Vibration analysis of visco-elastic clamped circular ...nopr.niscair.res.in/bitstream/123456789/19731/1/IJEMS 9(2) 103-108… · t SINGH & ARIF: VIBRATION ANALYSIS OF VISCO ELASTIC

Indian Journal of Engineering & M ateri al s Sciences VoL 9, Apri l 2002, pp. 103- 108

Vibration analysis of visco-elastic clamped circular plates subjected to thermal gradient

S B Si ngh" & M Arir'

"School of Basic and App lied Sc iences, Thapar Institute of Engi neering & Technology, Pati ala 147004, India

bDepartment of Civil Engineering, A ligarh Muslim Uni versity, A ligarh 202 002, India

Received 14 February 200 I; accepled 15 Jalluary 2002

An analysis of vibrati on of visco-elas tic circular plate of variable thi ckness subjected to thermal gradi ent is presented here. The governing dirferential equati on has been sol ved for free vibrat ions of visco-elast ic ci rcular plate, which is clamped along the boundary. Galerkin' s technique has been applied to obtain corresponding natural frequencies in the rorm or ex­pli cit formulae. Detlect ion, time period and logarithmic decrement at di frerent points for the first two modes of vibration are calculated for various types of thermal gradient and taper constant and are illustrated w ith tables and graphs.

Sufficiently high temperatures are encountered in various eng ineering branches such as nuclear, power generation, aeronautica l, chemical, e tc., where metals and their alloys exhibit visco-elas ti c behavior. For such conditions, vibration analyses has to be carried out using consti tuti ve equati ons of visco-elastic the­ory rather than those of elastic theory . Most of the inves tigations on vibration have been carried out us­ing theory of elas ti c ity. However, references are avail able l

-8 on the vibrati ons of uniform visco-elastic

isotropic beams and plates . Sobotka9 has considered free vibrations of uni fo rm visco-elastic orthotropic rectangular plates. Muki and Sternberg lO have investi ­gated the stress analysis of visco-elasti c plates at e le­vated temperatures .

The present investigation is aimed at a parametric study of the vibration of visco-elastic circul ar pl ate of variab le thickness subjected to thermal grad ient. The assu mptions of small deflection and linear, isotrop ic visco-elastic properties are made. It is further as­sumed that the visco-elastic properties of the pl ate are of the Kelvi n type. Numerical calculations have been done using the material constants of the Duralium alloy.

Analysis The eq uation of motion of a visco-elas ti c isotropic

plate of variab le thickness may be written in the form :

M x,xx +2M yx.xy +M v.yy= phw{{ .. . (1)

A comma followed by a suffix denotes partial differ­enti ation wi th respect to that variable. The expres-

sions for M x , M y and M yx are given by:

M x = - D D I (w ,xx + v W )'1' )

M y = - D D, (w,yy + v w.xx )

M yx = -D D, (1 - v)w,xy

... (2)

On substituting the values of M x' M y and M vx from

Eq. (2) in Eq . (I) , we get:

D[D1 (w. xxxx + 2w,xl'YY + W,yyyy ) + 2Dl,x(w,m + w XYY )

+ 2 Dl, y (w. yyy + w. yxx )

+ D1 ,xx (wrx + V W,yy ) + Dl, yy (w, yy + V w.xx )

x 2(1-v)DI ,xyw.xv ]+ phw,I( =0

... (3)

The so lution of Eq . (3) can be sought in the form of products of two functions as:

w(x, y,t) = w(x, y)T(t) ... (4)

Using Eq. (4) in Eq. (3) and simplifying, one gets:

[D 1 (w,xnx + 2w,xxyy + W, yyyy ) + 2 D1,x (w, xxx + W, Xyy )

+ 2DI, y (w, yyy + W,yxx )

+ D l,n (w,xx + v w.yy ) + Dl, yy (w.yy + v w xx )

+ 2(1 - v )Dl,xy W,xy ] / p h W = - i / DT ... (5)

104 INDIAN J. ENG. M ATER . SCI.. APR IL 2002

Relation (5) is sa ti sfied if both its sides are equal to a

constant. Denoting thi s consta nt: by p 2, we ge t:

D, ( lIl.u .u + 2w . .u".",' + IV,·.n,v) + 2D,.x(lV .. ux + lV,x."v )

+2Dl.y(w.rn' + W yxx )

+ D, .xx (w.xx + V w . ."v ) + D' ,.IT (IV ."." + V W xx )

+2(1-v)Dl.x."w,xy - plIp 2 w = 0

(6)

and, (7)

