Vibrations of cantilevered shallow cylindrical shells of rectangular planform

Journal of Sound and Vibration (1981) 78(3), 311-328

V I B R A T I O N S OF C A N T I L E V E R E D SHALLOW C Y L I N D R I C A L SHELLS OF R E C T A N G U L A R P L A N F O R M A. W. LEISSA, J. K. LEE AND A. J. WANG

Department of Engineering Mechanics, Ohio State University, Columbus, Ohio 43210, U.S.A. (Received 1 December 1980) A cantilevered, shallow shell of circular cylindrical curvature and rectangular planform exhibits free vibration behavior which differs considerably from that of a cantilevered beam or of a fiat plate. Some numerical results can be found for the problem in the previously published literature, mainly obtained by using various finite element methods. The present paper is the first definitive study of the problem, presenting accurate non-dimensional frequency parameters for wide ranges of aspect ratio, shallow~ness ratio and thickness ratio. The analysis is based upon shallow shell theory. Numerical results are obtained by using the Ritz method, with algebraic polynomial trial functions for the displacements. Convergence is investigated, with attention being given both to the number of terms taken for each co-ordinate direction and for each of the three components of displacement. Accuracy of the results is also established by comparison with finite element results- for shallow shells and with other accurate fiat plate solutions. 1. INTRODUCTION T h e cantilevered shallow shell is a structural element of considerable technical significance. One especially important application of this element is in the case of. turbomachinery blades. Literally hundreds of references can be found which relate directly to the turbine engine blade vibration p r o b l e m (cf. [1--4]). However, in the vast majority the blades are treated as cantilevered beams: that is, the problem is characterized as a one-dimensional one, with a single co-ordinate measured along the b e a m axis. Such a representation is highly inaccurate if (a) the blade has a small aspect ratio, (b) the blade is thin, or (c) m o r e than the first two or three frequencies and m o d e shapes are needed. In few studies attempts have been m a d e to model a blade as a cantilevered plate, but the important stiffening effect of chordwise c a m b e r cannot be represented by this model. M o r e than 30 references have been found in which cantilevered blades having chordwise c a m b e r are treated as shells. T h e majority of these have been discussed in a recent survey article [5]. In virtually all of them some form of finite element m e t h o d was used (e.g., with triangular or quadrilateral, plate or shell, conforming or non-conforming elements). A r good example of these efforts is the recent publication by Gill and Ucmakliogliu [6] in which the frequencies obtained by their finite element scheme for one particular shell are c o m p a r e d with results taken from eight other references, in seven of which finite elements were also used. A n excellent piece of work in which the finite element m e t h o d was not used is that of Beres [7], who analyzed cantilevered helicoidal shells using a form of the Ritz m e t h o d with algebraic polynomials as trial functions. Although helicoidal shells were studied, in the case of zero twist the helicoid becomes a circular cylinder, and shallow cylindrical shells are a further special case. 311 0022-460X/81/190311 + 18 $02.00/0 9 1981 Academic Press Inc. (London) Limited

312

A . W . LEISSA, J. K. LEE AND A. J. WANG

Although the various references mentioned above deal with the vibrations of canti= levered shallow cylindrical shells of rectangular planform, their primary emphasis is upon developing a particular analytical technique. While one, or even a few, different shell configurations have been used to demonstrate the application of the technique to this important problem area, none of these references (perhaps because of large computational costs) present comprehensive sets of results needed to describe the variations of the various geometrical parameters of the problem. The primary objectives of the present paper are two-fold: (I) to provide a definitive study of the free vibration problem for the cantilevered shell of rectangular planform and shallow, circular cylindrical curvature, providing accurate frequency information in nondimensional form for a sufficient number of modes and for ranges of values of aspect ratio (a/b), shallowness ratio (b/R) and thickness ratio (b/h); (2) to discuss the physical aspects of the problem and the numerical results, thereby aiding in the understanding of the significance of changing the various geometric parameters present. It is believed that the resulting work will he useful both to design engineers and to future researchers needing information in this problem area. The Ritz method has been used in this investigation, A sufficient number of trial functions were taken to guarantee a reasonable degree of accuracy in the numerical results. Classical thin, shallow shell theory was used, and the results obtained are for shells which are adequately described by this theory, The analytical procedure utilized in this work is particularly well suited for the large number of different shell parameters that are used in the extensive numerical results presented at the end of the work. 2. ANALYSIS The cantilevered shallow shell which is the subject of the present study is depicted in Figure 1. The shell has circular cylindrical curvature in the chordwise (i.e., y) direction.

Figure 1. Cantilevered shallow cylindrical shell having rectangular planform.

The planform is rectangular, having length a and width b and the shell thickness is denoted by h. One edge (x = O) is rigidly clamped while the other three are completely free. Displacements of the shell are entirely characterized by its two-dimensional middle surface which, in turn, can be related to the xy reference plane. The three orthogonal components of displacement are u (parallel to the xz-plane) and v, tangent to the shell midsurface, and w, normal to it.

