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VIBRATION ANALYSIS OF ORTHOTROPIC THIN CYLINDRICAL SHELLS WITH FREE ENDS BY THE RAYLEIGH-RITZ METHOD
Journal of Sound and Vibration (1996) 195(1), 117–135
VIBRATION ANALYSIS OF ORTHOTROPIC THIN CYLINDRICAL SHELLS WITH FREE ENDS BY THE RAYLEIGH-RITZ METHOD K. H. I, W. K. C, P. C. T T. C. L Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong (Received 4 May 1995, and in final form 29 December 1995) An analytical model is developed to predict the modal characteristics of thin-walled circular cylindrical laminated shells with free ends. The shell is orthotropic and has mid-plane symmetry. By using Love’s first-approximation shell theory, a strain energy functional containing both bending and stretching effects is formulated. The shell vibration mode shapes are then modelled by utilizing characteristic beam functions in the Rayleigh-Ritz variational procedure and the accuracy of the model is verified by test data. With the developed model, inextensional Rayleigh and Love modes can be identified having frequencies close to each other. The contributions to the strain energy due to various elastic properties are also investigated. Results show that the circumferential modulus provides a major portion of the flexural energy of the vibrating structure while the longitudinal and in-plane shear moduli contribute mostly to the stretching energy. It is also observed that reducing the shell thickness would result in a substantial increase in the ratio of the energies associated with the longitudinal and shear moduli, respectively. By rearranging the lamination stacking sequence, shells can be made to be more resilient to bending or twisting with only minor alterations in natural frequencies. 7 1996 Academic Press Limited
1. INTRODUCTION
The thin-walled circular cylindrical laminated shell has long been an important structural component due to its high stiffness-to-weight and strength-to-weight ratios. Extensive studies on the behavior of thin laminated anisotropic shells began three decades ago when Donnell-type displacement equations were adopted in formulating the equations of equilibrium and compatibility [1–3]. As to the vibration of composite thin shells, Weingarten [4] was the first to carry out theoretical and experimental investigations on the free vibration of multilayered cylindrical shells with various boundary conditions. The shell was assumed to consist of isotropic layers with equal Poisson’s ratios. Donnell-type equations were used and various modal properties of the shell including damping were considered. This was followed by the work of Dong [5] who studied a shell composed of an arbitrary number of layers with different thickness and orthotropic elastic properties. For boundary conditions other than freely supported edges, the resonance frequencies would be extracted with an iterative scheme. Bert, Baker and Egle [6] investigated theoretically anisotropic shells with freely supported ends. Displacement equations similar to Reissner’s were used. The significance of bending-stretching coupling was illustrated with a two-layered, cross-ply cylinder. Stavsky and Loewy [7] evaluated the natural frequencies of heterogeneous orthotropic 117 0022–460X/96/310117 + 19 $18.00/0
7 1996 Academic Press Limited
118
. . .
shells. Both the Donnell and Love type equations were employed and the ends of the cylinder were simply supported. Comparisons showed that inaccuracies in natural frequency calculations by using the former theory can be substantial and depend strongly on shell heterogeneity as the length-to-radius ratio becomes large. Rath and Das [8] derived equations of motion for shells having symmetrical general orthotropic layers and non-symmetrical cross-ply layers. Boundary conditions were restricted to simple supports. Effect of shear deformation were included and contrast was made to natural frequencies calculated without shear. Shivakumar and Krishna [9] used Love type equations to obtain frequency spectra for composite shells with simply supported ends. A comparison with isotropic steel shells was also made as a design guideline. Through considering shells made of bimodulus composite materials, Bert and Kuman [10] developed a theory taking into account shear deformation and rotatory inertia based on thin shell approximations. A closed form solution was given for a freely supported shell. Comparisons between theories, including those of Donnell, Love and Sanders, were presented. By incorporating a non-linear displacement equation which includes first-order and sinusoidal terms with respect to the radial co-ordinate of the shell, Lee [11] obtained natural frequencies of simply supported cylindrical orthotropic shells based on Donnell’s approximations. Although the Rayleigh-Ritz method was used, the complexity arising from the displacement functions nevertheless limited the calculations to only the fundamental frequencies. Sharma and Darvizeh [12–14] in a series of papers developed expressions for natural frequencies of specially orthotropic multi-layered thin shells. The boundary conditions considered included simply supported, clamped-free, clamped-supported and clamped-clamped. Combinations of single term characteristics beam functions were used as admissible functions in the Rayleigh-Ritz variational approach and envelopes of natural frequencies were plotted against some geometrical parameters. In these investigations, Love–Timoshenko strain-displacement relations were adopted. These equations had been established as the basic equations for dynamic shell analysis. The Rayleigh-Ritz method has been employed extensively to investigate the free vibration behaviour of cylindrical shells. In the formulation, the strain and kinetic energies of the elastic shell at resonance were expressed in terms of its mid-surface deflection and the Rayleigh quotient was formed. The deflection shapes appearing in the quotient were subsequently approximated by series of co-ordinate functions. Henceforth, the infinite degrees-of-freedom of the cylindrical shell were reduced to some finite generalized displacements. Stationarity of the Rayleigh quotient leads to an eigenvalue problem from which the natural frequencies and the co-ordinate displacements can be determined. The boundary conditions of the cylindrical shell are automatically satisfied when a sufficient number of terms is incorporated in the series. Recently Liew and Lim [15, 16] introduced as shape functions set of kinematially oriented admissible functions. Since these functions contain as factors the geometric equations of structure boundaries raised to an appropriate power according to different boundary conditions, they have been shown to be accurate, versatile and efficient. By employing the concept of artificial springs and the joining together of components, the Rayleigh-Ritz technique can be extended to model more complicated structures where simple separate functions can be used to represent the displacement of individual components, see, e.g., the paper by Yuan and Dickinson [17]. This paper describes the procedures for modelling the free vibration responses of fibre-reinforced composite cylindrical shells in a free-free configuration. Axial displacement of any point on the mid-surface of the shell is approximated by longitudinal free vibration
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shapes of unconstrained beams, while circumferential and radial displacement components both assume the modal forms of a beam under flexural vibration. The model incorporates both bending and membrane characteristics which are found to be important for obtaining accurate results. A study of energy distribution among the natural modes is also undertaken. This helps to shed some light on the possibility of using vibration testing as a non-destructive means of characterizing elastic properties of composite shells. 2. RAYLEIGH-RITZ MODELLING AND SHELL GEOMETRY
The shell element geometry is illustrated in Figure 1; x, u and z are co-ordinates along the axial, circumferential and radially inward directions respectively. Correspondingly, u, v and w are the displacement components of any point on the mid-plane of the shell in these direction (a list of symbols is given in Appendix B). The problem is formulated on the basis of thin shell (r q 20h) theory with the usual assumptions, based on the Love–Timoshenko theory. At resonance, the mid-surface displacements u, v and w with harmonic dependence on time can be expressed as u = U(x, u) ejvt,
v = V(x, u) ejvt,
w = W(x, u) ejvt.
The actual displacements U, V and W are approximated by sums of co-ordinate functions Dm (x) cos (nu), Cm (x) sin (nu) and Fm (x) cos (nu) with undetermined coefficients Im , Jm and Km . Thus p
p
U(x, u) = s [Im Dm (x)] cos (nu),
V(x, u) = s [Jm Cm (x)] sin (nu),
m=1
m=1 p
W(x, u) = s [Km Fm (x)] cos (nu).
(1)
m=1
Equations (1) represent a particular modal form of the vibrating cylinder with n circumferential whole waves. For free vibration analysis of the shell, the Rayleigh-Ritz approach seeks the stationary values of v in the Rayleigh quotient
$ >g g L
2p
v 2 = (2/rhr) Vmax
0
%
(U 2 + V 2 + W 2 ) du dx .
0
Figure 1. Geometry of shell element.
(2)
. . .
120
Vmax is the maximum strain energy of the shell, given by 1 2
Vmax =
gg$ 0 1 0 1 0 10 1 0 1 0 1 0 1 0 10 0 1% L
1U 1x
2p
A11
0
0
+ 2A12
1U 1x
+ D11
1 2W 1x 2
+ D66
1 1V W − r 1u r
2
+ A22
2
1 1V W 1V 1 1U − + A66 + r 1u r 1x r 1u
2
+ D22
4 1 2W 1V + r 2 1x1u 1x
1 1 2W 1V + 1u r 4 1u 2
2
+ 2D12
2
1 1 2W r 2 1x 2
1 2W 1V + 1u 1u 2
1
2
r du dx.
(3a)
The maximum and reduced kinetic energies are expressed as
gg L
Tmax = 12 rrhv 2
0
0
[U 2 + V 2 + W 2 ] du dx,
(3b)
0
gg L
T* = 12 rrh
2p
2p
[U 2 + V 2 + W 2 ] du dx.
(3c)
0
Following the Rayleight-Ritz procedure one then reduces the continuum structure of an infinite number of degrees-of-freedom to one having a finite number of degrees-of-freedom. In the usual case, each of the assumed functions Dm (x) cos (nu), Cm (x) sin (nu) and Fm (x) cos (nu) describing the mid-plane displacements should satisfy the geometric boundary conditions. Since the shell is free at both ends, no geometric restrictions are imposed and accurate modelling is secured by using more terms in the series of equations (1). Stationarity of expresson (2) requires that 1(Tmax − Vmax )/1Ii = 0,
1(Tmax − Vmax )/1Ji = 0,
1(Tmax − Vmax )/1Ki = 0,
for i = 1 . . . p. This results in a set of 3 × p simultaneous linear algebraic equations,
&
[A] [B] [C] [B]T [D] [E] [C]T [E]T [F]
'8 9
&
'8 9
{I} [G2] [0] [0] {J} = rrhLv 2 [0] [H1] [0] {K} [0] [0] [R1]
{I} {J} . {K}
The sub-matrices in equation (4) are of dimensions p × p. Their elements are Ai,m = G1i,m (r/L)A11 + G2i,m (n 2L/r)A66,
Bi,m = G3i,m (n)A12 − G4i,m (n)A66 ,
Ci,m = −G5i,m A12 , Di,m = H1i,m (n 2L/r)A22 + H2i,m (r/L)A66 + H1i,m (n 2L/r 3 )D22 + H2i,m (4/rL)D66 , Ei,m = −H3i,m (nL/r)A22 − H3i,m (n 3L/r 3 )D22 + H4i,m (n/rL)D12 − H5i,m (4n/rL)D66 , Fi,m = R1i,m (L/r)A22 + R2i,m (r/L 3 )D11 + R1i,m (n 4L/r 3 )D22 − [R3i,m + R3m,i ](n 2/rL)D12 + R4i,m (4n 2/rL)D66 .
