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The calculations of natural frequencies and forced vibration responses of conical shell using the Rayleigh–Ritz method
Mechanics Research Communications 36 (2009) 595–602
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The calculations of natural frequencies and forced vibration responses of conical shell using the Rayleigh–Ritz method Feng-Ming Li a,b,*, Kikuo Kishimoto b, Wen-Hu Huang a a b
School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, PR China Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan
a r t i c l e
i n f o
Article history: Received 9 October 2008 Received in revised form 15 January 2009 Available online 20 February 2009
Keywords: Conical shell Forced vibration analysis Hamilton’s principle Rayleigh–Ritz method
a b s t r a c t An effective method for analyzing the forced vibration of conical shell is presented. Hamilton’s principle with the Rayleigh–Ritz method is employed to derive the equation of motion of the conical shell. A set of simpler principal vibration modes of the conical shell with two simply supported boundaries are presented. The natural frequencies of conical shell can be obtained by solving eigenvalue problem of the equation of motion and the steady responses of forced vibration can also be obtained by solving the equation of motion. Numerical comparisons with the results in the open literature are made to validate the present methodology. Moreover, the forced vibration responses of a conical shell are also calculated. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Conical shells are widely used in many engineering applications. It is necessary to perform the vibration analysis of conical shells for the purpose of dynamical design, vibration control and so on. Up to now, many papers on the vibration problems of conical shells have been published (Goldberg et al., 1960; Serpico, 1963; Chang, 1978; Khatri and Asnani, 1995). Lam and Li (1999) and Li (2000) studied the frequency characteristics of free vibration of rotating conical shell using the Galerkin method. Fares et al. (2004) investigated the design and active vibration control of composite laminated conical shells. They used the Liapunov–Bellman theory to obtain the controlled deflections of the shells. Liew et al. (2005) analyzed the free vibration of thin conical shells using the element-free kp-Ritz method and discussed the frequency properties under different parameters. Chai et al. (2006) studied the spatially distributed microscopic control characteristics of distributed actuator patches on a rocket conical shell. Sofiyev et al. (2008) studied the vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure. It should be noted that Liew and Lim (Liew et al., 1995; Lim and Liew, 1996, 1995) systematically studied the free vibration of shallow conical shells using the pb-2 Ritz method. They employed the admissible pb-2 shape functions to approximate the three-dimensional displacements, and calculated comprehensively the dimensionless natural frequencies and the vibration modes of shallow conical shells. Although numerous studies on the dynamic problems of the conical shells have been published, the forced vibration problems of the conical shells should be thoroughly studied. Through the forced vibration analysis, an effective way for calculating the forced vibration responses can be obtained and further used in the vibration control and dynamic designs of the conical shells. Normally, the equation of motion of the conical shell is very complicated and some of the coefficients of the equation of motion are variables (Soedel, 1981), which makes it difficult to analytically solve the equation of motion of the conical shell. * Corresponding author. Address: School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, PR China. Tel.: +86 451 86414479. E-mail address: [email protected] (F.-M. Li). 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.02.003
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F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
In this study, an effective method for the forced vibration analysis of conical shells is presented. Using Hamilton’s principle with the Rayleigh–Ritz method, the equation of motion of the conical shell is derived. A set of simple formulations of the principal mode shapes is employed and verified to be effective by numerical simulations. Solving the eigenvalue problem of the equation of motion, the natural frequencies of the conical shell can be obtained. And the steady responses of the forced vibration of conical shells can also be obtained by solving the equation of motion. 2. Equation of motion A thin, homogeneous and isotropic conical shell with constant thickness is considered. Fig. 1 shows the schematic diagram of the conical shell. The two boundaries of the conical shell are simply supported (S-S). The corresponding Cartesian coordinates o-xyz and curvilinear surface coordinates O-nfg are also shown in Fig. 1. The curvilinear surface coordinates are limited to be orthogonal ones which coincide with the lines of principal curvature of the neutral surface. For conical shells, the lines of principal curvature of the neutral surface are the meridians (n-axis) and parallel circles (f-axis). For shell structures, the strain–displacement relationships are given by Soedel (1981)
1 @u1 u2 @A1 A1 þ þ u3 ; A1 ð1 þ g=R1 Þ @n A2 @f R1 1 @u2 u1 @A2 A2 ; ¼ þ þ u3 A2 ð1 þ g=R2 Þ @f A1 @n R2 @u3 A1 ð1 þ g=R1 Þ @ u1 A2 ð1 þ g=R2 Þ @ u2 þ ; ¼ ; e12 ¼ A2 ð1 þ g=R2 Þ @f A1 ð1 þ g=R1 Þ A1 ð1 þ g=R1 Þ @n A2 ð1 þ g=R2 Þ @g @ u1 1 @u3 ¼ A1 ð1 þ g=R1 Þ þ ; @ g A1 ð1 þ g=R1 Þ A1 ð1 þ g=R1 Þ @n @ u2 1 @u3 ¼ A2 ð1 þ g=R2 Þ þ ; @ g A2 ð1 þ g=R2 Þ A2 ð1 þ g=R2 Þ @f
e11 ¼ e22 e33 e13 e23
ð1Þ
where eij (i, j = 1, 2, 3) are the strains in which 1, 2 and 3 coincide with the n, f and g directions, ui (i = 1, 2, 3) are the displacements in the n, f and g directions, R1 and R2 are the radii of curvature of the two lines of principal curvatures in the neutral surface of the shell, and A1 and A2 are two variables. For conical shells, the radii of curvature R1 and R2 with regard to the meridian, i.e. n-axis, and the circle, i.e. f-axis, can be written as
R1 ¼ þ1;
R2 ¼ n tan a0 ;
ð2Þ
in which a0 is the semi-vertex cone angle of the conical shell. The two variables A1 and A2 are given by Soedel (1981)
A1 ¼ 1;
A2 ¼ n sin a0 :
ð3Þ
For thin shells, one can assume that the displacements in the n and f directions vary linearly through the shell thickness and the displacement in the g direction is independent of g, i.e.
u2 ðn; f; g; tÞ ¼ v ðn; f; tÞ þ gbðn; f; tÞ;
u1 ðn; f; g; tÞ ¼ uðn; f; tÞ þ gaðn; f; tÞ;
η
z O
u3 ðn; f; g; tÞ ¼ wðn; f; tÞ;
ζ
(a)
(b) l0
σ33
α0
ξ
ð4Þ
l
η
a1
ζ
O y
o
ξ
σ21 σ22
σ32 σ31 σ13 σ12
σ23 σ11
a2 x Fig. 1. The schematic diagram of a conical shell. (a) The geometry and the Cartesian and curvilinear surface coordinate systems; (b) the infinitesimal shell element and the corresponding stresses.
