Nonlinear hydroelastic vibration of a cylindrical tank with an elastic bottom containing liquid-analysis using harmonic balance method

Thin-Walled Structures 34 (1999) 233–260 www.elsevier.com/locate/tws

Nonlinear hydroelastic vibration of a cylindrical tank with an elastic bottom containing liquidanalysis using harmonic balance method M. Chibaa,*, K. Abeb a

Department of Mechanical Engineering, Iwate University, Morioka 020-8551, Japan b Tsukishima Engineering, Ichikawa 272-01, Japan

Abstract Theoretical analyses and experimental studies have been carried out on the non-linear hydroelastic vibration of a cylindrical tank with an elastic bottom. In this paper, nonlinear axisymmetric free vibration analysis of the bottom plate of the tank, coupled with that of the liquid contained within it, is presented by means of the harmonic balance method. In the analysis, the effect of an in-plane force in the plate due to static liquid pressure is taken into account. The effect of the liquid on the non-linearity of the backbone curve of the super-harmonic and the sub-harmonic resonances as well as the principal resonance of both sloshing- and bulgingtype responses was clarified, and it was found that with an increase of liquid height, the nonlinearity with a hard-spring type of the bottom plate decreased in degree, and became close to linear characteristics. The influence of the bottom plate motions on the free surface response amplitude in the first bulging-type resonance region was also investigated.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear vibration; Cylindrical tank; Bottom plate; Hydroelastic vibration

1. Introduction Although many studies have been conducted within small amplitude linear theory on the dynamic problem of a thin-walled shell filled with liquid, there are only a * Corresponding author. Tel.: ⫹ 81-19-621-6404; fax: ⫹ 81-19-621-6404; e-mail: [email protected] 0263-8231/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 9 ) 0 0 0 0 7 - 5

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few studies treating the non-linear range of the problem. Since the linear analysis cannot be applied when the amplitude of the response becomes large, non-linear analysis is needed. For the non-linear free vibration problem of a liquid–shell coupled system, Kana et al. [1], in 1966, conducted an experiment for pin-ended and cantilever shells with elastic walls, and found interesting instability phenomena. In 1967, Chu and Kana [2] analysed a simply supported cylindrical shell, employing linear theory for the shell and non-linear theory for the liquid, to examine the non-linear free surface response. Ramachandran [3], in 1979, analysed the non-linear transverse vibration of an orthotropic cylindrical shell with linearly varying thickness embedded in liquid. Goncalves and Batista [4], in 1988, treated a simply supported cylindrical shell completely filled with liquid. Concerning the parametric vibration of a liquid-filled cylindrical shell subjected to longitudinal excitation, some studies have been conducted by Kana and Craig [5], Obraztsova [6] and Kobayashi and Nagashima [7]. All these studies examined a thin walled cylindrical shell with a rigid bottom. Bauer et al. [8] analysed a rigid wall cylinder with an elastic bottom. In that study the bottom plate was treated with linear theory, since their concern was with the non-linear response of the liquid free surface. Bauer and Eidel [9] investigated two rectangular container systems; one was a rigid wall and rigid bottom container covered with membrane, and the second was a rigid wall container with membrane bottom and free surface. However, to the authors’ knowledge, there are no studies which have clarified the non-linear hydroelastic vibration of a liquid–shell coupled system, theoretically and experimentally. Therefore, the present study focuses on a cylindrical tank with an elastic bottom plate and a rigid wall, with a liquid free surface, as a primitive model of a cylindrical tank with an elastic bottom and an elastic wall, to clarify the nonlinear hydroelastic characteristics of the system systematically, both experimentally and theoretically, including the non-linearity of both the plate and the liquid in the analysis. It will provide fundamental engineering design data for the non-linear dynamic response analysis of such systems. The study has been tackled from experiments on this liquid–plate coupled system. In 1992, an experimental investigation was conducted [10] using a test tank of radius R ⫽ 144 mm, with two kinds of polyester bottom plates of thickness h ⫽ 0.254 mm and 0.357 mm. In that study, the influence of a liquid contained in the tank on the linear as well as on the non-linear hydroelastic characteristics of the liquidcoupled bottom plate was clarified. As the second step, Part II [11] of the study, before proceeding to the non-linear analysis, a linear axisymmetric free vibration analysis of the bottom plate coupled with a liquid was carried out. In the analysis, the effect of in-plane forces in the plate due to static deflection by the liquid was taken into consideration. The results were in good agreement with experimental ones [10]. Furthermore, this analysis has been developed for a more general cylindrical tank, taking an elastic foundation into account [12]. For the third step, Part III [13] of the study, a non-linear axisymmetric free vibration analysis of the liquid–plate coupled system was carried out, in which variations of the non-linearity with liquid height for only the principal resonance

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235

was clarified, using the Ritz-averaging method, which will be the first approximation for the non-linear analysis. The purpose of the present paper, which comprises Part IV of the study, is to analyse the non-linear hydroelastic vibration characteristics of the system not only for the principal resonance but also for the super-harmonic and the sub-harmonic resonances, which are typical responses in the non-linear dynamic system, using the harmonic balance method. The influence of the liquid height on the internal resonance of the system was clarified. Furthermore, the influence of the bottom plate motion, in the resonance, on the free surface amplitude was also investigated. The results obtained agreed well with the experimental results [10].

2. Formulation of the problem Let us consider the non-linear free axisymmetric vibration of the liquid–cylindrical tank system. The cylindrical tank, having a rigid wall with an inner radius R, has an elastic bottom of thickness h and is filled to a height H with an inviscid incompressible liquid. The bottom plate is supported on an elastic foundation of the Winkler type with spring constant K. The coordinate system is shown in Fig. 1. The bottom plate is isotropic, thin and elastic, and is assumed to perform axisymmetric bending motions around its static deformation. The motion of the liquid contained is assumed to be axisymmetric and irrotational.

Fig. 1.

Cylindrical container with elastic bottom.

