1、I- NASA TECHNICAL NOTE NASA TN D-3791 =m L, 1 -0 ms F -r: 0-m I+= h u-* 0- -2 T 0-g n ;L AFWL (WLIL-2) 3- TI I- -= E 4 -I o* LOAN COPY: RETURN T3 U-I - KIRTLAND AFB, N MEX ep M I - 0-g AFWL (WLIL-2) 3- TI ZP VIBRATION ANALYSIS OF CYLINDRICALLY CURVED PANELS WITH SIMPLY SUPPORTED OR CLAMPED EDGES AND
2、 COMPARISON WITH SOME EXPERIMENTS by John L. Sewull Langley Reseurcb Center Langley Station, Humpton, NATIONAL AERONAUTICS AND SPACE Va. ADMINISTRATION WASHINGTON, D. C. JANUARY 1967 J Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB
3、, NM I IIIII lllll II 11111 lllll lllll Ill1 llll 0 IJ 3 0 5 0 9 NASA TN D-3791 VIBRATION ANALYSIS OF CYLINDRICALLY CURVED PANELS WITH SIMPLY SUPPORTED OR CLAMPED EDGES AND COMPARISON WITH SOME EXPERIMENTS By John L. Sewall Langley Research Center Langley Station, Hampton, Va. NATIONAL AERONAUTICS A
4、ND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $2.00 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-VIBRATION ANALYSIS OF CYLINDRICALLY CURVED PANELS WITH SI
5、MPLY SUPPORTED OR CLAMPED EDGES AND COMPARISON WITH SOME EXPERIMENTS By John L. Sewall Langley Research Center SUMMARY The natural frequencies of cylindrically curved panels are calculated by application of the classical energy method employing the Rayleigh-Ritz procedure and are compared with measu
6、red frequencies of rectangular and square panels fastened at their edges by means of closely spaced bolts. The mode shapes used in the calculations are products of functions based on beam vibration mode shapes and satisfying geometrical edge condi- tions in both longitudinal and circumferential dire
7、ctions. General frequency equations are derived, and their applications to panels of small curvature lead to simplified Rayleigh-type frequency equations which were more satis- factory for certain modal combinations and edge conditions than for others. Measured frequencies were, for the most part, b
8、racketed by calculated frequencies based on simply supported or fully clamped edges and, at low-order circumferential modes, were gener- ally closer to calculated frequencies for simply supported edges but showed some tend- ency to move closer to calculated frequencies for clamped edges as the circu
9、mferential mode number increased. This behavior is attributed partly to membrane effects due to panel curvature and partly to complicated edge conditions due to the use of simple lap attachments which could not be represented adequately by theory. mental frequencies with inextensional frequencies, c
10、alculated with longitudinal and/or circumferential membrane strains omitted, indicated that actual panel mountings may have provided greater edge fixity along curved rather than along straight edges. Comparison of experi- Further efforts to bring experimental and theoretical frequencies into closer
11、agree- ment should include vibration tests on curved panels with edges designed to simulate simply supported or clamped conditions as closely as possible. In addition, the effects of modal functions other than beam functions should be examined, and other methods of analysis, such as those involving
12、finite difference or finite element techniques, should be applied. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INTRODUCTION Knowledge of the vibration characteristics of curved panels is important to designers confronted with aeroelastic and acou
13、stic problems in aerospace vehicle struc- tures in which curved panels are important structural components. Curved panels, like flat panels, possess many vibration modes which can respond to such dynamic forces as those due to acoustic excitation and unsteady aerodynamic flow. Panel frequencies and
14、mode shapes can be greatly affected by both curvature and edge support conditions. paper is concerned with both of these effects in comparisons of analytical and experi- mental frequencies of some thin curved panels. This Previous studies applicable to the vibrations of cylindrically curved panels a
15、re reported in references 1 to 6 for various combinations of edge conditions. In refer- ence 1, Reissner derives frequency equations for thin paraboloidal shallow shell elements with simply supported rectangular boundaries. Palmer (ref. 2) gives frequency equations for both simply supported and clam
16、ped-edge rectangular and square panels curved in both directions, but these equations are restricted to the mode with one half-wave in each direction. equations of motion of cylindrical shells (ref. 3), is applicable to cylindrically curved panels with simply supported straight edges and simply supp
17、orted, clamped, or free curved edges, and various combinations of these. In an extensive analytical treatment of the dynamic behavior of thin shallow shells of double curvature, Oniashvili in refer- ence 4, following an approach advocated by Vlasov in reference 5, considers the more general case of
