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Dynamics of cylindrical shells with variable curvature
Journal of Sound and Vibration
(1971)
19 (l), 39-48
DYNAMICS OF CYLINDRICAL
SHELLS WITH
VARIABLE CURVATURE T. J. MCDANIEL Department of Engineering Mechanics, University of Dayton, Dayton, Ohio 45409, U.S.A.
AND J. D. LOGAN Research Institute, University of Dayton, Dayton, Ohio 45409, U.S.A. (Received 8 June
1971)
The transfer matrix method is extended to the analysis of non-circular cylindrical panels. The exact solution for the transfer matrix of a panel with exponential curvature is obtained by solving exactly the variable coefficient differential equations of motion of the shell using a Laplace transform-difference equation technique. The results are compared with respect to accuracy and computer time with various approximate methods of computing the transfer matrix for the same panel. Natural frequencies and mode shapes for typical non-circular panels are computed and compared with a constant curvature panel to show the effects of variable curvature.
1. INTRODUCTION In spite of the many applications of non-circular cylindrical shells in structural mechanics, the literature on this subject is quite sparse when compared to the volumes written on circular cylindrical shells, e.g. see the compendium by Nash [l]. Undoubtedly, one of the major reasons for this scarcity of analysis lies in the difficulty in solving the variable coefficient differential equations of motion which arise from the variable shell geometry. In this paper, using the Donnell shell equations and Laplace transform techniques, we shall obtain an exact solution for the class of non-circular cylindrical shells having exponential curvature, and we shall indicate other curvatures for which this method can be used. Up to the present time, the usual analyses (buckling, statics, and dynamics) for unstiffened non-circular cylindrical shells have been carried out by various approximate or numerical techniques. To cite a few of these analyses, Marguerre [2] investigated the buckling characteristics of such shells, and Roman0 and Kempner [3] and Hedgren and Billington [4] studied the static behavior of such shells using a truncated double Fourier series analysis and a Runge-Kutta method, respectively. The static behavior of reinforced non-circular shells has been studied by Vafakos, Nissel, and Kempner [5] using an energy method. Klosner [6,7] obtained the free and forced vibrations of an oval cylinder by a perturbation method. Recently, Kurt and Boyd [8] examined the free vibration of non-circular cylindrical panels by a power series approach. Wunderlich [9] has developed several approximate methods of obtaining the transfer matrix for rotational shells. Only for special non-circular shells have exact solutions been obtained. For instance, Garnet [lo] solved for the uncoupled torsional vibrations of a conical shell, and Das [I l] obtained exact solutions for membrane shells having cross-sections which are segments of a lemniscate or cardioid. 39
40
T. J. MCDANIEL
2. TRANSFER
AND J. D. LOGAN
MATRIX
METHODS
The transfer matrix method, which has been employed in the study of the dynamics of circular cylindrical shells by Lin and Donaldson [12] and Henderson and McDaniel [13], will be applied in the present analysis to a cylindrical panel with varying curvature p(s) (see Figure 1). It is assumed that the dimensions and material of the panel are such that the usual thin shell assumptions are valid. In addition, to simplify the problem and to put it into a form amenable to transfer matrix analysis, i.e. a one-dimensional problem, it is assumed that the curved edges act as simple supports to the panel. This has been shown by Clarkson and Ford [14] and Simmons [15], from experimental measurements, to be a reasonable assumption for aircraft panels. On the stright edges, the boundary conditions may be arbitrary.
Figure 1. Cylindrical panel with varying curvature.
Figure 2 shows the coordinate system and the sign convention assigned to the displacements and forces in a non-circular shell element. A typical forced response of the panel can be represented by w(x, s, t), or equivalently by the Fourier transform of that response with respect to time, w(x, s, W) = r W(X,s, t) e-‘“’ dt. --oo The simple support assumption at the curved edges allows the Fourier transform of the displacements and forces to be expressed as follows. Displacements in the x, s and z directions are, respectively, m
24 =
nJ u,(s, w)
2,= 2 u,(s, w) sinE, cos=, b b n=l m
w = *?I w.(s, w) sin E!E. b
Rotation and bending moment about the line of constant s are, respectively,
Kirchhoff shear, inplane tension, and inplane shear along the line of constant s are, respectively,
v=
Q&
dMs.x w) + ----z sin !E,b dx
SHELLS WITH VARIABLE CURVATURE
41
N = nz, N,,(s, w) sin ‘9,
The state vector for this transfer matrix study, as a function of arc length s, is defined as {Z(s, ~)>n = MS, a), ~(3,w), w(s, w), KS, w), M(s, w), W, w), - W, w),J(s, w)>,
(2)
where { } denotes a column vector. Since the following equations apply to any given frequency and number of waves between frames, we will omit w and n and just write {Z(s)}. The transfer
Figure 2. Shell sign convention.
matrix analysis of the non-circular panel is formulated by determining the relationship between the initial state vector {Z(O)} and the state vector {Z(s)} at an arbitrary location on the panel. This relationship is of the form {Z(s)> = T(s)iZ(O)] - S(s),
(3)
where T(s) is a frequency dependent, complex valued transfer matrix which accounts for the structural elements between the initial and arbitrary location, s, and S(s) is an excitation vector which accounts for the applied excitation which is not coupled to the structural response. 3. THE GOVERNING EQUATIONS To determine the transfer matrix for a segment of the shell, the governing equations of a selected eighth-order shell theory must be solved. For the present analysis, the Donnell shell equations were chosen [16]. If we assume that the normal pressure can be decomposed in the same manner as the deflection W,that is p(x, s, w) = 5 PJS, W)sin y, n=l
then upon substitution of equations (1) into Donnell’s equations and an algebraic reduction of variables to those appearing in the state vector, we obtain, for each IZ,a first-order system of differential equations, ;{Z(s)}
= A(s){Z(s)} + (OOOOOp(s)OO}.
42
T.
