Frequency response of sloshing in an annular cylindrical tank subjected to pitching excitation

Journal of Sound and Vibration 331 (2012) 3199–3212

Contents lists available at SciVerse ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Frequency response of sloshing in an annular cylindrical tank subjected to pitching excitation Hiroki Takahara a,n, Koji Kimura b a b

Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552, Japan Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku Tokyo, 152-8552, Japan

a r t i c l e i n f o

abstract

Article history: Received 20 October 2011 Received in revised form 19 February 2012 Accepted 24 February 2012 Handling Editor: L.G. Tham Available online 17 March 2012

Frequency responses of stable planar and rotary motions in a partially filled annular cylindrical tank, subjected to a pitching excitation at a frequency in the neighborhood of the lowest resonant frequency, are investigated. The nonlinearity of the liquid surface oscillation and the nonlinear coupling between the dominant modes and other modes (e.g., an axisymmetric mode) are considered in the response analysis of the sloshing motion. The basic equations of the liquid motion are derived by using the variational principle and the nonlinear equations of motion of the liquid surface displacement are formulated. The characteristics of the liquid motion in an annular cylindrical tank are discussed. The equations governing the amplitude of the stable planar and rotary liquid motions are derived and the stability of each motion is analyzed. An experiment was carried out using a model tank. It is shown that the nonlinear characteristic of the liquid motion in an annular cylindrical tank is more complicated than that in a circular cylindrical tank. Furthermore, it is shown that the nonlinear analysis is important for estimating the sloshing responses. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction Nonlinear behavior of liquid in many types of moving containers such as rectangular, circular cylindrical and annular cylindrical tanks has often been the subject of research concerning several aerospace and seismic engineering problems [1,2]. Faltinsen et al. [3,4] focused on the nonlinear liquid motion in a rectangular tank performing an arbitrary threedimensional motion, i.e., horizontal, vertical and pitching motion, and studied the stable and unstable frequency domains of the planar resonant standing waves, the swirling waves and the square-like resonant standing waves in detail. Miles [5,6] investigated the nonlinear wave motion in a circular cylindrical tank subject to horizontal excitation by using a Hamiltonian system. Miles showed the internal resonance and the different types of wave motion, e.g., planar, rotary and chaotic motions. The characteristics of the nonlinear sloshing in a circular cylindrical tank subject to pitching excitation have been revealed by Kimura et al. [7]. There are tanks with inner structures such as various reaction vessels of chemical plants and boiling water reactor vessels. An annular cylindrical tank is often used to clarify the effect of the inner structure on the sloshing [8]. There are few studies of the sloshing in an annular cylindrical tank compared with those of sloshing in a rectangular or circular cylindrical tank. Meserole and Fortini [9] showed the nonlinear effects in sloshing motion in a toroidal tank by experiments. The nonlinear liquid sloshing in

n

Corresponding author. Tel.: þ81 3 5734 3599. E-mail address: [email protected] (H. Takahara).

0022-460X/$ - see front matter Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2012.02.023

3200

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

toroidal containers, which modeled the liquid tuned damper, subjected to horizontal excitation is studied by Modi and Welt [10] and Welt and Modi [11,12]. They proposed the optimal configurations for the toroidal damper. Most of the published works are concerned with the liquid motion in an annular cylindrical tank subjected to horizontal excitation, but the investigation of the dynamic behavior subjected to the pitching excitation is also important [3,7]. This paper focuses the nonlinear liquid sloshing in an annular cylindrical tank subjected to the pitching excitation whose frequency is in the vicinity of the lowest natural frequency. The liquid surface displacement and the velocity potential were expressed by the superposition of linear modes. The equations of motion of the liquid surface displacement are derived in the case that the planar and rotary (swirl) liquid motions are considered in the response analysis. The frequency response of those oscillations is analyzed. Liquid surface displacement responses of both motions are shown as the functions of the excitation frequency. Comparison with the experimental results demonstrates the usefulness of the present nonlinear analysis. It is noted that the nonlinear analysis is important for estimating the sloshing responses. 2. Theoretical analysis 2.1. Analytical model A rigid circular cylindrical tank partially filled with liquid is schematically shown in Figs. 1 and 2, in which O–XYZ is a fixed frame and both o–xyz and o2r yz are moving frames fastened to the bottom; a, b and h denote the outer and inner radius and the liquid depth of the tank, respectively. br ¼ b=a denotes the ratio of inner radius to outer radius. The tank is subjected to horizontal excitation FX(t) in the X-direction, vertical excitation FZ(t) in the Z-direction and pitching excitation xðtÞ around the pitching axis, which is parallel to y-axis and drawn through the point Cðxc ,0,zc Þ in o–xyz coordinate.

Z z

η

lp y

h

FZ (t)

Y

C o x

ξ (t) O FX (t)

X

Fig. 1. Analytical model: fixed frame O2XYZ.

z

2a 2b

η

h

O r

y

x

θ

Fig. 2. Analytical model: moving frames o2xyz and o2ryz.

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

3201

The liquid surface displacement, measured from the reference height h, is denoted by Z. Liquid motion is assumed to be incompressible, inviscid and irrotational. 2.2. Basic equations The nonlinear liquid motion can be formulated by using the variational principle. From the equality of the Lagrangian density with the liquid pressure P, Hamilton’s principle in the Eulerian description can be expressed [13] as Z t1 Z 2p Z a Z h þ Z d Pðr, y,z,tÞ dz r dr dy dt ¼ 0 (1) t0

0

0

b

With P expressed in terms of the velocity potential f that gives the absolute liquid velocity Eq. (1) can be written as  Z t 1 Z 2p Z a Z h þ Z  @f 1 _ vc  rf þ gZ þ rf  rf þ GðtÞ d dz r dr dy dt ¼ 0 (2) 2 @t t0 0 0 b _ where vc ¼ ðvr ,vy ,vz ÞT is the velocity of the tank and GðtÞ is the function obtained through integration of the Euler equation. _ GðtÞ depends only on time and not on position in the liquid. vr ¼ ½f_ X cos xf_ Z sin x þ x_ ðzzc Þ cos y vy ¼ ½f_ X cos x þ f_ Z sin xx_ ðzzc Þ sin y vz ¼ f_ X sin x þ f_ Z cos xx_ ðr cos yxc Þ

(3)

The variations with respect to f, Z and G in Eq. (2) lead to the equations @2 f @f @2 f @2 f þ þ þ 2 ¼0 2 2 2 r@r r @y @r @z    @f @f @f ¼ vr , ¼ vr , ¼ vz   @r r ¼ a @r r ¼ b @z z ¼ 0

r2 f ¼

Z 2p Z 0

Z 2p Z 0

Z 2p Z 0

a b

(4)

(5)

a

Zr dr dy ¼ 0

(6)

P9z ¼ h þ Z dZr dr dy ¼ 0

(7)

b a

b

 )    @Z   þ ðrfvc Þ   rðzhZÞ  df  @t z ¼ hþZ

(

r dr dy ¼ 0

(8)

z ¼ hþZ

Eq. (4) is the Laplace equation. Eqs. (5) are boundary conditions on the outer and inner sidewall and bottom, respectively. Eq. (6) is the condition of the incompressible fluid. Eqs. (7) and (8) are, respectively, nonlinear dynamic and kinematic boundary conditions on the free surface of the liquid. Eqs. (7) and (8) are expressed in integral forms for convenience in the subsequent analysis. 2.3. Admissible functions The solutions satisfying the nonlinear boundary conditions (7) and (8) are determined by using Galerkin’s method. The admissible functions of the velocity potential f and the liquid surface displacement Z are assumed to be represented by combining the modal functions obtained by the linearized analysis: 1 X 1  X r  coshðlm,n z=aÞ fðr, y,z,tÞ ¼ ðAm,n ðtÞcos my þ Bm,n ðtÞsin myÞQ m lm,n a coshðlm,n h=aÞ m¼0n¼1 þrvr þðzhÞvz þ x_ frðzhÞcos y þ Pðr, y,zÞg