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The governing equations and the boundary conditions of the liquid–plate coupled system are derived as follows. ¯ (r, t) and the stress function The basic equations for the bottom plate deflection W ¯F(r, t) are (c.f. Yanaki et al. [15], Chiba [16]): F¯,rr ⫹

1 1 Eh 2 ¯, , F¯,r ⫺ 2 F¯ ⫽ ⫺ W r r 2r r

¯ ⫹ ⵜ42W ⵜ22 ⫽

(1)

P 1 K ␳sh ¯ ,tt ⫹ g) ⫺ ¯ ,r),r ⫹ W ¯ ⫹ ⫽ 0, (W (F¯W D Dr D D

(2)

∂2 ∂ ⫹ , 2 ∂r r∂r

P ⫽ ⫺ ␳f [⌽,t ⫹

(3) 1 2 ¯ ⫺ H)] (⌽ ,r ⫹ ⌽2,z) ⫹ g(W 2

¯, at z ⫽ W

(4)

where P is the liquid pressure described by the velocity potential ⌽. The stress function F¯ corresponds to in-plane forces as, N␪ ⫽ F¯,r ,

(5)

1 F¯. r

(6)

Nr ⫽

The bottom plate is assumed to be clamped and there is no radial in-plane displacement U along the edge. ¯ ⫽0 W ¯ ,r ⫽ 0 W U⫽

at r ⫽ R,

(7)

at r ⫽ R,

r ␯ (F¯, ⫺ F¯) ⫽ 0 Eh r r

(8) at r ⫽ R.

(9)

In these equations, E, D, ␯ and ␳S denote Young’s modulus, flexural rigidity, Poisson’s ratio, and density of the plate, respectively, while ␳f denotes the density of the liquid, and t is time. Initially, due to the static liquid pressure, the bottom plate has a deflection ⫺ Wo(r), with a corresponding stress function Fo(r), and then vibrates with amplitude ¯ (r, t) and F¯(r, t) as W(r, t), with a corresponding stress function F(r, t), then we put W ¯ (r, t) ⫽ ⫺ Wo(r) ⫹ W(r, t), W

(10)

F¯(r, t) ⫽ Fo(r) ⫹ F(r, t).

(11)

2.1. Static deflection Wo(r) and stress function Fo(r) of the bottom plate Substituting Eqs. (10) and (11) into Eqs. (1), (2), (4), (7)–(9), the governing equations and the boundary conditions of the plate in the static deformed state, Wo(r) and Fo(r), are:

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F0,rr ⫹

1 1 Eh F, ⫺ F ⫽⫺ W , 2, r 0 r r2 o 2r o r

ⵜ42Wo ⫺

(12)

1 K Ps ␳sh (F W ) ⫹ W ⫺ ⫺ g ⫽ 0, Dr o o,r ,r D o D D

(13)

Ps ⫽ ␳f g(Wo ⫹ H),

(14)

Wo ⫽ Wo,r ⫽ 0 Uo ⫽

237

at r ⫽ R,

␯ r (F , ⫺ F ) ⫽ 0 Eh o r r o

(15) at r ⫽ R.

(16)

2.2. Large amplitude vibration W(r, t) and stress function F(r, t) And those for the vibration component, W(r, t) and F(r, t), are: F,rr ⫹

1 1 Eh F,r ⫺ 2 F ⫽ ⫺ (W,2r ⫺ 2Wo,rW,r), r r 2r

ⵜ42W ⫹ ⫹

(17)

␳sh 1 K W,tt ⫺ [ ⫺ (FWo,r),r ⫹ (FoW,r),r ⫹ (FW,r),r] ⫹ W D Dr D

(18)

Pd ⫽ 0, D

Pd ⫽ ⫺ ␳f [⌽,t ⫹

1 2 ¯] (⌽ ,r ⫹ ⌽2, z) ⫹ gW 2

W ⫽ W,r ⫽ 0 U⫽

¯, at z ⫽ W

at r ⫽ R,

r ␯ (F, ⫺ F) ⫽ 0 Eh r r

(19) (20)

at r ⫽ R.

(21)

While for the velocity potential ⌽(r, z, t) of the liquid, ⌽ must satisfy the Laplace equation: ⵜ21⌽ ⫽ 0 ⵜ21 ⫽

¯ ⬍ z ⬍ q, for 0 ⬍ r ⬍ R, W

∂2 ∂ ∂2 ⫹ ⫹ 2. 2 ∂r r∂r ∂z

(22) (23)

Then, at the liquid free surface, the kinematic and the dynamic conditions are to be satisfied. q,t ⫹ ⌽, r q,r ⫺ ⌽,z ⫽ 0 ⌽,t ⫹ gq ⫹

1 (⌽,2r ⫹ ⌽,2z ) ⫽ 0 2

at z ⫽ H ⫹ q,

(24)

at z ⫽ H ⫹ q,

(25)

where q(r, t) is the amplitude of the free surface. The cylindrical wall is rigid as follows:

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⌽,r ⫽ 0

at r ⫽ R.

(26)

At the interface between the bottom plate and the liquid, the velocity matching condition has to be satisfied as follows: ¯ ,t ⫹ ⌽,rW ¯ ,r ⫺ ⌽,z ⫽ 0 W

¯. at z ⫽ W

(27)

Here it should be noted that the liquid free surface conditions, Eqs. (24) and (25), are the ones that are satisfied at z ⫽ H ⫹ q. Then, these are transferred to the conditions satisfied at z ⫽ H using Taylor expansion around z ⫽ H. Similar treatment has to be considered for the dynamic liquid pressure Pd (Eq. (19)) and for the velocity matching condition (Eq. (27)) at z ⫽ 0. 2.3. Non-dimensional forms For convenience, we introduce the following non-dimensional parameters:

␰ ⫽ z/R, ␳ ⫽ r/R, ␩ ⫽ q/h, R¯ ⫽ h/R, (w, wo) ⫽ (W, Wo)/h, (f, fo) ⫽ (F, Fo)R/(Eh3), lo ⫽ H/R, ␳¯ ⫽ ␳f /␳s, ␶ ⫽ ⍀ot, ⍀o2 ⫽ D/(R4␳sh), g¯ ⫽ g/(h⍀o2), ␻ ⫽ ⍀/⍀o, k¯ ⫽ KR4/D, D ⫽ Eh3/c, c ⫽ 12(1 ⫺ ␯2),

(28)

∂ ∂ ∂ ␾ ⫽ ⌽/(⍀oRh), pd ⫽ PdR4/Dh, g ⫽ g␳sh/E, ⵜ¯ 12 ⫽ 2 ⫹ ⫹ 2, ∂␳ ␳ ∂␳ ∂␰ 2

2

∂2 ∂ ⵜ¯ 22 ⫽ 2 ⫹ . ∂␳ ␳ ∂␳ In these equations, R¯, ␳¯ , k¯, and lo are the non-dimensional parameters related to the plate thickness h, mass density, stiffness of the foundation K, and liquid height H, respectively. Then, the above equations are represented in the non-dimensional forms as follows. 2.3.1. Static deflection wo(␳) and stress function fo(␳) of the bottom plate f 0,␳␳ ⫹

1 1 1 2 f0,␳ ⫺ 2 fo ⫽ ⫺ w,, ␳ ␳ 2␳ o ␳

c ⵜ¯ 42wo ⫺ (fowo,␳),␳ ⫺ g¯[␳¯ (wo ⫹ 1o/R¯) ⫹ 1] ⫹ k¯wo ⫽ 0, ␳

(29) (30)

wo ⫽ wo⬘␳ ⫽ 0

at ␳ ⫽ 1,

(31)

f o,␳ ⫺ ␯ fo ⫽ 0

at ␳ ⫽ 1.

