The inconsistency of mass center and elastic center, i.e., eccentricity, widely exists in various engineering structures and components, including iced conductors, wind turbine blades, airfoils, and cable-supported bridges, and can also be extended to the application of energy harvesting and vibration control. Although the eccentricity influence on aerodynamic stability has already been recognized as important, the underlying mechanism remains rather unclear. This study established a comprehensive framework of explicit eigenvalue solutions, offering in-depth insights into the aerodynamic stability mechanism in a vertical-horizontal-torsional 3-degree-of-freedom linear system under all possible frequency scenarios. These solutions elucidated the coupling mechanism of eccentricity, aerodynamic damping, aerodynamic stiffness, and structural frequencies. Remarkably straightforward criteria to predict the promoting/suppressing effect of eccentricity were proposed, consisting of eccentric angle and aerodynamic forces, which not only align with the traditional principles but also offer a broader application range in terms of structure types and wind speeds. Validations of the proposed solutions were performed for all frequency cases via numerical studies. For instability tendency evaluation, numerical studies on a thin plate showed that eccentricity has a significant impact due to the strong aerodynamic effect and large ratio of width/gyration radius, highlighting the importance of considering small eccentricity errors during experiments. Examination for energy harvesting revealed that a small eccentricity can dramatically enhance power generation efficiency, and the explicit solutions successfully explained the rules governing performance changes against frequency ratio, mass center position, rotation center position, etc. The analysis of a bundled conductor showed that the double pendulum can control galloping because it diminishes the eccentricity effect by shifting the eccentric angle. The proposed explicit solution framework provides a systematic view and new insights into the aerodynamic instability mechanism influenced by eccentricity, serving as a reference for the design and studies of various engineering structures, energy harvesting, and vibration control.