Non-stationary nonlinear analysis of a composite rotating shaft passing through critical speed

Abstract

In this paper, nonlinear non-stationary dynamics of a nonlinear composite shaft passing through critical speed is studied. The nonlinearity is due to the large amplitude of shaft vibration. The equations of motion are obtained by three-dimensional constitutive relationships of composite materials. The gyroscopic effect, rotary inertia and coupling caused by material anisotropy are considered but shear deformation is neglected. Without any simplification, axial-flexural-flexural-torsional equations of motion (EOM) for the elastic composite shaft with variable rotational speed are obtained. The approximate analytical method namely asymptotic method is applied to analyze the nonstationary behavior of the composite shaft with constant acceleration. First, the EOMs are discretized using one and two-term Galerkin method. Then, the resulted equations are transformed to normal coordinates. Finally, the asymptotic method is applied to equations described in normal coordinates. Analytical expressions governing the amplitude and phase of motion during passage through critical speeds are obtained. By comparing the results obtained from analytical solutions, it is shown that discretization by one mode is not enough due to the existence of coupling in the equations and at least two modes are necessary for this purpose. Effects of damping, eccentricity, initial angular velocity and fiber angle on response amplitude are investigated. For verification, the results of perturbation theory are compared with numerical simulations and it is shown that there is good agreement between both methods.