0
Research Papers

Extremely Large Oscillations of Cantilevers Subject to Motion Constraints

[+] Author and Article Information
Hamed Farokhi

Department of Mechanical and
Construction Engineering,
Northumbria University,
Newcastle Upon Tyne NE1 8ST, UK
e-mail: hamed.farokhi@northumbria.ac.uk

Mergen H. Ghayesh

School of Mechanical Engineering,
University of Adelaide,
Adelaide 5005, South Australia, Australia
e-mail: mergen.ghayesh@adelaide.edu.au

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 22, 2018; final manuscript received November 7, 2018; published online December 17, 2018. Assoc. Editor: George Haller.

J. Appl. Mech 86(3), 031001 (Dec 17, 2018) (12 pages) Paper No: JAM-18-1434; doi: 10.1115/1.4041964 History: Received July 22, 2018; Revised November 07, 2018

The nonlinear extremely large-amplitude oscillation of a cantilever subject to motion constraints is examined for the first time. In order to be able to model the large-amplitude oscillations accurately, the equation governing the cantilever centerline rotation is derived. This allows for analyzing motions of very large amplitude even when tip angle is larger than π/2. The Euler–Bernoulli beam theory is employed along with the centerline inextensibility assumption, which results in nonlinear inertial terms in the equation of motion. The motion constraint is modeled as a spring with a large stiffness coefficient. The presence of a gap between the motion constraint and the cantilever causes major difficulties in modeling and numerical simulations, and results in a nonsmooth resonance response. The final form of the equation of motion is discretized via the Galerkin technique, while keeping the trigonometric functions intact to ensure accurate results even at large-amplitude oscillations. Numerical simulations are conducted via a continuation technique, examining the effect of various system parameters. It is shown that the presence of the motion constraints widens the resonance frequency band effectively which is particularly important for energy harvesting applications.