J. MCDANIELAND J.
D. LOGAN
This non-homogeneous set of differential equations has the well-known solution (see Pestel and Leckie [17], p. 148) (Z(s)} = T(s){Z(O)} + T(s) ~sT-i(~)(OOOOOp(~)OO~dcr,
(5)
0
where T(s) is the transfer matrix for the homogeneous system &Z(s)]
= A(s)iZ(s)).
(6)
Therefore, the problem reduces to solving the system (6). TABLE 1 Geometrical and material parameters D = E/z3/12(1 - v”) K = Hz/(1 - v2) q = n/b
rl = -2D(l - v)q2 r2 = D(l - v2)q4 - phw2 r3 = Kq2(1 - v’) - Phil
where b = 20.0 in (50.80 cm) E = 10.5 x lo6 Ibf/in2 (7.23 x lOlo N/m2) h = 044 in (1.016 mm) 1= 8.2 in (20.83 cm) ; 1;*:01/386 lb s2/in4 (1653 kg s2/in4)
width of panel Young’s Modulus thickness length of panel mass density Poisson’s ratio
The matrix A(s) is given as a function of the arc length s by -q
0
0
PCS> 0
0
0
0 0
0
-l/K
0001000
0 0
0 phw2
r3
0
‘0’
0 0
0
0
cl2 0 0
0
0
0
P(S) 0 vq
2/K(l - v)0 0 0 0 0 -4 0 -
(7)
J
where the geometrical and material parameters of the shell are defined in Table 1, and p(s) represents the curvature along the shell as a function of s. Inplane inertia terms are included in A(s) through the presence of phw2 in elements (8, 1) and (7, 2). Rotary inertia has been omitted. Hedgren and Billington [4] have developed an equation in rectangular coordinates similar to equation (6) but it is complicated by the fact that a metric coefficient multiplies each term of the matrix A. Furthermore, the A matrix used in that study, in contrast to A given by equation (7), does not satisfy the condition of cross-symmetry, and its elements a,, do not satisfy the condition a,, = 0 for i i-j even. If these requirements are satisfied, then the transfer matrix and its inverse are simply related. This relationship will be discussed in section 4. Several approximate methods for solving differential equation (6) to obtain the transfer matrix T(s) are available. These methods include solutions by power series, by perturbation techniques, by a Galerkin approach, by various numerical techniques of differential equations, and by approximating the variable curvature
for solving a system by a number of arcs
43
SHELLS WITH VARIABLE CURVATURE
of constant curvature. So that the accuracy and computational advantage of the various approximate techniques can be evaluated, it is also desirable to search for analytical solutions. Exact solutions for the governing differential equations for the non-circular shell are in general difficult to obtain. In this study, an analytical solution is sought for cylindrical shells having variable curvature cross-sections which reduce to constant curvature as a limiting case. The basic method used here to obtain a solution is to apply the Laplace transform to equation (6), solve the resulting problem in the transform domain, and return to the physical domain via the inverse transform. It is clear that the homogeneous part of equation (4) can be rewritten as
where A, and A2 are constant matrices. For certain special curvatures. viz. P(S) = f. ezs P(S) = PO+ Pl
(exponential curvature), (linear curvature)
s
and (periodic curvature),
P(S) = Po(1 f Scosrls)
the Laplace transform of equation (8) yields a differential or difference equation in the transform domain. For example, in the case of linear curvature the transform of equation (8) yields a second-order scalar differential equation in Nor w. In the case of periodic curvature, the transform of equation (8) yields a fourth-order scalar difference equation in N or w. The exact solutions to these equations in closed form have not yet been obtained. For the exponential curvature case, however, the resulting equation is a first-order matrix difference equation which can be reduced to a scalar first-order difference equation via a certain change of variable; this equation can then be solved by classical methods. In the next section, the exact solution for the exponential case is obtained. 4. EXPONENTIAL CHANGE OF CURVATURE We shall consider a panel of curvature changing exponentially with arc length, P(S) = p. ems,
(9)
where p. is a positive constant and CLis a real parameter which is not necessarily small. We note that when a = 0 the curvature is constant and the panel is circular. In the following section, we shall discuss the determination of the shape of an exponentially curved panel. In order to solve equation (8) with the curvature given by equation (9), we make the substitution fw, Is, M, v> = eoLs{w17 PI, MI, VI>. (10) If the new state vector is denoted by {Z,)={~,v,wi,P,,Mi,
I’,,-NJ)
then equation (6) can be rewritten as g{Z,} = B{Z,} $- poezoLs(O w, OOOOOO],
(11)
where the elements of A and B are identical except for the following elements : B(2,3) = 0,
B(6,7) = PO,
B(i, i) = -a,
and
i=
3,4,5,6.
Taking the Laplace transform of equation (1 I), we obtain ~12, (P)> - {Z, (0)) = B(Zi(p)) +
PO@
@I
(P
-
2a)
0 0 0 0 0 01,
44
T. J. MCDANIEL AND J. D. LOGAN
where Z,(p) = f Z,(s) eVpSds. 0
Therefore, {Z,(P)] = [PI -
W’(po@fldp - 240 0 0 0 0 01+ ~-WON.
(12)
It can be shown that [PI - RI-’ =
MP>l/PdP),
where P,(p) is an eighth-order polynomial and [R,,(p)] is an 8 x 8 matrix whose entries are polynomials of degrees which do not exceed seven. Moreover, the roots of P*(p) are all distinct. If the third row of equation (12) is written out, the result is a first-order difference equation in P?, of step-size 2cr: (13) where I] denotes a row vector. The solution to this difference equation is well known (e.g. see Mime-Thomson [18]) and can be written in terms of Gamma functions. For numerical studies, however, it is better to write the solution of equation (13) as a descending continued fraction expansion : bv’1(p- 2cz)=
lR& - 2a)l+ &(P - 2a) [R&P - 4a)lPo + . . . ~z,~o~l , Ps(P - 2a) Ps(P - 2cVs(P - 4a)
where the general term of the series for n > 2 is n-l - 2@1 P;-’ k2 R32(P - 2k4 lR3h
kq
{z~(0))
p8(P
-
(14)
2k4.
