- Email: [email protected]
Free vibration analysis of circular cylindrical shells
Journal of Sound and Vibration (1981) 74(3), 331-350
FREE
VIBRATION
ANALYSIS
CYLINDRICAL
OF
CIRCULAR
SHELLS
H. CHUNG Components Technology Division, Argonne National Laboratory, Argonne, Illinois 60439,
1I.S.A.
(Received 3 September 1979, and in revised form 14 July 1980)
A general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. The solution is obtained through a direct solution procedure in which Sanders’ shell equations are used with the axial modal displacements represented as simple Fourier series expressions. Stokes’ transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. An explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the method is checked against available data. The method is used to find the modal characteristics of the thermal liner model of the U.S. Fast Test Reactor (FTI?). The numerical results obtained are compared with finite element method solutions. 1. INTRODUCTION The free vibration problems of thin circular cylindrical shells have been of great interest to many structural engineers in recent years. Many investigations following the pioneering work of Arnold and Warburton [l, 21 have been summarized by Leissa [3]. Methods of solution for shell vibrations have ranged from the approximate energy methods used by Arnold and Warburton [l, 21 and by Sharma and Johns [4,5] to the exact solutions as studied by Forsberg [6-81, Warburton [9], Warburton and Higgs [ 101 and Goldman [l 11. Recently, the author [12, 131 used a Rayleigh-Ritz procedure in conjunction with Lagrange multipliers to study, mainly, the free vibration characteristics of constrained cylindrical shells. Simple Fourier series were used for the modal displacement functions along the axial direction and an explicit expression of the frequency equation was obtained. In this paper, the shell equations are solved directly with these Fourier series, and an exact frequency equation is obtained. Linear, elastic, first order shell theory due to Sanders [14,15] is employed in this paper. The modal displacement functions are expanded in Fourier series in the circumferential direction. The problem is then considered on a mode-by-mode basis associated with the circumferential Fourier index n. In the axial direction, the displacement functions are constructed to form a set of complete orthogonal functions in the form of simple Fourier series which are not required to satisfy the boundary conditions term-by-term. Stokes’ transformation [16, 171 is used to obtain correct series expressions for derivatives of the Fourier series. The boundary conditions which are not satisfied directly by the assumed series are enforced. The unwanted boundary conditions, that are originally unspecified but are implied by the nature of the assumed series, are released to remain unspecified. Through this process, all the boundary conditions of the problem become satisfied by the assumed displacement function in an overall sense. It should be emphasized that the present method can solve problems with any kind of boundary conditions. The direct solution method used here differs from those of other authors [6-111 in the selection of 331 0022-469X/81/030331+20$02.00/0
@ 1981 Academic
Press Inc. (London)
limited
332
H. CHUNG
general solutions of shell equations. They have chosen exponential functions for the modal displacements along the axial direction, substituted them into the equations of motion and then enforced the eight specified boundary conditions. This leads to an eighth-order algebraic equation and an eighth-order frequency determinant which are coupled together. The simultaneous solution of these two systems of equations involves extremely laborious computation as pointed out by Leissa [18]. In the method presented here, the use of simple Fourier series along the axial direction leads to an explicit expression for the frequency determinant, which is of low order (s 4), and this greatly simplifies the solution procedure. The numerical results from this method are checked with the available exact solutions reported by other authors. The natural frequencies of the thermal liner model of the U.S. Fast Test Reactor (FTR) are calculated and discussed. In particular, the variation of natural frequencies due to the change of boundary conditions is discussed in detail. 2. THEORETICAL
BACKGROUND
2.1. EQUATIONS OF MOTION The following shell equations due to Sanders [14, 151 are used:
,e -kw 1-v -Rku,x~ 2
3-u + ~Ru,, -Y~-R’~~J,,,o
+V,O+ w +k{R4W,,,,,
,888 = Y
2
V,m
+2R2w,xx00 + w,eeee-v,eee1 = -7
2
W,rr.
(1)
Here y2 = pR2(1 - y2)/E and k = h2/12R2. The geometry of a circular cylindrical shell is shown in Figure 1. Also shown on the figure are the forces and moments acting on a section of the shell parallel to the coordinate lines. A list of nomenclature is given in Appendix IV. The natural or geometric boundary conditions applicable to Sanders’ theory reduce to prescribing the following quantities at the ends of the cylindrical shell: I?*, or
N, or u,
V x =o,
6
1
QXorw,
M,org
1.
(2)
i
Here & = aM,/ax + (2/R)a&/aO.
$x0 = NXe+ (3/2R)tixe,
(3)
The effective forces per unit length gX and Q, are shown in Figure 1, The relationships between boundary forces and displacements are .&
u,,+iV,e+i~],
xe
3-v
=D(l-v) 2
1 [z(l-ik)
1
2-v
I
where D =Eh/(l-
v2) and K =Eh3/12(1
-
v2).
u,e+(l+$k)v,,-3kw,~e],
M, =
(4)
VIBRATION
OF
Figure
CIRCULAR
1. Cylindrical
CYLINDRICAL
SHELLS
333
shell with applied forces.
2.2. MODAL FUNCTIONS A general relation for the displacements in any mode of free vibrations may be written in the following form for any circumferential mode number n : u(x, 8, t) = &(x)
cos nt9 sin wf,
0(x, 8, t) = &(x) sin no sin wt,
w(x, 8, t) = $,(x) cos n@sin wt.
(5)
Here 1,4,,1,4”and I& are mode functions corresponding to axial, tangential and radial displacements, respectively. The crucial part of the analysis involves choosing appropriate series forms for these mode functions. The series should be simple in form and at the same time preserve orthogonality properties. It is not necessary that the series satisfy any particular boundary conditions since one is dealing with a general solution at this point. There are two convenient sets of Fourier series that meet all these requirements for the mode functions along the axial direction. The first set, designated “CSS”, can be written in the form
~,$,(x)=Ao,+ f A,,cos~, l?l=l
m=l
I&(X)=
f m=l
C,,siny,
(6)
334
H. CHUNG
while the second set, designated “SCC”, can be written as
vhCx)=m~l&. sin?, +bw(x)=Con+
l)U(X)=Bo”+
f
B,,
m=l
f WI=1
cosy?
c,,cos~.
(7)
The first set represents the exact solution for the shell when it has simply supported ends with no axial constraint (SNA-SNA). This “SNA” shell has boundary conditions at each end of the form N, =O,
A4, =o.
w = 0,
v =o,
(8)
The “CSS” set satisfies these boundary conditions term-by-term. The second set represents the exact solution for the shell when it has freely supported ends with no tangential constraint (FSNT-FSNT). This “FSNT” shell has boundary conditions at each end of the form u =o,
i& = 0,
awlax= 0.
&=O,
(9)
The “SCC” set satisfies the above boundary conditions term-by-term. By using Stokes’ transformation, it is possible to use these sets to find exact solutions for shell problems with any possible combination of homogeneous end conditions. There are sixteen possible sets of homgeneous boundary conditions that can be specified independently at each end [12, 131. 2.3. STOKES’ TRANSFORMATION Consider a function f(x) represented by a Fourier sine series in the open range 0
f(O)=fo,
mm
f(l) =fl,
f(x)= f a, sin m=l
O
I
(10)
’
Since it is not certain that the derivative f’(x) can be represented by term-by-term differentiation of the sine series, the derivative is instead represented by an independent cosine series of the form f’(x)=bo+
f m=l
b,cosyF 1
(11)
Stokes’ transformation then consists of integrating by parts in the basic definitions of the coefficients to obtain the relationship between b, and a,,,, as follows: 2 ’ b, = f’(x) cos ydx 1 I0 =
=;
f(x) cos ~]~+~~o’f(x)sin~dx
WO[(-l)“fi -f01+(m7dOam.
(12)
Similar care must be taken when finding the correct sine series corresponding complete set of formulas for the sine series is then written as f (0) = fo, fyx)
=
-f9-
f(l) =fl, mg,
mrrx a, sin 1,
f(x) = z
Ocx
m=l
mvx [~VO-(-lYfJ--~mam
tof”(x). The
I
I
cos-7-’
0~x61,
VIBRATION
OF CIRCULAR
f’W)=f&
CYLINDRICAL
335
SHELLS
f”(l) =fY,
p(x)=(y)m[, m[?lfo-(-l)mf~}-~mu,]
sin?,
O
(13)
If the function f(x) is represented by a Fourier cosine series, similar transformation formulas must be used to obtain the correct form of the successive derivatives of the series. These formulae for the sine and cosine series are used in the direct solution procedure of the shell equations. 3. GENERAL FORMULATIONS One can use either set, (6) or (7), of mode functions as a general solution of the shell equations (l), and can obtain an exact particular solution for a given shell with specific boundary conditions. In this paper, details are given only for the solution in which the first set (6) of mode functions, CSS, is used. However, results for frequencies are shown based on both sets of axial mode functions. The first set, CSS, identically satisfies all the boundary conditions (both geometrical and natural), only for the SNA (simple support without axial constraint) shell. Therefore, substitution of this set into the equations of .motion (1) lead to the frequency equation for an SNA shell extensively studied by Dym [19]. For all shells with other boundary conditions, the frequency equation is determined from the specified boundary conditions. For maximum generality of the formulation, the FSNT (freely supported without tangential constraint at both ends) shell will serve as the base problem for CSS. All other shell problems with different boundary conditions are simplified special cases of this base problem. For this base problem, none of the eight boundary conditions are satisfied by CSS on a term-by-term basis. Thus the constraint conditions to satisfy these boundary conditions lead to an eight-by-eight frequency determinant. Solutions for shells with other boundary conditions lead to a smaller size determinant. It is interesting to note that for the second set, SCC, all the boundary conditions of the FSNT shell are satisfied on a term-by-term basis and this problem is therefore simple to solve. The most complicated problem for SCC is the SNA shell. A CSS solution for the free vibration characteristics of an FSNT shell will now be outlined. The CSS set is given in equations (6) and their derivatives are obtained through the use of Stokes’ transformation (see Appendix I). Substitution of the CSS set and their derivatives into the equations of motion (1) leads to an explicit relation for A,,, and, in addition, a set of equations in which A,,, B,,, and C,, are coupled together. The solution of these coupled equations allows all the coefficients to be written explicitly in terms of the eight unspecified quantities ~20,z&,UO,~1,wo, WI,Go, Er and the frequency parameter R, as follows:
Aon =
2(u
1_n)[al(ao+oi)-a3(l;o+cr)-aa(wo+wi)l,
f2
#
(14)
a213
21
a,
u~[~o+~r(-l)“]-u3[2)0+V~(-l)ml -uagm[uo+ u,(-l)“]
[~1=[~~rnH
:J
1
-u~m[?Jo+U~(-l)m]
--a4[wo+ w-1)“l -u,m[wo+
wl(-l)“l
-(ugm3+u~om)[wo+wl(-l)ml+u+z[~o+~r(-l)”l
cl,, 1.
# 0.
(15)
336
H. CHUNG
Here 0 = (o/we)* and CJ~= E/pR’(l - v*). Other symbols are listed in Appendix II. The equations (14) and (15) include the four unspecified quantities VO, q, w. and wI, which are directly related to the geometrical boundary values: vo = - 7T,;
ne
o(O,a
sl’ ne, u(L (9,
f-J1= 7T
wo=---&wKw)’ Wf=?rs;ne wu,fv
(16)
The other four quantities, Q0, I&,Q. and r&, are associated with the unspecified end forces N, and moments M, as follows (see equations (4)):
Equations (14) and (15) may now be rewritten in terms of the eight unspecified boundary quantities ~0, UI,WO, wi, fiz, I?_ fi! and Z@f,for a FSNT shell:
Aon =
1 41(~o+ur)+q4(Wo+w)-
urr
fJ f
C
2&-R)
a0
(1%
UT. 41cvo+(-l)“u,l
b6 b, symm.
a21,
II
c,_ 42dvo+(-l)“vll i 43~r~o+(-l)mufl
+
45Lwo + (-l)"Wfl
+qsm[wo+(-l)“w,l-
,O,“#O. 2R2(1 - V2) m[ii: Ehl
(20)
+ (-l)“ti;]
Here fro = NXKJ,6) * cos t2e
fi’ _ ’
x
NX(&!) ’
c0sne
(21) The boundary conditions appropriate to the FSNT shell are listed in equations (9). The geometric boundary conditions that must be satisfied are associated with u and ~w/&K. Hence for arbitrary IZit follows that I/Iu(O)= Aon + : A,.,, = 0, ??I=1
$I,(Z)=Ao,+
:
m=l
A,“(-I)m=O,
&(O)=?[Jy +m;l c {WO+(-l)mW~+mCmn} =o, 1
,;,l,=;[Jy+m;, 1
{Wo+(-l)mW~+mCm”}(-l)m
I
=o.
