Parametric Resonances of a Three-Blade-Rotor System With Reference to Wind Turbines

2020 ◽  
Vol 142 (2) ◽  
Author(s):  
Gizem D. Acar ◽  
Mustafa A. Acar ◽  
Brian F. Feeny
Keyword(s):  
Multiple Scales ◽  
Rotor System ◽  
Degree Of Freedom ◽  
Rotor Speed ◽  
Horizontal Axis Wind Turbine ◽  
Direct Excitation ◽  
Parametric Resonances ◽  
Parametric Effects ◽  
Rotor Angle ◽  
First Order Method

Abstract Coupled blade-hub dynamics of a coupled three-blade-rotor system with parametric stiffness, which is similar to a horizontal-axis wind turbine, is studied. Blade equations have parametric and direct excitation terms due to gravity and are coupled through the hub equation. For a single degree-of-freedom blade model with only in-plane transverse vibrations, the reduced-order model shows parametric resonances. A small parameter is established for large blades, which enables us to treat the effect of blade motion as a perturbation on the rotor motion. The rotor speed is not constant, and the cyclic variations cannot be expressed as explicit functions of time. Therefore, it is more convenient to use the rotor angle as the independent variable. By expressing the system dynamics in the rotor angle domain and assuming small variations in rotor speed, the blade equations are decoupled from the rotor equation. The interdependent blade equations constitute a three-degree-of-freedom system with periodic parametric and direct excitation. The response is analyzed by using a first-order method of multiple scales (MMS). The system has a superharmonic and a subharmonic resonances due to direct and parametric effects introduced by gravity. Amplitude-frequency relations and stabilities of these resonances are studied. The MMS solutions are compared with numerical simulations for verification.

In-Plane Blade-Hub Dynamics in Horizontal-Axis Wind-Turbines

2016 ◽  
Author(s):  
Gizem Acar ◽  
Mustafa A. Acar ◽  
Brian F. Feeny
Keyword(s):  
Multiple Scales ◽  
Horizontal Axis ◽  
Rotor Speed ◽  
Horizontal Axis Wind Turbine ◽  
Direct Excitation ◽  
Horizontal Axis Wind Turbines ◽  
Parametric Effects ◽  
Cyclic Variations ◽  
Independent Variable ◽  
Three Degree Of Freedom

Blade-hub dynamics of a horizontal-axis wind turbine is studied. Blade equations are coupled through the hub equation, and have parametric terms due to cyclic aerodynamic forces, centrifugal effects and gravitational forces. Blade inertia is usually small compared to the rotor inertia, which enables us to treat the effect of blade motion as a perturbation on the rotor motion. The rotor speed is not constant, and the cyclic variations cannot be expressed as explicit functions of time. Therefore, it is more convenient to use the rotor angle as the independent variable. By doing so, and assuming small variations in rotor speed, the blade equations are decoupled from the rotor equation. The inter-dependent blade equations constitute a three-degree-of-freedom system with periodic parametric and direct excitation. The response is analyzed by using method of multiple scales. The system has a superharmonic and a subharmonic resonances due to direct and parametric effects introduced by gravity. Amplitude frequency relations and stabilities of these resonances are studied.

Numerical Study of Superharmonic Resonances in a Mistuned Horizontal-Axis Wind-Turbine Blade-Rotor Set

2019 ◽  
Author(s):  
Ayse Sapmaz ◽  
Gizem D. Acar ◽  
Brian F. Feeny
Keyword(s):  
Wind Turbine ◽  
Initial Conditions ◽  
Numerical Study ◽  
Horizontal Axis ◽  
Rotor Speed ◽  
Horizontal Axis Wind Turbine ◽  
First Order ◽  
Direct Excitation ◽  
Rotor Angle ◽  
Perturbation Solutions

Abstract This paper is on a simplified model of an in-plane blade-hub dynamics of a horizontal-axis wind turbine with a mistuned blade. The model has cyclic parametric and direct excitation due to gravity and aerodynamics. This work follows up a previous perturbation study applied to the blade equations written in the rotor-angle domain and decoupled from the hub, in which superharmonic and primary resonances were analyzed. In this work, the effects of mistuning, damping, and forcing level are illustrated. The first-order perturbation solutions are verified with comparisons to numerical simulations at superharmonic resonance of order two. Additionally, the effect of rotor loading on the rotor speed and blade amplitudes is investigated for different initial conditions and mistuning cases.

