1、c I cy U U cn U LOAN COPY: RETURN TO KIRTLAND AFB, N MEX AFWL MIL-2) A THEORETICAL ANALYSIS OF THE FREE VIBRATION OF DISCRETELY STIFFENED CYLINDRICAL SHELLS WITH ARBITRARY END CONDITIONS NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. JUNE 1969 . . . . . . I-“. . - Provided by IHSNot
2、 for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB. NR9 0069446 - NASA CR-1316 A THEORETICAL ANALYSIS OF THE FREE VIBRATION OF DISCRETELY STIFFENED CYLINDRICAL SHELLS WITH ARBITRARY END CONDITIONS By D. M. Egle and K. E. Soder, Jr. Distribution of this
3、report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it. Prepared under Grant No. NGR 37-003-035 by UNIVERSITY OF OKLAHOMA Norman, Okla. for Langley Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATIO
4、N For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ACKNOWLEDGEMENT The work reported herein was sponsored by NASA grant NGR
5、37-003-035, under the technical direction of the Dynamic Loads Division, Langley Research Center, with Mr. John L. Sewall acting as grant monitor, The authors are indebted to the University of Oklahoma Computation Center for financial support of a portion of this project. iv Provided by IHSNot for R
6、esaleNo reproduction or networking permitted without license from IHS-,-,-LIST OF FIGURES LIST OF TABLES NCMENCLATURE INTRODUCTION METHOD OF ANALYSIS NUMERICAL RESULTS CONCLUDING REMARKS REFERENCES APPENDIX I APPENDIX I1 APPENDIX I11 TABLE OF CONTENTS Page vi viii ix 1 3 27 55 57 62 65 72 V Provided
7、 by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-LIST OF FIGURES Figure 1 Geometry of Discretely Stiffened Cylinder 2 Geometric Detail of Eccentric Stiffeners 3 Circumferential and Longitudinal Radial Mode Shapes (w) of a Cylinder 4 Theoretical and Experiment
8、al Frequencies of an 5 Theoretical and Experimental Frequencies of an Unstiffened Clamped-Free Cylindrical Shell Unstiffened Freely-Supported Cylindrical Shell 6 Minimum Frequency of a Cylindrical Shell as a Function of the Number of Stringers with the Total Stringer Area and Torsional Stiffness Con
9、stant 7 Theoretical and Experimental Frequencies of a Freely-Supported Cylindrical Shell with Thirteen Equally Spaced Rings 8 Theoretical Axial Modes of a Freely-Supported Cylinder with Thirteen Equally Spaced Symmetric Rings (N=2) 9 Theoretical Axial Modes of a Freely-Supported Cylinder with Thirte
10、en Equally Spaced External Rings (N=2) 10 Theoretical Axial Modes of a Freely-Supported Cylinder with Thirteen Equally Spaced Symmetric Rings (N=lO) 11 Theoretical Axial Modes of a Freely-Supported Cylinder with Thirteen Equally Spaced External Rings (N=lO) 12 Theoretical and Experimental Frequencie
11、s of a Clamped-Free Cylindrical Shell with Three Rings and Sixteen Stringers 13 Theoretical Axial Modes of a Clamped-Free Cylinder with Three Rings and Sixteen Stringers (N=2)- 14 Theoretical Axial Modes of a Clamped-Free Cylinder with Three Rings and Sixteen Stringers (N=9) Page 6 -9 18 29 32 36 38
12、 42 43 44 45 49 52 53 vi Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Figure ,. Page 15 Theoretical Axial Modes of a Clamped-Free Cylinder with Three Rings and Sixteen Stringers (N=ll) 54 vi i Provided by IHSNot for ResaleNo reproduction or networ
13、king permitted without license from IHS-,-,-Table I I1 I11 IV V VI VI I LIST OF TABLES Shell Configurations used in Numerical Calculations Theoretical and Experimental Frequencies of an Unstiffened Clamped-Free Cylinder Theoretical and Experimental Frequencies of an Unstiffened Freely-Supported Cyli
14、nder Natural Frequencies of a Freely-Supported Cylindrical Shell with Four Internal Stringers Theoretical Frequencies of a Freely-Supported Cylinder with Thirteen Equally Spaced Rings Natural Frequencies of a Cylinder with Seven External Rings Theoretical and Expermental Frequencies of a Clamped-Fre
15、e Cylinder with Three Rings and Sixteen Stringers Page I 28 30 33 34 39 47 50 viii Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-a Ark7 D e IyzsE xzrk yysE xxrk cssE csrk K L m m* length of cylindrical shell. cross-sectional area of the kth ring ,
16、R stringer . th isotropic late flexural stiffness, t3/12 (l-v g 1 . strains. shell elastic modulus. elastic modulus of the kth ring, tth stringer. torsional stiffness of the kth ring, R stringer . th integers. moment of inertia of the kth stringer, kth ring cross-sectional area about an axis passing
17、 through the line of attachment and parallel to the z-axis. See Figure 2. product of inertia of the gth stringer, kth ring cross-sectional area about the yz, xz axes passing through the line of attachment. See Figure 2. moment of inertia of the kth stringer, kth ring cross-sectional area about the y