Cm
VIBRATION
OF CIRCULAR
CYLINDRICAL
337
SHELLS
The natural boundary conditions that must be forced to zero are fiXe = 0 and & = 0 at both ends. Substitution of the CSS set and their derivatives into these equations yields the following four equations: q1A,,+q~(y)+q3(-)
+ _,_, C [VIA,, +qz{~o+ ~(-1)”
+ m&z,}
+q3!wo+ wr(-1)“+mC,,}]=0, q~Ao, +qz(y)
+43(y)
123)
+ ,,_, C [qrA,,
+q3{wo+ wr(-1)” +mC,,}](-1)” q4Ao, +q5(y)
-q12(y)
+ qz{uo + ~(-1)”
+ mBmn} (24)
= 0,
+ag(F)
02 +
C ~q~Amn+~~~~~+~,(-l)~+m~mn~-~~~~~o+~~(-l)~+mCmn~ ??I=1
+ag{~,+~,I(-1)“-w~m2-W~m2(-l)m-Rz3Cmn}]=0,
(25)
llaAon+4s(~)-412(~)+~~(~) + :
[q~A,,+~~~~O+~r(-l)m+m~mn~-~~2~~O+~,(-~)m+~Cmr,~
Wl=l
+a~{~,o+~,(-l)m-wom2-w,m2(-1)“-m3C,,}](-1)”
=o.
(26)
The last two equations, (25) and (26), contain the inexpedient G+, and 6,. Therefore, equations (25) and (26) are rewritten in terms of i@ and &?f, as for the case of equations (14) and (15): 4~A~.+4.(~)-4is(~)-2R~~u2’(~0:8’)+~~~[q,A,., +~23m&n
+q3{UO+d-l)m
+~&,n}+~13~Cmn
+mC,“}m2-qq18{W~+W,(-l)m+mCmn}-
-C${wO+
Wl(-1)”
2R2;h-“2){A”
+n;l;(-l)mj]
=O,
(27)
4~A~.+43(~)-4~~(~)-2R~~Y2’(~0:s’)+~~l[4.Amn +423~~mn+43{~Of~I(-l)m+mB,“}+ql3mC~,-as(wO+wr(-l)m
x {ii?; +fi;
2R2&
wr(-1)” +mC,,}-
+mC,,}m2-q1*{W0+
(-l)“}]
V2) {j@
(-1)” = 0.
+#(-1)“)
(28)
The frequency determinant may now be constructed. Substitution of the relations (19) and (20) for the coefficrents A,,,, A,,, I?,, and C,, into the eight constraint conditions due to the geometric and the natural boundary conditions, expressions (22)-(24), (27) and (28), leads to the homogeneous matrix equation
[ei,jl{R7 *b R?, fi.L
uO9
z)l,
WO,
WI)=
=
(0)~
(29)
338
H. CHUNG
where i, j = 1,2, . . . , 8. It is interesting to compare this equation to the result previously obtained by the author [12, 131 through a Rayleigh-Ritz procedure in conjunction with Lagrange multipliers. By redefining the unspecified end forces EX, fiX as proportional to the Lagrange multipliers mp = AJTR, $I, = AJrrR, li?(t = AJRl and h?f, = A4/R1, equation (29) becomes identical to the matrix equation in references [12, 131. For a non-trivial solution of equation (29), the determinant of the matrix must vanish, lei,jI=
(30)
0,
resulting in a characteristic equation whose eigenvalues determine the natural frequencies of the shell. The corresponding eigenvectors determine the mode shapes. This frequency determinant is symmetric, and is based on the exact satisfaction of the boundary conditions. The natural frequency appears as a parameter (0 = (w/tic)*) in this equation. Each element of this frequency determinant is an infinite series, and these are listed in Appendix III. From the development of these equations it may be shown that there are two restrictions involved. The first restriction is n # 3(1 - v)(l + k/4)n *. For axisymmetric (n = 0) shell vibrations, this becomes R # 0, thereby invalidating expression (19) for rigid body motion. However, conditions associated with this rigid body motion are usually determined by inspection. For non-axisymmetric motion (n # O), the case of R = $(l - v)(l + k/4)n2 is automatically rejected on the grounds that it leads to rigid body motion with non-zero frequency. The second restriction, 0,” # 0, may be shown to imply that this CSS set cannot be used to solve the FSNT shell. This is not a serious drawback since it is easy to show that the second (SCC) set offers a very simple solution in this case.
4. FREQUENCY
DETERMINANT
In this section, the frequency determinant is derived for several cases of shell boundary conditions. The frequency determinant (30) is based on the boundary conditions of “freely supported with no tangential constraints” at both ends (FSNT-FSNT). None of the FSNT-FSNT boundary conditions are satisfied by the CSS set of displacement functions on a term-by-term basis. The requirement to satisfy these boundary conditions by the CSS set in an overall sense led to the eight-by-eight frequency determinant (30). From this determinant, one can also obtain the frequency equation for any combination of boundary conditions among sixteen possible cases [12, 131 without going through the detailed procedures required to formulate the corresponding algebraic eigenvalue problems. This can be done by deleting inapplicable rows and columns from the frequency determinant (30). Therefore, solutions for shells with other boundary conditions lead to a smaller size determinant than for the FSNT-FSNT shell. For certain boundary conditions, the second set, SCC, leads to a simpler frequency determinant than the CSS set. Indeed, if the order of the frequency determinant obtained by the use of one set is P the one obtained by the other set is 8-P for the same set of boundary conditions. Therefore, one can always obtain the exact frequency determinant with an order ~4 for any case of boundary conditions. 4.1.
SIMPLY
SUPPORTED
SHELL
WITH
NO
AXIAL
CONSTRAINTS
AT
BOTH
ENDS
(SNA-SNA)
The simplest case of boundary conditions is, of course, the SNA shell since the CSS set of axial mode functions satisfies all the boundary conditions, N,=O, on a term-by-term
v =o,
w = 0,
h4,=0,
x =o, 1,
(31)
basis. It should be noted that the boundary values U,fiXO,& and awlax
VIBRATION
OF CIRCULAR
CYLINDRICAL
339
SHELLS
at both ends are unspecified by this set. The frequency equation for the SNA shell is directly obtained from the equations of motion as a three-by-three determinant. The elements of the frequency determinant are no longer infinite series but reduce to algebraic terms. Past work on this problem includes that of Arnold and Warburton [l, 21 and Dym [193. The frequency equation is as follows: S 11
s12
S
s 22
s::
symm. 4.2.
SIMPLY
SUPPORTED
SHELL
= 0.
(32)
s33
WITH
AXIAL
CONSTRAINTS
AT
BOTH
ENDS
(SS-SSj
As a typical case involving the enforcement of geometric boundary conditions and the release of unwanted natural boundary conditions, consider the simply supported shell with axial constraints at both ends. The boundary conditions are u =o,
lJ =o,
w =o,
M,=O,
x =o, 1.
(33)
The CSS set automatically satisfies the last three equations and the unwanted natural boundary conditions N, = 0 at both ends (see equation (31)). The frequency equation can be found from the geometrical constraint conditions of u = 0 at both ends, with, at the same time, the end forces N, left unspecified. This is effectively done in equation (30) by retaining the rows and columns associated with fift and 6:. The end result is the exact two-by-two frequency determinant e1.1
el.2
symm.
e2.2
=
0.
(34)
I
The elements of this determinant are listed in Appendix III. It is interesting to note that the second set, SCC, would involve the enforcement and release of geometric and natural boundary conditions, and lead to a six-by-six determinant. 4.3.
SHELL
WITH
CLAMPED
ENDS
(C-C)
When the shell is clamped at both ends the boundary conditions are u =o,
U = 0,
w =o,
awlax = 0,
x =o, 1.
(35)
Among these conditions, the four geometric boundary conditions u = 0 and awlax = 0 at both ends are not automatically satisfied by the CSS set. Instead, this CSS set unnecessarily satisfies N, = 0 and M, = 0 at the ends. The frequency determinant is obtained from equation (30) by retaining the rows and columns associated with fi:, fi:, @ and Qk : e1.1
symm.
el,2
el.3
el.4
e2.2
e2,3
e2.4
e3,3
e3.4
(36)
e4.4
If the second set, SCC, is used, one would obtain a different four-by-four determinant.
frequency
4.4. SHELL WITH FREE ENDS (F-F) As a typical case involving the enforcement of natural boundary conditions and the release of unwanted geometrical boundary conditions consider a free-free shell with
340
H. CHUNG
boundary conditions N,=O,
&
= 0,
M,=O,
0X=0,
x =o, 1.
(37)
From the properties of CSS it is apparent that the tangential and radial displacements, u and w respectively, are identically zero at the ends. Therefore, the releasing procedure is required to remove these unwanted geometric boundary conditions. By defining u and w separately at the end points (17) and including these values in the subsequent differentiation via Stokes’ transformation, the exact frequency equation can be found from the enforcement of natural boundary conditions &., = 0 and Q, = 0 at the ends. The result is obtained from equation (30) by retaining the rows and columns associated with uo, vI, w. and wf: e5,5
e5.6
e5.7
e5.8
e6.6
e6,7
e6.8
e7,7
e-l;8
symm.
=
0.
(38)
e8.8
The use of the second set also leads to another four-by-four 4.5.
CLAMPED-FREE
SHELL
determinant.
(C-F)
The case of a clamped-free shell involves the enforcement and release of geometrical and natural boundary conditions. For this non-symmetric set of boundary conditions, u =o, N,=O,
v =o,
awlax
w =o,
fix* = 0,
= 0,
M,=O,
&=O,
x =o, x = 1,
the CSS set leads to a four-by-four frequency determinant. The conditions at x = 0 must be enforced with the release of the end force M,. The fix0 = 0 and & = 0 conditions at x = 1must be enforced with displacements v and w. Retaining the rows and columns in equation @, A.?:, VI and WIleads to el.1
el.3
e1,6
e1.8
e3.3
e3,6
e3,8
e6.6
e6.8
symm.
ZZ
5.1.
CONVERGENCE
AND
u = 0 and awlax = 0 and moment, N, and the release of the end (30) associated with
0.
(40)
e8.8
If the second set, SCC, were used, a different four-by-four be obtained. 5. DISCUSSION
(39)
frequency determinant
would
OF RESULTS
ACCURACY
The frequency equations derived in the preceding sections involve infinite series of algebraic terms. Before any numerical results are calculated, manipulations should be performed to make the rates of convergence of the sums of the infinite series terms as close to each other as possible. The convergence of certain sums can be accelerated by subtracting off a series with a known sum [20].
