Flutter instability and other singularity phenomena in symmetric systems via combination of mass distribution and weak damping

The local dynamic instability of autonomous conservative, lumped-mass (discrete) systems, is thoroughly discussed when negligibly small dissipative forces are included. It is shown that such small forces may change drastically the response of these systems. Hence, existing, widely accepted, findings based on the omission of damping could not be valid if damping, being always present in actual systems, is included. More specifically the conditions under which the above systems may experience dynamic bifurcations associated either with a degenerate or a generic Hopf bifurcation are examined in detail by studying the effect of the structure of the damping matrix on the Jacobian eigenvalues. The case whereby this phenomenon may occur before divergence is discussed in connection with the individual or coupling effect of non-uniform mass and stiffness distribution. Jump phenomena in the critical dynamic loading at a certain mass distribution are also assessed. Numerical results verified by a non-linear dynamic analysis using 2-DOF and 3-DOF models confirm the validity of the theoretical findings as well as the efficiency of the technique proposed herein.     

Damage induced structural instability

Multiaxial stress-strain damage relations, describing both time-independent and time-dependent strain and damage creation are postulated. The influence of damage creation on the load carrying capacity of simple structures is discussed. The time-independent deformation, and loss of stability, of a thick-walled cylinder under torque and internal pressure are analysed. The results are shown to be similar to previously found results for a beam under tension and bending.     

Flutter instability of damped plates under combined conservative and nonconservative loads

The combined flutter and divergence instability of plates of arbitrary geometry subjected to any type of boundary conditions under interior and edge conservative and nonconservative loads are solved in presence of external and internal damping. In contrast to previous investigations, the membrane stress resultants are not in general uniform, since they result from plane stress problem under the given body forces (conservative and nonconservative) and the prescribed inplane boundary conditions. The differential equations of the problem are derived using Hamilton’s principle. The resulted initial boundary value problem is solved using the analog equation method (AEM), which is a BEM-based domain meshless method. The combined action of conservative and nonconservative forces is also investigated. Several plates have been studied and useful conclusions on the effect of boundary conditions and damping on flutter load have been drawn. The obtained numerical results demonstrate the accuracy of the developed method and its capability to solve realistic engineering problems.     

Flutter instability of a fluid-conveying fluid-immersed pipe affixed to a rigid body

A set of simplified boundary conditions for a flexible beam connected to a rigid body at one end and free at the other end is presented and applied to the case of a fluid-conveying, fluid-immersed pipe. These boundary conditions represent an analytically tractable approximation to those of a submersible which uses a combination of jet action and flutter instability induced tail motion to produce thrust. The boundary conditions are made non-dimensional, and the effect of the non-dimensional mass of the rigid body on system stability is assessed. The neutral stability of this system is determined within a two-parameter space consisting of the velocity of the fluid within the tail, and the forward speed of the submersible. Equations in the literature, derived using slender-body theory, were used to compute the sign of the thrust produced by the tail and the tail's Froude efficiency for the neutrally stable waveforms of the beam.     

Harnessing structural instability for cell durotaxis

Cells were suggested to sense matrix rigidity by applying fluctuating forces, but the underlying mechanism remains elusive. Here with a generic filament-crosslinker modeling system for stress fibers, we demonstrate that high mechanical forces can be induced by specific protein-protein interactions with biased kinetics. Strikingly, we further find that there exist two patterns of force generation, a stable pattern and a fluctuated pattern, in agreement with previous experimental observations. Our analysis indicates that the fluctuated force profile is essentially due to force-induced structural instability during structural assembly. We suggest that how cells utilize or circumvent such stable forces or fluctuated forces may be important in other biological processes as well, though whether such forces should be regarded as passive or active is still tentative.     

Propagation of instability in electron beam systems

The nature of the normal mode instability of a plasma interacting with an electron beam [1] is studied. Dispersion equations of this type also occur for a whole series of physical phenomena. The velocities at which the initially localized disturbance propagates in an infinite system are determined. It is shown that the system is globally unstable [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 127–132, July–August, 1977.In conclusion, the author thanks A. G. Kulikovskii and A. A. Barmin for valuable advice and their interest in the work.     