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
597
where u, v and w denote the displacements in the n, f and g directions in the neutral surface, and a and b are the rotations of the normal to the neutral surface about the f and n-axis and given by Soedel (1981)
a¼
u 1 @w ; R1 A1 @n
b¼
v R2
1 @w : A2 @f
ð5Þ
Substituting Eqs. (2)–(4) into Eqs. (1) and (5) yields:
a¼
e22
@w ; @n
v
b¼
1 @w ; n sin a0 @f
@u @2w g 2 ; @n @n
e11 ¼
n tan a0 ! 1 @v u w 1 @v 1 @ 2 w 1 @w ; ¼ þg 2 þ þ n sin a0 @f n n tan a0 n sin a0 tan a0 @f n2 sin2 a0 @f2 n @n
e12 ¼
e33 ¼ 0;
1 @u @ v v 2 @2w 1 @v 2 2 @w ; 2 vþ 2 þ þ þg n sin a0 @f @n n n sin a0 @n@f n tan a0 @n n tan a0 n sin a0 @f
e13 ¼ 0; e23 ¼
v n tan a0
ð6Þ
:
The relationships between the stresses and strains are written by
1
1
e11 ¼ ½r11 lðr22 þ r33 Þ; e22 ¼ ½r22 lðr11 þ r33 Þ; E E 1 r12 r13 r23 e33 ¼ ½r33 lðr11 þ r22 Þ; e12 ¼ ; e13 ¼ ; e23 ¼ ; E
G
G
G
ð7Þ
where r11, r22 and r33 are the normal stresses acting in the n, f and g directions, r12, r13 and r23 are the shear stresses in the curvilinear coordinates O-nfg as shown in Fig. 1b, E is the modulus of elasticity, l is the Poisson’s ratio, and G = E/2(1 + l) is the shearing modulus. For unloaded outer shell surfaces, r33 = 0. For loaded shells, r33 is usually equivalent in magnitude to the external load, which is relatively small in most cases compared with the other two normal stresses (Soedel, 1981). In this analysis, the effects of normal stress r33 are neglected. From Eq. (6) we know that e33 and e13 are zero. Thus solving Eq. (7) for the stresses yields
9 8 r11 > > > > > > = <
2
E=ð1 l2 Þ 6 2 r22 6 lE=ð1 l Þ ¼6 > 4 r12 > 0 > > > > ; : r23 0
9 8 e11 > > > > > > < e22 = : 7¼ e12 > G 05 > > > > > ; : e23 0 G 3
lE=ð1 l2 Þ 0 0 E=ð1 l2 Þ 0 0 7 7 0 0
ð8Þ
Hamilton’s principle with the Rayleigh–Ritz method will be used to determine the equation of motion of the conical shell. Hamilton’s principle is written by
Z
t2
dðT UÞdt þ
t1
Z
t2
dWdt ¼ 0;
ð9Þ
t1
where d() denotes the first variation, T, U and W are the kinetic energy, strain energy (Khatri and Asnani, 1995; Mecitog˘lu, 1996) and work, and t1 and t2 are the integration time limits. For thin shells, the influence of rotatory inertia can be neglected. So the kinetic energy is written by
1 T¼ 2
Z V
" 2 2 # 2 @u @v @w dV; q þ þ @t @t @t
ð10Þ
where q and V are the mass density and volume of the conical shell. The strain energy of the conical shell can be written as (Soedel, 1981)
U¼
1 2
Z
ðr11 e11 þ r22 e22 þ r12 e12 þ r23 e23 ÞdV:
ð11Þ
V
The infinitesimal volume dV in Eqs. (10) and (11) is given by Soedel (1981)
dV ¼ A1 A2 dndfdg ¼ n sin a0 dndfdg:
ð12Þ
The virtual work can be written by
dW ¼
Z A
ðq1 du þ q2 dv þ q3 dwÞdA;
ð13Þ
where A is the surface area of the conical shell, and q1, q2 and q3 are the distributed load components per unit area along the n, f, and g directions and are assumed to act on the neutral surface of the shell. The units of q1, q2 and q3 are [N/m2]. The infinitesimal area dA is given by
dA ¼ A1 A2 dndf ¼ n sin a0 dndf:
ð14Þ
598
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
In order to use the Rayleigh–Ritz method, the displacements u, coordinates: m X n X
uðn; f; tÞ ¼
i¼1
i¼1
wðn; f; tÞ ¼
and w should be expressed in terms of generalized
U ij ðn; fÞpij ðtÞ ¼ U T ðn; fÞpðtÞ;
ð15Þ
V ij ðn; fÞr ij ðtÞ ¼ V T ðn; fÞrðtÞ;
ð16Þ
W ij ðn; fÞsij ðtÞ ¼ W T ðn; fÞsðtÞ;
ð17Þ
j¼1
m X n X
v ðn; f; tÞ ¼
v
j¼1
m X n X i¼1
j¼1
where p, r and s are the generalized coordinates or modal coordinates, and U, V and W are the displacement shape functions or the principal vibration modes which must satisfy the geometric boundary conditions. They are written by
p ¼ ½p11 ; . . . ; p1n ; p21 ; . . . ; p2n ; . . . ; pm1 ; . . . ; pmn T ; T
¼ ½s11 ; . . . ; s1n ; s21 ; . . . ; s2n ; . . . ; sm1 ; . . . ; smn ;
r ¼ ½r 11 ; . . . ; r1n ; r 21 ; . . . ; r2n ; . . . ; r m1 ; . . . ; rmn T ;
s T
U ¼ ½U 11 ; . . . ; U 1n ; U 21 ; . . . ; U 2n ; . . . ; U m1 ; . . . ; U mn ; T
¼ ½V 11 ; . . . ; V 1n ; V 21 ; . . . ; V 2n ; . . . ; V m1 ; . . . ; V mn ;
V
W ¼ ½W 11 ; . . . ; W 1n ; W 21 ; . . . ; W 2n ; . . . ; W m1 ; . . . ; W mn T :
ð18Þ
Then the kinetic energy, strain energy and work are expressed in terms of the generalized coordinates and displacement shape functions. Substituting Eqs. (12), (15), (16), and (17) into Eq. (10) to
T¼
1 dpT dp 1 dr T dr 1 dsT ds þ þ M1 M2 M3 ; 2 dt dt 2 dt dt 2 dt dt
ð19Þ
where M1, M2 and M3 are the modal mass matrices of the conical shell and they are listed in Appendix A. Substituting Eqs. (6), (8), (12), (15), (16), and (17) into Eq. (11), the strain energy is written by
U¼
1 T 1 1 1 1 1 1 1 1 p K 1 p þ pT K 2 r þ pT K 3 s þ r T K T2 p þ rT K 4 r þ r T K 5 s þ sT K T3 p þ sT K T5 r þ sT K 6 s; 2 2 2 2 2 2 2 2 2
ð20Þ
where K1, K2, . . ., K6 are the modal stiffness matrices which are also presented in Appendix A. Substituting Eqs. (14)–(17) into Eq. (13), the virtual work is expressed as
dW ¼ q1 F q1 dp þ q2 F q2 dr þ q3 F q3 ds;
ð21Þ
where Fq1, Fq2 and Fq3 are the forcing matrices which are also given in Appendix A. Substituting Eqs. (19)–(21) into Eq. (9) and performing the variation operation in terms of p, r and s, the equations of motion of the conical shell can be obtained as 2
d p þ K 1 p þ K 2 r þ K 3 s ¼ F Tq1 q1 ; dt 2 2 d r M 2 2 þ K T2 p þ K 4 r þ K 5 s ¼ F Tq2 q2 ; dt 2 d s M 3 2 þ K T3 p þ K T5 r þ K 6 s ¼ F Tq3 q3 : dt
M1
ð22Þ ð23Þ ð24Þ
Rearranging the generalized coordinates p, r and s as
X ¼ ½ pT
rT
T sT ;
ð25Þ
then Eqs. (22)–(24) can be integrated as 2
Mt
d X þ KtX ¼ Q ; dt 2
ð26Þ
where Mt, Kt and Q are the generalized mass matrix, stiffness matrix and forcing matrix and written by
2
M1 6 Mt ¼ 4 0 0
0 M2 0
0
3
7 0 5; M3
2
K1 6 T Kt ¼ 4 K2 K T3
K2 K4 K T5
3 K3 K5 7 5;
Q ¼ ½ F q1 q1
F q2 q2
F q3 q3 T :
ð27Þ
K6
The general solution of the homogeneous differential equation of Eq. (26) can be expressed as
XðtÞ ¼ X 0 ekt ;
ð28Þ
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
