Inverse problems constant frequency

The derivation of the elastic constant tensor to be used in RUS is an indirect iterative procedure. Provided the sample dimensions, density and elastic constants are known, the spectrum of resonant frequencies can be easily calculated. The inverse problem viz. calculating the elastic constants from a measured spectrum of mechanical resonances) has no known solution, however. For the indirect method, a starting resonant frequency spectrum, 7 (n = 1,2, ), can be calculated by elastic constants estimated from theory or from literature data for similar materials in addition to the known sample dimension and density. The difference between the calculated and measured resonance frequency spectrum, 7 (n = 1,2, ), is identified by a figure-of-merit function,... 

FIGURE 6.2.2 Comparison of the aggregation frequency from the inverse problems with the actual (constant) frequency for various values of the regularization parameter when the self-similar distribution is known exactly. Note that the most accurate estimate of the aggregation frequency is obtained with no regularization. (From Wright and Ramkrishna, 1992. Reprinted with permission from Elsevier Science.)... 

In practice, however, one has to solve the inverse vibrational problem (IVP), i.e. on the basis of known experimental frequencies ufxp for a given molecule to extract the values of the force-constants and thereby to... 

Finally, for this section we note that the valence interactions in Eq. [1] are either linear with respect to the force constants or can be made linear. For example, the harmonic approximation for the bond stretch, 0.5 (b - boV, is linear with respect to the force constant If a Morse function is chosen, then it is possible to linearize it by a Taylor expansion, etc. Even the dependence on the reference value bg can be transformed such that the force field has a linear term k, b - bo), where bo is predetermined and fixed, and is the parameter to be determined. The dependence of the energy function on the latter is linear. [After ko has been determined the bilinear form in b - bo) can be rearranged such that bo is modified and the term linear in b - bo) disappears.] Consequently, the fit of the force constants to the ab initio data can be transformed into a linear least-squares problem with respect to these parameters, and such a problem can be solved with one matrix inversion. This is to be distinguished from parameter optimizations with respect to experimental data such as frequencies that are, of course, complicated functions of the whole set of force constants and the molecular geometry. The linearity of the least-squares problem with respect to the ab initio data is a reflection of the point discussed in the previous section, which noted that the ab initio data are related to the functional form of empirical force fields more directly than the experimental data. A related advantage in this respect is that, when fitting the ab initio Hessian matrix and determining in this way the molecular normal modes and frequencies, one does not compare anharmonic and harmonic frequencies, as is usually done with respect to experimental results. 

The maximum torque depends on the strength of the magnetic field in the gap between the rotor and the coils on the stator. This depends on the current in the coils. A problem is that as the frequency increases the current reduces, if the voltage is constant, because of the inductance of the coils having an impedance that is proportional to the frequency. The result is that, if the inverter is fed from a fixed voltage, the maximum torque is inversely proportional to the speed. This is liable to be the case with a fuel cell system. 

The key issue in frequency spectroscopy is the knowledge of the quadratic force constants. In the past an enormous effort was made to obtain reliable sets of quadratic force constants conceptually by solving the inverse dynamical problem, i.e., using in Eq. (4) experimentally known frequencies and, as unknowns, the quadratic force constants [15,16]. 

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