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Natural frequencies of a linearly tapered non-homogeneous isotropic elastic circular plate resting on an elastic foundation
Journal of Sound and Vibration (1986) 111(1), 1-8
NATURAL FREQUENCIES OF A LINEARLY TAPERED NON-HOMOGENEOUS ISOTROPIC ELASTIC CIRCULAR PLATE RESTING ON AN ELASTIC FOUNDATION J. S.
TOMARt
Department of Mathematics, University of Roorkee, Roorkee 247672, India
D. C.
GUPTA
Department of Mathematics, 1. V. Jain Postgraduate College, Saharanpur, India AND
V.
KUMAR
Department of Mathematics, Regional Engineering College, Kuruksheta 132119, India (Received 19 September 1984, and in revised form 10 October 1985) An analysis is presented here for the design of engineering systems where composite materials and non-uniform structures are used. A simple model for free vibrations of a non-homogeneous isotropic elastic circular plate of linearly varying thickness resting on an elastic foundation is considered. The governing differential equation of motion is solved by the Frobenius method. The frequencies, deflections and moments corresponding to the first five modes of vibration are computed for clamped and simply supported edge conditions for various values of taper constant, non-homogeneity parameter and foundation modulus. 1. INTRODUCTION
An up-to-date account of problems involved in the vibrations of plates has been given by Leissa [1,2]. Jain [3] has discussed the axisymmetric vibrations of circular plates of linearly varying thickness. Later on, several other researchers have investigated the vibration problems of circular plates of variable thickness, taking different aspects into account such as in-plane forces [4] and elastic restraint of the edge against rotation [5]. Recently, Gupta [6] and Tomar et al. [7] have discussed the vibrations of tapered circular plates, including the effects of elastic foundation and non-homogeneity, respectively. Tomar et al. [8,9] have studied the vibration problems of non-homogeneous circular and infinite plates resting on elastic foundations. The non-homogeneity in these problems occurs due to the assumption of the linear variation in Young's modulus E and density p with one of the space variables. In the present investigation the Poisson's ratio is assumed to be constant. Statistical homogeneity and statistical isotropy are assumed, with exponential variations in Young's modulus and density with the x-co-ordinate. These assumptions would include glassspheres in epoxy resin and certain mixtures of metals, for example, lead/ aluminium. The numerical results for the first three modes of vibration are compared with those of reference [3] by allowing both the non-homogeneity parameter and the foundation modulus to be zero. t Now at Department of Mathematics, Al-fateh University, Tripoli, Libya. 1 0022-460X/86/220001 + 08 $03.00/0
© 1986 Academic Press Inc. (London) Limited
2
J . S. TOMAR, D . C. GUPTA AND V. KUMAR
2. FORMULATION OF THE PROBLEM AND ITS SOLUTION
The equation of axisymmetric motion of a non-homogeneous circular plate of variable thickness, resting on an elastic found ation , as derived after expressing the equation of motion [10] in polar co-ordinates and assuming that the axis of the plate coincides with the radial direction, and that the thickness of the plate h, Young's modulus E, density p and flexural rigidity D are functions of T, is
(1)
Here W is the transverse displacement, v the Poisson's ratio and kr the foundation modulus. On introducing the non-dimensional variables H = hi a, X = r]a and W = w] a, and with j3 = EI a and P= pi a, where a is the radius of the plate, equation (1) takes the form
EH3a4w4 +2[H3 aE-+{3H 2 (aH) +!!-3}] E a3W 3 ax
ax
ax
ax
X
3
H } -aE + [ H 3 -a2j32+ {(aH) 6H 2 +(2+v)ax
ax
ax
X
2
2W
(8H) 3H (a )-H } - J -a 2 +{ 3H2a2H -+6H - 2+(2+v)-H 2 E ax2 ax x ax x ax 3} 2 3 2 +[VH a E2 +{6VH (aH) _ H2 aE +{3lJH2(a 2~) + 6lJH(aH)2 2 X ax
X
2(aH)
3
3H
--2
X
-
ax
ax
x
ax
3
ax
ax
X
ax
2
H } E_] -+ aW 12H(I- v 2 )a 2 Paw +-3 2 + 12 (1- /I 2 )kfw == O. x
ax
at
(2)
Here the thickness of the plate H, modulus of elasticity E and mass density Pare assumed f3 x, fJ x to vary according to H = Ho(l- ax), E = Eo e and P= Po e where H o= H Ix=o, Eo = E Ix=o, Po = P Ix=o, a is a taper constant and (3 is the non-homogeneity parameter. For harmonic vibrations, the substitution W(X, t) = W(X ) eiwt, is made into equation (2), which reduces to ro
~
L a",B 1(A)Xc +.\ - 4 + L a",BiA)x cH
.1. = 0
"'=0
00
-
3
+
~
L a",E3 (A)Xc+.\ - 2 + L a",B 4(A )X +A - l C
A=O
A= O
(3) where <0
) W(x
= ",'"'"=' 0 aAXc+A,
(4)
3
TAPERED NON-HOMOGENEOUS SUPPORTED PLATE
is the assumed series solution for ~ c is the exponent of singularity, B 1(>..)
