1、LOAN C,OPY: RETURN TO AFWL (wLOL) KIX5AND AFl3, N MEX MODAL DENSITY OF THIN CIRCULAR CYLINDERS .I Prepared by NORTH CAROLINA STATE UNIVERSITY Raleigh, N. C. , fir 5 3 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION l WASHINGTON, D. C. l DECEtiiER 1967 I! Provided by IHSNot for ResaleNo reproduction or
2、 networking permitted without license from IHS-,-,-TECH LIBRARY KiJl=B, NM llnlllllllnlllllllllllllll 00bOL2b / NASA CR-897 MODbL DENSITY OF THIN CIRCULAR CYLINDERS ,., _- ,.,fii.l ! , w David%. Miller and Franklin D. Hart Distribution of this report i provided in the interest of information exchang
3、e. resides in the author r x NATIONAL AERONAUTICS D SPACE ADMINISTRATION ederol Scientific and Technical Information - - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without lice
4、nse from IHS-,-,-SUMMARY A combined analytical and experimental study is made of the modal density of a thin cylindrical shell Previous analytical work is dis- cussed and an integral form solution is presented and evaluated numeri- cally. Having cognizance of the experimental results, it is conclude
5、d that the integral form solution gives an accurate method for computing the cumulative number of resonant modes and the modal density of a thin cylindrical shell. iii Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo re
6、production or networking permitted without license from IHS-,-,-TABLE OF CONTENTS Page LISTOFTABLES . LISTOFFIGURES . INTRODUCTION REVIEWOFLITERATLJRE THE CONCEPT OF MODAL DENSITY General Discussion Method for Determining Modal Density . Example Modal Density Calculations Simply Supported Beam Simpl
7、y Supported Rectangular Plate . Clamped Circular Plate . Application of Modal Density . THEORETICAL DEVELOPMENT FOR SHELLS OF =VOLUTION Introduction . First Presentation . Second Presentation . Third Presentation . Summary EXPERIMENTAL PROGm1 Objective . Experimental Setup Operation . Reduction of D
8、ata . SUMMARY AND CONCLUSIONS . LIST OF REFERENCES . APPENDIX. LISTOFSYMBOLS . vi vii 1 4 7 i 12 12 -J-3 15 19 21 21 21 25 29 30 37 37 2 43 55 58 59 V Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-LIST OF TABLES Page 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
9、Tabulation of dimensionless number of natural frequencies and dimensionless modal density for a cylindrical shell using the modified Bolotin analysis . . . . . . . . . . . . . 31 Summary of analytical results for the number of natural frequencies and modal density of a cylindrical shell . . . . 33 E
10、xperimental cylinder specifications and nondinumsionalizing conversion factors for experimental data . . . . . . . . . . 38 Tabulation of standard one-third octave bands . . . . . . . . . 44 Experimental data for position 1 . . O . . . . . . . . . . . . 46 Experimental data for position 2 . . . . .
11、. . . . . . . . . . 47 Experimental data for position 3 . . . . . . . . . . . . . . . 48 Experimental data for position 4 . . . . . . . . . . . . . . . 49 Experimental data for position 5 . . O . . . . D . . . o . . . 50 Experimental data for position 6 . O . . . . . . . . . . . D . 51 vi Provided b
12、y IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-LIST OF FIGURES Page 1. Generalized re-ctangular region . . . . . . . . . . . . . . . . 10 2. k-space for generalized rectangular region . . . . . . . . . . 10 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. Simply
13、supported beam however, below the ring frequency there is sufficient differ- ence in the results presented to warrant further investigation of the matter. 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THE CONCEPT OF MODAL DENSITY General Discussi
14、on It is well known that any structure such as a beam, plate, or shell has an infinite number of resonant frequencies at which it may vibrate, with each frequency corresponding to each of the principal modes. In dealing with dynamic response problems wherein the input forcing quantity has a broad sp
15、ectral content, many modes will partici- pate in the overall motion of the vibrating system. In such cases, it is sometimes useful to introduce the concept of modal density. For a given structure, the modal density is defined as the asymptotic expres- sion for the density of the frequency distributi
16、on obtainable from the frequency equation of the structure. Thus, it is the continuous func- tion obtained by successively dividing the number of resonant frequen- cies contained in a frequency interval Aw by the interval width, AU. If AN(w) is the total number of resonances in the frequency band AU
17、, then the modal density at the center band frequency wc in the interval Ao may be written as, AN(w) n. (6) Since the waves in the beam are propagated only in a single direction (along the length of the beam) the k-space is one dimensional, Figure 3b, I.2 Provided by IHSNot for ResaleNo reproduction
18、 or networking permitted without license from IHS-,-,-and the equation for the number of resonant frequencies becomes N(w) = 1 / kl Akl 0 dkl which leads to N(w) = ;kl . But kl, from equations (4) and (5), becomes (7) (8) Therefore the expression for the number of resonant frequencies is . (10) Diff
19、erentiating the expression with respect to frequency yields, a n(w) = - 1 2a AZ? l L (11) Equation (11) is the expression for the modal density of a simply sup- ported beam. If the thickness of the beam is h, the radius of gyration is given by h/m. Simply Supported Rectanpular Plate A slightly bette
20、r example is given by the case of a rectangular plate, with simply supported edges, Figure 4a. The frequency equation for the plate may be expressed in the following form, Smith and Lyon (19651, 13 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-m27r
21、2 w = (- + c) KCL a; a; m,n = 1,2,3,. (12) where El and a2 are the length and width of the plate and K and CL are the same as in the case of the beam. Again the wave numbers are first defined as, k1 =? k2 = c l (13) The changes in the two wave numbers are then given by the expressions Akl = + 1 Ak2
22、= +- . 2 In this case two space variables are involved and therefore a two dimensional k-space is required, Figure 4b. The equation for the number of natural frequencies is written as in equation (31, N(w) f 1 I mp2 l AklAk2 s It is convenient in the case of the plate to integrate over the surface o
23、f the k-space using cylindrical coordinates. Therefore kf + kz = r2 and equation (3) becomes, N(w) = f$ Jr joI2 rdedr . II 0 0 (15) Integrating once with respect to r gives, V2 r/2 N(w) = - 2r2 0 I r2d0, (16) Then integrating with respect to 8 and substituting for r2, the follow- ing expression for
24、the number of resonant frequencies is obtained. 14 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-al!2 0 N(w)=Kc (17) L Finally equation (17) is differentiated with respect to frequency,w, to obtain the expression for the modal density of a rectangu
25、lar plate. (18) If the plate thickness is h, the radius of gyration is h/E and the expression becomes the same as that given by Heck1 (1962). n(w) = -g- hC . (19) L It is interesting to note that the modal density of a flat plate is a constant for a given plate and thus is independent of frequency.
