IACS Institutional Repository

Midya, S K

Sarkar, H

Indian J. Plm. 81 (2), 217-224 (2007) 
Variation of stratospheric ozone witty relative humidity and 
sharp depletion of absolute humidity associated with 
Nor'wester over Kolkata 
S K Midya1-2* and H Sarkar1-3 
department of Physics, Serampore College, Serampore, Hooghly-712 201, West Bengal, India 
2Centre for Space Physics, 43 Chalantika.Garia Station Road, Kolkata-700 084, India 
institute of Radiophysics and Electronics, University of Calcutta,92 A.RC Road, Kolkata-700 009, India 
E-mail :drskm06^'yahoo. co. in 
Received 4 July 2006, accepted 8 February 2007 
Abstract : The paper presents the variation of stratospheric ozone concentration with relative humidity 
over Kolkata. Analysis confirms that there is close correlation between them. The relative humidity in general, 
increases during commencement of severe thunderstorm. As drop in temperature occurs during thunderstorms, 
absolute humidity decreases during the onset of thunderstorm. The sharp depletion of absolute humidity at the 
time of starting of Nor'wester over Kolkata has been studied in the present paper. Possible explanations of such 
type of variation of humidity are presented. One possible method of forecasting of severe thunderstorm is also 
offered. 
Keywords : Relative humidity, absolute humidity, stratosphere, troposphere, Nor* wester. 
PACS Nos. : 92.60.hd, 92.60.hf, 82.33Tb, 92.60.jk 
1. Introduction 
Interrelation between stratospheric 0 3 concentration and different environmental parameters 
is now well established. Dobson et a/ [1] reported that average temperature of earth is 
directly related to ozone concentration of stratosphere and fall in ozone content in the 
stratosphere takes place over England before the arrival of warm front at the ground surface. 
They also reported that the rise in ozone content in the stratosphere takes place when 
cold front reaches near the ground level. Midya et al [2] showed that maximum surface 
temperature of Kolkata is directly related with stratospheric ozone concentration over 
Corresponding Author 
(D2007IACS 
218 S K Midya and H Sarkar 
Kolkata. Besides these, Mitra [3] mentioned that there is a close relation between barometric 
height, tropospheric weather and ionospheric parameters of the upper atmosphere. It was 
observed that minimum height of F region and average E ionization tend to follow the 
variation of barometric height. Mitra [3] also mentioned that correlation was observed 
between the lowest virtual height of E region and ground temperature at Standford, California, 
U.S.A. Bates [4] and Mackay et al [5] concluded that variation of solar UV radiation due 
to fall of stratospheric ozone concentration can influence tropospheric climate in many 
ways. This paper presents the variation of stratospheric ozone with relative humidity and 
sharp depletion of absolute humidity related with Nor'wester over Kolkata. Possible 
explanation based on physical considerations and chemical kinetics is offered. 
2. Observation 
Data for relative humidity is collected from Alipore. In the absence of ozone data over 
Alipore, that of Dumdum station which is very near to Alipore is taken from internet 
website (http://jwocky.gsfc.nasa.gov).The variation of ozone with relative humidity is nearly 
straight lines. A few selected variations of ozone concentration for particular time and 
year (1991, 1992, 1993, 1994 ) are shown in Figure U t is clear from this figure that the 
stratospheric ozone concentration increases with the increase of relative humidity over 
Relative Humidity (%) Relative Humidity (%) 
Figure 1. Variation of stratospheric ozone concentration with relative humidity for some selected dates 
over Kolkata. 
Variation of stratospheric ozone wih relative humidity and sharp depletion etc 219 