CANTILEVERED

SHALLOW

SHELL

VIBRATIONS

313

The maximum strain energy for a homogeneous and is0tropic shallow shell having a constant thickness h and a radius of curvature R is given by (cf. [8-10])

Vmax=VI-I-V2,

(I)

where Vl represents the membrane effectsdue to stretchingof the middle surface of the shell,

Eh

tb/2 t,, t

2

J- ,,lo and V2 represents the strain energy due to bending,

Eh'

f~;;

V'=24(I-va)

~

.r0h, a'w

(O:w'r

(3)

So { ( V ' w ) ' - 2 ( 1 - u ) [ 0 - - ~ 0y a \OxOy'J

The membrane strains are related to the displacements by ey = Ov/Oy + w / R ,

ex = O u / a x ,

y,y = Ov/Ox +

Ou/Oy,

(4)

where E is Young's modulus and v is Poisson's ratio. For small amplitude free vibration the three components of displacement of the shell midsurface are expressed as

u(x, y, t) = U(x, y) sin tot, v(x, y, t) = V(x, y) sin ~ot, w(x, y, t) = W(x, y) sin wt. fl

-

(5)

.

The maximum kinetic energy (which occurs at zero displacement, and maximum velocity) is

phoJa f b/2 Tm==

-~

f" (U2+ V2 + W 2) dx dy,

(6)

~-b/2 JO

where p is the mass density per unit volume. Assumed displacement functions are I

J

K

u(~:, 11) = Z Z A i i $ 1' 1 ,J i=1]=o

V(~, 11) = Z

L

M

~,, Bkl~k11 I,

W ( ~ , 11) = ~.

k = l I=O

N

~, Cmn~ " 11'~,

rn=2n=O

(7a-c) (where ~: = x/a and 11= 2y/b) which satisfy the clamped edge conditions exactly: that is,

U(O, n)= V(O, 11)= W(O, 11)= (OW/O~)(O, 11)= 0.

(8)

The displacement functions are taken to be the simple polynomials gi;cen by equations (7), rather than the oft-used eigenfunctions of vibrating beams, for several reasons, particularly to permit relatively easy subsequent generalization of the problem to include variable thickness, variable curvature and non-rectangular planforms. The Ritz method requires minimization of the functional T m ~ - Vm~. This is accomplished by setting a(Zmax-

Vmax)/OAii = 0

O(Tm~,- Vmax)/OBkt = 0

a(Tm~,-

Vm~,)/aCm. = 0

( i = 1. . . . , I ; ] = 0 . . . . . J),

(9a)

(k = 1. . . . . K ; l = 0 . . . . . L).

(9b)

(m = 2 . . . . . M ; n =.t~. . . . . N),

(9c)

which yields a total of I(J + 1) + K ( L + 1) + ( M - 1)(N + 1) simultaneous, linear, characteristic equations in an equal number of unknowns Aii, Bkt and C.,.. For a non-trivial solution, the determinant of the coefficient matrix is set equal to zero. The zeros of the

