Torsional vibration of uniform columns with arbitrarily shaped cross-sections partially submerged in water

Pergamon

0045-7!W(!M)EO263-2

CopyrighrCs 1994 Elsevicr.9ciena Ltd Printedin Great Brimin.All rights reserved cws-7949/w $7.00 + 0.00

TORSIONAL VIBRATION OF UNIFORM COLUMNS WITH ARBITRARILY SHAPED CROSS-SECTIONS PARTIALLY SUBMERGED IN WATER Zhou Ding Faculty of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210014,

People’s Republic of China (Received8 June 1993) Abstract-This paper presents an analytical-numerical solution to the torsional vibration of uniform columns with arbitrarily shaped cross-sections partially submerged in water, considering the effects of surface waves and the compressibility of water. First, the exact expression for the velocity potential of water motion is given by the method of separation of variables, whose coefficients are decided by utilizing the orthogonality of a group of generalized trigonometric series and the Fourier expansion collocation method. Then, the analytical solution of the differential equation of torsional motion of the column created by water pressure is obtained. The calculated formulae of frequency equations and mode shapes of column-water coupling torsional vibration, which are suitable for programming, are derived by the use of the boundary conditions of the column. The results may be numerically calculated by means of a computer.

1. INTRODUCMON

the coupling torsional vibration of uniform columns with arbitrarily shaped cross-sections partially submerged in water by the use of the analyticalnumerical method presented by author [ll]. The surface wave and compressibility of water are also simultaneously considered. The effect of soil on the torsional vibration characteristics of the column is equivalent to a mass and a torsional spring attached to the end of the column. The calculated formulae of eigen-frequency equations and mode shapes of column-water coupling torsional vibration, which are suitable for programming, are derived. The results may be numerically obtained by means of a computer.

Since. Westergaard published the first article [l] about the problem of coupling vibration of solid-liquid structures in 1933, many scientists and technicians have been working hard in this research field. Some achievements [2,3] have been made. With the rapid development of ocean and hydraulic engineering, more and more attention has been paid to the study of vibration of columns in water in recent years. As the governing equation of column-water coupling vibration is in essence a differential-integral equation, it has been quite difficult to find its analytical solution. Therefore, in previous studies, it has usually been assumed that the height of the column is equal to the depth of the water [4] and/or the surface wave, and the compressibility of water is neglected [5-81 and the approximate calculating method is commonly used [9]. More recently, the author presented a general method [lo] for solving the transverse vibration characteristics of uniform columns with arbitrarily shaped crosssections partially submerged in water, by combining the analytical method with the numerical method. Besides coupling transverse vibration, there also exists coupling torsional vibration for columns with non-circular cross-sections in water. In engineering, uniform columns with non-circular cross-sectional shapes, e.g. elliptical, quadrilateral, polyploid, etc. may also find their applications. To the author’s knowledge, there are no other reports on column-water coupling torsional vibration, except one of author’s papers [I 11, in which an analytical solution to coupling torsional vibration of elliptical columns in water was obtained. This paper studies

2. THE MOTION

OF WATER

Consider a uniform column with an arbitrarily shaped cross-section partially submerged in water as shown in Fig. 1. The height of the column is H, the depth of water is h (h < H). Cylindrical coordinates r, 0, z, whose originals are in the cross-section of the column, are established to analyse the motion of water. The outer wall of the column is described by r = a(0) (0 < 0 ,< 2x). The column vibrates torsionally. Assuming the water is an irrotational and invisic, but compressible, ideal liquid, the velocity potential of water exists and satisfies the Laplacian equation, i.e.

where c is the velocity of sound in water. 35

36

Zhou Ding eqn (1) can be reduced to the following three uncoupled equations: d*Z 2+1*z

=o,

(9)

dtO -$+n%

=o,

(‘0)

d*R

1 dR p+;z+(p+R

=o,

(11)

where 1 and n are unknown constants. The analytical solutions of these three ordinary differential equations can be obtained [12]: considering eqns (2)-(4) and the Sommerfield radialization condition, one has

Fig. 1. The column in water.

Considering the effect of the linearized surface wave of water, the Lagrangian integration is as follows:

gf+gg=o, z=h.