= T~I) b~) + T02 )b~)+ 1't)b ~I )+ 1(04) b~O),
v» n b(O) n )b(O)
(x) - 'T'll) b(3) + 'T'l2) b(2) + 'T'l 3) + j). Ii). Ij A B 2,,-I (3). ) T, ...,(2)b(2) + T(3)b(l)+ ( ' ) -- T(I)b B3/l 2 12 A 2 A
2l b (2) + T(3 )b(l) + (x ) -- T(l )b(3l + ...,( B4/l 3 ). i) A 3 A
+ r-2 T~3) b~l) +
Bs(>" ) := T~2l W B 6(A)
T~Il=l,
Tb2) =2,
:=
l b~1l
l
T~2)=-2a3f3,
T&3)=-1,
T~3) == 13 2+ 3a 2(5 + 2v) - 3a13 (4+ v), T~3)=3a2,B2-a3,B(8+v),
T~4)
= vf32+3a(f3 -a)(l-2v),
w
b ~O)
b~2)
T~I)=-a3,
T~2) :=2a2(313 -4a),
T~4)=l,
T\4)=_/3, 3
n,
3
T~4) = _a (3 2V,
TiS) == (aa 2/ 1*) -(,BFp / c*), (n
= 2,3, .. . , 11),
I*=H~/12,
4/ Eo, (), is the
C *==H~/12(1-v2),
frequency parameter,
b~\) == (c + A)(C+>"
== c+ A,
= (c+ >..)(c+ >.. -l)(c+ >.. -2)
A'
T~4) = -3a,B{,B1I+3a(1-4v)}+2a (l - 3 v),
(-,B)"Fp/j"C*
is the radian frequency, F p =
A'
b (Ol
n
T~3):=_a3f32,
Tl; )= -ur) 1*)+ (Fp / C*),
a2=a2w2po(l-v2)/Eo,
4) 3
T\3):::=3a+(f3-3a)(2+v), 2 3 T~3):::: -3a,B2+ 3a ,B (6 + v) - a (1l + 3 v),
T~4) == a 2f3{3 v,B + a(l- 6 v
T~S) ==
n
+ T~4l b~O) + ri), ) b~O) + T\S),
T~2)= 6a(3a - f3},
Ti 2):= 2(f3 -6a),
4 2
4
T~I ):=3a2,
Tp)=-3a,
4)
I).,
and
b~)
-ll,
= (c+ A)(c + >.. -1)(c+ A -2)(c+ >.. -3) .
In order that the series expression (4) for W be the solution for equation (3), the coefficients of different powers of X occurring in equation (3) must identically be equal to zero. By equating to zero the coefficient of the lowest power of X, the indicial roots c = 0, 0, 2, 2 are obtained and by applying the technique used in reference [10], the following solution, corresponding to c = 0, is obtained; (5)
°
where GbO) = G~2) "" 1, G~O) == G~O) = Gb2 ) = 0\2 ) = and OlO) and O~2) (A = 3,4, ...) are functions of a, /3, Fp , C *, 1* and a. It is evident that the solutions corresponding to the other values of c are included in equation (5). The convergence of the solution (5) is tested by applying the technique used by Lamb [l1J and it has been found convergent for all Ia I< 1. 3. BOUNDARY CONDITIONS AND FREQUENCY EQUATIONS
The frequency equations for clamped and simply supported circular plates are obtained by using the following boundary conditions.
4 3.1.
J. S. TOMAR, D. C. GUPTA AND V. KUMAR
CLAMPED PLATE
For a circular plate clamped at the edge r = a, the deflection wand slope of the plate element at the edge should be zero: i.e.,
w(r,t)lr=a=8w(r,t)/8rl r=a=0 or
WI X=1=aW/8xl x=1=0.
(6)
By using equation (5) and the boundary conditions (6) the frequency equation for a clamped circular plate is obtained as Ll(n)
IL
Lin) L 4(D)
3(n)
1=0
(7)
,
where 00
L1(n) = 1 +
L: A=3
G~°l,
co
L
L 2(D) = 1 +
L 3 ( f1 ) =
G~2),
co
L
AG~O),
A=3
A=3
cc
==2+
L
AG~2).
