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Natural frequencies of clamped orthotropic skew plates
Journal of Sound and Vibration (1982) 81(2), 287-298
NATURAL
FREQUENCIES
OF CLAMPED
ORTHOTROPIC
SKEW PLATES T.
SAKATA
AND
T.
HAYASHI
Chubu Institute of Technology, Department of Mechanical Engineering, Kasugai, Nagoya-sub., 487 Japan (Received 10 March 1981, and in
revised form 28 July 1981)
Approximate formulae are proposed for estimating natural frequencies of isotropic and specially orthotropic skew plates with clamped sides. It has been shown previously that one can estimate a natural frequency of a generally orthotropic skew plate with clamped sides by using an approximate formula for the isotropic plate which one can relate to the orthotropic one by applying a previously described reduction method. The accuracy of the proposed approximate formulae is demonstrated by comparing numerical and experimental results for several typical cases.
1. INTRODUCI’ION
Since a skew plate is an important structural member, many studies of free and forced vibrations of an isotropic skew plate have been reported. Durvasula [l] calculated natural frequencies and modes of a clamped skew plate, and Nair and Durvasula [2] calculated natural frequencies and modes of skew plates with various boundary conditions. Mizusawa, Kajita and Naruoka [3] proposed a Rayleigh-Ritz method with B-spline functions as co-ordinate functions for the determination of natural frequencies of skew plates and discussed the accuracy and convergence of the solution. Kuttler and Sigillito [4] calculated upper and lower bounds for natural frequencies of a clamped rhombic plate. Ban’erjee [5] discussed free and forced vibrations of a clamped skew plate with variable thickness. Nagaya [6] proposed an approach for free vibration of a plate with straight line boundaries and determined natural frequencies of clamped and simply supported rhombic plates. Work up to 1965 [7] and recent work over the period 1973-76 [8] have been reviewed by Leissa. For a simply supported polygonal plate one can easily estimate the natural frequency from the natural frequency of a clamped membrane with the same shape as the plate [7]. By using this fact, Bauer and Reiss [9] estimated natural frequencies of a simply supported rhombic plate. On the other hand, in spite of their importance, one can find only a few reports on vibration of orthotropic skew plates. Srinivasan and Munaswamy [lo] discussed free vibration of a point supported skew plate by using a finite strip method. Dokainish and Kumer [ll] determined natural frequencies of a clamped skew plate with variable thickness. Laura, Luisoni and Sarmiento [12] proposed a numerical approach for determining natural frequencies of an orthotropic polygonal plate with clamped or simply supported edges, although they did not apply the approach to a skew plate. Sakata showed that one can obtain an exact solution for the free vibration of a simply supported skew plate with particular flexural rigidities and skew angle [13]. Subsequently he proposed simple approximate formulae for estimating natural frequencies of simply supported skew plates from the natural frequency of clamped skew membranes [14]. 287 0022-460X/82/060287
+ 12 $02.00/O
@ 1982 Academic Press Inc. (London)
Limited
-1. SAKATA
288
AND
7‘. HAYASHI
These formulae are derived by applying his reduction method [15, 161, and using the fact that the natural frequency of a simply supported skew plate can be estimated quite easily from that of a clamped membrane with the same shape. In the present paper, by using the reduction method and the natural frequencies of clamped isotropic skew plates, simple approximate formulae are derived in the same manner as in reference [14]. The accuracy of the natural frequencies estimated from the approximate formulae is examined by comparison with those determined numerically and experimentally. A brief summary of the reduction method and description of how to apply it to a clamped skew plate are given in the Appendix.
2.
Consider co-ordinate
MATHEMATICAL
ANALYSIS
an orthotropic skew plate as shown in Figure 1, in which the rectangular axes, x and y, are parallel to the principal elastic axes of the plate material
Figure
1. The co-ordinate
systems
for isotropic
and specially
orthotropic
skew plates.
and the oblique co-ordinate axes, 5 and 5, are taken along the edges. The plate has uniform small thickness, and it is clamped along all the edges. According to the classical small deflection theory, the governing equation for free vibration of a thin orthotropic plate is (1)
D,(~4W/~x4)+2H(~4w/ax2ay’)+~,(a4W/ay4)-phOw2W=0, where W is the deflection, p is the mass density of the plate thickness and w is the radian frequency. The flexural rigidities
material, ho is the plate D,, H and D, are given
by D, = E&312(1
H = v,D, + Gh:/6,
- Y~Y~),
D, = E,h;/12(1-
vxv,),
where E,, E,, G, vx and v,, are elastic constants defined by the stress-strain relations the orthotropic material [7]. Since the plate is clamped along all the edges, the boundary conditions are
w =o,
d W/an = 0
along all the edges, where n denotes the direction perpendicular The relations between the rectangular and oblique co-ordinate 5=x--y
tan*,
l=
y/cos *.
(2) of
(3) to each edge. systems are (4)
ORTHOTROPIC
Under the transformation
PLATE
NATURAL
FREQUENCIES
289
(4), equations (1) and (3) become
sin’ CLcos2 1/1+(I&/D,) sin4 4,) a4 w/at4
[{cos4 IJ?+ 2(H/D,)
- 4{(H/D,) sin tj cos2 IJJ+ (Q/OX) sin3 $} a4 w/ae3 ag + 2{(H/D,)
- 4{(0,/0,) W=O,
~03~ 4 +
sin’ JI} a4 w/at2 al2
3(0,/D,)
sin +I) a4 w/a6 al3 + (D,/D,)
aW/&$=O
a4 w/al”] W=O,
at[=Oande=a,
- (phow2/D,) aW/a[=O
cos4 9 W = 0,
(5)
atl=Oandl=b. (6)
As in previous work [ 171, one can obtain a solution satisfying both the governing equation (5) and the boundary conditions (6). As a solution, one assumes the trigonometric form, W = E k=l
i
Wkl [sin
(k?r5/a)+A153+A2r2+A35+A41
I=1
(17T5/b)+8153+B252+B35+B4],
x[Sh
(7)
where the constants Ai and Bi (i = 1,2,3,4) are determined by requiring that the deflection W satisfies the boundary conditions (6). By substituting equation (7) into equation (5), one obtains f
i
wkl[Ekll
‘+hJJ2/~x)
CM4
$Ek/zl=
0,
k=ll=l
where the Ek,i (i = 1,2) are functions of LJand 5, and can be expanded in trigonometric series form as Ekli =
Fi
m=l n=l
Bmknli Sin
(mT[/U)
(nr[/b).
