Nonlinear dynamics near internal resonance in a beam-based resonator with spatially separated modal motions

Multimode mechanical resonators can exhibit strong nonlinear modal interactions that significantly influence response amplitude, stability, and effective operating range. Among these interactions, internal resonance is of particular interest because it can induce energy redistribution, modal activation, and complex bifurcation behaviour. For studies aimed at resolving activation of coupled modes and excitation-direction-dependent interaction, resonator platforms in which the dominant motions of the participating modes remain spatially distinguishable are especially advantageous. In this work, internal resonance is investigated theoretically and experimentally in a beam-based resonator with spatially separated modal motions. The first two modes are predominantly associated with the lateral beams and the cross beam, respectively, enabling the modal frequency ratio to be tuned close to 1:2 while preserving clear physical distinguishability of the modal motions. A continuous model is established and reduced to a two-degree-of-freedom nonlinear model, and a continuation method is used to determine the response branches, stability, and the associated bifurcation structure. Stepped steady-state frequency sweep experiments and single-frequency measurements are then conducted to examine the predicted responses. Under excitation at the first mode, the strongly coupled response is realized on a single-period branch, and quasi-periodic motion is observed experimentally in the central interaction region at higher excitation levels. Under excitation at the second mode, the stable single-period branch loses stability through period doubling, and the strongly coupled response is realized on a doubled-period branch. The experiments show good agreement with the theoretical predictions in terms of the main response structure, hysteretic behaviour, coupled-response region, and phase-space characteristics. These results clarify how the coupled response is realized under different excitation directions in this class of beam-based resonators with spatially separated modal motions.