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Natural frequencies of vibration of cantilever sandwich beams
Computers& Stmcfures,Vol. 6, pp. 34M53. PergamonPress 1976. Printed in Great Britain
NATURAL FREQUENCIES OF VIBRATION OF CANTILEVER SANDWICH BEAMS? N. A. RUBAYIS Department of Engineering Mechanics andMaterials, Southern Illinois University, Carbondale, IL 62901,U.S.A.
and S. CHAROENREEB Thai Melon Polyester Co., Tambol Klongnung,Pathumthani, Thailand (Received 30 November 1975)
Abstrae-Theoretical and experimental studies were made in obtaining the natural frequencies of cantilever sandwich beams subjected to only gravity forces. The method of minimizingthe total energy of the system was used for determining the frequencies. A vibration system made by Unholtz-Dickie was utilized to set the beam in vibration. Resonance occurred when the frequency of the shaker coincided with the natural frequency of the beam. The resonance frequencies were measured by transducers mounted at various locations on the beam. A total of sixteen beams of various lengths, thickness and core density were tested. It was found that the natural frequency of a cantilever sandwich beam depends largely upon the thickness, length, core density and stiffness of the beam. In addition, the natural frequency has a nonlinear variation with the mode and for any particular mode, the value of the frequency increases as the length of the beam decreases. Design factors were developed based upon the ratios of the theoretical frequencies of homogeneous beams having the same thicknesses and stiBnesses of that of sandwich beams and to the frequencies experimentally determined for similar sandwich beams.
NOMENCLATURE rectangular coordinates time coordinate length of sandwich panel thickness of core thickness of lower and upper facing respectively modulus of elasticity of facings Poisson’s ratio of facings modulus of elasticity of core modulus of rigidity of core in xz plane displacements of core in z direction respectively displacement in z direction of any point in sandwich panel normal and shear strains, respectively in core bending strains in lower and upper facings respectively shear stress in core normal forces of the lower and upper faces respectively strains in the x direction of lower and upper facings respectively normal stress in an arbitrary fiber at distance z elastic strain energy per unit width of core elastic strain energy per unit width associated with normal force in lower and upper facing
respectively totalelasticstrainenergyper unit width of panel kinetic energy per unit width of panel mass density of composite panel per unit length and width tpresented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, DC., 29-31 March 1976. *Professor. PEngineer. Formerly, Research Assistant, Dept. of Engineering Mechanics and Materials, Southern Illinois University, Carbondale, Illinois, U.S.A.
theoretical natural frequencies of vibration of cantilever sandwich beam, radians per unit time constants, configuration parameters experimental natural frequencies of cantilever sandwich beam cyclelsec natural frequencies of homogeneous beam equivalent to the sandwich beam cycle/set length of homogeneous beam mass per unit length of the homogeneous beam sandwich panel stiffness lb/in’ INTRODUCTION
The structure of sandwich plates generally consists of two relatively thin external materials called the facings separated by and bonded to a relatively thick internal structure called the core. The facings are usually of a material which has high strength and stiffness compared with that of the core which is normally of a lighter density and relatively low strength and stiffness. Due to the nature of their construction, sandwich plates are known to have an extremely high strength-weight ratio as compared to that of a single homogeneous plate. For this reason, their applications are very favorable in the construction of airplanes, guided missiles and space ships, where the weight is a major factor in the design of such structures. Recent improved techniques of bonding and fabrication have increased their application in many other industries and especially in the domestic appliances. In many instances, these structures are subjected to severe vibration and the designer must have a full understanding of the nature of vibration and the resulting frequencies. For the past two decades or so a great deal of theoretical and experimental effort has been made in the dynamic study of sandwich structures. The early attempt towards solving the natural frequencies of vibration of a
345
346
N. A. RUBA’I’Iand
simply supported sandwich beam has been made by Kimel et al.[l]. Later Yu[2] extended the new flexural theory developed by Mindlin[3] to include the effects of transverse-shear deformation and rotatory inertia in both the core and faces of the sandwich plates. The same author [4] used the new flexural theory to solve vibration problems of elastic sandwich plates. Hearmon[S] adopted Rayleigh’s method to derive closed formula for the frequencies of vibration of orthotropic plates with several combinations of clamped or supported edges. In later years Yu[6,7] extended his earlier work further to include flexural vibration of sandwich plate in plane strain. On the other hand Raville et a!.[81 carried out theoretical and experimental work on the natural frequencies of vibration of fixed-fixed sandwich beam. They used the energy approach in which Lagrangian multipliers were utilized to satisfy the boundary conditions of the problem. Ditaranto[9] developed the theory of vibratory bending for elastic and viscoelastic layered finite-length beam. He used the theory of elasticity to derive a sixth order differential equation for the deflection of sandwich beams. In the past decade Yu and Lai[lO] presented another theory of non linear vibration of sandwich plates where the transverse shear and edge condition were included. They extended the theory to solve dynamic buckling of homogeneous and sandwich plates. Later, Ueng[l l] derived the theory of natural frequencies of the all clamped rectangular sandwich plates using the energy method in which Lagrangian multipliers were utilized to satisfy the boundary conditions of the clamped edges. Nicholas and Heller[l2] used the theory of elasticity to determine the complex shear modulus of a filled elastomer from the vibration of sandwich beams. At the same time, Ditaranto and Blasingame[l3] extended the work of Ditaranto[9] for composite damping of vibrating sandwich beams by adding a damping factor term into the equilibrium equation. On the other hand, Mead and Markusll41 again extended the work of Ditaranto [9] to include force vibration of three layer damping sandwich beams with arbitrary boundary conditions. Lately, Yan and DowelIllS] presented a governing equation for the damped vibration of a constrained-layer sandwich plates and beams where the principle of virtual work was used. These preceding analyses do not reveal any rigorous solution for the natural frequencies of a vibrating cantilever sandwich beam subjected only to gravity forces. Therefore, the object of this study is to develop
Fig.