Sh
(9)
By substituting equation (9) into equation (8), one obtains ;
;
Wkl[&kn,r - (Phow2/o,)
cos4 ‘@mkn/z]= 0,
m=l,2
,...,
K,n=l,2
,...,
L.
k=ll=l (10)
The frequency equation for determining numerically orthotropic skew plate shown in Figure 1 is then det ]Bmknlr- (phow2/DX) cOs4 ti&,kn12]= 0,
the natural frequencies
of the
m, k = 1,2, . . . , K, n, 1= 1,2, . . . , L. (11)
The solutions of equation (11) converge to the exact natural frequencies when K and L approach infinity. However, it is impossible in practice to determine the solution when K and L are infinite. Thus the accuracy of equation (11) for finite K and L must be considered, and in particular the rate of convergence to exact values as K and L are increased. Table 1 shows the convergence of the natural frequency w (DX)Jphoa4/DX of isotropic skew plates with various skew angles (I and ratios of (b/u) sin $, together with values reported by other investigators. Table 2 shows the convergence for orthotropic skew plates. From the tables it appears that the convergence of the natural frequency is sufficient for a practical calculation when K and L are larger than 15. Hence, the numerical calculations reported in what follows were carried out by using 15 x 15 terms.
T. SAKATA AND T. HAYASHI
290
TABLE 1 Convergence
K=L
of the natural frequency w(D,)Jph0a4/D, of isotropic skew plates and the natural frequency as obtained by other investigators
(b/a) sin 4 *
0.5 30”
1.0 10” 23.2573 23.2589
15
Durvasula [l] Nagaya [61
1.0 60”
30.6754
110.2313
46.0875 46.0858
32.1906 32.1905
30.6875 30.6872
108.4811 108.2240
46.1404
32.2243
30.7043
32.1612
46.0248
23.2547
5
10
1.5 30”
I.0 30”
45.63
TABLE 2 Convergence of the natural frequency o(D,, H, D,)a H/D, = 1.0 and D,/Dx K=L\ 5
10 15
(b/a) sin $ 4
O-5 10” 23.7514 23.7679 237746
1.0
1.5
20” 25.7951 25.8063 25.8101
30” 29.5798 29.5853 29.5859
of orthotropic skew plates; = O-25
2-o
2.5
35.9372 35.9360 35.9342
46.9004 46.8895 46.8856
40”
50”
3.0
60”
68.3348 68.3075 68.2991
3. APPROXIMATE FORMULA FOR NATURAL FREQUENCY OF ISOTROPIC
SKEW PLATE Consider a set of isotropic skew plates made by enlarging or shortening the skew plate shown in Figure 1 proportionally in the y direction only. For such plates application of the reduction method [15, 161 yields an approximate formula for the natural frequency w(D,) as D41 w (D,)’ = (Dx/ph0a4)[B1
+ Bz tan’ (I + B3 tan4 $1,
(12)
where Bi (j = 1,2,3) are the constants (one can also derive this formula from equation (A13) of the Appendix by letting H = D, = D,). When the natural frequencies of at least three such plates are known, for a given value of (b/a) sin 4, the constants Bt (j = 1,2,3) can be determined from equation (12). Thus to derive an accurate approximate formula several natural frequencies are necessary. Table 3 shows the constants Bj determined for several values of (b/u) sin + and the corresponding errors in the natural frequency estimated from equation (12) by using these constants. The skew angles considered here are in the range 10”G 1,9< 60”. 4. NATURAL FREQUENCY OF ORTHOTROPIC
SKEW PLATE
As has been previously reported [14], one can estimate the natural frequency of an orthotropic plate as well as that of an isotropic one when the approximate formula for the isotropic plate obtained from the orthotropic one by the method of reduction is derived in the form of equation (12). In this section the procedure for estimating the natural frequency is explained briefly and the accuracy of the procedure is demonstrated
ORTHOTROPIC
PLATE
NATURAL
TABLE
291
FREQUENCIES
3
The constants Bi (j = 1,2,3) of equation (12) determined for the isotropic skew plate for various values of (b/a) sin (I, and the error of equation (12) when the constants Bi (j = 1,2,3) are used (b/u)
sin 4
Error (%)
BI
O-5
499.17 497.07 499.35 500.06 500.31 500.43
1-o l-5 2.0 2.5 3.0
2186.3 1388.6 1158-O 1081.8 1049.8 1033.7
8063.6 782.22 541.59 513.14 506.05 503.31
*0*21 *l-o0 *o*68 ztO.16 ho.08 *o-o5
by comparisons with numerical and experimental results. For the case of a skew plate, it is convenient to classify orthotropic plates into two types, as shown in Figures 1 and 2. The x and y axes of the figures are taken along the principal elastic axes of the plate material, and flexural rigidities of the plate are given by equation (2). Figure 1 shows a specially orthotropic skew plate, which has two edges parallel to one of principal elastic axes of the plate material. Figure 2 shows a generally orthotropic skew plate, which has no edges parallel to the principal elastic axes. Y t
Figure 2. The co-ordinate
4.1.
APPROXIMATE TROPIC
FORMULA
systems for isotropic and generally orthotropic
FOR
NATURAL
FREQUENCY
OF
skew plates.
