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Flexural vibration of rotating rectangular plates of variable thickness
Journal
of Sound
and Vibration (1988) 126(3), 373-385
FLEXURAL VIBRATION OF ROTATING RECTANGULAR PLATES
OF VARIABLE D. SHAW AND
National
K. Y.
THICKNESS SHEN
Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China AND
J. T. S.
WANG
Georgia Institute of Technology, Atlanta,
(Received
Georgia 30332, U.S.A.
18 June 1987, and in revised form 29 March 1988)
Free vibration of rotating rectangular plates with an exponentially varying thickness is studied. The plate is clamped at the rim of a central hub and is free along the other three edges. In the general formulation, the plate is considered to be made of orthotropic materials. The differential equations governing the in-plane displacements before vibration takes place are solved by a finite difference method. The curve-fitted previbratory in-plane stress distribution based on the numerical results is used in the equation for the flexural vibration of the plate, and the natural frequencies are subsequently determined by the extended Galerkin procedure. Some numerical results are presented for illustrative purposes.
1. INTRODUCTION Vibration
of rotating
blades
is one of the critical
problems
faced
in turbomachinery
design. Investigations of the vibration behavior of rotating blades have been systematically reviewed by Leissa [l]. In the vast majority of analyses the blade is modeled as a one-dimensional beam, and works on two- and three-dimensional models are limited in number. Over 150 references on the one-dimensional type model, with the blade treated as a cantilever beam, are given in reference [l]. The beam model provides a considerable amount of useful information about frequencies and mode shapes. Such one-dimensional models represent well some classes of blades used in rotating turbomachinery. For relatively thin blades having low aspect ratios, two-dimensional models are more desirable for providing better information. In a survey article, Leissa [2] gave a comprehensive review of two-dimensional models; most of these investigations were concerned with non-rotating blades. Recently, Leissa, Lee, Wang and Ewing [3-S] have spearheaded an exploratory comparison of beam, plate and shell theories for rotating blades. Lee, Leissa and Wang [6] also have published a paper in which the vibrations of variable thickness blades were studied by using shell theory. The simplest two-dimensional model is the rotating rectangular plate. This plate model has been used by a number of researchers such as Dokainish and Rawtani [7], Henry and Lalanne [8], Nagamatsu and Michimura [9], and Wang, Shaw and Mahrenholtz [lo]. Wang [ 1l] derived the equations of motion for rotating rectangular plates. While most of studies available in the open literature are for uniform plates, rotating plates having non-uniform thicknesses are considered in this study. The in-plane stress distribution due to the steady centrifugal acceleration is determined by using the finite difference method, and the frequencies are obtained by 373 0022-460X/88/210373+ 13 %03.00/O
@ 1988 Academic Press Limited
374
D. SHAW
using the Galerkin procedure. tion with the Galerkin method the elements of the frequency shape functions are complete,
ET AL.
Complete sets of orthogonal polynomials used in conjunclead to a simple and systematic procedure for generating determinant. Since the sets of polynomials used for the the convergence of the solutions is assured. 2. ANALYSIS
The geometry and some symbols are shown in Figure 1. Small oscillation about the prestressed equilibrium state due to the steady centrifugal acceleration is considered in the analysis. In this study, orthotropic composite laminates are considered in the general formulation, and only the flexural vibration is investigated.
Figure 1. Geometry.
Since it is extremely difficult to obtain closed form solutions for the in-plane stresses in a rotating plate with non-uniform thickness, the finite difference method is used. The numerical results of stress distribution are first curve-fitted. The curve-fitted previbrational stresses are then substituted in the differential equation governing the flexural vibration of the plate. The resulting equation is finally solved by using the Galerkin procedure for the determination of natural frequencies. 2.1.
IN-PLANE
The
DEFORMATION
AND
STRESS
RESULTANTS
equilibrium equations for in-plane stresses due to rotation of the plate are NX,X+ N,,,
The expression
u=v=O
= -Phfi2(X + r01r
NW + NW., = -phR2 cos2 ij~y.
(192)
for h is assumed to be h = h, eaxtaY. The boundary conditions are atx=O,
N,,=N,,=O
atx=L,
NXy=Nyy=O
aty=*b, (3-5)
where N,,, N,,,, and N,.. are stress resultants, p is the mass density, 0 is the rotating speed, JI is the setting angle, h is the thickness of the plate, ho is the plate thickness at the origin, which is placed at the root of the blade, (Y and /3 are the thickness factors describing the thickness variation of the plate, r,, is the distance between the axis of rotation and the root of the blade, L is the length of the plate and b is the half-width of the plate.
VIBRATION
OF
ROTATING
PLATES
375
;],
(6)
The constitutive law matrix form is [ ;;}=[
;;i
;;
;;;I{
where F, = au/ax,
yxy = au/aY
eyy = avlaY,
+ au/ax,
(7-9)
and A, are the extensional stiffnesses. Substitution of equations (6) in conjunction with equations (7) through (9) into equations (l), (2), (4) and (5) results in equilibrium equations and boundary conditions in terms of displacements. It is noted that the A, in equation (6) are linear functions of h and, since h = h(x, y), derivatives of A, must also be taken when deriving the equilibrium equations. Finally, the central difference approximations were used to express all equations in finite difference form. The equilibrium equations are written in the following form: {&A
,6~,+l,n-,+[A,,~2+~~~(~A~,+~A,6)1um+,,n+(-~SA,6)um+l,n-l
?? ~~66~~~(~~16~~~66)lUm,n+l+(-2m2All~2A66)Um.n +[A~~-~~(~A,~+PA~~)IU,,,+,+(-~SAI~)U,-,,~+I +EA,,s*-tsk(cuA,,+pA,,lu,-,,,+SsA,,u,-,,,-,}
+(A~6S2+tSk(~A,6+PA66)Vm+,,n +{~~(A,~+A,,)V,+I,,+I +~-~~(~,z+~,,)lV,+,,,-,+~~,,+~~(~~,,+~~,,)IV,,~+, +(-~~*A,,-~A~~)V,.,+[A~~-~~(~AI~+~A~~)IV,.,-I +[-fs(A,,+A,,)lV,-,,,+I +[A,6S2-~Sk(~A,6+~A66)1Vm-,,n +$S(&+A,,)V,_,~,_,
(10)
-fJhfi*(&~+X)k*,
{id&z + &&J ,+,,,+,+~~,6~*+~~k(~~,6+~~,~)1Um+,,n+t-~~(~,~+~66)1Um+,,n--1 +[A26+tk(~A66+~A26)lUm,n+,+[-2S2A,6-2Az6}U,., +[A26-~k(~A66+PA26)1U,,,-,+[-~S(A,2+A66)1Um-,,n+~ +[A,6S2-tSk(~A,6+PA,2)IUm-,,n+~S(A,2+A66)Um-,,n--1} +~~~A26~m+,,n+,+~~66~2+~~k(~~66+~~~6)lVm+~,n+(-~~A26)~~+~,n-, +[A~~+~~((YAz~+PAzz)~V,,,+I
+
(-2s2A66-2A22h?,n
+[A22-lk((YA26+PA22)Iv,,,-,+(-rSA26)v,~,,n+,
+[~66S2-~SSk((Y~66+~~26)]Vm-,,n+(~S~~~)V,_,,n_,}=-~h~22yCOS2J/k2,
(11)
in which u,,, and v,,, are the in-plane displacements at nodal points of the finite difference mesh, k = Ay and s = Ay/x. In the finite difference form, the boundary conditions for N,, = NXY= NY,,= 0 are given respectively as follows:
{sA,,Um+,,“+A,6Um,“+,
-A,6um.n-,
+{~46vm+,.n IsA
16Um+l,n
+ +
42~,.,+1
A66Um,n+,
-
-
+
+A22~m,n+1
-
A26Um.n+, -
-
A22vm,n-,
A,2vm..-,
A66Um,,-,
+{SA66hn+,,n+A26vm,n+, ~.4,2um+,.n
-sA,,um-,,,I
A26Um,n-, -
-
-
-
0,
(12)
O,
(13)
SA,6Um-,.J
A26b,.n-,
SA26vm-,.n
~A,6~m-,,nl=
-
SA66ht-,,n~
S&2um-,,n+ =
=
sA26%,+,,, 0.