I
Substituting equation (14) into equation (12) and performing the simplifying calculations, we obtain
G(P)>= (W
+Pa(p)p;op _ {R12W Za)
lR3h
-
241+ . . .
VI&W.
(15)
1
Therefore, the state vector {Z,(s)} is given by the inverse Laplace transform
(16) The integral in equation (16) is easily calculated by the method of residues for a # 0 since the integrand has only simple poles which occur at the zeros of P,(p), Ps(p - 2a), etc. Since each of these eighth-order polynomials can be factored into the product of two fourth-order polynomials, the roots can be obtained analytically. Numbering the roots of P&J) = 0 as pI through ps, the roots of Ps(p - 2a) = 0 as p9 throughp, 6, etc., then {Z,(s)> = 5 {Z:(p,)> eYZr(O)>, j=I where {Z&?J] = (P - P,) (Zr (A> IP’P, and j = (-1)“2. Finally, using equation (10) we can convert the vector (Z,(s)} to {Z(s)} and obtain an expression of the form (17) (Z(s)> = T(s){Z(O)],
SHELLS WITH
VARIABLE
45
CURVATURE
where T(s) is the desired transfer matrix. Numerical evaluation of the transfer matrix is obtained from the above expressions. Also, it is necessary in transfer matrix analysis to compute the inverse transfer matrix. The elements Rijof the inverse transfer matrix R can be computed from the elements T,, of the K x K transfer matrix T by the formula
Ri,= (-l)‘+’ Tklr where Z=K+l--i.
k=K+l-j, This result is developed in reference [ 191. 5. SHAPE
OF THE
PANEL
Given the curvature p(s) as a function of arc length S, it is not a simple matter to obtain in closed rectangular form the equation of the curve having that curvature; in general, the problem amounts to solving a non-linear integro-differential equation. For the case of exponential curvature, the equation reduces to a second-order non-linear differential equation. This can be seen as follows. In Cartesian coordinates, the exponential curvature is given by p(x) = p. exp CLX dl (
s 0
+ y’(x)2 dx
(18)
, 1
where J(X) is the structural shape. The curvature is also given by p(x) = y”(x)/(l + y’(x)Y.
(19)
Equating (18) and (19), taking logarithms of both sides, and then differentiating, one concludes that the desired structural shape y = y(x) must satisfy the differential equation -(l +y’2)y”’ + 3y’y”2 + a(1 +y’2)3’2y” = 0. By using the Runge-Kutta
(20)
method with initial conditions Y(O)= l/POP
Y’(0) = 0,
Y”(0) = PO
to solve equation (20), several panel shapes were obtained for p. = l/72, corresponding to different values of the parameter a, and these are displayed in Figure 3. Of particular interest in the following section (numerical results) are the panels for which u = 0, &O-l. One can see from the enlarged view of these panels that, over an 8.2 in (20.83 cm) panel, the CL= t-O.1 and TV = -0-l panels deviate by 0.17 in (4.32 mm) and 0.11 in (2.79 mm), respectively, from the a = 0 case. I
I
I
I
Section
20
40
I
I
I
I
A
60
Figure 3. Rectangular coordinate plot of exponentially curved panels with scales given in inches (1 in = 25.4 mm), p0 = l/72. The left-hand diagram shows an enlargement of a part of the right-hand diagram.
46
T. J. MCDANIEL AND J. D. LOGAN
6. NUMERICAL
RESULTS
As we mentioned previously, one of the reasons for obtaining an exact solution to the exponentially curved shell is for comparison with approximate methods, The accuracies of the Runge-Kutta method, of a circular arc approximation scheme, and of a power series solution are compared with the exact solution developed in section 4. These comparisons, with respect to transfer matrix elements and natural frequencies, are carried out for a typical aircraft panel having the properties listed in Table 1 and a shape, as a function of the shape parameter u, as shown in Figure 3A. In the numerical comparison of the transfer matrix elements, the form of the exact solution given by equations (15) and (16) was found to converge rapidly. For values of O!= f0.1, the transfer matrix elements calculated from the first two terms of the exact solution agree with the transfer matrix elements computed from the Runge-Kutta method to six significant figures. For cc= 0, the exact solution is numerically more difficult to obtain since multiple poles occur in equation (15); however, this form of the solution is not necessary since CL= 0 is the circular case. When a four-arc approximation of the u = +0-l panels is used, the transfer matrix generated deviates in some elements by 13 % from the exact result. Other elements which relate out-of-plane displacements and forces at the two edges of the panel deviated by less than 1%. On the other hand, in the power series solution, where nine terms were used to generate quarter-panel transfer matrices, the maximum deviation of the transfer matrix elements for the complete panel was 40%. This agrees with the conclusion of Kurt and Boyd [8] that the power series method converges slowly. If the power series is used to obtain transfer matrices over several short segments of panel, followed by a chain multiplication of these matrices, the accuracy of the resulting transfer matrix increases to the level of the exact solution. With respect to computer time for obtaining the transfer matrix at a given frequency, the exact solution and the power series solution were approximately three times faster than the Runge-Kutta method and the circular arcs method. The computer time for the approximate methods are linearly related to panel length whereas the computation time of the exact solution is independent of the panel length. A comparison of the natural frequencies predicted by the various approaches was obtained for boundary conditions of u = u = w = M = 0 at the straight edges and simple support at the curved edges of the panel. The frequencies predicted by the exact and the Runge-Kutta solutions were in agreement and were compared with the frequencies obtained from the four-arc approximation. Table 2 shows that the maximum deviation in frequencies of 9% is less than the maximum deviation of the transfer matrix elements. Table 2 also shows that the four-arc approximation gives higher frequencies for cc= 0.1 and lower frequencies for TV = -0.1. This effect is expected since the starting curvature of each segment is chosen as the circular curvature for that segment. Limited calculations in which the power series was used to obtain natural frequencies indicated deviations of about 15 % from the frequencies obtained from the exact solution. For two half-waves between frame edges, or n = 2, natural frequencies of 2205 and 3150 rad/s were obtained for u = 0.1. These frequencies, when compared with those in Table 2, indicate the possibility that n = 1 and n = 2 frequencies could be quite close for some natural frequencies. It is ofinterest to obtain the modal shapes of non-circular panels because of their usefulness in computing forced response. The procedures for obtaining the radial deflection mode shapes as well as the frequency determinant required for frequency computations are well known [17]. The first three modes of the 8.2 in (20.83 cm) panel for the values cc= 0, f0.1 are normalized to unit amplitude in Figure 4, where the boundary conditions are the same as for the natural frequency study. The first mode displayed in Figure 4, even for cc= 0, may
47
SHELLS WITH VARIABLE CURVATURE
appear unusual because it contains two half-sine-waves instead of one, but these results are consistent with the results reported in reference [20]. This shape is due to the membrane effects introduced by the boundary conditions. It is clear from Figure 4 that variable curvature has a pronounced effect upon the mode shapes. It would be interesting to experimentally verify these results as well as the previously mentioned possibility that some natural frequencies are closely spaced for non-circular panels.