I,
S. CHAROENREE
mathematical equations for the natural frequencies of a cantilever sandwich beam, clamped at one end and free at the other end. The method of minimizing the total energy of the system will be used. Furthermore. it is aimed to conduct experimental analysis for the purpose of verifying the theoretical results. An attempt will be made to develop design factors based upon the ratios of the theoretical natural frequencies of homogeneous beams having the same thicknesses and stiffnesses of that of sandwich beams and to the frequencies experimentally obtained for such sandwich beams. THEORETITAL ANALYSIS
(a) For the cantilever sandwich beurn Consider the sandwich beam shown in Fig. 1 which is only subjected to gravity force. In developing the theoretical analysis for the natural frequencies of such a beam, the following assumptions have been used. (1) The facings of the sandwich are homogeneous, isotropic and elastic thin plates: (2) The core consists of an elastic. orthotropic continuum whose load carrying capacity in the plane of the sandwich is negligible; (3) The modulus of elasticity of the core in the direction perpendicular to the facing is infinite; and (4) Perfect continuity exists at the interfaces where the facings and core are bonded together. The theoretical analysis will be outlined here as follows: The deflection of the beam is taken in a polynomial form[ 161 as:
W = W, =(A,.~~+A~xi+AixJ)sinot.
(1)
The strains in the facing which occur as a result of bending of the facings about their own middle surfaces are referred to as bending strains. The bending strains, ES and lL in the lower and upper facings respectively, can be expressed in terms of the core displacement as; (2)
and
Cantilever sandwich beam.
341
Natural frequencies of vibration of cantilever sandwichbeams
(4)
From beam bending theory and considering the neutral axis is in the middle of the sandwich beam, then the strains in the x erection of Iower and upper facings [ct. and eXurespectively] can be expressed as:
(5)
and
substitution of eqn (1) into eqns (2) and (3) yields: (2A,-r_6Ai~2~+12A,x*)sinoa and (2A, t6AZx t 12A3x2)sinot.
The elastic strain energy, VW associated with the strain caused by bending of the lower facing about its middle surface can be written as; E ’ dz o”&dx VBF= 2(1- v2) I -, J
Therefore,
(6)
or:
VBf
(16)
or:
=
24(l-
Ef3 [4&‘+ 12a2A,Az+ 16a3AiA, g)
+ 12u3AzZ+ 36u4AzA1+ 28.8~‘A>~lsin’ wt
VNF==-$
c+f
E 2(1
_
v2)
c J
(18)
or:
and for the upper facing as: vi,F =
t xiA;x*
(7)
J”[4A: t 24AiAzx +48A,A,x2 il + 144AtAXx’t 144A?x41sin’ ot dx
0
dzJo @a2dx
(8)
vNF=Efcf+c)* ~[4aA,*+
12u2A,A2+ 16n3A,Aj
8
t i2u3A:+36u4AzA~ +28.8usA?] sin’ ot.
(19)
01
“’
-ECf’)‘[4,A,2+ - 24(1 - v’)
In like mannef the strain energy V/W associated with the normal force in the upper facing may be expressed as:
12a2A,At+16a’A,A~
t 12a’Az t 36a4A2A, + 28.8u5Aj2]sin’ ot.
1 V~P = 2
(9)
The normal stress in an arbitrary fiber of the facing may be written as:
00)
J
(-NF”)(-&U)
dx
(20)
or: V& = EfU;
c)2 [4uA: + 12u2A1Azt 16u3AzA3
t 12u3A: t 36u4A2A3t 28,8uA:] sin* wt. f21) Let NFL and NW be the normal forces of the lower and upper faces. These forces may be expressed as: NFL=-
J-1 E(z-;)$$dz
The strain energy in the core may be written as:
J J
cm
1 yx2= G,, T=z
(23)
V, = f O’dz ‘G&dx.
(11)
0
Where
or
(12)
and from Fig. 2 it follows that:
and likewise: &,=-lr(‘r-E(z-;)?.$dz
(13)
TXZdx=-%?x*
(24)
Substi~t~g eqn (12) in (24) will give: or: 1 1 ~XL=-~i;Ef(ftc)(6At+24A~x)sinwt NFIJ= -E$Yf
cf’ + c).
The elastic strain energy associated with the normal force in the lower facing can be written as: (1.5) CAS Vol. 6, No. 4-F
(25)
(14) and the expression for the strain energy in the core becomes: V = cE2{z “)* [36uA? t 144u2A2Aj c XZ + 192a’A3*]sin2 wt.
(26)
348
A.
N.
RUBAYI
and S,
CHAROENKH
and AS as:
LOWER
NFL
FACING -
+ %=Ld, ax
i-
Fig. 2. Differential element of Iower facing and core.
Hence, the total elastic strain energy of the cantilever beam is expressed as: (17) The soIution of eqns (343-(361 other than the trivial one for which (AI = A2 = A3 = 0) requires that the determinant of these coefficients must be zero, that is:
or: v = cE’f’cf f c)’ (36aAz’+ 144a’A:A3+ 192a”A1’) 8G,: +&(4aAi’+
I2n’A,Az+ 16u’A,A1 = 0.
c 12aJAI+36a”A2A-I+28.8a’A,‘) Ecf’)’ +----_,(4aAi’+ 24( I - v-1
12a’Ai.++
16a’A,A>
+ 12a3Az’+36a3A1AI+28.8a’A~‘) Efcf+c,’ t-----(4aAI’+ 8
I?a’A,A,t
(37)
Iha’AIAq
Where:
The kinetic energy of the vibrating beam may be expressed as:
x2=
I2a’Ef’ 74(1+
12a’EOE’)1+ l?a’Ej(f i o )’ 24(1- v’) 8 i l!o’Ej’cf’+c)~ 8
(39)
(29) x3= or:
16a’EjJ 14(1-+
ku’ECf’)‘+ 2411- v’)
16n’Ej(f+ci-’ x (40)
a”
a’
_I
+ - AzAt + - A T-)co? ot. 4 9 The vibrating system is assumed to be conservative that:
(JO)
y1 _ 12a’Ej: ~ li?a’ECf’t’+ I2u’Ef(f + u)’ 24(1-I’-) 24(1 - 1,:) 8 + 12a ‘Ef’Cf’ + c I’ 8
so Y2=
72acE’f’Cf + cf’ f 24a ‘Ej: + 74a ‘E(f’)’ 24(1 - V-) Ml - I”) SC,: _t-240’Ejcf + c )’ + 24o’Ej’cf 8 8
Y3 = 144a2cE2f2Cf t c)’
8GX Equations (31&-(33)lead to the three equations in A /, A2
(41)
36a4Ej’
+ 24(1.+
+ c)’
(42)
36a4ECf’f3
24(1 - V7)
+ 36~4~~~ -i-F i’+ 36u4~~~’ + c )’ 8 8
143)
Natural frequencies of vibration of cantilever sandwich beams
where E and E, are the modulii of elasticity of facing and core respectively, b is the width of the sandwich beam and d is the distance between the center line of upper and lower faces, that is:
and: 16a’Ef’ ‘l=24(1-’
16a’E(f’)’ 16a’Ef(f t c)~ 8 24(1-v2) •t + 16a’Ef’(f’ t c)2 8
z2 = 144a2cE2f2(j t c)* 36a4Ef’ 36a4ECf’)’ +24(1-)+ 24(1-v*) 8GXZ + 36a’Ej(f t c)~+ 36a4Ej'(f' t c)* 8 8 z3
=
384a3cE2f2tj t c)* I 57.6a5E{I t 57.6a’ECf’)’ 24(1- v*) 24(1- v ) 8GXZ + 57.6aSEf(f t c)~+ 57.6a’Ef’Cf’ t c)’ 8 8 ’
(45)
where m = 1,2,3,4 ,.... IBM computer programs were developed for the (46) solutions of eqns (37) and (52) in order to determine the natural frequencies of the sandwich cantilever beam and of the homogeneous cantilever beam respectively.