(32)

2.3.2. Large amplitude vibration w(␳,␶) and stress function f(␳,␶) f ,␳␳ ⫹

1 1 1 f, ⫺ f ⫽ ⫺ (w2,␳ ⫺ 2wo,␳w,␳), ␳ ␳ ␳2 2␳

(33)

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c ⵜ¯ 42w ⫹ k¯w ⫺ [ ⫺ (fwo,␳),␳ ⫹ (fow,␳),␳ ⫹ (fw,␳),␳] ⫹ w,␶␶ ⫹ pd ⫽ 0 ␳

239

(34)

at ␰ ⫽ 0, pd ⫽ ⫺ ␳¯ [␾,␶/R¯ ⫹ 0.5(␾ 2,␳ ⫹ ␾ 2,␰ ) ⫹ (w ⫺ wo)(␾,␰␶ ⫹ R¯␾,␳␰ ␾,␳ ⫹ R¯␾,␰␰ ␾,␰ ) ⫹ g¯w ⫹ 0.5R¯(w ⫺ wo)2␾,␰␰␶]

at ␰ ⫽ 0,

(35)

w ⫽ w,␳ ⫽ 0

at ␳ ⫽ 1,

(36)

f , ␳ ⫺ ␯f ⫽ 0

at ␳ ⫽ 1.

(37)

2.3.3. Velocity potential ␾(␳, ␰, ␶) and free surface amplitude ␩(␳, ␶) ⵜ¯ 21␾ ⫽ 0,

(38)

␩,␶ ⫹ R¯␾,␳␩,␳ ⫺ ␾,␰ ⫹ R¯␩(R¯␾,␰␳␩,␳ ⫺ ␾,␰␰ ) ⫺ 0.5R¯2␩2␾,␰␰␰ ⫽ 0,

(39)

at ␰ ⫽ lo,

␾,␶/R¯ ⫹ 0.5[␾ 2,␳ ⫹ ␾,2␰ ] ⫹ ␩(␾,␰␶ ⫹ R¯␾,␳␰ ␾,␳ ⫹ R¯␾,␰␰ ␾,␰ ⫹ g¯)

(40)

⫹ 0.5R¯␾,␰␰␶␩ ⫽ 0 at ␰ ⫽ lo, 2

␾, ␳ ⫽ 0

at ␳ ⫽ 1,

(41)

w,␶ ⫺ ␾,␰ ⫹ R¯␾,␳(w,␳ ⫺ wo,␳) ⫹ (w ⫺ wo)[R¯ ␾,␰␳(w ⫺ wo),␳ ⫺ R¯␾,␰␰ ] 2

⫺ 0.5R¯2(w ⫺ wo)2␾,␰␰␰ ⫽ 0

(42)

at ␰ ⫽ 0.

3. Method of solution First of all, the static deflection of the bottom plate is determined, and then one proceeds to solve the large amplitude vibration of the bottom plate and liquid contained around wo(␳). 3.1. Static deflection due to liquid Considering the boundary condition Eq. (31), static deflection wo(␳) is assumed in the form as in [11], wo(␳) ⫽

冘

anwn(␳),

(43)

n

where the an values are unknown constants, and wn(␳) are spatial functions: wn(␳) ⫽ (1 ⫺ ␳2)2␳2(n ⫺ 1).

(44)

Substituting Eq. (43) into Eq. (29), and considering the boundary condition Eq. (32) and the regularity condition at ␳ ⫽ 0, one can obtain stress function as:

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f o ( ␳) ⫽

冘

coi␳[␳2i ⫺ (2i ⫹ 1 ⫺ ␯)/(1 ⫺ ␯)],

i

coi ⫽ ⫺

1 2i(i ⫹ 1)

冘冘 n

(45)

anaj [(n ⫺ 1)(j ⫺ 1)␦n ⫹ j, i ⫹ 3 ⫺ 2(2nj ⫺ n

j

⫺ j)␦n ⫹ j, i ⫹ 2 ⫹ 2(3nj ⫺ 1)␦n ⫹ j, i ⫹ 1 ⫺ 2(2nj ⫹ n ⫹ j)␦n ⫹ j, i

(46)

⫹ (n ⫹ 1)(j ⫹ 1)␦n ⫹ j, i ⫺ 1], where ␦i,j is the Kronecker delta. Hereafter, we will take only one term in Eq. (43), that is n ⫽ 1, which yields co1 ⫽ ⫺ a12, co2 ⫽ 2a12/3, co3 ⫽ ⫺ a12/6.

(47)

Then, applying the Galerkin method to Eq. (30), one obtains a cubic non-linear equation with respect to a1, from which one can determine a1. 3.2. Large amplitude vibration of bottom plate Considering the boundary condition Eq. (36), the vibration component w(␳, ␶) of the bottom plate is assumed in the form: w(␳, ␶) ⫽

冘

bm(␶)wm(␳).

(48)

m

For the first governing Eq. (33) of the plate, with respect to the stress function f(␳, ␶), substituting the above equation, and considering the boundary condition Eq. (37) and the regularity condition at ␳ ⫽ 0, one can obtain the stress function as: f (␳, ␶) ⫽

冘 k

Ck ⫽ ⫺

Ck␳[␳2k ⫺ (2k ⫹ 1 ⫺ ␯)/(1 ⫺ ␯)],

冘冘

1 [ 2k(k ⫹ 1) m

(49)

bmbj [(m ⫺ 1)(i ⫺ 1)␦m ⫹ i, k ⫹ 3 ⫺ 2(2mi

i

⫺ m ⫺ i)␦m ⫹ i, k ⫹ 2 ⫹ 2(3mi ⫺ 1)␦m ⫹ i, k ⫹ 1 ⫺ 2(2mi ⫹ m ⫹ i)␦m ⫹ i, k (50)