Kolkata. We have obtained almost the same type of observations during the period 1991-
1994. Detailed results are presented in Table 1.We have considered relative humidity for 
three times (8:30A.M., 11:30A.M., 17:30P.M.) of a particular day (03.04.1990, 18.05.1990, 
14.04.1990). 
Table 1. Stastical analysis of the variation of ozone concentration and relative humidity for different times 
of different years and their empirical relations. 
, — — • — • ' ' — * i 
Year Time Correlation Empirical equations 
coefficient 
O3=0.73(R.H.)+215.64 
5
 O3=0 652(R.H.)4219.66 
O3=0.818(R.H.)+198 32 
O3=0.15(R.H.)+258.18 
O3=0.498(R.H.)+238.12 
O3=0.36(R.H.)+247.99 
O3=0.55(R.H.)+227.50 
O3=0.63(R.H.)+215.13 
O3=0.708(R H.)+209.36 
O3=0.27(R.H.)+244.85 
O3=0.49(R.H.)+231.86 
O3=0.625(R.H.)+221.90 
The meteorological data of the year 1990 is used to study the association of transient 
changes of temperature, absolute humidity and relative humidity during the occurrence of 
Nor'wester over Kolkata. Liebe et al [6] presented one empirical equation which is helpful 
to calculate absolute humidity of a particular Nor'wester day. To calculate the absolute 
humidity we consider the atmospheric temperature T ( °C ) and relative humidity in 
percentage as input parameters. Using a relative inverse temperature variable( 0) defined 
as 0=3OO/(T+273.15) ; and water vapor pressure (e) defined as e =2.408 *108 * U * &4 
* exp(-22.64in gm/m cube. 
A typical record of the relative humidity, temperature and absolute humidity variations 
during two Nor*westers occurring on 03.04.1990 and 18.05.1990 are shown in Figure 2. 
Figure 3 shows the variation of the said parameters on a normal day 14.04.1990, with no 
report of thunderstorm. Comparison of the figures confirm the dip of absolute humidity at 
1430 1ST on 03.04.1990 and at 1370 1ST on18.051990 on two selected Nor'wester days 
around the time of onset of the thunderstorm. The drop in temperature is monotonic, 
showing no sharp dip, while the relative humidity also increases monotomically at the 
1991 
1991 
1991 
1992 
1992 
1992 
1993 
1993 
1993 
1994 
1994 
1994 
08:30 A.M. 
11:30 A.M. 
17:30 A.M. 
08:30 A.M. 
11:30 A.M. 
17:30 P.M. 
08:30 A.M. 
11:30 A.M. 
17:30 P.M. 
08:30 A.M. 
11:30 A.M. 
17:30-RM. 
0.39 
0.63 
0.55 
0.01 
0.64 
0.39 
0.39 
0.67 
0.52 
0.27 
0.61 
0.51 
220 S K Midya and H Sarkar 
same time. During the period of consideration, there were 26 cases of prominent 
Nor'westers of these 23 cases show a sharp drop of absolute humidity. Moreover, regarding 
the relative humidity variation during Nor'wester, it is generally monotonic and having no 
correlation with absolute humidity variations. From analysis we get the following results 
(i) Concentration of stratospheric 0 3 increases with the increase of relative humidity 
over Kolkata (Figure 1) 
(n) The rate of increase of 03 concentration depends on time of a particular day 
(Table 1) 
(in) Observations confirms a fall of temperature and a sharp drop in absolute humidity 
during the onset of a Nor'wester over Kolkata 
3. Discussion 
03 is a minor constituent of the atmosphere. It is mainly distributed in the stratosphere 
Maximum concentration of 0 3 occurs around 25 km altitude [7]. Ozone is formed in the 
Time (1ST) 
Date 03.041990 
Time (1ST) 
Date-18 5.1990 
36 
_ 32 
£ 28 
24F 
20 
Time (1ST) 
Date: 18.05.1990 
- O -0 
^ ^ CD i n y , 
«— t— c*> «q> m i n to 
Time (1ST) 
19 
12 
Date-03 04/1990 
TS-Thunderstorm onset 
Timing 
c> so in co 
in ir> u> N K 
<N 
e 5 
Time (1ST) 
I 20 
x 
< 10 
Date:18.05.1990 
TS-Thunderstorm onset 
Timing 
•fr-Ar-fc-A 
. * * i 11 i i 
o iS *Q ** 
J2 O * i CN «•> 
*— r* r» *•» 
U> ^ 00 
"< in A N 
Time (1ST) 
Figure 2. Variation of relative humidity, temperature and absolute humidity with time (iST) on two Nor'wester 
days over Kolkata 