Where (W’lg) is the mass per unit length of the beam. Solution of eqn (47) gives the natural frequency as
(W
where nm2
=
(2m-
W
4L2 and m = 1,2,3,4, . . .. In assuming the stiffness of the homogeneous beam (EI) equal to the stiffness of the sandwich beam, we may write as outlined in [17].
E~=D=&‘+Ebfd2 +E,$ 6
(51)
Where in this study the face thicknesses of the sandwich beam were the same, that is cf’ = f). Hence the frequency equation can be written as:
(47)
1
d=ctf.
(44)
(b) For the cantilever homogeneous beam The governing differential equation for the vibration of a cantilever homogeneous beam was derived from the state of equilibrium as:
an*EIg “* Wh=% [ w’
349
(50)
EXPERIMENTALTECHNIQUES
The experimental set up used for the vibration of the cantilever sandwich beams is shown in Fig. 3. The vibration system made by Unholtz-Dickie consists of three major parts, the shaker, the control console and the electronic power amplifier. Four sandwich panels made by Hexel Corporation of Casa Grande, Arizona having a width of 6in. each and of various lengths of 120, 100,80 and 60 in. respectively were tested. Sixteen different tests were carried out corresponding to different length, core, stiffness and thickness. The physical properties and dimensions of the sandwich panels are given in Table 1. One end of the beam was rigidly clamped to the head of the shaker while the other end was free. The beam was set in vibration by the shaker for various frequencies and when the frequency of the shaker coincided with the natural frequency of the beam, a resonance occurred. The resonating frequencies were measured by Quartz accelerometers which were mounted at various locations along the beam. The output of the accelerometers were fed simultaneously to oscilloscope, frequency counters and voltmeters. The mode number was obtained for each resonating frequency by counting the node1 points on the beam, and the number of modes is one less than the number of nodes.
Fig.3. Experimental set uii
350
N. A. RLJBAYI and S. CHAROENREE Table 1. Physical properties and dimensions of sandwich beams
l-2
3.743X10S
l-3
:.74axios
L.0
4.35x10*
0.0335
3,*-5052-.CO5
1-4
~3.74Rn105
1.0
4.35x10'
0.3335
3,8-5C51-.CO5
2-l
2.308x10*
0.25
4.35x:0*
0.0214
3,s-5052-.so5
2-2
;2.3Ooali)*
0.25
4.i5rlO"
o.t2:4
2-3
2.308X10"
cl.25
4.35x10'
0.3214
j
a0
3,3-535:-.I355
i , I i
ioo
3,8-5052-,055
;
"0
DISCUSSIONOF THEORETICAL AND EXPERIMENTAL RESULTS
The theoretical analysis yielded three real modes of natural frequencies for each sandwich panel, whereas it was experimentally possible to obtain ten modes for each beam tested as shown in Tables 2-5. It is clear from these tables that the theoretical values for modes Nos. 1 and 2 are in closer agreement with the experimental results than with those values obtained by the homogeneous beam approach. However for all investigated sandwich beams, the third theoretical mode was not in agreement with the value of the third experimental mode. This is a common pitfall with the energy method. Usually, as a rough rule of thumb, one may count on obtaining physical realistic results for a number of frequencies and mode shapes equal to or less than one half of the number of the
Go 120
unknown coefficients appearing in the assumed polynomial function of the beam deflection. Therefore it is suggested here when higher numbers of theoretical modes are desired, then the number of coefficients in the assumed function for beam deflection should be increased by at least twice the number of the desired modes. In general the values of the experimental results were lower than the theoretical ones for modes Nos. 1 and 2. This may be attributed to several factors such as the influence of the values of the modulus of rigidity G,,, which were supplied by the manufacturer, the damping resistance of the air surrounding the vibrating sandwich beam and the dynamic rotation which were neglected in the assumption used in this theory. The experimental results were presented in Figs. 4-7. In examining Figs. 4 and 6 for beam of 1 in. thickness it can be
Table 2. Natural frequencies of vibration of beams l-l, l-2,
‘and l-4. 7
914.722
I
213.668
516.032
^I/ 0 3.491 12.124 L7.428
7'
Wh ,!cmO. 3.830 24.001 G7.176 131.707 217.699 325.213 454.222 604.733 776.747 970.262
We
806.77
1019.6 1257.3
1.171
;.i90 1.256
exp. 3.450 21.590
62.645 122.~40 198.29 303.31 4LO.49 535.57 691.93 840.98
1.10
1
1.112 1.072 1.07* ?.I.09 1,072
2 3 r. 5 6
9.702
61.712 327.224
10.G37 bG.bb9 186.599 365.855 604.719 903.366
lb&.45 339.50 539.97 774.65
1.135 1.078 l.120 1.166
351
Natural frequencies of vibration of cantilever sandwich beams
-B!ZAM
1-l
-BEAM
l-2
-BEAM CI
1-3 BEAM l-4
-BEAM
3-l
MEEAt.
3-2
-BEAM
3-3
-BEAM
3-4
0 r ,--r 0123456769 2
3
4
5
MODE
6
7
6
9
10
NUMBER
Fig. 4. Experimental variation of frequency with mode for beams l-l, l-2.1-3 and l-4.
MODE
4oc--
-5EAM
2-1
--BEAM
2-2
-BEAM
2-3
-BEAM
2-4
10
1
NUMBER
Fig. 6. Experimental variation of frequency with mode for beams 3-1,3-Z, 3-3 and 3-4.