Eq. (6) is a differential equation of motion for elastic isotropic plate of variable thickness and Eq. (7) is a di fferential equation of ti me functi ons of free vibrations of visco-elastic plate. It is assumed that the visco-elasti c isotropic circul ar plate is subjected to a steady two dimensional temperature di stribution given by:

(8)

where, r denotes the temperature excess above the reference temperature at any point on concentric circles and r 0 denotes the temperature excess above

the reference temperature at the center of the circular plate (x = y =0),

Solution of Equation (6)

The temperature dependence of the modulus of elastici ty, for most engineering materials, is gi ven by a rel ation of the type:

E(r) = Eo(l-yr) ... (9)

Ii = IIo (I - (3 X) ... ( II )

where lio is the thickness of the plate at the center

(x = y = 0) . The flexural ri gidity of the plate can now

be written as:

.. , ( 12)

Here the Poisson's ratio is assumed constant. Using Eq. ( II ) and Eq. (12) in Eq. (6), one gets:

8 , (w.xxxx + 2w.xxl'y + w.V.I'.I 'v ) + 282 (w. xxx + w.xy." )

+ 283 (w . .".I'V + W,yxx)

+ 8 4 (w. xx + V Wyy ) + 8 5 (w,."v -'- V W,XX)

, .. ( 13) where,

8, =[I-a(l- X)](\ - {3 X) 2

2X 8 2 =-2 (1- {3 X)[a(l- (3 X)-3{3{I-a(1- X)}]

a y

8 , = 8 2 -x

8 4 = 2 / a 2 [(I - (3) {a (I - (3X) - I 2a{3 2 / a 2

-3{3{I-a(I-X)}}+12{3 X 2 /a 2{I-a(l-X)}]

8 5 = 21 a 2 [(1_ {3X){a(l- (3X) - 12a{3y 2 / a 2

- 3{3 (I - a(l- X)} } + 12{3 2 y 2 / a 2 (I - a(l- X)}]

8 6 = 24{3XY / a 4 [-a(l- (3X) + (I-a(l- X)}]

where, Eo is value of the modulus at some reference and,

temperature and y is the slope of the variation of E

with r. With the reference temperature taken as the temperature on the boundary, the modulus variations, in view of the expressions (8) and (9), become:

E = Eo[l- a (1- X)] . .. (10)

where a = y ro (0 ~ a ~ I) .

The thickness variation of the plate is assu med to be of the form :

pa 4

1=12(1-v 2 )pa -2

- where a 2 is a frequency pa-hoEo

rameter.

Free vibration of clamped circular plate

The deflection function W (x, y) of the plate is as-

sumed to be a finite sum of characteristic function Wk (x, y)

" W(x,y) = 2, CkWk(X,y) ... ( 14)

k='

•

t

SINGH & AR IF: VIBRATION ANALYSIS OF V ISCO ELASTIC CLAMPED CIRCU LAR PLATES 105

For a clamped plate, boundary conditions are that the

deflection W = 0 and the slope W x = w. y = 0 along

l-X=O. Using Galerkin's technique, it is required that:

If L[W(x, y )]W(x, y)dxdy = 0 ... (15)

A

where L[W(x,y)] is the left hand s ide of Eq. (1 3).

Taking the first two terms o f the sum Eq. (14), for the function Was a solution of Eq . ( 13), one has:

( 16)

where C1 and C2 are undetermined coefficients .

Substituting relation (16) into Eq. ( 15), and then

eliminating C1 and C2 , we get the frequency equa­

tion as:

... ( 17)

where,

F, = ~[( ~ -2a- 26 {3 + 59 {3 2 +~a (3 +~a (3 2) a 2 l 3 3 15 5 15

+ - - a + 2 --(3 --a{3 +-a{3 - p - x-(

2 (3 7 2 7 3 2 ) ? I] 5 5 15 5 20

The frequency Eq. (17) is a quadratic equation in p 2

from which the values of p 2 can be found.

Thus, deflection function w(x, y) can be obtained

from Eq. ( 16) after determining C1 and C2 from Eq.

