Free vibration of thin shells

Journal o f Sound and Vibration (1975) 39(3), 337-344

FREE VIBRATION OF THIN SHELLS C. T. F. Ross Department of Afechanical Enghteering and Natal Architecture, Portsmouth Polytechnic, Portsmouth PO1 3DJ, England

(Receired 17 May ! 974, and ht revisedform 27 September 1974) A partially conforming triangular fiat plate out-of-plane element was used in conjunction with an in-plane element to develop stiffness and mass matrices for application to free vibration of shells. Good agreement was found between theory and experiment for a curved fan blade, a curved panel and part of a model aircraft wing. Comparison of other theories with experiment was also found to be good. The fact, however, that the fiat elements predicted frequencies almost as accurately as the more sophisticated shell elements was found particularly encouraging, because of the difficulty of using the latter with smaller computers. 1. INTRODUCTION The vibration of thin shells is of interest in a number of branches of engineering, including aeronautical, civil and mechanical engineering, and naval architecture. Shells vibrate for a number of reasons, usually due to some fluctuating force, which may cause the shell to vibrate in various modes. The study reported in this paper was concerned with free or natural vibrations, and the theoretical solution used is based on the matrix displacement method. Much application of this method has already been made to the vibration of shells [1, 2, 3]. Although the use of fiat plates for thin shells is a well known and documented subject, [3, 4, 5] the present solution employs a partially-conforming plate bending element, which can be readily accommodated in smaller computers. 2. THEORY The partially-conforming triangular plate bending element [6] was made up from three triangular sub-elements, as shown in Figures 1, 2 and 3. The following displacement configuration was assumed for each sub-element, in which it can be seen that displacement and slope continuity was achieved along the base of each subelement:

where ~? is the displacement perpendicular to the ~-.~ plane and cq, u2, etc., are constants.

/o,

z,,,

t

z

~

"-->0~

Figure I. Triangular element with sub-elements. 337

338

C. T, F. ROSS

0, y

3

2 Figure 2. Triangular element in the x - y plane.

By making the outer boundary of the triangular element the Ycaxis ofeach sub-element in turn, as shown in Figure 3, it was possible to determine the stiffness and mass matrices of each sub-element in terms of their local axes. Transformation of these stiffness and mass matrices was then necessary in terms of the local axes of the element, and after summation, the nodal point C was eliminated through the Guyan reduction technique [7]. Thus, the resulting triangular element satisfied slope and deflection continuity along its boundary, although these were not achieved along the internal boundaries of the sub-elements.

c

3 Figure 3. The triangular sub-element 2-3-C.

The element had the following nine degrees of freedom: {wlOxlOyllt?2Ox2Oy2lV3Ox3Oy3) , where w is the displacement of the plate in a direction perpendicular to its plane. (A list of notation is given in the Appendix.) For the present solution, the nodal point C was taken at the centroid of the element, although this was not necessary. Comparison of theoretical frequencies obtained from the above element was made with experiment, and found to be good for a fiat cantilevered plate. 3. SHELL ANALYSIS To develop a "shell" type element (see Figure 4), it was necessary to adopt a method similar to that of Zienkiewicz and Cheung [4] and Argyris [5]. To achieve this, use was made of the triangular plane stress element of Turner et aL [8], which was superimposed with that of the plate bending element. The plane stress element (Figure 5) corresponded to the displacements {tQ Vl tl2V2ttaV3}. The resulting shell element was in terms of the 18 displacements:

{,11 vl wl 0xl 0yx 0zl u~ t,, w2... 0,~ 0z3}. The stiffness and mass matrices were then in terms of their local axes, the triangular element lying in the x - y plane of Figure 4. It was necessary therefore to transform the "shell"

339

FREE VIBRATION OF 'IHIN SltELLS

o, t w,Z

/,~0,

2:

>0,

X, U

Figure 4. Shell element in local axes.

ytV 1

3

I t.

2

) XoU

Figure 5. Plane stress element.

~Z0

Figure 6. Position of shell element with respect to the global axes.

element into the global system of Figure 6, and this was done by using the method of Argyris [5]. It should be noted, however, that any slope continuity that may have originally existed with the fiat plate element was lost on application to a curved surface, because of the discontinuity of the boundaries of the flat elements when joined to form the curved surface. A computer programme was written for this element, incorporating the continuous reduction technique of Irons [9]. It was necessary to use a continuous reduction technique of this type, as with 18 degrees of freedom per element, almost any practical shell vibration problem would have been too large for most computers. A similar, computer programme was also written for another triangular "shell" type element, in which the non-conforming plate bending element of Tocher [I0] was used, together with the plane stress element of Turner et al. [8].