SPECIALLY
ORTHO-
SKEW PLATE
Consider a specially orthotropic skew plate as shown in Figure 1. For the specially orthotropic plate, one can numerically determine the natural frequency by using equation (11). However, one can also derive, very easily, an approximate formula for the natural frequency from the approximate formula for the isotropic plate, equation (12), by using the reduction method [15,16]. According to the method, the approximate formula for the natural frequency w (OX,H, D,) of a set of orthotropic skew plates made by enlarging or shortening the skew plate shown in Figure 1 in the y direction only is given by [14] (also see equation (A13)) w(D,, H, D,)’ = (l/phoa4)[BIDx + B2H tan2 I,++ BsD, tan4 (11.
(13)
Here the constants Bi (j = 1,2,3) are exactly the same as those tabulated in Table 3. Table 4 shows the approximate values estimated from equation (13) and the values
292
AND 7‘. HAYASHI
7‘. SAKATA
TABLE 4 The accuracy (b/a) sin fJ
of equation
(13); HID,
= 1.0 and D,/Dx
ti
0.5 10”
1.0 20”
1.5 30”
2.0 40”
Equation (13) Equation (11) Error (%)
23.8557 23.7746 0.34
26.1621 25.8101 1.36
30.0066 29.5859 1.42
36.4052 35,9342 1.31
= 0.25
2.5 SO” 47.3974 46.8856 1.09
3.0 60” 68.8039 68.2991 0.74
determined numerically from equation (11) by using 15 x 15 terms, for which the convergence has been previously shown in Table 2. From Table 4 it follows that one can estimate accurately the natural frequency of the specially orthotropic skew plate from equation (13) by using the constants Bi (i = 1,2,3) tabulated in Table 3. FREQUENCY OF GENERALLY ORTHOTROPIC SKEW PLATE 4.2. NATURAL Consider now a generally orthotropic skew plate as shown in Figure 2, instead of the specially orthotropic plate shown in Figure 1. Since the co-ordinate axes x and y are parallel to the principal elastic axes of the plate material, the orthotropic skew plate shown in Figure 2 is indeed a generally orthotropic skew plate. According to the reduction method [ 15,161, one can easily estimate the natural frequency of the generally orthotropic skew plate in the same manner as for the specially orthotropic skew plate. The natural frequency of a set of isotropic plates made by enlarging or shortening the skew plate shown in Figure 2 in the y direction only is given by [14]
w(D,)*
= (D,/ph0e4)[C1
+ C, cot2 (8 - 4) + C, cot4 (@-4)],
(14)
where e = b cos (7~- 8 + 4) and Ci (i = 1,2,3) are the constants (one can also derive this formula from equation (A15) of the Appendix by letting H = D, = D,). The constants Cj (i = 1,2,3) can be determined easily from equation (14) when the natural frequencies of several isotropic skew plates are known, as explained in section 3. On the other hand, as obtained by application of the reduction method, an approximate formula for the natural frequency o(D,, H, DY) of the orthotropic plate shown in Figure 2 is [14] (see also equation (A15) of the Appendix) w(D,, H, D,)2 = (l/ph0e4)[C1D,
+ C2H cot2 (19- 4) + C,D,
cot4 (~3- 4)],
(15)
where e = b cos (7~- 8 + 4) and the constants Cj (i = 1,2,3) in equation (15) are exactly the same as the constants Cj (i = 1,2,3) in equation (14). By using equation (15) one can easily estimate the natural frequency of the generally orthotropic skew plate. The circled points in Figure 4 of section 4.3 are the numerical values estimated by using this approach. For these examples, the edge lengths were assumed to be equal and the angle 4 ranged from -30” to 60” in steps of 15”. The flexural rigidities were D, = O-871 Nm, H = 0*381D, and D, = 0.3920,. Since the natural frequency of a generally orthotropic skew plate cannot be determined by using equation (ll), experimental studies were carried out for a few orthotropic skew plates to examine how the natural frequency determined numerically differs from that obtained experimentally. The results are shown in the following section. 4.3.
JUSTIFICATION
OF NUMERICAL
RESULTS
To justify the numerical results experimental studies were carried out. First, to examine the effectiveness of the method of clamping, experiments were carried out for 10
ORTHOTROPIC
PLATE NATURAL
293
FREQUENCIES
/
ho
Figure 3. Cross section of the clamping device. Dimensions in mm.
