Research Papers

Exact Frequency Equation of a Linear Structure Carrying Lumped Elements Using the Assumed Modes Method

[+] Author and Article Information
Philip D. Cha

Department of Engineering,
Harvey Mudd College,
301 Platt Boulevard,
Claremont, CA 91711
e-mail: philip_cha@hmc.edu

Siyi Hu

Department of Engineering,
Harvey Mudd College,
301 Platt Boulevard,
Claremont, CA 91711

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 2, 2016; final manuscript received November 29, 2016; published online March 16, 2017. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 139(3), 031005 (Mar 16, 2017) (15 pages) Paper No: VIB-16-1436; doi: 10.1115/1.4035382 History: Received September 02, 2016; Revised November 29, 2016

Combined systems consisting of linear structures carrying lumped attachments have received considerable attention over the years. In this paper, the assumed modes method is first used to formulate the governing equations of the combined system, and the corresponding generalized eigenvalue problem is then manipulated into a frequency equation. As the number of modes used in the assumed modes method increases, the approximate eigenvalues converge to the exact solutions. Interestingly, under certain conditions, as the number of component modes goes to infinity, the infinite sum term in the frequency equation can be reduced to a finite sum using digamma function. The conditions that must be met in order to reduce an infinite sum to a finite sum are specified, and the closed-form expressions for the infinite sum are derived for certain linear structures. Knowing these expressions allows one to easily formulate the exact frequency equations of various combined systems, including a uniform fixed–fixed or fixed-free rod carrying lumped translational elements, a simply supported beam carrying any combination of lumped translational and torsional attachments, or a cantilever beam carrying lumped translational and/or torsional elements at the beam's tip. The scheme developed in this paper is easy to implement and simple to code. More importantly, numerical experiments show that the eigenvalues obtained using the proposed method match those found by solving a boundary value problem.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Özgüven, H. N. , and Çandir, B. , 1986, “Suppressing the First and Second Resonances of Beams by Dynamic Vibration Absorbers,” J. Sound Vib., 111(3), pp. 377–390. [CrossRef]
Cha, P. D. , and Wong, W. C. , 1999, “A Novel Approach to Determine the Frequency Equations of Combined Dynamical Systems,” J. Sound Vib., 219(4), pp. 689–706. [CrossRef]
Gürgöze, M. , 1996, “On the Eigenfrequencies of a Cantilever Beam With Attached Tip Mass and a Spring-Mass System,” J. Sound Vib., 190(2), pp. 149–162. [CrossRef]
Posiadała, B. , 1997, “Free Vibrations of Uniform Timoshenko Beams With Attachments,” J. Sound Vib., 204(2), pp. 359–369. [CrossRef]
Lueschen, G. G. G. , Bergman, L. A. , and McFarland, D. M. , 1996, “Green's Functions for Uniform Timoshenko Beams,” J. Sound Vib., 194(1), pp. 93–102. [CrossRef]
Kukla, S. , 1997, “Application of Green Functions in Frequency Analysis of Timoshenko Beams With Oscillators,” J. Sound Vib., 205(3), pp. 355–363. [CrossRef]
Chang, T. P. , Chang, F. I. , and Liu, M. F. , 2001, “On the Eigenvalues of a Viscously Damped Simple Beam Carrying Point Masses and Springs,” J. Sound Vib., 240(4), pp. 769–778. [CrossRef]
Wu, J. S. , and Lin, T. L. , 1990, “Free Vibration Analysis of a Uniform Cantilever Beam With Point Masses by an Analytical-and-Numerical-Combined Method,” J. Sound Vib., 136(2), pp. 201–213. [CrossRef]
Wu, J. S. , and Chou, H. M. , 1999, “A New Approach for Determining the Natural Frequencies and Mode Shapes of a Uniform Beam Carrying any Number of Sprung Masses,” J. Sound Vib., 220(3), pp. 451–468. [CrossRef]
Cha, P. D. , 2005, “A General Approach to Formulating the Frequency Equation for a Beam Carrying Miscellaneous Attachments,” J. Sound Vib., 286(4), pp. 921–939. [CrossRef]
Cha, P. D. , and Yoder, N. C. , 2007, “Applying Sherman–Morrison–Woodbury Formulas to Analyze the Free and Forced Responses of a Linear Structure Carrying Lumped Elements,” ASME J. Vib. Acoust., 129(3), pp. 307–316. [CrossRef]
Meirovitch, L. , 2001, Fundamentals of Vibrations, The McGraw-Hill Companies, New York.
Abramowitz, M. , and Segun, I. A. , 1972, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Dover Publications, New York.
Gonçalves, P. J. P. , Brennan, M. J. , and Elliott, S. J. , 2007, “Numerical Evaluation of Higher-Order Modes of Vibration in Uniform, Euler–Bernoulli Beams,” J. Sound Vib., 301(3), pp. 1035–1039. [CrossRef]


Grahic Jump Location
Fig. 2

Arbitrarily supported beam carrying various translational and torsional lumped elements

Grahic Jump Location
Fig. 1

Arbitrarily supported rod carrying various lumped translational elements

Grahic Jump Location
Fig. 3

The eigenfunction of a fixed–fixed rod evaluated at xa=1/5 squared

Grahic Jump Location
Fig. 4

The eigenfunction of a fixed–fixed rod evaluated at xa=3/8 squared

Grahic Jump Location
Fig. 5

The eigenfunction of a fixed-free rod evaluated at xa=4/7 squared

Grahic Jump Location
Fig. 6

The slope of the eigenfunction of a simply supported beam evaluated at xa=2/7 squared over the natural frequency of the simply supported beam

Grahic Jump Location
Fig. 7

The slope of the eigenfunction of a cantilever beam evaluated at xa=1 squared over the natural frequency of the cantilever beam

Grahic Jump Location
Fig. 8

The product of the eigenfunction and the slope of a cantilever beam evaluated at xi=xj=1 over the square root of the natural frequency of the cantilever beam

Grahic Jump Location
Fig. 9

ω4 versus n′ for Example 7



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