46 450--
I
’
C----G
BEAM
4-l
t
360--
:: lo 300-; 6 ;
250--
60--
0
0
10
0123456769
1
MODE NUMBER
Fig. 5. Ex~~rnen~ variationof frequencywithmodefor beams
2
3
4
5
MODE
6
7
6
Q
10
1
NUMBER
Fig. 7. Experimental variation of frequency with mode for beams 4-1,4-2,4-3 and 4-4.
Z-1,2-2,2-3 and 24. sandwich beams by developing ratio factors of oh 10, as shown in Tables 2-5.
seen that the variational trend of the frequency with mode is nearly the same for those beams of the same length. This trend is also evident in Figs. 5 and 7 for those beams of 0.25 in. thickness. Although, the experimental results do not agree directly with the homogeneous beam theory, however, the idea of in~odu~ing the homogen~us beam theory was intended to predict the natural freauencies of
It is concluded in this study that in general the natural frequencies have a nonlinear variation with the mode and for any particular mode, the values of frequencies increase as the length of the sandwich beam &creases. Furthermore, the experimental results indicate that the natural frequencies of a cantilever beam depend largely . upon the thickness, length, mass density and the stiffness
352
N. A. RUBAYI and S. CHAROENREE
Table 3. Natural frequencies of vibration of beams 2-1,2-2,2-3 and 2-4 3EdrO SO. 2-1
NO.2-3 ! De&n
c
Table 4. Natural frequencies of vibration of beams 3-1,3-2.3-3 and 3-4
r
353
Natural frequencies of vibration of cantilever sandwich beams Table 5. Natural frequencies of vibration of beams 4-1,4-2,4-3 and 4-4
wi;/w.
‘0. d
l.i28 7,133 37.895
233.i51
104.1
299.470
374.079
1:,r.as 176.21
2.17‘ 13.325 37.295 73.122 1ZO.OG liS.552 252.177 335.739 431.237 538.673
I..00 6.07 17.52 35.03 56.53 DC.'-)4 118.5 157.75 198.79 253.45
y
1.993 2.194 2.16 2.1*9 2.090
1 2
2.118
6
2.102
7 8
2.137 2.111 2.123
of the beam. Lastly, design factors were developed based upon the ratios of the theoretical frequencies of homogeneous beams having the same thickness and stiffness of that of sandwich beams and to the frequencies experimentally obtained for similar sandwich beams. These factors will aid the designer in obtaining the natural frequencies of cantilever sandwich beams without carrying out the experimental work. In addition, the experimental results presented here should help the designer in recognizing these natural frequencies in such vibrating structures. Ack~~w~edgeme~~s-This work was carried out in the Solid Mechanics Laboratory of the Department of Engineering Mechanics and Materials of Southern Illinois University. The authors are indebted to the Office of Research and Projects of Southern Illinois University at Carbondale for sponsoring this project. REFERENCES
1. W. R. Kimel, M. E. Raville, P. G. Kirmser and M. P. Pate], Natural frequencies of vibration of simply supported sandwich beams. Proc. 4th Midwestern Conf. on Solid and Fluid Mechanics. Austin, Texas (Sept. 1959). 2. Yi-Yuan Yu, A new theory of elastic sandwich plates onedimension case. J. Appl. Mechun. 26, Trans. ASME 81, Series E, 415-421 (1958). 3. R. D. Mindlin, AR in~~du~i~onto the mathemaficai theory of vibration of elastic plates. Monograph prepared for U.S. Army Signal Corps Engineering Laboratories (1955). 4. Yi-Yuan Yu, Flexural vibrations of elastic sandwich plate. J. AerolSpace Sci. 21, 212-282 (1960). 5. R. F. S. Hearmon, The frequency of flexural vibration of
Of
3 4 5
, -* meorv 1.762 ll.i49 59.204
9 10
2 3 4 5 G
7 8 9 10
3.133 19.841 105.32
"'h zo , 3.322 20.82 48.273 144.253 ica.wV m2.1:3 391.025 924.59 673.808 841.677
5.900 37.014
103.597 203.115 335.730 501.534 700.49 932.605 L197.68 1496.314
1.40
9.27 27.8 52.92 87.01 130.00 181.63 241.46 305.58 384.60
2.705 16.515 45.92 92.56 150.61 22V.17 314.s
2.183 2.241 2.256 2.19c 2.229 2.188 2.230
426.82
2.135
539.0 633.12
2.2*2 2.3‘3
rectangular orthotropic plates with clamped or supported edges. J. Appl. Mechan. 537-540 (1959). 6. Yi-Yuan Yu, Forced flexural vibrations of sandwich plates in plane strain. J. Appl. Mechan. 27, Trans. ASME 82, Series E, 535-540(1960). I. Yi-Yuan Yu, Simplifiedvibration analysis of elastic sandwich plate. J. Aero/Space Sci. 27, 894-900(l%O). 8. M. E. Raville, E. Ueng and M. Lei, Natural frequencies of vibration of fixed-fixed sandwich beams. J. Appl. Mechan. 367-371(Sept. 1961). 9. R. A. Ditaranto, Theory of vibratory bending for elastic. and viscoeiastic layered finite-le~h beams. L App~. Meckan. 881-886 (Dec. 1965). 10. Yi-Yuan Yu and Jai-Lue Lai, Influence of transverse shear and edge condition on nonlinear vibration and dynamic buckling of homogeneous and sandwich plates. I. AppZ. Meckan. 934-936 (Dec. 1966). 11. C. E. S. Ueng, Natural frequencies of vibration of an allclamped rectangular sandwich panel. J. Appl. Mechan. 683684 (Sept. 1966). 12. T. Nicholas and R. A. Heller, Determination of the complex shear modulus of a filled elastomer from a vibrating sandwich beam. Experiment. Mechan. 109-l 16 (Mar. 1%7). 13. R. A. Ditaranto and W. Blasingame, Composite damping of vibrating sandwich beams. 1 Engng Indust. 633-638 (Nov. 1967). 14. D. J. Mead and S. Markus, J; Sound V&rat. 2, 163-175 (Oct. 1969). 15. M. J. Yan and E. H. Doweli, Governing equations for vibrating constrained-layer damping sandwich plates and beams. .I Appl. Meckan. 1041-1046(Dec. 1972). 16. A. W. Leissa, Vibrations of plates. NASA SP-160, p. 44 (1969). 17. H. G. Allen, Analysis and Design of Structural Sandwich Panels, 1st Edn, p. 9. Pergamon Press, Oxford (1969).