18、arbitrary support conditions, with the mode shapes in each direction approximated by beam vibration functions. Some results are included in reference 4 showing the effects of curvature and edge clamping on the natural frequencies of spheri- cally curved panels, but most of the other examples, togeth
19、er with some experimental results on particular shell-roof configurations, are concerned with shell elements having simply supported edges. is to be found in these references. However, in a very recent paper (ref. 6) an extensive series of vibration experiments on cylindrically curved panels is repo
20、rted as part of a larger investigation concerned with sonic fatigue, and comparisons are made with the results of a vibration analysis based on a modal approach employing beam modes, as in reference 4. Forsbergs work, involving the direct numerical solution of the differential Very little, if any, c
21、orrelation between theory and experiment The present paper is exclusively concerned with the vibration of panels of small curvature, and its purpose is to report a study of the effects of curvature and edge con- ditions on the natural frequencies of some cylindrically curved panels for which exper-
22、imental frequency and mode shape data are known. The experimental data of reference 6 are used, together with the data of references 7 and 8, for panels of various radii of cur- vature, including the limiting case of infinite radius (that is, flat panel). The panels of 2 Provided by IHSNot for Resal
23、eNo reproduction or networking permitted without license from IHS-,-,-references 6 and 7 are rectangular and have very nearly the same dimensions but dif- ferent thicknesses. The panels of reference 8 are square, and the frequencies apply to just one panel thickness (namely, 0.020 in. (0.051 cm). Ex
24、perimental frequencies obtained in the same series of tests as reference 8 for other thicknesses (0.032 in. (0.081 cm) and 0.040 in. (0.10 cm), but not heretofore published, are included in the present paper and compared with calculated frequencies. Calculated frequencies in the present paper were o
25、btained by a straightforward application of the classical energy method employing the Rayleigh-Ritz procedure from which general frequency equations for arbitrary edge conditions are derived in the same manner as in references 9 and 10 for the vibrations of flat panels. Shell membrane and bending ef
26、fects are both included. Applications are based on the use of elementary beam vibration functions to approximate the mode shapes of the panel, as is done in refer- ences 4 to 6. Simplifications involving modal decoupling of the general Rayleigh-Ritz equations due to the use of beam functions are sho
27、wn to lead to a general approximate expression that is applicable to several different combinations of edge conditions. The main body of the paper begins with the development of the general frequency equations followed by applications involving the use of beam functions and leading to a general appr
28、oximate frequency equation and also to particular frequency equations for clamped-edge and simply supported curved panels. Although the main emphasis is on curvature and edge conditions, the effects of panel thickness are also discussed. SYMBOLS a radius of curvature to middle surface of panel (see
29、fig. 1) amplitude functions in assumed modal expansions (see eqs. (5) and (6) matrix elements in Rayleigh-Ritz frequency equations 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-C D E h I1 to Ifj 2 m n matrix elements in Rayleigh-Ritz frequency eq
30、uations Eh extensional (or membrane) stiffness, - bending stiffness, Youngs modulus 1 - lJ.2 Eh3 12(1 - p2) panel thickness surface integrals involving modal functions length of panel along straight edges (see fig. 1) integer denoting longitudinal modal component (number of axial half-waves for a pa
31、nel with simply supported curved edges) integer denoting circumferential modal component (number of circum- ferential half-waves for a panel with simply supported straight edges) eigenvalue of beam-mode approximation circumferential coordinate (see fig. 1) time kinetic energy strain energy displacem
32、ents of panel (see fig. 1) 4 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-X longitudinal coordinate (see fig. 1) xm(x) longitudinal mode shape component Yn(s circumferential mode shape component Q! central angle of panel (see fig. 1) pmyn=Nmt or N
33、naa A phw2 eigenvalue of frequency equation, - C A Ei9E2,E3 dimensionless eigenvalue, - Zaa 6 membrane strains (see, for example, eqs. (2) r2h coefficient in beam mode approximation Ym,n changes of curvature (see, for example, eqs. (2) K1 9 K29K12 P Poissons ratio P mass density of panel W angular f