VIBRATION
OF CIRCULAR
CYLINDRICAL
341
SHELLS
As a typical example, consider the following sequence of manipulations series:
for a simple
Through the above process, the order of the series has been improved from m m2to m “I, and accordingly the rate of convergence will be greatly accelerated. The infinite series in the shell frequency determinant have been treated in the same way as in the above example. In this way, fast convergence has been obtained and it was possible to use the same upper limit for all series in the frequency determinant; typically an upper limit of 50 terms was used. Frequency results were obtained from the frequency determinant by substituting values of w/o,, and monitoring the determinant until it vanished. Convergence and accuracy of the present method is described in Tables 1 and 2. In order to facilitate comparison with the exact solutions reported by other contributors who used Fliigge’s and Donnell’s theory, the present method was also applied to Donnell’s equations. In general, the solution approaches the exact frequency from above as the number of terms included in the series increases. However, as shown in Table 1, the use of more than 50 terms did not improve the solutions significantly. Table 2 compares the results of the present method with the available exact solutions. In all cases the comparison shows TABLE
1
Convergence of natural frequency parameters, w/w0
Number of terms 5 7 10 20 30 40 50 100 150 200 300 500 Reference r191 Case Case Case Case
Case 1 _PPY Sanders
0.01523
Case 3
Case 2
Case 4
Donnell
Sanders
Donnell
Sanders
Donnell
Sanders
0.01555 0.01552 0.01550 0.01547 0.01545 0.01544 0.01542
0.0684 0.0681 0.06792 0.06767 0.06760 0.06758 0.06758 0.06757 0.06757 0.06757 0.06757 0.06757
0.3161 0.3134 0.3122 0.3119 0.3119 0.3119 0.3118 0.3118 0.3118 0.3118 0.3118 0.3118
0.3230 0.3203 0.3191 0.3188 0.3188 0.3188 0.3188 0.3188 0.3188 0.3188 0.3188 0.3188
0.3589 0.3586 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 ___-
0.06757
0.3117t
0.3188
0.3583t
0.01520 0.01517 0.01515 0.01513 0.01512 0.01510 0.01509 0.01509 0.01509 0.01508 0.01508
0.01541 0.01541 0.01541 0.01541 0.01541
0.05880 0.05851 0.05823 0.05795 0.05788 0.05785 0.05784 0.05784 0.05784 0.05784 0.05784 0.05784
0.01508t
0*01541
0.057871
1: Clamped-clamped shell (l/R 2: Clamped-clamped shell (I/R 3: Clamped-clamped shell (I/R 4: Free-free shell (I/R = 5, R/h
= 10, R/h = 500, v = 0.3, n = 4). = 10, R/h = 20, v = 0.3, n = 2). = 2, R/h = 20, v = 0.3, n = 3). = 20, v = 0.3, n = 1).
t Solutions based on Fliigge’s theory [7, 81. $ No exact solutions are available from literature
Donnell
342
H. CHUNG TABLE 2 Comparison of natural frequency parameters, w/w0
Shell parameters n
m'
condition
'1fR
R/h
4 2 3 1 0 2 2 2 2 4 1 1 0 2 4
1 1 1 1 1 5 1 2 3 1 1 3 1 1 1
c-c c-c c-c
10 10 2 5 8.1 8.67 1.14 2.88 5.07 10 30 5 2n 2v 27F
500 20 20 20 20 500 20 20 20 500 500 500 10 10 10
F-F F-F F-F C-F C-F C-F ss-ss SNAT”-SNAT SNAT-SNAT SNA-SNA SNA-SNA SNA-SNA
Present method v’ 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 8:; 0.3 0.3
Otherst
Sanders
Donnell
Fliigge
Donnell
0.01508 0.05784 0.3118 0.3585 0.3669 0.4472 0.3076 0.3081 0.3079 0.01508 0.00702 0.3699 0.296081 0.095812 0.425921
0.01541 0.06757 0.3188 0.3590 0.295804 0.119730 0.454809
0.01508 0.05787 0.3117 0.3583 0.3671 0.4472 0.3082 0.3082 0.3082 0.01507 0~00700 0.3698 0.296081$ 0.0958121 0.425921t
0.01541 0.06757 0.3188 0.295804 0*119730 0.454808
-
t These results are due to Forsberg [6-81, Warburton [9, lo] and Dym [19]. $ These solutions are based on Sanders’ theoiy [ 191. 5 Simply supported without axial and tangential constraints.
excellent agreement for a wide variety of boundary conditions and geometric parameters. For shells clamped at both ends, the solutions of the present method with Donnell’s theory are the same as those reported by Forsberg [8]. The frequency parameters for the SNA-SNA shell for Sanders’ as well as Donnell’s theory are identical to the results given by Dym [19]. For other boundary conditions, F-F, C-F, SS-SS and SNAT-SNAT, the available solutions are all based on Fliigge’s shell theory. The differences between solutions obtained from different thin shell theories are usually small, as revealed in Table 2. Mode shapes for a particular natural frequency can always be investigated by substituting the calculated frequency parameter into the frequency matrix and finding the relative values of unspecified end forces and values, and finally evaluating the coefficients of the assumed modal displacement functions. Comparison of mode shapes between the present method and others is not attempted in this paper. VIBRATIONS OF FTR THERMAL LINER 5.2. APPLICATION: The thermal liner of the Fast Test Reactor (FTR) in the U.S.A. is a thin, circular cylindrical shell situated concentrically inside the main reactor vessel. The support conditions of the thermal liner may be best represented as rigidly fixed at the bottom and free at the top: clamped-free (C-F) boundary conditions. A series of experiments had been performed as a part of an effort to investigate the modal characteristics of the thermal liner [21]. The experimental model is approximately l/14 scale of the prototype thermal liner for the FTR, giving the following dimensions and material properties: thickness, h = 0.15 cm; radius (to the center of thickness), R = 21.62 cm; length, I= 51.12 cm; material density, p = 7492 kg/m3; Young’s modulus, E = 1.83 x 1O1l Pa; Poisson’s ratio, v = 0.3. The experimental model was intended to simulate a clamped-free shell; however, insufficient clamping (awl& Z 0) and imperfect axial constraint (u # 0) at the bottom has been judged a probable cause for disagreement between the experimental results and the analytical predictions of natural frequencies.
VIBRATION
OF CIRCULAR
CYLINDRICAL
SHELLS
343
Therefore, in this study the sensitivity of the natural frequencies to changes in the bottom-boundary conditions was investigated by using the present solution method. The accuracy of the solutions has been compared to the results obtained by two finite element computer codes: SAP IV [22] and NASTRAN [23]. The boundary conditions considered are as follows: (1) C-F: clamped-free; (2) SS-F: simply supported-free; (3) CNA-F: clamped with no axial constraint-free; (4) SNA-F: simply supported with no axial constraint-free. TABLE
3
Comparison of natural frequencies (in Hz) for the thermal liner model with clamped-free boundary conditions n
m’
Present method
SAP IV
1 1 1 1 1 1 1 1 1
855.10 403.72 223.34 171.77 199.16 268.86 361.92 472.54 599.03
-t
856.3
405.1 225.6 174.3 201.7 272.0 366.2 478.4
410.1 232.2 180.5 206.2 275.5 370.1 483.5 614.0
2 2 2 2 2 2 2
928.28 644.48 494.69 442.00 464.59 539.45 648.34
-
943.2 671.4 529.3 478.0 496.9 567.2 673.2
t Not calculated
5Oy.8 454.3 476.7 -
NASTRAN
in these cases.
In Table 3 the solutions of the present method are compared with the SAP IV and NASTRAN results for the thermal liner with clamped-free boundary conditions. The finite element grid used with the two computer codes is the same and consists of 10 divisions vertically and nine divisions over a quarter of the shell circumferentially. Apparently the present exact method always gives lower values compared to those given by the two finite element computer codes. The natural frequencies for the thermal liner model in vacuum have been calculated by the present method for the above four cases of boundary conditions, and the results are summarized in Table 4 and Figure 2. The frequencies become smaller as the boundary conditions change from Case (1) to Case (4) for a given pair of circumferential and axial mode number, n and m’. The effect of clamping (~w/&x = 0) can be evaluated by comparing the frequencies for the following two pairs of boundary conditions: C-F and SS-F, CNA-F and SNA-F. The natural frequencies for SS-F are smaller than those for C-F by less than about 1% ; thus they are not distinguishable in Figure 2. The differences between frequencies for CNA-F and SNA-F are also small (< 1*5%), except for the modes of n = 0 to 3 with m’ = 1. The influence of axial constraint (u = 0) is evident in Figure 2 and, in contrast to the effect of clamping, is significant throughout most of the region of interest. Note that the frequency curve for m’ = 1 is greatly shifted down for n < 5; moreover, the lowest natural frequencies for shell-type vibration (n 2 2) occur at n = 2 for CNA-F and
H. CHUNG
344
TABLE 4
Natural frequencies
(in Hz) of thermal liner model for various boundary conditions
n
m’
C-F
SS-F
0
1 1 1 1 1 1 1 1 1 1 1
1497.90 855.10 403.72 223.34 171.77 199.16 268.86 361.92 472.54 599.03 740.81
1497.90 853.85 402.59 222.61 171.34 198.63 268.72 361.82 472.45 598.94 740.77
0 12.50 27.79 61.81 112.92 179.68 261.69 358.81 470.97 598.15 740.32
0 21.57 58.93 111.32 178.68 261.03 358.34 470.63 597.89 740.12
2 2 2 2 2 2 2 2 2 2 2
2351.51 2318.98 1437.11 928.28 644.48 494.69 442.00 464.59 539.45 648.34 781.15
2346.69 2118.98 1435.35 925.39 641.43 491.99 439.89 463.06 538.31 647.41 780.37
1497.90 2077.48 1408.98 844.59 552.22 418.54 390.44 434.21 522.17 638.16 774.87
1497.90 2077.24 1407.79 842.44 548.87 414.14 385.77 430.04 518.70 635.28 772.44
3 3 3 3 3 3 3 3 3 3
3076.05 2487.60 1834.82 1367.64 1057.12 864.82 767.65 750.67 798.18 893.60
3070.64 2484.64 1830.35 1361.83 1050.67 858.43 761.85 745.75 7949 11 890.18
2723.53 2479.20 1811.52 1323.62 1001.38 808.93 720.27 715.49 774.21 877.72
2723.02 2476.84 1808.06 1318.80 995.01 801.14 711.69 707.05 766.53 871.04
1 2 3 4 5 6 7 8 9 10 0 1
2 3 4 5 6 7 8 9 10 1
2 3 4 : 7 8 9
10
CNA-F
SNA-F 0
in contrast to n = 4 for C-F and SS-F. However, for higher axial half-waves (m’ = 2,3), the differences are less than 20% throughout the entire range of circumferential wave number n. A comparative study of the analytically predicted and experimentally obtained natural frequencies can be found in reference [21]. SNA-F,
6. CONCLUSIONS
A general solution procedure has been presented for the free vibration of a circular cylindrical shell, and applied to study the free vibration characteristics of the thermal liner of the U.S. Fast Test Reactor. The present method offers distinct computational advantages over other exact solution methods [6-l 11. Table 5 compares the present method and other exact solution procedures. An uncoupled explicit expression of frequency determinant is obtained by directly solving Sanders’ shell equations. The frequency equation is obtained in the process of satisfying all boundary conditions exactly by the assumed simple
345
VIBRATION OF CIRCULAR CYLINDRICAL SHELLS
I2
- IO I” “0 t8 5 s ;
6
4
2
0 Figure
2. Natural
frequencies
2
4 Circumferential
of thermal
6 mode
liner model.
8
IO
no (n)
-,
C-F, SS-F; - - -, CNA-F, WA-F.
TABLE 5 Comparisons of the present method with the other exact solution method Items 1. Frequency determinant 2. Order of frequency determinant 3. Element of frequency determinant
: References
Present method Uncoupled explicit expression Varies (1 - 4) dependent on boundary conditions Infinite series (rate of convergence can be accelerated)
Others:
~-
Coupled with 8th order algebraic equation Fixed (8) Transcendental functions (also series in computer computation)
[6-111.
