• Structural Engineering and Mechanics
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• 제64권6권
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• pp.723-730
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• 2017

This paper studies on dynamic and stability behavior of a clamped-elastically restrained nanobeam under the action of a nonconservative force with an emphasis on the influence of surface properties on divergence and flutter instability. Using the Euler-Bernoulli beam theory incorporating surface effects, a governing equation for a clamped-elastically restrained nanobeam is derived according to Hamilton's principle. The characteristic equation is obtained explicitly and the force-frequency interaction curves are displayed to show the influence of the surface effects, spring stiffness of the elastic restraint end on critical loads including divergence and flutter loads. Divergence and flutter instability transition is analyzed. Euler buckling and stability of Beck's column are some special cases of the present at macroscale.

• Wind and Structures
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• 제28권4호
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• pp.255-270
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• 2019

Aerodynamic configurations of bridge decks have significant effects on the aerostatic torsional divergence and flutter forsuper long-span bridges, which are onset for selection of suitable bridge decksfor those bridges. Based on a cable-stayed bridge with double main spans of 1500 m, considering typical twin-box, stiffening truss and closed-box section, which are the most commonly used form of bridge decks and assumed that the rigidity of those section is completely equivalent, are utilized to investigate the effects of aerodynamic configurations of bridge decks on aerodynamic instability performance comprised of the aerostatic torsional divergence and flutter, by means of wind tunnel tests and numerical calculations, including three-dimensional (3D) multimode flutter analysis and nonlinear aerostatic analysis. Regarding the aerostatic torsional divergence, the results obtained in this study show twin-box section is the best, closed-box section the second-best, and the stiffening truss section the worst. Regarding the flutter, the flutter stability of the twin-box section is far better than that of the stiffening truss and closed-box section. Furthermore, wind-resistance design depends on the torsional divergence for the twin-box and stiffening truss section. However, there are obvious competitive relationships between the aerostatic torsional divergence and flutter for the closed-box section. Flutter occur before aerostatic instability at initial attack angle of $+3^{\circ}$ and $0^{\circ}$, while the aerostatic torsional divergence occur before flutter at initial attack angle of $-3^{\circ}$. The twin-box section is the best in terms of both aerostatic and flutter stability among those bridge decks. Then mechanisms of aerostatic torsional divergence are revealed by tracking the cable forces synchronous with deformation of the bridge decksin the instability process. It was also found that the onset wind velocities of these bridge decks are very similar at attack angle of $-3^{\circ}$. This indicatesthat a stable triangular structure made up of the cable planes, the tower, and the bridge deck greatly improves the aerostatic stability of the structure, while the aerodynamic effects associated with the aerodynamic configurations of the bridge decks have little effects on the aerostatic stability at initial attack angle of $-3^{\circ}$. In addition, instability patterns of the bridge depend on both the initial attack angles and aerodynamic configurations of the bridge decks. This study is helpful in determining bridge decksfor super long-span bridges in future.

• Structural Engineering and Mechanics
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• 제16권2호
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• pp.195-217
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• 2003

This paper, which is an extension of a previous work by Viola et al. (2002), deals with the dynamic stability of beams under a triangularly distributed sub-tangential forces when the effect of an elastically restrained end is taken into account. The sub-tangential forces can be realised by a combination of axial and tangential follower forces, that are conservative and non-conservative forces, respectively. The studied beams become unstable in the form of either flutter or divergence, depending on the degree of non-conservativeness of the distributed sub-tangential forces and the stiffness of the elastically restrained end. A non-conservative parameter ${\alpha}$ is introduced to provide all possible combinations of these forces. Problems of this kind are usually, at least in the first approximation, reduced to the analysis of beams according to the Bernoulli-Euler theory if shear deformability and rotational inertia are negligible. The equation governing the system may be derived from the extended form of Hamilton's principle. The stability maps will be obtained from the eigenvalue analysis in order to define the divergence and flutter domain. The passage from divergence to flutter is associated with a noticeable lowering of the critical load. A number of particular cases can be immediately recovered.

• 대한기계학회논문집A
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• 제28권2호
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• pp.143-148
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• 2004

This paper discussed on the stability of elastic material subjected to dry friction force for low boundary conditions: clamped free, clamped-simply supported, simply supported-simply supported, clamped-clamped. It is assumed in this paper that the dry frictional force between a tool stand and an elastic material can be modeled as a distributed follower force. The friction material is modeled for simplicity into a Winkler-type elastic foundation. The stability of beams on the elastic foundation subjected to distribute follower force is formulated by using finite element method to have a standard eigenvalue problem. It is found that the clamped-free beam loses its stability in the flutter type instability, the simply supported-simply supported beam loses its stability in the divergence type instability and the other two boundary conditions the beams lose their stability in the divergence-flutter type instability.