( 17). Choosing C1 = 1, one obtains C2 = - FI / F2

from first Eq. of (17). Therefore , w(x, y) becomes:

... ( 18)

Time functions of vibrations of visco-elastic plates Time functions of free vibrations of visco-elas ti c

plates are defined by the general ordinary differenti al Eq. (7) . Their form depends on the visco-elastic op-

erator D . For Kelvin 's model one can have:

i5=(1+!l~ 1 G dt )

... ( 19)

Taking temperature dependence of shear modulus and visco-elasti c constant in the same form as that of Young's modulus, we have:

G (r )= Go (I - Y IT): 17 (r )= 17 0(1 - Y 2 T)

... (20)

where Go is shear modulus and TJ o is visco-elastic

constant at some reference temperature i.e. at T =0,

Y I and Y 2 are slope variation of T with G and 17 re­

spectively. Using Eq . (8) in relation (20) , one can have:

... (2 1)

where a=YTo(O~al ~ 1)a1/da2 =Y2To (0 ~a2 ~ 1) Using Eq. (21) in relation(l9) , we get:

- d D= l+q-

dt

where,

Using Eq. (22) in Eq. (7), one obtains:

.. 2 . 2 T+p qT+p T=O

... (22)

... (23)

... (24)

Eq. (24) is a differential equation of second order for time function T. The time period of the vibration of the pl ate is g iven by:

106 IND IAN J. ENG . MATER. SC I. , APR IL 2002

K = 2n p

... (25)

where p is frequency given by Eq. ( 17). The logarith­mic decrement of the vibration is given by the stan­dard formulae:

( HI 1 A = log e _2

WI

... (26)

where WI is the de fl ec ti on at any point of the plate at a time peri od K= K I and W2 is the defl ec ti on at the sa me point at the ti me period succeed ing K I •

Results and Discussion Logarithmi c decrement, time period and defl ecti on

are computed for a clamped visco-elastic circul ar pl ate fo r different va lues of tem perature co nstants a, ai, a2; taper constan t f3 , and the di stance from the

center X. Results are presented th rough both graphs and tables. For numerical computati on, following material properties as reported for Duralium3

, are used:

Eo=7.08x I 0 10 N/m2 Go=2.632x 10 10 N/m2

170=14.612x I05N.S/m2, p=2.80x I0] kg/m3,

v=0 .345

The thickness of the plate at the center is taken as ho=O .O l 11'1 .• To study the effect of taper in plate, time

0.16 -,----------------~

'0 c:: o

0.14

0.12

~ 0.10 ~ :.:: '8 0.08 .~

Cl. Q)

0.06 E F

0.04

0.02

000

First Mode

Second Mode

0.0 0.2 0.4 0.6 0.8 1.0 1.2

fJ

Fig. I-Time peri od K versus taper constan t {3

peri od K and logarithmic decrement f3 have been computed fo r different va lues of f3 keeping a, a i, a:!

zero. I n other words, the plate is considered to be at a constant temperature. The variat ion of time peri od and logarithmi c decrement respec ti vely fo r both the first and the second mode of vibrations versus f3 , the

taper constant is shown in Figs 1 and 2. It is interes t­ing to see that wh ile the effect of increasi ng taper is more prominent fo r the time peri od in the first mode, on logarithmic decrement thi s effect is more promi­nent in the second mode. It has also been found that logarithmic decrement is independent of X in thi s case.

Effect of taper has also been studied fo r the case where there is temperature gradient. T: e values of temperature constants a ,a pa2 were assumed to be

0.2 , 0.3, 0.6 res pectively, and logarithmic decrement 1\ , ti me period K and defl ecti on w were calculated for the first two modes of vibrations for f3 =0.0 (uniform

thickness) and f3 =0.6 (varying thick ness) at different

values of X and are shown in Tables I a- I c. It can be seen from the Tables that logarithmic decrement and time period increases and defl ection decreases for the plate with taper for both the modes of vibration as compared to the plate with uniform thickness.

To study the effect of temperature gradient, A, K and ware calculated and compared in Tables 2a-2c for f3 =0.6 for the following two cases at different

values of X. (Case I: a=al=a2=0.0; Case II : a=0.2, al =0.3, a2=0.6). The effect of thermal gradient on

C Q)

E [1' u Q)

0 u E -5 .~

0> 0 -'

0.00

-0.01

-0.02

-0.03

-0.04

-0.05

-0.06

-0.07

-0.08

-0.09

-0.10

-0.11

-0. 12

0.0 0.2 0.4 0.6

fJ

First Mode

Second Mode

0.8 1.0 1.2

Fi g. 2-Logarithlllic decrement versus taper constant {3

SINGH & AR IF: VIBRATION ANALYS IS OF VISCO ELASTIC CLAMPED CIRCULAR PLATES 107

Table I a--Logarithmic dec rement !\ for a plate with thermal grad ient (1. = 0.2 ; (1.1 = 0.3; (1.2 = 0.6

