Effects of curvature upon the vibration characteristics of doubly curved shallow shells are assessed in this paper. Boundary conditions of the shell are generally specified in terms of distributed elastic restraints along the edges. The classical homogeneous boundary supports can be easily simulated by setting the stiffnesses of restraining springs to either zero or infinite. Vibration problems of the shell are solved by a modified Fourier series method that each of the displacements is invariably expressed as a simple trigonometric series which converges uniformly and acceleratedly over the solution domain. All the unknown expansion coefficients are treated equally as a set of independent generalized coordinates and solved using the Rayleigh-Ritz technique. The current method provides a unified solution to the vibration problems of curved shallow shells involving different geometric properties and boundary conditions with no need of modifying the formulations and solution procedures. Extensive tabular and graphical results are presented to show the curvature effects on the natural frequencies of the shell with various boundary conditions.
1. Introduction
Vibration problems of shell structures have long been of considerable attention by the researchers and engineers because they are widely used in structural, mechanical, and aerospace engineering applications. Leissa [1] wrote a monograph about the vibration of shells and summarized approximately 1000 related publications which had been done before 1973. Earlier investigations also have been reviewed by Qatu [2, 3] and Liew et al. [4].
Shallow shells can be considered as plates having small curvature in two perpendicular directions. It has three familiar types, spherical (Figure 1(a)), circular cylindrical (Figure 1(b)), and hyperbolic paraboloidal (Figure 1(c)) when the Gaussian curvature (1/RaRb) is positive, zero, and negative. The first theoretical study on the frequency analysis of shallow cylindrical shells was reported by Palmer [5]. Using cylindrical shell element, Olson and Lindberg [6] studied the vibratory behaviors of a cantilevered curved fan blade. They also investigated the dynamic characteristics of shallow shell structures using the conforming triangular shaped shell elements [7, 8]. Nath [9] determined the natural frequencies of a fully clamped cylindrical shell. Kantorovich’s method for reducing the partial differential equations to a set of ordinary differential equations was applied by Petyt and Nath [10] to study the free vibration characteristics of a singly curved rectangular plate. Petyt [11] also collected four theoretical methods for the vibration analysis of a singly curved rectangular plate and compared numerical results with the experimental results [9]. An approximate solution for the vibration analysis of open shallow cylindrical shells was presented by Elishakoff and Wiener [12]. The doubly curved right helicoidal shell elements were used by Walker [13] to investigate curved twisted fan blades. Although the finite element method has been widely used in solving various shell vibration problems, it is sometimes less desired as compared with an analytical solution because the parameters of concern are all digitized and their significance can be easily lost in the numerical or discretization process. From practical point of view, when a shell is elastically restrained, the springs will have to be manually created in a finite element model, which can easily become an overwhelming task, especially when spring rates vary along an edge. This concern will become more remarked when a stochastic process or field will have to be taken into account.
Shells of positive, zero, and negative Gaussian curvature: (a) spherical, (b) circular cylindrical, and (c) hyperbolic paraboloidal.
The Ritz method with algebraic polynomial trial functions was used by Leissa and his coworkers [15, 18, 19] to study the vibration characteristics of different types of shallow shells with various boundary conditions. Narita and Leissa [20] studied the vibration of corner point supported shallow shells. Lee et al. [21] compared the shallow and deep shell theories using cantilevered circular cylindrical shells with rectangular planform. In [15, 18–21], the boundary conditions had been imposed explicitly because the built-in basic function to satisfy the kinematic shell boundaries was not specified. Qatu and Leissa [16, 22–24] also used the Ritz method based on algebraic polynomial displacement functions to investigate the effects of edge constraints on the vibrations of shallow shells.
A numerical approximation with pb-2 functions was employed by Liew and Lim [14, 17, 25–27] to solve vibratory behaviors of shallow shells with different complicating factors. The limits of the shallow shell theory were investigated by Liew et al. [28] using the p-Ritz method. It was determined that the shallow shell theory was accurate enough if the subtended angle of a shell is no more than 40°. Recently, the spline finite strip method [29] and differential quadrature method (DQM) [30] were used to study the vibrations of circular curved panels.
Although vibration problems of shallow shells have been extensively studied for many years, most of the existing investigations are specifically dealt with shells having particular type and classical boundary conditions, while curvature effects on the vibration characteristics of doubly curved shallow shells with general elastic boundary supports have received little attention. Vlasov [31] pointed out that shells having negative Gaussian curvature will have the lowest frequencies, but he made no further study of this problem. For a shell that has a rectangular planform supported by shear diaphragms [32], it is shown that the frequencies of all modes became larger with positively increasing values of Rb/Ra, whereas negatively changing values of Rb/Ra cause first decreasing, then increasing frequencies. The primary objective of the present work is to assess the effects of curvature upon the natural frequencies of shallow shells with arbitrary elastic boundary conditions. For this purpose, four types of uniformly distributed elastic springs are specified along each edge to realize various boundary conditions. It represents a situation which will be much more practical in engineering applications. A modified Fourier series method in which all the displacements are expressed in the form of trigonometric functions is used in this investigation. Since the trigonometric functions are sufficient completeness and smoothness in the solution domain, Rayleigh-Ritz method is employed here instead of solving the series coefficients which simultaneously satisfy both the governing differential equations and boundary conditions. Extensive tabular and graphical results are presented to show the effects of curvature upon the natural frequencies of shallow shells with different boundary conditions. The changing trends of the frequencies varying with the curvatures are explained in detail.