TABLE
2
Comparison of naturalfrequencies -~
Natural frequency (rad/s)
a=O.l Natural frequency (rad/s)
a=-01 Natural frequency (rad/s)
Exact solution and Runge-Kutta method
1566 2462 3503 5763
1627 2954 4511 5806
1446 1863 3348
Four-arcs approximation
1566 2462 3503 5763
1597 2896 4182
1467 2019 3370
or=0
Method
Span
length
Figure 4. Normalized radial deflection mode shapes. (a) Mode 1; (b) mode 2; (c) mode 3. -, ---- ) LI= -0.1; ---, o(= 0.1.
OL = 0;
7. CONCLUDING REMARKS Analytic solutions of the equations of motion for general non-circular shells are essentially impossible to obtain with the available mathematical tools. The solution obtained in this paper for an exponentially curved panel is useful primarily for evaluation of the accuracy and computer time of approximate methods. These approximate methods apply not only to the exponential curvature case but to other curvatures as well, and it is the writers’ general .opinion that of the approximate methods evaluated, the Runge-Kutta method is the best method of obtaining numerical transfer matrices. As we mentionedin section 3, the homogeneous solution to the matrix differential equation (i.e. the transfer matrix) is the first step in computing forced response. Either through classical normal mode analysis or directly through equation (5), the forced response can be computed. An extension of the above study to non-circular skin-stringer structures paralleling the work of Lin [21] and Mercer[22] on cylindrical and flat skin-stringer structuresis thus possible.
48
T. J. MCDANIELAND J. D. LOGAN ACKNOWLEDGEMENTS
This work was sponsored by the U.S. Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio, on contract number F33615-70-C-1337. Thanks are due Mr Steven Protzik for his computer programming and Mrs Saundra Fusek for typing the manuscript. REFERENCES
1. W. A. NASH 1954 TMB Report 863, Washington: U.S. Navy Department. Bibliography on shells and shell-like structures. 2. K. MARGUERRE1951 NACA-TM-1302. Stability of the cylindrical shell of variable curvature. 3. F. ROMANOand J. KEMPNER1962 Journal of Applied Mechanics 29, 669-674. Stresses in short noncircular cylindrical shells under lateral pressure. 4. A. W. HEDGREN,JR and D. P. BILLINGTON1966 Journal of the Structural Division, American Society of Civil Engineers 92. Numerical analysis of translational shell roofs. 5. W. P. VAFAKOS,N. NI~SELand J. KEMPNER1966 American Institute of Aeronautics and Astronautics Journal 4, 338-345. Pressurized oval cylinders with closely spaced rings. 6. J. M. KLOSNERand F. V. POHLE1958 PZBAL No. 476. Natural frequencies of an infinitely long noncircular cylindrical shell. 7. J. M. KL~~NER1960 PZBAL No. 561. Free and forced vibrations of a long noncircular cylindrical shell. 8. C. E. KURT and D. E. BOYD 1971 American Institute qf Aeronautics and Astronautics Journal 9,239-244. Free vibrations of noncircular cylindrical shell segments. 9. W. WUNDERLICH1967 Ingenieur-Archiv 36, 262-279. Zur berechnung von rotationsschalen mit iibertragungsmatrizen. 10. H. GARNET, M. GOLDBERGand V. L. SALERNO1961 Journal of Applied Mechanics 28, 571-573. Torsional vibrations of shells of revolution. 11. A. DAS 1969 Journal of Science and Engineering Research 13, 201-208. Displacements in noncircular cylindrical shells. 12. Y. K. LIN and B. K. DONALDSON1969 Journal of Sound and Vibration 10, 103-143. A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels. 13. J. P. HENDERSONand T. J. MCDANIEL 1971 Journal of Sound and Vibration 18, 203-209. The analysis of curved multi-span structures. 14. B. L. CLARKSONand R. FORD 1962 Journal of the Royal Aeronautical Society 66, 31-45. The response of a typical aircraft structure to jet noise. 15. D. R. SIMMONSet al. 1969 Shock and Vibration Bulletin No. 39, Part 4. Multi-layer alternately anchored treatment for damping of skin-stringer structures. 16. W. FL~~GGE1962 Handbook of Engineering Mechanics. New York : McGraw-Hill. 17. E. C. PESTELand F. A. LECKIE1963 Matrix Methods in Elastomechanics. New York: McGrawHill. 18. L. M. MILNE-THOMSON 1965 The Calculus of Finite Differences. New York: MacMillan. Dynamics 19. Y. K. LIN and T. J. MCDANIEL 1969 JournalofEngineeringforZndustry91,1135-1141. of beam-type periodic structures. 20. T. J. MCDANIELand B. K. DONALDSON1966 AFML-TR-64-347, Part ZI, Ohio: Wright-Patterson Air Force Base. Free vibration of continuous skin stringer panels with nonuniform stringer spacing and panel thickness. 21. Y. K. LIN 1967 Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill. 22. C. A. MERCERand C. SEAVEY1967 Journal of Sound and Vibration 6, 149-162. Prediction of natural frequencies and normal modes of skin-stringer panel rows.
