Nonlinear vibration of stringer shell by means of extended Hamiltonian approach

In this paper, the nonlinear free vibration of a stringer shell is studied. The mathematical model of the string shell, which is the most convenient for frequency analysis, is considered. Due to the geometrical properties of the vibrating shell, strong nonlinearities are evident. Approximate analytical expressions for the nonlinear vibration are provided by introducing the extended version of the Hamiltonian approach. The method suggested in the paper gives the approximate solution for the differential equation with dissipative term for which the Lagrangian exists. The aim of this study is to provide engineers and designers with an easy method for determining the shell nonlinear vibration frequency and nonlinear behavior. The effects of different parameters on the ratio of nonlinear to linear natural frequency of shells are studied. This analytical representation gives excellent approximations to the numerical solutions for the whole range of the oscillation amplitude, reducing the respective error of the angular frequency in comparison with the Hamiltonian approach. This study shows that a first-order approximation of the Hamiltonian approach leads to highly accurate solutions that are valid for a wide range of vibration amplitudes.