Jounlal o f Sound and Vibration (1974) 34(3), 309326
VIBRATIONS OF UNSYMMETRICAL SANDWICH BEAMS AND PLATES WITH VISCOELASTIC CORES Y. V. K. SADASIVARAO Structural Engineering Division, Vikram Sarabhai Space Centre, Trivandrum695022, India AND
B. C. NAKRA Department of Mechanical Engineering, Indian Institute of Technology, Hauz Khas, New Delhi110029, htdia (Received 2 October 1973) Analysis of flexural vibration of unsymmetrical sandwich beams and plates, with viscoelastic cores, has been carried out. In addition to transverse inertia effects, longitudinal translatory and rotary inertia effects have been included. Influence of inclusion of inertia effects, other than transverse ones, on the response due to forced harmonic excitation, is reported. Damping effectiveness, for each of the resulting families of modes of vibration, is evaluated for various nondimensional parameters. !. INTRODUCTION Sandwich structures employing viscoelastic cores are particularly useful for vibration damping over a wide frequency range. Work on the analysis of flexural vibrations and prediction of damping effectiveness of these types of structures has been carried out by Ross, Ungar and Kerwin [I], Mead [2], Yu [3], Di Taranto [4, 5] and Nakra [6]. In most of the published work reported so far except that of Yu [3], only transverse inertia effects have been considered. Yu [3] has analysed a symmetrical sandwich plate with a viscoelastic core, taking into account inertia effects due to transverse, longitudinal and rotary motions. The inclusion of the above mentioned effects is shown to result in flexurai and other higher order families of modes of vibration. For unsymmetrical sandwich structures with viscoelastic cores, employing dissimilar elastic layers, work on the forced vibration response and evaluation of damping at the various modes of vibration, taking into account inertia effects due to transverse, longitudinal and rotary motions, has not been published so far and forms the subject matter of the present paper. It has been pointed out by Frankland [7] that inertia effects other than transverse ones are expected to be of considerable importance in sandwich structures, though these occur at relatively high frequencies in the case of homogeneous structures. In the case of sandwich structures with elastic cores, Yu [8], Chang and Fang [9], Nakra [6] and Rao and Nakra [10, 11] have considered the longitudinal and rotary inertia effects in additionto transverse inertia. It has been reported [10, 11, 12] that the inclusion of the above mentioned effects results in several families of modes, which occur at frequencies of practical interest. 2. RESPONSE OF SANDWICH BEAMS TO HARMONIC EXCITATION In order to determine the response of sandwich beams due to harmonic excitationf(x)g(t), certain assumptidns are made in the formulation: plane transverse sections of layers l and 3 309
Y. V. K. SADASIVA RAO AND B. C. NAKRA
310
(see Figure 1) remain plane and normal to the longitudinal fibres of the beam layers after bending; the transverse displacement w remains constant throughout the thickness of the beam while longitudinal displacements are assumed to vary as shown in Figure 1; all displacements are assumed to be small and perfect continuity at the interfaces is assumed. Both shear and extension in the core are included in the analysis. Inertia effects due to all the assumed displacements arc taken into account.
e~
i
Face I Core Z
.;t
Face 3
,"
2DX
b ~
3, u
Figure 1. Geometry of sandwich beams.
.The equations offlexural motion of a sandwich beam with an elastic core are [6, I0] (a list of symbols is given in the Appendix) (c~ 2
(2q + c3) wl"  2s ~ i 2 / =f(x)g(t)
2sc, i t '1 ] r
tt~ 
 
t2
2s
{
Pl tl iil +/3212
2sc
2s
= b
{
P3 I3 ii3 "F/7 212
q" 6
+ #' %
(1)
2s t~z u3  c2 u;
t~w'  c4 w" + t"~ ul  u~(2Tj + 2c2) 
= b
. tt 3 q  c s 113
 pii. + bLo2 t2(ea//1' + e, fij + fi,' es)],
2sc

2sc iv" q ~ t2
)]
'
(2)
2s
"F 3
)'r
,
(3)
311
VIBRATIONS OF S A N D W I C H BEAMS AND PLATES
where b
b
11 + t 3
q = ~  (E~ t] + E3 t]), c2 = ~ E2 t2, C t 2 + ~ , c3 =  ~  ( t ~ + t ~  t l t 3 ) , c , = c 2
el =
tl
t 3  t 1 tI + t3 t 3  2q 4 , /~2 = 2 , ~3 = 2 ' ~ = 3
pxtx
3 +p3t3
+p2t2~
t3
,csc2
,
2t3   tl
2
'
pzt2
2
b p = b(px t~ + P2 tz + Pa ta), Y, =~ E, t, (i = 1,3) and s = (b/2)G2Kz, where K2 is the shear coefficient due to nonuniform shear in the core, its value being taken as unity as a first approximation. Series forms for the displacement components u~, ua and w are assumed, which satisfy simply supported boundary conditions: ~) =
[ u3. ) cos ~
w=
IV. sin  n~l
sin rot,
sin cot,
L
and similarly the excitation is represented as f(x)g(t) =
f . sin ~ /I=1
sin wt.
