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A modal analysis of a rectangular plate floating on an incompressible liquid
Journal of Sound and Vibration (1990) 142(3), 453-460
A MODAL ANALYSIS OF A RECTANGULAR PLATE FLOATING ON AN INCOMPRESSIBLE LIQUID N.
J.
ROBINSON
AND
S.
C.
PALMER
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 lPZ, England (Received 25 July 1988, and in revised form 3 January 1990)
This paper presents a modal analysis of a flat horizontal plate floating on a body of liquid, for low-frequency, low-amplitude oscillations. A combined governing equation for the plate-liquid system is derived. Normal modes are considered, and the particular case of a plate constrained to have zero slope at its edges examined. The transfer function of a point-driven floating plate is obtained.
1.
INTRODUCTION
contains a modal analysis of a thin elastic plate floating on a body of liquid, and oscillating at low frequencies. The work arises as part of a project investigating the wind-induced vibration of floating roofs on oil storage tanks. Various types of roof exist, as referred to in the current tank design codes BS2654 [l] and AP1650 [2] and described by De Wit [3], but it is common to all types that they are fabricated from welded steel plate. One of the simplest types of floating roof in widespread use is the pontoon-type roof, which consists of a flat circular plate supported around its periphery by a buoyant pontoon ring. As discussed by Palmer [4], wind-induced vibration, and subsequent fatigue failure of welded joints of pontoon-type and other types of floating roof, has been observed. Repairs can be time-consuming and costly. In order to gain an insight into the above problem, the harmonic response of a thin elastic plate floating on an incompressible inviscid liquid has been studied. There has been work in the past on the related problem of an elastic plate with various edge conditions, loaded on one side by a compressible fluid [5-91. Generally, the displacement response of the plate is expanded in terms of orthogonal functions with unknown weighting coefficients. The velocity potential function, or the pressure distribution function, of the fluid is deduced by solving the wave equation with the unsteady Bernoulli equation applied at the fluid-plate interface. The fluid loading on the plate is then deduced. Appropriate weighting coefficients are determined for the given excitation. A preliminary investigation was presented by Dowel1 and Voss [5], who estimated the effect of fluid loading on the natural frequencies of a clamped rectangular plate. They find that the fluid can act as an aerodynamic spring or as an added mass attached to the plate. Pretlove [6] extended their approach considerably. By considering a simply supported plate, loaded on one side by a finite fluid-filled cavity, he addressed a problem in reconciling plate and fluid boundary conditions. The presence of the fluid side walls requires zero slope of the plate at its edges, while the simple supports require zero displacement at the plate edges. Normal modes of the fluid-plate system that satisfy these conditions are not readily obtained. Instead, Pretlove expressed the surface displacement This paper
453 0022-460X/90/210453+08
$03.00/O
@ 1990 Academic Press Limited
454
N. J. ROBINSON
AND
S. C. PALMER
as a weighted sum of the plate in uucuo modeshapes. These he expanded in terms of orthogonal functions which individually satisfy the fluid boundary conditions. Thus, fluid and plate boundary conditions are satisfied simultaneously. However, the displacement expansion is not in terms of normal modes of the combined system, as the in uucuo mode shapes are not eigenfunctions of the coupled governing equation. More recently, Qaisi [9] has tackled the simply supported plate and the clamped plate by similar methods, deriving the acoustic response as a sum of expressions involving all plate modes. His work extends previous works by the application of matrix methods to evaluate natural frequencies and mode shapes. Junger and Feit [7] also considered a simply supported fluid-loaded plate. In their analysis, the surface displacement was expanded as a weighted sum of in mzcw plate modes, but fluid side wall boundary conditions were not included. The fluid pressure distribution function in response to the surface displacement was deduced, and the near field and far field responses were discussed. An analysis of the basic problem in terms of plate modes requires the modes to be coupled by the fluid, as the normal modes of the fluid-loaded plate differ from the normal modes of the plate alone. Junger and Feit pointed out that the coupling is small for large plates, in which the only effect of the fluid is as an added mass. Davies discussed this coupling in some detail [8], examining the relation to radiation damping and added mass effects. The analysis presented in this paper is of a problem in which the plate and fluid modes are compatible; the plate mode shapes are not coupled by the fluid. Initially, a thin plate with unspecified edge conditions, floating on a body of incompressible liquid, is considered. A combined governing equation for the surface displacement of the plate-liquid system is derived. Unlike the above analyses, the hydrostatic pressure exerted by the displaced liquid is incorporated, as this is found to be a dominant term in problems of the scale of oil storage tank roofs. Free motion in a general combined mode is investigated, and constraints on the mode shapes developed. The particular case of a floating plate with edges constrained to have zero slope is then studied, and expressions for the mode shapes of the system found. These turn out to be orthogonal at the plate surface. There is no cross-coupling due to the liquid. Finally, following methods used by Newland [lo], the displacement response of the plate to an harmonic point load is obtained. The liquid is found to contribute inertia and stiffness to the plate, the latter effect due to the hydrostatic pressure exerted by the displaced liquid. The aerodynamic spring effects mentioned by Dowel1 and Voss are not observed here, as the liquid is assumed to be incompressible. The analysis in this paper has been extended by Robinson and Palmer [ll] to give the response of a floating plate to random correlated pressure loading. 2. GOVERNING EQUATIONS A uniform, flat, horizontal plate, floating on the surface of a body of liquid is considered. The weight of the plate is assumed to be supported by buoyancy forces. The dry surface of the plate is exposed to a randomly varying pressure loading p(x, t), while the pressure on the wet surface is p’(x, t), as shown in Figure 1. The plate has constant mass per unit area m, damping force per unit area, per unit velocity c, and flexural rigidity D, where D = Eh3/12( 1 - v*), in which E is the Young’s modulus of the plate material, Y is the Poisson ratio, and h is the thickness of the plate (a list of notation is given in the Appendix). For low-amplitude displacement, where membrane stresses in the plate are negligible, the response of the plate is governed by m d*y/dt*+cdy/dt+
DV4y=p’(x,
t)-p(x,
t),
(1)
MODAL
ANALYSIS
OF
A FLOATING
455
PLATE
Surface
Displaced
pressure
plate
Figure 1. Plate floating on liquid of constant depth.
