Natural vibration of pre-twisted shear deformable beam systems subject to multiple kinds of initial stresses

Abstract

Free vibration and buckling of pre-twisted beams exhibit interesting coupling phenomena between compression, moments and torque and have been the subject of extensive research due to their importance as models of wind turbines and helicopter rotor blades. The paper investigates the influence of multiple kinds of initial stresses due to compression, shears, moments and torque on the natural vibration of pre-twisted straight beam based on the Timoshenko theory. The derivation begins with the three-dimensional Green strain tensor. The nonlinear part of the strain tensor is expressed as a product of displacement gradient to derive the strain energy due to initial stresses. The Frenet formulae in differential geometry are employed to treat the pre-twist. The strain energy due to elasticity and the linear kinetic energy are obtained in classical sense. From the variational principle, the governing equations and the associated natural boundary conditions are derived. It is noted that the first mode increases together with the pre-twisted angle but the second decreases seeming to close the first two modes together for natural frequencies and compressions. The gaps close monotonically as the angle of twist increases for natural frequencies and buckling compressions. However, unlike natural frequencies and compressions, the closeness is not monotonic for buckling shears, moments and torques.