NATURAL FREQUENCIES OF VIBRATION OF CANTILEVER SANDWICH BEAMS? N. A. RUBAYIS Department of Engineering Mechanics andMaterials, Southern Illinois University, Carbondale, IL 62901,U.S.A.
and S. CHAROENREEB Thai Melon Polyester Co., Tambol Klongnung,Pathumthani, Thailand (Received 30 November 1975)
Abstrae-Theoretical and experimental studies were made in obtaining the natural frequencies of cantilever sandwich beams subjected to only gravity forces. The method of minimizingthe total energy of the system was used for determining the frequencies. A vibration system made by Unholtz-Dickie was utilized to set the beam in vibration. Resonance occurred when the frequency of the shaker coincided with the natural frequency of the beam. The resonance frequencies were measured by transducers mounted at various locations on the beam. A total of sixteen beams of various lengths, thickness and core density were tested. It was found that the natural frequency of a cantilever sandwich beam depends largely upon the thickness, length, core density and stiffness of the beam. In addition, the natural frequency has a nonlinear variation with the mode and for any particular mode, the value of the frequency increases as the length of the beam decreases. Design factors were developed based upon the ratios of the theoretical frequencies of homogeneous beams having the same thicknesses and stiBnesses of that of sandwich beams and to the frequencies experimentally determined for similar sandwich beams.
NOMENCLATURE rectangular coordinates time coordinate length of sandwich panel thickness of core thickness of lower and upper facing respectively modulus of elasticity of facings Poisson’s ratio of facings modulus of elasticity of core modulus of rigidity of core in xz plane displacements of core in z direction respectively displacement in z direction of any point in sandwich panel normal and shear strains, respectively in core bending strains in lower and upper facings respectively shear stress in core normal forces of the lower and upper faces respectively strains in the x direction of lower and upper facings respectively normal stress in an arbitrary fiber at distance z elastic strain energy per unit width of core elastic strain energy per unit width associated with normal force in lower and upper facing
respectively totalelasticstrainenergyper unit width of panel kinetic energy per unit width of panel mass density of composite panel per unit length and width tpresented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washington University, Washington, DC., 29-31 March 1976. *Professor. PEngineer. Formerly, Research Assistant, Dept. of Engineering Mechanics and Materials, Southern Illinois University, Carbondale, Illinois, U.S.A.
theoretical natural frequencies of vibration of cantilever sandwich beam, radians per unit time constants, configuration parameters experimental natural frequencies of cantilever sandwich beam cyclelsec natural frequencies of homogeneous beam equivalent to the sandwich beam cycle/set length of homogeneous beam mass per unit length of the homogeneous beam sandwich panel stiffness lb/in’ INTRODUCTION
The structure of sandwich plates generally consists of two relatively thin external materials called the facings separated by and bonded to a relatively thick internal structure called the core. The facings are usually of a material which has high strength and stiffness compared with that of the core which is normally of a lighter density and relatively low strength and stiffness. Due to the nature of their construction, sandwich plates are known to have an extremely high strength-weight ratio as compared to that of a single homogeneous plate. For this reason, their applications are very favorable in the construction of airplanes, guided missiles and space ships, where the weight is a major factor in the design of such structures. Recent improved techniques of bonding and fabrication have increased their application in many other industries and especially in the domestic appliances. In many instances, these structures are subjected to severe vibration and the designer must have a full understanding of the nature of vibration and the resulting frequencies. For the past two decades or so a great deal of theoretical and experimental effort has been made in the dynamic study of sandwich structures. The early attempt towards solving the natural frequencies of vibration of a
345
346
N. A. RUBA’I’Iand
simply supported sandwich beam has been made by Kimel et al.[l]. Later Yu[2] extended the new flexural theory developed by Mindlin[3] to include the effects of transverse-shear deformation and rotatory inertia in both the core and faces of the sandwich plates. The same author [4] used the new flexural theory to solve vibration problems of elastic sandwich plates. Hearmon[S] adopted Rayleigh’s method to derive closed formula for the frequencies of vibration of orthotropic plates with several combinations of clamped or supported edges. In later years Yu[6,7] extended his earlier work further to include flexural vibration of sandwich plate in plane strain. On the other hand Raville et a!.[81 carried out theoretical and experimental work on the natural frequencies of vibration of fixed-fixed sandwich beam. They used the energy approach in which Lagrangian multipliers were utilized to satisfy the boundary conditions of the problem. Ditaranto[9] developed the theory of vibratory bending for elastic and viscoelastic layered finite-length beam. He used the theory of elasticity to derive a sixth order differential equation for the deflection of sandwich beams. In the past decade Yu and Lai[lO] presented another theory of non linear vibration of sandwich plates where the transverse shear and edge condition were included. They extended the theory to solve dynamic buckling of homogeneous and sandwich plates. Later, Ueng[l l] derived the theory of natural frequencies of the all clamped rectangular sandwich plates using the energy method in which Lagrangian multipliers were utilized to satisfy the boundary conditions of the clamped edges. Nicholas and Heller[l2] used the theory of elasticity to determine the complex shear modulus of a filled elastomer from the vibration of sandwich beams. At the same time, Ditaranto and Blasingame[l3] extended the work of Ditaranto[9] for composite damping of vibrating sandwich beams by adding a damping factor term into the equilibrium equation. On the other hand, Mead and Markusll41 again extended the work of Ditaranto [9] to include force vibration of three layer damping sandwich beams with arbitrary boundary conditions. Lately, Yan and DowelIllS] presented a governing equation for the damped vibration of a constrained-layer sandwich plates and beams where the principle of virtual work was used. These preceding analyses do not reveal any rigorous solution for the natural frequencies of a vibrating cantilever sandwich beam subjected only to gravity forces. Therefore, the object of this study is to develop
Fig.