Zðr, y,tÞ ¼ where

 r þ r cos y tan x ðC m,n ðtÞcos my þDm,n ðtÞsin myÞQ m lm,n a m¼0n¼1 1 X

1 X

    r dn  r o r d n  r o r ¼ Y m lm,n J m lm,n Q m lm,n J m lm,n  Y m lm,n   a dr a a dr a a r¼a r¼a

(9) (10)

(11)

where m and n are, respectively, the circumferential and the radial mode numbers, and Jm and Ym are the Bessel functions of the first and second kind of order m and Am,n , Bm,n , C m,n and Dm,n are generalized coordinates of (m,n)-th order. Pðr, y,zÞ is

3202

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

introduced in Eq. (9) in order to satisfy the boundary condition on the bottom, Eq. (5). The eigenvalue lm,n is the n-th root satisfying   d r  Q m lm,n  ¼0 (12) dr a r¼b The natural frequency om,n is related to the eigenvalue lm,n by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g lm,n lm,n h tanh om,n ¼ a a

(13)

where g is the gravitational acceleration. As attention is paid to the response in the neighborhood of the lowest resonant frequency and by considering the internal resonances of other modes (See Section 5.1), the circumferential modes up to m¼3 and the radial mode number n ¼2 are considered here. 2.4. Equations of motion for liquid surface displacement The following dimensionless variables are introduced: r ¼ r n =an ,

t ¼ on1;1 t n ,

z ¼ zn =an ,

n

n

n

br ¼ b =an ,

h ¼ h =an

n n2 x€ ðtÞ ¼ x€ ðtn Þ=o1;1 , f ¼ f =an on1;1 , Z ¼ Zn =an , Pðr, y,zÞ ¼ Pn =an 2

2

2

2

o2m,n ¼ onm,n =on1;1 ¼ cm,n =c1;1 , lm,n ¼ lnm,n an , xc ¼ xnc =an , zc ¼ znc =an n 2 F€ X ðtÞ ¼ F€ X ðt n Þ=ðaon1;1 Þ,

n 2 F€ Z ðtÞ ¼ F€ Z ðt n Þ=ðaon1;1 Þ, 2

Am,n ðtÞ ¼ Anm,n ðt n Þ=ðan on1;1 Þ, C m,n ðtÞ ¼ C nm,n ðt n Þ=an ,

n

x€ ðtÞ ¼ x€ ðtn Þ=on1;1 2

2

Bm,n ðtÞ ¼ Bnm,n ðt n Þ=ðan on1;1 Þ Dm,n ðtÞ ¼ Dnm,n ðt n Þ=an

(14)

The variables with asterisks denote dimensional quantities and cm,n is defined by

cm,n ¼ lm,n tanhðlm,n hÞ

(15)

The equations of the following analysis are expressed by using these dimensionless quantities. A system of nonlinear differential equations of the second order for generalized coordinates C m,n ðtÞ and Dm,n ðtÞ are derived by substituting Eqs. (9) and (10) into Eqs. (7) and (8), performing the variational operation and eliminating Am,n ðtÞ and Bm,n ðtÞ [14] ð2Þ 3 _2 C€ 1;1 þ2z1;1 C_ 1;1 þð1 þ c1;1 f€ Z ðtÞ þxc c1;1 x€ ðtÞÞC 1;1 þ Pð1Þ 1;1 C 1;1 þ P 1;1 C 1;1 C 1;1 2

ð4Þ ð5Þ _ 2 _ _ þ P ð3Þ 1;1 C 1;1 D1;1 þ P 1;1 C 1;1 D 1;1 þP 1;1 C 1;1 D1;1 D 1;1

þ

2 X

Pð11Þ C C þ 1;1,0,k 1;1 0,k

k¼1

2 X

Pð12Þ C_ C_ þ 1;1,0,k 1;1 0,k

k¼1

þ

2 X

2 X

P ð11Þ ðC 1;1 C 2,k þD1;1 D2,k Þ 1;1,2,k

k¼1

_ 1;1 D _ 2,k Þ ¼ sðPÞ x€ ðtÞ þ sðHÞ f€ ðtÞ P ð12Þ ðC_ 1;1 C_ 2,k þ D 1;1 1;1 X 1;1,2,k

(16)

k¼1 ð1Þ 3 ð2Þ € € _ _2 € 1;1 þ 2z D D 1;1 1;1 þ ð1 þ c1;1 f Z ðtÞ þ xc c1;1 x ðtÞÞD1;1 þ P 1;1 D1;1 þP 1;1 D1;1 D 1;1 2

ð4Þ ð5Þ _ 2 _ _ þP ð3Þ 1;1 D1;1 C 1;1 þ P 1;1 D1;1 C 1;1 þ P 1;1 D 1;1 C 1;1 C 1;1

þ

2 X

Pð11Þ D C þ 1;1,0,k 1;1 0,k

k¼1

2 X k¼1

þ

2 X

_ C_ þ P ð12Þ D 1;1,0,k 1;1 0,k

2 X

Pð11Þ ðC 1;1 D2,k D1;1 C 2,k Þ 1;1,2,k

k¼1

_ 2,k D _ 1;1 C_ 2,k Þ ¼ P ð12Þ ðC_ 1;1 D 1;1,2,k

0

(17)

k¼1 2 _2 Þ ðC 21;1 þD21;1 Þ þ P ð22Þ ðC_ 1;1 þ D C€ 0,l þ2z0,l o0,l C_ 0,l þ o20,l C 0,l ¼ P ð21Þ 1;1 1;1,0,l 1;1,0,l

ðl ¼ 1; 2Þ

(18)

2 _2 Þ ðC 21;1 D21;1 Þ þ P ð22Þ ðC_ 1;1 D C€ 2,l þ2z2,l o2,l C_ 2,l þ o22,l C 2,l ¼ Pð21Þ 1;1 1;1,2,l 1;1,2,l

ðl ¼ 1; 2Þ

(19)

_ 2,l þ o2 D2,l ¼ 2fPð21Þ C 1;1 D1;1 þ Pð22Þ C_ 1;1 D _ 1;1 g € 2,l þ2z o2,l D D 2,l 2,l 1;1,2,l 1;1,2,l

ðl ¼ 1; 2Þ

(20)

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

3203

ð52Þ ð53Þ 3 2 _2 C€ 1;2 þ2z1;2 o1;2 C_ 1;2 þ o21;2 C 1;2 þ P ð51Þ 1;1,2 C 1;1 þ P 1;1,2 C 1;1 C 1;1 þ P 1;1,2 C 1;1 D1;1 2

ð55Þ _ _ _ þ Pð54Þ 1;1,2 C 1;1 D 1;1 þ P 1;1,2 C 1;1 D1;1 D 1;1 þ

2 X

P ð61Þ C C þ 1;1,0,k,2 1;1 0k

k¼1 2 X

þ

Pð61Þ ðC 1;1 C 2,k þ D1;1 D2,k Þ þ 1;1,2,k,2

k¼1

2 X

Pð62Þ C_ C_ 1;1,0,k,2 1;1 0,k

k¼1 2 X

_ 1;1 D _ 2,k Þ P ð62Þ ðC_ 1;1 C_ 2,k þ D 1;1,2,k,2

k¼1

¼ sP1;2 x€ ðtÞ þ sh1;2 f€ X ðtÞ

(21)

