Damping parameter

Methods for calculating undamped and damped critical speeds that closely follow the works of Prohl and Lund, respectively, are listed herein. Computer programs can be developed that use the equations shown in this section to provide estimations of the critical speeds of a given rotor for a range of bearing stiffness and damping parameters. 

In Fig. 4 we compare the adiabatic (dotted line) and the stabilized standard spectral densities (continuous line) for three values of the anharmonic coupling parameter and for the same damping parameter. Comparison shows that for a0 1, the adiabatic lineshapes are almost the same as those obtained by the exact approach. For aG = 1.5, this lineshape escapes from the exact one. That shows that for ac > 1, the adiabatic corrections becomes sensitive. However, it may be observed by inspection of the bottom spectra of Fig. 4, that if one takes for the adiabatic approach co0o = 165cm 1 and aG = 1.4, the adiabatic lineshape simulates sensitively the standard one obtained with go,, = 150 cm-1 and ac = 1.5. 

It is of importance to note that we shall consider, in the present section, that the fast and bending modes are subject to the same quantitative damping. Indeed, the damping parameter of the fast mode yG and that of the bending mode y will be supposed to be equal, so that we shall use in the following a single parameter, namely y (= yG = y5). This drastic restriction cannot be avoided when going beyond the adiabatic approximation. 

We must stress that the use of a single damping parameter y supposes that the relaxations of the fast and bending modes have the same magnitude. A more general treatment of damping has been proposed [22,23,71,72] however, this treatment (discussed in Section IV.D) requires the use of the adiabatic approximation, so that its application is limited to very weak hydrogen bonds. 

It was stated in Section IV that undertaking a nonadiabatic treatment prevents us from using different damping parameters for the fast and bending modes (i.e., Yo 7 Ys)- We shall see in the present section the role played by these damping... 

The account for the relaxations in (100) and (101) was made through the damping parameters yG of the fast mode and that ys of the bending mode. The... 

Figure 11. Pure Fermi coupling within the exchange approximation relative influence of the damping parameters. Common parameters

Of course, varying the damping parameters allows the bandshapes to evolve between these three limiting cases. 

Figure 12. Hydrogen bond involving a Fermi resonance damping parameters switching the intensities. The lineshapes were computed within the adiabatic and exchange approximations. Intensities balancing between two sub-bands are observed when modifying the damping parameters (a) with y0 =0.1, and y5 = 0.8 (b) with ya = ys = 0.8 (c) with yB — 0.8 and ys — 0.1. Common parameters oto = 1, A = 150cm, 2g)5 = 2850cm-1, and T = 30 K.

Figure 13. Hydrogen bond involving a Fermi resonance relative influence of the damping parameters. Spectral densities 7sf(co) computed from Eq. (81). Common parameters a0 = 1, A = 160cm-1, co0 = 3000cm-1, co00 = 150cm-1, 2t05 = 2790cm-1, and T = 300K.

Figure 15 gives the superposition of RR (full line) and RY (dotted plot) spectral densities at 300 K. For the RR spectral density, the anharmonic coupling parameter and the direct damping parameter were taken as unity (a0 = 1, y0 = ffioo), in order to get a broadened lineshape involving reasonable half-width (a = 1 was used systematically, for instance, in Ref. 72). For the RY spectral density, the corresponding parameters were chosen aD = 1.29, y00 = 0.85angular frequency shift (the RY model fails to obtain the low-frequency shift predicted by the RR model) and a suitable adjustment in the intensities that are irrelevant in the RR and RY models. 

The damping parameter is large enough to allow the smoothing of the fine structure. 

When pK > 4(32 holds, the singular point remains stable, Reei, 2 < 0, but the roots (2.1.16) have imaginary parts Imei = Im ei. In this case the phase portrait reveals a stable focus - Fig. 2.2. This regime results in damped oscillations around the equilibrium point (2.1.24). The damping parameter pK/(3 is small, for large 3, in which case the concentration oscillation frequency is just ui = y/pK. ... 

Figure 4.1. Comparison of the resonance fields as functions of the damping parameter in the Gilbert (1) and Landau-Lifshitz (2) representations.

In the last equation, y is the usual indirect damping parameter, whereas F° (t) is the fluctuating force resulting from the action of the surrounding molecules on the oscillator, the average of which and its corresponding ACF, are given, respectively, by... 

As seen, this SD is very near the semiclassical one (174) except for the presence of the decay term e >r 1 in place of the simple one e y° . The behavior of the additional decay term e n is the same as that of the direct damping that is e 7, although this additional term is of indirect nature, T being a function of the indirect damping parameter y via Eq. (183). 

Figure 8. Spectral analysis involving direct and indirect dampings at T = 300 K. The direct damping parameter has been chosen greater (y° = 0.25 f ) when the indirect damping is missing, than (Y° — 0.025Si) when it is present (y — 0.1 SI) in order to distinguish clearly the spectral densities. Dirac delta peaks are corresponding to the situation without any damping. co° = 3000 cm-1,...

The situations in Fig. 9a and c deal with approximate SDs computed without fitting the damping parameters, y° and y, whereas those in Fig. 9b and d are concerning approximate SDs computed with the aid of effective damping parameters, y°eff and yeff chosen in such a way as to get the best fit with the exact ones. 

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