Parametric Instability and Localization of Vibrations in Three-Blade Wind Turbines

2018 ◽  
Vol 13 (7) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Yukio Ishida
Keyword(s):  
Wind Turbine ◽  
Wind Turbines ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Vertical Wind Shear ◽  
Coupled System ◽  
Basins Of Attraction ◽  
Response Curves ◽  
Softening Behavior

Nonlinear vibration characteristics of three-blade wind turbines are theoretically investigated. The wind turbine is modeled as a coupled system, consisting of a flexible tower with two degrees-of-freedom (2DOF), and three blades, each with a single degree of freedom (SDOF). The blades are subjected to steady winds. The wind velocity increases proportionally with height due to vertical wind shear. The natural frequency diagram is calculated with respect to the rotational speed of the wind turbine. The corresponding linear system with parametric excitation terms is analyzed to determine the rotational speeds where unstable vibrations appear and to predict at what rotational speeds the blades may vibrate at high amplitudes in a real wind turbine. The frequency response curves are then obtained by applying the swept-sine test to the equations of motion for the nonlinear system. They exhibit softening behavior due to the nonlinear restoring moments acting on the blades. Stationary time histories and their fast Fourier transform (FFT) results are also calculated. In the numerical simulations, localization phenomena are observed, where the three blades vibrate at different amplitudes. Basins of attraction (BOAs) are also calculated to examine the influence of a disturbance on the appearance of localization phenomena.

Localization Phenomena of Nonlinear Vibrations in Three-Blade Wind Turbines

2013 ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Hisashi Takahashi ◽  
Yukio Ishida
Keyword(s):  
Wind Turbines ◽  
Degrees Of Freedom ◽  
Nonlinear Vibrations ◽  
Coupled System ◽  
Wind Pressure ◽  
Response Curves ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Frequency Responses ◽  
Time Histories

Vibration characteristics of three-blade wind turbines are investigated. The system is modeled by a coupled system of the flexible tower with two degrees of freedom and each blade with a single degree of freedom, and these blades are subjected to wind pressure which varies depending on the height from the ground. The vibrations of the three-blade wind turbines are theoretically analyzed to determine the natural frequency diagrams, frequency responses, stationary time histories and their FFT results. It is found that several peaks appear at the specific range of the rotational speed ω in the response curves because of both the wind pressure and the parametric excitation terms. In three-blade wind turbines, vibrations including predominant components of 3ω and its higher harmonics appear near these peaks. The response curves near the highest peak exhibit soft spring types due to the nonlinearities of the restoring moments of the blades. In the numerical simulations, “localization phenomena” in the blades, which vibrate at different amplitudes, are observed. The influence of an imperfection of the three blades is also examined.

Unstable Vibrations of a Wind Turbine Tower With Two Blades

2012 ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Yukio Ishida
Keyword(s):  
Wind Turbine ◽  
Equations Of Motion ◽  
Single Mode ◽  
Single Frequency ◽  
Dual Mode ◽  
Response Curves ◽  
Moments Of Inertia ◽  
System Parameters ◽  
Wind Turbine Tower ◽  
The Asymmetry

Unstable vibrations of a two-blade wind turbine tower are theoretically investigated. The theoretical model is a five-degree-of-freedom (5DOF) system, however, the equations of motion are derived separately for 3DOF subsystem (I) and 2DOF subsystem (II). Parametric excitation due to the asymmetry of the moments of inertia of the blade rotor is included only in subsystem (I). Frequency equations are derived and natural frequency diagrams are calculated to clearly demonstrate both the rotational speeds where unstable regions appear and which type of unstable vibrations may occur. It is found that at most, five unstable regions may appear depending on the values of the system parameters in subsystem (I). Two types of unstable vibrations may occur; single mode including a single frequency and dual mode including two frequencies. The influences of the asymmetry of moments of inertia, tower rigidity, and installation position of the blade rotor on the response of the system are also theoretically investigated. Van der Pol’s method is applied to determine the expressions for the response curves. The influences of the blade rotor unbalances on the translational, inclinational and torsional vibrations of the tower are shown. It is found that the amplitudes of the response curves corresponding to single and dual mode are infinite and finite at their boundaries, respectively. The validity of the theoretical analysis is confirmed by numerical simulations.