314

A . ~,V. L E I S S A ,

J. K . L E E

AND

A . J. W A N G

determinant are the eigenvalues (non-dimensional frequency parameters). Substituting each eigenvalue back into the original set of equations yields the corresponding eigenvector (amplitude ratio) in the usual manner. The mode shape corresponding to each frequency can then be determined by substituting the eigenvectors back into equations (7). In the problem depicted by Figure 1 the x z-plane is one of symmetry. The resulting free vibration modes are all either symmetric or antisymmetric with respect to this plane, and the characteristic equations described in the preceding paragraph uncouple into two sets. Even (or odd) increments are used for the second indices/, l and n for symmetric (or antisymmetric modes). The resulting two uncoupled characteristic determinants are each consequently only half as large, resulting in considerable savings in computation time (approximately 1/4 as much time is required). 3. CONVERGENCE STUDIES The displacement functions (7) used in the preceding section form a complete set, and therefore must converge to the exact solution as the number of terms used becomes sufficiently large [11-13]. However, because the computer time required to evaluate eigenvalues of large order systems increases approximately with the cube of the order, a proper size must be determined in order to yield sufficiently accurate frequencies without undue computational time. Tables 1 and 2 show typical results of convergence studies that were made for various configurations. In both cases the shell is moderately thin (a/h = 100), moderately shallow (b/R = 0.5) and has a Poisson ratio (v) of 0.3. In Table 1 the shell aspect ratio is relatively small (a/b = 1) and in Table 2 it is large (a/b = 5). Non-dimensional frequency parameters oJa2",/ph/Dare given in the tables for the lowest four frequencies of each symmetry class, where D = Eh3/12(1- ~,2)is the flexural rigidity usually used to describe fiat plates and shallow shells. Convergence is studied from two aspects: (1) increase in the determinant order and (2) the relative importance of polynomial terms used in the spanwise (x) and chordwise (y) directions. The displacements U, V and W were all given the same number of degrees of freedom. The most accurate results in Table 1 are for the 6 x 5 solution. That is, I = K = M - 1 = 6 in equations (7), and J = L = N -'- 8 or 9 for symmetric or antisymmetric modes, respectively. Beginning with those from the least accurate (4 x 2) solution, the frequencies of all modes decrease monotonically as polynomial terms are added to the displacements, thereby yielding successively more accurate upper bounds to the exact frequencies. For any given solution the higher frequency modes are generally less accurately determined than the lower ones, which is typical of approximate methods. " However, for this configuration (Table 1) it is seen that it is somewhat more important to give degrees of freedom to the x direction rather than to y. That is, the results of the 5 x 4 solution are better than for the 4 x 5, and those of the 6 x 3 are much better than for the 3• The 5 x 4 solution is virtually the same as the 5 • and is considered sufficiently accurate for determining the first eight modes for most practical purposes. Free vibration mode shapes corresponding to the eight frequency parameters listed in Table 1 are depicted by the nodal patterns shown in Figure 2. These are typical representations of shell nodal patterns in that the dashed lines are lines of zero normal (i.e., W) displacement and that, in general, the tangential components (U and V) are not simultaneously zero along the lines. An exception occurs along the line y = 0, where either U = 0 (for antisymmetric modes) or V = 0 (for symmetric modes). For the relatively long and narrow (a/b = 5) shell of Table 2, similar convergence behavior is seen, except that now the importance of additional terms in the x direction is

CANTILEVERED

~

0

SHALLOW

SHELL

315

VIBRATIONS

~

99~99~ 0 0 ~ 0 0 0 0

tt~

H

~q

H

o~

C~

~6&6~66

UJ ,J

9~9999~ ~ - ~

E t~

~6~6666

C~

~.~~

"~'~

~~

X

XXXXXXX

"~'~

o~&

_~

X

X X X X X X X

316

A . W . L E I S S A , J. K. L E E A N D

I

,

A . J. W A N G

5

i I

I

F i g u r e 2. N o d a l p a t t e r n s for a c a n t i l e v e r e d s h a l l o w shell.

even greater. The 6 x 3 results are now not only much better than for the 3 • 6, but even better than for the 5 x 4, which has a larger order characteristic determinant. Similar results were examined for other configurations. For a very short (a/b = 0.5), moderately shallow (b/R = 0.5) shell, for example, the 4 x 5 results were found to be the best of the 54 or 60 d.o.f, solutions indicating that, for this case, additional terms in the y direction are more important than in the x. It must be r e m e m b e r e d that, because of the symmetry and mode uncoupling in the present problem, the polynomials described above are generally of considerably higher degree in the y direction than in the x. Or if symmetry were not present, for example, the uncoupled 4 x 5 solutions mentioned above would correspond to coupled 4 x 10 solutions. Studies were also made to determine whether it was computationally more efficient to give additional degrees of freedom to one or another of the displacements, compared with the others. Typical results of this study are shown in Tables 3 and 4. The shell represented in Table 3 is the same one described previously by Table 1 (i.e., a/b = 1, a/h = 100, b/R = 0.5 and u = 0.3). The crudest solution, with four and two polynomial terms in the x and y directions, respectively, for all displacement components is depicted by the first row of Table 3 (and also of Table 1). The next three rows show the improvement in the solution obtained when the 4 • 2 = 8 terms are increased to 6 x 3 = 18 terms successively in each displacement function. It is seen that the greatest improvement is obtained in the S - 1 (i.e., first symmetric), S - 3 , S - 4 , A - 1 (first antisymmetric) and A - 2 modes when terms are added to U, compared with V and W. On the other hand the A - 3 and A - 4 modes are most improved by adding terms to V, and the S - 2 mode is most improved by adding freedom to W. Further comparison can be made in the table for the magnitudes of the improvements, for the effects of adding terms to both U and V (row 5), and with the most accurate solution present, having terms added to all three displacements (row 6). Similar comparisons can be made in Table 4 for another shell which is the same as the one for Table 3, except for being considerably thicker (a/h = 20). In this case the S - 1 and