(1971)
19 (l), 39-48
DYNAMICS OF CYLINDRICAL
SHELLS WITH
VARIABLE CURVATURE T. J. MCDANIEL Department of Engineering Mechanics, University of Dayton, Dayton, Ohio 45409, U.S.A.
AND J. D. LOGAN Research Institute, University of Dayton, Dayton, Ohio 45409, U.S.A. (Received 8 June
1971)
The transfer matrix method is extended to the analysis of non-circular cylindrical panels. The exact solution for the transfer matrix of a panel with exponential curvature is obtained by solving exactly the variable coefficient differential equations of motion of the shell using a Laplace transform-difference equation technique. The results are compared with respect to accuracy and computer time with various approximate methods of computing the transfer matrix for the same panel. Natural frequencies and mode shapes for typical non-circular panels are computed and compared with a constant curvature panel to show the effects of variable curvature.
1. INTRODUCTION In spite of the many applications of non-circular cylindrical shells in structural mechanics, the literature on this subject is quite sparse when compared to the volumes written on circular cylindrical shells, e.g. see the compendium by Nash [l]. Undoubtedly, one of the major reasons for this scarcity of analysis lies in the difficulty in solving the variable coefficient differential equations of motion which arise from the variable shell geometry. In this paper, using the Donnell shell equations and Laplace transform techniques, we shall obtain an exact solution for the class of non-circular cylindrical shells having exponential curvature, and we shall indicate other curvatures for which this method can be used. Up to the present time, the usual analyses (buckling, statics, and dynamics) for unstiffened non-circular cylindrical shells have been carried out by various approximate or numerical techniques. To cite a few of these analyses, Marguerre [2] investigated the buckling characteristics of such shells, and Roman0 and Kempner [3] and Hedgren and Billington [4] studied the static behavior of such shells using a truncated double Fourier series analysis and a Runge-Kutta method, respectively. The static behavior of reinforced non-circular shells has been studied by Vafakos, Nissel, and Kempner [5] using an energy method. Klosner [6,7] obtained the free and forced vibrations of an oval cylinder by a perturbation method. Recently, Kurt and Boyd [8] examined the free vibration of non-circular cylindrical panels by a power series approach. Wunderlich [9] has developed several approximate methods of obtaining the transfer matrix for rotational shells. Only for special non-circular shells have exact solutions been obtained. For instance, Garnet [lo] solved for the uncoupled torsional vibrations of a conical shell, and Das [I l] obtained exact solutions for membrane shells having cross-sections which are segments of a lemniscate or cardioid. 39
40
T. J. MCDANIEL
2. TRANSFER
AND J. D. LOGAN
MATRIX
METHODS
The transfer matrix method, which has been employed in the study of the dynamics of circular cylindrical shells by Lin and Donaldson [12] and Henderson and McDaniel [13], will be applied in the present analysis to a cylindrical panel with varying curvature p(s) (see Figure 1). It is assumed that the dimensions and material of the panel are such that the usual thin shell assumptions are valid. In addition, to simplify the problem and to put it into a form amenable to transfer matrix analysis, i.e. a one-dimensional problem, it is assumed that the curved edges act as simple supports to the panel. This has been shown by Clarkson and Ford [14] and Simmons [15], from experimental measurements, to be a reasonable assumption for aircraft panels. On the stright edges, the boundary conditions may be arbitrary.
Figure 1. Cylindrical panel with varying curvature.
Figure 2 shows the coordinate system and the sign convention assigned to the displacements and forces in a non-circular shell element. A typical forced response of the panel can be represented by w(x, s, t), or equivalently by the Fourier transform of that response with respect to time, w(x, s, W) = r W(X,s, t) e-‘“’ dt. --oo The simple support assumption at the curved edges allows the Fourier transform of the displacements and forces to be expressed as follows. Displacements in the x, s and z directions are, respectively, m
24 =
nJ u,(s, w)
2,= 2 u,(s, w) sinE, cos=, b b n=l m
w = *?I w.(s, w) sin E!E. b
Rotation and bending moment about the line of constant s are, respectively,
Kirchhoff shear, inplane tension, and inplane shear along the line of constant s are, respectively,
v=
Q&
dMs.x w) + ----z sin !E,b dx
SHELLS WITH VARIABLE CURVATURE
41
N = nz, N,,(s, w) sin ‘9,
The state vector for this transfer matrix study, as a function of arc length s, is defined as {Z(s, ~)>n = MS, a), ~(3,w), w(s, w), KS, w), M(s, w), W, w), - W, w),J(s, w)>,
(2)
where { } denotes a column vector. Since the following equations apply to any given frequency and number of waves between frames, we will omit w and n and just write {Z(s)}. The transfer
Figure 2. Shell sign convention.
matrix analysis of the non-circular panel is formulated by determining the relationship between the initial state vector {Z(O)} and the state vector {Z(s)} at an arbitrary location on the panel. This relationship is of the form {Z(s)> = T(s)iZ(O)] - S(s),
(3)
where T(s) is a frequency dependent, complex valued transfer matrix which accounts for the structural elements between the initial and arbitrary location, s, and S(s) is an excitation vector which accounts for the applied excitation which is not coupled to the structural response. 3. THE GOVERNING EQUATIONS To determine the transfer matrix for a segment of the shell, the governing equations of a selected eighth-order shell theory must be solved. For the present analysis, the Donnell shell equations were chosen [16]. If we assume that the normal pressure can be decomposed in the same manner as the deflection W,that is p(x, s, w) = 5 PJS, W)sin y, n=l
then upon substitution of equations (1) into Donnell’s equations and an algebraic reduction of variables to those appearing in the state vector, we obtain, for each IZ,a first-order system of differential equations, ;{Z(s)}
= A(s){Z(s)} + (OOOOOp(s)OO}.
42
T.
J. MCDANIELAND J.
D. LOGAN
This non-homogeneous set of differential equations has the well-known solution (see Pestel and Leckie [17], p. 148) (Z(s)} = T(s){Z(O)} + T(s) ~sT-i(~)(OOOOOp(~)OO~dcr,
(5)
0
where T(s) is the transfer matrix for the homogeneous system &Z(s)]
= A(s)iZ(s)).
