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Vibrations of orthotropic sandwich conical shells with free edges
Int. J . mech. Sei. Pergamon Press. 1969. Vol. 11, pp. 767-779. P r i n t e d in Great Britain
V I B R A T I O N S OF ORTHOTROPIC SANDWICH CONICAL SHELLS W I T H F R E E EDGES* CHARLES W . BERT a n d JOHN D. RAYt University of Oklahoma, Norman, Oklahoma
(Received 17 March 1969) S u m m a r y - - E x p e r i m e n t a l and analytical evaluations are presented for the vibrational characteristics of a truncated conical shell of sandwich construction. The shell consists of fiber glass-epoxy facings and aluminum honeycomb core. The shell is suspended by soft cords, to closely approximate free-free boundary conditions, and excited laterally by an electrodynamic shaker placed at various locations. The K e n n e d y - P a n c u method of data reduction is used to accurately separate vibrational modes having closely spaced frequencies. The two lowest unsymmetric-mode frequencies, for various values of circumferential wave number, agree quite closely with those predicted by a Rayleigh-Ritz inextensional analysis which includes the orthotropic nature of the composite material facings. NOTATION
A,B constants of integration appearing in equation (2) cit, d~t parameters defined by equations (7) Do shell flexural rigidity in circumferential direction; see equation (8) Dso shell twisting rigidity; see equation (9) Es, Eo Young's moduli of facing material in meridional and circumferential directions, respectively, psi Gso shear modulus of facings in sO plane, psi h distance between centroids of the facings, in. designation of general modes m circumferential wave n u m b e r n P,Q, R parameters defined b y equations (6) RI, R2 radii of small and large ends of shell, respectively, in. distance along meridian of middle surface of conical shell, measured from 8 apex, in. values of s at small and large ends of shell, respectively 81 , 8 2 t time, sec tl thickness of one facing, in. U , V, W meridional, circumferential and normal displacements, respectively, in. V, W normalized circumferential and normal displacements, given by equations (1 O) and (11) c~ semi-vertex angle of cone, degree completeness parameter = R1/R 2 EbO circumferential bending strain, dimensionless ~bo normalized value of ebO, defined b y equation (13), dimensionless 0 circumferential angular co-ordinate, degree ff change in curvature in circumferential direction, in-1 * This work is a part of research sponsored b y the U.S. Army Aviation Materiel Laboratories, Fort Eustis, Virginia, with James P. Waller as technical monitor. The assistance of W. C. Crisman in fabricating the shell and of B. L. Mayberry in instrumentation is gratefully acknowledged. t Presently at Memphis State University, Memphis, Tennessee. 767
768
CHARLES W . BERT a n d JOHN D. RAY
vao, vos
v p co toe, coi
e i g e n v a l u e ; see e q u a t i o n (4) P o i s s o n ' s r a t i o s of t h e facing m a t e r i a l , d i m e n s i o n l e s s P o i s s o n ' s r a t i o of isotropic m a t e r i a l , d i m e n s i o n l e s s m a t e r i a l d e n s i t y , lb-see~/in 4 frequency, rad./sec natural frequencies associated with purely extensional motion and with p u r e l y i n e x t e n s i o n a l m o t i o n , r e s p e c t i v e l y , rad./sec
1. I N T R O D U C T I O N THE OBJECTIVES of the present paper were (I) to measure resonant frequencies and associated modal strain distributions of a truncated conical sandwich shell suspended in the free-free condition and (2) to compare these quantities with theoretical predictions. A number of experimental investigations have been carried out to evaluate theories of vibration of sandwich-type beams and plates. 1 However, the authors do not know of any investigation on sandwich-type shell structures which are widely used in aircraft structures. The shell configuration used in the present investigation was that of a truncated cone. The scale selected was sufficiently large to be typical of an aircraft fuselage and to achieve resonant frequencies sufficiently low to enable accurate measurement of a large number of natural modes. Also, it is difficult to scale down a sandwich-type structure, i.e. if the core thickness is too thin, the sandwich effect is negligible. To be quite certain that the boundary conditions achieved in the experiments matched those used in analysis, the boundary conditions selected for the experiments were free edges. Also, these conditions were the least expensive to achieve experimentally and were used in previous experiments on homogeneous shells. ~ 6 Although the free-free condition is most easily achieved experimentally, it is perhaps the most difficult condition from an analytical viewpoint, particularly in the ease of a sandwich shell, because of the difficulty in finding functions that satisfy the boundary conditions. This is attested to b y the very limited number of analyses of homogeneous conical shells (or even homogeneous cylindrical shells) with these boundary conditions. The analysis presented here can be applied to truncated conical shells of either simple single-layer or sandwich construction with either isotropic or orthotropic (composite-material) facings. Inextensional motion, in which the middle-surface extensional strains and transverse shear strains are neglected, is assumed. Solution is carried out b y the Rayleigh-Ritz energy method for the two inextensional modes, which are the lowest unsymmetric modes. The circumferential wave number, n, may be any integer greater than one. The validity of the simplifying assumptions made in the analysis are verified b y the good agreement between the experimental and analytical results. 2. S P E C I M E N S
AND
MATERIAL
T a b l e 1 gives d a t a o n t h e s a n d w i c h c o n s t i t u e n t s , T a b l e 2 lists t h e d i m e n s i o n a l a n d w e i g h t c h a r a c t e r i s t i c s of t h e shell, a n d T a b l e 3 t a b u l a t e s t h e m a t e r i a l p r o p e r t i e s of t h e facings a n d core. T h e facing tensile p r o p e r t i e s a n d s h e a r m o d u l u s were d e t e r m i n e d f r o m t e s t s o n s m a l l tensile a n d t o r s i o n - t u b e specimens, w h i c h were s u b j e c t t o t h e s a m e c u r i n g cycle as t h e facing l a m i n a t e s .
V i b r a t i o n s of o r t h o t r o p i c s a n d w i c h conical shells w i t h free edges
769
TABLE 1. SANDWICH CONSTITUENTS Constituent
Description
Thickness
Facing
828-Z e p o x y , 181-E V o l a n A fiber glass
Two-ply (0.02 in. total)
W a r p parallel t o shell axis
Core
5052 A l u m i n u m , 1-mil n o n p e r f o r a t e d foil, 1/4 in. h e x a g o n a l cell
0.3 in.
Ribbon direction p a r a l l e l t o shell axis
Adhesive
A F - 110B filmsupported epoxy
0.015 in.
T A B L E 2.
Orientation
F I N A L SANDWICH S H E L L DIMENSIONS AND W E I G H T
R a d i u s t o m i d d l e o f cross-section (small end), R 1 R a d i u s t o m i d d l e of cross-section (large end), R 2 Axial length Conical h a l f a n g l e N o m i n a l surface a r e a of shell T o t a l w e i g h t o f shell
22.45 in. 28.86 in. 72.2 in. 5.06 ° 11,680 i n s 55 lb
TABLE 3. SHELL-CONSTITUENT MECHANICAL PROPERTIES
A. F a c i n g s ( a v e r a g e t e s t d a t a ) Initial tension modulus, parallel to warp Initial tension modulus, perpendicular to warp I n i t i a l P o i s s o n ' s ratio, l o a d i n g p a r a l l e l t o w a r p I n i t i a l P o i s s o n ' s ratio, l o a d i n g p e r p e n d i c u l a r to w a r p Initial shear modulus, shearing plane perpendicular to warp
E , = 3-64 x 10 e psi
Eo = 3.64 x 10 e psi v,o = 0.20 vo8 = 0"20 G,e = 1.33 x 106 psi
B. Core: S h e a r m o d u l u s , parallel t o r i b b o n d i r e c t i o n Shear modulus, perpendicular to ribbon direction
3. E X P E R I M E N T A L
0.0320 x 10 e psi 0.0183 x 106 psi
EQUIPMENT
T h e shells were s u s p e n d e d f r o m a large steel f r a m e b y six soft s p r i n g s {Fig. 1) so t h a t t h e s u s p e n s i o n r e s o n a n t f r e q u e n c y was b e l o w 1 cps. A n e l e c t r o d y n a m i c e x c i t e r (MB Model C l l , 50 lb m a x i m u m force c a p a c i t y ) w a s a t t a c h e d t o t h e shell b y a force l i n k t h a t was d e s i g n e d t o r e d u c e t h e c o u p l i n g b e t w e e n t h e s p e c i m e n a n d t h e steel f r a m e . T h e e x c i t e r was a t t a c h e d a t d i f f e r e n t l o c a t i o n s a r o u n d t h e shell a n d a l o n g t h e l e n g t h t o d e t e r m i n e t h e e x c i t e r effects o n t h e d a t a . A n a c e e l e r o m e t e r ( E n d e v c o , M o d e l F A - 7 2 ) w a s a t t a c h e d t o t h e a r m a t u r e of t h e e x c i t e r t o p r o v i d e a m e a s u r e m e n t of t h e e x c i t a t i o n a p p l i e d t o t h e shell.
