Natural frequencies of axial–torsional coupled motion in springs and composite bars

The natural frequencies corresponding to axial–torsional (extension–twist) coupled motion of a helical spring, or the corresponding motion induced through material coupling in a composite bar, are considered using an equivalent continuum approach. Closed form solution of the governing differential equations leads either to an exact dynamic stiffness matrix or to a number of exact relationships between the natural frequencies corresponding to coupled and uncoupled motion. The latter relationships both guarantee that the Wittrick–Williams root finding algorithm can still be used to converge on any required natural frequency, despite any lack of reciprocity arising from differential coupling, and for the case of symmetric material coupling coefficients, enable their value to be determined precisely from experimental results. A number of examples are then given to confirm the accuracy of the proposed theory and to indicate its range of application.