冘

⫹ (m ⫹ 1)(i ⫹ 1)␦m ⫹ i, k ⫺ 1] ⫹ 4a1

bm[(m ⫺ 1)␦m, k ⫹ 1

m

⫺ (3m ⫺ 1)␦m, k ⫹ (3m ⫹ 1)␦m, k ⫺ 1 ⫺ (m ⫹ 1)␦m, k ⫺ 2]]. So far, the stress function f(␳, ␶) is obtained, which satisfies the boundary condition exactly. For the next step, to the second governing Eq. (34) of the plate, we apply the Galerkin method, which yields the non-linear differential equation with quadratic and cubic non-linear terms with respect to bm:

冘 m

冘

{Mimb¨m ⫹ (Kim ⫹ k¯Mim)bm} ⫹ c兵a1

k

Ni1kCk[bj bn, a1bj ]

(51)

M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

⫺

冘冘 m

NimkbmCk[bj bn, a1bj ] ⫺

k

冘冘 m

241

Nimj bmcoj [a21]其 ⫹ PD ⫽ 0,

j

where PD is the dynamic liquid pressure. Actual expressions of it and those of Mim, Kim, Nimk are represented in Appendix A. 3.3. Large amplitude vibration of liquid-free surface Hereafter, we will proceed to the liquid-free surface condition. Considering the boundary condition Eq. (41), the velocity potential of the liquid ␾(␳, ␰, ␶) and free surface deflection ␩(␳, ␶) are assumed as:

冋

␾(␳, ␰, ␶) ⫽ ⫺ Ao(␶)␰ ⫹ Bo(␶) ⫹ ⫹

cos h(⑀k␰) B ( ␶) sin h(⑀klo) k

␩(␳, ␶) ⫽

冘

冎册

冘 k

Jo(⑀k␳)

再

sin h(⑀k␰) A ( ␶) cos h(⑀klo) k

(52)

,

hp(␶)Jo(⑀p␳),

(53)

p

where Ao(␶), Bo(␶), Ak(␶), Bk(␶), and hp(␶) are unknown time functions, Jo(⑀k␳) is the first-kind Bessel function of order zero, and ⑀k are the values which satisfy (Jo(⑀k␳)/␳)␳ ⫽ 1 ⫽ ⫺ J1(⑀k) ⫽ 0.

(54)

Substituting Eqs. (52) and (53) into Eqs. (39) and (40), and integrating over the free surface using Jo(⑀i ␳) as a weighting function:

冕

eq. (39) Jo(⑀i␳)␳ d␳ ⫽ 0,

冕

eq. (40) ␳ d␳ ⫽ 0,

冕

eq. (40) Jo(⑀i␳)␳ d␳ ⫽ 0.

1

i ⫽ 1,2,…

(55)

0 1

(56)

0 1

i ⫽ 1,2,…

(57)

0

These equations are the functions of Ao, Ak, Bk, bm and their time derivatives and products. The actual expressions are in Appendix A.

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3.4. Velocity matching condition at the bottom: Eq. (42) Next, for the velocity matching condition at the bottom Eq. (42), we apply the Galerkin method as:

冕

eq. (42) wi(␳)␳ d␳ ⫽ 0,

冕

eq. (42) ␳ d␳ ⫽ 0,

1

i ⫽ 1,2,…

(58)

0 1

(59)

0

which yields non-linear equations with Ao, Ak, Bk, Akbm, Bkbm, Akbmbp, b˙m. The actual expressions are in Appendix A. 3.5. Harmonic balance method So far, we have 2(k ⫹ m ⫹ 1) non-linear coupled differential equations in terms of Ao, Bo, Ak, Bk, hp, bm, that is, Eqs. (51), (55)–(59). Here we take three terms in the spatial coordinate of both liquid and plate, that is, k ⫽ p ⫽ m ⫽ 1, 2, 3, which yields 14 unknown time functions and 14 non-linear coupled differential equations. Fi ⫽ 0,

i ⫽ 1,2,…,14.

(60)

In the present study, we restrict our attention to investigating the effect of liquid height on the non-linear free vibration characteristics of not only the principal resonance but also the super-harmonic and the sub-harmonic resonance, at most up to the third order of both the sloshing and the bulging mode. Then, we use the harmonic balance method, assuming the time functions in the Fourier series as: {Ao(␶),…, b3(␶)}T ⫽ Si ⫹

冘

(d ki ␮ cos k␮␻␶ ⫹ eki ␮ sin k␮␻␶),

(61)

k⫽1

i ⫽ 1,…,14, where we put ␮ ⫽ 1 when considering the principal and the super-harmonic resonances, while ␮ ⫽ 1/2 and 1/3 when considering the sub-harmonic resonances of order 1/2 and 1/3. Furthermore, Si, dik␮, eik␮ are the Fourier coefficients, and subscript i(i ⫽ 1 to 14) is the order of unknown time function, while superscript k means the vibration order. Substituting the above equations into Eq. (60), each equation can be rearranged into the following Fourier series as: Fi ⫽

冘

(fi[Si] ⫹ gik␮[dik␮]cos k␮␻␶ ⫹ hik␮[eik␮]sin k␮␻␶) ⫽ 0.

(62)

k⫽1

Then, putting the Fourier coefficient of Fi, (i ⫽ 1,$,14) equal to zero, we get nonlinear coupled equations as:

M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

243

f i[Si] ⫽ 0, gik␮[d ki ␮] ⫽ 0,

(63)

hki ␮[eki ␮] ⫽ 0. From these equations, one obtains 14 unknown time functions, Ao(␶),$,b3(␶). Here, we take vibration order parameter k in Eq. (61) as k ⫽ 1,2,3, which yields 98 coupled non-linear equations. Eq. (63) can be obtained by applying the Fourier transform to Eq. (62). In the calculation, we employed the FFT (Fast Fourier Transform) method [17] to reduce the computational time. Whereas the Newton– Raphson method was employed to solve the non-linear Eq. (63), in this method most of the computational time is taken by the calculation of the inverse of the Jacobian matrix. To reduce the computational time in this calculation we used the Broyden method [18], after the first step of the inverse Jacobian matrix, i.e. from the second calculation, from which amount of calculation time can be reduced. The rms value of the bottom plate responses at the centre of the tank can be obtained as: wrms ⫽ ⫹(