where x represents the horizontal co-ordinates, Y is the upwards displacement of the plate from rest and t denotes time. The liquid, of density p, and constant depth d, is assumed to be homogeneous, incompressible, inviscid and its motion irrotational. Hence, for a velocity potential function 4(x, z, t), V’4(x, z, t) = 0,
(2)
where z is the vertical co-ordinate, measured upwards with its origin at the stationary surface. The liquid surface condition is derived from the unsteady Bernoulli equation. In this analysis, we are considering problems where there is heavy fluid loading, such that the weight of the liquid is significant. However, by confining the analysis to low-frequency, low-amplitude oscillations, in which particle velocities are small, the convective inertia terms can be omitted. Thus, the pressure at any point in the liquid is given by P, where -pa+/at+P+pgz=O.
(3)
At the surface of the liquid, this becomes (4)
-P Watl Z=o+P’+ PgY = 0, provided that the surface displacement is very small compared to the depth. Combining equations (4) and (1) yields an overall governing equation: m aZy/ar2-p
a~/atl,=,+~ay/at+DV~y+Pgy=
-p(x,
r).
(5)
Equating velocities at the surface of the liquid, with small displacements assumed, one has -a4/azl,=,
= ay/at.
(6)
3. MODAL RESPONSE
Consider now the response of the liquid and the plate in any one combined mode. For convenience, subscripts are omitted at this stage.
456
N. J. ROBINSON
AND
S. C. PALMER
Separable solutions are assumed for the displacement so that
response and velocity potential,
(7) Y(X, t) = ti(x)rl(t), where @I(x)describes the horizontal spatial variation of y and 77(f) is the time variation, and 4(x, z, r) = U(x)F(z)G(t), (8) where U(x), F(z) and G(t) respectively describe the horizontal, vertical and temporal variations of #J. Substituting expressions (7) and (8) into expression (6) gives 9(x, z, 2) = -~(x)F(z)7i(t)l{dF/dzl~=~} where the dot denotes differentiation equation (9) gives
(9)
with respect to time. Applying equation
(2) to
d2F/dz2 - pF = 0,
(10,11) where p is a real or complex constant. Solutions with p complex have no physical meaning in wave motion. It is necessary to consider separately solutions with positive and negative real p. Equation (11) has the general solutions v2*+/_L*=0,
F(z) = A, e”’+ BlepAz F(z) = A2 cos AZ+ B2 sin AZ
for p = A2 (A real), for p = -A2
(A real),
(12a) (I2b)
where A,, B, , A2 and B2 are constants. Now at the bed of the liquid there is no normal component of velocity, so &$/ezl,,_, = 0. Hence equations (12a) and (12b) become F(z) = D, cash A(z + d) for p = A2 and F(z) = D, cos A(z + d) for p = -A2, where D, and D, are constants. For free vibration in any one mode, equations (7) and (9) are substituted into equation (5) with p(x, t) = 0 to give DV4J/ + p& = W,
(m+m,)ij+c7j+K~=O,
(13,14)
where M, = pF(O)/{dF/dz(,,,} and K is a real or complex constant. Equation (13) with equation (10) gives K = Dp* + pg, so equation (13) is written in the form (v”-/.L)(v2+jL)~(x)=0.