340

c.T.F. ROSS 4. RESULTS

4.1.

FAN BLADE

A comparison is shown in Table 1 of the theoretical predictions of the triangular partiallyconforming element (FEPCT) and the triangular non-conforming element (FENCT) with the experimental observations of Olson and Lindberg [1 ] for a curved cantilevered fan blade, together with the theoretical values of Olson and Lindberg [2] obtained from a 28 degree of freedom curved cylindrical shell element. The FEPCT and FENCT solutions employed a 6 x 6 mesh with 42 nodal points and 72 elements. It was necessary to use a reduction technique [9] to accommodate the problem in the computer, and eventually only 29 displacements were left, these corresponding to the z ~ direction. Due to the fact that so few displacements were left, it was not possible to plot the eigenmodes; however, from Table 1, both the triangular element solutions appear to be satisfactory, showing good agreement with the solution o f Olson and Lindberg. The experimental values tend to verify the solutions, although there were probably some experimental errors due to imperfections in the fan blade because it was manufactured from an initially flat plate. TABLE I

Frequencies of curved fan blade (llz) Olson and Lindberg

Flat triangular elements ~

4.2.

,

Mode

3 x 3 Mesh

4 x 4 Mesh

FENCT

FEPCT

Exptl.

1 2 3

100.7 155.1 260.9

93.5 147-6 255.1

93.6 148.3 250.0

96-9 152.1 256-0

86.6 135.5 258.9

CURVED

PANEL

The second shell to be investigated was a machined thin panel ofcylindrical shape, clamped along its edges. This panel was manufactured by the Lockheed-Georgia Research Laboratories from a solid block of aluminium, and tested at Southampton University [11]. The panel was excited acoustically and Petyt [12] has presented the first ten frequencies, together with the theoretical predictions according to the Extended Rayleigh-Ritz solution (ERR), curved triangular finite element solution [2] (FET), rectangular finite element solution (FER) and the Kantorovich solution (K). The first four frequency values are shown in Table 2, together with the theoretical calculations by the non-conforming triangular finite element (FENCT) and the partially-conforming triangular element (FEPCT). The mesh is shown in Figure 7. TABLE 2

Frequencies of cylhldrical panel (ttz) tn, n

ERR

FET

FER

K

FENCT

FEPCT

Exptl.

1,2 -I,3 1,3 2,1

870 958 1288 1364

870 958 1288 1363

890 973 1311 1371

890 966 1295 1375

830 944 1288 1343

880 982 1348 1415

814 940 1260 1306

m = number of half-waves in yOdirection. n = number of half-wavesin x~ direction.

341

FREE VIBRATION OF THIN SHELLS

\\\\\\ \\\\\\

\\\\\\ \\\\\\ \\\\\\

~x 0

Figure 7. Mesh for curved cylindrical panel.

Once again, because o f the limited availability of core store, the number of degrees o f freedom were reduced to 25, and all of these displacements were in the z ~ direction. 4.3. MODEL WING The model wing was constructed from bright steel, because ofthe simplicity ofmanufacture from this material. It was knuckled in two directions and a small cut was made along the knuckle parallel t o t h e edge of the clamping plate to assist construction. This cut was later brazed and filed flat.

Figure 8. Model wing with clamping arrangement.

The model wing was clamped firmly to the table of an electromagnetically-driven shaker, as shown in Figure 8. Detection of the first three frequencies was made by the naked eye, although stroboscopic light Was later used to assist observation.

342

c.T.F. ROSS

/

\\\ \\\ \\\

\\ \\\ (a)

(b)

(c)

Figure 9. (a) Mesh 1 ; (b) mesh 2; (c) mesh 3.

Analysis of the wing was made by using the triangular partially conforming plate bending element. Three meshes were chosen, as shown in Figures 9(a), (b) and (c) where it can be seen that they vary quite appreciably in fineness. 9The first three frequencies for each case are given in Table 3, and the theoretical eigenmodes for Mesh 3 which appeared to accurately predict the experimental ones, are shown in Figures 10(a), (b) and (c).

