Torsional oscillation of vessels containing quantum fluids are one of oldest and most popular methods for the study of quantized vortices. One recent and very brilliant example is the discovery of the supersolidity of solid helium. In torsional oscillation experiments, a drop in the period of the oscillations is observed when some low temperature is reached. This effect has been attributed to the appearance of the superfluid component. It depends on many factors and has various explanations. But, assuming (at least hypothetically, at this stage) that the phenomenon of "supersolidity" (dissipationless flow) does occur, we must consider the relaxation of a vortex system (we can call it a vortex tangle, vortex fluid, chaotic set of vortices, etc.). This is necessary because the only way to involve the superfluid component in rotation is through polarized vortices (with nonzero mean polarization along their axis of rotation). Here we consider a vortex tangle relaxation model for the torsional oscillation response of quantum systems, with the aim of using it to study solid ~4He. It is shown that the rotation of the superfluid component occurs as a relaxation effect with a relaxation time that depends on the amplitude of the oscillations (as well as on temperature and pressure). This problem has a quasi-linear solution which explains the (amplitude dependent) shift in the period. There is also an imaginary shift of the frequency (also amplitude dependent), which represents an additional dissipation. The theoretical results are compared with recent measurements.
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