The destabilizing effect of external damping: Singular flutter boundary for the Pflüger column with vanishing external dissipation

Elastic structures loaded by non-conservative positional forces are prone to instabilities induced by dissipation: it is well-known that internal viscous damping destabilizes the marginally stable Ziegler's pendulum and Pflüger column (of which the Beck's column is a special case), two structures loaded by a tangential follower force. The result is the so-called ‘destabilization paradox’, where the critical force for flutter instability decreases by an order of magnitude when the coefficient of internal damping becomes infinitesimally small. Until now external damping, such as that related to air drag, is believed to provide only a stabilizing effect, as one would intuitively expect. Contrary to this belief, it will be shown that the effect of external damping is qualitatively the same as the effect of internal damping, yielding a pronounced destabilization paradox. Previous results relative to destabilization by external damping of the Ziegler's and Pflüger's elastic structures are corrected in a definitive way leading to a new understanding of the destabilizating role played by viscous terms.     

Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities

We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes—NNMs), as well as, asynchronous periodic motions (elliptic orbits—EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.     

Post-critical behavior of Beck's column with a tip mass

This study examines how a tip mass with rotary inertia affects the stability of a follower-loaded cantilevered column. Using nonlinear modeling and perturbation analysis, expressions are set up for determining the stability of the straight column and the amplitude of post-critical flutter oscillations. Bifurcation diagrams are given, showing how the vibration amplitude changes with follower load and other parameters. These results agree closely with numerical simulation. It is found that sufficiently large values of tip mass rotary inertia can change the primary bifurcation from supercritical into subcritical. This can imply very large motions for follower loads just beyond critical, contrasting the finite amplitude motions accompanying supercritical bifurcations. Also, the straight column may be destabilized by a sufficiently strong disturbance at loads far below the value of critical load predicted by linear theory. A similar change in bifurcation is found to occur with increased external (as compared to internal) damping, and with a shortening in column length. These effects are not revealed by linear modeling and analysis, which may consequently fail to predict even qualitatively the real critical load for a column with tip mass.     

On the stability of a deep beam subjected to nonconservative and dissipative forces

Summary For the case of a simply supported deep beam subjected to a transverse follower load applied at its center, the dependence of the critical flutter load upon the effects of internal and external damping and warping rigidity is considered. A Kelvin-Voigt solid is assumed, the external damping is assumed to be proportional to the velocity of the beam at a point, and, due to the nature of the nonconservative applied load, the flexural and torsional deformations of the beam are coupled. The resulting boundary value problem is nonself-adjoint in character, and the stability problem is solved in an approximate manner by means of an adjoint variational principle. Several graphs are presented to demonstrate the effect of the various damping and rigidity parameters on the value of the critical flutter load. The numerical results obtained here reveal that in the absence of external damping, the value of the critical flutter load becomes arbitrarily small as the internal damping parameter associated with flexure tends to zero.

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Sommario Per una trave alta, incernierata agli estremi e sollecitata da un carico rotante con la sezione cui sia applicato e distribuito simmetricamente rispetto alla mezzeria della trave, si considera la dipendenza del carico critico di flutter dagli effetti di smorzamento interno ed esterno e della rigidezza biflessionale. Si assnmono (i) un solido di tipo Kelvin-Voigt e (ii) uno smorzamento esterno che sia proporzionale alla velocità. Dovuta al genere del carico non conservativo, la deformazione consiste di spostamenti di flessione e torsione. Poichè il problema ai limiti che descrive il moto del sistema possiede coefficienti variabili e non è antoaggiunto, si risolve il problema di constatare il valore del carico critico per un procedimento approssimativo mediante un principio variazionale. Si mostrano grafici che rivelano gli effetti dei diversi parametri di smorzamento e rigidezza sul valore del carico critico. I risultati numerici ottenuti qui mostrano che nell'assenza di smorzamento esterno il valore del carico critico diviene arbitrariamente minnto qualora il parametro di smorzamento interim associate con flessione lenda a zero.