(4)
121
For a prescribed value of n, the indices i and m take on the values of 1 to p. The symbols G1 to G5, H1 to H5 and R1 to R4 are dimensionless integrals, defined in Appendix A. The elements of {I}, {J} and {K} in equation (4) are unknowns to be determined. When each of the functions Dm (x), Cm (x) and Fm (x) in equations (1) forms an orthogonal set in the interval 0 to L, that is,
g
L
0
6
L, m = i Dm (x)Di (x) dx = , 0, m $ i
g
L
0
7g
L
0
6
L, m = i Cm (x)Ci (x) dx = , 0, m $ i
6
L, m = i Fm (x)Fi (x) dx = , 0, m $ i
7
7
[G2], [H1] and [R1] become identity matrices and equation (4) is a symmetric eigenvalue problem represented by
&
[A] [B] [C] [B]T [D] [E] [C]T [E]T [F]
'8 9 8 9
{I} {I} {J} = l {J} . {K} {K}
(5)
The eigenvalue of this problem is l = rrhLv 2.
3. RESULTS AND DISCUSSIONS
Six laminated circular cylindrical shells were studied, all having orthotropic and mid-plane symmetric properties. Shells 1 and 2 were fabricated with E-glass/epoxy and consist of six and four layers of woven fabrics respectively. Shells 3 and 4 are graphite/epoxy shells each consisting of four plys of woven fabric laminae having the same geometric and elastic properties but different orientations with respect to the shell co-cordinates. Shells 5 and 6 are Shells 1 and 2 with their thicknesses halved. The physical properties of the specimen shells are given in Table 1. Modal testings were performed for Shells 1 and 2. The former has 72 grid points while the latter has 48. Each shell under test was suspended vertically by soft cord to simulate the free-free boundary conditions. An accelerometer was affixed to one of the grid points on the surface of the shell. Impact excitation was introduced to every grid point with an impact hammer. Frequency response functions were obtained and modal analysis performed. Resonance frequencies of the shells were determined and the corresponding modes shapes extracted. A schematic diagram of the experimental set-up is shown in Figure 2. T 1 Physical properties of orthotropic shells Shell
Material
Density (kg/m3 )
Length (mm)
Radius (mm)
Thickness (mm)
Angle of orientation (°)
1 2 3 4 5 6
E-glass/epoxy E-glass/epoxy graphite/epoxy graphite/epoxy E-glass/epoxy E-glass/epoxy
1662·03 1686·48 1455·87 1455·87 1662·03 1686·48
212 321 300 300 212 321
83·5 83·4 80·0 80·0 83·5 83·4
1·371 0·930 0·800 0·800 0·686 0·465
[0/90]6 [0/90]4 [0/90/ + 45/−45]]s [+45/−45/0/90]s [0/90]3 [0/90]2
. . .
122
There is a variety of choices for the functions Dm (x), Cm (x), Fm (x) in equations (1) which describe the variations of the mid-plane displacements along the x-direction. These functions were chosen as Dm (x) =
6
1,
m=1
z2 cos [(m − 1)px/L],
mq1
8
7
,
9
1, m=1 m=2 , Cm (x) = Fm (x) = z3(1 − 2x/L), transverse beam function m q 2
(6a, b)
where Dm (x)(m q 1) and Cm (x) = Fm (x)(m q 2) are respectively the longitudinal and transverse free vibration shapes of a free-free beam [18]. Each set of functions in equation (6) is orthogonal and hence equation (5) could be applied directly to solve for the modal responses of the shells. The Rayleigh-Ritz model was then used to predict the behaviour of the shells, for the dimensions shown in Table 1. The elastic properties of the material for these two shells were determined experimentally. The longitudinal Young’s modulus E1 and the major Poisson’s ratio n12 were obtained by measuring the axial and circumferential strains of an axially loaded cylinder with electrical resistance foil strain gauges. The shell was then evacuated and loaded by external pressure, and hence the circumferential Young’s modulus E2 could be determined. Two longitudinal strips were cut from each shell and are subjected to the Iosipescu test [19]. Strain gauge readings on a 45°-axis of the strip thus led to the value for the in-plane shear modulus G12 . For the purpose of investigating the dynamic behaviour and the effect of stacking sequence of the graphite/epoxy shells, their elastic properties are taken from specimen strips subjected to simple static tests. Table 2 summarizes the material properties of individual lamina in each shell. The values for n21 in the last column were deduced from the reciprocal relations E1 n21 = E2 n12 . By using these properties, the membrane and bending stiffnesses of the shells were calculated and these are given in Table 3. Consideration can now be given to the first thirteen natural modes of Shells 1–6. Tables 4 and 5 show comparisons between the predicted and measured frequencies for Shells 1 and 2. Good agreement is observed between the theoretical and experimental values. The discrepancies are mainly due to imperfections of the specimens and experimental errors incurred during static and modal testings. A study of the convergence of the predicted frequencies for all the shells can be achieved by using up to seven terms
Accelerometer B&K 4374
Shell
PCB hammer 086C80
HP dynamic signal analyzer 3562A
HP workstation
Ch1 Ch 2
Charge amp. B&K 2635
PCB sensor power unit 480E09
Operating program LMS CADA-X Modal analysis software Ver2.8
Figure 2. Experimental set-up for modal testing of free-free shell.
123
T 2 Elastic properties of individual lamina in each shell Shell no.
Material
E1 (GPa)
E2 (GPa)
G12 (GPa)
n12
n21
1 2 3 4 5 6
E-glass/epoxy E-glass/epoxy graphite/epoxy graphite/epoxy E-glass/epoxy E-glass/epoxy
18·274 18·166 44·188 44·188 18·274 18·166
18·157 18·779 44·188 44·188 18·157 18·779
2·930 2·955 5·739 5·739 2·930 2·955
0·210 0·220 0·128 0·128 0·210 0·220
0·209 0·227 0·128 0·128 0·209 0·227
T 3 Membrane and bending stiffnesses of orthotropic shells Stiffness (N/m) ZXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXV Shell 1 Shell 2 Shell 3 Shell 4 Shell 5 Shell 6 A11 [N/m] A22 [N/m] A12 [N/m] A66 [N/m] D11 [N/m] D22 [N/m] D12 [N/m] D66 [N/m]
26·204E6 26·079E6 5·477E6 4·017E6 4·104 4·085 0·8578 0·6292
17·782E6 18·348E6 4·037E6 2·748E6 1·282 1·322 0·2909 0·1981
30·396E6 30·396E6 10·133E6 10·131E6 1·843 1·843 0·3188 0·3187
30·396E6 30·396E6 10·133E6 10·131E6 1·400 1·400 0·7621 0·7620
13·102E6 13·040E6 2·739E6 2·009E6 0·5130 0·5106 0·1072 0·0787
8·891E6 9·174E6 2·019E6 1·374E6 0·1603 0·1653 0·0364 0·0248
T 4 Comparison of Rayleigh-Ritz predictions and experimental results for Shell 1 Rayleigh-Ritz frequency (Hz) Mode indices ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV Exp. (m, n) p=2 p=3 p=4 p=5 p=6 p=7 freq. (Hz) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 1,6 2,6 3,5 3,4 3,6
81·94 88·91 231·75 243·59 444·36 458·47 718·63 733·90 1054·21 1070·14 – – –
81·94 88·91 231·74 243·59 444·31 458·47 718·47 733·90 1053·86 1070·14 1079·47 1161·01 1254·07
81·94 88·91 231·74 243·59 444·31 458·46 718·46 733·90 1053·86 1070·14 1079·14 1160·89 1253·70
81·93 88·91 231·73 243·59 444·27 458·46 718·38 733·89 1053·70 1070·14 1070·72 1138·95 1250·23
81·93 88·91 231·73 243·57 444·27 458·43 718·38 733·82 1053·70 1070·00 1070·68 1138·84 1250·22
81·93 88·91 231·71 243·57 444·23 458·43 718·30 733·82 1053·55 1070·00 1069·26 1134·77 1249·82
79·57 88·63 225·64 241·78 433·37 455·03 710·08 739·02 1036·43 1068·02 – 1150·22 1249·99
in the admissible functions in equations (1) and the results are displayed in Tables 4–9. The index pair (m, n) indicates that the mode has m-1 node lines and n sinusoidal wave patterns around the circumference. Review of the convergence rate for the natural frequencies indicates that a series with five terms for the admissible function provides sufficient accuracy. Thus subsequent analytical results are based on five-term series for the displacement functions.
. . .