599
where X0 is the eigenvector and k is the eigenvalue. Substituting Eq. (28) into the homogeneous differential equation of Eq. (26) leads to the following eigenvalue problem:
ðM t k2 þ K t ÞX 0 ¼ 0;
ð29Þ
from which the eigenvectors and eigenvalues can be obtained. The imaginary parts of the eigenvalues are the natural frequencies of the conical shell. In order to solve Eq. (26), we must present the formulations of the principal mode shapes U, V and W in Eqs. (15)–(17). Some types of vibration mode shapes of conical shells have been applied. For example, Liew et al. (2005) employed the kernel particle functions in hybridized form with harmonic functions to approximate the vibration modes. Lam and Li (1999) and Li (2000) employed a kind of displacement formulations of conical shells which are similar to those of cylindrical shells (Soedel, 1981). In their study, however, the displacement field has been expressed as the single term of trigonometric function, which can actually be regarded as a specific principal vibration, i.e. the vibration (m, n). In practice, when a structure is motivated by arbitrary external forces, any principal vibration may be motivated. The displacement field should be expressed as the superposition of all the principal vibrations (Soedel, 1981; Shabana, 1997; Clough and Penzien, 1993). So the displacements of conical shells can be written by Eqs. (15)–(17). In this paper, the vibration modes of conical shells similar to those of cylindrical shells (Soedel, 1981) are used. In the curvilinear surface coordinates O-nfg as shown in Fig. 1a, the principal mode shapes of conical shells with S-S boundaries can be expressed as
ipðn l0 Þ cosðjfÞ; U ij ðn; fÞ ¼ cos l l0 i ¼ 1; 2; . . . ; m;
ipðn l0 Þ V ij ðn; fÞ ¼ sin sinðjfÞ; l l0
ipðn l0 Þ W ij ðn; fÞ ¼ sin cosðjfÞ; l l0
j ¼ 1; 2; . . . ; n;
ð30Þ
where i and j are the wave numbers in the meridional and circumferential directions. For simply supported conical shells, the boundary conditions at both ends can be written as
v ðl0 ; f; tÞ ¼ v ðl; f; tÞ ¼ 0;
wðl0 ; f; tÞ ¼ wðl; f; tÞ ¼ 0;
N11 ðl0 ; f; tÞ ¼ N11 ðl; f; tÞ ¼ 0;
M 11 ðl0 ; f; tÞ ¼ M 11 ðl; f; tÞ ¼ 0;
ð31Þ ð32Þ
where N11 and M11 are the force and bending moment per unit length of neutral surface. They are given by
N11 ¼
Z
h=2
r11 dg; M11 ¼
h=2
Z
h=2
r11 gdg;
ð33Þ
h=2
where h is the thickness of the shell. Substituting r11 from Eq. (8) into Eq. (33), then Eq. (32) is changed as
@u 1 @v u w þl þ þ ¼ 0; at n ¼ l0 ; l; @n n sin a0 @f n n tan a0 " !# @2w 1 @v 1 @ 2 w 1 @w ¼ 0; at n ¼ l0 ; l; ¼D 2 þl 2 @n n sin a0 tan a0 @f n2 sin2 a0 @f2 n @n
N11 ¼ K M11
ð34Þ
3
where K ¼ Eh=ð1 l2 Þ and D ¼ Eh =½12ð1 l2 Þ are the membrane stiffness and bending stiffness. It is seen from Eqs. (30) and (31) that the principal mode shapes of conical shells satisfy accurately the geometric boundary conditions for the S-S boundaries. For the force boundary conditions, from Eqs. (30) and (34), it is observed that there are two terms which can not be satisfied to be zero. Namely, the force boundary conditions should be expressed as the following formulations:
u N11 ¼ K l ; n
M11 ¼ Dl
1 @w ; at n ¼ l0 ; l; n @n
ð35Þ
which are not satisfied to be exactly zero for the principal mode shapes, Eq. (30), at the two boundaries. are also smaller for n = l0 and l. Especially, when the conical shell is But for thin shells with small deformation, un and 1n @w @n not subjected to forces along n direction, i.e. q1 = 0, the displacement u can also be neglected. So based on the above analysis, we see that the mode shapes, Eq. (30), satisfies approximately the force boundary conditions for the S-S boundaries. However, the mode shapes given by Eq. (30) are very simple in form and convenient in implementation in engineering. The distributed loads are assumed to be harmonic and written by
q1 ðn; f; tÞ ¼ q10 sinxt;
q2 ðn; f; tÞ ¼ q20 sinxt; q3 ðn; f; tÞ ¼ q30 sinxt;
ð36Þ
where q10, q20 and q30 are the amplitudes and x is the frequency of the dynamic loads. Under the application of the external dynamic loads, the steady state solution of Eq. (26) can be written as
X i ðtÞ ¼ Ai sin xt;
i ¼ 1; 2; . . . ; 3mn;
ð37Þ
where Ai is the amplitude to be determined. Substituting Eq. (37) into Eq. (26) results in
ðK t x2 M t ÞA ¼ Q 0 ;
ð38Þ
600
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
where A ¼ ½A1 ; A2 ; :::; A3mn T and Q 0 ¼ ½F q1 q10 ; F q2 q20 ; F q3 q30 T . By solving Eq. (38), the amplitudes Ai of Xi(t) can be determined. Then, from Eqs. (15)–(17), (25), (30), (37), the steady state responses of the conical shell can be finally obtained. 3. Numerical simulations and discussions 3.1. Validation of the present method In order to validate the present methodology, comparisons with the results available in the open literature are made. In the numerical calculations, the non-dimensional frequency parameter is defined as (Lam and Li, 1999; Liew et al., 2005; Irie et al., 1984) Table 1 Comparisons of frequency parameter f for the conical shell with S-S boundaries (m = 1). n
2 3 4 5 6 7 8
a0 = 30°
a0 = 45°
a0 = 60°
Irie et al. (1984)
Lam and Li (1999)
Present
Irie et al. (1984)
Lam and Li (1999)
Present
Irie et al. (1984)
Lam and Li (1999)
Present
0.7910 0.7284 0.6352 0.5531 0.4949 0.4653 0.4645
0.8420 0.7376 0.6362 0.5528 0.4950 0.4661 0.4660
0.8431 0.7416 0.6419 0.5590 0.5008 0.4701 0.4687
0.6879 0.6973 0.6664 0.6304 0.6032 0.5918 0.5992
0.7655 0.7212 0.6739 0.6323 0.6035 0.5921 0.6001
0.7642 0.7211 0.6747 0.6336 0.6049 0.5928 0.6005
0.5722 0.6001 0.6054 0.6077 0.6159 0.6343 0.6650
0.6348 0.6238 0.6145 0.6111 0.6171 0.6350 0.6660
0.6342 0.6236 0.6146 0.6113 0.6172 0.6347 0.6653
Fig. 2. The amplitudes of the displacements in the n, f and g directions at the position (l0 + s/2, p/4) of the neutral surface of conical shell varying with the frequency x of the dynamical loads. (a) In the n direction; (b) in the f direction; (c) in the g direction.