770
CHARLES W. BERT a n d JOHN D. RAY
The instrumentation transducers used to measure the strain distribution at the r e s o n a n t f r e q u e n c i e s were 600 metallic-foil s t r a i n gages ( B u d d Model C6-141B) l o c a t e d o n t h e o u t e r facing of t h e shell in a n a r r a y as s h o w n o n Fig. 2. Two gages were l o c a t e d a t e a c h grid p o i n t , one t o m e a s u r e c i r c u m f e r e n t i a l s t r a i n a n d one t o m e a s u r e m e r i d i o n a l s t r a i n . T h e a s s o c i a t e d electronic i n s t m m ~ e n t a t i o n is s h o w n o n t h e p h o t o g r a p h in Fig. 1.
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PJIIIIII~III/I///J LOCATIONS
FIG. 2. L a y o u t o f shell s h o w i n g s t r a i n - g a g e locations. A t e a c h i n t e r i o r i n t e r s e c t i o n , t h e r e are t w o gages, one m e r i d i o n a l a n d one c i r c u m f e r e n t i a l . T h e o u t p u t s of t h e s t r a i n - g a g e s y s t e m a n d t h e a c c e l e r o m e t e r were a p p l i e d t o e a c h side of t h e digital p h a s e m e t e r ( A D - Y U D i g i t a l P h a s e M e t e r , Model 524A a n d E A I D i g i t a l V o l t m e t e r , Model 5002A), w h i c h i n d i c a t e d t h e r e l a t i v e p h a s e b e t w e e n t h e i n p u t a n d s t r a i n distribution. O t h e r i n s t r u m e n t s i n c l u d e d in t h e d a t a a c q u i s i t i o n s y s t e m i n c l u d e d a s t r o b o s c o p e ( G e n e r a l R a d i o , Model 1531-AB), u s e d s o m e t i m e s to help define t h e m o d a l shape, a n d a s e c o n d oscilloscope t o d i s p l a y a Lissajous p a t t e r n as a c h e c k o n t h e p h a s e - a n g l e q u a d r a n t obtained from the phase meter.
4. E X P E R I M E N T A L
PROCEDURE
T h e p r o c e d u r e u s e d to a c q u i r e r e s o n a n t - f r e q u e n c y a n d m o d a l - s t r a i n d i s t r i b u t i o n d a t a a t e a c h r e s o n a n c e was (1) t o m a k e a n a p p r o x i m a t e s u r v e y of t h e r e s o n a n t frequencies, (2) t o p i n p o i n t t h e r e s o n a n t f r e q u e n c y u s i n g a modified K e n n e d y - P a n c u t e c h n i q u e , 8 a n d (3) to define t h e m o d a l - s t r a i n d i s t r i b u t i o n a n d a p p r o x i m a t e n o d a l lines a t t h i s f r e q u e n c y . T h e first r e s o n a n c e s u r v e y was m a d e b y m o n i t o r i n g t h e i n p u t acceleration, s t r a i n o u t p u t s a n d frequencies for a few selected s t r a i n gages. F r e q u e n c y i n t e r v a l s were t a k e n a t 5 cps across t h e e n t i r e b a n d , w i t h 1 cps n e a r r e s o n a n c e . Closer i n t e r v a l s were t a k e n t o define t h e r e s o n a n t p o i n t m o r e precisely. Once t h e r e s o n a n t f r e q u e n c i e s were l o c a t e d , d a t a were t a k e n a t t h e s e frequencies to m a k e t h e m o d i f i e d K e n n e d y - P a n c u plot. T h e K e n n e d y - P a n c u d a t a were t a k e n b y v a r y i n g t h e e x c i t a t i o n f r e q u e n c y a n d b y m o n i t o r i n g t h e p h a s e a n g l e b e t w e e n t h e i n p u t a c c e l e r a t i o n a n d t h e s t r a i n - g a g e signal. T h e gage selected for t h e s e d a t a was t h e one t h a t g a v e t h e largest o u t p u t in t h e p r e l i m i n a r y p e a k - a m p l i t u d e s u r v e y . T h e e x c i t a t i o n f r e q u e n c y was v a r i e d f r o m a p o i n t b e l o w r e s o n a n c e t h a t i n d i c a t e d n e a r - z e r o p h a s e to a p o i n t a b o v e r e s o n a n c e t h a t i n d i c a t e d t h a t t h e p h a s e was r e t u r n i n g to zero. Since t h e p h a s e m e t e r w o u l d n o t f u n c t i o n below a t h r e s h o l d i n p u t v o l t a g e (0.35 v), d a t a were t a k e n o n l y in t h e g e n e r a l v i c i n i t y of r e s o n a n c e s . T h e s e d a t a were sufficient t o define t h e u n c o u p l e d r e s o n a n c e s b y m e a n s of t h e c h a r a c t e r i s t i c c i r c u l a r arcs o n t h e m o d i f i e d K e n n e d y - P a n c u plots. To m i n i m i z e t h e n o n l i n e a r effects of t h e shell, t h e s t r a i n level e x h i b i t e d o n t h e s t r a i n gage was held c o n s t a n t a n d t h e i n p u t level i n t o t h e shell was v a r i e d . T h e e x c i t a t i o n f r e q u e n c y was c h a n g e d u n t i l a p p r o x i m a t e l y a 10 ° v a r i a t i o n in p h a s e was n o t e d . T h e i n p u t force was v a r i e d u n t i l t h e gages e x h i b i t e d a p r e d e t e r m i n e d o u t p u t ; t h e n , t h e i n p u t acceleration, o u t p u t - g a g e signal, p h a s e a n g l e a n d f r e q u e n c y were recorded. T h e modified K e n n e d y - P a n c u p l o t was d r a w n f r o m t h e s e d a t a a n d t h e u n c o u p l e d r e s o n a n t frequencies were d e t e r m i n e d . T h e e x c i t a t i o n f r e q u e n c y of t h e shell was set a t t h e u n c o u p l e d r e s o n a n t f r e q u e n c y a n d t h e e x c i t a t i o n level was n o t c h a n g e d f r o m t h e p r e v i o u s d a t a . E a c h s t r a i n - g a g e o u t p u t a n d t h e c o r r e s p o n d i n g p h a s e were r e a d .