     

Post-critical behavior of damped beam columns with variable cross section subjected to distributed follower forces

In this paper the post-critical behavior of beam columns with variable mass and stiffness properties subjected to follower forces arbitrarily distributed along their length in the presence of damping (both internal and external) is investigated using a complete nonlinear dynamic analysis. Although the static nonlinear analysis is more economical in computational cost, it is associated only with the loss of local stability via flutter or divergence. Thus, the nonlinear dynamic analysis is adopted in order to examine the global stability of the system. The governing equations of hyperbolic type are derived in terms of the displacements by considering (a) nonlinear response including the axial deformation, (b) nonlinear response excluding the axial deformation and (c) linear response. Moreover, as the cross-sectional properties of the beam vary along its axis, the resulting coupled nonlinear differential equations have variable coefficients. Their solution is achieved using the analog equation method (AEM) of Katsikadelis. Besides its accuracy and effectiveness, this method overcomes the shortcoming of a possible FEM solution which may experience a lack of convergence. The problems treated in this investigation include beam columns with various load distributions, such as constant, linear and parabolic. Some of the conclusions detected in studying the nonlinear dynamic stability of Beck’s column with variable cross section (Katsikadelis and Tsiatas, Nonlinear dynamic stability of damped Beck’s column with variable cross section. Int. J. Non-linear Mech. 42, 164–171, 2007), are also valid for the case of distributed loads. The important, however, finding is that the post-critical response under distributed loads depends on the law of distribution of mass and stiffness properties, which may lead also to explosive flutter (unbounded amplitude), in contrast to Beck’s column (end-tip load) where the motion is always bounded.     

A theory of finite deformation magneto-viscoelasticity

This paper deals with the mathematical modelling of large strain magneto-viscoelastic deformations. Energy dissipation is assumed to occur both due to the mechanical viscoelastic effects as well as the resistance offered by the material to magnetisation. Existence of internal damping mechanisms in the body is considered by decomposing the deformation gradient and the magnetic induction into ‘elastic’ and ‘viscous’ parts. Constitutive laws for material behaviour and evolution equations for the non-equilibrium fields are derived that agree with the laws of thermodynamics. To illustrate the theory the problems of stress relaxation, magnetic field relaxation, time dependent magnetic induction and strain are formulated and solved for a specific form of the constitutive law. The results, that show the effect of several modelling parameters on the deformation and magnetisation process, are illustrated graphically.     

Experimental evidence of flutter and divergence instabilities induced by dry friction

Flutter and divergence instabilities have been advocated to be possible in elastic structures with Coulomb friction, but no direct experimental evidence has ever been provided. Moreover, the same types of instability can be induced by tangential follower forces, but these are commonly thought to be of extremely difficult, if not impossible, practical realization. Therefore, a clear experimental basis for flutter and divergence induced by friction or follower-loading is still lacking. This is provided for the first time in the present article, showing how a follower force of tangential type can be realized via Coulomb friction and how this, in full agreement with the theory, can induce a blowing-up vibrational motion of increasing amplitude (flutter) or an exponentially growing motion (divergence). In addition, our results show the limits of a treatment based on the linearized equations, so that nonlinearities yield the initial blowing-up vibration of flutter to reach eventually a steady state. The presented results give full evidence to potential problems in the design of mechanical systems subject to friction, open a new perspective in the realization of follower-loading systems and of innovative structures exhibiting ‘unusual’ dynamical behaviors.     

On damping properties of a frictionless physical pendulum with a moving mass

The damping effects are generated in a frictionless oscillating physical pendulum by a continuous motion of an auxiliary mass. The main parameters affecting the damping properties of the pendulum-mass system are identified. In particular, the effective damping ratio for a cycle is introduced and derived in a closed form from the energy considerations and then independently from Mathieu's equation. It is shown that a continuous damping can be achieved if the mass motion is synchronized with the pendulum rotation. Otherwise the system becomes prone to ‘beating’ phenomenon. The results presented may be useful for design of active control strategy of autonomous systems with negligible passive damping.     

FLUTTER INSTABILITY OF SUPPORTED PIPES CONVEYING FLUID SUBJECTED TO DISTRIBUTED FOLLOWER FORCES

In the past decades,it has been reported that divergence is the expected form of instability for fluid-conveying pipes with both ends supported.In this paper,the form of instability of supported pipes ...     

