Non-conservative stability of multi-step non-uniform columns

The non-conservative stability of non-uniform columns under the combined action of concentrated and variably distributed forces is solved analytically. Two types of follower force system are considered: (i) concentrated follower forces and variably distributed follower forces, (ii) concentrated follower forces and variably distributed conservative forces. The exact solutions for stability of four kinds of one-step non-uniform columns subjected to the two types of follower force system are derived for the first time. Then a new exact approach, which combines the exact solutions of one-step columns and the transfer matrix method, is presented for the non-conservative stability analysis of multi-step non-uniform columns. The advantage of the proposed method is that the resulting eigenvalue equation for a multi-step non-uniform column with any kinds of two end support configurations, an arbitrary number of spring supports and concentrated masses can be conveniently determined from a second order determinant. The decrease in the determinant order, as compared with previously developed procedures, leads to significant savings in the computational effort. A numerical example shows that the results obtained from the proposed method are in good agreement with those determined from the finite element method (FEM), but the proposed method takes less computational time than FEM.     

Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis

The nonlinear coupled longitudinal-transverse vibrations and stability of an axially moving beam, subjected to a distributed harmonic external force, which is supported by an intermediate spring, are investigated. A?case of three-to-one internal resonance as well as that of non-resonance is considered. The equations of motion are obtained via Hamilton??s principle and discretized into a set of coupled nonlinear ordinary differential equations using Galerkin??s method. The resulting equations are solved via two different techniques: the pseudo-arclength continuation method and direct time integration. The frequency-response curves of the system and the bifurcation diagrams of Poincaré maps are analyzed.     

Stability and bifurcations of an axially moving beam with an intermediate spring support

The forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically in this paper. The equation of motion is obtained via Hamilton??s principle and constitutive relations. This equation is then discretized via the Galerkin method using the eigenfunctions of a hinged-hinged beam as appropriate basis functions. The resultant non-linear ordinary differential equations are then solved via either the pseudo-arclength continuation technique or direct time integration. The sub-critical response is examined when the excitation frequency is set near the first natural frequency for both the systems with and without internal resonances. Bifurcation diagrams of Poincaré maps obtained from direct time integration are presented as either the forcing amplitude or the axial speed is varied; as we shall see, a sequence of higher-order bifurcations ensues, involving periodic, quasi-periodic, periodic-doubling, and chaotic motions.     

Natural frequencies of the elastically end restrained non-uniform Timoshenko beam using the power series method

In recent years, the transverse vibration of the uniform Timoshenko beam has become a highly significant research topic. The Timoshenko beam (TB) model is suitable for describing the behavior of the beams under high-frequency excitation load when the wavelength approaches the thickness of the beam because it considers the shear deformation and the rotational inertia effects. In this paper, the natural frequencies of the elastically end constrained nonuniform TB is discussed. For this reason, the nonuniform TB is modeled by an exact and direct modeling technique. The semi-analytical solution based on the power series method is adopted in technique. The effect of the spring supports, different boundary conditions, the non-uniformity in the cross-section of the TB, and other parameters are assessed. The results are shown that the nonuniformity in the cross-section of the beam is impressed on the natural frequencies. On the other hand, the presented formulation not only effectively is easy to use, but also provides the reasonable solution of non-uniform TB, carrying various boundary conditions.     

Transition radiation in an elastically supported string. Abrupt and smooth variations of the support stiffness

Generation of waves due to a constant load moving uniformly along an infinite string resting on an inhomogeneous elastic foundation is studied. Two types of inhomogeneity are considered: (1) an abrupt change of the foundation stiffness; (2) a smooth change of the foundation stiffness. It is shown that transition radiation arises as the load passes the region of inhomogeneity. Expressions for the spectral density of the radiation energy forwards (in the direction of the load motion) and backwards are found and analysed as a function of the load velocity, the ratio of the foundation stiffness and a characteristic length of the inhomogeneity. To visualise the process of radiation the transient vibrations of the string are determined for an abrupt change of the stiffness.     

