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Vibration analysis of orthotropic shells
Journal
of Sound and Vibration
(1989) 133(3), 510-514
VIBRATION
ANALYSIS 1.
OF ORTHOTROPIC
SHELLS
INTRODUCTION
The growing importance of composite materials in modem aircraft and missile structures has encouraged the study of vibrational behavior of composite-structure especially thinshell structures. Many papers have been published in this area, but orthotropic shells with variable thickness have been studied by very few researchers. In references [l-7] isotropic shells of constant thickness have been studied, and Sharma [8] analyzed multilayered shells for different boundary conditions. In the present study a cantilever orthotropic shell of variable thickness has been dealt with. The mass of the shell is kept constant for a given l/a ratio. 2.
FORMULATION
Love’s first approximation shell theory is used to solve the problem. The thickness variation is given by h = h&(x), where 4(x) = 1 - kx’ (linear asymmetric), = 1 - /42x’- 11 (linear symmetric), = 1 - kx” (quadratic asymmetric), = 1-/42x’1)2 (quadratic symmetric), and h, = h,,/( 1 - k/2) (linear), = h,,/( 1 - k/3) (quadratic). Here h,, is the equivalent thickness, k is the thickness variation parameter, x’= x/l, x is the axial co-ordinate, 1 is the length of the shell, and I( )I d enotes the absolute value of ( ). The strain-displacement relationships used are those from reference [8]. The expressions for strain and kinetic energies are
where {s]T= {&Ox, aoe,ro,e, Xl 9x2,2x,2~, h/2 {A,,
B,,
Qjl=
(1,
z,
z’K+i
dz,
I -h/2 Ql
being the stress-strain
relation, K.E.=;
IA
pT(ti2+ d2+ ti2) dA,
where ( * ) denotes differentiation with respect to time, and pr is the mass per unit surface area. Two sets of mode shapes are assumed u= f F V,, sin (ax) sin no sin (ot), In=, n=,
u= f z U,, sin (ax) cos n0 sin (wt), In=, n=, f W,,@,(x) In=, n=l U= f E U,,@~(x)cos In=, n=,
cos nt9 sin (wt);
(1)
u= f t V,,@,(x)sin m=, n=,
&sin(&),
w= f T W,,@,(x) In=, n=,
nOsin(
cos ne sin (wt).
(2)
@
1989 Academic Press Limited
510 0022460X/89/180510+05
$03.00/O
LETTERS
TO THE
511
EDITOR
Here Q,(x) is the beam function for a clamped-free boundary, (Y= [(2m - 1)/2](7r/l) and (‘) denotes differentiation with respect to x. Substituting and minimizing the strain and kinetic energies gives the stiffness matrix [K,], and mass matrix [M,,] respectively for the nth circumferential mode. The frequencies are obtained by solving the eigenvalue problem, [ K,]-w*[M,,]=O. 3. RESULTS
AND
DISCUSSION
The properties used for calculating the frequencies are: Young’s modulus (longitudinal direction), 1.81 x 10” N/m*, Young’s modulus (transverse direction) 1.03 x 10” N/m*, shear modulus 7.17 x lo9 N/m*, mass density 1600.0 kg/m3, major Poisson ratio 0.28, radius of the shell 0.0625 m, equivalent thickness 5.08 x 10e4 m. The minor Poisson ratio can be obtained by the inverse law. A comparison of natural frequencies with those presented in reference [3] is shown in Figure 1. It can be seen that the present results agree well with the reference values. The second type of mode shape was used for the analysis, because it gave better values than the first set of mode shapes. Sharma [3] has suggested that the second set of mode shapes give practically comparable results. The variation of the normalized frequencies (w/w,), where w is the frequency of the variable-thickness shell of thickness variation parameter S and w. is the corresponding frequency of a shell of equivalent thickness, are shown in Figures 2-4 for various l/a ratios and various thickness profiles.
600
-
12345670
4
9
10
11
12
13
2 2500 Lk
tbl
??