(14)
376
D.
SHAW
ET AL.
These finite difference algebraic equations are written in matrix form which are easily solved by using the computer. The in-plane stress resultants are then calculated from equations (6) through (9) by using the central difference scheme. The numerical results for the in-plane stress resultants are curve-fitted for subsequent use in the vibration analysis.
2.2. NATURAL FREQUENCIES With respect to the prestressed state of N,,, NV,,, and NXy derived previously, the flexural vibration for the harmonic motion of the plate is governed by the following differential equation: H(w) = ~,1~,,,,+~~,~~,,~+2(~,~+2~~~)~.xx~~+~~~~~,x~,~~+~~~~,~~~~ +2(D*,,, +~16,y)~,xxx+2(~12,y+2~66.y+3~16,x)~,xxy +2(&,,+2&l,x
+ 3~*,,,)w,,
+ 2(&,X + 4*,y)w,yy.v
+(~,,,,+2~,,,,+~1*,y)~)~,xx+(~12.xx+2~26,x.”+~22..“y)~,yy + 2(&5,, +
+ 2&.sy + &,.~~)w,,?, - @2*(x + 4w.x - M2* cos* clvw,,
N,,w,,,~ + 2 NX_Vw,,y + N,,yw,yy+ (phR* sin* Ic,+ A*ph) w = 0.
(15)
Here D, are the flexural stiffnesses, w is the transverse displacement, H is the linear differential operator, and o is the natural circular frequency. The boundary conditions are w = w,, = 0 M,,,=
&=O
along x = 0, alongy=+6,
M.,, = V, = 0 Mx.“=O
along x = L,
atx=Landy=*b,
(16)
where M,,, M,,, and Mxy are the stress couples, and V, and V, are transverse shear stress resultants. As the thickness of the plate is assumed to vary exponentially in both directions the flexural stiffnesses Dij may be written as D, = DU e3(ax+p.v),
(17)
where & are the stiffnesses at the origin of the plate. Clearly, D, = Eh3/12(1 - v*) for isotropic materials where E is Young’s modulus and v is Poisson ratio. For convenience of analysis, non-dimensionalized co-ordinates of 6 = xl L and 17= y/b are introduced in equations (15) and (16). The general solution for w is represented in the form 2J w=
c m=*.4.6...
n=().,.*...
m=*.‘$...
I! “=@,.*...
An,~,n(Wn(~),
(18)
in which
~m(5)=~f?l(5)-~‘,(0),
(19)
and 9, is the Legendre polynomial. The subscript m in equation (18) takes on even integer values while the subscript n takes on both even and odd integer values. The shape functions in equation (18) in both directions belong to complete sets of orthogonal functions of Legendre polynomials. The expression for w in equation (18) satisfies geometric boundary conditions only. Substituting equation (18) into equation (15) and applying the extended Galerking procedure which accounts for the stress boundary conditions along the free edges, one arrives at a system of simultaneous algebraic equations which may be written in the matrix form
lIKimjnl{Cmn~ =O*
(20)
VIBRATION
OF
ROTATING
PLATES
377
It may be noted that the derivation for the elements of the coefficient matrix is lengthy and complex but manageable as the properties of Legendre polynomials are well established. Detailed derivations leading to equation (20) may be found in references [12]. For non-trivial solutions of equation (20), the eigenvalues corresponding to the natural frequencies of the plate are obtained by requiring the determinant of the coefficient matrix to vanish. 3. UNIFORM
PLATES
For partially checking and verifying the mathematical derivations and the computer program, some numerical results for uniform plates based on the present analysis can be compared to existing results. Frequencies for the first five modes which are essentially the first bending, first torsion, second bending, third bending and second torsion respectively have been computed. The first five mode shapes are plotted in Figure 2 for isotropic square plates. In Figure 2, the solid line represents the mode shape of the plate and the dashed line represents the neutral plane of the plate without deformation. The number
Figure 2. Mode shapes for isotropic 3; (d) mode 4; (e) mode 5.
square
plate (0 = 5000 rpm, a = 1.0). (a) mode 1; (b) mode 2; (c) mode
D.
378
SHAW
ET AL.
of terms with various combinations of J x K used in the series given in equation (18) to represent the mode shapes has been examined. inasmuch as all numerical results show rapid convergence, results obtained by using 4 x 5, 5 x 6, and 6 x 7 terms are listed in Table 1 to illustrate convergence and for comparison with existing results for square plates. The discrepancies between the present results and existing analytical results for the frequency parameter A = wL2m are about 5%. As obtained by using 4 x 5 terms in the series of equation (18), results for the frequency parameter for the first five modes for various aspect ratios are listed in Table 2. The present results compare well with solutions given in reference [13] as may be seen in Table 2. Dokainish and Rawtani [7] used the Southwell representation W2= w;+ s&22
(21)
for the frequencies of a rotating rectangular plate, in which w0 is the natural circular frequency for the stationary plate and SRD is the Southwell factor. A comparison of results for the frequency parameter based on the present analysis and those given in reference [7] for an isotropic square plate rotating at an angular velocity R = 5000 rpm is given in Table 3. The plate considered in Table 3 is set at the angle J, = 45” and located at r, = O-1 m, and has a Poisson ratio Y= O-3. While the discrepancy for the fourth mode is about 6%, discrepancies for all other modes are below 1%. Using various analytical techniques, Jensen and Crawley [14] presented frequencies of stationary composite plates having an aspect ratio of 4. The results were compared with their experimental findings to show the effectiveness of different techniques. The TABLE
1
Frequency parameter A for a square plate Mode
Table 4-45 [ 131 Table 4-47 [ 131 Table 4-47t [ 131 Table [lo] 4x5 5x6 6x7
1
2
3
3.494 344 3.33 3.41 3.949 3.486 3,482
8.547 8.21 8.17 8.28 8.634 8.564 a.544
2144 21.09 19.97 21.45 21.901 21.554 21.432
4
5’
27.46
31.17
26.67 27.772 27.615 27.295
31.12 32.247 31.43 31.234
t Experimental. TABLE 2
Comparison of frequency parameter L/b=1 I Ref. Mode
[W
1 2 3 4 5
3.508 5.372 21.96 10.26 24.85
t Experimental.