(4)
Z
Upon substituting equations (4) in equations (I) to (3) and using complex modulii G2(1 + iq2) and E2(I + if12) for G2 and E2, respectively, the solution for a beam with a viscoelastic core is given by u , . ( f ~ + ifJ)  u3.(g~ + ig~) = W.(h~ + ih~) fin, tq.(fR2 + if~) + ua.(gR2 + ig~) = W.(il~ + ih~) fin,
(5)
ux.(f~ + ifJ) + u3.(g~ + ig~) + W.(hR3 + ih~) f n = g,, where b62.3 flR~
b
02.3f12n2 + b~l'30t'3['3"
b f 2R  ff a2.3 02.s
b 62. 3 02.3 f2 n2
b f ~ = ~" ct2.302.3(89  013)
0~2"302"3
b201 "3
b 02.3 ~. 72.3
fl21/2
3 f12n2
b Y2.3 2 6 71.3 f12n2 02.3,
02.3 fl2//2
02"3 ~
13
,
71,3'
312
Y. V. K. SADASIVARAO AND B. C. NAKRA b62 "3 b f l = 02.3//2 n 2 t12 + ~" ~2.3 02.3//2,
b b 32.3 fg = ~ ~2.3 02.3//2  02.3//2,?' tl~,
b 62.3 (
b f~1 =  h ,/" = g ~2.~ 02.~({  o, .3)//, b(
32. 3
).
g~=h 02.3//2n~ 6 b 62.3 gl
~2"302"3
//2112
62.3
)'2.__2.302.3), ]/1'3
b
b2
b )'2.3
)'13//2 n2
)
 n2, 3 _)'1_ "3 02. 3 //2
b
62.3
g~" = 02.3 //2 n2 72 + ~ ~2.3 02.3//2,
23/ 2,2(i o3)
g~=h~ = b
1"['01.3) 2 112'
~2.3 0~.3//2,
 g~2  02.3 //2112 "{ b + ~ ~23 023
b
02"3 "~
b
02.3 //2n~ ' t 2  ~
b
//2112
02. 3
1+o1 )
02.3//2n2,~02.3+ ~
(0,.3)
+~~2.302.3 I 
2
6 m)'1.3 02.3
).
b
62.3 [
I + 01.3
g~ = h; ~ 02.3 //2,2 ~023 +   T   hR = b
(l 1(Xl. 3 013.3) ..[.. r
62.3 + b 02.3//21'~2
(
3
0.2.3(02.3 01.3 "[" 1)
1 0 2 3 "["
13
}.b ( /'J4 ll4

01"3 "["
~2.3 1 ) 02. 3 "[  "F1.3 71.3
b). [ 01.3 1 ~'2.3 02.3 02.3 "~2.3 (I ..[.. 0 1 . 3 ) 2 / , 4//2 n z [ 3  3)'1."'~4 "F~.3 4 (1  02.3) + IV~q.3
J
b
62.3
[
h~ ~~~'~ ~23 023//2(02"3  01.3 "[" 1) + b 02.3//2 112 ?]2 ~ 02.3 "["
t, , G2 E1 E2 o1.3=7~ o2.3=~,~2.3N,~,.3=E,~,.3=e2,
)'13
P1 P2 rot3 '~2"3 = ~ ' // : P3 P3 'L' g n  
: ~'
p~ 032 t32 ) . =  E3"
f. L ' 7tE3//2113 
1 + 01 ~ 2 2
.3 ] ,
VIBRATIONS OF S A N D W I C I I BEAMS A N D PLATES
313
Explicit expressions f o r f , can be obtained for any distribution of excitation. For forced harmonic excitation of uniform intensity Fsincot, it can easily be shown that 4F f.=
for
n = 1,3,5, ...
= 0
for
n = 2, 4, 6 . . . .
It
(6)
The expressions for displacement response can be obtained from equations (4) and (5). When the ends of the simply supported beam are subjected to a sinusoidal displacements of amplitude xosincot, F c a n be taken as F=
pcO 2 Xo
and thus
g. = ~ 3 n4 "L
0~ 3 + 71.3 0z.3 +
L.
Further, (longitudinal strain at the outer fibre of layer 3)
(7) n1,3.5,...
