# Linearization with OpenFAST, off-diagonal elements of the mass and stiffness matrix

Hello everyone,

I would like to inquire about an issue regarding obtaining the A matrix from linearization within OpenFAST.

I am currently conducting a linearization analysis on a wind turbine in parked state considering yaw misalignment. In the process of linearization analysis, with a yaw misalignment of 30 degrees and a wind speed of 30 m/s. Throughout the process of linearization analysis, a yaw misalignment of 30 degrees and a wind speed of 30 m/s have been considered. Specifically, only the first-order degrees of freedom in the tower’s alongwind and crosswind directions have been taken into account, with all other degrees of freedom being disabled. The state-space equation A of the wind turbine can be easily obtained through linearization. The A matrix I obtained is as follows:

I have noticed that the off-diagonal elements of the -M^(-1) K matrix are non-zero. My understanding is that, with other degrees of freedom disabled, the tower’s alongwind and crosswind directions should be decoupled. The paper “Modeling of the UAE Wind Turbine for Refinement of FAST_AD” also mentions the independence of the tower in the alongwind and crosswind directions.

Could you kindly explain why the off-diagonal elements of the -M^(-1) K matrix obtained through linearization are non-zero? Could you shed light on the underlying reasons for this phenomenon.

Thank you in advance.

Dear @Xing.Tan,

I’m not sure I fully understand your simulation set-up, but coupling between tower fore-aft and side-side motions could come from asymmetries in the tower-top inertia and aerodynamics. It may also arise through the geometric nonlinearities of the model, like radial shortening of the tower. Regardless, in the example you show, the off-diagonal elements in the -M^-1*K matrix are small relative to the diagonal elements, so, the coupling appears to be small in your case.

Best regards,