A=3
3.2. SIMPLY SUPPORTED PLATE For a circular plate simply supported at r = a, the boundary conditions are
w(r,t)lr=a=Mr(r,t)lr=a=o or
2
WIX=l =[82W/8X + (v/ X ) aw/aXJIX=l=O. (8)
By applying the boundary conditions (8) for an equation (5), the frequency equation in this case comes out to be Ll(n)
IL (f1 ) 5
L 2 ( f1 ) L6 ( f1 )
I= 0,
(9)
where,
L 5(D)=
L
A(A+V-l)G~O)
and
L6(D)=2(l+v)+
L
A(A+v-l)G\2 l .
A=3
A~3
4. RESULTS AND DISCUSSION
The numerical results have been computed by using the DEC 20 high speed digital computer at the University of Roorkee, Roorkee. The Poisson's ratio II and initial thickness H o have been taken throughout to be 0·3 and 0,1, respectively. For both edge condition cases the frequency parameters D( = aw.J(l- v 2)/(Eo/ Po» for different modes of vibration, have been computed for various combinations of taper constant a, non-homogeneity parameter {3 and foundation modulus Fp • In computing the series, terms up to an accuracy of 10-8 in their absolute values have been retained. Figure 1 shows the effect of the taper constant a on n. For {3 = 1·0 and F p = 0,0, the decrements in the frequencies of a simply supported plate are 30,53,29,67,28,40,28·43 and 27·53 per cent, while, for a clamped plate, they are 41'67,32,43, 29·74, 28·67 and 28·37 per cent, for the first five modes, respectively, when a varies from 0·0 to 0·5. For the same value of {3 and Fp = 0,02, the frequency parameter n, for a simply supported plate increases by 8·25 per cent for the first mode and decreases by 22,88,27'13,27,57 and 27·35 per cent for the next four modes, respectively; on the other hand, for a clamped plate, it decreases by 10,82, 28,00, 28,85, 28·41 and 28·23 per cent, respectively, for the first five modes as a increases correspondingly. It is also observed that D for a clamped plate is higher than for a simply supported plate for all the cases and modes discussed here.
5
TAPERED NON-HOMOGENEOUS SUPPORTED PLATE
8·0,-----,----r----,----.,.--_-----,
~ 4 ·0
3 ·0
0·0
0 ·\
0 ·2
0 '3
0 ·4
0 '5
a Figure 1. Variation of frequency parameter .n with taper constant c ; f3 = 1,0, Ho ={}'1, v = {)·3. Simply supported plate: - - , Fp = 0·0; - . - . -. Fp = 0·02. Clamped plate : - - - -, Fp = 0·0; - . . . -, Fp = 0·02 .
The results for the variation of f3 and different combinations of a and Fp , are shown in Figure 2. For a = 0·3 and Fp = 0,0, the frequency parameter n for the simply supported case increases by 6'33 ,3-13, 1,77, 1·09 and 0·8 per cent, and for the clamped case by 30,59, 7,76, 3,67, 2'21 and 1·66 per cent for the first five modes, respectively. For a: = 0·3 and Fp = 0'001, for a simply supported plate n decreases by 13·52 per cent for the first mode and increases by 1,99, 1'55, 1·02 and 0·72 per cent for the next four modes, respectively, and for a clamped plate it increases by 17,03, 7·02, 3,53,2'14 and 1·60 per cent, respectively, for the modes considered here. The graphs for the third, fourth and fifth modes in the latter case almost coincide with the graphs of the corresponding modes of the previous case, for both the edge conditions, and hence have not been shown in the figure. The frequency parameter n has also been computed for a = 0'3 and Fp = 0'01 . and is shown in the figure. It can be noted that.n decreases by 30,59,5'89 and 0·25 per cent for the first three modes, respectively, and increases by 0·53 per cent each for the next two modes for a simply supported plate, while for a clamped plate it decreases by 14·16 per cent in the first mode, while in the second mode it decreases by 1·05 per cent as f3 varies from -1,0 to 0·0 and increases by 1·85 per cent as (3 goes from 0·0 to 1'0; for the next three modes there is a continuous increase. Figure 3 shows the significant effects of the foundation modulus on the frequencies of a linearly tapered non-homogeneous isotropic elastic circular plate. The transverse displacement Wand moment parameter lVI, for the first five modes and for both edge conditions, are shown in Figures 4 and 5 respectively.
6
J. S. TOMAR, D. C. GUPTA AND V. KUMAR 7-0 , . - - - - - - - - , -
-,--
.......,.
---,
6-0
5-0
Fifth mode Fourth mode
3·0
Third mode 2-0
Second mode 1-0
0·0
..
:.~,.~~*~~:.~.~.~::::::..=~~-:::....:::::-'-":'
-1-0
-0-5
..
0-0
1·0
0·5
,8 Figure 2. Variation of frequency parameter n with non-homogeneity parameter f3;a = 0,3, H o = 0'1, v = 0·3. Simply supported plate: - - , Fp = 0'0; - ..... -, Fp = 0·001; - . -, Fp = 0·01. Clamped plate: - - - -, Fp = 0·0; ... " F p = 0·001; - ... -, Fp = 0·01.