It has been shown that the transverse displacement is not appreciably affected when the rotary and longitudinal translatory inertia terms are included [6]. However, the longitudinal strain in the outer layer 3 is found to be considerably affected with the inclusion ofthese effects. This is shown in Figure 2 where the longitudinal strain response at the outer fibre of layer 3 at the centre ofthe beam, is plotted for both the cases: i.e., when all inertia effects are taken into account and when only transverse inertia is included, the former being shown by firm lines and the latter by dotted lines. The core is assumed to be plasticised P.V.C., whose properties are taken from reference [13] and are presented in Figure 3. 0"I
.c~
I0~
'~
103
...J
i
i ii
t
i iI IOa
I
i
I
i
I
10 4 5xtO 5 2x105
I05
109
I 107
I 10"6
I 105
I I0 4
103 4x103
), Figure 2. V a r i a t i o n o f l o n g i t u d i n a l strain with frequency. =1.~ = 1.4; 01a = 0.64; 0~.3 = 25; Yl3 = 2.64; 7.~ = 0.5; xolL = 002; fl = 001(MT.
It was shown by the authors earlier [10] that, in the case of unsymmetrical sandwich beams with elastic cores, inclusion of all the inertia effects in the analysis of flexural vibrations gives three families of modes of vibration for each modal number, n. These are predominantly
314
Y. io ~
10 4
i
V. s
K.
SADASIVA
J i ill,
P, A O
AND
I
B.
C.
NAKRA
i

I8 ._c 1"6 1"4 12 IO 3 tO 08 06 04 I
iO z IO
I
t I IIIll
I iO 3'
iO z
02 I04
Frequency (Hz)
Figure 3. Propcrtics ofviscoelastic material.
flexural, extensional and thickness shear types of modes, designated as the I, II and III families of modes, respectively. It is seen that if the core of the sandwich beam of Figure 2 were to be elastic, with its shear modulus equal to the inphase shear modulus of the viscoelastic core, the nondimensional natural frequencies of the II and III families of modes corresponding to n = 1 would be around 1.3 x 104 and 103. Thus, the peaks near these values of frequency parameters in Figure 2 can be due to the resonant modes of the II and III families. 3. DAMPING OF SANDWICtt BEAMS T h e displacement response to sinusoidal excitation at a normal mode is obtained, in general, from the equation  m o ) Z q + K(I + iq~)q =fo sinpt, where q is the generalised displacement coordinate for any mode, fo is the amplitude of harmonic loading, m is the generalised mass, Kis the generalised stiffness and qs is the damping loss factor of the system for the nth mode, equal to the ratio of the imaginary to the real part of the complex generalised stiffness. Equations (5), after simplification, can be reduced to a single equation of the form b W, tRr, + ilr,] = f , .
(8)
Both Rr, and l r , involve the frequency parameter), since all inertia terms are included. As the present analysis involves three families of modes, equation (8) can further be written in the form
f. W . = b(R.l + iI.1  $1 ).I)(R.2 + iI.2  $22.2)(R.3 + ii.3  $3 23)'
(9)
where 21, 22, and 23 are the nondimensional frequency parameters of the three modes of vibration. Thus,.the system loss factor for the three families of modes of vibration can be taken as I . J R . I , 1.2/R.z and I,,JR.3, respectively.
VIBRATIONS OF SANDWIC[!
BEAMS AND PLATES
315
The calculation o f !1~ on a digital computer is carried out by equating each factor in the d e n o m i n a t o r o f equation (9) to zero, replacing the nondimensional frequency parameter by a complex one and taking the ratio o f the imaginary to the real term o f tile same. Thus, for the first family o f modes, R.I + il.l  $1 )1 = O. $1).1 can now be replaced by $1 )1 = )x + i2y which gives R,I + ii,1 = 2x + i)r, or
2y ).x"
Int tl~ = R.I
(10)
Curves have been plotted in Figures 4 to 7 to show the variation of q, with other nondimensional parameters. D a m p i n g is found to increase with modal number till it reaches a m a x i m u m value for the case of the I and II families of modes, which are predominantly 0"5
0"1
102 o
103 ~JtlO " 4 2xlO 4
104
T I
T 2
I 3
I
T
T
4
,5
6
I 7
I 8
I
9
I I0
I II
12
/7
Figure 4. Variation of system loss factor with modal number n. cq.3 = 71.3 = I0; 72.3= 0"5; =23= 0.00026; Oa.3= 05; 02.3 = 2"5; ~2.3 = 00001 ; p = 0"0125; ~z = 0'5; 82 = 0'5.
