For the relative rotation dynamic equation of the cylinder between random twain cross sections, we gain an analytic solution for the equation by using the coupling method. Due to the particularity of the equation, we gain an invertible matrix by utilizing the Jordan canonical. Thereby, we obtain an integral solution for the relative rotation dynamic equation of the cylinder. According to the reguirements in engineering applications, we give some analytic solutions of two types of load action, i.e.the impact and periodic types.
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