(15)
Mode shapes must satisfy equation (15) and (IO) simultaneously. However, all solutions of equation (10) also satisfy equation (15), so equation (10) is a sufficient condition on the mode shapes. Equation (15) yields four independent general solutions. These are solutions of V2$ + PI&= 0 where p takes both positive and negative sign. The mode shapes of a plate in uacuo in general must satisfy an equation of the form of equation (15). However, adding the constraint (2) reduces the number of independent general solutions to two; the mode shapes satisfy V2Jl + p+ = 0, where p takes either a positive or a negative sign, but not both. Considering a separable solution for $(x) in Cartesian co-ordinates. ccl(xr9x2) = 51(a-2(x2), and applying equation (IO) gives, for non-trivial solutions,
(16)
d2&ldx: + r:& = 0, d2&/dx: + r:& = 0, where y1 and y2 are real or imaginary constants such that 7: + y: = I_L,These have general solutions &(x1) = a, eyIxi+ b, e-yIXI,
&(x2) = a2 eYzX2+b2 e-y2x2,
(1% b)
MODAL
ANALYSIS
OF
A FLOATING
457
PLATE
where a,, b,, a2 and b2 are constants. By substituting appropriate edge conditions, particular solutions are obtained. The configuration described below yields simple modeshapes, and is used for illustration. A rectangular (L, x LJ plate floating on an (L, x L2 x d) tank of liquid is considered. The plate edges are constrained to have zero slope, as shown in Figure 2, and there is no component of liquid velocity normal to the tank walls. Thus, a+/ax, = ay/ax, = 0 at = ay/dxz = 0 at x2 = 0 or L?. Applying these conditions to x,=0 or L,, and a4/ax, equations (7) and (9) with $(x) defined by equations (16), (17a) and (17b), yields the surface displacement mode shapes $!ljk(X)=Ujk
These happen to be orthogonal.
R
A is
(k?TXz/L2).
Normalizing so that
I where
COS (&TX,/LI)COS
(18)
‘+!‘jk(X)d’qr(X) dx = A$qakr,
the area of the plate and Sj, is the Kronecker delta function, gives 2 cos (j,xJ
for j, k > 0 forj>O, k=O fork>O,j=O forj= k=O
L,) cos (knx,/ LJ
(19)
’ 1
---
i
~-_-__-_-__
Liquid
d--
__~_-__-_~
g_--------g_______--------__ g______-------____
-------
- --_??,m?m?
Figure 2. Mode shape and edge restraint assumed for the analysis of a rectangular plate floating in a rectangular tank.
4.
FREQUENCY RESPONSE FUNCTION
The general displacement response and velocity potential functions are written as modal expansions,
-f
Y(K f)=
?
j=O
I?
(20)
$jk(X)l)jk(f)r
k=O
.
F $jk jk(x)COShAjk(Z+d) Ajk
sinh
Ajkd
%k(
r,
for p = h$
j=O k=O
4(x, z, r) =
f
?
,
COS Aj\jk(Z+d)
hkjk(X) Ajk sinhjkd
$k(f)
for p = -hfk
j=O k=O
where the Gjk, njk and Ajk replace the $, q and A of the previous section.
Wa,b)
458
N. J. ROBINSON
AND
S. C. PALMER
Substituting expression (20), (21a) and (21b) into equation (5) gives jj.O kf, [I”jktjk+ Cjktijk + KjkrJikl+jk(X) = -PCxvt)~
(22)
where Cj, =
Mjk=m+m,
C,
DATk+Pg
Kjk=
(22a-c)
(224 e) Multiplying equation (22) by $jk(X) and integrating over the surface of the plate, R, gives f j=O
f k=O
{Mjkyjk
+
qk+jk
+
t) dx. 1 I $jk(X)P(X
Kjkvjk)$jk(x)
dx
=
(23)
-
R
It is possible to simplify the analysis at this stage by assuming that the mode shapes of the liquid loaded plate are orthogonal. Those given in equation (19) happen to be orthogonal, but in general, this is not so. However, for a sufficiently thin plate with free edges, oscillating at low frequency, the motion is dominated by the liquid. Liquid-coupled plate modes can then be approximated by the surface response of a body of liquid with a smooth and continuous surface. For problems of this nature, equation (18) is assumed to hold. Thus equation (23) becomes a series of decoupled modal expressions:
R
Consider now the time response at x to an unit, harmonic, point load at s, of frequency w. Then p(x, t) = e’“‘6(x -s) and Mjk’fjk+ Cjkrijk + Kjkvjk = -(I/A)$jk(S)
If it
iS
assumed that vjk( r) = &(w)
e’“‘; then
&(W) = -$jk(S)/A[-W’Mjk So, from equation
e’“‘.
+iWCjk + Kjk].
(20), the response at x to an unit, harmonic, point load at s is: Yb,
cl=-
;
(Clik(X)qjk(S)
if
j=o k=OA[-W2Mjk +iuCjk + Kjk]
e ior.