aluminum square plates with a thickness of 0.8 mm and an edge length of 134.5 mm. These plates can be considered to be isotropic. Figure 3 shows the cross section of the clamping device. For the vibration pick-up a “GAP-SENCER” was located at the mid-point of the plate, this being a non-contacting small displacement measuring system in which the principle of high frequency eddy current loss is utilized. Table 5 shows the results obtained for the natural frequencies together with the theoretical clamped natural frequency. One can conclude from these results that the clamping device is sufficiently effective. Second, the natural frequency of an isotropic rhombic plate with a skew angle of 30” was determined. Four aluminum rhombic plates with a thickness of 0.8 mm and a side TABLE
5
The natural frequency f (Hz) of isotropic square plates determined experimentally and calculated accurutely by using equation (11); a = 134.5 mm, ho = O-8 mm and D, = 3.44 x 10e2 Nm Test piece
f U-W
No. 1 No. 2
387
No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10 Equation
(11)
388 392 391 387 391 390 385 388 387 391.8
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t7*9ZOI
(ST) uoynba 8ysn rCqpaup23q6 iInsaJ px+raurnN
00
OS1
.OE
6SOI
LPlI
2811
= ‘(7 PUv “QT8E.O= H
‘WN IL8*0= 27
IInsal laiuauyadxa @
‘,OE= !i’ ‘uuu 8Z.E
= oy tutu 0.0~~ = v f @.+aurnu puv hllvluau+adxa pau!waaap salvld 3!qtuo+ D!dorloyjro h]p~auai3 40 (ZH) J hwanbaJj pmlvu aye
103 umoqs Su!aq 0sIa A~pzy~aurnu paurwaiap sayanbaq IaJnieu aqi ‘L alqeL u! uhtoqs a.w A1IaluayJadxa pau!unaiap sapuanbaq lamleu aqL ‘“aZ6c.O = “0 pue “ar8c.O = H ‘uIN 1L8.0 = “a ale rCIpwau+adxa pau!waiap dad aqi 30 say!p!t$ Iernxaj3 au *uru~0.0~1 Jo qi8ual a%pa ue pue uu.u 8Z.c 30 ssauy3rql e aAeq pue ‘(dad) apstqd paDJo3u!aJ .xaqy 30 iaaqs B 30 aplew alaw .Os 30 aI%ulrMaJs e qym sawqd 3!qmoql asaqL -s!xe ~ysala Isd!3upd aql pue aleid aql 30 a%pa aql uaamlaq ‘,OE pue &T ‘OO= + ‘sa@xe aaJqi ~03 pau!uuaiap SBM aleId D!quroql D!do.xioqlJo ua 30 huanbaq ~empu aqi lC~pzu!d ‘(Z1) uoyenba ‘tqntu~o3 awuqxoldda aq$ uxo.13buanbal3 Iemleu aie.xnDw ue uyqo ue3 auo ~eq~smo~~03I! aIqa$ aqi I.IIOI~*l(lpz)uauqladxa pue blp?~!Jauxnu paupuialap se saguanbaq IeJnieu aqi SMoqs 9 a!qeL *pau!wexa alaM unu 0.0~1 30 q$Sual
asard IsaL
(ZH)/
P “‘N E ‘ON Z “‘N I ‘ON
19P 19P 199 9SP
(11) uwnba
T.E9P T.f.9P
(ZI) =o!wnba
~N,_OTU’~~=“apuv .OE = f/I ‘UIUI8.0 = Ot/ ‘UIUIO.OpI = v :(z~) pun (11) suoyvnba Bu!sn hq h1pxyAatunu pa~vtuysa h~~v~uau+adxa paulu4 Puv -.ialap sawid xqtuoy 3!dorlos! Jo (ZH) J hmanbarJ pmlvu aq~ 9 IHSV.\VH
3lfIV~
‘.L aNV
VLVXVS
‘L
P6Z
ORTHOTROPIC PLATE NATURAL FREQUENCIES
I-0
Frequency I
295
I-I
(k&l
Figure 4. The natural frequency f (kHz) of the generally orthotropic skew plate shown in Figure 2 for variousangles 4; D, = 0.871 Nm, H = O-3810,, D, = 0*392D,, a = b = 140-O mm, ho = 3.28 mm and 19= 120”. -O-,
Numerical results; Cl, experimental results.
5. CONCLUSIONS To estimate the natural frequency of a clamped orthotropic skew plate a numerical approach has been proposed and the accuracy of the approach has been examined numerically and experimentally for typical plates. First, for the estimation of the natural frequency of an isotropic plate an approximate formula with three terms has been derived, the three coefficients being determined by solving the governing differential equation with boundary conditions in at least three particular cases. Numerical discussions for typical plates with various aspect ratios and skew angles have shown that one can accurately estimate the natural frequency by using the approximate formula. Second, for estimating the natural frequency of a specially orthotropic plate (one pair of edges coinciding with a principal elastic axis) an approximate formula with three terms has been directly derived from the approximate formula for the isotropic plate by applying the reduction method. Numerical examination of the accuracy of the approximate formula has shown that one can obtain good results by using it. Next, it has been shown that one can estimate the natural frequency of a generally orthotropic plate from the proposed approximate formula for an isotropic plate, but that this approach can be somewhat inconvenient. By using the reduction method one can derive a more convenient approximate formula for the orthotropic plate in which one uses the natural frequencies of the isotropic plate reduced from the orthotropic one. Finally, typical natural frequencies of an orthotropic plate as well as an isotropic one were determined experimentally in order to examine the differences between the natural frequencies determined numerically from the approximate formulae and experimentally. Results have shown that the natural frequency estimated numerically is in good agreement with that determined numerically.
REFERENCES 1. S. DURVASULA 1969 American Institute of Aeronautics and Astronautics Journal 7, 11641167. Natural frequencies and modes of clamped skew plates. 2. P. S. NAIR and S. DURVASULA 1973 Journal of Sound and Vibration 26, 1-19. Vibration of skew plates. 3. T. MIZUSAWA, T. KAJITA and M. NARUOKA 1979 Journal of Sound and Vibration 62, 301-308. Vibration of skew plates by using B-spline functions. 4. J. R. KUTTLER and V. G. SIGILLITO 1980 Journal of Sound and Vibration 68, 597-607.
Upper and lower bounds for frequencies of clamped rhombical plates.
7‘. SAKATA
296
AND
T.