CANTILEVERED

SHALLOW

SHELL

317

VIBRATIONS

0

0

mq

II

II

Q I

1

<

q~

~

<

i #.,

.-

X X X X X X

X X X X X X

m'g X X X X X X

X X X X X X

~

.~.~ x. o~ ~

X X X X X X

X X X X X X

318

A . W . L E I S S A , J. K . L E E

AND

A . J. W A N G

A - 3 modes are most greatly improved by adding terms in the U (row 2) and V (row 3) directions, respectively, whereas all six other modes are most improved by adding terms to W (row 4). The general conclusion that can be drawn from Tables 3 and 4, therefore, is that no particular preference should be given among the U, V and W displacements if the scope of a study is to include both thin and relatively thick shells (the second category not meaning "thick" shells, requiring higher order "thick shell theory"). Therefore, equal importance was assigned to each of the displacements in studies performed subsequently. 4. ELIMINATION OF TANGENTIAL INERTIA Considerable computational time can be saved by neglecting the inertia of the shell in the tangential directions, thereby reducing the size of the frequency determinant considerably and eliminating two-thirds of the eigenvalues. The resulting mathematical model of the physical vibration problem will often be sufficiently accurate for practical purposes provided that (1) the shell is sufficiently shallow, (2) the vibration modes sought do consist of displacements which are predominantly represented by W, (3) or conversely, mode shapes involving significant tangential displacements are not needed from the analysis. As a computational procedure the terms U 2 and V 2 are simply set equal to zero in equation (6) for the kinetic energy. The minimizing equations (9) can then be written in the characteristic form

where X is the vector of coefficients Aii and Bk! associated with the tangential displacements U and V in equations (7), and Y is the vector of Cmn coefficients associated with W. With U 2 and V 2 being dropped from the kinetic energy, the eigenvalues A (=oj2a4ph/D) appear then only in the submatrix ~. If c~ is of order M x N, then s/will typically be of the order 2(M x N) (i.e., the value when it is assumed that there are equal numbers of terms in equation (7) for all three displacements). Then by eliminating X the characteristic equation (10) of order 3 ( M x N ) can b e reduced to

(ce- ~ r d - l ~ ) Y = 0

(11)

from which, for a non-trivial solution, the characteristic determinant is set equal to z e r o ; i.e.,

I~e- ~Ts~-l~l = 0.

(12)

The resulting characteristic determinant is of order M x N, having M x N eigenvalues. The accuracy of the resulting frequency parameters when tangential inertia is e l i m i n a t e d may be seen in Table 5. Four sets of results are presented. The first set corresponds to t h e moderately thin (a/h =100), moderately shallow (b/R =0.5) shell having s q u a r e (a/b = 1) planform already described by Tables 1 and 3. Of course, when tangential i n e r t i a is neglected, the resulting frequency is greater. However, the error thus induced is s e e n t o be less than one percent in each of the first eight modes of the first set. However, when the aspect ratio (a/b) is increased to 5, significantly larger e r r o r s in frequency occur, as the second data set shows. This is especially true for the s y m m e t r i c modes. But, worst of all, the second antisymmetric mode is not found at all w h e n t a n g e n t i a l

CANTILEVERED SHALLOW SHELL VIBRATIONS

~

e

I lg/

.

.

~

~

0

666

~ A

.

.

~

0

~

0

11

o~

&&6

AA6

319

320

, A . W . L E I S S A , J. K. L E E A N D

A . J. W A N G

inertia is eliminated. This mode is predominantly sideways bending (i.e., displacements in the y-direction) coupled with torsion. When the shell is made very shallow (b/R =0.1) the errors induced by neglecting tangential inertia are substantially decreased, as shown in the third and fourth sets of data. However, again the second antisymmetrie mode for the shell with the larger aspect ratio. The errors caused by neglecting tangential inertia in the present problem are found to be approximately the same as in the case of the shallow shell of rectangular planform supported on all sides by shear diaphragms [14]. The latter case also results from the problem of the closed, circular cylindrical shell supported on both ends by shear diaphragms for modes having large circumferential wave numbers. The increase in frequencies in Table 5 due to neglect of tangential inertia for b/R = 0.5 can, for example, be compared directly with results for closed shells having a circumferential wave number (n) of six (el. [15], pp. 76-79).

5. COMPARISON WITH FINITE ELEMENT SOLUTIONS Olson and Lindberg [16] in a relatively early paper applied the finite element method to a model of turbofan blade having dimensions R =24.0 in, h =0.120 in, and a = b = 12.0 in. The analysis was based upon a non-conforming, cylindrical shell element of rectangular planform, having 28 degrees of freedom. In a subsequent reference [17] they solved the same problem using a conforming, doubly curved shallow shell element of triangular shape having 38 degrees of freedom. Walker [18] also analyzed the problem using doubly curved, right helicoidal shell elements, each having 40 degrees of freedom. In each of the three works numerical results were presented for several sizes of finite element grids. Frequencies of the most accurate solutions (i.e., most degrees of freedom) from each reference are summarized in Table 6. Also given in Table 6 are the frequencies the present method gives from one of the better solutions. Degrees of freedom listed in the table are all based upon the assumption that no reduction is used for either symmetry or for eliminating tangential inertia. Thus the results of the present method (120 d.o.f.) correspond exactly to those of the 5 x4 solution (60 d.o.f.) already described in Table 1 for the same shell, with non-dimensional values of the parameters. Because all four methods yield frequencies which converge monotonically from above as degrees of freedom are added, the results given by the present method are the most accurate. Similar comparisons have been made [19] for three other cantilevered shells analyzed by Walker [18], having various combinations of radius and thicknehs. Results from the present method with 120 degrees of freedom were found to be generally better than those from the finite element approach with 205 d.o.f.