18、, x axis passing through the line of attachment. See Figure 2. cross stiffening parameters for the stringer, kth ring. See Appendix I. “ total number of rings. total number of stringers. axial wave number. maximum number of terms used in the axial dis- placement series. n number of circumferential f
19、ull waves. ix Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-n* R t T v u, v, w X Z rk rk A V w T Subscripts maximum number of terms used in the circumferential displacement series. radius of shell middle surface. shell thickness kinetic energy ; R+
20、t/2 In (-1 potential energy. shell middle surface displacements in the x, e, z directions. Bernoulli-Euler beam eigenfunctions. axial mode functions representing displacements in the x, 8, z directions. coordinates of kth stringer. kth ring centroidal axis referred to line 0f“atfachment. See Figure
21、2. coordinates of kth stringer, kth ring elastic axis referred to line of attachment. See Figure 2. v Kronecker delta function. frequency parameter, (1-v2) pC2w2/C. Poissons ratio of shell material. shell density. density of kth ring, kth stringer. angle of twist of a ring, stringer cross section ab
22、out its elastic axis. circular frequency. time refers to cylinder refers to the kth ring refers to the gth stringer X Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-r S refers to rings. refers to stringers. A coma before a subscript denotes partial
23、differentation with res ect to that subscript; e.g., v, denotes av/ay and w, denotes a2w/ax B . xi Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INTRODUCTION The vibration analysis of stiffened cylindrical shells has been and continues to be of con
24、siderable interest to structural analysts because of the wide spread use of this or similar type structures in air, space, and water craft. The degree of interest and the complexity of the problem are reflected in the number of publications in the literature devoted to this topic. The investigative
25、efforts may be divided into two broad classes: those which consider the stiffeners to be closely spaced and which average or “smear“ the stiffening effects over the entire surface of the shell thus effectively replacing the stiffened shell by an orthotropic shell; and those which do not consider the
26、 stiffeners to be closely spaced and do not take advantage of the simplification of averaging the stiffener effects. References (1-15) apply the averaging technique to the analysis of stiffened shells while the discrete approach is used in references (16- 47). The more recent studies using the avera
27、ged stiffener approach (10-15) have been concerned with the effect of stiffener eccentricity and have included that effect explicitly. Of those investigations using the discrete approach, references (16-20) are concerned with stringer stiffened shells, references (21-40) deal with ring stiffened cyl
28、inders, and references (41- 47) have considered both ring and stringer stiffeners. The present effort may be considered an extension of the work in reference (46) and the theory and part of the numerical results are, in essence, the same as that of reference (47). In this report, an analysis 1 Provi
29、ded by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-of the free vibrational characteristics of a thin uniform cylindrical shell with arbitrary end conditions and with an arbitrary number of ring and stringer stiffeners is developed. The stiffeners may be arbi
30、trarily spaced and need not be identical but are assumed to be uniform along the stiffener axis. The analysis considers the. effects of the flexure and extension of the shell; the flexure (about two perpendicular axes), extension, and torsion of the stiffeners, including the possibility of nonsymmet
31、ric stif- fener cross sections. Stiffener flexural cross stiffening is also in- cluded in an approximate manner. The three translational shell inertia components and all six of the stiffener inertia components are considered. The problem is formulated by the energy method and the Rayleigh-Ritz techn
32、ique is used to obtain an approximate solution. Numerical results for several configurations of stiffened cylinders are presented and compared to existing theoretical and experimental fre- quencies. The stiffened shells considered include freely supported stringer-stiffened and ring-stiffened cylind
33、ers, a clamped-free ring and stringer-stiffened shell and a clamped-clamped ring stiffened shell. 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-“DD OF ANALYSIS The method of analysis utilized is the Rayleigh-Ritz energy technique. The general app
