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Dynamic analysis of ring-stiffened circular cylindrical shells
Journal of Sound and Vibration (1981) 75(l), 1-15
DYNAMIC
ANALYSIS
CIRCULAR
OF RING-STIFFENED
CYLINDRICAL
SHELLS
D. E. BESKOS Department of Civil and Mineral Engineering, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A. AND
J. B. OATES Convair Division, General Dynamics Corporation, San Diego, California 92138,
U.S.A.
(Received 7 July 1980)
A numerical method is developed for the dynamic analysis of ring-stiffened circular cylindrical thin elastic shells. Only circular symmetric vibrations of the shell segments and radial and torsional vibrations of the rings are considered. The geometric and material properties of the shell segments and the rings may vary from segment to segment. Free vibrations or forced vibrations due to harmonic pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings. Forced vibrations due to transient pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings defined in the Laplace transform domain. The time domain response is then obtained by a numerical inversion of the transformed solution. The effect of external viscous or internal viscoelastic damping is also investigated by the proposed method. In all the cases, the dynamic problem is reduced to a static-like form and the “exact” solution of the problem is numerically obtained.
1. INTRODUCTION Cylinders stiffened with rings are used in many structural applications such as pipes conveying fluids or gases and various aerospace structures. A vibration analysis of these structures in problems of large dynamic pressures, resonance and noise control is very important. In particular, one is interested in the modal characteristics of these shells as well as their response to various dynamic loads. The free vibration analysis of linear elastic ring-stiffened cylinders has been dealt with by various authors. Bleich [l] approximately determined the frequencies of ring-stiffened cylinders corresponding to flexural modes of vibration in which the shell cross-section does not remain circular. Baron [2] dealt analytically with free circular symmetric vibrations of infinitely long, stiffened cylinders. Wah [3] treated, again analytically, by the finite difference calculus, the free circular symmetric vibration problem of ring-stiffened cylinders of finite length under various end boundary conditions and considered both radial and torsional ring motion. Wah [4] and Wah and Hu [5] later used the same analytical approach to study the general free vibration problems of these cylindrical shells, in an approximate and an exact way, respectively. A unified description of references [3-51 can be found in the book of Wah and Calcote [6]. Wilken and Soedel[7,8] considered exact and approximate ways of studying the modal characteristics of ring-stiffened cylinders with the aid of the receptance method, a dynamic flexibility technique. A more advanced treatment of free vibration analysis of ring-stiffened cylindrical shells can be found in the paper by Forsberg [9]. 00;?.-4&~x/81/050001+
15 $02.00/O
@ 1981 Academic Press Inc. (London) Limited
2
D. E. BESKOS
AND
J. B. OATES
In this paper free and forced vibrations of ring-stiffened, linear elastic, thin, circular cylindrical shells of finite length are studied on the basis of the following assumptions. (i) Only circular symmetric modes of vibration are considered so that the cross-section remains circular at all times. This case is of importance in problems of cylindrical vessels or pipes under uniform internal fluid pressure or in problems involving end-on loads, such as a missile re-entry. (ii) The rings experience radial and torsional motion only. No flexural motion of the rings is considered, and the centroidal line of the ring section experiences no deformation and the relative twist between the sections is zero during torsional ring motion. Furthermore, the ring eccentricity is assumed to be zero. (iii) The longitudinal inertia is considered negligible, an assumption which, after Baron [2], is justified when the spacing of the rings is less than or equal to the cylinder radius. These assumptions are the same as those made by Wah [3]. In this work, however, the geometric and material properties of the rings and the shell segments are not the same as in reference [3] but may vary from segment to segment and intermediate rigid or elastic supports of the shell system besides the end ones can exist. Furthermore, this study is not confined to free vibrations only [l-9], forced vibrations being considered as well. The effect of external viscous or internal viscoelastic damping is also studied. All these are accomplished by employing dynamic stiffness influence coefficients (dynamic stiffness approach), in the frequency domain for free vibrations or forced vibrations due to harmonic forces, and in the Laplace transform domain for general transient forced vibrations. These coefficients for the cylindrical elements? and the circular rings are constructed on the basis of displacement functions which represent the “exact” solution of the corresponding equations of motion in the frequency or the Laplace transform domain. Thus, use of these dynamic stiffness influence coefficients in-a finite element formulation reduces the dynamic problem to a static-like form and leads to its “exact” solution in numerical form. Of course, a numerical Laplace inversion of the transform solution is required to obtain the dynamic response in the time domain. This dynamic stiffness approach has been successfully used for free and forced vibrations of frameworks by utilizing stiffness coefficients defined in the frequency domain (see, e.g., references [lo-121) or the Laplace transform domain (see, e.g., references [13-171). The proposed numerical method of solution of dynamic problems of ring-stiffened cylindrical shells undergoing circular symmetric vibrations is illustrated by four numerical examples which also demonstrate the advantages of the method against other analytical or numerical methods. The same method can be easily extended to cover the more general case of flexural vibrations of ring-stiffened cylindrical shells and this is presently under investigation.
2. FREE
VIBRATION
PROBLEM
The free vibration problem involves finding the natural frequencies and the modal shapes of the ring-stiffened cylindrical shell structure of Figure l(a), which can be broken down to shell and ring elements with interaction forces acting on them as shown in Figure 1:b). One must start with the equations of motion for each of the constituent parts of the system. The equation of radially symmetric motion for the shell segment (r, r + 1) of Figure + A cylindrical
segment
may consist of one or more cylindrical
elements.
RING-STIFFENED
CYLINDRICAL
SHELLS
3
h
r;
-----_ m
8 ---(b)
Figure 1. Ring-stiffened, circular cylindrical shell. (a) General shell-ring geometry; (b) nodal shell and ring forces, mechanics convention; (c) nodal shell and ring forces, matrix convention.
l(b) under harmonic end forces is [ .8] D, a4w/ax4+(Eh/a2)w
+ph d2w/at2= 0,
(1)
where w = w (x, t) is the radial deflection, a function of the longitudinal distance x and the time t, E is the modulus of elasticity, p is the mass density of the material, h is the thickness of the shell, (Yis the mean radius of the shell and D, is the flexural rigidity of the shell given by D, = Eh3/12(1 -v’),
(2)
with v being the Poisson’s ratio. Assuming a solution of the form w(x, t) = W(x) e’“‘, where i=a obtains
(3)
an d o is the radian frequency, and substituting it into equation (l), one d4 W/dx4 - K4 W = 0,
(4)
where K4= (h/D&m’-E/a’).
(5)
The solution of equation (4) is . W(x) = A cash Kx +B sinh Kx + C cos Kx + D sin Kx,
(6)
4
D.
E. BESKOS
AND
J. B. OATES
where A, B, C and D are constants of integration. Using the displacement function W(x) given by expression (6) in conjunction with the relations [18] M(x) = -D,
O(x) = d W/dx,
F(x)=
d2 W/dx’,
-D,, d3WfdX3,
(7)
for the slope O(x), the bending moment M(x) and shear force Fix) per unit length of the shell circumference and with sign notation that of Figure l(bi, one can construct the dynamic stiffness influence coefficients (Dii) for a shell element (r, r + 1) of length b and nodal lines (circles) at x = 0 and x = b, by standard techniques. Thus, the nodal forcedisplacement relation, on the basis of the sign convention of Figure l(c), takes the following form for a shell element (I, r + 1) in circular symmetric harmonic motion:
[;J=J;;
;:;
“D::
D~~[;~LJ~
(8)
where Dll=
033 = NK*(s ch + c sh),
Dzl = -043 = NK(s sh),
022 = 044 = N(s ch - c sh), D31= -NK*(s
+ sh),
D3* = -D41=
NK(c-ch),
042 = -N(s-sh),
(9)
with N=D,K/(l-cch),
c = cos Kb,
s = sin Kb,
ch = cash Kb,
sh = sinh Kb. (10)
On the basis of the assumption (ii) mentioned in the introduction, the equations of radial and torsional motion of a circular ring element (Figure l(b)) are [19] (AE/a*)w, +Apo d2w,/dt2 = fr,
(EI/(r*)t% +p&
d*&/dt* = m,,
(11,121
respectively. In equations (11) and (12) w, and 8, are the radial deflection and torsional angle, respectively, of the ring of nodal line r, p. is the ring mass density, A is the cross-sectional area of the ring, I is the moment of inertia and 1, is the polar moment of inertia of the ring cross-section, and fr and m, are the external radial force and twisting moment, respectively, per unit length of the circumference at the node r. For harmonic nodal external forces
f, = F, the deformation
components
e’“‘,
m, = M, e’“‘,
(13)
are also harmonic, i.e., w, = W, e’“‘,
er = 0, e’“‘,
(14)
and equations (11) and (12) become [(AE/a2) -Ap,02]
W, = F,,
[(EI/a2) -poIpa2]Or
= M,,
(15,16)
indicating that the dynamic stiffness influence coefficients of a ring element are D, = [(AE/a2) -Apow2],
for radial and torsional motion, respectively.