The Role of Modal Coupling on the Non-Linear Response of Cylindrical Shells Subjected to Dynamic Axial Loads

2000 ◽  
Author(s):  
Paulo B. Gonçalves ◽  
Zenón J. G. N. Del Prado
Keyword(s):  
Dynamic Instability ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Parametric Instability ◽  
Compressive Load ◽  
Basins Of Attraction ◽  
Dynamic Component ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Low Dimensional

Abstract This paper discusses the dynamic instability of circular cylindrical shells subjected to time-dependent axial edge loads of the form P(t) = P0+P1(t), where the dynamic component p1(t) is periodic in time and P0 is a uniform compressive load. In the present paper a low dimensional model, which retains the essential non-linear terms, is used to study the non-linear oscillations and instabilities of the shell. For this, Donnell’s shallow shell equations are used together with the Galerkin method to derive a set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge-Kutta method. To study the non-linear behavior of the shell, several numerical strategies were used to obtain Poincaré maps, stable and unstable fixed points, bifurcation diagrams and basins of attraction. Particular attention is paid to two dynamic instability phenomena that may arise under these loading conditions: parametric instability and escape from the pre-buckling potential well. The numerical results obtained from this investigation clarify the conditions, which control whether or not instability may occur. This may help in establishing proper design criteria for these shells under dynamic loads, a topic practically unexplored in literature.

Dynamic Behavior of Damaged Bridge with Multi-Cracks Under Moving Vehicular Loads

2017 ◽  
Vol 17 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Xinfeng Yin ◽  
Yang Liu ◽  
Lu Deng ◽  
Xuan Kong
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Coupled System ◽  
Interaction Force ◽  
Vehicle Model ◽  
Coupled Equations ◽  
Contact Points ◽  
Local Flexibility ◽  
Bridge Structures ◽  
Road Surface Roughness

When studying the vibration of a bridge–vehicle coupled system, most researchers mainly focus on the intact or original bridge structures. Nonetheless, a large number of bridges were built long ago, and most of them have suffered serious deterioration or damage due to the increasing traffic loads, environmental effect, material aging, and inadequate maintenance. Therefore, the effect of damage of bridges, such as cracks, on the vibration of vehicle–bridge coupled system should be studied. The objective of this study is to develop a new method for considering the effect of cracks and road surface roughness on the bridge response. Two vehicle models were introduced: a single-degree-of-freedom (SDOF) vehicle model and a full-scale vehicle model with seven degrees of freedom (DOFs). Three typical bridges were investigated herein, namely, a single-span uniform beam, a three-span stepped beam, and a non-uniform three-span continuous bridge. The massless rotational spring was adopted to describe the local flexibility induced by a crack on the bridge. The coupled equations for the bridge and vehicle were established by combining the equations of motion for both the bridge and vehicles using the displacement relationship and interaction force relationship at the contact points. The numerical results show that the proposed method can rationally simulate the vibrations of the bridge with cracks under moving vehicular loads.

Application of Generalized Newton-Euler Equations and Recursive Projection Methods to Dynamics of Deformable Multibody Systems

1989 ◽  
Author(s):  
R. A. Wehage ◽  
A. A. Shabana
Keyword(s):  
Euler Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Coupled System ◽  
Projection Methods ◽  
Open Loop ◽  
System Of Equations ◽  
Kinematic Chains ◽  
Deformable Bodies ◽  
Generalized Newton

Abstract A general symbolic-based method is presented for solving equations of motion for open-loop kinematic chains consisting of interconnected rigid and deformable bodies. The method utilizes matrix partitioning, recursive projection based on optimal block U-L factorization and generalized Newton-Euler equations to obtain an order n solution for the constrained equations of motion. Kinematic relationships between the absolute reference, joint and elastic coordinates are used with the generalized Newton-Euler equations for deformable bodies to obtain a large, loosely coupled system of equations. Taking advantage of the inertia matrix structure associated with elastic coordinates yields a recursive solution algorithm whose dimension is independent of the elastic degrees of freedom. The above solution techniques applied to this system of equations yield a much smaller operations count and can more effectively exploit vectorization and parallel processing. The algorithms presented in this paper are illustrated with the aid of cylindrical joints which are easily extended to revolute, prismatic, rigid and other joint types.