Die Wirkung äu\erer und innerer Dämpfung auf die Stabilität des Prandtlschen Kragträgers unter mitgehender Momentenbelastung

übersicht über das Prinzip von Hamilton wird das nichtlineare Randwertproblem eines kippenden Kragträgers hergeleitet, der durch ein mitgehendes, zeitunabhängiges Endmoment belastet wird. Die Stabilität des sich einstellenden Gleichgewichtszustandes wird untersucht. Der Einflu\ sowohl äu\erer als auch innerer Dämpfung auf die kritische Last wird eingehend diskutiert.

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Effects of external and internal damping on the stability of Prandtl's cantilever subjected to a follower bending moment

Summary Using Hamilton's principle the non-linear boundary value problem for lateral buckling of a cantilever beam subjected to a follower time-independent bending moment is derived. The stability of the resulting state of equilibrium is studied. The influence of external and internal damping on the critical load is discussed in detail.

     

Dynamic stability analysis of shear-flexible composite beams

Dynamic stability behavior of the shear-flexible composite beams subjected to the nonconservative force is intensively investigated based on the finite element model using the Hermitian beam elements. For this, a formal engineering approach of the mechanics of the laminated composite beam is presented based on kinematic assumptions consistent with the Timoshenko beam theory, and the shear stiffness of the thin-walled composite beam is explicitly derived from the energy equivalence. An extended Hamilton’s principle is employed to evaluate the mass-, elastic stiffness-, geometric stiffness-, damping-, and load correction stiffness matrices. Evaluation procedures for the critical values of divergence and flutter loads of the nonconservative system with and without damping effects are then briefly introduced. In order to verify the validity and the accuracy of this study, the divergence and flutter loads are presented and compared with the results from other references, and the influence of various parameters on the divergence and flutter behavior of the laminated composite beams is newly addressed: (1) variation of the divergence and flutter loads with or without the effects of shear deformation and rotary inertia with respect to the nonconservativeness parameter and the fiber angle change, (2) influence of the internal and external damping on flutter loads whether to consider the shear deformation or not.     

Lumped damping and stability of Beck column with a tip mass

Lumped external damping, acting on the concentrated mass at the tip of Beck’s column, is taken into account. This additional external damping removes the prolonged inconsistency between theory and experiment for a column with a tip mass, subjected to follower force.In Sugiyama et al. [J. Aerospace Eng. 8 (1995) 9] and Wood et al. [Proc. Roy. Soc. Lond. Ser. A. 313 (1969) 239] the experimental critical loads for a beam with a tip mass agree well with theoretical values for an undamped system. On the other hand, the critical loads, calculated in Sugiyama et al. with account for the internal and distributed external damping, were 50% lower than the experimental values. Since the actual experimental system is subject to damping, the experiment and theory come into conflict. This conflict is resolved in this paper by observing that the experimental system was subjected not only to the distributed external damping (due to distributed mass), but also to the lumped damping, acting on the tip mass. When the lumped external damping included in the analysis, analytical critical forces are in good agreement with experimental ones.     

Enhancing flutter and buckling capacity of column by piezoelectric layers

This paper illustrates the use of a pair of piezoelectric layers in increasing the flutter and buckling capacity of a column subjected to a follower force. The column is fixed at one end while the other one is free to rotate but constrained transversely by a spring. The mathematical formulation is presented and solved numerically. The effect of the spring stiffness on the capacity and type of instability of the column is first illustrated numerically for the case without any piezoelectric actuators. A transition value for the stiffness can be identified, below which the column fails by flutter and above which the column buckles. Next, an external voltage is applied on the piezoelectric layers bonded on the surfaces of the column, which induces locally a pair of tensile follower force. This has the effect of increasing the capacity of the column as the voltage increases while the transition stiffness remains virtually unchanged for a given size and location of piezoelectric actuators. It is also shown that the capacity of the column increases with longer layers for a fixed voltage. However, the location of the layers along the column determines the transition stiffness and hence has an effect on the type of failure for a fixed spring constant. Positioning towards the fixed end increases the flutter capacity whereas positioning away will result in an increase in buckling capacity.     