D, =
W/a2)-d,w21,
(17918)
RING-STIFFENED
CYLINDRICAL
3
SHELLS
Thus, for a shell-ring structural system (Figure l(a)) under harmonic nodal forces, the equation of motion in the frequency domain is of the static-like form
[m~)l{w = m,
(19)
where {W} and {F} are the amplitudes of the nodal displacement and force vectors, respectively, and [D(W)] is the dynamic stiffness matrix of the system obtained by an appropriate superposition of the Dij, D, and D, coefficients of the various shell and ring elements of the system. For the free vibration problem one has {F} = 0 and equation (19) yields
(20)
[D(~)lIWl= KU. For equation (20) to have non-trivial solutions det [D(w)] = 0.
(21j
The values of o that satisfy equation (21) are the natural frequencies of the system, which are usually determined numerically due to the transcendental nature of the Dii coefficients. The method of “false position” is usually employed for this numerical determination [20,21]. Notice that equation (21) provides the “exact” values of the natural frequencies of the system in the framework of the theory used, because the Dij, D, and D, coefficients have been constructed on the basis of the exact solution of the corresponding governing equations of motion in the frequency domain. Furthermore, equation (21) provides all (infinitely many) the natural frequencies of the system due to the presence of the trigonometric and hyperbolic functions in the expressions for the Dij coefficients. The system modal shapes {W}i can be obtained from equation (20) for known values of Wi(i= 1,2,. . . , 001, and represent also the “exact” solution. For a single shell element, equation (6) can be written as (22)
W(x) = {&T{CL where {d}‘= {cash Kx, sinh Kx, cos Kx, sin Kx},
(23)
{Cl’= {A, B, C, DL
and T denotes transposition. The boundary conditions for the general case are W(0) = w,,
(d W/dx )(O) = O,,
W(b) = W+1,
(d W/dx)(b) = @+I.
(24)
When equation (22) is used in equations (24) they become I WI =
[mcl,
(25)
where
IWI’={W,
@, Wr+l, @+I],
:H]=[;ih
ih
-is
;I.
(26327)
Elimination of {C} between equations (22) and (25) yields W(x) = WTWl-lW’),
(28)
which expresses the radial deflection at any point x along the length of the shell element in terms of the nodal displacement vector {W} of the element, which is the element modal shape vector for a free vibration problem.
6
D. E. BESKOS
3. FORCED
AND
J. B. OATES
VIBRATION
PROBLEM
In the forced vibration problem of the shell-ring system of Figure l(a), the external dynamic forces may be either harmonic or general forcing functions of time. In the case of harmonic forces of the form {f) = {F} eiRr, the response of the system (steady-state) is {w} = {IV} e’“‘, with {W} obtained from expression (19) by a simple matrix inversion: i.e., {WI=
[mww.
(29)
In the case of general dynamic forces, there are basically two ways for determining the response. One way is to employ modal analysis in a manner analogous to that used for obtaining the dynamic response of frameworks [ll, 121. This method, however, requires prior knowledge of natural frequencies and modal shapes and always provides an approximate solution due to the finite number of the modes considered; furthermore, for shell problems, a great number of modes is usually required for acceptable results. The other way for obtaining the dynamic response is based on the application of the Laplace transform. This second approach, which is adopted in this paper, presents distinct advantages over the method of modal analysis, as will become apparent in the following. Application of Laplace transform with respect to time under zero initial conditions on equation (1) yields d4@/dx4 +4p4+ = 0,
(30)
where co
E(x,P)=
I
4P4 = (W~S)[(W~2) +Lv*1,
w (x, t) eepf dt,
0
and p is the Laplace transform parameter.
(31932)
The general solution of equation (30) is
fi(x,p)=eex(Acospx+Bsinpx)+e-P”(Ccos~x+Dsin~x),
(33)
where A, B, C and D are constants of integration. On the basis of the displacement function iit(x, p) given by expression (33) and by following a procedure quite analogous to that of the free vibration problem, the nodal force-displacement relation with sign convention that of Figure l(c), takes the following form in the Laplace transform domain for a shell element (r, r + 1) in circular symmetric motion:
(34)
where the overbars indicate Laplace transformed with respect to time quantities and the dynamic stiffness influence coefficients (D,) in the Laplace transform domain are given by 61, =fi33=2N/32(y2-4ysc+1),
D’21=-D43=Np[y2-2y(1-2s2)+1],
&=1544=zv(y2-4ycs-l), zY31 = -41vp2J&(s3
+ fi4*
s c2 + c) + (s3+ s c* -c)], = 2N<7[r(S3
+S C2-C)
rS32=2N@~~(S3+SC2+S)(l-~),
(35)
+ (S3 +S C* +C)],
with N=2D$/[y2-2r(l+2s2)+1],
y = cab,
s = sin pb,
c = cos pb.
(36)
RING-STIFFENED
CYLINDRICAL~
SHELLS
7
Application of the Laplace transform with respect to time under zero initial conditions on equations (11) and (12) eventually leads to the following expression for the dynamic stiffnesses of a ring element in radial and torsional motion defined in the Laplace domain: D, = [(AE/a*)
+&o~~l,
6, = [(EI/a*) +Ipp,,p*].
(37,3S)
Thus, for a shell-ring structural system (Figure l(a)) under general transient nodal forces, the equation of motion in the Laplace domain is of the static like form
ma@}
= PI,
(39)
where {G} and {F} are the nodal transformed displacement and force vectors, respectively: and [D(p)] is the transformed dynamic stiffness matrix of the system obtained by an appropriate superposition of the Dij, Dr and Dt coefficients of the various shell and ring elements of the system. Equation (39) is solved for {@} numerically for a sequence of values of the transform parameter p and the response {w} in the time domain is obtained by a numerical inversion of the transformed solution. Notice that if the dynamic forces are complicated functions of time, then their Laplace transform has to be determined numerically. A comprehensive account on direct and inverse numerical Laplace transform algorithms can be found in references [16, 171. Of course, the most important problem is the numerical inversion of Laplace transform and this is taken up in the next section. 4. NUMERICAL INVERSION OF LAPLACE TRANSFORM Letf(t) be a real function of r, withf(t) = 0 for t < 0. The Laplace transform f(p) of f(t) is defined by f(-(p) =
Ifff(r) e-“‘dt
(40)
and its inversion formula is given by (41) where t > 0 is arbitrary, but greater than the real part of all the singularities off(p) and p is a complex number with Re (p) 2 5 > 0. In many engineering applications f(p) is either too complicated to be inverted analytically by evaluating the integral (41) or is given in numerical form. In those cases a numerical inversion of the Laplace transform is imperative. References [16] and [17] discuss this problem in detail and investigate the accuracy, computational efficiency and range of applicability of eight methods of numerical Laplace inversion as applied to dynamic problems. Two of those methods have been used in this work and are briefly presented below. The first method is due to Papoulis [22] and determines{(t) in terms of the values off(p) on a finite sequence of equidistant points pn on the real axis by the formula f(4) = z C, sin (2n + l)$, “=O where N is a finite positive integer, the coefficients C,, are given in reference functions of the values of T(p) and i.j = cos-l (e-63,
(42) [22] as (43)
8
D. E. BESKOS
AND
J. B. OATES
where S 7 0 is a constant selected by the user. This method has the advantage that it works with real data and provides easy to use expressions for the evaluation of the Cm’s, thus requiring small amounts of computer time. Unfortunately, the accuracy of the method is acceptable only for early times and improvement of its accuracy by increasing N is not possible because the computation of the Cn’s becomes an unstable process for large N’s. The second method of inversion is due to Durbin [23] and combines both finite Fourier cosine and sine transforms to obtain the inversion formula f(r,)=2(e”“‘/T)[
-$Re{f([)}+Re
(1::
(A(a)+iB(n))Z”}],
(44)
where -
pn = 4 + in (2Ir/ T), J-1
n=0,1,2,3
,...,
= i, Z = ei2rr’N,
1= 0, 1,2,3, . . . ) L
N,
(45)
and f(t) is computed for N equidistant points ti = j At = jT/N, j = 0, 1,. . . , N - 1. It is suggested that for L x N ranging from 50 to 5000 one should select CT = 5-10 for good results, where T is the total time interval of interest. The computations involved in expression (44) are performed by employing the Fast Fourier Transform algorithm of Cooley and Tukey [24]. The accuracy of Durbin’s method [23] is very high, even for late times and its only disadvantage is the fact that it works with complex data (complex values of p) and is therefore more time consuming than the method of Papoulis [22].