124
T 5 Comparison of Rayleigh-Ritz predictions and experimental results for Shell 2 Rayleigh-Ritz frequency (Hz) Mode indices ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV Exp. (m, n) p=2 p=3 p=4 p=5 p=6 p=7 freq. (Hz) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 3,4 3,5 1,6 2,6 3,6
56·43 58·58 159·62 163·12 306·05 310·18 494·95 499·41 – – 726·08 730·72 –
56·43 58·58 159·61 163·12 306·03 310·18 494·90 499·41 605·13 621·29 725·96 730·72 788·91
56·43 58·58 159·61 163·12 306·03 310·18 494·90 499·40 604·44 620·58 725·96 730·72 788·39
56·43 58·58 159·61 163·12 306·03 310·18 494·88 499·40 599·87 619·25 725·92 730·72 787·90
56·43 58·58 159·61 163·12 306·03 310·18 494·88 499·39 599·85 619·24 725·92 730·69 787·90
56·43 58·58 159·61 163·12 306·02 310·18 494·87 499·39 598·79 618·97 725·89 730·69 787·90
55·99 59·11 158·21 164·20 301·92 310·69 496·38 503·87 502·09 629·22 722·62 733·66 790·83
The vibratory motion of all the modes tabulated is mainly radial. Resonance frequencies associated with the longitudinal or torsional modes are too high to be considered here. Such modes involve predominantly extensional vibration of which stretching energy as compared to reduced kinetic energy is large. The analytical frequencies of Shells 1 and 2 predicted by the Rayleigh-Ritz model satisfactorily matched those measured by modal testing. It is noticed from Tables 4–9 that higher values of the circumferential wave number do not necessarily imply higher frequencies for cylindrical shells. This phenomenon can be explained through energy consideration. To accomplish this, the terms stretching energy factor hs and bending energy factor hb , are introduced. They are defined as respectively the ratios of the strain energies associated with the stretching and bending stiffnesses to the reduced kinetic energy given by equation (3c). It was found in Shell 1 that as the n-index increases through the modes from (3, 4) to (3, 5) there is an increase in hb .
T 6 Rayleigh-Ritz predictions for Shell 3 Mode indices (m, n) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 3,5 3,4 1,6 2,6 3,6
Rayleigh-Ritz frequency (Hz) ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV p=2 p=3 p=4 p=5 p=6 p=7 83·94 88·06 237·41 243·92 455·21 462·81 736·18 744·33 – – 1079·96 1088·43 –
83·94 88·06 237·41 243·92 455·21 462·81 736·15 744·33 970·09 1006·45 1079·88 1088·43 1194·45
83·94 88·06 237·41 243·92 455·21 462·81 736·15 744·32 966·22 1002·45 1079·87 1088·41 1191·77
83·94 88·06 237·41 243·92 455·20 462·81 736·14 744·32 965·32 999·86 1079·85 1088·41 1191·26
83·94 88·06 237·41 243·92 455·20 462·81 736·14 744·32 965·32 999·84 1079·85 1088·41 1191·26
83·94 88·06 237·41 243·92 455·20 462·81 736·13 744·32 965·11 999·09 1079·83 1088·41 1191·19
125
T 7 Rayleigh-Ritz predictions for Shell 4 Mode indices (m, n) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 1,6 3,5 2,6 3,4 3,6
Rayleigh-Ritz frequency (Hz) ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV p=2 p=3 p=4 p=5 p=6 p=7 73·16 86·70 206·92 226·21 396·75 418·50 641·63 664·61 941·26 – 964·93 – –
73·16 86·80 206·91 226·21 396·68 418·50 641·41 664·61 940·83 945·32 964·93 1006·11 1127·19
73·16 86·70 206·91 226·20 396·68 418·49 641·41 664·59 940·83 941·35 964·87 1002·12 1124·36
73·16 86·70 206·90 226·20 396·66 418·49 641·34 664·59 940·67 939·89 964·87 999·19 1122·97
73·16 86·70 206·90 226·20 396·66 418·48 641·34 664·56 940·67 939·89 964·82 999·17 1122·97
73·15 86·70 206·89 226·20 396·63 418·48 641·28 664·56 940·56 939·64 964·82 988·41 1122·85
Nevertheless, at the same time, hs is reduced by a greater amount. This results is a net decrease in the Rayleigh quotient. Further increasing in the n-index results in a rise in natural frequency for mode (3, 6) becasuse the increase in hb is greater than the reduction of hs . Similar behaviour has been reported in the free vibration of freely supported isotropic shells by Arnold and Warburton [20]. This explains the occasional choice of inextensional shell models to describe modes dominated by bending, which normally have a high circumferential wave number (see, e.g., the paper by Sharma [21]). As obtained by using the Rayleigh-Ritz model, the computed values of the natural frequencies corresponding to m = 1 to 5 for all six shells are plotted against the n-index in Figures 3–8. In Figures 3 and 4, Rayleigh modes (m = 1) and Love modes (m = 2) with the characteristic of an inextensional mid-surface are identified. Their resonance frequencies always increase with the n-index owing to the increasing bending energy factor. For other
T 8 Rayleigh-Ritz predictions for Shell 5 Mode indices (m, n) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 1,6 2,6 3,6 3,5 3,4
Rayleigh-Ritz frequency (Hz) ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV p=2 p=3 p=4 p=5 p=6 p=7 41·00 44·49 115·96 121·89 222·35 229·41 359·59 367·23 527·51 535·48 – – –
41·00 44·49 115·96 121·89 222·34 229·41 359·56 367·23 527·44 535·48 778·15 828·37 1068·53
41·00 44·49 115·96 121·89 222·34 229·41 359·56 367·23 527·44 535·48 777·59 827·97 1068·41
41·00 44·49 115·96 121·89 222·34 229·41 359·55 367·23 527·42 535·48 773·19 817·47 1044·26
41·00 44·49 115·96 121·89 222·34 229·40 359·55 367·22 527·42 535·49 773·18 817·42 1044·13
41·00 44·49 115·96 121·34 222·33 229·40 359·54 367·22 527·40 535·46 772·30 815·22 1039·33
. . .
126
T 9 Rayleigh-Ritz predictions for Shell 6 Mode indices (m, n) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 1,6 2,6 3,5 3,6 3,4
Rayleigh-Ritz frequency (Hz) ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV p=2 p=3 p=4 p=5 p=6 p=7 28·22 29·29 79·81 81·56 153·03 155·09 247·48 249·71 363·04 365·37 – – –
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·71 363·02 365·37 429·99 448·37 534·24
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·71 363·02 365·37 428·99 447·48 533·47
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·71 363·02 365·37 427·25 446·90 528·37
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·70 363·02 365·36 427·25 446·90 528·34
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·70 363·01 365·36 426·80 446·76 527·08
modes (m q 2), the natural frequencies are highest when n is unity. As the number of circumferential nodes is increased, the frequency values decrease until a minimum is reached. Before this point, vibration modes other than the Rayleigh and Love modes are in general stretching dominated. As the n-index increases, the stretching energy factor of a given mode which tends to be inextensional decreases. Further increase in the circumferential wave number causes a rise in the natural frequencies due to the increasing bending energy factor. The watersheds in the frequency variation are modes (3, 5), (4, 6) and (5, 6) in Shell 1 and modes (3, 4), (4, 5) and (5, 6) in Shell 2. Similar behaviour is observed in Figures 5 and 6 for the graphite/epoxy shells. Furthermore, Figures 7 and 8 show that reduction of shell thickness did not alter this trend. It is also worth noting that the energy distributions of Shells 3 and 4 are different and are significantly affected by the lamination configurations.
Figure 3. Theoretical frequency variations for shell 1, m values: —q—, 1; +2; *, 3; —Q—, 4; ×, 5.
127
Figure 4. Theoretical frequency variations for Shell 2. Key as Figure 3.
Scrutiny of the energy distributions of these modes will cast some light on the behaviour of the vibrating shells. The strain energy of the shell is composed of eight separate terms associated with the four extensional and four flexural stiffnesses respectively. Their contributions are expressed as percentages of the total strain energy and are shown in Figures 9–12. It is evident from these figures that the Rayleigh and Love modes, i.e. modes (1, 2) and (2, 2) consist of little membrane energy and hence negligible energy contribution from the stretching stiffnesses. These inextensional modes have frequencies close to each other as shown in Figures 3–8; the former undergoes predominantly bending motion while the latter has bending and a certain degree of twisting. This conclusion is drawn in the view of the bending energy related to D22 in these modes with the additional torsion energy associated with D66 in the latter.
Figure 5. Theoretical frequency variations for Shell 3. Key as Figure 3.
128
. . .
Figure 6. Theoretical frequency variations for Shell 4. Key as Figure 3.
For modes with m q 2, the energies developed from stiffnesses A11 and A66 cover a major portion of the total stretching energy while that from stiffness D22 dominates the total bending energy. The existence of such stretching energy accounts for the inaccuracy of inextensional shell models for the prediction of the modal frequencies of these modes. By increasing the m-index from three to higher values, the energy contribution associated with A66 increases progressively whereas that of A11 diminishes. This is a clear indication that the shell is subjected to significant in-plane shear deformation for modes with a large m-index. Energy distributions in Figures 9–12 also reveal that strain energy associated with the membrane stiffness A12 can be negative for some of the extensional modes. It is seen that when the axial and circumferential strains ox0 and ou0 have opposite signs, the component in the strain energy which involves integration of the product of these strains over the entire shell would then be negative.
Figure 7. Theoretical frequency variations for Shell 5. Key as Figure 3.
129
5
Frequency (kHz)
4
3
* 2
* 1
0
* 1
2
3
*
*
*
*
4 5 6 7 Circumferential wave no. (n)
*
*
8
9
*
10
Figure 8. Theoretical frequency variations for Shell 6. Key as Figure 3.
As seen from Table 3, Shell 3 is much stiffer in terms of bending whereas Shell 4 is stiffer in twisting. This is highlighted by a large energy proportion associated with the bending stiffness D22 for Shell 3 and a more significant twisting energy associated with D66 in Shell 4. This points to the possibility of achieving different characteristics of the shell through manipulation of the lamination sequence. Such an optimization procedure have been attempted by Nshanian and Pappes [22]. A study of the effect of thickness change on the energy distribution has been made with Shells 5 and 6. An immediate observation is the general reduction of the natural frequency values when comparing Figure 7 to Figure 3 and Figure 8 to Figure 4. More modes come under the category of stretching dominated because the minima of the frequency curves have shifted to the right in Figures 7 and 8 due to the decrease
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A22
A12
A66
D 11
Figure 9. Energy contribution of Shell 1. Modes:q, (1, 2);
D22
D12
, (2, 2);
D66 , (3, 5);
, (4, 6); Q, (5, 6).
