Some aspects of non-linear free vibration of structures behaving dissimilarly in tension and in compression are elucidated. It is shown that, for bimodular structures, the character of the vibration is governed by the position of the breakpoint in the restoring force versus displacement graph: when the breakpoint is outside the co-ordinate origin, the vibration is non-isochronous, i.e., dependent on the non-linear vibration amplitude; when the breakpoint is at the origin, there is no such dependence. An exact solution is given for the case in which the graph consists of a straight-line portion and a cubic-curve portion. In addition, for both problems mentioned, a bilinear approximation is used in conjunction with a modification of Panovko's direct linearization method. This modification both retains the non-isochronicity property and agrees quite well with the exact solution. © 1991.