The extended and improve version of the investigation described in [1-2] and the firstly presented in  is proposed in this work. Distinguish features of these investigations are applications of the theory of multi-component dry friction [4-16] in the problem of the ball rolling on boundaries of two identical frames. Father development of this model can be used in investigation of the dynamic of the a so-called 'Butterfly' robot, consisting of two identical shaped plates rigidly placed parallel to each other on a small distance aimed at manipulating a ball that can freely roll on the plates' boundaries as on rails . The difficult control systems considered in [18-20] are another engineering application of the combined dry theory. Following developed in previous investigation approach, the friction force and torque are computed by the integration over the contact area so that the exact dynamically coupled integral model accounting the relationship of all the components of friction is obtained. This exact model is replaced by approximated analytical model [1-2] which is completely satisfy to all analytical properties of integral model as function kinematics parameters without increasing the number of coefficients.
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