Now, the response to an harmonic point load can also be written as y(x, t) = H(x, s, w) e’“‘, where H(x, s, w) is the transfer function for the system. Thus,
H(x,s,fLJ)=-: f A[-U2Mjk
$jk(x)$jk(s)
j-0
k=O
+kJCjk + Kjk]’
(24)
Equation (24) can be rewritten as H(x,%
w)’
5 j=O
f k=O
+jk(x)Hjk(@)+jk(s)s
(25)
where
(254
MODAL
The undamped pjk
=
ANALYSIS
OF A FLOATING
modal natural frequency is
wjk
=
X/[&/i!+],
459
PLATE
and the damping factor is
~k/2’dMjkKjkl*
The
frequency response function for the plate supported by liquid is similar to the standard result obtained for a plate in air, with added or subtracted mass and stiffness due to the liquid loading. For p = A$, there is an added mass per unit area equal to iS iS tXpiVdent to a layer of liquid, of depth coth (Ajkd), Th (P/Ajk) coth (Ajkd)* moving with the plate. In deep liquid, the thickness of this layer is approximately l/Ajk. As the mode number increases, the thickness of this liquid layer decreases. For p = -ATk, there iS an added mass per unit area term equal to (-P/A,,) Cot (Ajkd). This term is negative for certain Ajkd, in which case it tends to destabilize the system, reducing the general mass Mjk, and increasing the natural frequencies. For zero or negative general mass, there are no natural modes, and the analysis is no longer valid. There is an added stiffness term equal to pg for all p. This represents the hydrostatic pressure exerted by the displaced liquid, and remains constant for all modes. As the mode number increases, the plate stiffness becomes larger, and the added stiffness of the liquid becomes less significant. The transfer function (25) is derived as an infinite sum of modal terms. However, the analysis is valid only for a finite number of lower-frequency modes. At higher frequencies, the convective inertia of the liquid may become a significant part of the unsteady Bernoulli equation, so equation (3) is not correct. Compressibility effects in the liquid may emerge, invalidating equation (1). However, in problems of the form outlined in the introduction, it is the response in the low-frequency modes that is important, as the nature of the loading tends to inhibit the high-frequency response. (l/hjk)
ACKNOWLEDGMENTS
The authors would like to thank the Science and Engineering Research Council and British Petroleum for their financial support of this work.
REFERENCES 1. BS2654 1984 Specification for Vertical Steel Welded Storage Tanks with Butt-welded Shells for the Petroleum Industry. British Standards Institute. 2. API 650 1978 Welded Steel Tanks for Oil Storage. American Petroleum Institute.
3. J. DE WIT 1970 Engineering 210, 55-58. Floating roof tanks. 4. S. C. PALMER1986 Institution of Mechanical Engineers Paper C257/86. Design of floating roofs on oil storage tanks to withstand wind loading-a review with recommendations. 5. E. H. DOWELL and H. M. VOSS 1963 American Institute of Aeronautics and Astronautics Journal 1, 476-477. The effect of a cavity on panel vibration. 6. A. J. PRETLOVE1965 Journal of Sound and Vibration 2,197-209. Free vibrations of a rectangular panel backed by a closed rectangular cavity. 7. M. JUNGER and FEIT 1972 Sound, Structures and Their Interaction. Cambridge, Mass.: MIT Press. 8. H. G. DAVIES 1971 Journal of Sound and Vibration 15, 107-126. Low frequency random excitation of water-loaded rectangular plates. 9. M. I. QAISI 1988 Applied Acoustics 24, 49-61. Free vibrations of a rectangular plate-cavity system. 10. D. E. NEWLAND 1984 An Introduction to Random Vibrations and Spectral Analysis. New York: Longman, second edition. 11. N. J. ROBINSONand S. C. PALMER1987 Cambridge University Engineering Department, Report no. CUED/ C-mech/ Tr.38. Vibration of a rectangular plate floating on an inviscid liquid, subject
to correlated random pressure loading.
460
N. J. ROBINSON
APPENDIX: aI9 a2
ajk,
b, , b, : g h
j, k m
ml Pb,
t) t)
P'h 4,
r
S
XI
X
t)
Yb,
i 4,4,B,rB2 cjk D
L G(t) Hjk(a) H(x, s,w) Kjk L, L2 Mjk w4 Yjk(u) pjk
YI9
Y2
a(
)
17(t) 2 7
z, t)
@jk (x) hjk CL
iI9
P 0 Wik
52
AND
S. C. PALMER
NOTATION
constants constants damping force per unit area of plate, per unit velocity depth of liquid gravitational constant plate thickness integers mass per unit area of plate added mass per unit area due to liquid pressure loading pressure on wet surface of plate integers horizontal co-ordinate of loading point (similar to x) time horizontal Cartesian co-ordinate shorthand for (x, , x2) vertical displacement of surface vertical co-ordinate area of plate constants generalized damping flexural rigidity of plate Young’s modulus of plate material vertical variation of 4 time variation of $J frequency response function frequency response function of displacement at x, due to point load at s generalized stiffness length breadth generalized mass horizontal variation of 4 frequency response function damping factor constants such that y: + y: = *A* Dirac delta function time variation of y Kronecker delta function velocity potential function j, kth mode shape of plate constant such that V2tijk = *th$~& constant Poisson ratio of plate material separable components of JI density of liquid frequency j, kth natural frequency
A MODAL ANALYSIS OF A RECTANGULAR PLATE FLOATING ON AN INCOMPRESSIBLE LIQUID N.