6-0
5·0
Fifth mode
Fourth mode 4-0
3·0
Third mode
Second mode
. -:-::.='..-:--:,..:-:-:-.'.=:.".:-=-:- ..
First mode 0·01
0·02
0·03
0-04
Fp
n
Figure 3. Variation of frequency parameter with foundation modulus Fp ; rx = 0'3, H o = 0·1, u = 0·3. Simply supported plate: - - , ,B == 0·0; - ..• -, ,B = 1'0. Clamped plate: - - - -, f3 =0·0; - . - . -, f3 = 1'0.
7
TAPERED NON-HOMOGENEOUS SUPPORTED PLATE 1·0 ~to::"''''''';;':;;:::::T----,----....,.....----r----,
0·4
Four th mode
0·2
0·0
Third mode
0·6
0·4
0·8
1·0
x Figure 4. Transverse displacement W corresponding to the first five modes of vibration; H o = 0,1, a O· 3, f3 = 1'0, Fp = 0·01. - - - -, Clamped plate; - - , simply supported plate.
= 0,3,
IJ =
140
120
100
80
60
40
I~
20
0
-20
-40
-60
-80
-100
1·0
X
Figure 5. Moment parameters M corresponding to the first five modes of vibration; ct = 0,3, f3 = 1'0, Fp - - - -, Clamped plate; - - - , simply supported plate.
'"
O-Ol.
8
J. S. TOMAR, D. C. GUPTA AND V. KUMAR
To make a comparison with the results in reference [3], the value of {l was replaced by -/{12a 2 w 2po(1- :v 2 ) } / EoH~ and the frequencies were computed for the first three modes of vibration for both clamped and simply supported edge conditions, for some values of the taper constant given in reference [3]. These results are shown in Table 1. It can be seen from this table that the results are in good agreement.
TABLE
1
Values offrequency parameter n, for different values of a; v = 0·3 Simply-supported circular plate
Clamped circular plate A
a
First mode
0·1
A
Second mode
Third mode
First mode
Second mode
Third mode
9·40578 (9·4058)t
37·37705 (37·3771)
84·11987 (84'1199)
4·66707 (4·6671)
28·07944 (28·0895)
70·22492 (70'2250)
0·3
7·80705 (7,8071)
32·47527 (32,4753)
73'99077 (73'9908)
4·14676 (4'1468)
24·74401 (24'7440)
70·07055 (62'0706)
0·5
6·23384 (6,2339)
27·34073 (27·3408)
63'10695 (63'1070)
3·63707 (3'6371)
21·28773 (21'2878)
53·46863 (53'4687)
0·7
4·66457 (4,6646)
21·77643 (21'7765)
51·01941 (51'0195)
3·09938 (3·0994)
17·57303 (17·5731)
44·03882 (44·0389)
t The results in brackets are from reference [3].
REFERENCES 1. A. W. LEISSA 1969 NASA SP.160. Vibration of plates. A. W. LEISSA 1977 Shock and Vibration Digest 9, 13-24. Recent research in plate vibrations. 3. R. K.. JAIN 1972 Zeitschrift fur Angewandte Mathematik und Physik 23, 941-948. AXisymmetric vibrations of circular plates of linearly varying thickness. 4. R. K. JAIN 1972 Journal of Sound and Vibration 23, 407-414. Vibration of circular plates of variable thickness under an in-plane force. 5. P. A. A. LAURA and R. O. GROSSI 1978 Journal ofSound and Vibration, 60, 587-590. Influence of Poisson's ratio on the lower natural frequencies of transverse vibration of a circular plate of linearly varying thickness and with an edge elastically restrained against rotation. 6. D. C. GUPTA 1974 Ph.D. Thesis, Meerut University. Some vibration problems of elastic plates. 7. J. S. TOMAR, D. C. GUPTA and N. C. JAIN 1982 Journal of Sound and Vibration 85, 365-370. Axisymmetric vibrations of an isotropic elastic non-homogeneous circular plate of linearly varying thickness. 8. J. S. TOMAR, D. C. GUPTA and V. KUMAR 1983 Journal of Engineering Design, 49-54. Free vibrations of non-homogeneous circular plates of variable thickness resting on an elastic foundation. 9. J. S. TOMAR, D. C. GUPTA and V. KUMAR 1984 MECCANICA, Journal of the Italian Association of Theoretical and Applied Mechanics 19, 320-324. Natural frequencies of a nonhomogeneous isotropic elastic infinite plate of variable thickness resting on elastic foundation. 10. J. S. TOMAR and D. C. GUPTA 1976 Journal ofSound and Vibration 47, 143-145. Free vibrations of an infinite plate of parabolically varying thickness on elastic foundation. 11. H. LAMB 1945 Hydrodynamics. New York: Dover. See p. 335. 2.
