Non linear dynamics of a thin narrow ribbon in an airflow

The nonlinear aeroelastic vibrations of the Windbelt System are described in the paper. The system works based on aeroelastic flutter principle. The long narrow ribbon with pinned supports at the ends is used as mechanical model. The analysis of aerodynamic forces for a thin rectangular cross-section of the ribbon is provided. Lateral-torsional oscillations of the system is derived, considering aerodynamic forces. Conditions for generation of oscillations are determined using Poincaré-Andronov-Hopf bifurcation. The critical speed, which defines dynamic instability (aeroelastic flutter), is determined as well. The influence of the tension force and other ribbon parameters on the critical speed is considered. The supercritical behavior of the system is investigated.


Introduction
Wind energy is one of the main areas of alternative energy. It is based on conversion of kinetic energy of the air masses into electrical, mechanical or thermal power. The transformation of airflow energy into electrical energy is usually accomplished through wind turbines. Recently, the possibility of producing electric energy using a wind generator, called "Wind Belt" [1][2][3][4][5], is being intensively studied.
The schematic diagram of the device is shown in Fig. 1; taken from [3]. A permanent magnet 3 is fixed on a flexible tape 2 stretched between two rigid supports 1 (constructions with two or more magnets are also known [1]). Inductance coils 5 are arranged coaxially with the magnet on top and bottom of the ribbon on the frame 4. The airflow can disturb the static balance of the ribbon and lead to the excitation of its oscillation as an aeroelastic flutter [6] under certain conditions. The vibrating movement of the magnets induces current in nearby pickup coils, which subsequently can be rectified to a direct voltage. There are a few studies related to the generation of electric current in such devices in the existing scientific literature. However, investigation of the oscillations of a thin ribbon depending on its tension and the airspeed has been studied insufficiently. Known computational models are based on various simplified assumptions that limit investigation of supercritical behavior based on bifurcation analysis.
The purpose of this work is to model the dynamics of the ribbon in the air flow, analyze its movement and the possibility of structure optimization.

Problem statement
A ribbon of length with hinged supports on both ends is positioned in the oncoming air flow (Fig. 2). It is stretched by the longitudinal force along the axis. The cross section of the ribbon is constant along the length. The ribbon in horizontal air flow of constant rate has lifting force (wind load), which is evenly distributed along the length.
The net lifting force is applied along a line that passes through the pressure centers of cross section. This line is parallel to the -axis at a distance 0 4 ⁄ (Fig. 2). a) b) Fig. 2

. a) Model scheme and b) cross-section loading diagram
The lifting force is distributed over length and applied to the center of pressure line (point in Fig. 2 , where is the density of the environment, is the width of the cross section of the ribbon, -is lifting force coefficient. The lifting force point is relocated to the center of the cross section, point . To compensate this transition, the force system is supplemented by linear torques 8 ⁄ . The lifting force coefficient for a thin rectangular profile of the ribbon is given by a known solution for a flow of an ideal incompressible fluid ⁄ is the effective angle of attack, is the vertical movement of the ribbon axis (pole).

Equations of motion
The motion of the considered model is described by a system of equations for lateral-torsional oscillations of a string on hinged supports, which is loaded with distributed force and distributed moment [7]. Taking into account the relocation of lifting forces to the center of gravity and the expression for the lifting coefficient, the system becomes: where is the density of the ribbon, is the shear modulus, ℎ is area of cross section, ℎ ℎ 12 ⁄ is the polar moment of inertia, ℎ 3 ⁄ -is geometric stiffness of the section (strip) during torsion, , -is linear viscous Rayleigh damping coefficients in bending and torsion, respectively.
Further, Eq. (1) is reduced to a dimensionless form. Whereupon, dimensionless parameters are introduced: time * The systems of Eq. (1) take the form: (2) All of them are expressed in terms of two fixed dimensionless parameters = ℎ⁄ and = ⁄ : Damping coefficients are equal to 5 % of its critical value ( is mode number).

The calculation of the critical speed
The system of Eq. (2) is reduced to a system with a finite number of degrees of freedom using the Galerkin method. The solution is represented in the form of an expansion on the basis ≈ ∑ , ≈ ∑ , where = sin are functions satisfying the boundary conditions for the considered model (Fig. 2). The constant is determined from the condition of basis property and orthonormality of functions ( ).
After applying the Ritz-Galerkin method Eq. (2) will take the form: Solutions of the Eq. (4) are sought in the form = . Then, the characteristic equation to determine takes the form det( + + ) = 0. Further, the solution is considered only for the first mode ( = 1) with the following parameter values = 0.002, = 0.03.
The dimensionless tension force is set in a way that the dynamic loss of stability occurs earlier than the static one (divergence). Otherwise, there is a divergence phenomenon when torsional rigidity of the ribbon is completely lost. Fig. 3(a) shows the dependence of the critical velocity Λ * on the tension force . When a certain value of the tension force * is reached, the critical speed becomes equal to and does not grow with further increase of . This value of the critical velocity Λ * corresponds to divergence. This is confirmed by Fig. 3(b) and Fig. 3(c) when > * (○ -beginning of trajectory, × -trajectory end). The first trajectory intersects the Re( ) = 0 axis at in the value Λ * = ( Fig. 3(b)) that corresponds to the transition of the same trajectory through the origin at the Argand diagram (Fig. 3(c)). It corresponds to static loss of stability. Fig. 3. a) The dependence of the critical speed Λ * on the tension force ; b) the dependence Re( )on Λ; c) static stability loss on the Argand diagram An analysis of the results shows that the parameter must be in the range of (0, * ) in order to observe dynamic loss of stability before static one. Here, it is defined as = 0.002. Fig. 4(a) shows trajectories of roots for selected parameters. The point of their transition to the right half-plane is determined. The system has a critical velocity value Λ * at this point.
The value of the imaginary part has the meaning of the oscillation frequency of the system at Λ = Λ * . The value of corresponds to the ordinate of the marked point Λ = Λ * in Fig. 4(a). The value of Λ * is defined by dependence of the real part of the roots Re( ) on Λ (Fig. 4(b)), at the point Re( ) equals zero Λ * = 2.203. a) b) Fig. 4. a) The Argand diagram and b) dependence of the real part of the roots on the dimensionless velocity Λ; ○ -beginning of trajectory, × -trajectory end

Analysis of supercritical behavior
To study supercritical behavior the nonlinear system (see Eq. (3)) is integrated. A speed value Λ greater than Λ * by 10 % is used. The motion is considered at the dimensionless time interval = 30 , where = 1⁄ is the wave number for the frequency . Numerical simulation of the Eq. (3) shows that periodic solutions can be established, i.e., there are both unstable equilibrium position and stable periodic motions in the supercritical region.
To evaluate the establishment of a self-oscillatory regime, phase portraits in Fig. 5(a) and Fig. 5 The analysis of stability of the equilibrium rectilinear position of the tape allows to plot a bifurcation diagram (Fig. 6). It was shown above that there are stable periodic motions in the supercritical, region that corresponds to a stable branch on the bifurcation diagram. Numerical integration by establishing method is utilized to plot the diagram, [8]. The results for the amplitudes and are presented in Fig. 6(a) and Fig. 6(b), respectively. a) b) Fig. 7. The bifurcation diagram a) for and b) for : • -stable motion, × -unstable motion

Conclusions
The mathematical model for the lateral-torsional oscillations of a flexible ribbon under the influence of oncoming air flow is derived and analyzed. The value of the critical velocity, the trajectory of motion, and the bifurcation diagram are obtained. Small lateral-torsional oscillations of the system are analyzed. They describe the dynamics of a Windbelt-type wind generator in a