Variation of stratospheric ozone wih relative humidity and sharp depletion etc 221 
atmosphere due to the decomposition of water molecule by solar UV ray as given below 
[8]: 
H20 -> H + OH, 
OH -> O + H, 
O + O + M -> 02+ M . 
0 3 is formed in the atmosphere from the combination of atomic and molecular oxygen. 
There are also other ozone formation and destruction processes The net results are as 
follows: 
Reactions Rate Constants 
1. 0 3 + hv ,abs > 03* labs 
2. 03* _ * ! _ » 0 2 + 0 K, 
3. O + 0 3 K 2 > 2 0 2 K2 
4. O + 0 3 * * 3 > 202 K3 
5. 0 + 0 K4 > 0 2 K4 
6. 0 + 0 2 _ * ! _ > 0 3 K5 
All rate constants are calculated by Midya et al [9] in a previous communication. 
Solar UV rays contain high energy photons which decompose water molecules into 
atomic oxygen and OH radical. Thus we can expect that if stratospheric ozone is depleted, 
the high energy solar UV photons fall directly on troposphere and as a result, tropospheric 
water molecules are decomposed resulting the fall of relative humidity and increase of 
tropospheric ozone which acts as Greenhouse gas and plays important role in global 
warming. 
Again for high stratospheric ozone concentration, ozone layer absorbs maximum of 
the incoming UV photons and resist falling on the lower atmospheric water molecules. 
Thus increase of relative humidity with the increase of stratospheric ozone concentration 
is expected. It is clear from the Figure 1 which is obtained from the meteorological data 
of Alipur and 0 3 data over Kolkata. WMO bulletin [10] confirmed that 0 3 is depleted 
everywhere in a smaller amount, but dramatic decrease of 0 3 concentration takes place 
only over Antarctica during spring time. But for Indian regions for the latitude range 25-
34°N, we have shown that ozone concentration gradually increases from the month of 
January, attains its maximum value for the period of May and June, then gradually decreases 
and attains a minimum value for the month of December [11]. Thus variation stratospheric 
0 3 concentration with relative humidity is quite expected. 
222 S K Midya and H Sarkar 
The fall in surface temperature and increase in relative humidity with thunderstorm is a 
normal trend. From the observations of change of surface meteorological parameters 
associated with thunderstorms many models have been proposed by various researchers 
in this field. Maddox et al [12] in their model postulated that low level warm and moist air 
from the forward sector slides over the warm and dry moving air mass during the formation 
(%)
 H 
DC 
75 
70 
65 
60 
55 
50 
11 
Date 14 04 1990 
12 13 14 15 
Time (1ST) 
16 17 
38 
P 36 
Q. 34 
^ 32 
7 
i i 
Date 14 04 1990 
i i i i i i i 1 1 1 
11 12 13 14 15 
Time (1ST) 
16 17 
26.5 
B 26 
6 
3 25.5 
I 25 
24.5 , 
11 
Date* 14 04 1990 
—-•• • • i i i, i i i i 
12 13 14 15 16 
Time (1ST) 
— 6 
, f 
17 
Figure 3. Variation of relative humidity, temperature and absolute humidity with time (1ST) on one normal 
day over Kolkata 
stage of thunderstorms. As the thunderstorms overtake into maturity the whole field of 
thunderstorm gets filled up with regions of updraft and downdraft. Doppler radar observations 
confirmed that downdrafts of air beginning from lower levels and not accompanied with 
rain exhibit weak reflectivity meaning thereby impinging precipitation free air on the surface, 
indicating low absolute humidity at the surface during such events. 
It may be realized that at least two mechanisms operate in a moving thunderstorm 
for the sudden drop of absolute humidity during the onset of a thunderstorm over a station 
and these mechanisms may be explained as follows: 
Variation of stratospheric ozone wih relative humidity and sharp depletion etc 223 
(i) The turbulence produced at the leading edge of the wind front mixes dry air at 
