Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid

doi:10.1016/j.jsv.2013.11.049

Journal of Sound and Vibration

Volume 333, Issue 9, 28 April 2014, Pages 2543-2555

https://doi.org/10.1016/j.jsv.2013.11.049 Get rights and content

Highlights

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Modeling of cantilever beam attached to an axially moving base in fluid is given.
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Coordinate transformation is introduced to fix the boundaries.
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Vibration and stability of the cantilever beam are studied by Galerkin method.
•
Stability boundaries are obtained by numerical analysis.

Abstract

Free vibration and stability are investigated for a cantilever beam attached to an axially moving base in fluid. The equations of motion of the slender cantilever beam affiliated to an axially moving base at a known rate while immersed in an incompressible fluid are derived first. An “axially added mass coefficient” is taken into account in the obtained equations. Then, a coordinate transformation is introduced to fix the boundaries. Based on the Galerkin approach, the natural frequencies of the beam system are numerically analyzed. The effects of moving speed of the base and several other system parameters on the dynamics and stability of the beam are discussed in detail. It is found that when the moving speed exceeds a certain value the beam becomes unstable and the instability type is sensitive to the system parameters. When the values of system parameters, such as mass ratio and axially added mass coefficient, are big enough, however, no instabilities are detected. The variations of the lowest unstable critical moving speed with respect to several key parameters are also investigated.

Introduction

Dynamical behavior of axially moving system has attracted much attention because of their applications in many fields. The boundaries of the axially moving system are either free or supported. According to the supports of such system being fixed or not, the axially moving systems can be divided into two classes: first, systems with fixed and unmovable support; and second, systems that the axial motion of the structures is induced by the axial movements of the supports. There are big differences between these two types of axially moving system although these two kinds of systems seem to be similar.

It is noted that the axially moving system with fixed support can be further classified into two types, namely, the cantilever system and the supported system (both two ends are supported). Some examples of engineering applications of the cantilever system include, but are not limited to, deploying/extruding process [1], [2], [3]. The effective length of the axial moving cantilever beam is time-dependent in the moving process since the base is fixed [2], [3]. In fact, the axially moving systems with both ends supported are just similar to the control volume of fluid dynamics because the particles contained in the supported system are changed in the moving process. Related engineering applications of the supported system include power transmission band, belt saws, aerial cable tramways, and elevator cables.

The linear and nonlinear dynamics of the supported axially moving materials with fixed support ends have been investigated by many researchers. Some extensive reviews on this subject can be found in [4], [5]. The linear vibrations of band saw [6], [7], axially moving elastic beams [8], [9], [10], [11] were obtained by various methods. The pipe conveying fluid can be modeled as an axially moving string if the flexural stiffness vanishes. Based on this, the nonlinear vibrations and stabilities of tensioned pipes conveying fluid with variable velocity were explored by Öz [12]. Recently, the nonlinear dynamics of axially moving beams have attracted much attention [13], [14], [15], [16], [17], [18]. Bağdatli et al. [19] studied the dynamics of axially moving beams using the method of multiple scales, with the consideration of the intermediate support.

In connection with a number of diverse disciplines, such as robotics and aircraft dynamics, the studies of slender beam attached to axially moving supports, which belongs to the axially moving system with axially moving support (i.e. the second types of axially moving system), have been vigorously pursued for several decades. Kane et al. [20] explored the dynamics of a cantilever beam attached to a moving base. Yoo and his co-works [21], [22] investigated the linear stability of a cantilever beam subjected to axially oscillating base by Kane's method. Feng and Hu [23], [24] developed a set of nonlinear differential equations by using Kane's method for the planar oscillation of slender beams subjected to a parametric excitation through the base movement, with the consideration of cubic geometrical and inertia nonlinearities.

To date, only few works can be found on the axially moving system with the consideration of the fluid–structure interaction. Taleb and Misra [2] investigated the dynamics of a cantilever beam being deployed in a dense incompressible fluid. Subsequently, Gosselin et al. [3] proved that the fluid-dynamic forces were not correctly accounted for in the analysis performed by Taleb and Misra [2]. Gosselin et al. [3] introduced an “axially added mass coefficient”, which was proved to better approximate the force of the surrounding fluid acting on the beam. Recently, Wang and Ni [25] presented numerical results for the natural frequency for a supported axially moving beam in fluid based on the differential quadrature method. However, the axially moving systems presented in [2], [3], [25] belong to the first types of axially moving systems. In other words, the supports of the axially moving systems investigated in [2], [3], [25] were fixed and unmovable. In the case of the second types of axially moving systems surrounded by fluid, there are many engineering examples, such as slender structures towed underwater to the desired location [26] and the probe and drogue aerial refueling [27]. In the first stage of aerial refueling process (i.e. before the probe and drogue connected), a probe attached to the receiving aircraft can be modeled as a cantilever beam attached to an axially moving base immersed in fluid. Once connected, both aircraft continue to fly in formation until the fuel transfer is complete. The last stage of the aerial refueling is that when refueling is finished, the receiving aircraft slows down to disconnect the probe from the drogue basket. To the authors' knowledge, the dynamics of an axially moving system subjected to axially moving support (i.e. the second types of axially moving systems) with consideration of fluid–structure interactions have not been reported yet. Motivated by this, we will investigate the dynamics of the cantilever beam attached to an axially moving base immersed in fluid.