冘 冘

1 [( √2

冘

dm ⫹ 111␮wm(0))2 ⫹ (

m

dm ⫹ 113␮wm(0))2 ⫹ (

m

⫹(

冘

冘

dm ⫹ 112␮wm(0))2

m

em ⫹ 111␮wm(0))2 ⫹ (

m

em ⫹ 113␮wm(0))2]1/2,

冘

em ⫹ 112␮wm(0))2

m

(64)

m

⫽

1 [(d 1␮)2 ⫹ (d122␮)2 ⫹ (d123␮)2 √2 12

⫹ (e121␮)2 ⫹ (e122␮)2 ⫹ (e123␮)2]1/2. 4. Numerical results Numerical results are presented to see the variation of the non-linear vibration characteristics of the liquid–bottom plate coupled system with liquid height. Initially, we consider the bulging-type response in which bottom plate motion is predominant. Next, we consider the sloshing-type response. 4.1. Bulging-type response Before proceeding to investigate the liquid–plate coupled system, the bottom plate without liquid is considered. Fig. 2 shows backbone curves for the plate at the centre when k¯ ⫽ 0. In the figure, the two numerals in parentheses, (i, j), indicate the mode of vibration, i, and the dominant order of harmonics, j, respectively, i.e. j ⫽ 1 is the principal resonance, j ⫽ 2, 3 are the super-harmonic resonances of order 2 and

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Fig. 2.

Backbone curves of the bottom plate without liquid: lo ⫽ 0, k¯ ⫽ 0.

3, and j ⫽ 1/2, 1/3 are the sub-harmonic resonances of order 1/2 and 1/3. The results for the principal resonances are shown with thick lines. When we see the non-linearity of these responses, natural frequencies increase with amplitude wrms, that is, backbone curves bend to the right with wrms, which indicates that the responses have hard-spring non-linearity. And the higher order modes have the greater non-linearity. Note that one can see a discontinuity in the principal resonance curve of the first mode. This is the internal resonance with (1, 1) and (2, 3) modes, which has been explained detail by Yamaki et al. [15]. Similar internal resonances can be seen in (1, 1/2) and (1, 1/3) response curves. To explain this more clearly from the viewpoint of the non-linear dynamical system, backbone curves are compared with those calculated with three temporal terms and those with reduced number of the temporal terms, i.e. one term. In Fig. 3, the dashed lines correspond to the results when only one temporal term was taken, while solid lines are those when three terms were considered. That is, by considering the higher order temporal terms, the effect of the super-harmonic resonance of order 3 of the 2nd mode, (2, 3), can be recognised in the (1, 1) response curve as an internal resonance. From the figure it is clear that a one-term approximation cannot demonstrate the internal resonance with the super-harmonic type. Next, we consider the liquid–plate coupled system, when R¯ ⫽ 0.005, ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0. Linear natural frequency variations with liquid height, lo, are presented in Fig. 4. Each frequency suddenly decreases with lo and gradually tends to some values. In Fig. 5, the corresponding backbone curves of the principal resonances when lo ⫽ 0.1 and 0 are presented with solid and dashed lines, respectively. With an increase of liquid height, the linear natural frequencies, which correspond to the roots of the backbone curves in the abscissa decrease, and the hardspring non-linearity becomes a little weak, which can appear as a decrease of the

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245

Fig. 3. Effect of temporal term number on the first principal resonance response of the bottom plate, lo ⫽ 0, k¯ ⫽ 0: 앶앶앶, three-term; - - -, one-term.

Fig. 4. Linear bulging-type natural frequency variation with liquid height lo: R¯ ⫽ 0.005, ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0.

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Fig. 5. Backbone curves of the bulging-type response when lo ⫽ 0.1 (앶앶앶) and lo ⫽ 0 (- - - -), principal resonances, R¯ ⫽ 0.005, ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0.

Fig. 6. Backbone curves g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0.

of

the

bulging-type

response

when

lo ⫽ 0.1, R¯ ⫽ 0.005, ␳¯ ⫽ 0.1,

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247

curvature of the backbone curves. Overall backbone curves including the sub-harmonic and the super-harmonic responses are shown in Fig. 6, when lo ⫽ 0.1. Comparing this figure with Fig. 2 when lo ⫽ 0, due to the change of the linear natural frequencies, the regions in which internal resonances occur change, in (1, 1), (1, 1/2), and (1, 1/3) modes. To see the variations of the non-linearity of the backbone curve with a smaller liquid height step, the results for the first, second and third modes are shown in Fig. 7(a–d), respectively. In the first mode, shown in Fig. 7(a), with an increase of lo, the hard-spring non-linearity begins to weaken, and the internal resonances, which can be seen when lo ⫽ 0, 0.05, 0.1, and 0.2, disappear when lo ⫽ 0.4 and 1.0. A similar tendency can be seen in the second mode, in Fig. 7(b). In this mode, however, though there seems no internal resonance when lo ⫽ 0, with an increase of lo, the internal resonances appear when lo ⫽ 0.4, 0.6 and 1.0. The details of these response areas are shown in Fig. 7(c). These are the internal resonances between (2, 1) and (3, 3) modes. The variations of the non-linearity of the backbone curve calculated here agree well with the experimental results [10]. Next, we will consider the effect of density ratio ␳¯ on the non-linearity. Linear

Fig. 7. Variation of the non-linearity of the bulging-type response with liquid height lo: (a) first mode; (b) second mode; (d) third mode. (c) Internal resonance in the second mode, R¯ ⫽ 0.005, ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0.