J.
ROBINSON
AND
S.
C.
PALMER
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 lPZ, England (Received 25 July 1988, and in revised form 3 January 1990)
This paper presents a modal analysis of a flat horizontal plate floating on a body of liquid, for low-frequency, low-amplitude oscillations. A combined governing equation for the plate-liquid system is derived. Normal modes are considered, and the particular case of a plate constrained to have zero slope at its edges examined. The transfer function of a point-driven floating plate is obtained.
1.
INTRODUCTION
contains a modal analysis of a thin elastic plate floating on a body of liquid, and oscillating at low frequencies. The work arises as part of a project investigating the wind-induced vibration of floating roofs on oil storage tanks. Various types of roof exist, as referred to in the current tank design codes BS2654 [l] and AP1650 [2] and described by De Wit [3], but it is common to all types that they are fabricated from welded steel plate. One of the simplest types of floating roof in widespread use is the pontoon-type roof, which consists of a flat circular plate supported around its periphery by a buoyant pontoon ring. As discussed by Palmer [4], wind-induced vibration, and subsequent fatigue failure of welded joints of pontoon-type and other types of floating roof, has been observed. Repairs can be time-consuming and costly. In order to gain an insight into the above problem, the harmonic response of a thin elastic plate floating on an incompressible inviscid liquid has been studied. There has been work in the past on the related problem of an elastic plate with various edge conditions, loaded on one side by a compressible fluid [5-91. Generally, the displacement response of the plate is expanded in terms of orthogonal functions with unknown weighting coefficients. The velocity potential function, or the pressure distribution function, of the fluid is deduced by solving the wave equation with the unsteady Bernoulli equation applied at the fluid-plate interface. The fluid loading on the plate is then deduced. Appropriate weighting coefficients are determined for the given excitation. A preliminary investigation was presented by Dowel1 and Voss [5], who estimated the effect of fluid loading on the natural frequencies of a clamped rectangular plate. They find that the fluid can act as an aerodynamic spring or as an added mass attached to the plate. Pretlove [6] extended their approach considerably. By considering a simply supported plate, loaded on one side by a finite fluid-filled cavity, he addressed a problem in reconciling plate and fluid boundary conditions. The presence of the fluid side walls requires zero slope of the plate at its edges, while the simple supports require zero displacement at the plate edges. Normal modes of the fluid-plate system that satisfy these conditions are not readily obtained. Instead, Pretlove expressed the surface displacement This paper
453 0022-460X/90/210453+08
$03.00/O
@ 1990 Academic Press Limited
454
N. J. ROBINSON
AND
S. C. PALMER
as a weighted sum of the plate in uucuo modeshapes. These he expanded in terms of orthogonal functions which individually satisfy the fluid boundary conditions. Thus, fluid and plate boundary conditions are satisfied simultaneously. However, the displacement expansion is not in terms of normal modes of the combined system, as the in uucuo mode shapes are not eigenfunctions of the coupled governing equation. More recently, Qaisi [9] has tackled the simply supported plate and the clamped plate by similar methods, deriving the acoustic response as a sum of expressions involving all plate modes. His work extends previous works by the application of matrix methods to evaluate natural frequencies and mode shapes. Junger and Feit [7] also considered a simply supported fluid-loaded plate. In their analysis, the surface displacement was expanded as a weighted sum of in mzcw plate modes, but fluid side wall boundary conditions were not included. The fluid pressure distribution function in response to the surface displacement was deduced, and the near field and far field responses were discussed. An analysis of the basic problem in terms of plate modes requires the modes to be coupled by the fluid, as the normal modes of the fluid-loaded plate differ from the normal modes of the plate alone. Junger and Feit pointed out that the coupling is small for large plates, in which the only effect of the fluid is as an added mass. Davies discussed this coupling in some detail [8], examining the relation to radiation damping and added mass effects. The analysis presented in this paper is of a problem in which the plate and fluid modes are compatible; the plate mode shapes are not coupled by the fluid. Initially, a thin plate with unspecified edge conditions, floating on a body of incompressible liquid, is considered. A combined governing equation for the surface displacement of the plate-liquid system is derived. Unlike the above analyses, the hydrostatic pressure exerted by the displaced liquid is incorporated, as this is found to be a dominant term in problems of the scale of oil storage tank roofs. Free motion in a general combined mode is investigated, and constraints on the mode shapes developed. The particular case of a floating plate with edges constrained to have zero slope is then studied, and expressions for the mode shapes of the system found. These turn out to be orthogonal at the plate surface. There is no cross-coupling due to the liquid. Finally, following methods used by Newland [lo], the displacement response of the plate to an harmonic point load is obtained. The liquid is found to contribute inertia and stiffness to the plate, the latter effect due to the hydrostatic pressure exerted by the displaced liquid. The aerodynamic spring effects mentioned by Dowel1 and Voss are not observed here, as the liquid is assumed to be incompressible. The analysis in this paper has been extended by Robinson and Palmer [ll] to give the response of a floating plate to random correlated pressure loading. 2. GOVERNING EQUATIONS A uniform, flat, horizontal plate, floating on the surface of a body of liquid is considered. The weight of the plate is assumed to be supported by buoyancy forces. The dry surface of the plate is exposed to a randomly varying pressure loading p(x, t), while the pressure on the wet surface is p’(x, t), as shown in Figure 1. The plate has constant mass per unit area m, damping force per unit area, per unit velocity c, and flexural rigidity D, where D = Eh3/12( 1 - v*), in which E is the Young’s modulus of the plate material, Y is the Poisson ratio, and h is the thickness of the plate (a list of notation is given in the Appendix). For low-amplitude displacement, where membrane stresses in the plate are negligible, the response of the plate is governed by m d*y/dt*+cdy/dt+
DV4y=p’(x,
t)-p(x,
t),
(1)
MODAL
ANALYSIS
OF
A FLOATING
455
PLATE
Surface
Displaced
pressure
plate
Figure 1. Plate floating on liquid of constant depth.