(6)
Therefore, the problem reduces to solving the system (6). TABLE 1 Geometrical and material parameters D = E/z3/12(1 - v”) K = Hz/(1 - v2) q = n/b
rl = -2D(l - v)q2 r2 = D(l - v2)q4 - phw2 r3 = Kq2(1 - v’) - Phil
where b = 20.0 in (50.80 cm) E = 10.5 x lo6 Ibf/in2 (7.23 x lOlo N/m2) h = 044 in (1.016 mm) 1= 8.2 in (20.83 cm) ; 1;*:01/386 lb s2/in4 (1653 kg s2/in4)
width of panel Young’s Modulus thickness length of panel mass density Poisson’s ratio
The matrix A(s) is given as a function of the arc length s by -q
0
0
PCS> 0
0
0
0 0
0
-l/K
0001000
0 0
0 phw2
r3
0
‘0’
0 0
0
0
cl2 0 0
0
0
0
P(S) 0 vq
2/K(l - v)0 0 0 0 0 -4 0 -
(7)
J
where the geometrical and material parameters of the shell are defined in Table 1, and p(s) represents the curvature along the shell as a function of s. Inplane inertia terms are included in A(s) through the presence of phw2 in elements (8, 1) and (7, 2). Rotary inertia has been omitted. Hedgren and Billington [4] have developed an equation in rectangular coordinates similar to equation (6) but it is complicated by the fact that a metric coefficient multiplies each term of the matrix A. Furthermore, the A matrix used in that study, in contrast to A given by equation (7), does not satisfy the condition of cross-symmetry, and its elements a,, do not satisfy the condition a,, = 0 for i i-j even. If these requirements are satisfied, then the transfer matrix and its inverse are simply related. This relationship will be discussed in section 4. Several approximate methods for solving differential equation (6) to obtain the transfer matrix T(s) are available. These methods include solutions by power series, by perturbation techniques, by a Galerkin approach, by various numerical techniques of differential equations, and by approximating the variable curvature
for solving a system by a number of arcs
43
SHELLS WITH VARIABLE CURVATURE
of constant curvature. So that the accuracy and computational advantage of the various approximate techniques can be evaluated, it is also desirable to search for analytical solutions. Exact solutions for the governing differential equations for the non-circular shell are in general difficult to obtain. In this study, an analytical solution is sought for cylindrical shells having variable curvature cross-sections which reduce to constant curvature as a limiting case. The basic method used here to obtain a solution is to apply the Laplace transform to equation (6), solve the resulting problem in the transform domain, and return to the physical domain via the inverse transform. It is clear that the homogeneous part of equation (4) can be rewritten as
where A, and A2 are constant matrices. For certain special curvatures. viz. P(S) = f. ezs P(S) = PO+ Pl
(exponential curvature), (linear curvature)
s
and (periodic curvature),
P(S) = Po(1 f Scosrls)
the Laplace transform of equation (8) yields a differential or difference equation in the transform domain. For example, in the case of linear curvature the transform of equation (8) yields a second-order scalar differential equation in Nor w. In the case of periodic curvature, the transform of equation (8) yields a fourth-order scalar difference equation in N or w. The exact solutions to these equations in closed form have not yet been obtained. For the exponential curvature case, however, the resulting equation is a first-order matrix difference equation which can be reduced to a scalar first-order difference equation via a certain change of variable; this equation can then be solved by classical methods. In the next section, the exact solution for the exponential case is obtained. 4. EXPONENTIAL CHANGE OF CURVATURE We shall consider a panel of curvature changing exponentially with arc length, P(S) = p. ems,
(9)
where p. is a positive constant and CLis a real parameter which is not necessarily small. We note that when a = 0 the curvature is constant and the panel is circular. In the following section, we shall discuss the determination of the shape of an exponentially curved panel. In order to solve equation (8) with the curvature given by equation (9), we make the substitution fw, Is, M, v> = eoLs{w17 PI, MI, VI>. (10) If the new state vector is denoted by {Z,)={~,v,wi,P,,Mi,
I’,,-NJ)
then equation (6) can be rewritten as g{Z,} = B{Z,} $- poezoLs(O w, OOOOOO],
(11)
where the elements of A and B are identical except for the following elements : B(2,3) = 0,
B(6,7) = PO,
B(i, i) = -a,
and
i=
3,4,5,6.
Taking the Laplace transform of equation (1 I), we obtain ~12, (P)> - {Z, (0)) = B(Zi(p)) +
PO@
@I
(P
-
2a)
0 0 0 0 0 01,
44
T. J. MCDANIEL AND J. D. LOGAN
where Z,(p) = f Z,(s) eVpSds. 0
Therefore, {Z,(P)] = [PI -
W’(po@fldp - 240 0 0 0 0 01+ ~-WON.
(12)
It can be shown that [PI - RI-’ =
MP>l/PdP),
where P,(p) is an eighth-order polynomial and [R,,(p)] is an 8 x 8 matrix whose entries are polynomials of degrees which do not exceed seven. Moreover, the roots of P*(p) are all distinct. If the third row of equation (12) is written out, the result is a first-order difference equation in P?, of step-size 2cr: (13) where I] denotes a row vector. The solution to this difference equation is well known (e.g. see Mime-Thomson [18]) and can be written in terms of Gamma functions. For numerical studies, however, it is better to write the solution of equation (13) as a descending continued fraction expansion : bv’1(p- 2cz)=
lR& - 2a)l+ &(P - 2a) [R&P - 4a)lPo + . . . ~z,~o~l , Ps(P - 2a) Ps(P - 2cVs(P - 4a)
where the general term of the series for n > 2 is n-l - 2@1 P;-’ k2 R32(P - 2k4 lR3h
kq
{z~(0))
p8(P
-
(14)
2k4.
I
Substituting equation (14) into equation (12) and performing the simplifying calculations, we obtain
G(P)>= (W
+Pa(p)p;op _ {R12W Za)
lR3h
-
241+ . . .
