Non linear dynamics of a thin narrow ribbon in an airflow

1.1. Problem statement

A ribbon of length $l$ with hinged supports on both ends is positioned in the oncoming air flow (Fig. 2). It is stretched by the longitudinal force $T$ along the $Z$ axis. The cross section of the ribbon is constant along the length. The ribbon in horizontal air flow of constant rate $U$ 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 $Z$-axis at a distance $x_0=b/4$ (Fig. 2).

Fig. 2a) Model scheme and b) cross-section loading diagram

a)

b)

The lifting force $q_y$ is distributed over length and applied to the center of pressure line (point $B$ in Fig. 2) $q_y=\left(\rho U^2/2\right)b{C}_y$ [6], where $\rho$ is the density of the environment, $b$ is the width of the cross section of the ribbon, $C_y$ – is lifting force coefficient. The lifting force point $B$ is relocated to the center of the cross section, point $C$. To compensate this transition, the force system is supplemented by linear torques ${\mu }_z=C_y\rho U^2{b}^2/8$.

The lifting force coefficient $C_y$ for a thin rectangular profile of the ribbon is given by a known solution for a flow of an ideal incompressible fluid $C_y=2\pi \mathrm{s}\mathrm{i}\mathrm{n}\left({\tilde{\phi }}_z\right)\approx A_1\left({\tilde{\phi }}_z-A_3{\tilde{\phi }}_z^3\right)$, where $A_1=2\pi$, $A_3=2\pi /6$ are the expansion coefficients, ${\tilde{\phi }}_z={\phi }_z-\left({\dot{\phi }}_{z}b/4+\dot{v}\right)/U$ is the effective angle of attack, $v$ is the vertical movement of the ribbon axis (pole).

1.2. 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 ${\rho }_1$ is the density of the ribbon, $G$ is the shear modulus, $A=bh$ is area of cross section, $I_0=bh\left(b^2+h^2\right)/12$ is the polar moment of inertia, $I_k=b{h}^3/3$ – is geometric stiffness of the section (strip) during torsion, $d_v$, $d_{\phi }$ – 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 $\tau =t/t^{\mathrm{*}}$, deflection $\xi =v/V^{\mathrm{*}}$, coordinate $\zeta =z/Z^{\mathrm{*}}$, velocity $\mathrm{\Lambda }=U/U^{\mathrm{*}}$, force $\theta =T/T^{\mathrm{*}}$, and scales $t^{\mathrm{*}}=l/h\sqrt{{\rho }_1\left(b^2+h^2\right)/4G}$, $U^{\mathrm{*}}=\sqrt{8G{h}^3/\left(3{\rho }_0{a}_{0}b{l}^2\right)}$, $V^{\mathrm{*}}=h$, $Z^{\mathrm{*}}=l$, $T^{\mathrm{*}}=G{h}^2$. The systems of Eq. (1) take the form:

All of them are expressed in terms of two fixed dimensionless parameters $\epsilon =h/b$ and $\kappa =\sqrt{\rho /{\rho }_1}$:

Damping coefficients are equal to 5 % of its critical value ($k$ is mode number).

1.3. 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 $\xi \approx \sum _{k=1}^n{p}_k\left(\tau \right)\mathrm{}{u}_k\left(\zeta \right)\text{,}$$\phi \approx \sum _{k=1}^n{q}_k\left(\tau \right)\mathrm{}{u}_k\left(\zeta \right)\text{,}$ where $u_k\left(\zeta \right)=C_k\mathrm{s}\mathrm{i}\mathrm{n}\left(k\pi \zeta \right)$ are functions satisfying the boundary conditions for the considered model (Fig. 2). The constant $C_k$ is determined from the condition of basis property and orthonormality of functions $u_k\left(\zeta \right)$.

After applying the Ritz-Galerkin method Eq. (2) will take the form:

where $B=\underset{0}{\overset{1}{\int }}{u}_k^{4}d\zeta$.

The existing trivial solution of the Eq. (3) corresponds to the equilibrium position, which is defined as an unperturbed motion. The Eq. (3) is linearized near the equilibrium position for stability analysis. Then, the Eq. (3) in matrix form:

Solutions of the Eq. (4) are sought in the form $\mathbf{x}=e^{\lambda \tau }\mathbf{u}$. Then, the characteristic equation to determine $\lambda$ takes the form $\mathrm{d}\mathrm{e}\mathrm{t}\left({\lambda }^2\mathbf{M}+\lambda \mathbf{D}+\mathbf{C}\right)=0$. Further, the solution is considered only for the first mode ($k=$1) with the following parameter values $\epsilon =$ 0.002, $\kappa =$ 0.03.

