Vibrations of hemi-spherical shells of revolution with eccentricity from a three-dimensional theory

Abstract

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of hemi-spherical shells of revolution with eccentricity having uniform thickness. Unlike conventional shell theories, which are mathematically two-dimensional, the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_r, u_θ, and u_z in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to three or four-digit exactitude is demonstrated for the first five frequencies of the shells of revolution. Numerical results are presented for a variety of hemi-spherical shells of revolution with eccentricity.

Keywords

Eccentricity hemi-spherical shell of revolution three-dimensional analysis vibration

1. Introduction

Even though many researchers have analyzed by 3-D approaches the free vibration problems of solid spheres(Poisson, 1829; Jaerisch, 1880; Lamb, 1882; Chree, 1889; Sato and Usami, 1962; Buchanan and Rich, 2002) and complete hollow spheres (Sato and Usami, 1962; Shah et al., 1969a,b; Cohen and Shah, 1972; Grigorenko and Kilina, 1990; Chang and Demkowicz, 1995; Ding and Chen, 1996; Jiang et al., 1996; McGee and Spry, 1997; Buchanan and Rich, 2002), there is little published literature on the problem of hemispherical shells by a 3-D approach. Some researchers (Leissa and Kang, 1999; Kang and Leissa, 2000) analyzed hemispherical shells with variable thickness by a 3-D Ritz method, but they did not give results for hemispherical shells without a top opening due to the singularities arising at the peak of the shell. A review article by Qatu (2002) summarizes research on vibrations of homogeneous shells, including spherical ones, published between 1989 and 2000. Buchanan and Rich (2002) formulated a nine-node Lagrangian finite element based on a 3-D analysis and reported some numerical results for thick hemispherical shells with uniform thickness of various boundary conditions. Kang and Leissa (2004) analyzed hemi-spherical shells with variable thickness without a top opening by a 3-D method. All the existing researches on hemi-spherical shells are related to the shells with no eccentricity. It is known thatthere is no published literature on the problem of hemi-spherical shells of revolution with eccentricity. According to the size of eccentricity in hemi-spherical shells of revolution, diverse shapes of hemi-spherical shells can be obtained.

Conventional shell analysis, which is mathematically 2-D, makes simple kinematic assumptions about the variation of displacements through the thickness. Almost all such shell analyses assume, at most, constant plus linear variations. Such assumptions reduce the 3-D theory to a 2-D one, characterized by the middle surface displacements. And such analyses aretypically accurate for thin or moderately thick shells of homogeneous and isotropic materials, and at least for the lower frequencies (i.e., long wave lengths of mode shapes). But for thicker shells or higher frequencies, a 3-D analysis becomes necessary for accurate frequencies.

In the present work hemi-spherical shells of revolution with eccentricity having uniform thickness are analyzed by a 3-D approach. Instead of attempting to solve equations of motion, an energy approach is followed which, as sufficient freedom is given to the three displacement components, yields frequency values as close to the exact ones as desired. To evaluate the energy integrations over the shell volume, displacementsand strains are expressed in terms of the circular cylindrical coordinates, instead of related 3-D shell coordinates which are normal and tangent to the shell midsurface. Natural frequencies are obtained for 16 free hemi-spherical shells of revolution with eccentricity having uniform thickness.

2. Method of analysis

A representative cross-section of a hemi-spherical shell of revolution with eccentricity of e and a radius of R of the mid-surface (z_m) of the hemi-spherical shell segment having uniform thickness (H) is shown in Figure 1. The hemi-spherical shells of revolution with eccentricity are obtained by rotating the cross-section for $r \geq 0$ in Figure 1 360° about the z-axis. The eccentricity (e) isthe distance from the axis of revolution of the shell (z-axis) to the center of the hemi-spherical shell segment. Thus the mid-surface (z_m) of the shell for $r \geq 0$ has the equation of $(r - e)^{2} + z^{2} = R^{2}$ . The cylindrical coordinate system (r,θ, z), also shown in the figure, is used in the analysis, where θ is the circumferential angle. The domain (Λ) of the shell is obtained by subtracting the inner portion

R_{i} \leq r \leq e + R - H / 2, 0 \leq z \leq z_{i}, 0 \leq θ \leq 2 π

(1)

from the outer portion

R_{i} \leq r \leq e + R + H / 2, 0 \leq z \leq z_{o}, 0 \leq θ \leq 2 π

(2)

where R_i is the radius of the inner hole of the shell of revolution and

z_{i, o}

are the coordinates of the inner and outer surface of the cross-section for

r \geq 0

in Figure 1, respectively,

z_{i, o} = \sqrt{(R \mp H / 2)^{2} - (r - e)^{2}}

(3)

For mathematical convenience, the radial (r) and axial (z) coordinates are made dimensionless as

ψ = r / R, ζ = z / R

(4)

Thus the domain (Λ) of the shell in terms of thenondimensional circular cylindrical coordinates ( $ψ, θ, ζ$ ) are given by subtracting the inner portion

R_{i}^{*} \leq ψ \leq e^{*} + 1 - H^{*} / 2, 0 \leq ζ \leq ζ_{i}, 0 \leq θ \leq 2 π

(5)

from the outer portion

R_{i}^{*} \leq ψ \leq e^{*} + 1 + H^{*} / 2, 0 \leq ζ \leq ζ_{o}, 0 \leq θ \leq 2 π

(6)

where

ζ_{i, o} = \sqrt{(1 \mp H^{*} / 2)^{2} - (ψ - e^{*})^{2}}

(7)

with nondimensional parameters of

R_{i}^{*}

H^{*}

, and

e^{*}

defined by

R_{i}^{*} = R_{i} / R, H^{*} = H / R, e^{*} = e / R

(8)

Utilizing tensor analysis, the three equations of motion in terms of the circular cylindrical coordinate system (r,θ, z) are found to be (Sokolnikoff, 1956)

σ_{rr, r} + σ_{rz, z} + \frac{1}{r} (σ_{rr} - σ_{θ θ} + σ_{r θ, θ}) = ρ {\overset{··}{u}}_{r}

(9a)

σ_{rz, r} + σ_{zz, z} + \frac{1}{r} (σ_{rz} + σ_{z θ, θ}) = ρ {\overset{··}{u}}_{z}

(9b)

σ_{r θ, r} + σ_{z θ, z} + \frac{1}{r} (2 σ_{r θ} + σ_{θ θ, θ}) = ρ {\overset{··}{u}}_{θ}

(9c)

where the

σ_{ij}

are the normal (

i = j

) and shear (

i \neq j

) stress components; u_r, u_z, and

u_{θ}

are the displacement components in the r, z, and θ directions, respectively; ρ is mass density per unit volume; the commas indicate spatial derivatives; and the dots denote time derivatives.

