Abstract:

Natural frequencies and the corresponding modes of an elastically supported cantilever Timoshenko beam carrying a tip mass partly immersed in water are investigated. Taking into account the effects of the cross-section rotation, shear distortion and compression axial force due to the tip mass, the mathematical model for free vibration of a cantilever beam is presented, in which constraints of the elastic horizontal displacement and elastic rotation are coupled. Due to the inertial force and inertial moment of the tip mass, boundary conditions of the mathematical model are frequency dependent. The frequency equation and the generalized orthogonal condition for the vibration modes are established. Influences of tip mass,moment of inertia, location of tip mass and coupled springstiffness coefficient, etc., on the natural frequencies are presented numerically and discussed. It is shown that, compared with the model of Euler-Bernoulli beam, due to the crosssection rotation and shear distortion of the beam, the natural frequencies of the Timoshenko beam decrease, especially for the higher order natural frequencies. The addition of coupled spring-stiffness coefficient will reduce the natural frequencies to different extent, and the axial force reduces the lower order natural frequencies while has a negligible influence on the higher order ones.

Key words: Timoshenko beam; natural frequency; orthogonality condition; coupled stiffness