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Vibration of a viscoelastic plate having a circular outer boundary and an eccentric circular inner boundary for various edge conditions
Journal
of Sound and Vibration (1979) 63( I), 73-85
VIBRATION
OF AVISCOELASTIC
PLATF HAVING A CIRCULAR
OUTER BOUNDARY AND AN ECCENTRIC CIRCULAR INNER BOUNDARY FOR VARIOUS EDGE CONDITIONS K. NAGAYA Department of Mechanical Engineering, Faculty of Engineering, Yamagata Universit?,, Yonezawa, Japan (Received 20 March 1978, and in revisedform 7 September 1978)
In this paper, vibration problems of a circular viscoelastic plate having an eccentric circular inner edge are investigated. The frequency equations in complex forms for various edge conditions are obtained. Numerical calculations are carried out for both elastic and viscoelastic plates, and the non-dimensional natural frequencies and the logarithmic decrements are given for a number of cases.
1. INTRODUCTION
A circular plate having an eccentric inner edge is an important element of machines and structures, such plates being widely used in many branches of engineering. The dynamic behaviour of this type of plate therefore needs to be understood, as well as that of a plate with a concentric inner edge. These problems however have not been thoroughly discussed although a few theoretical [l, 21 and experimental [3] results for elastic plates have been reported. In these studies the effect of the eccentricity of the inner edge on the natural frequencies of the plate has been discussed in references [2] and [3]. Maruyama and Ichinomiya [3] investigated an elastic plate with fixed circular outside and free circular inside edges in their experiments, and they concluded that the effect of the eccentricity can be neglected from an engineering point of view. However, in the author’s previous paper [2] concerning an elastic plate with free outside and fixed inside edges it was reported that the effect of the eccentricity is large and cannot be neglected in general. These discrepancies are due to the differences between the boundary conditions. In this paper the vibrations of a circular viscoelastic plate with an eccentric circular inner edge are discussed for various boundary conditions. An analytical method for solving the problems has been previously given by the author [2]. In that study the terms having small influence on the natural frequencies were neglected in treating the outer boundary conditions. In this paper the transformed boundary conditions are given in exact forms, and the influence of neglecting terms is investigated. The exact solution is obtained by applying the Laplace transform to the equation of motion for the viscoelastic plate. The boundary conditions along the outer edge of the plate are satisfied by means of the Fourier expansion method.
2. ANALYSIS A viscoelastic plate with a circular outer boundary and an eccentric circular inner boundary of an eccentricity d is shown in Figure 1, in which the origin 0 of Cartesian co-ordinates x and y is chosen in the center of the plate, and the z axis is normal to the plate surface. The polar co-ordinates r, 8 and r’, 8’ are taken as shown in the figure, and the radii of the
73 0022-460X/79/050073+ 13$02.00/O
01979 Academic Press Inc. (London) Limited
74
K. NAGAYA
outer edge, r, and the inner edge, r’, are denoted by R and a, respectively. If one denotes the displacement of the plate in the z direction by w, the equation of motion for the viscoelastic plate, in the co-ordinates I’ and 8’, is [4]
[(2PQ' + P'Q)Q(h3/12)V2V2 + (PQ'+ 2P'Q)Pph a2/at2 + (PQ’ + 2P’Q)Pc$/&]w
= 0 (I)
where P = 5
p, d’ldt’, p’ = ;
I=0
p; d’ldt’, Q =
I=0
i q, d’ldt’, Q’ = t q; d’/dt’, I=0 I -- 0
(2)
V2 is the two-dimesnional Laplacian operator, p is the mass density, h is the thickness, co is the coefficient of the external viscou’s damping, t is the time and pl, q,, pi and q[ are constant
Figure 1. Geometry ofa viscoelastic plate with a circular outer boundary and an eccentric circular inner boundary.
coefficients describing the properties of the materials. [(2PQ’ + P’Q)Q/(PQ’ + 2QP')] x (h3/12P) (denoted by D(t) later) corresponds to the flexural rigidity 9 = Eh3/12( 1 - v2) in the elastic plate, where E is the Young’s modulus and v is the Poisson’s ratio of the elastic plate. The application of the Laplace transformation f(r’, 8’, s) =
m
f(r’, 0’, t) e-“’ dt
(3)
s0
to equation (1) turns D(t) into b(s). To identify the frequency parameter 1, the flexural rigidity 9 of an elastic plate which has the same value as D(t), but in which the effects of viscous damping are omitted, is introduced in this analysis. The solution of the transformed viscoelastic plate equation is obtained, by using the transformed flexural rigidity D(s), as W = (l/S’) i j=l
f
Fn[AjnJn(tlr’)+ BjnY,(ar’) + CjJn(ar’) + Dj,,K”(ar’)]djjn,
(4)
n=O
where E,, = l/2 for n = 0 and E,, = 1 for n 2 1, aIn= cos nel, Qp,”= sin nt?, a4 = (A25 - c,s&3),
2 = -is(ph/S?)f,
[ = ~/D(S)
(5)
75
ECCENTRIC ANNULAR VISCOELASTIC PLATE
and i = m The coefficients A,,Bi,, Cjn and Djn are constants of integration to be determined from the boundary conditions, J”(ar’) and Y,(ar’) are the complex Bessel functions of first and second kinds of order n, and I”(ar’) and K,(ar’) are the modified Bessel functions. The boundary conditions along the inner edge are (G), = D = (&/lar’)l. = II = for the clamped edge, (H,,),, = (1= (TV),,= (I = 0 for the free edge, (W),, _- a = (RI.)*. = a = 0 for the simply supported edge
(6)
where Mu, is the bending moment and V,.is the Kirchhoff shear in the plate [S]. Substituting equations (4) into equations (6) and eliminating the constants Cjn and D,,, one obtains w = (l/9)
i
5
j=l
sn[Ajn{Jn(ar’) + y&W)
+ ?,.K,(ar’)) + Bj,{Y,(ar’) + Y&(u~‘)
n=O
Y2nKnW)Il
+
(7)
@j+
where Yi, = 0’&” - j&)/F, ~4”
=
-
-
(y&z,
Y2”= (Yl”f& - Y,,il.)E
y,,k,,YF7F
=
ilnk2n
-
Y3”= - (il”k2, -j&,)/F (8)
klnL,
and the coefficients jln - k,, are given, for the clamped edge, by
’ = J,,@4jZn = Jn_ 1(au) - (n/au)J,(aa), yin = Y,(w),
Jl”
’ = 11”
Y,” = Yn _ ,(a4
-
Wa0’,,@4
I,(au), i,” = In _ 1(au) - (n/au)I,(au), k,” = Kn(au), k,, = - Kn _ ,(au) -(n/au) K,,(au),
(9)
for the free edge, by ’ = rnTn = [n(n + l)(l - v)/a2u2 - l]J,(au) - [(l - v)/aa]J”_ i(cru), Jl” Yl” = mm = [n(n + l)(l - v)/a2u2 - l]Y”(au) - [(l - v)/aa]Yn _ ,(aa), - v)/ a2u2 + l]I”(au) - [( 1 - v)/au]In _ ,(au),
’ = rn$ = [n(n + l)(l 5” k,” =mX,= [n(n + l)(l
- v)/a’a” + l]K”(au) + [(l - v)/aa]K” _ ,(au),
j2” = Cn2(n + l)(l - v)/ a3u3 + n/au]J”(au) - [n2(1 - v)/a2u2 + l]J, _ ,(~a),
y,” = [n2(n + 1) (1 - v)/a3u3 + n/au]Yn(au) - [n2(1 - v)/a2u2 + l]Y, _ ,(aa), ’ 12”
=
[n2(n
+
l)(l - v)/a3u3 - n/au]In(au) - [n2(1 - v)/a2u2 - l]I,_ ,(~a),
k,” = [n”(n + 1) (1 - v)/a3u3 - n/au]Kn(au) + [n2(l - v)/a’u’ -
l]K, _ ,@a), (IO)
and, for the simply supported edge, by pi, = Jn@4j2n = mFny Y,”
= Y.(m),
Y,” = mfn,iln
k,, = m4:.
= I&4,
i2n = m:,,
kin = Knb).