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

References

Liu, H. , Lee, C. , Kobayashi, T. , Tay, C. J. , and Quan, C. , 2012, “ Piezoelectric MEMS-Based Wideband Energy Harvesting Systems Using a Frequency-Up-Conversion Cantilever Stopper,” Sens. Actuators A: Phys., 186, pp. 242–248. [CrossRef]
Huicong, L. , Chengkuo, L. , Takeshi, K. , Cho Jui, T. , and Chenggen, Q. , 2012, “ Investigation of a MEMS Piezoelectric Energy Harvester System With a Frequency-Widened-Bandwidth Mechanism Introduced by Mechanical Stoppers,” Smart Mater. Struct., 21(3), p. 035005. [CrossRef]
Moon, F. C. , and Shaw, S. W. , 1983, “ Chaotic Vibrations of a Beam With Non-Linear Boundary Conditions,” Int. J. Non-Linear Mech., 18(6), pp. 465–477. [CrossRef]
Shaw, S. W. , and Holmes, P. J. , 1983, “ A Periodically Forced Piecewise Linear Oscillator,” J. Sound Vib., 90(1), pp. 129–155. [CrossRef]
Choi, Y. S. , and Noah, S. T. , 1988, “ Forced Periodic Vibration of Unsymmetric Piecewise-Linear Systems,” J. Sound Vib., 121(1), pp. 117–126. [CrossRef]
Heiman, M. S. , Bajaj, A. K. , and Sherman, P. J. , 1988, “ Periodic Motions and Bifurcations in Dynamics of an Inclined Impact Pair,” J. Sound Vib., 124(1), pp. 55–78. [CrossRef]
Shaw, S. W. , and Rand, R. H. , 1989, “ The Transition to Chaos in a Simple Mechanical System,” Int. J. Non-Linear Mech., 24(1), pp. 41–56. [CrossRef]
Li, G. X. , Rand, R. H. , and Moon, F. C. , 1990, “ Bifurcations and Chaos in a Forced Zero-Stiffness Impact Oscillator,” Int. J. Non-Linear Mech., 25(4), pp. 417–432. [CrossRef]
Natsiavas, S. , 1990, “ On the Dynamics of Oscillators With Bi-Linear Damping and Stiffness,” Int. J. Non-Linear Mech., 25(5), pp. 535–554. [CrossRef]
Natsiavas, S. , 1993, “ Dynamics of Multiple-Degree-of-Freedom Oscillators With Colliding Components,” J. Sound Vib., 165(3), pp. 439–453. [CrossRef]
Nigm, M. M. , and Shabana, A. A. , 1983, “ Effect of an Impact Damper on a Multi-Degree of Freedom System,” J. Sound Vib., 89(4), pp. 541–557. [CrossRef]
Björkenstam, U. , 1977, “ Impact Vibration of a Bar,” Int. J. Mech. Sci., 19(8), pp. 471–481. [CrossRef]
Masri, S. , Mariamy, Y. , and Anderson, J. , 1981, “ Dynamic Response of a Beam With a Geometric Nonlinearity,” ASME J. Appl. Mech., 48(2), pp. 404–410. [CrossRef]
Metallidis, P. , and Natsiavas, S. , 2000, “ Vibration of a Continuous System With Clearance and Motion Constraints,” Int. J. Non-Linear Mech., 35(4), pp. 675–690. [CrossRef]
Liu, S. , Cheng, Q. , Zhao, D. , and Feng, L. , 2016, “ Theoretical Modeling and Analysis of Two-Degree-of-Freedom Piezoelectric Energy Harvester With Stopper,” Sens. Actuators A: Phys., 245, pp. 97–105. [CrossRef]
Soliman, M. S. M. , Abdel-Rahman, E. M. , El-Saadany, E. F. , and Mansour, R. R. , 2008, “ A Wideband Vibration-Based Energy Harvester,” J. Micromech. Microeng., 18(11), p. 115021. [CrossRef]
Halim, M. A. , and Park, J. Y. , 2014, “ Theoretical Modeling and Analysis of Mechanical Impact Driven and Frequency Up-Converted Piezoelectric Energy Harvester for Low-Frequency and Wide-Bandwidth Operation,” Sens. Actuators A: Phys., 208, pp. 56–65. [CrossRef]
Wu, Y. , Badel, A. , Formosa, F. , Liu, W. , and Agbossou, A. , 2014, “ Nonlinear Vibration Energy Harvesting Device Integrating Mechanical Stoppers Used as Synchronous Mechanical Switches,” J. Intell. Mater. Syst. Struct., 25(14), pp. 1658–1663. [CrossRef]
Nayfeh, A. H. , and Pai, P. F. , 2008, Linear and Nonlinear Structural Mechanics, Wiley, Hoboken, NJ.
Farokhi, H. , Ghayesh, M. H. , and Hussain, S. , 2016, “ Large-Amplitude Dynamical Behaviour of Microcantilevers,” Int. J. Eng. Sci., 106, pp. 29–41. [CrossRef]
Abdalla, M. M. , Reddy, C. K. , Faris, W. F. , and Gürdal, Z. , 2005, “ Optimal Design of an Electrostatically Actuated Microbeam for Maximum Pull-In Voltage,” Comput. Struct., 83(15–16), pp. 1320–1329. [CrossRef]
Allgower, E. L. , and Georg, K. , 2003, Introduction to Numerical Continuation Methods, Society for Industrial and Applied Mathematics, Philadelphia, PA.

Figures

Grahic Jump Location
Fig. 1

(a) Schematic of a cantilever under transverse base excitation subject to motion constraints and (b) deformed configuration of the system

Grahic Jump Location
Fig. 2

Frequency-amplitude plots of the cantilever without constraint: (a) maximum tip transverse displacement, (b) maximum tip longitudinal displacement, and (c) maximum tip angle. The solid line showing stable solution, while the dotted line showing unstable one. P1, P2, and P3 are points of interest which are examined in more detail in Fig. 3.