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.

Second-Order Perturbation Analysis of In-Plane Blade-Hub Dynamics of Horizontal-Axis Wind Turbines

2018 ◽  
Author(s):  
Ayse Sapmaz ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Perturbation Expansion ◽  
Second Order ◽  
Horizontal Axis ◽  
Dynamic Responses ◽  
Order Perturbation ◽  
Resonance Case ◽  
Horizontal Axis Wind Turbine ◽  
Rotor Angle

This paper deals with a second-order perturbation analysis of the in-plane dynamic responses of both tuned and mistuned three-blade-hub horizontal-axis wind-turbine equations. The blades are under effect of gravitational and cyclic aerodynamics forces and centrifugal forces. Although the blades and hub equations are coupled, they can be decoupled by changing the independent variable from time to rotor angle and by using a small parameter approximation. A second-order method of multiple scales is applied in the rotor-angle domain to analyze in-plane blade-hub dynamics. A superharmonic resonance case at one third the natural frequency was revealed. This resonance case was not captured by a first-order perturbation expansion. The relationship between response amplitude and frequency is studied. The effect of blade mistuning on the coupled blade-hub dynamics are taken into account.

PARAMETRICALLY EXCITED NONLINEAR TWO-DEGREE-OF-FREEDOM SYSTEMS WITH NONSEMISIMPLE ONE-TO-ONE RESONANCE

1995 ◽  
Vol 05 (03) ◽  
pp. 725-740 ◽  
Author(s):  
C. CHIN ◽  
A.H. NAYFEH
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Degree Of Freedom ◽  
Parametrically Excited ◽  
Parametric Resonances ◽  
Quadratic Nonlinearities ◽  
One To One ◽  
Two Degree Of Freedom

The response of a parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities and a nonsemisimple one-to-one internal resonance is investigated. The method of multiple scales is used to derive four first-order differential equations governing the modulation of the amplitudes and phases of the two modes for the cases of fundamental and principal parametric resonances. Bifurcation analysis of the case of fundamental parametric resonance reveals that the quadratic nonlinearities qualitatively change the response of the system. They change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.

scholarly journals Influences of Randomly Uncertain Factors on Dynamic Coefficients of an Interlocking Labyrinth Seal-Rotor System

Machines ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 39
Author(s):  
Xin Xiong ◽  
Yanfei Zhou ◽  
Yiqun Wang
Keyword(s):  
Pressure Difference ◽  
Gas Flow ◽  
Labyrinth Seal ◽  
Momentum Equation ◽  
Rotor System ◽  
Rotor Speed ◽  
Bounded Noise ◽  
Dynamic Coefficients ◽  
Noise Perturbation ◽  
Rotor Center

Many randomly uncertain factors inevitably arise when gas flows through a labyrinth seal, and the orbit of the rotor center will not rotate along a steady trajectory, as previously studied. Here, random uncertainty is considered in an interlocking labyrinth seal-rotor system to investigate the fluctuations of dynamic coefficients. The bounded noise excitation is introduced into the momentum equation of the gas flow, and as a result, the orbit of the rotor center is expressed as the combination of an elliptic trajectory with the bounded noise perturbation. Simulation results of the coefficients under randomly uncertain perturbations with various strengths are comparatively investigated with the traditional predictions under ideal conditions, from which the influences of random uncertain factors on dynamic coefficients are analyzed in terms of the rotor speed, pressure difference, and inlet whirl velocity. It is shown that the deviation levels of the dynamic coefficients are directly related to the random perturbations and routinely increase with such perturbation strengths, and the coefficients themselves may exhibit distinct variation patterns against the rotor speed, pressure difference, and inlet whirl velocity.

scholarly journals Bifurcation Analysis of a Transversely Cracked Nonlinear Jeffcott Rotor System at Different Resonance Cases