FIG. 1. Experimental vibration test set-up (A, conical sandwich shell; B. soft suspension springs; C, vibration exciter; D, instrumentation cart).
f. p. 770
Vibrations of orthotropic sandwich conical shells with free edges
771
From these data, the modal strain distribution was determined. Once the data were acquired, the shell nodal lines were investigated and checked approximately by feel and also visually using the stroboscope. The same procedure was repeated for each resonant frequency throughout the frequency range. Resonant frequencies were investigated from 5 to 400 cps. The lower limit was the lowest frequency of the exciter; the upper limit was reached when the strain signal-to. noise ratio was too low to obtain sufficiently reliable data. 5. ANALYSIS Most of the analyses for sandwich-type shells have assumed simply supported edges, and a few assumed clamped edges. As stated in Section 1, this is probably due to the increased mathematical complexity of the free-edge condition. In fact, only a very limited number of analyses of homogeneous, isotropic, conical shells with free edges have been carried out, and these only quite recently. After a careful review of previous analyses, it was decided to use the Rayleigh-Ritz method of analysis, but to include the inextensional modes only. This choice was moti· vated by these practical considerations: (1) the best agreement reported to date for the unsymmetric vibrations of free-free, homogeneous, isotropic shells was obtained by the inextensional analysis of Hu et al. ;9 (2) in a sandwich shell, the magnitude of the membrane strain energy relative to the bending strain energy is considerably smaller than that for a homogeneous shell. (This is due to the difference in the ratio of membrane stiffness to flexural stiffness.) Thus, in a sandwich shell, the error introduced by neglecting membrane effects is small except for the axisymmetric case (n = 0). This is clearly shown by an approximate relation presented by Platus lO for the natural frequencies, w, of a shell: (1)
where We is the frequency calculated by considering only extensional effects and Wi is the frequency calculated for the same n value but considering inextensional effects only. In deriving equation (1), it is assumed that the modal functions associated with the two kinds of modes are geometrically similar, so that they affect the frequency only through their effect on the stiffness. The hypothesis that the extensional effects are relatively small for a sandwich conical shell was verified, for the case of freely supported edges, by the analysis of Bacon and Bert.n The analysis presented in this section considers sandwich shells in the form of truncated cones with orthotropic facings and a perfectly rigid core. All components of translational inertia are included, but rotatory inertia is neglected. Ref. 9 presented an inextensional analysis of a homogeneous, isotropic, conical shell with free edges. Following Rayleigh's inextensional concept, the authors set the middlesurface strain components equal to zero and solved the resulting set of first-order differential equations to obtain the following expressions for the inextensional displacements: u = A sin ex cos ex sin nO cos Wi t, V
= (A+B8/82)ncosexcosnOcoswit,
w = [A(n 2-sin 2 ex) + Bn 2 8/8 2] sin nO cos Wi t.
(2)
J
Here the same general approach as used in ref. 9 is employed, except that an orthotropic sandwich conical shell is considered (see Fig. 3). The homogeneous, linear algebraic equations for the constants A and B appearing in equation (2) obtained from application of the Rayleigh-Ritz technique are as follows: (,\2 Cll - d ll ) A + (,\2 C12 - d 12 ) B = 0, (,\2 C12 - d 12 ) A + (,\2 C 12 - d 22 ) B = O.
(3)
The eigenvalues, '\, for these equations are defined as follows:
(4)
772
CHARLES
W.
BERT
and
JOHN
D.
RAY
The solutions of equations (3) are
A = [Q ± (Q2 - 4PR)i]/2P and the corresponding ratios of B to A are (5)
where (6)
SHELL NOTATION
CROSS SECTION
NOTATION
FIG. 3. Notation used in inextensional analysis.
The values for the factors in equations (6) are as follows: Cu = C12
[(I-y2)/2]{1-[(2n2-I)sin2();]/[n2(n2+cos2();)]),
= [(I-y3)/3][1-(sin2 ();)/(n 2 +cos 2 ();)],
= (l-y4)/4, du = {[(I/y2) -1]/2} {I + (Dse/D e) [(sin ();)/n 2 ]},
C 22
(7)
d 12 = (y-l_l),
d 22 = log. (l/y), where y is the completeness parameter (R 1 /R 2 ), and for a sandwich with thin facings, (8)
and (9)
Equations (7) are identical to those given in ref. 9 except the expression for du' However, for the isotropic case considered in ref. 9, Dae/De becomes 2(I-v) and the expression for d u coincides with that in ref. 9. When the expression for the B/A ratio, equation (5), is substituted into the expression for the circumferential displacement, the second of equations (2), the resulting expression for the normalized modal shape is (10)
Vibrations of orthotropic sandwich conical shells with free edges
773
Similarly, when equation (5) is substituted into the expression for normal displacement, the third of equations (2), the resulting expression for the normalized modal shape is (11)
Since the middle surface extensional strains and the meridional bending strain are zero, the only strain existing is the circumferential bending strain, ebB' The circumferential surface bending strain is (12)
where KB is the change in circumferential curvature. Thus, the meridional distribution of the dimensionless circumferential surface bending strain is (13)
6. EXPERIMENTAL AND ANALYTICAL RESULTS Resonant frequencies for various meridional modes are shown in Fig. 4 as a function of circumferential mode number. On this figure, the two curves drawn through the points calculated by the inextensional analysis presented in Section 5 are shown along with the experimental resonant-frequency data. Each resonant point was obtained from a modified Kennedy-Pancu plot. A typical plot that was used to separate the resonant frequencies is shown in Fig. 5. As can be seen from the figure, separation of the resonant frequencies which were very close together was accomplished quite adequately by the KennedyPancu technique. It can be seen in Fig. 4 that the experimentally measured resonantfrequency points associated with the two lowest unsymmetric modes agree quite closely with the frequencies calculated by the inextensional analysis over the range of circumferential wave numbers. The resonant-frequency curves for the higher modes indicate the same trend as those for the two lowest modes. In the region of low circumferential wave numbers, n, the higher modes were difficult to detect experimentally because the resonant frequency coincided with the resonant frequency of another mode. At these lower resonant frequencies, the response of the mode having a higher n would dominate that of the lower-n mode. This problem was one of the major difficulties encountered in applying the KennedyPancu technique in this investigation. It was beyond the scope of this investigation because of the limitation of equipment, but a solution to this problem would be to introduce additional excitation systems. By placing the exciters near a point that is a node of the strong mode, the system would exhibit a larger response in the weak mode. 4 80 r--,--,----,r-.,.--...,----,-,..-..,.-~-,-___, INEXTENSIONAL MODES
400
,GENERAL MODES
320
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FIG. 4. Experimental and analytical resonant frequency variations.
774
W.
CHARLES
BERT
and
JOHN
D.
RAY
At the lower circumferential wave numbers, it is believed that the higher modes will not indicate as severe a hook in the resonant frequency vs. n plot as those reported in refs. 2-4 for homogeneous-material conical shells. The reason for this was discussed in Section 5 in connexion with equation (1).
60°
120°
30°
'PHASE ANGLE, '"
0·8
0'4 RESPONSE
1·2
FACTOR, Xo
FIG. 5. Kennedy-Pancu plot for 38·47 and 38·29 cps. Shown as Figs. 6 and 7 are the plots of circumferential strain associated with the two lowest meridional modes. Also included in Figs. 6 and 7 are the analytical, circumferential strain distributions derived from the inextensional analysis presented in Section 5. In general, there is very good agreement between the analytical and experimental strain distributions for these modes. The effect of exciter mass and location is noticeable in the meridional modal shapes. However, this is to be expected since it was observed by Mixson. 5 The circumferential modal shapes are symmetric about the exciter location. Fig. 8 shows a plot of the meridional strain distribution associated with the two lowest meridional modes. The meridional strain distribution is identically zero in the inextensional analysis presented in Section 5. Although the curves indicate a meridional strain, the values are small and can be attributed to restraint of the shell from completely free movement at the shaker-attachment location and at the suspension points. n=2
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FIG. 6. Normalized circumferential strain distribution along meridian, inextensional modes, n = 2-5.
Vibrations of orthotropic sandwich conical shells with free edges
1.0~n=6
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SMALL END
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LARGE END
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FIG. 7. Normalized circumferential strain distribution along meridian, inextensional modes, n = 6-9.
o n=2 '" n=3
D n=6 v n=8
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FIG. 8. Meridional strain distribution along meridian, inextensional modes, n = 2, 3, 6, 8.
52
775
776
CHARLES W. BERT and JOHN D. RAY
Since the experimental data and the values derived by the inextensional analysis are in close agreement, it is assumed that the lowest experimental unsymmetrical modes are the inextensional modes. The agreement is close for the resonant frequencies as a function of circumferential wave number, for the circumferential strain as a function of meridional position and for the meridional strain as a function of meridional position. Therefore, hereafter, these two lowest experimental modes will be referred to as the inextensional modes. The other modes which have higher resonant frequencies will be referred to as general modes, since they exhibit both extensional and inextensional deformation as discussed in Section 5. The circumferential strain distributions associated with the circumferential modes for the inextensional and general modes are shown in Figs. 9 and 10. The circumferential strain distributions associated with the general modes are shown in Fig. 11, and the
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FIG. 9. Cross-section of circumferential waveforms, n = 0, 2, 3, 4.