Stability analysis of pipes conveying fluid with fractional viscoelastic model

Divergence and flutter instabilities of pipes conveying fluid with fractional viscoelastic model has been investigated in the present work. Attention is concentrated on the boundaries of the stability. Based on the Euler–Bernoulli beam theory for structural dynamics, viscoelastic fractional model for damping and, plug flow model for fluid flow, equation of motion has been derived. The effects of gravity, and distributed follower forces are also considered. By transferring the equation of motion to the Laplace domain and using the Galerkin method, the characteristic equations are obtained. By solving the eigenvalue problem, frequencies and dampings of the system have been obtained versus flow velocity. Some numerical test cases have been studied with viscoelastic fractional model and the effect of the fractional derivative order and the retardation time is investigated for various boundary conditions.

     

Hysteresis in multi-stable lattices with non-local interactions

We study a model for martensitic phase transitions represented by a lattice of mass points connected by bi-stable nearest neighbor (NN) springs and harmonic next to nearest neighbor (NNN) springs. Our main assumption of weak NNN interactions allows us to obtain a fully analytical representation of the quasistatic evolution of the overdamped system, including both the ‘non-local’ interaction with the external load and the presence of imperfections.This simple model reproduces the experimental observation of different evolution strategies, with internal or boundary nucleation and with the possibility of one or more coherently propagating phase fronts. The model describes also the observation of a Peierls stress higher or lower than the nucleation stress. We show that all these properties are also preserved in the ‘continuum’ limit.     

泥岩损伤特性试验研究

以南京长江三桥地基中的泥岩为对象,对泥岩进行三轴试验。试验结果表明：随着侧压的增大,破坏荷载增大,塑性变形明显增大,岩石破坏后,残余强度随侧压增大而提高。在此基础上研究分析了泥岩微元强度服从Weibull分布,泥岩微元体破坏服从莫尔-库仑岩石强度判据时的损伤软化参数与围压的关系特征。结合岩石破裂过程应力 应变全过程曲线,讨论了初始损伤特性,分析结果表明：泥岩初始损伤时的主应力差对数随围压增大而增大,两者呈线性关系; 分析了泥岩损伤变量随主应力差变化关系,结果表明泥岩损伤变量与主应力差呈双曲线数学关系,通过对双曲线模型作线性化处理,结合试验数据采用回归分析法确定模型参数,分析结果发现F 0随围压的增大而增大,而m则随压的增大而减小,反映泥岩随围压的增大,脆性度降低。     

Gradient plasticity constitutive model reflecting the ultrafine micro-structure scale: the case of severely deformed copper

A further development of the mechanism-based strain gradient plasticity model well established in literature is reported. The major new element is the inclusion of the cell size effect in dislocation cell forming materials. It is based on a ‘phase mixture’ approach in which the dislocation cell interiors and dislocation cell walls are treated as separate ‘phases’. The model was applied to indentation testing of copper severely pre-strained by equal channel angular pressing. The deformation behaviour and the intrinsic length scale parameter of the gradient plasticity model were related to the micro-structural characteristics, notably the dislocation cell size, resulting from the deformation history of the material.     

Influence of boundary conditions on the stability of a column under non-conservative load

The cantilever follower force problem with external damping is extended to a three-parameter case, including a concentrated mass, a linear elastic spring, and a partial follower force at the free end. As a result of the study an unexpected, hitherto unrecognized feature of stability/vibration is identified.Normally, the boundary conditions have great influence on the stability limits. However, it is proved that there exists a force at which the critical frequency is independent of all three boundary parameters. The characteristics of this force/frequency combination are discussed in detail, especially in relation to the corresponding eigenfunctions.Also a direct study of the onset of flutter as a function of the boundary parameters is included.     

Active feedback control of a beam subjected to a nonconservative force

This study examines the possibility of controlling through feedback a thin cantilevered beam subjected to a nonconservative follower force. A converging frequency flutter instability which occurs in this model is similar to classical bending-torsion flutter of an aircraft wing. Because of the similar nature of the instabilities, the beam under the follower force can be a useful vehicle for investigating the fundamental aspects of stabilization of wing flutter by feedback control. A modal approach is used for obtaining the mathematical model and control laws. A standard root locus technique for simple analytical models is also used to understand and explain the control of the beam. Experiments are carried out to verify the validity of this theoretical model. Good correlation is shown between theoretically and experimentally determined stability boundaries as well as for modal frequency and damping variation with follower force.