2. Theoretical Formulations
A doubly curved shell on rectangular planform with uniform thickness h is illustrated in Figure 2. The shell is described in a curvilinear coordinate system and considered shallow for its small rise compared to the minimum radii of curvature. The curvilinear lengths of edges are denoted as a and b while la and lb are the lengths of planform. Ra and Rb are constant principal radii of curvature in x and y direction, respectively. Boundary conditions of the shallow shell are specified as general elastic restraints which are described in terms of flexural, longitudinal, tangential, and rotational springs of arbitrary stiffnesses. For simplicity, it is assumed that the restraining springs have uniform stiffness distributions along each edge. All the classic homogeneous boundary conditions can be obtained by setting the stiffness coefficients equal to either zero or infinity. Other more complex boundary conditions, such as partial and nonuniform elastic supports, can be readily dealt with [33].
Geometrical parameters and coordinate system of a doubly curved shallow shell with elastically restrained edges.
Vibration of the doubly curved shallow shell is considered three-dimensional: u(x,y), v(x,y), and w(x,y), respectively, denotes the displacement at a given point on the middle surface of the shell in x, y, and z directions. The strain-displacement relationships for thin shells which are based on Love’s first approximation assumptions are adopted in the present study. Strain components in an arbitrary point on the shell are defined by the relations
(1)εx=εx0+zτx,εy=εy0+zτy,εxy=εxy0+zτxy,
where z is the distance of the shell from the middle surface,
(2)εx0=∂u∂x+wRa,εy0=∂v∂y+wRb,εxy0=∂u∂y+∂v∂x
are the middle surface strain-displacement relationships, and
(3)τx=-∂2w∂x2,τy=-∂2w∂y2,τxy=-2∂2w∂x∂y
are the changes in the curvature and torsion of the middle surface. Neglecting σz, the strain potential energy of the deformed shallow shell is given as
(4)Vsh=12∫0a∫0b∫-h/2h/2σε×Ra+zRaRb+zRbdxdydz,
where
(5)σ=σxσyσxy,ε=εxεyεxyT,σ=εTE, and the material constitutive matrix E of the shell is
(6)E=E1-μ21μ0μ1000(1-μ)2,
where E and μ are Young’s modulus and Poisson ratio of the shell material, respectively.
Combining (1)–(6), one can obtain
(7)Vsh=12D∫0a∫0b∂2w∂x22+∂2w∂y22+2μ∂2w∂x2∂2w∂y2+1-μ22∂2w∂x∂y2dxdy+12G∫0a∫0b∂u∂x+wRa2+∂v∂y+wRb2+2μ∂u∂x+wRa∂v∂y+wRb+1-μ2∂u∂y+∂v∂x2dxdy-D1Ra+1Rb×∫0a∫0b∂u∂x+wRa∂2w∂x2+μ∂2w∂y2+∂v∂y+wRb∂2w∂y2+μ∂2w∂x2+1-μ2∂u∂y+∂v∂x2∂2w∂x∂ydxdy+Oh4,
where O(h4) is a higher-order in h and can be neglected here and D=Eh3/[12(1-μ2)] and G=Eh/(1-μ2) are the bending rigidity and extensional rigidity of the shell, respectively. Equation (7) includes three terms which can be interpreted as bending energy, membrane energy, and coupling energy.
The potential energies stored in the boundary springs can be written as
(8)Vsp=12∫0aky0fw2+ky0lu2+ky0tv2+Ky0∂w∂y2y=0dx+12∫0akybfw2+kyblu2+kybtv2+Kyb∂w∂y2y=bdx+12∫0bkx0fw2+kx0lu2+kx0tv2+Kx0∂w∂x2x=0dy+12∫0bkxafw2+kxalu2+kxatv2+Kxa∂w∂x2x=ady.
The definitions for all the boundary springs are given in the Nomenclature.
The total kinetic energy of the doubly curved shallow shell, by neglecting rotary inertia, is given as
(9)T=12ρh∫0a∫0b∂u∂t+∂v∂t+∂w∂tdxdy,
where ρ is the mass density of the shell.
The displacements of a doubly curved shallow shell can be expressed as [34]
(10)ux,y=∑m=-2∞∑n=-2∞Um,nφmxφny,(11)vx,y=∑m=-2∞∑n=-2∞Vm,nφmxφny,(12)wx,y=∑m=-4∞∑n=-4∞Wm,nφmxφny,
where Um,n, Vm,n, and Wm,n denote the unknown trigonometric series coefficients to be determined and the basis functions are defined as
(13)φm(x)=cosλamxm≥0,φm(x)=sinλamxm<0,φn(y)=cosλbnyn≥0,φn(y)=sinλbnyn<0,
where λam=mπ/a and λbn=nπ/b. Since trigonometric series are “invariants” under differential and integral operations, the current displacement expressions are much more attractive. It can be mathematically proven that the trigonometric series expansions, (10)–(12), are better suited for expanding a sufficiently smooth function defined over a compact interval, respectively, and converge uniformly over the solution domain. As a matter of fact, (10) is able to expand to any function f(x,y)∈C3 for ∀(x,y)∈D:([0,a]⊗[0,b]). So the current displacement solutions are simply those elements in the vector space which simultaneously satisfy both the governing differential equations and the boundary conditions on a point-wise basis. It can be seen that the in-plane displacements u(x,y) and v(x,y) have less sine terms than the out-of-plane displacement w(x,y) because they are only required to have C1 continuity over the shell.