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
f ¼ x0 a2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qð1 l2 Þ=E;
601
ð39Þ
where x0 is the natural frequency of the conical shell in radians per second. The conical shell is aluminum whose modulus of elasticity E = 70 GPa, mass density q = 2710 kg/m3 and Poisson’s ratio l = 0.3. The structural parameters are h = 0.004 m, h/ a2 = 0.01, (l l0)sin a0/a2 = 0.25. For the case of the meridional wave number m = 1 and semi-vertex cone angle a0 = 30°, 45° and 60°, the frequency parameters calculated by Eq. (39) are listed in Table 1 for different circumferential wave numbers. The corresponding results by Lam and Li (1999) and Irie et al. (1984) are also listed in Table 1. From (Table 1) we can see that the frequency parameters obtained by the present method are in good agreement with those in the open literature, which verifies the validity of the present analytical method. And the principal mode shapes expressed by Eq. (30) can be used for the conical shells with two simply supported boundaries. 3.2. The forced vibration responses The forced vibration responses of conical shell with two simply supported boundaries are calculated. The parameters of the conical shell model are the same as those used in Section 3.1. The semi-vertex cone angle is a0 = 30°. The position coordinates of the conical shell in the curvilinear surface coordinates O-nfg are l0 = 0.6 m and l = 0.8 m. So the length of the conical shell is s = l l0 = 0.2 m. The radii at the two ends are a1 = 0.3 m and a2 = 0.4 m. The amplitudes q10, q20 and q30 of the external dynamic loads are all set to be 1.0 MPa. In the calculation, m and n in Eqs. (15)–(17) are all set to be 3. The amplitudes of the displacements in the n, f and g directions at position (l0 + s/2, p/4) of the neutral surface of conical shell varying with the frequency x (Hz) of external dynamic loads are shown in Fig. 2. It is seen from Fig. 2 that there exist some peak values of the displacements in the frequency-response curves, which correspond to the resonant responses of the conical shell under the external dynamic loads. 4. Conclusions In this paper, Hamilton’s principle with the Rayleigh–Ritz method is used to derive the equation of motion of the conical shell. A set of simpler principal vibration modes of the conical shell are presented. By solving the eigenvalue problem of the equation of motion, the natural frequencies of the conical shell are obtained. By solving the equation of motion, the steady responses of forced vibration are gotten. Numerical comparisons with the results in the open literature are made to verify the validity of the present method. And the forced vibration responses of conical shell varying with the external dynamical loads are calculated. This method can also be used for other kinds of boundary conditions of the conical shell. By means of the present analytical method, we can further study the structural vibration control in the conical shells. Acknowledgments The authors appreciate the supports provided by the National Natural Science Foundation of China under Grant No. 10672017 for this work. Feng-Ming Li also acknowledges the supports provided by the China Postdoctoral Science Foundation, Heilongjiang Province Postdoctoral Science Foundation and the support provided by Japan Society for the Promotion of Science (JSPS). Appendix A.
The expressions of the modal mass, modal stiffness and forcing matrices in Eqs. (19)–(21) are given by
M 1 ¼ qh sin a0
Z 2p Z 0
M 3 ¼ qh sin a0
UU T ndndf;
l0
M 2 ¼ qh sin a0
Z 2p Z 0
l0
Z 2p Z 0
l
l
l
VV T ndndf;
l0
WW T ndndf;
! Z Z Ehsina0 2p l @U @U T @U T @U T T1 dndf K1 ¼ U þ lU n þ UU þ l @n @n n @n 1 l2 0 @n l0 Z 2p Z l Gh @U @U T 1 þ dndf; sina0 0 l0 @f @f n ! Z 2p Z l Eh @V T 1 @U @V T þ dndf U l K2 ¼ 1 l2 0 @n @f @f n l0 ! Z 2p Z l @U @V T @U T 1 dndf; þ Gh V @f @n @f n l0 0 Z Z Ehcosa0 2p l @U T T1 K3 ¼ UW l þ W dndf; n @n 1 l2 0 l0
602
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
# Z 2p Z l " 2 Eh 1 @V @V T 1 h @V @V T 1 dndf þ 1 l2 sina0 0 @f @f n 12tan2 a0 @f @f n3 l0 ! Z 2p Z l @V @V T @V T @V T T1 dndf þ Ghsina0 V V nþVV @n @n n @n @n l0 0 " Z Z Ghsina0 2p l 1 @V @V T 1 2 T1 þ VV þ h n 12 @n @n n tan2 a0 0 l0 !# T 1 1 @V T 1 1 @V 1 T 1 þ VV 3 dndf; V 2 V 3 6 @n 6 @n n2 n n Z 2p Z l Eh 1 @V W T K5 ¼ dndf 1 l2 tana0 0 n l0 @f ! Z 2p Z l 3 Eh @V @W T 1 @V @ 2 W T 1 1 @V @ 2 W T 1 þ dndf l þ @f @n n2 @f @n2 n sin2 a0 @f @f2 n3 12ð1 l2 Þtana0 0 l0 ! Z 2p Z l 3 Gh @V @W T 1 @V @ 2 W T 1 2V @ 2 W T 2V @W T dndf; þ þ 3 @n @f n2 @n @n@f n n2 @n@f 6tana0 0 @f n l0 ! Z 2p Z l 3 Eh sina0 @2W @2W T @W @W T 1 @ 2 W @W T @W @ 2 W T K6 ¼ dndf nþ þl þl @n @n n @n @n2 12ð1 l2 Þ 0 @n2 @n2 @n2 @n l0 Z 2p Z l 3 Eh 1 @2W @2W T 1 @2W @2W T 1 @2W @2W T 1 þ þ l l þ 2 2 12ð1 l2 Þsina0 0 @f2 n3 @n2 @f2 n @f2 @n2 n sin a0 @f l0 ! Z Z 2p l @ 2 W @W T 1 @W @ 2 W T 1 Eh sina0 WW T dndf þ þ þ dndf 2 2 2 2 2 2 @n 1 @n l n tan a0 0 @f n @f n l0 ! Z 2p Z l 3 Gh @2W @2W T @W @W T 1 @ 2 W @W T @W @ 2 W T 1 nþ dndf; þ @n@f @n@f @f @f n @n@f @f @f @n@f n2 3sina0 0 l0 Z 2p Z l Z 2p Z l Z 2p Z l U T ndndf; F q2 ¼ sina0 V T ndndf; F q3 ¼ sina0 W T ndndf: F q1 ¼ sina0 K4 ¼
0
l0
0
l0
0
ð40Þ
l0
References Chai, W.K., Han, Y., Higuchi, K., Tzou, H.S., 2006. Micro-actuation characteristics of rocket conical shell sections. Journal of Sound and Vibration 293, 286– 298. Chang, C.H., 1978. Membrane vibrations of conical shells. Journal of Sound and Vibration 60, 335–343. Clough, R.W., Penzien, J., 1993. Dynamics of Structures, second ed. McGraw-Hill Inc.. Fares, M.E., Youssif, Y.G., Alamir, A.E., 2004. Design and control optimization of composite laminated truncated conical shells for minimum dynamic response including transverse shear deformation. Composite Structures 64, 139–150. Goldberg, J.E., Bogdanoff, J.L., Marcus, L., 1960. On the calculation of the axisymmetric modes and frequencies of conical shells. The Journal of the Acoustical Society of America 32, 738–742. Irie, T., Yamada, G., Tanaka, K., 1984. Natural frequencies of truncated conical shells. Journal of Sound and Vibration 92, 447–453. Khatri, K.N., Asnani, N.T., 1995. Vibration and damping analysis of multilayered conical shells. Composite Structures 33, 143–157. Lam, K.Y., Li, Hua, 1999. Influence of boundary conditions on the frequency characteristics of a rotating truncated circular conical shell. Journal of Sound and Vibration 223, 171–195. Li, Hua, 2000. Frequency analysis of rotating truncated circular orthotropic conical shells with different boundary conditions. Composites Science and Technology 60, 2945–2955. Liew, K.M., Lim, M.K., Lim, C.W., Li, D.B., Zhang, Y.R., 1995. Effects of initial twist and thickness variation on the vibration behaviour of shallow conical shells. Journal of Sound and Vibration 180, 271–296. Liew, K.M., Ng, T.Y., Zhao, X., 2005. Free vibration analysis of conical shells via the element-free kp-Ritz method. Journal of Sound and Vibration 281, 627– 645. Lim, C.W., Liew, K.M., 1995. Vibratory behaviour of shallow conical shells by a global Ritz formulation. Engineering Structures 17, 63–70. Lim, C.W., Liew, K.M., 1996. Vibration of shallow conical shells with shear flexibility: a first-order theory. International Journal of Solids and Structures 33, 451–468. Mecitog˘lu, Z., 1996. Vibration characteristics of a stiffened conical shell. Journal of Sound and Vibration 197, 191–206. Serpico, J.C., 1963. Elastic stability of orthotropic conical and cylindrical shells subjected to axisymmetric loading conditions. AIAA Journal 1, 128–137. Shabana, A.A., 1997. Vibration of Discrete and Continuous Systems. Springer-Verlag Inc., New York. Soedel, W., 1981. Vibrations of Shells and Plates. Marcel Dekker Inc., New York. Sofiyev, A.H., Omurtag, M.H., Schnack, E., 2008. The vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure. Journal of Sound and Vibration. doi:10.1016/j.jsv.2008.06.033.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
