Trajectory-based modal veering, thermally induced branch reordering, and mode-weighted active control of graphene origami-reinforced joined conical–cylindrical shells

This study develops an integrated modeling and active-control framework for the parametric instability of graphene-origami (GOri)-reinforced joined conical–cylindrical shells, focusing on three coupled aspects: trajectory-based characterization of modal veering, temperature-induced modal loss and branch reordering, and mode-dependent redistribution of piezoelectric control authority. The structural formulation employs higher-order shear deformation theory with von Kármán nonlinearity and a curvature-consistent geometric stiffness that fully preserves membrane–bending coupling across the cone–cylinder junction. Surface-bonded PZT-4 patches are incorporated as fully electromechanically coupled layers, delivering stiffness-level actuation rather than equivalent force inputs. Parametric resonance under harmonic axial excitation is mapped using Bolotin’s method, and a POD-reduced model is incorporated into a mode-weighted linear quadratic regulator (LQR) to determine how mass fraction and folding degree of GOri, and temperature jointly modify modal interaction, instability boundaries, and piezoelectric control authority. The veering trajectory analysis demonstrates that increasing GOri folding systematically shifts the onset of modal interaction: the critical GOri mass fraction required for strong coupling drops by 30% (from 1.23% to 0.85%) as the folding degree rises from 0.3 to 0.9, while the minimum spectral gap narrows by 18%. Mode-shape evolution and strain-energy localization confirm that folding does not merely scale frequencies but actively restructures the coupling topology. Temperature introduces a more profound alteration: at 400 K, the first eigenvalue crosses into negativity, triggering mode-selective thermal instability, followed at elevated temperatures by replacement of the lowest stable modal branch, as verified through signed-eigenvalue tracking and mass-weighted modal assurance criterion (MAC) assignment. Under closed-loop control, the mode-weighted LQR achieves robust stabilization with moderate voltages (≤500 V), widening near-coalescent frequency gaps by three orders of magnitude (e.g., from 0.0041 Hz open-loop to 8.95 Hz closed-loop). Most notably, veering provokes a dramatic reallocation of control authority across both modes and actuators, revealing that conventional uniform-gain strategies are fundamentally inadequate; effective stabilization requires mode-weighted, multi-input architectures. Collectively, this trajectory-anchored framework merges folding-induced heterogeneity, modal hybridization, and smart actuation into a unified material–control co-design platform for lightweight shell structures.