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Research Papers

Spectral Finite Element for Wave Propagation in Curved Beams

[+] Author and Article Information
Namita Nanda

Department of Applied Mechanics,
Indian Institute of Technology Delhi,
Hauz Khas, New Delhi 110016, India
e-mail: nanda.namita@gmail.com

Santosh Kapuria

Department of Applied Mechanics,
Indian Institute of Technology Delhi,
Hauz Khas, New Delhi 110016, India
e-mail: kapuria@am.iitd.ac.in

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 19, 2014; final manuscript received February 18, 2015; published online March 12, 2015. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 137(4), 041005 (Aug 01, 2015) (10 pages) Paper No: VIB-14-1352; doi: 10.1115/1.4029900 History: Received September 19, 2014; Revised February 18, 2015; Online March 12, 2015

In this paper, spectral finite elements (SFEs) are developed for wave propagation analysis of isotropic curved beams using three different beam models: (1) the refined third-order shear deformation theory (TOT), (2) the first-order shear deformation theory (FSDT), and (3) the classical shell theory (CST). The formulation is validated by comparing the results for the wavenumber dispersion relations and natural frequencies with the published results based on the FSDT. The numerical study reveals that even for a very thin curved beam with radius-to-thickness ratio of 1000, the wavenumbers predicted by the CST at high frequencies show significant deviation from those of the shear deformable theories, FSDT and TOT. The FSDT results for the wavenumber of the flexural displacement mode differ significantly from the TOT results at high frequencies even for thin beams. The deviation increases and occurs at lower frequencies with the decrease in the radius-to-thickness ratio. The results for wave propagation response show that the CST yields highly erroneous response for flexural mode wave propagation even for thin beams and at a relatively low frequency of 20 kHz. The FSDT results too differ by unacceptably high margin from the TOT results for flexural wave response of thin beams at frequencies greater than 100 kHz, which are typically used for structural health monitoring (SHM) applications. For thick beams, FSDT results for the tangential wave response also show large deviation from the TOT results.

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References

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Figures

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Fig. 1

Curved beam section

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Fig. 2

Wavenumber dispersion relations for the curved beam of Kang and Riedel [12]

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Fig. 3

Wavenumber dispersion relations for a very thin curved beam with R/h = 1000

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Fig. 4

Wavenumber dispersion relations for a thin curved beam with R/h = 50

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Fig. 5

Wavenumber dispersion relations for a thick curved beam with R/h = 5

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Fig. 6

Comparison of phase speed relations for the curved beam with Kang and Riedel [12]

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Fig. 7

Group speed of tangential wave mode for a thin curved beam with R/h = 50

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Fig. 8

Group speed of flexural displacement mode for a thin curved beam with R/h = 50

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Fig. 9

Group speed of flexural rotation mode for a thin curved beam with R/h = 50

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Fig. 10

Group speed of tangential wave mode for a thick curved beam with R/h = 5

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Fig. 11

Group speed of flexural displacement mode for a thick curved beam with R/h = 5

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Fig. 12

Group speed of flexural rotation mode for a thick curved beam with R/h = 5

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Fig. 13

Five-cycle Hanning window modulated sinusoidal tone burst excitation with 200 kHz central frequency. Frequency spectrum of the load is shown in the inset.

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Fig. 14

Clamped–clamped curved beam. Excited at point P and the response measured at point Q.

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Fig. 15

Transverse velocity at point Q due to 20 kHz modulated five-cycle tone burst applied transversely at point P for R/h = 50

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Fig. 16

Transverse velocity at point Q due to 100 kHz modulated five-cycle tone burst applied transversely at point P for R/h = 50

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Fig. 17

Tangential velocity at point Q due to 200 kHz modulated five-cycle tone burst applied transversely at point P for R/h = 5

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