Non-conservative stability of intermediate spring supported columns with an elastically restrained end support

The non-conservative stability of an intermediate spring supported uniform column clastically restrained at one end and subjected to a follower force at the other unsupported end is studied. It is found that when the intermediate spring support is far from the unsupported end, the instability mechanism is flutter. As the intermediate spring support approaches the unsupported end, the instability mechanism is changed from flutter to divergence with the increase of intermediate spring stiffness. For the hinged-intermediate and guided-intermediatc spring supported columns, the critical buckling load of flutter instability will first decrease, then increase as the intermediate spring stiffness is increased. Nevertheless, when the instability mechanism is divergence, the critical buckling load depends on the location of the intermediate spring support only, whereas for the clamped-intermediate spring supported column the critical buckling load of divergence instability decreases monotonically to a fixed value as the intermediate spring stiffness is increased. Finally, the influence of elastic end restraints on the stability of the column is also investigated.     

Elastic stability of laminated elastomeric columns

The effect of axial load on the behavior of columns consisting of alternate layers of rubber and curved or flat rigid shims is analyzed. First, finite difference equations governing the column response are derived. Relative movement between adjacent shims is assumed to be linear with respect to shearing force and bending moment, but the relation between axial load and this movement is not required to be linear. The equations are then approximated by letting the number of layers become infinite while the column length remains finite. The resulting differential equations for this so-called continuum column are solved for critical loads and mode shapes for globally homogeneous columns with three different end conditions. A general discrete analysis of the same problem, which is more appropriate for columns with a small number of layers and for nonhomogeneous columns, is also presented. Comparison of the two analyses establishes the limits of applicability of the approximate continuum analysis. The theory is then shown to agree quite well with experimental results for critical loads and mode shapes.     

Finding the buckling load of non-uniform columns using the iteration perturbation method

The aim of this study is to calculate the critical load of variable inertia columns. The example studied in this paper can be used as a paradigm for other non-uniform columns. The wavelength of equivalent vibratory system is used to calculate the critical load of the trigonometrically varied inertia column. In doing so, the equilibrium equation of the column is theoretically studied using the perturbation method. Accuracy of the calculated results is evaluated by comparing the solution with numerical results. Effect of improving the initial guess on the solution accuracy is investigated. Effects of varying parameters of the trigonometrically varied inertia and the uniformly tapered columns on their stability behavior are studied. Finally, using the so-called “perfectibility” parameter, two design goals, i.e., being lightweight and being strong, are studied for the discussed columns.     

Exact solutions for buckling of non-uniform columns under axial concentrated and distributed loading

In this paper, the buckling problem of non-uniform columns subjected to axial concentrated and distributed loading is studied. The expression for describing the distribution of flexural stiffness of a non-uniform column is arbitrary, and the distribution of axial forces acting on the column is expressed as a functional relation with the distribution of flexural stiffness and vice versa. The governing equation for buckling of a non-uniform column with arbitrary distribution of flexural stiffness or axial forces is reduced to a second-order differential equation without the first-order derivative by means of functional transformations. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 12 cases, several of which are important in engineering practice. The exact solutions that represent a class of exact functional solutions for the buckling problem of non-uniform columns subjected to axial concentrated and distributed loading are obtained. In order to illustrate the proposed method, a numerical example is given in the last part of this paper.     

Elastica type dynamic stability analysis of viscoelastic columns

Summary The dynamic stability analysis of a uniform, homogeneous, simply supported column, subjected to a periodic axial force, is presented. The viscoelastic behaviour is given in terms of the Boltzmann superposition principle. The equation of motion, derived within theelastica and including variations in the column's length, is in the form of a nonlinear integro-differential equation. The stability analysis of this equation is carried out within the Lyapunov exponents concept, which is also used together with the Fourier power spectrum, in order to examine the possibility of a chaotic situation.

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Dynamische Stabilitätsanalyse vom Elastica-Typ für viskoelastische Säulen

Übersicht Im Rahmen der Elastica-Annahmen wird eine dynamische Stabilitätsanalyse einer gleichmäßigen, homogenen, einfach gestützten Säule für periodische axiale Kraftanregung durchgeführt. Das viskoelastische Verhalten des Werkstoffes wird nach dem Boltzmann'schen Superpositionsprinzip modelliert. Die Bewegungsgleichung berücksichtigt die Längenänderung der Säule und wird in integro-differentialer Form aufgestellt. Ihre Störungsanalyse beruht auf dem Lyapunov'schen Exponenten-Konzept. Das Spektrum der Fourier'schen Potenzen wird zur Überprüfung der Möglichkeit einer chaotischen Lösung verwendet.

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This research was jointly supported by the French-Israeli Scientific Cooperation Exchange and by the Ben-Gurion University of the Negev. They are greatefully acknowledged.     

Non-conservative oscillatory systems with periodic solutions

This note extends the work of previous authors to the closed orbits of non-linear oscillatory systems. The simple phase plane analysis is used.     