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natural frequency variations with liquid height, lo, are presented in Fig. 8(a), when R ⫽ 0.005, g¯ ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0, changing ␳¯ ⫽ 0.1, 0.4, 0.8. These values nearly correspond to the density ratios of water/steel (0.13), water/aluminium (0.37) and water/polyester (0.72). From the figure, with an increase of ␳¯ , the linear natural frequency of the bottom plate decreased. It should also be noted that when ␳¯ ⫽ 0.4 and 0.8, the linear natural frequency variation with lo changes in the second and third modes, i.e. after the sudden decrease with lo it begins to increase slightly due to the increase of the in-plane stress in the plate, which comes from the increase of the static deflection of the plate (see [11]). Non-linearity variations with ␳¯ for the principal resonances are shown in Fig. 8(b). From the figure, one can see that the increase in ␳¯ weakens the non-linearity of the hard-spring type and the degree of weakness is significant in the lower order mode. Then, the effect of thickness ratio R¯ is considered, when ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0. The effect of R¯ on the linear natural frequency variations with liquid height, lo, are presented in Fig. 9(a), taking R¯ ⫽ 0.005, 0.01, 0.02. Since a plate with a large value of R¯ corresponds to a thicker one, and with an increase of R¯ the linear natural frequency of the bottom plate increases. Non-linearity variations with R¯ for the principal resonances are shown in Fig. 9(b). From the figure, the increase in R¯ (i.e. the plate becomes thicker) makes the hard-spring non-linearity strong. Finally, the effect of stiffness of the foundation k¯ is considered when ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, R¯ ⫽ 0.005, lo ⫽ 0.5. Non-linearity variations for the principal resonances are shown in Fig. 10, changing k¯ ⫽ 0, 100, 1000 and shown by solid, dashed and dotted lines, respectively. From the figure, one can see that the influence of the stiffness both on the natural frequency [11], and on the non-linearity is significant for the lower mode, and with an increase in k¯, the hard-spring non-linearity becomes weak in the first mode. 4.2. Sloshing-type response In this section, we consider the non-linearity of the sloshing response in which free surface amplitude is predominant. In this type, linear natural frequency increases with lo, e.g. see [11]. Backbone curves with different liquid height for the first sloshing mode are shown in Fig. 11, when R¯ ⫽ 0.005, g ⫽ 1.0 ⫻ 10−10, ␳¯ ⫽ 0.1, and k¯ ⫽ 0. In the figure, although the dashed lines and the solid lines are the calculated response curves, the solid lines are rigorous results considering the height of liquid contained lo. In this case, the non-linearity is a soft spring type for higher liquid height, e.g. lo ⫽ 1.0, 0.6, 0.4, and becomes a hard-soft spring type with decrease of lo. This tendency agrees well with the experimental results [10]. Furthermore, when the amplitude of the free surface becomes large, the influence of the higher order temporal component becomes significant and sometimes it is difficult to obtain numerical results, e.g. when lo ⫽ 0.2. To show this more clearly, comparison with the present results used temporal terms up to the third order harmonic and those with one term approximation [13], is shown in Fig. 12(a) and (b), when lo ⫽ 1.0 and 0.1, respectively. In the figure, the results with three and one temporal terms are presented with solid and dashed lines, respectively. From these figures, although

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249

Fig. 8. (a) Effect of density ratio ␳¯ on the linear natural frequency with lo. (b) Effect of density ratio ␳¯ on the backbone curves of the bulging-type response: lo ⫽ 0.5; R¯ ⫽ 0.005, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0, ␳¯ ⫽ 0.1, 0.4, 0.8.

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M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

Fig. 9. (a) Effect of thickness ratio R¯ on the linear natural frequency with lo. (b) Effect of thickness ratio R¯ on the backbone curves of the bulging-type response: lo ⫽ 0.05; ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0, R¯ ⫽ 0.005, 0.01, 0.02.

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251

Fig. 10. Effect of foundation stiffness k¯ on the backbone curves of the bulging-type response: R¯ ⫽ 0.005, ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, lo ⫽ 0.5.

Fig. 11. Variation of the non-linearity of the sloshing-type response with liquid height lo, first mode: R¯ ⫽ 0.005, ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0.

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M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

Fig. 12. Comparison of the sloshing-type response with present results and those with one temporal term approximation [13], first mode: R¯ ⫽ 0.005, ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0; (a) lo ⫽ 1.0; (b) lo ⫽ 0.1.

M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

253

a similar tendency can be seen in the lo ⫽ 1.0 case, i.e. soft spring type, a hard-soft spring characteristic observed in the curve when lo ⫽ 0.1, in Fig. 12(b), cannot be obtained from a one-term approximation. This indicates the necessity of higher temporal terms in the calculation especially for the lower liquid height. 4.3. Free surface response in the bulging-type resonance The present analysis employed only the lowest three spatial coordinates of the free surface. We will examine how much influence the bottom plate motion has on the free surface response in the bulging-type resonance, i.e. when the bottom plate is in the resonance. If we consider it precisely, it is obvious that additional higher order terms should be used, because the frequency range of the bulging-type and that of the sloshing-type is usually far. In this moment, however, we will see it within the present three mode system. Backbone curves of the free surface are shown in Fig. 13, varying the liquid height lo ⫽ 0.05, 0.2, 0.4, 0.6, 0.8, 1.0, when the bottom plate vibrates with amplitude wrms ⫽ 1.0 at the centre. Roots of the curves, which correspond to the linear natural frequencies of the bulging-type, decrease with lo, and with an increase of lo from lo ⫽ 0.2 the amplitude of the free surface becomes small, which means that the influence of the bottom plate motion becomes insignificant with liquid height. In addition, the liquid free surface response is at its maximum when lo ⫽ 0.2, which is nearly the same value as that obtained in the experiment (lo ⫽ 0.3, see [11]). The influence of the bottom plate motion on the free surface response in the lower sloshing frequency range has been studied by Chiba [14].

Fig. 13. Backbone curves of the free surface in the first bulging-type frequency range with lo: R¯ ⫽ 0.005, ␳¯ ⫽ 0.1, g ⫽ 1.0 ⫻ 10−10, k¯ ⫽ 0, wrms ⫽ 1.0.

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5. Conclusions Theoretical analyses have been carried out on the non-linear axisymmetric free vibration of an elastic bottom plate coupled with a liquid in a rigid wall cylindrical tank. In the analysis, the effect of the static deflection of the bottom plate was taken into consideration. In this study, the effect of liquid contained in the tank on the non-linear characteristics of liquid–plate coupled system for the super-harmonic and the sub-harmonic resonances as well as for the principal resonance of both the sloshing- and bulging-type responses was investigated. The results may be summarised as follows. 1. With an increase of liquid height, the hard-spring non-linearity, which the bottom plate itself possesses, decreased in intensity. These conclusions agree well with experimental results. 2. The appearance of the internal resonance depends on the liquid height lo, since the linear natural frequency of the bottom plate varies with liquid height. 3. Effects of the density ratio, the thickness ratio, and foundation stiffness on the non-linearity of the bulging-type response were clarified. 4. With an increase of liquid height, the hard-spring type non-linearity changes to the soft-spring type in the sloshing-type response. 5. Influence of the bottom plate motion on the liquid free surface amplitude in the bulging-type resonance was clarified, which agrees with the experimental results.

Acknowledgements This study forms part of a systematic investigation started at the Universista¨t der Bundeswehr Mu¨nchen, in 1988. The author would like to thank Professor Dr. rer. nat. H.F. Bauer, Institut fu¨r Raumfahrttecknik, Universita¨t der Bundeswehr Mu¨nchen, for his encouragement during the study.