5. EFFECT OF DAMPING The effect of both the external viscous damping and the internal viscoelastic damping on the dynamic response are considered in this work. In the case where there is external viscous damping, equations (l), (11) and (12) are replaced by D, a4w/ax4+(Eh/a2)w
+c,
aw/at+ph a2w/at2 = 0,
(46)
(AE/(u2)w, +c, dw,/dt +Ap,, d2wJdt2 = fi,
(47)
(EI/a2)& + cr d&/dt + p& d* 8,/dt* = m,,
(48)
respectively, where c,, cI and ct are the viscous damping coefficients for the shell, the radial ring and torsional ring motion, respectively. Application of the Laplace transform to equation (46) yields d4Q’ldX4+4&r?
= 0,
4P:
=(hlD,)[(E/a2)+pp2+c,plhl,
(49)
indicating that in the presence of external damping the Fiji coefficients of equation (35) are computed with & given by the second of equations (49) instead of equation (32). Similarly, application of the Laplace transform to equations (47) and (48) eventually leads to the following values of the coefficients Dr and 6,: 6, = [(AEla2) +Apop2 + crpl,
fit = [(EI/a*)
+ &pop2 + ctp].
(50751)
The case of internal viscoelastic damping is treated with the aid of the correspondence principle, as described, for example, in the book by Boley and Weiner [25]. Thus, on the
RING-STIFFENED
CYLINDRICAL
9
SHELLS
assumption that a material obeys the constitutive law of the Kelvin viscoelastic model sii = 2/.~(cii + f deij/dt),
(52)
i, j = 1,2,3,
where b is the shear modulus, f is the internal damping coefficient and the deviatoric stress and strain tensors Sij and eij, respectively, are defined in terms of the stress and strain tensors CQ and &ii,respectively, by Sij= ‘Tij- Sij(1/3)Vii,
(53)
eij=&ij_Sij(l/3)&ii,
with 6ij being Kronecker’s delta and repeated indices indicating summation, the correspondence principle can be stated as follows: the Laplace transform of the viscoelastic solution can be obtained from the Laplace transform of the elastic solution by replacing the: elastic constants k and A (Lame constants) by [25] A’=h+(2/3)(p-p’),
~‘=wu+fPL or
(54’1
the elastic constants E and v by E’ = EC1+fp)llIl+
(l/3)(1
v’=[3v-(1-2Y)fp]/[3+(1-2v)fp],
-2v)fpl,
(55)
where p is the Laplace transform parameter. If the observation is made that one can go from the Laplace domain to the frequency domain (compare, for example, equations (5) and (32)) by simply replacing p by iw, then the results of this section can also be extended to the case of steady-state vibrations. 6. NUMERICAL
6.1.
EXAMPLE
EXAMPLES
1
For the simply supported cylindrical shell of Figure 2(a), the natural frequencies and the response to internal suddenly applied pressure have been determined by the proposed method and compared with known analytic solutions.
0
I
2
I
I
6
I
,
,
I
I
3
4
5
6
7
8
9
IO
Time
I
I
I
I
I,,
II
12
13
14
15
,
16
I7
I8
I
19
(tom5 s)
Figure 2. (a) Simply supported, circular cylindrical shell under suddenly applied internal history of rotation B(O, t) of the shell; a--, analytical results; --V--, present approach -x--, present approach (Papoulis results).
pressure; (Durbin
(b) time results);
10
D. E. BESKOS
AND J. B. OATES
WI = WZ = 0, equation
After application of the boundary conditions
(20) becomes (56)
leading to the frequency equation D& - D;d = 0.
(57)
Use of equations (9), (10) and (5) permits one to solve equation (57) numerically for the first five frequencies o, which are given in Table 1 together with the corresponding values of w obtained from the analytic solution [26]
UW~2)1,
o’, = (llpMDs(m&)4+
m = 1,2,
. . . , co.
(58)
As expected, the two results in Table 1 are identical. TABLE
1
Natural frequencies (rad/s) of the shell of Figure 2(a) m
Analytical
1 2 3 4 5
66 66 66 66 67
Present
684 700 768 950 333
66 66 66 66 67
method 684 700 768 950 333
Consider now the response of the shell to a step pressure of intensity PO = 2000 psi. After application of the boundary conditions riir = E$ = 0 and use of the fixed end moment expressions of reference [18] in the Laplace domain, equation (39) becomes (59) leading to the following solution for I%:
The response f&(t) has been obtained by a numerical inversion of 8;(p) on the basis of the Papoulis and Durbin algorithms. The results are shown in Figure 2(b) together with the analytic solution e,(t) = [dw/ax],,,,, where N
w(x, t)=
1 m=1,3,s
sin-
mvx b
4P0 mTm2,hp (I-
C0SCf.&t),
(61)
was obtained in reference [26] by modal analysis and w,,, is given by equation (58). It was determined that N = 250 gives results accurate to the fourth significant digit. Figure 2(b) clearly shows that the results obtained by the method of Durbin are very accurate for short as well as long times, while those of Papoulis are accurate for short times only. The CP computer time (CDC Cyber 74 machine) was 1.096 s for Papoulis and 1.885 s for Durbin for 80 time steps and a total time interval of 0*0002 s.
RING-STIFFENED
CYLINDRICAL
11
SHELLS
6.2. EXAMPLE 2 For the cylindrical shell of Figure 3(a) with a step change in thickness, the dynamic response to an internal suddenly applied pressure has been determined by the proposed method and compared with the known analytic solution.
294 Q P c E -A
b
soI.6
-
I.2 -
Time (IO-‘s)
Figure 3. (a) Two piece circular cylindrical shell with step change in thickness under suddenly applied internal pressure; h=O.Sin; a=3.0in; b=20.212in; v=O.3; E=29x106psi; p=7+2463x10-41bs2/in4; (b) time history of stress r&(0, t) of the two piece shell: -0-, Suzuki results; -V-, present method (Durbin results’l; -x-, present method (Papoulis results).
The equation of motion of the free-free shell structure of Figure 3(a) in the Laplace transform domain is given by expression (39) where
with indices 1, 2 and 3 indicating nodal points, and EF}= (&Ip)I-1IP1,
-l/P:,
(-1IP1)-(1IP2)r
ww:)-ww~),
-1//32,1/&3;1=, (63)
with indices 1 and 2 indicating shell element numbers. The solution of equation (39) has been obtained numerically for a sequence of values of p (real p for Papoulis and complex p for Durbin algorithms) and inverted numerically by both the Papoulis and the Durbin algorithms to yield the time domain response. Figure 3(b) depicts the circumferential stress ~~(0, r) = Ew~(~)/(Y as obtained by the proposed method and compares it with the exact results of Suzuki [27], which were obtained by employing Laplace transform and complicated analytical-numerical computations lacking generality and restricted to the particular problem of interest. As can be seen from Figure 3(b), Durbin’s inversion algorithm, which required 2.196 s of CP computer time for 80 time steps, yields very accurate results for early as well as later times, while Papoulis’ method, which required 1.293 s, is good for early times only.
D. E. BESKOS
12 6.3.
AND
J. B. OATES
EXAMPLE
3 For the thin, elastic, ring-stiffened, simply supported, circular cylindrical shell of Figure 4 the natural frequencies have been computed by the proposed method and the results compared with those of Wah [3].
h
_”
IFigure 4. Simply supported, h = 0.25 in; p = 7.246377E-4 A = 0.02586 in2.