I,
S. CHAROENREE
mathematical equations for the natural frequencies of a cantilever sandwich beam, clamped at one end and free at the other end. The method of minimizing the total energy of the system will be used. Furthermore. it is aimed to conduct experimental analysis for the purpose of verifying the theoretical results. An attempt will be made to develop design factors based upon the ratios of the theoretical natural frequencies of homogeneous beams having the same thicknesses and stiffnesses of that of sandwich beams and to the frequencies experimentally obtained for such sandwich beams. THEORETITAL ANALYSIS
(a) For the cantilever sandwich beurn Consider the sandwich beam shown in Fig. 1 which is only subjected to gravity force. In developing the theoretical analysis for the natural frequencies of such a beam, the following assumptions have been used. (1) The facings of the sandwich are homogeneous, isotropic and elastic thin plates: (2) The core consists of an elastic. orthotropic continuum whose load carrying capacity in the plane of the sandwich is negligible; (3) The modulus of elasticity of the core in the direction perpendicular to the facing is infinite; and (4) Perfect continuity exists at the interfaces where the facings and core are bonded together. The theoretical analysis will be outlined here as follows: The deflection of the beam is taken in a polynomial form[ 161 as:
W = W, =(A,.~~+A~xi+AixJ)sinot.
(1)
The strains in the facing which occur as a result of bending of the facings about their own middle surfaces are referred to as bending strains. The bending strains, ES and lL in the lower and upper facings respectively, can be expressed in terms of the core displacement as; (2)
and
Cantilever sandwich beam.
341
Natural frequencies of vibration of cantilever sandwichbeams
(4)
From beam bending theory and considering the neutral axis is in the middle of the sandwich beam, then the strains in the x erection of Iower and upper facings [ct. and eXurespectively] can be expressed as:
(5)
and
substitution of eqn (1) into eqns (2) and (3) yields: (2A,-r_6Ai~2~+12A,x*)sinoa and (2A, t6AZx t 12A3x2)sinot.
The elastic strain energy, VW associated with the strain caused by bending of the lower facing about its middle surface can be written as; E ’ dz o”&dx VBF= 2(1- v2) I -, J
Therefore,
(6)
or:
VBf
(16)
or:
=
24(l-
Ef3 [4&‘+ 12a2A,Az+ 16a3AiA, g)
+ 12u3AzZ+ 36u4AzA1+ 28.8~‘A>~lsin’ wt
VNF==-$
c+f
E 2(1
_
v2)
c J
(18)
or:
and for the upper facing as: vi,F =
t xiA;x*
(7)
J”[4A: t 24AiAzx +48A,A,x2 il + 144AtAXx’t 144A?x41sin’ ot dx
0
dzJo @a2dx
(8)
vNF=Efcf+c)* ~[4aA,*+
12u2A,A2+ 16n3A,Aj
8
t i2u3A:+36u4AzA~ +28.8usA?] sin’ ot.
(19)
01
“’
-ECf’)‘[4,A,2+ - 24(1 - v’)
In like mannef the strain energy V/W associated with the normal force in the upper facing may be expressed as:
12a2A,At+16a’A,A~
t 12a’Az t 36a4A2A, + 28.8u5Aj2]sin’ ot.
1 V~P = 2
(9)
The normal stress in an arbitrary fiber of the facing may be written as:
00)
J
(-NF”)(-&U)
dx
(20)
or: V& = EfU;
c)2 [4uA: + 12u2A1Azt 16u3AzA3
t 12u3A: t 36u4A2A3t 28,8uA:] sin* wt. f21) Let NFL and NW be the normal forces of the lower and upper faces. These forces may be expressed as: NFL=-
J-1 E(z-;)$$dz
The strain energy in the core may be written as:
J J
cm
1 yx2= G,, T=z
(23)
V, = f O’dz ‘G&dx.
(11)
0
Where
or
(12)
and from Fig. 2 it follows that:
and likewise: &,=-lr(‘r-E(z-;)?.$dz
(13)
TXZdx=-%?x*
(24)
Substi~t~g eqn (12) in (24) will give: or: 1 1 ~XL=-~i;Ef(ftc)(6At+24A~x)sinwt NFIJ= -E$Yf
cf’ + c).
The elastic strain energy associated with the normal force in the lower facing can be written as: (1.5) CAS Vol. 6, No. 4-F
(25)
(14) and the expression for the strain energy in the core becomes: V = cE2{z “)* [36uA? t 144u2A2Aj c XZ + 192a’A3*]sin2 wt.
(26)
348
A.
N.
RUBAYI
and S,
CHAROENKH
and AS as:
LOWER
NFL
FACING -
+ %=Ld, ax
i-
Fig. 2. Differential element of Iower facing and core.
Hence, the total elastic strain energy of the cantilever beam is expressed as: (17) The soIution of eqns (343-(361 other than the trivial one for which (AI = A2 = A3 = 0) requires that the determinant of these coefficients must be zero, that is:
or: v = cE’f’cf f c)’ (36aAz’+ 144a’A:A3+ 192a”A1’) 8G,: +&(4aAi’+
I2n’A,Az+ 16u’A,A1 = 0.
c 12aJAI+36a”A2A-I+28.8a’A,‘) Ecf’)’ +----_,(4aAi’+ 24( I - v-1
12a’Ai.++
16a’A,A>
+ 12a3Az’+36a3A1AI+28.8a’A~‘) Efcf+c,’ t-----(4aAI’+ 8
I?a’A,A,t
(37)
Iha’AIAq
Where:
The kinetic energy of the vibrating beam may be expressed as:
x2=
I2a’Ef’ 74(1+
12a’EOE’)1+ l?a’Ej(f i o )’ 24(1- v’) 8 i l!o’Ej’cf’+c)~ 8
(39)
(29) x3= or:
16a’EjJ 14(1-+
ku’ECf’)‘+ 2411- v’)
16n’Ej(f+ci-’ x (40)
a”
a’
_I
+ - AzAt + - A T-)co? ot. 4 9 The vibrating system is assumed to be conservative that:
(JO)
y1 _ 12a’Ej: ~ li?a’ECf’t’+ I2u’Ef(f + u)’ 24(1-I’-) 24(1 - 1,:) 8 + 12a ‘Ef’Cf’ + c I’ 8
so Y2=
72acE’f’Cf + cf’ f 24a ‘Ej: + 74a ‘E(f’)’ 24(1 - V-) Ml - I”) SC,: _t-240’Ejcf + c )’ + 24o’Ej’cf 8 8
Y3 = 144a2cE2f2Cf t c)’
8GX Equations (31&-(33)lead to the three equations in A /, A2
(41)
36a4Ej’
+ 24(1.+
+ c)’
(42)
36a4ECf’f3
24(1 - V7)
+ 36~4~~~ -i-F i’+ 36u4~~~’ + c )’ 8 8
143)
Natural frequencies of vibration of cantilever sandwich beams
where E and E, are the modulii of elasticity of facing and core respectively, b is the width of the sandwich beam and d is the distance between the center line of upper and lower faces, that is:
and: 16a’Ef’ ‘l=24(1-’
16a’E(f’)’ 16a’Ef(f t c)~ 8 24(1-v2) •t + 16a’Ef’(f’ t c)2 8
z2 = 144a2cE2f2(j t c)* 36a4Ef’ 36a4ECf’)’ +24(1-)+ 24(1-v*) 8GXZ + 36a’Ej(f t c)~+ 36a4Ej'(f' t c)* 8 8 z3
=
384a3cE2f2tj t c)* I 57.6a5E{I t 57.6a’ECf’)’ 24(1- v*) 24(1- v ) 8GXZ + 57.6aSEf(f t c)~+ 57.6a’Ef’Cf’ t c)’ 8 8 ’
(45)
where m = 1,2,3,4 ,.... IBM computer programs were developed for the (46) solutions of eqns (37) and (52) in order to determine the natural frequencies of the sandwich cantilever beam and of the homogeneous cantilever beam respectively.