热释电体非保守动力学系统的守恒定律

以Noether定理为基础,系统地研究热机电耦合的热释电体非保守动力学系统的守恒定律,引进熵流矢量和温度耗散函数描述热释电体系统的耗散现象,提出了热释电体非保衬动力学系统的Lagrange函数以及广义Hamilton最小作用量原理,论证了不变性变换群的存在条件,提出了并证明了广义Noether定理,由此得到了一组守恒定律及J。M积分。     

多介质流体非守恒律欧拉方程组的数值计算方法

对多介质流体在界面处满足的Euler方程进行了探讨 ,方程组中增加了描述材料参数间断性质的对流形式非守恒律方程组。以波传播算法为基础 ,通过Roe方程近似求解Riemann问题 ,同时采用相同的数值差分格式求解流体动力学Euler方程组和界面方程组。该方法可以有效消除多介质流体在界面处压力、速度可能出现的非物理振荡。给出了部分典型一维和二维数值计算结果。     

多介质流体非守恒律欧拉方程组的数值计算方法

对多介质流体在界面处满足的Euler方程进行了探讨 ,方程组中增加了描述材料参数间断性质的对流形式非守恒律方程组。以波传播算法为基础 ,通过Roe方程近似求解Riemann问题 ,同时采用相同的数值差分格式求解流体动力学Euler方程组和界面方程组。该方法可以有效消除多介质流体在界面处压力、速度可能出现的非物理振荡。给出了部分典型一维和二维数值计算结果。     

非保守力作用下FGM矩形板的稳定性分析

对受均布随从力作用的功能梯度材料(FGM)矩形板,引入应力函数,得到了以应力函数和挠度函数表示的耦合运动微分方程组。用Fourier级数法研究了四边简支FGM非保守矩形板的稳定性,给出了不同边长比和不同梯度指标下频率和发散载荷的变化曲线,以及梯度指标变化对频率和发散载荷的影响。     

The nonlinear stability of a heavy rigid plate supported by flexible columns

A heavy rigid platform is supported by thin elastic legs. The governing equations for large deformations are formulated and solved numerically by homotopy and quasi-Newton methods. Nonlinear phenomena such as nonuniqueness, catastrophe and hysteresis are found. A global critical load for nonlinear stability is introduced.     

Hypersonic flow over a multi-step afterbody

Effect of a multi-step base on the total drag of a missile shaped body was studied in a shock tunnel at a hypersonic Mach number of 5.75. Total drag over the body was measured using a single component accelerometer force balance. Experimental results indicated a reduction of 8% in total drag over the body with a multi-step base in comparison with the base-line (model with a flat base) configuration.The flow fields around the above bodies were simulated using a 2-D axisymmetric Navier–Stokes solver and the simulated results on total drag were compared with the measured results. The simulated flow field pictures give an insight into the involved flow physics. Communicated by K. Takayama PACS 47.40.Ki     

The effect of non-uniform bending deformation on the stability of circumferential growth of through-wall cracks in brass tubes

Experimental results on the stability of circumferential growth of through-wall cracks in brass tubes, show that non-uniform bending can adversely affect the crack stability criterion. The results are relevant to the important technological problem of crack stability in 304 stainless steel pipes used in Boiling Water Reactors.     

A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t _f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p₁ >0) and the duration t _f does not exceed 2, i.e., p₁t _f <2. The proposed criterion (p₁t _f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p₁t _f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.     

Conditions of stability and instability for a pair of arbitrarily stratified compressible fluids in an arbitrary non-uniform gravity field

The work continues and develops authors’ previous investigation of stability in the small for a two-layer system of inhomogeneous compressible fluids in the uniform gravity field. Here we present a solution of a similar problem in the case of arbitrary non-uniform potential gravity field. The equilibrium stratification of both density and elastic properties of the fluids is supposed arbitrary, as well as the shape of open on top reservoir filled by the fluids. The problem of stability of equilibrium is analyzed as the corresponding problem for the non-linearly elastic bodies, basing on the static energy criterion with regard for the boundary conditions at all parts of the boundary. The crucial element of the analysis is conversion of the quadratic functional of second variation of total potential energy of the system into a “canonical” form that enables to determine its sign. Making use of this canonical form, we obtain almost coinciding with each other necessary and sufficient conditions for stability (those being valid also for an arbitrary number of layers).     

Non-conservative pressure-based compressible formulation for multiphase flows with heat and mass transfer

A pressure-based compressible multiphase flow solver has been developed based on non-conservative discretization of the mixture continuity equation. The formulation is an extension of the single phase incompressible pressure-correction approach, such that it can be applied to both two-phase flows using interface resolving methods and general n-phase ensemble-averaged mixture flows. The formulation is currently presented with the single pressure and single temperature assumption, but extension to multiple temperatures is straightforward. A robust treatment of phase change allows the method to model conditions with rapid phase change such as expansion through nozzles and valves. The method has been validated thoroughly using canonical single phase problems such as the shock tube, tank filling and sudden valve closure problems. Multiphase flow validation has been carried out for sound propagation in mixtures using the ensemble-averaged model and pressure wave transmission and reflection across an air-water interface, using the level set interface tracking method. The method has been used to study sound propagation in saturated steam-water systems under thermodynamic non-equilibrium, where the expected drastic reduction in the speed of sound is reproduced. Finally the method is applied to the problem of critical (choked) flow in a nozzle for a saturated steam-water system.     

On non-inflectional solutions of non-uniform elasticae

We study buckled states of non-uniform elasticae and show that solutions having pairs of loops not separated by inflection points appear if and only if there is an (isolated) minimum in the bending stiffness. Such solutions lie on branches disconnected from the trivial solution in the global bifurcation diagram and so we obtain partial information regarding the qualitative structure of the latter.