I
ring-stiffened, circular cylindrical shell in free motion. E = 29 000 000 psi; lb s2 ine4; v = 0.3; Z = 0.03541 in4; I,, = 0.07083 in4; 6 = 2.0687 in; a = 4.0 in;
After application of the boundary conditions WI = W7 = 0, the structural dynamic stiffness matrix [D(w)] in the frequency equation (21) consisting of coefficients of the type Dii, D, and D, as given by equations (9), (17) and (18), respectively, becomes of order 12 x 12. Notice that one half of the ring stiffnesses are applied at nodes 1 and 7 to conform with the assumptions set forth by Wah [3]. The determination of the roots of equation (21) has been done numerically, as explained in section 2, with the aid of a general computer program which also forms the matrix [D(w)] and which can be used for any shell-ring system with a finite number of shell segments with different geometric and material properties. A listing of that program can be found in reference [21]. The first seven natural frequencies of the structure of Figure 4, as well as of structures with the same geometrical and material properties but with 6,4, 3 and 2 shell segments, computed by the proposed method, are given in Table 2. The same table also provides for comparison the corTABLE
Natural frequencies
Number of segments
(radls) of the shells of Example
4
6 I
3
2
3
\PPF
Wah
Present
Wah
Present
Wah
Present
50795 51502
50795 51502
50262 51503
50262 51503
51503
51503
2
60 071
3 4 5 6 7
60071 61758 65289 68913 71734
61758 65289 68913 71734
60 071 63428 68913 72724 73974
63428 68913 72724 73974
1 Natural frequency number
2
65290 71734 73974 122030 125509
Wah 51503
60 071 65290 71734 73974 122030 125509
68913 73974 122030 129850 142430
Present
51530 68913 73974 122030 129850
RING-STIFFENED
CYLINDRICAL
‘13
SHELLS
responding results of Wah [3], obtained on the basis of the finite difference calculus. It is apparent that the present results are identical with those of Wah [3], which is to be expected since both methods are based on the exact solution of the equations of motion. The frequencies 60 071 rad/s and 142 430 rad/s were not recovered by the dynamic stiffness method due to the fact that Wah [3] considered active half rings at the end supports. 6.4. EXAMPLE 4 For a thin, elastic, ring-stiffened, simply supported, circular cylindrical shell, subjected to a suddenly applied internal pressure PO as shown in Figure 5(a), the effect of external viscous damping on the maximum midspan radial deflection has been studied. The geometric and material properties of the shell and ring elements are the same as those of Figure 4, and the end rings are half rings. The intensity of the internal suddenly applied pressure is PO= 1750 psi. A general computer program capable of treating any shell-ring system with a finite number of shell segments with different geometric and material properties was used for the computations. A listing of this program is available in reference
Pll.
6in
6in
Time (10e5 s)
Figure 5. Effect of external viscous damping shownintheinset.-,c=O;---,c=l;...,c=8;--,c=18.
on the time history
of the midspan
radial deflection
of the shell
It has been assumed that c, = c,/a = cJa3 = c and the deflection wZ(t) has been computed for c = 0, 1,8 and 18 lb s/in3; results are shown in Figure 5. Table 3 contains the maximum values of wZ(t) for values of c equal to 0, 1,2,4,8, 12,16 and 18 lb s/in3. Figure 5 and Table 3 clearly indicate that increased damping in the shell-ring system significantly reduces the response. This effect becomes more pronounced as c increases until a damping coefficient of c = 18 is reached for which the oscillatory effects are completely removed.
14
D. E. BESKOS AND J. B. OATES TABLE 3 Maximum
midspan radial deflection of the shell of
Figure 5 as affected by damping
Damping coefficient c (lb s/in3) 0
1 2 4 8 12 16 18
Maximum midspan radial deflection (in) 7.37 x 1o-3 6.80 x 1O-3 6.31 x 1O-3 5.53 x 1o-3 4.50 x 1o-3 3.92x 1O-3 3.66 x lo-? 3.61 x 1O-3
7. CONCLUSIONS On the basis of the preceding discussion one can draw the following conclusions. (1) Use of the dynamic stiffness method for the dynamic analysis of circular symmetrical vibrations of ring-stiffened cylindrical shells leads to reducing the dynamic problem to a static-like form and to obtaining, apart from errors due to numerical computations, the exact solution of the problem. No knowledge of natural frequencies and modal shapes is required in forced vibration problems solved by this method. (2) For free vibration problems or forced ones due to harmonic forces, in the dynamic stiffness method one utilizes D, coefficients defined in the frequency domain, while for general forced vibration problems, Fiji coefficients defined in the Laplace transform domain are used. Thus, in the latter case, the response is obtained by a numerical inversion of the transformed solution. (3) Two methods of Laplace transform inversion, namely those of Durbin and Papoulis, were used and checked in this work with respect to their accuracy and computational efficiency. Durbin’s method is more accurate for early as well as later times, while Papoulis’ method is good for early times only; however, Papoulis’ method requires much less CP computer time. (4) The effect of damping on the dynamic response of a shell-ring system can be very easily taken into account in the dynamic stiffness approach; thus, external viscous damping can be very easily incorporated into the Dij or bij coefficients, while internal viscoelastic damping can be accounted for by invoking the correspondence principle of linear viscoelasticity in the frequency or the Laplace domain. (5) The use of Dii and rSij coefficients with the dynamic stiffness approach allows for a dynamic analysis of more general problems of varied boundary conditions, varied dimensions and properties of ring and shell elements, and types of loading more easily than laborious analytical methods or other numerical techniques such as the conventional finite element analysis in conjunction with either nodal analysis or numerical integration. ACKNOWLEDGMENT The authors express their appreciation to the University of Minnesota Computer Center for making its facilities available to them. The second author also acknowledges the support of the General Dynamics Corporation.
RING-STIFFENED
CYLINDRICAL
SHELLS
15
REFERENCES 1. H. H. BLEICH 1961 Bsterreichisches Ingenieur-Archiv 15, 1-4. Approximate determination of the frequencies of ring stiffened cylindrical shells. 23,316-318. Circular symmetric vibrations 2. M. L. BARON 1958 JournalofAppliedMechanics of infinitely long cylindrical shells with equidistant stiffeners. 3. T. WAH 1964 Journal of the Society of Industrial and Applied Mathematics 12, 649-662. Circular symmetric vibrations of ring-stiffened cylindrical shells. 4, T. WAH 1966 Journal ofSound and Vibration 3, 242-251. Flexural vibration of ring-stiffened cylindrical shells. 5. T. WAH and W. C. L. HU 1968 Journal of the Acoustical Society of America 43, 1005-1016. Vibration analysis of stiffened cylinders including inter-ring motion. 6. T. WAH and L. R. CALCOTE 1970 Structure Analysis by Finite Difference Calculus. New York: j Van Nostrand Reinhold Co. 7. I. D. WILKEN and W. SOEDEL 1976 Journal of Sound and Vibration 44, 563-576. The receptance method applied to ring-stiffened cylindrical shells: analysis of modal characteristics. 8. I. D. WILKEN and W. SOEDEL 1976 Journal of Sound and Vibration 44, 577-589. Simplified prediction of the modal characteristics of ring-stiffened cylindrical shells. 9. K. FORSBERG 1969 AIAAfASME 10th Structures, Structural Dynamics and Materials Con .’ ference, New Orleans, Louisiana, 18-30. Exact solution for natural frequencies of ringstiffened cylinders. 10. R. W. CLOUGH and J. PENZIEN 1975 Dynamics of Structures. New York: McGraw-Hill. 1974 Computers and Structures 4, 1061-1089. Dynamics of frameworks by 11. B. A. OVUNK continuous mass method. 12. D. E. BESKOS 1979 Computers and Structures 10,785-795. Dynamics and stability of plane trusses with gusset plates. Use of dynamic 13. D. E. BESKOS and B. A. BOLEY 1975 Computers and Structures 5,263-269. influence coefficients in forced vibration problems with the aid of Laplace transform. 14. G. D. MANOLIS and D. E. BESKOS 1979 Developments in Mechanics 10,85-89. Dynamic response of beam structures with the aid of numerical Laplace transform. 15. G. D. MANOLIS and D. E. BESKOS 1980 Computer Methods in Applied Mechanics and Engineering 21, 337-355. Thermally induced vibrations of beam structures. 16. G. V. NARAYANAN 1979 Ph.D. Thesis, University of Minnesota, Minneapolis. Numerical operational methods in structural dynamics. 17. D. E. BESKOS 1980 Proceedings of the 2nd International Symposium on Innovative Numerical Analysis in Applied Engineering Science, Montreal, Canada, 16-20 June. Numerical operational methods for time-dependent linear problems. TheoryofPIatesandShells.New York: 18. S. TIMOSHENKO~~~~WOINOWSKY-KRIEGER~~~~ McGraw-Hill, second edition. D. H. YOUNG and W. WEAVER, JR 1974 Vibration Problems in 19. S. P. TIMOSHENKO, Engineering. New York: John Wiley & Sons, fourth edition. 20. H. KRAUS 1967 Thin Elastic Shells. New York: John Wiley & Sons. 21. J. B. OATES 1979 MS. Thesis, University of Minnesota, Minneapolis. Dynamic analysis of ring-stiffened circular cylindrical shells. 14,405-414. A new method of inversion 22. A. PAPOULIS 1957 QuarterlyofAppliedMathematics of the Laplace transform. 23. F. DURBIN 1974 The Computer Journal 17, 371-376. Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method. 24. J. W. COOLEY and J. W. TUKEY 1965 Mathematics of Computation 19, 297-301. An algorithm for the machine calculation of complex Fourier series. 25. B. A. BOLEY and J. H. WEINER 1960 Theory of ThermalStresses. New York: John Wiley & Sons. 26. C. L. DYM 1974 Introduction to the Theory of Shells. New York: Pergamon Press. 27. S. I. SUZUKI 1976 Journal of Sound and Vibration 44, 169-178. Dynamic behaviour of thin cylindrical shells with a step change in thickness subjected to inner impulsive loads.
