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Research Papers

Mode Count and Modal Density of Isotropic Circular Cylindrical Shells Using a Modified Wavenumber Space Integration Method

[+] Author and Article Information
Mohammad H. Farshidianfar

Master of Applied Science Candidate
Department of Mechanical and
Mechatronics Engineering,
University of Waterloo,
Waterloo, ON N2L 3G1, Canada
e-mail: mhfarshi@uwaterloo.ca

Anooshiravan Farshidianfar

Professor
Department of Mechanical Engineering,
Ferdowsi University of Mashhad,
Mashhad, Iran
e-mail: farshid@um.ac.ir

Mahdi Mazloom Moghadam

Master of Applied Science Candidate
Department of Mechanical Engineering,
Ferdowsi University of Mashhad,
Mashhad, Iran
e-mail: mah.mazloom.mo@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 9, 2013; final manuscript received March 13, 2014; published online April 18, 2014. Assoc. Editor: Marco Amabili.

J. Vib. Acoust 136(4), 041004 (Apr 18, 2014) (13 pages) Paper No: VIB-13-1313; doi: 10.1115/1.4027212 History: Received September 09, 2013; Revised March 13, 2014

A new method based on the wavenumber space integration algorithm is proposed in order to obtain mode count and modal density of circular cylindrical shells. Instead of the simplified equation of motion, the exact equation is applied in mode count calculations. Modal plots are changed significantly in the k-space when using the exact equation. Mode repetition in cylindrical shells is represented by additional mode count curves in the k-space. On the other hand, a novel technique is presented in order to implement boundary condition effects in mode count and modal density calculations. Integrating these two significant corrections, a modified wavenumber space integration (MWSI) method is developed. Mode count and modal densities of three shells with different geometrical and acoustical properties are obtained using the MWSI method and conventional WSI. Results are verified using the exact mode count calculations. Moreover, effects of geometrical properties are studied on mode count plots in the k-space. Modal densities are obtained for cylindrical shells of different lengths, radii, and thicknesses. Finally, modal densities of cylindrical shells are compared to flat plates of the same size and boundary condition. Interesting results are obtained which will contribute in calculation of acoustic radiation efficiency and sound transmission in cylindrical shells.

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Figures

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Fig. 1

Circular cylindrical shell: coordinate system and dimensions

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Fig. 2

Modes of a simply supported (SS–SS) flat plate shown in k-space

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Fig. 3

Modes of a simply supported cylindrical (SS–SS) shell in the mn-space (analogue to k-space)

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Fig. 4

Modal plot of shell no. 1 in the mn-space: (a) low frequency Ω1 = 0.1 (F1 = 1112 Hz), (b) medium frequency Ω2 = 0.5 (F2 = 5560 Hz), and (c) high frequency Ω3 = 1.5 (F3 = 16680 Hz)

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Fig. 5

Mode repetition identified by different markers: (1) once “•,” (2) twice “○,” and (3) trice “*” repeated modes

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Fig. 8

Modal densities of the three shells according to different theories: (a) shell no. 1, (b) shell no. 2, and (c) shell no. 3

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Fig. 9

Modal densities of the three shells for simply supported (SS–SS) and clamped (C–C) boundary conditions: (a) shell no. 1, (b) shell no. 2, and (c) shell no. 3

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Fig. 10

Modification of mode plots with length variation in mn-space: (a) l = 1.73 m, (b) l = 4 m, and (c) l = 6 m (each column is obtained for a similar frequency and each row is for a fixed shell length)

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Fig. 11

Modification of mode plots with thickness variation in mn-space: (a) h = 0.00147 m, (b) h = 0.004 m, and (c) h = 0.006 m (each column is obtained for a similar frequency and each row is for a fixed shell length)

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Fig. 12

Modification of mode plots with radius variation in mn-space: (a) R = 0.0762 m, (b) R = 0.2 m, and (c) R = 0.4 m (each column is obtained for a similar frequency and each row is for a fixed shell length)

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Fig. 13

Modal densities of shells with different lengths

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Fig. 14

Modal densities of shells with different thicknesses

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Fig. 15

Modal densities of shells with different radii

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Fig. 16

Modal density of a cylindrical shell compared to a flat plate at two different boundary conditions

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