Summary: An elastic helicoidal structure modelled as a plate twisted around its axis is studied in this paper. Accurate strain-displacement relationships for the shell are derived by the Green strain tensor in general shell theory and first-order shear deformation theory. An energy equilibrium equation of free vibration is introduced by the principle of virtual work. Applying the Rayleigh-Ritz method, an analytical eigenvalue equation is formulated and solved via an efficient computational approach for vibration characteristics of the helicoidal structure. A set of normalized orthogonal polynomials generated by the Gram-Schmidt procedure is presented to approximate the admissible functions. The first polynomial is taken as a kinematically compliant geometric equation of boundary conditions of the shell. The convergence and the accuracy of the present method, and the effects of geometric parameters and boundary conditions on vibration of the helicoidal structure are investigated.