The Lagrangian L for the doubly curved shallow shell can be generally expressed as
(14)L=V-T=Vsh+Vsp-T,
where V is the total potential energy of the shell. Substituting (7)–(9) into (14) and minimizing Lagrangian with respect to all the unknown series coefficients, one can obtain a system of linear algebraic equations in matrix form
(15)K-ω2MA=0,
where A is a vector that contains all the unknown series expansion coefficients and is defined as
(16)A=UTVTWTT,
where
(17)U=U-2,-2,U-2,-1,…,U-2,n,…,U-2,N,U-1,-2,…,U-1,N,…,Um,N,…,UM-1,N,UM,-2,…,UM,NT,V=V-2,-2,V-2,-1,…,V-2,n,…,V-2,N,V-1,-2,…,V-1,N,…,Vm,N,…,VM-1,N,VM,-2,…,VM,NT,W=W-4,-4,W-4,-3,…,W-4,n,…,W-4,N,W-3,-4,…,W-3,N,…,Wm,N,…,WM-1,N,WM,-4,…,WM,NT.M and N are truncation numbers of the trigonometric expansion series. K and M stand for the stiffness and mass matrices, respectively. The detailed expressions for these matrices are given in the Appendix.
By solving a standard matrix eigenvalue problem, modal properties of the doubly curved shallow shell can be readily and directly determined. Since each of the eigenvectors actually contains the trigonometric series coefficients, the corresponding physical mode shape of the shell can be simply obtained by using the displacement expressions, (10)–(12). It should be pointed out that although this paper is focused on the free vibration of doubly curved shallow shells, its response to an applied load can be easily calculated by including the work done by this load in the Lagrangian, eventually leading to a force term on the right side of (15). Once the primary solution variables, displacements, are determined over the shell, other dynamic variables of interest can be readily calculated by directly apply appropriate mathematical operations to the displacement functions.
3. Results and Discussions
Accuracy and convergence of the current method will be demonstrated in this section by numerical results firstly. Consider a doubly curved shallow shell with fully clamped along four edges (C-C-C-C). The clamped edge can be regarded as a special case when the stiffnesses for all the boundary restraining springs become infinitely large (which is actually represented by a very lager number, 2.0×1012, in the numerical calculations). The first six nondimensional frequency parameters, Ω=ωlalbρh/D, which are determined using different numbers of expansion terms M and N are listed in Table 1 with the following geometric parameters: Rb/Ra=1,Rb/lb=10,la/lb=2,lb/h=100, and μ=0.3. The results compare very well with those obtained from the finite element method and [14]. Since it can be seen that the current results converge rapidly with a small number of expansion terms and have great numerical stability, the displacement series expansions will be truncated to M=N=12 in all the following calculations. The corresponding mode shapes are plotted in Figure 3. It is shown that the in-plane two directional displacements are coupled together at any field point. The solution times between the current method and finite element method are also compared in Table 1. Due to the models that are both small, the time given in Table 1 is not conclusive regarding which method is more effective computationally. However, the effectiveness of the current method over the FEA has been adequately demonstrated in a previous support [35] in which the modified Fourier series method is shown to cut computing time by two orders of magnitude as compared with the FEM model used to simulate a box-like structure in frequency response analysis.
Frequency parameters for a completely clamped doubly curved shallow shell.
Mode number
Mode frequencies
M=N
FEM^{*}
FEM^{#}
[14]
6
7
8
9
10
11
12
1
102.17
102.15
102.14
102.13
102.13
102.13
102.13
102.11
102.12
102.22
2
103.02
103.01
103.00
103.00
103.00
103.00
103.00
103.00
103.01
103.08
3
118.73
118.65
118.64
118.62
118.61
118.60
118.60
118.56
118.59
118.76
4
144.73
144.72
144.71
144.71
144.71
144.71
144.71
144.63
144.66
144.82
5
145.47
145.45
145.38
145.37
145.36
145.35
145.35
145.26
145.30
145.64
6
158.54
158.53
158.52
158.51
158.51
158.51
158.51
158.43
158.46
158.67
Time (s)
4.41
6.11
8.71
12.11
16.35
22.31
28.35
23.20
134.60
*100 × 100 elements, ^{#}200 × 200 elements.
The mode shapes (above: out-of-plane mode shapes; below: in-plane mode shapes) for a doubly curved shallow shell with completely clamped edges: The (a) first, (b) second, (c) third, (d) fourth, (e) fifth, and (f) sixth mode.
Next example concerns completely free (F-F-F-F) shallow shells having square and rectangular planform. The completely free boundary condition represents a classical, but quite challenging, case for testing a shell solution. Under the current framework, the free edge condition is easily realized by setting all the stiffness constants to zero. Eight frequency parameters, Ω=ωla2ρh/D, are listed in Table 2 for different curvature ratios (where S or A is used to indicate that a vibration mode is symmetric (S) or antisymmetric (A) with respect to the x- or y-axis). Two sets of reference results are also given there for comparison, and these three sets of solutions agree well with each other.
Frequency parameters of completely free shallow shells having square and rectangular planform (la/Ra=0.2, la/h=100, and μ=0.3).