L/b=4 ,
Ref.
Present (4x5)
[wt
3.503 5.373 22.14 10.27 25.08
3.472 14.93 21.61 94.49 48.71
L/b=10 Present (4x5)
[wt
Ref.
Present (4x5)
3.48 15.18 22.1 96.03 50.52
3.45 34.73 21.52 563.9 105.9
3.442 34.99 21.61 565.05 107.62
VIBRATION
OF ROTATING TABLE
Comparison
“0 Ref. [ 71 Present
offrequency
PLATES
379
3
parameter A (R = 5000 t-pm)
1
2
3
4
3.494 4.334 4.345
8.634 9.044 8.996
21.901 22.81 22.744
21.772 29.85 28.02
TABLE
5 32.247 32.90 32.92
4
Material properties of laminae
&(GW
&(GPa)
98.0
7.9
GLT(GPa)
VL-T 0.28
5.6
ply thickness = O-134 mm;
G,, = G3 = GT;
TABLE
density
5
Comparison of natural frequencies Laminate
(Hz)
(a)
(b)
(~1
Cd)
(e)
(El
Present
2 3
11.1 39.5 69.5
11.1 39.6 69.3
11.1 39.6 69.4
11.1 39.7 72.00
11.1 39.7 72.1
11.1 42.4 70.5
Il.07 40.27 81.27
[15*/01,
1 2 3
8.9 42.9 62.7
8.7 49.2 59.9
8.8 48.8 60.5
9.0 43.5 65.0
9.0 44.5 66.9
9.4 45.8 66.2
9.09 43.74 66.18
r3wo1,
1 2 3
6.3 56.9 37.3
6.2 60.7 42.0
6.2 69.0 42.0
6.3 58.5 38.9
6.4 70.9 39.4
6.6 59.1 40.0
6.63 62.83 38.99
r45s/f&
1 2 3
4.9 49.4 30.1
4.8 56.3 32.6
4.9 73.9 32.7
4.9 51.2 31.5
4.9 71.4 31.6
4.8 51.3 29.8
5.18 54.14 31.85
[6WOls
1 2 3
4.2 41.7 26.1
4.2 47.1 26.8
4.2 65.4 27.0
4.2 42.7 27.2
4.2 65.4 27.2
4.3 47,l 27.1
4.32 45.50 27.21
[75*/01,
1 2 3
3.9 36.7 24.3
3.9 39.2 24.4
3.9 47.2 24.5
3.9 37.0 25.3
3.9 47.4 25.3
3.8 38.9 25.1
3.91 39.25 25.06
[9WOls
1 2 3
3.8 35.1 23.9
3.8 35.1 23.9
3-8 35.1 23.9
3.8 35.2 23.8
3.8 35.2 23.8
3.7 38.2 24.3
3.81 36.70 24.55
iw901,
Mode
p = 1520 kg/m3
1
(a) 365 degree of freedom (finite element); (b) 5 mode Rayleigh-Ritz; (c) 4 mode Rayleigh-Ritz; (d) 3 mode partial Ritz; (e) 2 mode partial Ritz; (E) Experiment.
380
D. SHAW
ET AL.
material properties for the laminated plates used in reference [14] are listed in Table 4. As may be seen in Table 5, results based on the present analysis compare well with the experimental measurements given in reference [ 141. 4.
PLATES WITH VARIABLE THICKNESS
Square isotropic and composite plates with thickness varying exponentially in the longitudinal direction have been investigated. The setting angle was taken to be zero. The length of the plate L and the radius of the central hub r, were each considered to be 10 cm. E = 200 GPa and Y= 0.3 were used for isotropic plates, and the data for material properties given in Table 4 were used for composite plates. In this study, layers of 0 plies are considered for the composite laminates. The variation of thickness of the plates is represented in the form h = h,, eax,
(22)
where h,, is the plate thickness at x = 0. If hL is the plate thickness at x = L and the thickness ratio a = hJ h,, , then aL = In (a). The root thickness ho is related to the average thickness h, as ho = h, In (a)/(~ - 1). Clearly, h, is the parameter for determining the total mass of the plate. In the numerical examples, 0.002 m and 0.001 m for h, have been used for isotropic and composite plates respectively. The in-plane stress distributions in plates rotating at the speed of 5000 rpm were first computed. Numerical results for isotropic plates along x = O-0125 m, O-0375 m, O-0625 m and O-0875 m, identified as sections A, B, C, and D respectively, are plotted in Figure 3
0
10000
20000
30000
40000
5OooO
60000
70000
NI (N/m)
3 ?? -0.01
-
-0.05
1 -14CGO
1 -1Ooo0
1
’ -6000
’
’ -2000
2000 Nxr
-0.05
-4000
0
4000
I 6000
I
I IOCKQ
/
I 14000
(N/m)
8000 NY (N/m)
Figure
3. Stresses
in isotropic
plate (0 = 5000 rpm, CI= 1.0).
VIBRATION
OF ROTATING
381
PLATES
for a = 1.0. It is found that the stress distribution for uniform plates corresponding to a = 1 obtained from the present finite difference solution compare favourably with the analytical solutions given in reference [lo]. Results for composite plates for a = 0.5 are shown in Figure 4. 4.1. IN-PLANE STRESS DISTRIBUTION Some results for the in-plane stress were presented in Figures 3 and 4, and one can make a few observations based on the calculated results. While the forms of the stress distributions for uniform and non-uniform plates are similar for plates made of the same materials, the stress distributions in isotropic and composite plates show noticeable differences. The largest value of N,, for all cases occurs at the section A which is closest to the root of blade as the maximum N,, should occur at the root. As expected, smallest values of N,, all occur at the section D. The shear stress resultant NXYis generally smaller than the normal stress resultant in the transverse direction NYYfor the isotropic plate. Magnitudes of NXYare comparable to N, at various locations in the composite plate. 4.2.