where x represents the horizontal co-ordinates, Y is the upwards displacement of the plate from rest and t denotes time. The liquid, of density p, and constant depth d, is assumed to be homogeneous, incompressible, inviscid and its motion irrotational. Hence, for a velocity potential function 4(x, z, t), V’4(x, z, t) = 0,
(2)
where z is the vertical co-ordinate, measured upwards with its origin at the stationary surface. The liquid surface condition is derived from the unsteady Bernoulli equation. In this analysis, we are considering problems where there is heavy fluid loading, such that the weight of the liquid is significant. However, by confining the analysis to low-frequency, low-amplitude oscillations, in which particle velocities are small, the convective inertia terms can be omitted. Thus, the pressure at any point in the liquid is given by P, where -pa+/at+P+pgz=O.
(3)
At the surface of the liquid, this becomes (4)
-P Watl Z=o+P’+ PgY = 0, provided that the surface displacement is very small compared to the depth. Combining equations (4) and (1) yields an overall governing equation: m aZy/ar2-p
a~/atl,=,+~ay/at+DV~y+Pgy=
-p(x,
r).
(5)
Equating velocities at the surface of the liquid, with small displacements assumed, one has -a4/azl,=,
= ay/at.
(6)
3. MODAL RESPONSE
Consider now the response of the liquid and the plate in any one combined mode. For convenience, subscripts are omitted at this stage.
456
N. J. ROBINSON
AND
S. C. PALMER
Separable solutions are assumed for the displacement so that
response and velocity potential,
(7) Y(X, t) = ti(x)rl(t), where @I(x)describes the horizontal spatial variation of y and 77(f) is the time variation, and 4(x, z, r) = U(x)F(z)G(t), (8) where U(x), F(z) and G(t) respectively describe the horizontal, vertical and temporal variations of #J. Substituting expressions (7) and (8) into expression (6) gives 9(x, z, 2) = -~(x)F(z)7i(t)l{dF/dzl~=~} where the dot denotes differentiation equation (9) gives
(9)
with respect to time. Applying equation
(2) to
d2F/dz2 - pF = 0,
(10,11) where p is a real or complex constant. Solutions with p complex have no physical meaning in wave motion. It is necessary to consider separately solutions with positive and negative real p. Equation (11) has the general solutions v2*+/_L*=0,
F(z) = A, e”’+ BlepAz F(z) = A2 cos AZ+ B2 sin AZ
for p = A2 (A real), for p = -A2
(A real),
(12a) (I2b)
where A,, B, , A2 and B2 are constants. Now at the bed of the liquid there is no normal component of velocity, so &$/ezl,,_, = 0. Hence equations (12a) and (12b) become F(z) = D, cash A(z + d) for p = A2 and F(z) = D, cos A(z + d) for p = -A2, where D, and D, are constants. For free vibration in any one mode, equations (7) and (9) are substituted into equation (5) with p(x, t) = 0 to give DV4J/ + p& = W,
(m+m,)ij+c7j+K~=O,
(13,14)
where M, = pF(O)/{dF/dz(,,,} and K is a real or complex constant. Equation (13) with equation (10) gives K = Dp* + pg, so equation (13) is written in the form (v”-/.L)(v2+jL)~(x)=0.