Dynamic coefficient of an elastically supported,pre-stressed beam

Summary An analysis is made of the problem of vibrations of a beam with an axial force resting on eltstic foundation, when the beam is uniform and of finite length and is subjected to an impulsive load. The solution is presented within the framework of a beam theory which includes the effects of shear deformation and rotary inertia. An example is provided where the dynamic coefficient for the bending moment is calculated. From the results of theoretical analysis, it becomes evident that the axial force in the beam and the stiffness of the foundation have considerable effect upon the dynamical behaviour of the system.

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Übersicht Die Schwingungen eines auf elastischer Grundlage liegenden Balkens, der durch eine Axialkraft beansprucht ist, werden für den Fall analysiert, daß der Balken gleichförmig und von endlicher Länge ist, und daß eine Schlagbelastung an dem Balken angreift. Die Lösung wird innerhalb der Balkentheorie gegeben, die die Wirkung der Schubverformung und die Drehträgheit berücksichtigt. In einem Beispiel wird der Schwing-beiwert für das Biegemoment ermittelt. Die Ergebnisse zeigen, da\ die Axialkraft und die Steifigkeit der elastischen Grundlage auf das dynamische Verhalten des Systems erheblichen Einfluß haben.

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The author wishes to acknowledge the consistent guidance and encouragent of Professor H, Saito of Tohoku University.     

Stability and optimization of a clamped beam elastically restrained against translation on one end resting on Winkler foundation

In this study, stability and bimodal optimization of clamped beam elastically restrained against translation on one end subjected to a constant axially load are analyzed. The beam is positioned on elastic Winkler type foundation. The Euler method of adjacent equilibrium configuration is used in deriving the nonlinear governing equations. The critical load parameters, axial force and stiffness of foundation, are obtained for beam with the unit cross-sectional area.The shape of the beam stable against buckling that has minimal volume is determined by using Pontryagin’s maximum principle. The optimality conditions for the case of bimodal optimization are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. A first integral (Hamiltonian) is used to monitor accuracy of integration. It is shown that there is the saving in material for the same buckling force.     

Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array

In this paper, the three-dimensional (3-D) non-linear dynamics of a cantilevered pipe conveying fluid, constrained by arrays of four springs attached at a point along its length is investigated. In the theoretical analysis, the 3-D equations are discretized via Galerkin's technique. The resulting coupled non-linear differential equations are solved numerically using a finite difference method. The dynamic behaviour of the system is presented in the form of bifurcation diagrams, along with phase-plane plots, time-histories, PSD plots, and Poincaré maps for five different spring configurations. Interesting dynamical phenomena, such as 2-D or 3-D flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were performed for the cases studied theoretically, and good qualitative and quantitative agreement was observed. The experimental behaviour is illustrated by video clips (electronic annexes). The effect of the number of beam modes in the Galerkin discretization on accuracy of the results and on convergence of the numerical solutions is discussed.     

Formal analysis of elastically supported beam-columns

Summary A method is presented for the systematic analysis of elastically supported beam-columns or tie-bars in which formal notational devices simplify the handling of complicated discontinuous lateral loads. Various relationships between axial load and support modulus are treated and three particular cases (fixed-fixed, pinned-pinned, and free-free ends) are exhibited.

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Übersicht Es wird ein Verfahren zur systematischen Untersuchung von elastisch gelagerten Balken-Säulen oder Verbindungs-Stäben angegeben, wobei die Berücksichtigung komplizierter Querlasten durch einen besonderen Rechenformalismus erleichtert wird. Verschiedenartige Beziehungen zwischen der Längslast und der Bettungsziffer werden untersucht; drei Spezialfälle sind diskutiert, wobei die Enden fest-fest, gelenkiggelenkig bzw. frei-frei gelagert sind.

     

Asymptotic long wave models for a pre-stressed elastic layer with elastically restrained boundaries

Long wave dispersion phenomena is investigated in respect of a pre-stressed incompressible elastic layer subject to elastically restrained boundary conditions (ERBC). Such conditions can be treated as a generalisation of classical free and fixed-face boundary conditions, allowing investigating of the transition between the Neumann and Dirichlet statements of the problem. Symmetric elastically restrained boundary conditions are introduced, followed by both a numerical investigation and a multi-parameter asymptotic analysis of the dispersion relations. All possible asymptotic regimes are grouped into classes based on the magnitude of the associated restraint parameter. A long wave low frequency model is developed to describe motion associated with the fundamental modes for small values of the restraint parameters. Four high frequency models are developed describing asymptotic regimes connected with vibration within the vicinity of the thickness resonances.     