FREQUENCIES
OF SQUARE
PLATES
first five modes of non-rotating isotropic plates are given in Table 6 for various thickness ratios ranging from a = 0.4 to 1. One can see from Table 6 that the frequencies decrease as the thickness ratio increases for the first three modes, and the trend is reversed for the fourth and fifth modes. The effect of the thickness ratio is greatest on the first mode and becomes insignificant for higher modes. The fundamental frequency for the uniform plate corresponding to a = 1 is only 62.4% of the frequency for a = 0.4. The
0.03 3 \ -0.01
-0.05.
C
D
-
0
1
1 EGO
1
1
1600
1 1 1 2400
4
’ 3200
1
4Ocm
4800
5600
120
200
280
480
560
Nz (N/m)
-0.05 -280
-200
-I20
-40
40
NW (N/m)
-0.05
0
80
160
240
320
400
NY (N.lm)
Figure 4. Stresses
in orthotropic
plate (0 = 5000 rpm, o = 0.5).
382
D. SHAW
ET AL.
TABLE 6
Frequencies of non-rotating isotropic square plate (Hz) Mode a
1
2
3
4
5
0.4 0.5 0.6 0.7 0.8 0.9 1.0
273 247 226 208 194 181 170
500 484 469 455 442 431 421
1145 1134 1118 1103 1090 1078 1068
1270 1286 1303 1318 1332 1344 1354
1535 1537 1541 1546 1558 1565 1572
Frequencies for the first five modes of isotropic square plates rotating at various speeds are given in Table 7. At the same rotational speed, the frequency generally increases as the thickness ratio decreases for the first three modes. The trend reverses for mode 4 and mode 5. The effect of a becomes less significant for higher modes. The frequencies also increase as the rotational speed increases, as anticipated. The frequencies of the first modes corresponding to various rotational speeds are presented in Figure 5 for a = 0.7 and 1.0. It is noticeable in Figure 5 that the difference between the respective frequencies of these two plates decreases as the rotational speed increases. Clearly, a smaller value of thickness ratio implies smaller flexural stiffness and less mass inertia resulting in less stiffening effect due to centrifugal influence. The curve for a = 0.7 shown in Figure 5 for the first mode frequencies is above the curve for a = 1, but flatter. The results presented in Figure 5 indicate for the assumed form of thickness variation is smaller than the effect of the transverse inertia on the fundamental frequency. TABLE
7
Frequencies for isotropic plate (Hz)
a
-~fi
(rpm)
Mode
1
2
3
4
5
1.0
0 1000 5 000 10000
170 173 220 326
421 422 443 503
1068 1070 1110 1215
1355 1355 1371 1428
1573 1514 1606 1701
0.8
0.0 1000 5 000 10 000
194 196 240 341
443 444 464 522
1090 1092 1128 1221
1332 1333 1350 1412
1559 1560 1590 1680
0.6
0.0 1000 5 000 10 000
226 228 267 363
469 470 490 546
1119 1120 1150 1226
1304 1304 1324 1396
1541 1542 1571 1656
0.4
0.0 1000 5 000 10 000
273 275 310 400
501 502 521 571
1145 1146 1166 1214
1270 1271 1298 1390
1535 1536 1562 1642
VIBRATION
OF ROTATING
383
PLATES
__-1
160
I
0
1 1 1 1 1 1 ’ 1 IO!xc 8000 6000 4000 2000
Sl (rpm) Figure
5. Fundamental
frequency
TABLE
of isotropic
plate.
8
Frequencies of non-rotating orthotropic square plate (Hz)
Mode a
1
2
3
4
5
0.4 0.5 0.6 0.7 0.8 0.9 1.0
210 190 173 160 148 139 130
246 229 216 204 195 187 181
920 901 884 870 858 848 840
381 376 373 370 368 367 366
886 914 921 917 913 908 904
Numerical results for the orthotropic composite plate are given in Tables 8 and 9. The effect of the thickness ratio on the frequencies for rotating and non-rotating plates is similar to that for isotropic plates.
5.
CONCLUSIONS
Free vibration of rotating rectangular plates made of isotropic and composite materials has been studied. The plates were considered to have an exponentially varying thickness. The previbratory in-plane stress resultants due to the rotation of the plate have been calculated by a finite difference method. Frequencies for the small oscillations with respect to the prestressed plate under the centrifugal acceleration have been determined by the use of the Galerkin procedure. The coupling of the numerical method of finite difference and the analytical Galerkin procedure with complete sets of orthogonal functions of Legendre polynomials represents a simple and effective scheme for investigating the free vibration of rotating plates. Based on the numerical results computed in this study, frequencies for all modes increase as the rotational speed increases, which confirms the known stiffening effect under rotation. The effect of the thickness ratio characterizing the present form of thickness variation is small for frequencies of higher modes. The decrease
384
D.
SHAW
ET AL.
TABLE 9 Frequencies
oforthotropic
square plate (Hz)
Mode a
fi (rpm)
1
2
3
4
5
1.0
0 1000 5 000 10 000
130 133 190 306
181 183 228 330
830 842 902 1059
366 367 400 487
903 906 953 1065
0.8
0 1000 5 000 10 000
148 151 204 316
195 197 240 340
858 861 916 1072
368 378 402 489
913 914 955 1037
0.6
0 1000 5 000 10 000
173 176 224 332
216 217 258 355
884 886 938 1083
373 374 407 493
920 922 948 1005
0.4
0 1000 5 000 10 000
210 212 256 358
246 248 286 380
920 922 969 1102
381 383 417 506
976 978 1026 1165
of the thickness ratio generally causes the frequencies of lower modes to increase, and the trend reverses at higher modes. 6. ACKNOWLEDGMENTS The work has been sponsored by the National Science Council of the Republic of China under grant No. NSC-75-0401-E007-09. The advice and financial support of the National Science Council are gratefully acknowledged.
REFERENCES 1. A. W. LEISSA turbomachinery 2. A. W. LEISSA shell analysis. 3. A. E. LEISSA,
1981 Applied Mechanics Reviews 34, 629-635. Vibrational aspects of rotating blades. 1980 The Shock and Vibration Digest 12, 3-10. Vibrations of turbine blades of
J. K. LEE and A. J. WANG 1982 American Society of Mechanical Engineers Journal of Engineering for Power 104,296-302. Rotating blade vibration analysis using shells. 4. A. W. LEISSA, J. K. LEE and A. J. WANG 1984 American Society of Mechanical Engineers Journal of Vibration, Acoustics, Stress and Reliability in Design 106, 251-257. Vibrations of
twisted rotating blades. 5. A. W. LEISSA and M. S. EWING 1983 American Society of Mechanical Engineers Journal of Engineering for Power 105,383-392. Comparison of beam and shell theories for the vibrations of thin turbomachinery blades. 6. J. K. LEE, A. W. LEISSA and A. J. WANG 1984American Society of Mechanical Engineers Journal of Engineeringfor Gas Turbines and Power 106, 1l-16. Vibrations of blades with variable thickness and curvature by shell theory. 7. M. A. DOKAINISH and S. RAWTANI 1971 International Journal for Numerical Methods in Engineering 3, 233-248. Vibration analysis of rotating cantilever plates. 8. R. HENRY and M. LALANNE 1974 American Society of Mechanical Engineers Journal of Engineering for Industry, 1028-1035. Vibration analysis of rotating compressor blades.
