Finding the strongest stable weightless column with a follower load and relocatable concentrated masses

We consider the problem of optimal placement of concentrated masses along a massless elastic column that is clamped at one end and loaded by a nonconservative follower force at the free end. The goal is to find the largest possible interval such that the variation in the loading parameter within this interval preserves stability of the structure. The stability constraint is nonconvex and nonsmooth, making the optimization problem quite challenging. We give a detailed analytical treatment for the case of two masses, arguing that the optimal parameter configuration approaches the flutter and divergence boundaries of the stability region simultaneously. Furthermore, we conjecture that this property holds for any number of masses, which in turn suggests a simple formula for the maximal load interval for $n$ masses. This conjecture is strongly supported by extensive computational results, obtained using the recently developed open-source software package GRANSO (GRadient-based Algorithm for Non-Smooth Optimization) to maximize the load interval subject to an appropriate formulation of the nonsmooth stability constraint. We hope that our work will provide a foundation for new approaches to classical long-standing problems of stability optimization for nonconservative elastic systems arising in civil and mechanical engineering.

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 Pfluger 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 non-zero 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 Pfluger column. Thus, the Whitney umbrella singularity is experimentally confirmed, responsible for discontinuities accompanying dissipation-induced instabilities in a broad range of physical contexts.

Short wavelength local instabilities of a circular Couette flow with radial temperature gradient

We perform a linearized local stability analysis for short-wavelength perturbations of a circular Couette flow with the radial temperature gradient. Axisymmetric and nonaxisymmetric perturbations are considered and both the thermal diffusivity and the kinematic viscosity of the fluid are taken into account. The effect of the asymmetry of the heating both on the centrifugally unstable flows and on the onset of the instabilities of the centrifugally stable flows, including the flow with the Keplerian shear profile, is thoroughly investigated. It is found that the inward temperature gradient destabilizes the Rayleigh stable flow either via Hopf bifurcation if the liquid is a very good heat conductor or via steady state bifurcation if viscosity prevails over the thermal conductance.

Determining role of Krein signature for 3D Arnold tongues of oscillatory dynamos

Using a homotopic family of boundary eigenvalue problems for the mean-field $alpha^2$-dynamo with helical turbulence parameter $alpha(r)=alpha_0+gammaDeltaalpha(r)$ and homotopy parameter $beta in [0,1]$, we show that the underlying network of diabolical points for Dirichlet (idealized, $beta=0$) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, $beta=1$) boundary conditions. In the $(alpha_0,beta,gamma)-$space the Arnold tongues of oscillatory solutions at $beta=1$ end up at the diabolical points for $beta=0$. In the vicinity of the diabolical points the space orientation of the 3D tongues, which are cones in first-order approximation, is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones. The Krein space induced geometry of the resonance zones explains the subtleties in finding $alpha$-profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.

The spherically symmetric alpha2-dynamo and some of its spectral peculiarities

A brief overview is given over recent results on the spectral properties of spherically symmetric MHD alpha2-dynamos. In particular, the spectra of sphere-confined fluid or plasma configurations with physically realistic boundary conditions (BCs) (surrounding vacuum) and with idealized BCs (super-conducting surrounding) are discussed. The subjects comprise third-order branch points of the spectrum, self-adjointness of the dynamo operator in a Krein space as well as the resonant unfolding of diabolical points. It is sketched how certain classes of dynamos with a strongly localized alpha-profile embedded in a conducting surrounding can be mode decoupled by a diagonalization of the dynamo operator matrix. A mapping of the dynamo eigenvalue problem to that of a quantum mechanical Hamiltonian with energy dependent potential is used to obtain qualitative information about the spectral behavior. Links to supersymmetric Quantum Mechanics and to the Dirac equation are indicated.