(11)
The boundary conditions along the outer edge are expressed by using equations (6) with is expressed in terms of the co-ordinates r’ and 6’, and hence it is convenient to treat the outer boundary conditions when the derivatives in the equations of the bending slope, bending moment and shearing force are transformed in terms of r’ and 0’. The transformed expressions are r’ replaced by r and a by R. The displacement
76
K. NAGAYA
aiqar Rr = -D(s)[(b,+
= b,aw/ar’ + b,aw/aw,
ve,/r2)a2i?/W2
+ ve,Jr2)&/iY
+ (b4 + vb,/r
+ (b, + ve,/r2) a2i’i+%‘%’ + (b6 + ve,/r2)t32w/jlde’2 + (b7 + vb,/r + ve,jr2)&/W], y = -B(S)
Gf,iT3i@r’3
+ f2a2i7/W2
+ fea3ic/ar’ae’2
f,ai@tJ +
+
+ f,aicqau
+ f,a%iqae’2
f,a3w/arJ2ae’
+ f5a2w,iar’ae
+ f,a%/ae’3],
(12)
where
b, =
b, = WJar, b, = (W/iYr)2, b, = a2r’/ar2, b, = 2(&‘/h)
W/i%,
b,
= (aO’/ar)*, b, = a2B’/ar2, b, = (&t/&)3,
b,,
= a3rr/ar3, b, 1 = 3(W/ar)’
b,,
= 3(W/ar)
(W/&)2,
b,,
b, = 3(&‘/h)
(d2&r2),
(LX?/&), b, 2 = 3[(a2r’/ar2)
= a3B’/ar3, b,,
= 3(W/ar)
(at?/&),
(a@/&+) + (W/h)
(aze;‘ar2),
b16 = (aO’/ar)3, e, = ar’/ae, e2 = ael/ae, e3 = (a@/atI)2, e4 = 2(aeyae)
e5 e9
= aWfae2, =
(a+/ae),
e6 = a2+/ae2, e7 = (ar’lae)2, es = (ae’pr) (aepep,
z(ael/ae)
(a28’/arae)
(aW/ar), e, o = (W/h)
+ (aVyae2)
(8x8)2 x
e, 1 = 2(aW/arae) e,, = 2(ae’/ae)
(a+/ae)
+ 2(aeyae)
(a2r’/ade)
e, h = a38’/hW2,fl
e, 5 = a3r’/arae2, b,/r
f3 = b,, -
- (3 -
(ar’/ar) + (a@/&) (@+/ae2),
+ z(w/ae)
(a2r’/arae),
= b, + (2 - v)e13/r2,
v)e7/r3 + (2 - v)e14/r2,
b,/r2 + b,/r - (3 - v)e,/r3 + (2 - v)e,,/r2,f,
f5
=b,,
+ b,/r - (3 - v)e,/r”
f7
=b,,
- bJr2
fs = b,,
+ (aW/ae’)
+ 2caeyae) (&‘/?e) (aV/ae2),
(art/a@ (ar’/ar) + (ar’/ae)2 (ae’pr),
e13 = (art/a@’ (ar’/ar), e14 = (W/&j (a2+/ae2)
f2 = b, +
(a2@/ar2)],
+ b,/r
+ (2 - v)eIl/r2,f6
One has the following
v)e,/r3
relations
+ (2 -
v)elo/r2,
v)e,/r” + (2 - v)e16/r2,
+ b7/r - (3 - (3 -
= b,,
= b, 1 + (2 - v)e12/r2,
+ (2 -
between
v)e,/r2,f9’=
b,,
the co-ordinates
+ (2 -
v)e,/r’.
(13)
r, 8 and r’, 8’:
r cos 8 + d = r’ cos 8’, r sin 0 = r’ sin 19, r’ = (r2 + 2rd cos 8 + d2)1’2, 8’ = cos- l[(r/r’) cos e + a/r’].
(14)
The coefficients b 1 - f, in equation s (13) are evaluated by using equations (14). The bending slope, the bending moment and the shearing force are then obtained from equations (12) (13) and (14): &$?r
= (E/9) f:
t
En[Ajn(Xji
+ Y3nXj” + Y,,X,~) + B,.(X,f
+ Y,,Xj,” + r,,xj~)],
E,[Aj,(mj;
+ Y3”Mj.j + Ylnmjf) + Bjn(mji + Y4nrnji + Y2,mj~119
j=ln=O RI = -(cr2/0.
9 = -Ca3/1)
i: f .i=ln=O i
f
j-r
n=o
En[Aj~(Uj~ + Y3n”jn3+ Yln’jz) + Bj”(Vj~ + Y4,‘j,3 + ‘,?“‘jt)]3
(15)
ECCENTRIC
ANNULAR VISCOELASTIC
PLATE
77
where
(c::)=b,* [(; ::I;;)- (n/at) (::)*k:l)]cos n@- nb,*kg)))sin I
X,2 = b,* [In Xl:
=
-
I(~~‘) - (n/ar’)In(ar’)]
m1f =
(m,nz )
[(b3* + ve7*/a2r2){(n2
x (n/ar’) cos n@ + (n*/ar’)(b,* -n(b,*
+ vb,*/ar +(b,*
cos nd’ - nb?*I”(ar’) sin d’,
l(ar’) + (n/ar’)K,(ar’)]cos
b,*[K,
n@ - nbz*Kn(d)sin
+ n)/(ar’)* - l> cos n$’ + ve,*/a2r2)sin
+ ve,*/cc2r2)sin
no’,
(b,* + vb,*/ar
+ [ - (b: + veT/a”r’) (I /ar’)cos nO
nO’]
ml;3 = [(bj + ve3/a2r2){(n2 + n)/(ar’)* +
+ veh*/a2r2)
nH’- n2(b,* + ve3*/a2r2) cos nt)
+ vb:far + vez/a2r2)cos n@ - rt(bS + ve$‘a2r2)sin
1 } cos
n@]
’
n@ - (bl + vb:Jar + veg/a2r2) (n/rr’pos n6’
+(n2/ar’) (b: + ve,*/a2r2)sin n@ - n2(b,* + veT/a’r’)cos x sin n@]I,,(ar’) + [-(b:
nt?,
nQ’- n(bT + vbz/ar + ve:idr2)
+ veT/a2r2) (l/ar’)cos n0’ + (bz + vbT/ar + veg/a*r’)cos n6 -n(bz
+ vez/a2r2)sin
l(ar’),
nB’]In
rn,: = [(bz + ve3/a2r2){(n2 + n)/(ar’)2+ I} cos n@ - (b,* + vb:/ar + vez/a2r2) (n/at) x cos nt?’+ (n2/ar’) (b: + veX/a2r2)sin no’ - n2(bz + ve:/ctzr2)cos n0 - n(b3 + vbT/ar + ve:/a2r2) sin nO’]Kn(ar’) + [(b: + veT/a2r2) (1jar’) cos no -(bX + vb:/ar + vez/a2r2) cos n@ + n(bz + veq*/r’r’) sin ne’]Kn _ l(xr’),
(
1 Ul” 2 = uln > cos
vF{
-
(n3 + 3n2
+
2n)/(ar’)3 + (n + l)/ar’} cos n@ + f,*((n2 + n)/(ar’)2 -- 1)
n@ - ,fT(n/ar’) cos n@ - nf,*{(n2 + n)/(ar’)2 - 1} sin nd’ + (,f5*n2/ar’) sin no’ +(f6*n3/ar’) cos ne - nf; sin n& - n’f:
cos no + n3fz sin nd]
:irr!) C n ‘)
+Cf;F{(n’ + 2)/(ar’)2 - 1) cos n@ - V;*/ar’) cos n@ + ,fT cos n8’ + (nfk*/ar’) sin nH - nf,* sin n@ - n”f,* cos ne’]
0,: = v,*{ -(n3 X COS n@ -
ar’ ’
+ 3n2 + 2n)/(ar’)3 - (n + l)/ar’} cos nf7 +fi*{(n2
+ n)/(ar’)2 + 1 :
Cf,*n/ar’) COS n@ - nf,*{(n’ + n)/(ar’)* + 1) sin nl? + (&*n*/ar’) sin n@
+Cf$n3/ar’) + Cf:{(n2
tar’)
(::1:t ,!