VIBRATION OF ROTATING PLATES
385
9. A. NAGAMATSU and S. MICHIMURA 1978 Theoretical and Applied Mechanics 26, 327-338. Vibration of blades of a rotating impeller. 10. J. T. S. WANG, D. SHAW and 0. MAHRENHOLTZ 1987 Journal of Sound and Vibration 112, 455-468. Vibration of rotating plates. 11. J. T. S. WANG 1974 Mechanics Research Communications 1,341-346. Equations for the vibration of a rotating blade. 12. D. SHAW and K. Y. SHEN 1986 National Science Council Technical Report NSC75-0401-E00709. Vibration of rotating plates of variables thickness. 13. A. W. LEISSA 1969 NASA-SP-160. Vibration of plates. 14. D. W. JENSEN and E. F. CRAWLEY 1984 American Institute of Aeronautics and Astronautics Journal 22, 415-420. Determination techniques for cantilevered plates with bending-torsion coupling.
of Sound
and Vibration (1988) 126(3), 373-385
FLEXURAL VIBRATION OF ROTATING RECTANGULAR PLATES
OF VARIABLE D. SHAW AND
National
K. Y.
THICKNESS SHEN
Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China AND
J. T. S.
WANG
Georgia Institute of Technology, Atlanta,
(Received
Georgia 30332, U.S.A.
18 June 1987, and in revised form 29 March 1988)
Free vibration of rotating rectangular plates with an exponentially varying thickness is studied. The plate is clamped at the rim of a central hub and is free along the other three edges. In the general formulation, the plate is considered to be made of orthotropic materials. The differential equations governing the in-plane displacements before vibration takes place are solved by a finite difference method. The curve-fitted previbratory in-plane stress distribution based on the numerical results is used in the equation for the flexural vibration of the plate, and the natural frequencies are subsequently determined by the extended Galerkin procedure. Some numerical results are presented for illustrative purposes.
1. INTRODUCTION Vibration
of rotating
blades
is one of the critical
problems
faced
in turbomachinery
design. Investigations of the vibration behavior of rotating blades have been systematically reviewed by Leissa [l]. In the vast majority of analyses the blade is modeled as a one-dimensional beam, and works on two- and three-dimensional models are limited in number. Over 150 references on the one-dimensional type model, with the blade treated as a cantilever beam, are given in reference [l]. The beam model provides a considerable amount of useful information about frequencies and mode shapes. Such one-dimensional models represent well some classes of blades used in rotating turbomachinery. For relatively thin blades having low aspect ratios, two-dimensional models are more desirable for providing better information. In a survey article, Leissa [2] gave a comprehensive review of two-dimensional models; most of these investigations were concerned with non-rotating blades. Recently, Leissa, Lee, Wang and Ewing [3-S] have spearheaded an exploratory comparison of beam, plate and shell theories for rotating blades. Lee, Leissa and Wang [6] also have published a paper in which the vibrations of variable thickness blades were studied by using shell theory. The simplest two-dimensional model is the rotating rectangular plate. This plate model has been used by a number of researchers such as Dokainish and Rawtani [7], Henry and Lalanne [8], Nagamatsu and Michimura [9], and Wang, Shaw and Mahrenholtz [lo]. Wang [ 1l] derived the equations of motion for rotating rectangular plates. While most of studies available in the open literature are for uniform plates, rotating plates having non-uniform thicknesses are considered in this study. The in-plane stress distribution due to the steady centrifugal acceleration is determined by using the finite difference method, and the frequencies are obtained by 373 0022-460X/88/210373+ 13 %03.00/O
@ 1988 Academic Press Limited
374
D. SHAW
using the Galerkin procedure. tion with the Galerkin method the elements of the frequency shape functions are complete,
ET AL.
Complete sets of orthogonal polynomials used in conjunclead to a simple and systematic procedure for generating determinant. Since the sets of polynomials used for the the convergence of the solutions is assured. 2. ANALYSIS
The geometry and some symbols are shown in Figure 1. Small oscillation about the prestressed equilibrium state due to the steady centrifugal acceleration is considered in the analysis. In this study, orthotropic composite laminates are considered in the general formulation, and only the flexural vibration is investigated.
Figure 1. Geometry.
Since it is extremely difficult to obtain closed form solutions for the in-plane stresses in a rotating plate with non-uniform thickness, the finite difference method is used. The numerical results of stress distribution are first curve-fitted. The curve-fitted previbrational stresses are then substituted in the differential equation governing the flexural vibration of the plate. The resulting equation is finally solved by using the Galerkin procedure for the determination of natural frequencies. 2.1.
IN-PLANE
The
DEFORMATION
AND
STRESS
RESULTANTS
equilibrium equations for in-plane stresses due to rotation of the plate are NX,X+ N,,,
The expression
u=v=O
= -Phfi2(X + r01r
NW + NW., = -phR2 cos2 ij~y.
(192)
for h is assumed to be h = h, eaxtaY. The boundary conditions are atx=O,
N,,=N,,=O
atx=L,
NXy=Nyy=O
aty=*b, (3-5)
where N,,, N,,,, and N,.. are stress resultants, p is the mass density, 0 is the rotating speed, JI is the setting angle, h is the thickness of the plate, ho is the plate thickness at the origin, which is placed at the root of the blade, (Y and /3 are the thickness factors describing the thickness variation of the plate, r,, is the distance between the axis of rotation and the root of the blade, L is the length of the plate and b is the half-width of the plate.