Appendix A

Notation Ao, Bo, Ak, Bk a n, b m C¯k, coi c D E F¯

unknown time functions in Eq. (52) coefficients of Eqs. (43) and (48) coefficients of Eqs. (49) and (45) parameter defined in Eq. (28) flexural rigidity Young’s modulus total stress function, ⫽ F ⫹ Fo

M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

F(f), Fo(fo) g(g¯, g) H(lo ⫽ H/R) h(R¯ ⫽ h/R) hp Jo K(k¯) N␪, Nr Pd(pd) q(␩) R r(␳), z(␰) t(␶) U(u), Uo(uo) ¯ W W(w) Wo(wo) ␦i,j ⑀k ␯ ␳s, ␳f ␳¯ ⌽ ( ␾) ⍀ ( ␻) ⍀o

255

stress function (non-dimensional form) gravitational acceleration liquid height thickness of bottom plate unknown time function in Eq. (53) Bessel function of the first kind with order zero stiffness of foundation in-plane forces dynamic liquid pressure free surface amplitude radius of tank coordinate system time radial in-plane displacement of plate total plate deflection, ⫽ W ⫺ Wo dynamic component of plate deflection static deflection of plate Kronecker delta parameter satisfied by Eq. (54) Poisson’s ratio mass density of plate and liquid mass density ratio, ⫽ ␳f /␳s velocity potential circular frequency circular frequency parameter

Eq. (51):

冋

冘

Qi Qi 2 PD ⫽ ⫺ ␳¯ ⫺ ¯ B˙o ⫹ A ⫹ a1MilA˙o ⫹ (g¯ ⫺ A˙o) Mimbm R 2 o m ⫹ a1

冘 k

⫹

冘 冘⑀ ⑀ 冉 冊 冘冘

1 ⑀kR8ilk ˙ Ak ⫹ cos h(⑀klo) 2

k p

k

p

R11 ikpAkAp ⫺ a1R¯ cos h(⑀klo)cos h(⑀plo)

⫹ ⑀kR13 ilkp)ApBk ⫹ Ao ⫹ R¯

冘冘冘 m

k

p

冘冉 ⑀k

k

k

p

R10 ikpBkBp sin h(⑀klo)sin h(⑀plo)

⑀k⑀p (⑀ R12 sin h(⑀klo)cos h(⑀plo) p ilkp

冊

R ⑀kR8ilk Ak ⫺ a1R¯ B cos h(⑀klo) sin h(⑀plo) k 1 ik

⑀k⑀p (⑀ R12 ⫹ ⑀kR13 imkp)ApBkbm sin h(⑀klo)cos h(⑀plo) p imkp

(A1)

256

⫺

M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

R¯ 2

1 ⫺ ¯ R

冘冘冘 ⑀ ⑀ 冘 ⑀ 冉

冘冘

R9impk B˙ b b ⫹ R¯Ao sin h( klo) k m p m 2 k

m

k

p

k

⑀k2R8imk Bb sin h(⑀klo) k m

冊 冘冘 冉

a1 R¯ 1 R1ik ⫹ ⑀k2R9i11k B˙k ⫺ sin h( klo) 2

k

冊册

⑀kR9im1k ˙ B b , sin h(⑀klo) k m

⫺ R¯a1

冕

2

2

⑀k

m

k

R8imk A˙ cos h(⑀klo) k

i ⫽ 1,2,3

1

Mim ⫽ wi(␳)wm(␳)␳ d␳ ⫽

1 2 3 2 ⫺ ⫹ ⫺ 2(i ⫹ j ⫺ 1) i ⫹ j i ⫹ j ⫹ 1 i ⫹ j ⫹ 2

0

⫹

1 , 2(i ⫹ j ⫹ 3)

冕 1

Qi ⫽ wi(␳)␳ d␳ ⫽

1 1 1 ⫺ ⫹ , 2i i ⫹ 1 2(i ⫹ 2)

0

冕 1

Kim ⫽ ⵜ¯ 42wi(␳)wm(␳)␳ d␳

冋

⫽ 16 ⫹

0

(j ⫺ 1)2(j ⫺ 2)2 2(j ⫺ 1)2(j2 ⫺ 2j ⫹ 2) ⫺ 2(i ⫹ j ⫺ 3) i⫹j⫺2

Nijk ⫽ 4 ⫺

冋

(j ⫺ 1)(j ⫹ k ⫺ 1) 2j2 ⫹ 2jk ⫺ 2j ⫺ k ⫹ 1 3j2 ⫹ 3jk ⫹ 1 ⫺ ⫹ 2(i ⫹ j ⫹ k ⫺ 2) i⫹j⫹k⫺1 i⫹j⫹k

册

2j2 ⫹ 2jk ⫹ 2j ⫹ k ⫹ 1 (j ⫹ 1)(j ⫹ k ⫹ 1) 2k ⫹ 1 ⫺ ␯ ⫺ Uij , ⫹ i⫹j⫹k⫹1 2(i ⫹ j ⫹ k ⫹ 2) 1⫺␯

Uij ⫽ 4 ⫹

册

3j4 ⫺ 6j3 ⫹ 9j2 ⫺ 6j ⫹ 2 2j2(j2 ⫹ 1) j2(j ⫹ 1)2 , ⫺ ⫺ i⫹j⫺1 i⫹j 2(i ⫹ j ⫹ 1)

冋

(j ⫺ 1)2 2j2 ⫺ 2j ⫹ 1 3j2 ⫹ 1 2j2 ⫹ 2j ⫹ 1 ⫺ ⫹ ⫺ 2(i ⫹ j ⫺ 2) i⫹j⫺1 i⫹j i⫹j⫹1

册

(j ⫹ 1)2 , 2(i ⫹ j ⫹ 2)

冕 1

R1ik ⫽ wi(␳)Jo(⑀k␳)␳ d␳, 0

M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

冕

257

冕

1

1

R8imk ⫽ wi(␳)wm(␳)Jo(⑀k␳)␳ d␳,

R9impk ⫽ wi(␳)wm(␳)wp(␳)Jo(⑀k␳)␳ d␳,

0

0

冕

冕

1

1

R10 wi(␳)J1(⑀k␳)J1(⑀p␳)␳ d␳, ikp ⫽

R11 wi(␳)J0(⑀k␳)J0(⑀p␳)␳ d␳, ikp ⫽

0

0

冕

冕

1

1

R12 wi(␳)wm(␳)J1(⑀k␳)J1(⑀p␳)␳ d␳, imkp ⫽

R13 wi(␳)wm(␳)J0(⑀k␳)J0(⑀p␳)␳ d␳, imkp ⫽

0

0

Eq. (55): 1 2

冘 ⑀ ␦ 冘⑀ 冘 冘⑀ ⑀ ⑀ ␴ 冘 冘 冘⑀ ⑀ Jo2( p)h˙p

i,p

p

⫺ R¯

⫹

1 2

( pdkip ⫺

k

k

J 2(⑀k)(Ak ⫹ Bk)␦k,i

k o

k

k

kip

)(Aktan h(⑀klo) ⫹ Bkcot h(⑀klo))hp

p

⫺ R¯2

2 k

k

p

q

( qkikpq ⫺

1 ⑀ l )(A ⫹ Bk)hphq ⫽ 0, 2 k ikpq k

i ⫽ 1, 2, 3.

