Journal of Sound and Fibration (1975) 39(2), 229-235

FREE VIBRATION OF ELASTICALLY CONNECTED CIRCULAR

PLATE SYSTEMS

A. S. J. SWAMIDAS

Department of Applied l~fechanic~, btdian lnstitttte of Technology, ]~ IadraJ-600036, btdia

A N D

V. X. KUNq.JKKASSERIL

Department of Aeronautical Enghteerhzg, btdian btstitute of Technology, i~ladras-600036, bldia

(Received 28 May 1974, and ill revised form I October 1974)

In a previous paper [I], the authors have presented the solutions for the normal mode vibrations of an n-plate system. In that paper the numerical results were limited to the frequencies and mode shapes of some representative models of double plate systems. In the present paper additional numerical results are presented to illustrate the various other features such as the influence of the thickness ratio and foundation stiffness on the frequencies. The omission of a normal mode of the free-fixed case in the previously published paper is also pointed out.

1. INTRODUCTION

In a recent paper [l], the authors presented an analysis for the free vibration o f systems comprising n elastically connected isotropic plates. The numerical results presented in that paper were mainly confined to the normal modes and the corresponding frequencies. For fully understanding the practical feasibility o f such systems, it is further seen to be necessary to know the influences o f the thickness ratio and foundat ion stiffness on the vibrational

I I

' I

~ :'1 r (a)

!~/~'

I i

(b) Figure 1. Example problems solved. (a) Free-free plate system; (b) free-fixed plate system.

229

230 A. S. J. SWAMIDAS AND V. X. KUNUKKASSERIL

characteristics. Therefore, additional results have been obtained for the double-plate systems shown in Figure 1 and are presented in this paper to exemplify further the behaviour o f such systems. Details o f the analytical par t have already been given in reference [1] and are not repeated here. The eigen-determinants o f the physical systems considered here and a discussion o f the numerical results are given in the following sections.

2. FREQUENCY DETERMINANTS

For a double-plate system with free-free edges, the edge conditions to be satisfied at p = 1 are

M , - - 0 ,

v, = 0, (1)

for both the top and bo t tom plates. By using the appropriate solutions given in reference [1], the frequency equat ion corresponding to the above conditions can be seen to be the following (a list o f symbols is given in the Appendix) :

Ra J.(2-1) + R3 I,.(2-i) --

-b R 2 J,.+1(2-1) -- R2 Ira+x(2-,)

R 7 J,n(Al) "b R9 Ira(2,) -I-

R4 Jm(22) +

q" R5 Jm+l(2-2)

R I I Jm(2-2) +

+ RI2 Jm+1(2-2)

P2[Rls J,,,0.2) +

+ R19 J,.+i(;~9]

?~[R~ J,.(2-9 +

"[- R26 Jm+1(2-2) ]

where

+ Rs Jm+x(2-t)

PI [R15 J,.(2-d +

+ RI6 Jm+1(2-1)]

PI[R21J~0.1) +

+ R ~ J.+x(,h)]

+ Rio Im+1(21)

Px [R,71.1(21) -

- RI6 Irn+l(2-1) ]

Px [R23 Im(2-x) +

q- R2,1.1..+1(21)]

P I = (2~ - 0 2 + q l ) /q l , / '2 = (2-~ - I22 + ql)[q~,

RI = (I - vl)(m 2 - m) - 25, R2 = (1 - vl) 21,

R 3

R$ ----

R7 =

R 9 =

R l l - -

R13 =

R15

R17

R19

R21

R23

R2s

R27

R6 Im(~2) --

-- R 5 Is+t(2-2)

R13 ]m(~2) -I-

-Jr R 14 Irn+l(2-2)

P2 [R2 o Im(2-2) - -

- R19 I,.+1(2-2)]

~'~[R~ I,.(22) +

+ R2a Im+10-2)]

(1 -- vl)(m 2 -- m ) + 2-2, R 4 = (1 -- vl)(m 2 - m ) - 22,

(1 - - Vi) 2-2, R 6 = (1 - - V l ) ( m 2 - - m ) + 2-~.