NATURAL FREQUENCIES OF A LINEARLY TAPERED NON-HOMOGENEOUS ISOTROPIC ELASTIC CIRCULAR PLATE RESTING ON AN ELASTIC FOUNDATION J. S.
TOMARt
Department of Mathematics, University of Roorkee, Roorkee 247672, India
D. C.
GUPTA
Department of Mathematics, 1. V. Jain Postgraduate College, Saharanpur, India AND
V.
KUMAR
Department of Mathematics, Regional Engineering College, Kuruksheta 132119, India (Received 19 September 1984, and in revised form 10 October 1985) An analysis is presented here for the design of engineering systems where composite materials and non-uniform structures are used. A simple model for free vibrations of a non-homogeneous isotropic elastic circular plate of linearly varying thickness resting on an elastic foundation is considered. The governing differential equation of motion is solved by the Frobenius method. The frequencies, deflections and moments corresponding to the first five modes of vibration are computed for clamped and simply supported edge conditions for various values of taper constant, non-homogeneity parameter and foundation modulus. 1. INTRODUCTION
An up-to-date account of problems involved in the vibrations of plates has been given by Leissa [1,2]. Jain [3] has discussed the axisymmetric vibrations of circular plates of linearly varying thickness. Later on, several other researchers have investigated the vibration problems of circular plates of variable thickness, taking different aspects into account such as in-plane forces [4] and elastic restraint of the edge against rotation [5]. Recently, Gupta [6] and Tomar et al. [7] have discussed the vibrations of tapered circular plates, including the effects of elastic foundation and non-homogeneity, respectively. Tomar et al. [8,9] have studied the vibration problems of non-homogeneous circular and infinite plates resting on elastic foundations. The non-homogeneity in these problems occurs due to the assumption of the linear variation in Young's modulus E and density p with one of the space variables. In the present investigation the Poisson's ratio is assumed to be constant. Statistical homogeneity and statistical isotropy are assumed, with exponential variations in Young's modulus and density with the x-co-ordinate. These assumptions would include glassspheres in epoxy resin and certain mixtures of metals, for example, lead/ aluminium. The numerical results for the first three modes of vibration are compared with those of reference [3] by allowing both the non-homogeneity parameter and the foundation modulus to be zero. t Now at Department of Mathematics, Al-fateh University, Tripoli, Libya. 1 0022-460X/86/220001 + 08 $03.00/0
© 1986 Academic Press Inc. (London) Limited
2
J . S. TOMAR, D . C. GUPTA AND V. KUMAR
2. FORMULATION OF THE PROBLEM AND ITS SOLUTION
The equation of axisymmetric motion of a non-homogeneous circular plate of variable thickness, resting on an elastic found ation , as derived after expressing the equation of motion [10] in polar co-ordinates and assuming that the axis of the plate coincides with the radial direction, and that the thickness of the plate h, Young's modulus E, density p and flexural rigidity D are functions of T, is
(1)
Here W is the transverse displacement, v the Poisson's ratio and kr the foundation modulus. On introducing the non-dimensional variables H = hi a, X = r]a and W = w] a, and with j3 = EI a and P= pi a, where a is the radius of the plate, equation (1) takes the form
EH3a4w4 +2[H3 aE-+{3H 2 (aH) +!!-3}] E a3W 3 ax
ax
ax
ax
X
3
H } -aE + [ H 3 -a2j32+ {(aH) 6H 2 +(2+v)ax
ax
ax
X
2
2W
(8H) 3H (a )-H } - J -a 2 +{ 3H2a2H -+6H - 2+(2+v)-H 2 E ax2 ax x ax x ax 3} 2 3 2 +[VH a E2 +{6VH (aH) _ H2 aE +{3lJH2(a 2~) + 6lJH(aH)2 2 X ax
X
2(aH)
3
3H
--2
X
-
ax
ax
x
ax
3
ax
ax
X
ax
2
H } E_] -+ aW 12H(I- v 2 )a 2 Paw +-3 2 + 12 (1- /I 2 )kfw == O. x
ax
at
(2)
Here the thickness of the plate H, modulus of elasticity E and mass density Pare assumed f3 x, fJ x to vary according to H = Ho(l- ax), E = Eo e and P= Po e where H o= H Ix=o, Eo = E Ix=o, Po = P Ix=o, a is a taper constant and (3 is the non-homogeneity parameter. For harmonic vibrations, the substitution W(X, t) = W(X ) eiwt, is made into equation (2), which reduces to ro
~
L a",B 1(A)Xc +.\ - 4 + L a",BiA)x cH
.1. = 0
"'=0
00
-
3
+
~
L a",E3 (A)Xc+.\ - 2 + L a",B 4(A )X +A - l C
A=O
A= O
(3) where <0
) W(x
= ",'"'"=' 0 aAXc+A,
(4)
3
TAPERED NON-HOMOGENEOUS SUPPORTED PLATE
is the assumed series solution for ~ c is the exponent of singularity, B 1(>..)
