Technical Brief

Modeling Geometric Nonlinearities in the Free Vibration of a Planar Beam Flexure With a Tip Mass

[+] Author and Article Information
Hamid Moeenfard

Department of Engineering,
School of Mechanical Engineering,
Ferdowsi University of Mashhad,
Vakil Abad Boulevard,
Mashhad, Khorasan Razavi 9177948974, Iran
e-mail: h_moeenfard@um.ac.ir

Shorya Awtar

Department of Mechanical Engineering,
University of Michigan,
2350 Hayward Street,
Ann Arbor, MI 48109
e-mail: awtar@umich.edu

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received August 27, 2012; final manuscript received November 27, 2013; published online February 26, 2014. Assoc. Editor: Craig Lusk.

J. Mech. Des 136(4), 044502 (Feb 26, 2014) (8 pages) Paper No: MD-12-1432; doi: 10.1115/1.4026147 History: Received August 27, 2012; Revised November 27, 2013

The objective of this work is to analytically study the nonlinear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton's principle is utilized to derive the equations governing the nonlinear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these nonlinear partial differential equations are reduced to two coupled nonlinear ordinary differential equations. These equations are solved analytically using the multiple time scales perturbation technique. Parametric analytical expressions are presented for the time domain response of the beam around and far from its internal resonance state. These analytical results are compared with numerical ones to validate the accuracy of the proposed analytical model. Compared with numerical solution methods, the proposed analytical technique shortens the computational time, offers design insights, and provides a broader framework for modeling more complex flexure mechanisms. The qualitative and quantitative knowledge resulting from this effort is expected to enable the analysis, optimization, and synthesis of flexure mechanisms for improved dynamic performance.

Copyright © 2014 by ASME
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Howell, L. L., 2001, Compliant Mechanisms, Wiley, New York.
Smith, S. T., 2000, Flexures: Elements of Elastic Mechanisms, Gordon & Breach, Amsterdam, Netherlands.
Awtar, S., Slocum, A., and Sevincer, E., 2007, “Characteristics of Beam-Based Flexure Modules,” ASME J. Mech. Des., 129(6), pp. 625–639. [CrossRef]
Su, W., and Cesnik, C. E. S., 2011, “Strain-Based Geometrically Nonlinear Beam Formulation for Modeling Very Flexible Aircraft,” Int. J. Solids Struct., 48, pp. 2349–2360. [CrossRef]
Awtar, S., 2004, “Synthesis and Analysis of Parallel Kinematic XY Flexure Mechanisms,” Sc.D., Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA.
Awtar, S., and Parmar, G., 2013, “Design of a Large Range XY Nanopositioning System,” ASME J. Mech. Rob., 5(2), p. 021008. [CrossRef]
Fowler, A. G., Laskovski, A. N., Hammond, A. C., and Moheimani, S. O. R., 2012, “A 2-DOF Electrostatically Actuated MEMS Nanopositioner for On-Chip AFM,” J. Microelectromech. Syst., 21, pp. 771–773. [CrossRef]
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, New York.
Rao, S. S., 2007, Vibration of Continuous Systems, Wiley, Hoboken, NJ.
Crespo Da Silva, M. R. M., 1988, “Non-Linear Flexural-Flexural-Torsional-Extensional Dynamics of Beams-I. Formulation,” Int. J. Solids Struct., 24, pp. 1225–1234. [CrossRef]
Crespo Da Silva, M. R. M., 1998, “A Reduced-Order Analytical Model for the Nonlinear Dynamics of a Class of Flexible Multi-Beam Structures,” Int. J. Solids Struct., 35, pp. 3299–3315. [CrossRef]
Nayfeh, A. H., 1973, “Nonlinear Transverse Vibrations of Beams With Properties That Vary Along the Length,” J. Acoust. Soc. Am., 53, pp. 766–770. [CrossRef]
Zaretzky, C. L., and Crespo da Silva, M. R. M., 1994, “Experimental Investigation of Non-Linear Modal Coupling in the Response of Cantilever Beams,” J. Sound Vib., 174, pp. 145–167. [CrossRef]
Hijmissen, J. W., and van Horssen, W. T., 2008, “On the Weakly Damped Vibrations of a Vertical Beam With a Tip-Mass,” J. Sound Vib., 310, pp. 740–754. [CrossRef]
Zavodney, L. D., and Nayfeh, A. H., 1989, “The Non-Linear Response of a Slender Beam Carrying a Lumped Mass to a Principal Parametric Excitation: Theory and Experiment,” Int. J. Non-Linear Mech., 24, pp. 105–125. [CrossRef]
Yu, Y. Q., Howell, L. L., Lusk, C., Yue, Y., and He, M. G., 2005, “Dynamic Modeling of Compliant Mechanisms Based on the Pseudo-Rigid-Body Model,” ASME J. Mech. Des., 127(4), pp. 760–765. [CrossRef]
Awtar, S., and Sen, S., 2010, “A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation,” ASME J. Mech. Des., 132(8), p. 081009. [CrossRef]
MATLAB Product Help, R2010a, ode45 command.


Grahic Jump Location
Fig. 1

A parallel-kinematic XY flexure mechanism [5]

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Fig. 2

Schematic view of a beam with a tip mass

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Fig. 3

Comparison of analytical results with numerical simulations for an undamped system with initial conditions w(0) = 0.1 and u(0) = −0.006. (a) Tip transverse displacement and (b) tip axial displacement.

Grahic Jump Location
Fig. 4

Comparison of the analytical results with numerical simulations for a damped system with C1 = 0.001 and initial conditions w(0) = 0.1 and u(0) = −0.006. (a) Tip transverse displacement and (b) tip axial displacement.

Grahic Jump Location
Fig. 5

Zoomed view of the normalized axial displacement: (a) undamped system, and (b) damped system, C1 = 0.001




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