2019 ◽  
Vol 24 (2) ◽  
pp. 284-302 ◽  
Author(s):  
Nasser A. Saeed ◽  
Mostafa Eissa
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Amplitude Spectrum ◽  
Orientation Angle ◽  
Dynamical Behaviour ◽  
Rotor System ◽  
Resonance Case ◽  
Restoring Force ◽  
Jeffcott Rotor ◽  
Harmonic Resonance

This work focuses on the dynamical behaviour and bifurcations of a vertically supported Jeffcott rotor system having a transverse crack and nonlinear stiffness characteristics at the primary, sub-harmonic, and super-harmonic resonance cases. The nonlinear restoring force due to the bearing-clearance, the crack breathing, the disc eccentricity, and the orientation angle between the crack and imbalance direction are considered in the system model. The equations governing the system motion are derived and solved analytically by applying the Multiple Scales Perturbation Technique (MSPT). The slow-flow modulating equations are obtained and the spinning speed response curve is plotted. The whirling orbit and amplitude spectrum are constructed in the three considered resonance cases. The acquired results provide a better understanding of the main reasons of the super- and sub-harmonic resonance excitations. In additions, we concluded that the suitable resonance case that can be used for early detections of the cracks in the rotating shafts is the sub-harmonic resonance case. Finally, the obtained results are confirmed numerically and compared with the work published in the literature

Response of a Stationary, Damped, Circular Plate Under a Rotating Slider Bearing System

1993 ◽  
Vol 115 (1) ◽  
pp. 65-69 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Circular Plate ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Slider Bearing ◽  
Numerical Examples ◽  
Elastic Circular Plate ◽  
Parametric Resonances ◽  
Supercritical Speed ◽  
Bearing System

This paper is to demonstrate that axisymmetric plate damping will suppress unbounded response of a stationary, elastic, circular plate excited by a rotating slider. Use of the method of multiple scales shows that the axisymmetric plate damping will suppress parametric resonances excited by slider stiffness and slider inertia at supercritical speed. In addition, the plate damping will increase the onset speed above which slider damping destabilizes the elastic circular plate. Moreover, numerical examples show that the plate damping could stabilize the plate/slider system at discrete rotation speeds above the onset speed.

Parametric Stiffness in Large-Scale Wind-Turbine Blades and the Effects on Resonance and Speed Locking

2020 ◽  
Author(s):  
Ayse Sapmaz ◽  
Brian F. Feeny
Keyword(s):  
Wind Turbine ◽  
Large Scale ◽  
Turbine Blades ◽  
Relative Strength ◽  
Wind Turbine Blades ◽  
Fixed Position ◽  
Horizontal Axis Wind Turbine ◽  
Parametric Effect ◽  
Parametric Effects ◽  
Blade Tip

Abstract This paper is on parametric effect in large scale horizontal-axis wind-turbine blades and speed locking phenomenon for a simplified model of the in-plane blade-hub dynamics. The relative strength of the parametric stiffness is evaluated for actual and scaled-length blades. Fixed-position natural frequencies are found at different rotation angles to show the significance of the gravity’s parametric effect. The ratio of the parametric and elastic modal stiffness is then estimated for the scaled versions of the NREL’s blades for four models to present the relation between the blade size and the parametric effects. The parametric effect on blade tip placements are investigated for superharmonic resonances at orders two and three for blades of various lengths. An analysis of speed-locking is presented, and interpreted for the various blades.

Deceleration of an Unbalanced Rotor Through a Critical Speed

1967 ◽  
Vol 89 (4) ◽  
pp. 582-585
Author(s):  
W. K. Bodger
Keyword(s):  
Critical Speed ◽  
Degree Of Freedom ◽  
Bending Stress ◽  
Rotor Speed ◽  
Single Degree Of Freedom ◽  
Closed Solution ◽  
Rotor Shaft ◽  
Single Degree ◽  
Unbalanced Rotor ◽  
Maximum Increment

The problem of a single-degree-of-freedom rotor decelerating slowly through its critica speed is solved by an energy approach; a closed solution is obtained. A small discontinuous downward jump of rotor speed across the critical speed is shown to be required, either with or without damping in the system. The maximum increment of deflection, hence bending stress, in the rotor shaft is shown to be small, provided the rotor is carefully balanced and/or the system is sufficiently damped.

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