Ra/Rb
Mode
la/lb=1
la/lb=2
Current
[15]
FEM
Current
[15]
FEM
1
SS-1
19.691
19.757
19.733
22.721
22.794
22.709
SS-2
42.235
42.353
42.127
98.601
99.041
98.593
SA-1
35.764
35.880
35.790
58.439
58.574
58.450
SA-2
73.624
73.890
73.544
160.51
172.23
160.53
AS-1
35.764
35.880
35.790
62.120
62.345
62.126
AS-2
73.624
73.890
73.544
110.65
110.83
110.58
AA-1
13.480
13.524
13.491
26.521
26.577
26.531
AA-2
69.342
69.598
69.480
101.68
102.00
101.69
0
SS-1
21.831
21.904
21.810
21.559
21.631
21.538
SS-2
38.495
38.473
38.510
100.28
100.55
100.28
SA-1
34.771
34.852
34.737
58.310
58.425
58.252
SA-2
75.278
75.298
75.281
159.78
171.56
159.74
AS-1
37.625
37.643
37.623
59.613
59.845
59.585
AS-2
60.914
61.154
60.886
109.53
109.60
109.53
AA-1
13.461
13.483
13.467
26.517
26.562
26.520
AA-2
70.765
70.952
70.740
101.24
101.53
101.20
−1
SS-1
24.659
24.741
25.216
25.294
SS-2
52.538
52.574
102.11
102.19
SA-1
36.842
36.957
58.407
58.543
SA-2
76.865
77.063
160.72
172.64
AS-1
36.842
36.957
66.341
66.576
AS-2
76.865
77.063
108.95
109.07
AA-1
13.418
13.425
26.508
26.564
AA-2
77.349
77.647
101.69
102.00
To further validate the accuracy and reliability of the proposed analytical method, Table 3 shows frequency parameters Ω=ωla2ρh/D for a few more classical cases (F-F-F-F, C-F-F-F, C-C-F-F, C-F-C-F, C-C-C-F and C-C-C-C). The reference results from [16] and FEM models are also given there for comparison. Traditionally, the displacement expressions and the subsequent solution algorithms and implementations are dictated by the intended boundary condition. Consequently, most studies are specifically related to a particular type of boundary conditions. In the above examples, it has been demonstrated that the proposed analytical method can be universally applied to different boundary conditions with no need of making any algorithm or procedural modifications; the modifying boundary conditions are as simple as changing shell parameters such as geometrical and material properties. Consider the simply supported case (SSSS) for example. It can be produced easily by letting the stiffnesses of the three linear springs be infinitely large and the stiffness of the rotational spring zero. Similarly, singly curved shells can be considered as special cases when one of curvatures becomes zero, as illustrated in Table 4 for various boundary conditions.
Frequency parameters of shallow shells with different boundary conditions (la/Rb=0.2, la/lb=1, la/h=20, and μ=0.3).
Ra/Rb
Mode number
Boundary conditions
F-F-F-F
C-F-F-F
C-C-F-F
C-F-C-F
C-C-C-F
C-C-C-C
1
1
13.41
3.739
7.860
25.28
26.79
40.27
(13.46^{a})
(3.754)
(7.894)
(25.30)
(26.62)
(40.26)
[13.43^{b}]
[3.756]
[7.891]
[25.32]
[26.84]
[40.25]
2
19.50
8.456
23.80
25.28
42.58
74.43
(19.56^{a})
(8.492)
(23.90)
(25.30)
(42.71)
(74.17)
[19.54^{b}]
[8.467]
[23.84]
[25.32]
[42.57]
[74.39]
3
25.92
21.44
27.83
45.44
63.39
74.43
(25.99^{a})
(21.53)
(27.92)
(45.53)
(63.56)
(74.17)
[25.82^{b}]
[21.46]
[27.78]
[45.45]
[63.41]
[74.39]
4
34.72
28.17
49.08
61.32
77.62
108.7
(34.85^{a})
(28.26)
(49.27)
(61.34)
(77.64)
(108.7)
[34.73^{b}]
[28.12]
[49.09]
[61.33]
[77.60]
[108.7]
∞
1
13.44
3.802
7.201
22.72
24.44
37.75
(13.46^{a})
(3.806)
(7.218)
(22.36)
(24.34)
(37.56)
[13.45^{b}]
[3.807]
[7.216]
[22.73]
[24.46]
[37.75]
2
20.08
8.507
24.23
26.46
40.47
73.31
(20.12^{a})
(8.526)
(24.30)
(26.43)
(40.53)
(72.59)
[20.07^{b}]
[8.509]
[24.23]
[26.47]
[40.46]
[73.30]
3
24.74
21.96
27.03
43.63
63.65
74.28
(24.77^{a})
(21.98)
(27.08)
(43.59)
(62.56)
(72.66)
[24.73^{b}]
[21.96]
[27.02]
[43.63]
[63.69]
[74.30]
4
34.70
27.21
48.09
61.67
76.58
108.3
(34.79^{a})
(27.29)
(48.21)
(61.59)
(76.47)
(108.4)
[34.67^{b}]
[27.19]
[48.08]
[61.70]
[76.57]
[108.3]
−1
1
13.41
3.794
7.041
25.06
26.68
38.59
(13.46^{a})
(3.814)
(7.071)
(25.07)
(26.78)
(38.51)
2
21.84
8.444
25.47
28.57
41.54
74.02
(21.90^{a})
(8.480)
(25.58)
(28.62)
(41.60)
(74.09)
3
24.16
22.28
26.92
44.82
63.48
74.02
(24.24^{a})
(22.38)
(27.03)
(44.92)
(63.92)
(73.97)
4
34.81
27.80
48.02
61.44
77.31
108.1
(34.94^{a})
(27.89)
(48.22)
(61.47)
(77.26)
(108.0)
aResults from [16].
^{
b}Results are obtained from FEM models.
Frequency parameters for different boundary condition singly curved shallow shells with thickness ratio h/lb=100 (μ=0.3).