COS
n@ - nf; sin nt7 - nff,* cos nfY + n”f,* sin nO’]In(ar’)
+ 2)/(ar’)2 + 1) cos n@ - (f2*/ar’) cos n@ + j: - nf: sin nB’ - n”fg” cos nellIn
cos nt7 + V;*n/ar’) sin nt)
1(ar’),
78 u,f
K. NAGAYA =
rT{--(n3
X COS n8’
-
+
(f~qhr’)
3n2 + 2n)/(ar’)3 - (n + l)/crr’} cosn@ +f:{(n’ c0S n&
-
nf4*{(n2
+(f,*n3/ar’) cos n0’ - nf:
+
sin nf3’ -
+ n)/(ar’)2 + Ii
n)/(ar’)2 + l} sin rz@+ Qn2/ar’)
sin
n0’
n’fff cos n0’ + n”fc sin n@]K,,(ar’)
+ [ -f;“{(n” + 2)/(ar’)2 + 1} cos n@ + (f;c/ar’) cos n@ - f3* cos n@ - (fk*n/ar’) sin ntI +nfz sin n0’ + n’fz cos nO’]Kn _ I(ar’).
(16)
In equations (16), b:, bz, . . . are the non-dimensional values which correspond to b,, b,, . . . in equations (13), and these are expressed as follows : br
=
b:
= (l/ar’) sin 2(0 - B’),b,* = (lja’r”)
cos(8
-
b: = (l/ar’) sin(8 - O’),b: = cos2(8 - O’),bz = (l/ar’)
el),
sin’(6 - O’),b; = -(l/a2r’2)
sin2(B
-
@),
sin 2(0 - 8’),
bz = cos3(8 - el), bg = (3/ar’) sin2(6’- 0’) cos(8 - O’), by,-,= -(3/a2r’2)
sin2(0 - 0’) cos(8 - O’),bf, = (3/ar’) sin(8 - 0’) cos2(8 - 0’)
bT2 = (3/a2r’2) sin(0 - 8’)[sin2(8 - 0’) - 2 cos2(B - O’)], bT3 = (3/a2r’2) sin2(6 - 13’cos(8 ) - O’),bT4 = (2/a3r’3) sin(0 - B’)[3cos2(8 - @)- sin2(f3-@)I, bT5 = (- 6/a3r’3) sin2(8 - 0) cos(8 - O’),bT6 = (l/a3r’3) sin3(8 - O’),
e:
= -ar sin(8 - el), er = (r/r’) cos(8 - CT),ez = (r/r’)2 cos2(B - el),
e*4
= -ar(r/r’)sin
e6* =
2(8 - el),ez = (r/r’) sin(8 - @)[2(r/r’) cos(8 - &) - 11,
ar cos(8 - @)[(r/r’) cos(0 - el) - 11, e: = a2r2 sin2(8 - el),
e;t = (l/ar’)(r/r’)2 sin(0 - @)cos2(B - el), eX = (l/ar’)(r/r’)[2cos2(0
- el) - 2(r/r’) cos3(0 - el) + 4(r/r’) sin2(8 - &)
x cos(6 - el) - sin2(8 - P)], er, = (r/r’)2 cos(8 - 8’)[cos2(8 - @) - 2sin2(8 - @)I, e:, = (r/r’) sin(8 - e’)[ - 6~0.548- 6’) + 7(r/r’) cos2(fI - 0’) - 2(r/r’) sin2(8 - @‘)I, eT2 = ar(r/r’) sin(B - e’)[ - 2cos2(8 - fY) + sin2(8 - &)I, eT3 = a2r2 sin2(6 - el) cos($ - 19h ey, = -ar[cos2(8 e:,=
-cos(8-8’)-(/)
- @) - 2sin2(0 - e’)][l - (r/r’) cos(8 - el)], r r’ sin2(8 - I?) + 2(r/r’) cos’(8 - el) - (r/r’)2 cos3(8 - tY)
+ 2(r/r’)2 sin2(8 - V) cos(8 - @),
eF6 = (l/ar’) sin(0 - 0’)[6(r/r’) cos(B - 0’) - 6(r/r’)2 cos2(B - el) + 2(r/r’)2 sin2(8 - @) - 11,
f: = bg + (2 - v)eT3/a2r2, f t = b,* + bt/ar - (3 - v)eq/a3r3 + (2 - v)erJa2r2, f:
= b:, - bl/a2r2 + bX/ar - (3 - v)eg*/a3r3 + (2 - v)et5/a2r2,
fz = b:, + (2 - v)eT2/a2r2, f* 5 = br2 + b:/ar - (3 - v)et/a”r” + (2 - v)eTI/a2r2, f E = bT3 + (2 - v) e:,,/a’r’, f: = bT4 - b;/a2r2 + b:/ar - (3 - v)e:/a3r3 + (2 - v)e:,la2r2, fg = by5 + bEjar - (3 - v)ez/a3r3 + (2 - v)ez/a2r2, fz = by, + (2 - v)eE/a2r2.
(17)
ECCENTRIC
ANNULAR VISCOELASTIC
PLATE
79
The expressions for j = 2 have the same forms as equations (16), but cos no’is replaced by sin nB' and also sin ne’isreplaced by - cos nfl'.In the author’s previous report [2], the bending moment and the shearing force were obtained by neglecting :he terms atI’/& and M/k%. If these terms are neglected and with B(s) replaced by ~3 in equations (15) and (16), the bending moment and the shearing force for the elastic plate reduce to the author’s previous results. The outer boundary conditions cannot be satisfied directly, and hence Fourier expansion is applied to the expressions for the outer boundary conditions. The boundary conditions therefore are expressed, by using the symmetry, as
where$, ,,,= cos m0 and (I/,,= sin me, and slj““, = (2/z)
Silj, - TfL are as follows:
“[{J”(W + ~j”In(af)+.?il”Kn(~(r’))~j”],=R~j,“de, s0
” = (2!11) “[{Y~(rr’) + Y~.I~(c~J)+ ;(2nKn(‘Y’)}~jn]r=Rl(/jl,ld’, s “ill s0
+y4,Xfn+ y2nX;,,)r__R$j,, d6’ T:,!,= (2/7c)“(x;n
(19)
s0
for the clamped edge,
(20) for the free edge, and
S” “,” = (2/e)
“[{J”(d) + Y3”I,,W’)+ j,l”K”(ar’)}~jnl*=R~j,” de, s0
s2j “,,,= (2/7r) “[{ x(ar’) + Y,&ar’) +
s0
i’z”K”(ar’)j~j”I,=R~j,” de,
80
K. NAGAYA
for the simply supported edge. The integration in equations (19), (20) and (21) cannot be performed analytically ,and hence the integration is performed numerically. The frequency equation is obtained from equations (18) for the symmetric modes with respect to the x-axis by taking j = 1, and when the terms n and m are truncated to N + 1, one obtains
(22)
The equation for the antisymmetric modes with respect to the x-axis can also be obtained bytakingj = 2,n,m = 1,2,3 ,..., N. In this case the following substitutions should be made in equations (16): cos n0’ + sin ntl’and sin nB’+ - cos ntJ’. The natural frequencies therefore can be evaluated from equation (22) for various combinations of the outer and the inner boundary conditions. In vibration problems of viscoelastic plates, since the roots of the frequency equation have complex forms in the complex s-plane, ofone puts s = io, and A* = ~(phR~/@“~, which is the root of the frequency equation, then o represents the natural frequency in a complex form and 1* represents the nondimensional complex frequency. Therefore the logarithmic decrement 6 of the first mode can be given by 6 = 2nB,Q2,, where 52, represents the real part and Sz,the imaginary part of the frequency A* of the first mode. The imaginary parts Q, of the frequencies vanish, of course, in the case of elastic plates.
3. NUMERICAL
RESULTS
In this numerical calculation, a three-element standard viscoelastic solid as shown in Figure 2 is considered. The coefficients p[, ql, pi and q; are readily obtained in the same way as in references [6,7]. If the Poisson’s ratio is assumed to be a constant [8,9], the flexural rigidity of the viscoelastic plate is given by D(s) =:
9G(l + t&h3 g,[iA*(3s + t&i) + 3 + 9(1 + iA*t;) 12(1 - v2) [s(3t, + t,g,) + 3 + 911 Or i =
s,l ’
(23)
where t, = cl/El, t, = & =
[(El + E,)/E,](c,/E,), g1
(cl/El)(G9/phR4p2,t; = e(E1 + E,)/E,,
=
G/K 33 = E,,
g2 =
E,IG
A* = -is(phR4/9)‘12,
E, and E, are the spring constants, c1 is the coefficient of viscosity of the viscoelastic model, G is the shear modulus and K is the bulk modulus. The parameter 54 is introduced to obtain
the non-dimensional
frequency parameter A* as mentioned before, and is selected as CS=
81
ECCENTRIC ANNULAR MSCOELASTIC PLATE
Figure 2. Three-element standard viscoelastic solid.