HAYASHI
5. M. M. BANERJEE 1979 Journal of Sound and Vibration 63, 377-383. On the vibration of skew plates of variable thickness. 6. K. NAGAYA 1980 BulletinofJapanSocietyofMechanicalEngineers 46, 1017-1023. Vibration of a plate with straightline boundaries. 7. A. W. LEISSA 1969 NASA W-160. Vibration of plates. 8. A. W. LEISSA 1977 The Shock and Vibration Digest 9(10), 13-24. Recent research in plate vibrations: classical theory. 9. L. BAUER and E. L. REISS1973 Journal of the Acoustical Society of America 54, 1373-1375. Free vibration of rhombic plates and membranes. 10. R. S. SRINIVASANand K. MUNASWAMY 1975 Journal of Sound and Vibration 39,207-216. Frequency analysisof skew orthotropic point supported plates. 11. M. A. DOKAINISHand K. KUMER 1973 American Institute of Aeronautics and Astronautics Journal 11, 1618-1621. Vibrations of orthotropic parallelogramic plates with variable thickness. 12. P. A. A. LAURA, L. E. LUISONI and G. SANCHEZ SARMIENTO 1980 Journal of Sound and Vibration 70, 77-84. A method for the determination of the fundamental frequency of orthotropic plates of polygonal boundary shapes. 13. T. SAKATA 1976 Journal of Sound and Vibration 48, 405-412. A reduction method for problems of vibration of orthotropic plates. 14. T. SAKATA (to appear) International Journal of Mechanical Sciences. Approximate formulae for natural frequencies of simply supported skew plates. 15. T. SAKATA 1979 The Shock and Vibration Digest 11(6), 19-22. Reduction methods for problems of vibration of orthotropic plates. 16. T. SAKATA 1977 Journal of the Franklin Institute 303, 415-424. Generalized procedure of the reduction method for vibrating problems of orthotropic plates. 17. T. SAKATA and Y. SAKATA 1977 Journal of the Acoustical Society of America 61,982-985. Approximate formulas for natural frequencies of rectangular plates with linearly varying thickness.
APPENDIX By using the reduction method one can derive approximate formulae for estimating the natural frequencies of both orthotropic and isotropic plates. The reduction method has been explained in detail in references [13] and [16], and summarized in reference [15]. The method has been previously applied to a simply supported orthotropic skew plate [14]. In this appendix the reduction method and how it is applied to a skew plate are briefly explained. Consider two kinds of clamped plates. The first is an orthotropic plate with flexural rigidities Ox, H and D,, with boundary shape given by T(x, y) = 0.
(AI)
The typical lengths of the plate in the x and y directions second kind of plate is also orthotropic and has flexural boundary shape is given by
are e and d, respectively. rigidities D,, H* and 0;.
T*(X, y*) = 0.
The The
(AZ)
The typical lengths of this plate in the x and y* directions are e and d*, respectively. The co-ordinate axes are taken along the principal elastic axes of the respective plate materials. The thickness and mass density of both plates are assumed to be respectively equal, and are denoted by ho and p, respectively. The natural frequency is denoted by w(D,, H, D,) for the first plate, and by o*(D,, H*, 0:) for the second. When the relations H* = (d*/d)*H,
0;
= (d*/d)4D,,
r*(x,
y*) = r(x,
dy*/d*)
(A3, A4)
ORTHOTROPIC
PLATE
NATURAL
297
FREQUENCIES
are satisfied the governing equation and boundary conditions of the first plate reduce to those of the second under the transformation y* = (d*/d)y.
(W
Thus one has w*(Dx,H*,D~)=w(D,,H,D,).
(Ah)
On the other hand, the natural frequency o(D,, H, DY) of the first plate can be considered to be a function of the flexural rigidities D,, I-Z and D, when the remaining material properties, boundary shape and boundary conditions are given. By using Taylor’s theorem and neglecting higher order differentiations, one has o(D*, H, D$
= (l/phoe4)[C;D,
+ c;IY+ GO,],
(A7)
where Ci (j = 1,2,3) are constants. Applying the transformation (AS) to the first plate reduces it to the second under the conditions given by equations (A3) and (A4). Hence the natural frequency w*(D,, H*, 0:) of the second plate is given by w*(D,, H*, 0:)” = (l/phoe4)[C;D,
+ C;H*(d/d*)2+
C;D; (d/d*)4].
(AS)
Since the second plate is made by lengthening or shortening the first plate in the y direction only, the value of e/d is a constant. Equation (AB) therefore can be rewritten as w*(D,, H*, 0;)’ = (l/phoe4)[CTD,
+ CzH*(e/d*)2+
CTD;(e/d*)4],
(449)
where CT (j = 1,2,3) are constants. Equation (A9) shows that the natural frequency w(D,, H, D,) of a set of orthotropic plates made by lengthening or shortening a given orthotropic plate in the y direction only is approximated by w(D,, H, D,)2 = (ll&oe4)[GDX
+ C2H(eld)2+
C9,(eld)41,
(A101
where Cj (j = 1,2,3) are constants. The reduction method can be applied to a clamped skew plate and approximate formulae for the natural frequency desired as follows. First, consider the specially orthotropic skew plate shown in Figure 1. As the typical lengths in the n and y directions, one can choose e = b sin 4,
d = b cos t,k
(All)
From equations (AlO) and (Al 1) the natural frequency o (D,, H, D,) of the plate is given by w(D,, H, D,)2 = (l/phoe”)[B;D,
+BkH tan2 $+BiD,
tan4 $1.
(A12)
Since the value of e is a constant in this case, one can rewrite equation (A12) as w (Ox, H, D,)2 = (l/ph0a4)[BIDX
+ B2H tan2 $ + BsD, tan4 ~1.
(A13)
Second, consider the generally orthotropic skew plate shown in Figure 2. As the typical lengths in the x and y directions one can choose e=bcos(r-0+4),
d=bsin(r-8+r$).
(A14)
By substituting e and d into equation (AlO), the natural frequency o(D,, H, D,) of the plate can be expressed as w(D,, H, D,)’ = (l/phoe4)[GD,
+ GH cot2 (0 -4) + GD,
cot4 (8 -4)].
(A15)
298
T. SAKATA
AND
T. HAYASHI
In this case 8 and C$are not independent variables, but there is a relation between them as follows. Assume 8 = &, 9 = &, a = a0 and b = b. for the given plate. Between the given plate and the plate made by enlarging or shortening the given plate in the y direction only, there are the relationships a cos C$= a0 cos do,
bc0s(~-~+~)=b0c0s(~-~0+~0),
a sin O/a0 sin &, = b sin (7~- 0 + d)/bo
sin (T -
Bo+
~$0).
(‘416)
From equations (A16) one has tan (4 - 0) = tan 4 tan (40- eo)/tan CJ~~.