34、roach of the method is outlined in the following steps. First, the expressions for the kinetic and potential energies are written for the cylinder, stringers, and rings. These six expressions are then used to give one expression for the total kinetic energy and one for the total potential energy of
35、the stiffened cylinder, which are then expressed in terms of the displacement of the middle surface of the cylinder. Next, deflection shapes are assumed in the form of a finite series with un- determined coefficients, where each term satisfies the appropriate end conditions. These assumed displaceme
36、nt series are substituted into the energy expressions, and Hamiltons principle is used to develop a linear eigenvalue problem in the undetermined coefficients. This eigenvalue problem is solved, allowing the calculation of the desired natural fre- quencies and mode shapes. Detailed Analysis The ener
37、gy expressions are written first in terms of the strain energy and then the strains are written in terms of the displacements of the middle surface of the shell to give the energy expressions as functions of the displacements. Only the strain energy due to the normal strain in the direction of.the s
38、tiffener axis and shear strain due to twisting about the stiffener axis are considered for the stiffeners. The normal strain in- cludes the effects due to extension of the stiffener and bending of the stiffener about two axes. The rotatory inertia of the shell is considered 3 Provided by IHSNot for
39、ResaleNo reproduction or networking permitted without license from IHS-,-,-negligible; however, the rotatory inertia is included in the stiffener kinetic energy terms. Potential Energies The strain displacement relations for a cylindrical shell with the coordinates shown in Figure 1 are given by Flu
40、gge (48) as en - u, - z w - xx - v, e zwee W eee R R(R+) R+Z “- +- where a comma before the subscript indicates differentiation with to the subscript (w, = =). These relationships are referred a 2w (la- c) respect to as Flugges exact strain relations, and assume that normals to the middle surface re
41、main normal after straining and that the displacements are small. The strain energy or the potential energy of the shell is found by considering a small element in a thin shell. Since the shell is considered thin, it is assumed that the normal stress uzz is zero throughout the element and that the o
42、ut of plane shear stresses are negligible (uxz = u = 0). Hookes law for an isotropic material in a state of plane stress is BZ E = l“ (en + ve 88 ) E o= ee (eee + (2a- c) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I-“ dVvol = u,de, + ueedeee + a
43、xe dexe (3) Substituting (2a-c) into (3) and integrating gives the strain energy per unit volume as - E lek e 2 “ + ve vvol - -(1“3 7 + 2 mee e +- 4 xe The total energy of the shell is then the integral over the volume of the shell Vc = V.Vold(Vol) ,To 1 or +- e (R+z) dxdedz 2 xe (6) where d (Vol) =
44、(R+z) dxdedz, and Ec is Youngs modulus of the cylinder. The strain energy of the cylinder is obtained as a function of t.he displace- ment of the middle surface by substituting (la-c) into equation (6) and integrating over the shell thichess. The potential energy for the cylin- drical shell may then
45、 be written as Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Figure 1. Geometry of Discretely Stiffened Cylinder 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-where T=ln( R + t/2 R“ Ect and = 12
46、(1“) This form of the shell potential energy can be shown to be equivalent to that developed by Miller (5) if the approximation is used in equation (7). The potential energy expressions for the stringers and rings will be developed with the assumption that these stiffeners are uniform along their le
47、ngth and have an asymmetric cross section. Further, it is assumed that only normal strains in the direction of the stiffener axis and shear- ing strains due to twisting about the stiffener axis are important. It is also assumed that the cross sectional planes do not warp. The elastic axis is chosen
48、as a reference line for the stiffener since it remains undeformed in a state of pure torsion, and the deforma- tions in this state may be described by a single variable, 4, the angular displacement of the cross section about the elastic axis. Since the elastic axis is chosen as the reference line, there is no coupling of the displacements of the elastic axis (u, vE, w,), which describe the flex- ural