_ 1;2 þ o2 D1;2 þ P ð51Þ D3 þ Pð52Þ D1;1 D _ 2 þ Pð53Þ D1;1 C 2 € 1;2 þ 2z1;2 o1;2 D D 1;2 1;1 1;1 1;1,2 1;1 1;1,2 1;1,2 ð55Þ _ _2 _ þ Pð54Þ 1;1,2 D1;1 C 1;1 þ P 1;1,2 D 1;1 C 1;1 C 1;1 þ

2 X

P ð61Þ D C þ 1;1,0,k,2 1;1 0,k

k¼1

þ

2 X

2 X

Pð61Þ ðC 1;1 D2,k D1;1 C 2,k Þ þ 1;1,2,k,2

k¼1

2 X

_ C_ Pð62Þ D 1;1,0,k,2 1;1 0,k

k¼1

_ 1;1 C_ 2,k Þ ¼ 0 _ 2,k D P ð62Þ ðC_ 1;1 D 1;1,2,k,2

(22)

k¼1

ð32Þ ð33Þ ð34Þ ð35Þ _ 3 2 _2 _2 _ C€ 3,n þ 2z3,n o3,n C_ 3,n þ o23,n C 3,n ¼ Pð31Þ 1;1,n C 1;1 þP 1;1,n C 1;1 C 1;1 þ P 1;1,n C 1;1 D1;1 þP 1;1,n C 1;1 D 1;1 þP 1;1,n C 1;1 D1;1 D 1;1

þ

2 X

P ð41Þ ðC 1;1 C 2,k D1;1 D2,k Þ þ 1;1,n,k

k¼1

2 X

_ 2,k Þ _ 1;1 D Pð42Þ ðC_ 1;1 C_ 2,k D 1;1,n,k

ðn ¼ 1; 2Þ

(23)

k¼1

_ _ 3,n þ o2 D3,n ¼ P ð31Þ D3 Pð32Þ D1;1 D _ 2 P ð33Þ D1;1 C 2 P ð34Þ D1;1 C_ 2 P ð35Þ D _ D€ 3,n þ 2z3,n o3,n D 3,n 1;1 1;1 1;1 1;1,n 1;1 1;1,n 1;1,n 1;1,n 1;1,n 1;1 C 1;1 C 1;1 þ

2 X k¼1

Pð41Þ ðC 1;1 D2,k þD1;1 C 2,k Þ þ 1;1,n,k

2 X

_ 1;1 C_ ,2k Þ _ 2,k þ D Pð42Þ ðC_ 1;1 D 1;1,n,k

ðn ¼ 1; 2Þ

(24)

k¼1

where the equivalent damping terms are introduced to take account of the energy dissipation. (See Appendix A for the ðPÞ ð1Þ ð11Þ ð21Þ mathematical expressions for the influence factors s1;1 , sðHÞ 1;1 and the coefficients such as P 11 ,P 110k ,P 112n .) The following

assumption of the order of the generalized coordinates and the excitations is used and the terms up to Oðe3=3 Þ are retained [15]. 9 > A1;1 , A_ 1;1 , A€ 1;1 ,B1;1 , B_ 1;1 , B€ 1;1 > > > 1=3 > _ € _ € > C 1;1 , C 1;1 , C 1;1 ,D1;1 , D 1;1 , D 1;1  Oðe Þ > > > > > > Am,l , A_ m,l , A€ m,l ,Bm,l , B_ m,l , B€ m,l > > > > 2=3 > _ € _ € C m,l , C m,l , C m,l ,Dm,l , D m,l , D m,l  Oðe Þ > > > > > > ðm ¼ 0; 2,l ¼ 1; 2Þ > = _ € _ € A3,l , A 3,l , A 3,l ,B3,l , B 3,l , B 3,l (25) > > _ 3,l , D € 3,l > C 3,l , C_ 3,l , C€ 3,l ,D3,l , D  Oðe3=3 Þ > > > > > > ðl ¼ 1; 2Þ > > > > _ € _ € > A1;2 , A 1;2 , A 1;2 ,B1;2 , B 1;2 , B 1;2 > > > > 3=3 _ 1;2 , D € 1;2  Oðe Þ > > C 1;2 , C_ 1;2 , C€ 1;2 ,D1;2 , D > > > 3=3 > _ € _ € _ € x, x , x ,F X , F X , F X ,F Z , F Z , F Z  Oðe Þ ;

2.5. Frequency response We consider the response to the harmonic pitching excitation xðtÞ ¼ x0 cos ot. First, the solutions of the dominant modes C 1;1 and D1;1 can be assumed in the form C ð1Þ 1;1 ¼ a1 ðtÞcos ot þa2 ðtÞsin ot

(26)

Dð1Þ 1;1 ¼ b1 ðtÞcos ot þ b2 ðtÞsin ot

(27)

where ai and bi ði ¼ 1; 2Þ are slowly varying amplitudes and t is a dimensionless slow time (Oðt=tÞ ¼ e2=3 ) [15]. The superscript ð1Þ denotes the first approximation. The solutions of secondary modes C 0,l , C 2,l and D2,l are obtained by substituting Eqs. (26) and (27) into Eqs. (18)–(20). Moreover, by substituting these solutions into Eqs. (16) and (17), retaining the terms up to Oðe3=3 Þ, and considering the periodicity condition for C 1;1 and D1;1 , equations governing the amplitude of the liquid surface oscillation are derived: 2oa_ 2 þ ð1o2 Þa1 þ 2z1;1 oa2 þ F ð1Þ a31 þ F ð2Þ a21 a2 þF ð1Þ a1 a22 þ F ð2Þ a32

3204

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212 2

2

2

2

þ F ð1Þ a1 b1 þF ð3Þ a1 b2 þF ð4Þ a2 b1 þ F ð2Þ a2 b2 þ F ð5Þ a1 b1 b2 þF ð6Þ a2 b1 b2 ¼ o2 sp x0

(28)

2oa_ 1 þ ð1o2 Þa2 2z1;1 oa1 F ð2Þ a31 þ F ð1Þ a21 a2 F ð2Þ a1 a22 þ F ð1Þ a32 2

2

2

2

F ð2Þ a1 b1 F ð4Þ a1 b2 þ F ð3Þ a2 b1 þ F ð1Þ a2 b2 þ F ð6Þ a1 b1 b2 F ð5Þ a2 b1 b2 ¼ 0

(29)

3 2 2 3 2ob_ 2 þð1o2 Þb1 þ 2z1;1 ob2 þ F ð1Þ b1 þ F ð2Þ b1 b2 þ F ð1Þ b1 b2 þF ð2Þ b2

þF ð1Þ b1 a21 þF ð3Þ b1 a22 þ F ð4Þ b2 a21 þ F ð2Þ b2 a22 þ F ð5Þ b1 a1 a2 þ F ð6Þ b2 a1 a2 ¼ 0

(30)

2ob_ 1 þð1o2 Þb2 2z1;1 ob1 F ð2Þ b1 þ F ð1Þ b1 b2 F ð2Þ b1 b2 þF ð1Þ b2 3

2

2

3

F ð2Þ b1 a21 F ð4Þ b1 a22 þ F ð3Þ b2 a21 þF ð1Þ b2 a22 þF ð6Þ b1 a1 a2 F ð5Þ b2 a1 a2 ¼ 0