. . .
130
100
Energy contribution (%)
80
60
40
20
0
–20
A 11
A22
A12
A66
D11
D 22
D 12
Figure 10. Energy contributions of Shell 2. Modes: q, (1, 2); .
D 66
, (2, 2);
, (3, 4);
, (4, 5); Q, (5, 6).
in thickness. While the bending dominated modes show substantial decreases in natural frequencies, the stretching dominated modes have virtually no changes in these values. Numerical calculations show that the Rayleigh quotient of the bending dominated modes is approximately proportional to h 2 while that associated with stretching dominated modes is more or less independent of the shell thickness. The natural modes corresponding to the minima of the frequency curves in Figure 7 are modes (3, 6), (4, 7) and (5, 8) and that of Figure 8 are modes (3, 5), (4, 7) and (5, 8). Figures 13 and 14 have been drawn to illustrate the energy contributions of the various stiffnesses for these modes.
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A 22
A 12
A 66
D11
Figure 11. Energy contributions of Shell 3. Modes: q, (1, 2);
D22
D12
, (2, 2);
D66 , (3, 5);
, (4, 6); Q, (5, 7).
131
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A22
A12
A66
D 11
D22
D12
D66
Figure 12. Energy contributions of Shell 4. Key as Figure 11.
As pointed out earlier, the stretching energy of the vibration modes with m q 2 is mainly due to A11 and A66 . This kind of energy is a large proportion of the total strain energy for modes having a smaller n-index. This phenomenon is more pronounced when the shell thickness is reduced. An important application of the foregoing observations is in the field of material characterization of structures through the use of modal frequencies. To characterize composite shells successfully, it is desirable to have their modal frequencies sensitive to each of the elastic properties. Hence, the Rayleigh modes will always be useful for the determination of D22 . For the characterization of stiffnesses A11 and A66 , it is preferable
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A22
A12
A66
D11
D 22
D 12
D 66
Figure 13. Energy contributions of Shell 5. Key as Figure 11.
. . .
132
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A22
A12
A66
D 11
D 22
Figure 14. Energy contributions of Shell 6. Modes: q, (1, 2);
D 12 , (2, 2);
D 66 , (3, 5);
, (4, 7); Q, (5, 8).
to have a thin specimen and to make use of its extensional modes associated with a small n-index. For the shells in this study, laminate stiffnesses Q 11 , Q 22 and Q 66 can be obtained once the laminate stiffnesses A11 , D22 and A66 have been characterized. The laminate stiffnesses will then yield the values of the material properties E1 , E2 and G12 of the shells. With reference to Figures 9–14, the strain energy associated with stiffnesses A12 and D12 constitutes only a minor portion of the total strain energy. As a result, the values of these stiffnesses have a negligible effect on the total strain energy and hence on the frequencies of the natural modes. Since both these stiffnesses are primarily related to the Poisson’s ratios, the modal frequencies of the shells are thus insensitive to these elastic constants.
4. CONCLUSIONS
A Rayleigh-Ritz vibration model has been developed to evaluate the dynamic responses of orthotropic shells under free vibration. The shells have symmetric laminations with completely free ends. The natural frequencies and mode shapes of the shells are predicted and the frequencies match well those extracted from modal tests, with typical errors of less than 3%. For the inextensional Rayleigh and Love modes, natural frequencies are close to each other and increase with the n-index. The major part of the strain energy comes from bending and is related to D22 . A considerable amount of twisting energy associated with D66 is also observed for the Love modes. For the extensional modes considered, with the m-index taking a value between 3 and 5, the resonance frequencies change through a minimum as the circumferential wave number n increases. The stretching energy of these modes is mainly due to membrane stiffnesses A11 and A66 while the bending energy is attributed to D22 . For modes with a
133
large m-index, the shell would be under significant shearing indicated by a progressively larger energy term associated with A66 . On the other hand, for modes with a small n-index, stretching and shearing deformations become dominant as the bending energy term due to D22 subsides. This effect is more apparent for thin shells. When natural frequencies of an orthotropic shell are used in the application of material characterization, thin cylinders are the ideal specimens for an accurate determination of the elastic constants E1 , E2 and G12 .
REFERENCES 1. V. V. N 1959 The Theory of Thin Shells. Groningen, The Netherlands: P. Noordhoff Ltd. 2. S. A. A 1961 Theory of Anisotropic Shells. Moscow: State Publishing House for Physical and Mathematical Literautre. 3. S. B. D, K. S. P and R. L. T 1962 Journal of Aerospace Sciences, 969–975. On the theory of laminated anisotropic shells and plates. 4. V. I. W 1964 Experimental Mechanics (July), 200–205. Free vibrations of multilayered cylindrical shells. 5. S. B. D 1968 Journal of Acoustical Society of America 44, 1628–1635. Free vibrations of laminated orthotropic cylindrical shells. 6. C. W. B, J. L. B and D. M. E 1969 Journal of Composite Materials 3, 480–499. Free vibration of multilayer anisotropic cylindrical shells. 7. Y. S and R. L 1971 Journal of Sound and Vibration 15, 235–256. On vibrations of heterogeneous orthotropic cylindrical shells. 8. B. K. R and Y. C. D 1973 Journal of Sound and Vibration 28, 737–757. Vibration of layered shells. 9. K. N. S and A. V. K M 1976 Journal of Structural Mechanics 4, 379–393. Vibrations of multifiber composite shells—some numerical results. 10. C. W. B and M. K 1982 Journal of Sound and Vibration 81, 107–121. Vibration of cylindrical shells of bimodulus composite materials. 11. D. G. L 1988 Journal of Composite Materials 22, 1102–1115. Calculation of natural frequencies of vibration of thin orthotropic composite shells by energy method. 12. C. B. S and M. D 1984 Proceedings of the 2nd Internatinal Conference on Recent Advances in Structural Dynamics, University of Southampton, England, 9–13 April 1, 15–24. Calculation of natural frequencies of specially orthotropic multilayered thin circular cylinder. 13. M. D and C. B. S 1984 Thin-walled Structures 2, 207–217. Natural frequencies of laminated orthotropic thin circular cylinders. 14. C. B. S and M. D 1987 Composite Structures 7, 123–138. Free vibration of specially orthotropic, multilayered, thin cylindrical shells with various end conditions. 15. K. M. L and C. W. L 1994 AIAA Journal 32, 387–396. Vibratory characteristics of cantilevered rectangular shallow shells of variable thickness. 16. C. W. L and K. M. L 1994 Journal of Sound and Vibration 173, 343–375. A pb-2 Ritz formulation for flexural vibration of shallow cylindrical shells of rectangular planform. 17. J. Y and S. M. D 1994 Journal of Sound and Vibration 175, 241–263. The free vibration of circularly cylindrical shell and plate systems. 18. R. D. B 1979 Formulas for Natural Frequency and Mode Shape. New York: Van Nostrand Reinhold Ltd. 19. ASTM Standards D5379/D 5379M—93 Standard test method for shear properties of composite materials by the V-notched beam method, 229–241. 20. R. N. A and G. B. W 1949 Proceedings of the Royal Society A, 197, 238–256. Flexural vibrations of the walls of thin cylindrical shells having freely supported ends. 21. C. B. S 1974 Journal of Sound and Vibration 35, 55–76. Calculation of natural frequencies of fixed-free circular cylindrical shells. 22. Y. S. N and M. P 1983 AIAA Journal 21, 430–437. Optimal laminated composite shells for buckling and vibration.
. . .