DYNAMIC
ANALYSIS
CIRCULAR
OF RING-STIFFENED
CYLINDRICAL
SHELLS
D. E. BESKOS Department of Civil and Mineral Engineering, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A. AND
J. B. OATES Convair Division, General Dynamics Corporation, San Diego, California 92138,
U.S.A.
(Received 7 July 1980)
A numerical method is developed for the dynamic analysis of ring-stiffened circular cylindrical thin elastic shells. Only circular symmetric vibrations of the shell segments and radial and torsional vibrations of the rings are considered. The geometric and material properties of the shell segments and the rings may vary from segment to segment. Free vibrations or forced vibrations due to harmonic pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings. Forced vibrations due to transient pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings defined in the Laplace transform domain. The time domain response is then obtained by a numerical inversion of the transformed solution. The effect of external viscous or internal viscoelastic damping is also investigated by the proposed method. In all the cases, the dynamic problem is reduced to a static-like form and the “exact” solution of the problem is numerically obtained.
1. INTRODUCTION Cylinders stiffened with rings are used in many structural applications such as pipes conveying fluids or gases and various aerospace structures. A vibration analysis of these structures in problems of large dynamic pressures, resonance and noise control is very important. In particular, one is interested in the modal characteristics of these shells as well as their response to various dynamic loads. The free vibration analysis of linear elastic ring-stiffened cylinders has been dealt with by various authors. Bleich [l] approximately determined the frequencies of ring-stiffened cylinders corresponding to flexural modes of vibration in which the shell cross-section does not remain circular. Baron [2] dealt analytically with free circular symmetric vibrations of infinitely long, stiffened cylinders. Wah [3] treated, again analytically, by the finite difference calculus, the free circular symmetric vibration problem of ring-stiffened cylinders of finite length under various end boundary conditions and considered both radial and torsional ring motion. Wah [4] and Wah and Hu [5] later used the same analytical approach to study the general free vibration problems of these cylindrical shells, in an approximate and an exact way, respectively. A unified description of references [3-51 can be found in the book of Wah and Calcote [6]. Wilken and Soedel[7,8] considered exact and approximate ways of studying the modal characteristics of ring-stiffened cylinders with the aid of the receptance method, a dynamic flexibility technique. A more advanced treatment of free vibration analysis of ring-stiffened cylindrical shells can be found in the paper by Forsberg [9]. 00;?.-4&~x/81/050001+
15 $02.00/O
@ 1981 Academic Press Inc. (London) Limited
2
D. E. BESKOS
AND
J. B. OATES
In this paper free and forced vibrations of ring-stiffened, linear elastic, thin, circular cylindrical shells of finite length are studied on the basis of the following assumptions. (i) Only circular symmetric modes of vibration are considered so that the cross-section remains circular at all times. This case is of importance in problems of cylindrical vessels or pipes under uniform internal fluid pressure or in problems involving end-on loads, such as a missile re-entry. (ii) The rings experience radial and torsional motion only. No flexural motion of the rings is considered, and the centroidal line of the ring section experiences no deformation and the relative twist between the sections is zero during torsional ring motion. Furthermore, the ring eccentricity is assumed to be zero. (iii) The longitudinal inertia is considered negligible, an assumption which, after Baron [2], is justified when the spacing of the rings is less than or equal to the cylinder radius. These assumptions are the same as those made by Wah [3]. In this work, however, the geometric and material properties of the rings and the shell segments are not the same as in reference [3] but may vary from segment to segment and intermediate rigid or elastic supports of the shell system besides the end ones can exist. Furthermore, this study is not confined to free vibrations only [l-9], forced vibrations being considered as well. The effect of external viscous or internal viscoelastic damping is also studied. All these are accomplished by employing dynamic stiffness influence coefficients (dynamic stiffness approach), in the frequency domain for free vibrations or forced vibrations due to harmonic forces, and in the Laplace transform domain for general transient forced vibrations. These coefficients for the cylindrical elements? and the circular rings are constructed on the basis of displacement functions which represent the “exact” solution of the corresponding equations of motion in the frequency or the Laplace transform domain. Thus, use of these dynamic stiffness influence coefficients in-a finite element formulation reduces the dynamic problem to a static-like form and leads to its “exact” solution in numerical form. Of course, a numerical Laplace inversion of the transform solution is required to obtain the dynamic response in the time domain. This dynamic stiffness approach has been successfully used for free and forced vibrations of frameworks by utilizing stiffness coefficients defined in the frequency domain (see, e.g., references [lo-121) or the Laplace transform domain (see, e.g., references [13-171). The proposed numerical method of solution of dynamic problems of ring-stiffened cylindrical shells undergoing circular symmetric vibrations is illustrated by four numerical examples which also demonstrate the advantages of the method against other analytical or numerical methods. The same method can be easily extended to cover the more general case of flexural vibrations of ring-stiffened cylindrical shells and this is presently under investigation.
2. FREE
VIBRATION
PROBLEM
The free vibration problem involves finding the natural frequencies and the modal shapes of the ring-stiffened cylindrical shell structure of Figure l(a), which can be broken down to shell and ring elements with interaction forces acting on them as shown in Figure 1:b). One must start with the equations of motion for each of the constituent parts of the system. The equation of radially symmetric motion for the shell segment (r, r + 1) of Figure + A cylindrical
segment
may consist of one or more cylindrical
elements.
RING-STIFFENED
CYLINDRICAL
SHELLS
3
h
r;
-----_ m
8 ---(b)
Figure 1. Ring-stiffened, circular cylindrical shell. (a) General shell-ring geometry; (b) nodal shell and ring forces, mechanics convention; (c) nodal shell and ring forces, matrix convention.
l(b) under harmonic end forces is [ .8] D, a4w/ax4+(Eh/a2)w
+ph d2w/at2= 0,
(1)
where w = w (x, t) is the radial deflection, a function of the longitudinal distance x and the time t, E is the modulus of elasticity, p is the mass density of the material, h is the thickness of the shell, (Yis the mean radius of the shell and D, is the flexural rigidity of the shell given by D, = Eh3/12(1 -v’),
(2)
with v being the Poisson’s ratio. Assuming a solution of the form w(x, t) = W(x) e’“‘, where i=a obtains
(3)
an d o is the radian frequency, and substituting it into equation (l), one d4 W/dx4 - K4 W = 0,
(4)
where K4= (h/D&m’-E/a’).
(5)
The solution of equation (4) is . W(x) = A cash Kx +B sinh Kx + C cos Kx + D sin Kx,
(6)
4
D.
E. BESKOS
AND
J. B. OATES
where A, B, C and D are constants of integration. Using the displacement function W(x) given by expression (6) in conjunction with the relations [18] M(x) = -D,
O(x) = d W/dx,
F(x)=
d2 W/dx’,
-D,, d3WfdX3,
(7)
for the slope O(x), the bending moment M(x) and shear force Fix) per unit length of the shell circumference and with sign notation that of Figure l(bi, one can construct the dynamic stiffness influence coefficients (Dii) for a shell element (r, r + 1) of length b and nodal lines (circles) at x = 0 and x = b, by standard techniques. Thus, the nodal forcedisplacement relation, on the basis of the sign convention of Figure l(c), takes the following form for a shell element (I, r + 1) in circular symmetric harmonic motion:
[;J=J;;
;:;
“D::
D~~[;~LJ~
(8)
where Dll=
033 = NK*(s ch + c sh),
Dzl = -043 = NK(s sh),
022 = 044 = N(s ch - c sh), D31= -NK*(s
+ sh),
D3* = -D41=
NK(c-ch),
042 = -N(s-sh),
(9)
with N=D,K/(l-cch),
c = cos Kb,
s = sin Kb,
ch = cash Kb,
sh = sinh Kb. (10)
On the basis of the assumption (ii) mentioned in the introduction, the equations of radial and torsional motion of a circular ring element (Figure l(b)) are [19] (AE/a*)w, +Apo d2w,/dt2 = fr,
(EI/(r*)t% +p&
d*&/dt* = m,,
(11,121
respectively. In equations (11) and (12) w, and 8, are the radial deflection and torsional angle, respectively, of the ring of nodal line r, p. is the ring mass density, A is the cross-sectional area of the ring, I is the moment of inertia and 1, is the polar moment of inertia of the ring cross-section, and fr and m, are the external radial force and twisting moment, respectively, per unit length of the circumference at the node r. For harmonic nodal external forces
f, = F, the deformation
components
e’“‘,
m, = M, e’“‘,
(13)
are also harmonic, i.e., w, = W, e’“‘,
er = 0, e’“‘,
(14)
and equations (11) and (12) become [(AE/a2) -Ap,02]
W, = F,,
[(EI/a2) -poIpa2]Or
= M,,
(15,16)
indicating that the dynamic stiffness influence coefficients of a ring element are D, = [(AE/a2) -Apow2],
for radial and torsional motion, respectively.