(31)

ðÞ

where the mathematical expressions for the coefficients F are shown in Appendix A. 2.5.1. Planar liquid motions In the case of steady-state planar motion, only C 1;1 mode is considered a_ 1 ¼ a_ 2 ¼ b_ 1 ¼ b_ 2 ¼ b1 ¼ b2 ¼ 0

(32)

By setting the amplitudes ai ði ¼ 1; 2Þ of C 1;1 mode as a1 ¼ a cos g,

a2 ¼ a sin g

(33)

where a and g are the amplitude of C 1;1 mode and the phase angle between the excitation xðtÞ and C 1;1 mode, respectively. Substituting Eqs. (32) and (33) into Eqs. (28)–(31), the amplitude a and the phase angle g are determined by the following equations: 2

fF ð1Þ a3 þ ð1o2 Þag2 þ fF ð2Þ a3 þ2z1;1 oag2 ¼ o4 s2p x0 cos g ¼

sin g ¼

fF ð1Þ a3 þð1o2 Þago2 sp x0

o2 Þag2 þfF ð2Þ a3 þ 2z1;1 oag2

fF ð1Þ 3 þ ð1

a

fF ð2Þ a3 þ 2z1;1 oago2 sp x0

o2 Þag2 þfF ð2Þ a3 þ 2z1;1 oag2

fF ð1Þ 3 þ ð1

a

(34)

(35)

(36)

Stability of the steady-state planar motion obtained by solving Eqs. (34)–(36) is investigated by imposing a small perturbation from this steady-state solution. We set st a1 ¼ að0Þ 1 þ b1 e ,

st a2 ¼ að0Þ 2 þ b2 e

(37)

að0Þ i

ði ¼ 1; 2Þ denote the steady-state solution and b1 , b2 are small perturbations. By substituting Eq. (37) into Eqs. where (28) and (29), the following simultaneous equations for b1 and b2 are obtained: " #" #   q11 q12 þ2os b1 0 ¼ (38) q22 q21 2os b2 0 where 2

2

ð2Þ ð0Þ ð0Þ ð1Þ ð0Þ q11 ¼ ð1o2 Þ þ 3F ð1Þ að0Þ 1 þ 2F a1 a2 þF a2 2

2

ð2Þ ð0Þ ð0Þ ð2Þ ð0Þ q12 ¼ 2z1;1 o þ F ð2Þ að0Þ 1 þ 2F a1 a2 þ 3F a2 2

2

ð1Þ ð0Þ ð0Þ ð2Þ ð0Þ q21 ¼ 2z1;1 o3F ð2Þ að0Þ 1 þ 2F a1 a2 F a2 2

2

ð2Þ ð0Þ ð0Þ ð1Þ ð0Þ q11 ¼ ð1o2 Þ þF ð1Þ að0Þ 1 2F a1 a2 þ 3F a2

(39) (40) (41) (42)

For the nontrivial solution of b1 and b2 , the determinant of a coefficient matrix must be 0, so s is determined. The stability condition of the steady-state solution is the negative real part of s. Thus,

z1;1 o þ F ð2Þ r2 4 0

(43)

fð1o2 Þ þ F ð1Þ r 2 gfð1o2 Þ þ 3F ð1Þ r 2 g þ f2z1;1 o þ F ð2Þ r 2 gf2z1;1 o þF ð2Þ r 2 g 40

(44)

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

3205

2.5.2. Rotary (swirl) liquid motions In the case of steady-state rotary motion, both C 1;1 and D1;1 modes are considered a_ 1 ¼ a_ 2 ¼ b_ 1 ¼ b_ 2 ¼ 0

(45)

By considering that the rotary motion is the traveling wave on the liquid surface due to C 1;1 and D1;1 modes, we set the amplitudes ai and bi ði ¼ 1; 2Þ of C 1;1 and D1;1 modes a1 ¼ a1 cos g, b1 ¼ a2 sin ðg þ DgÞ,

a2 ¼ a1 sin g b2 ¼ a2 cos ðg þ DgÞ

(46) (47)

where a1 , a2 , g and Dg are the amplitude of C 1;1 mode, the amplitude of D1;1 mode, the phase angle between the excitation xðtÞ and C 1;1 mode and the phase angle between C 1;1 and D1;1 modes respectively. Substituting Eqs. (45)–(47) into Eqs. (28)–(31), the amplitudes a1 , a2 and the phase angles g, Dg are determined by the following equations: 1

a21 ¼ ~ f2z1;1 oF ð1Þ ð1o2 ÞF ð2Þ g D

(48)

a22 ¼ ~ ½2z1;1 oF ð1Þ þ ð1o2 ÞF ð2Þ þ f2z1;1 oF ð6Þ ð1o2 ÞF ð5Þ gcos2 Dgf2z1;1 oF ð5Þ þ ð1o2 ÞF ð6Þ gcos Dg sin Dg D

(49)

1

cos g ¼

A~ o2 sp x0 , 2 2 a1 ðA~ þ B~ Þ 2

sin g ¼

B~ o2 sp x0 2 2 a1 ðA~ þ B~ Þ

2

a21 ðA~ þ B~ Þ ¼ o4 s2p x20

(50)

(51)

D~ ¼ ðF ð2Þ F ð3Þ F ð1Þ F ð4Þ Þcos2 Dg þ ðF ð1Þ F ð6Þ þ F ð2Þ F ð5Þ Þsin Dg cos Dg A~ ¼ ð1o2 Þ þ F ð1Þ a21 þ a22 ðF ð3Þ cos2 Dg þF ð1Þ sin2 DgF ð3Þ cos2 Dgsin2 DgÞ B~ ¼ 2z1;1 o þ F ð2Þ a21 þ a22 ðF ð2Þ cos2 Dg þ F ð4Þ sin2 Dg þF ð6Þ sin Dg cos DgÞ Stability of the steady-state rotary motion is investigated by using the same procedure as for the planar motion. 3. Nonlinear characteristics In Eqs. (16)–(24) only C 1,l mode with one nodal diameter is directly related to the pitching excitation xðtÞ, whereas other modes, e.g., C 0,l , C 2,l and D1,l , are coupled with C 1;1 mode through nonlinear terms. 3.1. Internal resonance Since C 0,l and C 2,l modes are coupled with C 1;1 and D1;1 modes through the quadratic terms as shown in Eqs. (18) and (19), they are apparently subjected to the excitation with double frequency of C 1;1 and D1;1 modes. Thus, under the condition that the natural frequency of C 0,l mode or C 2,l mode approximates twice that of C 1;1 mode, the internal resonance of C 0,l mode or C 2,l mode occurs in the case of the fundamental resonance of C 1;1 mode. 3.2. Planar and rotary (swirl) liquid motions It is noted that D1;1 mode, closely related to the rotary sloshing, is only excited through the parametric terms (e.g., C 21;1 D1;1 ) as shown in Eq. (17). Since D1;1 and C 1;1 modes have equal natural frequency and the frequency of coefficient C 21;1 is twice that of D1;1 mode, the parametric resonance of D1;1 mode occurs in the case of the fundamental resonance of C 1;1 mode. In the case that the liquid surface is perfectly flat initially, only C i,l modes are excited and the D1;1 mode is not excited. As a result, the liquid motion is planar. In the case of the liquid surface with the initial disturbance, D1;1 mode becomes large due to the parametric resonance. As a result, the liquid motion can be rotary. 4. Experiments The block diagram for instrumentation is shown in Fig. 3. The experiment was conducted using two tanks (Tank A has n outer radius an ¼ 140 mm, inner radius b ¼ 27:5 mm (br ¼0.196), Tank B has outer radius an ¼ 140 mm, inner radius n b ¼ 55 mm (br ¼0.393)). The model tank is mounted on the pitching table, which is pitched by the hydraulic servo vibrator

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

Fig. 3. Experimental apparatus.