Fourier series. This frequency equation involves infinite series which are evaluated by the use of appropriate numerical techniques and a computer. The order of frequency determinant may vary from 1 to 8 depending on the boundary conditions to be satisfied. However, an appropriate choice of displacement functions (i.e., the CSS or SCC set) can always furnish a frequency determinant of low order (~4) for any given boundary conditions. The frequency determinant obtained by others [6-l l] is always eight-by-eight and coupled with eighth-order algebraic equations resulting from the shell equations. The elements of the frequency determinant consist of transcendental functions in place of the infinite series of the present method. However, it should be noted that the transcendental function is evaluated as a series in the present day computer. In conclusion, the method presented in this paper is a relatively simple, exact-solution method for the free vibration problem of circular cylindrical shells.
346
H. CHUNG ACKNOWLEDGMENT
This work was performed Energy, Division of Reactor
under the sponsorship of the United Research and Technology.
States Department
of
REFERENCES 1. R. N. ARNOLD and G. B. WARBURTON 1948 Proceedings ofthe Royal Society, London A197, 238-256. Flexural vibrations of the walls of thin cylindrical shells having freely supported ends. 2. R. N. ARNOLD and G. B. WARBURTON 1953 Proceedings of the Institution of Mechanical Engineers A167, 62-80. The flexural vibrations of thin cylinders. 3. A. W. LEISSA 1973 NASA SP-288. Vibration of shells. 4. C. B. SHARMA and D. J. JOHNS 1972 Journal of Sound and Vibration 25, 433-449. Free
vibration of cantilever circular cylindrical shells-a comparative study. 5. C. B. SHARMA 1974 Journal of Sound and Vibration 35, 55-76. Calculation of natural frequencies of fixed-free circular cylindrical shells. 6. K. FORSBERG 1964 American Institute of Aeronautics and Astronautics Journal 2,2150-2157. Influence of boundary conditions on the modal characteristics of thin cylindrical shells. 7. K. FORSBERG 1969 American Institute of Aeronautics and Astronautics Journal 7, 221-227. Axisymmetric and beam-type vibrations of thin cylindrical shells. 8. K. FORSBERG 1965 NASA CR-613. A review of analytical methods used to determine the modal characteristics of cylindrical shells. 9. G. B. WARBURTON 1965 Journal of Mechanical Engineering Science 7,399-407. Vibration of thin cylindrical shells. Natural 10. G. B. WARBURTON and J. HIGGS 1970 Journal of Sound and Vibration l&335-338. frequencies of thin cantilever cylindrical shells. 11. R. L. GOLDMAN 1974 American Institute of Aeronautics and Astronautics Journal 12, 1755-1756. Mode shapes and frequencies of clamped-clamped cylindrical shells. 12. H. CHUNG 1974 Ph.D. Thesis, Tufts University. A general method of solution for vibrations of cylindrical shells. 13. R. GREIF and H. CHUNG 1975 American Institute of Aeronautics and Astronautics Journal 13, 1190-l 198. Vibrations of constrained cylindrical shells. 14. J. L. SANDERS, JR 1959 NASA Report 24. An improved first approximation theory for thin shells. Prager 15. B. BUDIANSKY and J. L. SANDERS, JR 1963 Progress in Applied Mechanics-The Anniversary Volume 129-140. On the best first-order linear shell theory. 16. T. J. 1’~ BROMWICH 1955 An Introduction to the theory of Infinite Series. London: MacMillan Publications, second edition revised. 17. B. BUDIANSKY and R. C. DIPRIMA 1960 Journal of Mathematics and Physics 39,237-245. Bending vibrations of uniform twisted beams. 18. H. KRAUS 1967 Thin Elastic Shells. New York: John Wiley & Sons, Inc. 19. C. L. DYM 1973 Journal of Sound and Vibration 29, 189-205. Some new results for the vibrations of circular cylinders. 20. L. B. W. JOLLEY 1961 Summation of Series. New York: Dover Publications, second edition revised. 21. H. CHUNG, P. TURULA, T. M. MULCAHY and J. A. JENDRZEJCZYK 1981 (to appear) Nuclear Engineering and Design. Analysis of a cylindrical shell vibrating in a cylindrical fluid region. 22. E. L. WILSON, K.-J. BATHE, F. E. PETERSON and H. H. DOVEY 1973 Nuclear Engineering and Design 25, 257-274. SAP-a structural analysis program for linear systems. 23. C. W. MCCORMICK 1972 The NASTRAN User’s Manual (level 15). NASA publications.
APPENDIX
I: FIRST
SET
(CSS) OF MODAL DISPLACEMENT THEIR DERIVATIVES
Aon + : A,, l?l=l
cos
cos nt3,
FUNCTIONS
OGXGl,
AND
VIBRATION
OF CIRCULAR
CYLINDRICAL
Amnm sin 7
u,x= - 7 El
0
cos ne,
~,,(o,e) = -(T2/21)n, cos ne, ?T 2 I&+c& -+ 1 2 (>[
U,XX= -
m 1 {tiO+J~(-l)“’
ITI=1
u(x, f3) = ? B,,
m=l
~(0, e) = -(7~/2)~
O
m7rx cos cos ne,
-m*A,,}
mrx sin sin 1
sin ne,
tie,
O
v(l, e) = (n/2)2+ sin ne,
u.,,(O, e) = -(7~~/21*)27~sin ne,
mrrx 3
I1
1
O[lJ-
w,,,(O,
W[ cos ne,
mrrx cos cos no, 1 I
{wo+ wl(-1)” + mC,,}
m7Tx worn + w[m(-1)” + m2Cmn} sin __ cos ne, I
*
e) = -(~~/2/*)6~
773 l&+G, -----+ w,,,, = 1 2 (>[ -worn*-
cos ne,
m=l
+ m4C,,}
E w,,,,,(O, e) = W. cos ne,
al = (di TR a3=--..1
USED
vo*, l+u -_p 2
O
w,,,(l, e) = (7~~/21*)G~cos ne,
rnrx wlm2(-l)m -m3Cmn} cos 1
II: SYMBOLS
3
OGXSi,
m c {&)+lq-l)m
+ worn3 + wrm3(-l)m
APPENDIX
O
O
w (z, e) = cT/2)
m=l
w,,, = - -
OSXSl,
u,,,(I, e) = (7~~/21*)25sin t2e;
m7rx
f
sin ne,
m7rx sin sin 4,
W(X,8) = f C,, sin cos ne, 1 Wl=l w. cos ne,
OSXGI:
1 I
uom + vrm(--1)” +m2B,,}
w (0, e) = +/2)
341
~,,(f, e) = cT2/2z)fi1 cos ne,
{vo+vl(-l)“‘+mBmn}cosT v,,, = -- T * (HI
SHELLS
ne,O~x~l, Icos
1Icos
m7rx
sin -
ne,
O
w,,,,,(I, e) = R, cos ne.
IN EQUATIONS
azl=f(l-v)(l+k/4)n*,
(14)-(28)
AND APPENDIX
a2=azI-II,
(1-v), I
v-ykn*
III
348
H. CHUNG a.51 = (1 + k)n*, a9
=
a6=a61-fi?,
u7
al0 = 2(?rR/l)*kn*,
(7rR/1)4k,
q1=yF(l-?k)n,
q6
=
all=1+kn4-0,
q2 =
q4 =
-(a9m2
45 =
v)kn*,
s11= alm2+a21-0, s22 =a5m2+a61-f&
a, = ba
(s22s33 =
(sns33
-
s;3
S*3
)/n,n,
a,
-S:3)/0mn,
=
= b,
=
APPENDIX
a3m,
a7m + aa, -
s13 = 4
s33=a9m
sl2s33)/bnr -
ql3
=
a,
slls23)/~mn,
a4m,
+aIom =
c,
(s12s23 =
el.2
=
el,4=
-I_+ 2a2
f m=l
m=l
elp5=~+ f IqlU, +q2apm ITI=1
2
?I m=l
2
e1,7
=
Q,(-l)m,
g
c+
=
ff+
e2.2
=
+ q5apm
+
q5apm
e2.6
e3,3
=
=
e1,5,
f m=l
e3,5
=
=
f m=l
e2.7
c,m*,
? m=l
ha,m
ha,m
+q3a,m},
+
+ q6a,.m},
;
q&m}(-1)”
e2.4= el.3,
e2,3= el.4,
el,l,
e3.6
{q4a,
f m=l
2
{q4a,
? m=l
a,m(-l)m,
+q2apm +q3a,m}(-l)“,
{qla,
2
el,8
-
=
e2.8
el.8,
e3,4=
f m=l
e2.5= =
el.7;
cym2(-l)“,
+ q2bvm2 + q3c,m}, +q2b,m2+q3cym}(-I)“,
e1.6,
0;
s13s22)/k1,
DETERMINANT
m
C qm,
e1,6=$+
+a12-
(slls22-s:2)/0m”,
m=l
=
2
=
III: ELEMENTS OF FREQUENCY
el.3
q23n,
qz3 = -(rR/l)*vkn; s12 =
(s12sl3
CL”
- v)kn ,
2
(sl3s23
-a7,
2
q12 = (~R/l)*(2
41s = 2(7rR/Z)*(l-
a12=l+kn4;
-a5,
-
+ ql2),
Qg = (1 + kn2)n,
(?rR/r)*$(3 - v)kn,
=
VIBRATION e3,7=$+
OF CIRCULAR ?
CYLINDRICAL
349
SHELLS
{q4a,m+q5b~m2+q6c,m2+1},
m=l
es,s=$+
?
(q4a,m-q5b,m2+q6c,m2+1}(-1)“;
m=l e4.4
=
2
q2
2
2
e5.5 = e+-+
e3.3,
*
1
m=l
e4.5
=
e3.6,
{qfaa + qzbpm* 2
2a2
q1q4
q3
2a2
m=l
+q6kTl%m q1q4
e5,a = -+yf 2a2
O”
1
43
+
=
e3,5,
e4.7
q&m* + 2qIq2apm
=
e3.8,
e4.8
=
e3,7;
+ 2q2q3bvm2 + 2qlq3a,m
00
e5,6=41+9z+
f3.7 = -+y+
e4,6
2
1
m=~
{qfa,
{q4(qlarr +qzapm
+q:bpm2+q:c,mL
+q3a,m)+q5(qtapm
+q2bpm2+q3bvm2)
+q2b,m2+q3c,m2)+q3}, m 1 m=l
+q3bvm2)+q6(q~a,m
{44(41a,
+ q&m
+
q3a,m) + q5(qlLygm + q2bam2
+q~b~rn*+q~c,m*)+q~}(-1)“‘; e6,6= e5.5,
e6.8 = e5.7;
e6.7 = e5.8,
2 e7,7=E-@+
!Z
2
2
+46(44%m
4: e7.8 =G-y+
m=l
{q4(q4aa+q5apm+q6a,m)+q5(q4aam+q5bam2+q6b,m2)
+45bvm2+q6cym2)
- a9m2-q18},
{q 4(q 4ffe +q5apm f WI=1
+q6(q4aym
+q6a,m)
+q5(q4asm
+q5bym2+q6cym2)-a9m2-qIs}(-l)m; e8.8 = e7,7.
APPENDIX
IV: NOMENCLATURE
thickness, radius and length of shell Fourier component in axial direction number of axial mode number number of circumferential waves axial, circumferential and radial displacement axial, and circumferential co-ordinates time unspecified end values Fourier coefficient for u, v and w extensional rigidity, Hz/(1 - v’) Young’s modulus shear modulus flexural rigidity, Eh ‘/ 12( 1 - v’) bending and twisting moment per unit length membrane force per unit length effective membrane shear force per unit length effective transverse shear force per unit length
+q5bpm2+q6b,m2)
+ q2),
350 2 k’ Y P 462,Icl”,4L w WI
R
H. CHUNG
l/c& = pR2(1 -2)/E h2/12R2 Poisson’s ratio mass density axial mode shapes corresponding to axial, circumferential, and radial directions, respectively radian frequency of shell lowest extensional frequency of a ring in plain strain (=JE/bR*(l - v”)}) frequency parameter (=o’/~J:)
FREE
VIBRATION
ANALYSIS
CYLINDRICAL
OF
CIRCULAR
SHELLS
H. CHUNG Components Technology Division, Argonne National Laboratory, Argonne, Illinois 60439,
1I.S.A.