(15)
Mode shapes must satisfy equation (15) and (IO) simultaneously. However, all solutions of equation (10) also satisfy equation (15), so equation (10) is a sufficient condition on the mode shapes. Equation (15) yields four independent general solutions. These are solutions of V2$ + PI&= 0 where p takes both positive and negative sign. The mode shapes of a plate in uacuo in general must satisfy an equation of the form of equation (15). However, adding the constraint (2) reduces the number of independent general solutions to two; the mode shapes satisfy V2Jl + p+ = 0, where p takes either a positive or a negative sign, but not both. Considering a separable solution for $(x) in Cartesian co-ordinates. ccl(xr9x2) = 51(a-2(x2), and applying equation (IO) gives, for non-trivial solutions,
(16)
d2&ldx: + r:& = 0, d2&/dx: + r:& = 0, where y1 and y2 are real or imaginary constants such that 7: + y: = I_L,These have general solutions &(x1) = a, eyIxi+ b, e-yIXI,
&(x2) = a2 eYzX2+b2 e-y2x2,
(1% b)
MODAL
ANALYSIS
OF
A FLOATING
457
PLATE
where a,, b,, a2 and b2 are constants. By substituting appropriate edge conditions, particular solutions are obtained. The configuration described below yields simple modeshapes, and is used for illustration. A rectangular (L, x LJ plate floating on an (L, x L2 x d) tank of liquid is considered. The plate edges are constrained to have zero slope, as shown in Figure 2, and there is no component of liquid velocity normal to the tank walls. Thus, a+/ax, = ay/ax, = 0 at = ay/dxz = 0 at x2 = 0 or L?. Applying these conditions to x,=0 or L,, and a4/ax, equations (7) and (9) with $(x) defined by equations (16), (17a) and (17b), yields the surface displacement mode shapes $!ljk(X)=Ujk
These happen to be orthogonal.
R
A is
(k?TXz/L2).
Normalizing so that
I where
COS (&TX,/LI)COS
(18)
‘+!‘jk(X)d’qr(X) dx = A$qakr,
the area of the plate and Sj, is the Kronecker delta function, gives 2 cos (j,xJ
for j, k > 0 forj>O, k=O fork>O,j=O forj= k=O
L,) cos (knx,/ LJ
(19)
’ 1
---
i
~-_-__-_-__
Liquid
d--
__~_-__-_~
g_--------g_______--------__ g______-------____
-------
- --_??,m?m?
Figure 2. Mode shape and edge restraint assumed for the analysis of a rectangular plate floating in a rectangular tank.
4.
FREQUENCY RESPONSE FUNCTION
The general displacement response and velocity potential functions are written as modal expansions,
-f
Y(K f)=
?
j=O
I?
(20)
$jk(X)l)jk(f)r
k=O
.
F $jk jk(x)COShAjk(Z+d) Ajk
sinh
Ajkd
%k(
r,
for p = h$
j=O k=O
4(x, z, r) =
f
?
,
COS Aj\jk(Z+d)
hkjk(X) Ajk sinhjkd
$k(f)
for p = -hfk
j=O k=O
where the Gjk, njk and Ajk replace the $, q and A of the previous section.
Wa,b)
458
N. J. ROBINSON
AND
S. C. PALMER
Substituting expression (20), (21a) and (21b) into equation (5) gives jj.O kf, [I”jktjk+ Cjktijk + KjkrJikl+jk(X) = -PCxvt)~
(22)
where Cj, =
Mjk=m+m,
C,
DATk+Pg
Kjk=
(22a-c)
(224 e) Multiplying equation (22) by $jk(X) and integrating over the surface of the plate, R, gives f j=O
f k=O
{Mjkyjk
+
qk+jk
+
t) dx. 1 I $jk(X)P(X
Kjkvjk)$jk(x)
dx
=
(23)
-
R
It is possible to simplify the analysis at this stage by assuming that the mode shapes of the liquid loaded plate are orthogonal. Those given in equation (19) happen to be orthogonal, but in general, this is not so. However, for a sufficiently thin plate with free edges, oscillating at low frequency, the motion is dominated by the liquid. Liquid-coupled plate modes can then be approximated by the surface response of a body of liquid with a smooth and continuous surface. For problems of this nature, equation (18) is assumed to hold. Thus equation (23) becomes a series of decoupled modal expressions:
R
Consider now the time response at x to an unit, harmonic, point load at s, of frequency w. Then p(x, t) = e’“‘6(x -s) and Mjk’fjk+ Cjkrijk + Kjkvjk = -(I/A)$jk(S)
If it
iS
assumed that vjk( r) = &(w)
e’“‘; then
&(W) = -$jk(S)/A[-W’Mjk So, from equation
e’“‘.
+iWCjk + Kjk].
(20), the response at x to an unit, harmonic, point load at s is: Yb,
cl=-
;
(Clik(X)qjk(S)
if
j=o k=OA[-W2Mjk +iuCjk + Kjk]
e ior.
