Buckling and post-buckling of long pressurized elastic thin-walled tubes under in-plane bending

The present paper focuses on the structural stability of long uniformly pressurized thin elastic tubular shells subjected to in-plane bending. Using a special-purpose non-linear finite element technique, bifurcation on the pre-buckling ovalization equilibrium path is detected, and the post-buckling path is traced. Furthermore, the influence of pressure (internal and/or external) as well as the effects of radius-to-thickness ratio, initial curvature and initial ovality on the bifurcation moment, curvature and the corresponding wavelength, are examined. The local character of buckling in the circumferential direction is also demonstrated, especially for thin-walled tubes. This observation motivates the development of a simplified analytical formulation for tube bifurcation, which considers the presence of pressure, initial curvature and ovality, and results in closed-form expressions of very good accuracy, for tubes with relatively small initial curvature. Finally, aspects of tube bifurcation are illustrated using a simple mechanical model, which considers the ovalized pre-buckling state and the effects of pressure.     

Discontinuous fold bifurcations in mechanical systems

Summary  This paper treats discontinuous fold bifurcations of periodic solutions of discontinuous systems. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. A Floquet multiplier of a discontinuous system can jump through the unit circle, causing a discontinuous bifurcation. Numerical examples are treated, which show discontinuous fold bifurcations. A discontinuous fold bifurcation can connect stable branches to branches with infinitely unstable solutions. Received 20 September 2000; accepted for publication 26 June 2001     

Complete solution of the stability problem for elastica of Euler's column

The stability of postcritical equilibrium forms of a simply supported column loaded with an axial force is analyzed. Investigating the sign of the second variation of the column's total energy, we obtain the Sturm-Liouville boundary-value problem, which is solved numerically. The stability conditions are formulated in terms of eigenvalues of the problem. The complete solution to the column plane elastica is given. The ranges of the compressive force corresponding to stable equilibrium configurations of the column are established.     

Flexible Wiper System Dynamic Instabilities: Modelling and Experimental Validation

The optimization of wiper systems under various conditions and the creation of a product which is as robust as possible are the main objectives for an equipment supplier. However, in certain conditions, instabilities can appear and generate wiping defects due to the rubber-glass contact. To improve wiping quality and to reduce the number of test stages for design, this study proposes a wiper system modeling method. The wiper system is represented by a rigid blade holder on which a rubber blade is fitted. This rigid blade system is used on a flat test bench at constant wiping velocity. The model is based on modal synthesis methods and will be validated through comparison with experimental tests under various conditions. The right correlation obtained allows the same modelling method to be applied to the new generation of flexible wiper blades which take account of the degree of freedom of the wiper blade flexions. So, a new computation tool will be developed and validated through experimentation on a specific test bench.     

Rain-wind-induced vibrations of a simple oscillator

In this paper a relatively simple mechanical oscillator which may be used to study rain-wind-induced vibrations of stay cables of cable-stayed bridges is considered. In recent publications, mention is made of vibrations of (inclined) stay cables which are excited by a wind field containing rain drops. The rain drops that hit the cables generate a rivulet on the surface of the cable. The presence of flowing water on the cable changes the cross section of the cable experienced by the wind field. A symmetric flow pattern around the cable with circular cross section may become asymmetric due to the presence of the rivulet and may consequently induce a lift force as a mechanism for vibration. During the motion of the cable the position of rivulet(s) may vary as the motion of the cable induces an additional varying aerodynamic force perpendicular to the direction of the wind field. It seems not too easy to model this phenomenon, several author state that there is no model available yet.The idea to model this problem is to consider a horizontal cylinder supported by springs in such a way that only one degree of freedom, i.e. vertical vibration is possible. We consider a ridge on the surface of the cylinder parallel to the axis of the cylinder. Additionally, let the cylinder with ridge be able to oscillate, with small amplitude, around the axis such that the oscillations are excited by an external force.It may be clear that the small amplitude oscillations of the cylinder and hence of the ridge induce a varying lift and drag force. In this approach it is assumed that the motion of the ridge models the dynamics of the rivulet(s) on the cable. By using a quasi-steady approach to model the aerodynamic forces, one arrives at a non-linear second-order equation displaying three different kinds of excitation mechanisms: self-excitation, parametric excitation and ordinary forcing. The first results of the analysis of the equation of motion show that even in a linear approximation for certain values of the parameters involved, stable periodic motions are possible. In the relevant cases where in linear approximation unstable periodic motions are found, results of an analysis of the non-linear equation are presented.     