VIBRATION
OF
ROTATING
PLATES
375
;],
(6)
The constitutive law matrix form is [ ;;}=[
;;i
;;
;;;I{
where F, = au/ax,
yxy = au/aY
eyy = avlaY,
+ au/ax,
(7-9)
and A, are the extensional stiffnesses. Substitution of equations (6) in conjunction with equations (7) through (9) into equations (l), (2), (4) and (5) results in equilibrium equations and boundary conditions in terms of displacements. It is noted that the A, in equation (6) are linear functions of h and, since h = h(x, y), derivatives of A, must also be taken when deriving the equilibrium equations. Finally, the central difference approximations were used to express all equations in finite difference form. The equilibrium equations are written in the following form: {&A
,6~,+l,n-,+[A,,~2+~~~(~A~,+~A,6)1um+,,n+(-~SA,6)um+l,n-l
?? ~~66~~~(~~16~~~66)lUm,n+l+(-2m2All~2A66)Um.n +[A~~-~~(~A,~+PA~~)IU,,,+,+(-~SAI~)U,-,,~+I +EA,,s*-tsk(cuA,,+pA,,lu,-,,,+SsA,,u,-,,,-,}
+(A~6S2+tSk(~A,6+PA66)Vm+,,n +{~~(A,~+A,,)V,+I,,+I +~-~~(~,z+~,,)lV,+,,,-,+~~,,+~~(~~,,+~~,,)IV,,~+, +(-~~*A,,-~A~~)V,.,+[A~~-~~(~AI~+~A~~)IV,.,-I +[-fs(A,,+A,,)lV,-,,,+I +[A,6S2-~Sk(~A,6+~A66)1Vm-,,n +$S(&+A,,)V,_,~,_,
(10)
-fJhfi*(&~+X)k*,
{id&z + &&J ,+,,,+,+~~,6~*+~~k(~~,6+~~,~)1Um+,,n+t-~~(~,~+~66)1Um+,,n--1 +[A26+tk(~A66+~A26)lUm,n+,+[-2S2A,6-2Az6}U,., +[A26-~k(~A66+PA26)1U,,,-,+[-~S(A,2+A66)1Um-,,n+~ +[A,6S2-tSk(~A,6+PA,2)IUm-,,n+~S(A,2+A66)Um-,,n--1} +~~~A26~m+,,n+,+~~66~2+~~k(~~66+~~~6)lVm+~,n+(-~~A26)~~+~,n-, +[A~~+~~((YAz~+PAzz)~V,,,+I
+
(-2s2A66-2A22h?,n
+[A22-lk((YA26+PA22)Iv,,,-,+(-rSA26)v,~,,n+,
+[~66S2-~SSk((Y~66+~~26)]Vm-,,n+(~S~~~)V,_,,n_,}=-~h~22yCOS2J/k2,
(11)
in which u,,, and v,,, are the in-plane displacements at nodal points of the finite difference mesh, k = Ay and s = Ay/x. In the finite difference form, the boundary conditions for N,, = NXY= NY,,= 0 are given respectively as follows:
{sA,,Um+,,“+A,6Um,“+,
-A,6um.n-,
+{~46vm+,.n IsA
16Um+l,n
+ +
42~,.,+1
A66Um,n+,
-
-
+
+A22~m,n+1
-
A26Um.n+, -
-
A22vm,n-,
A,2vm..-,
A66Um,,-,
+{SA66hn+,,n+A26vm,n+, ~.4,2um+,.n
-sA,,um-,,,I
A26Um,n-, -
-
-
-
0,
(12)
O,
(13)
SA,6Um-,.J
A26b,.n-,
SA26vm-,.n
~A,6~m-,,nl=
-
SA66ht-,,n~
S&2um-,,n+ =
=
sA26%,+,,, 0.
(14)
376
D.
SHAW
ET AL.
These finite difference algebraic equations are written in matrix form which are easily solved by using the computer. The in-plane stress resultants are then calculated from equations (6) through (9) by using the central difference scheme. The numerical results for the in-plane stress resultants are curve-fitted for subsequent use in the vibration analysis.
2.2. NATURAL FREQUENCIES With respect to the prestressed state of N,,, NV,,, and NXy derived previously, the flexural vibration for the harmonic motion of the plate is governed by the following differential equation: H(w) = ~,1~,,,,+~~,~~,,~+2(~,~+2~~~)~.xx~~+~~~~~,x~,~~+~~~~,~~~~ +2(D*,,, +~16,y)~,xxx+2(~12,y+2~66.y+3~16,x)~,xxy +2(&,,+2&l,x
+ 3~*,,,)w,,
+ 2(&,X + 4*,y)w,yy.v
+(~,,,,+2~,,,,+~1*,y)~)~,xx+(~12.xx+2~26,x.”+~22..“y)~,yy + 2(&5,, +
+ 2&.sy + &,.~~)w,,?, - @2*(x + 4w.x - M2* cos* clvw,,
N,,w,,,~ + 2 NX_Vw,,y + N,,yw,yy+ (phR* sin* Ic,+ A*ph) w = 0.
(15)
Here D, are the flexural stiffnesses, w is the transverse displacement, H is the linear differential operator, and o is the natural circular frequency. The boundary conditions are w = w,, = 0 M,,,=
&=O
along x = 0, alongy=+6,
M.,, = V, = 0 Mx.“=O
along x = L,
atx=Landy=*b,
(16)
where M,,, M,,, and Mxy are the stress couples, and V, and V, are transverse shear stress resultants. As the thickness of the plate is assumed to vary exponentially in both directions the flexural stiffnesses Dij may be written as D, = DU e3(ax+p.v),
(17)
where & are the stiffnesses at the origin of the plate. Clearly, D, = Eh3/12(1 - v*) for isotropic materials where E is Young’s modulus and v is Poisson ratio. For convenience of analysis, non-dimensionalized co-ordinates of 6 = xl L and 17= y/b are introduced in equations (15) and (16). The general solution for w is represented in the form 2J w=
c m=*.4.6...
n=().,.*...
m=*.‘$...
I! “=@,.*...
An,~,n(Wn(~),
(18)
in which
~m(5)=~f?l(5)-~‘,(0),
(19)
and 9, is the Legendre polynomial. The subscript m in equation (18) takes on even integer values while the subscript n takes on both even and odd integer values. The shape functions in equation (18) in both directions belong to complete sets of orthogonal functions of Legendre polynomials. The expression for w in equation (18) satisfies geometric boundary conditions only. Substituting equation (18) into equation (15) and applying the extended Galerking procedure which accounts for the stress boundary conditions along the free edges, one arrives at a system of simultaneous algebraic equations which may be written in the matrix form
lIKimjnl{Cmn~ =O*
(20)
VIBRATION
OF
ROTATING
PLATES
377
It may be noted that the derivation for the elements of the coefficient matrix is lengthy and complex but manageable as the properties of Legendre polynomials are well established. Detailed derivations leading to equation (20) may be found in references [12]. For non-trivial solutions of equation (20), the eigenvalues corresponding to the natural frequencies of the plate are obtained by requiring the determinant of the coefficient matrix to vanish. 3. UNIFORM
PLATES
For partially checking and verifying the mathematical derivations and the computer program, some numerical results for uniform plates based on the present analysis can be compared to existing results. Frequencies for the first five modes which are essentially the first bending, first torsion, second bending, third bending and second torsion respectively have been computed. The first five mode shapes are plotted in Figure 2 for isotropic square plates. In Figure 2, the solid line represents the mode shape of the plate and the dashed line represents the neutral plane of the plate without deformation. The number
Figure 2. Mode shapes for isotropic 3; (d) mode 4; (e) mode 5.
square
plate (0 = 5000 rpm, a = 1.0). (a) mode 1; (b) mode 2; (c) mode
D.
