Explicit solution framework and new insights of 3-DOF linear flutter considering various frequency relationships

Different engineering structures, e.g., long-span bridges, bundled conductors, and cable-supported photovoltaic modules, exhibit various frequency relationships among different degrees of freedom (DOFs). Despite extensive research, the initiation mechanisms of flutter remain somewhat ambiguous for a 3-DOF system considering heaving-lateral-torsional motions, especially when frequencies are close among different DOFs. This study introduced an explicit eigenvalue solution framework for tackling the 3-DOF linear flutter problem by utilizing matrix perturbation methods, enabling the extraction of modal damping and frequency solutions for all possible frequency scenarios. These solutions explicitly clarified the influence mechanism of all flutter derivatives (aerodynamic damping and stiffness), structural frequencies, and mechanical dampings. Important flutter derivatives, flutter risk level, and sensitivity level to mechanical damping and to frequency detuning were summarized in a table for all frequency scenarios, providing a panoramic perspective on flutter instability and the coupling mechanism. Numerical studies were conducted on a thin plate, the Akashi Kaikyo Bridge, and an eight-bundled conductor to examine the proposed explicit solutions. The results showed that approaching frequencies can amplify the coupling effect among different DOFs by half an order, but this does not necessarily mean the system is more prone to flutter. A system where the torsional frequency approximates the heaving/lateral frequency is expected to face a higher flutter risk if given significant torsional-related aerodynamic stiffness (H3*, P3*). Evidently, this high risk exhibits insensitivity to mechanical damping and small frequency-detuning. Though usually ignored in bridge flutter analysis, drag force and lateral motion could exert noticeable influence in a minority of cases, e.g., more than 10 m/s decrease of critical wind speed for the Akashi Kaikyo Bridge at large angles of attack. The proposed explicit solution framework provides a systematic view and new insights into the flutter initiation mechanism, serving as a reference in the design and studies of various engineering structures with different frequencies and coupling features.