flexural and extensional modes o f vibration, respectively, and it decreases with increase in modal n u m b e r for the III family o f modes, which is o f predominantly thicknessshear type, as shown in Figure 4. M o d e s relating to the I and I I families have m a x i m u m d a m p i n g at particular values o f core stiffness parameter as shown in Figure 5, while the loss factor for the I I I family o f modes is seen to be increasing with the core stiffness parameter. It can also be seen from Figure 6 that, with the increase of thickness o f outer elastic layer, 01,3, the system loss factor for the II and I I I families o f modes decreases whereas it increases in the case o f
316
Y. V. K. SADASIVA RAO A N D B. C. N A K R A
10 ~
IOZ
~0~ 5]~10 4
2=tlO 4 I0  4
106
I
I
i0~
i0 4
i0 "3
az/ e3 F i g u r e 5. V a r i a t i o n o f s y s t e m loss f a c t o r w i t h (52.a. =ta = 1"0; Yla ~ 1 "0; )',.3 = 0"5 ; 013 = 0"5 ; n = I ' 0 ; fl = 0 . 0 1 2 5 ; tl, =,82 = 0 " 5 .
I0
i
i
l
l
i
l
Oz.3 =
2"5 ;
i
10,
~,.~
~0~
103 5 xlO'4
2xlO ~ I0 4 0
0"1
I 0"2



I
0 3 04 05 0 6
!
I
[
07 o B o9
Io
0a3 F i g u r e 6. V a r i a t i o n o f s y s t e m loss f a c t o r w i t h 0=.3. (zt.~ = ] ' 0 ; # = 0 " 0 1 2 5 ; Yl.3 = J0; 72.3 = 05; d~,.~ = 104; ,'/z = #z = 05; n = 10.
VIBRATIONSOF SANDWICHBEAMSAND PLATES i'O
I
i
I
i
i
I
I
I
I
I
I
I 8
I I0
I 12
I 14
I 16
I 18
20
3 17
10" /2 5 x l O "~
2 x l O 13
10.3 0
I 2
I 4
6
I 22
823 Figure 7. Variation o f system loss factor with 02,3. ~x:.3 = 10; fl = 00125; }'1.3 = 1'0; 7z.a = 05; Oza = 0"5;
32a = 104; n = 1"0; th =f12=0"5.
the I family of modes. Further, loss factors for the I and II families of modes increase as the core thickness is increased while the reverse happens for the I I I family of modes, as seen from Figure 7. In all the above mentioned cases, the II family of modes has the least value of loss factor. 4. RESPONSE OF SANDWICH PLATES TO HARMONIC EXCITATION In order to determine the response of a sandwich plate due to harmonic excitation
Q(x,y)g(t), the assumptions are identical to those for sandwich beams, as given in section 2 except that the extensional effect in the core has been ignored for the purpose ofsimplification. This effect was seen to be of importance only for a very stiff core [6]. The variation of the longitudinal displacements in the x and y directions, along the thickness of sandwich plate, is shown in Figure 8. The equations for flexural vibrations of an unsymmetrical sandwich plate, with an elastic core [l 1, 12] are
~,
{
(1 + vl) v'~* +
u; +    y  
Pltlfilp2t2
(l+h) 71 vl**+  " ~ 
plll~Jllp2t2
5
up
)(c +~


t~
w'


t~
)
+ "~+ f/" ~3 = 0 ,
u~,+~v~
2
+~'2
)
d ~ q (r t~3 = 0 ,
(11)
 w*tg
tg
02)
318
Y. V. K. SADASIVA RAO AND B. C. NAKRA Y
z
/
ul >~? I
I:/!:
~ v~,~ Figure 8. Geometry of sandwich plate.
{
~,3 u;~ (I + v~._._._~)v;* +
u~*
2
}
 ~,~ ~ w'
,.,}  
t~
03)
~, v~,+(z),,3+ pataO3p2t 2
v;~2
~ + T  i 
# * t4
,Tw*~ ,~ /j (14)
:0,
c (D 1 + D3) V" Iv  ~2 t~ { c ( w " + w * * )  u'l + u ~  v* + v*}
pl t~ + p3 tJ 12
(19'" "k 19'**)  P2 t2{~3(U; t V~) Jr"s
+ I)~)
( r
+ 4+ff
(15)
where
E,t,
E,t?
Y~= 1  v~'
D i   12(1  v~)'
t3  2tx sa =
12
2t3  tl '
e4 
12
i = 1, 3,
~'2 = G2 t~,
VIBRATIONS OF SANDWICII BEAMS AND PLATES
319
Series forms for the displacement c o m p o n e n t s lq, u3, vl, v3 and w are assumed, which satisfy the simply supported boundary conditions:
{Ul } = It3
2 2 n.l{ } 22/ rn~l
{'1
=
v3
,..,
U 1,~.
cos
mgx
Uaran
r
W,.. sin
cos
a
sin
a
mI n1
ngy
sin cot,
T

.1 t V3"" )
w=
sin
sin cot, b
sin wt
T
and similarly, for the excitation,
Q(x,y)g(t) =
~. ~
m.I n=l
,,mx ,my. Q,,, sin  sin ~ sln ogt.