- - ( I - - v l ) ( m 3 - - m 2) - - m2-12,

--(1 - - V l ) (m 3 --/7' /2) dr ?1l), 2,

--(1 - - v l ) ( m 3 - - m 2) - - m2-~,

- - (1 -- Vl) (m 3 - - m 2) -Jr m ~ ,

= - (1 - v2)(m 3 - m 2) + m21,

= - (1 - v2)(m 3 - - m 2) - m2-, 2,

= - (1 -- v2)(m 3 -- m 2) q- m2-22,

Rs = +(1 - vl) 21 m 2 q- 213,

Rio = - ( I -- vl)2-1 m2 q- 2-3,

R12 = +(1 - Vl) 2-2 m 2 + ).2 a,

R14 = - (1 - vl) 2-2m 2 -1- ).23,

(1 - v2)(m 2 - m) - 22, gx6 = (1 - v2)2.1,

( 1 - v2)(m 2 - m) + 2-2, Raa = (1 - v2)(/'n 2 - - / / ' 1 ) - - ,~2,

( 1 - - V2) 22, R2o = (1 -- v2)(m 2 -- m) + 22,

--(1 -- v2)(m 3 -- m 2) -- m2-], R22 = +(1 -- v2) 2-1 m 2 + 25,

R24 = - (1 -- v2)21 m 2 + 25,

R26 "= + ( I -- V2)2-2 m2 Jr 23,

R ~ = - ( i - vD 2-2 m" + 2-~,.

=0 , (2)

(3)

ELASTICALLY CONNECTED PLATE SYSTEMS 231

The 4 • 4 determinant given in equation (2) is solved to obtain the frequencies and their corresponding mode shapes. When the plates are identical, equation (2) can be reduced into two 2 x 2 determinants, one giving the in-phase motion and the other giving the out-of-phase motion. While the in-phase mode shapes and frequencies are the same as those of a free plate, the out-of-phase mode shapes are the same as those of a plate on an elastic foundation of modulus 2K. The frequency determinant for the fixed-free plate system is given in refer- ence [l].

3. NUMERICAL RESULTS AND DISCUSSION

The results, obtained for the specific problems shown in Figure 1, are given in Figures 2 through 5. The program was written in Fortran language and the numerical computations were done on an IBM 360 computer. For lower values of the frequency parameter, f2., Bessel functions with complex arguments are obtained while for the higher values of f2 Bessel functions with real arguments are obtained. The accuracy imposed upon the calcula- tion of Bessel functions w a s 10 -9. The iteration process for calculating each eigenvalue and the corresponding eigenmode took 4 seconds on the average.

ct

6 0 * t \ , i , = I ' ' h

'~ \ \ I s s ,~

~\~ I~ o "I

"~.._~I~Z (s; 4)

2o \ ' ~ . . . . " ~ / " - " ~ f " ' -

io _ _ ~ .._..G~I mode(s=l) 0i

2 5 I | 1 t I i I i 0 .25 0 50 I .O0 I. 50 2 O0 2 5

hllhz

I (b) h~, j ,/Steel

i �9 I I'

I< a >1

~ ' "~-....._._..~

\ . . i , / ~ _.._...i-

]2

I I I I I I I ,= O S O I (30 1 . 5 0 2 0 0

Figure 2. Influence of thickness ratio on the symmetric vibration o f elastically embedded double-pldte system with (a) free-free edges and (b) free-fixed e d g e s . - - , Steel-aluminium; . . . . . . . , steel-steel; Vs = 0.286; va = 0.333; "/s/Ya = 2.89; Es/E,4 = 3.00; qt = K l a 4 / D t = 170-0; f2 :-: (7~ h ia4to2/Dl) t l ' .

Figures 2(a) and (b) give the influence of the thickness ratio of a free-free and free-fixed double-plate system on the symmetric frequencies of vibration. In comparing the two figures, it is important to note that the lower values for the first mode are given by a free-fixed plate system rather than by a free-free plate system. The reason for this apparent contradiction is due to the fact that lower modes which are not possible for a free-free plate system are possible for a free-fixed plate system. Similar behaviour was noted in the case of the vibration of a continuous double-plate system in reference [2]. In Figure 2(a), the behaviour of two types of free-free plate systems with different combinations of materials are given: viz.,

232 A. S. J. SWAMIDAS AND V. X. KUNUKKASSERIL

C:I

200

160

120

80

40

) o :'I . / ~ ~...1. ~ " ~ .

t J/ /I/if s'3

,i

T I T I I I T I" I ' 0 2 0 5 0 4 0

LcXJ~o q~

Figure 3. Influence o f founda t ion stiffness on the symmet r i c v ibra t ion ofelast ical ly embedded double-plate sys tem with f ree-free ends . , In-phase ; . . . . . , ou t -of -phase ; vl = v2 = 0-286; ql = K~a'/D, = 170.0; -Q = (71 hi a4co2/Di) t/2, (steel-steel).