E,h3/12(1 - v’) for the three-element standard viscoelastic solid. The modulus shown in equation (23) is found from the three dimensionaltheory of elasticity, and there exists a little discrepancy between the result of this analysis for c1 = 0 and that of the thin elastic plate theory. However if one puts [ = 1, the results for the elastic plate can be evaluated directly by using this analysis. The convergence of the Fourier series is good, and n and m are taken up to 6 in this numerical calculation. Tables l-7 show the relation between the fundamental non-dimensional natural frequency a*[ = $7 = (phR4~2/9)1’4] of symmetric modes of the elastic plate and the nondimensional eccentricity d/R with c,, = 0, c1 = 0 and v = 0.3, for various combinations of TABLE
1
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with fixed outside and fixed inside edges 4R alR
0
@1
0.2
03
0.4
@5
0.6
0.1 0.2 0.3
5.220 5.885 6.730
4900 5.432 6.092
4.551 4.986 5.514
4.264 4.615 5.041
4.014 4.307 4.658
3.807 4.049 4.338
3.63 I 3.830 4.012
0.5
9.445
8.079
7.024
6.223
5.604
5.107
TABLE 2
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with j?xed outside and free inside edges for v = 0.3
4R alR
0
@I
0.2
0.3
0.4
0.5
0.6
0.1
3.187 3.227 3.380 4.205
2.189 3.227 3.363 3.977
3.190 3.226 3.321 3.701
3.195 3.217 3.214 3.485
3.199 3.209 3.227 3.330
3.199 3,199 3.185 3.216
3.199 3.185 3.152 --
0.2 0.3 0.5
82
K. NAGAYA TABLE 3
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with fixed outside and simply supported inside edges for v = 03
4R
0
0
0.1
Q2
0.3
0.4
0.5
0.6
0.1
4.758 5.173 5.805 7.997
4.540 4.863 5.351 6.934
4.259 4.512 4.891 6.078
4.015 4.216 4.516 5.434
3.809 3.968 4.207 4.934
3.635 3.762 3.954 4.534
3.490 3.593 3.743
0.2 0.3 0.5
TABLE 4
Fundamental non-dimensional natural frequencies a* for a circular elastic plate with simply supported outside and fixed inside edges for v = 0.3
dlR 0
0
0.1
0.2
0.3
0.4
0.5
@6
0.1 02 03 0.5
4,223 4.769 5.468 7,733
3.973 4.421 4.978 6.645
3.692 4.059 4.509 5.777
3448 3.753 4.122 5.121
3.236 3.497 3.797 4.605
3.055 3.272 3.528 4.191
2.895 3.084 3.297
TABLE 5
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with simply supported outside and simply supported inside edges for v = @3
W 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.1
3.812 4.096 4.598 6.324
3.641 3.879 4.267 5,536
3.418 3.605 3.911 4.867
3.214 3.371 3.608 4.352
3.036 3.160 3.357 3.948
2.882 2.988 3.146 3.627
2.746 2.832 2.966 ~
0.2 0.3 0.5
TABLE 6
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with simply supported outside and free inside edges for v = 03
dlR alR
0
0.1
0.2
0.3
0.4
0.5
0.6
01 @2 0.3 0.5
2.203 2.173 2.160 2.254
2204 2173 2158 2.242
2.204 2.173 2.158 2.214
2.204 2.177 2.158 2.177
2207 2.180 2.155 2.136
2.210 2.183 2.154 2.105
2221 2186 2.154
ECCENTRIC
ANNULAR
VISCOELASTIC
TABLE
83
PLATE
7
Fundamental non-dimensional naturalfrequencies a* of a circular elastic plate withfree outside andfixed inside edges for v = 0.3
dlR alR
0
0.1
0.2
0.3
0.4
05
0.6
@I
I.865 2.194 2.515 3644
1.770 2.022 2.299 3.101
I .634
1512 1.685 I.873 2.374
1.399 1 .I549 I.709 2.123
1,300 1,432 1.572 I.918
I.212 I.329
0.2 0.3 0.5
I.840 2.068 2.693
1.45I
the outer and the inner boundary conditions. Figure 3 shows the frequency versus d/R for an elastic plate with clamped inside and free outside edges with a/R = 0.3. In :he figure, s, - S, show the symmetric vibrations for first - third modes, and A,, A, show the antisymmetric modes. The solid line is the result obtained from this analysis and the dashed line is that from the approximate analysis [2]. It can be seen that the discrepancy between the present and the previous approximate results is small, and hence the approximation in the previous paper is valid in practical use. The effects of the eccentricity on the natural frequencies are small for small values of a/R (a/R = 0.1) in the case of a clamped circular plate with a circular hole (Table 2) as was found by Maruyama and Ichinomiya [3]. However, as the ratio a/R increases (a/R = O-5),the frequency varies significantly, and the effects of the eccentricity therefore cannot be neglected for large values of a/R. Especially when the inner edge is a clamped or a supported boundary, these effects cannot be neglected even if the plate
4
I
I
I
I
I
1
2.58 2.52
*CI
0
I
I 0.1
I
I
I
I
02
0.3
0.4
0.5
0.6
d/R
Figure 3. Non-dimensional natural frequencies a* versus eccentricities d/R for an elastic plate having a free circular outside and a fixed circular inside edge with n/R = 0.3 and v = 0.3. --, Present analysis: ---, approximate analysis.
84
K. NAGAYA
has a small eccentric inner boundary. Therefore it can be concluded that the effects of the eccentricity on the frequency increases as the rigidity of the inner edge increases, and in general, those effects cannot be neglected. Figure 4 shows the logarithmic decrement 6 versus the viscous parameter E with a/R = @3 (E, + E,)/E, = l-01, c0 = 0 and v = 0.33, for a clamped circular viscoelastic plate with an eccentric clamped inside edge. It is observed that the location of the maximum value of the logarithmic decrement moves with d/R although the maximum values are almost the same for various values of d/R.
Figure 4. Logarithmic decrements 6 versus viscous parameters E for a viscoelastic plate having a lixed circular outside edge and a fixed circular inside edge with a/R = @3, (E, + E&E1 = 1.01, c0 = 0 and v = 0.33.
4. CONCLUSIONS
In this paper, a method for solving vibration problems of a viscoelastic plate having a circular outer boundary and an eccentric circular inner boundary has been presented. The frequency equation of the plate has been obtained, and numerical calculations have been carried out for the plate with various boundary conditions. It is concluded that the effects of the eccentricity of the inner edge on the natural frequency increase as the rigidity of the inner boundary becomes large, and in general, these effects cannot be neglected.
ACKNOWLEDGMENT
The author wishes to thank heartily Professor Y. Hirano, Yamagata University, for his kind guidance in the present work.
ECCENTRIC ANNULAR VISCOELASTIC PLATE
85
REFERENCES 1.
2. 3.
4. 5. 6. 7.
8. 9.
H. SAITO and F. OHTSUGA 1973 Preprint in the Meeting of Japan Society of Mechanical Engineer. Yonezawa, No. 731-1, 5-8, (in Japanese). Vibration analysis of a circular plate with eccentric circular holes by the finite element method. K. NAGAYA 1977 Journal of Applied Mechanics 44, 165-166. Transverse vibration of a plate having an eccentric inner boundary. K. MARUYAMA and 0. ICHINOMIYA1976 Preprint in the Meeting of Japan Society oj Mechanical Engineer Nagoya No. 76G14, 137-140 (in Japanese). Vibration of a clamped circular plate with an eccentric circular hole. H. H. PAN 1966 Journal de M&unique 5, 355-374. Vibration of viscoelastic plates. S. TIMOSHENKOand WOINOWSKY-KRIEGER 1959 Theory of Plate and Shells. New York : McGrawHill. See pp. 282-284. T. C. HUANG and C. C. HUANG 1971 Journal of Applied Mechanics 38, 515-521. Free vibration of viscoelastic Timoshenko beams. K. NAGAYA 1977 Journal of Engineering for Industry Transaction of the American Society of Mechanical Engimers Series B 994041109. Vibrations and dynamic response of viscoelastic plates on nonperiodic elastic supports. S. R. ROBERTSON1971 Journal of Soundand Vibration 14,263-278. Solving the problem of forced motion of viscoelastic plates by Valanis’ method with an application to a cricular plate. S. R. ROBERTSON1971 Journalof Soundand Vibration 17.363-381. Forced axis! mmetric motion of circular viscoelastic plates.
of Sound and Vibration (1979) 63( I), 73-85
VIBRATION
OF AVISCOELASTIC
PLATF HAVING A CIRCULAR
OUTER BOUNDARY AND AN ECCENTRIC CIRCULAR INNER BOUNDARY FOR VARIOUS EDGE CONDITIONS K. NAGAYA Department of Mechanical Engineering, Faculty of Engineering, Yamagata Universit?,, Yonezawa, Japan (Received 20 March 1978, and in revisedform 7 September 1978)
In this paper, vibration problems of a circular viscoelastic plate having an eccentric circular inner edge are investigated. The frequency equations in complex forms for various edge conditions are obtained. Numerical calculations are carried out for both elastic and viscoelastic plates, and the non-dimensional natural frequencies and the logarithmic decrements are given for a number of cases.