Grahic Jump Location
Fig. 3

((a)–(c)) Oscillation of the system of Fig. 2 at points P1, P2, and P3, respectively

Grahic Jump Location
Fig. 4

Comparison between the frequency-amplitude plots of the constrained cantilever to those of a cantilever with no constraint: (a) maximum tip transverse displacement, (b) maximum tip longitudinal displacement, and (c) maximum tip angle. The solid line showing stable solution, while the dotted line showing unstable one. g0 = 0.03, K1 = 2.0 × 104, and w0 = 0.018.

Grahic Jump Location
Fig. 5

Details of the motion of the system of Fig. 4 at ωe/ω1 = 0.9775 (i.e., in the resonance region before hitting the motion constraint), showing, respectively, the time trace and phase-plane portrait of: ((a), (b)) tip transverse displacement, ((c), (d)) tip longitudinal displacement, and ((e), (f)) tip angle. tn denotes normalized time with respect to the period of the oscillation.

Grahic Jump Location
Fig. 6

Details of the motion of the system of Fig. 4 at ωe/ω1 = 1.1030 (i.e., in the resonance region after hitting the motion constraint), showing, respectively, the time trace and phase-plane portrait of ((a), (b)) tip transverse displacement, ((c), (d)) tip longitudinal displacement, and ((e), (f)) tip angle. tn denotes normalized time with respect to the period of the oscillation.

Grahic Jump Location
Fig. 7

Comparison between the frequency-amplitude plots of the cantilever constrained at both sides to those of a cantilever constrained at one side: (a) maximum tip transverse displacement and (b) maximum tip longitudinal displacement. The solid line showing stable solution, while the dotted line showing unstable one. g0 = 0.03, K1 = 2.0 × 104, and w0 = 0.018.

Grahic Jump Location
Fig. 8

Effect of the motion constraint stiffness on frequency-amplitude plots of the constrained cantilever: (a) maximum tip transverse displacement and (b) maximum tip longitudinal displacement. The solid line showing stable solution, while the dotted line showing unstable one. g0 = 0.03 and w0 = 0.018.

Grahic Jump Location
Fig. 9

Effect of the motion constraint gap-width on the frequency-amplitude plots of the constrained cantilever: (a) maximum tip transverse displacement and (b) maximum tip longitudinal displacement. The solid line showing stable solution, while the dotted line showing unstable one. K1 = 2.0 × 104 and w0 = 0.018.

Grahic Jump Location
Fig. 10

Effect of base-excitation amplitude on the frequency-amplitude plots of the constrained cantilever: (a) maximum tip transverse displacement and (b) maximum tip longitudinal displacement. The solid line showing stable solution, while the dotted line showing unstable one. g0 = 0.03 and K1 = 2.0 × 104.

Grahic Jump Location
Fig. 11

Effect of the position of the motion constraint on the frequency-amplitude plots of the cantilever: (a) maximum tip transverse displacement and (b) maximum tip longitudinal displacement. The solid line showing stable solution, while the dotted line showing unstable one. g0 = 0.03, K1 = 2.0 × 104, and w0 = 0.018.

Grahic Jump Location
Fig. 12

Effect of material damping coefficient on the frequency-amplitude plots of the constrained cantilever: (a) maximum tip transverse displacement and (b) maximum tip longitudinal displacement. The solid line showing stable solution, while the dotted line showing unstable one. g0 = 0.03 and K1 = 2.0 × 104, and w0 = 0.018.

Grahic Jump Location
Fig. 13

Comparison between the cantilever tip deflections under perpendicular tip load: (a) transverse deflection and (b) longitudinal deflection. Solid line: present study; symbols: 3D FEA.

Grahic Jump Location
Fig. 14

Deformed configuration of the cantilever at various forcing amplitudes. Solid line: present study; symbols: 3D FEA.

Grahic Jump Location
Fig. 15

Strain distribution as a function of x at z = −h/2 for cantilever under static load fs = 7.68 (the case with the largest force in Fig. 14)

Grahic Jump Location
Fig. 16

Effect of number of degrees-of-freedom on the frequency-amplitude diagrams of the system

Tables

Errata

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