C~

2.0 3 0 4.0 5 0

LOg~O qt

Figure 4. Influence o f founda t ion elasticity on the symmet r i c vibrat ion o f elastically embedded double- plate sys tem with free-fixed ends. vl = v, = 0.286; ql = Kta4[Di = 170.0 (steel-steel); 12 = (ylhlaCt.o2/D~)tt2; 1, 2, 3, 4, 5 s imilar m o t i o n ; I, II, 11I, diss imilar mot ion .

ELASTICALLY CONNECTED PLATE SYSTEMS 233

steel-steel and steel-aluminium. For the modes in which the vibrations are similar (i.e., the deflections at any instant are in the same direction for both plates), viz. modes I and III, the steel-~/luminium plate system gives lower frequencies than the steel-steel system when hJh2 is less than I. For higher values of h~/h2, the behaviour is reversed. For dissimilar motions (i.e., the deflection at any instant are in the opposite directions for both plates), viz. modes

Fixed - free Fixed - free dissimilar plates similar plates

h~ J = t / Mode = ~.~ = = =

h2 h,

Free -free dissimilar plates

hz

Free- free similar plates

,~- =~ ~-~- ~- ~', h,

0 0

(5'936) 0 (6-750) 0 9.=6.66;) .~. :7.848

0 0 0 0

�9 (14.4111 0 ,('/,=14.538 6

U3

o q~

s

(29-04~) .0. : 20"929

t~3

4

0 ~D (37.067) F,- D:34-333 6

0 gO

oa ,~- O~ 0')

0 ~ I~.

(43 810} 0 .0.=41-245 6

0 0 0 0

175"2011 0 ~ ~=72"710 6 6

0 o % o

0 O~ 0 0 (8 620) ~ (B.997) co ~D D. :8.190 6 9.=8.972 6

0 0

0 0 0 0

~ o (29.070) ~ 127.2911 .0.=21.153 0 .0.=20-506 6

re) cO 0 6 6 ~ ~ ~

�9 138.4sl1 o ~ (D ~=33.040 ~3) 0 ~,:38.4,3 6 0

0 tO. 00 0 tO .~

(27-369) ,0. = 20.393

0 ~ q)

0 139.204) ~t" (43-944) 0 r~ ~ ~ (46-289) 0 r~ co 9,=39-329 6 ,0,=41.362 6 6 ~=42-610 6 0

~. c0 oJ co 0 ~ . cp.

~ :43-077 6 .0,=71.583 6 o 6 ,0,=87.721 O O O

~ ~o~ ~ 060 0 000 0 OO 0 0

1~.,o5) o~ ~ 188.9ss) o~ ~ =~ (9,.,;,;,;) o~ ~ ~0 ~ . 0 . = 8 8 . 2 7 0 6 o ~ = 8 8 . 7 5 9 6 6 6 D.--896~ 6 6 6

Figure 5. N o r m a l modes (symmetric ofelastically embedded double-plate system steel-steel plates (values in brackets for steel-aluminium plates). Edge conditions: hJh2 = 1.25; q, = ] 70.0. Steel at top .

234 A . S . J . SWAMIDAS AND V. X. KUNLIKKASSERIL

II and IV, the steel-aluminium system always gives the higher frequencies. For a free-fixed double-plate system as shown in Figure 2(b), the steel-aluminium system gives the lower value for the first two similar motions of the plates (modes I and II). For the dissimilar motion of the system (mode III) the steel-aluminium system gives higher values than the steel- steel system.

