Extension of Kirchhoff’s Kinetic Analogy to Cylindrical Springs Carrying Distributed Loads

The use of cylindrical spiral springs as flexible connectors, screws and tools for milling and sifting different materials leads to complex geometric non-linear boundary value problems in mechanics of flexible rods. In general, such problems can only be solved by numerical methods, which make the results subject to errors that are difficult to control.  This paper proposes a method of estimating these errors by using the conservation law, based on a modification of Kirchhoff’s kinetic analogy. The identical relation, following the analogy of the equations for rigid body motion with one fixed point and the equations of large displacements in flexible rods, applies to the case of helical cylindrical springs carrying distributed loads. Accounting for the distributed loads significantly expands the range of application of Kirchhoff’s kinetic analogy. It is shown that the proposed modification results from the equilibrium equations and the ratio of elasticity of the spring. The feasibility of using the proposed methodology for controlling numerical calculation of large displacements of the springs carrying the distributed load is demonstrated by an example.

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