Static and dynamic analysis of conical helices based on exact geometry via mixed FEM

Ermis M., OMURTAG M. H.

International Journal of Mechanical Sciences, vol.131-132, pp.296-304, 2017 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 131-132
  • Publication Date: 2017
  • Doi Number: 10.1016/j.ijmecsci.2017.07.010
  • Journal Name: International Journal of Mechanical Sciences
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.296-304
  • Keywords: Archimedean spiral, Conical helix, Exact geometry, Lancert helix, Logarithmic spiral, Mixed finite element
  • Istanbul Medipol University Affiliated: Yes


The first objective of this research is to indicate the steps that are necessary to generate an exact non-circular (Lancert type) helix geometry as an alternative to the widely used approximate geometrical functions obtained by exploiting exact circular helix geometry. A parametric analysis is performed in order to determine the range of the geometric parameters in which the approximate geometry is insufficient to provide the precision required by exact geometry. This investigation is considered important since the validity of approximate helix geometry is satisfied in a limited range of geometric parameters. The exact conical helix geometry is based on associated plane curves, namely either the Archimedean spiral or logarithmic spiral. The second objective of this research is to determine the necessary conditions to use the Archimedean spiral and logarithmic spiral to supply nearly the same geometry of the conical helix. The final aim is to compare the numerical performance of curved mixed type finite elements with straight displacement type elements over exact conical helix geometry. This is based on the Archimedean spiral. For this purpose, static and free vibration analyses are performed. Curved mixed finite element formulation is based on Timoshenko beam theory which considers shear angle and rotary inertias. The degrees of freedom at a node is twelve: three translational displacements, three cross-sectional rotations, three force components, two bending moments and torque. Approaching the exact geometry of the conical helix with approximate geometry assumptions is only possible under some limitations of the taper ratio and the number of active turns of the conical helix. In free vibration analysis, the mixed-curved elements are considerably faster than straight displacement type elements. In the case of static analysis, convergence performance the two different FE formulations differ in terms of boundary conditions.