FIG. 10. Cross-section of circumferential waveforms, n = 5, 6, 7, 8. distributions of meridional strain along the meridian are shown in Fig. 12. It is noted that the meridional strain is not uniform along the meridian, in contrast to the cases of the two inextensional modes. It can be seen in Fig. 4 that the resonant frequency for the m = 1 general mode is close to inextensional mode B at the low circumferential wave numbers, but as the wave number increases, the curves separate and the m = 1 mode converges to the m = 2 general mode. Also, it can be noted in this figure that all of the general modes tend to converge at the higher frequencies. This convergence is characteristic of both cylindrical and conical shells.
Vibrations of orthotropic sandwich conical shells with free edges
777
The resonant-frequency data and the associated strain distributions indicated that some coupling between modes occurred. These coupled modes were a combination of two circumferential modes having the same meridional mode when the resonant frequencies were close. Pronounced coupling phenomena were observed at resonant frequencies of 33, 64 and 94 cps. In all of these cases, the strain distributions were typically the same except for the difference in circumferential wave numbers. Fig. 13 is a plot of a nodal pattern typical of those associated with the above frequencies. C/)
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n. Normalized circumferential strain distribution along meridian, general modes.
The phenomenon of U-shaped nodal lines, i.e. the existence of more circumferential waves at the large end of a free-free conical-frustum shell than at the small end, was observed in experiments reported by Watkins and Clary.2,3 This phenomenon has been controversial, since it was not observed in any of the similar experiments reported by Hu et al.' Hu12 suggested that the U-shaped modes were caused by shaker effects, since his experiments' were excited by magnetic fields (no direct contact with shell). Additional comments were made by Koval,13, 14 who showed photographs of a similar phenomenon in vibrational experiments on cylindrical shells. He attributed the U-shaped-mode phenomenon to the dynamic asymmetries caused by the spot-welded longitudinal seams present in the shells of refs. 2, 3. According to the theory originated by Tobias15 and applied by Arnold and Warburton,16 the presence of small imperfections causes each of the two close resonant frequencies to have a preferential nodal pattern. Then, under excitation at a frequency between the two component frequencies, the two different component modes combine to produce the complicated mixed-mode pattern. As a result of the controversy, following the suggestion of Watkins and Clary,17 Mixson5,6 conducted an extensive series of additional experiments on shells with and without longitudinal seams. In these experiments, he used air shakers, as well as electrodynamic ones, and varied the ratio ofthe frequency of the axial rigid-body mode (controlled by the suspension stiffness) to that of the lowest elastic shell mode. He concluded that imperfections were probably the principal cause of the mixed modes, since they occurred most often in the cones with seams and smaller wall thickness (in which it is more difficult to control imperfections). In the present investigation, the coupling observed was probably caused by the presence of the four equally spaced longitudinal lap joints on the inside and outside facings. (i.e. a total of eight lap joints; one every 45°). However, it is interesting to note that the only modes which coupled were inextensional modes (A and B). There was no observed
778
CHARLES W. BERT and
JOHN
D.
RAY
coupling of the general modes (m = 1,2, 3) with each other or with the inextensional modes. This may have been because, for the particular shell construction and geometry, the inextensional modes were the only ones sufficiently close together (in frequency as well as in circumferential wave number) to couple.
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FIG. 12. Meridional strain distribution along meridian, general modes.
SHELL
LAYOUT
FIG. 13. Nodal pattern for coupled mode at 94 cps. At present, the authors know of no analytically based criterion for determining a priori whether or not two adjacent modes will couple to form a combined mode. Since the available experimental data are limited and expensive to obtain, an analytically based criterion, rather than an empirically based one, would be quite useful. For calculating the frequencies of certain higher modes peculiar to sandwich structures, Yu's analysis18 was used. The calculated frequencies for all three of these modes (shear modes in both the meridional and circumferential planes and the thickness-normal mode) were much higher than the highest frequencies attained in the experiments. Therefore, it is presumed that no modes of these types were excited in the experiments reported. 7. CONCLUSIONS
For free-free sandwich shells with orthotropic facings, an inextensional analysis predicts quite accurately the resonant frequencies associated with the first two unsymmetric modes. Under the conditions of the present experiments, extensional, core shear, and core normal effects can be neglected for these two modes.
Vibrations of orthotropic sandwich conical shells with free edges
779
Combined modes with V-shaped nodal patterns, as reported in previous experiments on free-free cones, were present. However, in all cases observed here they represented coupling between inextensional modes only. REFERENCES
1. N. L. ROUST, Vibration and Fatigue Sandwich Bibliography. Rexcel Corp., Dublin, Calif. (20 January, 1968). 2. J. D. WATKINS and R. R. CLARY, Vibrational Oharacteristics of Some Thin-walled Oylindrical and Oonical Frustum Shells, NASA TN D·2729 (1965). 3. J. D. WATKINS and R. R. CLARY, Am. Inst. Aer. Astr. J. 2, 1815 (1964). 4. W. C. L. Ru, J. F. GORMLEY and U. S. LINDHOLM, Flexural Vibrations of Oonical Shells with Free Edges, NASA CR-384 (1966). 5. J. S. MIXSON, J. Spacecraft and Rkts. 4, 414 (1967). 6. J. S. MIXSON, Experimental Modes of Vibration of 14° Oonicaljrustum Shells, NASA TN D-4428 (1968). 7. C. W. BERT, W. C. CRISMAN and G. M. NORDBY, J. Aircraft 5, 27 (1968). 8. J. D. RAY, C. W. BERT and D. M. EGLE, The Application of the Kennooy-Pancu Method to Experimental Vibration Studies of Oomplex Shell Structures, presented at the 39th Shock and Vibration Symposium, Monterey, Calif. (22-24 October, 1968). 9. W. C. L. Ru, J. F. GORMLEY and U. S. LINDHOLM, Int. J. mech. Sci. 9, 123 (1967). 10. D. R. PLATUS, Oonical Shell Vibrations, NASA TN D·2767 (1965). n. M. D. BACON and C. W. BERT, Am. Inst. Aer. Astr. J. 5, 413 (1967). 12. W. C. L. Ru, discussion of Ref. 3, Am. Inst. Aer. Astr. J. 3, 1213 (1965). 13. L. R. KOVAL, J. acoust. Soc. Am. 35, 252 (1963). 14. L. R. KOVAL, Am. Inst. Aer. Astr. J. 4, 571 (1966). 15. S. A. TOBIAS, Engineering 172, 409 (1951). 16. R. N. ARNOLD and G. B. WARBURTON, Proc. R. Soc. 197A, 238 (1949). 17. J. D. WATKINS and R. R. CLARY, authors' reply to discussion of Ref. 3, Am. Inst. Aer. Astr. J. 3, 1213 (1965). 18. Y.-Y. Yu, J. appl. Mech. 27, 653 (1960).
