This paper investigates the nonlinear dynamic responses of the rotating blade with varying rotating speed under high-temperature supersonic gas flow. The varying rotating speed and centrifugal force are considered during the establishment of the analytical model of the rotating blade. The aerodynamic load is determined using first-order piston theory. The rotating blade is treated as a pretwist, presetting, thin-walled rotating cantilever beam. Using the isotropic constitutive law and Hamilton’s principle, the nonlinear partial differential governing equation of motion is derived for the pretwist, presetting, thin-walled rotating beam. Based on the obtained governing equation of motion, Galerkin’s approach is applied to obtain a two-degree-of-freedom nonlinear system. From the resulting ordinary equation, the method of multiple scales is exploited to derive the four-dimensional averaged equation for the case of 1:1 internal resonance and primary resonance. Numerical simulations are performed to study the nonlinear dynamic response of the rotating blade. In summary, numerical studies suggest that periodic motions and chaotic motions exist in the nonlinear vibrations of the rotating blade with varying speed.

Nonlinear vibrations of blade with varying rotating speed

Dynamic model for free vibration and response analysis of rotating beams

The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli–Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced by the rotation of the aerodynamic load and the curvature, as well as inertial induced non-linearities caused by the support point motion. The non-linear partial differential equations of motion in the moving frame of reference have been discretized, using the fixed base eigenmodes as a functional basis. Important non-linear couplings between the fundamental blade mode and edgewise modes have been identified based on a resonance excitation of the wing, caused by a harmonically varying support point motion with the circular frequency ω . Assuming that the fundamental blade and edgewise eigenfrequencies have the ratio of ω 2 / ω 1 ≃ 2 , internal resonances between these modes have been studied. It is demonstrated that for ω / ω 1 ≃ 0.66 , 1.33 , 1.66 and 2.33 coupled periodic motions exist brought forward by parametric excitation from the support point in addition to the resonances at ω / ω 1 ≃ 1.0 and ω / ω 2 ≃ 1.0 partly caused by the additive load term.

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J.B. Yang

Papers

Dynamic modelling and control of a rotating Euler–Bernoulli beam