134
APPENDIX A
The dimensionless integrals defined for the elements of equation (4) are as follows:
g$ L
G1i,m = L
0
% g$ g$ g$
1Di (x) 1Dm (x) dx, 1x 1x
0
L
Di (x)
G4i,m =
0
0
H1i,m =
1 L
g
L
[Ci (x)Cm (x)] dx,
0
g$ g g$
Ci (x)
0
R1i,m =
1 L
g
[Ci (x)Fm (x)] dx,
0
%
1Fm2 (x) dx, 1x 2
0
0
R2i,m = L 3
%
1Fm2 (x) dx, 1x 2
L
R4i,m = L
% %
0
1Fi (x) 1Fm (x) dx. 1x 1x
APPENDIX B: LIST OF SYMBOLS
Im , Jm , Km L, r, h Aij , Dij E1 , E2 G12 Q ij x, u, z u, v, w U(x, u) V(x, u) W(x, u) Vmax , Tmax T* A, B, C, D, E, F a l
1
1Ci (x) 1Fm (x) dx 1x 1x
1Fi2 (x) 1Fm2 (x) dx, 1x 2 1x 2
0
Fi (x)
0
g$ g$ g$ L
H5i,m = L
L
[Fi (x)Fm (x)] dx,
L
%
1Ci (x) 1Cm (x) dx, 1x 1x
L
L
1 L
R3i,m = L
H2i,m = L
0
L
H4i,m = L
% % % g$
1Cm (x) dx, 1x
L
H3i,m =
[Di (x)Dm (x)] dx,
0
1Di (x) Fm (x) dx, 1x
L
G5i,m =
L
1Di (x) Cm (x) dx, 1x
L
G3i,m =
g
1 L
G2i,m =
deflection coefficients length, radius and thickness of shell stretching and bending stiffnesses of laminate Young’s moduli along the perpendicular to fibers in-plane shear modulus transformed reduced lamina stiffnesses cylindrical coordinates of shell displacement components at mid-plane maximum of longitudinal mid-surface deflection maximum of circumferential mid-surface deflection maximum of radial mid-surface deflection maximum strain and kinetic energies reduced kinetic energy elements of system matrix lamination angle of lamina with x-axis theoretical eigenvalue
r n12 , n21 Dm (x) Cm (x), Fm (x) v hs hb
density of shell material major and minor Poisson’s ratios longitudinal free vibration shape of beam free-free characteristic beam functions angular frequency of shell stretching energy factor bending energy factor
135
VIBRATION ANALYSIS OF ORTHOTROPIC THIN CYLINDRICAL SHELLS WITH FREE ENDS BY THE RAYLEIGH-RITZ METHOD K. H. I, W. K. C, P. C. T T. C. L Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong (Received 4 May 1995, and in final form 29 December 1995) An analytical model is developed to predict the modal characteristics of thin-walled circular cylindrical laminated shells with free ends. The shell is orthotropic and has mid-plane symmetry. By using Love’s first-approximation shell theory, a strain energy functional containing both bending and stretching effects is formulated. The shell vibration mode shapes are then modelled by utilizing characteristic beam functions in the Rayleigh-Ritz variational procedure and the accuracy of the model is verified by test data. With the developed model, inextensional Rayleigh and Love modes can be identified having frequencies close to each other. The contributions to the strain energy due to various elastic properties are also investigated. Results show that the circumferential modulus provides a major portion of the flexural energy of the vibrating structure while the longitudinal and in-plane shear moduli contribute mostly to the stretching energy. It is also observed that reducing the shell thickness would result in a substantial increase in the ratio of the energies associated with the longitudinal and shear moduli, respectively. By rearranging the lamination stacking sequence, shells can be made to be more resilient to bending or twisting with only minor alterations in natural frequencies. 7 1996 Academic Press Limited
1. INTRODUCTION
The thin-walled circular cylindrical laminated shell has long been an important structural component due to its high stiffness-to-weight and strength-to-weight ratios. Extensive studies on the behavior of thin laminated anisotropic shells began three decades ago when Donnell-type displacement equations were adopted in formulating the equations of equilibrium and compatibility [1–3]. As to the vibration of composite thin shells, Weingarten [4] was the first to carry out theoretical and experimental investigations on the free vibration of multilayered cylindrical shells with various boundary conditions. The shell was assumed to consist of isotropic layers with equal Poisson’s ratios. Donnell-type equations were used and various modal properties of the shell including damping were considered. This was followed by the work of Dong [5] who studied a shell composed of an arbitrary number of layers with different thickness and orthotropic elastic properties. For boundary conditions other than freely supported edges, the resonance frequencies would be extracted with an iterative scheme. Bert, Baker and Egle [6] investigated theoretically anisotropic shells with freely supported ends. Displacement equations similar to Reissner’s were used. The significance of bending-stretching coupling was illustrated with a two-layered, cross-ply cylinder. Stavsky and Loewy [7] evaluated the natural frequencies of heterogeneous orthotropic 117 0022–460X/96/310117 + 19 $18.00/0
7 1996 Academic Press Limited
118
. . .
shells. Both the Donnell and Love type equations were employed and the ends of the cylinder were simply supported. Comparisons showed that inaccuracies in natural frequency calculations by using the former theory can be substantial and depend strongly on shell heterogeneity as the length-to-radius ratio becomes large. Rath and Das [8] derived equations of motion for shells having symmetrical general orthotropic layers and non-symmetrical cross-ply layers. Boundary conditions were restricted to simple supports. Effect of shear deformation were included and contrast was made to natural frequencies calculated without shear. Shivakumar and Krishna [9] used Love type equations to obtain frequency spectra for composite shells with simply supported ends. A comparison with isotropic steel shells was also made as a design guideline. Through considering shells made of bimodulus composite materials, Bert and Kuman [10] developed a theory taking into account shear deformation and rotatory inertia based on thin shell approximations. A closed form solution was given for a freely supported shell. Comparisons between theories, including those of Donnell, Love and Sanders, were presented. By incorporating a non-linear displacement equation which includes first-order and sinusoidal terms with respect to the radial co-ordinate of the shell, Lee [11] obtained natural frequencies of simply supported cylindrical orthotropic shells based on Donnell’s approximations. Although the Rayleigh-Ritz method was used, the complexity arising from the displacement functions nevertheless limited the calculations to only the fundamental frequencies. Sharma and Darvizeh [12–14] in a series of papers developed expressions for natural frequencies of specially orthotropic multi-layered thin shells. The boundary conditions considered included simply supported, clamped-free, clamped-supported and clamped-clamped. Combinations of single term characteristics beam functions were used as admissible functions in the Rayleigh-Ritz variational approach and envelopes of natural frequencies were plotted against some geometrical parameters. In these investigations, Love–Timoshenko strain-displacement relations were adopted. These equations had been established as the basic equations for dynamic shell analysis. The Rayleigh-Ritz method has been employed extensively to investigate the free vibration behaviour of cylindrical shells. In the formulation, the strain and kinetic energies of the elastic shell at resonance were expressed in terms of its mid-surface deflection and the Rayleigh quotient was formed. The deflection shapes appearing in the quotient were subsequently approximated by series of co-ordinate functions. Henceforth, the infinite degrees-of-freedom of the cylindrical shell were reduced to some finite generalized displacements. Stationarity of the Rayleigh quotient leads to an eigenvalue problem from which the natural frequencies and the co-ordinate displacements can be determined. The boundary conditions of the cylindrical shell are automatically satisfied when a sufficient number of terms is incorporated in the series. Recently Liew and Lim [15, 16] introduced as shape functions set of kinematially oriented admissible functions. Since these functions contain as factors the geometric equations of structure boundaries raised to an appropriate power according to different boundary conditions, they have been shown to be accurate, versatile and efficient. By employing the concept of artificial springs and the joining together of components, the Rayleigh-Ritz technique can be extended to model more complicated structures where simple separate functions can be used to represent the displacement of individual components, see, e.g., the paper by Yuan and Dickinson [17]. This paper describes the procedures for modelling the free vibration responses of fibre-reinforced composite cylindrical shells in a free-free configuration. Axial displacement of any point on the mid-surface of the shell is approximated by longitudinal free vibration
119
shapes of unconstrained beams, while circumferential and radial displacement components both assume the modal forms of a beam under flexural vibration. The model incorporates both bending and membrane characteristics which are found to be important for obtaining accurate results. A study of energy distribution among the natural modes is also undertaken. This helps to shed some light on the possibility of using vibration testing as a non-destructive means of characterizing elastic properties of composite shells. 2. RAYLEIGH-RITZ MODELLING AND SHELL GEOMETRY
The shell element geometry is illustrated in Figure 1; x, u and z are co-ordinates along the axial, circumferential and radially inward directions respectively. Correspondingly, u, v and w are the displacement components of any point on the mid-plane of the shell in these direction (a list of symbols is given in Appendix B). The problem is formulated on the basis of thin shell (r q 20h) theory with the usual assumptions, based on the Love–Timoshenko theory. At resonance, the mid-surface displacements u, v and w with harmonic dependence on time can be expressed as u = U(x, u) ejvt,
v = V(x, u) ejvt,
w = W(x, u) ejvt.
The actual displacements U, V and W are approximated by sums of co-ordinate functions Dm (x) cos (nu), Cm (x) sin (nu) and Fm (x) cos (nu) with undetermined coefficients Im , Jm and Km . Thus p
p
U(x, u) = s [Im Dm (x)] cos (nu),
V(x, u) = s [Jm Cm (x)] sin (nu),
m=1
m=1 p
W(x, u) = s [Km Fm (x)] cos (nu).
(1)
m=1
Equations (1) represent a particular modal form of the vibrating cylinder with n circumferential whole waves. For free vibration analysis of the shell, the Rayleigh-Ritz approach seeks the stationary values of v in the Rayleigh quotient
$ >g g L
2p
v 2 = (2/rhr) Vmax
0
%
(U 2 + V 2 + W 2 ) du dx .
0
Figure 1. Geometry of shell element.
(2)
. . .
120
Vmax is the maximum strain energy of the shell, given by 1 2
Vmax =
gg$ 0 1 0 1 0 10 1 0 1 0 1 0 1 0 10 0 1% L
1U 1x
2p
A11
0
0
+ 2A12
1U 1x
+ D11
1 2W 1x 2
+ D66
1 1V W − r 1u r
2
+ A22
2
1 1V W 1V 1 1U − + A66 + r 1u r 1x r 1u
2
+ D22
4 1 2W 1V + r 2 1x1u 1x
1 1 2W 1V + 1u r 4 1u 2
2
+ 2D12
2
1 1 2W r 2 1x 2
1 2W 1V + 1u 1u 2
1
2
r du dx.
(3a)
The maximum and reduced kinetic energies are expressed as
gg L
Tmax = 12 rrhv 2
0
0
[U 2 + V 2 + W 2 ] du dx,
(3b)
0
gg L
T* = 12 rrh
2p
2p
[U 2 + V 2 + W 2 ] du dx.
(3c)
0
Following the Rayleight-Ritz procedure one then reduces the continuum structure of an infinite number of degrees-of-freedom to one having a finite number of degrees-of-freedom. In the usual case, each of the assumed functions Dm (x) cos (nu), Cm (x) sin (nu) and Fm (x) cos (nu) describing the mid-plane displacements should satisfy the geometric boundary conditions. Since the shell is free at both ends, no geometric restrictions are imposed and accurate modelling is secured by using more terms in the series of equations (1). Stationarity of expresson (2) requires that 1(Tmax − Vmax )/1Ii = 0,
1(Tmax − Vmax )/1Ji = 0,
1(Tmax − Vmax )/1Ki = 0,
for i = 1 . . . p. This results in a set of 3 × p simultaneous linear algebraic equations,
&
[A] [B] [C] [B]T [D] [E] [C]T [E]T [F]
'8 9
&
'8 9
{I} [G2] [0] [0] {J} = rrhLv 2 [0] [H1] [0] {K} [0] [0] [R1]
{I} {J} . {K}
The sub-matrices in equation (4) are of dimensions p × p. Their elements are Ai,m = G1i,m (r/L)A11 + G2i,m (n 2L/r)A66,
Bi,m = G3i,m (n)A12 − G4i,m (n)A66 ,
Ci,m = −G5i,m A12 , Di,m = H1i,m (n 2L/r)A22 + H2i,m (r/L)A66 + H1i,m (n 2L/r 3 )D22 + H2i,m (4/rL)D66 , Ei,m = −H3i,m (nL/r)A22 − H3i,m (n 3L/r 3 )D22 + H4i,m (n/rL)D12 − H5i,m (4n/rL)D66 , Fi,m = R1i,m (L/r)A22 + R2i,m (r/L 3 )D11 + R1i,m (n 4L/r 3 )D22 − [R3i,m + R3m,i ](n 2/rL)D12 + R4i,m (4n 2/rL)D66 .