• International Journal of Aeronautical and Space Sciences
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• 제12권2호
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• pp.134-148
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• 2011

In general, forces acting on aerospace structures can be divided into two categories-a) conservative forces and b) nonconservative forces. Aeroelastic effects occur due to highly flexible nature of the structure, coupled with the unsteady aerodynamic forces, causing unbounded static deflection (divergence) and dynamic oscillations (flutter). Flexible wing panels subjected to jet thrust and missile type of structures under end rocket thrust are nonconservative systems. Here the structural elements are subjected to follower kind of forces; as the end thrust follow the deformed shape of the flexible structure. When a structure is under a constant follower force whose direction changes according to the deformation of the structure, it may undergo static instability (divergence) where transverse natural frequencies merge into zero and dynamic instability (flutter), where two natural frequencies coincide with each other resulting in the amplitude of vibration growing without bound. However, when the follower forces are pulsating in nature, another kind of dynamic instability is also seen. If certain conditions are satisfied between the driving frequency and the transverse natural frequency, then dynamic instability called 'parametric resonance' occurs and the amplitude of transverse vibration increases without bound. The present review paper will discuss the aeroelastic behaviour of aerospace structures under nonconservative forces.

• 한국지진공학회논문집
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• 제9권6호
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• pp.1-12
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• 2005

비보존력(non-conservative force)을 받는 외팔기둥의 동적 안정성 거동특성을 조사하기 위하여, 전단변형 및 감쇠효과가 고려된 Hamilton의 원리를 적용하고 무차원화 된 운동방정식 및 유한요소 정식화 과정을 제시한다. 유도된 행렬운동방정식을 이용하여 외팔보(Beck's column)의 고유치해석에 의한 정적좌굴(divergence) 및 동적 좌굴하중(flutter load)을 산정하고 $Newmark-{\beta}$법에 의해서 시간응답해석을 실시한다. 이러한 해석법을 이용한 매개변수연구를 통하여 전단변형 및 회전관성효과, 비보존력의 방향파라미터에 대한 임계하중의 영향, 그리고 내적 및 외적 감쇠하중의 영향이 비보존력계의 동적 안정성에 미치는 영향을 분석한다.

• 대한기계학회논문집
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• 제18권6호
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• pp.1496-1502
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• 1994

The influences of spring position and spring stiffness on the critical force of a cantilevered beam subjected to a follower force are investigated. The spring attatched to the beam is assumed to be a translational one and can be located at arbitrary positions of the beam as it has not been assumed so far. The effects of transeverse shear deformation and rotary intertia of the beam are also included in this analysis. The charateristic equation for the system is derived and a finite element model of the beam using local coordinates is formulated through extended Hamilton's principle. It is found that when the spring is located at position less than that of 0.5L, the flutter type instability only exists. It is shown that the spring position approaches to the free end of the beam from its midpoint, instability type is changed from flutter to divergence through the jump phenomina according to the increase of spring stiffness.

• 한국소음진동공학회:학술대회논문집
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• 한국소음진동공학회 2005년도 추계학술대회논문집
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• pp.903-906
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• 2005

The purpose of this paper is to investigate free vibrations and critical loads of the uniform Beck's columns with a tip spring, carrying a tip mass. The ordinary differential equation governing free vibrations of such Beck's column subjected to a follower force is derived based on the Bernoulli-Euler beam theory. Both the divergence and flutter critical loads are calculated from the load-frequency curves that are obtained by solving the differential equation numerically. The critical loads are presented in the figures as functions of various non-dimensional system parameters such as the mass moment of inertia and spring parameter.

• 한국전산구조공학회논문집
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• 제17권4호
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• pp.351-363
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• 2004

비보존력을 받는 보-부재의 질량행렬, 탄성강도행릴, circulatory비보존력의 방향변화로 인한 load correction강도행력, 그리고 Winkler 및 Pasternak지반강도행렬을 고려한 운동방정식을 유도하고 divergence 및 flutter에 의한 안정성 해석을 수행한다. 또한 내적 및 외적 감쇠계수를 운동방정식에 포함시킴으로써 감쇠효과를 고려하고, 2차 고유치문제의 해법(quadratic eigen problem solution)을 적용하여 flutter에 미치는 영향을 조사한 후, Beck's column, Leipholz's column 및 Hauger's column에 대하여 비보존력의 방향파라미터 ${\alpha}$에 대한 임계하중의 영향, 내적 및 외적 감쇠계수 및 Winkler 및 Pasternak지반에 의한 임계하중의 영향을 각각 조사한다.

• Wind and Structures
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• 제25권4호
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• pp.343-360
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• 2017

The post-flutter state of streamlined steel box girder is studied in this paper. Firstly, the nonlinear aerodynamic self-excited forces of the bridge deck cross section were investigated by CFD dynamic mesh technique and then the nonlinear flutter derivatives were identified on this basis. Secondly, based on the 2-degree-of-freedom (DOF) coupling flutter theory, the torsional amplitude and the nonlinear flutter derivatives were introduced into the traditional direct flutter calculation method, and the original program was improved to the "post-flutter state analysis program" so that it can predict not only the critical flutter velocity but also the movement of the girder in the post-flutter state. Finally, wind tunnel tests were set to verify the method proposed in this paper. The results show that the effect of vertical amplitude on the nonlinear flutter derivatives is negligible, but the torsional amplitude is not; with the increase of wind speed, the post-flutter state of streamlined steel box girder includes four stages, namely, "little amplitude zone", "step amplitude zone", "linearly growing amplitude zone" and "divergence zone"; damping ratio has limited effect on the critical flutter velocity and the steady state response in the post-flutter state; after flutter occurs, the vibration form is a single frequency vibration coupled with torsional and vertical DOF.