The Influence of Mass on Dynamic Response of Cracked Timoshenko Beam with Restrained End Conditions: The Truncated Theory

In this paper, the dynamic response of the Timoshenko cracked beam subjected to a mass is investigated. In turn, it is assumed that the beam has its ends restrained with both transverse and rotational elastic springs. Based on an alternative beam theory, truncated Timoshenko theory (TTT), the governing equations of motion of the cracked beam are derived and the influence of a mass on the behavior of free vibrations is investigated. The novelty of the proposed approach lies in the fact that the variational method used in the truncated theory simplifies the derivation of the equation of motion via the classical theory, and the perfect analogy between the two theories is shown. The objective of the present formulation lies in finding the equations of the truncated Timoshenko model with their corresponding boundary conditions and establishing their mathematical similarity with the geometric approach. It is shown that the differential equations with their corresponding boundary conditions, used to solve the dynamic problem of Timoshenko truncated beams through variational formulations, have the same form as those obtained through the direct method. Finally, some numerical examples are carried out to evaluate the influence of a mass and its position on the vibration performances of the cracked Timoshenko model. Additionally, the effects of the crack positions, the shear deformation and rotational inertia, and the yielding constraints on the natural frequencies are also discussed in some numerical examples.