VI&W.
(15)
1
Therefore, the state vector {Z,(s)} is given by the inverse Laplace transform
(16) The integral in equation (16) is easily calculated by the method of residues for a # 0 since the integrand has only simple poles which occur at the zeros of P,(p), Ps(p - 2a), etc. Since each of these eighth-order polynomials can be factored into the product of two fourth-order polynomials, the roots can be obtained analytically. Numbering the roots of P&J) = 0 as pI through ps, the roots of Ps(p - 2a) = 0 as p9 throughp, 6, etc., then {Z,(s)> = 5 {Z:(p,)> eYZr(O)>, j=I where {Z&?J] = (P - P,) (Zr (A> IP’P, and j = (-1)“2. Finally, using equation (10) we can convert the vector (Z,(s)} to {Z(s)} and obtain an expression of the form (17) (Z(s)> = T(s){Z(O)],
SHELLS WITH
VARIABLE
45
CURVATURE
where T(s) is the desired transfer matrix. Numerical evaluation of the transfer matrix is obtained from the above expressions. Also, it is necessary in transfer matrix analysis to compute the inverse transfer matrix. The elements Rijof the inverse transfer matrix R can be computed from the elements T,, of the K x K transfer matrix T by the formula
Ri,= (-l)‘+’ Tklr where Z=K+l--i.
k=K+l-j, This result is developed in reference [ 191. 5. SHAPE
OF THE
PANEL
Given the curvature p(s) as a function of arc length S, it is not a simple matter to obtain in closed rectangular form the equation of the curve having that curvature; in general, the problem amounts to solving a non-linear integro-differential equation. For the case of exponential curvature, the equation reduces to a second-order non-linear differential equation. This can be seen as follows. In Cartesian coordinates, the exponential curvature is given by p(x) = p. exp CLX dl (
s 0
+ y’(x)2 dx
(18)
, 1
where J(X) is the structural shape. The curvature is also given by p(x) = y”(x)/(l + y’(x)Y.
(19)
Equating (18) and (19), taking logarithms of both sides, and then differentiating, one concludes that the desired structural shape y = y(x) must satisfy the differential equation -(l +y’2)y”’ + 3y’y”2 + a(1 +y’2)3’2y” = 0. By using the Runge-Kutta
(20)
method with initial conditions Y(O)= l/POP
Y’(0) = 0,
Y”(0) = PO
to solve equation (20), several panel shapes were obtained for p. = l/72, corresponding to different values of the parameter a, and these are displayed in Figure 3. Of particular interest in the following section (numerical results) are the panels for which u = 0, &O-l. One can see from the enlarged view of these panels that, over an 8.2 in (20.83 cm) panel, the CL= t-O.1 and TV = -0-l panels deviate by 0.17 in (4.32 mm) and 0.11 in (2.79 mm), respectively, from the a = 0 case. I
I
I
I
Section
20
40
I
I
I
I
A
60
Figure 3. Rectangular coordinate plot of exponentially curved panels with scales given in inches (1 in = 25.4 mm), p0 = l/72. The left-hand diagram shows an enlargement of a part of the right-hand diagram.
46
T. J. MCDANIEL AND J. D. LOGAN
6. NUMERICAL
RESULTS
As we mentioned previously, one of the reasons for obtaining an exact solution to the exponentially curved shell is for comparison with approximate methods, The accuracies of the Runge-Kutta method, of a circular arc approximation scheme, and of a power series solution are compared with the exact solution developed in section 4. These comparisons, with respect to transfer matrix elements and natural frequencies, are carried out for a typical aircraft panel having the properties listed in Table 1 and a shape, as a function of the shape parameter u, as shown in Figure 3A. In the numerical comparison of the transfer matrix elements, the form of the exact solution given by equations (15) and (16) was found to converge rapidly. For values of O!= f0.1, the transfer matrix elements calculated from the first two terms of the exact solution agree with the transfer matrix elements computed from the Runge-Kutta method to six significant figures. For cc= 0, the exact solution is numerically more difficult to obtain since multiple poles occur in equation (15); however, this form of the solution is not necessary since CL= 0 is the circular case. When a four-arc approximation of the u = +0-l panels is used, the transfer matrix generated deviates in some elements by 13 % from the exact result. Other elements which relate out-of-plane displacements and forces at the two edges of the panel deviated by less than 1%. On the other hand, in the power series solution, where nine terms were used to generate quarter-panel transfer matrices, the maximum deviation of the transfer matrix elements for the complete panel was 40%. This agrees with the conclusion of Kurt and Boyd [8] that the power series method converges slowly. If the power series is used to obtain transfer matrices over several short segments of panel, followed by a chain multiplication of these matrices, the accuracy of the resulting transfer matrix increases to the level of the exact solution. With respect to computer time for obtaining the transfer matrix at a given frequency, the exact solution and the power series solution were approximately three times faster than the Runge-Kutta method and the circular arcs method. The computer time for the approximate methods are linearly related to panel length whereas the computation time of the exact solution is independent of the panel length. A comparison of the natural frequencies predicted by the various approaches was obtained for boundary conditions of u = u = w = M = 0 at the straight edges and simple support at the curved edges of the panel. The frequencies predicted by the exact and the Runge-Kutta solutions were in agreement and were compared with the frequencies obtained from the four-arc approximation. Table 2 shows that the maximum deviation in frequencies of 9% is less than the maximum deviation of the transfer matrix elements. Table 2 also shows that the four-arc approximation gives higher frequencies for cc= 0.1 and lower frequencies for TV = -0.1. This effect is expected since the starting curvature of each segment is chosen as the circular curvature for that segment. Limited calculations in which the power series was used to obtain natural frequencies indicated deviations of about 15 % from the frequencies obtained from the exact solution. For two half-waves between frame edges, or n = 2, natural frequencies of 2205 and 3150 rad/s were obtained for u = 0.1. These frequencies, when compared with those in Table 2, indicate the possibility that n = 1 and n = 2 frequencies could be quite close for some natural frequencies. It is ofinterest to obtain the modal shapes of non-circular panels because of their usefulness in computing forced response. The procedures for obtaining the radial deflection mode shapes as well as the frequency determinant required for frequency computations are well known [17]. The first three modes of the 8.2 in (20.83 cm) panel for the values cc= 0, f0.1 are normalized to unit amplitude in Figure 4, where the boundary conditions are the same as for the natural frequency study. The first mode displayed in Figure 4, even for cc= 0, may
47
SHELLS WITH VARIABLE CURVATURE
appear unusual because it contains two half-sine-waves instead of one, but these results are consistent with the results reported in reference [20]. This shape is due to the membrane effects introduced by the boundary conditions. It is clear from Figure 4 that variable curvature has a pronounced effect upon the mode shapes. It would be interesting to experimentally verify these results as well as the previously mentioned possibility that some natural frequencies are closely spaced for non-circular panels.