V I B R A T I O N S OF ORTHOTROPIC SANDWICH CONICAL SHELLS W I T H F R E E EDGES* CHARLES W . BERT a n d JOHN D. RAYt University of Oklahoma, Norman, Oklahoma
(Received 17 March 1969) S u m m a r y - - E x p e r i m e n t a l and analytical evaluations are presented for the vibrational characteristics of a truncated conical shell of sandwich construction. The shell consists of fiber glass-epoxy facings and aluminum honeycomb core. The shell is suspended by soft cords, to closely approximate free-free boundary conditions, and excited laterally by an electrodynamic shaker placed at various locations. The K e n n e d y - P a n c u method of data reduction is used to accurately separate vibrational modes having closely spaced frequencies. The two lowest unsymmetric-mode frequencies, for various values of circumferential wave number, agree quite closely with those predicted by a Rayleigh-Ritz inextensional analysis which includes the orthotropic nature of the composite material facings. NOTATION
A,B constants of integration appearing in equation (2) cit, d~t parameters defined by equations (7) Do shell flexural rigidity in circumferential direction; see equation (8) Dso shell twisting rigidity; see equation (9) Es, Eo Young's moduli of facing material in meridional and circumferential directions, respectively, psi Gso shear modulus of facings in sO plane, psi h distance between centroids of the facings, in. designation of general modes m circumferential wave n u m b e r n P,Q, R parameters defined b y equations (6) RI, R2 radii of small and large ends of shell, respectively, in. distance along meridian of middle surface of conical shell, measured from 8 apex, in. values of s at small and large ends of shell, respectively 81 , 8 2 t time, sec tl thickness of one facing, in. U , V, W meridional, circumferential and normal displacements, respectively, in. V, W normalized circumferential and normal displacements, given by equations (1 O) and (11) c~ semi-vertex angle of cone, degree completeness parameter = R1/R 2 EbO circumferential bending strain, dimensionless ~bo normalized value of ebO, defined b y equation (13), dimensionless 0 circumferential angular co-ordinate, degree ff change in curvature in circumferential direction, in-1 * This work is a part of research sponsored b y the U.S. Army Aviation Materiel Laboratories, Fort Eustis, Virginia, with James P. Waller as technical monitor. The assistance of W. C. Crisman in fabricating the shell and of B. L. Mayberry in instrumentation is gratefully acknowledged. t Presently at Memphis State University, Memphis, Tennessee. 767
768
CHARLES W . BERT a n d JOHN D. RAY
vao, vos
v p co toe, coi
e i g e n v a l u e ; see e q u a t i o n (4) P o i s s o n ' s r a t i o s of t h e facing m a t e r i a l , d i m e n s i o n l e s s P o i s s o n ' s r a t i o of isotropic m a t e r i a l , d i m e n s i o n l e s s m a t e r i a l d e n s i t y , lb-see~/in 4 frequency, rad./sec natural frequencies associated with purely extensional motion and with p u r e l y i n e x t e n s i o n a l m o t i o n , r e s p e c t i v e l y , rad./sec
1. I N T R O D U C T I O N THE OBJECTIVES of the present paper were (I) to measure resonant frequencies and associated modal strain distributions of a truncated conical sandwich shell suspended in the free-free condition and (2) to compare these quantities with theoretical predictions. A number of experimental investigations have been carried out to evaluate theories of vibration of sandwich-type beams and plates. 1 However, the authors do not know of any investigation on sandwich-type shell structures which are widely used in aircraft structures. The shell configuration used in the present investigation was that of a truncated cone. The scale selected was sufficiently large to be typical of an aircraft fuselage and to achieve resonant frequencies sufficiently low to enable accurate measurement of a large number of natural modes. Also, it is difficult to scale down a sandwich-type structure, i.e. if the core thickness is too thin, the sandwich effect is negligible. To be quite certain that the boundary conditions achieved in the experiments matched those used in analysis, the boundary conditions selected for the experiments were free edges. Also, these conditions were the least expensive to achieve experimentally and were used in previous experiments on homogeneous shells. ~ 6 Although the free-free condition is most easily achieved experimentally, it is perhaps the most difficult condition from an analytical viewpoint, particularly in the ease of a sandwich shell, because of the difficulty in finding functions that satisfy the boundary conditions. This is attested to b y the very limited number of analyses of homogeneous conical shells (or even homogeneous cylindrical shells) with these boundary conditions. The analysis presented here can be applied to truncated conical shells of either simple single-layer or sandwich construction with either isotropic or orthotropic (composite-material) facings. Inextensional motion, in which the middle-surface extensional strains and transverse shear strains are neglected, is assumed. Solution is carried out b y the Rayleigh-Ritz energy method for the two inextensional modes, which are the lowest unsymmetric modes. The circumferential wave number, n, may be any integer greater than one. The validity of the simplifying assumptions made in the analysis are verified b y the good agreement between the experimental and analytical results. 2. S P E C I M E N S
AND
MATERIAL
T a b l e 1 gives d a t a o n t h e s a n d w i c h c o n s t i t u e n t s , T a b l e 2 lists t h e d i m e n s i o n a l a n d w e i g h t c h a r a c t e r i s t i c s of t h e shell, a n d T a b l e 3 t a b u l a t e s t h e m a t e r i a l p r o p e r t i e s of t h e facings a n d core. T h e facing tensile p r o p e r t i e s a n d s h e a r m o d u l u s were d e t e r m i n e d f r o m t e s t s o n s m a l l tensile a n d t o r s i o n - t u b e specimens, w h i c h were s u b j e c t t o t h e s a m e c u r i n g cycle as t h e facing l a m i n a t e s .
V i b r a t i o n s of o r t h o t r o p i c s a n d w i c h conical shells w i t h free edges
769
TABLE 1. SANDWICH CONSTITUENTS Constituent
Description
Thickness
Facing
828-Z e p o x y , 181-E V o l a n A fiber glass
Two-ply (0.02 in. total)
W a r p parallel t o shell axis
Core
5052 A l u m i n u m , 1-mil n o n p e r f o r a t e d foil, 1/4 in. h e x a g o n a l cell
0.3 in.
Ribbon direction p a r a l l e l t o shell axis
Adhesive
A F - 110B filmsupported epoxy
0.015 in.
T A B L E 2.
Orientation
F I N A L SANDWICH S H E L L DIMENSIONS AND W E I G H T
R a d i u s t o m i d d l e o f cross-section (small end), R 1 R a d i u s t o m i d d l e of cross-section (large end), R 2 Axial length Conical h a l f a n g l e N o m i n a l surface a r e a of shell T o t a l w e i g h t o f shell
22.45 in. 28.86 in. 72.2 in. 5.06 ° 11,680 i n s 55 lb
TABLE 3. SHELL-CONSTITUENT MECHANICAL PROPERTIES
A. F a c i n g s ( a v e r a g e t e s t d a t a ) Initial tension modulus, parallel to warp Initial tension modulus, perpendicular to warp I n i t i a l P o i s s o n ' s ratio, l o a d i n g p a r a l l e l t o w a r p I n i t i a l P o i s s o n ' s ratio, l o a d i n g p e r p e n d i c u l a r to w a r p Initial shear modulus, shearing plane perpendicular to warp
E , = 3-64 x 10 e psi
Eo = 3.64 x 10 e psi v,o = 0.20 vo8 = 0"20 G,e = 1.33 x 106 psi
B. Core: S h e a r m o d u l u s , parallel t o r i b b o n d i r e c t i o n Shear modulus, perpendicular to ribbon direction
3. E X P E R I M E N T A L
0.0320 x 10 e psi 0.0183 x 106 psi
EQUIPMENT
T h e shells were s u s p e n d e d f r o m a large steel f r a m e b y six soft s p r i n g s {Fig. 1) so t h a t t h e s u s p e n s i o n r e s o n a n t f r e q u e n c y was b e l o w 1 cps. A n e l e c t r o d y n a m i c e x c i t e r (MB Model C l l , 50 lb m a x i m u m force c a p a c i t y ) w a s a t t a c h e d t o t h e shell b y a force l i n k t h a t was d e s i g n e d t o r e d u c e t h e c o u p l i n g b e t w e e n t h e s p e c i m e n a n d t h e steel f r a m e . T h e e x c i t e r was a t t a c h e d a t d i f f e r e n t l o c a t i o n s a r o u n d t h e shell a n d a l o n g t h e l e n g t h t o d e t e r m i n e t h e e x c i t e r effects o n t h e d a t a . A n a c e e l e r o m e t e r ( E n d e v c o , M o d e l F A - 7 2 ) w a s a t t a c h e d t o t h e a r m a t u r e of t h e e x c i t e r t o p r o v i d e a m e a s u r e m e n t of t h e e x c i t a t i o n a p p l i e d t o t h e shell.
770
CHARLES W. BERT a n d JOHN D. RAY
The instrumentation transducers used to measure the strain distribution at the r e s o n a n t f r e q u e n c i e s were 600 metallic-foil s t r a i n gages ( B u d d Model C6-141B) l o c a t e d o n t h e o u t e r facing of t h e shell in a n a r r a y as s h o w n o n Fig. 2. Two gages were l o c a t e d a t e a c h grid p o i n t , one t o m e a s u r e c i r c u m f e r e n t i a l s t r a i n a n d one t o m e a s u r e m e r i d i o n a l s t r a i n . T h e a s s o c i a t e d electronic i n s t m m ~ e n t a t i o n is s h o w n o n t h e p h o t o g r a p h in Fig. 1.