冕 1

dkip ⫽ J1(⑀k␳)J0(⑀i␳)J1(⑀p␳)␳ d␳,

(A2)

0

冕 1

␴kip ⫽ J0(⑀k␳)J0(⑀i␳)J0(⑀p␳)␳ d␳, 0

冕 1

kkipq ⫽ J0(⑀k␳)J1(⑀i␳)J0(⑀p␳)J1(⑀q␳)␳ d␳, 0

冕 1

lkipq ⫽ J0(⑀k␳)J0(⑀i␳)J0(⑀p␳)J0(⑀q␳)␳ d␳, 0

Eq. (56):

冘 冘⑀ ⑀ ⑀ ␦ 冘 冘 冘⑀ ⑀ ⑀

1 1 1 ⫺ ¯ (loA˙o ⫹ B˙o) ⫹ A2o ⫹ 2R 4 4 ⫺

1 2

冘 冘⑀

J (⑀k)(A˙k ⫹ B˙k)hp

2 k o

k

p

J ( k)(Ak ⫹ Bk)(Ap ⫹ Bp)␦k,p

2 k p 0

k

p

k,p

⫹ R¯

k q

k

p

q

( kdkpq

258

M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

⫹ ⑀q␴kpq)(Ak ⫹ Bk){Aqtan h(⑀qlo) ⫹ Bqcot h(⑀qlo)}hp ⫹ ⫺

冘 冘⑀ ⑀ 冘 冘 冘⑀ ␴

R¯Ao 2 R¯ 2

(A3)

J ( k){Aktan h(⑀klo) ⫹ Bkcot h(⑀klo)}hp

2 2 k o

k

p

k

k

p

2

{A˙ktan h(⑀klo) ⫹ B˙kcot h(⑀klo)}hphq ⫽ 0.

kpq

q

Eq. (57):

冘 ⑀ 冘 ⑀ 冘 冘⑀ ⑀

1 ⫺ ¯ 2R

k

A˙o ⫺ 2 ⫹

1 2

g¯ J02( k){A˙ktan h(⑀klo) ⫹ B˙k cot h(⑀klo)}␦i,k ⫹ 2

J2o( p)hp␦i,p ⫺

p

1 2

冘 冘⑀ ⑀ ␴ 冘 冘 冘⑀ ⑀ ⑀ 冘 冘⑀ ␴ 冘 冘 冘⑀ k p

k

(A˙k ⫹ B˙k)hp

p

kpi

k q

k

p

k

p

冘⑀

(A4)

k

(Ak ⫹ Bk)(Ap ⫹ Bp)

( kkikpq ⫹ ⑀qlikpq)(Ak ⫹ Bk){Aqtan h(⑀qlo) ⫹ Bq cot h(⑀qlo)}hp

kpi

{Aktan h(⑀klo) ⫹ Bk cot h(⑀klo)}hpAo

2 k ikpq

l

k

p

{A˙ktanh (⑀klo)A˙k ⫹ B˙k cot h(⑀klo)}hphq ⫽ 0,

冘⑀

k

k

冘 冘 ⑀⑀ 冘冘冘 ⑀⑀ 冘冘 ⑀⑀

⫹ R¯

k

m

sin h( klo)

⫹ R¯2

m

p

冘 k

cos h( klo) 2 k

⫺ a1R¯2

k

m

Ak 1 {R1 ⫹ a21R¯2⑀k(R7i11k ⫹ ⑀kR9i11k)} cos h(⑀klo) ik 2

(R6imk ⫹ ⑀kR8imk)bmBk ⫺ a1R¯

2 k

k

i ⫽ 1,2,3.

q

Mimb˙m ⫹ AoQi ⫹

k

J (⑀k)(Ak ⫹ Bk)␦i,k

2 k o

q

2 k

R¯ 2

Ao 2

p

Eq. (58):

m

p

d {Aktan h(⑀klo) ⫹ Bk cot h(⑀klo)}{Aptan h(⑀plo)

p

⫹ R¯

冘

kpi

J2o(⑀p)hp␦p,i

k p kip

k

⫹ R¯

⫺

k

k

⫹ Bp cot h(⑀plo)} ⫹ ⫹

冘 冘⑀ ␴

冘

cos h( klo)

(R7impk ⫹

⑀k (R6 ⫹ ⑀kR8i1k)Bk sin h(⑀klo) i1k

1 ⑀ R9 )A b b 2 k impk k m p

(R7im1k ⫹ R7i1mk ⫹ ⑀kR9im1k)Akbm ⫽ 0,

(A5) i ⫽ 1, 2, 3.

M. Chiba, K. Abe / Thin-Walled Structures 34 (1999) 233–260

259

Eq. (59):

冘 m

1 a12R¯2 b˙mQm ⫹ Ao ⫹ 2 2

冘 ⑀⑀ 冘冘 ⑀ ⑀ 冘冘冘 ⑀ ⑀

⫺ a1R¯k

sin h( klo)

2 k

⫺ a1R¯2

m

⫹ R¯2

cos h( klo)

k

k

p

k

冘冘 m

k

⑀k (R17 ⫹ ⑀kR1mk)Bkbm sin h(⑀klo) mk

(R6m1k ⫹ R61mk ⫹ ⑀kR8m1k)Akbm

2 k

m

⑀k 2 (2R611k ⫹ ⑀kR811k)Ak cos h(⑀klo)

1 ¯ (R17 1k ⫹ ⑀kR1k)Bk ⫹ R

k

k

冘

cos h( klo)

(R6mpk ⫹

(A6)

1 ⑀ R8 )A b b ⫽ 0. 2 k mpk k m p

冕 1

6 imk

R

⫽ wi(␳)wm(␳),␳J1(⑀k␳)␳ d␳, 0

冕 1

17 mk

R

⫽ wm(␳),␳J1(⑀k␳)␳ d␳. 0

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