Abstract

Free lateral vibration of stepped shafts is investigated in this paper using the Timoshenko beam theory and the finite element method. Beam finite elements having two nodes and 16 degrees of freedom were employed to model flexural vibration of a stepped shaft for a total four field variables — two lateral displacements and two bending angles. Within each uniform segment, the stepped shaft is modeled as a substructure for which a system of equations of motion may be easily formulated using the Galerkin method. The global equations of motion for the entire stepped shaft are subsequently formulated by enforcing the displacement continuity and force equilibrium conditions across the interfaces between two adjacent substructures. The second order governing differential equations for a non self-adjoint dynamic system are then reduced to the equivalent first order differential equations for which eigenvalue problem is formulated and solved using the Matlab® program. Values of natural frequencies are in excellent agreement with those available in the literature. Effects of rotational springs attached to the end of a stepped shaft, used to simulate the non-classical boundary constraints of chuck on a work piece in a typical turning process, are also investigated. The bi-orthogonal conditions for modal vectors, which are useful in chatter analysis during turning processes, are given in this paper.

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