(2)

Also, I$ must satisfy the boundary conditions:

+“fl,j$o

!?Lo

+,=F+,

aZ

d=O,

’

citHY'(ujrlJ;(z)

i

z=o,

c/,Kn(ujrlf;(z)

r-+co.

(4)

84

a*

r = a(O).

sinW), (12)

where Cj, and (7j, are unknown constants, Hlf’ and K, are the Hankle and modified Bessel functions of second order, n, respectively,

The restraint condition of the motion of water on the outer wall of the column is - = ;j~ r cos(q, e), atl

1

CIO = (k; + &/c2)“2,

GI,= ]w*/c2 - kf] 1’2,

I!. = {maxj I k, < o/c}

(5)

and

where $(z, t) is the torsion angle of the column, q is the vector normal to the outer wall of the column, 0 is the unit angular vector in polar-coordinates. When the column vibrates freely in water, the velocity potential of the water can be written as follows:

‘(‘)

J2k, cosh(k,z) = (k,h + f sinh(2k,h))1/2 ’

J2k, cos(k,z) r;(‘) = (k,h + f sin(2kih))l/2 ’ J = ” 2’ 37

.’ (13)

d(r,

e, z, t) =

iw@(r, 6, z) eiw’,

(6)

and the torsion angle of the column can also be written as follows:

in which, k, is the wave number, decided by o* = gk,tanh(k,h)

= -gk, tan(k,h), j=l,2,3

$ (z, t ) = Y(z) eiwr,

(7)

where w is the natural radian frequency of torsional vibration of the column in water and i = ,/--r. By applying the method of separation of iariables, i.e. by taking

Wr, 0, z) = R(r)@WZ(z),

(8)

,....

(14)

It is easily demonstrated that {f;(z)}, j = 0, 1,2, 3, . . . are a group of orthogonal functions in the interval [0, h] [13], i.e.

s h

0

where S(i -j)

f;(z)f;(z)

dz = S(i -j).

is the Dirac delta function.

(‘3

Torsional vibration of columns in water Substituting gives

eqn (12) into restraint equation

(5)

37

and k(e) = du(e)/de.

Ci,g’,(Q+

5 "==I

=-g

‘P(z),

(16)

where

By using the orthogonality of the generalized trigonometric series {J;(z)}, j = 0, 1,2,3, . . . , expanding eqn (16) into a generalized Fourier series in interval [0, h] gives

“z.Cd,(e)

+ f, %d,(e) =

g,‘,(e) = ajci~*~(aja(~))cos(nO) + n -‘(‘) H~*)(a,f@)) a*(e) xsin(n@,

j=O,1,2

,...,

L,

where

av

g’,(e) =

j=o,1,2

)...,

aj$(aja(0))cOs(nO)

+

n

Gj=

-$$

G(aj4e))

- nKdaN)) a*(e)

j=L+l,L+2,L+3

n$os$c$+ f ~Z~=T;G~, "=I

i,j=O,1,2,...,

,....

(2Oa) (17)

Here

f S;jCj,+f ac;= n=-0

fi(*)(x) = dHI;?‘(x)/dx, c(x)

(19)

w9

q;(e) = aj$(aja(0))sin(nO) xcos(n0),

‘y(C)J(C)dr. I0

Next, the columns with symmetric cross-sections about the original o are considered. In this case the torsional vibration of the columns in water does not cause the column-water coupling transverse vibration because the resultant force of water pressure on the outer wall of the columns in every unit length equals zero. Expanding eqn (18) into a Fourier series gives

L,

j = L + 1, L + 2, L + 3,. . . ,

x sin(&),

-$$ Gj,

j=O,1,2,3 ,..., (18)

q{(e) = ajA~*)(aja(8))sin(n6) - n H(*)(a.@)) u*(e) n 1 xcos(ne),

(17)

i=l,2,3

= dK,(x)/dx

T;G,,

n=l

,...,

j=O,1,2

,...,

(20b)

where

II

g-j

=

.