The unsteady aerodynamics of floating wind turbine under platform pitch motion

2018 ◽  
Vol 232 (8) ◽  
pp. 1019-1036 ◽  
Author(s):  
Xin Shen ◽  
Ping Hu ◽  
Jinge Chen ◽  
Xiaocheng Zhu ◽  
Zhaohui Du
Keyword(s):  
Wind Turbine ◽  
Wind Turbines ◽  
Degrees Of Freedom ◽  
Unsteady Aerodynamics ◽  
Aerodynamic Performance ◽  
Floating Platform ◽  
Pitch Motion ◽  
Floating Wind Turbine ◽  
Fixed Base ◽  
Turbine Rotors

The aerodynamic performance of floating platform wind turbines is much more complex than fixed-base wind turbines because of the flexibility of the floating platform. Due to the extra six degrees-of-freedom of the floating platform, the inflow of the wind turbine rotors is highly influenced by the motions of the floating platform. It is therefore of interest to study the unsteady aerodynamics of the wind turbine rotors involved with the interaction of the floating platform induced motions. In the present work, a lifting surface method with a free wake model is developed for analysis of the unsteady aerodynamics of wind turbines. The aerodynamic performance of the NREL 5 MW floating wind turbine under the prescribed floating platform pitch motion is studied. The unsteady aerodynamic loads, the transient of wind turbine states, and the instability of the wind turbine wakes are discussed in detail.

Dynamic Modeling and Stability Analysis of a Dual-Rotor Wind Turbine

2018 ◽  
Author(s):  
Oliver T. Filsoof ◽  
Morten H. Hansen ◽  
Anders Yde ◽  
Xuping Zhang
Keyword(s):  
Wind Turbine ◽  
Dynamic Modeling ◽  
Wind Turbines ◽  
Floquet Theory ◽  
Equations Of Motion ◽  
Response Analysis ◽  
Modal Response ◽  
Asymmetric Rotor ◽  
Modal Property ◽  
Fidelity Model

Various modal analysis methods are available for single-rotor wind turbines, but there is no report and guidance on the modal property analysis of multi-rotor wind turbines. This paper presents a dynamic modeling method for the modal response analysis of a wind turbine with two three-bladed isotropic rotors. The equations of motion are derived using Lagrange’s equations and are further linearized at a steady-state equilibrium. To avoid using Floquet Theory to remove the periodic coefficients, multi-blade coordinates are utilized. Comparison between the numerical simulations and a high-fidelity model in HAWC2 shows agreements in terms of modal frequencies. The results shows that the whirling modes splits into symmetric and asymmetric rotor modes.

Non-Linear Aeroelastic Stability of Wind Turbines

2013 ◽  
Vol 569-570 ◽  
pp. 531-538 ◽  
Author(s):  
Z.L. Zhang ◽  
M.T. Sichani ◽  
Jie Li ◽  
J.B. Chen ◽  
S.R.K. Nielsen
Keyword(s):  
Wind Turbine ◽  
Wind Turbines ◽  
Degrees Of Freedom ◽  
Dynamic Instability ◽  
Maximum Lyapunov Exponent ◽  
Moving Frames ◽  
Internal Resonances ◽  
Wind Turbine System ◽  
Non Linear ◽  
The Stability

As wind turbines increase in magnitude without a proportional increase in stiffness, the risk of dynamic instability is believed to increase. Wind turbines are time dependent systems due to the coupling between degrees of freedom defined in the fixed and moving frames of reference, which may trigger off internal resonances. Further, the rotational speed of the rotor is not constant due to the stochastic turbulence, which may also influence the stability. In this paper, a robust measure of the dynamic stability of wind turbines is suggested, which takes the collective blade pitch control and non-linear aero-elasticity into consideration. The stability of the wind turbine is determined by the maximum Lyapunov exponent of the system, which is operated directly on the non-linear state vector differential equations. Numerical examples show that this approach is robust for stability identification of the wind turbine system.