6. NUMERICAL RESULTS AND DISCUSSION Extensive numerical results were obtained for the non-dimensional frequency parameters om 2",/ph/D of cantilevered shallow shells having circular cylindrical curvature in the chordwise (y-direction) and rectangular planform, as depicted in Figure 1. These results are presented in the Appendix as Tables A1-A8. Each table corresponds to a particular value of a thickness ratio (b/h = 100,20) and of an aspect rato (a/b = 0.5, 1, 2, 5), the first four tables being for the thinner shell. The value b/h = 100 corresponds to a shell well within the accepted limits of applicability of thin shell theory.

CANTILEVERED

SHALLOW

SHELL

VIBRATIONS

u

H

0

H I1

AA~

0

-6-6.6 ~

~ x X X X X

"O o (D

o

321

322

A . w . LEISSA, J. K. LEE AND A. J. WANG

However, b/h = 20 exceeds the limits somewhat for a/b = 0.5 (i.e., a/h = 10) in the case of modes which involve considerable spanwise bending. Smaller thickness ratios would necessitate using a thick shell theory. All tables are for ~ = 0-3. A typical table is designed to answer the question " W h a t effect does curvature have upon the frequencies of a cantilevered shell?" Thus, with all other parameters fixed (b]h, a/b, v), one may observe the increases in frequencies as the shallowness parameter b/R increases from 0 to 0.5 in increments of 0.1. The upper limit of 0.5 was set to avoid significant inaccuracies due to the use of shallow shell theory.

b

o m= b

.q

"-~

H

24

-~ = 0.5 R

I

a

Ib

--a=5 b

R

~" = O.5

TOP VIEW

END VIEW

Figure 3. Examples of moderately shallow shells.

Figure 3 is presented to provide better understanding in the use of Tables A 1 - A 8 . T h e figure shows typical shells, having square (a/b = 1) and slender (a/b = 5) planforms. Both shells are drawn for the upper limit of shallowness ratio (i.e., b/R = 0.5) used in the tables. The shallowness of these shells is readily apparent in the figure; it can also be seen in the corresponding values of rise-to-width ratio (H/b = 0.0318) and the included angle (2~b = 28.96~ For complete (i.e., .closed) circular cylindrical shells the shallow shell (or DonnellMushtari) theory is known to give accurate results for the lower frequency modes having six circumferential waves (i.e., 2~b = 30 ~ or more [15]. Therefore, the value b/R = 0.1 used in the tables corresponds to a very shallow shell, having 2~b = 5.73 ~ and H/b = 0.0013, the rise being essentially imperceptible to the eye. This very slight curvature is seen to cause significant increases in the frequencies corresponding to the spanwise (x-direction) bending modes. Figure 3 also serves to demonstrate the significances of the various parameters and the proper sequence in which they should be designated in order to determine their effects upon frequency changes. Since the non-dimensional parameter toa2,/ph/D contains the length (a), the a/b ratio is varied by changing the width. Thus, a/b = 5 should be regarded as a narrower (rather than longer) shell than a/b = 1, as seen in the figure. This narrower shell has typically higher frequencies, except for those modes which are predominantly spanwise bending (having lesser frequencies due to the greater shallowness of the shell). Thus, for fixed a/b, the b/R ratio is changed by varying R. Direct comparison of