Where (W’lg) is the mass per unit length of the beam. Solution of eqn (47) gives the natural frequency as
(W
where nm2
=
(2m-
W
4L2 and m = 1,2,3,4, . . .. In assuming the stiffness of the homogeneous beam (EI) equal to the stiffness of the sandwich beam, we may write as outlined in [17].
E~=D=&‘+Ebfd2 +E,$ 6
(51)
Where in this study the face thicknesses of the sandwich beam were the same, that is cf’ = f). Hence the frequency equation can be written as:
(47)
1
d=ctf.
(44)
(b) For the cantilever homogeneous beam The governing differential equation for the vibration of a cantilever homogeneous beam was derived from the state of equilibrium as:
an*EIg “* Wh=% [ w’
349
(50)
EXPERIMENTALTECHNIQUES
The experimental set up used for the vibration of the cantilever sandwich beams is shown in Fig. 3. The vibration system made by Unholtz-Dickie consists of three major parts, the shaker, the control console and the electronic power amplifier. Four sandwich panels made by Hexel Corporation of Casa Grande, Arizona having a width of 6in. each and of various lengths of 120, 100,80 and 60 in. respectively were tested. Sixteen different tests were carried out corresponding to different length, core, stiffness and thickness. The physical properties and dimensions of the sandwich panels are given in Table 1. One end of the beam was rigidly clamped to the head of the shaker while the other end was free. The beam was set in vibration by the shaker for various frequencies and when the frequency of the shaker coincided with the natural frequency of the beam, a resonance occurred. The resonating frequencies were measured by Quartz accelerometers which were mounted at various locations along the beam. The output of the accelerometers were fed simultaneously to oscilloscope, frequency counters and voltmeters. The mode number was obtained for each resonating frequency by counting the node1 points on the beam, and the number of modes is one less than the number of nodes.
Fig.3. Experimental set uii
350
N. A. RLJBAYI and S. CHAROENREE Table 1. Physical properties and dimensions of sandwich beams
l-2
3.743X10S
l-3
:.74axios
L.0
4.35x10*
0.0335
3,*-5052-.CO5
1-4
~3.74Rn105
1.0
4.35x10'
0.3335
3,8-5C51-.CO5
2-l
2.308x10*
0.25
4.35x:0*
0.0214
3,s-5052-.so5
2-2
;2.3Ooali)*
0.25
4.i5rlO"
o.t2:4
2-3
2.308X10"
cl.25
4.35x10'
0.3214
j
a0
3,3-535:-.I355
i , I i
ioo
3,8-5052-,055
;
"0
DISCUSSIONOF THEORETICAL AND EXPERIMENTAL RESULTS
The theoretical analysis yielded three real modes of natural frequencies for each sandwich panel, whereas it was experimentally possible to obtain ten modes for each beam tested as shown in Tables 2-5. It is clear from these tables that the theoretical values for modes Nos. 1 and 2 are in closer agreement with the experimental results than with those values obtained by the homogeneous beam approach. However for all investigated sandwich beams, the third theoretical mode was not in agreement with the value of the third experimental mode. This is a common pitfall with the energy method. Usually, as a rough rule of thumb, one may count on obtaining physical realistic results for a number of frequencies and mode shapes equal to or less than one half of the number of the
Go 120
unknown coefficients appearing in the assumed polynomial function of the beam deflection. Therefore it is suggested here when higher numbers of theoretical modes are desired, then the number of coefficients in the assumed function for beam deflection should be increased by at least twice the number of the desired modes. In general the values of the experimental results were lower than the theoretical ones for modes Nos. 1 and 2. This may be attributed to several factors such as the influence of the values of the modulus of rigidity G,,, which were supplied by the manufacturer, the damping resistance of the air surrounding the vibrating sandwich beam and the dynamic rotation which were neglected in the assumption used in this theory. The experimental results were presented in Figs. 4-7. In examining Figs. 4 and 6 for beam of 1 in. thickness it can be
Table 2. Natural frequencies of vibration of beams l-l, l-2,
‘and l-4. 7
914.722
I
213.668
516.032
^I/ 0 3.491 12.124 L7.428
7'
Wh ,!cmO. 3.830 24.001 G7.176 131.707 217.699 325.213 454.222 604.733 776.747 970.262
We
806.77
1019.6 1257.3
1.171
;.i90 1.256
exp. 3.450 21.590
62.645 122.~40 198.29 303.31 4LO.49 535.57 691.93 840.98
1.10
1
1.112 1.072 1.07* ?.I.09 1,072
2 3 r. 5 6
9.702
61.712 327.224
10.G37 bG.bb9 186.599 365.855 604.719 903.366
lb&.45 339.50 539.97 774.65
1.135 1.078 l.120 1.166
351
Natural frequencies of vibration of cantilever sandwich beams
-B!ZAM
1-l
-BEAM
l-2
-BEAM CI
1-3 BEAM l-4
-BEAM
3-l
MEEAt.
3-2
-BEAM
3-3
-BEAM
3-4
0 r ,--r 0123456769 2
3
4
5
MODE
6
7
6
9
10
NUMBER
Fig. 4. Experimental variation of frequency with mode for beams l-l, l-2.1-3 and l-4.
MODE
4oc--
-5EAM
2-1
--BEAM
2-2
-BEAM
2-3
-BEAM
2-4
10
1
NUMBER
Fig. 6. Experimental variation of frequency with mode for beams 3-1,3-Z, 3-3 and 3-4.