378
SHAW
ET AL.
of terms with various combinations of J x K used in the series given in equation (18) to represent the mode shapes has been examined. inasmuch as all numerical results show rapid convergence, results obtained by using 4 x 5, 5 x 6, and 6 x 7 terms are listed in Table 1 to illustrate convergence and for comparison with existing results for square plates. The discrepancies between the present results and existing analytical results for the frequency parameter A = wL2m are about 5%. As obtained by using 4 x 5 terms in the series of equation (18), results for the frequency parameter for the first five modes for various aspect ratios are listed in Table 2. The present results compare well with solutions given in reference [13] as may be seen in Table 2. Dokainish and Rawtani [7] used the Southwell representation W2= w;+ s&22
(21)
for the frequencies of a rotating rectangular plate, in which w0 is the natural circular frequency for the stationary plate and SRD is the Southwell factor. A comparison of results for the frequency parameter based on the present analysis and those given in reference [7] for an isotropic square plate rotating at an angular velocity R = 5000 rpm is given in Table 3. The plate considered in Table 3 is set at the angle J, = 45” and located at r, = O-1 m, and has a Poisson ratio Y= O-3. While the discrepancy for the fourth mode is about 6%, discrepancies for all other modes are below 1%. Using various analytical techniques, Jensen and Crawley [14] presented frequencies of stationary composite plates having an aspect ratio of 4. The results were compared with their experimental findings to show the effectiveness of different techniques. The TABLE
1
Frequency parameter A for a square plate Mode
Table 4-45 [ 131 Table 4-47 [ 131 Table 4-47t [ 131 Table [lo] 4x5 5x6 6x7
1
2
3
3.494 344 3.33 3.41 3.949 3.486 3,482
8.547 8.21 8.17 8.28 8.634 8.564 a.544
2144 21.09 19.97 21.45 21.901 21.554 21.432
4
5’
27.46
31.17
26.67 27.772 27.615 27.295
31.12 32.247 31.43 31.234
t Experimental. TABLE 2
Comparison of frequency parameter L/b=1 I Ref. Mode
[W
1 2 3 4 5
3.508 5.372 21.96 10.26 24.85
t Experimental.
L/b=4 ,
Ref.
Present (4x5)
[wt
3.503 5.373 22.14 10.27 25.08
3.472 14.93 21.61 94.49 48.71
L/b=10 Present (4x5)
[wt
Ref.
Present (4x5)
3.48 15.18 22.1 96.03 50.52
3.45 34.73 21.52 563.9 105.9
3.442 34.99 21.61 565.05 107.62
VIBRATION
OF ROTATING TABLE
Comparison
“0 Ref. [ 71 Present
offrequency
PLATES
379
3
parameter A (R = 5000 t-pm)
1
2
3
4
3.494 4.334 4.345
8.634 9.044 8.996
21.901 22.81 22.744
21.772 29.85 28.02
TABLE
5 32.247 32.90 32.92
4
Material properties of laminae
&(GW
&(GPa)
98.0
7.9
GLT(GPa)
VL-T 0.28
5.6
ply thickness = O-134 mm;
G,, = G3 = GT;
TABLE
density
5
Comparison of natural frequencies Laminate
(Hz)
(a)
(b)
(~1
Cd)
(e)
(El
Present
2 3
11.1 39.5 69.5
11.1 39.6 69.3
11.1 39.6 69.4
11.1 39.7 72.00
11.1 39.7 72.1
11.1 42.4 70.5
Il.07 40.27 81.27
[15*/01,
1 2 3
8.9 42.9 62.7
8.7 49.2 59.9
8.8 48.8 60.5
9.0 43.5 65.0
9.0 44.5 66.9
9.4 45.8 66.2
9.09 43.74 66.18
r3wo1,
1 2 3
6.3 56.9 37.3
6.2 60.7 42.0
6.2 69.0 42.0
6.3 58.5 38.9
6.4 70.9 39.4
6.6 59.1 40.0
6.63 62.83 38.99
r45s/f&
1 2 3
4.9 49.4 30.1
4.8 56.3 32.6
4.9 73.9 32.7
4.9 51.2 31.5
4.9 71.4 31.6
4.8 51.3 29.8
5.18 54.14 31.85
[6WOls
1 2 3
4.2 41.7 26.1
4.2 47.1 26.8
4.2 65.4 27.0
4.2 42.7 27.2
4.2 65.4 27.2
4.3 47,l 27.1
4.32 45.50 27.21
[75*/01,
1 2 3
3.9 36.7 24.3
3.9 39.2 24.4
3.9 47.2 24.5
3.9 37.0 25.3
3.9 47.4 25.3
3.8 38.9 25.1
3.91 39.25 25.06
[9WOls
1 2 3
3.8 35.1 23.9
3.8 35.1 23.9
3-8 35.1 23.9
3.8 35.2 23.8
3.8 35.2 23.8
3.7 38.2 24.3
3.81 36.70 24.55
iw901,
Mode
p = 1520 kg/m3
1
(a) 365 degree of freedom (finite element); (b) 5 mode Rayleigh-Ritz; (c) 4 mode Rayleigh-Ritz; (d) 3 mode partial Ritz; (e) 2 mode partial Ritz; (E) Experiment.
380
D. SHAW
ET AL.
material properties for the laminated plates used in reference [14] are listed in Table 4. As may be seen in Table 5, results based on the present analysis compare well with the experimental measurements given in reference [ 141. 4.
PLATES WITH VARIABLE THICKNESS
Square isotropic and composite plates with thickness varying exponentially in the longitudinal direction have been investigated. The setting angle was taken to be zero. The length of the plate L and the radius of the central hub r, were each considered to be 10 cm. E = 200 GPa and Y= 0.3 were used for isotropic plates, and the data for material properties given in Table 4 were used for composite plates. In this study, layers of 0 plies are considered for the composite laminates. The variation of thickness of the plates is represented in the form h = h,, eax,
(22)
where h,, is the plate thickness at x = 0. If hL is the plate thickness at x = L and the thickness ratio a = hJ h,, , then aL = In (a). The root thickness ho is related to the average thickness h, as ho = h, In (a)/(~ - 1). Clearly, h, is the parameter for determining the total mass of the plate. In the numerical examples, 0.002 m and 0.001 m for h, have been used for isotropic and composite plates respectively. The in-plane stress distributions in plates rotating at the speed of 5000 rpm were first computed. Numerical results for isotropic plates along x = O-0125 m, O-0375 m, O-0625 m and O-0875 m, identified as sections A, B, C, and D respectively, are plotted in Figure 3
0
10000
20000
30000
40000
5OooO
60000
70000
NI (N/m)
3 ?? -0.01
-
-0.05
1 -14CGO
1 -1Ooo0
1
’ -6000
’
’ -2000
2000 Nxr
-0.05
-4000
0
4000
I 6000
I
I IOCKQ
/
I 14000
(N/m)
8000 NY (N/m)
Figure
3. Stresses
in isotropic
plate (0 = 5000 rpm, CI= 1.0).
