On dualities in the oscillations of naturally curved and pretwisted rods

The pertinent equations of naturally curved and pretwisted rods, in the form of compatibility, equilibrium and constitutive relations are obtained under the assumptins of infinitesimal deformations and material isotropy. Then by forming the expressions for various energy terms, the equations of motion of the rod are obtained via Hamilton's principle and the complementary energy principle. On comparing these two forms of equations of motion, and the associated boundary conditions certain dualities are exposed. Finally the equations of some special rods, including the plane arch and the straight pretwisted rod, are examined.