(Received 3 September 1979, and in revised form 14 July 1980)
A general analytical method is presented for evaluating the free vibration characteristics of a circular cylindrical shell with classical boundary conditions of any type. The solution is obtained through a direct solution procedure in which Sanders’ shell equations are used with the axial modal displacements represented as simple Fourier series expressions. Stokes’ transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. An explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the method is checked against available data. The method is used to find the modal characteristics of the thermal liner model of the U.S. Fast Test Reactor (FTI?). The numerical results obtained are compared with finite element method solutions. 1. INTRODUCTION The free vibration problems of thin circular cylindrical shells have been of great interest to many structural engineers in recent years. Many investigations following the pioneering work of Arnold and Warburton [l, 21 have been summarized by Leissa [3]. Methods of solution for shell vibrations have ranged from the approximate energy methods used by Arnold and Warburton [l, 21 and by Sharma and Johns [4,5] to the exact solutions as studied by Forsberg [6-81, Warburton [9], Warburton and Higgs [ 101 and Goldman [l 11. Recently, the author [12, 131 used a Rayleigh-Ritz procedure in conjunction with Lagrange multipliers to study, mainly, the free vibration characteristics of constrained cylindrical shells. Simple Fourier series were used for the modal displacement functions along the axial direction and an explicit expression of the frequency equation was obtained. In this paper, the shell equations are solved directly with these Fourier series, and an exact frequency equation is obtained. Linear, elastic, first order shell theory due to Sanders [14,15] is employed in this paper. The modal displacement functions are expanded in Fourier series in the circumferential direction. The problem is then considered on a mode-by-mode basis associated with the circumferential Fourier index n. In the axial direction, the displacement functions are constructed to form a set of complete orthogonal functions in the form of simple Fourier series which are not required to satisfy the boundary conditions term-by-term. Stokes’ transformation [16, 171 is used to obtain correct series expressions for derivatives of the Fourier series. The boundary conditions which are not satisfied directly by the assumed series are enforced. The unwanted boundary conditions, that are originally unspecified but are implied by the nature of the assumed series, are released to remain unspecified. Through this process, all the boundary conditions of the problem become satisfied by the assumed displacement function in an overall sense. It should be emphasized that the present method can solve problems with any kind of boundary conditions. The direct solution method used here differs from those of other authors [6-111 in the selection of 331 0022-469X/81/030331+20$02.00/0
@ 1981 Academic
Press Inc. (London)
limited
332
H. CHUNG
general solutions of shell equations. They have chosen exponential functions for the modal displacements along the axial direction, substituted them into the equations of motion and then enforced the eight specified boundary conditions. This leads to an eighth-order algebraic equation and an eighth-order frequency determinant which are coupled together. The simultaneous solution of these two systems of equations involves extremely laborious computation as pointed out by Leissa [18]. In the method presented here, the use of simple Fourier series along the axial direction leads to an explicit expression for the frequency determinant, which is of low order (s 4), and this greatly simplifies the solution procedure. The numerical results from this method are checked with the available exact solutions reported by other authors. The natural frequencies of the thermal liner model of the U.S. Fast Test Reactor (FTR) are calculated and discussed. In particular, the variation of natural frequencies due to the change of boundary conditions is discussed in detail. 2. THEORETICAL
BACKGROUND
2.1. EQUATIONS OF MOTION The following shell equations due to Sanders [14, 151 are used:
,e -kw 1-v -Rku,x~ 2
3-u + ~Ru,, -Y~-R’~~J,,,o
+V,O+ w +k{R4W,,,,,
,888 = Y
2
V,m
+2R2w,xx00 + w,eeee-v,eee1 = -7
2
W,rr.
(1)
Here y2 = pR2(1 - y2)/E and k = h2/12R2. The geometry of a circular cylindrical shell is shown in Figure 1. Also shown on the figure are the forces and moments acting on a section of the shell parallel to the coordinate lines. A list of nomenclature is given in Appendix IV. The natural or geometric boundary conditions applicable to Sanders’ theory reduce to prescribing the following quantities at the ends of the cylindrical shell: I?*, or
N, or u,
V x =o,
6
1
QXorw,
M,org
1.
(2)
i
Here & = aM,/ax + (2/R)a&/aO.
$x0 = NXe+ (3/2R)tixe,
(3)
The effective forces per unit length gX and Q, are shown in Figure 1, The relationships between boundary forces and displacements are .&
u,,+iV,e+i~],
xe
3-v
=D(l-v) 2
1 [z(l-ik)
1
2-v
I
where D =Eh/(l-
v2) and K =Eh3/12(1
-
v2).
u,e+(l+$k)v,,-3kw,~e],
M, =
(4)
VIBRATION
OF
Figure
CIRCULAR
1. Cylindrical
CYLINDRICAL
SHELLS
333
shell with applied forces.
2.2. MODAL FUNCTIONS A general relation for the displacements in any mode of free vibrations may be written in the following form for any circumferential mode number n : u(x, 8, t) = &(x)
cos nt9 sin wf,
0(x, 8, t) = &(x) sin no sin wt,
w(x, 8, t) = $,(x) cos n@sin wt.
(5)
Here 1,4,,1,4”and I& are mode functions corresponding to axial, tangential and radial displacements, respectively. The crucial part of the analysis involves choosing appropriate series forms for these mode functions. The series should be simple in form and at the same time preserve orthogonality properties. It is not necessary that the series satisfy any particular boundary conditions since one is dealing with a general solution at this point. There are two convenient sets of Fourier series that meet all these requirements for the mode functions along the axial direction. The first set, designated “CSS”, can be written in the form
~,$,(x)=Ao,+ f A,,cos~, l?l=l
m=l
I&(X)=
f m=l
C,,siny,
(6)
334
H. CHUNG
while the second set, designated “SCC”, can be written as
vhCx)=m~l&. sin?, +bw(x)=Con+
l)U(X)=Bo”+
f
B,,
m=l
f WI=1
cosy?
c,,cos~.
(7)
The first set represents the exact solution for the shell when it has simply supported ends with no axial constraint (SNA-SNA). This “SNA” shell has boundary conditions at each end of the form N, =O,
A4, =o.
w = 0,
v =o,
(8)
The “CSS” set satisfies these boundary conditions term-by-term. The second set represents the exact solution for the shell when it has freely supported ends with no tangential constraint (FSNT-FSNT). This “FSNT” shell has boundary conditions at each end of the form u =o,
i& = 0,
awlax= 0.
&=O,
(9)
The “SCC” set satisfies the above boundary conditions term-by-term. By using Stokes’ transformation, it is possible to use these sets to find exact solutions for shell problems with any possible combination of homogeneous end conditions. There are sixteen possible sets of homgeneous boundary conditions that can be specified independently at each end [12, 131. 2.3. STOKES’ TRANSFORMATION Consider a function f(x) represented by a Fourier sine series in the open range 0
f(O)=fo,
mm
f(l) =fl,
f(x)= f a, sin m=l
O
I
(10)
’
Since it is not certain that the derivative f’(x) can be represented by term-by-term differentiation of the sine series, the derivative is instead represented by an independent cosine series of the form f’(x)=bo+
f m=l
b,cosyF 1
(11)
Stokes’ transformation then consists of integrating by parts in the basic definitions of the coefficients to obtain the relationship between b, and a,,,, as follows: 2 ’ b, = f’(x) cos ydx 1 I0 =
=;
f(x) cos ~]~+~~o’f(x)sin~dx
WO[(-l)“fi -f01+(m7dOam.
(12)
Similar care must be taken when finding the correct sine series corresponding complete set of formulas for the sine series is then written as f (0) = fo, fyx)
=
-f9-
f(l) =fl, mg,
mrrx a, sin 1,
f(x) = z
Ocx
m=l
mvx [~VO-(-lYfJ--~mam
tof”(x). The
I
I
cos-7-’
0~x61,
VIBRATION
OF CIRCULAR
f’W)=f&
CYLINDRICAL
335
SHELLS
f”(l) =fY,
p(x)=(y)m[, m[?lfo-(-l)mf~}-~mu,]
sin?,
O
(13)
If the function f(x) is represented by a Fourier cosine series, similar transformation formulas must be used to obtain the correct form of the successive derivatives of the series. These formulae for the sine and cosine series are used in the direct solution procedure of the shell equations. 3. GENERAL FORMULATIONS One can use either set, (6) or (7), of mode functions as a general solution of the shell equations (l), and can obtain an exact particular solution for a given shell with specific boundary conditions. In this paper, details are given only for the solution in which the first set (6) of mode functions, CSS, is used. However, results for frequencies are shown based on both sets of axial mode functions. The first set, CSS, identically satisfies all the boundary conditions (both geometrical and natural), only for the SNA (simple support without axial constraint) shell. Therefore, substitution of this set into the equations of .motion (1) lead to the frequency equation for an SNA shell extensively studied by Dym [19]. For all shells with other boundary conditions, the frequency equation is determined from the specified boundary conditions. For maximum generality of the formulation, the FSNT (freely supported without tangential constraint at both ends) shell will serve as the base problem for CSS. All other shell problems with different boundary conditions are simplified special cases of this base problem. For this base problem, none of the eight boundary conditions are satisfied by CSS on a term-by-term basis. Thus the constraint conditions to satisfy these boundary conditions lead to an eight-by-eight frequency determinant. Solutions for shells with other boundary conditions lead to a smaller size determinant. It is interesting to note that for the second set, SCC, all the boundary conditions of the FSNT shell are satisfied on a term-by-term basis and this problem is therefore simple to solve. The most complicated problem for SCC is the SNA shell. A CSS solution for the free vibration characteristics of an FSNT shell will now be outlined. The CSS set is given in equations (6) and their derivatives are obtained through the use of Stokes’ transformation (see Appendix I). Substitution of the CSS set and their derivatives into the equations of motion (1) leads to an explicit relation for A,,, and, in addition, a set of equations in which A,,, B,,, and C,, are coupled together. The solution of these coupled equations allows all the coefficients to be written explicitly in terms of the eight unspecified quantities ~20,z&,UO,~1,wo, WI,Go, Er and the frequency parameter R, as follows:
Aon =
2(u
1_n)[al(ao+oi)-a3(l;o+cr)-aa(wo+wi)l,
f2
#
(14)
a213
21
a,
u~[~o+~r(-l)“]-u3[2)0+V~(-l)ml -uagm[uo+ u,(-l)“]
[~1=[~~rnH
:J
1
-u~m[?Jo+U~(-l)m]
--a4[wo+ w-1)“l -u,m[wo+
wl(-l)“l
-(ugm3+u~om)[wo+wl(-l)ml+u+z[~o+~r(-l)”l
cl,, 1.
# 0.
(15)
336
H. CHUNG
Here 0 = (o/we)* and CJ~= E/pR’(l - v*). Other symbols are listed in Appendix II. The equations (14) and (15) include the four unspecified quantities VO, q, w. and wI, which are directly related to the geometrical boundary values: vo = - 7T,;
ne
o(O,a
sl’ ne, u(L (9,
f-J1= 7T
wo=---&wKw)’ Wf=?rs;ne wu,fv
(16)
The other four quantities, Q0, I&,Q. and r&, are associated with the unspecified end forces N, and moments M, as follows (see equations (4)):
Equations (14) and (15) may now be rewritten in terms of the eight unspecified boundary quantities ~0, UI,WO, wi, fiz, I?_ fi! and Z@f,for a FSNT shell:
Aon =
1 41(~o+ur)+q4(Wo+w)-
urr
fJ f
C
2&-R)
a0
(1%
UT. 41cvo+(-l)“u,l
b6 b, symm.
a21,
II
c,_ 42dvo+(-l)“vll i 43~r~o+(-l)mufl
+
45Lwo + (-l)"Wfl
+qsm[wo+(-l)“w,l-
,O,“#O. 2R2(1 - V2) m[ii: Ehl
(20)
+ (-l)“ti;]
Here fro = NXKJ,6) * cos t2e
fi’ _ ’
x
NX(&!) ’
c0sne
(21) The boundary conditions appropriate to the FSNT shell are listed in equations (9). The geometric boundary conditions that must be satisfied are associated with u and ~w/&K. Hence for arbitrary IZit follows that I/Iu(O)= Aon + : A,.,, = 0, ??I=1
$I,(Z)=Ao,+
:
m=l
A,“(-I)m=O,
&(O)=?[Jy +m;l c {WO+(-l)mW~+mCmn} =o, 1
,;,l,=;[Jy+m;, 1
{Wo+(-l)mW~+mCm”}(-l)m
I
=o.