Systematic description of imperfect bifurcation behavior of symmetric systems

A systematic method is presented for describing experimental curves of force vs strain of a system with regular polygonal (dihedral group) symmetry subject to bifurcation behavior, with an aim toward overcoming the following problems : (1) it is difficult to judge whether the system is undergoing bifurcation or not ; (2) the perfect behavior of the system cannot be known due to the presence of initial imperfections ; (3) those curves are often qualitatively different from bifurcation diagrams predicted by mathematics. The tools employed are : the asymptotic theory for imperfect bifurcation, such as the Koiter law, and the stochastic theory of initial imperfections. The former theory is extended in this paper to the system with regular-polygonal symmetry to present asymptotic laws for recovering perfect curves with reference to the experimental ones. These laws are formulated for physically observable displacements, instead of the variables in the mathematical bifurcation diagrams, in order to make them readily applicable to the experimental curves. The stochastic theory is combined with an asymptotic law to develop a means to identify the multiplicity of the bifurcation point. The systematic method for describing the experimental curves developed in this manner is applied to the bifurcation analysis of regular-polygonal truss domes to testify its validity. Furthermore, this method is applied to the shear behavior of cylindrical sand specimens to show that they, in fact, are undergoing bifurcation, and, in turn, to demonstrate the importance of a viewpoint of bifurcation in the study of shear behavior of materials. The need of a dual viewpoint of bifurcation and plasticity in the study of constitutive relationship of materials is emphasized to conclude the paper.     

Solitary waves in dissipative-dispersive systems with instability

The wave processes in a system described by a fourth-order partial differential equation with Burgers-Korteweg-de Vries nonlinearity are considered. The initial equation is reduced to a dynamical system of three equations, which is analyzed by means of a numerical method. It is shown that the equation for the waves in dissipative-dispersive systems with instability has solutions in the form of solitary waves and wave fronts.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 99–104, March–April, 1989.     

Localization and shear-crippling (kinkband) instability in a thick imperfect laminated composite ring under hydrostatic pressure

A fully nonlinear finite elements analysis for prediction of localization representing shear-crippling (kinkband) instability in a thick laminated composite (plane strain) ring (infinitely long cylindrical shell) under applied hydrostatic pressure is presented. The primary accomplishment of the present investigation is prediction of meso(lamina)-structure-related equilibrium paths, which are often unstable in the presence of local imperfections and/or material nonlinearity, and which are considered to “bifurcate” from the primary equilibrium paths, representing periodic buckling patterns pertaining to global or structural level stability of the thick cross-ply ring with modal or harmonic imperfection. The present nonlinear finite elements solution methodology, based on the total Lagrangian formulation, employs a quasi-three-dimensional hypothesis, known as layerwise linear displacement distribution theory (LLDT) to capture the three-dimensional interlaminar (especially, shear) deformation behavior, associated with the localized interlaminar shear-crippling failure.A thick laminated composite [90/0/90] imperfect (plane strain) ring is investigated with the objective of analytically studying its premature compressive failure behavior. Numerical results suggest that interlaminar shear/normal deformation (especially, the former) is primarily responsible for the appearance of a limit (maximum pressure) point on the post-buckling equilibrium path associated with a periodic (modal or harmonic) buckling pattern, for which a modal imperfection serves as a perturbation. Localization of the buckling pattern results from “bifurcation” at or near this limit point, and can be viewed as a symmetry breaking phenomenon.In order to investigate a localization of the buckling pattern, a local or dimple shaped imperfection superimposed on a fixed modal one is selected. With the increase of local imperfection amplitude, the limit load (hydrostatic pressure) decreases, and also the limit point appears at an increased normalized deflection. Additionally, the load–deflection curves tend to flatten (near-zero slope) to an undetermined lowest pressure level, signaling the onset of “phase transition” in the localized region, and coexistence of two “phases”, i.e., a highly localized band of shear crippled (kinked) phase and its unshear-crippled (unkinked) counterpart along the circumference of the ring. Interlaminar shear-crippling triggered by the combined effect of imperfection, material nonlinearity and interlaminar shear/normal deformation appears to be the dominant compressive failure mode. A three-dimensional or quasi-three-dimensional theory, such as the afore-mentioned LLDT is essential in order to capture the meso-structure-related instability failure such as localization of the interlaminar shear crippling, triggered by the combined presence of local imperfection and material nonlinearity.     

Transition to instability of the interface in geothermal systems

High-temperature geothermal reservoir in porous media is under consideration, consisting of two high-permeability layers, which are separated by a low-permeability stratum. The thermodynamic conditions are assumed to imply that the upper and lower high-permeability layers are filled in by water and by vapour, respectively. In these circumstances the low-permeability stratum possesses the phase transition interface, separating domains occupied by water and vapour. The stable stationary regimes of vertical phase flow between water and vapour layers in the low-permeability stratum may exist. Stability of such regimes where the heavier fluid is located over the lighter one is supported by a heat transfer, caused by a temperature gradient in the Earth's interior. We give the classification of the possible types of transition to instability of the vertical flows in such a system under the condition of smallness of the advective heat transfer in comparison with the conductive one. It is found that in the non-degenerate case there exist three different scenarios of the onset of instability of the stationary vertical phase transition flows. Two of them are accompanied by the bifurcations of the destabilizing vertical flow, leading to appearance of horizontally non-homogeneous regimes with non-constant shape of the interface. The bifurcations correspond to the simple resonance and $1:1$-resonance, which typically arise in reversible systems.     

