Nonlinear lateral vibrations of a deploying EulerBernoulli beamwith a spinning motion

Kefei Zhu, Jintai Chung n

Department of Mechanical Engineering, Hanyang University, 1271 Sa-3-dong, Sangnok-gu, Ansan 425-791, Kyeonggi-do, Republic of Korea

a r t i c l e i n f o

Article history:Received 10 July 2014Received in revised form23 October 2014Accepted 11 November 2014Available online 20 November 2014

Keywords:Nonlinear lateral vibrationsDeploying beam with spinEulerBernoulli beamBeat phenomenon

a b s t r a c t

The nonlinear lateral vibrations of a beam deploying from a xed rigid hub are analyzed when the beam hasa spinning motion. Previous studies do not consider the nonlinear coupled effect between axial and lateraldisplacements when analyzing the vibration behavior of a deploying beamwith spin. In contrast, the presentstudy considers the nonlinear coupling effect of the axial and lateral displacements for a deploying beamwith spin to investigate the vibration behavior more exactly and comprehensively. The EulerBernoulli beamtheory and von Karman nonlinear strain theory are used together to develop a new nonlinear model. Thedifferences in dynamic response between the linear and nonlinear models are compared, and the effects ofphysical parameters on the differences are investigated. In addition, the beat phenomenon of a deployingbeamwith spin is studied. Signicant differences in dynamic responses are observed between the proposednonlinear and previous linear models. It is found that the beat phenomenon is caused by the interference oftwo close natural frequencies: the rst and second natural frequencies of a spinning beam.

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1. Introduction

Since deploying beams and spinning beams are used in variousindustrial applications, many researchers have paid considerableattention to the vibration problems of these beams. Some exam-ples of engineering applications are drilling machines in produc-tion, drill-string systems in oil wells, manipulators for industrialrobot systems, and spinning appendages attached to space vehi-cles. There are numerous studies related to deploying beams andspinning beams from various perspectives.

Some papers have been published regarding deploying beamswithout a spinning motion, which are applicable to robotic manip-ulators, deployable space appendages, and machine tools [19].Wang and Wei [1] considered the vibration in a exible robot armmodeled by a moving slender beam. They found that the extensionmotion has a destabilizing effect on the vibration motion, andcontracting motion conversely result in a stabilizing effect. Yuh andYoung [2] derived a linear time-dependent differential equation todescribe the lateral motion of an axially-moving and rotating beam.Al-Bedoor and Khulief [3] presented approximate analytical solutionsfor the lateral vibrations of a beam during axial deployment withdifferent end conditions. Gosselin et al. [4] obtained the equations ofmotion for axially moving beams immersed in uid. Chang et al. [5]

investigated the vibration and stability of a deploying Rayleigh beamusing the nite element method with variable-domain elements.Park et al. [6] analyzed the longitudinal and transverse vibrations of adeploying beam when the beam was in a deploying or retractingmotion. The longitudinal and transverse displacements were simul-taneously considered, and the nonlinear effect was included byadopting the von Karman nonlinear strain theory. Zhang et al. [7]studied the nonlinear dynamical behaviors of deploying-and-retreating wings in super-sonic airow. They modeled a cantileverlaminated composite beam, which is axially moving at a known rate,for a deploying-and-retreating wing. Kim and Chung [8] presented amethod for reducing the residual vibration of a exible beamdeployed from a translating hub. Most researchers focused on thedynamic problems of a deploying uniform beam; however, Duanet al. [9] investigated the transverse vibration of a deploying nestedcantilever beam with a tip mass.

Other papers were published for the vibration analyses ofspinning beams without an axial (deploying or retracting) motion.According to a beam theory used for analysis, the studies on thevibrations of spinning beams can be categorized into three groups:the spinning EulerBernoulli, Rayleigh, and Timoshenko beams.Both the rotary inertia effect and shear deformation are neglectedin the EulerBernoulli beam, only the shear deformation isneglected in the Rayleigh beam, and both the rotary inertia effectand shear deformation are considered in the Timoshenko beam.

