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Research Papers

Nonlinear Response of an Inextensible, Cantilevered Beam Subjected to a Nonconservative Follower Force

[+] Author and Article Information
Kevin A. McHugh

Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27708
e-mail: kevin.mchugh@duke.edu

Earl H. Dowell

Professor
Department of Mechanical Engineering and
Materials Science,
Duke University,
Durham, NC 27708
e-mail: earl.dowell@duke.edu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 10, 2018; final manuscript received December 5, 2018; published online January 11, 2019. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 14(3), 031004 (Jan 11, 2019) (9 pages) Paper No: CND-18-1399; doi: 10.1115/1.4042324 History: Received September 10, 2018; Revised December 05, 2018

The dynamic stability of a cantilevered beam actuated by a nonconservative follower force has previously been studied for its interesting dynamical properties and its applications to engineering designs such as thrusters. However, most of the literature considers a linear model. A modest number of papers consider a nonlinear model. Here, a system of nonlinear equations is derived from a new energy approach for an inextensible cantilevered beam with a follower force acting upon it. The equations are solved in time, and the agreement is shown with published results for the critical force including the effects of damping (as determined by a linear model). This model readily allows the determination of both in-plane and out-of-plane deflections as well as the constraint force. With this novel transparency into the system dynamics, the nonlinear postcritical limit cycle oscillations (LCO) are studied including a concentration on the force which enforces the inextensibility constraint.

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Figures

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Fig. 1

Schematic of cantilever beam with follower force

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Fig. 2

Time histories for follower force of: (a) 20.045 and ((b) and (c)) 20.055 EI/L2. Here (c) is a zoomed in plot of (b).

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Fig. 3

Limit cycle oscillation: transverse tip deflection versus time

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Fig. 4

Fast Fourier transform of postcritical force limit cycle (w/L) at steady-state (13.5 < t < 15 s)

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Fig. 5

RMS of transverse tip deflection versus follower force for multiple damping coefficients

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Fig. 6

Modal convergence of (a) u modes, (b) w modes, and (c) λ modes

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Fig. 7

RMS tip deflection in (a) u and (b) w and (c) RMS λ at tip versus follower force with varying numbers of u and λ modes

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Fig. 8

RMS tip deflection in (a) u and (b) w and (c) RMS λ at tip versus follower force with varying numbers of w modes

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Fig. 9

RMS tip deflection in (a) u and (b) w and (c) RMS λ at tip versus follower force with varying numbers of λ modes

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Fig. 10

Distribution of λ values across beam at corresponding beam deflections

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