(A17)
Equations (AH) are valid only for a set of skew plates for which the angles 0 and 4 are related by equation (A17).
NATURAL
FREQUENCIES
OF CLAMPED
ORTHOTROPIC
SKEW PLATES T.
SAKATA
AND
T.
HAYASHI
Chubu Institute of Technology, Department of Mechanical Engineering, Kasugai, Nagoya-sub., 487 Japan (Received 10 March 1981, and in
revised form 28 July 1981)
Approximate formulae are proposed for estimating natural frequencies of isotropic and specially orthotropic skew plates with clamped sides. It has been shown previously that one can estimate a natural frequency of a generally orthotropic skew plate with clamped sides by using an approximate formula for the isotropic plate which one can relate to the orthotropic one by applying a previously described reduction method. The accuracy of the proposed approximate formulae is demonstrated by comparing numerical and experimental results for several typical cases.
1. INTRODUCI’ION
Since a skew plate is an important structural member, many studies of free and forced vibrations of an isotropic skew plate have been reported. Durvasula [l] calculated natural frequencies and modes of a clamped skew plate, and Nair and Durvasula [2] calculated natural frequencies and modes of skew plates with various boundary conditions. Mizusawa, Kajita and Naruoka [3] proposed a Rayleigh-Ritz method with B-spline functions as co-ordinate functions for the determination of natural frequencies of skew plates and discussed the accuracy and convergence of the solution. Kuttler and Sigillito [4] calculated upper and lower bounds for natural frequencies of a clamped rhombic plate. Ban’erjee [5] discussed free and forced vibrations of a clamped skew plate with variable thickness. Nagaya [6] proposed an approach for free vibration of a plate with straight line boundaries and determined natural frequencies of clamped and simply supported rhombic plates. Work up to 1965 [7] and recent work over the period 1973-76 [8] have been reviewed by Leissa. For a simply supported polygonal plate one can easily estimate the natural frequency from the natural frequency of a clamped membrane with the same shape as the plate [7]. By using this fact, Bauer and Reiss [9] estimated natural frequencies of a simply supported rhombic plate. On the other hand, in spite of their importance, one can find only a few reports on vibration of orthotropic skew plates. Srinivasan and Munaswamy [lo] discussed free vibration of a point supported skew plate by using a finite strip method. Dokainish and Kumer [ll] determined natural frequencies of a clamped skew plate with variable thickness. Laura, Luisoni and Sarmiento [12] proposed a numerical approach for determining natural frequencies of an orthotropic polygonal plate with clamped or simply supported edges, although they did not apply the approach to a skew plate. Sakata showed that one can obtain an exact solution for the free vibration of a simply supported skew plate with particular flexural rigidities and skew angle [13]. Subsequently he proposed simple approximate formulae for estimating natural frequencies of simply supported skew plates from the natural frequency of clamped skew membranes [14]. 287 0022-460X/82/060287
+ 12 $02.00/O
@ 1982 Academic Press Inc. (London)
Limited
-1. SAKATA
288
AND
7‘. HAYASHI
These formulae are derived by applying his reduction method [15, 161, and using the fact that the natural frequency of a simply supported skew plate can be estimated quite easily from that of a clamped membrane with the same shape. In the present paper, by using the reduction method and the natural frequencies of clamped isotropic skew plates, simple approximate formulae are derived in the same manner as in reference [14]. The accuracy of the natural frequencies estimated from the approximate formulae is examined by comparison with those determined numerically and experimentally. A brief summary of the reduction method and description of how to apply it to a clamped skew plate are given in the Appendix.
2.
Consider co-ordinate
MATHEMATICAL
ANALYSIS
an orthotropic skew plate as shown in Figure 1, in which the rectangular axes, x and y, are parallel to the principal elastic axes of the plate material
Figure
1. The co-ordinate
systems
for isotropic
and specially
orthotropic
skew plates.
and the oblique co-ordinate axes, 5 and 5, are taken along the edges. The plate has uniform small thickness, and it is clamped along all the edges. According to the classical small deflection theory, the governing equation for free vibration of a thin orthotropic plate is (1)
D,(~4W/~x4)+2H(~4w/ax2ay’)+~,(a4W/ay4)-phOw2W=0, where W is the deflection, p is the mass density of the plate thickness and w is the radian frequency. The flexural rigidities
material, ho is the plate D,, H and D, are given
by D, = E&312(1
H = v,D, + Gh:/6,
- Y~Y~),
D, = E,h;/12(1-
vxv,),
where E,, E,, G, vx and v,, are elastic constants defined by the stress-strain relations the orthotropic material [7]. Since the plate is clamped along all the edges, the boundary conditions are
w =o,
d W/an = 0
along all the edges, where n denotes the direction perpendicular The relations between the rectangular and oblique co-ordinate 5=x--y
tan*,
l=
y/cos *.
(2) of
(3) to each edge. systems are (4)
ORTHOTROPIC
Under the transformation
PLATE
NATURAL
FREQUENCIES
289
(4), equations (1) and (3) become
sin’ CLcos2 1/1+(I&/D,) sin4 4,) a4 w/at4
[{cos4 IJ?+ 2(H/D,)
- 4{(H/D,) sin tj cos2 IJJ+ (Q/OX) sin3 $} a4 w/ae3 ag + 2{(H/D,)
- 4{(0,/0,) W=O,
~03~ 4 +
sin’ JI} a4 w/at2 al2
3(0,/D,)
sin +I) a4 w/a6 al3 + (D,/D,)
aW/&$=O
a4 w/al”] W=O,
at[=Oande=a,
- (phow2/D,) aW/a[=O
cos4 9 W = 0,
(5)
atl=Oandl=b. (6)
As in previous work [ 171, one can obtain a solution satisfying both the governing equation (5) and the boundary conditions (6). As a solution, one assumes the trigonometric form, W = E k=l
i
Wkl [sin
(k?r5/a)+A153+A2r2+A35+A41
I=1
(17T5/b)+8153+B252+B35+B4],
x[Sh
(7)
where the constants Ai and Bi (i = 1,2,3,4) are determined by requiring that the deflection W satisfies the boundary conditions (6). By substituting equation (7) into equation (5), one obtains f
i
wkl[Ekll
‘+hJJ2/~x)
CM4
$Ek/zl=
0,
k=ll=l
where the Ek,i (i = 1,2) are functions of LJand 5, and can be expanded in trigonometric series form as Ekli =
Fi
m=l n=l
Bmknli Sin
(mT[/U)
(nr[/b).
