scholarly journals Resonances of a Forced Mathieu Equation With Reference to Wind Turbine Blades

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Wind Turbine ◽  
Multiple Scales ◽  
Turbine Blades ◽  
Mathieu Equation ◽  
Axial Loading ◽  
Wind Turbine Blades ◽  
Horizontal Axis Wind Turbine ◽  
Response Level ◽  
Subharmonic Response ◽  
Steady Rotation

A horizontal axis wind turbine blade in steady rotation endures cyclic transverse loading due to wind shear, tower shadowing and gravity, and a cyclic gravitational axial loading at the same fundamental frequency. These direct and parametric excitations motivate the consideration of a forced Mathieu equation. This equation with cubic nonlinearity is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. The order-two superharmonic resonance persists for the linear system. While the order-two subharmonic response level is dependent on the ratio of parametric excitation and damping, nonlinearity is essential for the order-two subharmonic resonance. Order-three resonances are present in the system as well and, to first order, they are similar to those of the Duffing equation.

scholarly journals In-Plane Nonlinear Dynamics of Wind Turbine Blades

2011 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wind Turbine ◽  
Multiple Scales ◽  
Turbine Blades ◽  
Mathieu Equation ◽  
Plane Motion ◽  
Wind Turbine Blades ◽  
Partial Differential ◽  
Direct Excitation

The partial differential equation that governs the in-plane motion of a wind turbine blade subject to gravitational loading and which accommodates for aerodynamic loading is developed using the extended Hamilton principle. This partial differential equation includes nonlinear terms due to nonlinear curvature and nonlinear foreshortening, as well as parametric and direct excitation at the frequency of rotation. The equation is reduced using an assumed cantilevered beam mode to produce a single second-order ordinary differential equation (ODE) as an approximation for the case of constant rotation rate. Embedded in this ODE are terms of a nonlinear forced Mathieu equation. The forced Mathieu equation is analyzed for resonances by using the method of multiple scales. Superharmonic and subharmonic resonances occur. The effect of various parameters on the response of the system is demonstrated using the amplitude-frequency curve. A superharmonic resonance persists for the linear system as well.

In-Plane Blade-Hub Dynamics of Horizontal-Axis Wind Turbine With Mistuned Blades

2017 ◽  
Author(s):  
Ayse Sapmaz ◽  
Gizem D. Acar ◽  
Brian Feeny
Keyword(s):  
Wind Turbine ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Turbine Blades ◽  
Horizontal Axis ◽  
Wind Turbine Blades ◽  
Horizontal Axis Wind Turbine ◽  
Direct Excitation ◽  
Independent Variable ◽  
Rotor Angle

Understanding vibration of the wind turbine blades is of fundamental importance. This paper regards the effect of blade mistuning on the coupled blade-hub dynamics. Unavoidably, at any stage of the wind turbine, the set of blades will not be precisely identical due to the inhomogeneous material, manufacturer tolerances, etc. This paper is based on blade-hub dynamics of a horizontal-axis wind turbine with mistuned blade. The equations of motion are derived for the wind turbine blades and hub exposed to centrifugal effects and gravitational and cyclic aerodynamic forces. The equations are coupled. To decoupled them, the independent variable is changed from time to rotor angle. The resulting blade equations include parametric and direct excitation terms. The method of multiple scales is applied to examine response of the system. This analysis shows that superharmonic and primary resonances exist and are influenced by the mistuning. Resonance cases and the relations between response amplitude and frequency are studied.

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