The well-known relationships between the tensorial stresses ( $σ_{ij}$ ) and strains ( $ɛ_{ij}$ ) of isotropic, linear elasticity are

σ_{ij} = λ ɛ δ_{ij} + 2 G ɛ_{ij},

(10)

where λ and G are the Lamé parameters, expressed interms of Young’s modulus (E) and Poisson’s ratio (ν) for an isotropic solid as:

λ = \frac{E ν}{(1 + ν) (1 - 2 ν)}, G = \frac{E}{2 (1 + ν)},

(11)

ɛ = ɛ_{rr} + ɛ_{zz} + ɛ_{θ θ}

is the trace of the strain tensor, and

δ_{ij}

is Kronecker’s delta.

The 3-D tensorial strains ( $ɛ_{ij}$ ) are found to be related to the three displacements u_r, u_z, and $u_{θ}$ , by (Sokolnikoff, 1956)

ɛ_{rr} = u_{r, r}, ɛ_{zz} = u_{z, z}, ɛ_{θ θ} = \frac{u_{r} + u_{θ, θ}}{r}

(12a)

\begin{matrix} 2 ɛ_{rz} = u_{r, z} + u_{z, r}, 2 ɛ_{r θ} = u_{θ, r} + \frac{u_{r, θ} - u_{θ}}{r}, \\ 2 ɛ_{z θ} = u_{θ, z} + \frac{u_{z, θ}}{r} . \end{matrix}

(12b)

Substituting equations (10) and (12) into equations (9), a set of three second-order partial differential equations is obtained in u_r, u_z, and $u_{θ}$ governing free vibrations. However, in the case of hemi-spherical shells with eccentricity, exact solutions are intractable because of the variable coefficients that appear in many terms. Alternatively, the problem may be approached from an energy perspective.

Because the strains are related to the displacement components by equations (12), unacceptable strain singularities may be encountered exactly at r = 0 due to the term $1 / r$ when R_i = 0 (no hole). Since a negligibly small hole ( $R_{i} \approx 0$ ) does not affect the frequencies (Kang and Leissa, 2004), such singularities may be avoided by replacing the range for $ψ (= r / R)$ in equations (5) and (6), $R_{i}^{*} \leq ψ \leq e^{*} + 1 \mp H^{*} / 2$ , with $10^{- 10} \leq ψ \leq e^{*} + 1 \mp H^{*} / 2$ .

During vibratory deformation of the body, its strain(potential) energy (V) is the integral over the domain (Λ):

\begin{matrix} V = \frac{1}{2} \int_{Λ} (σ_{rr} ɛ_{rr} + σ_{zz} ɛ_{zz} + σ_{θ θ} ɛ_{θ θ} + 2 σ_{rz} ɛ_{rz} + 2 σ_{r θ} ɛ_{r θ} \\ + 2 σ_{z θ} ɛ_{z θ}) r dr dz d θ \end{matrix}

(13)

Substituting equations (10) and (12) into equation (13) results in the strain energy in terms of the three displacements:

\begin{matrix} V = \frac{1}{2} \int_{Λ} [λ (ɛ_{rr} + ɛ_{zz} + ɛ_{θ θ})^{2} + 2 G {ɛ_{rr}^{2} + ɛ_{zz}^{2} + ɛ_{θ θ}^{2} \\ + 2 (ɛ_{rz}^{2} + ɛ_{z θ}^{2} + ɛ_{r θ}^{2})}] r dr dz d θ \end{matrix},

(14)

where the tensorial strains

ɛ_{ij}

are expressed in terms of the three displacements by equations (12).

The kinetic energy (T) is simply

T = \frac{1}{2} \int_{Λ} ρ ({\overset{\cdot}{u}}_{r}^{2} + {\overset{\cdot}{u}}_{z}^{2} + {\overset{\cdot}{u}}_{θ}^{2}) r dr dz d θ

(15)

For the free, undamped vibration, the time (t) response of the three displacements is sinusoidal and, moreover, the circular symmetry of the body of revolution allows the displacements to be expressed by

u_{r} (ψ, ζ, θ, t) = U_{r} (ψ, ζ) \cos n θ \sin (ω t + α),

(16a)

u_{z} (ψ, ζ, θ, t) = U_{z} (ψ, ζ) \cos n θ \sin (ω t + α),

(16b)

u_{θ} (ψ, ζ, θ, t) = U_{θ} (ψ, ζ) \sin n θ \sin (ω t + α),

(16c)

where U_r, U_z, and

U_{θ}

are displacement functions of ψ and ζ, ω is a natural frequency, and α is an arbitrary phase angle determined by the initial conditions. The circumferential wave number is taken to be an integer (n = 0, 1, 2, … ,∞), to ensure periodicity in θ. It may be verified by substituting the displacements into the 3-D equations of motion (Kang and Leissa, 2001) that the variables separable form of equations (16) does apply. Then equations (16) account for all free vibration modes except for the torsional ones. These modes arise from an alternative set of solutions which are the same as equations (16), except that

\cos n θ

and

\sin n θ

are interchanged. For

n \geq 1

, this set duplicates the solutions of equations (16), with the symmetry axesof the mode shapes being rotated. But for n = 0 the alternative set reduces to

u_{r} = u_{z} = 0

u_{θ} = U_{θ}^{*} (ψ, ζ) \sin (ω t + α)

, which corresponds to the torsional modes. The displacements uncouple by circumferential wave number (n), leaving only coupling in r (or ψ) and z(or ζ).