(16)
(7
U p o n substitution of equation (16) in equations (11) to (15) and replacing G2 and E2 by complex moduli G2(I + h12) and E2(I + i/~2), respectively, the following equations are obtained for a sandwich plate with a viscoelastic core:
 I V , . . ( a f + ial) + V,m.d. + U,.,.(d~ + ia~.)  u3,..(a~ + ia'.)=o,
(17)
lv,..(ff
u,,..(d~ + i a l ) + u3,..(f~ +if~2)+ v 3 , . . f f = o ,
(18)
 W , . . ( e f + ieO + U,,..d4 + Va,..(e~ + ie~2) V3,..(d ~ + ida) = O,
(19)
 W , . . ( g f + ig~) + U3,..f3  v,,..(a~ + ida) + V3,..(g~ + igt2) = O,
(20)
+ifl)
W.,.(h• + ih~)  U,.,.(hR2 + ih~2)  Ua,..(h~ + ihta)  Vl,..(h~ + ih 1)  Va.,.(hf + ih~) = Q ~ . ,
(21) where
(,+0,3) _ ..0.
df=62.3 I+ 202.3 +./13 12 (I  20,.3), ( d[ = f 2 z =  f t z = /,~ = 62.3
0~1"301'3 d2a = h ~ =
(1

2 v3) 2 ~,3
62 3
d~ = d3t = e~ =  
0,.3 pm
{liIp'~ '12,
._I_I(62.3 )~'2.302.3) d3a= #m~02.3t ./13 6 n/3./ f ~  2(1  v3)' Q~"
Q,.. a E3nm
1I"01.3/ 1+
202.3 ] I]2'
(1~tl.3V3)
2
./2p2
,,.
_~..
m
02.3~m
, ( 01.3 , 2+3 )
m
"/13
,
320
Y. V. K. SADASIVA RAO AND B. C. NAKRA
2
e~= ([~1.3v~)
m 3' f l +
flDI +
2
O,.3flm
tim
01. 3 +  71.3 02.3
'
n
e~ = h~ = m ]/d~,
tl I el = hl =   3'dl, 1/'l f~
ZY2.37x.302.312(2  01.3)  62.3
1
f _ a = (v2) I
raft+,
0v3)
I+
1 dr 013 ~ ~ 20~.3 ] da~ = h~,
3'~fln ~ + G . 3
2
m
I + ~ 202.3

+
7, .3
02.3
'
II R
R 171
1 gtt = hst = II ]/fx,
Ill
)" )" ]/2"3 02.3 + (1 v3) mfl } + ~ 023 G . 3 f l m  ]/l.3/hn 3 ]/13  ' ~ ' 2
1 [s g~ = v3~ (1 t m
I
I
g2 = d J,
hlR
_
3 3 [ f cq.303.3 . 1 ]/*} + 1,/ /j i t ]2(1 __ ~2.3 V2) _l_ 12(i __ V2) ){ i _i_ 23')2/H~ /'l'n 2 q_ (~7/) 4'1l
1
+
0L3 +  
hl = mfl
( 1 + ~,2 y2
]/1.3
IH 2
~t .30a .3 tiny a', = 2(1  4,,.3 v3)'
Pl o~2 t~ , E3
2 = ~
~'2 3
+
){
3'13
14
02.3 ( 1 + 012.3  013) {
01.3 +
]/, 3
023 "~
)]
3'1.3
(1 + 01.3) "12 2023 / 62.3q2, nt3
/~ = 7 '
a 7 =~,
vz ~,.3 =  . v3
Expressions for 01.3, 02.3, 62.3, ]/~.3, 3'2.3 and cq.3 are the same as for beams. Expressions for displacement response are also obtained from equations (16) to (21), in the same manner as for sandwich beams.. It can be shown that, in the case of forced harmonic excitation of uniform intensity Fsinwt, 16F Q,,,,,
mnlt 2
111, n
I, 3, 5 . . . . .
321
VIBRATIONS OF S A N D W I C H BEAMS A N D PLATES
When the edges of the plate are subjected to a sinusoidal displacement of amplitude xosincot, F c a n be taken in the same form as in the case of beams and Qg, in nondimensional form can be written as
(
O~"  ~ m ~ n~q ~ 0~.~ + ~',~ 0,.~ + ~
l)
(22)
~ a,
where XO O~I ~   ~ 
a
The stress trx~ at the outer fibre of layer 3 of the plate is trx~ = 1  v2

+ v3 w**)
.