The calculations of natural frequencies and forced vibration responses of conical shell using the Rayleigh–Ritz method Feng-Ming Li a,b,*, Kikuo Kishimoto b, Wen-Hu Huang a a b
School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, PR China Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan
a r t i c l e
i n f o
Article history: Received 9 October 2008 Received in revised form 15 January 2009 Available online 20 February 2009
Keywords: Conical shell Forced vibration analysis Hamilton’s principle Rayleigh–Ritz method
a b s t r a c t An effective method for analyzing the forced vibration of conical shell is presented. Hamilton’s principle with the Rayleigh–Ritz method is employed to derive the equation of motion of the conical shell. A set of simpler principal vibration modes of the conical shell with two simply supported boundaries are presented. The natural frequencies of conical shell can be obtained by solving eigenvalue problem of the equation of motion and the steady responses of forced vibration can also be obtained by solving the equation of motion. Numerical comparisons with the results in the open literature are made to validate the present methodology. Moreover, the forced vibration responses of a conical shell are also calculated. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Conical shells are widely used in many engineering applications. It is necessary to perform the vibration analysis of conical shells for the purpose of dynamical design, vibration control and so on. Up to now, many papers on the vibration problems of conical shells have been published (Goldberg et al., 1960; Serpico, 1963; Chang, 1978; Khatri and Asnani, 1995). Lam and Li (1999) and Li (2000) studied the frequency characteristics of free vibration of rotating conical shell using the Galerkin method. Fares et al. (2004) investigated the design and active vibration control of composite laminated conical shells. They used the Liapunov–Bellman theory to obtain the controlled deflections of the shells. Liew et al. (2005) analyzed the free vibration of thin conical shells using the element-free kp-Ritz method and discussed the frequency properties under different parameters. Chai et al. (2006) studied the spatially distributed microscopic control characteristics of distributed actuator patches on a rocket conical shell. Sofiyev et al. (2008) studied the vibration and stability of orthotropic conical shells with non-homogeneous material properties under a hydrostatic pressure. It should be noted that Liew and Lim (Liew et al., 1995; Lim and Liew, 1996, 1995) systematically studied the free vibration of shallow conical shells using the pb-2 Ritz method. They employed the admissible pb-2 shape functions to approximate the three-dimensional displacements, and calculated comprehensively the dimensionless natural frequencies and the vibration modes of shallow conical shells. Although numerous studies on the dynamic problems of the conical shells have been published, the forced vibration problems of the conical shells should be thoroughly studied. Through the forced vibration analysis, an effective way for calculating the forced vibration responses can be obtained and further used in the vibration control and dynamic designs of the conical shells. Normally, the equation of motion of the conical shell is very complicated and some of the coefficients of the equation of motion are variables (Soedel, 1981), which makes it difficult to analytically solve the equation of motion of the conical shell. * Corresponding author. Address: School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, PR China. Tel.: +86 451 86414479. E-mail address: [email protected] (F.-M. Li). 0093-6413/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2009.02.003
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F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
In this study, an effective method for the forced vibration analysis of conical shells is presented. Using Hamilton’s principle with the Rayleigh–Ritz method, the equation of motion of the conical shell is derived. A set of simple formulations of the principal mode shapes is employed and verified to be effective by numerical simulations. Solving the eigenvalue problem of the equation of motion, the natural frequencies of the conical shell can be obtained. And the steady responses of the forced vibration of conical shells can also be obtained by solving the equation of motion. 2. Equation of motion A thin, homogeneous and isotropic conical shell with constant thickness is considered. Fig. 1 shows the schematic diagram of the conical shell. The two boundaries of the conical shell are simply supported (S-S). The corresponding Cartesian coordinates o-xyz and curvilinear surface coordinates O-nfg are also shown in Fig. 1. The curvilinear surface coordinates are limited to be orthogonal ones which coincide with the lines of principal curvature of the neutral surface. For conical shells, the lines of principal curvature of the neutral surface are the meridians (n-axis) and parallel circles (f-axis). For shell structures, the strain–displacement relationships are given by Soedel (1981)
1 @u1 u2 @A1 A1 þ þ u3 ; A1 ð1 þ g=R1 Þ @n A2 @f R1 1 @u2 u1 @A2 A2 ; ¼ þ þ u3 A2 ð1 þ g=R2 Þ @f A1 @n R2 @u3 A1 ð1 þ g=R1 Þ @ u1 A2 ð1 þ g=R2 Þ @ u2 þ ; ¼ ; e12 ¼ A2 ð1 þ g=R2 Þ @f A1 ð1 þ g=R1 Þ A1 ð1 þ g=R1 Þ @n A2 ð1 þ g=R2 Þ @g @ u1 1 @u3 ¼ A1 ð1 þ g=R1 Þ þ ; @ g A1 ð1 þ g=R1 Þ A1 ð1 þ g=R1 Þ @n @ u2 1 @u3 ¼ A2 ð1 þ g=R2 Þ þ ; @ g A2 ð1 þ g=R2 Þ A2 ð1 þ g=R2 Þ @f
e11 ¼ e22 e33 e13 e23
ð1Þ
where eij (i, j = 1, 2, 3) are the strains in which 1, 2 and 3 coincide with the n, f and g directions, ui (i = 1, 2, 3) are the displacements in the n, f and g directions, R1 and R2 are the radii of curvature of the two lines of principal curvatures in the neutral surface of the shell, and A1 and A2 are two variables. For conical shells, the radii of curvature R1 and R2 with regard to the meridian, i.e. n-axis, and the circle, i.e. f-axis, can be written as
R1 ¼ þ1;
R2 ¼ n tan a0 ;
ð2Þ
in which a0 is the semi-vertex cone angle of the conical shell. The two variables A1 and A2 are given by Soedel (1981)
A1 ¼ 1;
A2 ¼ n sin a0 :
ð3Þ
For thin shells, one can assume that the displacements in the n and f directions vary linearly through the shell thickness and the displacement in the g direction is independent of g, i.e.
u2 ðn; f; g; tÞ ¼ v ðn; f; tÞ þ gbðn; f; tÞ;
u1 ðn; f; g; tÞ ¼ uðn; f; tÞ þ gaðn; f; tÞ;
η
z O
u3 ðn; f; g; tÞ ¼ wðn; f; tÞ;
ζ
(a)
(b) l0
σ33
α0
ξ
ð4Þ
l
η
a1
ζ
O y
o
ξ
σ21 σ22
σ32 σ31 σ13 σ12
σ23 σ11
a2 x Fig. 1. The schematic diagram of a conical shell. (a) The geometry and the Cartesian and curvilinear surface coordinate systems; (b) the infinitesimal shell element and the corresponding stresses.
