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s u g g e s t i o n s a b o u t t h e m a t h e m a t i c a l f o u n d a t i o n s of o p t i m a l l a y o u tp r o b l e m s .APPENDIX.-REFERENCES1. Kohn, R., and Strang, G., "Optimal Design for Torsional Rigidity," Conf.M ixed and H ybrid Finite Element Methods, Atlanta, Ga., Apr., 1981.2. Olhoff, N., and Rozvany, G. I. N., "Optimal Gril lage Layout for Given Natural Freq uenc y," Journal of the Engineering Mechanics Division, ASCE, Vol. 108,N o. EM4, 1982.3 . Prager, W., and Shield, R. T., "A General Theory of Optimal Plastic Design,"Journal of Applied Mechanics, ASME, Vol. 35, No. 1, 1967, pp. 184-186.4. Prager, W., and Rozvany, G. I. N., "Optimal Layout of Gril lages," Journalof Structural Mechanics, Vol. 5, No. 1, 1977, pp. 1-18.5. Prager, W., and Rozvany, G. I. N., "Optimization of Structural Geometry,"Proceedings Conference on 'Dynamic Systems,' A. R. Bednarek and L. Cesari ,

eds. , Academic Press, New York, N.Y., 1977, pp. 265-294.6. Rozvany, G. I. N., Optimization of Flexural Systems, Pergamon Press, Oxford,England, 1976.7. Rozvany, G. I. N., "Variational Methods and Optimality Criteria," Proceed-ings of NA TO ASI Optimization of Distributed Parameter Structures, E. J. Haugand J. Cea, ed s., Sijthoff & Noordhoff, A ph en a an den Rijn, The N eth erlands , 1981, p p . 82-111.8. Rozvany, G. I. N., "Optimality Criteria for Grids, Shells and Arches," Proceedings of NATO ASI Optimization of Distributed Parameter Structures, E. J. Haug,and J. Cea , ed s. , Sijthoff & Noordhoff, A ph en aa n d e n Rijn, The N eth erlands, 1981, pp . 112-151.9. Rozvany, G. I. N., Wang. C.-M., and Dow, M., "Prager-Structures: Arch-Grids and Cable Networks of Optimal Layout," Computer Methods in AppliedMechanics and Engineering, Vol. 31, 1982.10. Rozvany, G. I. N., Olhoff, N., Cheng, K.-T., and Taylor, J. E. "On the SolidPlate Paradox in Structural O ptim ization ," DCAMM Report No. 212, also Jour-nal of Structural Mechanics, Vol. 10, No. 1, 1982, pp. 1-32.

TECHNIQUE FOR ESTIMATING NATURAL FREQUENCIESBy Robert Schmidt ,1 M. ASCE

INTRODUCTIONTh e R a y l e i g h m e t h o d fo r e s t i m a t i n g fu n d a m e n t a l f r e q u e n c i e s o f b e a m s ,p l a t e s , a n d o t h e r s im i la r s t r u c t u r e s m a k e s u s e of a p p r o x i m a t i n g f u n c t ions of the form'Prof, of Engrg. Mech., Dept. of Civ. Engrg., Univ. of Detroit, Detroit, Mich.48221.Note.Discussion open until September 1, 1983. To extend the closing dateone month, a written request must be filed with the ASCE Manager of Technicaland Professional Publications. The manuscript for this paper was submitted forreview and possible publication on September 2, 1981. This paper is part of theJournal of Engineering Mechanics, Vol. 109, No. 2, April, 1983. ASCE, ISSN0733-9399/83/0002-0654/$01.00. Paper No. 17841.

