Torsional oscillation of the vessels with quantum fluids is one of oldest and most popular methods for the study of quantized vortices. The recent and very bright example is the discovery of the supersolidity of the solid helium. In the torsion oscillation experiments the drop in the period of oscillations with achievement of some small temperature has been observed. This effect was attributed to the appearance of the superfluid component. This phenomenon depends on many various factors and has various explanations. But, if to adopt (at least hypothetically, at this stage) that the phenomenon of "supersolidity" (dissipativeless flow) is realized, we must consider the relaxation of the vortex system (we can call it as vortex tangle, vortex fluid, chaotic set of vortices, etc.). We have to do it for the very simple reason, that the only way to involve the superfluid component into rotation is the presence of the polarized vortices (with nonzero mean polarization along the axis of rotation). In the present work we submit the approach describing the vortex tangle relaxation model for the torsional oscillation responses of quantum systems, having in mind to apply it for the study of solid <^>4He. It is shown that the rotation of the superfluid component occurs in the relaxation-like manner with the relaxation time dependent on the amplitude of oscillation (as well as on the temperature and pressure). The study of this problem shows that there is a quasi-linear solution explaining the (amplitude dependent) shift of period. There is also an imaginary shift of the frequency (also the amplitude-dependent), which describes an additional dissipation. The results of the theory are compared with the recent measurements.

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