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
597
where u, v and w denote the displacements in the n, f and g directions in the neutral surface, and a and b are the rotations of the normal to the neutral surface about the f and n-axis and given by Soedel (1981)
a¼
u 1 @w ; R1 A1 @n
b¼
v R2
1 @w : A2 @f
ð5Þ
Substituting Eqs. (2)–(4) into Eqs. (1) and (5) yields:
a¼
e22
@w ; @n
v
b¼
1 @w ; n sin a0 @f
@u @2w g 2 ; @n @n
e11 ¼
n tan a0 ! 1 @v u w 1 @v 1 @ 2 w 1 @w ; ¼ þg 2 þ þ n sin a0 @f n n tan a0 n sin a0 tan a0 @f n2 sin2 a0 @f2 n @n
e12 ¼
e33 ¼ 0;
1 @u @ v v 2 @2w 1 @v 2 2 @w ; 2 vþ 2 þ þ þg n sin a0 @f @n n n sin a0 @n@f n tan a0 @n n tan a0 n sin a0 @f
e13 ¼ 0; e23 ¼
v n tan a0
ð6Þ
:
The relationships between the stresses and strains are written by
1
1
e11 ¼ ½r11 lðr22 þ r33 Þ; e22 ¼ ½r22 lðr11 þ r33 Þ; E E 1 r12 r13 r23 e33 ¼ ½r33 lðr11 þ r22 Þ; e12 ¼ ; e13 ¼ ; e23 ¼ ; E
G
G
G
ð7Þ
where r11, r22 and r33 are the normal stresses acting in the n, f and g directions, r12, r13 and r23 are the shear stresses in the curvilinear coordinates O-nfg as shown in Fig. 1b, E is the modulus of elasticity, l is the Poisson’s ratio, and G = E/2(1 + l) is the shearing modulus. For unloaded outer shell surfaces, r33 = 0. For loaded shells, r33 is usually equivalent in magnitude to the external load, which is relatively small in most cases compared with the other two normal stresses (Soedel, 1981). In this analysis, the effects of normal stress r33 are neglected. From Eq. (6) we know that e33 and e13 are zero. Thus solving Eq. (7) for the stresses yields
9 8 r11 > > > > > > = <
2
E=ð1 l2 Þ 6 2 r22 6 lE=ð1 l Þ ¼6 > 4 r12 > 0 > > > > ; : r23 0
9 8 e11 > > > > > > < e22 = : 7¼ e12 > G 05 > > > > > ; : e23 0 G 3
lE=ð1 l2 Þ 0 0 E=ð1 l2 Þ 0 0 7 7 0 0
ð8Þ
Hamilton’s principle with the Rayleigh–Ritz method will be used to determine the equation of motion of the conical shell. Hamilton’s principle is written by
Z
t2
dðT UÞdt þ
t1
Z
t2
dWdt ¼ 0;
ð9Þ
t1
where d() denotes the first variation, T, U and W are the kinetic energy, strain energy (Khatri and Asnani, 1995; Mecitog˘lu, 1996) and work, and t1 and t2 are the integration time limits. For thin shells, the influence of rotatory inertia can be neglected. So the kinetic energy is written by
1 T¼ 2
Z V
" 2 2 # 2 @u @v @w dV; q þ þ @t @t @t
ð10Þ
where q and V are the mass density and volume of the conical shell. The strain energy of the conical shell can be written as (Soedel, 1981)
U¼
1 2
Z
ðr11 e11 þ r22 e22 þ r12 e12 þ r23 e23 ÞdV:
ð11Þ
V
The infinitesimal volume dV in Eqs. (10) and (11) is given by Soedel (1981)
dV ¼ A1 A2 dndfdg ¼ n sin a0 dndfdg:
ð12Þ
The virtual work can be written by
dW ¼
Z A
ðq1 du þ q2 dv þ q3 dwÞdA;
ð13Þ
where A is the surface area of the conical shell, and q1, q2 and q3 are the distributed load components per unit area along the n, f, and g directions and are assumed to act on the neutral surface of the shell. The units of q1, q2 and q3 are [N/m2]. The infinitesimal area dA is given by
dA ¼ A1 A2 dndf ¼ n sin a0 dndf:
ð14Þ
598
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
In order to use the Rayleigh–Ritz method, the displacements u, coordinates: m X n X
uðn; f; tÞ ¼
i¼1
i¼1
wðn; f; tÞ ¼
and w should be expressed in terms of generalized
U ij ðn; fÞpij ðtÞ ¼ U T ðn; fÞpðtÞ;
ð15Þ
V ij ðn; fÞr ij ðtÞ ¼ V T ðn; fÞrðtÞ;
ð16Þ
W ij ðn; fÞsij ðtÞ ¼ W T ðn; fÞsðtÞ;
ð17Þ
j¼1
m X n X
v ðn; f; tÞ ¼
v
j¼1
m X n X i¼1
j¼1
where p, r and s are the generalized coordinates or modal coordinates, and U, V and W are the displacement shape functions or the principal vibration modes which must satisfy the geometric boundary conditions. They are written by
p ¼ ½p11 ; . . . ; p1n ; p21 ; . . . ; p2n ; . . . ; pm1 ; . . . ; pmn T ; T
¼ ½s11 ; . . . ; s1n ; s21 ; . . . ; s2n ; . . . ; sm1 ; . . . ; smn ;
r ¼ ½r 11 ; . . . ; r1n ; r 21 ; . . . ; r2n ; . . . ; r m1 ; . . . ; rmn T ;
s T
U ¼ ½U 11 ; . . . ; U 1n ; U 21 ; . . . ; U 2n ; . . . ; U m1 ; . . . ; U mn ; T
¼ ½V 11 ; . . . ; V 1n ; V 21 ; . . . ; V 2n ; . . . ; V m1 ; . . . ; V mn ;
V
W ¼ ½W 11 ; . . . ; W 1n ; W 21 ; . . . ; W 2n ; . . . ; W m1 ; . . . ; W mn T :
ð18Þ
Then the kinetic energy, strain energy and work are expressed in terms of the generalized coordinates and displacement shape functions. Substituting Eqs. (12), (15), (16), and (17) into Eq. (10) to
T¼
1 dpT dp 1 dr T dr 1 dsT ds þ þ M1 M2 M3 ; 2 dt dt 2 dt dt 2 dt dt
ð19Þ
where M1, M2 and M3 are the modal mass matrices of the conical shell and they are listed in Appendix A. Substituting Eqs. (6), (8), (12), (15), (16), and (17) into Eq. (11), the strain energy is written by
U¼
1 T 1 1 1 1 1 1 1 1 p K 1 p þ pT K 2 r þ pT K 3 s þ r T K T2 p þ rT K 4 r þ r T K 5 s þ sT K T3 p þ sT K T5 r þ sT K 6 s; 2 2 2 2 2 2 2 2 2
ð20Þ
where K1, K2, . . ., K6 are the modal stiffness matrices which are also presented in Appendix A. Substituting Eqs. (14)–(17) into Eq. (13), the virtual work is expressed as
dW ¼ q1 F q1 dp þ q2 F q2 dr þ q3 F q3 ds;
ð21Þ
where Fq1, Fq2 and Fq3 are the forcing matrices which are also given in Appendix A. Substituting Eqs. (19)–(21) into Eq. (9) and performing the variation operation in terms of p, r and s, the equations of motion of the conical shell can be obtained as 2
d p þ K 1 p þ K 2 r þ K 3 s ¼ F Tq1 q1 ; dt 2 2 d r M 2 2 þ K T2 p þ K 4 r þ K 5 s ¼ F Tq2 q2 ; dt 2 d s M 3 2 þ K T3 p þ K T5 r þ K 6 s ¼ F Tq3 q3 : dt
M1
ð22Þ ð23Þ ð24Þ
Rearranging the generalized coordinates p, r and s as
X ¼ ½ pT
rT
T sT ;
ð25Þ
then Eqs. (22)–(24) can be integrated as 2
Mt
d X þ KtX ¼ Q ; dt 2
ð26Þ
where Mt, Kt and Q are the generalized mass matrix, stiffness matrix and forcing matrix and written by
2
M1 6 Mt ¼ 4 0 0
0 M2 0
0
3
7 0 5; M3
2
K1 6 T Kt ¼ 4 K2 K T3
K2 K4 K T5
3 K3 K5 7 5;
Q ¼ ½ F q1 q1
F q2 q2
F q3 q3 T :
ð27Þ
K6
The general solution of the homogeneous differential equation of Eq. (26) can be expressed as
XðtÞ ¼ X 0 ekt ;
ð28Þ
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
