0
Research Papers

A Wave-Based Analytical Solution to Free Vibrations in a Combined Euler–Bernoulli Beam/Frame and a Two-Degree-of-Freedom Spring–Mass System

[+] Author and Article Information
C. Mei

Department of Mechanical Engineering,
The University of Michigan—Dearborn,
4901 Evergreen Road,
Dearborn, MI 48128
e-mail: cmei@umich.edu

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 18, 2017; final manuscript received April 9, 2018; published online May 7, 2018. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 140(6), 061001 (May 07, 2018) (8 pages) Paper No: VIB-17-1324; doi: 10.1115/1.4039961 History: Received July 18, 2017; Revised April 09, 2018

In this paper, natural frequencies and modeshapes of a transversely vibrating Euler–Bernoulli beam carrying a discrete two-degree-of-freedom (2DOF) spring–mass system are obtained from a wave vibration point of view in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external forces applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam and 2DOF spring–mass system, the vibrating discrete spring–mass system injects waves into the distributed beam through the spring forces at the two spring attached points. Assembling these wave relations in the beam provides an analytical solution to vibrations of the combined system. Accuracy of the proposed wave analysis approach is validated through comparisons to available results. This wave-based approach is further extended to analyze vibrations in a planar portal frame that carries a discrete 2DOF spring–mass system, where in addition to the transverse motion, the axial motion must be included due to the coupling effect at the angled joint of the frame. The wave vibration approach is seen to provide a systematic and concise technique for solving vibration problems in combined distributed and discrete systems.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Wu, J. J. , and Whittaker, A. R. , 1999, “The Natural Frequencies and Mode Shapes of a Uniform Cantilever Beam With Multiple Two-DOF Spring–Mass Systems,” J. Sound Vib., 227(2), pp. 361–381. [CrossRef]
Wu, J. J. , 2004, “Free Vibration Analysis of Beams Carrying a Number of 2DOF Spring-Damper-Mass Systems,” Finite Elem. Anal. Des., 40(4), pp. 363–381. [CrossRef]
Wu, J. J. , 2006, “Use of Equivalent Mass Method for Free Vibration Analyses of a Beam Carrying Multiple Two-DOF-Spring–Mass Systems With Inertia Effect of the Helical Springs Considered,” Int. J. Numer. Methods Eng., 65(5), pp. 653–678. [CrossRef]
Wu, J. J. , 2005, “Use of Equivalent-Damper Method for Free Vibration Analysis of a Beam Carrying Multiple Two-Degree-of-Freedom Spring–Damper–Mass Systems,” J. Sound Vib., 281(1–2), pp. 275–293. [CrossRef]
Wu, J. J. , 2002, “Alternative Approach for Free Vibration of Beams Carrying a Number of Two-Degree of Freedom Spring–Mass Systems,” J. Struct. Eng., 128(12), pp. 1604–1616. [CrossRef]
Chen, D. W. , 2006, “The Exact Solution for Free Vibration of Uniform Beams Carrying Multiple Two-Degree-of-Freedom Spring–Mass Systems,” J. Sound Vib., 295(1–2), pp. 342–361. [CrossRef]
Jen, M. U. , and Magrab, E. B. , 1993, “Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System,” ASME J. Vib. Acoust., 115(2), pp. 202–209. [CrossRef]
Chang, T. P. , and Chang, C. Y. , 1998, “Vibration Analysis of Beams With a Two Degree-of-Freedom Spring-Mass System,” Int. J. Solids Struct., 35(5–6), pp. 383–401. [CrossRef]
Banerjee, J. R. , 2003, “Dynamic Stiffness Formulation and Its Application for a Combined Beam and a Two Degree-of-Freedom System,” ASME J. Vib. Acoust., 125(3), pp. 351–358. [CrossRef]
Chen, J. , Dong, D. , Yan, B. , and Hua, C. , 2016, “An Analytical Study on Forced Vibration of Beams Carrying a Number of Two Degrees-of-Freedom Spring–Damper–Mass Subsystems,” ASME J. Vib. Acoust., 138(6), p. 061011. [CrossRef]
Cha, P. D. , 2007, “Free Vibration of a Uniform Beam With Multiple Elastically Mounted 2DOF Systems,” J. Sound Vib., 307(1–2), pp. 386–392. [CrossRef]
Graff, K. F. , 1975, Wave Motion in Elastic Solids, Ohio State University Press, New York.
Cremer, L. , Heckl, M. , and Ungar, E. E. , 1987, Structure-Borne Sound, Springer-Verlag, Berlin.
Doyle, J. F. , 1989, Wave Propagation in Structures, Springer-Verlag, New York. [CrossRef]
Mace, B. R. , 1984, “Wave Reflection and Transmission in Beams,” J. Sound Vib., 97(2), pp. 237–246. [CrossRef]
Mei, C. , and Mace, B. R. , 2005, “Wave Reflection and Transmission in Timoshenko Beams and Wave Analysis of Timoshenko Beam Structures,” ASME J. Vib. Acoust., 127(4), pp. 382–394. [CrossRef]
Mei, C. , 2010, “In-Plane Vibrations of Classical Planar Frame Structures—An Exact Wave-Based Analytical Solution,” J. Vib. Control, 16(9), pp. 1265–1285. [CrossRef]
Mei, C. , 2012, “Wave Analysis of In-Plane Vibrations of L-Shaped and Portal Planar Frame Structures,” ASME J. Vib. Acoust., 134(2), p. 021011. [CrossRef]
Mei, C. , 2013, “Free Vibration Analysis of Classical Single-Story Multi-Bay Planar Frames,” J. Vib. Control, 19(13), pp. 2022–2035. [CrossRef]
Mei, C. , and Sha, H. , 2015, “An Exact Analytical Approach for Vibrations in Built-Up Space Frames,” ASME J. Vib. Acoust., 137(3), p. 031005. [CrossRef]
Meirovitch, L. , 2001, Fundamentals of Vibrations, McGraw-Hill, New York. [PubMed] [PubMed]
Inman, D. J. , 1994, Engineering Vibrations, Prentice Hall, Upper Saddle River, NJ.
Chang, C. H. , 1978, “Vibrations of Frames With Inclined Members,” J. Sound Vib., 56(2), pp. 201–214. [CrossRef]
Lin, H. P. , and Ro, J. , 2003, “Vibration Analysis of Planar Serial-Frame Structures,” J. Sound Vib., 262(5), pp. 1113–1131. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Sign convention with positive parameters shown

Grahic Jump Location
Fig. 2

(a) A beam carrying a 2DOF spring–mass system and (b) its free body diagram

Grahic Jump Location
Fig. 3

Waves generated by external forces and moments

Grahic Jump Location
Fig. 4

Wave in a beam carrying a 2DOF spring–mass system

Grahic Jump Location
Fig. 5

Wave in a portal frame carrying a 2DOF spring–mass system

Grahic Jump Location
Fig. 6

The first five modeshapes of the clamped-clamped beam with (___) and without (…) the 2DOF spring–mass system

Grahic Jump Location
Fig. 7

The first five modeshapes of the simply supported and simply supported beam with (___) and without (…) the 2DOF spring–mass system

Grahic Jump Location
Fig. 8

Responses of the characteristic polynomial of frame from Ref. [23] with (___) and without (…) the 2DOF spring–mass system

Grahic Jump Location
Fig. 9

Responses of the characteristic polynomial of frame from Ref. [24] with (___) and without (…) the 2DOF spring–mass system

Tables

Errata

Discussions

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