the top resulting in a drop in surface moisture. 
(II) In the initial stage of downdraft, the air is evaporating and thus cooled but devoid 
of any moisture. 
It is felt that in the case of Nor'wester over Kolkata, both the above mechanisms were 
being operated causing fall in surface temperature an$j sharp fall in absolute humidity 
With the progress of thunderstorm activity the downdraft reaches the ground and is 
embedded with rain causing the surface air to cool with rise in relative humidity and 
absolute humidity 
It is now well established that Doppler radar study ill the thunderstorm region reveals 
a very complex nature of wind It has been observed that at different stages of thunderstorm, 
there is a shift in the location of regimes of updraft and downdraft with consequential 
influence on environmental condition The main point emerges from the present study is 
that the absolute humidity is more realizing than that of relative humidity during the 
occurrence of Nor'westers For the sharp drop in temperature associated with tends to 
increase the relative humidity while the absolute humidity really drops at much higher rate 
and it is absolute humidity which is more important parameter for modeling of such 
thunderstorms As absolute humidity starts to drop at about forty to sixty minutes before 
the starting time of Nor'wester, so this parameter may be utilized for the forecasting of 
Nor wester events Also, observing the nature of rate of fall of the absolute humidity (fast 
rate or slow rate ), we may conclude about the different types of thunderstorms i.e squall 
type or ordinary thunderstorm, which may be left for our next research work It may be 
mentioned here that a faster response absolute humidity sensor would be more desirable 
for such studies and a Lyman- hygrometers with having a time constant of 15 milli-
second would be ideal for this purpose It is also expected that depletion of stratospheric 
03 increases tropospheric temperature and decreases presence water molecules of the 
troposphere due to unabsorbed solar UV ray by the stratosphere As a result, rainfall will 
be decreased and nature of Nor'wester will be changed due to depletion of stratospheric 
ozone. Midya et al [13] showed that N20 plays important role in the depletion of ozone in 
the stratosphere So it may be concluded that in order to maintain suitable condition of 
our different environmental parameters, we have to give more attention to control 
concentration of N20 with respect to other ozone depleting substances (ODS) 
References 
[1] G M B Dobson, A W Brewer and B M Cwilong Proc Roy Soc A185 144 (1946) 
[2] S K Midya, H Sarkar and A Manna 'Covariation of Daily Maximum Temperature and ozone 
concentration over Calcutta' Published m Recent Trends in Astro and Plasma Physics in India, 
p241 (2003) 
I3] S K Mitra The Upper Atmosphere (Published by the Asiatic Society) 351 (1992) 
[4] J R Bates Solar Physics 74(2) 399 (1981) 
(5) R M Mackay, K W Ko Malcom, Run-lie Shia and Yajalng Yang J Climate, 10 774 (1997) 
224 SK Midya and HSark 
[6] H J Liebe and G A Hufford Intl. J. Infrared and Millimeter Wave 10 631 (1989) 
[7] S K Midya and P K Jana Indian J. Phys 76B 107 (2002) 
[8] S N Ghosh and S K Midya Indian 1 Phys. 68B 473 (1994) 
[9] S K Midya, P K Jana and T Lahiri Earth, Moon and Planets (Netherlands) 66 279 (1995) 
[10] R D Bojkov WMO Bulletin 41 171 (1992) 
[11] P K Jana, S K Midya and U K De Indian J. Radio & Space Phys. 30 176 (2001) 
[12] R A Maddox, F Caracena, L R Hoxit and C F Chappel NOAA, Technical Report.ERL 388-APCL 41 
(1977) 
[13] S K Midya, S C Ganda and S N Sahu Indian J. Phys. 74B 337 (2000) R A Maddox, F Caracena, 
L R Hoxit and C F Chappel NOAA, Technical Report, ERL 388-APCL 41 (1977) 
[13] S K Midya, S C Ganda and S N Sahu Indian J. Phys. 74B 337 (2000)