D, =
W/a2)-d,w21,
(17918)
RING-STIFFENED
CYLINDRICAL
3
SHELLS
Thus, for a shell-ring structural system (Figure l(a)) under harmonic nodal forces, the equation of motion in the frequency domain is of the static-like form
[m~)l{w = m,
(19)
where {W} and {F} are the amplitudes of the nodal displacement and force vectors, respectively, and [D(W)] is the dynamic stiffness matrix of the system obtained by an appropriate superposition of the Dij, D, and D, coefficients of the various shell and ring elements of the system. For the free vibration problem one has {F} = 0 and equation (19) yields
(20)
[D(~)lIWl= KU. For equation (20) to have non-trivial solutions det [D(w)] = 0.
(21j
The values of o that satisfy equation (21) are the natural frequencies of the system, which are usually determined numerically due to the transcendental nature of the Dii coefficients. The method of “false position” is usually employed for this numerical determination [20,21]. Notice that equation (21) provides the “exact” values of the natural frequencies of the system in the framework of the theory used, because the Dij, D, and D, coefficients have been constructed on the basis of the exact solution of the corresponding governing equations of motion in the frequency domain. Furthermore, equation (21) provides all (infinitely many) the natural frequencies of the system due to the presence of the trigonometric and hyperbolic functions in the expressions for the Dij coefficients. The system modal shapes {W}i can be obtained from equation (20) for known values of Wi(i= 1,2,. . . , 001, and represent also the “exact” solution. For a single shell element, equation (6) can be written as (22)
W(x) = {&T{CL where {d}‘= {cash Kx, sinh Kx, cos Kx, sin Kx},
(23)
{Cl’= {A, B, C, DL
and T denotes transposition. The boundary conditions for the general case are W(0) = w,,
(d W/dx )(O) = O,,
W(b) = W+1,
(d W/dx)(b) = @+I.
(24)
When equation (22) is used in equations (24) they become I WI =
[mcl,
(25)
where
IWI’={W,
@, Wr+l, @+I],
:H]=[;ih
ih
-is
;I.
(26327)
Elimination of {C} between equations (22) and (25) yields W(x) = WTWl-lW’),
(28)
which expresses the radial deflection at any point x along the length of the shell element in terms of the nodal displacement vector {W} of the element, which is the element modal shape vector for a free vibration problem.
6
D. E. BESKOS
3. FORCED
AND
J. B. OATES
VIBRATION
PROBLEM
In the forced vibration problem of the shell-ring system of Figure l(a), the external dynamic forces may be either harmonic or general forcing functions of time. In the case of harmonic forces of the form {f) = {F} eiRr, the response of the system (steady-state) is {w} = {IV} e’“‘, with {W} obtained from expression (19) by a simple matrix inversion: i.e., {WI=
[mww.
(29)
In the case of general dynamic forces, there are basically two ways for determining the response. One way is to employ modal analysis in a manner analogous to that used for obtaining the dynamic response of frameworks [ll, 121. This method, however, requires prior knowledge of natural frequencies and modal shapes and always provides an approximate solution due to the finite number of the modes considered; furthermore, for shell problems, a great number of modes is usually required for acceptable results. The other way for obtaining the dynamic response is based on the application of the Laplace transform. This second approach, which is adopted in this paper, presents distinct advantages over the method of modal analysis, as will become apparent in the following. Application of Laplace transform with respect to time under zero initial conditions on equation (1) yields d4@/dx4 +4p4+ = 0,
(30)
where co
E(x,P)=
I
4P4 = (W~S)[(W~2) +Lv*1,
w (x, t) eepf dt,
0
and p is the Laplace transform parameter.
(31932)
The general solution of equation (30) is
fi(x,p)=eex(Acospx+Bsinpx)+e-P”(Ccos~x+Dsin~x),
(33)
where A, B, C and D are constants of integration. On the basis of the displacement function iit(x, p) given by expression (33) and by following a procedure quite analogous to that of the free vibration problem, the nodal force-displacement relation with sign convention that of Figure l(c), takes the following form in the Laplace transform domain for a shell element (r, r + 1) in circular symmetric motion:
(34)
where the overbars indicate Laplace transformed with respect to time quantities and the dynamic stiffness influence coefficients (D,) in the Laplace transform domain are given by 61, =fi33=2N/32(y2-4ysc+1),
D’21=-D43=Np[y2-2y(1-2s2)+1],
&=1544=zv(y2-4ycs-l), zY31 = -41vp2J&(s3
+ fi4*
s c2 + c) + (s3+ s c* -c)], = 2N<7[r(S3
+S C2-C)
rS32=2N@~~(S3+SC2+S)(l-~),
(35)
+ (S3 +S C* +C)],
with N=2D$/[y2-2r(l+2s2)+1],
y = cab,
s = sin pb,
c = cos pb.
(36)
RING-STIFFENED
CYLINDRICAL~
SHELLS
7
Application of the Laplace transform with respect to time under zero initial conditions on equations (11) and (12) eventually leads to the following expression for the dynamic stiffnesses of a ring element in radial and torsional motion defined in the Laplace domain: D, = [(AE/a*)
+&o~~l,
6, = [(EI/a*) +Ipp,,p*].
(37,3S)
Thus, for a shell-ring structural system (Figure l(a)) under general transient nodal forces, the equation of motion in the Laplace domain is of the static like form
ma@}
= PI,
(39)
where {G} and {F} are the nodal transformed displacement and force vectors, respectively: and [D(p)] is the transformed dynamic stiffness matrix of the system obtained by an appropriate superposition of the Dij, Dr and Dt coefficients of the various shell and ring elements of the system. Equation (39) is solved for {@} numerically for a sequence of values of the transform parameter p and the response {w} in the time domain is obtained by a numerical inversion of the transformed solution. Notice that if the dynamic forces are complicated functions of time, then their Laplace transform has to be determined numerically. A comprehensive account on direct and inverse numerical Laplace transform algorithms can be found in references [16, 171. Of course, the most important problem is the numerical inversion of Laplace transform and this is taken up in the next section. 4. NUMERICAL INVERSION OF LAPLACE TRANSFORM Letf(t) be a real function of r, withf(t) = 0 for t < 0. The Laplace transform f(p) of f(t) is defined by f(-(p) =
Ifff(r) e-“‘dt
(40)
and its inversion formula is given by (41) where t > 0 is arbitrary, but greater than the real part of all the singularities off(p) and p is a complex number with Re (p) 2 5 > 0. In many engineering applications f(p) is either too complicated to be inverted analytically by evaluating the integral (41) or is given in numerical form. In those cases a numerical inversion of the Laplace transform is imperative. References [16] and [17] discuss this problem in detail and investigate the accuracy, computational efficiency and range of applicability of eight methods of numerical Laplace inversion as applied to dynamic problems. Two of those methods have been used in this work and are briefly presented below. The first method is due to Papoulis [22] and determines{(t) in terms of the values off(p) on a finite sequence of equidistant points pn on the real axis by the formula f(4) = z C, sin (2n + l)$, “=O where N is a finite positive integer, the coefficients C,, are given in reference functions of the values of T(p) and i.j = cos-l (e-63,
(42) [22] as (43)
8
D. E. BESKOS
AND
J. B. OATES
where S 7 0 is a constant selected by the user. This method has the advantage that it works with real data and provides easy to use expressions for the evaluation of the Cm’s, thus requiring small amounts of computer time. Unfortunately, the accuracy of the method is acceptable only for early times and improvement of its accuracy by increasing N is not possible because the computation of the Cn’s becomes an unstable process for large N’s. The second method of inversion is due to Durbin [23] and combines both finite Fourier cosine and sine transforms to obtain the inversion formula f(r,)=2(e”“‘/T)[
-$Re{f([)}+Re
(1::
(A(a)+iB(n))Z”}],
(44)
where -
pn = 4 + in (2Ir/ T), J-1
n=0,1,2,3
,...,
= i, Z = ei2rr’N,
1= 0, 1,2,3, . . . ) L
N,
(45)
and f(t) is computed for N equidistant points ti = j At = jT/N, j = 0, 1,. . . , N - 1. It is suggested that for L x N ranging from 50 to 5000 one should select CT = 5-10 for good results, where T is the total time interval of interest. The computations involved in expression (44) are performed by employing the Fast Fourier Transform algorithm of Cooley and Tukey [24]. The accuracy of Durbin’s method [23] is very high, even for late times and its only disadvantage is the fact that it works with complex data (complex values of p) and is therefore more time consuming than the method of Papoulis [22].