1. INTRODUCTION
A circular plate having an eccentric inner edge is an important element of machines and structures, such plates being widely used in many branches of engineering. The dynamic behaviour of this type of plate therefore needs to be understood, as well as that of a plate with a concentric inner edge. These problems however have not been thoroughly discussed although a few theoretical [l, 21 and experimental [3] results for elastic plates have been reported. In these studies the effect of the eccentricity of the inner edge on the natural frequencies of the plate has been discussed in references [2] and [3]. Maruyama and Ichinomiya [3] investigated an elastic plate with fixed circular outside and free circular inside edges in their experiments, and they concluded that the effect of the eccentricity can be neglected from an engineering point of view. However, in the author’s previous paper [2] concerning an elastic plate with free outside and fixed inside edges it was reported that the effect of the eccentricity is large and cannot be neglected in general. These discrepancies are due to the differences between the boundary conditions. In this paper the vibrations of a circular viscoelastic plate with an eccentric circular inner edge are discussed for various boundary conditions. An analytical method for solving the problems has been previously given by the author [2]. In that study the terms having small influence on the natural frequencies were neglected in treating the outer boundary conditions. In this paper the transformed boundary conditions are given in exact forms, and the influence of neglecting terms is investigated. The exact solution is obtained by applying the Laplace transform to the equation of motion for the viscoelastic plate. The boundary conditions along the outer edge of the plate are satisfied by means of the Fourier expansion method.
2. ANALYSIS A viscoelastic plate with a circular outer boundary and an eccentric circular inner boundary of an eccentricity d is shown in Figure 1, in which the origin 0 of Cartesian co-ordinates x and y is chosen in the center of the plate, and the z axis is normal to the plate surface. The polar co-ordinates r, 8 and r’, 8’ are taken as shown in the figure, and the radii of the
73 0022-460X/79/050073+ 13$02.00/O
01979 Academic Press Inc. (London) Limited
74
K. NAGAYA
outer edge, r, and the inner edge, r’, are denoted by R and a, respectively. If one denotes the displacement of the plate in the z direction by w, the equation of motion for the viscoelastic plate, in the co-ordinates I’ and 8’, is [4]
[(2PQ' + P'Q)Q(h3/12)V2V2 + (PQ'+ 2P'Q)Pph a2/at2 + (PQ’ + 2P’Q)Pc$/&]w
= 0 (I)
where P = 5
p, d’ldt’, p’ = ;
I=0
p; d’ldt’, Q =
I=0
i q, d’ldt’, Q’ = t q; d’/dt’, I=0 I -- 0
(2)
V2 is the two-dimesnional Laplacian operator, p is the mass density, h is the thickness, co is the coefficient of the external viscou’s damping, t is the time and pl, q,, pi and q[ are constant
Figure 1. Geometry ofa viscoelastic plate with a circular outer boundary and an eccentric circular inner boundary.
coefficients describing the properties of the materials. [(2PQ’ + P’Q)Q/(PQ’ + 2QP')] x (h3/12P) (denoted by D(t) later) corresponds to the flexural rigidity 9 = Eh3/12( 1 - v2) in the elastic plate, where E is the Young’s modulus and v is the Poisson’s ratio of the elastic plate. The application of the Laplace transformation f(r’, 8’, s) =
m
f(r’, 0’, t) e-“’ dt
(3)
s0
to equation (1) turns D(t) into b(s). To identify the frequency parameter 1, the flexural rigidity 9 of an elastic plate which has the same value as D(t), but in which the effects of viscous damping are omitted, is introduced in this analysis. The solution of the transformed viscoelastic plate equation is obtained, by using the transformed flexural rigidity D(s), as W = (l/S’) i j=l
f
Fn[AjnJn(tlr’)+ BjnY,(ar’) + CjJn(ar’) + Dj,,K”(ar’)]djjn,
(4)
n=O
where E,, = l/2 for n = 0 and E,, = 1 for n 2 1, aIn= cos nel, Qp,”= sin nt?, a4 = (A25 - c,s&3),
2 = -is(ph/S?)f,
[ = ~/D(S)
(5)
75
ECCENTRIC ANNULAR VISCOELASTIC PLATE
and i = m The coefficients A,,Bi,, Cjn and Djn are constants of integration to be determined from the boundary conditions, J”(ar’) and Y,(ar’) are the complex Bessel functions of first and second kinds of order n, and I”(ar’) and K,(ar’) are the modified Bessel functions. The boundary conditions along the inner edge are (G), = D = (&/lar’)l. = II = for the clamped edge, (H,,),, = (1= (TV),,= (I = 0 for the free edge, (W),, _- a = (RI.)*. = a = 0 for the simply supported edge
(6)
where Mu, is the bending moment and V,.is the Kirchhoff shear in the plate [S]. Substituting equations (4) into equations (6) and eliminating the constants Cjn and D,,, one obtains w = (l/9)
i
5
j=l
sn[Ajn{Jn(ar’) + y&W)
+ ?,.K,(ar’)) + Bj,{Y,(ar’) + Y&(u~‘)
n=O
Y2nKnW)Il
+
(7)
@j+
where Yi, = 0’&” - j&)/F, ~4”
=
-
-
(y&z,
Y2”= (Yl”f& - Y,,il.)E
y,,k,,YF7F
=
ilnk2n
-
Y3”= - (il”k2, -j&,)/F (8)
klnL,
and the coefficients jln - k,, are given, for the clamped edge, by
’ = J,,@4jZn = Jn_ 1(au) - (n/au)J,(aa), yin = Y,(w),
Jl”
’ = 11”
Y,” = Yn _ ,(a4
-
Wa0’,,@4
I,(au), i,” = In _ 1(au) - (n/au)I,(au), k,” = Kn(au), k,, = - Kn _ ,(au) -(n/au) K,,(au),
(9)
for the free edge, by ’ = rnTn = [n(n + l)(l - v)/a2u2 - l]J,(au) - [(l - v)/aa]J”_ i(cru), Jl” Yl” = mm = [n(n + l)(l - v)/a2u2 - l]Y”(au) - [(l - v)/aa]Yn _ ,(aa), - v)/ a2u2 + l]I”(au) - [( 1 - v)/au]In _ ,(au),
’ = rn$ = [n(n + l)(l 5” k,” =mX,= [n(n + l)(l
- v)/a’a” + l]K”(au) + [(l - v)/aa]K” _ ,(au),
j2” = Cn2(n + l)(l - v)/ a3u3 + n/au]J”(au) - [n2(1 - v)/a2u2 + l]J, _ ,(~a),
y,” = [n2(n + 1) (1 - v)/a3u3 + n/au]Yn(au) - [n2(1 - v)/a2u2 + l]Y, _ ,(aa), ’ 12”
=
[n2(n
+
l)(l - v)/a3u3 - n/au]In(au) - [n2(1 - v)/a2u2 - l]I,_ ,(~a),
k,” = [n”(n + 1) (1 - v)/a3u3 - n/au]Kn(au) + [n2(l - v)/a’u’ -
l]K, _ ,@a), (IO)
and, for the simply supported edge, by pi, = Jn@4j2n = mFny Y,”
= Y.(m),
Y,” = mfn,iln
k,, = m4:.
= I&4,
i2n = m:,,
kin = Knb).