II

g',(e)c0s(ie) de,

both n and i are even or odd numbers either n or i is an odd number

0:

ri II

3’4 n.r=

g;(e)cOs(ie) de,

both n and i are even or odd numbers either n or i is an odd number

0:

I

SZ=

.

i’

gi,(O)sin(iO) de,

either n or i is an odd number

0:

7xgi(e)sin(i@

q-j= (I0:

both n and i are even or odd numbers

de,

both n and i are even or odd numbers either n or i is an odd number i is an even number

,

i is an odd number i is an even number i is an odd number

(21)

Zhou Ding

38

From eqns (20) and (21), one has c:, =o,

C!, = 0,

W, Rr, 1) = iw ei”” ,tO G,f;(z) “tO H!,2’(air)

n = 1,3,5,.

. .)

(22) x [P’, cos(2nB) + & sin(2nB)]

and when the terms n and i are truncated to 2N + 1, respectively, the following equation is obtained

+

f

GjJ(z)

2 K&r)

j=L+I

W_,v+ ,lC’= TG,,

“=O

(23) x [P!, cos(2nB) + Pi sin(2&)]

.

(27)

where

C’0 . C’2 ..

Pm+,1 =

Sl” 0.2N 9”

.

pi

$;G,ZN

p 0.2

.

SJ0.2N

s2.2,

. . .

9” 2N.2

. ..

:

2.2

0.4

f

s::j,

.’

”

’

22.2,

c$,,=piG,,

n =

,...,

1,2,3,..

i$N

. . ’

. (24)

.

N)

>

3. THE DIFFERENTIAL EQUATION OF TORSIONAL MOTION OF COLUMN IN WATER

From eqn (27), the resultant torque of water pressure on the outer wall of the column in a unit length is

N, .,N,

J

(25)

na4wh

I

0

x Jm ,

T=

%,2N

m(z, 1) = Go

IT:,

,

;

CJ4

. ..

where

+ : ID!t+I,,+N+I ,= I

d

S,.

S2d

From eqn (23) the constants C$, (n = 0, 1,2, and c$‘:n(n = 1,2,3, . ,N) are obtained: C<,,=Pi,G,,n=0,1,2

cj=

YP4.2

32,’ 2.4

.

,

.

n=0,1,2,.

at

0, f)

wvcos(tl, 0)

d0 = M(z) ehur, (28)

.3

1

where p. is the density of water. If the damping of water is neglected, the differential equation of torsional motion of the O-h segment of the column is [14]

GJ a2bh(zT f) ITT--

pJ av, (z, t) Pp= at2

-m(z, t), (29)

W-9 in which, ID+,,]= (- l)‘+‘]&) is the cofactor of the determinant ]k2N+ , 1 and ]kj’,,]is the minor which is given by removing the s th row and r th column in the determinant ]kiN +,I. From the above analysis, the velocity potential of water is therefore obtained from eqns (12), (22) and (25)

where 1(1,(z, t) is the torsion angle of the O-h segment of the column, G is the shear modulus of elasticity, J, is the torsion constant dependent on the geometry of the cross-section of the column, which can be obtained by the use of elastic mechanics method about torsion problem (151, p is the mass density of the column, Jr, is the polar moment of inertia of the cross-section.

39

Torsional vibration of columns in water Considering the column does free vibration, can assume that Y,(z,

Substituting

t) = Y,(z)e’“‘.

4. THE SOLUTION OF TORSIONAL MOTION THE COLUMN IN WATER

one

According to eqn (32), the differential eqn (31) can be written as

(30)

the above equation into eqn (29) gives y+k’Y,(z)=

r

OF

GJ d’Y,(z) p+ ’ dz2

pJ,wV,(z)

= -M(z),

(31)

f R,G,f;(z), /=a

(36)

where

k2=pJ,w2

GJ,

’ ‘,cos(2n@)+~~sin(2nB)Jdf?, K,(cxja(0))b(e)a(0)[Pi,cos(2n0)

j=O,1,2,...,L

+ pi sin(2n0)] de,

j = L + 1, L + 2, L + 3,. . . . (37)

where

The solution for the homogeneous equation of eqn (36) . , is

M(z) = 2p,w2

i G,h(z) I j=O

5 n=O

Y,,,(z) = D, cos(kz) + D, sin(kZ),

’ HS)(cr,a(B)) i0

where D, and D, are the unknown constants. particular solution of eqn (36) is expressed as OCI

x ri(B)a(B)[Pi cos(2nB) + Fi sin(2nB)] d0

Y,,~(z)= 1 E, Gjh(z).