The progress of the present study is described as follows. First, the equation of motions of the cantilever beam attached to an axially moving base immersed fluid will be derived. Second, the discrete equation of motion is obtained by the Galerkin method. Third, the effects of axially moving speed of base, mass ratio, and several other system parameters on the free vibration and stability are analyzed by calculating the natural frequencies and the lowest critical moving speed.

Section snippets

Problem formulation

The system to be analyzed, shown in Fig. 1, consists of a cantilever beam attached to an axially moving base with a known motion $L (t)$ . Let this beam be of diameter $D$ , with length $l$ , area moment of inertia $I$ , mass per unit length $m$ and modulus of elasticity $E$ . Consider this system to be immersed in an incompressible fluid of density $ρ$ , with boundaries sufficiently distant to have negligible effect on the fluid forces on the beam. Only small lateral motions are considered in the present study. It

Solution by Galerkin approach

It is convenient to expand the transverse displacement in series form in terms of a set of admissible functions: $η (ξ, τ) = \sum_{j = 1}^{N} ϕ_{j} (ξ) q_{j} (τ)$ where $N$ is the number of terms for Galerkin truncation. The comparison functions $ϕ_{j} (ξ)$ can be somewhat arbitrary, as long as they satisfy the boundary conditions. In the present analysis, the cantilevered beam eigenfunctions are utilized: $ϕ_{j} (ξ) = [\cosh (λ_{j} ξ) - \cos (λ_{j} ξ)] - σ_{j} [\sinh (λ_{j} ξ) - \sin (λ_{j} ξ)]$ where $σ_{j} = (\cosh λ_{j} + \cos λ_{j}) / (\sinh λ_{j} + \sin λ_{j})$ and $λ_{j}$ are the eigenvalues given by the

Result

In this section, the dimensionless system parameters need to be determined before numerical studies are conducted. The range of the drag coefficient are 0.005< $C_{N}$ <0.04 and 0.01< $C_{T}$ <0.025 [1]. Moreover, the order of ${\bar{C}}_{N}$ is lower than $C_{N}$ or $C_{T}$ [1], [2], [3], [25]. It should be pointed out that the drag coefficients given in [3], [25] may be wrong physically $(C_{N} = C_{T} = 1.0)$ . As for the added mass coefficient, a reasonable varied range for $β$ may be 0–0.2 [3]. In this study, the moving speed of the base

Conclusion

The equations of motion of a flexible slender cantilever beam with uniform circular cross-section, attached an axially moving base at a known rate while immersed in an incompressible fluid are derived. To formulate the added inertia forces on the moving beam, an axially added mass coefficient is introduced in these equations. Numerical studies are conducted to investigate the effect of moving speed of the base on the natural frequency and stability of the system by Galerkin method. It is found

Acknowledgment

The financial support of the National Natural Science Foundation of China (Nos. 11172109 and 11172107), and the Program for New Century Excellent Talents in University of Ministry of Education of China (No. NCET-11-0183) to this work are gratefully acknowledged.

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Free vibration and stability of a cantilever beam attached to an axially moving base immersed in fluid

Highlights

Abstract

Introduction

Section snippets

Problem formulation

Solution by Galerkin approach

Result

Conclusion

Acknowledgment

Journal of Sound and Vibration

Journal of Franklin Institute

Journal of Sound and Vibration

Journal of Sound and Vibration

Journal of Sound and Vibration

International Journal of Mechanical Sciences

International Journal of Solids and Structures

Applied Mathematical Modelling

International Journal of Mechanical Sciences

Journal of Sound and Vibration

Journal of Sound and Vibration

Dynamics of an axially moving beam submerged in a fluid

AIAA Journal of Hydronautics

Dynamic stability of axially moving materials

Shock and Vibration Digest

Current research on vibration and stability of axially-moving materials

Shock and Vibration Digest