BC
la/lb
lb/Rb
Mode number
1
2
3
4
5
6
S-F-S-F
0.5
0.1
22.199
24.194
36.289
55.883
79.506
83.045
(22.194^{a})
(24.178)
(36.262)
(55.824)
(79.487)
(83.002)
0.2
26.397
27.512
39.769
56.755
82.575
84.809
(26.375^{a})
(27.487)
(39.788)
(56.783)
(82.540)
(84.756)
1.0
0.1
15.704
17.062
37.872
43.834
48.108
72.459
(15.693^{a})
(17.042)
(37.853)
(43.811)
(48.065)
(72.396)
0.2
19.498
24.284
42.410
51.796
53.554
75.946
(19.460^{a})
(24.234)
(42.446)
(51.713)
(53.469)
(76.087)
[19.446^{b}]
[24.240]
[42.378]
[51.695]
[53.464]
[75.851]
C-F-C-F
0.5
0.1
45.776
47.172
55.584
71.384
97.130
123.52
(45.782^{a})
(47.176)
(55.602)
(71.414)
(97.187)
(123.53)
0.2
48.365
48.688
57.865
72.110
97.314
125.52
(48.373^{a})
(48.697)
(57.905)
(72.210)
(97.514)
(125.53)
1.0
0.1
25.463
26.982
44.584
64.445
68.153
79.963
(25.463^{a})
(26.982)
(44.608)
(64.437)
(68.152)
(80.016)
0.2
28.587
31.671
48.595
70.853
71.644
80.405
(28.589^{a})
(31.667)
(48.653)
(70.850)
(71.626)
(80.608)
[28.576^{b}]
[31.673]
[48.580]
[70.813]
[71.617]
[80.372]
S-S-S-S
0.5
0.1
29.363
41.242
64.852
85.445
99.035
99.911
(29.351^{a})
(41.161)
(64.753)
(85.406)
(98.912)
(99.747)
0.2
40.018
45.813
66.603
89.807
99.491
102.93
(40.053^{a})
(45.810)
(66.625)
(89.789)
(99.568)
(102.83)
1.0
0.1
36.804
51.608
58.413
82.449
99.526
103.72
(36.841^{a})
(51.576)
(58.383)
(82.302)
(99.527)
(103.66)
0.2
57.644
63.712
79.167
91.551
102.61
117.24
(57.708^{a})
(63.834)
(79.217)
(91.542)
(102.84)
(117.23)
[57.581^{b}]
[63.799]
[79.184]
[91.414]
[102.53]
[117.19]
C-S-C-S
0.5
0.1
50.190
59.063
78.651
109.62
128.06
139.39
(50.200^{a})
(59.082)
(78.688)
(109.69)
(128.07)
(139.41)
0.2
57.074
62.404
80.129
110.05
131.02
141.58
(57.107^{a})
(62.467)
(80.267)
(110.31)
(131.04)
(141.64)
1.0
0.1
42.398
56.809
76.001
97.409
102.95
132.92
(42.445^{a})
(56.845)
(76.022)
(97.448)
(103.03)
(132.94)
0.2
62.563
67.119
92.949
105.38
105.88
143.75
(62.678^{a})
(67.233)
(93.012)
(105.52)
(106.18)
(143.80)
[62.559^{b}]
[67.201]
[92.982]
[105.40]
[105.87]
[143.76]
aResults from [17].
^{
b}Results are obtained from FEM models.
All the examples considered thus far have been limited to the classical boundary conditions which are viewed as the special cases of elastically restrained edges. We now turn to elastically restrained shells. The stiffnesses of the linear and rotational restraints are set equal to k=106 and K=107, respectively. The first eight frequency parameters Ω=ωla2ρh/D are listed in Table 5 with the following geometric parameters: Ra/Rb=1, Ra/la=10, la/lb=1, lb/h=200, and μ=0.3. The corresponding mode shapes are plotted in Figure 4. It can be seen from Figure 4 that these lower order modes exhibit complicated spatial patterns. Moreover, the in-plane patterns tend to be more local and are typically more complicated than their out-of-plane counterpart. Through those modes, one can easily understand the “unpredictable” behaviors of a doubly curved shallow shell and the effects of curvatures and boundary conditions.
Frequency parameters of an elastically supported shallow shell.
Mode number
1
2
3
4
5
6
7
8
33.008
41.067
41.067
46.364
46.364
57.091
58.083
64.769
The mode shapes (left: out-of-plane mode shapes; right: in-plane mode shapes) for a doubly curved shallow shell with elastic restraints, k=106 and K=107, along each edge. The (a) first, (b) second, (c) third, (d) fourth, (e) fifth, (f) sixth (g) seventh, and (h) eighth mode.
Effects of curvature on the vibration characteristics of shallow shells are studied in this section. It is assumed that the geometric parameters of the shell are la/lb=1 and lb/h=100 and Poisson’s ratio μ=0.3 in the following calculations. Tables 6, 7, 8, 9, and 10 show the first frequency parameters Ω of shallow shells having different x-direction curvature (1/Ra) and y-direction curvature (1/Rb) with C-C-C-C, C-C-C-F, C-F-C-F, C-F-F-F, and F-F-F-F boundary conditions. The free edge condition (F) is easily simulated by setting the stiffnesses for the boundary restraining springs to zero.
The first frequency parameters of C-C-C-C shallow shells.