English

Variation of Stratospheric Ozone with Relative Humidity and Sharp Depletion of Absolute Humidity Associated with Nor' Wester Over Kolkata

Al-Awfi, Saud

Effect of Phase Fluctuations of a Laser on the Dynamic Process of an Atom Near a Metallic Layer

Choudhury, D K

Dhar, Pijush Kanti

An Improved Analysis of Deep Inelastic Neutrino Data with the DGLAP Equations at Low and High x

Goel, Sumant

Bansal, Vakul

Kumar, Rajendra

InJuwJ. Phys. 81 (4) 477-484(2007) 
Damped vibrations of a linearly tapered homogeneous 
orthotropic circular plate 
Sumant God*, Vakul Bansal and Rajendraf Kumar1 
Department of PG Studies and Research in Physics, J V Jain College, Sahaianpur-24700, Uttar Pradesh, India 
Department of PG Studies and Research m Mathematics, J V Jain College Saharanpur-24700, Uttar Pradesh, India 
E-mail sumantgoeK" yahoo com 
Received 7 September 2006, accepted 8 May 2007 
Abstract : The effect of damping on free vibrations of an orthotropic homogeneous elastic circular plate of 
linearly varying thickness has been analysed in present research work The governing differential equation of 
motion has been solved by Frobenius method The frequencies corresponding to the first two modes of vibrations 
kiave been obtained for an orthotropic circular plate with different combinations of boundary conditions for 
various values of damping constant and taper constant 
Keywords : Orthotropic circular plate, damped vibration, taper constant 
PACS Nos. : 46 05 +b, 46 70 De 
1 Introduction 
For more than last two decades, interest has been highly developed in the effect of 
clamping and temperature on solid bodies because of rapid development in space 
technology, high speed atmospheric flights efc In mechanical systems where certain 
parts of machine have to operate under damping, its effect is far from negligible 
The engineering materials are of three types in terms of elastic symmetry viz 
isotropic, anisotropic and orthotropic type The isotropic material has an infinite number 
of symmetry §e. every plane is plane of symmetry and it requires only two elastic 
constants for its characterization On the other hand, a material without any plane of 
symmetry is called fully anisotropic it requires 21 independent elastic constants for its 
characterization Finally, orthotropic materials are special case of anisotropic materials. 
By definition, an orthotropic material has two orthogonal planes of symmetry where 
material properties are independent of direction within each plane such materials 
requires 9 independent elastic constants for their characterization These studies have 
Responding Author © 2 0 0 7 , A C S 
478 SumantGoel, Vakul Bansal and Ra/endra Kumar 
been intensively made by Timoshenko and Kneger [1] and Ghosh [2] Several 
authors [3-5] have studied the natural frequency and free vibration on some circular 
plate Non-linear interactions in asymmetric vibrations of a circular plate was studied by 
Lee and Yeo [8] Gupta and Goel [12] studied the forced asymmetric response of 
linearly tapered circular plates Singh and Saxena [13] studied the transverse vibration 
of a quarter of a circular plate with variable thickness Celep [15] has studied the free 
vibration on some circular plate on arbitrary thickness Mc Nitt [18] introduced damping 
factor in free vibration of a damped elliptical plate The analysis presented here pertdins 
to the damping effect on frequencies of a circular orthotropic homogenous plate of 
linearly varying thickness with different boundary conditions The first two modes of 
vibrations with clamped and simply supported edge conditions for various values of 
damping constant and taper constant have been derived 
2. Theory and computation 
The axisymmetnc motion of a circular plate of radius la\ in polar coordinates (rt)) is 
governed by the following equations 
(rQrr) = phwjt, (1 v 
and 
Or = [(rMrU - MfA/r, 0 2) 
where h is thickness of the plate, p is the mass density per unit volume of the plate 
w the transverse deflection, t is the time, Qr is the shears resultant, Mr and M0 are the 