Sh
(9)
By substituting equation (9) into equation (8), one obtains ;
;
Wkl[&kn,r - (Phow2/o,)
cos4 ‘@mkn/z]= 0,
m=l,2
,...,
K,n=l,2
,...,
L.
k=ll=l (10)
The frequency equation for determining numerically orthotropic skew plate shown in Figure 1 is then det ]Bmknlr- (phow2/DX) cOs4 ti&,kn12]= 0,
the natural frequencies
of the
m, k = 1,2, . . . , K, n, 1= 1,2, . . . , L. (11)
The solutions of equation (11) converge to the exact natural frequencies when K and L approach infinity. However, it is impossible in practice to determine the solution when K and L are infinite. Thus the accuracy of equation (11) for finite K and L must be considered, and in particular the rate of convergence to exact values as K and L are increased. Table 1 shows the convergence of the natural frequency w (DX)Jphoa4/DX of isotropic skew plates with various skew angles (I and ratios of (b/u) sin $, together with values reported by other investigators. Table 2 shows the convergence for orthotropic skew plates. From the tables it appears that the convergence of the natural frequency is sufficient for a practical calculation when K and L are larger than 15. Hence, the numerical calculations reported in what follows were carried out by using 15 x 15 terms.
T. SAKATA AND T. HAYASHI
290
TABLE 1 Convergence
K=L
of the natural frequency w(D,)Jph0a4/D, of isotropic skew plates and the natural frequency as obtained by other investigators
(b/a) sin 4 *
0.5 30”
1.0 10” 23.2573 23.2589
15
Durvasula [l] Nagaya [61
1.0 60”
30.6754
110.2313
46.0875 46.0858
32.1906 32.1905
30.6875 30.6872
108.4811 108.2240
46.1404
32.2243
30.7043
32.1612
46.0248
23.2547
5
10
1.5 30”
I.0 30”
45.63
TABLE 2 Convergence of the natural frequency o(D,, H, D,)a H/D, = 1.0 and D,/Dx K=L\ 5
10 15
(b/a) sin $ 4
O-5 10” 23.7514 23.7679 237746
1.0
1.5
20” 25.7951 25.8063 25.8101
30” 29.5798 29.5853 29.5859
of orthotropic skew plates; = O-25
2-o
2.5
35.9372 35.9360 35.9342
46.9004 46.8895 46.8856
40”
50”
3.0
60”
68.3348 68.3075 68.2991
3. APPROXIMATE FORMULA FOR NATURAL FREQUENCY OF ISOTROPIC
SKEW PLATE Consider a set of isotropic skew plates made by enlarging or shortening the skew plate shown in Figure 1 proportionally in the y direction only. For such plates application of the reduction method [15, 161 yields an approximate formula for the natural frequency w(D,) as D41 w (D,)’ = (Dx/ph0a4)[B1
+ Bz tan’ (I + B3 tan4 $1,
(12)
where Bi (j = 1,2,3) are the constants (one can also derive this formula from equation (A13) of the Appendix by letting H = D, = D,). When the natural frequencies of at least three such plates are known, for a given value of (b/a) sin 4, the constants Bt (j = 1,2,3) can be determined from equation (12). Thus to derive an accurate approximate formula several natural frequencies are necessary. Table 3 shows the constants Bj determined for several values of (b/u) sin + and the corresponding errors in the natural frequency estimated from equation (12) by using these constants. The skew angles considered here are in the range 10”G 1,9< 60”. 4. NATURAL FREQUENCY OF ORTHOTROPIC
SKEW PLATE
As has been previously reported [14], one can estimate the natural frequency of an orthotropic plate as well as that of an isotropic one when the approximate formula for the isotropic plate obtained from the orthotropic one by the method of reduction is derived in the form of equation (12). In this section the procedure for estimating the natural frequency is explained briefly and the accuracy of the procedure is demonstrated
ORTHOTROPIC
PLATE
NATURAL
TABLE
291
FREQUENCIES
3
The constants Bi (j = 1,2,3) of equation (12) determined for the isotropic skew plate for various values of (b/a) sin (I, and the error of equation (12) when the constants Bi (j = 1,2,3) are used (b/u)
sin 4
Error (%)
BI
O-5
499.17 497.07 499.35 500.06 500.31 500.43
1-o l-5 2.0 2.5 3.0
2186.3 1388.6 1158-O 1081.8 1049.8 1033.7
8063.6 782.22 541.59 513.14 506.05 503.31
*0*21 *l-o0 *o*68 ztO.16 ho.08 *o-o5
by comparisons with numerical and experimental results. For the case of a skew plate, it is convenient to classify orthotropic plates into two types, as shown in Figures 1 and 2. The x and y axes of the figures are taken along the principal elastic axes of the plate material, and flexural rigidities of the plate are given by equation (2). Figure 1 shows a specially orthotropic skew plate, which has two edges parallel to one of principal elastic axes of the plate material. Figure 2 shows a generally orthotropic skew plate, which has no edges parallel to the principal elastic axes. Y t
Figure 2. The co-ordinate
4.1.
APPROXIMATE TROPIC
FORMULA
systems for isotropic and generally orthotropic
FOR
NATURAL
FREQUENCY
OF
skew plates.
