Nonlinear vibration, stability, and bifurcation of rotating axially moving conical shells

The nonlinear vibration characteristics of rotating axially moving conical shells are investigated in the current paper. The nonlinear equations of motion and strain compatibility equation based on Donnell’s nonlinear shell theory are obtained. Three nonlinear equations of motion are reduced to a radial equation by applying the appropriate Airy stress function, forming a set of equations with the compatibility equation. The compatibility equation is solved by employing the seven degrees of freedom with respect to the system’s flexural mode shape. By substituting the flexural mode shape into the equation of motion and applying the Galerkin method, seven nonlinear coupled nonhomogeneous ODEs are achieved, then the set of equations is transformed into the normal form where it has been solved by the numerical method. The effects of the axial and rotational velocity on bifurcation diagrams, frequency response curves, time history, and the phase portraits of the system are discussed. The results of the present paper are validated against available data, and good agreements are achieved.