(4)
121
For a prescribed value of n, the indices i and m take on the values of 1 to p. The symbols G1 to G5, H1 to H5 and R1 to R4 are dimensionless integrals, defined in Appendix A. The elements of {I}, {J} and {K} in equation (4) are unknowns to be determined. When each of the functions Dm (x), Cm (x) and Fm (x) in equations (1) forms an orthogonal set in the interval 0 to L, that is,
g
L
0
6
L, m = i Dm (x)Di (x) dx = , 0, m $ i
g
L
0
7g
L
0
6
L, m = i Cm (x)Ci (x) dx = , 0, m $ i
6
L, m = i Fm (x)Fi (x) dx = , 0, m $ i
7
7
[G2], [H1] and [R1] become identity matrices and equation (4) is a symmetric eigenvalue problem represented by
&
[A] [B] [C] [B]T [D] [E] [C]T [E]T [F]
'8 9 8 9
{I} {I} {J} = l {J} . {K} {K}
(5)
The eigenvalue of this problem is l = rrhLv 2.
3. RESULTS AND DISCUSSIONS
Six laminated circular cylindrical shells were studied, all having orthotropic and mid-plane symmetric properties. Shells 1 and 2 were fabricated with E-glass/epoxy and consist of six and four layers of woven fabrics respectively. Shells 3 and 4 are graphite/epoxy shells each consisting of four plys of woven fabric laminae having the same geometric and elastic properties but different orientations with respect to the shell co-cordinates. Shells 5 and 6 are Shells 1 and 2 with their thicknesses halved. The physical properties of the specimen shells are given in Table 1. Modal testings were performed for Shells 1 and 2. The former has 72 grid points while the latter has 48. Each shell under test was suspended vertically by soft cord to simulate the free-free boundary conditions. An accelerometer was affixed to one of the grid points on the surface of the shell. Impact excitation was introduced to every grid point with an impact hammer. Frequency response functions were obtained and modal analysis performed. Resonance frequencies of the shells were determined and the corresponding modes shapes extracted. A schematic diagram of the experimental set-up is shown in Figure 2. T 1 Physical properties of orthotropic shells Shell
Material
Density (kg/m3 )
Length (mm)
Radius (mm)
Thickness (mm)
Angle of orientation (°)
1 2 3 4 5 6
E-glass/epoxy E-glass/epoxy graphite/epoxy graphite/epoxy E-glass/epoxy E-glass/epoxy
1662·03 1686·48 1455·87 1455·87 1662·03 1686·48
212 321 300 300 212 321
83·5 83·4 80·0 80·0 83·5 83·4
1·371 0·930 0·800 0·800 0·686 0·465
[0/90]6 [0/90]4 [0/90/ + 45/−45]]s [+45/−45/0/90]s [0/90]3 [0/90]2
. . .
122
There is a variety of choices for the functions Dm (x), Cm (x), Fm (x) in equations (1) which describe the variations of the mid-plane displacements along the x-direction. These functions were chosen as Dm (x) =
6
1,
m=1
z2 cos [(m − 1)px/L],
mq1
8
7
,
9
1, m=1 m=2 , Cm (x) = Fm (x) = z3(1 − 2x/L), transverse beam function m q 2
(6a, b)
where Dm (x)(m q 1) and Cm (x) = Fm (x)(m q 2) are respectively the longitudinal and transverse free vibration shapes of a free-free beam [18]. Each set of functions in equation (6) is orthogonal and hence equation (5) could be applied directly to solve for the modal responses of the shells. The Rayleigh-Ritz model was then used to predict the behaviour of the shells, for the dimensions shown in Table 1. The elastic properties of the material for these two shells were determined experimentally. The longitudinal Young’s modulus E1 and the major Poisson’s ratio n12 were obtained by measuring the axial and circumferential strains of an axially loaded cylinder with electrical resistance foil strain gauges. The shell was then evacuated and loaded by external pressure, and hence the circumferential Young’s modulus E2 could be determined. Two longitudinal strips were cut from each shell and are subjected to the Iosipescu test [19]. Strain gauge readings on a 45°-axis of the strip thus led to the value for the in-plane shear modulus G12 . For the purpose of investigating the dynamic behaviour and the effect of stacking sequence of the graphite/epoxy shells, their elastic properties are taken from specimen strips subjected to simple static tests. Table 2 summarizes the material properties of individual lamina in each shell. The values for n21 in the last column were deduced from the reciprocal relations E1 n21 = E2 n12 . By using these properties, the membrane and bending stiffnesses of the shells were calculated and these are given in Table 3. Consideration can now be given to the first thirteen natural modes of Shells 1–6. Tables 4 and 5 show comparisons between the predicted and measured frequencies for Shells 1 and 2. Good agreement is observed between the theoretical and experimental values. The discrepancies are mainly due to imperfections of the specimens and experimental errors incurred during static and modal testings. A study of the convergence of the predicted frequencies for all the shells can be achieved by using up to seven terms
Accelerometer B&K 4374
Shell
PCB hammer 086C80
HP dynamic signal analyzer 3562A
HP workstation
Ch1 Ch 2
Charge amp. B&K 2635
PCB sensor power unit 480E09
Operating program LMS CADA-X Modal analysis software Ver2.8
Figure 2. Experimental set-up for modal testing of free-free shell.
123
T 2 Elastic properties of individual lamina in each shell Shell no.
Material
E1 (GPa)
E2 (GPa)
G12 (GPa)
n12
n21
1 2 3 4 5 6
E-glass/epoxy E-glass/epoxy graphite/epoxy graphite/epoxy E-glass/epoxy E-glass/epoxy
18·274 18·166 44·188 44·188 18·274 18·166
18·157 18·779 44·188 44·188 18·157 18·779
2·930 2·955 5·739 5·739 2·930 2·955
0·210 0·220 0·128 0·128 0·210 0·220
0·209 0·227 0·128 0·128 0·209 0·227
T 3 Membrane and bending stiffnesses of orthotropic shells Stiffness (N/m) ZXXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXXV Shell 1 Shell 2 Shell 3 Shell 4 Shell 5 Shell 6 A11 [N/m] A22 [N/m] A12 [N/m] A66 [N/m] D11 [N/m] D22 [N/m] D12 [N/m] D66 [N/m]
26·204E6 26·079E6 5·477E6 4·017E6 4·104 4·085 0·8578 0·6292
17·782E6 18·348E6 4·037E6 2·748E6 1·282 1·322 0·2909 0·1981
30·396E6 30·396E6 10·133E6 10·131E6 1·843 1·843 0·3188 0·3187
30·396E6 30·396E6 10·133E6 10·131E6 1·400 1·400 0·7621 0·7620
13·102E6 13·040E6 2·739E6 2·009E6 0·5130 0·5106 0·1072 0·0787
8·891E6 9·174E6 2·019E6 1·374E6 0·1603 0·1653 0·0364 0·0248
T 4 Comparison of Rayleigh-Ritz predictions and experimental results for Shell 1 Rayleigh-Ritz frequency (Hz) Mode indices ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV Exp. (m, n) p=2 p=3 p=4 p=5 p=6 p=7 freq. (Hz) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 1,6 2,6 3,5 3,4 3,6
81·94 88·91 231·75 243·59 444·36 458·47 718·63 733·90 1054·21 1070·14 – – –
81·94 88·91 231·74 243·59 444·31 458·47 718·47 733·90 1053·86 1070·14 1079·47 1161·01 1254·07
81·94 88·91 231·74 243·59 444·31 458·46 718·46 733·90 1053·86 1070·14 1079·14 1160·89 1253·70
81·93 88·91 231·73 243·59 444·27 458·46 718·38 733·89 1053·70 1070·14 1070·72 1138·95 1250·23
81·93 88·91 231·73 243·57 444·27 458·43 718·38 733·82 1053·70 1070·00 1070·68 1138·84 1250·22
81·93 88·91 231·71 243·57 444·23 458·43 718·30 733·82 1053·55 1070·00 1069·26 1134·77 1249·82
79·57 88·63 225·64 241·78 433·37 455·03 710·08 739·02 1036·43 1068·02 – 1150·22 1249·99
in the admissible functions in equations (1) and the results are displayed in Tables 4–9. The index pair (m, n) indicates that the mode has m-1 node lines and n sinusoidal wave patterns around the circumference. Review of the convergence rate for the natural frequencies indicates that a series with five terms for the admissible function provides sufficient accuracy. Thus subsequent analytical results are based on five-term series for the displacement functions.
. . .