= T~I) b~) + T02 )b~)+ 1't)b ~I )+ 1(04) b~O),
v» n b(O) n )b(O)
(x) - 'T'll) b(3) + 'T'l2) b(2) + 'T'l 3) + j). Ii). Ij A B 2,,-I (3). ) T, ...,(2)b(2) + T(3)b(l)+ ( ' ) -- T(I)b B3/l 2 12 A 2 A
2l b (2) + T(3 )b(l) + (x ) -- T(l )b(3l + ...,( B4/l 3 ). i) A 3 A
+ r-2 T~3) b~l) +
Bs(>" ) := T~2l W B 6(A)
T~Il=l,
Tb2) =2,
:=
l b~1l
l
T~2)=-2a3f3,
T&3)=-1,
T~3) == 13 2+ 3a 2(5 + 2v) - 3a13 (4+ v), T~3)=3a2,B2-a3,B(8+v),
T~4)
= vf32+3a(f3 -a)(l-2v),
w
b ~O)
b~2)
T~I)=-a3,
T~2) :=2a2(313 -4a),
T~4)=l,
T\4)=_/3, 3
n,
3
T~4) = _a (3 2V,
TiS) == (aa 2/ 1*) -(,BFp / c*), (n
= 2,3, .. . , 11),
I*=H~/12,
4/ Eo, (), is the
C *==H~/12(1-v2),
frequency parameter,
b~\) == (c + A)(C+>"
== c+ A,
= (c+ >..)(c+ >.. -l)(c+ >.. -2)
A'
T~4) = -3a,B{,B1I+3a(1-4v)}+2a (l - 3 v),
(-,B)"Fp/j"C*
is the radian frequency, F p =
A'
b (Ol
n
T~3):=_a3f32,
Tl; )= -ur) 1*)+ (Fp / C*),
a2=a2w2po(l-v2)/Eo,
4) 3
T\3):::=3a+(f3-3a)(2+v), 2 3 T~3):::: -3a,B2+ 3a ,B (6 + v) - a (1l + 3 v),
T~4) == a 2f3{3 v,B + a(l- 6 v
T~S) ==
n
+ T~4l b~O) + ri), ) b~O) + T\S),
T~2)= 6a(3a - f3},
Ti 2):= 2(f3 -6a),
4 2
4
T~I ):=3a2,
Tp)=-3a,
4)
I).,
and
b~)
-ll,
= (c+ A)(c + >.. -1)(c+ A -2)(c+ >.. -3) .
In order that the series expression (4) for W be the solution for equation (3), the coefficients of different powers of X occurring in equation (3) must identically be equal to zero. By equating to zero the coefficient of the lowest power of X, the indicial roots c = 0, 0, 2, 2 are obtained and by applying the technique used in reference [10], the following solution, corresponding to c = 0, is obtained; (5)
°
where GbO) = G~2) "" 1, G~O) == G~O) = Gb2 ) = 0\2 ) = and OlO) and O~2) (A = 3,4, ...) are functions of a, /3, Fp , C *, 1* and a. It is evident that the solutions corresponding to the other values of c are included in equation (5). The convergence of the solution (5) is tested by applying the technique used by Lamb [l1J and it has been found convergent for all Ia I< 1. 3. BOUNDARY CONDITIONS AND FREQUENCY EQUATIONS
The frequency equations for clamped and simply supported circular plates are obtained by using the following boundary conditions.
4 3.1.
J. S. TOMAR, D. C. GUPTA AND V. KUMAR
CLAMPED PLATE
For a circular plate clamped at the edge r = a, the deflection wand slope of the plate element at the edge should be zero: i.e.,
w(r,t)lr=a=8w(r,t)/8rl r=a=0 or
WI X=1=aW/8xl x=1=0.
(6)
By using equation (5) and the boundary conditions (6) the frequency equation for a clamped circular plate is obtained as Ll(n)
IL
Lin) L 4(D)
3(n)
1=0
(7)
,
where 00
L1(n) = 1 +
L: A=3
G~°l,
co
L
L 2(D) = 1 +
L 3 ( f1 ) =
G~2),
co
L
AG~O),
A=3
A=3
cc
==2+
L
AG~2).
A=3
3.2. SIMPLY SUPPORTED PLATE For a circular plate simply supported at r = a, the boundary conditions are
w(r,t)lr=a=Mr(r,t)lr=a=o or
2
WIX=l =[82W/8X + (v/ X ) aw/aXJIX=l=O. (8)
By applying the boundary conditions (8) for an equation (5), the frequency equation in this case comes out to be Ll(n)
IL (f1 ) 5
L 2 ( f1 ) L6 ( f1 )
I= 0,
(9)
where,
L 5(D)=
L
A(A+V-l)G~O)
and
L6(D)=2(l+v)+
L
A(A+v-l)G\2 l .