CANTILEVERED

SHALLO~,V S H E L L V I B R A T I O N S

323

frequencies from ma2x/ph/D for changing b/h is improper, for D = Eh3/12 (1 - v2): that is, it contains the thickness. The parameter ~oa2~/ph/D is independent of b/h for fiat plates, but not for shells, which can be seen by comparing values between, for example, Tables A1 and AS. Numerical results given in the tables were obtained by using the best of the polynomial fits for the determinants of order 54 or 60, as discussed in section 3. These were 6 x 3 solutions (see Table 1) for a/b =5 and 2, and 5 x 4 solutions for a/b = 1 and 0.5, with certain exceptions, as indicated in the tables. The accuracy of the numerical results in the case of a fiat plate (b/R = 0) can be established by comparison with other values in the literature. Lower bounds for the symmetric modes were calculated by Bazley, Fox and Stadter [20], and are presented in Table 7 for a/b = 0.5, 1 and 2. They also obtained relatively accurate upper bounds by the Ritz method, using the first 50 terms which are the products of beam functions. However, Sigillito [21] obtained generally more accurate upper bounds by using 30 terms which are the products of beam functions and Legendre functions, and those are given in Table 6. Corresponding values of ~oa 2pN/-~ obtained by the present method, taken from Tables A1 through A8, are also given for comparison. These are, of course, upper bounds as well. Less accurate upper bound results obtained by Young [22] and Barton [23] using the Ritz method and only 18 terms, along with accurate results by Claassen and Thorne [24] using a variant of the series method, are also presented for comparison of the antisymmetric frequencies. Comparing the various results listed in Table 7 shows that the present method, with either 54 or 60 terms, is very accurate, yielding the most accurate upper bounds known to date for all of the antisymmetric and some of the symmetric modes. Comparing the results for fiat plates between the first set (A1-A4) and second set (A5-A8) of reference tables, one notes that bne set of antisymmetric mode frequencies appears among the first four of the thicker (b/h = 20) plate frequencies which does not appear for the thinner (b/h = 100) plate. These are antisymmetric modes 4, 3, 3 and 2 for a/b = 0.5, 1, 2 and 5, respectively, of the thicker case. These modes are antisymmetric in-plane modes (i.e., displacements predominantly in the y-direction), and are consequently omitted from Table 7. On the other hand the first symmetric in-plane mode occurs at a frequency too great to be included among the first four modes in any of the cases of Tables A1-AS. Tables A1-A8 show that, although the lowest frequency mode of a fiat plate is always symmetric (i.e., the first spanwise bending mode) for a/b >>0.5, - increasing the chordwise curvature causes the spanwise bending frequency to increase more rapidly than that of the first antisymmetric mode (torsion), so that the torsional mode can bec6me the fundamental one, especially for thinner (large b/h) and less shallow (large b/R) stiells. This was already observed in Figure 2, which shows nodal patterns corresponding to b/R = 0.5 in Table A2. Walker [18] obtained results, both theoretically and experimentally, which show that if the shell of Table A2 were increased in depth to b/R = 1.0, the first spanwise bending mode would not appear among the first eight modes of the shell. One observes in Tables A1-A8 that increasing curvature causes increases in all of the frequencies in every case, and that the percent increases for the symmetric mode frequencies are generally greater than those of the antisymmetric modes. One also observes greater percent increases of frequency with b/R for the thinner (b/h = 100) shells than for the thicker (b/h = 20) ones. Except for the first spanwise bending and the first torsional mode, the mode shapes are generally rather complicated. The other relatively simple beam-like and plate-like modes (e.g., chordwise bending, combined bendfng-torsion, higher spanwise bending and

324

A. W . L E I S S A , .I. K. L E E

AND

A. ,I. W A N G

~1 I I ~ ~ 1 I I ~ ~ 1 I I

6 11

~1

I ~ ~1 I I ~ ~1 I I ~

~I I I ~ ~

I ~ (""-I I ~

~ I I ~ ~-~I I~ ~ I

~

I~

<

[... t~

"r~

8~

~E o

~

~

~ 0

~

9

~

~ 0

~

~

~

~ 0

o

CANTILEVERED SHALLOW SHELL VIBRATIONS

325

torsional modes) often do not exist for the shells. The change in nodal patterns with increasing curvature for a/b = 1 and b/h = 100 was excellently depicted by Olson and Lindberg [16].

ACKNOWLEDGMENT This work was carried out with the support of the National Aeronautics and Space Administration, Lewis Research Center, under Grant No. N A G 3-36.

REFERENCES 1. J. S. RAO 1973 Shock and Vibration Digest 5, 3-16. Natural frequencies of turbine bladingma survey. 2. J. S. RAO 1977 Shock attd Vibration Digest 9, 15-22. Turbine blading excitation and vibration. 3. J. S. RAO 1980 Shock and Vibration Digest 12, 19-26, Turbomachine blade vibration. 4. A. W. LEISSA 1981 Applied Mechanics Reviews 34, 629-635. Vibrational aspects of rotating turbomachinery blades. 5. A. W. LEISSA 1980 Shock and Vibration Digest 12, 3-10. Vibrations of turbine engine blades by shell analysis. 6. P. A. T. GILL and M. UCMAKLIOGLU 1979 Journal of Sound and Vibration 65, 259-273. Isoparametric finite elements for free vibration analysis of shell segments and non-axisymmetric shells. 7. D. P. BERES 1974 Ph.D. Dissertation, Ohio State University. Vibration analysis of skewed cantilever systems--helicoidal shells and plates. 8. N. V. NOVOZHOLOV 1959 The Theory of Thin Shells (English translation). Groningen: P. Noordhoff Limited. 9. A. A. NAZAROV 1956 NACA TM 1426. On the theory of thin shallow shells (English translation). 10. A. W. LEISSA and A. S. KADI 1969 AirForce Flight Dynamics Laboratory Report TR-69-71. Analysis of shallow shells by the method of point matching. 11. L. V. KANTOROVICH and V. I. KRYLOV 1958 Approximate Methods of Higher Analysis (English translation). Groningen: P. NoordhotI Limited. 12. S. G. MIKHLIN 1964 VariationaIMethods in MathematicalPhysics (English translation). New York: The MacMillan Co. 13. A. T. HOPPER, A. W. LEISSA, L. E. HULBERT and W. E. CLAUSEN 1967 Air Force Flight Dynamics Laboratory Report TR-67-121. Numerical analysis of equilibrium and eigenvalue problems. 14. A. W. LEISSA and A. S. KADI 1971 Journal of Sound and Vibration 16, 173-187. Curvature effects on shallow shell vibrations. 15. A. W. LEISSA 1973 NASA SP-288. Vibration of shells. 16. M. D. OLSON and G. M. LINDBERG 1969 Proceedings of the 2nd Conference on Matrix