(39)

,=o

Substituting x [Pj” cos(2nB) + l@” sin(2n6)] de.

(32)

a2J/2(zT

t)

t~ a2

-

pJ&,(z,

t) = 0.

the above equation into eqn (36) gives

Eo=&J(k2+k3, j=l,2,3

The differential equation of torsional motion of the h-H segment of the column is

,....

(33)

Y,(z) = Y,.,(z)+

(34)

Y,,,(z)=D,cWz)

G, = i GJ d2Y’,(z) ~ + pJ,w2Y2 (z) = 0. ’ dz2

s ‘h h

C? =

fo(5)cos(k5)

s0

WsWt,

d5

=

(35)

- E,),

0

f;(<)cos(k<)

d< =

fi(<)sin(k[)

d{ =

(42)

where

’

2ko k. sinh(k,h)sm(kh) -k cosh(k,h)cos(kh) tk2 + k2)tk h + isinhc2k hjj,,2 d5 = J-(

+ k)

,

0

&(kjsin(kjh)cos(kh)-k’cos(k.h)lin(kh)) (kf

h

s 0

D#‘/(l

J2k,60 sWkoh)cos(kh)+kcosMkh)sin(kh))

s 0

1j2)=

i=,

(ki + k2)(koh + i sinh(2koh))1’2

h

I;‘) =

(41)

Substituting above equation into eqn (19) and using the orthogonality of the generalized trigonometric series {f;(z)}, j = 0, 1,2,. . . gives

the above equation into eqn (33) gives

=

(40)

+ D2 sin(kz) + 2 E,G,f;(z). ,=o

tj2 (z, t ) = YZ(z) e”‘.

Icj

E,=R,/(k2-kf),

The general solution is given by the addition of the solution for the homogeneous equation and the partitular solution

Similarly, one can assume that

Substituting

The

n=o Jo

j=L+l

GJ

(38)

J2ki

-

k2)(kjh + 4 sin(2kjh;)l’2

(kjsin(kjh)sin(kh)

'

+ k cos(k,h)cos(kh) - k)

(kf - k2)(kjh + f sin(2kjh))1’2

.

(43)

40

Zhou Ding

So eqn (41) becomes

Substituting eqns (44) and (45) into eqns (47) and (50) gives four simultaneous linear equations about Di (i = 1,2,3,4) as follows: kD2=(lk2-K,)

O
+f

(4)

Yu,(z) = D, cos(k(z -h))

+ D, sin(k(z - h)) h
ayY,(z, t) = -k,Y,(z, t ~ aZ

GJay2(z’t)=0 t

az

t)-Jo

1

at2

+ sin(kh)+ ,,0, f 2 _E Yf;(h)

(46)

’

where J,, is the moment of inertia of the attached mass and k0 is the stiffness of the torsional spring. Substituting eqns (30) and (34) into above equations gives

dl//,(z) = -K,Y,(z)+Ak2YY,(z),

z =O,

dz

D2,

I

-k

+

sin(kh)+ ,=o, f x _ E rj”f;(h) I

k cos(kh) + f - ’ ,,0l-E,

I!“f’ (h) D,,

’

’

where J(h) = df;(z)/dz 11=A. The frequency equation can be obtained by putting the determinant of the coefficients of eqn (51) to zero

[

f

E;Ij2’

j=O

xcos(kh)

- 5 TI;” J=o

I[ 1

sin(kh) + tan(k(H -h))

=(~k2-K,)(I+~oF,I;1~)

x cos(kh) - tan(k(H - h)sin(kh) + f

,=o

[ dY,(z) -=O,

z=H,

dz

4

(51)

k -(lk2-Ku)

’

- h))D, + cos(k(H - h))D, = 0,

(45)

a2y’,tz,t) , z=o,

z=H

D,

I;2’l;,o,D2,

D,=

where D, and D, are also the unknown constants. The four unknown constants Di (i = 1,2, 3,4) can be decided by the boundary conditions of the column and the connecting conditions of the column between Y,(z) and Yz(z) at z = h. In this paper, the effect of the soil on the torsional motion of the column is equivalent to a mass and a torsional spring attached to the end (z = 0) of the column. In this case, the boundary conditions of the column can be written as GJ