5. EFFECT OF DAMPING The effect of both the external viscous damping and the internal viscoelastic damping on the dynamic response are considered in this work. In the case where there is external viscous damping, equations (l), (11) and (12) are replaced by D, a4w/ax4+(Eh/a2)w
+c,
aw/at+ph a2w/at2 = 0,
(46)
(AE/(u2)w, +c, dw,/dt +Ap,, d2wJdt2 = fi,
(47)
(EI/a2)& + cr d&/dt + p& d* 8,/dt* = m,,
(48)
respectively, where c,, cI and ct are the viscous damping coefficients for the shell, the radial ring and torsional ring motion, respectively. Application of the Laplace transform to equation (46) yields d4Q’ldX4+4&r?
= 0,
4P:
=(hlD,)[(E/a2)+pp2+c,plhl,
(49)
indicating that in the presence of external damping the Fiji coefficients of equation (35) are computed with & given by the second of equations (49) instead of equation (32). Similarly, application of the Laplace transform to equations (47) and (48) eventually leads to the following values of the coefficients Dr and 6,: 6, = [(AEla2) +Apop2 + crpl,
fit = [(EI/a*)
+ &pop2 + ctp].
(50751)
The case of internal viscoelastic damping is treated with the aid of the correspondence principle, as described, for example, in the book by Boley and Weiner [25]. Thus, on the
RING-STIFFENED
CYLINDRICAL
9
SHELLS
assumption that a material obeys the constitutive law of the Kelvin viscoelastic model sii = 2/.~(cii + f deij/dt),
(52)
i, j = 1,2,3,
where b is the shear modulus, f is the internal damping coefficient and the deviatoric stress and strain tensors Sij and eij, respectively, are defined in terms of the stress and strain tensors CQ and &ii,respectively, by Sij= ‘Tij- Sij(1/3)Vii,
(53)
eij=&ij_Sij(l/3)&ii,
with 6ij being Kronecker’s delta and repeated indices indicating summation, the correspondence principle can be stated as follows: the Laplace transform of the viscoelastic solution can be obtained from the Laplace transform of the elastic solution by replacing the: elastic constants k and A (Lame constants) by [25] A’=h+(2/3)(p-p’),
~‘=wu+fPL or
(54’1
the elastic constants E and v by E’ = EC1+fp)llIl+
(l/3)(1
v’=[3v-(1-2Y)fp]/[3+(1-2v)fp],
-2v)fpl,
(55)
where p is the Laplace transform parameter. If the observation is made that one can go from the Laplace domain to the frequency domain (compare, for example, equations (5) and (32)) by simply replacing p by iw, then the results of this section can also be extended to the case of steady-state vibrations. 6. NUMERICAL
6.1.
EXAMPLE
EXAMPLES
1
For the simply supported cylindrical shell of Figure 2(a), the natural frequencies and the response to internal suddenly applied pressure have been determined by the proposed method and compared with known analytic solutions.
0
I
2
I
I
6
I
,
,
I
I
3
4
5
6
7
8
9
IO
Time
I
I
I
I
I,,
II
12
13
14
15
,
16
I7
I8
I
19
(tom5 s)
Figure 2. (a) Simply supported, circular cylindrical shell under suddenly applied internal history of rotation B(O, t) of the shell; a--, analytical results; --V--, present approach -x--, present approach (Papoulis results).
pressure; (Durbin
(b) time results);
10
D. E. BESKOS
AND J. B. OATES
WI = WZ = 0, equation
After application of the boundary conditions
(20) becomes (56)
leading to the frequency equation D& - D;d = 0.
(57)
Use of equations (9), (10) and (5) permits one to solve equation (57) numerically for the first five frequencies o, which are given in Table 1 together with the corresponding values of w obtained from the analytic solution [26]
UW~2)1,
o’, = (llpMDs(m&)4+
m = 1,2,
. . . , co.
(58)
As expected, the two results in Table 1 are identical. TABLE
1
Natural frequencies (rad/s) of the shell of Figure 2(a) m
Analytical
1 2 3 4 5
66 66 66 66 67
Present
684 700 768 950 333
66 66 66 66 67
method 684 700 768 950 333
Consider now the response of the shell to a step pressure of intensity PO = 2000 psi. After application of the boundary conditions riir = E$ = 0 and use of the fixed end moment expressions of reference [18] in the Laplace domain, equation (39) becomes (59) leading to the following solution for I%:
The response f&(t) has been obtained by a numerical inversion of 8;(p) on the basis of the Papoulis and Durbin algorithms. The results are shown in Figure 2(b) together with the analytic solution e,(t) = [dw/ax],,,,, where N
w(x, t)=
1 m=1,3,s
sin-
mvx b
4P0 mTm2,hp (I-
C0SCf.&t),
(61)
was obtained in reference [26] by modal analysis and w,,, is given by equation (58). It was determined that N = 250 gives results accurate to the fourth significant digit. Figure 2(b) clearly shows that the results obtained by the method of Durbin are very accurate for short as well as long times, while those of Papoulis are accurate for short times only. The CP computer time (CDC Cyber 74 machine) was 1.096 s for Papoulis and 1.885 s for Durbin for 80 time steps and a total time interval of 0*0002 s.
RING-STIFFENED
CYLINDRICAL
11
SHELLS
6.2. EXAMPLE 2 For the cylindrical shell of Figure 3(a) with a step change in thickness, the dynamic response to an internal suddenly applied pressure has been determined by the proposed method and compared with the known analytic solution.
294 Q P c E -A
b
soI.6
-
I.2 -
Time (IO-‘s)
Figure 3. (a) Two piece circular cylindrical shell with step change in thickness under suddenly applied internal pressure; h=O.Sin; a=3.0in; b=20.212in; v=O.3; E=29x106psi; p=7+2463x10-41bs2/in4; (b) time history of stress r&(0, t) of the two piece shell: -0-, Suzuki results; -V-, present method (Durbin results’l; -x-, present method (Papoulis results).
The equation of motion of the free-free shell structure of Figure 3(a) in the Laplace transform domain is given by expression (39) where
with indices 1, 2 and 3 indicating nodal points, and EF}= (&Ip)I-1IP1,
-l/P:,
(-1IP1)-(1IP2)r
ww:)-ww~),
-1//32,1/&3;1=, (63)
with indices 1 and 2 indicating shell element numbers. The solution of equation (39) has been obtained numerically for a sequence of values of p (real p for Papoulis and complex p for Durbin algorithms) and inverted numerically by both the Papoulis and the Durbin algorithms to yield the time domain response. Figure 3(b) depicts the circumferential stress ~~(0, r) = Ew~(~)/(Y as obtained by the proposed method and compares it with the exact results of Suzuki [27], which were obtained by employing Laplace transform and complicated analytical-numerical computations lacking generality and restricted to the particular problem of interest. As can be seen from Figure 3(b), Durbin’s inversion algorithm, which required 2.196 s of CP computer time for 80 time steps, yields very accurate results for early as well as later times, while Papoulis’ method, which required 1.293 s, is good for early times only.
D. E. BESKOS
12 6.3.
AND
J. B. OATES
EXAMPLE
3 For the thin, elastic, ring-stiffened, simply supported, circular cylindrical shell of Figure 4 the natural frequencies have been computed by the proposed method and the results compared with those of Wah [3].
h
_”
IFigure 4. Simply supported, h = 0.25 in; p = 7.246377E-4 A = 0.02586 in2.
I
ring-stiffened, circular cylindrical shell in free motion. E = 29 000 000 psi; lb s2 ine4; v = 0.3; Z = 0.03541 in4; I,, = 0.07083 in4; 6 = 2.0687 in; a = 4.0 in;
After application of the boundary conditions WI = W7 = 0, the structural dynamic stiffness matrix [D(w)] in the frequency equation (21) consisting of coefficients of the type Dii, D, and D, as given by equations (9), (17) and (18), respectively, becomes of order 12 x 12. Notice that one half of the ring stiffnesses are applied at nodes 1 and 7 to conform with the assumptions set forth by Wah [3]. The determination of the roots of equation (21) has been done numerically, as explained in section 2, with the aid of a general computer program which also forms the matrix [D(w)] and which can be used for any shell-ring system with a finite number of shell segments with different geometric and material properties. A listing of that program can be found in reference [21]. The first seven natural frequencies of the structure of Figure 4, as well as of structures with the same geometrical and material properties but with 6,4, 3 and 2 shell segments, computed by the proposed method, are given in Table 2. The same table also provides for comparison the corTABLE
Natural frequencies
Number of segments
(radls) of the shells of Example
4
6 I
3
2
3
\PPF
Wah
Present
Wah
Present
Wah
Present
50795 51502
50795 51502
50262 51503
50262 51503
51503
51503
2
60 071
3 4 5 6 7
60071 61758 65289 68913 71734
61758 65289 68913 71734
60 071 63428 68913 72724 73974
63428 68913 72724 73974
1 Natural frequency number
2
65290 71734 73974 122030 125509
Wah 51503
60 071 65290 71734 73974 122030 125509
68913 73974 122030 129850 142430
Present
51530 68913 73974 122030 129850
RING-STIFFENED
CYLINDRICAL
‘13
SHELLS
responding results of Wah [3], obtained on the basis of the finite difference calculus. It is apparent that the present results are identical with those of Wah [3], which is to be expected since both methods are based on the exact solution of the equations of motion. The frequencies 60 071 rad/s and 142 430 rad/s were not recovered by the dynamic stiffness method due to the fact that Wah [3] considered active half rings at the end supports. 6.4. EXAMPLE 4 For a thin, elastic, ring-stiffened, simply supported, circular cylindrical shell, subjected to a suddenly applied internal pressure PO as shown in Figure 5(a), the effect of external viscous damping on the maximum midspan radial deflection has been studied. The geometric and material properties of the shell and ring elements are the same as those of Figure 4, and the end rings are half rings. The intensity of the internal suddenly applied pressure is PO= 1750 psi. A general computer program capable of treating any shell-ring system with a finite number of shell segments with different geometric and material properties was used for the computations. A listing of this program is available in reference
Pll.
