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Research Papers

Linear Versus Nonlinear Response of a Cantilevered Beam Under Harmonic Base Excitation: Theory and Experiment

[+] Author and Article Information
Michal Raviv Sayag

Department of Mechanical Engineering &
Materials Science,
Duke University,
Durham, NC 27708
e-mail: michal.raviv.sayag@gmail.com

Earl H. Dowell

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

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 23, 2016; final manuscript received July 8, 2016; published online July 28, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(10), 101002 (Jul 28, 2016) (8 pages) Paper No: JAM-16-1316; doi: 10.1115/1.4034117 History: Received June 23, 2016; Revised July 08, 2016

A computational and experimental study of a uniform cantilever beam with a tip mass under base excitation was performed. The beam was excited at various levels of base displacement to provoke tip displacements greater than 15% of the beam length. Damping and yield stress of the beam were both considered. It was found that a large tip displacement causes nonlinear inertial (NLI) and structural (NLS) effects to arise. Each of the structural and inertial nonlinearities has an opposite effect on the resulting resonance frequency, which are nearly mutually canceling. The result was that resonant frequency calculated using the full nonlinear (FNL) model was essentially equal to the value calculated by linear (LIN) theory, and the tip displacement amplitude varied only modestly from the LIN value. It was also observed that the damping in this system is likely nonlinear, and depends on tip displacement amplitude. A theoretical model for fluid damping is suggested. Initial investigation shows encouraging agreement between the theoretical fluid damping and the measured values.

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Figures

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Fig. 2

Measurements of a typical run: (a) acceleration measurements and (b) displacement measurements

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Fig. 1

Experimental system: shaker, two sided beam, and three accelerometers

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Fig. 3

Damping coefficient for various nominal base displacement values, half power method

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Fig. 6

Stress variation at the root of the beam: ζ = 0.006 and W0 = 0.5 mm

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Fig. 7

Transfer function gain versus frequency: experimental and numerical results. (a) W0 = 0.5 mm and (b) W0 = 1.2 mm.

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Fig. 10

Tip displacement/base displacement at resonance versus tip displacement: experimental and numerical results. (a) Dimensional tip displacement and (b) normalized tip displacement.

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Fig. 11

Tip displacement/base displacement at resonance versus base displacement: experimental and numerical results. 0.006 ≤ ζ ≤ 0.02. (a) Beam 1 and (b) beam 2.

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Fig. 4

Tip displacement amplitude for various numerical models: FNL, NLS, NLI, and LIN. All the calculations performed with ζ = 0.006 and W0 = 0.5 mm.

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Fig. 5

Tip displacement amplitude for various numerical models, increasing/decreasing frequency: ζ = 0.006 and W0 = 0.5 mm

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Fig. 8

Transfer function gain versus normalized frequency: experimental and numerical results. (a) W0 = 0.5 mm and (b) W0 = 1.2 mm.

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Fig. 9

Tip displacement/base displacement at resonance versus base displacement: numerical results

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Fig. 12

Variation of damping with tip displacement: experiments and theory

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