**6. Conclusions**

In this chapter we introduce a vibration method which is suitable to analyze the stability of a constrained elastica. A planar elastica constrained by a space-fixed point constraint is used to demonstrate the method. Generally speaking, static analysis allows one to find all the possible equilibrium configurations of a constrained elastica. In order to predict how the elastica behaves in reality, the stability of these equilibrium configurations needs to be determined. The key of the vibration method is to take into account the sliding between the elastica and the unilateral constraint during vibration. In order to accomplish this, Eulerian coordinates are defined to specify the positions of the material points on the elastica. After transforming the governing equations and the boundary conditions from the Lagrangian description to the Eulerian one, the natural frequencies and the vibration mode shapes of the constrained elastica can be calculated. The vibration method is applied to an elastica constrained by a point constraint in this chapter. The same principles can be extended to other similar problems as well, for instance; multiple point constraints (Chen et al., 2010) and plane constraints (Ro et al., 2010).