6in
6in
Time (10e5 s)
Figure 5. Effect of external viscous damping shownintheinset.-,c=O;---,c=l;...,c=8;--,c=18.
on the time history
of the midspan
radial deflection
of the shell
It has been assumed that c, = c,/a = cJa3 = c and the deflection wZ(t) has been computed for c = 0, 1,8 and 18 lb s/in3; results are shown in Figure 5. Table 3 contains the maximum values of wZ(t) for values of c equal to 0, 1,2,4,8, 12,16 and 18 lb s/in3. Figure 5 and Table 3 clearly indicate that increased damping in the shell-ring system significantly reduces the response. This effect becomes more pronounced as c increases until a damping coefficient of c = 18 is reached for which the oscillatory effects are completely removed.
14
D. E. BESKOS AND J. B. OATES TABLE 3 Maximum
midspan radial deflection of the shell of
Figure 5 as affected by damping
Damping coefficient c (lb s/in3) 0
1 2 4 8 12 16 18
Maximum midspan radial deflection (in) 7.37 x 1o-3 6.80 x 1O-3 6.31 x 1O-3 5.53 x 1o-3 4.50 x 1o-3 3.92x 1O-3 3.66 x lo-? 3.61 x 1O-3
7. CONCLUSIONS On the basis of the preceding discussion one can draw the following conclusions. (1) Use of the dynamic stiffness method for the dynamic analysis of circular symmetrical vibrations of ring-stiffened cylindrical shells leads to reducing the dynamic problem to a static-like form and to obtaining, apart from errors due to numerical computations, the exact solution of the problem. No knowledge of natural frequencies and modal shapes is required in forced vibration problems solved by this method. (2) For free vibration problems or forced ones due to harmonic forces, in the dynamic stiffness method one utilizes D, coefficients defined in the frequency domain, while for general forced vibration problems, Fiji coefficients defined in the Laplace transform domain are used. Thus, in the latter case, the response is obtained by a numerical inversion of the transformed solution. (3) Two methods of Laplace transform inversion, namely those of Durbin and Papoulis, were used and checked in this work with respect to their accuracy and computational efficiency. Durbin’s method is more accurate for early as well as later times, while Papoulis’ method is good for early times only; however, Papoulis’ method requires much less CP computer time. (4) The effect of damping on the dynamic response of a shell-ring system can be very easily taken into account in the dynamic stiffness approach; thus, external viscous damping can be very easily incorporated into the Dij or bij coefficients, while internal viscoelastic damping can be accounted for by invoking the correspondence principle of linear viscoelasticity in the frequency or the Laplace domain. (5) The use of Dii and rSij coefficients with the dynamic stiffness approach allows for a dynamic analysis of more general problems of varied boundary conditions, varied dimensions and properties of ring and shell elements, and types of loading more easily than laborious analytical methods or other numerical techniques such as the conventional finite element analysis in conjunction with either nodal analysis or numerical integration. ACKNOWLEDGMENT The authors express their appreciation to the University of Minnesota Computer Center for making its facilities available to them. The second author also acknowledges the support of the General Dynamics Corporation.
RING-STIFFENED
CYLINDRICAL
SHELLS
15
REFERENCES 1. H. H. BLEICH 1961 Bsterreichisches Ingenieur-Archiv 15, 1-4. Approximate determination of the frequencies of ring stiffened cylindrical shells. 23,316-318. Circular symmetric vibrations 2. M. L. BARON 1958 JournalofAppliedMechanics of infinitely long cylindrical shells with equidistant stiffeners. 3. T. WAH 1964 Journal of the Society of Industrial and Applied Mathematics 12, 649-662. Circular symmetric vibrations of ring-stiffened cylindrical shells. 4, T. WAH 1966 Journal ofSound and Vibration 3, 242-251. Flexural vibration of ring-stiffened cylindrical shells. 5. T. WAH and W. C. L. HU 1968 Journal of the Acoustical Society of America 43, 1005-1016. Vibration analysis of stiffened cylinders including inter-ring motion. 6. T. WAH and L. R. CALCOTE 1970 Structure Analysis by Finite Difference Calculus. New York: j Van Nostrand Reinhold Co. 7. I. D. WILKEN and W. SOEDEL 1976 Journal of Sound and Vibration 44, 563-576. The receptance method applied to ring-stiffened cylindrical shells: analysis of modal characteristics. 8. I. D. WILKEN and W. SOEDEL 1976 Journal of Sound and Vibration 44, 577-589. Simplified prediction of the modal characteristics of ring-stiffened cylindrical shells. 9. K. FORSBERG 1969 AIAAfASME 10th Structures, Structural Dynamics and Materials Con .’ ference, New Orleans, Louisiana, 18-30. Exact solution for natural frequencies of ringstiffened cylinders. 10. R. W. CLOUGH and J. PENZIEN 1975 Dynamics of Structures. New York: McGraw-Hill. 1974 Computers and Structures 4, 1061-1089. Dynamics of frameworks by 11. B. A. OVUNK continuous mass method. 12. D. E. BESKOS 1979 Computers and Structures 10,785-795. Dynamics and stability of plane trusses with gusset plates. Use of dynamic 13. D. E. BESKOS and B. A. BOLEY 1975 Computers and Structures 5,263-269. influence coefficients in forced vibration problems with the aid of Laplace transform. 14. G. D. MANOLIS and D. E. BESKOS 1979 Developments in Mechanics 10,85-89. Dynamic response of beam structures with the aid of numerical Laplace transform. 15. G. D. MANOLIS and D. E. BESKOS 1980 Computer Methods in Applied Mechanics and Engineering 21, 337-355. Thermally induced vibrations of beam structures. 16. G. V. NARAYANAN 1979 Ph.D. Thesis, University of Minnesota, Minneapolis. Numerical operational methods in structural dynamics. 17. D. E. BESKOS 1980 Proceedings of the 2nd International Symposium on Innovative Numerical Analysis in Applied Engineering Science, Montreal, Canada, 16-20 June. Numerical operational methods for time-dependent linear problems. TheoryofPIatesandShells.New York: 18. S. TIMOSHENKO~~~~WOINOWSKY-KRIEGER~~~~ McGraw-Hill, second edition. D. H. YOUNG and W. WEAVER, JR 1974 Vibration Problems in 19. S. P. TIMOSHENKO, Engineering. New York: John Wiley & Sons, fourth edition. 20. H. KRAUS 1967 Thin Elastic Shells. New York: John Wiley & Sons. 21. J. B. OATES 1979 MS. Thesis, University of Minnesota, Minneapolis. Dynamic analysis of ring-stiffened circular cylindrical shells. 14,405-414. A new method of inversion 22. A. PAPOULIS 1957 QuarterlyofAppliedMathematics of the Laplace transform. 23. F. DURBIN 1974 The Computer Journal 17, 371-376. Numerical inversion of Laplace transforms: an efficient improvement to Dubner and Abate’s method. 24. J. W. COOLEY and J. W. TUKEY 1965 Mathematics of Computation 19, 297-301. An algorithm for the machine calculation of complex Fourier series. 25. B. A. BOLEY and J. H. WEINER 1960 Theory of ThermalStresses. New York: John Wiley & Sons. 26. C. L. DYM 1974 Introduction to the Theory of Shells. New York: Pergamon Press. 27. S. I. SUZUKI 1976 Journal of Sound and Vibration 44, 169-178. Dynamic behaviour of thin cylindrical shells with a step change in thickness subjected to inner impulsive loads.