Non-Ideal System With Quadratic Nonlinearities Containing a Two-to-One Internal Resonance

2016 ◽  
Author(s):  
Rodrigo T. Rocha ◽  
Jose M. Balthazar ◽  
D. Dane Quinn ◽  
Angelo M. Tusset ◽  
Jorge L. P. Felix
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Coupled System ◽  
Dynamical Behaviour ◽  
Main System ◽  
Weakly Coupled System ◽  
Ideal System ◽  
Quadratic Nonlinearities

The dynamical behaviour of a non-ideal three-degrees-of-freedom weakly coupled system associated with the quadratic nonlinearities in the equations of motion is investigated. The main system consists of two nonlinear mechanical oscillators coupling with quadratic nonlinearities and in which possess a 2:1 internal resonance between their translational movements. Under these conditions, we analyzed the response when a DC unbalanced motor with limited power supply (non-ideal system) excites the main system. When the excitation frequency is near to second natural frequency of the main system, saturation and jump phenomena are presented. Then, this work will analyze some torques of the motor, which causes the phenomena, and due to high amplitudes of motion will be possible to look for a way to harvest energy in a future work.

Mitigation of Ship Motion Using Passive and Active Anti-Roll Tanks

2007 ◽  
Author(s):  
Osama A. Marzouk ◽  
Ali H. Nayfeh
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Behavior ◽  
Coupled System ◽  
Nonlinear Phenomena ◽  
Response Curves ◽  
Horizontal Pipe ◽  
Require Power ◽  
Tank System ◽  
Frequency Relationship ◽  
Rough Sea

Because excessive roll motions of ships in rough seas badly affect their performance, there is a continuous interest in efficient ways to mitigate these undesirable motions. There are different devices to mitigate the roll of ships with different levels of performance and operating limits. Anti-roll tanks are more effective than other roll stabilization devices when the ship is not underway or moves slowly. Here, we investigate the application of passive and active anti-roll tank systems. The tank system consists of three tanks: each one consists of two columns connected at the bottom via a horizontal pipe equipped with a pump. The tanks are arranged along the length of the ship, symmetrically located about its center of gravity. The motion of the liquid in the tank is 1-D, but it exerts loads on all degrees of freedom of the ship. The equation governing the tank-liquid motion is coupled with the equations governing the 6-DOF motion of the ship in waves; the coupled system is solved simultaneously in time. First, we derive expressions for the forces and moments exerted on the ship by the tanks. Then, we study the roll response at different sea heading angles in rough sea conditions in the absence of the tanks to identify the critical heading angles where the roll is large. We demonstrate the nonlinear behavior of roll through frequency-response curves for different beam wave amplitudes. These curves exhibit typical nonlinear phenomena (jumps and hysteresis) for high wave amplitudes. Spectral analysis shows a two-to-one frequency relationship between the roll and pitch in rough head and follower seas, which are the most critical sea headings. For passive and active tanks, we study the effect of the frequency of the tank system on its effectiveness. We consider active anti-roll tanks in which the pump power is controlled via a proportional-derivative (PD) control law using the roll angle and its rate. We compare the performances of passive and active tanks in rough sea for the critical heading angles. We found that active anti-roll tanks outperform passive ones in terms of roll reduction and size and weight, but they require power consumption. To achieve a specified roll reduction, the weight of a passive tank might be as large as five times that of an active tank. We also found that the performance of passive tanks depends strongly on their frequencies, in contrast with active tanks, which are insensitive to their frequencies.

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