Modal analysis of non-diagonalizable continuous systems with application to wind turbine blades

This work represents an investigation of the modal analysis of distributed parameter systems whose stiffness or damping terms are non-diagonalizable with an undamped modal-coordinate transformation. The non-diagonalizability may be caused by nonmodal damping or stiffness that includes parametric excitation. The modal properties for these kinds of problems will be investigated.;An approach for analyzing the complex modes of continuous systems with nonmodal damping is first developed. As an example, a cantilevered beam with damping at the free end is studied. Assumed modes are applied to discretize the eigenvalue problem in state-variable form, to then obtain estimates of the natural frequencies and state-variable modal vectors. The finite-element method is also used to get the mass, stiffness, and damping matrices for the state-variable eigenvalue problem. A comparison between the complex modes and eigenvalues obtained from the assumed-mode analysis and the finite-element analysis shows that the methods produce consistent results. The assumed-mode method is then used to study the effects of the end-damping coefficient on the estimated normal modes and modal damping. Most modes remain underdamped regardless of the end-damping coefficient. There is an optimal end-damping coefficient for vibration decay, which correlates with the maximum modal nonsynchronicity.;As an experimental example of a non-modally damped continuous system, an end-damped cantilevered beam is studied for its complex modal behavior. An eddy-current damper is applied considering its noncontact and linear properties. The state-variable modal decomposition method (SVMD) is applied to extract the modes from impact responses. Characteristics of the mode shapes and modal damping are examined for various values of the damping coefficient. The eigenvalues and mode shapes obtained from the experiments are consistent with the numerical analysis of the model, although there is variation relative to sampling parameters. Over the range of damping coefficients studied in the experiments, we observe a maximum damping ratio in the lowest underdamped mode, which correlates with the maximum modal nonsynchronicity.;The vibration model of a horizontal-axis wind turbine blade can be approximated as a rotating pretwisted nonsymmetric beam, with damping and gravitational and aeroelastic loading. The out-of-plane (flapwise) and in-plane (edgewise) motion of a wind turbine blade are examined with simple aeroelastic damping effects. Hamilton's principle is applied to derive the in-plane and out-of-plane equations of motion, and the partial differential equation is linearized and then discretized by the assumed-mode method. A simple quasi-steady blade-element airfoil theory is applied to obtain the aeroelastic damping. The analysis is performed on three blades of different size. The effects of nonproportional damping are not strong, but are seen to become more significant as the blade size increases. The results provide some experience for the validity of making modal damping assumptions in blade analyses.;A perturbation approach is developed to analyze the perturbation effect of the parametric excitation on the unperturbed linear modes. The method is applied in the examples of a three-mass system and a wind turbine blade. In wind-turbine blades, the parametric excitation has a weak effect on the non-resonant unperturbed linear modal responses.