Rb/Ra
lb/Rb
0
0.1
0.2
0.3
0.4
0.5
−1.0
50.707
79.487
110.59
135.19
155.65
(50.750)
(110.80)
(157.35)
−0.9
49.523
76.526
106.07
127.62
147.57
−0.8
48.480
73.848
101.82
120.15
138.48
−0.7
47.589
71.504
97.941
113.01
129.39
−0.6
46.862
69.540
93.579
106.44
120.72
−0.5
46.305
67.999
89.835
100.65
112.89
(46.335)
(90.225)
(113.65)
−0.4
45.926
66.921
86.814
95.871
106.27
−0.3
45.730
66.332
84.612
92.317
101.26
−0.2
45.719
66.250
83.309
90.176
98.185
−0.1
45.894
66.678
82.958
89.578
97.291
0
35.985
46.253
67.607
83.577
90.571
98.675
(35.985^{a})
(46.281)
(83.923)
(99.263)
0.1
46.791
69.014
85.151
93.116
102.27
0.2
47.502
70.867
87.629
97.095
107.85
0.3
48.377
73.128
90.937
102.34
115.14
0.4
49.408
75.751
94.986
108.66
123.82
0.5
50.583
78.693
99.680
115.87
133.60
(50.618)
(100.00)
(134.13)
0.6
51.892
81.906
104.93
123.80
144.22
0.7
53.323
85.346
110.64
132.30
155.48
0.8
54.867
88.969
116.74
141.23
167.19
0.9
56.511
92.732
123.15
150.48
179.18
1.0
58.247
96.593
129.81
159.95
191.28
(58.297)
(130.16)
(191.99)
aResults from [14].
The first frequency parameters of C-C-C-F shallow shells.
Rb/Ra
lb/Rb
0
0.1
0.2
0.3
0.4
0.5
−1.0
23.931
38.344
63.123
84.048
94.427
105.48
−0.9
36.441
58.579
80.959
92.572
102.83
−0.8
34.643
54.101
74.174
90.733
100.16
−0.7
32.971
49.744
67.323
84.768
97.506
−0.6
31.446
45.573
60.539
75.735
90.829
−0.5
30.093
41.670
53.971
66.684
79.600
−0.4
28.937
38.139
47.805
57.919
68.363
−0.3
28.005
35.111
42.294
49.825
57.684
−0.2
27.320
32.739
37.783
42.954
48.332
−0.1
26.901
31.180
34.701
38.086
41.478
0
26.762
30.563
33.463
36.095
38.603
0.1
26.906
30.945
34.266
37.441
40.566
0.2
27.3293
32.284
36.958
41.755
46.668
0.3
28.017
34.461
41.134
48.161
55.387
0.4
28.951
37.315
46.347
55.819
65.426
0.5
30.106
40.689
52.223
64.112
75.889
0.6
31.456
44.446
58.483
72.595
86.092
0.7
32.976
48.473
64.917
80.907
88.735
0.8
34.640
52.683
71.352
83.633
90.079
0.9
36.428
57.004
77.637
84.550
91.489
1.0
38.318
61.376
78.413
85.521
92.934
The first frequency parameters of C-F-C-F shallow shells.
Rb/Ra
lb/Rb
0
0.1
0.2
0.3
0.4
0.5
−1.0
37.096
61.134
79.890
88.221
97.623
(37.125)
(80.303)
(98.611)
−0.9
35.166
56.586
78.103
86.839
95.828
−0.8
33.343
52.092
71.450
85.579
94.183
−0.7
31.647
47.709
64.679
82.005
92.660
−0.6
30.103
43.503
57.930
73.143
88.445
−0.5
28.735
39.560
51.359
64.192
77.442
(28.737)
(51.411)
(77.473)
−0.4
27.572
35.993
45.163
55.473
66.370
−0.3
26.643
32.944
39.611
47.389
55.807
−0.2
25.972
30.583
35.084
40.527
46.549
−0.1
25.580
29.090
32.067
35.734
39.817
0
22.176
25.481
28.607
31.029
33.973
37.170
(22.171^{a})
(25.463)
(30.979)
(36.952)
0.1
25.676
29.183
32.166
35.703
39.464
0.2
26.160
30.757
35.252
40.439
45.888
0.3
26.915
33.178
39.803
47.205
54.850
0.4
27.918
36.264
45.335
55.140
65.061
0.5
29.141
39.848
51.472
63.644
75.653
(29.129)
(51.478)
(75.723)
0.6
30.557
43.789
57.945
72.297
85.965
0.7
32.138
47.977
64.559
80.756
87.010
0.8
33.858
52.329
71.154
81.636
87.562
0.9
35.695
56.778
75.744
81.814
88.035
1.0
37.629
61.270
75.682
81.957
88.407
(37.636)
(76.008)
(89.358)
aResults from [14].
The first frequency parameters of C-F-F-F shallow shells.