46 450--
I
’
C----G
BEAM
4-l
t
360--
:: lo 300-; 6 ;
250--
60--
0
0
10
0123456769
1
MODE NUMBER
Fig. 5. Ex~~rnen~ variationof frequencywithmodefor beams
2
3
4
5
MODE
6
7
6
Q
10
1
NUMBER
Fig. 7. Experimental variation of frequency with mode for beams 4-1,4-2,4-3 and 4-4.
Z-1,2-2,2-3 and 24. sandwich beams by developing ratio factors of oh 10, as shown in Tables 2-5.
seen that the variational trend of the frequency with mode is nearly the same for those beams of the same length. This trend is also evident in Figs. 5 and 7 for those beams of 0.25 in. thickness. Although, the experimental results do not agree directly with the homogeneous beam theory, however, the idea of in~odu~ing the homogen~us beam theory was intended to predict the natural freauencies of
It is concluded in this study that in general the natural frequencies have a nonlinear variation with the mode and for any particular mode, the values of frequencies increase as the length of the sandwich beam &creases. Furthermore, the experimental results indicate that the natural frequencies of a cantilever beam depend largely . upon the thickness, length, mass density and the stiffness
352
N. A. RUBAYI and S. CHAROENREE
Table 3. Natural frequencies of vibration of beams 2-1,2-2,2-3 and 2-4 3EdrO SO. 2-1
NO.2-3 ! De&n
c
Table 4. Natural frequencies of vibration of beams 3-1,3-2.3-3 and 3-4
r
353
Natural frequencies of vibration of cantilever sandwich beams Table 5. Natural frequencies of vibration of beams 4-1,4-2,4-3 and 4-4
wi;/w.
‘0. d
l.i28 7,133 37.895
233.i51
104.1
299.470
374.079
1:,r.as 176.21
2.17‘ 13.325 37.295 73.122 1ZO.OG liS.552 252.177 335.739 431.237 538.673
I..00 6.07 17.52 35.03 56.53 DC.'-)4 118.5 157.75 198.79 253.45
y
1.993 2.194 2.16 2.1*9 2.090
1 2
2.118
6
2.102
7 8
2.137 2.111 2.123
of the beam. Lastly, design factors were developed based upon the ratios of the theoretical frequencies of homogeneous beams having the same thickness and stiffness of that of sandwich beams and to the frequencies experimentally obtained for similar sandwich beams. These factors will aid the designer in obtaining the natural frequencies of cantilever sandwich beams without carrying out the experimental work. In addition, the experimental results presented here should help the designer in recognizing these natural frequencies in such vibrating structures. Ack~~w~edgeme~~s-This work was carried out in the Solid Mechanics Laboratory of the Department of Engineering Mechanics and Materials of Southern Illinois University. The authors are indebted to the Office of Research and Projects of Southern Illinois University at Carbondale for sponsoring this project. REFERENCES
1. W. R. Kimel, M. E. Raville, P. G. Kirmser and M. P. Pate], Natural frequencies of vibration of simply supported sandwich beams. Proc. 4th Midwestern Conf. on Solid and Fluid Mechanics. Austin, Texas (Sept. 1959). 2. Yi-Yuan Yu, A new theory of elastic sandwich plates onedimension case. J. Appl. Mechun. 26, Trans. ASME 81, Series E, 415-421 (1958). 3. R. D. Mindlin, AR in~~du~i~onto the mathemaficai theory of vibration of elastic plates. Monograph prepared for U.S. Army Signal Corps Engineering Laboratories (1955). 4. Yi-Yuan Yu, Flexural vibrations of elastic sandwich plate. J. AerolSpace Sci. 21, 212-282 (1960). 5. R. F. S. Hearmon, The frequency of flexural vibration of
Of
3 4 5
, -* meorv 1.762 ll.i49 59.204
9 10
2 3 4 5 G
7 8 9 10
3.133 19.841 105.32
"'h zo , 3.322 20.82 48.273 144.253 ica.wV m2.1:3 391.025 924.59 673.808 841.677
5.900 37.014
103.597 203.115 335.730 501.534 700.49 932.605 L197.68 1496.314
1.40
9.27 27.8 52.92 87.01 130.00 181.63 241.46 305.58 384.60
2.705 16.515 45.92 92.56 150.61 22V.17 314.s
2.183 2.241 2.256 2.19c 2.229 2.188 2.230
426.82
2.135
539.0 633.12
2.2*2 2.3‘3
rectangular orthotropic plates with clamped or supported edges. J. Appl. Mechan. 537-540 (1959). 6. Yi-Yuan Yu, Forced flexural vibrations of sandwich plates in plane strain. J. Appl. Mechan. 27, Trans. ASME 82, Series E, 535-540(1960). I. Yi-Yuan Yu, Simplifiedvibration analysis of elastic sandwich plate. J. Aero/Space Sci. 27, 894-900(l%O). 8. M. E. Raville, E. Ueng and M. Lei, Natural frequencies of vibration of fixed-fixed sandwich beams. J. Appl. Mechan. 367-371(Sept. 1961). 9. R. A. Ditaranto, Theory of vibratory bending for elastic. and viscoeiastic layered finite-le~h beams. L App~. Meckan. 881-886 (Dec. 1965). 10. Yi-Yuan Yu and Jai-Lue Lai, Influence of transverse shear and edge condition on nonlinear vibration and dynamic buckling of homogeneous and sandwich plates. I. AppZ. Meckan. 934-936 (Dec. 1966). 11. C. E. S. Ueng, Natural frequencies of vibration of an allclamped rectangular sandwich panel. J. Appl. Mechan. 683684 (Sept. 1966). 12. T. Nicholas and R. A. Heller, Determination of the complex shear modulus of a filled elastomer from a vibrating sandwich beam. Experiment. Mechan. 109-l 16 (Mar. 1%7). 13. R. A. Ditaranto and W. Blasingame, Composite damping of vibrating sandwich beams. 1 Engng Indust. 633-638 (Nov. 1967). 14. D. J. Mead and S. Markus, J; Sound V&rat. 2, 163-175 (Oct. 1969). 15. M. J. Yan and E. H. Doweli, Governing equations for vibrating constrained-layer damping sandwich plates and beams. .I Appl. Meckan. 1041-1046(Dec. 1972). 16. A. W. Leissa, Vibrations of plates. NASA SP-160, p. 44 (1969). 17. H. G. Allen, Analysis and Design of Structural Sandwich Panels, 1st Edn, p. 9. Pergamon Press, Oxford (1969).