(23)
By substituting the assumed distribution of u3, v3 and w in equation (23), the longitudinal stress at the middle of the outer fibre of layer 3 in nondimensional form can be obtained as cr~
_
22[
~
yhi2
m.I
1711~"
m
I1 ( m ~ + v~ n ~ , f l ) 
r.J~%..
n=l
I1~
v3 ny Va~ sin 7 " sin ~ sin cot,

(24)
where
u3,.. u~.,. =
V3m. , v~mo=
G
Q
iv,.. , w~
Figure 9 is plotted for both the cases: i.c., whcn thc cffccts of longitudinal and rotary inertia are taken into account as well as transverse inertia and when transverse inertia alone is included. The corresponding curves are shown by firm and dotted lines, respectively. The I'0
I
' l~'hq
i
i
i
i
I0 "I
~
102
~
~,
iO'S 5 x l O 4 2 x l O "4 104 I r llttl~T i09 2xlO'~ 5xlO'~lO'a
I i 0 r
I lOe
1 lO'S
I 104
i0~
k Figure 9. V a r i a t i o n o f nondimensional longitudinal stress w i t h frequency. ~,.a = 7~a  V~.a = Y = ] ' 0 ; Xo 0 1 . 3 : 0  7 ; 02.5 : 10.0; f l : 0'0125; va : 0  3 ;   : 0.04; Y2a = 0 . 5 . a
322
Y. V. K. SADASIVA RAO AND B. C, NAKRA
p r o p e r t i e s o f t h e c o r e m a t e r i a l a r e t a k e n f r o m F i g u r e 3. It c a n be seen t h a t c o n s i d e r a b l e difference in tr~ at h i g h e r f r e q u e n c i e s exists b e t w e e n the t w o c u r v e s in F i g u r e 9, the difference b e i n g n e g l i g i b l e at l o w e r f r e q u e n c i e s . It was s h o w n by t h e a u t h o r s [1 I] that, for an u n s y m m e t r i c a l s a n d w i c h p l a t e with a n elastic core, five families o f m o d e s are e n c o u n t e r e d f o r e a c h set o f m o d a l n u m b e r s m and n. T h e s e w e r e d e s i g n a t e d as t h o s e b e l o n g i n g to f a m i l i e s I to V ; the I f a m i l y o f m o d e s was seen to be p r e d o m i n a n t l y flexural, the II a n d I V f a m i l i e s o f m o d e s w e r e o f e x t e n s i o n a l types w h i l e the I I I a n d V o n e s w e r e o f t h i c k n e s s  s h e a r types. A s in t h e case o f b e a m s , t h e c h a n g e in l o n g i t u d i n a l stress at high f r e q u e n c i e s , arising f r o m l o n g i t u d i n a l a n d r o t a r y inertia, is a s s o c i a t e d with t h e h i g h e r f a m i l i e s o f m o d e s o f v i b r a t i o n .
5. D A M P I N G
OF SANDWICH PLATES
C o r r e s p o n d i n g to e q u a t i o n (9) f o r b e a m s , the f o l l o w i n g e q u a t i o n is o b t a i n e d for s a n d w i c h plates: O""
,
(25)
IV. = (R.I + il.x  S t 21)(R.z + iln2   $2 ),2) (Rn3 '1i/n3  S a 23) • x ( R , , + i l , ,  S , ;.,) x ( g , 5 + il,5  $5).5) w h e r e 21, 22, 2a, 2 4 a n d 25 c o r r e s p o n d to the f r e q u e n c y p a r a m e t e r s o f t h e five f a m i l i e s o f m o d e s o f v i b r a t i o n . R,1, 1,1, R,2, 1,2, etc., a r e the coefficients o b t a i n e d o n f a c t o r i s a t i o n . T h e c a l c u l a t i o n o f q s is d o n e in t h e s a m e w a y as e x p l a i n e d for b e a m s . T h e results a r e p r e s e n t e d in F i g u r e s 10 to 14. T a b l e I s h o w s the v a l u e s o f s y s t e m s loss f a c t o r , q,, at d i f f e r e n t m o d e s o f v i b r a t i o n . It is f o u n