3

Ω1,1 Ω0,1 Ω2,1 Ω2,2 2 Ω1,1

2.5

m,n

2 1.5 1 0.5 0

m,n = 0

0.5

m,n 1,1⏐h→∞ 1 h

1.5

2

Fig. 4. Natural frequency vs. depth of liquid (br ¼ 0.196).

3 2.5 2 m,n

3206

1.5 1

Ω1,1 Ω0,1 Ω2,1 Ω2,2 2 Ω1,1

0.5 0

0

0.5

1 h

1.5

Fig. 5. Natural frequency vs. depth of liquid (br ¼ 0.393).

2

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

3207

driven by the signal of the harmonic oscillator. The liquid surface displacement is measured by the ultrasonic probe mounted on the fixture on the top of the tank. 5. Discussion 5.1. Natural frequency and internal resonance The natural frequency of each mode depends on the liquid depth h as shown in Eq. (13). In the cases of Tank A and Tank B, the relations between the natural frequency and the liquid depth h are shown, respectively, in Figs. 4 and 5. The horizontal and vertical axes denote, respectively, the dimensionless liquid depth h and the dimensionless natural frequencies Om,n which is defined by

Om,n ¼

om,n om,n 9h-1

(52)

50

50

40

40

30

30

* (mm)

* (mm)

The solid line, the broken line, the short chain line and the long chain line indicate respectively the natural frequencies of C 1;1 , C 0;1 , C 2;1 and C 2;2 modes and twice the natural frequency of C1 mode is also plotted as the dotted line for comparison. In the low liquid depth region, twice the natural frequency of C 1;1 mode approximates the natural frequency of C 0;1 mode in Fig. 4 and that of C 2;1 mode in Fig. 5. In the high liquid depth region, twice the natural frequency of C 1;1 mode approximates the natural frequency of C 2;2 mode in Fig. 4 and that of C 0;1 mode in Fig. 5. By the internal resonance mentioned in Section 3.1, when the frequency of the excitation is in the vicinity of the natural frequency of C 1;1 mode, in

20

10

0 −0.3 −0.2

20

10

−0.1

0

ν

0.1

0.2

0 −0.3

0.3

−0.2

−0.1

0

ν

0.1

0.2

0.3

50

50

40

40

30

30

* (mm)

* (mm)

Fig. 6. Frequency responses of liquid surface displacement in Tank A ðbr ¼ 0:196,ðxc ,zc Þ ¼ ð0; 0ÞÞ: (a) h ¼ 0:3, x0 ¼ 0:251, zi,j ¼ 0:009 and (b) h ¼ 1:0, x0 ¼ 0:181, zi,j ¼ 0:006; — theory (stable planar); –  – theory (unstable planar); – – – theory (rotary); J experiment (planar); W experiment (rotary).

20

10

10

0 −0.3 −0.2

20

−0.1

0

ν

0.1

0.2

0.3

0 −0.3

−0.2

−0.1

0

ν

0.1

0.2

0.3

Fig. 7. Frequency responses of liquid surface displacement in Tank B ðbr ¼ 0:393,ðxc ,zc Þ ¼ ð0; 0ÞÞ: (a) h ¼ 0:3, x0 ¼ 0:261, zi,j ¼ 0:011 and (b) h ¼ 1:0, x0 ¼ 0:181, zi,j ¼ 0:007; — theory (stable planar); –  – theory (unstable planar); – – – theory (rotary); J experiment (planar); W experiment (rotary).

3208

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

the case of Tank A, C 0;1 mode may become large in the low liquid depth region and C 2;2 mode may become large in the high liquid depth region and, in the case of Tank B, C 2;1 mode may become large in the low liquid depth region and C 0;1 mode may become large in the high liquid depth region. 5.2. Results The frequency responses of the liquid surface displacement in the neighborhood of the side wall ðr ¼ 1:0, y ¼ 0:0Þ to the pitching excitation xðtÞ ¼ x0 cos ot are shown in Figs. 6 and 7, which correspond to the cases of Tank A and Tank B, respectively. 5.2.1. Tank A: br ¼0.196 When liquid depth h¼0.3 and h¼1.0, the responses are shown in Fig. 6(a) and (b), respectively. The vertical axis shows the amplitude of the liquid surface oscillation and the horizontal axis shows the detuning parameter between the excitation frequency and the natural frequency of the dominant mode, defined by

o2 ¼ o21;1 þ n

(53)

The damping ratio of the lowest mode (z1;1 ¼ 0:009,0:006) at each liquid depth (h¼0.3, h¼1.0) was obtained from the experimental results of the time history of damped free vibration and damping ratios of modes used for theoretical calculation were approximated using the obtained damping ratios. The solid, chain and broken lines, respectively, denote the amplitudes of the stable planar, unstable planar and rotary motions obtained by theoretical analysis. Experimental results of the amplitudes of the stable planar and rotary motions are plotted as the symbol J and n, respectively. Good agreement is found between the theoretical and experimental results. It is noted in Fig. 6(a) that the response curve of the planar motion hardly has the hardening characteristic that is well-known in a circular cylindrical tank at this liquid depth and that the curve reaches a peak due to the internal resonance of the axisymmetric mode C 0:1 at n ¼ 0:116. On the other hand, it is noted in Fig. 6(b) that the response curve of the planar motion has the softening characteristic and that the curve shifts slightly upward owing to the internal resonance of C 2;2 mode at n ¼ 0:04. The internal resonance that appears in Fig. 6(a) and (b) is consistent with the discussion in the preceding section (Section 5.1). 5.2.2. Tank B: br ¼ 0.393 Fig. 7(a) and (b) represents the responses when liquid depth h ¼0.3 and h ¼1.0, respectively. The damping ratio of the lowest mode (z1;1 ¼ 0:011,0:007) at each liquid depth (h¼0.3, h ¼1.0) was obtained from the experimental results of the time history of damped free vibration and damping ratios of modes used for theoretical calculation were approximated using the obtained damping ratios. Good agreement is found between the theoretical and experimental results at low liquid depth h¼0.3 (Fig. 7(a)). It is noted in Fig. 7(a) that the response curve of the planar motion has the hardening characteristic and that the curve shifts slightly upward owing to the internal resonance of C 2;1 mode at n ¼ 0:187. The internal resonance at n ¼ 0:187 is consistent with the discussion in the preceding section (Section 5.1). At high liquid depth h ¼1.0 (Fig. 7(b)), the amplitude of the axisymmetric mode C 0;1 becomes large due to the internal resonance in the vicinity of the natural frequency of the dominant mode C 1;1 . Therefore, in the neighborhood of n ¼ 0, the effect of the axisymmetric mode C 0;1 on the frequency response curve becomes large. In the theoretical calculation, damping ratios of all modes are approximated to be the same, which results in underestimate of the damping ratio of the axisymmetric mode C 0;1 and causes the difference between theoretical and experimental results in the neighborhood of n ¼ 0: 6. Conclusion Frequency response of nonlinear sloshing in an annular cylindrical tank subjected to pitching excitation is investigated. Conditions for occurrence of internal resonance were discussed on the basis of natural frequencies of liquid motion modes. The equations governing the amplitude of the liquid surface oscillation are derived. Thus, the steady-state responses of both planar and rotary liquid motions are obtained and their stabilities are investigated. It is shown that the nonlinear characteristic of the liquid motion in an annular cylindrical tank is more complicated than that in a circular cylindrical tank. The theoretical results are in fairly good agreement with experimental results obtained using a model tank. It is shown that the nonlinear analysis is important for estimating the sloshing responses. From analytical and experimental results, it was confirmed that internal resonance effects on the frequency response. Appendix A. Influence factors and coefficients 0 sðPÞ sðHÞ 0 ¼ Si,k0 ci,k0 fSi,k0 ðhzc Þ þ R 0 g, 0 ¼ ci,k0 Si,k0 k i,k i,k