E, ,=01-E,

-sin(k(H

Similarly, the general solution of eqn (35) is

1 + ,=o f -% 1 _ E l;“f;(O) /

F,Ij2’ ,

I (52)

(47) in which,

where (48) The connecting conditions at z = h are as follows:

F,= &J(O)>

q. = A/‘(h)

$ - tan(k(H -h))

I

(

. >

(53)

yy,(z, t) = Y,(z, t), GJ

awz,0 =

GJ

‘az Substituting gives

ay2(z, 0 ,-, az

z=h.

(49)

eqns (30) and (34) into above equations

‘y,(z) = Ydz),

dyY,(z) dy’,(z)

7

“l

=

-

“I

rt7

z= ,

h.

(50)

By using the method of determinant search, the eigenfrequencies can be calculated from eqn (52) by means of a computer. By substituting the eigenfrequencies into eqns (50), the constants Di (i = 1,2,3,4) can be obtained. The mode shapes of the column and the resultant torque distribution of water pressure on the outer wall of the column can be obtained from eqns (44), (45) and (32), respectively.

Torsional vibration of columns in water 5.CONCLUSION This paper presents an analytical-numerical method for solving the torsional vibration problem of a column in water, which is suitable for uniform columns with arbitrarily shaped non-circular crosssections symmetric about the centre. In analysis, the effects of surface waves and the compressibility of water on the torsional vibration characteristics of the column in water are also simultaneously considered, and the effect of the soil on the torsional vibration of the column is equivalent to a mass and a torsional spring attached to the end of the column. The calculated formulae of frequency equations and mode shapes of column-water coupling torsional vibration, which are suitable for programming, are derived. The method may also be further extended to the coupling torsional vibration problems of water-shell and soil-water-shell. REFERENCES H. M. Westergaard, Water pressures on dams during earthquakes. Trans. Am. Sot. civ. Engng 98, 418-433 (1933). N. Shim&

Advances and trends in seismic design of cylindrical liquid storage tanks. JSME Int. J. Ser. III 33, III-124 (1990). Ju Rong-chu and Zen Xin-zhuan, Theory on Coupling Vibration of Elastic Structures with Liquids. Earthquake Press, Beijing (1983) (in Chinese).

41

4. H. Goto and K. Toki, Vibrational characteristics and aseismic design of submerged bridge piers. Proc. 3rd World Conf Earthquake Engng, Vol. II (1965). 5. Zhang Xi-de, Free bending vibration of circular column partially submerged in water. Appl. Math. Mech. 3, 537-546 (1983) (in Chinese). 6, Chang H&h,. Earthquake response of circular columns partially submerged in water. Appl. Math. Mech. 4, 847-852 (1983) (in Chinese). 7. Zhou Ding, The free bending vibration of cylindrical tank partially filled with liquid and submerged in water. Appl. Math. Mech. 11,469-477(1990). g Zhou Ding, The free bending vibration of elliptical reservoir partially filled with liquid and submerged in water. Chin. J. appl. Mech. 6, 83-90 (in Chinese). Cheng Che-min and Ma Zong-kui, Vibration of 9. cantilever beam placed against water with free surface. Acta mech. sin. 3, 1I l-l 19 (1959) (in Chinese). 10. Zhou Ding, Vibration of uniform columns with arbitrarily shaped cross-sections partially submerged in water considering the effects of surface wave and compressibility of water. Comput. Struct. 46, 104991054 (1993).

11. Zhou Ding, The analysis of torsional vibration of elliptic column submerged in water. Mech. Pratt. 4, 32-35 (1991) (in Chinese). 12. Liang Kun-miao, Mathematical Methoak in Physics. Peonle’s Education Press. Beiiina (1979) (in Chinese). 13. G. Kreisel, Surface waves. Q.- J-appl. h4ech. 7, 21-44 (1949).

14. S. P. Timoshenko et al., Vibration Problems in Engineering, 4th edn. John Wiley, New York (1974). 15. S. P. Timoshenko and N. J. Goodier, Theory of Elasticity, 2nd edn. McGraw-Hill, New York (1951).