Magnetic stiffness

The rotordynamic study of a machine with magnetic bearings is quite different from the one of either oil-film or rolling-element bearings. The stiffness and damping properties of magnetic bearings can... 

Improved rotor dynamic stability. The active magnetic bearing s ability to vary stiffness and damping permits rotation about the rotor s inertial axis, eliminating vibration and noise. 

T. E. Oliphant, A. Manduca, R. L. Ehman and J. F. Greenleaf, Complex-valued stiffness reconstruction for magnetic resonance elastography by algebraic inversion of the differential equation, Magn. Reson. Med., 2001, 45, 299-310. 

By using two traps, it is possible to maintain a constant force [91]. This is also possible with magnetic tweezers. However, because of the low stiffness of the magnetic trap, the spatial resolution due to thermal flucmations is limited to a few tens of nanometers. 

Aromatic ring currents[45, 46] or the charge stiffness[47] provide a related application of correction vectors, now associated with magnetic fields. Ring currents are the diamagnetic susceptibility of 4n+2 systems. The correction vector in monocycles is obtained from... 

Figure 3. Exchange energy as a function of the wave vector of the magnetization inhomogenity Lindhard function (solid line) and exchange-stiffness or continuum approximation (dashed line). For late 3d elements (Cu), k/2kf = 1 corresponds to modulation wavelength of 0.23 nm...

In noncubic materials, A must be replaced by the 3x3 exchange-stiffness tensor A v, and the energy is X J A v dM/dxy dM/dxv dV. Here the indices ju and v denote the spatial coordinates x, y, and z of the bonds. The energy is anisotropic with respect to the nabla operator = d/d (bond anisotropy) but isotropic with respect to the magnetization M. By contrast, the relativistic anisotropic exchange Xa i Ha(i VMa VMP dV is isotropic with respect to V but anisotropic with respect to M. 

Here Ms(r) is the spontaneous magnetization, A)(r) is the first uniaxial anisotropy constant, A(r) denotes the exchange stiffness, and n(r) is the unit vector of the local anisotropy direction. H is the external magnetic field, and Hd is the magnetostatic self-interaction field. The latter can be written as... 

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