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Research Articles

Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

[+] Author and Article Information
D. Zhou

School of Civil Engineering,
Nanjing University of Technology,
Nanjing 210009, China
e-mail: dingzhou57@yahoo.com

S. H. Lo

Department of Civil Engineering,
University of Hong Kong,
Pokfulam Road,
Hong Kong, China

1To whom correspondence should be addressed.

Manuscript received September 22, 2011; final manuscript received July 4, 2012; accepted manuscript posted August 27, 2012; published online January 25, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 80(2), 021016 (Jan 25, 2013) (10 pages) Paper No: JAM-11-1342; doi: 10.1115/1.4007475 History: Received September 22, 2011; Revised July 04, 2012; Accepted August 27, 2012

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

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Figures

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Fig. 1

A twisted cylinder with sectorial cross section: shape, sizes, and coordinates

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Fig. 6

The first six modes of cantilevered twisted cylinder with semicircular cross section for twist angle φ0=90 deg

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Fig. 7

The first six modes of a cantilevered twisted cylinder of sectorial cross section with subtended angle θ0 = 270 deg for twist angle φ0=90 deg

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Fig. 5

The effect of twist angle on the first four frequency parameters of a cantilevered twisted cylinder with semicircular cross section, ○ the first frequency parameter; □ the second frequency parameter; Δ the third frequency parameter; × the fourth frequency parameter

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Fig. 4

The effect of twist angle on the first four frequency parameters of a complete free twisted cylinder with a radial crack, □ the first frequency parameter; □ the second frequency parameter; Δ the third frequency parameter; × the fourth frequency parameter

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Fig. 3

The effect of twist angle on the first four frequency parameters of a fixed-fixed twisted cylinder with a radial crack, ○ the first frequency parameter; □ the second frequency parameter; Δ the third frequency parameter; × the fourth frequency parameter

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Fig. 2

The effect of twist angle on the first four frequency parameters of a cantilevered twisted cylinder with a radial crack, ○ the first frequency parameter; □ the second frequency parameter; Δ the third frequency parameter; × the fourth frequency parameter

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