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y = Cf(x1,x 2, ,. ., xM) (1)in which C = undetermined constant which may be set equal to unity;/ = assumed shape function; and x's represent the M independent var i ables. Herein, the utility of approximating functions of the formy = Cf(x ltx2, ...,xM; rti,n2, . . . , nN) (2)will be demonstrated by means of several relatively simple examples. InEq. 2, the constant parameters n are to be adjusted in such a way as tominimize (or nearly minimize) the approximate value of the natural frequency. (It should be recalled that Rayleigh's method yields an upperbound for the frequency.)EXAMPLE ONE

Consider a prismatic cantilever beam vibrating freely. Taking the origin of coordinates at the clamped end and assuming for the deflectionfunction y(x)y = cx2 (3)Thomson (3) obtains from Rayleigh's formula

"'['^"fA^h

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y = C{Ln-lx - *") (8)which satisfies the imposed boundary conditions y(0) = y(L) = 0. Eqs.4 and 8 yielda 2 = P A L V _ 3n2(n + 2)(2n + l)

EI 2(2n - 3)Jacobsen and Ayre (2) take n = 2 and obtain w2 = 120, w = 10.9545.Ho we ver, the m inimization of Eq. 9 leads to12n3 - 4n2 - 41n - 12 = 0 (10)from which n = 2.1442, so tha t Eq. 9 yields u>2 = 117.31, and w = 10.831.The exact value is w = IT2 = 9.8696 (Ref. 2).EXAMPLE THREE

Consider a w edg e-sha ped cant ilever bea m of constant thickness b a n dvariable depth h = Hx/L, in which the distance x is measured along thebeam from its t ip. Timoshenko, Young, and Weaver (4) assume for theshape function

* = c H ) 2

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+ 2(x/Lf, yields d>2 = 238.737 = (15.4511)2. However, even this accuratevalue can still be improved by assumingy = x2 - Bx" + Dx4 (14)in which B = 10/[12 - n(n - 1)], D = [n(n - 1) - 2]/[1 2 - n(n - 1)],if the conditions y(L) = y"(L) = 0 are to be satisfied. Substitution of Eq.14 in Eq. 4 yields a fairly long expression for w2 as a function of n. A napproximate minimization of this long expression by trial and improvement gives n 3.2 and w2 238.234 = (15.4348)2, a very accurate value(about 0.1% error).EXAMPLE FIVE

Consider a clamped circular plate of radius a, uniform thickness h, an dbending rigidity D. Assume for the shape functionw = C(a" - r"f (15)which satisfies the boundary conditions. The Rayleigh condition for thisproblem can be written in the form

, 2 w'2 u> 2phw 2\u/12 + - w'w" + )rdr = 0 (16)r r2 D /in which w' = dw/dr; an d r = the radial distance. Eqs. 15 and 16 yield3 ( - l) ( 2 - l)(3 n - 2) - - = 2(n + 1) (n + 2)(2n + 1 ) ( 3 M + 2)(2n2 - 3n + 2) (17)Differentiation of this equation is difficult, but a simple trial-and-correc-tion procedure leads to an approximate least value of n = 1.85, withwhich u2p/w 4/D = 104.979 = (10.2459)2, as compared to the exact valueof (10.21)2 which is found in Ref. 4.If n = 2 in Eq. 15, as in Ref. 4, Eq. 17 yields &)2p/w4/D = 106.667 =(10.328)2, a worse value than the preceding approximate one.CONCLUSIONS

The foregoing examples attest the usefulness of the proposed technique for estimating fundamental frequencies of nondiscrete systems.Moreover, they successfully introduce nonintegral powers of variablesinto the Rayleigh-Ritz analysis.APPENDIX.-REFERENCES1. Hutton, D. V., Applied Mechanical Vibrations, McGraw-Hill Book Co., Inc., NeYork, N.Y., 19812. Jacobsen, L. S., and Ayre, R. S., Engineering Vibrations, McGraw-Hill BookCo., Inc., New York, N.Y., 1958.3. Thomson, W. T., Theory of Vibration with Applications, 2nd ed., Prentice-HalInc., Englewood Cliffs, N.J., 1981.4. Timoshenko, S., Young, D. H., and Weaver, W., Jr., Vibration Problems inEngineering, 4th ed., John Wiley & Sons, New York, N.Y., 1974

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