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FIG. 2. L a y o u t o f shell s h o w i n g s t r a i n - g a g e locations. A t e a c h i n t e r i o r i n t e r s e c t i o n , t h e r e are t w o gages, one m e r i d i o n a l a n d one c i r c u m f e r e n t i a l . T h e o u t p u t s of t h e s t r a i n - g a g e s y s t e m a n d t h e a c c e l e r o m e t e r were a p p l i e d t o e a c h side of t h e digital p h a s e m e t e r ( A D - Y U D i g i t a l P h a s e M e t e r , Model 524A a n d E A I D i g i t a l V o l t m e t e r , Model 5002A), w h i c h i n d i c a t e d t h e r e l a t i v e p h a s e b e t w e e n t h e i n p u t a n d s t r a i n distribution. O t h e r i n s t r u m e n t s i n c l u d e d in t h e d a t a a c q u i s i t i o n s y s t e m i n c l u d e d a s t r o b o s c o p e ( G e n e r a l R a d i o , Model 1531-AB), u s e d s o m e t i m e s to help define t h e m o d a l shape, a n d a s e c o n d oscilloscope t o d i s p l a y a Lissajous p a t t e r n as a c h e c k o n t h e p h a s e - a n g l e q u a d r a n t obtained from the phase meter.
4. E X P E R I M E N T A L
PROCEDURE
T h e p r o c e d u r e u s e d to a c q u i r e r e s o n a n t - f r e q u e n c y a n d m o d a l - s t r a i n d i s t r i b u t i o n d a t a a t e a c h r e s o n a n c e was (1) t o m a k e a n a p p r o x i m a t e s u r v e y of t h e r e s o n a n t frequencies, (2) t o p i n p o i n t t h e r e s o n a n t f r e q u e n c y u s i n g a modified K e n n e d y - P a n c u t e c h n i q u e , 8 a n d (3) to define t h e m o d a l - s t r a i n d i s t r i b u t i o n a n d a p p r o x i m a t e n o d a l lines a t t h i s f r e q u e n c y . T h e first r e s o n a n c e s u r v e y was m a d e b y m o n i t o r i n g t h e i n p u t acceleration, s t r a i n o u t p u t s a n d frequencies for a few selected s t r a i n gages. F r e q u e n c y i n t e r v a l s were t a k e n a t 5 cps across t h e e n t i r e b a n d , w i t h 1 cps n e a r r e s o n a n c e . Closer i n t e r v a l s were t a k e n t o define t h e r e s o n a n t p o i n t m o r e precisely. Once t h e r e s o n a n t f r e q u e n c i e s were l o c a t e d , d a t a were t a k e n a t t h e s e frequencies to m a k e t h e m o d i f i e d K e n n e d y - P a n c u plot. T h e K e n n e d y - P a n c u d a t a were t a k e n b y v a r y i n g t h e e x c i t a t i o n f r e q u e n c y a n d b y m o n i t o r i n g t h e p h a s e a n g l e b e t w e e n t h e i n p u t a c c e l e r a t i o n a n d t h e s t r a i n - g a g e signal. T h e gage selected for t h e s e d a t a was t h e one t h a t g a v e t h e largest o u t p u t in t h e p r e l i m i n a r y p e a k - a m p l i t u d e s u r v e y . T h e e x c i t a t i o n f r e q u e n c y was v a r i e d f r o m a p o i n t b e l o w r e s o n a n c e t h a t i n d i c a t e d n e a r - z e r o p h a s e to a p o i n t a b o v e r e s o n a n c e t h a t i n d i c a t e d t h a t t h e p h a s e was r e t u r n i n g to zero. Since t h e p h a s e m e t e r w o u l d n o t f u n c t i o n below a t h r e s h o l d i n p u t v o l t a g e (0.35 v), d a t a were t a k e n o n l y in t h e g e n e r a l v i c i n i t y of r e s o n a n c e s . T h e s e d a t a were sufficient t o define t h e u n c o u p l e d r e s o n a n c e s b y m e a n s of t h e c h a r a c t e r i s t i c c i r c u l a r arcs o n t h e m o d i f i e d K e n n e d y - P a n c u plots. To m i n i m i z e t h e n o n l i n e a r effects of t h e shell, t h e s t r a i n level e x h i b i t e d o n t h e s t r a i n gage was held c o n s t a n t a n d t h e i n p u t level i n t o t h e shell was v a r i e d . T h e e x c i t a t i o n f r e q u e n c y was c h a n g e d u n t i l a p p r o x i m a t e l y a 10 ° v a r i a t i o n in p h a s e was n o t e d . T h e i n p u t force was v a r i e d u n t i l t h e gages e x h i b i t e d a p r e d e t e r m i n e d o u t p u t ; t h e n , t h e i n p u t acceleration, o u t p u t - g a g e signal, p h a s e a n g l e a n d f r e q u e n c y were recorded. T h e modified K e n n e d y - P a n c u p l o t was d r a w n f r o m t h e s e d a t a a n d t h e u n c o u p l e d r e s o n a n t frequencies were d e t e r m i n e d . T h e e x c i t a t i o n f r e q u e n c y of t h e shell was set a t t h e u n c o u p l e d r e s o n a n t f r e q u e n c y a n d t h e e x c i t a t i o n level was n o t c h a n g e d f r o m t h e p r e v i o u s d a t a . E a c h s t r a i n - g a g e o u t p u t a n d t h e c o r r e s p o n d i n g p h a s e were r e a d .
FIG. 1. Experimental vibration test set-up (A, conical sandwich shell; B. soft suspension springs; C, vibration exciter; D, instrumentation cart).
f. p. 770
Vibrations of orthotropic sandwich conical shells with free edges
771
From these data, the modal strain distribution was determined. Once the data were acquired, the shell nodal lines were investigated and checked approximately by feel and also visually using the stroboscope. The same procedure was repeated for each resonant frequency throughout the frequency range. Resonant frequencies were investigated from 5 to 400 cps. The lower limit was the lowest frequency of the exciter; the upper limit was reached when the strain signal-to. noise ratio was too low to obtain sufficiently reliable data. 5. ANALYSIS Most of the analyses for sandwich-type shells have assumed simply supported edges, and a few assumed clamped edges. As stated in Section 1, this is probably due to the increased mathematical complexity of the free-edge condition. In fact, only a very limited number of analyses of homogeneous, isotropic, conical shells with free edges have been carried out, and these only quite recently. After a careful review of previous analyses, it was decided to use the Rayleigh-Ritz method of analysis, but to include the inextensional modes only. This choice was moti· vated by these practical considerations: (1) the best agreement reported to date for the unsymmetric vibrations of free-free, homogeneous, isotropic shells was obtained by the inextensional analysis of Hu et al. ;9 (2) in a sandwich shell, the magnitude of the membrane strain energy relative to the bending strain energy is considerably smaller than that for a homogeneous shell. (This is due to the difference in the ratio of membrane stiffness to flexural stiffness.) Thus, in a sandwich shell, the error introduced by neglecting membrane effects is small except for the axisymmetric case (n = 0). This is clearly shown by an approximate relation presented by Platus lO for the natural frequencies, w, of a shell: (1)
where We is the frequency calculated by considering only extensional effects and Wi is the frequency calculated for the same n value but considering inextensional effects only. In deriving equation (1), it is assumed that the modal functions associated with the two kinds of modes are geometrically similar, so that they affect the frequency only through their effect on the stiffness. The hypothesis that the extensional effects are relatively small for a sandwich conical shell was verified, for the case of freely supported edges, by the analysis of Bacon and Bert.n The analysis presented in this section considers sandwich shells in the form of truncated cones with orthotropic facings and a perfectly rigid core. All components of translational inertia are included, but rotatory inertia is neglected. Ref. 9 presented an inextensional analysis of a homogeneous, isotropic, conical shell with free edges. Following Rayleigh's inextensional concept, the authors set the middlesurface strain components equal to zero and solved the resulting set of first-order differential equations to obtain the following expressions for the inextensional displacements: u = A sin ex cos ex sin nO cos Wi t, V
= (A+B8/82)ncosexcosnOcoswit,
w = [A(n 2-sin 2 ex) + Bn 2 8/8 2] sin nO cos Wi t.
