The harmonic analysis often used for the analysis of axisymmetric bodies under asymmetric loads is shown to be efficient for elastic contact problems. The contact surface is assumed frictionless and unbonded. By a Fourier series expansion of displacements in the circumferential direction, the contact problem is formulated as a sequence of decoupled partial differential equations with inequalities induced from contact conditions on the contact surface. For solution, an equivalent minimization problem with inequality constraints is formulated as a sum of decoupled contributions from each harmonic.
An axisymmetric finite element modelling is adopted as a numerical approximation. Potential contact nodes along the nodal circumferences are defined to impose the contact conditions. The resulting finite element discretization of the equivalent minimization problem becomes a quadratic programming problem. A dual problem formulation in terms of contact force is used for programming.
Based on the present formulation, the effect of various geometric and loading factors on the contact problem solution is studied. The resulting formulas are useful for practical design applications, minimizing the number of analyses which depends on the number of design factors.
Two examples are solved to test the proposed method and to compare accuracy and computational time with the full three dimensional finite element analysis. In the first example of two dimensional contact problem of pin and piston rod, the computational time is saved by 43%. Comparison of rigid body displacements and maximum stress components is shown to be in agreement within 14% difference. The second example is a three dimensional contact problem of conical fit assembly. The computational time is reduced by factors of 1/6 to 1/9. The contact region is predicted well, but rigid body displacements and maximum stress components show somewhat large difference of up to 26%.
The numerical difference between harmonic and full three dimensional finite element analysis comes from many sources, mainly from coarse mesh division for both analyses and the way that the stress is recovered. It is concluded however that the present harmonic analysis will give better results when a similar mesh is used because of smoothness of displacement representation.
A detailed study of conical fit assembly in terms of cone angle and its misfit, and clearance or interference, shows relative changes in contact region and stress concentration at both ends. Although friction should be considered to have an absolute meaning of the magnitudes obtained, the study has presented important preliminary data for design purposes.