124
T 5 Comparison of Rayleigh-Ritz predictions and experimental results for Shell 2 Rayleigh-Ritz frequency (Hz) Mode indices ZXXXXXXXXXXXXXCXXXXXXXXXXXXXV Exp. (m, n) p=2 p=3 p=4 p=5 p=6 p=7 freq. (Hz) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 3,4 3,5 1,6 2,6 3,6
56·43 58·58 159·62 163·12 306·05 310·18 494·95 499·41 – – 726·08 730·72 –
56·43 58·58 159·61 163·12 306·03 310·18 494·90 499·41 605·13 621·29 725·96 730·72 788·91
56·43 58·58 159·61 163·12 306·03 310·18 494·90 499·40 604·44 620·58 725·96 730·72 788·39
56·43 58·58 159·61 163·12 306·03 310·18 494·88 499·40 599·87 619·25 725·92 730·72 787·90
56·43 58·58 159·61 163·12 306·03 310·18 494·88 499·39 599·85 619·24 725·92 730·69 787·90
56·43 58·58 159·61 163·12 306·02 310·18 494·87 499·39 598·79 618·97 725·89 730·69 787·90
55·99 59·11 158·21 164·20 301·92 310·69 496·38 503·87 502·09 629·22 722·62 733·66 790·83
The vibratory motion of all the modes tabulated is mainly radial. Resonance frequencies associated with the longitudinal or torsional modes are too high to be considered here. Such modes involve predominantly extensional vibration of which stretching energy as compared to reduced kinetic energy is large. The analytical frequencies of Shells 1 and 2 predicted by the Rayleigh-Ritz model satisfactorily matched those measured by modal testing. It is noticed from Tables 4–9 that higher values of the circumferential wave number do not necessarily imply higher frequencies for cylindrical shells. This phenomenon can be explained through energy consideration. To accomplish this, the terms stretching energy factor hs and bending energy factor hb , are introduced. They are defined as respectively the ratios of the strain energies associated with the stretching and bending stiffnesses to the reduced kinetic energy given by equation (3c). It was found in Shell 1 that as the n-index increases through the modes from (3, 4) to (3, 5) there is an increase in hb .
T 6 Rayleigh-Ritz predictions for Shell 3 Mode indices (m, n) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 3,5 3,4 1,6 2,6 3,6
Rayleigh-Ritz frequency (Hz) ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV p=2 p=3 p=4 p=5 p=6 p=7 83·94 88·06 237·41 243·92 455·21 462·81 736·18 744·33 – – 1079·96 1088·43 –
83·94 88·06 237·41 243·92 455·21 462·81 736·15 744·33 970·09 1006·45 1079·88 1088·43 1194·45
83·94 88·06 237·41 243·92 455·21 462·81 736·15 744·32 966·22 1002·45 1079·87 1088·41 1191·77
83·94 88·06 237·41 243·92 455·20 462·81 736·14 744·32 965·32 999·86 1079·85 1088·41 1191·26
83·94 88·06 237·41 243·92 455·20 462·81 736·14 744·32 965·32 999·84 1079·85 1088·41 1191·26
83·94 88·06 237·41 243·92 455·20 462·81 736·13 744·32 965·11 999·09 1079·83 1088·41 1191·19
125
T 7 Rayleigh-Ritz predictions for Shell 4 Mode indices (m, n) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 1,6 3,5 2,6 3,4 3,6
Rayleigh-Ritz frequency (Hz) ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV p=2 p=3 p=4 p=5 p=6 p=7 73·16 86·70 206·92 226·21 396·75 418·50 641·63 664·61 941·26 – 964·93 – –
73·16 86·80 206·91 226·21 396·68 418·50 641·41 664·61 940·83 945·32 964·93 1006·11 1127·19
73·16 86·70 206·91 226·20 396·68 418·49 641·41 664·59 940·83 941·35 964·87 1002·12 1124·36
73·16 86·70 206·90 226·20 396·66 418·49 641·34 664·59 940·67 939·89 964·87 999·19 1122·97
73·16 86·70 206·90 226·20 396·66 418·48 641·34 664·56 940·67 939·89 964·82 999·17 1122·97
73·15 86·70 206·89 226·20 396·63 418·48 641·28 664·56 940·56 939·64 964·82 988·41 1122·85
Nevertheless, at the same time, hs is reduced by a greater amount. This results is a net decrease in the Rayleigh quotient. Further increasing in the n-index results in a rise in natural frequency for mode (3, 6) becasuse the increase in hb is greater than the reduction of hs . Similar behaviour has been reported in the free vibration of freely supported isotropic shells by Arnold and Warburton [20]. This explains the occasional choice of inextensional shell models to describe modes dominated by bending, which normally have a high circumferential wave number (see, e.g., the paper by Sharma [21]). As obtained by using the Rayleigh-Ritz model, the computed values of the natural frequencies corresponding to m = 1 to 5 for all six shells are plotted against the n-index in Figures 3–8. In Figures 3 and 4, Rayleigh modes (m = 1) and Love modes (m = 2) with the characteristic of an inextensional mid-surface are identified. Their resonance frequencies always increase with the n-index owing to the increasing bending energy factor. For other
T 8 Rayleigh-Ritz predictions for Shell 5 Mode indices (m, n) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 1,6 2,6 3,6 3,5 3,4
Rayleigh-Ritz frequency (Hz) ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV p=2 p=3 p=4 p=5 p=6 p=7 41·00 44·49 115·96 121·89 222·35 229·41 359·59 367·23 527·51 535·48 – – –
41·00 44·49 115·96 121·89 222·34 229·41 359·56 367·23 527·44 535·48 778·15 828·37 1068·53
41·00 44·49 115·96 121·89 222·34 229·41 359·56 367·23 527·44 535·48 777·59 827·97 1068·41
41·00 44·49 115·96 121·89 222·34 229·41 359·55 367·23 527·42 535·48 773·19 817·47 1044·26
41·00 44·49 115·96 121·89 222·34 229·40 359·55 367·22 527·42 535·49 773·18 817·42 1044·13
41·00 44·49 115·96 121·34 222·33 229·40 359·54 367·22 527·40 535·46 772·30 815·22 1039·33
. . .
126
T 9 Rayleigh-Ritz predictions for Shell 6 Mode indices (m, n) 1,2 2,2 1,3 2,3 1,4 2,4 1,5 2,5 1,6 2,6 3,5 3,6 3,4
Rayleigh-Ritz frequency (Hz) ZXXXXXXXXXXXXXXXCXXXXXXXXXXXXXXXV p=2 p=3 p=4 p=5 p=6 p=7 28·22 29·29 79·81 81·56 153·03 155·09 247·48 249·71 363·04 365·37 – – –
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·71 363·02 365·37 429·99 448·37 534·24
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·71 363·02 365·37 428·99 447·48 533·47
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·71 363·02 365·37 427·25 446·90 528·37
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·70 363·02 365·36 427·25 446·90 528·34
28·22 29·29 79·81 81·56 153·02 155·09 247·47 249·70 363·01 365·36 426·80 446·76 527·08
modes (m q 2), the natural frequencies are highest when n is unity. As the number of circumferential nodes is increased, the frequency values decrease until a minimum is reached. Before this point, vibration modes other than the Rayleigh and Love modes are in general stretching dominated. As the n-index increases, the stretching energy factor of a given mode which tends to be inextensional decreases. Further increase in the circumferential wave number causes a rise in the natural frequencies due to the increasing bending energy factor. The watersheds in the frequency variation are modes (3, 5), (4, 6) and (5, 6) in Shell 1 and modes (3, 4), (4, 5) and (5, 6) in Shell 2. Similar behaviour is observed in Figures 5 and 6 for the graphite/epoxy shells. Furthermore, Figures 7 and 8 show that reduction of shell thickness did not alter this trend. It is also worth noting that the energy distributions of Shells 3 and 4 are different and are significantly affected by the lamination configurations.
Figure 3. Theoretical frequency variations for shell 1, m values: —q—, 1; +2; *, 3; —Q—, 4; ×, 5.
127
Figure 4. Theoretical frequency variations for Shell 2. Key as Figure 3.
Scrutiny of the energy distributions of these modes will cast some light on the behaviour of the vibrating shells. The strain energy of the shell is composed of eight separate terms associated with the four extensional and four flexural stiffnesses respectively. Their contributions are expressed as percentages of the total strain energy and are shown in Figures 9–12. It is evident from these figures that the Rayleigh and Love modes, i.e. modes (1, 2) and (2, 2) consist of little membrane energy and hence negligible energy contribution from the stretching stiffnesses. These inextensional modes have frequencies close to each other as shown in Figures 3–8; the former undergoes predominantly bending motion while the latter has bending and a certain degree of twisting. This conclusion is drawn in the view of the bending energy related to D22 in these modes with the additional torsion energy associated with D66 in the latter.
Figure 5. Theoretical frequency variations for Shell 3. Key as Figure 3.
128
. . .
Figure 6. Theoretical frequency variations for Shell 4. Key as Figure 3.
For modes with m q 2, the energies developed from stiffnesses A11 and A66 cover a major portion of the total stretching energy while that from stiffness D22 dominates the total bending energy. The existence of such stretching energy accounts for the inaccuracy of inextensional shell models for the prediction of the modal frequencies of these modes. By increasing the m-index from three to higher values, the energy contribution associated with A66 increases progressively whereas that of A11 diminishes. This is a clear indication that the shell is subjected to significant in-plane shear deformation for modes with a large m-index. Energy distributions in Figures 9–12 also reveal that strain energy associated with the membrane stiffness A12 can be negative for some of the extensional modes. It is seen that when the axial and circumferential strains ox0 and ou0 have opposite signs, the component in the strain energy which involves integration of the product of these strains over the entire shell would then be negative.
Figure 7. Theoretical frequency variations for Shell 5. Key as Figure 3.
129
5
Frequency (kHz)
4
3
* 2
* 1
0
* 1
2
3
*
*
*
*
4 5 6 7 Circumferential wave no. (n)
*
*
8
9
*
10
Figure 8. Theoretical frequency variations for Shell 6. Key as Figure 3.
As seen from Table 3, Shell 3 is much stiffer in terms of bending whereas Shell 4 is stiffer in twisting. This is highlighted by a large energy proportion associated with the bending stiffness D22 for Shell 3 and a more significant twisting energy associated with D66 in Shell 4. This points to the possibility of achieving different characteristics of the shell through manipulation of the lamination sequence. Such an optimization procedure have been attempted by Nshanian and Pappes [22]. A study of the effect of thickness change on the energy distribution has been made with Shells 5 and 6. An immediate observation is the general reduction of the natural frequency values when comparing Figure 7 to Figure 3 and Figure 8 to Figure 4. More modes come under the category of stretching dominated because the minima of the frequency curves have shifted to the right in Figures 7 and 8 due to the decrease
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A22
A12
A66
D 11
Figure 9. Energy contribution of Shell 1. Modes:q, (1, 2);
D22
D12
, (2, 2);
D66 , (3, 5);
, (4, 6); Q, (5, 6).