Pð1Þ ¼ P ð51Þ , i,j i,j,j

P ð2Þ ¼ Pð52Þ , i,j i,j,j

P ð3Þ ¼ Pð53Þ , i,j i,j,j

P ð4Þ ¼ P ð54Þ , i,j i,j,j

Pð5Þ ¼ P ð55Þ i,j i,j,j

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

P ð11Þ ¼ P ð61Þ , i,j,m,k i,j,m,k,j 1

Pð21Þ ¼ cm,k Hð22Þ  i,j,m,k i,j,m,k

ci,j

P ð32Þ i,j,l

¼

c3i,l

Hð31Þ þ i,j,l c2i,j

2

ci,j

Hð36Þ þ i,j,l

2 X k¼1

(

P ð12Þ ¼ P ð62Þ i,j,m,k i,j,m,k,j

Hð23Þ , i,j,m,k

Pð31Þ ¼ c3i,l Hð32Þ  i,j,l i,j,l

1

ci,j

P ð22Þ ¼ i,j,m,k

Hð36Þ þ i,j,l

2 X

1

ci,j

¼

1

ci,j

1

ci,j

Hð23Þ i,j,m,k

Hð44Þ Hð22Þ i,j,l,k i,j,2i,k

k¼1

P ð34Þ ¼ P ð32Þ , i,j,l i,j,l 1

ci,j

1 Hð41Þ Hð23Þ  i,j,l,k i,j,2i,k ci,j c2i,k

) Hð44Þ Hð23Þ i,j,l,k i,j,2i,k

P ð35Þ ¼ 2P ð32Þ i,j,l i,j,l

ðHð43Þ þ Hð44Þ Þ i,j,l,k i,j,l,k

1

ci,j

Hð56Þ  i,j,l

2 X

fHð64Þ Hð22Þ þ Hð64Þ Hð22Þ g i,j,0,k,l i,j,0,k i,j,2i,k,l i,j,2i,k

k¼1

!

!

2 X ci,l ð51Þ ci,l ð61Þ 1 þ Hð23Þ Hi,j,l þ 2Hð56Þ Hi,j,0,k,l þ Hð64Þ i,j,l i,j,0,k,l i,j,0,k ci,j c c c i,j 0,k i,j k¼1

2 X 1  Hð64Þ Hð21Þ  i:j,0,k,l i,j,0,k 2 2 c c k ¼ 1 i:j k ¼ 1 i,j

Pð54Þ i,j,l

Hð21Þ þ i,j,m,k

c3i,l 1 ð43Þ 1 Hð41Þ þ H þ Hð44Þ ci,j c2i,k i,j,l,k ci,j i,j,l,k c2i,k i,j,l,k

Pð51Þ ¼ P ð53Þ ¼ ci,l Hð52Þ þ i,j,l i,j,l i,j,l

Pð52Þ ¼ i,j,l

c2i,j

c Hð44Þ Hð21Þ  2 3i,l i,j,l,k i,j,2i,k c2i,j ci,j c2i,k

P ð41Þ ¼ c3i,l Hð42Þ  i,j,l,k i,j,l,k

¼ Pð42Þ i,j,l,k

cm,k

2 X

1

¼ 3Pð31Þ , P ð33Þ i,j,l i,j,l

3209

1

Hð64Þ Hð21Þ þ i,j,2i,k,l i,j,2i,k

2 X

1

k¼1

ci,j c2,k

!

ci,l ð61Þ Hð23Þ H þ Hð64Þ i,j,2i,k,l i,j,2i,k ci,j i,j,2i,k,l

!

!

2 2 2 X X X ci,l ð53Þ ci,l ð61Þ 1 ð64Þ 1 ð64Þ 1 Hi,j,l þ Hð58Þ Hi,j,0,k,l Hð21Þ þ Hi,j,2i,k,l Hð21Þ þ Hi,j,2i,k,l þ Hð64Þ  Hð23Þ i,j,l i,j,0,k i,j,2i,k i,j,2i,k,l i,j,2i,k 2 2 ci,j c c c c c i,j k ¼ 1 i,j k ¼ 1 i,j k ¼ 1 i,j 2i,k

P ð55Þ ¼ P ð52Þ P ð54Þ i,j,l i,j,l i,j,l P ð61Þ ¼ ci,l Hð62Þ þ i,j,m,k,l i,j,m,k,l

¼ Pð62Þ i,j,m,k,l Hð1Þ ¼ Hð51Þ , i,j i,j,j

1

cm,k

1

ci,j

ðHð63Þ þHð64Þ Þ i,j,m,k,l i,j,m,k,l !

ci,l ð61Þ 1 ð63Þ H þHð64Þ H  i,j,m,k,l ci,j i,j,m,k,l ci,j i,j,m,k,l

Hð2Þ ¼ Hð52Þ , i,j i,j,j

Hð3Þ ¼ Hð53Þ , i,j i,j,j

ð56Þ ð7Þ ð57Þ Hð6Þ i,j ¼ H i,j,j ; H i,j ¼ H i,j,j ,

Hð4Þ ¼ Hð54Þ , i,j i,j,j

Hð5Þ ¼ Hð55Þ i,j i,j,j

ð58Þ Hð8Þ i,j ¼ H i,j,j

Hð11Þ ¼ Hð61Þ , i,j,m,k i,j,m,k,j

Hð12Þ ¼ Hð62Þ i,j,m,k i,j,m,k,j

Hð13Þ ¼ Hð63Þ , i,j,m,k i,j,m,k,j

Hð14Þ ¼ Hð64Þ i,j,m,k i,j,m,k,j 2

ðrÞ ðyÞ Hð21Þ ¼ 14fK i,j,i,j,k,l akðcÞ i2 K i,j,i,j,k,l ci,j K ðzÞ g i,j,k,l i,j,i,j,k,l

¼ 12K ðzÞ Hð22Þ i,j,k,l i,j,i,j,k,l 2

ðrÞ yÞ Hð23Þ ¼ 12fK i,j,i,j,k,l akðcÞ i2 K ði,j,i,j,k,l þ li,j K ðzÞ g i,j,k,l i,j,i,j,k,l 2 1 1X 2 ðrÞ ðyÞ ¼  ci,j ðK i,j,i,j,i,j,3i,l i2 K i,j,i,j,i,j,3i,l þ li,j K ðzÞ Þ c K ðzÞ Hð21Þ Hð31Þ i,j,l i,j,i,j,i,j,3i,l 4 2 k ¼ 1 2i,k i,j,2i,k,3i,l i,j,2i,k