d t h a t the v a l u e s of~Is c o r r e s p o n d i n g to the I f a m i l y o f m o d e s , o b t a i n e d by c o n s i d e r i n g all inertia t e r m s a n d i n c l u d i n g o n l y the t r a n s v e r s e i n e r t i a t e r m , a r e identical. T h e loss f a c t o r o f the I I I a n d V f a m i l i e s o f m o d e s o f v i b r a t i o n (i.e., the t h i c k n e s s  s h e a r types) increases w i t h c o r e stiffness p a r a m e t e r w h e r e a s t h e s a m e increases first a n d t h e n d e c r e a s e s in the cases o f e x t e n s i o n a l o n e s (i.e., II a n d I V families) a n d the flexural o n e (i.e., I f a m i l y ) , as s h o w n in F i g u r e 10. T h e s y s t e m loss factors o f the II, III, IV TABLE 1
L o s s f a c t o r s o f sandwich p l a t e s ~' = cq.3 = 7t.3 = ~13 = 1"0, v3 = 0"3, fl = 0"0125, Y2.3 = 0"5, 013 = 0'7, 02..3 = 1 0 " 0 , ~2"3 = 10 5. ~1, at different modes of vibration when all inertia terms are included .A,
t
m
n
I
I 1 1 1 1 2 2 2 2 3 3 3 4 4 5
1 2 3 4 5 2 3 4 5 3 4 5 4 5 5
0"373 0"273 0"189 0132 0"095 0215 0"160 0"117 0"087 0"126 0"098 0"076 0'080 0"065 0055
II 0624 • 0"261 x 0132 • 0"783 • 0'514 x 0"165 x 0"1 x 0"665 x 0"461 x 0"738 x 0'534 x 0392 x 0418 • 0324x 0268 x
III I0 s 10 3 10 a 10 4 10 4 10 3 IOa 10 4 10 4 10 4 10 '~ 10 4 10 4 10 4 lO 4
0928 • 0"375 x 0"188 • 0.11 • 0"725 x 0"235 x 0"145 x 0'941 x 0'649 x 0"105 x 0"753 x 0"554 x 0"589 • 0"459 x 0"377 x
9
IV 10 2 10 2 10 2 10 2 10 3 10 z 10 2 IO3 10 3 10 2 10 3 10 a 10 a I0 a I0 a
0"229 • 0"932 • 0469 • 0277 • 0"181 x 0'584 x 0"361 x 0"236 x 0"163 x 0"261 x 0"189 x 0"139 x 0148 x OI15x 0"949 x
V
10 3 10 4 10 4 10 4 10 4 10 4 IO 4 10 4 10 4 10 4 10 4 10 4 10 4 10 '~ lO s
0328 0132 0'658 0"387 0"253 0"824 0"506 0"329 0"226 0"366 0263 0"194 0205 0"161 0131
• x x • x x x x x x x x x x x
10 2 10" 10 3 10 3 10 a I0 3 I0 a 10 3 10 3 10 a 10 3 10 a 10 a 10 3 I0 a
VIBRATIONS OF SANDWICH BEAMS A N D PLATES 1.0
I
323
I
I01
aoz
103 5 x l O 4
2x10 "4 I0 ~
I
I05
l
i04
i03
i02
~2.3
Figure ]0. V a r i a t i o n o f system loss factor w i t h ~2.5.0q.5  I . O ; # = 0.0125; 7~.3 = l'O; 72.5 = 0"5; {71.5 = 0.5; 029=2"5; ~13 = l ' O ; VZ = 0 " 3 ; q2 = 0 ' 5 ; 7 = 1 . O ; m = n = 1. 5xlO ~
f
I
I 02
I 03
I
!
I
t
i
I 0,5
I 06
I 0.7
I 08
i01
IO~
i0~
104 5xZO S
2xlO 5 iOS 0.1
I 04
09
e j3 Figure ] ] . V a r i a t i o n o f system loss factor with 01.3. ~ . 3 = ] ' 0 ; f l = 0  0 ] 2 5 ; 7 = 10; )q.3 = ]  0 ; 72.3 = 0"5; ~2.3 = I OS; 0z9 = 0"5; ~:/131"0; v3 = 0"3; qz = 0"5; m = n = I.
Y. V.K. SADASIVARAOANDB. C. NAKRA
324
I'0
I
I
i
i
I
I
I .'5
I I0
I 15
I 20
I 25
tot
102
103 ~
104
t
0
30
02.~t
F i g u r e 12. V a r i a t i o n o f s y s t e m loss f a c t o r w i t h 02.~. r
= ] ' 0 ;,6' = 0"0125; 7~.3 = ] '0; 723 = 05; ~,.3 = l 0  s ;
0=.3 = 0 . 5 ; VI.3  1  0 ; v3  0 ' 3 ; r/2 = 0  5 ; 7 = l ' 0 ; m = n = 1.
I0
I
I
i
I
i0 I
I0 "z
IO:S 5xlO "4
2xlO 4 10"4
I
0
00I
I
I
I
0.02
0.03
0.04
0 05
F i g u r e 13. V a r i a ! i o n o f systcm loss f a c t o r w i t h ft. ~,3 = 1"0; Yr3 = ! ' 0 ; Y23 = 0"5; 62.3 = l OS; 0~.~ = 0"5; 02a  2"5 ; Vt, .a  l ' O ; i'a  0"3 ; 17,  05 ; y = 1"0; m   n = I .
