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Research Papers

An Exact Analytical Approach for Free Vibration Analysis of Built-Up Space Frames

[+] 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

H. Sha

Department of Mechanical Engineering,
The University of Michigan–Dearborn,
4901 Evergreen Road,
Dearborn, MI 48128

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 15, 2013; final manuscript received November 23, 2014; published online January 27, 2015. Assoc. Editor: Guilhem Michon.

J. Vib. Acoust 137(3), 031005 (Jun 01, 2015) (12 pages) Paper No: VIB-13-1318; doi: 10.1115/1.4029380 History: Received September 15, 2013; Revised November 23, 2014; Online January 27, 2015

Coupled in- and out-of-plane bending, axial, and torsional vibrations exist in a built-up space frame, which greatly challenges the conventional modal approach. A wave-based exact analytical method is applied to study vibrations in built-up multistory space frames. From the wave vibration standpoint, vibrations are described as waves propagating along uniform structural elements and being reflected and transmitted at structural discontinuities. Free vibration responses are studied based on classical vibration theories. Numerical examples are given with comparison to results available in the literature. Good agreements have been reached. This study not only provides an exact analytical approach to complex vibration problems in built-up multistory space frames that have mostly been studied numerically but also provides a benchmark to existing numerical tools.

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References

Petyt, M., 2010, Introduction to Finite Element Vibration Analysis, Cambridge University Press, New York.
Tu, T. H., Yu, J. F., Lien, H. C., Tsai, G. L., and Wang, B. P., 2008, “Free Vibration Analysis of Frames Using the Transfer Dynamic Stiffness Matrix Method,” ASME J. Vib. Acoust., 130(2), p. 024501. [CrossRef]
Graff, K. F., 1975, Wave Motion in Elastic Solids, Oxford University Press, London, UK.
Cremer, L., Heckl, M., and Petersson, B. A. T., 2014, Structure-Borne Sound, Springer-Verlag, Berlin, Germany.
Doyle, J. F., 1989, Wave Propagation in Structures, Springer-Verlag, New York.
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., 2013, “Free Vibration Analysis of Classical Single-Story Multi-Bay Planar Frames,” J. Vib. Control, 19(13), pp. 2022–2035. [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., 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., 2008, “Wave Analysis of In-Plane Vibrations of H- and T-Shaped Planar Frame Structures,” ASME J. Vib. Acoust., 130(6), p. 061004. [CrossRef]
Meirovitch, L., 2001, Fundamentals of Vibrations, McGraw-Hill, New York.
Mei, C., 2011, “Free Vibration Studies of Classical Beams/Rods With Lumped Masses at Boundaries Using a Wave Vibration Based Approach,” Int. J. Mech. Eng. Edu., 39(3), pp. 256–268. [CrossRef]
Mei, C., and Sha, H., 2014, “Analytical and Experimental Study of Vibrations in Simple Spatial Structures,” J. Vib. Control (in press).

Figures

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Fig. 1

An n-story space frame

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Fig. 2

Definition of positive signs for shear forces, longitudinal force, bending moments, and torque

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Fig. 3

The spatial K joint and the definition of coordinates

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Fig. 4

(a) Front view of FBD of the K joint, (b) left view of FBD of the K joint, and (c) top view of FBD of the K joint

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Fig. 5

Wave transmission and reflection at the K joint with waves: (a) Incident from beam 1, (b) incident from beam 2, (c) incident from beam 3, and (d) incident from beam 4

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Fig. 6

Linear magnitude response of the characteristic polynomial of the three-story frame

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Fig. 7

dB magnitude response of the characteristic polynomial of the three-story frame

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Fig. 8

Linear magnitude response of the characteristic polynomial of the two-story frame

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Fig. 9

dB magnitude response of the characteristic polynomial of the two-story frame

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