Methods in StructuralMechanics, Wright-PattersonAir ForceBase, Ohio AFFDL-TR-69-150, 17. 18. 19. 20. 21. 22.

247-270. Vibration analysis of cantilevered curved plates using a new cylindrical shell finite element. M. D. OLSON and G. M. LINDBERG 1971 Journal of Sound and Vibration 19, 299-318. Dynamic analysis of shallow shells with a doubly-curved triangular finite element. K. P. WALKER 1978 Journal of Sound and Vibration 59, 35-37. Vibrations of cambered helicoidal fan blades. A. W. LEISSA, J. K. LEE and A. J. WANe 1981 PresentedatASME Gas Turbine Conference, Houston, Texas, March 1981. Rotating blade vibration analysis using shells. N. W. BAZLEY, D. W. FOX and J. T. 8TADTER 1965 Johns Hopkins University, Applied Physics Laboratory ReportNo. TO-705. Upper and lower bounds for frequencies of rectangular cantilever plates. V. G. SIGILLITO 1965 Johns Hopkins University, Applied Physics Laboratory Report No. EM-4012. Improved upper bounds for frequencies of rectangular free and cantilever plates. D. YOUNG 1950JoumalofAppliedMechanics 17,448-453.Vibrationofrectangularplatesby the Ritz method.

A. W. LEISSA. J. K. LEE AND A. J. WANG

326

23. M. V. BARTON 1949 Universityo[ Texas, De[enseResearch Laboratory Report No. DRL-222, CM 570. Free vibration characteristics of cantilever plates. 24. R. W. CLAASSEN and C. J. THORNE 1962 PacificMissile Range, TechnicalReport PMR-TR61-1. Vibrations of a rectangular cantilever plate. 25. A. W. LEISSA 1973 Journal of Sound and Vibration 31, 257-293. The free vibration of rectangular plates.

APPENDIX: REFERENCE VALUES OF FREQUENCY PARAMETERS TABLE A 1

Frequency parameters t o a 2 ~

for b/h = 100, a/b = 0.5 (5 • 4 solution)

Symmetric modes

Antisymmetric modes

b/R

1

2

3

4

1

2

3

4

0 0.1 0"2 0"3 0.4 0.5

3"4948 4.4944 6"3309 7-7331 8"4670 8"8785

10"187 10.355 11"004 12.502 14.816 17.513

21.851 22.625 24.597 26"880 28.714 30.079

31"547f 31"810t 32'639t 33"6109 36"0149 36"2889

5.3551 5.4654 5"7790 6"2532 6"8377 7"48159

19.085 19"124 19"236 19"414 19.6499 19.9469

24.686 24.965 25.763 26.977 28.458 30.047

43-127 43"266 43"696 44"442 45"544 47"046

t More accurate values from 4 x 5 solution.

TABLE A 2

Frequency parameters toa

2

p~

for b/ h = 100, a/ b = 1 (5 x 4 solution)

Symmetric modes

Antisymmetric modes

b/R

1

2

3

4

0 0.1 0.2 0.3 0.4 0.5

3.4739 5.2198 8.3724 11.613 14.563 17.000

21.301 24-785 26.831 27.721 28.924 30.664

27.203 28.237 35.147 41.778 45.508 47.780

54.250 55.194 58.765 66.710 78.601 90.807

1

8.5131 8.6156 8.9082 9.3576 9.9300 10.598

2

3

4

30.980 31.578 33.272 35.819 38.920 42.296

64.273 64.403 64.682 64.983 65.283 65.602

72-448 73.469 76.420 80.847 85-994 91.056

TAnLE A 3

Frequency parameters waZ~/ph/D for b/h = 100, a/b = 2 (6 x 3 solution) Symmetric modes

Antisymmetrie modes

b/R

1

2

3

4

1

2

3

4

0 0"1 0"2 0"3 0"4 0.5

3"4436 5"4524 9"0930 13.023 16.993 20.926

21"453 30"939 48.247 65"158t 78"395t 86.464t

60.212 77.051 93"424t 94"445* 95-765t 98"008t

93"263t 93"837t 107.94 126.91* 137.91 147"99

14.811 14.926 15"251 15.734 16.322 16"975

48"206 48.827 50"607 53"357 56"876 60.992

92"595 94"617 100.27 108.67 118.97 130"45

160.17 163.66 173"48 188"00 205"37 223"96

1"More accurate results from 5 x4 solution.