A=3
A~3
4. RESULTS AND DISCUSSION
The numerical results have been computed by using the DEC 20 high speed digital computer at the University of Roorkee, Roorkee. The Poisson's ratio II and initial thickness H o have been taken throughout to be 0·3 and 0,1, respectively. For both edge condition cases the frequency parameters D( = aw.J(l- v 2)/(Eo/ Po» for different modes of vibration, have been computed for various combinations of taper constant a, non-homogeneity parameter {3 and foundation modulus Fp • In computing the series, terms up to an accuracy of 10-8 in their absolute values have been retained. Figure 1 shows the effect of the taper constant a on n. For {3 = 1·0 and F p = 0,0, the decrements in the frequencies of a simply supported plate are 30,53,29,67,28,40,28·43 and 27·53 per cent, while, for a clamped plate, they are 41'67,32,43, 29·74, 28·67 and 28·37 per cent, for the first five modes, respectively, when a varies from 0·0 to 0·5. For the same value of {3 and Fp = 0,02, the frequency parameter n, for a simply supported plate increases by 8·25 per cent for the first mode and decreases by 22,88,27'13,27,57 and 27·35 per cent for the next four modes, respectively; on the other hand, for a clamped plate, it decreases by 10,82, 28,00, 28,85, 28·41 and 28·23 per cent, respectively, for the first five modes as a increases correspondingly. It is also observed that D for a clamped plate is higher than for a simply supported plate for all the cases and modes discussed here.
5
TAPERED NON-HOMOGENEOUS SUPPORTED PLATE
8·0,-----,----r----,----.,.--_-----,
~ 4 ·0
3 ·0
0·0
0 ·\
0 ·2
0 '3
0 ·4
0 '5
a Figure 1. Variation of frequency parameter .n with taper constant c ; f3 = 1,0, Ho ={}'1, v = {)·3. Simply supported plate: - - , Fp = 0·0; - . - . -. Fp = 0·02. Clamped plate : - - - -, Fp = 0·0; - . . . -, Fp = 0·02 .
The results for the variation of f3 and different combinations of a and Fp , are shown in Figure 2. For a = 0·3 and Fp = 0,0, the frequency parameter n for the simply supported case increases by 6'33 ,3-13, 1,77, 1·09 and 0·8 per cent, and for the clamped case by 30,59, 7,76, 3,67, 2'21 and 1·66 per cent for the first five modes, respectively. For a: = 0·3 and Fp = 0'001, for a simply supported plate n decreases by 13·52 per cent for the first mode and increases by 1,99, 1'55, 1·02 and 0·72 per cent for the next four modes, respectively, and for a clamped plate it increases by 17,03, 7·02, 3,53,2'14 and 1·60 per cent, respectively, for the modes considered here. The graphs for the third, fourth and fifth modes in the latter case almost coincide with the graphs of the corresponding modes of the previous case, for both the edge conditions, and hence have not been shown in the figure. The frequency parameter n has also been computed for a = 0'3 and Fp = 0'01 . and is shown in the figure. It can be noted that.n decreases by 30,59,5'89 and 0·25 per cent for the first three modes, respectively, and increases by 0·53 per cent each for the next two modes for a simply supported plate, while for a clamped plate it decreases by 14·16 per cent in the first mode, while in the second mode it decreases by 1·05 per cent as f3 varies from -1,0 to 0·0 and increases by 1·85 per cent as (3 goes from 0·0 to 1'0; for the next three modes there is a continuous increase. Figure 3 shows the significant effects of the foundation modulus on the frequencies of a linearly tapered non-homogeneous isotropic elastic circular plate. The transverse displacement Wand moment parameter lVI, for the first five modes and for both edge conditions, are shown in Figures 4 and 5 respectively.
6
J. S. TOMAR, D. C. GUPTA AND V. KUMAR 7-0 , . - - - - - - - - , -
-,--
.......,.
---,
6-0
5-0
Fifth mode Fourth mode
3·0
Third mode 2-0
Second mode 1-0
0·0
..
:.~,.~~*~~:.~.~.~::::::..=~~-:::....:::::-'-":'
-1-0
-0-5
..
0-0
1·0
0·5
,8 Figure 2. Variation of frequency parameter n with non-homogeneity parameter f3;a = 0,3, H o = 0'1, v = 0·3. Simply supported plate: - - , Fp = 0'0; - ..... -, Fp = 0·001; - . -, Fp = 0·01. Clamped plate: - - - -, Fp = 0·0; ... " F p = 0·001; - ... -, Fp = 0·01.