The Ritz method uses the maximum potential (strain) energy ( $V_{max}$ ) and the maximum kinetic energy ( $T_{max}$ ) functionals in a cycle of vibratory motion. The functionals for the shell are obtained by setting $\sin^{2} (ω t + α)$ and $\cos^{2} (ω t + α)$ equal to unity in equations (14) and (15) after the displacements (16) are substituted, and by using the nondimensional coordinates ψ and ζ, as follows:

V_{max} = \frac{RG}{2} [\int_{R_{i}^{*}} e^{*} + 1 + H^{*} / 2 \int_{0} ζ_{o} I_{V} ψ d ζ d ψ - \int_{R_{i}^{*}} e^{*} + 1 - H^{*} / 2 \int_{0} ζ_{i} I_{V} ψ d ζ d ψ]

(17)

T max = \frac{R^{3} ρ ω^{2}}{2} [\int_{R_{i}^{*}} e^{*} + 1 + H^{*} / 2 \int_{0} ζ_{o} I_{T} ψ d ζ d ψ - \int_{R_{i}^{*}} e^{*} + 1 - H^{*} / 2 \int 0 ζ_{i} I_{T} ψ d ζ d ψ]

(18)

where

\begin{matrix} I_{V} = [\frac{λ}{G} {(κ_{1} + κ_{2} + κ_{3})}^{2} + 2 (κ_{1}^{2} + κ_{2}^{2} + κ_{3}^{2}) + κ_{4}^{2}] Γ_{1} \\ + (κ_{5}^{2} + κ_{6}^{2}) Γ_{2}, \end{matrix}

(19)

I_{T} = (U_{r}^{2} + U_{z}^{2}) Γ_{1} + U_{θ}^{2} Γ_{2},

(20)

and

κ_{1} = \frac{U_{r} + {nU}_{θ}}{ψ}, κ_{2} = U_{r, ψ}, κ_{3} = U_{z, ζ},

(21a)

\begin{matrix} κ_{4} = U_{z, ψ} + U_{r, ζ}, κ_{5} = \frac{{nU}_{z}}{ψ} - U_{θ, ζ}, \\ κ_{6} = \frac{{nU}_{r} + U_{θ}}{ψ} - U_{θ, ψ}, \end{matrix}

(21b)

and

Γ_{1}

and

Γ_{2}

are constants, defined by

Γ_{1} = \int_{0} 2 π \cos^{2} n θ d θ = {\begin{matrix} 2 π if n = 0 \\ π if n \geq 1 \end{matrix} Γ_{2} = \int_{0} 2 π \sin^{2} n θ d θ = {\begin{matrix} 0 if n = 0 \\ π if n \geq 1 \end{matrix}

(22)

From equations (11) it is seen that the nondimensional constant $λ / G$ in equation (19) involves only ν as follows

\frac{λ}{G} = \frac{2 ν}{1 - 2 ν}

(23)

The displacement functions U_r, U_z and

U_{θ}

in equations (16) are further assumed as algebraic polynomials,

U_{r} (ψ, ζ) = η_{r} (ψ, ζ) \sum_{i = 0}^{I} \sum_{j = 0}^{J} A_{ij} ψ^{i} ζ^{j}

(24a)

U_{z} (ψ, ζ) = η_{z} (ψ, ζ) \sum_{k = 0}^{K} \sum_{l = 0}^{L} B_{kl} ψ^{k} ζ^{l}

(24b)

U_{θ} (ψ, ζ) = η_{θ} (ψ, ζ) \sum_{m = 0}^{M} \sum_{n = 0}^{N} C_{mn} ψ^{m} ζ^{n}

(24c)

and similarly for

U_{θ}^{*}

, where i, j, k, l, m, and n are integers; I, J, K, L, M, and N are the highest degrees taken in the polynomial terms;

A_{ij}

B_{kl}

and

C_{mn}

are arbitrary coefficients to be determined, and the

η (ψ, ζ)

are functions depending upon the geometric boundary conditions to be enforced. For example:

completely free: $η_{r} = η_{z} = η_{θ} = 1$ ,

the bottom edge fixed: $η_{r} = η_{z} = η_{θ} = ζ$ ,

The functions of η shown above, impose only the necessary geometric constraints related to displacement boundary conditions. Together with the algebraic polynomials in equations (24), they form function sets which are mathematically complete (Kantorovich and Krylov, 1958, pp. 266–268). Thus, the function sets are capable of representing any 3-D motion of the shell with increasing accuracy as the indices I, J, … , N are increased. In the limit, as sufficient terms are taken, all internal kinematic constraints vanish, and the functions (24) will approach the exact solution as closely as desired.

The eigenvalue problem is formulated by minimizing the free vibration frequencies with respect to the arbitrary coefficients $A_{ij}$ , $B_{kl}$ , and $C_{mn}$ , thereby minimizingthe effects of the internal constraints present, when the upper limits (I, J,…, N) become large. This corresponds to the equations (Ritz, 1909):

\begin{matrix} \frac{\partial}{\partial A_{ij}} (V_{max} - T_{max}) = 0 (i = 0, 1, 2, \dots, I; \\ j = 0, 1, 2, \dots, J), \end{matrix}

(25a)

\begin{matrix} \frac{\partial}{\partial B_{kl}} (V_{max} - T_{max}) = 0 (k = 0, 1, 2, \dots, K; \\ l = 0, 1, 2, \dots, L), \end{matrix}

(25b)

\begin{matrix} \frac{\partial}{\partial C_{mn}} (V_{max} - T_{max}) = 0 (m = 0, 1, 2, \dots, M; \\ n = 0, 1, 2, \dots, N) . \end{matrix}

(25c)

Equations (25) yield a set of (I + 1)(J + 1) + (K + 1)(L + 1) + (M + 1)(N + 1) linear, homogeneous, algebraic equations in the unknowns $A_{ij}$ , $B_{kl}$ , and $C_{mn}$ . The equations can be written in the form

(K - Ω M) x = 0,

(26)

where K and M are stiffness and mass matrices resulting from the maximum strain energy (

V_{max}

) and the maximum kinetic energy (

T_{max}

), respectively, and Ω is an eigenvalue of the vibrating system, expressed as the square of non-dimensional frequency,

Ω \equiv ω^{2} R^{2} ρ / G

, and the vector x takes the form

\begin{matrix} x = (A_{00}, A_{01}, \dots, A_{IJ}; B_{00}, B_{01}, \dots, \\ B_{KL}; C_{00}, C_{01}, \dots, C_{MN})^{T} \end{matrix}

(27)

For a nontrivial solution, the determinant of the coefficient matrix of equation (26) is set equal to zero, which yields the frequencies (eigenvalues); that is to say $| K - Ω M | = 0$ . The roots of the determinant are the eigenvalues Ω. These frequencies are upper bounds on the exact values. The mode shape (eigenfunction) corresponding to each frequency is obtained, in the usual manner, by substituting each Ω back into the set of algebraic equations, and solving for the ratios of coefficients.