(11)
The boundary conditions along the outer edge are expressed by using equations (6) with is expressed in terms of the co-ordinates r’ and 6’, and hence it is convenient to treat the outer boundary conditions when the derivatives in the equations of the bending slope, bending moment and shearing force are transformed in terms of r’ and 0’. The transformed expressions are r’ replaced by r and a by R. The displacement
76
K. NAGAYA
aiqar Rr = -D(s)[(b,+
= b,aw/ar’ + b,aw/aw,
ve,/r2)a2i?/W2
+ ve,Jr2)&/iY
+ (b4 + vb,/r
+ (b, + ve,/r2) a2i’i+%‘%’ + (b6 + ve,/r2)t32w/jlde’2 + (b7 + vb,/r + ve,jr2)&/W], y = -B(S)
Gf,iT3i@r’3
+ f2a2i7/W2
+ fea3ic/ar’ae’2
f,ai@tJ +
+
+ f,aicqau
+ f,a%iqae’2
f,a3w/arJ2ae’
+ f5a2w,iar’ae
+ f,a%/ae’3],
(12)
where
b, =
b, = WJar, b, = (W/iYr)2, b, = a2r’/ar2, b, = 2(&‘/h)
W/i%,
b,
= (aO’/ar)*, b, = a2B’/ar2, b, = (&t/&)3,
b,,
= a3rr/ar3, b, 1 = 3(W/ar)’
b,,
= 3(W/ar)
(W/&)2,
b,,
b, = 3(&‘/h)
(d2&r2),
(LX?/&), b, 2 = 3[(a2r’/ar2)
= a3B’/ar3, b,,
= 3(W/ar)
(at?/&),
(a@/&+) + (W/h)
(aze;‘ar2),
b16 = (aO’/ar)3, e, = ar’/ae, e2 = ael/ae, e3 = (a@/atI)2, e4 = 2(aeyae)
e5 e9
= aWfae2, =
(a+/ae),
e6 = a2+/ae2, e7 = (ar’lae)2, es = (ae’pr) (aepep,
z(ael/ae)
(a28’/arae)
(aW/ar), e, o = (W/h)
+ (aVyae2)
(8x8)2 x
e, 1 = 2(aW/arae) e,, = 2(ae’/ae)
(a+/ae)
+ 2(aeyae)
(a2r’/ade)
e, h = a38’/hW2,fl
e, 5 = a3r’/arae2, b,/r
f3 = b,, -
- (3 -
(ar’/ar) + (a@/&) (@+/ae2),
+ z(w/ae)
(a2r’/arae),
= b, + (2 - v)e13/r2,
v)e7/r3 + (2 - v)e14/r2,
b,/r2 + b,/r - (3 - v)e,/r3 + (2 - v)e,,/r2,f,
f5
=b,,
+ b,/r - (3 - v)e,/r”
f7
=b,,
- bJr2
fs = b,,
+ (aW/ae’)
+ 2caeyae) (&‘/?e) (aV/ae2),
(art/a@ (ar’/ar) + (ar’/ae)2 (ae’pr),
e13 = (art/a@’ (ar’/ar), e14 = (W/&j (a2+/ae2)
f2 = b, +
(a2@/ar2)],
+ b,/r
+ (2 - v)eIl/r2,f6
One has the following
v)e,/r3
relations
+ (2 -
v)elo/r2,
v)e,/r” + (2 - v)e16/r2,
+ b7/r - (3 - (3 -
= b,,
= b, 1 + (2 - v)e12/r2,
+ (2 -
between
v)e,/r2,f9’=
b,,
the co-ordinates
+ (2 -
v)e,/r’.
(13)
r, 8 and r’, 8’:
r cos 8 + d = r’ cos 8’, r sin 0 = r’ sin 19, r’ = (r2 + 2rd cos 8 + d2)1’2, 8’ = cos- l[(r/r’) cos e + a/r’].
(14)
The coefficients b 1 - f, in equation s (13) are evaluated by using equations (14). The bending slope, the bending moment and the shearing force are then obtained from equations (12) (13) and (14): &$?r
= (E/9) f:
t
En[Ajn(Xji
+ Y3nXj” + Y,,X,~) + B,.(X,f
+ Y,,Xj,” + r,,xj~)],
E,[Aj,(mj;
+ Y3”Mj.j + Ylnmjf) + Bjn(mji + Y4nrnji + Y2,mj~119
j=ln=O RI = -(cr2/0.
9 = -Ca3/1)
i: f .i=ln=O i
f
j-r
n=o
En[Aj~(Uj~ + Y3n”jn3+ Yln’jz) + Bj”(Vj~ + Y4,‘j,3 + ‘,?“‘jt)]3
(15)
ECCENTRIC
ANNULAR VISCOELASTIC
PLATE
77
where
(c::)=b,* [(; ::I;;)- (n/at) (::)*k:l)]cos n@- nb,*kg)))sin I
X,2 = b,* [In Xl:
=
-
I(~~‘) - (n/ar’)In(ar’)]
m1f =
(m,nz )
[(b3* + ve7*/a2r2){(n2
x (n/ar’) cos n@ + (n*/ar’)(b,* -n(b,*
+ vb,*/ar +(b,*
cos nd’ - nb?*I”(ar’) sin d’,
l(ar’) + (n/ar’)K,(ar’)]cos
b,*[K,
n@ - nbz*Kn(d)sin
+ n)/(ar’)* - l> cos n$’ + ve,*/a2r2)sin
+ ve,*/cc2r2)sin
no’,
(b,* + vb,*/ar
+ [ - (b: + veT/a”r’) (I /ar’)cos nO
nO’]
ml;3 = [(bj + ve3/a2r2){(n2 + n)/(ar’)* +
+ veh*/a2r2)
nH’- n2(b,* + ve3*/a2r2) cos nt)
+ vb:far + vez/a2r2)cos n@ - rt(bS + ve$‘a2r2)sin
1 } cos
n@]
’
n@ - (bl + vb:Jar + veg/a2r2) (n/rr’pos n6’
+(n2/ar’) (b: + ve,*/a2r2)sin n@ - n2(b,* + veT/a’r’)cos x sin n@]I,,(ar’) + [-(b:
nt?,
nQ’- n(bT + vbz/ar + ve:idr2)
+ veT/a2r2) (l/ar’)cos n0’ + (bz + vbT/ar + veg/a*r’)cos n6 -n(bz
+ vez/a2r2)sin
l(ar’),
nB’]In
rn,: = [(bz + ve3/a2r2){(n2 + n)/(ar’)2+ I} cos n@ - (b,* + vb:/ar + vez/a2r2) (n/at) x cos nt?’+ (n2/ar’) (b: + veX/a2r2)sin no’ - n2(bz + ve:/ctzr2)cos n0 - n(b3 + vbT/ar + ve:/a2r2) sin nO’]Kn(ar’) + [(b: + veT/a2r2) (1jar’) cos no -(bX + vb:/ar + vez/a2r2) cos n@ + n(bz + veq*/r’r’) sin ne’]Kn _ l(xr’),
(
1 Ul” 2 = uln > cos
vF{
-
(n3 + 3n2
+
2n)/(ar’)3 + (n + l)/ar’} cos n@ + f,*((n2 + n)/(ar’)2 -- 1)
n@ - ,fT(n/ar’) cos n@ - nf,*{(n2 + n)/(ar’)2 - 1} sin nd’ + (,f5*n2/ar’) sin no’ +(f6*n3/ar’) cos ne - nf; sin n& - n’f:
cos no + n3fz sin nd]
:irr!) C n ‘)
+Cf;F{(n’ + 2)/(ar’)2 - 1) cos n@ - V;*/ar’) cos n@ + ,fT cos n8’ + (nfk*/ar’) sin nH - nf,* sin n@ - n”f,* cos ne’]
0,: = v,*{ -(n3 X COS n@ -
ar’ ’
+ 3n2 + 2n)/(ar’)3 - (n + l)/ar’} cos nf7 +fi*{(n2
+ n)/(ar’)2 + 1 :
Cf,*n/ar’) COS n@ - nf,*{(n’ + n)/(ar’)* + 1) sin nl? + (&*n*/ar’) sin n@
+Cf$n3/ar’) + Cf:{(n2
tar’)
(::1:t ,!
