In this work, we review a recently developed method for the characterization and identification of strongly nonlinear dynamical systems, including the detection of strongly nonlinear modal interactions, directly from transient response data. The method synergistically combines the proper orthogonal decomposition and the Rayleigh quotient to create estimated frequency-energy plots (FEPs) that capture the rich and interesting nonlinear dynamical interactions. The method is first applied to the experimentally measured response of a cantilever beam with a local, smooth nonlinearity. In this application, the estimated FEP reveals the presence of nonsmooth perturbations that connect different nonlinear normal modes (NNMs) of the system. The wavelet-bounded empirical mode decomposition and slow-flow analysis are used to demonstrate that the nonsmooth perturbations correspond to strongly nonlinear internal resonances between two NNMs. In the second example, the method is applied to the experimentally measured response of a cantilever beam with a local, nonlinear attachment in the form of a nonlinear energy sink (NES). An estimated frequency-displacement plot for the NES is created, and an optimization routine is then used to identify the unknown parameters for a given model of the nonlinearity. Ultimately, the method is conceptually and computationally simple compared to traditional methods while providing significant insight into the nonlinear physics governing dynamical systems with strong, local nonlinearity directly from measured time series data.