Free Vibration of Helical Springs Using a Dynamic Finite Element Mesh Reduction Technique


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Abstract


A Dynamic Finite Element (DFE) is developed to analyze the vibration characteristics of helical springs. The Dynamic Trigonometric Shape Functions (DTSF’s) of approximation space are developed from the exact solutions to the uncoupled equations governing axial and torsional vibrations of the system. By exploiting the principle of virtual work and the DTSF’s, the Dynamic Stiffness Matrix (DSM) of a uniform spring element is produced. The element matrices, exhibiting both mass and stiffness properties, are then assembled and the boundary conditions are applied to form the eigenproblem of the system. Based on the Sturm Sequence properties of the overall stiffness matrix, the well-known Wittrick-Williams algorithm is then used to evaluate the natural frequencies and modes of vibration of the system. Numerical checks are performed to confirm the accuracy and to ensure confidence for practical applicability and the performance of the DFE technique. The vibration characteristics of a uniform, cylindrical, helical spring with different boundary conditions are investigated. Numerical results on natural frequencies and convergence tests demonstrate the higher accuracy and superconvergent characteristics of the DFE and its superiority over the classical Finite Element Methods (FEM). Based on the numerical results, the DFE formulation can be justifiably called a Mesh Reduction Method (MRM).
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Keywords


Extension-Torsion Couplings; Dynamic Finite Element (DFE); Dynamic Stiffness Matrix (DSM); Finite Element Method (FEM); Geometrically Coupled Vibrations; Helical Springs

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