COS
n@ - nf; sin nt7 - nff,* cos nfY + n”f,* sin nO’]In(ar’)
+ 2)/(ar’)2 + 1) cos n@ - (f2*/ar’) cos n@ + j: - nf: sin nB’ - n”fg” cos nellIn
cos nt7 + V;*n/ar’) sin nt)
1(ar’),
78 u,f
K. NAGAYA =
rT{--(n3
X COS n8’
-
+
(f~qhr’)
3n2 + 2n)/(ar’)3 - (n + l)/crr’} cosn@ +f:{(n’ c0S n&
-
nf4*{(n2
+(f,*n3/ar’) cos n0’ - nf:
+
sin nf3’ -
+ n)/(ar’)2 + Ii
n)/(ar’)2 + l} sin rz@+ Qn2/ar’)
sin
n0’
n’fff cos n0’ + n”fc sin n@]K,,(ar’)
+ [ -f;“{(n” + 2)/(ar’)2 + 1} cos n@ + (f;c/ar’) cos n@ - f3* cos n@ - (fk*n/ar’) sin ntI +nfz sin n0’ + n’fz cos nO’]Kn _ I(ar’).
(16)
In equations (16), b:, bz, . . . are the non-dimensional values which correspond to b,, b,, . . . in equations (13), and these are expressed as follows : br
=
b:
= (l/ar’) sin 2(0 - B’),b,* = (lja’r”)
cos(8
-
b: = (l/ar’) sin(8 - O’),b: = cos2(8 - O’),bz = (l/ar’)
el),
sin’(6 - O’),b; = -(l/a2r’2)
sin2(B
-
@),
sin 2(0 - 8’),
bz = cos3(8 - el), bg = (3/ar’) sin2(6’- 0’) cos(8 - O’), by,-,= -(3/a2r’2)
sin2(0 - 0’) cos(8 - O’),bf, = (3/ar’) sin(8 - 0’) cos2(8 - 0’)
bT2 = (3/a2r’2) sin(0 - 8’)[sin2(8 - 0’) - 2 cos2(B - O’)], bT3 = (3/a2r’2) sin2(6 - 13’cos(8 ) - O’),bT4 = (2/a3r’3) sin(0 - B’)[3cos2(8 - @)- sin2(f3-@)I, bT5 = (- 6/a3r’3) sin2(8 - 0) cos(8 - O’),bT6 = (l/a3r’3) sin3(8 - O’),
e:
= -ar sin(8 - el), er = (r/r’) cos(8 - CT),ez = (r/r’)2 cos2(B - el),
e*4
= -ar(r/r’)sin
e6* =
2(8 - el),ez = (r/r’) sin(8 - @)[2(r/r’) cos(8 - &) - 11,
ar cos(8 - @)[(r/r’) cos(0 - el) - 11, e: = a2r2 sin2(8 - el),
e;t = (l/ar’)(r/r’)2 sin(0 - @)cos2(B - el), eX = (l/ar’)(r/r’)[2cos2(0
- el) - 2(r/r’) cos3(0 - el) + 4(r/r’) sin2(8 - &)
x cos(6 - el) - sin2(8 - P)], er, = (r/r’)2 cos(8 - 8’)[cos2(8 - @) - 2sin2(8 - @)I, e:, = (r/r’) sin(8 - e’)[ - 6~0.548- 6’) + 7(r/r’) cos2(fI - 0’) - 2(r/r’) sin2(8 - @‘)I, eT2 = ar(r/r’) sin(B - e’)[ - 2cos2(8 - fY) + sin2(8 - &)I, eT3 = a2r2 sin2(6 - el) cos($ - 19h ey, = -ar[cos2(8 e:,=
-cos(8-8’)-(/)
- @) - 2sin2(0 - e’)][l - (r/r’) cos(8 - el)], r r’ sin2(8 - I?) + 2(r/r’) cos’(8 - el) - (r/r’)2 cos3(8 - tY)
+ 2(r/r’)2 sin2(8 - V) cos(8 - @),
eF6 = (l/ar’) sin(0 - 0’)[6(r/r’) cos(B - 0’) - 6(r/r’)2 cos2(B - el) + 2(r/r’)2 sin2(8 - @) - 11,
f: = bg + (2 - v)eT3/a2r2, f t = b,* + bt/ar - (3 - v)eq/a3r3 + (2 - v)erJa2r2, f:
= b:, - bl/a2r2 + bX/ar - (3 - v)eg*/a3r3 + (2 - v)et5/a2r2,
fz = b:, + (2 - v)eT2/a2r2, f* 5 = br2 + b:/ar - (3 - v)et/a”r” + (2 - v)eTI/a2r2, f E = bT3 + (2 - v) e:,,/a’r’, f: = bT4 - b;/a2r2 + b:/ar - (3 - v)e:/a3r3 + (2 - v)e:,la2r2, fg = by5 + bEjar - (3 - v)ez/a3r3 + (2 - v)ez/a2r2, fz = by, + (2 - v)eE/a2r2.
(17)
ECCENTRIC
ANNULAR VISCOELASTIC
PLATE
79
The expressions for j = 2 have the same forms as equations (16), but cos no’is replaced by sin nB' and also sin ne’isreplaced by - cos nfl'.In the author’s previous report [2], the bending moment and the shearing force were obtained by neglecting :he terms atI’/& and M/k%. If these terms are neglected and with B(s) replaced by ~3 in equations (15) and (16), the bending moment and the shearing force for the elastic plate reduce to the author’s previous results. The outer boundary conditions cannot be satisfied directly, and hence Fourier expansion is applied to the expressions for the outer boundary conditions. The boundary conditions therefore are expressed, by using the symmetry, as
where$, ,,,= cos m0 and (I/,,= sin me, and slj““, = (2/z)
Silj, - TfL are as follows:
“[{J”(W + ~j”In(af)+.?il”Kn(~(r’))~j”],=R~j,“de, s0
” = (2!11) “[{Y~(rr’) + Y~.I~(c~J)+ ;(2nKn(‘Y’)}~jn]r=Rl(/jl,ld’, s “ill s0
+y4,Xfn+ y2nX;,,)r__R$j,, d6’ T:,!,= (2/7c)“(x;n
(19)
s0
for the clamped edge,
(20) for the free edge, and
S” “,” = (2/e)
“[{J”(d) + Y3”I,,W’)+ j,l”K”(ar’)}~jnl*=R~j,” de, s0
s2j “,,,= (2/7r) “[{ x(ar’) + Y,&ar’) +
s0
i’z”K”(ar’)j~j”I,=R~j,” de,
80
K. NAGAYA
for the simply supported edge. The integration in equations (19), (20) and (21) cannot be performed analytically ,and hence the integration is performed numerically. The frequency equation is obtained from equations (18) for the symmetric modes with respect to the x-axis by taking j = 1, and when the terms n and m are truncated to N + 1, one obtains
(22)
The equation for the antisymmetric modes with respect to the x-axis can also be obtained bytakingj = 2,n,m = 1,2,3 ,..., N. In this case the following substitutions should be made in equations (16): cos n0’ + sin ntl’and sin nB’+ - cos ntJ’. The natural frequencies therefore can be evaluated from equation (22) for various combinations of the outer and the inner boundary conditions. In vibration problems of viscoelastic plates, since the roots of the frequency equation have complex forms in the complex s-plane, ofone puts s = io, and A* = ~(phR~/@“~, which is the root of the frequency equation, then o represents the natural frequency in a complex form and 1* represents the nondimensional complex frequency. Therefore the logarithmic decrement 6 of the first mode can be given by 6 = 2nB,Q2,, where 52, represents the real part and Sz,the imaginary part of the frequency A* of the first mode. The imaginary parts Q, of the frequencies vanish, of course, in the case of elastic plates.
3. NUMERICAL
RESULTS
In this numerical calculation, a three-element standard viscoelastic solid as shown in Figure 2 is considered. The coefficients p[, ql, pi and q; are readily obtained in the same way as in references [6,7]. If the Poisson’s ratio is assumed to be a constant [8,9], the flexural rigidity of the viscoelastic plate is given by D(s) =:
9G(l + t&h3 g,[iA*(3s + t&i) + 3 + 9(1 + iA*t;) 12(1 - v2) [s(3t, + t,g,) + 3 + 911 Or i =
s,l ’
(23)
where t, = cl/El, t, = & =
[(El + E,)/E,](c,/E,), g1
(cl/El)(G9/phR4p2,t; = e(E1 + E,)/E,,
=
G/K 33 = E,,
g2 =
E,IG
A* = -is(phR4/9)‘12,
E, and E, are the spring constants, c1 is the coefficient of viscosity of the viscoelastic model, G is the shear modulus and K is the bulk modulus. The parameter 54 is introduced to obtain
the non-dimensional
frequency parameter A* as mentioned before, and is selected as CS=
81
ECCENTRIC ANNULAR MSCOELASTIC PLATE
Figure 2. Three-element standard viscoelastic solid.