500 -
-d
a-add011 1
Figure [3]; -+-,
1. Comparison experiment
of natural [3]; -*-,
14
11 3
I1 11(* 5 7 9 Circumferential mode no.
1111 11
13
frequencies. (a) First axial mode; (b) second axial mode. -O-, present (1); --Cl--, present (2).
Reference
512
LETTERS
TO THE
EDITOR
(b)
k=0.8
k=0.8
?! 1.4
(d)
(cl
P 2 ;b 2
k=0.8
12kz0.6 1.0
1-I
o.8,!
Circumfereniiol mode no.
Figure 2. First axial mode normalized frequencies (c) quadratic symmetric; (d) quadratic asymmetric.
(l/a = 0.5). (a) L mear symmetric;
(b) linear asymmetric;
(b)
0.6. 1
5
10
15 Circumferential
mode no.
Figure 3. Variations of normalized frequencies (I/a = 2, m = 1). (a) Linear symmetric; (c) quadratic symmetric; (d) quadratic asymmetric. -, k = 0.2; -. .-, k = 0.4; -.-,
(b) linear asymmetric; k = 0.6; - - -, k = 0.8.
LETI-ERS
TO THE
513
EDITOR
(d)
Circumferential Figure 4. First axial mode normalized frequencies (c) quadratic symmetric; (d) quadratic asymmetric.
mode no
(I/a = 10). (a) Linear symmetric;
(b) linear asymmetric;
All of the figures show an initial increase in frequency as the circumferential mode number increases. The frequency subsequently decreases and eventually falls below unity: i.e., the frequency of the variable thickness shell becomes smaller than that of the corresponding equivalent thickness shell. From the figures it is observed that as k increases the percentage difference of the frequency from the equivalent thickness shell frequency is increasing. It can also be seen that as l/a increases the percentage difference decreases. For a thickness varying from one end to the other (maximum thickness at fixed end) the normalized frequencies are higher than for the symmetrically varying case. It is observed that as l/a is increased the circumferential mode at which the normalized frequency falls below unit shifts from a higher circumferential mode to a lower one. The lowest natural frequencies of the shells are shown in Table 1. The numbers in brackets indicate the corresponding circumferential modes. It can be seen that the lowest frequencies of the shells increase as the thickness variation parameter is increased, in most of the cases. Maximum variation is obtained for a short shell with an asymmetric thickness variation. When l/a = 2 the percentage variation is less compared with the other two cases. From this one can say that thickness variation is more effective in raising the lowest natural frequency for relatively short shells. 4. CONCLUSIONS 1. Lower circumferential corresponding frequencies
mode frequencies of variable-thickness of equivalent-thickness shells.
shells are higher than
514
LETTERS TO THE EDITOR TABLE 1 Lowest frequency
Length to radius ratio
of the shell (in Hz)
Asymmetric
Symmetric A k
Linear
I Quadratic
\ Linear
Quadratic
0.5
0.0 0.2 0.4 0.6 0.8
2647.5 2662.1 2696.7 2768.2 2907.5
(9) (9) (9) (9) (9)
2647.5 2662.1 2694.4 2751.0 2838.2
(9) (9) (9) (9) (9)
2647.5 2697.7 2783.9 2941.5 3306.7
(9) (9) (10) (10) (11)
2647.5 2699.4 2781.7 2913.0 3155.3
(9) (9) (10) (10) (10)
2.0
0.0 0.2 0.4 0.6 0.8
769.9 (6) 766.9 (6) 768.5 (6) 778.6 (6) 803.1 (6)
769.9 767.1 769.0 777.0 791.0
(6) (6) (6) (6) (6)
769.9 764.9 766.7 781.0 794.8
(6) (6) (6) (7) (7)
769.9 765.0 769.2 788.4 810.3
(6) (6) (6) (6) (7)
10.0
0.0 0.2 0.4 0.6 0.8
158.7 157.0 156.4 157.4 160.9
(3) (3) (3) (3) (3)
158.7 156.9 156.7 160.3 173.0
(3) (3) (3) (3) (3)
158.7 156.7 156.7 160.9 170.3
(3) (3) (3) (3) (3)
158.7 156.9 156.2 157.5 163.2
(3) (3) (3) (3) (3)
