Detecting singular weak-dissipation limit for flutter onset in reversible systems

A “flutter machine” is introduced for the investigation of a singular interface between the classical and reversible Hopf bifurcations that is theoretically predicted to be generic in nonconservative reversible systems with vanishing dissipation. In particular, such a singular interface exists for the Pflüger viscoelastic column moving in a resistive medium, which is proven by means of the perturbation theory of multiple eigenvalues with the Jordan block. The laboratory setup, consisting of a cantilevered viscoelastic rod loaded by a positional force with nonzero curl produced by dry friction, demonstrates high sensitivity of the classical Hopf bifurcation onset to the ratio between the weak air drag and Kelvin-Voigt damping in the Pflüger column. Thus, the Whitney umbrella singularity is experimentally confirmed, responsible for discontinuities accompanying dissipation-induced instabilities in a broad range of physical contexts.

Figure 1

The Pflüger column [53] clamped at $x=0$ with a point mass $M$ at $x=l$. The column is loaded at $x=l$ with a constant compressing circulatory force $P$ inclined to the tangent to the elastic line of the column, so that $v^{\prime }\left(l\right)\overline{\chi }=\text{const}.$ (equal to 0.092 in all the experiments).

Figure 2

(a) Stability boundary for (green dash-dot curve) internally and (blue dashed curve) externally damped discretized model of the Pflüger column with $N=2$ modes and $\chi =1$, when one of the damping coefficients is zero and another one tends to zero. The red solid curve shows the stability boundary of the nondamped discretized model of the Pflüger column according to Eq. (13). (b) The eigenvalue movement when $p$ increases from 0 (circle) to 70 (diamond) for $N=2$, $\chi =1$, $\alpha =0.1$, and (red solid curves) $\gamma =0$, $\eta =0$, (blue dashed curves) $\gamma =4.5$, $\eta =0$, and (green dash-dotted curves) $\gamma =0$, $\eta =0.015$.

Figure 3

(a) For $N=2$, ${\chi }_0=1$, and ${\alpha }_0=0.1$ the linear approximation Eq. (23) to the classical-Hopf bifurcation onset in the $(\eta ,\gamma )$ plane for (black dotted line) $p=p_0-0.1$, (blue dashed line) $p=p_0-0.04$, (green dot-dashed line) $p=p_0-0.02$, and (red solid line) $p=p_0$. The stability region for every $p$ is inside the narrow angle-shaped regions in the first quadrant; flutter instability in the complement. (b) The critical flutter load in the limit of vanishing dissipation as a function of the damping ratio $\beta =\gamma /\eta$ according to the (blue dashed curve) exact expression Eq. (23) and (red solid curve) its quadratic approximation Eq. (24). The maximum of the limit coincides with the critical flutter load $p_0\approx 17.83368$ of the undamped system at $\beta ={\beta }_0\approx 1478.074$ that is determined from Eq. (25). (c) The stabilizing ratio ${\beta }_0$ as a function of ${\alpha }_0$ according to Eq. (25) with vertical asymptotes at ${\alpha }_0=0$ (Beck's column) and ${\alpha }_0\approx 0.342716$.

Figure 4

For $N=2$, ${\chi }_0=1$, stability boundary of the discretized model for the Pflüger column in the plane of internal, $\eta$, and external, $\gamma$, damping for (a) ${\alpha }_0=0$ with ${\beta }_0\to +\infty$, (b) ${\alpha }_0=0.1$ with ${\beta }_0\approx 1478.074$, (c) ${\alpha }_0\approx 0.3427$ with ${\beta }_0\to +\infty$, (d) ${\alpha }_0=0.5$ with ${\beta }_0\approx -1856.099$. The red solid lines correspond to the undamped critical load $p=p_0\left({\alpha }_0\right)$, which depends on ${\alpha }_0$, the blue dashed lines to $p=p_0\left({\alpha }_0\right)+0.02$, and the green dash-dotted lines to $p=p_0\left({\alpha }_0\right)-0.02$.

Figure 5

Each curve, computed with the use of the Eq. (23), shows the critical flutter load in the limit of vanishing dissipation as a function of the damping ratio $\beta$ for the discretized model with $N=2$ and $\chi =1$ and corresponds to a different mass ratio $\alpha$ (reported in the legend). Note that at large mass ratios $0.7\lesssim \alpha \le \pi /2$ the curves form a dense family.

Figure 6

The sketch of the “flutter machine” producing the frictional partial follower load $P$ at the free end of the cantilevered viscoelastic Pflüger column.

Figure 7

Pulsation (red solid curves) and growth rates (blue dashed curves) for the Pflüger column versus the dimensionless load $p$ (a) without damping and (b) in the presence of a Kelvin-Voigt damping for the material ($\eta$) and air drag ($\gamma$), demonstrating the drop in the onset of flutter. The plots were obtained with the parameters representative of sample 5 in Table 1.

Figure 8

Critical flutter load $p$ versus mass ratio $\alpha$. Theoretical predictions based on Eq. (7) are plotted (the upper dashed curve) when damping is absent, when only external ($\gamma$, dot-dashed lines) or internal ($\eta$, lower dashed lines) damping is present, and (solid lines) when both damping mechanisms are present. Experimental results are marked by diamonds with error bars. The tested samples are numerated and their characteristics reported in Table 1.

Figure 9

Solid curves mark the critical flutter load versus damping ratio $\beta =\gamma /\eta$ at different values of mass ratio $\alpha$ and corresponding fixed values of $\eta$; see Table 1. The experimental data are shown by spots with error bars. Dashed lines indicate the critical flutter load of the undamped Pflüger column for the same values of $\alpha$.

Figure 10

Real (blue dashed curve) and imaginary (red solid curve) part of the eigenfrequencies associated to the first (lower frequency) flutter branch. Each number corresponds to a value of the tangential load $p$ for which the relevant eigenvector is computed and reported on the right in separate boxes. The vibrations numbered 1 to 3 are stable. Flutter instability first occurs at the load for which the mode numbered 4 is reported.