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Uncertainty quantification (UQ) is a well-established category of methods to estimate the effect of parameter variations on a quantity of interest based on a solid mathematical foundation. In the wind energy field most UQ studies focus on the sensitivity of turbine loads. This article presents a framework, wrapped around a modern Python UQ library, to analyze the impact of uncertain turbine properties on aeroelastic stability. The UQ methodology applies a polynomial chaos expansion surrogate model. A comparison is made between different wind turbine simulation tools on the engineering model level (alaska/Wind, Bladed, HAWC2/HAWCStab2, and Simpack). Two case studies are used to demonstrate the effectiveness of the method to analyze the sensitivity of the aeroelastic damping of an unstable turbine mode to variations of structural blade cross-section parameters. The code-to-code comparison shows good agreement between the simulation tools for the reference model, but also significant differences in the sensitivities.