2. The lowest natural frequency of a cantilever shell of asymmetrically varying thickness is higher than for one with symmetrically varying thickness. 3. Thickness variation for short shells can give appreciable increases in the lowest natural frequencies. N. GANESAN K. R. SIVADAS
Department of Applied Mechanics, Indian Institute of Technology, Madras 36, India (Received
16 February 1989)
REFERENCES 1. K. FORSBERG 1964 American Institute of Aeronautics and Astronautics Journal 2( 12), 2150-2157. Influence of boundary condition on the modal characteristics of thin shells. 2. G. B. WARBURTON 1970 Journal of Sound and Vibration 11, 335-338.Natural frequencies of thin cantilever cylindrical shells. 3. C. B. SHARMA 1974 Journal of Sound and Vibration 35,55-76. Calculation of natural frequencies of fixed free circular cylindrical shells. 4. C. B. SHARMA and D. J. JOHNS 1972 Journal of Sound and Vibration 25,433-449. Free vibration of cantilever circular cylindrical shells-a comparative study. 5. J. T. S. WANG, J. H. ARMSTRONG and D. V. HO 1979 Journal of Sound and Vibration 64, 529-538. Axisymmetric vibration of prestressed non-uniform cantilever cylindrical shells. 6. R. F. JONIN and P. A. DIES 1979 Journal of Sound and Vibration 62, 165-180. Free vibration of circular cylinders of variable thickness. 7. K. SUZUKI and A. W. LEISSA 1986 Journal of Sound and Vibration 107, 1-15. Exact solution for the free vibration of open cylindrical shells with circumferentially varying curvature and thickness. 8. C. B. SHARMA and M. DARVIZEH 1987 Composite Structures 7, 123-138. Free vibration of specially orthotropic, multilayered, thin cylindrical shells with various end conditions.
of Sound and Vibration
(1989) 133(3), 510-514
VIBRATION
ANALYSIS 1.
OF ORTHOTROPIC
SHELLS
INTRODUCTION
The growing importance of composite materials in modem aircraft and missile structures has encouraged the study of vibrational behavior of composite-structure especially thinshell structures. Many papers have been published in this area, but orthotropic shells with variable thickness have been studied by very few researchers. In references [l-7] isotropic shells of constant thickness have been studied, and Sharma [8] analyzed multilayered shells for different boundary conditions. In the present study a cantilever orthotropic shell of variable thickness has been dealt with. The mass of the shell is kept constant for a given l/a ratio. 2.
FORMULATION
Love’s first approximation shell theory is used to solve the problem. The thickness variation is given by h = h&(x), where 4(x) = 1 - kx’ (linear asymmetric), = 1 - /42x’- 11 (linear symmetric), = 1 - kx” (quadratic asymmetric), = 1-/42x’1)2 (quadratic symmetric), and h, = h,,/( 1 - k/2) (linear), = h,,/( 1 - k/3) (quadratic). Here h,, is the equivalent thickness, k is the thickness variation parameter, x’= x/l, x is the axial co-ordinate, 1 is the length of the shell, and I( )I d enotes the absolute value of ( ). The strain-displacement relationships used are those from reference [8]. The expressions for strain and kinetic energies are
where {s]T= {&Ox, aoe,ro,e, Xl 9x2,2x,2~, h/2 {A,,
B,,
Qjl=
(1,
z,
z’K+i
dz,
I -h/2 Ql
being the stress-strain
relation, K.E.=;
IA
pT(ti2+ d2+ ti2) dA,
where ( * ) denotes differentiation with respect to time, and pr is the mass per unit surface area. Two sets of mode shapes are assumed u= f F V,, sin (ax) sin no sin (ot), In=, n=,
u= f z U,, sin (ax) cos n0 sin (wt), In=, n=, f W,,@,(x) In=, n=l U= f E U,,@~(x)cos In=, n=,
cos nt9 sin (wt);
(1)
u= f t V,,@,(x)sin m=, n=,
&sin(&),
w= f T W,,@,(x) In=, n=,
nOsin(
cos ne sin (wt).
