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

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.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Beck, M. , 1952, “ Die Knicklast Des Einseitig Eingespannten, tangential Gedrückten Stabes,” Z. Angew. Math. Phys., 3(3), pp. 225–228. [CrossRef]
Bolotin, V. , 1963, Nonconservative Problems of the Theory of Elastic Stability, Pergamon Press, Oxford, UK.
Bolotin, V. , and Zhinzher, N. , 1969, “ Effects of Damping on Stability of Elastic Systems Subjected to Nonconservative Forces,” Int. J. Solids Struct., 5(9), pp. 965–989. [CrossRef]
Chen, M. , 1987, “ Hopf Bifurcation in Beck's Problem,” Nonlinear Anal., Theory, Methods Appl., 11(9), pp. 1061–1073. [CrossRef]
Di Egidio, A. , Luongo, A. , and Paolone, A. , 2007, “ Linear and Non-Linear Interactions Between Static and Dynamic Bifurcations of Damped Planar Beams,” Int. J. Non-Linear Mech., 42(1), pp. 88–98. [CrossRef]
Luongo, A. , and Di Egidio, A. , 2005, “ Bifurcation Equations Through Multiple-Scales Analysis for a Continuous Model of a Planar Beam,” Nonlinear Dyn., 41(1–3), pp. 171–190. [CrossRef]
Luongo, A. , and Di Egidio, A. , 2006, “ Divergence, HOPF and Double-Zero Bifurcations of a Nonlinear Planar Beam,” Comput. Struct., 84(24–25), pp. 1596–1605. [CrossRef]
Luongo, A. , and D'Annibale, F. , 2017, “ Nonlinear Hysteretic Damping Effects on the Post-Critical Behaviour of the Visco-Elastic Beck's Beam,” Math. Mech. Solids, 22(6), pp. 1347–1365. [CrossRef]
Stanciulescu, I. , Virgin, L. , and Laursen, T. , 2007, “ Slender Solar Sail Booms: Finite Element Analysis,” J. Spacecr. Rockets, 44(3), pp. 528–537. [CrossRef]
Langthjem, M. , and Sugiyama, Y. , 2000, “ Dynamic Stability of Columns Subjected to Follwer Loads: A Survey,” J. Sound Vib., 238(5), pp. 809–851. [CrossRef]
Ziegler, H. , 1952, “ Die Stabilitätskriterien Der Elastomechanik,” Ing.-Arch., 20(1), pp. 49–56. [CrossRef]
Luongo, A. , and D'Annibale, F. , 2014, “ On the Destabilizing Effect of Damping on Discrete and Continuous Circulatory Systems,” J. Sound Vib., 333(24), pp. 6723–6741. [CrossRef]
Hagedorn, P. , 1970, “ On the Destabilizing Effect of Non-Linear Damping in Non-Conservative Systems With Follower Forces,” Int. J. Non-Linear Mech., 5(2), pp. 341–358. [CrossRef]
Thomson, J. , 1995, “ Chaotic Dynamics of the Partially Follower-Loaded Elastic Double Pendulum,” J. Sound Vib., 188(3), pp. 385–405. [CrossRef]
Crespo da Silva, M. R. M. , 1978, “ Harmonic Non-Linear Response of Beck's Column to a Lateral Excitation,” Int. J. Solids Struct., 14(12), pp. 987–997. [CrossRef]
Crespo da Silva, M. , 1978, “ Flexural-Flexural Oscillations of Beck's Column Subjected to a Planar Harmonic Excitation,” J. Sound Vib., 60(1), pp. 133–144. [CrossRef]
Luongo, A. , and D'Annibale, F. , 2013, “ Double Zero Bifurcation of Non-Linear Viscoelastic Beams Under Conservative and Non-Conservative Loads,” Int. J. Non-Linear Mech., 55(Suppl. C), pp. 128–139. [CrossRef]
D'Annibale, F. , Ferretti, M. , and Luongo, A. , 2016, “ Improving the Linear Stability of the Beck's Beam by Added Dashpots,” Int. J. Mech. Sci., 110, pp. 151–159. [CrossRef]
Crespo da Silva, M. , and Glynn, C. , 1978, “ Non-Linear Flexural-Flexural-Torsional Dynamics of Inextensional Beams—I: Equations of Motion,” J. Struct. Mech., 6(4), pp. 437–448. [CrossRef]
Crespo da Silva, M. , and Glynn, C. , 1978, “ Non-Linear Flexural-Flexural-Torsional Dynamics of Inextensional Beams—II: Forced Motions,” J. Struct. Mech., 6(4), pp. 449–461. [CrossRef]
Dowell, E. , and McHugh, K. , 2016, “ Equations of Motion for an Inextensible Beam Undergoing Large Deflections,” ASME J. Appl. Mech., 83(5), p. 051007. [CrossRef]
Hamdan, M. , and Dado, M. , 1997, “ Large Amplitude Free Vibrations of a Uniform Cantilever Beam Carrying an Intermediate Lumped Mass and Rotary Inertia,” J. Sound Vib., 206(2), pp. 151–168. [CrossRef]
Mahmoodi, S. N. , Jalili, N. , and Khadem, S. E. , 2008, “ An Experimental Investigation of Nonlinear Vibration and Frequency Response Analysis of Cantilever Viscoelastic Beams,” J. Sound Vib., 311(3–5), pp. 1409–1419. [CrossRef]
McHugh, K. , and Dowell, E. , 2018, “ Nonlinear Responses of Inextensible Cantilever and Free-Free Beams Undergoing Large Deflections,” ASME J. Appl. Mech., 85(5), p. 051008. [CrossRef]
Raviv Sayag, M. , and Dowell, E. , 2016, “ Linear Versus Nonlinear Response of a Cantilevered Beam Under Harmonic Base Excitation: Theory and Experiment,” ASME J. Appl. Mech., 83(10), p. 101002. [CrossRef]
Tang, D. , Zhao, M. , and Dowell, E. , 2014, “ Inextensible Beam and Plate Theory: Computational Analysis and Comparison With Experiment,” ASME J. Appl. Mech., 81(6), p. 061009. [CrossRef]
Villanueva, L. , Karabalin, R. B. , Matheny, M. H. , Chi, D. , Sader, J. , and Roukes, M. , 2013, “ Nonlinearity in Nanomechanical Cantilevers,” Phys. Rev. B, 87(2), p. 024304. [CrossRef]
Novozhilov, V. , 1953, Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Rochester, NY.
Tang, D. , Gibbs, S. , and Dowell, E. , 2015, “ Nonlinear Aeroelastic Analysis With Inextensible Plate Theory Including Correlation With Experiment,” AIAA J., 53(5), pp. 1299–1308. [CrossRef]
Kirillov, O. N. , and Seyranian, A. O. , 2005, “ The Effect of Small Internal and External Damping on the Stability of Distributed Non-conservative Systems,” J. Appl. Math. Mech., 69(4), pp. 529–552. [CrossRef]


Grahic Jump Location
Fig. 1

Schematic of cantilever beam with follower force

Grahic Jump Location
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).

Grahic Jump Location
Fig. 3

Limit cycle oscillation: transverse tip deflection versus time

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
Fig. 10

Distribution of λ values across beam at corresponding beam deflections



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In