3. Convergence studies

To guarantee the accuracy of frequencies obtained by the procedure described above, it is necessary to conduct some convergence studies to determine the number of terms required in the power series of equations (24). A convergence study is based upon the fact that, if thedisplacements are expressed as power series, all the frequencies obtained by the Ritz method should converge to their exact values in an upper bound manner. If the results do not converge properly, or converge too slowly, it would be likely that the assumed displacement functions chosen are poor ones, or be missing some functions from a minimal complete set of polynomials.

Tables 1 to 3 are such a study for free hemi-spherical shells of revolution with eccentricity of e/R = −0.5 (Table 1), 1.5 (Table 2), and 0.5 (Table 3) having uniform thickness ( $H / R$ = 0.05), depicted as the configurations in Figures 2(b), 3(b), and 4(b), respectively. The radius R_o in Table 2 and Figure 3(b) stands for theouter one of the shell. The tables list the first five nondimensional frequencies in $ω R \sqrt{ρ / G}$ for ν = 0.3, for torsional (n = 0^T), axisymmetric (n = 0^A), and bending (n = 2) modes, respectively. The bending modes (n = 2) have two circumferential waves in their mode shapes.

Table 1.

Convergence of frequencies in $ω R \sqrt{ρ / G}$ of a completely free, closed (R_i = 0) hemi-spherical shell of revolution with eccentricity (e/R = −0.5) for the five lowest torsional modes(n = 0^T) for H/R = 0.05 (ν = 0.3)

TZ ^a	TR ^b	DET ^c	Five lowest torsional modes
TZ ^a	TR ^b	DET ^c	1	2	3	4	5
2	2	4	5.219	25.59	64.93	–	–
	4	8	4.888	9.297	19.24	61.49	69.25
	6	12	4.862	8.480	14.29	22.72	38.82
	8	16	4.848	8.393	13.02	18.80	27.44
	10	20	4.841	8.362	12.59	17.48	23.94
3	2	6	4.727	9.967	34.59	66.38	69.97
	4	12	4.648	8.357	12.74	20.55	66.18
	6	18	4.640	8.083	11.72	16.85	23.76
	8	24	4.635	8.013	11.56	15.26	20.16
	10	30	4.634	7.992	11.48	14.94	19.17
4	2	8	4.661	8.234	15.91	44.85	67.71
	4	16	4.627	7.858	11.68	16.66	25.70
	6	24	4.627	7.828	11.03	14.80	20.49
	8	32	4.627	7.822	10.97	14.41	18.27
	10	40	4.627	7.821	10.94	14.21	17.87
5	2	10	4.634	7.983	11.96	23.20	55.75
	4	20	4.627	7.829	10.99	14.93	20.47
	6	30	4.627	7.821	10.93	14.10	17.95
	8	40	4.627	7.820	10.91	14.00	17.30
	10	50	4.627	7.820	10.91	13.96	17.08
6	2	12	4.634	7.874	11.37	16.03	31.82
	4	24	4.627	7.820	10.94	14.16	18.28
	6	36	4.627	7.820	10.91	14.00	17.26
	8	48	4.627	7.820	10.90	13.96	17.05
7	2	14	4.632	7.867	11.09	14.97	20.50
	4	28	4.627	7.820	10.91	14.04	17.40
	6	42	4.627	7.820	10.90	13.95	17.07
	8	56	4.627	7.820	10.90	13.95	16.98

TZ = Total numbers of polynomial terms used in the z (or ζ) direction.

TR = Total numbers of polynomial terms used in the r (or ψ) direction.

DET = Frequency determinant order.

Table 2.

Convergence of frequencies in $ω R \sqrt{ρ / G}$ of a completely free hemi-spherical shell of revolution with eccentricity (e/R = 1.5) for the five lowest axisymmetric modes (n = 0^A) for H/R = 0.05 and R_o/R = 2 (ν = 0.3)

TZ ^a	TR ^b	DET ^c	Five lowest axisymmetric modes
TZ ^a	TR ^b	DET ^c	1	2	3	4	5
2	2	8	0.8462	1.783	3.619	5.777	6.146
	4	16	0.2268	1.196	2.186	3.331	3.599
	6	24	0.1775	0.8837	1.501	2.411	3.336
	8	32	0.1640	0.7213	1.242	1.812	2.821
	10	40	0.1629	0.6954	1.188	1.671	2.410
3	2	12	0.6908	1.6160	3.389	4.572	5.936
	4	24	0.1921	0.9194	1.709	2.769	3.331
	6	36	0.1568	0.7314	1.300	1.895	2.630
	8	48	0.1491	0.6681	1.132	1.577	2.220
	10	60	0.1468	0.6474	1.094	1.511	2.041
4	2	16	0.5452	1.5270	3.176	3.404	5.378
	4	32	0.1804	0.8292	1.505	2.283	3.091
	6	48	0.1508	0.6875	1.203	1.726	2.370
	8	64	0.1469	0.6501	1.096	1.511	2.018
	10	80	0.1458	0.6431	1.073	1.476	1.922
5	2	20	0.5239	1.502	3.156	3.374	4.658
	4	40	0.1743	0.819	1.444	2.148	2.996
	6	60	0.149	0.6608	1.137	1.608	2.181
	8	80	0.1462	0.6465	1.084	1.489	1.943
	10	100	0.1456	0.6417	1.067	1.461	1.892
	12	120	0.1456	0.6413	1.065	1.454	1.870
6	2	24	0.4810	1.4790	3.059	3.364	4.402
	4	48	0.1703	0.8006	1.411	2.061	2.867
	6	72	0.1482	0.6534	1.117	1.567	2.093
	8	96	0.1458	0.6436	1.074	1.469	1.911
	10	120	0.1456	0.6414	1.065	1.456	1.875
	11	132	0.1456	0.6413	1.065	1.454	1.868
7	2	28	0.4387	1.4330	2.932	3.316	3.466
	4	56	0.1688	0.7861	1.378	1.990	2.766
	6	84	0.1477	0.6494	1.107	1.551	2.049
	8	112	0.1457	0.6423	1.070	1.463	1.890
	10	140	0.1456	0.6413	1.065	1.454	1.870
	11	154	0.1456	0.6413	1.065	1.454	1.867
8	2	32	0.4346	1.431	2.913	3.314	3.448
	4	64	0.1682	0.783	1.365	1.956	2.703
	6	96	0.1472	0.6485	1.098	1.534	2.024
	8	128	0.1457	0.6418	1.068	1.461	1.882
	9	144	0.1456	0.6413	1.065	1.457	1.873