3210

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

Hð32Þ ¼ i,j,l

2 l2i,j ðzÞ 1X K i,j,i,j,i,j,3i,l  c K ðzÞ Hð22Þ 2 k ¼ 1 2i,k i,j,2i,k,3i,l i,j,2i,k 8ci,j

Hð33Þ ¼ Hð31Þ , i,j,l i,j,l

Hð34Þ ¼ 2Hð31Þ , i,j,l i,j,l

Hð35Þ ¼ 3Hð32Þ i,j,l i,j,l

2 X 1 1 2 1 ðrÞ yÞ Hð36Þ ¼  ðci,j þ c3i,l ÞðK i,j,i,j,i,j,3i,l i2 K ði,j,i,j,i,j,3i,l Þ þ li,j ðci,j þ2c3i,l ÞK ðzÞ  c3i,l K ðzÞ Hð23Þ i,j,l i,j,i,j,i,j,3i,l i,j,2i,k,3i,l i,j,2i,k 4 8 2 k¼1

Hð37Þ ¼ Hð36Þ , i,j,l i,j,l

Hð38Þ ¼ 2Hð36Þ i,j,l i,j,l

ðrÞ yÞ Hð41Þ ¼ 12ðK i,j,2i,k,3i,l 2i2 K ði,j,2i,k,3i,l þ ci,j c2i,k K ðzÞ Þ i,j,l,k i,j,2i,k,3i,l

Hð42Þ i,j,l,k

! c2i,k ðzÞ 1 1þ ¼ K i,j,2i,k,3i,l 2 ci,j 2

ðrÞ yÞ Hð43Þ ¼ 12ðK i,j,2i,k,3i,l þ 2i2 K ði,j,2i,k,3i,l þ li,j K ðzÞ Þ i,j,l,k i,j,2i,k,3i,l 2

ðrÞ yÞ ¼ 12ðK i,j,2i,k,3i,l þ 2i2 K ði,j,2i,k,3i,l þ l2i,k K ðzÞ Þ Hð44Þ i,j,l,k i,j,2i,k,3i,l 2 2 X 1 1X 2 2 yÞ ðrÞ ¼  ci,j ð3K i,j,i,j,i,j,i,l þ i K ði,j,i,j,i,j,i,l þ 3li,j K ðzÞ Þ c0,k K ðzÞ Hð21Þ  c K ðzÞ Hð21Þ Hð51Þ i,j,l i,j,i,j,i,j,i,l 0,k,i,j,i,l i,j,0,k 4 2 k ¼ 1 2i,k i,j,2i,k,i,l i,j,2i,k k¼1

Hð52Þ ¼ i,j,l

2 2 X 3 2 ðzÞ 1X l K  c K ðzÞ Hð22Þ  c K ðzÞ Hð22Þ 8ci,j i,j i,j,i,j,i,j,i,l k ¼ 1 0,k 0,k,i,j,i,l i,j,0,k 2 k ¼ 1 2i,k i,j,2i,k,i,l i,j,2i,k

2 2 X 1 1X 2 ðrÞ yÞ Hð53Þ ¼  ci,j ðK i,j,i,j,i,j,i,l þ 3i2 K ði,j,i,j,i,j,i,l þ li,j K ðzÞ Þ c0,k K ðzÞ Hð21Þ þ c K ðzÞ Hð21Þ i,j,l i,j,i,j,i,j,i,l 0,k,i,j,i,l i,j,0,k 4 2 k ¼ 1 2i,k i,j,2i,k,i,l i,j,2i,k k¼1

Hð54Þ ¼ Hð51Þ Hð53Þ , i,j,l i,j,l i,j,l

Hð55Þ ¼ Hð52Þ i,j,l i,j,l



2 2 X 1 3 1 2 1X ðrÞ yÞ Hð56Þ ¼  ðci,j þ ci,l Þð3K i,j,i,j,i,j,i,l þi2 K ði,j,i,j,i,j,i,l Þþ li,j ci,j þ l2i,j ci,l K ðzÞ  ci,l K ðzÞ Hð23Þ  c K ðzÞ Hð23Þ i,j,l i,j,i,j,i,j,i,l 0,k,i,j,i,l i,j,0,k 4 4 2 2 k ¼ 1 i,l i,j,2i,k,i,l i,j,2i,k k¼1

2 1 1 1 2 1X 2 2 ðyÞ ðrÞ ðzÞ ð Hð57Þ ¼  c þ c ÞðK i K Þ þ l c þ l c c K ðzÞ Hð23Þ i,l i,j i,l K i,j,i,j,i,j,i,l  i,j,l i,j,i,j,i,j,i,l i,j,i,j,i,j,i,l 4 i,j 4 2 i,j i,j 2 k ¼ 1 i,l i,j,2i,k,i,l i,j,2i,k Hð58Þ ¼ Hð56Þ Hð57Þ i,j,l i,j,l i,j,l Hð61Þ ¼ bm ðK ðrÞ þ i,j,m,k,l i,j,m,k,i,l

m2 ðyÞ K þ ci,j cm,k K ðzÞ Þ i,j,m,k,i,l 2 i,j,m,k,i,l !

¼ bm 1 þ Hð62Þ i,j,m,k,l

cm,k ðzÞ K i,j,m,k,i,l ci,j



m2 ðyÞ 2 ðzÞ ðrÞ K Hð63Þ ¼ b K  þ l K m i,j i,j,m,k,i,l i,j,m,k,l i,j,m,k,i,l 2 i,j,m,k,i,l

m2 ðyÞ 2 ðrÞ K i,j,m,k,i,l þ lm,k K ðzÞ Hð64Þ ¼ bm K i,j,m,k,i,l  i,j,m,k,l i,j,m,k,i,l 2 (

akðcÞ ¼ ðrÞ K a,b,c,d,e,f ¼ ,g,h

ðk ¼ 0Þ,

1

Z

ðyÞ K a,b,c,d,e,f ¼ ,g,h

1

1

r

br

Z

1 br

ðk ¼ 2iÞ,

(

bm ¼

1

ðm ¼ 0Þ

1 2

ðm ¼ 2iÞ

dQ a ðla,b rÞ dQ c ðlc,d rÞ Q e ðle,f rÞQ g ðlg,h rÞdr=Ig,h dr dr 1 Q ðl rÞQ c ðlc,d rÞQ e ðle,f rÞQ g ðlg,h rÞdr=Ig,h r a a,b

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212

K ðzÞ ¼ a,b,c,d,e,f ,g,h

Z

ðrÞ K a,b,c,d,e,f ¼

1

rQ a ðla,b rÞQ c ðlc,d rÞQ e ðle,f rÞQ g ðlg,h rÞdr=Ig,h

br

Z

1

r br

Z

yÞ K ða,b,c,d,e,f ¼

dQ a ðla,b rÞ dQ c ðlc,d rÞ Q e ðle,f rÞ dr=Ie,f dr dr

1

br

Z

K ðzÞ ¼ a,b,c,d,e,f

1 Q ðl rÞQ c ðlc,d rÞQ e ðle,f rÞ dr=Ie,f r a a,b

1

br

rQ a ðla,b rÞQ c ðlc,d rÞQ e ðle,f rÞ dr=Ie,f

Ia,b ¼

Sa,b ¼

Z

1 br

3211

Z

1 br

rfQ a ðla,b rÞg2 dr

r 2 Q a ðla,b rÞ dr=Ia,b ,

R0j ¼ R1,j =coshðl1,j hÞ

3 ð1Þ 1 ð2Þ 2 P þ P o 4 1,1 4 1,1 ( ) 2  1  X 1  ð21Þ ð21Þ ð1Þ 2 ð22Þ 2 ð22Þ P þ þ P ð11Þ P þ o P  o P D 1,1,0,n 1,1,0,n 1,1,0,n 0,n 4 1,1,0,n 2o20,n 1,1,0,n n¼1

