Reference frame theory

In the sequel, some general issues concerning AC-machines modeling, mainly change of variables through coordinate transformation (reference-frame theory, see [8]) are discussed. This is followed by the presentation of machine schematics in machine variables and equivalent circuits in transformed variables of both the synchronous and the induction machines, under detailed modeling assumptions. This is accompanied by the corresponding BG models of both machines in transformed variables. Finally, simplified models of the induction motor usually encountered in control system applications are addressed. 

After this formal discussion of chemical diffusion, let us now turn to some more practical aspects. In order to compare the formal theory with experiment, we have to carefully define the reference frame for the diffusion process, which is not trivial in the case of binary or multicomponent diffusion. To become acquainted with the philosophy of this problem, we deal briefly with defining a suitable reference frame in a binary system. Since only one (independent) transport coefficient is needed to describe chemical diffusion in a binary system, then according to Eqn. (4.57) we have in a one-dimensional system... 

Fig. 9. Plot of normalized approach to equilibrium mass against the square root of time for a temperature-sensitive 10 x 4 PNIPAAm gel sheet swelling and shrinking between 10 and 25 °C-Shown are the curve fits to the kinetic data of theory developed from Fick s law of diffusion in a polymer-fixed reference frame [149]. The equilibrium degree of swelling is 17.0 at 10 °C and 11.1 at 25 °C the diffusion coefficients obtained from the curve fits are 2.3 x 10 7 cm2/s for swelling and 3.6 x 10 7 cm2/s for shrinking [121]...

Whether the volume transition occurs with respect to temperature, salt or solvent composition, each model works well. Furthermore, neither model outperforms the other in terms of statistical quality of fit of data to theory. Also, the diffusion coefficients obtained from either model are normally comparable, even if a correction for the difference in reference frames is applied [119, 121, 153]. Theoretically, the value of D obtained from the different models differ significantly only for very large volume changes. Thus, if the desire is to correlate different experiments or to reduce kinetic data to a single parameter either model can be used satisfactorily. 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

Linear viscoelastic behavior is actually observed with polymers only in very restricted circumstances involving homogeneous, isotropic, amorphous specimens subjected to small strains at temperatures near or above Tg and under test conditions that are far removed from those in which the sample may be broken. Linear viscoelasticity theory is of limited use in predicting service behavior of polymeric articles, because such applications often involve large strains, anisotropic objects, fracture phenomena, and other effects which result in nonlinear behavior. The theory is nevertheless valuable as a reference frame for a wide range of applications, just as the thermodynamic equations for ideal solutions help organize the observed behavior of real solutions. 

Michelson interferometer—An instrument designed to divide a beam of visible light into two beams which travel along different paths until they recombine for observation of the interference fringes that are produced. Interferometers are used to make precision measurements of distances. Special relativity—The part of Einstein s theory of relativity that deals only with nonaccelerating (inertial) reference frames. 

The suggestion in view is that when volume is lost by diffusive mass transfer, the consequent shortening rate along some direction n is controlled by regardless of the spatial variations in other stress components. The nature of the argument advanced is comparable with the one on which the theory of relativity is based At two separate points in a universe, it is not reasonable to suppose that the fundamental laws of behavior will be different at one point from the other. If it is only in respect to some reference frame set up by an observer that point P differs from point Q, one should not expect behavior at P to differ from behavior at Q. It is convenient to use anthropomorphic phrasing If there is nothing intrinsic about point P to tell the material there to behave differently, the material at P will behave in the same way as the material at Q. ... 

The set of transformations of the spacetime coordinates that project the laws of electrodynamics from any observer s reference frame to any other (continuously connected) inertial frame such that the laws remain unchanged is the symmetry group of the theory of special relativity. It was discovered that this is... 

The idea of covariance is then that the same set of spacetime transformations that leave the differential metric (13) in special relativity, or (14) in general relativity, unchanged (invariant) also leave all the laws of nature covariant (unchanged in form) under these transformations between reference frames. The metric (13) in special relativity, or (14) in general relativity, then guides one to the forms of the covariant laws of nature, in accordance with the theory of (special or general) relativity. This is the role of the differential metrics—they are not to be considered as observables on their own ... 

In section 2, the formalism behind the separation of nuclear and electronic systems is sketched. The theory leading to an approach complementing the BO scheme used hitherto is introduced. The driving idea is to eliminate the use of the dynamical coordinates in defining reference frames. A rigged BO scheme is obtained where a one-to-one mapping between chemical species and electronic... 

The integration of this set of coupled first-order differential equation can be done in a number of ways. Care must be taken since there are basically rather two different time scales involved, i.e. that of the nuclear dynamics and that of the normally considerably faster electron dynamics. It should be observed that this END takes place in a Cartesian laboratory reference frame, which means that the overall translation as well as overall rotation of the molecular system is included. This offers no complications since the equations of motion satisfy basic conservation laws and, thus, total momentum and angular momentum are conserved. At any time in the evolution of the molecular system can the overall translation be isolated and eliminated if so should be deemed necessary. This level of theory [16,19] is implemented in the program system ENDyne [20], and has been applied to atomic and molecular reactive collisions. Calculations of cross sections, differential as well as integral, yield results in excellent agreement with the best experiments. 

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