TZ = Total numbers of polynomial terms used in the z (or ζ) direction.

TR = Total numbers of polynomial terms used in the r (or ψ) direction.

DET = Frequency determinant order.

Table 3.

Convergence of frequencies in $ω R \sqrt{ρ / G}$ of a completely free, closed (R_i = 0) hemi-spherical shell of revolution with eccentricity (e/R = 0.5) for the five lowest bending modes (n = 2) for H/R = 0.05 (ν = 0.3)

TZ ^a	TR ^b	DET ^c	Five lowest bending modes (n = 2)
TZ ^a	TR ^b	DET ^c	1	2	3	4	5
2	2	12	0.70340	1.533	3.024	3.993	5.152
	4	24	0.12760	1.139	1.574	2.508	3.144
	6	36	0.10840	1.064	1.347	1.789	2.585
	8	48	0.10030	1.052	1.133	1.532	2.032
	10	60	0.09604	1.049	1.115	1.456	1.819
3	2	18	0.44080	1.237	2.128	3.064	4.692
	4	36	0.06286	1.081	1.218	1.695	2.555
	6	54	0.05693	1.045	1.076	1.361	1.713
	8	72	0.05549	1.036	1.044	1.250	1.563
	10	90	0.05519	1.024	1.038	1.225	1.498
4	2	24	0.37180	1.209	1.727	2.852	4.283
	4	48	0.05667	1.058	1.161	1.516	2.155
	6	72	0.05417	1.038	1.051	1.275	1.591
	8	96	0.05395	1.024	1.037	1.203	1.478
	10	120	0.05384	1.021	1.036	1.189	1.453
5	2	30	0.27070	1.181	1.637	2.683	3.379
	4	60	0.05462	1.055	1.133	1.403	1.931
	6	90	0.05384	1.033	1.043	1.230	1.520
	8	120	0.05374	1.022	1.037	1.193	1.452
	10	150	0.05370	1.020	1.036	1.185	1.433
	12	180	0.05366	1.020	1.036	1.185	1.431
6	2	36	0.26410	1.171	1.497	2.428	2.944
	4	72	0.05434	1.055	1.119	1.384	1.814
	6	108	0.05374	1.027	1.040	1.210	1.488
	8	144	0.05365	1.021	1.036	1.189	1.436
	10	180	0.05363	1.020	1.036	1.185	1.431
	12	216	0.05361	1.020	1.036	1.185	1.430
7	2	42	0.23090	1.167	1.478	2.094	2.830
	4	84	0.05413	1.054	1.111	1.370	1.741
	6	126	0.05365	1.025	1.039	1.200	1.469
	8	168	0.05362	1.021	1.036	1.186	1.433
	10	210	0.05360	1.020	1.036	1.185	1.430
8	2	48	0.21050	1.161	1.409	1.953	2.786
	4	96	0.05407	1.053	1.100	1.365	1.678
	6	144	0.05361	1.024	1.038	1.197	1.465
	8	192	0.0536	1.020	1.036	1.186	1.432
	10	240	0.05360	1.020	1.036	1.185	1.430

TZ = Total numbers of polynomial terms used in the z (or ζ) direction.

TR = Total numbers of polynomial terms used in the r (or ψ) direction.

DET = Frequency determinant order.

The range of ψ in equations (5) and (6), $R_{i}^{*} \leq ψ \leq e^{*} + 1 \mp H^{*} / 2$ , must be replaced by $e^{*} - 1 \pm H^{*} / 2 \leq ψ \leq R_{o} / R$ for the shell configurations in Figures 3(a) and (b).

To make the study of convergence less complicated, equal numbers of polynomial terms were taken in both the r (or ψ) coordinate (i.e., I = K = M) and z (or ζ) coordinate (i.e., J = L = N), although some computational optimization could be obtained for some configurations and some mode shapes by using unequal numbers of polynomial terms.

The symbols TZ and TR in the table indicate the total numbers of polynomial terms used through the axial (z or ζ) and the radial (r or ψ) directions, respectively. Note that the frequency determinant order DET is related to TZ and TR as follows:

DET = {\begin{matrix} TZ \times TR & for torsional modes (n = 0), \\ 2 \times TZ \times TR & for axisymmetric modes (n = 0), \\ 3 \times TZ \times TR & for general modes (n \geq 1) . \end{matrix}

(28)

Tables 1 to 3 show the monotonic convergence of all five frequencies as TZ (=J + 1, L + 1, and N + 1 in equations (24)) are increased, as well as TR (=I + 1, K + 1, and M + 1 in equations (24)). It is seen in Table 3, for example, that the first nondimensional frequency in $ω R \sqrt{ρ / G}$ for n = 2 converges to four digits (0.05360) when ( TZ , TR ) = (8,8) terms are used, which results in DET = 3 × (8 × 8) = 192.

