The estimation of a model of the foundation of a rotary machine has been recently attempted by using the difference between two sets of response data at some of the bearing locations from two consecutive rundowns of the machine, with and without known unbalance weights at certain positions on the two balance discs of each rotor respectively. However, it would be a great advantage to be able to perform the estimation with a single rundown. Due to practical restrictions in performing such tests (accessibility, costs etc.), there are cases in which data for only one rundown are available. In this case, the unbalance configuration is unknown and has, therefore, to be estimated, in addition to the unknown foundation model. Due to the special form of the unbalance force, this overall inverse problem can be solved by eliminating the unbalance configuration from the model estimation process. The remaining equation to estimate the foundation model consists of the projection of the response data, where the associated projector depends on the foundation model parameter. First results using the method, applied to a laboratory test rig and to a commercial turbo-generator, are presented.