6-0
5·0
Fifth mode
Fourth mode 4-0
3·0
Third mode
Second mode
. -:-::.='..-:--:,..:-:-:-.'.=:.".:-=-:- ..
First mode 0·01
0·02
0·03
0-04
Fp
n
Figure 3. Variation of frequency parameter with foundation modulus Fp ; rx = 0'3, H o = 0·1, u = 0·3. Simply supported plate: - - , ,B == 0·0; - ..• -, ,B = 1'0. Clamped plate: - - - -, f3 =0·0; - . - . -, f3 = 1'0.
7
TAPERED NON-HOMOGENEOUS SUPPORTED PLATE 1·0 ~to::"''''''';;':;;:::::T----,----....,.....----r----,
0·4
Four th mode
0·2
0·0
Third mode
0·6
0·4
0·8
1·0
x Figure 4. Transverse displacement W corresponding to the first five modes of vibration; H o = 0,1, a O· 3, f3 = 1'0, Fp = 0·01. - - - -, Clamped plate; - - , simply supported plate.
= 0,3,
IJ =
140
120
100
80
60
40
I~
20
0
-20
-40
-60
-80
-100
1·0
X
Figure 5. Moment parameters M corresponding to the first five modes of vibration; ct = 0,3, f3 = 1'0, Fp - - - -, Clamped plate; - - - , simply supported plate.
'"
O-Ol.
8
J. S. TOMAR, D. C. GUPTA AND V. KUMAR
To make a comparison with the results in reference [3], the value of {l was replaced by -/{12a 2 w 2po(1- :v 2 ) } / EoH~ and the frequencies were computed for the first three modes of vibration for both clamped and simply supported edge conditions, for some values of the taper constant given in reference [3]. These results are shown in Table 1. It can be seen from this table that the results are in good agreement.
TABLE
1
Values offrequency parameter n, for different values of a; v = 0·3 Simply-supported circular plate
Clamped circular plate A
a
First mode
0·1
A
Second mode
Third mode
First mode
Second mode
Third mode
9·40578 (9·4058)t
37·37705 (37·3771)
84·11987 (84'1199)
4·66707 (4·6671)
28·07944 (28·0895)
70·22492 (70'2250)
0·3
7·80705 (7,8071)
32·47527 (32,4753)
73'99077 (73'9908)
4·14676 (4'1468)
24·74401 (24'7440)
70·07055 (62'0706)
0·5
6·23384 (6,2339)
27·34073 (27·3408)
63'10695 (63'1070)
3·63707 (3'6371)
21·28773 (21'2878)
53·46863 (53'4687)
0·7
4·66457 (4,6646)
21·77643 (21'7765)
51·01941 (51'0195)
3·09938 (3·0994)
17·57303 (17·5731)
44·03882 (44·0389)
t The results in brackets are from reference [3].
REFERENCES 1. A. W. LEISSA 1969 NASA SP.160. Vibration of plates. A. W. LEISSA 1977 Shock and Vibration Digest 9, 13-24. Recent research in plate vibrations. 3. R. K.. JAIN 1972 Zeitschrift fur Angewandte Mathematik und Physik 23, 941-948. AXisymmetric vibrations of circular plates of linearly varying thickness. 4. R. K. JAIN 1972 Journal of Sound and Vibration 23, 407-414. Vibration of circular plates of variable thickness under an in-plane force. 5. P. A. A. LAURA and R. O. GROSSI 1978 Journal ofSound and Vibration, 60, 587-590. Influence of Poisson's ratio on the lower natural frequencies of transverse vibration of a circular plate of linearly varying thickness and with an edge elastically restrained against rotation. 6. D. C. GUPTA 1974 Ph.D. Thesis, Meerut University. Some vibration problems of elastic plates. 7. J. S. TOMAR, D. C. GUPTA and N. C. JAIN 1982 Journal of Sound and Vibration 85, 365-370. Axisymmetric vibrations of an isotropic elastic non-homogeneous circular plate of linearly varying thickness. 8. J. S. TOMAR, D. C. GUPTA and V. KUMAR 1983 Journal of Engineering Design, 49-54. Free vibrations of non-homogeneous circular plates of variable thickness resting on an elastic foundation. 9. J. S. TOMAR, D. C. GUPTA and V. KUMAR 1984 MECCANICA, Journal of the Italian Association of Theoretical and Applied Mechanics 19, 320-324. Natural frequencies of a nonhomogeneous isotropic elastic infinite plate of variable thickness resting on elastic foundation. 10. J. S. TOMAR and D. C. GUPTA 1976 Journal ofSound and Vibration 47, 143-145. Free vibrations of an infinite plate of parabolically varying thickness on elastic foundation. 11. H. LAMB 1945 Hydrodynamics. New York: Dover. See p. 335. 2.