VIBRATION
OF ROTATING
381
PLATES
for a = 1.0. It is found that the stress distribution for uniform plates corresponding to a = 1 obtained from the present finite difference solution compare favourably with the analytical solutions given in reference [lo]. Results for composite plates for a = 0.5 are shown in Figure 4. 4.1. IN-PLANE STRESS DISTRIBUTION Some results for the in-plane stress were presented in Figures 3 and 4, and one can make a few observations based on the calculated results. While the forms of the stress distributions for uniform and non-uniform plates are similar for plates made of the same materials, the stress distributions in isotropic and composite plates show noticeable differences. The largest value of N,, for all cases occurs at the section A which is closest to the root of blade as the maximum N,, should occur at the root. As expected, smallest values of N,, all occur at the section D. The shear stress resultant NXYis generally smaller than the normal stress resultant in the transverse direction NYYfor the isotropic plate. Magnitudes of NXYare comparable to N, at various locations in the composite plate. 4.2.
FREQUENCIES
OF SQUARE
PLATES
first five modes of non-rotating isotropic plates are given in Table 6 for various thickness ratios ranging from a = 0.4 to 1. One can see from Table 6 that the frequencies decrease as the thickness ratio increases for the first three modes, and the trend is reversed for the fourth and fifth modes. The effect of the thickness ratio is greatest on the first mode and becomes insignificant for higher modes. The fundamental frequency for the uniform plate corresponding to a = 1 is only 62.4% of the frequency for a = 0.4. The
0.03 3 \ -0.01
-0.05.
C
D
-
0
1
1 EGO
1
1
1600
1 1 1 2400
4
’ 3200
1
4Ocm
4800
5600
120
200
280
480
560
Nz (N/m)
-0.05 -280
-200
-I20
-40
40
NW (N/m)
-0.05
0
80
160
240
320
400
NY (N.lm)
Figure 4. Stresses
in orthotropic
plate (0 = 5000 rpm, o = 0.5).
382
D. SHAW
ET AL.
TABLE 6
Frequencies of non-rotating isotropic square plate (Hz) Mode a
1
2
3
4
5
0.4 0.5 0.6 0.7 0.8 0.9 1.0
273 247 226 208 194 181 170
500 484 469 455 442 431 421
1145 1134 1118 1103 1090 1078 1068
1270 1286 1303 1318 1332 1344 1354
1535 1537 1541 1546 1558 1565 1572
Frequencies for the first five modes of isotropic square plates rotating at various speeds are given in Table 7. At the same rotational speed, the frequency generally increases as the thickness ratio decreases for the first three modes. The trend reverses for mode 4 and mode 5. The effect of a becomes less significant for higher modes. The frequencies also increase as the rotational speed increases, as anticipated. The frequencies of the first modes corresponding to various rotational speeds are presented in Figure 5 for a = 0.7 and 1.0. It is noticeable in Figure 5 that the difference between the respective frequencies of these two plates decreases as the rotational speed increases. Clearly, a smaller value of thickness ratio implies smaller flexural stiffness and less mass inertia resulting in less stiffening effect due to centrifugal influence. The curve for a = 0.7 shown in Figure 5 for the first mode frequencies is above the curve for a = 1, but flatter. The results presented in Figure 5 indicate for the assumed form of thickness variation is smaller than the effect of the transverse inertia on the fundamental frequency. TABLE
7
Frequencies for isotropic plate (Hz)
a
-~fi
(rpm)
Mode
1
2
3
4
5
1.0
0 1000 5 000 10000
170 173 220 326
421 422 443 503
1068 1070 1110 1215
1355 1355 1371 1428
1573 1514 1606 1701
0.8
0.0 1000 5 000 10 000
194 196 240 341
443 444 464 522
1090 1092 1128 1221
1332 1333 1350 1412
1559 1560 1590 1680
0.6
0.0 1000 5 000 10 000
226 228 267 363
469 470 490 546
1119 1120 1150 1226
1304 1304 1324 1396
1541 1542 1571 1656
0.4
0.0 1000 5 000 10 000
273 275 310 400
501 502 521 571
1145 1146 1166 1214
1270 1271 1298 1390
1535 1536 1562 1642
VIBRATION
OF ROTATING
383
PLATES
__-1
160
I
0
1 1 1 1 1 1 ’ 1 IO!xc 8000 6000 4000 2000
Sl (rpm) Figure
5. Fundamental
frequency
TABLE
of isotropic
plate.
8
Frequencies of non-rotating orthotropic square plate (Hz)
Mode a
1
2
3
4
5
0.4 0.5 0.6 0.7 0.8 0.9 1.0
210 190 173 160 148 139 130
246 229 216 204 195 187 181
920 901 884 870 858 848 840
381 376 373 370 368 367 366
886 914 921 917 913 908 904
Numerical results for the orthotropic composite plate are given in Tables 8 and 9. The effect of the thickness ratio on the frequencies for rotating and non-rotating plates is similar to that for isotropic plates.
5.
CONCLUSIONS
Free vibration of rotating rectangular plates made of isotropic and composite materials has been studied. The plates were considered to have an exponentially varying thickness. The previbratory in-plane stress resultants due to the rotation of the plate have been calculated by a finite difference method. Frequencies for the small oscillations with respect to the prestressed plate under the centrifugal acceleration have been determined by the use of the Galerkin procedure. The coupling of the numerical method of finite difference and the analytical Galerkin procedure with complete sets of orthogonal functions of Legendre polynomials represents a simple and effective scheme for investigating the free vibration of rotating plates. Based on the numerical results computed in this study, frequencies for all modes increase as the rotational speed increases, which confirms the known stiffening effect under rotation. The effect of the thickness ratio characterizing the present form of thickness variation is small for frequencies of higher modes. The decrease
384
D.
SHAW
ET AL.
TABLE 9 Frequencies
oforthotropic
square plate (Hz)
Mode a
fi (rpm)
1
2
3
4
5
1.0
0 1000 5 000 10 000
130 133 190 306
181 183 228 330
830 842 902 1059
366 367 400 487
903 906 953 1065
0.8
0 1000 5 000 10 000
148 151 204 316
195 197 240 340
858 861 916 1072
368 378 402 489
913 914 955 1037
0.6
0 1000 5 000 10 000
173 176 224 332
216 217 258 355
884 886 938 1083
373 374 407 493
920 922 948 1005
0.4
0 1000 5 000 10 000
210 212 256 358
246 248 286 380
920 922 969 1102
381 383 417 506
976 978 1026 1165
of the thickness ratio generally causes the frequencies of lower modes to increase, and the trend reverses at higher modes. 6. ACKNOWLEDGMENTS The work has been sponsored by the National Science Council of the Republic of China under grant No. NSC-75-0401-E007-09. The advice and financial support of the National Science Council are gratefully acknowledged.
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