(2)
@
1989 Academic Press Limited
510 0022460X/89/180510+05
$03.00/O
LETTERS
TO THE
511
EDITOR
Here Q,(x) is the beam function for a clamped-free boundary, (Y= [(2m - 1)/2](7r/l) and (‘) denotes differentiation with respect to x. Substituting and minimizing the strain and kinetic energies gives the stiffness matrix [K,], and mass matrix [M,,] respectively for the nth circumferential mode. The frequencies are obtained by solving the eigenvalue problem, [ K,]-w*[M,,]=O. 3. RESULTS
AND
DISCUSSION
The properties used for calculating the frequencies are: Young’s modulus (longitudinal direction), 1.81 x 10” N/m*, Young’s modulus (transverse direction) 1.03 x 10” N/m*, shear modulus 7.17 x lo9 N/m*, mass density 1600.0 kg/m3, major Poisson ratio 0.28, radius of the shell 0.0625 m, equivalent thickness 5.08 x 10e4 m. The minor Poisson ratio can be obtained by the inverse law. A comparison of natural frequencies with those presented in reference [3] is shown in Figure 1. It can be seen that the present results agree well with the reference values. The second type of mode shape was used for the analysis, because it gave better values than the first set of mode shapes. Sharma [3] has suggested that the second set of mode shapes give practically comparable results. The variation of the normalized frequencies (w/w,), where w is the frequency of the variable-thickness shell of thickness variation parameter S and w. is the corresponding frequency of a shell of equivalent thickness, are shown in Figures 2-4 for various l/a ratios and various thickness profiles.
600
-
12345670
4
9
10
11
12
13
2 2500 Lk
tbl
??
500 -
-d
a-add011 1
Figure [3]; -+-,
1. Comparison experiment
of natural [3]; -*-,
14
11 3
I1 11(* 5 7 9 Circumferential mode no.
1111 11
13
frequencies. (a) First axial mode; (b) second axial mode. -O-, present (1); --Cl--, present (2).
Reference
512
LETTERS
TO THE
EDITOR
(b)
k=0.8
k=0.8
?! 1.4
(d)
(cl
P 2 ;b 2
k=0.8
12kz0.6 1.0
1-I
o.8,!
Circumfereniiol mode no.
Figure 2. First axial mode normalized frequencies (c) quadratic symmetric; (d) quadratic asymmetric.
(l/a = 0.5). (a) L mear symmetric;
(b) linear asymmetric;
(b)
0.6. 1
5
10
15 Circumferential
mode no.
Figure 3. Variations of normalized frequencies (I/a = 2, m = 1). (a) Linear symmetric; (c) quadratic symmetric; (d) quadratic asymmetric. -, k = 0.2; -. .-, k = 0.4; -.-,
(b) linear asymmetric; k = 0.6; - - -, k = 0.8.
LETI-ERS
TO THE
513
EDITOR
(d)
Circumferential Figure 4. First axial mode normalized frequencies (c) quadratic symmetric; (d) quadratic asymmetric.
mode no
(I/a = 10). (a) Linear symmetric;
(b) linear asymmetric;
All of the figures show an initial increase in frequency as the circumferential mode number increases. The frequency subsequently decreases and eventually falls below unity: i.e., the frequency of the variable thickness shell becomes smaller than that of the corresponding equivalent thickness shell. From the figures it is observed that as k increases the percentage difference of the frequency from the equivalent thickness shell frequency is increasing. It can also be seen that as l/a increases the percentage difference decreases. For a thickness varying from one end to the other (maximum thickness at fixed end) the normalized frequencies are higher than for the symmetrically varying case. It is observed that as l/a is increased the circumferential mode at which the normalized frequency falls below unit shifts from a higher circumferential mode to a lower one. The lowest natural frequencies of the shells are shown in Table 1. The numbers in brackets indicate the corresponding circumferential modes. It can be seen that the lowest frequencies of the shells increase as the thickness variation parameter is increased, in most of the cases. Maximum variation is obtained for a short shell with an asymmetric thickness variation. When l/a = 2 the percentage variation is less compared with the other two cases. From this one can say that thickness variation is more effective in raising the lowest natural frequency for relatively short shells. 4. CONCLUSIONS 1. Lower circumferential corresponding frequencies
mode frequencies of variable-thickness of equivalent-thickness shells.
