1.4.1 Free vibrations of a spherical shell

Product: ABAQUS/Standard  

The first papers on the vibration of thin, elastic, spherical shells precede the general formulation of the classical bending theory of shells. The problem of free vibration of a “complete” spherical shell was first examined by Lamb (1882). More detailed treatments were given by Baker (1961) and Silbiger (1962). The problem has many interesting features and serves well as a good test case for the shell elements in ABAQUS.

Problem description

Analytical solution

Based on the membrane theory of shells, it is known that the natural frequency spectrum of a hollow, thin, elastic sphere consists of two infinite sets of modes and that one set of an infinite number of modes is spaced within a finite frequency interval. The mode shapes of the shell are expressed in terms of Legendre polynomials of degree n. For each value of n there are two distinct frequencies. The smaller of the two frequencies forms the “lower branch.” The second or “upper branch” modes are primarily extensional. The first 10 frequencies are given in Table 1.4.1–1.

The 0 mode consists of purely radial vibration. Its frequency lies well above all of the frequencies associated with modes in the lower branch. It can be seen in the table that the frequencies of the upper branch increase without limit as n increases but that those of the lower branch approach the limit:

where f is the frequency of vibration, E is the modulus of elasticity, is the mass density, and R is the radius of the sphere. Such a limiting situation is a result of the membrane theory employed (Kalnins, 1964). Membrane theory is accurate only for very thin shells and for low mode numbers. The ABAQUS shell elements account for membrane and bending effects, so we should expect good agreement only in membrane-type modes.

If only axisymmetric modes are considered, there is a distinct mode shape for each value of frequency. However, a model based on general shell elements allows for nonaxisymmetric modes. Interestingly, for the spherical shell the frequencies corresponding to nonaxisymmetric modes are identical to the frequencies of the axisymmetric modes. This is a consequence of the spherical symmetry of the shell. Corresponding to each value of n there are +1 linearly independent modes. To verify this, we have chosen to model the entire sphere, although the problem can be analyzed more economically by modeling a partial sphere using symmetry and antisymmetry boundary conditions. In addition, because of the multiple modes of identical frequency, this problem serves as a good test for the eigenvalue-eigenvector algorithms.

Results and discussion

Input files

References

Tables

Table 1.4.1–1 Natural frequencies in cycles/sec based on membrane theory. ( 180.0 × 109,  1/3,  7670.0.)

ModeLower spectrumHigher spectrum
0445.0
10.0545.18
2187.34748.02
3222.57995.37
4236.561256.58
5239.561522.62
6247.371791.24
7249.802060.92
8251.412331.42
9252.542602.36
10253.352873.62

Table 1.4.1–2 Natural frequencies with axisymmetric shell elements.

Mode(n)Membrane theory=0.01=0.05
SAX1SAX2 SAX1SAX2
2187.34187.26187.36187.72187.82
3222.57222.30222.69225.19225.57
4236.56236.15236.95245.35246.09
5239.56243.12244.41264.61265.76
6247.37247.43249.30289.13290.66
7249.80250.76253.29321.84323.68
8251.41253.99257.25364.00366.02
9252.54257.66261.69415.81417.88
10253.35262.18267.00445.14445.14

Table 1.4.1–3 Natural frequencies with first-order asymmetric-axisymmetric shell elements.

Eigenvalue numberSAXA11SAXA12SAXA13SAXA14
4187.26187.26187.26187.26
5187.35187.35187.35187.35
6222.30187.41187.41187.41
7222.53222.30222.30222.30
8236.15222.53222.53222.53
9236.51222.73222.73222.73
10243.12236.15222.76222.76
11243.59236.51236.15236.15
12247.43236.84236.51236.51
13248.01243.12236.83236.83
14250.76243.59237.03237.03
15251.45244.03243.12237.04

Table 1.4.1–4 Natural frequencies with second-order asymmetric-axisymmetric shell elements.

Eigenvalue numberSAXA21SAXA22SAXA23SAXA24
4187.36187.36187.36187.36
5187.36187.36187.36187.36
6222.69187.36187.36187.36
7222.69222.69222.69222.69
8236.94222.69222.69222.69
9236.95222.69222.69222.69
10244.41236.94222.69222.69
11244.41236.95236.95236.95
12249.29236.95236.95236.95
13249.30244.41236.95236.95
14253.29244.41236.95236.95
15253.30244.41244.41236.95

Table 1.4.1–5 Natural frequencies with second-order general shell elements S8R, S8R5, S9R5, and STRI65.

Eigenvalue numberS8RS8R5S9R5STRI65
7187.37187.36187.36187.38
8187.37187.36187.36187.38
9187.38187.36187.36187.38
10187.38187.37187.37187.38
11187.38187.37187.37187.38
12222.66222.63222.63222.74
13222.66222.63222.63222.75
14222.66222.63222.63222.75
15222.74222.70222.70222.76
16222.74222.70222.70222.81
17222.74222.70222.70222.81
18222.81222.77222.77222.84
19236.81236.66236.68237.14
20236.93236.80236.80237.24

Table 1.4.1–6 Natural frequencies with first-order general shell elements S4R, S4R5, S4, STRI3, and S3R.

Eigenvalue numberS4RS4R5S4STRI3S3R
7189.97189.97189.86187.32190.19
8189.97189.97189.86188.76190.66
9190.05190.05190.04188.76190.66
10190.05190.05190.06189.97192.25
11190.05190.05190.06189.97192.25
12223.71223.70225.66223.85229.55
13223.71223.70225.74224.16230.82
14223.71223.70225.74224.16230.82
15227.90227.89228.59227.51233.47
16227.90227.89228.59228.71234.32
17227.90227.89228.61228.71234.82
18231.43231.37233.57229.06234.82
19233.48233.45237.24239.45252.14
20233.59233.45242.00239.50252.14


Figures

Figure 1.4.1–1 Spherical shell model, with second-order quadrilaterals.

Figure 1.4.1–2 Modes 2,3 of spherical shell.