It is interesting to note in Tables 1 to 3 that the modes for n = 2 require much larger size of DET compared with the torsional (n = 0^T) and the axismmetric modes (n = 0^A). This is primarily because only the circumferential displacement components ( $u_{θ}$ ) are involved in the torsional modes, and the radial (u_r) and axial (u_z) displacement components are involved in the axisymmetrical modes, whereas all three components enter into the modes having $n \geq 1$ , as seen in equations (28).

Frequencies in underlined, bold-faced type in Tables 1 to 3 are the most accurate values (to four significant figures) achieved with the smallest determinant sizes.

4. Numerical results and discussion

Table 4 compares the first nondimensional frequencies in

ω R \sqrt{ρ / G}

for each n of a completely free, hemispherical shell of revolution with no eccentricity (e = 0) having constant thickness (H/R = 0.05) for ν = 0.3 from the present 3-D Ritz solutions and other 3-D ones (Kang and Leissa, 2004). They analyzed the free vibrations of solid and hollow hemispherical shells of revolution of arbitrary wall thickness with and without axial conical holes by a 3-D approach. In their analysis, 3-D shell coordinates are used which are normal and tangent to the shell midsurface, while the circular cylindrical coordinates (r,θ, z) are used in the present analysis. It is observed in Table 4 that the values of frequency parameters agree well with those presented previously by Kang and Leissa (2004).

Table 4.

Comparisons of the first nondimensional frequencies in $ω R \sqrt{ρ / G}$ for each n of a completely free, hemispherical shell of revolution with no eccentricity (e = 0) having constant thickness (H/R = 0.05) for ν = 0.3 from the present 3-D Ritz solutions and other 3-D ones (Kang and Leissa, 2004)

Methods	n
Methods	0^T	0^A	1	2	3	4	5	6
Kang and Leissa (2004)	3.161	1.418	1.425	0.09848	0.2597	0.4724	0.7270	1.019
Present	3.161	1.418	1.425	0.09846	0.2596	0.4723	0.7269	1.019

Tables 5 to 7 present the nondimensional frequencies in $ω R \sqrt{ρ / G}$ of free hemi-spherical shells of revolution with various range of eccentricity of $- 0.75 R \leq e \leq 2 R$ . Each table is for the shells of uniform thickness of $H / R = 0.05$ . Poisson’s ratio (ν) was taken to be 0.3. The shell configurations for Tables 5 to 7 are depicted in Figures 2 and 3, Figures 4 and 5, and Figure 6, respectively.

Table 5.

Frequencies in $ω R \sqrt{ρ / G}$ of completely free, hemispherical shells of revolution with eccentricity ( $- R < e < 0$ , $e > R$ ) having constant thickness (H/R = 0.05) for ν = 0.3

n	s	$- R < e < 0$ ( $R_{i} = 0$ )			e > R
		e = -0.75R	e = -0.5R	e = -0.25R	e = 1.3R R_o = 1.3R	e = 1.5R R_o = 2R	e = 2R
		Figure 2(a)	Figure 2(a)	Figure 2(a)	Figure 3(a)	Figure 3(b)	Figure 3(c)
0^T	1	6.609	4.627	3.718	2.540	1.921	1.296
	2	11.19	7.820	6.256	4.191	3.141	2.096
	3	15.60	10.90	8.708	6.126	4.591	3.060
	4	19.94	13.95	11.13	8.097	6.069	4.045
	5	24.27	16.97	13.54	10.08	7.556	5.036
0^A	1	6.118	3.094	2.017	0.4864(5)	0.1456 (3)	0.1042(4)
	2	6.715	3.378	2.202	1.277	0.6413	0.4771
	3	8.373	4.188	2.634	1.999	1.065	0.5433
	4	10.11	5.298	3.287	2.661	1.454	0.6790
	5	10.86	6.081	4.146	3.298	1.869	0.8592
1	1	5.402(4)	2.867	1.932	0.552	0.1777 (4)	0.1193(5)
	2	6.011	3.553	2.313	1.363	0.6795	0.4614
	3	7.233	4.248	2.832	1.990	1.132	0.5438
	4	9.139	4.580	3.073	2.130	1.339	0.7092
	5	9.678	5.686	3.599	2.583	1.580	0.8471
2	1	1.186(1)	0.3265(1)	0.1584(1)	0.2068(1)	0.07407(1)	0.02144(1)
	2	3.710(3)	2.197(4)	1.772	0.3759(3)	0.2887	0.1755
	3	6.707	3.482	2.346	0.7831	0.3480	0.1950
	4	8.209	4.061	2.645	1.260	0.6968	0.4649
	5	9.552	4.685	3.017	1.905	1.0580	0.5543
3	1	3.086(2)	0.8631(2)	0.4198(2)	0.3271(2)	0.1401 (2)	0.05158(2)
	2	5.716	2.609	1.924	0.7937	0.5610	0.3420
	3	8.806	3.965	2.581	1.301	0.6872	0.3628
	4	11.56	5.417	3.407	1.865	0.9775	0.5897
	5	12.32	5.909	3.935	2.285	1.153	0.6635
4	1	5.538(5)	1.576(3)	0.7683(3)	0.4574(4)	0.2156 (5)	0.09117(3)
	2	8.401	3.263	2.163	1.139	0.7837	0.5232
	3	11.70	4.795	2.961	1.781	1.004	0.5535
	4	14.96	6.510	3.960	2.513	1.308	0.6982
	5	15.73	7.685	5.145	3.260	1.583	0.7533
5	1	8.408	2.442(5)	1.188(4)	0.6141	0.3013	0.1372
	2	11.53	4.160	2.518	1.481	0.9751	0.7051
	3	15.02	5.870	3.468	2.250	1.258	0.7201
	4	18.48	7.783	4.615	3.124	1.604	0.8277
	5	19.39	9.471	5.991	4.065	2.004	0.9159
6	1	11.60	3.449	1.671(5)	0.7986	0.3968	0.1884
	2	14.94	5.253	2.989	1.834	1.154	0.8185
	3	18.60	7.122	4.079	2.734	1.482	0.8818
	4	22.12	9.199	5.356	3.742	1.880	0.9852
	5	23.17	11.25	6.852	4.832	2.369	1.1340

Table 7.