Rb/Ra
lb/Rb
0
0.1
0.2
0.3
0.4
0.5
−1.0
4.9390
6.4878
7.3699
7.8426
8.0919
(4.9410)
(7.4051)
(8.2255)
[4.9440]
[6.5038]
[7.4128]
[8.2429]
−0.9
4.9825
6.6661
7.6655
8.2239
8.5387
−0.8
5.0252
6.8587
7.9971
8.6576
9.0487
−0.7
5.0663
7.0653
8.3697
9.1546
9.6369
−0.6
5.1048
7.2840
8.7874
9.5269
9.8504
−0.5
5.1396
7.5104
9.2176
9.6185
10.010
(5.1385)
(9.2357)
(10.056)
[5.1414]
[7.5071]
[10.062]
−0.4
5.1694
7.7375
9.2555
9.7052
10.170
−0.3
5.1931
7.9543
9.2880
9.7834
10.322
−0.2
5.2095
8.1458
9.3140
9.8483
10.456
−0.1
5.2176
8.2943
9.3323
9.8949
10.556
0
3.4719
5.2167
8.3822
9.3424
9.9193
10.608
(3.4714^{a})
(5.2146)
(9.3516)
(10.589)
[3.4730^{b}]
[5.2174]
[8.3683]
[10.595]
0.1
5.2066
8.3962
9.3441
9.9203
10.606
0.2
5.1873
8.3318
9.3379
9.8996
10.557
0.3
5.1596
8.1954
9.3246
9.8612
10.472
0.4
5.1241
8.0020
9.3052
9.8097
10.365
0.5
5.0823
7.7709
9.2807
9.7488
10.246
(5.0815)
(9.3020)
(10.284)
[5.0840]
[7.7655]
[10.295]
0.6
5.0354
7.5207
9.2520
9.6815
10.121
0.7
4.9848
7.2667
9.0767
9.6096
9.9934
0.8
4.9318
7.0193
8.5747
9.5342
9.8641
0.9
4.8777
6.7852
8.1283
8.9808
9.5183
1.0
4.8234
6.5675
7.7347
8.4410
8.8618
(4.8259)
(7.7733)
(9.0054)
[4.8282]
[6.5854]
[7.7836]
[9.0327]
aResults from [14].
^{
b}Results from [18].
The first frequency parameters of F-F-F-F shallow shells.
Rb/Ra
lb/Rb
0
0.1
0.2
0.3
0.4
0.5
−1.0
13.468
13.456
13.419
13.356
13.267
13.151
−0.9
13.457
13.423
13.367
13.287
13.182
−0.8
13.458
13.428
13.377
13.305
13.211
−0.7
13.459
13.433
13.387
13.322
13.237
−0.6
13.461
13.437
13.397
13.338
13.261
−0.5
13.462
13.442
13.406
13.354
13.283
−0.4
13.463
13.446
13.415
13.368
13.304
−0.3
13.464
13.450
13.423
13.382
13.323
−0.2
13.465
13.454
13.431
13.394
13.341
−0.1
13.466
13.458
13.439
13.406
13.356
0
13.467
13.462
13.446
13.416
13.369
0.1
13.468
13.465
13.453
13.425
13.380
0.2
13.469
13.468
13.458
13.432
13.388
0.3
13.470
13.471
13.463
13.437
13.392
0.4
13.471
13.474
13.466
13.440
13.392
0.5
13.472
13.476
13.469
13.440
13.389
0.6
13.473
13.478
13.470
13.438
13.381
0.7
13.474
13.480
13.469
13.433
13.369
0.8
13.474
13.481
13.467
13.425
13.352
0.9
13.475
13.481
13.464
13.414
13.330
1.0
13.476
13.481
13.459
13.400
13.304
Figures 5(a)–5(e) show the changes in the frequency parameters Ω=ωlalbρh/D for the first modes of shallow shells with five different boundary conditions, which are listed in Tables 6–10, as the curvature ratio Rb/Ra is varied from −1 to 1, respectively. The curves are drawn beginning with a flat plate (lb/Rb=0) and changing the y-direction curvature. Thus, the change in circular frequency ω with changing 1/Rb is observed for fixed la,ρ,h,E, and μ (D=Eh3/[12(1-μ2)]) and fixed lb (la/lb=1) by changing lb/Rb.
The first mode frequency as a function of curvature ratio for shells with different boundary conditions: (a) C-C-C-C, (b) C-C-C-F, (c) C-F-C-F, (d) C-F-F-F, and (e) F-F-F-F.
It can be seen from Figure 5(a) that, for a C-C-C-C shell, increasing y-direction curvature can causes a considerable increase in the first mode frequency, and giving the shell significant additional x-direction curvature, either positive or negative, can cause an increase in the first mode frequency too. It also can be found from Figure 5(a) that the first mode frequency for the shell of positive Gaussian curvature is greater than that having negative Gaussian curvature with the same absolute value. With increasing the y-direction curvature, difference of the first mode frequencies between spherical shell (Rb/Ra=1) and hyperbolic paraboloidal shell (Rb/Ra=-1) becomes more and more large. Figure 5(a) also shows that the minimum value of the first mode frequency occurs for the shell having slightly negative Gaussian curvature and the point of minima is shifted to the left with decreasing lb/Rb.
For a C-C-C-F shallow shell, which means that the stiffness constants of the restraining springs at edge y=0 are equal to zero, the first mode frequency increases gradually with increasing the y-direction curvature as shown in Figure 5(b). Frequency of the first mode also increases rapidly through adding the x-direction curvature. But the increase trend of the first mode frequency becomes slow when the absolute value of curvature ratio Rb/Ra is comparatively large. It is more obvious if the Gaussian curvature is positive. The curvature effects on the first mode frequency for the C-F-C-F shell and the C-C-C-F shell are almost the same, which can be seen in Figure 5(c).
The shapes of curves in Figure 5(d) are different. Although the first mode frequency of a C-F-F-F shell increases gradually with the increasing of y-direction curvature, it decreases if giving the shell significant additional x-direction curvature, which is just the opposite of the three boundary conditions mentioned above. Figure 5(d) also shows that, with lb/Rb decreasing from 0.5 to 0.1, the difference of the first mode frequencies between spherical shell (Rb/Ra=1) and hyperbolic paraboloidal shell (Rb/Ra=-1) is varied gradually from positive number to negative number.