The dimensionless tension force $\theta$ 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 ${\mathrm{\Lambda }}_{\mathrm{*}}$ on the tension force $\theta$. When a certain value of the tension force ${\theta }_{\mathrm{*}}$ is reached, the critical speed becomes equal to $\pi$ and does not grow with further increase of $\theta$. This value of the critical velocity ${\mathrm{\Lambda }}_{\mathrm{*}}$ corresponds to divergence. This is confirmed by Fig. 3(b) and Fig. 3(c) when $\theta >{\theta }_{\mathrm{*}}$ (○ – beginning of trajectory, × – trajectory end). The first trajectory intersects the $\mathrm{R}\mathrm{e}\left(\lambda \right)=0$ axis at in the value ${\mathrm{\Lambda }}_{\mathrm{*}}=\pi$ (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. 3a) The dependence of the critical speed Λ* on the tension force θ; b) the dependence Reλon Λ; c) static stability loss on the Argand diagram

An analysis of the results shows that the parameter $\theta$ must be in the range of $\left(0,{\theta }_{\mathrm{*}}\right)$ in order to observe dynamic loss of stability before static one. Here, it is defined as $\theta =$ 0.002. Fig. 4(a) shows trajectories of roots $\lambda$ for selected parameters. The point of their transition to the right half-plane is determined. The system has a critical velocity value ${\mathrm{\Lambda }}_{\mathrm{*}}$ at this point.

The value of the imaginary part has the meaning of the oscillation frequency ${\omega }_f$ of the system at $\mathrm{\Lambda }={\mathrm{\Lambda }}_{\mathrm{*}}$. The value of ${\omega }_f$ corresponds to the ordinate of the marked point $\mathrm{\Lambda }={\mathrm{\Lambda }}_{\mathrm{*}}$ in Fig. 4(a). The value of ${\mathrm{\Lambda }}_{\mathrm{*}}$ is defined by dependence of the real part of the roots $\mathrm{R}\mathrm{e}\left(\lambda \right)$ on $\mathrm{\Lambda }$ (Fig. 4(b)), at the point $\mathrm{R}\mathrm{e}\left(\lambda \right)$ equals zero ${\mathrm{\Lambda }}_{\mathrm{*}}=$ 2.203.

Fig. 4a) The Argand diagram and b) dependence of the real part of the roots λ on the dimensionless velocity Λ; ○ – beginning of trajectory, × – trajectory end

a)

b)

1.4. Analysis of supercritical behavior

To study supercritical behavior the nonlinear system (see Eq. (3)) is integrated. A speed value $\mathrm{\Lambda }$ greater than ${\mathrm{\Lambda }}_{\mathrm{*}}$ by 10 % is used. The motion is considered at the dimensionless time interval $\tau =30T_f$, where $T_f={1/\omega }_f$ is the wave number for the frequency ${\omega }_f$.

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(b) of vertical dimensionless movement $\xi \left(\tau \right)$ and dimensionless rotation of the cross-section $\phi \left(\tau \right)$ from the Eq. (3) are derived for different initial conditions: 1) $\left\{\xi ,\dot{\xi },\phi ,\dot{\phi }\right\}=\left\{\mathrm{0,0.85,0},0\right\}$, 2) $\left\{\xi ,\dot{\xi },\phi ,\dot{\phi }\right\}=\left\{\mathrm{130,0},\mathrm{0,0}\right\}$.

The attractions of the trajectories to the limit cycles are clearly visible in Fig. 5.

Fig. 5Phase portraits for coordinate functions at a) ζ= 1/2 lateral oscillations ξ˙,ξ and b) torsional oscillations φ˙, φ; “ –– “ – trajectory with initial conditions No. 1; “– – –“ – trajectory with initial conditions No. 2; ○ – beginning of trajectory, × – trajectory end

a)

b)

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 $A_{\xi }$ and $A_{\phi }$ are presented in Fig. 6(a) and Fig. 6(b), respectively.

Fig. 7The bifurcation diagram a) for ξ and b) for φ: ● – stable motion, × – unstable motion

a)

b)