Cm
VIBRATION
OF CIRCULAR
CYLINDRICAL
337
SHELLS
The natural boundary conditions that must be forced to zero are fiXe = 0 and & = 0 at both ends. Substitution of the CSS set and their derivatives into these equations yields the following four equations: q1A,,+q~(y)+q3(-)
+ _,_, C [VIA,, +qz{~o+ ~(-1)”
+ m&z,}
+q3!wo+ wr(-1)“+mC,,}]=0, q~Ao, +qz(y)
+43(y)
123)
+ ,,_, C [qrA,,
+q3{wo+ wr(-1)” +mC,,}](-1)” q4Ao, +q5(y)
-q12(y)
+ qz{uo + ~(-1)”
+ mBmn} (24)
= 0,
+ag(F)
02 +
C ~q~Amn+~~~~~+~,(-l)~+m~mn~-~~~~~o+~~(-l)~+mCmn~ ??I=1
+ag{~,+~,I(-1)“-w~m2-W~m2(-l)m-Rz3Cmn}]=0,
(25)
llaAon+4s(~)-412(~)+~~(~) + :
[q~A,,+~~~~O+~r(-l)m+m~mn~-~~2~~O+~,(-~)m+~Cmr,~
Wl=l
+a~{~,o+~,(-l)m-wom2-w,m2(-1)“-m3C,,}](-1)”
=o.
(26)
The last two equations, (25) and (26), contain the inexpedient G+, and 6,. Therefore, equations (25) and (26) are rewritten in terms of i@ and &?f, as for the case of equations (14) and (15): 4~A~.+4.(~)-4is(~)-2R~~u2’(~0:8’)+~~~[q,A,., +~23m&n
+q3{UO+d-l)m
+~&,n}+~13~Cmn
+mC,“}m2-qq18{W~+W,(-l)m+mCmn}-
-C${wO+
Wl(-1)”
2R2;h-“2){A”
+n;l;(-l)mj]
=O,
(27)
4~A~.+43(~)-4~~(~)-2R~~Y2’(~0:s’)+~~l[4.Amn +423~~mn+43{~Of~I(-l)m+mB,“}+ql3mC~,-as(wO+wr(-l)m
x {ii?; +fi;
2R2&
wr(-1)” +mC,,}-
+mC,,}m2-q1*{W0+
(-l)“}]
V2) {j@
(-1)” = 0.
+#(-1)“)
(28)
The frequency determinant may now be constructed. Substitution of the relations (19) and (20) for the coefficrents A,,,, A,,, I?,, and C,, into the eight constraint conditions due to the geometric and the natural boundary conditions, expressions (22)-(24), (27) and (28), leads to the homogeneous matrix equation
[ei,jl{R7 *b R?, fi.L
uO9
z)l,
WO,
WI)=
=
(0)~
(29)
338
H. CHUNG
where i, j = 1,2, . . . , 8. It is interesting to compare this equation to the result previously obtained by the author [12, 131 through a Rayleigh-Ritz procedure in conjunction with Lagrange multipliers. By redefining the unspecified end forces EX, fiX as proportional to the Lagrange multipliers mp = AJTR, $I, = AJrrR, li?(t = AJRl and h?f, = A4/R1, equation (29) becomes identical to the matrix equation in references [12, 131. For a non-trivial solution of equation (29), the determinant of the matrix must vanish, lei,jI=
(30)
0,
resulting in a characteristic equation whose eigenvalues determine the natural frequencies of the shell. The corresponding eigenvectors determine the mode shapes. This frequency determinant is symmetric, and is based on the exact satisfaction of the boundary conditions. The natural frequency appears as a parameter (0 = (w/tic)*) in this equation. Each element of this frequency determinant is an infinite series, and these are listed in Appendix III. From the development of these equations it may be shown that there are two restrictions involved. The first restriction is n # 3(1 - v)(l + k/4)n *. For axisymmetric (n = 0) shell vibrations, this becomes R # 0, thereby invalidating expression (19) for rigid body motion. However, conditions associated with this rigid body motion are usually determined by inspection. For non-axisymmetric motion (n # O), the case of R = $(l - v)(l + k/4)n2 is automatically rejected on the grounds that it leads to rigid body motion with non-zero frequency. The second restriction, 0,” # 0, may be shown to imply that this CSS set cannot be used to solve the FSNT shell. This is not a serious drawback since it is easy to show that the second (SCC) set offers a very simple solution in this case.
4. FREQUENCY
DETERMINANT
In this section, the frequency determinant is derived for several cases of shell boundary conditions. The frequency determinant (30) is based on the boundary conditions of “freely supported with no tangential constraints” at both ends (FSNT-FSNT). None of the FSNT-FSNT boundary conditions are satisfied by the CSS set of displacement functions on a term-by-term basis. The requirement to satisfy these boundary conditions by the CSS set in an overall sense led to the eight-by-eight frequency determinant (30). From this determinant, one can also obtain the frequency equation for any combination of boundary conditions among sixteen possible cases [12, 131 without going through the detailed procedures required to formulate the corresponding algebraic eigenvalue problems. This can be done by deleting inapplicable rows and columns from the frequency determinant (30). Therefore, solutions for shells with other boundary conditions lead to a smaller size determinant than for the FSNT-FSNT shell. For certain boundary conditions, the second set, SCC, leads to a simpler frequency determinant than the CSS set. Indeed, if the order of the frequency determinant obtained by the use of one set is P the one obtained by the other set is 8-P for the same set of boundary conditions. Therefore, one can always obtain the exact frequency determinant with an order ~4 for any case of boundary conditions. 4.1.
SIMPLY
SUPPORTED
SHELL
WITH
NO
AXIAL
CONSTRAINTS
AT
BOTH
ENDS
(SNA-SNA)
The simplest case of boundary conditions is, of course, the SNA shell since the CSS set of axial mode functions satisfies all the boundary conditions, N,=O, on a term-by-term
v =o,
w = 0,
h4,=0,
x =o, 1,
(31)
basis. It should be noted that the boundary values U,fiXO,& and awlax
VIBRATION
OF CIRCULAR
CYLINDRICAL
339
SHELLS
at both ends are unspecified by this set. The frequency equation for the SNA shell is directly obtained from the equations of motion as a three-by-three determinant. The elements of the frequency determinant are no longer infinite series but reduce to algebraic terms. Past work on this problem includes that of Arnold and Warburton [l, 21 and Dym [193. The frequency equation is as follows: S 11
s12
S
s 22
s::
symm. 4.2.
SIMPLY
SUPPORTED
SHELL
= 0.
(32)
s33
WITH
AXIAL
CONSTRAINTS
AT
BOTH
ENDS
(SS-SSj
As a typical case involving the enforcement of geometric boundary conditions and the release of unwanted natural boundary conditions, consider the simply supported shell with axial constraints at both ends. The boundary conditions are u =o,
lJ =o,
w =o,
M,=O,
x =o, 1.
(33)
The CSS set automatically satisfies the last three equations and the unwanted natural boundary conditions N, = 0 at both ends (see equation (31)). The frequency equation can be found from the geometrical constraint conditions of u = 0 at both ends, with, at the same time, the end forces N, left unspecified. This is effectively done in equation (30) by retaining the rows and columns associated with fift and 6:. The end result is the exact two-by-two frequency determinant e1.1
el.2
symm.
e2.2
=
0.
(34)
I
The elements of this determinant are listed in Appendix III. It is interesting to note that the second set, SCC, would involve the enforcement and release of geometric and natural boundary conditions, and lead to a six-by-six determinant. 4.3.
SHELL
WITH
CLAMPED
ENDS
(C-C)
When the shell is clamped at both ends the boundary conditions are u =o,
U = 0,
w =o,
awlax = 0,
x =o, 1.
(35)
Among these conditions, the four geometric boundary conditions u = 0 and awlax = 0 at both ends are not automatically satisfied by the CSS set. Instead, this CSS set unnecessarily satisfies N, = 0 and M, = 0 at the ends. The frequency determinant is obtained from equation (30) by retaining the rows and columns associated with fi:, fi:, @ and Qk : e1.1
symm.
el,2
el.3
el.4
e2.2
e2,3
e2.4
e3,3
e3.4
(36)
e4.4
If the second set, SCC, is used, one would obtain a different four-by-four determinant.
frequency
4.4. SHELL WITH FREE ENDS (F-F) As a typical case involving the enforcement of natural boundary conditions and the release of unwanted geometrical boundary conditions consider a free-free shell with
340
H. CHUNG
boundary conditions N,=O,
&
= 0,
M,=O,
0X=0,
x =o, 1.
(37)
From the properties of CSS it is apparent that the tangential and radial displacements, u and w respectively, are identically zero at the ends. Therefore, the releasing procedure is required to remove these unwanted geometric boundary conditions. By defining u and w separately at the end points (17) and including these values in the subsequent differentiation via Stokes’ transformation, the exact frequency equation can be found from the enforcement of natural boundary conditions &., = 0 and Q, = 0 at the ends. The result is obtained from equation (30) by retaining the rows and columns associated with uo, vI, w. and wf: e5,5
e5.6
e5.7
e5.8
e6.6
e6,7
e6.8
e7,7
e-l;8
symm.
=
0.
(38)
e8.8
The use of the second set also leads to another four-by-four 4.5.
CLAMPED-FREE
SHELL
determinant.
(C-F)
The case of a clamped-free shell involves the enforcement and release of geometrical and natural boundary conditions. For this non-symmetric set of boundary conditions, u =o, N,=O,
v =o,
awlax
w =o,
fix* = 0,
= 0,
M,=O,
&=O,
x =o, x = 1,
the CSS set leads to a four-by-four frequency determinant. The conditions at x = 0 must be enforced with the release of the end force M,. The fix0 = 0 and & = 0 conditions at x = 1must be enforced with displacements v and w. Retaining the rows and columns in equation @, A.?:, VI and WIleads to el.1
el.3
e1,6
e1.8
e3.3
e3,6
e3,8
e6.6
e6.8
symm.
ZZ
5.1.
CONVERGENCE
AND
u = 0 and awlax = 0 and moment, N, and the release of the end (30) associated with
0.
(40)
e8.8
If the second set, SCC, were used, a different four-by-four be obtained. 5. DISCUSSION
(39)
frequency determinant
would
OF RESULTS
ACCURACY
The frequency equations derived in the preceding sections involve infinite series of algebraic terms. Before any numerical results are calculated, manipulations should be performed to make the rates of convergence of the sums of the infinite series terms as close to each other as possible. The convergence of certain sums can be accelerated by subtracting off a series with a known sum [20].