movement resultants A comma followed by a suffix denotes partial differentiation with 
respect to that variable 
The moment resultants Mr and M0 for homogeneous and polar orthotropic material 
of the plate, are given by [17] 
Mr = -Dr\W,rr + (ve/r)Wtr] 
and
 ( 1 3 ) 
Me = ~De [0 / 0 W, r + vrW> rr], 
where Dr and De are the flexural rigidities in r- and 6 directions respectively 
The equation for transfer motion of a polar orthotropic circular plate of variable 
thickness of damping effect has been obtained from Eqs (1 1), (1 2) and (13) as 
DrWtrrrr + 2[{Df + rDht)fiWjrr 
+[{-0* + /(2 + v0)Dnr + r2Dnn)f}Wjr 
+[{D„ - rDff,r + ^D^/r^Wj 
+ phW.tt + KWJ = 0, (1) 
introducing the following non-dimensional variables 
Damped vibrations of a linearly tapered homogeneous orthotropic circular plate 479 
H - hfa, W = w /a, R = r/a, D„ = Dr/a3 and D„ = 0,,/a3. 
DR = E„H3/\2, D„ = E22 H3 /12, 
E„ = E, / (1 - vrv»), E22 = E 2 / (1 - vrv,l 
Eq (1) can be written as 
r
 rffl4 R dR j (iff3 Z?2 R ,)R dtf 
DR dW ZLJ<)2W , [§L . . 19 l)D» + v<> <f_?_B d™_ + p a 2 H - ^ + K - 1 - 0 
3. Solution 
1 he solution of eq. (2) we assume 
W = w(r)e '"cospf 
and thickness variation of the plate as 
On substitution of eq. (3) into eq. (2) the Differential Equation takes the form. 
s
 ' ' dR* R 'OR3 V dR3 R2 E / * ' df?2 
H ()H hi uri orf 
i f <1-/W)« -^ > - ^ S ^ - 6 / ^ ( i - ^ ) 2 f i£ 
fl3 E, tifl /T E, dff ft E, <m 
(3) 
-(1-PRfQ2rW-D^r2W = 0, (4) 
where 
5 
'* = VH, 2 
(5) 
^ _ 3(1-vfvfl)/<2 
P -> Circular frequency, i2 - * frequency parameter, DK -» Damping parameter. 
Now the solution of eq. (4) is taken as 
W = S a k f l c + * , 3 ) * 0 , ' (6) 
K«0 
480 Sumant Goel, VakulBansalandRajendra Kumar 
where C is exponent of singularity . 
Substituting (6) in (4) and putting E2 /E, = /2, v0 = m2, we get 
K 0 
+ I a* r ^ O ) + T*bK(2)+7X0) + r x ] ^ * 3 
K 0 L 
+ £ aj7:X(3)+ 7:f6K(2)+ r/dKd)+ r346K]flc^ 2 
+ 2>* [TX(3) + 7X<2) + 7X0) + 7i46k]flc,K ' 
+ £ ««[>X<3H TX(2) + * X 0 ) + T<bK + 7*]flc"f 
K 0 
•JUKI***41 
+ £a,[r7']Rc^2=0 
Eq. (7) can be written as 
i a ^ W ^ + Xa^fl0^ 3 
K 0 K-0 
K-0 /f-0 
+ S^/?ir)flc+^2«0. (8) 
For eq. (6) to be the solution, the coefficients of various powers of R in the expression 
obtained by substituting (6) in (8) must be identically zero. Thus by equating the 
coefficients of the lowest power of R to zero, one get the identical equations : 
a0Fi(0) = 0 since a0 * 0 i.e. F^O) = 0, 
^(0) = T,%(3) + T?bo(2) + TfoO) + TfttO) = 0, 
IdC - 1)(C - 2){C - 3)] + 2[QC - 1)(C - 2)] - /2[C(C - 1)] + /2C = 0, 
C = 0, 2, 1+/, 1- / . 
Damped vibrations of a linearly tapered homogeneous orthotopic circular plate 481 
It may be seen that the series corresponding to C = 0 will also contain the series 
corresponding to C = 2. Also, the series corresponding to C = 1-/ , / > 1 vanishes 
because of its singularity at R = 0 and the series for C = 1-/, / < 1 will be contained 
m the series corresponding to C = 1+/. 
Further equating the coefficient of the next subsequent power of R to zero, it is 
found that a, = 0, a2 is indeterminant, so we can choose any value of a2 and aA 
(A = 3, 4, 5 ) can be written in terms of a0 and a%. 
Hence assuming \ 
aA = AAao + BAa2(A = 0,1,2,3 ), l (9) 
the following solution, corresponding to C = 0 and C fe 1+/ is obtained : 
W ^ h + ZV?* +^h + 2^«A, , , / (10) 
It is also evident that the solution corresponding to the other value of C is induced in 
(10) Hence no new solution can exist for this value of C. 
Application of the test of Lamb [19] shows that the solution (10) is convergent 
tor ft < 1, | / / |<1. 
4. Boundary conditions and frequency equations 
The following combinations of boundary conditions have been considered : 
(i) Clamped (Q at the edge R = 1. 
(ii) Simply supported (S) at the edge R = 1. 
The boundary conditions for different edge conditions are : 
For clamped edge 
dW W ^ O a n d - — = 0. (11) OR 
For simply supported edge, 
W = 0 and MR = 0, 
i.e. 
W = 0 and —5- + — vv, = 0. (12) 
dR2 R & dR 
C-Plates : 
After applying boundary conditions, the frequency equation for clamped plate is 