TABLE
2
Comparison of naturalfrequencies -~
Natural frequency (rad/s)
a=O.l Natural frequency (rad/s)
a=-01 Natural frequency (rad/s)
Exact solution and Runge-Kutta method
1566 2462 3503 5763
1627 2954 4511 5806
1446 1863 3348
Four-arcs approximation
1566 2462 3503 5763
1597 2896 4182
1467 2019 3370
or=0
Method
Span
length
Figure 4. Normalized radial deflection mode shapes. (a) Mode 1; (b) mode 2; (c) mode 3. -, ---- ) LI= -0.1; ---, o(= 0.1.
OL = 0;
7. CONCLUDING REMARKS Analytic solutions of the equations of motion for general non-circular shells are essentially impossible to obtain with the available mathematical tools. The solution obtained in this paper for an exponentially curved panel is useful primarily for evaluation of the accuracy and computer time of approximate methods. These approximate methods apply not only to the exponential curvature case but to other curvatures as well, and it is the writers’ general .opinion that of the approximate methods evaluated, the Runge-Kutta method is the best method of obtaining numerical transfer matrices. As we mentionedin section 3, the homogeneous solution to the matrix differential equation (i.e. the transfer matrix) is the first step in computing forced response. Either through classical normal mode analysis or directly through equation (5), the forced response can be computed. An extension of the above study to non-circular skin-stringer structures paralleling the work of Lin [21] and Mercer[22] on cylindrical and flat skin-stringer structuresis thus possible.
48
T. J. MCDANIELAND J. D. LOGAN ACKNOWLEDGEMENTS
This work was sponsored by the U.S. Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio, on contract number F33615-70-C-1337. Thanks are due Mr Steven Protzik for his computer programming and Mrs Saundra Fusek for typing the manuscript. REFERENCES
1. W. A. NASH 1954 TMB Report 863, Washington: U.S. Navy Department. Bibliography on shells and shell-like structures. 2. K. MARGUERRE1951 NACA-TM-1302. Stability of the cylindrical shell of variable curvature. 3. F. ROMANOand J. KEMPNER1962 Journal of Applied Mechanics 29, 669-674. Stresses in short noncircular cylindrical shells under lateral pressure. 4. A. W. HEDGREN,JR and D. P. BILLINGTON1966 Journal of the Structural Division, American Society of Civil Engineers 92. Numerical analysis of translational shell roofs. 5. W. P. VAFAKOS,N. NI~SELand J. KEMPNER1966 American Institute of Aeronautics and Astronautics Journal 4, 338-345. Pressurized oval cylinders with closely spaced rings. 6. J. M. KLOSNERand F. V. POHLE1958 PZBAL No. 476. Natural frequencies of an infinitely long noncircular cylindrical shell. 7. J. M. KL~~NER1960 PZBAL No. 561. Free and forced vibrations of a long noncircular cylindrical shell. 8. C. E. KURT and D. E. BOYD 1971 American Institute qf Aeronautics and Astronautics Journal 9,239-244. Free vibrations of noncircular cylindrical shell segments. 9. W. WUNDERLICH1967 Ingenieur-Archiv 36, 262-279. Zur berechnung von rotationsschalen mit iibertragungsmatrizen. 10. H. GARNET, M. GOLDBERGand V. L. SALERNO1961 Journal of Applied Mechanics 28, 571-573. Torsional vibrations of shells of revolution. 11. A. DAS 1969 Journal of Science and Engineering Research 13, 201-208. Displacements in noncircular cylindrical shells. 12. Y. K. LIN and B. K. DONALDSON1969 Journal of Sound and Vibration 10, 103-143. A brief survey of transfer matrix techniques with special reference to the analysis of aircraft panels. 13. J. P. HENDERSONand T. J. MCDANIEL 1971 Journal of Sound and Vibration 18, 203-209. The analysis of curved multi-span structures. 14. B. L. CLARKSONand R. FORD 1962 Journal of the Royal Aeronautical Society 66, 31-45. The response of a typical aircraft structure to jet noise. 15. D. R. SIMMONSet al. 1969 Shock and Vibration Bulletin No. 39, Part 4. Multi-layer alternately anchored treatment for damping of skin-stringer structures. 16. W. FL~~GGE1962 Handbook of Engineering Mechanics. New York : McGraw-Hill. 17. E. C. PESTELand F. A. LECKIE1963 Matrix Methods in Elastomechanics. New York: McGrawHill. 18. L. M. MILNE-THOMSON 1965 The Calculus of Finite Differences. New York: MacMillan. Dynamics 19. Y. K. LIN and T. J. MCDANIEL 1969 JournalofEngineeringforZndustry91,1135-1141. of beam-type periodic structures. 20. T. J. MCDANIELand B. K. DONALDSON1966 AFML-TR-64-347, Part ZI, Ohio: Wright-Patterson Air Force Base. Free vibration of continuous skin stringer panels with nonuniform stringer spacing and panel thickness. 21. Y. K. LIN 1967 Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill. 22. C. A. MERCERand C. SEAVEY1967 Journal of Sound and Vibration 6, 149-162. Prediction of natural frequencies and normal modes of skin-stringer panel rows.