Saturation phenomena and nonlinear resonances of rotating pretwisted laminated composite blade under subsonic air flow excitation

The primary resonance and nonlinear vibrations of the functionally graded graphene platelet (FGGP)-reinforced rotating pretwisted composite blade under combined the external and multiple parametric excitations are investigated with three different distribution patterns The FGGP-reinforced rotating pretwisted composite blade is simplified to the rotating pretwisted composite cantilever plate reinforced by the functionally graded graphene platelet It is novel to simplify the leakage of the airflow in the tip clearance to the non-uniform axial excitation The rotating speed of the steady state adding a small periodic perturbation is considered The aerodynamic load subjecting to the surface of the plate is simulated as the transverse excitation Utilizing the first-order shear deformation theory, von Karman nonlinear geometric relationship, Lagrange equation and mode functions satisfying the boundary conditions, three-degree-of-freedom nonlinear ordinary differential equations of motion are derived for the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations The primary resonance and nonlinear dynamic behaviors of the FGGP-reinforced rotating pretwisted composite cantilever plate are analyzed by Runge–Kutta method The amplitude–frequency response curves, force–frequency response curves, bifurcation diagrams, maximum Lyapunov exponent, phase portraits, waveforms and Poincare map are obtained to investigate the nonlinear dynamic responses of the FGGP-reinforced rotating pretwisted composite cantilever plate under combined the external and multiple parametric excitations

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Nonlinear Vibrations of Rotating Pretwisted Composite Blade Reinforced by Functionally Graded Graphene Platelets Under Combined Aerodynamic Load and Air Flow in Tip Clearance

The primary resonances of a pre-deformed rotating beam model including the quadratic and cubic nonlinearities are investigated in the presence of the 3:1 internal resonance. The steady state responses of the beam are analyzed in two cases of the primary resonance with the method of multiple scales. The original dynamic equation is integrated numerically in two frequency sweep directions. The theoretical results are consistent with those obtained in the numerical simulation. The contributions of quadratic nonlinearities and cubic nonlinearities to the primary resonances behavior of the rotating beam are clarified. The frequency response curves are discussed by considering different model parameters such as the thermal gradient, the rotating speed, the damping coefficient and the gas pressure. The stability regions of coupled mode solutions are compared between the models with different nonlinearities in the case of the primary resonance of the second mode. A series of interesting findings are presented.

Three to one internal resonances of a pre-deformed rotating beam with quadratic and cubic nonlinearities

Subharmonic and Combination Resonance of Rotating Pre-deformed Blades Subjected to High Gas Pressure

Turbo-machineries, as key components, have a wide utilization in fields of civil, aerospace, and mechanical engineering. By calculating natural frequencies and dynamical deformations, we have explained the rationality of the series form for the aerodynamic force of the blade under the subsonic flow in our earlier studies. In this paper, the subsonic aerodynamic force obtained numerically is applied to the low pressure compressor blade with a low constant rotating speed. The blade is established as a pre-twist and presetting cantilever plate with a rectangular section under combined excitations, including the centrifugal force and the aerodynamic force. In view of the first-order shear deformation theory and von-Karman nonlinear geometric relationship, the nonlinear partial differential dynamical equations for the warping cantilever blade are derived by Hamilton’s principle. The second-order ordinary differential equations are acquired by the Galerkin approach. With consideration of 1:3 internal resonance and 1/2 sub-harmonic resonance, the averaged equation is derived by the asymptotic perturbation methodology. Bifurcation diagrams, phase portraits, waveforms, and power spectrums are numerically obtained to analyze the effects of the first harmonic of the aerodynamic force on nonlinear dynamical responses of the structure.

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Bifurcation and dynamic behavior analysis of a rotating cantilever plate in subsonic airflow

The present work investigates the super-harmonic resonances combined with 2:1 internal resonance of a rotating blade subjected to strong gas pressure. The dynamic model includes the effect of the initial curved axis of the blade induced by the thermal gradient. The dimensionless gas excitation amplitude is assumed to be the same magnitude of the dimensionless vibration displacement. The vibration of the blade in the plane of rotation and the plane perpendicular to it are described by a set of coupled ordinary differential equations with quadratic and cubic nonlinearities. The steady-state response of the rotating blade is calculated via the method of multiple scales. The stabilities of the steady-state responses are determined via Lyapunov theory. Parametric studies are performed to clarify the influences of system parameters on dynamic response and unstable regions. The various typical phenomena including jump, hysteresis and saturation are observed in the dynamic model. The stable and unstable regions of the solution are analyzed in the plane of external detuning parameter and excitation amplitude. The evaluation of the blade dynamic response is revealed in the unstable region. The theoretical results obtained via the method of multiple scales coincide with the numerical solutions.

Bo Zhang

Papers

Super-harmonic resonances of a rotating pre-deformed blade subjected to gas pressure

Three to one internal resonances of a pre-deformed rotating beam with quadratic and cubic nonlinearities

Subharmonic and Combination Resonance of Rotating Pre-deformed Blades Subjected to High Gas Pressure