A comprehensive study on the coupled multi-mode vibrations of cylindrical shells

As is discussed in the previous work (Dong et al. 2021), the functions of beam-like modes are not suitable for simulating the cylindrical shells with the clamped and free ends. Also, ignoring the influences of the in-plane inertias can lead to large relative differences of the solutions of vibration characteristics. This work aims to accurately calculate the vibration characteristics of the cylindrical shells under various classical boundary conditions comprised of movable simply-supported, immovable simply-supported, sliding, clamped and free ends. Meanwhile, this work deals with the nonlinear coupled multi-mode vibrations of the cylindrical shell under different boundary conditions with the in-plane inertias taken into consideration. Firstly, several modified Chebyshev polynomials in terms of the axial coordinate and new harmonic functions in terms of the circumferential coordinate are employed to establish the equations of natural frequency in the frame of Lagrange equations for all the single-modes, where the polynomials are independent of geometries and material properties of the shells. And then, the circumferential-wave-dependent mode functions of the cylindrical shells are proposed, based on these mode functions coming from the linear vibration, a unified nonlinear differential equation of motion of the cylindrical shells under different boundary conditions is established for the first endeavor, in which the coupled multi-mode vibrations containing various vibration waves are considered. Finally, the high dimensional differential equation with the square and cubic nonlinearities is investigated by employing the incremental harmonic balance method. Results shown that polynomials accurately satisfy various boundary conditions, and the contribution of the single-mode to the coupled multi-mode vibration is related to the loading patterns.