During dynamic atomic force microscopy (AFM), the deflection of a scanning cantilever generates multiple frequency terms due to the nonlinear nature of AFM tip-sample interactions. Even though each frequency term is reasonably expected to encode information about the sample, only the fundamental frequency term is typically decoded to provide topographic mapping of the measured surface. One of main reasons for discarding higher harmonic signals is their low signal-to-noise ratio. Here, we introduce a new design concept for multi-harmonic AFM, exploiting intentional nonlinear internal resonance for the enhancement of higher harmonics. The nonlinear internal resonance, triggered by the non-smooth tip-sample dynamic interactions, results in nonlinear energy transfers from the directly excited fundamental bending mode to the higher-frequency mode and, hence, enhancement of the higher harmonic of the measured response. It is verified through detailed theoretical and experimental study that this AFM design can robustly incorporate the required internal resonance and enable high-frequency AFM measurements. Measurements on an inhomogeneous polymer specimen demonstrate the efficacy of the proposed design, namely that the higher harmonic of the measured response is capable of enhanced simultaneous topography imaging and compositional mapping, exhibiting less crosstalk with an abrupt height change.