The vibration analyses of spinning beams, based on the EulerBernoulli theory, were reported in some papers [1016]. Lee [10]analyzed the motion of an orthotropic rotating shaft moving

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International Journal of Mechanical Sciences

http://dx.doi.org/10.1016/j.ijmecsci.2014.11.0090020-7403/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: 82 31 400 5287; fax: 82 31 406 696.E-mail addresses: kefeizhu@gmail.com (K. Zhu),

jchung@hanyang.ac.kr (J. Chung).

International Journal of Mechanical Sciences 90 (2015) 200212

longitudinally over supports, related to the dynamic analysis ofrobotic manipulators with prismatic joints and combinations ofdirect drive actuators which can set a robot arm in translation,spinning and rotation. Fung and Lee [11] presented a parametricvariable structure control for a spinning simple-exure beam via acontrol force applied along the axial direction to suppress thetransverse vibrations. Banerjee and Su [12] derived the governingdifferential equations of a spinning beam in free vibration usingHamilton's principle. They showed that the torsional motion is notcoupled with the lateral bending motion when the cross section of abeam is doubly symmetric. Firouz-Abadi and Haddadpour [13] dealtwith the exural instability of exible spinning cylinders partiallylled with viscous uid. They examined the effect of materialviscoelasticity and structural damping on the stability margins ofthe exible cylinder. Using the spectral element method, Mustaphaand Zhong [14] presented a mathematical model for the freevibration analysis of a circular doubly-symmetric spinningmicrobeam embedded in an elastic medium. Yu and Huang [15]investigated the buckling and the critical speed of a microdrill in adrilling process, and considered the effects of rotating speed, pre-twisted angle and thrust force on the buckling and critical speed ofthe microdrill. Hosseini [16] studied the stability and bifurcations in asimply supported rotating shaft, and modeled the shaft as an in-extensional spinning beam with large amplitude.

Related to the Rayleigh beam theory, some researchers investi-gated the vibrations of spinning beams [1721]. Liao and Huang [17]studied the parametric instability of a cantilever pretwisted beamrotating around its longitudinal axis at a time-dependent speedwhich contains a steady state part and a small periodically uctuat-ing component. Sheu and Yang [18] analyzed the whirl speed,critical speed and mode shape of a spinning Rayleigh beam in sixgeneral boundary conditions. Das et al. [19] made an attempt toactively control the transverse vibration due to bending of a exiblerotorshaft relative to the moving stator under simultaneous inu-ences of mass unbalance and base motion. Xiang et al. [20]developed a class of wavelet-based Rayleigh rotating beam elementusing B-spline wavelets on the interval to analyze rotor-bearingsystem. Pai et al. [21] presented linear and nonlinear models ofspinning Rayleigh beams and investigated dynamic characteristics ofdownward vertical spinning Rayleigh beams with six different setsof boundary conditions.

On the other hand, some research papers for vibration analyses ofspinning Timoshenko beams can be found in the literature [2226].Liao and Dang [22] investigated the structural characteristics ofspinning pretwisted orthotropic beams by the nite element method,using the Euler and Timoshenko beam theories. Chen [23] derivedtwo sets of lateral vibration equations for a spinning axially loadedtwisted Timoshenko beam, and studied the free vibration andbuckling stability of various twisted Timoshenko beams. Chan et al.[24] studied the revolving superposed standing waves in a spinningTimoshenko beam. Lee and Jang [25] developed a spectral elementmodel for a spinning uniform shaft represented by the uniformTimoshenko beam model. Mustapha and Zhong [26] presented adistributed parameter model for the transverse vibration analysis of amicroend mill. They found the spinning rate to have the mostsignicant effect on the frequency value of the microtool followedby the taper ratio and the complex geometry of the microute.

Although many studies have investigated deploying beams with-out a spinning motion as well as spinning beams without an axialmotion, few studies have been published regarding the vibration of abeam with both deploying and spinning motions. The authors'literature survey found a single paper related to this type of beam[27]. In this paper, the lateral vibration of a spinning, deploying beamwas analyzed. However, this study did not address the axial deforma-tion of the beam, nor the coupling effect between the axial and lateraldeformations. In fact, the deploying motion of a spinning beam has

not yet been widely studied. Since the deploying or retracting motionmay initiate an entire work process, a drilling process for example, thevibration of a deploying beam with a spinning motion can affect thequality of the resulting work. Hence, it is important to study thedeploying/retracting procedure of a spinning beam. In contrast toaxially-moving spinning beams with constant length, spinning beamswith deployment yield time-dependent beam lengths. The naturalfrequencies and modes depend on time, as the beam length varieswith time.