First Mode Second Mode X {3=0.0 {3=0.6 {3=0.0 {3=0.6

0.0 - 0.0149 - 0.00S4 - 0.0637 - 0.0408 0.5 - 0.02 15 - 0.0121 - 0.0918 - 0.05S8 0.9 - 0.0253 - 0.01 42 - 0.1080 - 0.0692

Table I b - Time peri od K (sec.) for a plate with thermal gradient

(1. = 0.2 ; (1.1 = 0.3; (1. 2 = 0.6 (K is constant for all va lues of X)

p

0.0 0.6

Fi rst Mode

0.042 0.075

Second Mode

0.010 0.015

Table I c - Denection IV for a plate with thermal grad ient

(1. = 0.2 ; (1. 1 = 0.3: (1.2 = 0.6

Tempe- X First Mode Second Mode rature {3=0.0 P=0.6 {3=0.0 {3=0.6

0.0 1.4890 0.7388 - 0.1 902 - 0.2096 OK 0.5 0.3111 0.2174 0.1012 0.098S

0.9 0.0105 0.0097 0.0088 0.00S8

0.0 1.38 17 0.7085 - 0.1383 - 0.1709 5K 0.5 0.2794 0.2046 0.0640 0.0736

0.9 0.0092 0.0091 0.0051 0.0062

logarithmic decrement, time period and deflection is presented in Tables 2a-2c . It is interesting to observe from Table 2a that the logarithmic decrement is con­stant across the plate for constant temperature case. For the plate with temperature gradient, the logarith­mic increment increases as compared to the plate at constant temperature for all values of X. Table 2b shows that the effect on period K is much smaller, in fact for the second mode the effect was so small that the difference in the values occurred at the fourth place of decimal. Table 2c shows that the deflection increases for a plate with temperature gradient com­pared to a plate at constant temperature at all values of X.

Conclusions On the basis of these limited analytical investiga­

tions, it is found that: (i) The effect of increasing taper is more prominent for the time period in the first mode, on logarithmic decrement this effect is more prominent in the second mode; (ii) The logarithmic decrement and time period increases and deflection decreases for the pl ate with taper for both the modes of vibration as compared to the plate with uniform

Table 2a - Logarithmic decrement 1\ for a plate with vari able thi ckness, p = 0.6

Case I-zero therm al gradient (1.= (1.1 =(1.2=0.0 Case II -thermal gradient represented by (1.=0.2, (1.1=0.3, (1.2=0.6

X First Mode Second Mode (1.=(1.1= (1.=0.2. (1.=(1.1= (1.=0.2.

(1.2=0 (1.1=0.3 (1.~=0 (1.\=0.3 (1.2=0.6 (1.2=0.6

0.0 - 0.0155 - 0.0084 - 0.0742 - 0.040S 0.5 - 0.0155 - 0.0127 - 0.0742 - 0.0588 0.9 - 0.0 155 - 0.0142 - 0.0742 - 0.0692

Table 2b - Time peri od K (secol/ds) for a plate with va ri able thickness. p = 0.6

Case I-zero thermal gradi ent (1. = (1.\ = (1.2 = 0.0

Case II -thermal gradient represented by (1.=0.2. (1.1 =0.3, (1.! = 0.6.

(1. =(1.\ =(1.2 = 0.0

(1. =0.2,(1.1 =0.3, (1.2 = 0.6

First Mode

0.071

0.075

Second Mode

0.015

0.015

Table 2c-Denecti on IV for a plate with variable thickness, p=0.6: Case I-zero thermal gradient (1.=(1.1 =(1.2=0.0 Case II -thermal gradient represented by (1.=0.2, (1.\=0.3 , (1.2=0.6

Tempe- X First Mode Second Mode rature (1.=(1.1= (1.=0.2, (1.=(1.1= (1.=0.2,

(1.2=0 (1.1=0.3 (1.2=0 (1.1=0.3 (1.2=0.6 (1.2=0.6

0.0 0.6823 0.7388 - 0.2125 - 0.2096 OK 0.5 0.2103 0.2174 0.0984 0.0988

0.9 0.0097 0.0097 0.0088 0.0088

0.0 0.6313 0.7085 -0.1466 -0.1709 5K 0.5 0.1946 0.2046 0.0679 0.0736

0.9 0.0090 0.0091 0.0061 0.0062

thickness ; (iii) The logarithmic decrement is constant across the plate for constant temperature case; (iv) For the plate with temperature gradient, the logarithmic increment increases as compared to the plate at con­stant temperature; and, (v) The deflection increases for a plate with temperature gradient compared to a plate at constant temperature.

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