. . .
130
100
Energy contribution (%)
80
60
40
20
0
–20
A 11
A22
A12
A66
D11
D 22
D 12
Figure 10. Energy contributions of Shell 2. Modes: q, (1, 2); .
D 66
, (2, 2);
, (3, 4);
, (4, 5); Q, (5, 6).
in thickness. While the bending dominated modes show substantial decreases in natural frequencies, the stretching dominated modes have virtually no changes in these values. Numerical calculations show that the Rayleigh quotient of the bending dominated modes is approximately proportional to h 2 while that associated with stretching dominated modes is more or less independent of the shell thickness. The natural modes corresponding to the minima of the frequency curves in Figure 7 are modes (3, 6), (4, 7) and (5, 8) and that of Figure 8 are modes (3, 5), (4, 7) and (5, 8). Figures 13 and 14 have been drawn to illustrate the energy contributions of the various stiffnesses for these modes.
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A 22
A 12
A 66
D11
Figure 11. Energy contributions of Shell 3. Modes: q, (1, 2);
D22
D12
, (2, 2);
D66 , (3, 5);
, (4, 6); Q, (5, 7).
131
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A22
A12
A66
D 11
D22
D12
D66
Figure 12. Energy contributions of Shell 4. Key as Figure 11.
As pointed out earlier, the stretching energy of the vibration modes with m q 2 is mainly due to A11 and A66 . This kind of energy is a large proportion of the total strain energy for modes having a smaller n-index. This phenomenon is more pronounced when the shell thickness is reduced. An important application of the foregoing observations is in the field of material characterization of structures through the use of modal frequencies. To characterize composite shells successfully, it is desirable to have their modal frequencies sensitive to each of the elastic properties. Hence, the Rayleigh modes will always be useful for the determination of D22 . For the characterization of stiffnesses A11 and A66 , it is preferable
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A22
A12
A66
D11
D 22
D 12
D 66
Figure 13. Energy contributions of Shell 5. Key as Figure 11.
. . .
132
100
Energy contribution (%)
80
60
40
20
0
–20
A11
A22
A12
A66
D 11
D 22
Figure 14. Energy contributions of Shell 6. Modes: q, (1, 2);
D 12 , (2, 2);
D 66 , (3, 5);
, (4, 7); Q, (5, 8).
to have a thin specimen and to make use of its extensional modes associated with a small n-index. For the shells in this study, laminate stiffnesses Q 11 , Q 22 and Q 66 can be obtained once the laminate stiffnesses A11 , D22 and A66 have been characterized. The laminate stiffnesses will then yield the values of the material properties E1 , E2 and G12 of the shells. With reference to Figures 9–14, the strain energy associated with stiffnesses A12 and D12 constitutes only a minor portion of the total strain energy. As a result, the values of these stiffnesses have a negligible effect on the total strain energy and hence on the frequencies of the natural modes. Since both these stiffnesses are primarily related to the Poisson’s ratios, the modal frequencies of the shells are thus insensitive to these elastic constants.
4. CONCLUSIONS
A Rayleigh-Ritz vibration model has been developed to evaluate the dynamic responses of orthotropic shells under free vibration. The shells have symmetric laminations with completely free ends. The natural frequencies and mode shapes of the shells are predicted and the frequencies match well those extracted from modal tests, with typical errors of less than 3%. For the inextensional Rayleigh and Love modes, natural frequencies are close to each other and increase with the n-index. The major part of the strain energy comes from bending and is related to D22 . A considerable amount of twisting energy associated with D66 is also observed for the Love modes. For the extensional modes considered, with the m-index taking a value between 3 and 5, the resonance frequencies change through a minimum as the circumferential wave number n increases. The stretching energy of these modes is mainly due to membrane stiffnesses A11 and A66 while the bending energy is attributed to D22 . For modes with a
133
large m-index, the shell would be under significant shearing indicated by a progressively larger energy term associated with A66 . On the other hand, for modes with a small n-index, stretching and shearing deformations become dominant as the bending energy term due to D22 subsides. This effect is more apparent for thin shells. When natural frequencies of an orthotropic shell are used in the application of material characterization, thin cylinders are the ideal specimens for an accurate determination of the elastic constants E1 , E2 and G12 .
REFERENCES 1. V. V. N 1959 The Theory of Thin Shells. Groningen, The Netherlands: P. Noordhoff Ltd. 2. S. A. A 1961 Theory of Anisotropic Shells. Moscow: State Publishing House for Physical and Mathematical Literautre. 3. S. B. D, K. S. P and R. L. T 1962 Journal of Aerospace Sciences, 969–975. On the theory of laminated anisotropic shells and plates. 4. V. I. W 1964 Experimental Mechanics (July), 200–205. Free vibrations of multilayered cylindrical shells. 5. S. B. D 1968 Journal of Acoustical Society of America 44, 1628–1635. Free vibrations of laminated orthotropic cylindrical shells. 6. C. W. B, J. L. B and D. M. E 1969 Journal of Composite Materials 3, 480–499. Free vibration of multilayer anisotropic cylindrical shells. 7. Y. S and R. L 1971 Journal of Sound and Vibration 15, 235–256. On vibrations of heterogeneous orthotropic cylindrical shells. 8. B. K. R and Y. C. D 1973 Journal of Sound and Vibration 28, 737–757. Vibration of layered shells. 9. K. N. S and A. V. K M 1976 Journal of Structural Mechanics 4, 379–393. Vibrations of multifiber composite shells—some numerical results. 10. C. W. B and M. K 1982 Journal of Sound and Vibration 81, 107–121. Vibration of cylindrical shells of bimodulus composite materials. 11. D. G. L 1988 Journal of Composite Materials 22, 1102–1115. Calculation of natural frequencies of vibration of thin orthotropic composite shells by energy method. 12. C. B. S and M. D 1984 Proceedings of the 2nd Internatinal Conference on Recent Advances in Structural Dynamics, University of Southampton, England, 9–13 April 1, 15–24. Calculation of natural frequencies of specially orthotropic multilayered thin circular cylinder. 13. M. D and C. B. S 1984 Thin-walled Structures 2, 207–217. Natural frequencies of laminated orthotropic thin circular cylinders. 14. C. B. S and M. D 1987 Composite Structures 7, 123–138. Free vibration of specially orthotropic, multilayered, thin cylindrical shells with various end conditions. 15. K. M. L and C. W. L 1994 AIAA Journal 32, 387–396. Vibratory characteristics of cantilevered rectangular shallow shells of variable thickness. 16. C. W. L and K. M. L 1994 Journal of Sound and Vibration 173, 343–375. A pb-2 Ritz formulation for flexural vibration of shallow cylindrical shells of rectangular planform. 17. J. Y and S. M. D 1994 Journal of Sound and Vibration 175, 241–263. The free vibration of circularly cylindrical shell and plate systems. 18. R. D. B 1979 Formulas for Natural Frequency and Mode Shape. New York: Van Nostrand Reinhold Ltd. 19. ASTM Standards D5379/D 5379M—93 Standard test method for shear properties of composite materials by the V-notched beam method, 229–241. 20. R. N. A and G. B. W 1949 Proceedings of the Royal Society A, 197, 238–256. Flexural vibrations of the walls of thin cylindrical shells having freely supported ends. 21. C. B. S 1974 Journal of Sound and Vibration 35, 55–76. Calculation of natural frequencies of fixed-free circular cylindrical shells. 22. Y. S. N and M. P 1983 AIAA Journal 21, 430–437. Optimal laminated composite shells for buckling and vibration.
. . .
134
APPENDIX A
The dimensionless integrals defined for the elements of equation (4) are as follows:
g$ L
G1i,m = L
0
% g$ g$ g$
1Di (x) 1Dm (x) dx, 1x 1x
0
L
Di (x)
G4i,m =
0
0
H1i,m =
1 L
g
L
[Ci (x)Cm (x)] dx,
0
g$ g g$
Ci (x)
0
R1i,m =
1 L
g
[Ci (x)Fm (x)] dx,
0
%
1Fm2 (x) dx, 1x 2
0
0
R2i,m = L 3
%
1Fm2 (x) dx, 1x 2
L
R4i,m = L
% %
0
1Fi (x) 1Fm (x) dx. 1x 1x
APPENDIX B: LIST OF SYMBOLS
Im , Jm , Km L, r, h Aij , Dij E1 , E2 G12 Q ij x, u, z u, v, w U(x, u) V(x, u) W(x, u) Vmax , Tmax T* A, B, C, D, E, F a l
1
1Ci (x) 1Fm (x) dx 1x 1x
1Fi2 (x) 1Fm2 (x) dx, 1x 2 1x 2
0
Fi (x)
0
g$ g$ g$ L
H5i,m = L
L
[Fi (x)Fm (x)] dx,
L
%
1Ci (x) 1Cm (x) dx, 1x 1x
L
L
1 L
R3i,m = L
H2i,m = L
0
L
H4i,m = L
% % % g$
1Cm (x) dx, 1x
L
H3i,m =
[Di (x)Dm (x)] dx,
0
1Di (x) Fm (x) dx, 1x
L
G5i,m =
L
1Di (x) Cm (x) dx, 1x
L
G3i,m =
g
1 L
G2i,m =
deflection coefficients length, radius and thickness of shell stretching and bending stiffnesses of laminate Young’s moduli along the perpendicular to fibers in-plane shear modulus transformed reduced lamina stiffnesses cylindrical coordinates of shell displacement components at mid-plane maximum of longitudinal mid-surface deflection maximum of circumferential mid-surface deflection maximum of radial mid-surface deflection maximum strain and kinetic energies reduced kinetic energy elements of system matrix lamination angle of lamina with x-axis theoretical eigenvalue
r n12 , n21 Dm (x) Cm (x), Fm (x) v hs hb
density of shell material major and minor Poisson’s ratios longitudinal free vibration shape of beam free-free characteristic beam functions angular frequency of shell stretching energy factor bending energy factor
135