The influence of the elasticity of the foundation (foundation modulus) on the symmetric vibration of an elastically embedded double-plate system (identical plates) with free-free and free-fixed edges is shown in Figures 3 and 4. For the free-free system shown in Figure 3, the motions are separable into in-phase and out-of-phase motions. The frequency values of in-phase motions are the same as those of a single free plate while those of the out-of-phase motions are the same as those of a single plate on an elastic foundation of modulus 2K. The foundation modulus influences the out-of-phase motion considerably; as qt increases beyond 10 000 all the different modes tend to come nearer to one another. For the free-fixed system shown in Figure 4, there are similar motions and dissimilar motions of the system. For similar motions, the frequencies tend to approach their maximum values asymptotically. Hence for the lower mode with similar motion a value of ql = 100 000(iogtoql = 5.00) gives a rigid and stiff foundation. The lowest mode is a similar motion mode with no nodal circles in both the plates. Also it could be observed that, in the case of similar motions, when the values ofq l are very small the frequencies are nearly the same as those of a single free plate while for larger values ofql the frequencies approximate to those of a single fixed plate (except for the lower mode). For dissimilar motions a behaviour similar to that of the free- free plate system mentioned before is observed. For both the free-free and free-fixed systems it can be seen that for very large values o fq l the out-of-phase or dissimilar modes tend to disappear giving only in-phase or similar modes.

Figure 5 gives a comparative study of the symmetric vibration behaviour when the two plates along with their boundary conditions are similar and dissimilar. From Figure 5, it is seen that the boundary conditions do not affect much the symmetric frequencies of vibration for the dissimilar motions of the plate system. A similar observation could be made for the similar motions of the plate system too, except for the first mode. An additional mode (second symmetric mode) which was not given in the previous paper [1 ] due to some error in the Bessel functions with complex arguments, is given in the figure for the case of a free-fixed plate system. Moreover the mode shapes are affected very negligibly by the change of material, steel-steel to steel-aluminium. In the case of identical plates with free-free edges (shown in the fourth column), it may be seen that for each mode there is a corresponding in-phase and out-of-phase motion, the radii of nodal circles ofboth plates being the same in both cases. Also the loci of nodal circles for this system are the same as those of a single plate [3]. In

, comparing Figure 2(b) of reference [l] with the present Figure 5, it must be noted that the .change in Poisson's ratio changes the in-phase or similar motion frequencies more than the out-of-phase or dissimilar motion frequencies.

4. CONCLUSIONS

When the two plates and their boundary conditions are identical, the system executes in- phase and out-of-phase motions; the loci of nodal circles are the same for both the plates and are the same as those of a single plate. When the plates are not identical (of the same material) the frequencies of the similar motions are more sensitive to the change in thickness than the frequencies of dissimilar motions of the plate system. The changes in the thickness ratio or in the modulus of foundation (in certain ranges) or in the material ofthe plates affect the frequencies more considerably than the change in the boundary conditions, viz. free-

ELASTICALLY CONNECTED PLATE SYSTEMS 235

free to free-fixed. When the values o f the foundat ion modulus, q~, become very large (more than I00 000), the out-of-phase or dissimilar motions tend to disappear, and only in-phase or similar motions are observed.

REFERENCES

!. V. X. KUNUKKASSERIL and A. S. J. S~,VAMIDAS 1973 Journal o f Sotmd and Vibration 30, 99-I08. Normal modes of elastically connected circular plates.

2. A. S. J. SWA~,DAS and V. X. KUNUKKASSERIL 1974 Report, Department o f Applied Mechanics, hldian Institute o f Technology, Madras-36 (sent for external publication). Vibration of continuous double-plate systems.

3. A. W. LEISSA 1969 NASA SP-160, 8-10. Vibration of plat6s.

a

DI Dl, D, El, E, hi, h2

J,,(;-P), ym(;.p), Im().p), Km( ).p)

KI I / ' |

M, ql

r, 0 v,

7~, 72 VI, V2

p O9

12 2

APPENDIX: LIST OF SYMBOLS

outer radius of the complete plate Elh3/12(l - v~) flexural rigidities of the top and bottom plates, respectively Young's moduli of elasticity of the top and bottom plates thickness of the top and bottom plates ordinary and modified Bessel functions of first and second kinds of argument 2p and of order m the foundation modulus of the embedding material the number of nodal diameters the radial moment per unit width KI a~/Dl polar co-ordinates the Kirchhoff shear per unit width mass densities of the top and bottom plates, respectively Poisson's ratios of the top and bottom plates r/a circular frequency in radians per second 7t hi to 2 a4[ Dl