Positive solutions for Lotka–Volterra competition system with large cross-diffusion in a spatially heterogeneous environment

In this paper, we study the stationary problem for the Lotka–Volterra competition system with cross-diffusion in a spatially heterogeneous environment. Although some sufficient conditions for the existence of positive solutions are obtained by using global bifurcation theory, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we focus on the asymptotic behaviour of positive solutions and derive two shadow systems as the cross-diffusion coefficient tends to infinity, moreover, the structure of positive solutions of the limiting system is analysed. The result of asymptotic behaviour also reveals different phenomena from that studied in Wang and Li (2013).     

Multicloud Convective Parametrizations with Crude Vertical Structure

Recent observational analysis reveals the central role of three multi-cloud types, congestus, stratiform, and deep convective cumulus clouds, in the dynamics of large scale convectively coupled Kelvin waves, westward propagating two-day waves, and the Madden–Julian oscillation. The authors have recently developed a systematic model convective parametrization highlighting the dynamic role of the three cloud types through two baroclinic modes of vertical structure: a deep convective heating mode and a second mode with low level heating and cooling corresponding respectively to congestus and stratiform clouds. The model includes a systematic moisture equation where the lower troposphere moisture increases through detrainment of shallow cumulus clouds, evaporation of stratiform rain, and moisture convergence and decreases through deep convective precipitation and a nonlinear switch which favors either deep or congestus convection depending on whether the troposphere is moist or dry. Here several new facets of these multi-cloud models are discussed including all the relevant time scales in the models and the links with simpler parametrizations involving only a single baroclinic mode in various limiting regimes. One of the new phenomena in the multi-cloud models is the existence of suitable unstable radiative convective equilibria (RCE) involving a larger fraction of congestus clouds and a smaller fraction of deep convective clouds. Novel aspects of the linear and nonlinear stability of such unstable RCE’s are studied here. They include new modes of linear instability including mesoscale second baroclinic moist gravity waves, slow moving mesoscale modes resembling squall lines, and large scale standing modes. The nonlinear instability of unstable RCE’s to homogeneous perturbations is studied with three different types of nonlinear dynamics occurring which involve adjustment to a steady deep convective RCE, periodic oscillation, and even heteroclinic chaos in suitable parameter regimes.     

Diffuse mode bifurcation of soil causing convection-like shear investigated by group-theoretic image analysis

Diffuse mode bifurcation of soil under plane-strain compression test is shown, by means of an image analysis based on group-theoretic bifurcation theory, to trigger convection-like shear and to precede shear band formation. First digital photos of Toyoura sand specimens are processed by PIV (particle image velocimetry) to gather digitized images of deformation. Next bifurcation from a uniform state is detected by expanding these images into the double Fourier series and finding a predominant harmonic diffuse bifurcation mode based on that theory. This harmonic bifurcation mode, which is the mixture of a few harmonic functions, expresses complex convection-like shear. Last bifurcation from a non-uniform state is detected by decomposing each image into a few images with different symmetries to extract non-harmonic diffuse bifurcation modes. Diffuse modes of bifurcation, which hitherto were hidden behind predominant uniform compressive deformation, have thus been made transparent by virtue of the group-theoretic image analysis proposed. A possible course of deformation suggested herein is the evolution of diffuse mode bifurcation with a convection-like bifurcation mode breaking uniformity and symmetry, followed by the formation of shear bands through localization.     

Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation

This paper concerns the quadratically-damped Mathieu equation:

     

Effect of Lateral Linear Stiffness on Nonlinear Characteristics of Hunting Motion of a Railway Wheelset

Railway wheelset experiences the problem of hunting above a critical speed, which is a kind of self-excited oscillation. At the critical speed, it is known that the system undergoes a subcritical Hopf bifurcation. Therefore, for clarifying the nonlinear characteristics of hunting it is very important to detect, for example, the nonlinear forces in the wheelset due to the creep forces acting between the wheels and rails, and the nonlinear component of the resorting forces by the suspensions. However, it is impossible to determine each force quantitatively. In the present paper, it is first shown, by using the center manifold theory and the method of normal form, that the nonlinear characteristics of the bifurcation in a wheelset model with two degrees of freedom are governed by a single parameter, hence each nonlinear force need not be detected when examining the nonlinear characteristics. Also, a method of determining the governing parameter from experimentally observed radiuses of the unstable limit cycle is proposed. Next, we experimentally investigate the variation of the parameter due to the presence of linear spring suspensions in the lateral direction and discuss the variation of the nonlinear characteristics of the hunting motion, which depends on the lateral stiffness. As a result, the improvement of the stability of the wheelset against the disturbance by the linear spring suspensions is clarified.