Liubov Klimina

Parametrical Analysis of the behavior of an Aerodynamic Pendulum with Vertical Axis of Rotation

Parametric analysis of the behavior of an aerodynamic pendulum is carried out, motivated by the study of a small wind power generator with a vertical axis. A mathematical model for the free rotation of this pendulum is constructed, leading to a system of nonlinear ODEs and transcendental algebraic equations. A qualitative analysis of the phase portrait is carried out: all equilibrium solutions are found, their stability is studied, characteristics of a stable rotational regime are determined; and domains of attraction for equilibrium solutions and for the rotational regime are also found. The mathematical model is used to study the operational regimes of the system “wind turbine + generator”. Estimations of the trapped power as a function of the external load in the circuit are obtained; optimal values for the power and the load are found. A pitch angle control mechanism is proposed in order to increase this power.

Periodical motions of a hinge mechanism type wind power system

A wind power system based on an antiparallel link mechanism is proposed. Mathematical model is constructed. Sufficient conditions of existence of attractive periodical motions are derived using the Poincare-Pontryagin theorem. Asymptotic bifurcation diagrams are constructed. Trapped power at periodical motions is estimated.

On wind turbine blade design optimization

A wind turbine with a horizontal rotation axis is considered in the stationary airflow. A problem of choosing the law of change of the setting angle of the cross section of the wind turbine blade and the value of the angular speed so that the wind energy utilization coefficient is maximal is discussed. Maximization of the respective functional is considered as a variational problem with a priori unknown parameter. Once it is solved numerically or analytically, the optimal value of this parameter is determined, with the sought variables given by rather simple formulas. The results are analyzed on a qualitative level. In particular, the setting angle is found to change monotonically along the blade, which is confirmed by practical wind turbine design. Examples of using the proposed approach to wind turbine blade design are considered.

On unstable periodic regime of small HAWT

Dynamics of a small HAWT is studied. The closed mathematical model involving phenomenological description of both aerodynamic load upon turbine blades and permanent magnet electric generator is developed, in order to take into account the inductive reactance of the electric circuit. A series of experiments is performed in the subsonic wind tunnel of the LMSU Institute of Mechanics that allowed verifying the model and identifying its parameters. Parameters of dynamic model are identified, such as the coefficient of electromechanical interaction, the active internal resistance of generator, the circuit reactance. Parametric analysis of steady regimes is performed. The model prediction that HAWT operating dynamic system has two stable steady regimes (high speed regime and low speed one) is confirmed by experiments. Transient regimes are registered depending on parameters of the system, which allows estimating the unstable steady regime. The characteristics of the unstable regime are experimentally determined...

Rotational modes of motion for an aerodynamic pendulum with a vertical rotation axis

An aerodynamic pendulum placed in a steady horizontal air flow is studied in connection with modeling a wind-receiving element of a vertical-axis wind turbine (VAWT). When modeling the medium effects on the pendulum, the following two approaches are combined: the quasi-static one based on stationary wind tunnel experiments and the unsteady one based on the added mass effects. The existence of stable and unstable rotational modes is analyzed analytically (via the Poincaré-Pontryagin method) and numerically. The dependence on the parameters responsible for the viscous friction at the rotation axis and for the added mass effects is taken into account.

Wind Car Driven by the Magnus Force

A mathematical model of a wind car is introduced. The car is equipped by the Magnus-type horizontal axis wind rotor. The shaft of the rotor is connected by a reduction gear with driving wheels of a vehicle. The wheels move without slipping along the line that is parallel to the wind flow. Steady motions of the vehicle are studied. It is shown that the vehicle can move upwind, as well as downwind faster than the wind. The speed of the vehicle is estimated depending on the transmission gear ratio. The maximal speed is compared with that obtained for different types of wind powered vehicles.

A Vehicle Driven Upwind by the Horizontal Axis Wind Turbine

A mathematical model of a wind powered vehicle is constructed. The vehicle is driven by a horizontal axis wind turbine. It is supposed that the velocity of the center of mass of the vehicle is directed upwind. Conditions of existence and stability of the steady upwind motion are discussed. The maximum speed of the upwind motion is estimated. The maximum speed is compared with that obtained for different types of wind powered vehicles.

Two-Frequency Averaging in the Problem of Motion of a Counter-Rotating Vertical Axis Wind Turbine

Motion of a small-scale Darrieus counter-rotating vertical axis wind turbine (VAWT) in a steady wind flow is studied. The system consists of two turbines that rotate in opposite directions. The shaft of the first turbine is rigidly joined to the rotor of a generator and the shaft of the second turbine is rigidly joined to the stator. A closed few-parametric mathematical model that takes into account the changeable electrical load in the local circuit of the generator is constructed. The corresponding dynamical system is a two-frequency system. In order to describe operating modes of the model, the system is averaged over two angles under the assumption that both frequencies are bounded away from zero. It is shown that passage through resonances has no crucial effect on the system behavior in the considered range of the parameters of the model.

Application of the Poincare-Pontryagin theorem to analysis of a dynamical model of a wind powered car

The mathematical model of a wind-powered car is constructed. The car moves against the wind flow using energy produced by an oscillation type wind turbine. The Poincare-Pontryagin theorem is used to obtain sufficient conditions of existence of attractive periodic solutions of the equations of motion. Bifurcation diagrams are constructed.

Preferable Operation Modes of a Wind Turbine with a Differential Planetary Gearbox

The paper is devoted to studying behaviour of a small-scale wind turbine with a differential planetary gearbox. Control torque is applied to an external ring of a gearbox, a wind receiving element is rigidly joined with a carrier, and a rotor of an electrical generator is rigidly joined with a sun gear. The closed mathematical model of the mechanical system in question is constructed to find effective control strategies. This model takes into account aerodynamic load upon blades of the turbine as well as electromagnetic torque acting on the rotor of the generator. An operation mode of the mechanical system corresponds to a steady point of model equations. We performed parametrical analysis of the dynamical system, found the set of possible operation modes and suggested different control algorithms. Constructed control laws allow the mechanical system to approach a preferable operation mode from any initial conditions. We solved this control problem by piecewise constant control and linear feedback control. We discussed different purposes of control for the mechanical system in question.