Standing waves can exist as stable vibrating patterns in perfect structures, such as spherical shells. Any imperfections, such as mass-stiffness anisotropy, destroy the standing waves. In this paper we consider internal prestress, which stipulates a stiffness asymmetry and hence a natural frequency splitting in the structure. Damping inhomogeneity, which is phenomenologically similar to stiffness asymmetry, is introduced. Frequency splitting generates beats of the vibrating pattern and substantially affects the precession of standing waves because of inertial rotation of the sphere. The combined effects of frequency, damping splitting and rotation are considered in a special system of variables. This system is introduced by means of the introduction of principal (or main) and quadrature vibrating patterns, the angle of their rotation and the phase shift of the vibrating pattern. It is shown that, in the absence of frequency splitting, the energy of the original standing wave is maintained in the principal pattern and inertial rotation does not influence its amplitude. At the beats, the energy of the original standing wave is periodically transformed into a quadrature wave, while the phase shift and the rotation angle are also subjected to this periodic influence. If the inertial rate of rotation is less than or equal to the frequency splitting, "the capture effect" of the first kind is observed. If the inertial angular rate is less than or equal to the splitting of the damping coefficient, "the capture effect" of the second kind is observed. In the absence of "the capture effect", the vibrating pattern rotates relative to the rotating structure according to "Bryan's effect" with a rate that is less than the inertial angular rate of the structure. "The capture effects" consist of rotation of the vibrating pattern with the angular rate of the structure. These effects are analyzed by the method of averaging.
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