Now, the response to an harmonic point load can also be written as y(x, t) = H(x, s, w) e’“‘, where H(x, s, w) is the transfer function for the system. Thus,
H(x,s,fLJ)=-: f A[-U2Mjk
$jk(x)$jk(s)
j-0
k=O
+kJCjk + Kjk]’
(24)
Equation (24) can be rewritten as H(x,%
w)’
5 j=O
f k=O
+jk(x)Hjk(@)+jk(s)s
(25)
where
(254
MODAL
The undamped pjk
=
ANALYSIS
OF A FLOATING
modal natural frequency is
wjk
=
X/[&/i!+],
459
PLATE
and the damping factor is
~k/2’dMjkKjkl*
The
frequency response function for the plate supported by liquid is similar to the standard result obtained for a plate in air, with added or subtracted mass and stiffness due to the liquid loading. For p = A$, there is an added mass per unit area equal to iS iS tXpiVdent to a layer of liquid, of depth coth (Ajkd), Th (P/Ajk) coth (Ajkd)* moving with the plate. In deep liquid, the thickness of this layer is approximately l/Ajk. As the mode number increases, the thickness of this liquid layer decreases. For p = -ATk, there iS an added mass per unit area term equal to (-P/A,,) Cot (Ajkd). This term is negative for certain Ajkd, in which case it tends to destabilize the system, reducing the general mass Mjk, and increasing the natural frequencies. For zero or negative general mass, there are no natural modes, and the analysis is no longer valid. There is an added stiffness term equal to pg for all p. This represents the hydrostatic pressure exerted by the displaced liquid, and remains constant for all modes. As the mode number increases, the plate stiffness becomes larger, and the added stiffness of the liquid becomes less significant. The transfer function (25) is derived as an infinite sum of modal terms. However, the analysis is valid only for a finite number of lower-frequency modes. At higher frequencies, the convective inertia of the liquid may become a significant part of the unsteady Bernoulli equation, so equation (3) is not correct. Compressibility effects in the liquid may emerge, invalidating equation (1). However, in problems of the form outlined in the introduction, it is the response in the low-frequency modes that is important, as the nature of the loading tends to inhibit the high-frequency response. (l/hjk)
ACKNOWLEDGMENTS
The authors would like to thank the Science and Engineering Research Council and British Petroleum for their financial support of this work.
REFERENCES 1. BS2654 1984 Specification for Vertical Steel Welded Storage Tanks with Butt-welded Shells for the Petroleum Industry. British Standards Institute. 2. API 650 1978 Welded Steel Tanks for Oil Storage. American Petroleum Institute.
3. J. DE WIT 1970 Engineering 210, 55-58. Floating roof tanks. 4. S. C. PALMER1986 Institution of Mechanical Engineers Paper C257/86. Design of floating roofs on oil storage tanks to withstand wind loading-a review with recommendations. 5. E. H. DOWELL and H. M. VOSS 1963 American Institute of Aeronautics and Astronautics Journal 1, 476-477. The effect of a cavity on panel vibration. 6. A. J. PRETLOVE1965 Journal of Sound and Vibration 2,197-209. Free vibrations of a rectangular panel backed by a closed rectangular cavity. 7. M. JUNGER and FEIT 1972 Sound, Structures and Their Interaction. Cambridge, Mass.: MIT Press. 8. H. G. DAVIES 1971 Journal of Sound and Vibration 15, 107-126. Low frequency random excitation of water-loaded rectangular plates. 9. M. I. QAISI 1988 Applied Acoustics 24, 49-61. Free vibrations of a rectangular plate-cavity system. 10. D. E. NEWLAND 1984 An Introduction to Random Vibrations and Spectral Analysis. New York: Longman, second edition. 11. N. J. ROBINSONand S. C. PALMER1987 Cambridge University Engineering Department, Report no. CUED/ C-mech/ Tr.38. Vibration of a rectangular plate floating on an inviscid liquid, subject
to correlated random pressure loading.
460
N. J. ROBINSON
APPENDIX: aI9 a2
ajk,
b, , b, : g h
j, k m
ml Pb,
t) t)
P'h 4,
r
S
XI
X
t)
Yb,
i 4,4,B,rB2 cjk D
L G(t) Hjk(a) H(x, s,w) Kjk L, L2 Mjk w4 Yjk(u) pjk
YI9
Y2
a(
)
17(t) 2 7
z, t)
@jk (x) hjk CL
iI9
P 0 Wik
52
AND
S. C. PALMER
NOTATION
constants constants damping force per unit area of plate, per unit velocity depth of liquid gravitational constant plate thickness integers mass per unit area of plate added mass per unit area due to liquid pressure loading pressure on wet surface of plate integers horizontal co-ordinate of loading point (similar to x) time horizontal Cartesian co-ordinate shorthand for (x, , x2) vertical displacement of surface vertical co-ordinate area of plate constants generalized damping flexural rigidity of plate Young’s modulus of plate material vertical variation of 4 time variation of $J frequency response function frequency response function of displacement at x, due to point load at s generalized stiffness length breadth generalized mass horizontal variation of 4 frequency response function damping factor constants such that y: + y: = *A* Dirac delta function time variation of y Kronecker delta function velocity potential function j, kth mode shape of plate constant such that V2tijk = *th$~& constant Poisson ratio of plate material separable components of JI density of liquid frequency j, kth natural frequency