34、requency, radians per second n d d2 ds2 ds YA(S) = - Yn(s); Yi(S) = - y (s) Subscripts: E. extensional j ,m longitudinal modal integers I inextensional k,n circumferential modal integers 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-PA X S genera
35、l modal integers denotes differentiation with respect to x denotes differentiation with respect to s Dots over quantities denote differentiation with respect to time. METHOD OF ANALYSIS The method of analysis used in this paper is the classical energy method employing the Rayleigh-Ritz procedure in
36、which the longitudinal, circumferential, and radial (or normal) displacements u, v, and w indicated in figure 1 are each repre- sented in terms of an arbitrary number of products of longitudinal and circumferential mode shapes. Two general frequency equations are derived, one in which the strain ene
37、rgy is written in terms of the complete strain-displacement relations given by Sanders in reference 11, and the other in which the simplifying relations of Donne11 (ref. 12) are employed. These equations are given in terms of general longitudinal and circumferential mode-shape components. In the app
38、lication of the method of analysis, functions for these components are chosen to satisfy desired edge conditions, and in the present paper the choice of beam vibration functions leads to a simplified frequency equation for curved panels with various combinations of edge conditions. Particular forms
39、of this equation are also included for curved panels with all edges fully clamped and with all edges simply supported. Derivation of General Frequency Equations The essential steps in the derivation are the same for both general frequency equa- tions and are presented for the equation based on the c
40、omplete strain-displacement rela- tions of reference 11 with in-plane inertias retained. Strain energy and strain-displacement . .- relations.- The strain energy for a thin isotropic cylindrically curved panel of thickness h may be written as 6 Provided by IHSNot for ResaleNo reproduction or network
41、ing permitted without license from IHS-,-,-where Z is the length along the straight edges, aa the length along the curved edges as shown in figure 1, C is the extensional (or membrane) stiffness - and D is Eh a 1 - ph E being Youngs modulus and Eh3 the inextensional (or bending) stiffness p Poissons
42、 ratio. 12(1 - p2) The quantities E 1, 2, . . . 12 in equation (1) are strains and changes of curva- ture which may be written in terms of Sanders displacement functions in reference 11: El=UX W 2 = vs +- a 1 where 1 longitudinal membrane strain 2 circumferential membrane strain middle surface shear
43、 strain 12 K1 longitudinal change of curvature circumferential change of curvature K2 torsional change of curvature K12 Subscripts on the displacements denote differentiation with respect to x and/or s. The strain-displacement relations due to Donne11 (ref. 12) are the same as equa- tions (2) except
44、 for the neglect of circumferential and longitudinal contributions to the changes of curvature; thus K and K are reduced to 7 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ss 1 K2 % -W (3) J K12 =-wxs Kinetic energy.- The total kinetic energy may b
45、e written as (Ya 1 T = (ti2 + C2 + w2)dx ds (4) 20 0 where p is the mass density of the panel and where dots over the displacements denote differentiation with respect to time. Modal functions.- The displacements u, v, and w are assumed to be of the form I mn J mn where amn(t), bmn(t), and cmn(t) ar
46、e time-dependent amplitude functions and m and n are integers identifying the longitudinal and circumferential mode shape components Xm and Yn, which are chosen to satisfy desired edge conditions. Primes on % and Yn denote dx Xm and - Yn, respectively. d d ds General equations of motion and frequenc
47、y equation.- With the use of equations (2) in equation (l), the strain energy becomes a function of u, v, and w and their spatial derivatives. Next, both strain and kinetic energies are obtained in terms of the modal functions in equations (5). With the assumption of simple harmonic motion of fre- q
48、uency w . 8 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- the total energy of the system is minimized with respect to the amplitudes Em, bmn, and cmn in accordance with the Rayleigh-Ritz procedure to obtain the equations of motion. kth circumferential modal combination may be written as - The basic relations involved in this minimization for the jth longitudinal and - thus, extensional and inextensional contributions to the frequency uncouple and lead to equation (12) as before. When the longitudinal strain 1 is set