327

CANTILEVERED SHALLOW SHELL VIBRATIONS TABLE A 4

Frequency parameters toa 2x/ph/D for b/h = 100, a/b = 5 (6 • 3 solution) Symme~ic modes

blR

1

0 0.1 0.2 0.3 0.4 0.5

2

3.4086 21.314 5.5088 33.845 9.2874 56.578 13.369 81.121 17.518 105.92 21.665 130.48

Antisymmetric modes ,<

3

59.785 93.412 9 215.39 275.76 331.08

4

1

2

3

4

125.09 183.03 288.95 392.81 476.83 534.67

33.971 34.105 34.401 34.828 35.358 35.963

103.87 104.57 106.16 108.49 111.44 114.87

179.55 181-52 186.42 193.82 203.27 214.34

269.42 281-85 295.23 311.65 326.67 326.74

TABLE A 5 2

Frequency parameters toa x/ph/D for b/h = 20, a/b = 0.5 (5 x 4 solution) Symmetric modes

b/R 0 0"1 0"2 0"3 0"4 0"5

Antisymmetric modes 9

1

2

3'4948 9 10"187 3"5407 10"192 3"6741 10.208 3"8289 10'236 4.1508 10"275 4"4611 10"328

r

3

4

1

2

3

4

21.851 21"881 21"972 22"122 22"327 22"584

31"547t 31"556t 31-583t 31"629t 31"693t 31"777t

5"3551 5-3591 5.3709 5"3905 5"4178 3"4524

19.085 19.085 19.087 19.089 19.091 19.094

24.686 24.695 24.723 24.768 24.831 24.912

27'046 27"047 27"050 27"055 27"062 27"072

I"More accurate results from 4 x 5 solution.

TABLE A 6

Frequency parameters wa

2

px/p-~for b/h = 20, a/b = 1 (5 x 4 solution)

Symmetric modes

b/R 0 0.1 0.2 0.3 0'4 0.5

Antisymmetric modes

r

9

r

1

2

3

4

1

2

3

4

3.4739 3"5613 3.8079 4"1814 4.6451 5"1682

21.301 21.477 21.984 22"764 23.711 24"663

27"203 27"222 27.287 27.416 27.656 28.106

54.250 54"280 54.371 54.524 54.744 55.034

8.5131 8"5164 8.5264 8.5429 8.5659 8.5950

30.980 31.001 31.064 31-167 31.312 31.495

43.557 43.558 43.560 43.563 43.567 43.573

64.273 64.275 64.281 64.290 64.300 64.311

328

A. W. LEISSA. J- K. LEE AND A. J. WANG TABLE A 7

Freqttency parameters taa2p~f~t~dfor b/h b/R 0 0.1 0.2 0.3 0.4 0-5

Symme~icmodes

.

= 20,

a/b

r

= 2 (6 x 3

solution)

Antisymmetriemodes

1

2

3

4

1

2

3

4

3.4436 3"5451 3'8321 4.2641 4"7977 5"3980

21"453 21.915 23.234 25.249 27.772 30.636

60.212 60"997 63.270 66.813 71.332 76.509

93.263t 93.279t 93.321t 93"377t 93.443t 93"539t

14.811 14.814 14"824 14"840 14"863 14.891

48.206 48.226 48.286 48.386 48.525 48.701

57"514 57"515 57"518 57"524 57"531 57.540

92"595 92"669 92"888 93"250 92"752 94'387

= 5 (6 x 3

soh, tion)

t More accurate results from 5 x4 solution.

TABLE A 8

Frequency parameters toa 2 ~

[or b/h = 20, a/b

Symme~ic modes

b/R 0 0.1 0.2 0.3 0.4 0"5

Antisymmetric modes -r

1

2

3

4

3"4086 3.5156 3.8175 4.2705 4.8283 5.4544

21.314 21.947 23.737 26"431 29.760 33"506

59.785 61.615 66.589 73.828 82.632 92.477

125.09 127.90 135.96 148.27 163.72 181.27

2 33.971 33.975 33.984 33.995 34.007 34.019

3

65"353 103.87 65.353 " 103.90 65.355 103.97 65.358 104.07 65-364 104.19 65.373 104-32

4 179.55 179.63 179"85 180"18 180.59 181.08