VIBRATION
OF CIRCULAR
CYLINDRICAL
341
SHELLS
As a typical example, consider the following sequence of manipulations series:
for a simple
Through the above process, the order of the series has been improved from m m2to m “I, and accordingly the rate of convergence will be greatly accelerated. The infinite series in the shell frequency determinant have been treated in the same way as in the above example. In this way, fast convergence has been obtained and it was possible to use the same upper limit for all series in the frequency determinant; typically an upper limit of 50 terms was used. Frequency results were obtained from the frequency determinant by substituting values of w/o,, and monitoring the determinant until it vanished. Convergence and accuracy of the present method is described in Tables 1 and 2. In order to facilitate comparison with the exact solutions reported by other contributors who used Fliigge’s and Donnell’s theory, the present method was also applied to Donnell’s equations. In general, the solution approaches the exact frequency from above as the number of terms included in the series increases. However, as shown in Table 1, the use of more than 50 terms did not improve the solutions significantly. Table 2 compares the results of the present method with the available exact solutions. In all cases the comparison shows TABLE
1
Convergence of natural frequency parameters, w/w0
Number of terms 5 7 10 20 30 40 50 100 150 200 300 500 Reference r191 Case Case Case Case
Case 1 _PPY Sanders
0.01523
Case 3
Case 2
Case 4
Donnell
Sanders
Donnell
Sanders
Donnell
Sanders
0.01555 0.01552 0.01550 0.01547 0.01545 0.01544 0.01542
0.0684 0.0681 0.06792 0.06767 0.06760 0.06758 0.06758 0.06757 0.06757 0.06757 0.06757 0.06757
0.3161 0.3134 0.3122 0.3119 0.3119 0.3119 0.3118 0.3118 0.3118 0.3118 0.3118 0.3118
0.3230 0.3203 0.3191 0.3188 0.3188 0.3188 0.3188 0.3188 0.3188 0.3188 0.3188 0.3188
0.3589 0.3586 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 0.3585 ___-
0.06757
0.3117t
0.3188
0.3583t
0.01520 0.01517 0.01515 0.01513 0.01512 0.01510 0.01509 0.01509 0.01509 0.01508 0.01508
0.01541 0.01541 0.01541 0.01541 0.01541
0.05880 0.05851 0.05823 0.05795 0.05788 0.05785 0.05784 0.05784 0.05784 0.05784 0.05784 0.05784
0.01508t
0*01541
0.057871
1: Clamped-clamped shell (l/R 2: Clamped-clamped shell (I/R 3: Clamped-clamped shell (I/R 4: Free-free shell (I/R = 5, R/h
= 10, R/h = 500, v = 0.3, n = 4). = 10, R/h = 20, v = 0.3, n = 2). = 2, R/h = 20, v = 0.3, n = 3). = 20, v = 0.3, n = 1).
t Solutions based on Fliigge’s theory [7, 81. $ No exact solutions are available from literature
Donnell
342
H. CHUNG TABLE 2 Comparison of natural frequency parameters, w/w0
Shell parameters n
m'
condition
'1fR
R/h
4 2 3 1 0 2 2 2 2 4 1 1 0 2 4
1 1 1 1 1 5 1 2 3 1 1 3 1 1 1
c-c c-c c-c
10 10 2 5 8.1 8.67 1.14 2.88 5.07 10 30 5 2n 2v 27F
500 20 20 20 20 500 20 20 20 500 500 500 10 10 10
F-F F-F F-F C-F C-F C-F ss-ss SNAT”-SNAT SNAT-SNAT SNA-SNA SNA-SNA SNA-SNA
Present method v’ 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 8:; 0.3 0.3
Otherst
Sanders
Donnell
Fliigge
Donnell
0.01508 0.05784 0.3118 0.3585 0.3669 0.4472 0.3076 0.3081 0.3079 0.01508 0.00702 0.3699 0.296081 0.095812 0.425921
0.01541 0.06757 0.3188 0.3590 0.295804 0.119730 0.454809
0.01508 0.05787 0.3117 0.3583 0.3671 0.4472 0.3082 0.3082 0.3082 0.01507 0~00700 0.3698 0.296081$ 0.0958121 0.425921t
0.01541 0.06757 0.3188 0.295804 0*119730 0.454808
-
t These results are due to Forsberg [6-81, Warburton [9, lo] and Dym [19]. $ These solutions are based on Sanders’ theoiy [ 191. 5 Simply supported without axial and tangential constraints.
excellent agreement for a wide variety of boundary conditions and geometric parameters. For shells clamped at both ends, the solutions of the present method with Donnell’s theory are the same as those reported by Forsberg [8]. The frequency parameters for the SNA-SNA shell for Sanders’ as well as Donnell’s theory are identical to the results given by Dym [19]. For other boundary conditions, F-F, C-F, SS-SS and SNAT-SNAT, the available solutions are all based on Fliigge’s shell theory. The differences between solutions obtained from different thin shell theories are usually small, as revealed in Table 2. Mode shapes for a particular natural frequency can always be investigated by substituting the calculated frequency parameter into the frequency matrix and finding the relative values of unspecified end forces and values, and finally evaluating the coefficients of the assumed modal displacement functions. Comparison of mode shapes between the present method and others is not attempted in this paper. VIBRATIONS OF FTR THERMAL LINER 5.2. APPLICATION: The thermal liner of the Fast Test Reactor (FTR) in the U.S.A. is a thin, circular cylindrical shell situated concentrically inside the main reactor vessel. The support conditions of the thermal liner may be best represented as rigidly fixed at the bottom and free at the top: clamped-free (C-F) boundary conditions. A series of experiments had been performed as a part of an effort to investigate the modal characteristics of the thermal liner [21]. The experimental model is approximately l/14 scale of the prototype thermal liner for the FTR, giving the following dimensions and material properties: thickness, h = 0.15 cm; radius (to the center of thickness), R = 21.62 cm; length, I= 51.12 cm; material density, p = 7492 kg/m3; Young’s modulus, E = 1.83 x 1O1l Pa; Poisson’s ratio, v = 0.3. The experimental model was intended to simulate a clamped-free shell; however, insufficient clamping (awl& Z 0) and imperfect axial constraint (u # 0) at the bottom has been judged a probable cause for disagreement between the experimental results and the analytical predictions of natural frequencies.
VIBRATION
OF CIRCULAR
CYLINDRICAL
SHELLS
343
Therefore, in this study the sensitivity of the natural frequencies to changes in the bottom-boundary conditions was investigated by using the present solution method. The accuracy of the solutions has been compared to the results obtained by two finite element computer codes: SAP IV [22] and NASTRAN [23]. The boundary conditions considered are as follows: (1) C-F: clamped-free; (2) SS-F: simply supported-free; (3) CNA-F: clamped with no axial constraint-free; (4) SNA-F: simply supported with no axial constraint-free. TABLE
3
Comparison of natural frequencies (in Hz) for the thermal liner model with clamped-free boundary conditions n
m’
Present method
SAP IV
1 1 1 1 1 1 1 1 1
855.10 403.72 223.34 171.77 199.16 268.86 361.92 472.54 599.03
-t
856.3
405.1 225.6 174.3 201.7 272.0 366.2 478.4
410.1 232.2 180.5 206.2 275.5 370.1 483.5 614.0
2 2 2 2 2 2 2
928.28 644.48 494.69 442.00 464.59 539.45 648.34
-
943.2 671.4 529.3 478.0 496.9 567.2 673.2
t Not calculated
5Oy.8 454.3 476.7 -
NASTRAN
in these cases.
In Table 3 the solutions of the present method are compared with the SAP IV and NASTRAN results for the thermal liner with clamped-free boundary conditions. The finite element grid used with the two computer codes is the same and consists of 10 divisions vertically and nine divisions over a quarter of the shell circumferentially. Apparently the present exact method always gives lower values compared to those given by the two finite element computer codes. The natural frequencies for the thermal liner model in vacuum have been calculated by the present method for the above four cases of boundary conditions, and the results are summarized in Table 4 and Figure 2. The frequencies become smaller as the boundary conditions change from Case (1) to Case (4) for a given pair of circumferential and axial mode number, n and m’. The effect of clamping (~w/&x = 0) can be evaluated by comparing the frequencies for the following two pairs of boundary conditions: C-F and SS-F, CNA-F and SNA-F. The natural frequencies for SS-F are smaller than those for C-F by less than about 1% ; thus they are not distinguishable in Figure 2. The differences between frequencies for CNA-F and SNA-F are also small (< 1*5%), except for the modes of n = 0 to 3 with m’ = 1. The influence of axial constraint (u = 0) is evident in Figure 2 and, in contrast to the effect of clamping, is significant throughout most of the region of interest. Note that the frequency curve for m’ = 1 is greatly shifted down for n < 5; moreover, the lowest natural frequencies for shell-type vibration (n 2 2) occur at n = 2 for CNA-F and
H. CHUNG
344
TABLE 4
Natural frequencies
(in Hz) of thermal liner model for various boundary conditions
n
m’
C-F
SS-F
0
1 1 1 1 1 1 1 1 1 1 1
1497.90 855.10 403.72 223.34 171.77 199.16 268.86 361.92 472.54 599.03 740.81
1497.90 853.85 402.59 222.61 171.34 198.63 268.72 361.82 472.45 598.94 740.77
0 12.50 27.79 61.81 112.92 179.68 261.69 358.81 470.97 598.15 740.32
0 21.57 58.93 111.32 178.68 261.03 358.34 470.63 597.89 740.12
2 2 2 2 2 2 2 2 2 2 2
2351.51 2318.98 1437.11 928.28 644.48 494.69 442.00 464.59 539.45 648.34 781.15
2346.69 2118.98 1435.35 925.39 641.43 491.99 439.89 463.06 538.31 647.41 780.37
1497.90 2077.48 1408.98 844.59 552.22 418.54 390.44 434.21 522.17 638.16 774.87
1497.90 2077.24 1407.79 842.44 548.87 414.14 385.77 430.04 518.70 635.28 772.44
3 3 3 3 3 3 3 3 3 3
3076.05 2487.60 1834.82 1367.64 1057.12 864.82 767.65 750.67 798.18 893.60
3070.64 2484.64 1830.35 1361.83 1050.67 858.43 761.85 745.75 7949 11 890.18
2723.53 2479.20 1811.52 1323.62 1001.38 808.93 720.27 715.49 774.21 877.72
2723.02 2476.84 1808.06 1318.80 995.01 801.14 711.69 707.05 766.53 871.04
1 2 3 4 5 6 7 8 9 10 0 1
2 3 4 5 6 7 8 9 10 1
2 3 4 : 7 8 9
10
CNA-F
SNA-F 0
in contrast to n = 4 for C-F and SS-F. However, for higher axial half-waves (m’ = 2,3), the differences are less than 20% throughout the entire range of circumferential wave number n. A comparative study of the analytically predicted and experimentally obtained natural frequencies can be found in reference [21]. SNA-F,
6. CONCLUSIONS
A general solution procedure has been presented for the free vibration of a circular cylindrical shell, and applied to study the free vibration characteristics of the thermal liner of the U.S. Fast Test Reactor. The present method offers distinct computational advantages over other exact solution methods [6-l 11. Table 5 compares the present method and other exact solution procedures. An uncoupled explicit expression of frequency determinant is obtained by directly solving Sanders’ shell equations. The frequency equation is obtained in the process of satisfying all boundary conditions exactly by the assumed simple
345
VIBRATION OF CIRCULAR CYLINDRICAL SHELLS
I2
- IO I” “0 t8 5 s ;
6
4
2
0 Figure
2. Natural
frequencies
2
4 Circumferential
of thermal
6 mode
liner model.
8
IO
no (n)
-,
C-F, SS-F; - - -, CNA-F, WA-F.
TABLE 5 Comparisons of the present method with the other exact solution method Items 1. Frequency determinant 2. Order of frequency determinant 3. Element of frequency determinant
: References
Present method Uncoupled explicit expression Varies (1 - 4) dependent on boundary conditions Infinite series (rate of convergence can be accelerated)
Others:
~-
Coupled with 8th order algebraic equation Fixed (8) Transcendental functions (also series in computer computation)
[6-111.