|F(1,A>) G ( I A * ) L ^ 
F<(1.A2) G\U> ( *' 
482 Sumant Goel, Vakul Bansaf and Ra/endra Kumar 
S-Plates : 
Applying boundary condition, the frequency equation is 
\F(U2) G(1,A^ 
F,UA2) (3,(1, A2)| l = ~0, (14) 
where 
F,(1, A2) = F ' (1 , A2) + i'„F'(1, ^2 
and 
G1(1, /i2) = G (1, A2) + ^ G ( 1 , /I2) 
Here, dash denotes the partial differentiation with respect to ft 
5. Result and discussion 
Frequency equations (13) and (14) are transcendental equations in A2 from which 
infinite! roots can be determined The frequency parameter A corresponding to the first 
two modes of vibrations of a clamped and simply supported orthotopic circular plate 
has been computed for different values of damping constant and taper constant The 
orthotropic materials are taken as [14] 
I2 = 1.44 and m2 = 0.3. 
It is concluded that the frequencies in the first two modes of vibration decrease with 
the increasing values of damping constant as well as taper constant in both the cases 
of boundary conditions, considered here. The frequencies corresponding to the first two 
modes of vibration for various values of damping constant (DK) and taper constant \\ for 
both the boundary conditions have been plotted in Figures 1 and 2 For comparing the 
numerical values of the frequency with those of Tomar and Tiwan [14], A has also been 
0 01 0 02 0 03 
Damping Constant 
0 04 
*- Series 1 
• Serles2 
-+- Series3 
- Series4 
* Serles5 
-*~ Senes6 
r Series7 
- Serws8 
Figure 1. Variation of frequency parameter 
with damping constant for a circular plate of 
linearly varying thickness (colour on line). 
0.01 0.02 0,03 
Taper Constant (0) 
Figure 2. Variation of frequency parameter with taper 
constant for a circular plate of linearly varying 
thickness (colour on line). 
Damped vibrations of a linearly tapered homogeneous orthotropic circular plate 483 
computed for the corresponding elastic constant for a clamped and simply supported 
orthotropic circular plate DK = 0 = ft and it is found that the results are in satisfactory 
agreement for the first two modes of vibration. 
Values of transverse deflection W corresponding to first two modes of vibration 
for O and S-plates at different points, have been calculated for ft = 0.2 and DK = 0.01. 
These results are plotted in Figure 3 for a circular plafe of linearly varying thickness. 
1 2 
1 
08 
06 
04 
02 
0 
-0 2 
04 
-0 6 
-0 8 
Figure 3. Transverse deflection for a circular plate 
of linearly varying thickness (colour on line). 
References 
[1] Stephen P Timoshenko and S Woinowsky Krieger 'Theory of Plates and Shells' (New York . Mcgraw-Hill 
Kogakusha) Second international student edition p364 (1959) 
[2] P K Ghosh 'The Mathematics of Waves and Vibration' (India : Mechmillion Company) Appendix F p381 
(1975) 
[3] D Zhou, S H Lo, F T K Au and Y K Cheung J. Sound and Vibration 292 726 (2006) 
[4] Vinayak Ranjan and M K Ghosh J. Sound and Vibration 292 999 (2006) 
[5] Helmut F Bauer and Werner Eidel J. Sound and Vibration 292 742 (2006) 
[6] Shahruz, S M and P Kessler J. Sound and Vibration 276 1093 (2004) 
[7] C S Kim J. Sound and Vibration 259 733 (2003) 
[8] W K Lee and H M Yeo J. Sound and Vibration 263 1017 (2003) 
[9] P A A Laura. R H Gutierrez and R E Rossi J. Sound and Vibration 254 175 (2002) 
[10] C Touze, O Thomas and A Chaigne J. Sound and Vibration 258 649 (2002) 
[11] C Y Wang J. Sound and Vibrations 243 945 (2001) 
[12] A P Gupta and N Goel Intl. J. Mechanical Sciences 220 641 (1999) 
[13] B Singh and V Saxena J. Sound and Vibration 183 49 (1995) 
[14] J S Tomar and V S Tewari J. Non-Equilibrium Thermodynamics $ 115 (1981) 
[15] Z Celep J. Sound and Vibration 70 379 (1980) 
f 0.1 0^ 0.3 0.4 Vs 0.« 0J 0,8 4# ' 1 
// 
-Senesl 
~Sort6&2 
-Senes3 
Senes4 
484 Sumant Goel, Vakul Bansal and Rajendra Kumar 
[16] S H Crandall J. Sound and Vibration I I3 (1970) 
[17] A W Leissa 'Vibration of plates! NASA.SP-160 (1969) 
[18] Mc R P Nitt J. Aerospace Science 29 1124 (1962) 
[19] H Lamb 'Hydrodynamics' (New York; Dover Publication) 335 (1945)

