Resonance Problem

1 Resonance Problem. - In the asymptotic region the equation (1) effeetively reduces to 

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The heightened appreciation of resonance problems, in particular, has been quite recent [63, 62], and contrasts the more systematic error associated with numerical stability that grows systematically with the discretization size. Ironically, resonance artifacts are worse in the modern impulse multiple-timestep methods, formulated to be symplectic and reversible the earlier extrapolative variants were abandoned due to energy drifts. 

A fan blade is continuously vibrating millions of cycles up and down ia operatioa over a short period of time. Each time a blade tip moves past an obstmction it is loaded and then unloaded. If forced by virtue of tip speed and number of blades to vibrate at its natural frequency, the ampHtude is greatly iacreased and internal stresses result. It is very important when selecting or rating a fan to avoid operation near the natural frequency. The most common method of checking for a resonance problem is by usiag the relatioa ... 

These pumps have the following disadvantages intermediate or line bearings are generahy required when the shaft length exceeds about 3 m (10 ft) in order to avoid shaft resonance problems these... 

Noise and resonance problems (Section 23.5.2(C)) in the electrical distribution and communication networks. 

As illustrated in Figure 44.42, a resonance peak represents a large amount of energy. This energy is the result of both the amplitude of the peak and the broad area under the peak. This combination of high peak amplitude and broad-based energy content is typical of most resonance problems. The damping system associated with a resonance frequency is indicated by the sharpness or width of the response curve, ci) , when measured at the half-power point. i MAX is the maximum resonance and Rmax/V is the half-power point for a typical resonance-response curve. 

CAS F calculations are not a universal solution to symmetry-breaking of the wave functions, and for such weak resonance problems it is far more reliable to start from state average solutions which treat on an equal footing the two configurations which interact weakly. 

Ignoring the damping term in Eq. (17), the resonance is described by dM/dt = KMxHeff). For homogeneously magnetized ellipsoids of revolution, the effective field is equal to the applied field H = H ez plus the anisotropy field Ha, and the resonance problem is solved by the diagonalization of a 2x2 matrix. This uniform or ferromagnetic resonance (FMR) yields resonance frequencies determined by [17]... 

L. D. Hall, Solutions to the hidden resonance problem in proton nuclear magnetic resonance spectroscopy, Adv. Carbohydr. Chem., 29 (1974) 11 -0. 

The resonator problems that we have discussed are of limited interest until we couple to a source and load in order to examine the response of... 

The above problem is the so-called resonance problem when the positive eigen-energies lie under the potential barrier. We solve this problem, using the technique fully described in refs. 12, 26, 30-32. 

The numerical results obtained for the five methods, with several number of function evaluations (NFE), were compared with the anal5hic solution of the Woods-Saxon potential resonance problem, rounded to six decimal places. Fig. 20 show the errors Err = -logic calculated - analytical of the highest eigenenergy 3 = 989.701916 for several values of NFE (Fig. 21-23). 

Following the analysis of section 6, we will solve the so-called resonance problem. This problem consists either of finding the phase-shift 5/ or finding those E, for... 

E e [1,1000], at which 5i =. We actually solve the latter problem, known as the resonance problem when the positive eigenenergies lie under the potential barrier. The boundary conditions for this problem are ... 

We compute the approximate positive eigenenergies of the Woods-Saxon resonance problem using ... 

In order to illustrated their efficiency, we have applied the new methods to the resonance problem of the one-dimensional Schrodinger equation. 

For these methods numerical results on the resonance problem of the radial Schrodinger equation are given and analysed. 

In Section 4 we present Four-step P-stable Methods with minimal Phase-Lag. We give a new procedure for the construction of such methods. This procedure is based on the requirement that the roots of the characteristic equation associated with the methods must have specific forms. For these methods numerical results on the resonance problem of the radial Schrodinger equation are given and analysed. 

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