Classical force appropriation methods are used in the identi"cation of linear systems to determine the multi-point force vector that will induce single-mode behaviour, thus allowing each normal mode to be identi"ed in isolation. This paper presents an extension to this linear approach that will enable the approximate identi"cation of multi-degree-of-freedom (d.o.f.) systems with weak non-linearity on a similar basis. The classical linear modal model is used, with additional terms included to represent the direct non-linear restoring forces of the system. Using this force appropriation for non-linear systems (FANS) method, a force vector with harmonics present is derived using an optimization approach such that the response of the system is restricted to that of a target mode, but in the non-linear region. The response obtained from several force levels is then curve "tted using the restoring force method applied in linear modal space so as to yield the direct linear and non-linear modal parameters for the target mode. The method is applied to a simulated two-d.o.f. example and good agreement is found between estimated and true parameters. An extension to the identi"cation of critical non-linear modal cross-coupling terms is proposed.