Damped Vibrations of a Linearly Tapered Homogeneous Orthotropic Circular Plate

Indian J. Phys. 81 (4), 501-502 (2007) 
d^)RTHCOMING PUBLICATIONS 
Volume 81, (2007) 
Astrophysics, Atmospheric & Space Physics 
Unusual behavior of D-region ionization time at 18.2kHz during seismically 
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Notes 
Superposition model analysis for the zero-field splitting of Mn2+ in 
La2Co3(N03)12.24H20 
MANU JAIN 
The study of interplay of charge density wave and spin density wave in 
cuprates systems 
S K PANDA AND G C ROUT 


Forthcoming Publications

Jain, Manu

Superposition Model Analysis for the Zero-Field Splitting of Mn2+ in La2Co3 (NO3)12.24H2O

From Editor's Desk

Choudhary, S N

Prasad, K

Kumar, A

Choudhary, R N P

Vogel-Fulcher Like Dielectric Response in Pb[(Mg,Zn) 1/3Nb2/3]O3 Ceramic

Front Title

Indian J. Phys. 81 (10), 1097-1100(2007) l l i M AM 
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[c] U Fano and A R P Rao Atomic Collisions and Spactra (New York : Academic) Vol 1, Ch 2, Sec 4, p 25 
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171 T Atsumi, T Isihara, M Koyama and M Matsuzawa Phys. Rev. A42 6391 (1990) 
[III T Le-Brun, M Lavollee* and P Morin X-ray and Inner Shell Processes (AlP Conf. Proc. 215) (eds) T A 
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Notes for Contributors

Indian Association for the Cultivation of Science

Not a member yet

Variation of Stratospheric Ozone with Relative Humidity and Sharp Depletion of Absolute Humidity Associated with Nor' Wester Over Kolkata

Effect of Phase Fluctuations of a Laser on the Dynamic Process of an Atom Near a Metallic Layer

An Improved Analysis of Deep Inelastic Neutrino Data with the DGLAP Equations at Low and High x

Damped Vibrations of a Linearly Tapered Homogeneous Orthotropic Circular Plate

Forthcoming Publications

Superposition Model Analysis for the Zero-Field Splitting of Mn<SUP>2+ </SUP>in La<SUB>2</SUB>Co<SUB>3</SUB> (NO<SUB>3</SUB>)<SUB>12</SUB>.24H<SUB>2</SUB>O

From Editor's Desk

Vogel-Fulcher Like Dielectric Response in Pb[(Mg,Zn) <SUB>1/3</SUB>Nb<SUB>2/3</SUB>]O<SUB>3</SUB> Ceramic

Front Title

Notes for Contributors

6,371

8,071