The Transfer Matrix Method for the Vibration of Compressed Helical Springs

The partial differential equations of motion are obtained for a helical spring subject to a static axial force, the typical element of the spring having six degrees of freedom. The wire cross-section can be any doubly symmetrical shape. The overall transfer matrix is calculated and its application is discussed for obtaining the response to forced sinusoidal vibration. Natural frequencies are found from the transfer matrix by iteration. Comparisons are made with published experiments on the natural frequencies of helical springs, made from round wire, with and without a static axial force. Comparison is also made with published theory for the static buckling of helical springs. Information is given on the effect on the natural frequencies of the static axial force, helix angle, number of active turns, ratio of helix to wire diameter, Poisson's ratio, shear coefficient, and the end conditions. The calculation of the normal modes is discussed.