the case of the CFRP sample, the sound velocity V is about 3 x 10 m/s. Equation (31) gives the value of the distance L as 1.25 mm, which is close to the value of the distance between the Teflon sheet and the bottom surface of the CFRP sample tested here (1.2 mm). The power spectrum at this point C exhibits 10 MHz frequency intervals Af, which correspond to a peak at about 100 ns in the autocorrelation function in addition to the peak at about 0.8 ps. If the value of 1.2 X 10 m/s is employed for the sound velocity in the Teflon sheet, the peak at 100 ns results in the value of 60 pm for the theoretical thickness of the Teflon sheet this roughly agrees with the 80 pm thickness of the actual Teflon sheet. The difference of 20 pm is probably due to the increase in the sound velocity in the Teflon sheet or a decrease in the thickness of the actual Teflon sheet due to compressional force upon composite formation. 

Figure 4.11 shows a typical sample of the cutting force and noise from the micro-milling experiment. As can be seen, the SNR is very low and the absolute value of the noise is nearly one-third that of the force. The noise statistics are computed as mean = -0.1093, variance = 0.2631, skew = 0.0004, kurtosis = 1.2621. It is super-Gaussian, with a longer tail than Gaussian distribution and is well fitted with Laplacian distribution (see Figure 4.11b). The power spectrum density (PSD) and autocorrelation...

Figure 4.12 shows the three force signals and their corresponding power spectrum at medium flank wear in precision milling. As can be seen from Figure 4.12, the low-frequency components have relatively larger peaks... 

PRBS exciting forces are applied to the trial system, the forces and the responses are measured simultaneously. Through power spectrum cuialysis, the transfer functions of the system can be obtained, and by computer processing of these functions characteristics of the bearing can be deteimiined. 

The ice force records are digitized based on the Nyquist frequency criterion. Fluid-structure interaction is considered in an approximate manner using the frequency-independent added mass concept, i.e. added mass equal to the mass of water displaced. The equivalent evolutionary white noise, is based on the maximum peak of the averaged power spectrum and the evolutionary white noise is considered as a two-segment piece-wise linear function, as shown in Fig 3. The random excitation, Y(t), can be decomposed as follows ... 

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