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Research Papers

Nonlinear Responses of Inextensible Cantilever and Free–Free Beams Undergoing Large Deflections

[+] Author and Article Information
Kevin McHugh

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

Earl 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 Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 9, 2017; final manuscript received February 27, 2018; published online March 19, 2018. Assoc. Editor: George Haller.

J. Appl. Mech 85(5), 051008 (Mar 19, 2018) (8 pages) Paper No: JAM-17-1629; doi: 10.1115/1.4039478 History: Received November 09, 2017; Revised February 27, 2018

A theoretical and computational model has been developed for the nonlinear motion of an inextensible beam undergoing large deflections for cantilevered and free–free boundary conditions. The inextensibility condition was enforced through a Lagrange multiplier which acted as a constraint force. The Rayleigh–Ritz method was used by expanding the deflections and the constraint force in modal series. Lagrange's equations were used to derive the equations of motion of the system, and a fourth-order Runge–Kutta solver was used to solve them. Comparisons for the cantilevered beam were drawn to experimental and computational results previously published and show good agreement for responses to both static and dynamic point forces. Some physical insights into the cantilevered beam response at the first and second resonant modes were obtained. The free–free beam condition was investigated at the first and third resonant modes, and the nonlinearity (primarily inertia) was shown to shift the resonant frequency significantly from the linear natural frequency and lead to hysteresis in both modes.

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References

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McHugh, K. , and Dowell, E. , 2017, “ Nonlinear Responses of Inextensible Cantilever and Free-Free Beams Undergoing Large Deflections,” ASME Paper No. DSCC2017-5113.

Figures

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Fig. 1

Configuration and coordinate system identification for (a) cantilevered and (b) free–free boundary conditions

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Fig. 2

Deflection of beam tip in (a) w and (b) u directions due to varying mass loads on tip

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Fig. 3

First mode RMS response diagram of cantilevered beam to sinusoidal excitation force of F = 0.147 N at xF = 0.7∗L for (a) forward and (b) backward frequency sweeps

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Fig. 4

Second mode RMS response diagram of cantilevered beam to sinusoidal excitation force of F = 1.96 N at xF = 0.3 ∗ L for (a) forward and (b) backward frequency sweeps

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Fig. 5

Second mode linear versus nonlinear cantilever RMS responses to varying loads acting at xF = 0.3 ∗ L, backward frequency sweep

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Fig. 6

w Deflection versus static load of free–free beam in first bending mode for current formulation and ANSYS solution

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Fig. 7

First mode RMS response diagram of free–free beam to sinusoidal excitation force of F = 1.47 N at xF = L/2 for (a) forward and (b) backward frequency sweeps

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Fig. 8

Third mode RMS response diagram of free–free beam to sinusoidal excitation force of F = 12.27 N at xF = L/2 for (a) forward and (b) backward frequency sweeps

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Fig. 9

Linear versus nonlinear free–free beam RMS responses to varying loads acting at xF = L/2, backward frequency sweep: (a) first resonant and (b) third resonant

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