599
where X0 is the eigenvector and k is the eigenvalue. Substituting Eq. (28) into the homogeneous differential equation of Eq. (26) leads to the following eigenvalue problem:
ðM t k2 þ K t ÞX 0 ¼ 0;
ð29Þ
from which the eigenvectors and eigenvalues can be obtained. The imaginary parts of the eigenvalues are the natural frequencies of the conical shell. In order to solve Eq. (26), we must present the formulations of the principal mode shapes U, V and W in Eqs. (15)–(17). Some types of vibration mode shapes of conical shells have been applied. For example, Liew et al. (2005) employed the kernel particle functions in hybridized form with harmonic functions to approximate the vibration modes. Lam and Li (1999) and Li (2000) employed a kind of displacement formulations of conical shells which are similar to those of cylindrical shells (Soedel, 1981). In their study, however, the displacement field has been expressed as the single term of trigonometric function, which can actually be regarded as a specific principal vibration, i.e. the vibration (m, n). In practice, when a structure is motivated by arbitrary external forces, any principal vibration may be motivated. The displacement field should be expressed as the superposition of all the principal vibrations (Soedel, 1981; Shabana, 1997; Clough and Penzien, 1993). So the displacements of conical shells can be written by Eqs. (15)–(17). In this paper, the vibration modes of conical shells similar to those of cylindrical shells (Soedel, 1981) are used. In the curvilinear surface coordinates O-nfg as shown in Fig. 1a, the principal mode shapes of conical shells with S-S boundaries can be expressed as
ipðn l0 Þ cosðjfÞ; U ij ðn; fÞ ¼ cos l l0 i ¼ 1; 2; . . . ; m;
ipðn l0 Þ V ij ðn; fÞ ¼ sin sinðjfÞ; l l0
ipðn l0 Þ W ij ðn; fÞ ¼ sin cosðjfÞ; l l0
j ¼ 1; 2; . . . ; n;
ð30Þ
where i and j are the wave numbers in the meridional and circumferential directions. For simply supported conical shells, the boundary conditions at both ends can be written as
v ðl0 ; f; tÞ ¼ v ðl; f; tÞ ¼ 0;
wðl0 ; f; tÞ ¼ wðl; f; tÞ ¼ 0;
N11 ðl0 ; f; tÞ ¼ N11 ðl; f; tÞ ¼ 0;
M 11 ðl0 ; f; tÞ ¼ M 11 ðl; f; tÞ ¼ 0;
ð31Þ ð32Þ
where N11 and M11 are the force and bending moment per unit length of neutral surface. They are given by
N11 ¼
Z
h=2
r11 dg; M11 ¼
h=2
Z
h=2
r11 gdg;
ð33Þ
h=2
where h is the thickness of the shell. Substituting r11 from Eq. (8) into Eq. (33), then Eq. (32) is changed as
@u 1 @v u w þl þ þ ¼ 0; at n ¼ l0 ; l; @n n sin a0 @f n n tan a0 " !# @2w 1 @v 1 @ 2 w 1 @w ¼ 0; at n ¼ l0 ; l; ¼D 2 þl 2 @n n sin a0 tan a0 @f n2 sin2 a0 @f2 n @n
N11 ¼ K M11
ð34Þ
3
where K ¼ Eh=ð1 l2 Þ and D ¼ Eh =½12ð1 l2 Þ are the membrane stiffness and bending stiffness. It is seen from Eqs. (30) and (31) that the principal mode shapes of conical shells satisfy accurately the geometric boundary conditions for the S-S boundaries. For the force boundary conditions, from Eqs. (30) and (34), it is observed that there are two terms which can not be satisfied to be zero. Namely, the force boundary conditions should be expressed as the following formulations:
u N11 ¼ K l ; n
M11 ¼ Dl
1 @w ; at n ¼ l0 ; l; n @n
ð35Þ
which are not satisfied to be exactly zero for the principal mode shapes, Eq. (30), at the two boundaries. are also smaller for n = l0 and l. Especially, when the conical shell is But for thin shells with small deformation, un and 1n @w @n not subjected to forces along n direction, i.e. q1 = 0, the displacement u can also be neglected. So based on the above analysis, we see that the mode shapes, Eq. (30), satisfies approximately the force boundary conditions for the S-S boundaries. However, the mode shapes given by Eq. (30) are very simple in form and convenient in implementation in engineering. The distributed loads are assumed to be harmonic and written by
q1 ðn; f; tÞ ¼ q10 sinxt;
q2 ðn; f; tÞ ¼ q20 sinxt; q3 ðn; f; tÞ ¼ q30 sinxt;
ð36Þ
where q10, q20 and q30 are the amplitudes and x is the frequency of the dynamic loads. Under the application of the external dynamic loads, the steady state solution of Eq. (26) can be written as
X i ðtÞ ¼ Ai sin xt;
i ¼ 1; 2; . . . ; 3mn;
ð37Þ
where Ai is the amplitude to be determined. Substituting Eq. (37) into Eq. (26) results in
ðK t x2 M t ÞA ¼ Q 0 ;
ð38Þ
600
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
where A ¼ ½A1 ; A2 ; :::; A3mn T and Q 0 ¼ ½F q1 q10 ; F q2 q20 ; F q3 q30 T . By solving Eq. (38), the amplitudes Ai of Xi(t) can be determined. Then, from Eqs. (15)–(17), (25), (30), (37), the steady state responses of the conical shell can be finally obtained. 3. Numerical simulations and discussions 3.1. Validation of the present method In order to validate the present methodology, comparisons with the results available in the open literature are made. In the numerical calculations, the non-dimensional frequency parameter is defined as (Lam and Li, 1999; Liew et al., 2005; Irie et al., 1984) Table 1 Comparisons of frequency parameter f for the conical shell with S-S boundaries (m = 1). n
2 3 4 5 6 7 8
a0 = 30°
a0 = 45°
a0 = 60°
Irie et al. (1984)
Lam and Li (1999)
Present
Irie et al. (1984)
Lam and Li (1999)
Present
Irie et al. (1984)
Lam and Li (1999)
Present
0.7910 0.7284 0.6352 0.5531 0.4949 0.4653 0.4645
0.8420 0.7376 0.6362 0.5528 0.4950 0.4661 0.4660
0.8431 0.7416 0.6419 0.5590 0.5008 0.4701 0.4687
0.6879 0.6973 0.6664 0.6304 0.6032 0.5918 0.5992
0.7655 0.7212 0.6739 0.6323 0.6035 0.5921 0.6001
0.7642 0.7211 0.6747 0.6336 0.6049 0.5928 0.6005
0.5722 0.6001 0.6054 0.6077 0.6159 0.6343 0.6650
0.6348 0.6238 0.6145 0.6111 0.6171 0.6350 0.6660
0.6342 0.6236 0.6146 0.6113 0.6172 0.6347 0.6653
Fig. 2. The amplitudes of the displacements in the n, f and g directions at the position (l0 + s/2, p/4) of the neutral surface of conical shell varying with the frequency x of the dynamical loads. (a) In the n direction; (b) in the f direction; (c) in the g direction.