(2)
J
Here the same general approach as used in ref. 9 is employed, except that an orthotropic sandwich conical shell is considered (see Fig. 3). The homogeneous, linear algebraic equations for the constants A and B appearing in equation (2) obtained from application of the Rayleigh-Ritz technique are as follows: (,\2 Cll - d ll ) A + (,\2 C12 - d 12 ) B = 0, (,\2 C12 - d 12 ) A + (,\2 C 12 - d 22 ) B = O.
(3)
The eigenvalues, '\, for these equations are defined as follows:
(4)
772
CHARLES
W.
BERT
and
JOHN
D.
RAY
The solutions of equations (3) are
A = [Q ± (Q2 - 4PR)i]/2P and the corresponding ratios of B to A are (5)
where (6)
SHELL NOTATION
CROSS SECTION
NOTATION
FIG. 3. Notation used in inextensional analysis.
The values for the factors in equations (6) are as follows: Cu = C12
[(I-y2)/2]{1-[(2n2-I)sin2();]/[n2(n2+cos2();)]),
= [(I-y3)/3][1-(sin2 ();)/(n 2 +cos 2 ();)],
= (l-y4)/4, du = {[(I/y2) -1]/2} {I + (Dse/D e) [(sin ();)/n 2 ]},
C 22
(7)
d 12 = (y-l_l),
d 22 = log. (l/y), where y is the completeness parameter (R 1 /R 2 ), and for a sandwich with thin facings, (8)
and (9)
Equations (7) are identical to those given in ref. 9 except the expression for du' However, for the isotropic case considered in ref. 9, Dae/De becomes 2(I-v) and the expression for d u coincides with that in ref. 9. When the expression for the B/A ratio, equation (5), is substituted into the expression for the circumferential displacement, the second of equations (2), the resulting expression for the normalized modal shape is (10)
Vibrations of orthotropic sandwich conical shells with free edges
773
Similarly, when equation (5) is substituted into the expression for normal displacement, the third of equations (2), the resulting expression for the normalized modal shape is (11)
Since the middle surface extensional strains and the meridional bending strain are zero, the only strain existing is the circumferential bending strain, ebB' The circumferential surface bending strain is (12)
where KB is the change in circumferential curvature. Thus, the meridional distribution of the dimensionless circumferential surface bending strain is (13)
6. EXPERIMENTAL AND ANALYTICAL RESULTS Resonant frequencies for various meridional modes are shown in Fig. 4 as a function of circumferential mode number. On this figure, the two curves drawn through the points calculated by the inextensional analysis presented in Section 5 are shown along with the experimental resonant-frequency data. Each resonant point was obtained from a modified Kennedy-Pancu plot. A typical plot that was used to separate the resonant frequencies is shown in Fig. 5. As can be seen from the figure, separation of the resonant frequencies which were very close together was accomplished quite adequately by the KennedyPancu technique. It can be seen in Fig. 4 that the experimentally measured resonantfrequency points associated with the two lowest unsymmetric modes agree quite closely with the frequencies calculated by the inextensional analysis over the range of circumferential wave numbers. The resonant-frequency curves for the higher modes indicate the same trend as those for the two lowest modes. In the region of low circumferential wave numbers, n, the higher modes were difficult to detect experimentally because the resonant frequency coincided with the resonant frequency of another mode. At these lower resonant frequencies, the response of the mode having a higher n would dominate that of the lower-n mode. This problem was one of the major difficulties encountered in applying the KennedyPancu technique in this investigation. It was beyond the scope of this investigation because of the limitation of equipment, but a solution to this problem would be to introduce additional excitation systems. By placing the exciters near a point that is a node of the strong mode, the system would exhibit a larger response in the weak mode. 4 80 r--,--,----,r-.,.--...,----,-,..-..,.-~-,-___, INEXTENSIONAL MODES
400
,GENERAL MODES
320
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m;3
)/
,
~
240
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m=2~ /
(f)
,
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u
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0
0
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<>
PRESENT THEORY
<>
0/ f. /
160
::J
m=ly,
o
llJ
/'
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cr
lJ..
IBIA
80
O'--'-----.L----..J_-'---'----'_L----L----'-_l.-...J
o
2
4
CIRCUMFERENTIAL
6
8
WAVE
NUMBER, n
10
FIG. 4. Experimental and analytical resonant frequency variations.
774
W.
CHARLES
BERT
and
JOHN
D.
RAY
At the lower circumferential wave numbers, it is believed that the higher modes will not indicate as severe a hook in the resonant frequency vs. n plot as those reported in refs. 2-4 for homogeneous-material conical shells. The reason for this was discussed in Section 5 in connexion with equation (1).
60°
120°
30°
'PHASE ANGLE, '"
0·8
0'4 RESPONSE
1·2
FACTOR, Xo
FIG. 5. Kennedy-Pancu plot for 38·47 and 38·29 cps. Shown as Figs. 6 and 7 are the plots of circumferential strain associated with the two lowest meridional modes. Also included in Figs. 6 and 7 are the analytical, circumferential strain distributions derived from the inextensional analysis presented in Section 5. In general, there is very good agreement between the analytical and experimental strain distributions for these modes. The effect of exciter mass and location is noticeable in the meridional modal shapes. However, this is to be expected since it was observed by Mixson. 5 The circumferential modal shapes are symmetric about the exciter location. Fig. 8 shows a plot of the meridional strain distribution associated with the two lowest meridional modes. The meridional strain distribution is identically zero in the inextensional analysis presented in Section 5. Although the curves indicate a meridional strain, the values are small and can be attributed to restraint of the shell from completely free movement at the shaker-attachment location and at the suspension points. n=2
rn
IO~ o 0
o
f-
Z
:::J
>a: a:
<[
f-
aJ
a:
<[
z
-1,0 EXPERIMENTAL o MODE A "'MODE B
0
DATA
POINTS
0
n=3
1'0
°r~~~:-z~:== MODE A
-1,0 PRESENT THEORY'
MODE B
<[
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rn o W N
::::i
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o z
n=5
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=a-",_
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IHGFEDC
SMALL END
BA
LARGE END BODY STATIONS
FIG. 6. Normalized circumferential strain distribution along meridian, inextensional modes, n = 2-5.
Vibrations of orthotropic sandwich conical shells with free edges
1.0~n=6
o
o'-""==3::::~----'---"='P-..l...:o::!::-........L'" '"
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-1,0
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EXPERIMENTAL DATA MODE A MODE B
z
::>
POINTS n=7
1·0
>It: <:I
It:
tID ~
-1-0 PRESENT THEORY;
I':
Z <:I
It:
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a
'" 0
n=8 t::.
t::.
0
-1-0
t::.
0
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n=9 0
0
0
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SMALL END
BODY
LARGE END
STATIONS
FIG. 7. Normalized circumferential strain distribution along meridian, inextensional modes, n = 6-9.
o n=2 '" n=3
D n=6 v n=8
100
t)
MODE A
z .....
w
I
v
50
t)
V
~
z
V Z
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<:I
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J
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STATIONS
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100 MODE
B
<:I
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v
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BODY STATIONS
FIG. 8. Meridional strain distribution along meridian, inextensional modes, n = 2, 3, 6, 8.
52
775
776
CHARLES W. BERT and JOHN D. RAY
Since the experimental data and the values derived by the inextensional analysis are in close agreement, it is assumed that the lowest experimental unsymmetrical modes are the inextensional modes. The agreement is close for the resonant frequencies as a function of circumferential wave number, for the circumferential strain as a function of meridional position and for the meridional strain as a function of meridional position. Therefore, hereafter, these two lowest experimental modes will be referred to as the inextensional modes. The other modes which have higher resonant frequencies will be referred to as general modes, since they exhibit both extensional and inextensional deformation as discussed in Section 5. The circumferential strain distributions associated with the circumferential modes for the inextensional and general modes are shown in Figs. 9 and 10. The circumferential strain distributions associated with the general modes are shown in Fig. 11, and the
EXCITER
EXCITER
FIG. 9. Cross-section of circumferential waveforms, n = 0, 2, 3, 4.
