The given book is devoted to the solution of the problem which almost doesn’t have any researches. It is also devoted to the research of dynamics of shells rotating on supports. These constructions are widely used in different branches of industry, and on the stage of designing the mathematical means are necessary for the analysis of their behavior. In the monograph the author presented theoretical bases of mathematical modeling of dynamics of shells rotating on supports. The feature of the analysis of such systems is the necessity of accounting of precessional movement of the stationary waves, raised under the influence of revolting factors. In the book you can find the way of accounting of these features in the equations of dynamics of shells; theorems of possible character of precessional movements of the raised stationary waves are proved here. The material is a basis for the decision of a more difficult problem - the creation of the theory of multibasic rotating shells. The monograph is intended for science workers and experts who are engaged in researches of questions of dynamics and in working out the designs containing shells rotating on supports.