Technical Briefs

Numerical Evaluation of High-Order Modes for Stepped Beam

[+] Author and Article Information
Wei Xu

Research Assistant
Department of Engineering Mechanics,
College of Mechanics and Materials,
Hohai University,
Nanjing 210098, China

Maosen Cao

e-mail: cmszhy@hhu.edu.cn

Qingwen Ren

Department of Engineering Mechanics,
College of Mechanics and Materials,
Hohai University,
Nanjing 210098, China

Zhongqing Su

Associate Professor
Department of Mechanical Engineering,
The Hong Kong Polytechnic University,
Hung Hom, Kowloon,
Hong Kong

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 21, 2012; final manuscript received September 18, 2013; published online November 6, 2013. Assoc. Editor: Olivier A. Bauchau.

J. Vib. Acoust 136(1), 014503 (Nov 06, 2013) (6 pages) Paper No: VIB-12-1323; doi: 10.1115/1.4025696 History: Received November 21, 2012; Revised September 18, 2013

The numerical evaluation of high-order modes of a uniform Euler–Bernoulli beam has been studied by reformatting the classical expression of mode shapes. That method, however, is inapplicable to a stepped beam due to the nonuniform expressions of the mode shape for each beam segment. Given that concern, this study develops an alternative method for the numerical evaluation of high-order modes for stepped beams. This method effectively expands the space of high-order modal solutions by introducing local coordinate systems to replace the conventional global coordinate system. This set of local coordinate systems can significantly simplify the frequency determinant of vibration equations of a stepped beam, in turn, largely eliminating numerical round-off errors and conducive to high-order mode evaluation. The efficacy of the proposed scheme is validated using various models of Euler–Bernoulli stepped beams. The principle of the method has the potential for extension to other types of Euler–Bernoulli beams with discontinuities in material and geometry. (The Matlab code for the numerical evaluation of high-order modes for stepped beams can be provided by the corresponding author upon request.)

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Bai, R. B., Cao, M. S., Su, Z. Q., Ostachowicz, W., and Xu, H., 2012, “Fractal Dimension Analysis of Higher-Order Mode Shapes for Damage Identification of Beam Structures,” Math. Probl. Eng., 2012, p. 454568. [CrossRef]
Wei, G. W., Zhao, Y. B., and XiangY., 2002, “A Novel Approach for the Analysis of High-Frequency Vibrations,” J. Sound Vib., 257(2), pp. 207–246. [CrossRef]
Zhang, W. G., Wang, A. M., Vlahopoulos, N., and Wu, K. C., 2003, “High-Frequency Vibration Analysis of Thin Elastic Plates Under Heavy Fluid Loading by an Energy Finite Element Formulation,” J. Sound Vib., 263, pp. 21–46. [CrossRef]
Lau.E., Al-Dujaili, S., Guenther, A., Liu, D., Wang, L., and You, L., 2010, “Effect of Low-Magnitude, High-Frequency Vibration on Osteocytes in the Regulation of Osteoclasts,” Bone, 46, pp. 1508–1515. [CrossRef] [PubMed]
Tang, Y., 2003, “Numerical Evaluation of Uniform Beam Modes,” J. Eng. Mech., 12, pp. 1475–1477. [CrossRef]
Goncalves, P. J. P., Brennan, M. J., and Elliott, S. J., 2007, “Numerical Evaluation of High-Order Modes of Vibration in Uniform Euler–Bernoulli Beams,” J. Sound Vib., 301, pp. 1035–1039. [CrossRef]
Shankar, K., and Keane, A. J., 1994, “Energy Flow Predictions in a Structure of Rigidly Joined Beams Using Receptance Theory,” J. Sound Vib., 185, pp. 867–890. [CrossRef]
Low, K. H., 1993, “A Reliable Algorithm for Solving Frequency Equations Involving Transcendental Functions,” J. Sound Vib., 161, pp. 369–377. [CrossRef]
Michael, A. K., Abhijit, B., and Brain, P. M., 2006, “Closed Form Solutions for the Dynamic Response of Euler–Bernoulli Beams With Step Changes in Cross Section,” J. Sound Vib., 295, pp. 214–225. [CrossRef]
Naguleswaran, S., 2002, “Natural Frequencies Sensitivity and Mode Shape Details of an Euler–Bernoulli Beam With One-Step Change in Cross-Section and With Ends on Classical Supports,” J. Sound Vib., 254(4), pp. 751–767. [CrossRef]
G.Failla, 2011, “Closed-Form Solutions for Euler–Bernoulli Arbitrary Discontinuous Beams,” Arch. Appl. Mech., 81, pp. 605–608. [CrossRef]
Kukla, S., and Zamojska, I., 2007, “Frequency Analysis of Axially Loaded Stepped Beams by Green's Function Method,” J. Sound Vib., 300, pp. 1034–1041. [CrossRef]
Institute of Electrical and Electronics Engineers, 2008, “754-2008-IEEE Standard for Floating-Point Arithmetic,” IEEE, NewYork.


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Fig. 1

Two-step cantilever beam in the global coordinate system

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Fig. 2

The 12th-mode bottleneck in numerically evaluating high-order modes of the uniform beam illustrated by model I: (a) profile of D(ω) versus ω, and (b) the 11th and 12th mode shapes

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Fig. 3

The 8th-mode bottleneck in numerically evaluating high-order modes of the uniform beam illustrated by model II

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Fig. 4

Two-step cantilever beam in the local coordinate systems

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Fig. 5

Profile of Y1(ω) versus ω for model II

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Fig. 6

Profiles of (a),(c),(e) Y1(ω) versus ω and (b),(d),(f) D(ω) versus ω arising from the local and global coordinate systems, respectively




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