Both axial displacements and coupled terms between the axialand lateral displacements were neglected in many of previousstudies, when analyzing the vibration of an axially-moving spin-ning beam. Neglecting these axial displacements and couplingterms is appropriate for many applications, as the natural fre-quencies of the axial vibration are signicantly larger than those ofthe lateral vibration. However, in the case of a drill string in an oilwell, the natural frequencies of axial vibration are of the sameorder of magnitude as the natural frequencies of lateral vibration[28]. In such case, it is no longer adequate to neglect the couplingterms. When a spinning beam has an axial motion, the axialdisplacement plays a more important role, and a precise check ofthe coupling effects on lateral vibration should be considered.

This study analyzes the dynamic behavior of a deploying EulerBernoulli beam with a spinning motion, improving the accuracy ofcomputational results via full consideration of the axial and lateraldisplacements with their nonlinear coupled effects. The investiga-tion procedure of this study is as follows. The equations of motionfor a deploying beam with spin are derived, considering both axialand lateral displacements. During derivation of the equations, it isassumed that the beam has a constant spinning speed and time-dependent axial deploying speed. Since the present study isprimarily concerned with the nonlinear coupling effects of theaxial and lateral vibrations, the nonlinear von Karman straintheory is adopted. After the equations are transformed into theweak forms, the transformed equations are discretized using theGalerkin method. Applying the generalized- time integrationmethod [29] to the discretized equations, the dynamic responsesof the beam are obtained for the axial and lateral motions. Thedifferences in dynamic response between the linear and nonlinearmodels are analyzed, and the effects of physical parameters onthese differences are also studied. Finally, the beat phenomenon isinvestigated when a spinning beam is axially deploying.

2. Dynamic modeling and equations of motion

The dynamic model of a spinning circular beam deployingoutside a rigid hub is shown in Fig. 1. The beam is axisymmetricand uniform and has Young's modulus E, beam diameter, d, and totallength, L. The length of the beam deployed outside the rigid hub, l, isa function of time. It is assumed that there is no friction between thebeam and rigid hub and the spinning speed, , is constant. Anexternal axial force, F, exerted at the left end, pushes the beamdeploying axially from the hub. This force induces the moving speed,V, which is dependent on time. The external force, moving speed,and deployed beam length have the following relations:

_l V ; F AL _V 1where the superposed dot represents differentiation with respect totime, A is cross-sectional area, and is mass density of the beam.

The present study adopts the EulerBernoulli beam theory tomodel the deploying beam with spin, because the beam is slenderenough. As shown in Fig. 1, the displacements of point P on theneutral axis in the x, y and z directions are expressed by u, v and w,respectively. The displacement, u, is referred to as the axial dis-placement, while the displacements v and w are referred to as the

K. Zhu, J. Chung / International Journal of Mechanical Sciences 90 (2015) 200212 201

lateral displacements. These displacements are functions of position,x, and time, t.

Since the lateral motion of the beam inside the rigid hub isrestricted by the hub, the beam inside the rigid hub has only theaxial displacement, while the beam outside the hub has lateraldisplacements as well. For this reason, the position vector of pointP at a distance x away from the hub can be written as

rxu i for lLrxr0xu ivjwk for 0rxr l

(2

where i, j and k are unit vectors in the x, y and z directions. Notethat the torsional displacement is neglected because a beam withan axisymmetric cross section has no coupling between torsionaland lateral vibrations. Applying the material differentiation to theposition vector of Eq. (2), the velocity vector is obtained as

vV utVux

i for lLrxr0V utVux

i vtVvxw

j wt Vwx v

k for 0rxr l

(

3The von Karman nonlinear strain theory is used to take into

account the geometric nonlinearity caused by large deection. Thisnonlinear strain theory is more appropriate for a nonlinear systemthan the conventional innitesimal strain theory. Using the vonKarman theory, the axial normal strain in the x direction may beexpressed as

x ux for lLrxr0uxy

2vx2z

2wx2 12 vx

212 wx 2 for 0rxr l8