Singular configurations of structural systems

A singular configuration of a structural system is characterized by rank deficiency of the equilibrium matrix and kinematic matrix (the rank is lower than both the number of degrees of freedom and the number of constraints) . Such configurations exist only in systems that are not geometrically invariant (underconstrained structural systems) . Most interesting among them are systems with infinitesimal mobility which attracted attention of many prominent researchers. This paper puts the entire issue in a different perspective by addressing a critical, yet so far unexplored, aspect of singular configurations—their realizability. It turns out that the only generic, physically realizable type of a singular configuration is a system with first-order infinitesimal mobility, and even this cannot be constructed without inducing prestress of finite magnitude. All other singular configurations (unprestressable first-order mechanisms; higher-order mechanisms; and singular configurations of finite mechanisms) are unrealizable. Moreover, short of exact or symbolic calculation, they are also noncomputable and are just formal analytical constructs.     

Three-dimensional instability problems for viscoelastic composite materials and structural members

Investigations of viscoelastic composite materials and structural members carried out using the TDLTSDB and initial-imperfection method are reviewed. The investigations address the internal and surface loss of stability of layered and fibrous composites, the loss of stability of plates, and the delamination (buckling) of plates with cracks. Each of these problems is reviewed separately. New areas of further research are proposed. The review focuses on the investigations carried out by the author and his students Published in Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 3–27, October 2007.     

Gyroscopic modes decoupling method in parametric instability analysis of gyroscopic systems

Traditional procedures to treat vibrations of gyroscopic continua involve direct application of perturbation methods to a system with both a strong gyroscopic term and other weakly coupled terms. In this study, a gyroscopic modes decoupling method is used to obtain an equivalent system with decoupled gyroscopic modes having only weak couplings. Taking the axially moving string as an example, the instability boundaries in the vicinity of parametric resonances are detected using both the traditional coupled gyroscopic system and our system with decoupled gyroscopic modes, and the results are compared to show the advantages and disadvantages of each method.     

Probabilistic interval reliability of structural systems

The probabilistic reliability approach is the most widely used method for reliability analysis. The recent research shows that the reliabilities of structural systems strongly depend on the parameters of the probability model. It is possible that the little error in the estimation of the parameters may lead to the remarkable error of the resulting probability. In this study, we introduce the interval approach into the conventional reliability theory. We present a novel approach which allows us to obtain the system failure probability interval from the statistical parameter intervals of the basic variables. This approach is a combination of the two techniques, namely the classical reliability theory and the interval analysis. In the end of this paper, we show the feasibility of the proposed approach through two examples of the truss systems.     

The role of boundary conditions in the instability of one-dimensional systems

We investigate the instability properties of one-dimensional systems of finite length that can be described by a local wave equation and a set of boundary conditions. A method to quantify the respective contributions of the local instability and of wave reflections in the global instability is proposed. This allows to differentiate instabilities that emanate from wave propagation from instabilities due to wave reflections. This is illustrated on three different systems, that exhibit three different behaviors. The first one is a model system in fluid mechanics (Ginzburg–Landau equation), the second one is the fluid-conveying pipe (Bourrières equation), the third one is the fluid-conveying pipe resting on an elastic foundation (Roth equation).     

Criteria of hydrodynamic instability of a phase interface in hydrothermal systems

The loss of stability of a vertical phase flow in a geothermal system in which a liquid layer overlies a vapor layer is considered. The loss of stability criteria are obtained in explicit form. It is found that when the physical parameters of the system are varied the transition to phase interface instability can be realized by means of one of the following mechanisms: the transition occurs spontaneously for any perturbation wavenumber (degenerate case); an unstable wavenumber arises at infinity; the instability threshold is determined by a double zero wavenumber. In the latter case the transition to instability is accompanied by simple resonance bifurcation. As a result of this bifurcation, secondary regimes dependent on the horizontal coordinate branch off from the basic regime describing the horizontally-homogeneous vertical phase flows.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 100–109. Original Russian Text Copyright © 2004 by Ilichev and Tsypkin.     

On equilibrium instability for conservative and partially dissipative systems

The investigation brings some contributions to the classical problem of inverting the Lagrange-Dirichlet stability theorem. First, an example is given of a conservative holonomic mechanical system with a stable equilibrium at the origin, although the potential function is strictly negative along some rays issuing from the origin. Then, one establishes a new instability result in the conservative case. Last, by means of a vector auxiliary function, one proves an instability theorem for holonomic systems with partial dissipation.