An approach to development of reduced order models for systems with local nonlinearities is presented. The key of this approach is the augmentation of conventional basis functions with others having appropriate discontinuities at the locations of nonlinearity. A Galerkin solution using the above combination of basis functions appears to capture the dynamics of the system very efficiently—employing small basis sets. This method is particularly useful for problems of structural dynamics, but may have application in other fields as well. For problems involving small amplitude dynamics, when one employs as a basis the eigenmodes of a reference linear system plus the discontinuous (joint) modes, the resulting predictions, though still nonlinear, are approximated well as linear combinations of the eigenmodes. This is in good agreement with the experimental observation that jointed structures, though demonstrably nonlinear, manifest kinematics that are well described using eigenmodes of a corresponding system where the joints are replaced by linear springs.