Fourier series. This frequency equation involves infinite series which are evaluated by the use of appropriate numerical techniques and a computer. The order of frequency determinant may vary from 1 to 8 depending on the boundary conditions to be satisfied. However, an appropriate choice of displacement functions (i.e., the CSS or SCC set) can always furnish a frequency determinant of low order (~4) for any given boundary conditions. The frequency determinant obtained by others [6-l l] is always eight-by-eight and coupled with eighth-order algebraic equations resulting from the shell equations. The elements of the frequency determinant consist of transcendental functions in place of the infinite series of the present method. However, it should be noted that the transcendental function is evaluated as a series in the present day computer. In conclusion, the method presented in this paper is a relatively simple, exact-solution method for the free vibration problem of circular cylindrical shells.
346
H. CHUNG ACKNOWLEDGMENT
This work was performed Energy, Division of Reactor
under the sponsorship of the United Research and Technology.
States Department
of
REFERENCES 1. R. N. ARNOLD and G. B. WARBURTON 1948 Proceedings ofthe Royal Society, London A197, 238-256. Flexural vibrations of the walls of thin cylindrical shells having freely supported ends. 2. R. N. ARNOLD and G. B. WARBURTON 1953 Proceedings of the Institution of Mechanical Engineers A167, 62-80. The flexural vibrations of thin cylinders. 3. A. W. LEISSA 1973 NASA SP-288. Vibration of shells. 4. C. B. SHARMA and D. J. JOHNS 1972 Journal of Sound and Vibration 25, 433-449. Free
vibration of cantilever circular cylindrical shells-a comparative study. 5. C. B. SHARMA 1974 Journal of Sound and Vibration 35, 55-76. Calculation of natural frequencies of fixed-free circular cylindrical shells. 6. K. FORSBERG 1964 American Institute of Aeronautics and Astronautics Journal 2,2150-2157. Influence of boundary conditions on the modal characteristics of thin cylindrical shells. 7. K. FORSBERG 1969 American Institute of Aeronautics and Astronautics Journal 7, 221-227. Axisymmetric and beam-type vibrations of thin cylindrical shells. 8. K. FORSBERG 1965 NASA CR-613. A review of analytical methods used to determine the modal characteristics of cylindrical shells. 9. G. B. WARBURTON 1965 Journal of Mechanical Engineering Science 7,399-407. Vibration of thin cylindrical shells. Natural 10. G. B. WARBURTON and J. HIGGS 1970 Journal of Sound and Vibration l&335-338. frequencies of thin cantilever cylindrical shells. 11. R. L. GOLDMAN 1974 American Institute of Aeronautics and Astronautics Journal 12, 1755-1756. Mode shapes and frequencies of clamped-clamped cylindrical shells. 12. H. CHUNG 1974 Ph.D. Thesis, Tufts University. A general method of solution for vibrations of cylindrical shells. 13. R. GREIF and H. CHUNG 1975 American Institute of Aeronautics and Astronautics Journal 13, 1190-l 198. Vibrations of constrained cylindrical shells. 14. J. L. SANDERS, JR 1959 NASA Report 24. An improved first approximation theory for thin shells. Prager 15. B. BUDIANSKY and J. L. SANDERS, JR 1963 Progress in Applied Mechanics-The Anniversary Volume 129-140. On the best first-order linear shell theory. 16. T. J. 1’~ BROMWICH 1955 An Introduction to the theory of Infinite Series. London: MacMillan Publications, second edition revised. 17. B. BUDIANSKY and R. C. DIPRIMA 1960 Journal of Mathematics and Physics 39,237-245. Bending vibrations of uniform twisted beams. 18. H. KRAUS 1967 Thin Elastic Shells. New York: John Wiley & Sons, Inc. 19. C. L. DYM 1973 Journal of Sound and Vibration 29, 189-205. Some new results for the vibrations of circular cylinders. 20. L. B. W. JOLLEY 1961 Summation of Series. New York: Dover Publications, second edition revised. 21. H. CHUNG, P. TURULA, T. M. MULCAHY and J. A. JENDRZEJCZYK 1981 (to appear) Nuclear Engineering and Design. Analysis of a cylindrical shell vibrating in a cylindrical fluid region. 22. E. L. WILSON, K.-J. BATHE, F. E. PETERSON and H. H. DOVEY 1973 Nuclear Engineering and Design 25, 257-274. SAP-a structural analysis program for linear systems. 23. C. W. MCCORMICK 1972 The NASTRAN User’s Manual (level 15). NASA publications.
APPENDIX
I: FIRST
SET
(CSS) OF MODAL DISPLACEMENT THEIR DERIVATIVES
Aon + : A,, l?l=l
cos
cos nt3,
FUNCTIONS
OGXGl,
AND
VIBRATION
OF CIRCULAR
CYLINDRICAL
Amnm sin 7
u,x= - 7 El
0
cos ne,
~,,(o,e) = -(T2/21)n, cos ne, ?T 2 I&+c& -+ 1 2 (>[
U,XX= -
m 1 {tiO+J~(-l)“’
ITI=1
u(x, f3) = ? B,,
m=l
~(0, e) = -(7~/2)~
O
m7rx cos cos ne,
-m*A,,}
mrx sin sin 1
sin ne,
tie,
O
v(l, e) = (n/2)2+ sin ne,
u.,,(O, e) = -(7~~/21*)27~sin ne,
mrrx 3
I1
1
O[lJ-
w,,,(O,
W[ cos ne,
mrrx cos cos no, 1 I
{wo+ wl(-1)” + mC,,}
m7Tx worn + w[m(-1)” + m2Cmn} sin __ cos ne, I
*
e) = -(~~/2/*)6~
773 l&+G, -----+ w,,,, = 1 2 (>[ -worn*-
cos ne,
m=l
+ m4C,,}
E w,,,,,(O, e) = W. cos ne,
al = (di TR a3=--..1
USED
vo*, l+u -_p 2
O
w,,,(l, e) = (7~~/21*)G~cos ne,
rnrx wlm2(-l)m -m3Cmn} cos 1
II: SYMBOLS
3
OGXSi,
m c {&)+lq-l)m
+ worn3 + wrm3(-l)m
APPENDIX
O
O
w (z, e) = cT/2)
m=l
w,,, = - -
OSXSl,
u,,,(I, e) = (7~~/21*)25sin t2e;
m7rx
f
sin ne,
m7rx sin sin 4,
W(X,8) = f C,, sin cos ne, 1 Wl=l w. cos ne,
OSXGI:
1 I
uom + vrm(--1)” +m2B,,}
w (0, e) = +/2)
341
~,,(f, e) = cT2/2z)fi1 cos ne,
{vo+vl(-l)“‘+mBmn}cosT v,,, = -- T * (HI
SHELLS
ne,O~x~l, Icos
1Icos
m7rx
sin -
ne,
O
w,,,,,(I, e) = R, cos ne.
IN EQUATIONS
azl=f(l-v)(l+k/4)n*,
(14)-(28)
AND APPENDIX
a2=azI-II,
(1-v), I
v-ykn*
III
348
H. CHUNG a.51 = (1 + k)n*, a9
=
a6=a61-fi?,
u7
al0 = 2(?rR/l)*kn*,
(7rR/1)4k,
q1=yF(l-?k)n,
q6
=
all=1+kn4-0,
q2 =
q4 =
-(a9m2
45 =
v)kn*,
s11= alm2+a21-0, s22 =a5m2+a61-f&
a, = ba
(s22s33 =
(sns33
-
s;3
S*3
)/n,n,
a,
-S:3)/0mn,
=
= b,
=
APPENDIX
a3m,
a7m + aa, -
s13 = 4
s33=a9m
sl2s33)/bnr -
ql3
=
a,
slls23)/~mn,
a4m,
+aIom =
c,
(s12s23 =
el.2
=
el,4=
-I_+ 2a2
f m=l
m=l
elp5=~+ f IqlU, +q2apm ITI=1
2
?I m=l
2
e1,7
=
Q,(-l)m,
g
c+
=
ff+
e2.2
=
+ q5apm
+
q5apm
e2.6
e3,3
=
=
e1,5,
f m=l
e3,5
=
=
f m=l
e2.7
c,m*,
? m=l
ha,m
ha,m
+q3a,m},
+
+ q6a,.m},
;
q&m}(-1)”
e2.4= el.3,
e2,3= el.4,
el,l,
e3.6
{q4a,
f m=l
2
{q4a,
? m=l
a,m(-l)m,
+q2apm +q3a,m}(-l)“,
{qla,
2
el,8
-
=
e2.8
el.8,
e3,4=
f m=l
e2.5= =
el.7;
cym2(-l)“,
+ q2bvm2 + q3c,m}, +q2b,m2+q3cym}(-I)“,
e1.6,
0;
s13s22)/k1,
DETERMINANT
m
C qm,
e1,6=$+
+a12-
(slls22-s:2)/0m”,
m=l
=
2
=
III: ELEMENTS OF FREQUENCY
el.3
q23n,
qz3 = -(rR/l)*vkn; s12 =
(s12sl3
CL”
- v)kn ,
2
(sl3s23
-a7,
2
q12 = (~R/l)*(2
41s = 2(7rR/Z)*(l-
a12=l+kn4;
-a5,
-
+ ql2),
Qg = (1 + kn2)n,
(?rR/r)*$(3 - v)kn,
=
VIBRATION e3,7=$+
OF CIRCULAR ?
CYLINDRICAL
349
SHELLS
{q4a,m+q5b~m2+q6c,m2+1},
m=l
es,s=$+
?
(q4a,m-q5b,m2+q6c,m2+1}(-1)“;
m=l e4.4
=
2
q2
2
2
e5.5 = e+-+
e3.3,
*
1
m=l
e4.5
=
e3.6,
{qfaa + qzbpm* 2
2a2
q1q4
q3
2a2
m=l
+q6kTl%m q1q4
e5,a = -+yf 2a2
O”
1
43
+
=
e3,5,
e4.7
q&m* + 2qIq2apm
=
e3.8,
e4.8
=
e3,7;
+ 2q2q3bvm2 + 2qlq3a,m
00
e5,6=41+9z+
f3.7 = -+y+
e4,6
2
1
m=~
{qfa,
{q4(qlarr +qzapm
+q:bpm2+q:c,mL
+q3a,m)+q5(qtapm
+q2bpm2+q3bvm2)
+q2b,m2+q3c,m2)+q3}, m 1 m=l
+q3bvm2)+q6(q~a,m
{44(41a,
+ q&m
+
q3a,m) + q5(qlLygm + q2bam2
+q~b~rn*+q~c,m*)+q~}(-1)“‘; e6,6= e5.5,
e6.8 = e5.7;
e6.7 = e5.8,
2 e7,7=E-@+
!Z
2
2
+46(44%m
4: e7.8 =G-y+
m=l
{q4(q4aa+q5apm+q6a,m)+q5(q4aam+q5bam2+q6b,m2)
+45bvm2+q6cym2)
- a9m2-q18},
{q 4(q 4ffe +q5apm f WI=1
+q6(q4aym
+q6a,m)
+q5(q4asm
+q5bym2+q6cym2)-a9m2-qIs}(-l)m; e8.8 = e7,7.
APPENDIX
IV: NOMENCLATURE
thickness, radius and length of shell Fourier component in axial direction number of axial mode number number of circumferential waves axial, circumferential and radial displacement axial, and circumferential co-ordinates time unspecified end values Fourier coefficient for u, v and w extensional rigidity, Hz/(1 - v’) Young’s modulus shear modulus flexural rigidity, Eh ‘/ 12( 1 - v’) bending and twisting moment per unit length membrane force per unit length effective membrane shear force per unit length effective transverse shear force per unit length
+q5bpm2+q6b,m2)
+ q2),
350 2 k’ Y P 462,Icl”,4L w WI
R
H. CHUNG
l/c& = pR2(1 -2)/E h2/12R2 Poisson’s ratio mass density axial mode shapes corresponding to axial, circumferential, and radial directions, respectively radian frequency of shell lowest extensional frequency of a ring in plain strain (=JE/bR*(l - v”)}) frequency parameter (=o’/~J:)