F ð1Þ ¼

2   1 2X ð21Þ ð1Þ 2 ð22Þ o P ð12Þ 1,1,0,n P 1,1,0,n o P 110n D0,n 2 n¼1 ( ) 2  1  X 1  ð21Þ ð11Þ ð21Þ ð1Þ 2 ð22Þ 2 ð22Þ P þ P 1,1,2,n P þ o P 1,1,2,n þ o P 1,1,2,n D2,n 4 1,1,2,n 2o22,n 1,1,2,n n¼1

þ

þ

2   1 2X ð21Þ ð1Þ 2 ð22Þ o Pð12Þ 1,1,2,n P 1,1,2,n o P 1,1,2,n D2,n 2 n¼1

F ð2Þ ¼ 

2   1X ð21Þ ð2Þ 2 ð22Þ P ð11Þ 1,1,0,n P 1,1,0,n þ o P 1,1,0,n D0,n 4 n¼1

2   X 1 ð21Þ ð2Þ 2 ð22Þ  o2 P ð12Þ 1,1,0,n P 1,1,0,n o P 1,1,0,n D0,n 2 n¼1



2   1X ð2Þ P ð11Þ P ð21Þ o2 P ð22Þ 1,1,2,n D2,n 4 n¼1 1,1,2,n 1,1,2,n

2   X 1 ð21Þ ð2Þ 2 ð22Þ  o2 Pð12Þ 1,1,2,n P 1,1,2,n o P 1,1,2,n D2,n 2 n¼1

1 ð1Þ 1 ð2Þ 2 2 P  P o þ Pð4Þ 1,1 o 4 1,1 4 1,1 ( ) 2  1  X 1  ð21Þ ð11Þ ð21Þ ð1Þ 2 ð22Þ 2 ð22Þ P þ P 1,1,0,n P þ o P1,1,0,n  o P1,1,0,n D0,n 4 1,1,0,n 2o20,n ,1,1,0,n n¼1

F ð3Þ ¼

2   X 1 P ð12Þ P ð21Þ o2 P ð22Þ Dð1Þ  o2 1,1,0,n 1,1,0,n 1,1,0,n 0,n 2 n¼1 ( ) 2  3  X 1  ð21Þ ð21Þ ð1Þ 2 ð22Þ 2 ð22Þ P þ P ð11Þ  P þ o P  o P D þ 1,1,2,n 1,1,2,n 1,1,2,n 2,n 4 1,1,2,n 2o22,n 1,1,2,n n¼1

þ F ð4Þ ¼

2   3 2X ð21Þ ð1Þ 2 ð22Þ o Pð12Þ 1,1,2,n P 1,1,2,n o P 1,1,2,n D2,n 2 n¼1

2   1X ð2Þ P ð11Þ P ð21Þ þ o2 Pð22Þ 1,1,0,n D0,n 4 n¼1 1,1,0,n 1,1,0,n

þ

2   1 2X ð21Þ ð2Þ 2 ð22Þ o P ð12Þ 1,1,0,n P 1,1,0,n o P 1,1,0,n D0,n 2 n¼1



2   3X ð2Þ Pð11Þ Pð21Þ o2 P ð22Þ 1,1,2,n D2,n 4 n¼1 1,1,2,n 1,1,2,n

3212

H. Takahara, K. Kimura / Journal of Sound and Vibration 331 (2012) 3199–3212 2   X 3 ð21Þ ð2Þ 2 ð22Þ  o2 P ð12Þ 1,1,2,n P 1,1,2,n o P 1,1,2,n D2,n 2 n¼1

F ð5Þ ¼ F ð2Þ F ð4Þ

Dð1Þ m,n ¼ Dð2Þ m,n ¼

,

F ð6Þ ¼ F ð1Þ F ð3Þ

o2m,n 4o2 ðo2m,n 4o2 Þ2 þ ð4zm,n om,n oÞ2 4zm,n om,n o ðo2m,n 4o2 Þ2 þ ð4zm,n om,n oÞ2

ðm ¼ 0,2,n ¼ 1,2Þ ðm ¼ 0,2,n ¼ 1,2Þ

References [1] H.N. Abramson, The Behavior of Liquids in Moving Containers with Application to Space Vehicle Technology, National Aeronautics and Space Administration Special Publication 106, 1966. [2] R.A. Ibrahim, V.N. Pilipchuk, T. Ikeda, Recent advances in liquid sloshing dynamics, Transactions of American Society of Mechanical Engineers, Applied Mechanics Reviews 54 (2) (2001) 133–199. [3] O.M. Faltinsen, O.F. Rognebakke, I.A. Lukovsky, A.N. Timokha, Multidimensional model analysis of nonlinear sloshing in a rectangular tank with finite water depth, Journal of Fluid Mechanics 407 (2000) 201–234. [4] O.M. Faltinsen, O.F. Rognebakke, A.N. Timokha, Resonant three-dimensional nonlinear sloshing in a square-base basin analysis of nonlinear sloshing in a rectangular tank, Journal of Fluid Mechanics 487 (2003) 1–42. [5] J.W. Miles, Internally resonant surface waves in a circular cylinder, Journal of Fluid Mechanics 149 (1984) 1–14. [6] J.W. Miles, Resonantly surface waves in a circular cylinder, Journal of Fluid Mechanics 149 (1984) 15–31. [7] K. Kimura, H. Takahara, M. Sakata, Nonlinear liquid oscillation in a circular cylindrical tank subjected to pitching excitation, Proceedings of AsiaPacific Vibration Conference’93, Kitakyushu, vol. 1, 1993, pp. 294–299. [8] K. Fujita, T. Ito, K. Okada, Seismic response of liquid sloshing in the annular region formed by coaxial circular cylinder, Bulletin of the JSME 29 (258) (1986) 4318–4325. [9] J.S. Meserole, A. Fortini, Sloshing dynamics in a toroidal tank, Journal of Spacecraft 24 (6) (1987) 523–531. [10] V.J. Modi, F. Welt, An investigation of liquid sloshing in toroidal containers, Journal of Ship Research 34 (4) (1990) 253–261. [11] F. Welt, V.J. Modi, Vibration damping through liquid sloshing, Part I: a nonlinear analysis, Transactions of American Society of Mechanical Engineers, Journal of Vibration and Acoustics 114 (1992) 10–16. [12] F. Welt, V.J. Modi, Vibration damping through liquid sloshing, Part II: experimental results, Transactions of American Society of Mechanical Engineers, Journal of Vibration and Acoustics 114 (1992) 17–23. [13] R.L. Seliger, G.B. Whitham, Variational principles in continuum mechanics, Proceedings of the Royal Society of London A 305 (1968) 1–25. [14] M. Sakata, K. Kimura, M. Utsumi, Non-Stationary response of non-linear liquid motion in a circular cylindrical tank subjected to random base excitation, Journal of Sound and Vibration 94 (3) (1984) 351–363. [15] E.R. Hutton, Investigation of resonant, non-linear, non-planar free surface oscillations of a fluid, National Aeronautics and Space Administration Technical Note D-1870, 1963.