".|...[/I/

Figure 10. (a) First mode--bending; (b) second mode--twisting; (c) third mode--combined bending and

twisting. From Table 3, it can be seen that the element was not completely conforming, but as the differences in frequencies between Mesh I and Mesh 2 were considerably larger than that between Mesh 2 and Mesh 3, it appears that the element was partially conforming. Some

343

FREE VIBRATION OF TIIlN SHELLS TABLE 3

Natural frequencies of model whig (Hz) Mode 1 2 3

Mesh 1 66.5 190.2 376.7

Mesh 2 64.1 187"6 364.8

Mesh 3 65.5 181.8 368.7

Exptl. 49 172 308

Type B T C

B--bending; T--twisting; C---combined bending and twisting. error may have occurred in determining the frequencies for Mesh 3, as the problem was reduced from 186 degrees of freedom to 21. In any case, where non-conformity appeared to occur for modes l and 3, the frequency increased only by about 2 ~ and l ~ , respectively. As far as comparison with experimental frequencies was concerned, it appeared that the predictions for the fundamental bending mode were the worst o f the first three modes and that the combined bending and twisting mode was the second worst. This appeared to indicate that the end connection was the probable source of experimental error, any lack of end fixity affecting the bending modes more than the twisting modes. Other sources of error could have been in the assumptions for the elastic modulus and density, which were taken as 30 x l06 lbf/in 2 and 7.33 x l0 -4 Ibfs2/in 4, respectively. After a considerable amount of testing was carried out, a small crack became visible in the region where the two knuckles met and in a direction parallel to the damping plate edge. This crack appeared to cause a decrease in the fundamental bending frequency from 49 Hz to 42 Hz, but had little effect on the other two frequencies. This latter observation reinforced the argument that the fundamental frequency was very sensitive to the clamping method. 5. CONCLUSIONS The results have shown that when the partially conforming triangular plate bending element was used in conjunction with the plane stress element, the theoretical predictions for frequencies and mode shapes were quite successful for the lower modes. This was found particularly encouraging, because whereas the flat plate bending element had slope and deflection along its external boundaries, these properties tended to be lost on application to a curved surface. Thus, some loss in accuracy due to representing a curved surface with a flat triangular "shell-type" element, appeared to be offset by the advantages of using the simpler element. The results have also shown that end connections play an important role for cantilever shells, as the fundamental bending frequency for both the curved fan blade and the model wing showed the poorest correlation in the results presented. Although the use of fiat plate elements to represent thin shells has been in existence for a number of years, and has had varying degrees of success, the flat plate element presented in this paper has shown that it is very useful for the vibration of thin shells, particularly for the smaller computer, which cannot accommodate elements with a large number of degrees of freedom. ACKNOWLEDGMENTS The author would like to thank Dr W. Davey for permitting this work to be carried out. REFERENCES 1. M. D. OLSONand G. M. LINDBERG1969 Proceedings of the 2nd Conference on Afatrix Methods ht Structural Mechanics, Wright-PattersonAir ForceBase, Ohio, AFFDL-TR-68-150, pp. 247-269. Vibration analysis of cantilevered curved plates using a new cylindrical shell finite element.

344

c.T.F.

ROSS

2. G. M. LINDBERG and M. D. OLSON 1970 NRCC Aeronautical Memo ST-122. Finite element analysis of Lockheed cylindrical shell panel. 3. M. PETYT 1965 lnstilttle of Sound and Vibration Research Report No. 120. The application of finite element techniques to plate and shell problems. 4. O. C. ZIENKIE~,VICZand Y. K. CHEONO 1964 htternational S)'mposiltm oll Theory of Arch Dams, Southampton, pp. 123-140. Finite element method of analysis for arch dams and comparison with finite difference procedures. 5. J. H. ARGYRIS 1966 lngeniettr-Archiv. 35, 102-142. Matrix displacement analysis of plates and shells. 6. C. T. F. Ross 1973 Journal of Strain Analysis 8, 260-263. Partially conforming plate bending elements for static and dynamic analyses. 7. R. J. GUYAN 1965 American hlstitttte of Aeronautics and Astronautics Journal 3, 380. Reduction of stiffness and mass matrices. 8. M. J. TURNER, R. W. CLOUGtl, H. C. MARTIN and L. J. TopP 1956 Journal of the Aeronautical Sciences 23, 805-823. Stiffness and deflection analysis o f complex structures. 9. B. IRONS 1965 American htstitute of Aeronautics and Astronautics Journal 3, 961-962. Structural eigenvalue problems: elimination of unwanted variables. 10. J. L. TOCltER 1962 Ph.D. Thesis, University of California, Berkeley. Analysis of plate bending using triangular elements. I I. J. M. DEB NATH 1969 Ph.D. Thesis, Unicersity of Soztthampton. Dynamics of rectangular curved plates. 12. M. PETYT 1970 Conference on Recent Developments ht Sonic Fatigue, University of Southampton, July 1970. Vibration of curved plates. APPENDIX: NOTATION co-ordinates of triangular sub-element X, y, Z co-ordinates of triangular element xO yO Zo global co-ordinates of shell structure

ip displacement perpendicular to .~-p plane displacements in x, y and z directions, respectively o~,o,,o= rotations in x, y and z directions, respectively, according to the right-hand screw rule constants ~ 2 p 9 9 9 0'9 II, V~ It'

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