Frequencies in $ω R \sqrt{ρ / G}$ of completely free, hemispherical shells of revolution with eccentricity (e = R) having constant thickness (H/R = 0.05) for ν = 0.3

n	S	e = R
		R_o = R	R_o = 1.5R	R_i = 0	R_i = 0.5R
		Figure 6(a)	Figure 6(b)	Figure 6(c)	Figure 6(d)
0^T	1	4.306	3.161	2	2.044
	2	6.487	4.848	3.162	3.389
	3	8.411	6.414	4.239	4.782
	4	9.863	7.891	5.282	6.218
	5	11.24	9.198	6.302	7.676
0^A	1	0.6243(3)	0.2049(3)	0.1829 (4)	0.2121 (4)
	2	1.688	0.8211	0.7029	0.7033
	3	2.747	1.429	0.7945	0.796
	4	3.759	2.047	0.9632	0.969
	5	4.408	2.669	1.256	1.284
1	1	0.7535	0.2737(4)	0.2483	0.2787
	2	1.77	0.8921	0.6777	0.6816
	3	2.257	1.519	0.7894	0.7925
	4	2.653	1.784	1.005	1.002
	5	3.139	2.052	1.227	1.256
2	1	0.2922(1)	0.1117(1)	0.03674(1)	0.03583(1)
	2	0.6925(5)	0.5982	0.5863	0.4821
	3	1.533	0.7571	0.6491	0.5515
	4	2.666	1.324	0.7823	0.7717
	5	3.271	1.973	0.9366	0.8875
3	1	0.4413(2)	0.2025(2)	0.09020(2)	0.09016(2)
	2	1.226	0.8998	0.8587	0.787
	3	2.275	1.211	0.9026	0.8661
	4	3.597	1.821	1.046	0.9721
	5	4.041	2.555	1.219	1.132
4	1	0.6460(4)	0.3181(5)	0.1611 (3)	0.1610 (3)
	2	1.746	1.151	1.001	0.994
	3	3.012	1.587	1.094	1.071
	4	4.538	2.289	1.261	1.231
	5	4.932	3.169	1.475	1.361
5	1	0.9042	0.4526	0.2445 (5)	0.2444 (5)
	2	2.287	1.389	1.127	1.127
	3	3.763	1.939	1.255	1.249
	4	5.481	2.745	1.435	1.426
	5	5.865	3.757	1.682	1.626
6	1	1.214	0.6054	0.3384	0.3383
	2	2.862	1.63	1.248	1.248
	3	4.539	2.294	1.394	1.393
	4	6.438	3.206	1.588	1.585
	5	6.807	4.334	1.883	1.862

The ranges of ψ in equations (5) and (6), $R_{i}^{*} \leq ψ \leq e^{*} + 1 \mp H^{*} / 2$ , must be replaced by $e^{*} - 1 \pm H^{*} / 2 \leq ψ \leq e^{*} + 1 \mp H^{*} / 2$ for the shells in Figure 3(c), by $H^{*} / 2 \leq ψ \leq R_{o} / R$ and $0 \leq ψ \leq R_{o} / R$ for the shells in Figures 6(a) and (b), and by $H^{*} / 2 \leq ψ \leq e^{*} + 1 - H^{*} / 2$ and $0 \leq ψ \leq e^{*} + 1 + H^{*} / 2$ for the shells in Figure 6(c), respectively.

Forty frequencies are given for each configuration, which arise from eight circumferential wave numbers (n = 0^T, 0^A, 1, 2, 3, 4, 5, 6) and the first five modes (s = 1, 2, 3, 4, 5) for each value of n, where the superscripts T and A indicate torsional and axisymmetric modes, respectively. The numbers in parentheses identify the first five frequencies for each configuration. For example, in the case of

R_{o} = R

in Table 6, the first five frequencies are modes for (n,s) = (2,1), (3,1), (0^A,1), (4,1), and (2,2) in this order. The zero frequencies of rigid body modes are omitted from the tables.

Table 6.

Frequencies in $ω R \sqrt{ρ / G}$ of completely free, hemispherical shells of revolution with eccentricity ( $0 \leq e < R$ ) having constant thickness ( $H / R$ = 0.05) for ν = 0.3

n	s	0 < e < R (R_i = 0)			$e = 0$
		e = 0.25R	e = 0.5R	e = 0.75R	R_i = 0	R_i = 0.2R	R_i = 0.6R
		Figure 4(a)	Figure 4(b)	Figure 4(c)	Figure 5(a)	Figure 5(b)	Figure 5(c)
0^T	1	2.768	2.467	2.219	3.161	3.172	3.758
	2	4.599	4.060	3.602	5.289	5.351	6.988
	3	6.367	5.594	4.926	7.345	7.511	10.30
	4	8.113	7.111	6.232	9.376	9.695	13.65
	5	9.851	8.619	7.533	11.40	11.91	17.01
0^A	1	0.7258	0.3750(4)	0.2485(4)	1.418	1.435	1.530
	2	1.1490	0.9392	0.8026	1.581	1.589	1.682
	3	1.283	1.060	0.9073	1.744	1.772	2.133
	4	1.530	1.264	1.091	2.136	2.229	2.305
	5	2.017	1.663	1.429	2.767	2.986	3.579
1	1	0.9937	0.5857	0.3695	1.425	1.435	1.508
	2	1.170	0.9293	0.7796	1.621	1.621	1.707
	3	1.363	1.077	0.9089	1.912	1.889	2.342
	4	1.739	1.366	1.155	2.369	2.368	2.487
	5	1.928	1.624	1.401	2.424	2.401	3.621
2	1	0.06968(1)	0.05360(1)	0.04356(1)	0.09846(1)	0.09808(1)	0.08897(1)
	2	1.0265	1.020	0.7707	1.516	1.062	0.2772(3)
	3	1.339	1.036	0.8343	1.747	1.648	1.692
	4	1.540	1.185	0.9419	1.954	1.853	1.791
	5	1.687	1.430	1.152	2.144	2.080	2.453
3	1	0.1817(2)	0.1373 (2)	0.1093(2)	0.2596(2)	0.2597(2)	0.2556(2)
	2	1.398	1.189	1.008	1.619	1.592	0.5996(5)
	3	1.546	1.297	1.088	1.925	1.878	1.878
	4	1.868	1.496	1.240	2.438	2.358	2.521
	5	2.363	1.849	1.480	2.962	2.949	2.698
4	1	0.3290(3)	0.2474 (3)	0.1960(3)	0.4723(3)	0.4722(3)	0.4740(4)
	2	1.519	1.325	1.149	1.752	1.751	0.9864
	3	1.728	1.466	1.271	2.163	2.158	2.028
	4	2.135	1.724	1.456	2.793	2.775	2.843
	5	2.735	2.151	1.751	3.642	3.601	3.660
5	1	0.5046(4)	0.3781 (5)	0.2985(5)	0.7269(4)	0.7269(4)	0.7191
	2	1.647	1.452	1.279	1.932	1.932	1.463
	3	1.933	1.623	1.423	2.461	2.462	2.220
	4	2.427	1.949	1.640	3.204	3.199	3.136
	5	3.105	2.442	1.992	4.150	4.135	4.453
6	1	0.7046(5)	0.5263	0.4143	1.019(5)	1.019(5)	1.022
	2	1.795	1.574	1.402	2.170	2.169	1.927
	3	2.171	1.794	1.561	2.819	2.819	2.515
	4	2.748	2.189	1.826	3.665	3.663	3.484
	5	3.499	2.746	2.236	4.701	4.689	4.827

