Dynamic analysis of a rotating planar Timoshenko beam using an accurate global spatial discretization method

A new formulation is developed for dynamic analysis of a rotating planar Timoshenko beam. The configuration of Timoshenko beam is described using its slope angle and axial and shear strains; hence, the shear locking problem can be naturally avoided. Nonlinear partial differential equations of the rotating hub–beam system and associated boundary conditions are derived using Hamilton's principle. While six boundary conditions are needed for choices of trial functions of three dependent variables, there are only four boundary conditions that can be determined and two boundary conditions are undetermined. An accurate global spatial discretization method is used, where dependent variables are divided into internal and boundary-induced terms. Internal terms only need to satisfy homogeneous boundary conditions, which can be easily chosen as trigonometric functions. Boundary-induced terms are interpolated using dependent variables at boundaries that are taken as generalized coordinates. Nonlinear ordinary differential governing equations of the system are obtained using Lagrange's equations. When the hub rotates at a constant angular velocity, nonlinear governing equations can be linearized for vibration analysis, and dimensionless vibration equations of the beam are obtained. Natural frequencies and mode shapes of the beam with a constant angular velocity are calculated and compared with available results in the literature. Frequency veering and mode shift phenomena occur. Nonlinear dynamic responses of the system are then calculated and compared with those from the commercial software ADAMS, and they are in good agreement. Axial and shear strains of the beam and their spatial derivatives are also calculated. Since trial functions in the assumed modes method cannot satisfy undetermined boundary conditions, inaccurate results of strains and their spatial derivatives are obtained using the assumed modes method. Hence, use of the accurate global spatial discretization method in the current formulation is essential.