E,h3/12(1 - v’) for the three-element standard viscoelastic solid. The modulus shown in equation (23) is found from the three dimensionaltheory of elasticity, and there exists a little discrepancy between the result of this analysis for c1 = 0 and that of the thin elastic plate theory. However if one puts [ = 1, the results for the elastic plate can be evaluated directly by using this analysis. The convergence of the Fourier series is good, and n and m are taken up to 6 in this numerical calculation. Tables l-7 show the relation between the fundamental non-dimensional natural frequency a*[ = $7 = (phR4~2/9)1’4] of symmetric modes of the elastic plate and the nondimensional eccentricity d/R with c,, = 0, c1 = 0 and v = 0.3, for various combinations of TABLE
1
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with fixed outside and fixed inside edges 4R alR
0
@1
0.2
03
0.4
@5
0.6
0.1 0.2 0.3
5.220 5.885 6.730
4900 5.432 6.092
4.551 4.986 5.514
4.264 4.615 5.041
4.014 4.307 4.658
3.807 4.049 4.338
3.63 I 3.830 4.012
0.5
9.445
8.079
7.024
6.223
5.604
5.107
TABLE 2
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with j?xed outside and free inside edges for v = 0.3
4R alR
0
@I
0.2
0.3
0.4
0.5
0.6
0.1
3.187 3.227 3.380 4.205
2.189 3.227 3.363 3.977
3.190 3.226 3.321 3.701
3.195 3.217 3.214 3.485
3.199 3.209 3.227 3.330
3.199 3,199 3.185 3.216
3.199 3.185 3.152 --
0.2 0.3 0.5
82
K. NAGAYA TABLE 3
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with fixed outside and simply supported inside edges for v = 03
4R
0
0
0.1
Q2
0.3
0.4
0.5
0.6
0.1
4.758 5.173 5.805 7.997
4.540 4.863 5.351 6.934
4.259 4.512 4.891 6.078
4.015 4.216 4.516 5.434
3.809 3.968 4.207 4.934
3.635 3.762 3.954 4.534
3.490 3.593 3.743
0.2 0.3 0.5
TABLE 4
Fundamental non-dimensional natural frequencies a* for a circular elastic plate with simply supported outside and fixed inside edges for v = 0.3
dlR 0
0
0.1
0.2
0.3
0.4
0.5
@6
0.1 02 03 0.5
4,223 4.769 5.468 7,733
3.973 4.421 4.978 6.645
3.692 4.059 4.509 5.777
3448 3.753 4.122 5.121
3.236 3.497 3.797 4.605
3.055 3.272 3.528 4.191
2.895 3.084 3.297
TABLE 5
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with simply supported outside and simply supported inside edges for v = @3
W 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.1
3.812 4.096 4.598 6.324
3.641 3.879 4.267 5,536
3.418 3.605 3.911 4.867
3.214 3.371 3.608 4.352
3.036 3.160 3.357 3.948
2.882 2.988 3.146 3.627
2.746 2.832 2.966 ~
0.2 0.3 0.5
TABLE 6
Fundamental non-dimensional natural frequencies a* of a circular elastic plate with simply supported outside and free inside edges for v = 03
dlR alR
0
0.1
0.2
0.3
0.4
0.5
0.6
01 @2 0.3 0.5
2.203 2.173 2.160 2.254
2204 2173 2158 2.242
2.204 2.173 2.158 2.214
2.204 2.177 2.158 2.177
2207 2.180 2.155 2.136
2.210 2.183 2.154 2.105
2221 2186 2.154
ECCENTRIC
ANNULAR
VISCOELASTIC
TABLE
83
PLATE
7
Fundamental non-dimensional naturalfrequencies a* of a circular elastic plate withfree outside andfixed inside edges for v = 0.3
dlR alR
0
0.1
0.2
0.3
0.4
05
0.6
@I
I.865 2.194 2.515 3644
1.770 2.022 2.299 3.101
I .634
1512 1.685 I.873 2.374
1.399 1 .I549 I.709 2.123
1,300 1,432 1.572 I.918
I.212 I.329
0.2 0.3 0.5
I.840 2.068 2.693
1.45I
the outer and the inner boundary conditions. Figure 3 shows the frequency versus d/R for an elastic plate with clamped inside and free outside edges with a/R = 0.3. In :he figure, s, - S, show the symmetric vibrations for first - third modes, and A,, A, show the antisymmetric modes. The solid line is the result obtained from this analysis and the dashed line is that from the approximate analysis [2]. It can be seen that the discrepancy between the present and the previous approximate results is small, and hence the approximation in the previous paper is valid in practical use. The effects of the eccentricity on the natural frequencies are small for small values of a/R (a/R = 0.1) in the case of a clamped circular plate with a circular hole (Table 2) as was found by Maruyama and Ichinomiya [3]. However, as the ratio a/R increases (a/R = O-5),the frequency varies significantly, and the effects of the eccentricity therefore cannot be neglected for large values of a/R. Especially when the inner edge is a clamped or a supported boundary, these effects cannot be neglected even if the plate
4
I
I
I
I
I
1
2.58 2.52
*CI
0
I
I 0.1
I
I
I
I
02
0.3
0.4
0.5
0.6
d/R
Figure 3. Non-dimensional natural frequencies a* versus eccentricities d/R for an elastic plate having a free circular outside and a fixed circular inside edge with n/R = 0.3 and v = 0.3. --, Present analysis: ---, approximate analysis.
84
K. NAGAYA
has a small eccentric inner boundary. Therefore it can be concluded that the effects of the eccentricity on the frequency increases as the rigidity of the inner edge increases, and in general, those effects cannot be neglected. Figure 4 shows the logarithmic decrement 6 versus the viscous parameter E with a/R = @3 (E, + E,)/E, = l-01, c0 = 0 and v = 0.33, for a clamped circular viscoelastic plate with an eccentric clamped inside edge. It is observed that the location of the maximum value of the logarithmic decrement moves with d/R although the maximum values are almost the same for various values of d/R.
Figure 4. Logarithmic decrements 6 versus viscous parameters E for a viscoelastic plate having a lixed circular outside edge and a fixed circular inside edge with a/R = @3, (E, + E&E1 = 1.01, c0 = 0 and v = 0.33.
4. CONCLUSIONS
In this paper, a method for solving vibration problems of a viscoelastic plate having a circular outer boundary and an eccentric circular inner boundary has been presented. The frequency equation of the plate has been obtained, and numerical calculations have been carried out for the plate with various boundary conditions. It is concluded that the effects of the eccentricity of the inner edge on the natural frequency increase as the rigidity of the inner boundary becomes large, and in general, these effects cannot be neglected.
ACKNOWLEDGMENT
The author wishes to thank heartily Professor Y. Hirano, Yamagata University, for his kind guidance in the present work.
ECCENTRIC ANNULAR VISCOELASTIC PLATE
85
REFERENCES 1.
2. 3.
4. 5. 6. 7.
8. 9.
H. SAITO and F. OHTSUGA 1973 Preprint in the Meeting of Japan Society of Mechanical Engineer. Yonezawa, No. 731-1, 5-8, (in Japanese). Vibration analysis of a circular plate with eccentric circular holes by the finite element method. K. NAGAYA 1977 Journal of Applied Mechanics 44, 165-166. Transverse vibration of a plate having an eccentric inner boundary. K. MARUYAMA and 0. ICHINOMIYA1976 Preprint in the Meeting of Japan Society oj Mechanical Engineer Nagoya No. 76G14, 137-140 (in Japanese). Vibration of a clamped circular plate with an eccentric circular hole. H. H. PAN 1966 Journal de M&unique 5, 355-374. Vibration of viscoelastic plates. S. TIMOSHENKOand WOINOWSKY-KRIEGER 1959 Theory of Plate and Shells. New York : McGrawHill. See pp. 282-284. T. C. HUANG and C. C. HUANG 1971 Journal of Applied Mechanics 38, 515-521. Free vibration of viscoelastic Timoshenko beams. K. NAGAYA 1977 Journal of Engineering for Industry Transaction of the American Society of Mechanical Engimers Series B 994041109. Vibrations and dynamic response of viscoelastic plates on nonperiodic elastic supports. S. R. ROBERTSON1971 Journal of Soundand Vibration 14,263-278. Solving the problem of forced motion of viscoelastic plates by Valanis’ method with an application to a cricular plate. S. R. ROBERTSON1971 Journalof Soundand Vibration 17.363-381. Forced axis! mmetric motion of circular viscoelastic plates.