26、Clamped Circular Plate As a final example the case of a flat circular plate will be examined, Figure 5a. For high frequencies, the frequency equation for a plate with either clamped or free edge conditions is given by, Rayleigh (1945), 2 ll W”- 4a2 KCL(n + 2m)2 where a is the plate radius, K the rad
27、ius ity of wave propagation along the plate. tion as 2 w = (E + L KCL , m,n = 1,2,3,. (20) of gyration, and C the veloc- L Rewriting the frequency equa- (21) the wave numbers may be defined in the following manner: 15 Provided by IHSNot for ResaleNo reproduction or networking permitted without licen
28、se from IHS-,-,-kl -E, k2=$. (22) Hence, the change in the wave numbers from one mode of vibration to the next is given by Akl=+, Ak2=% . (23) The wave propagation in the circular plate is again two dimensional, as in the case of the rectangular plate, so that the k-space is also two dimensional, Fi
29、gure 5b. The number of eigenvalues is again given by equation (3). Substituting the values for the change in wave numbers and converting to cylindrical coordinates yields, N(w) = ?- nE e se2 r2de . 1 (24) Noting that the frequency equation may be written in the form (kl + k212 = $- , L (25) and conv
30、erting the equation to cylindrical coordinates and solving for r2 yields, r2 = + 1 L (sine + 088) 0 (26) Hence, r2 may be eliminated from the above integral expressions for the number of eigenvalues. The limits on the integral are from 0 to r/2 since the argument of the integral exists for all value
31、s of 8 in the first quadrant. Therefore, (27) 16 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-(a) (b) FIGURE 3. SIMPLY SUPPORTED BEAM 8 k-SPACE S.S. El T S.S. S.S. II A- t- 12 ,y (a) (b) FIGURE 4. SIMPLY SUPPORTED RECTANGULAR PLATE S k-SPACE (b) F
32、IGURE 5. CLAMPED CIRCULAR PLATE a k-SPACE 17 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-which may be written in the form I“ da 1 + sina l Integration yields 2 N(w)=% how- ever, the above examples are sufficient to point out the usefulness of thi
33、s concept with regard to engineering applications. 20 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL DEVELOPMENT FOR SHELLS OF REVOLUTION Introduction In this section three different derivations of the modal density expression for the th
34、in cylindrical shell will be presented in detail. The first presentation is that of Bolotin (1963) in which the general shell of revolution is discussed and then -adapted to the case of the thin cylindrical shell. The second presentation is that of Heck1 (1962) in which the expressions are found for
35、 the cylindrical shell alone, The final presentation is essentially a modification of Bolotins work for the specific case of the cylindrical shell. First Presentation In his general derivation for thin shells of revolution, Bolotin (1963) examined the case of shells with two principal radii of curva
36、- ture, neglecteding the effects of tangential and inertial forces. The presentation is also restricted to shells which are simply supported at their edges. However, it is mentioned that the effects of boundary con- ditions on the vibrational modes are limited, and that only the first few modes of v
37、ibration are affected significantly. Hence the edge conditions are of little significance in the modal density expression development. In his work, Bolotin used the method described by Courant and Hilbert (1953) which has already been discussed to determine the number of resonant modes. Therefore th
38、e number of resonant modes in the shell is given by equation (3) N(w)= /jdkldk2 AklAk2 8 21 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-where k 1 and k 2 are the wave numbers. In this case Akl and Ak2 are given by n/al and v/a2 respectively where
39、 al and a2 are the principal dimen- sions of the shell surface. The surface integral is then put into cylindrical coordinates to obtain the following expression: N b) = y 1 $ rdrd0 = q se2 r2d0 . 8 271 e1 (34) The value of r which appears in equation (34) may be found from the frequency equation for
40、 the general shell of revolution. The frequency equation was obtained by solving the following differential equations for the shell, - phw2 = 0 (35) 2 i a2w -$-AA+- = 2 ax; Ri ax; 0 where xl and x 2 are the general curvilinear coordinates;R 1 and R 2 are the principal radii of curvature, D is the pl
41、ate st,iffness 3 (D = .-AL+-, ii ) p is the density , h the thickness of the shell, 12 Cl-IJ) E the modulus of elasticity, w the normal deflection, I# the stress function for the middle surface, and w the frequency of vibration. The solution of these equations results in the following frequency equation, 2 = 5, (k; + kg) + Eh (k;X. + kf) R1 W DR; (k; + kg) 1; “ii;. (36) 22 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