It is interesting to note in Tables 5 to 7 that, irrespective of shell configurations, the fundamental (lowest) frequencies and the second ones are for modes having two (n = 2) and three (n = 3) circumferential waves in their modes, respectively, and the torsional frequencies (n = 0^T) are all for higher modes. It is also seen that the axisymmetric modes (n = 0^A) are important for positive eccentricity ( $e > 0$ ). That is, they are among the lowest frequencies of the shells.

Table 8 presents the nondimensional frequencies in

ω R \sqrt{ρ / G}

of the bottom edge fixed hemi-spherical shells of revolution with eccentricity. The shell configurations for Table 8 are the same as for Table 7.

Table 8.

Frequencies in $ω R \sqrt{ρ / G}$ of bottom edge fixed, hemispherical shells of revolution with eccentricity (e = R) having constant thickness (H/R = 0.05) for ν = 0.3

n	s	e = R
		R_o = R	R_o = 1.5R	R_i = 0	R_i = 0.5R
		Figure 6(a)	Figure 6(b)	Figure 6(c)	Figure 6(d)
^T	1	2.063	2.690	3.156	3.850
	2	2.997	4.363	5.28	5.953
	3	4.144	6.229	7.333	7.211
	4	5.129	8.169	9.360	8.790
	5	6.080	10.13	11.38	10.70
0^A	1	0.09804(2)	0.4053	1.445(5)	0.4712(4)
	2	0.7453	0.7405	1.647	1.273
	3	0.7739	0.8566	2.165	2.142
	4	0.9014	1.017	3.117	3.117
	5	1.154	1.284	3.330	4.202
1	1	0.1369(4)	0.3986	1.452	0.5246(5)
	2	0.6403	0.7240	1.844	1.309
	3	0.7473	0.8630	2.365	1.971
	4	0.9224	1.228	2.598	2.588
	5	1.176	1.297	3.726	2.718
2	1	0.04863(1)	0.04810(1)	0.1872(1)	0.3290(1)
	2	0.4841	0.1347(2)	1.617	0.6138
	3	0.5989	0.4454	1.952	1.240
	4	0.729	0.7126	2.185	2.191
	5	0.8904	0.8971	3.132	3.269
3	1	0.1361(3)	0.1359(3)	0.4783(2)	0.3795(2)
	2	0.7672	0.2721(5)	1.867	0.9042
	3	0.8081	0.6635	2.630	1.660
	4	1.038	0.9481	2.963	2.772
	5	1.060	1.046	3.751	4.098
4	1	0.2501(5)	0.2499(4)	0.8528(3)	0.4595(3)
	2	0.9374	0.3894	2.216	1.176
	3	0.9911	0.8578	3.174	2.080
	4	1.216	1.057	3.888	3.336
	5	1.417	1.476	4.439	4.850
5	1	0.3844	0.3842	1.298(4)	0.564
	2	1.094	0.5124	2.670	1.452
	3	1.157	1.043	3.796	2.502
	4	1.387	1.180	4.793	3.901
	5	1.586	1.586	5.184	5.586
6	1	0.5365	0.5361	1.809	0.6907
	2	1.245	0.6442	3.217	1.740
	3	1.314	1.214	4.484	2.933
	4	1.534	1.317	5.692	4.474
	5	1.706	1.712	5.979	6.329

5. Concluding remarks

Extensive and accurate frequency data determined by the 3-D Ritz analysis have been presented for hemi-spherical shells of revolution with eccentricity having uniform thickness. The analysis uses the 3-D equations of the theory of elasticity in their general forms for isotropic materials. They are only limited to small strains. No other constraints are placed upon the displacements. This is in stark contrast with the classical 2-D thin shell theories, which make very limiting assumptions about the displacement variation through the shell thickness.

The method is straightforward, but it is capable of determining frequencies and mode shapes as close to the exact ones as desired. Therefore, the data in Tables 4 to 6 may be regarded as benchmark results against which 3-D results obtained by other methods, such as finite elements and finite differences, may be compared to determine the accuracy of the latter. Moreover, the frequency determinants required by the present method are at least an order of magnitude smaller than those needed by finite element analyses of comparable accuracy. This was demonstrated extensively in a paper by McGee and Leissa (1991). The Ritz method guarantees upper bound convergence of the frequencies in terms of functions sets that are mathematically complete, such as algebraic polynomials. Some finite element methods can also accomplish this, but at much greater costs, and others cannot.

The method presented could also be extended to circumferentially open ( $0 \leq θ \leq θ_{0}$ ) hemi-spherical shells with eccentricity, instead of circumferentially closed ( $0 \leq θ \leq 2 π$ ) hemi-spherical shells of revolution with eccentricity considered in the present work. However, the periodicity in θ would not be present. It would be necessary then to replace the double sums of algebraic polynomials in equations (24) by triple sums, with polynomials in θ being included.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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