On the compliance of coiled springs

The 6×6 spatial compliance matrix for a helical spring is computed. The method used is to find the compliance of infinitesimal elements along the length of the spring. Then integration is used to sum the compliances and produce the total compliance of the spring. A key point in the method is that the compliance matrix for each element must be expressed in a common coordinate frame. The results produced are significantly different from the those given in elementary texts. However, it is shown how these results can be recovered by making the standard “closely coiled” approximation. Next, the principal wrenches and eigencompliances of the spring are studied, both for the closely coiled spring and for the case where these assumptions are not made. Finally, some numerical examples are given.