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.     

弹性约束边界圆环板面内自由振动的二维弹性解

基于二维线弹性理论,应用哈密顿原理导出弹性约束边界圆环板面内自由振动的控制微分方程。采用微分求积法(DQM)数值研究了弹性约束边界圆环板面内自由振动的频率特性。通过设置弹性刚度系数为0或∞,问题退化为四种典型边界圆环板的面内自由振动,与已有文献的计算数值结果进行比较,证实本文的分析求解方法行之有效。最后全面考虑了圆环板边界条件、几何系数及刚度系数对自振频率的影响。     

Hygroelasticity analysis of an elastically restrained functionally graded porous metamaterial circular plate resting on an auxetic material circular plate

The main objective of this research is to investigate the hygroelastic behavior of a non-homogeneous circular plate made up of porous metamaterial resting on an auxetic material plate. The mechanical properties of the main plate, as well as moisture concentration, vary as an exponential function in the transverse direction. Poisson’s ratio is constant. The elastic supporting medium is developed by considering the structurestructure coupling. Based on the linear hygroelasticity theory, the govern...     

Differential quadrature method to stability analysis of pipes conveying fluid with spring support

It is a new attempt to extend the differential quadrature method (DQM) to stability analysis of the straight and curved centerline pipes conveying fluid. Emphasis is placed on the study of the influences of several parameters on the critical flow velocity. Compared to other methods, this method can more easily deal with the pipe with spring support at its boundaries and asks for much less computing effort while giving aceptable precision in the numerical results. Supported by National Key Project of China (No. PD9521907) and the National Science Foundation of China (No. 19872025).     

弹性约束梁参数失稳分析的离散奇异卷积法

针对具有重要的学术价值和工程意义的弹性约束梁在动轴向载荷作用下的参数失稳问题,本文采用离散奇异卷积法对其进行研究。在离散奇异卷积法中,采用正则化的Shannon核进行空间离散,同时采用四阶龙格-库塔法进行时间离散。在采用界面与边界匹配技术处理弹性边界条件后,计算了四种不同弹性约束梁的参数失稳区。研究结果表明:计算结果和采用假定模态法得到的结果基本一致,从而说明了采用离散奇异卷积法求解弹性约束梁的参数失稳问题是可行和有效的;同时采用离散奇异卷积法可以更加准确地处理具有较大轴向力时的结构参数失稳问题。     

Nonlinear dynamic behavior analysis of an elastically restrained double-beam connected through a mass-spring system that is nonlinear

Some complex engineering structures can be modeled as multiple beams connected through coupling elements. When the coupling element is elastic, it can be simplified as a mass-spring system. The existing studies mainly concentrated on the double-beam coupled through elastic connectors, where the connector is simplified as the equivalent linear stiffness element or linear mass-spring system. Furthermore, many researches ignore rotational boundary restraints in analyzing dynamic behavior of the double-beam connected through elastic connectors, limiting their engineering generality. Considering the above limitations, this study attempts to employ the cubic nonlinear stiffness in the coupling mass-spring system and study the potential application of the mass-spring system that is nonlinear on the vibration control of the double-beam system. Using the variational method and the generalized Hamiltonian method build the corresponding system’s governing functions. Applying the Galerkin truncation method (GTM) obtains the dynamic behavior of the double-beam connected through a mass-spring system that is nonlinear. According to this study, the change of the mass-spring system that is nonlinear significantly influences the dynamic behavior of the double-beam system, where the complex dynamic behavior occurs under certain parameters of the mass-spring system that is nonlinear. Suitable parameters of the mass-spring system that is nonlinear are good at the vibration suppression at the boundary of the vibration system. Furthermore, the mass-spring system that is nonlinear can change the characteristics of the double-beam system’s kinetic energy transfer. For the vibration model established in this work, a quasi-periodic vibration state can be regarded as a sign of the occurrence of the targeted energy transfer of the double-beam connected through a mass-spring system that is nonlinear.