A rational dimension reduction method based on a new type of complex modal analysis is developed in order to accurately analyze nonlinear vibrations generated in large-scale structures with local strong nonlinearity, global weak nonlinearity and non-proportional damping at low computation cost. In the proposed method, first, the state variables of weakly nonlinear nodes are transformed into modal coordinates using complex constrained modes obtained by fixing strongly nonlinear nodes. Next, a reduced model is derived by selecting a small number of modal coordinates that have a significant effect on the computational accuracy of the solution, and coupling them with the state variables of strongly nonlinear nodes expressed in physical coordinates. In that process, the remaining modal coordinates that have little effect on the computational accuracy are appropriately approximated and integrated into the equations of motion for strongly nonlinear nodes as correction terms. Furthermore, in order to improve computational efficiency, the global weakly nonlinear forces are directly computed from modal coordinates. It was confirmed that periodic solutions and their stability can be computed from the reduced model constructed by these procedures with a very high computational accuracy and at a high computational speed.