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
f ¼ x0 a2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qð1 l2 Þ=E;
601
ð39Þ
where x0 is the natural frequency of the conical shell in radians per second. The conical shell is aluminum whose modulus of elasticity E = 70 GPa, mass density q = 2710 kg/m3 and Poisson’s ratio l = 0.3. The structural parameters are h = 0.004 m, h/ a2 = 0.01, (l l0)sin a0/a2 = 0.25. For the case of the meridional wave number m = 1 and semi-vertex cone angle a0 = 30°, 45° and 60°, the frequency parameters calculated by Eq. (39) are listed in Table 1 for different circumferential wave numbers. The corresponding results by Lam and Li (1999) and Irie et al. (1984) are also listed in Table 1. From (Table 1) we can see that the frequency parameters obtained by the present method are in good agreement with those in the open literature, which verifies the validity of the present analytical method. And the principal mode shapes expressed by Eq. (30) can be used for the conical shells with two simply supported boundaries. 3.2. The forced vibration responses The forced vibration responses of conical shell with two simply supported boundaries are calculated. The parameters of the conical shell model are the same as those used in Section 3.1. The semi-vertex cone angle is a0 = 30°. The position coordinates of the conical shell in the curvilinear surface coordinates O-nfg are l0 = 0.6 m and l = 0.8 m. So the length of the conical shell is s = l l0 = 0.2 m. The radii at the two ends are a1 = 0.3 m and a2 = 0.4 m. The amplitudes q10, q20 and q30 of the external dynamic loads are all set to be 1.0 MPa. In the calculation, m and n in Eqs. (15)–(17) are all set to be 3. The amplitudes of the displacements in the n, f and g directions at position (l0 + s/2, p/4) of the neutral surface of conical shell varying with the frequency x (Hz) of external dynamic loads are shown in Fig. 2. It is seen from Fig. 2 that there exist some peak values of the displacements in the frequency-response curves, which correspond to the resonant responses of the conical shell under the external dynamic loads. 4. Conclusions In this paper, Hamilton’s principle with the Rayleigh–Ritz method is used to derive the equation of motion of the conical shell. A set of simpler principal vibration modes of the conical shell are presented. By solving the eigenvalue problem of the equation of motion, the natural frequencies of the conical shell are obtained. By solving the equation of motion, the steady responses of forced vibration are gotten. Numerical comparisons with the results in the open literature are made to verify the validity of the present method. And the forced vibration responses of conical shell varying with the external dynamical loads are calculated. This method can also be used for other kinds of boundary conditions of the conical shell. By means of the present analytical method, we can further study the structural vibration control in the conical shells. Acknowledgments The authors appreciate the supports provided by the National Natural Science Foundation of China under Grant No. 10672017 for this work. Feng-Ming Li also acknowledges the supports provided by the China Postdoctoral Science Foundation, Heilongjiang Province Postdoctoral Science Foundation and the support provided by Japan Society for the Promotion of Science (JSPS). Appendix A.
The expressions of the modal mass, modal stiffness and forcing matrices in Eqs. (19)–(21) are given by
M 1 ¼ qh sin a0
Z 2p Z 0
M 3 ¼ qh sin a0
UU T ndndf;
l0
M 2 ¼ qh sin a0
Z 2p Z 0
l0
Z 2p Z 0
l
l
l
VV T ndndf;
l0
WW T ndndf;
! Z Z Ehsina0 2p l @U @U T @U T @U T T1 dndf K1 ¼ U þ lU n þ UU þ l @n @n n @n 1 l2 0 @n l0 Z 2p Z l Gh @U @U T 1 þ dndf; sina0 0 l0 @f @f n ! Z 2p Z l Eh @V T 1 @U @V T þ dndf U l K2 ¼ 1 l2 0 @n @f @f n l0 ! Z 2p Z l @U @V T @U T 1 dndf; þ Gh V @f @n @f n l0 0 Z Z Ehcosa0 2p l @U T T1 K3 ¼ UW l þ W dndf; n @n 1 l2 0 l0
602
F.-M. Li et al. / Mechanics Research Communications 36 (2009) 595–602
# Z 2p Z l " 2 Eh 1 @V @V T 1 h @V @V T 1 dndf þ 1 l2 sina0 0 @f @f n 12tan2 a0 @f @f n3 l0 ! Z 2p Z l @V @V T @V T @V T T1 dndf þ Ghsina0 V V nþVV @n @n n @n @n l0 0 " Z Z Ghsina0 2p l 1 @V @V T 1 2 T1 þ VV þ h n 12 @n @n n tan2 a0 0 l0 !# T 1 1 @V T 1 1 @V 1 T 1 þ VV 3 dndf; V 2 V 3 6 @n 6 @n n2 n n Z 2p Z l Eh 1 @V W T K5 ¼ dndf 1 l2 tana0 0 n l0 @f ! Z 2p Z l 3 Eh @V @W T 1 @V @ 2 W T 1 1 @V @ 2 W T 1 þ dndf l þ @f @n n2 @f @n2 n sin2 a0 @f @f2 n3 12ð1 l2 Þtana0 0 l0 ! Z 2p Z l 3 Gh @V @W T 1 @V @ 2 W T 1 2V @ 2 W T 2V @W T dndf; þ þ 3 @n @f n2 @n @n@f n n2 @n@f 6tana0 0 @f n l0 ! Z 2p Z l 3 Eh sina0 @2W @2W T @W @W T 1 @ 2 W @W T @W @ 2 W T K6 ¼ dndf nþ þl þl @n @n n @n @n2 12ð1 l2 Þ 0 @n2 @n2 @n2 @n l0 Z 2p Z l 3 Eh 1 @2W @2W T 1 @2W @2W T 1 @2W @2W T 1 þ þ l l þ 2 2 12ð1 l2 Þsina0 0 @f2 n3 @n2 @f2 n @f2 @n2 n sin a0 @f l0 ! Z Z 2p l @ 2 W @W T 1 @W @ 2 W T 1 Eh sina0 WW T dndf þ þ þ dndf 2 2 2 2 2 2 @n 1 @n l n tan a0 0 @f n @f n l0 ! Z 2p Z l 3 Gh @2W @2W T @W @W T 1 @ 2 W @W T @W @ 2 W T 1 nþ dndf; þ @n@f @n@f @f @f n @n@f @f @f @n@f n2 3sina0 0 l0 Z 2p Z l Z 2p Z l Z 2p Z l U T ndndf; F q2 ¼ sina0 V T ndndf; F q3 ¼ sina0 W T ndndf: F q1 ¼ sina0 K4 ¼
0
l0
0
l0
0
ð40Þ
l0
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