shells are higher than
514
LETTERS TO THE EDITOR TABLE 1 Lowest frequency
Length to radius ratio
of the shell (in Hz)
Asymmetric
Symmetric A k
Linear
I Quadratic
\ Linear
Quadratic
0.5
0.0 0.2 0.4 0.6 0.8
2647.5 2662.1 2696.7 2768.2 2907.5
(9) (9) (9) (9) (9)
2647.5 2662.1 2694.4 2751.0 2838.2
(9) (9) (9) (9) (9)
2647.5 2697.7 2783.9 2941.5 3306.7
(9) (9) (10) (10) (11)
2647.5 2699.4 2781.7 2913.0 3155.3
(9) (9) (10) (10) (10)
2.0
0.0 0.2 0.4 0.6 0.8
769.9 (6) 766.9 (6) 768.5 (6) 778.6 (6) 803.1 (6)
769.9 767.1 769.0 777.0 791.0
(6) (6) (6) (6) (6)
769.9 764.9 766.7 781.0 794.8
(6) (6) (6) (7) (7)
769.9 765.0 769.2 788.4 810.3
(6) (6) (6) (6) (7)
10.0
0.0 0.2 0.4 0.6 0.8
158.7 157.0 156.4 157.4 160.9
(3) (3) (3) (3) (3)
158.7 156.9 156.7 160.3 173.0
(3) (3) (3) (3) (3)
158.7 156.7 156.7 160.9 170.3
(3) (3) (3) (3) (3)
158.7 156.9 156.2 157.5 163.2
(3) (3) (3) (3) (3)
2. The lowest natural frequency of a cantilever shell of asymmetrically varying thickness is higher than for one with symmetrically varying thickness. 3. Thickness variation for short shells can give appreciable increases in the lowest natural frequencies. N. GANESAN K. R. SIVADAS
Department of Applied Mechanics, Indian Institute of Technology, Madras 36, India (Received
16 February 1989)
REFERENCES 1. K. FORSBERG 1964 American Institute of Aeronautics and Astronautics Journal 2( 12), 2150-2157. Influence of boundary condition on the modal characteristics of thin shells. 2. G. B. WARBURTON 1970 Journal of Sound and Vibration 11, 335-338.Natural frequencies of thin cantilever cylindrical shells. 3. C. B. SHARMA 1974 Journal of Sound and Vibration 35,55-76. Calculation of natural frequencies of fixed free circular cylindrical shells. 4. C. B. SHARMA and D. J. JOHNS 1972 Journal of Sound and Vibration 25,433-449. Free vibration of cantilever circular cylindrical shells-a comparative study. 5. J. T. S. WANG, J. H. ARMSTRONG and D. V. HO 1979 Journal of Sound and Vibration 64, 529-538. Axisymmetric vibration of prestressed non-uniform cantilever cylindrical shells. 6. R. F. JONIN and P. A. DIES 1979 Journal of Sound and Vibration 62, 165-180. Free vibration of circular cylinders of variable thickness. 7. K. SUZUKI and A. W. LEISSA 1986 Journal of Sound and Vibration 107, 1-15. Exact solution for the free vibration of open cylindrical shells with circumferentially varying curvature and thickness. 8. C. B. SHARMA and M. DARVIZEH 1987 Composite Structures 7, 123-138. Free vibration of specially orthotropic, multilayered, thin cylindrical shells with various end conditions.