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Vibration analysis of a circular cylinder closed with a hemi-spheroidal cap having a hole

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Abstract

The natural frequencies and mode shapes of a circular cylindrical shell closed with a hemi-spheroidal dome are determined by the Ritz method using a three-dimensional (3-D) analysis instead of two-dimensional (2-D) thin shell theories or higher-order thick shell theories. The present analysis is based upon the circular cylindrical coordinates, while in the traditional shell analyses 3-D shell coordinates have been usually used. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. Natural frequencies are presented for different boundary conditions. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the combined shell. The frequencies from the present 3-D method are compared with those from 2-D thin shell theories. The present method is applicable to very thick shells as well as thin shells.

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Abbreviations

a :

Length of semimajor axis of the mid-surface of a hemi-ellipsoidal cap radius of the mid-surface of a circular cylindrical shell

\(A_{ij},B_{kl},C_{mn} \) :

Arbitrary coefficients

b :

Length of semiminor axis of the mid-surface of a hemi-ellipsoidal cap

\(D_{A }\) :

Hilbert space

DET :

Size of determinant

E :

Young’s modulus

H :

Uniform thickness of a joined shell

\(H^{*}\) :

\({\equiv } H/a\)

i, j, k, l, m, n :

Indices for double summation (nonnegative integer)

\(I_V, I_T \) :

Defined by Eqs. (21) and (22), respectively

I, J, K, L, M, N :

Highest degrees of the Legendre polynomial terms

\({k}'\) :

\({\equiv } b/a\)

K :

Stiffness matrix

\(\hbox {K}_{\alpha \beta \hat{{\alpha }}\hat{{\beta }}} ,\hbox {M}_{\alpha \beta \hat{{\alpha }}\hat{{\beta }}} \) :

Submatrix of K and M \( (\alpha =i,k,m;\;\;\beta \hbox {=}j,l,n:\quad \hat{{\alpha }}=\hat{{i}},\hat{{k}},\hat{{m}};\;\;\hat{{\beta }}\hbox {=}\hat{{j}},\hat{{l}},\hat{{n}})\)

L :

Height of the mid-surface of a circular cylindrical shell

\(L^{*}\) :

\({\equiv } L/a\)

M :

Mass matrix

n :

Circumferential wave number (\(n=0,1,2,\ldots \))

\(P_n\) :

Legendre polynomial (\(n=0,1,2,\ldots \))

\(P_{\alpha \beta }\) :

\({\equiv } P_\alpha (\psi )P_\beta (\zeta )\; (\alpha =i,k,m,\;\beta =j,l,n)\)

r :

Radial coordinate

\(r,\theta , z\) :

Circular cylindrical coordinate system

\(R_i \) :

Radius of an axially circular hole of hemi-ellipsoidal cap

\(R_i^*\) :

\(R_i /a\)

s :

Mode number

t :

Time

T :

Kinetic energy

\(T_{\max }\) :

Maximum kinetic energy

TR :

Total number of the Legendre polynomial terms used in r or \(\psi \) direction

TZ :

Total number of the Legendre polynomial terms used in z or \(\zeta \) direction

\(u_r,u_z,u_\theta \) :

Displacements in the directions of \(r,z,\theta \), respectively

\(U_r, U_z, U_\theta \) :

Displacement functions of \(\psi \) and \( \zeta \)

V :

Strain energy

\(V_{\max }\) :

Maximum strain energy

x :

Vector of unknown coefficients

z :

Axial coordinate

\(z_{i,o} \) :

Coordinates of the inner and outer hemi-ellipsoidal surfaces for \(r\ge 0\), respectively

\(\alpha \) :

Arbitrary phase angle

\(\Gamma _1, \Gamma _2 \) :

Constants, defined by Eq. (24)

\(\delta _{ij} \) :

Kronecker delta

\(\varepsilon \) :

\({\equiv } \varepsilon _{rr} +\varepsilon _{zz} +\varepsilon _{\theta \theta }\)

\(\varepsilon _{ij} \) :

Tensorial strain

\( \zeta \) :

Non-dimensional axial coordinate (\({\equiv } z/L)\)

\( \zeta _{i,o}\) :

\({\equiv } z_{i,o} /L\) \(\upeta _{r,\theta ,z}\) functions of \(\psi \) and \(\zeta \) depending upon the geometric boundary conditions

\(\theta \) :

Circumferential coordinate

\(\kappa _i \) :

Functions defined by Eq. (23) (\(i=1,2,\ldots ,6\))

\(\lambda ,G\) :

Lamé parameters

\(\Lambda \) :

Domain of a joined shell

\(\upnu \) :

Poisson’s ratio

\(\pi \) :

3.1415926535...

\(\rho \) :

Mass density per unit volume

\(\sigma _{ij} \) :

Tensorial stress

\(\psi \) :

Non-dimensional radial coordinate \(\left( {{\equiv } r/a} \right) \)

\(\psi ,\theta ,\zeta \) :

Non-dimensional circular cylindrical coordinates

\(\omega \) :

Natural frequency

\(\Omega \) :

Square of non-dimensional frequency (\({\equiv } \omega ^{2}a^{2}\rho /G)\)

\(0^\mathrm{A}\) :

Circumferential wave number for axisymmetric modes

\(0^\mathrm{T}\) :

Circumferential wave number for torsional modes

2DS:

2-D thin shell theory

\({\bullet }\) :

Time derivative

, :

Spatial derivative

[n]:

The largest integer \({\le } n\)

\(\left\langle {f,g} \right\rangle \) :

\({\equiv } \eta (\psi ,\zeta ){\mathop {\iint }\limits _{\Lambda }} {f(\psi ,\zeta ) g(\psi ,\zeta )\; \psi d\zeta d\psi }\)

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  1. Department of Architectural Engineering, Chung-Ang University, 221 Heukseok-Dong, Dongjak-Gu, Seoul, 156-756, South Korea

    Jae-Hoon Kang

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Kang, JH. Vibration analysis of a circular cylinder closed with a hemi-spheroidal cap having a hole. Arch Appl Mech 87, 183–199 (2017). https://doi.org/10.1007/s00419-016-1186-9

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