VIBRATIONS IO
OF SANDWICH
BEAMS
AND
325
PLATES
I
I
I
i

I 1.0
I 20
I 30 ),
I 40
I .50
i01
,oz
103 5xlO4
2xlO4 104
6.0
Figure 14. Variation of system loss factor with y. el~ = 1"0; 01.a = 05; 02.3= 2"5; Y~3= 1.0; Yz.3= 05; ~2.3= 105; ~t.3 = 1.0;va=0'3;t12=0.5;fl=00125;m=n= 1.
and V families decrease with increase of 01.3 whereas the change for the I family of modes is not pronounced as can be seen from Figure 11. The loss factors for all the modes of vibration except the flexural one decrease when the thickness of the core is increased, as shown in Figure 12. Increase in fl, i.e., nta/a, as seen from Figure 13, is found to decrease the loss factor corresponding to all the five families of modes of vibration. Similar observations can be made from Figure 14 when ),, the ratio of length to breadth of the plate, is increased. It is also observed, for all the above cases, that the predominantly extensional modes of vibration are always the least damped. 6. CONCLUSIONS The effects of inertia other than that due to transverse motion have been clearly established for the case of flexural vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores. The longitudinal strain response in the case of beams and the longitudinal stress response in the case of plates are considerably affected by the inclusion of all these inertia terms. Loss factors for three families of modes for beams, and five families of modes for plates have been obtained and the dependence of these loss factors on appropriate nondimensional parameters has been established. REFERENCES 1. D. Ross, E. E. UNGARand E. M. KERWiN 1959 Colloquium on Structural Damping, American Society of MechanicalEngineers. Damping o f flexural vibrations by means o f viscoelastic laminae. 2. D.J. MEAD 1962 University of Southampton A.A.S.U. Report No. 160. The double skin damping configuration. 3. Y. Y. Yu 1962 Journal of Aerospace Sciences 29, 790803. Damping of flexural vibrations of sandwich plates.
326
Y. V. K. SADAS[VARAO AND B. C. NAKRA
4. R. A. Dl TARAm'O 1965 doltrna[ of Applied Mechanics 32, 881886. Theory of vibratory bending for elastic and viscoelastic layered finitelength beams. 5. R. A. Dx TARANIO and J. R. MCGRAw 1969 Journal of Engineerhtgfor Industry, Transactions of the American Society of Alechanical Engineers 91, 10811090. Vibratory bending of damped laminated plates. 6. B. C. NAKRA 1966 Ph.D. Thesis, Unit'ersity of London. Vibrations of viscoelastically damped laminated structures. 7. J. M. FRANKLAHO 1960 Journal of Applied Mechanics 27, 374375. Discussion on the paper by Y. Y. Yu. 8. Y. Y. YLr 1959 Journal of Applied AIechanics 26, 415421. A new theory of elastic sandwich platesone dimensional case. 9. C. C. CHANG and B. T. FAnG 1961 Journal of Aerospace Sciences 28, 382396. Transient and periodic response of a loaded sandwich panel. 10. Y. V. K. SADASlVARAO and B. C. NAKRA 1970 Proceedings of the 15th Conference LS.T.A.M., 301314. Influence o f rotary and longitudinal translatory inertia on the vibrations o f unsymmetrical sandwich beams. 11. Y. V. K. SAOASWARAO and B. C. NAKRA 1973 Archires of AIechanics 25, 213225. Theory of vibratory bending o f unsymmetrical sandwich plates. 12. Y. V. K. SADASIVARAO 1972 Ph.D. Thesis, hMian htstitttte of Technology, DelhL Vibrations of unsymmetrical sandwich structures. 13. Chesapeake Instrument Corporation 1962 Technical Report No. 2. Dynamic mechanical properties o f materials for noise and vibration control. APPENDIX
LIST OF SYMBOLS a b
length of sandwich plate along xaxis width of sandwich beam, or width of sandwich plate
c tz qf(x) m,n t t, u, v, w x Xo y z E, G, L ,02 t12 th v, p Ps ~o 9 *
tl I t3
2
intensity o f d y n a m i c l o a d i n g modal numbers time variable thickness o f layer i longitudinal displacement in xdirection in layer i longitudinal displacement in ydirection in layer i transverse displacement longitudinal coordinate in beam; space variable in plate along xaxis amplitude of harmonic excitation space variable in plate along yaxis space variable in transverse direction Young's modulus of layer i; inphase component of Young's modulus for viscoelasticlayer shear modulus of layer i; inphase component of shear modulus for viscoelastic layer length of sandwich beam material loss factor of core material in direct strain material loss factor of core material in shear system loss factor Poisson's ratio of layer i mass of sandwich beam per unit length; mass of sandwich plate per unit area mass density o f layer i per unit volume circular frequency in radians per second differentiation with respect to x differentiation with respect to y differentiation with respect to t