Figure 5(e) shows the first mode frequency as a function of curvature ratio for an F-F-F-F shell. It can be seen that changing the curvature of a completely free shell does not have much effect on the first mode frequency. The difference between the maximum frequency and minimum frequency is no more than 0.35 Hz. In general, increasing y-direction curvature causes a decreasing in the first mode frequency, which is different from the other four boundary conditions. The maximum value of the first mode frequency occurs for the shell having significant positive Gaussian curvature.
It is known that the value of the mode frequency depends on the mode stiffness and mode mass
(18)ωm2=KmMm,
where ωm, Km, and Mm denote the mth mode frequency, mode stiffness, and mode mass, respectively. The mode frequency can be directly calculated since the mode stiffness and mode mass are the corresponding diagonal elements of the diagonal stiffness matrix and mass matrix of the shell which can be easily obtained through mathematical operations. Differentiating the square of mode frequency ωm2 with respect to curvature ratio Rb/Ra,
(19)∂ωm2∂Rb/Ra=∂Km/Mm∂Rb/Ra.
When (19) is equal to zero, one can obtain
(20)KmMm=∂Km∂Mm.
So, one can say that the extremum value of the mth mode frequency occurs when Km/Mm is equal to ∂Km/∂Mm.
Two examples are given here to verify the conclusion. The first example is one case in Figure 5(a). The change in the first frequency parameter of a fully clamped shell with lb/Rb=0.5 is shown in Figure 6, as the curvature ratio Rb/Ra is varied from −0.3 to 0.1. The first mode frequency of the shell gets its minimal (minimum value of the discrete data) when the curvature ratio Rb/Ra is equal to −0.12, and the frequency increases with either increasing or decreasing the x-direction curvature. The changes in K1/M1 and ∂K1/∂M1 of this shell are shown in Figure 7. It can be seen that these two curves have an intersection point when curvature ratio Rb/Ra is approximately equal to −0.12. The error is caused by numerical computation.
The first mode frequency as a function of curvature ratio for a C-C-C-C shell.
The value of K1/M1 and ∂K1/∂M1 as functions of curvature ratio for a C-C-C-C shell.
The second example is one case in Figure 5(e). The change in the first frequency parameter of an F-F-F-F shell with lb/Rb=0.5 is shown in Figure 8, as the curvature ratio Rb/Ra is varied from 0.3 to 0.5. The first mode frequency of the shell gets its maximal when the curvature ratio Rb/Ra is equal to 0.36, and the frequency decreases with either increasing or decreasing the x-direction curvature. The K1/M1 and ∂K1/∂M1 of this shell as functions of curvature ratio are shown in Figure 9. The value of horizontal ordinate for the intersection point of these two curves is approximately equal to 0.36.
The first mode frequency as a function of curvature ratio for an F-F-F-F shell.
The value of K1/M1 and ∂K1/∂M1 as functions of curvature ratio for an F-F-F-F shell.
4. Conclusions
The primary purpose of the current work is to assess the effects of curvature on the natural frequencies for the shallow shells with elastic edge restraints. Each of the displacement fields is generally expressed as a modified Fourier series function. The sine function is used to remove the potential discontinuities in related spatial partially differentials. Although the current solution is sought in a weak form by using the Rayleigh-Ritz method, it is mathematically equivalent to the strong from through solving the governing equations and the boundary conditions because the displacement functions represented by the series expansions are adequately smooth throughout the entire solution domain. The effectiveness of current method is verified through the comparison of numerical results. The first mode frequencies of shallow shells with various x-direction and y-direction curvature under five kinds of boundary conditions are calculated. The results show that increasing y-direction curvature causes a considerable increase in the first mode frequency for the shell at least having one clamped edge while the other edges are free; increasing y-direction curvature causes a decrease in the first mode frequency of a completely free shell when lb/Rb is greater than 0.2; giving significant x-direction curvature, either positive or negative, can cause an increase in the first mode frequency for the shell at least having two clamped edges while the other edges are free, which is just the opposite for a cantilever shell; the minimum value of the first mode frequency occurs for a completely clamped shell with negative Gaussian curvature and the maximal value of the first mode frequency occurs of a completely free shell with positive Gaussian curvature.
AppendixMatrix Definitions
The stiffness matrix in (15) can be expressed as
(A.1)K=Kbe+Kin+Kco+Ksp,
where Kbe, Kin, Kco, and Ksp are stiffness matrix of the shell corresponding to the bending energy, the membrane energy, the coupling energy between out-of-plane and in-plane motions, and potential energy stored in the boundary restraining springs
(A.2)Kbe=000⋮00S⋯Kbew,w,Kin=Kinu,uKinu,vKinu,w⋮Kinv,vKinv,wS⋯Kinw,w,Kco=00Kcou,w⋮0Kcov,wS⋯Kinw,w,Ksp=Kspu,u00⋮Kspv,v0S⋯Kspw,w.
The shell mass matrix in (15) is given by
(A.3)M=Mu00⋮Mv0S⋯Mw.
NomenclatureKx0,Kxa(Ky0,Kyb):
Stiffnesses for rotational springs, respectively, at x=0 and a (y=0 and b)
kx0f,kxaf(ky0f,kybf):
Stiffnesses for flexural springs, respectively, at x=0 and a (y=0 and b)
kx0l,kxal(ky0l,kybl):
Stiffnesses for longitudinal springs, respectively, at x=0 and a (y=0 and b)
kx0t,kxat(ky0t,kybt):
Stiffnesses for tangential springs, respectively, at x=0 and a (y=0 and b)
ω:
Frequency in radian.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work was supported by the National Natural Science Foundation of China (Grant no. 51375103).
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