Abstract: The Wilberforce pendulum is a special coupled system,which is a combination of a vertically suspended cylindrical helical spring and an object fixed at the end of it. The block has two degrees of freedom: the displacement stretching along the central axis of the spring the angular displacement twisting around the central axis of the spring. Its translational kinetic energy and rotational kinetic energy can transform periodically during the viberation at a special set of parameters. The Wilberforce pendulum in question 12 of iypt2021 is studied. In this work，the Lagrange function of the system is given, and the motion equation describing the coupling phenomenon is obtained. Using the natural coordinate system of cylindrical helix, the model of cylindrical helix spring is established. From the perspective of theory of elasticity, the spring model is analyzed by microelement. The geometric meaning of the curvature and torsion of the cylindrical helix is connected with the physical meaning of the spring material strain, and the analytical solution of the spring elastic potential energy is obtained. The mathematical calculation of this method is strict and universal, and can be extended to all kinds of precision calculation related to homogeneous spring mechanics. The static experiment is also carried out, and the theoretical value of the elastic coefficient falls within the range of the synthetic standard uncertainty of the experimental value, which verifies the correctness of the cylindrical coil spring model.

Key words:  cylindrical helical spring model, Wilberforce pendulum, coupled system, Theory of elasticity, The neutral surface