FIG. 10. Cross-section of circumferential waveforms, n = 5, 6, 7, 8. distributions of meridional strain along the meridian are shown in Fig. 12. It is noted that the meridional strain is not uniform along the meridian, in contrast to the cases of the two inextensional modes. It can be seen in Fig. 4 that the resonant frequency for the m = 1 general mode is close to inextensional mode B at the low circumferential wave numbers, but as the wave number increases, the curves separate and the m = 1 mode converges to the m = 2 general mode. Also, it can be noted in this figure that all of the general modes tend to converge at the higher frequencies. This convergence is characteristic of both cylindrical and conical shells.
Vibrations of orthotropic sandwich conical shells with free edges
777
The resonant-frequency data and the associated strain distributions indicated that some coupling between modes occurred. These coupled modes were a combination of two circumferential modes having the same meridional mode when the resonant frequencies were close. Pronounced coupling phenomena were observed at resonant frequencies of 33, 64 and 94 cps. In all of these cases, the strain distributions were typically the same except for the difference in circumferential wave numbers. Fig. 13 is a plot of a nodal pattern typical of those associated with the above frequencies. C/)
I-
Z
::::>
>a: <[ a:
ICD
'0]
-I'~
m=1
~~ 0
a: <[
z
-
<[
a:
IC/)
0 W N
'0]
+~
m=2
't>.e~, 0
'~~ 0
...J <[ ~
a: 0
z
m=3
-::]
~~~~ !
.J
o
0
!
I
BODY
FIG.
!
I
I
1
1
HGFEDCBA STATIONS
n. Normalized circumferential strain distribution along meridian, general modes.
The phenomenon of U-shaped nodal lines, i.e. the existence of more circumferential waves at the large end of a free-free conical-frustum shell than at the small end, was observed in experiments reported by Watkins and Clary.2,3 This phenomenon has been controversial, since it was not observed in any of the similar experiments reported by Hu et al.' Hu12 suggested that the U-shaped modes were caused by shaker effects, since his experiments' were excited by magnetic fields (no direct contact with shell). Additional comments were made by Koval,13, 14 who showed photographs of a similar phenomenon in vibrational experiments on cylindrical shells. He attributed the U-shaped-mode phenomenon to the dynamic asymmetries caused by the spot-welded longitudinal seams present in the shells of refs. 2, 3. According to the theory originated by Tobias15 and applied by Arnold and Warburton,16 the presence of small imperfections causes each of the two close resonant frequencies to have a preferential nodal pattern. Then, under excitation at a frequency between the two component frequencies, the two different component modes combine to produce the complicated mixed-mode pattern. As a result of the controversy, following the suggestion of Watkins and Clary,17 Mixson5,6 conducted an extensive series of additional experiments on shells with and without longitudinal seams. In these experiments, he used air shakers, as well as electrodynamic ones, and varied the ratio ofthe frequency of the axial rigid-body mode (controlled by the suspension stiffness) to that of the lowest elastic shell mode. He concluded that imperfections were probably the principal cause of the mixed modes, since they occurred most often in the cones with seams and smaller wall thickness (in which it is more difficult to control imperfections). In the present investigation, the coupling observed was probably caused by the presence of the four equally spaced longitudinal lap joints on the inside and outside facings. (i.e. a total of eight lap joints; one every 45°). However, it is interesting to note that the only modes which coupled were inextensional modes (A and B). There was no observed
778
CHARLES W. BERT and
JOHN
D.
RAY
coupling of the general modes (m = 1,2, 3) with each other or with the inextensional modes. This may have been because, for the particular shell construction and geometry, the inextensional modes were the only ones sufficiently close together (in frequency as well as in circumferential wave number) to couple.
..
0 x :I: C> Z
..... en w
:I: C> Z
I~:t I~:t
o n=3
m=1 l>
8
l>
.s
l>n=4 l> 0
!
B
0
A
•
~
0
0
3
l>n=6
4
l>
~
on=5
m=2
&
l>
on=5
&
B
b
A
Z
a:
I-
en
C>
:::;;
I~;t
o n=3
m=3
B.
&
J
I
B.
.&
H
G
l> n=6
&
{l
.&
&
.8.
F
E
0
C
B
>-
0
BODY
A
STATIONS
FIG. 12. Meridional strain distribution along meridian, general modes.
SHELL
LAYOUT
FIG. 13. Nodal pattern for coupled mode at 94 cps. At present, the authors know of no analytically based criterion for determining a priori whether or not two adjacent modes will couple to form a combined mode. Since the available experimental data are limited and expensive to obtain, an analytically based criterion, rather than an empirically based one, would be quite useful. For calculating the frequencies of certain higher modes peculiar to sandwich structures, Yu's analysis18 was used. The calculated frequencies for all three of these modes (shear modes in both the meridional and circumferential planes and the thickness-normal mode) were much higher than the highest frequencies attained in the experiments. Therefore, it is presumed that no modes of these types were excited in the experiments reported. 7. CONCLUSIONS
For free-free sandwich shells with orthotropic facings, an inextensional analysis predicts quite accurately the resonant frequencies associated with the first two unsymmetric modes. Under the conditions of the present experiments, extensional, core shear, and core normal effects can be neglected for these two modes.
Vibrations of orthotropic sandwich conical shells with free edges
779
Combined modes with V-shaped nodal patterns, as reported in previous experiments on free-free cones, were present. However, in all cases observed here they represented coupling between inextensional modes only. REFERENCES
1. N. L. ROUST, Vibration and Fatigue Sandwich Bibliography. Rexcel Corp., Dublin, Calif. (20 January, 1968). 2. J. D. WATKINS and R. R. CLARY, Vibrational Oharacteristics of Some Thin-walled Oylindrical and Oonical Frustum Shells, NASA TN D·2729 (1965). 3. J. D. WATKINS and R. R. CLARY, Am. Inst. Aer. Astr. J. 2, 1815 (1964). 4. W. C. L. Ru, J. F. GORMLEY and U. S. LINDHOLM, Flexural Vibrations of Oonical Shells with Free Edges, NASA CR-384 (1966). 5. J. S. MIXSON, J. Spacecraft and Rkts. 4, 414 (1967). 6. J. S. MIXSON, Experimental Modes of Vibration of 14° Oonicaljrustum Shells, NASA TN D-4428 (1968). 7. C. W. BERT, W. C. CRISMAN and G. M. NORDBY, J. Aircraft 5, 27 (1968). 8. J. D. RAY, C. W. BERT and D. M. EGLE, The Application of the Kennooy-Pancu Method to Experimental Vibration Studies of Oomplex Shell Structures, presented at the 39th Shock and Vibration Symposium, Monterey, Calif. (22-24 October, 1968). 9. W. C. L. Ru, J. F. GORMLEY and U. S. LINDHOLM, Int. J. mech. Sci. 9, 123 (1967). 10. D. R. PLATUS, Oonical Shell Vibrations, NASA TN D·2767 (1965). n. M. D. BACON and C. W. BERT, Am. Inst. Aer. Astr. J. 5, 413 (1967). 12. W. C. L. Ru, discussion of Ref. 3, Am. Inst. Aer. Astr. J. 3, 1213 (1965). 13. L. R. KOVAL, J. acoust. Soc. Am. 35, 252 (1963). 14. L. R. KOVAL, Am. Inst. Aer. Astr. J. 4, 571 (1966). 15. S. A. TOBIAS, Engineering 172, 409 (1951). 16. R. N. ARNOLD and G. B. WARBURTON, Proc. R. Soc. 197A, 238 (1949). 17. J. D. WATKINS and R. R. CLARY, authors' reply to discussion of Ref. 3, Am. Inst. Aer. Astr. J. 3, 1213 (1965). 18. Y.-Y. Yu, J. appl. Mech. 27, 653 (1960).