SPECIALLY
ORTHO-
SKEW PLATE
Consider a specially orthotropic skew plate as shown in Figure 1. For the specially orthotropic plate, one can numerically determine the natural frequency by using equation (11). However, one can also derive, very easily, an approximate formula for the natural frequency from the approximate formula for the isotropic plate, equation (12), by using the reduction method [15,16]. According to the method, the approximate formula for the natural frequency w (OX,H, D,) of a set of orthotropic skew plates made by enlarging or shortening the skew plate shown in Figure 1 in the y direction only is given by [14] (also see equation (A13)) w(D,, H, D,)’ = (l/phoa4)[BIDx + B2H tan2 I,++ BsD, tan4 (11.
(13)
Here the constants Bi (j = 1,2,3) are exactly the same as those tabulated in Table 3. Table 4 shows the approximate values estimated from equation (13) and the values
292
AND 7‘. HAYASHI
7‘. SAKATA
TABLE 4 The accuracy (b/a) sin fJ
of equation
(13); HID,
= 1.0 and D,/Dx
ti
0.5 10”
1.0 20”
1.5 30”
2.0 40”
Equation (13) Equation (11) Error (%)
23.8557 23.7746 0.34
26.1621 25.8101 1.36
30.0066 29.5859 1.42
36.4052 35,9342 1.31
= 0.25
2.5 SO” 47.3974 46.8856 1.09
3.0 60” 68.8039 68.2991 0.74
determined numerically from equation (11) by using 15 x 15 terms, for which the convergence has been previously shown in Table 2. From Table 4 it follows that one can estimate accurately the natural frequency of the specially orthotropic skew plate from equation (13) by using the constants Bi (i = 1,2,3) tabulated in Table 3. FREQUENCY OF GENERALLY ORTHOTROPIC SKEW PLATE 4.2. NATURAL Consider now a generally orthotropic skew plate as shown in Figure 2, instead of the specially orthotropic plate shown in Figure 1. Since the co-ordinate axes x and y are parallel to the principal elastic axes of the plate material, the orthotropic skew plate shown in Figure 2 is indeed a generally orthotropic skew plate. According to the reduction method [ 15,161, one can easily estimate the natural frequency of the generally orthotropic skew plate in the same manner as for the specially orthotropic skew plate. The natural frequency of a set of isotropic plates made by enlarging or shortening the skew plate shown in Figure 2 in the y direction only is given by [14]
w(D,)*
= (D,/ph0e4)[C1
+ C, cot2 (8 - 4) + C, cot4 (@-4)],
(14)
where e = b cos (7~- 8 + 4) and Ci (i = 1,2,3) are the constants (one can also derive this formula from equation (A15) of the Appendix by letting H = D, = D,). The constants Cj (i = 1,2,3) can be determined easily from equation (14) when the natural frequencies of several isotropic skew plates are known, as explained in section 3. On the other hand, as obtained by application of the reduction method, an approximate formula for the natural frequency o(D,, H, DY) of the orthotropic plate shown in Figure 2 is [14] (see also equation (A15) of the Appendix) w(D,, H, D,)2 = (l/ph0e4)[C1D,
+ C2H cot2 (19- 4) + C,D,
cot4 (~3- 4)],
(15)
where e = b cos (7~- 8 + 4) and the constants Cj (i = 1,2,3) in equation (15) are exactly the same as the constants Cj (i = 1,2,3) in equation (14). By using equation (15) one can easily estimate the natural frequency of the generally orthotropic skew plate. The circled points in Figure 4 of section 4.3 are the numerical values estimated by using this approach. For these examples, the edge lengths were assumed to be equal and the angle 4 ranged from -30” to 60” in steps of 15”. The flexural rigidities were D, = O-871 Nm, H = 0*381D, and D, = 0.3920,. Since the natural frequency of a generally orthotropic skew plate cannot be determined by using equation (ll), experimental studies were carried out for a few orthotropic skew plates to examine how the natural frequency determined numerically differs from that obtained experimentally. The results are shown in the following section. 4.3.
JUSTIFICATION
OF NUMERICAL
RESULTS
To justify the numerical results experimental studies were carried out. First, to examine the effectiveness of the method of clamping, experiments were carried out for 10
ORTHOTROPIC
PLATE NATURAL
293
FREQUENCIES
/
ho
Figure 3. Cross section of the clamping device. Dimensions in mm.
aluminum square plates with a thickness of 0.8 mm and an edge length of 134.5 mm. These plates can be considered to be isotropic. Figure 3 shows the cross section of the clamping device. For the vibration pick-up a “GAP-SENCER” was located at the mid-point of the plate, this being a non-contacting small displacement measuring system in which the principle of high frequency eddy current loss is utilized. Table 5 shows the results obtained for the natural frequencies together with the theoretical clamped natural frequency. One can conclude from these results that the clamping device is sufficiently effective. Second, the natural frequency of an isotropic rhombic plate with a skew angle of 30” was determined. Four aluminum rhombic plates with a thickness of 0.8 mm and a side TABLE
5
The natural frequency f (Hz) of isotropic square plates determined experimentally and calculated accurutely by using equation (11); a = 134.5 mm, ho = O-8 mm and D, = 3.44 x 10e2 Nm Test piece
f U-W
No. 1 No. 2
387
No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10 Equation
(11)
388 392 391 387 391 390 385 388 387 391.8
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t7*9ZOI
(ST) uoynba 8ysn rCqpaup23q6 iInsaJ px+raurnN
00
OS1
.OE
6SOI
LPlI
2811
= ‘(7 PUv “QT8E.O= H
‘WN IL8*0= 27
IInsal laiuauyadxa @
‘,OE= !i’ ‘uuu 8Z.E
= oy tutu 0.0~~ = v f @.+aurnu puv hllvluau+adxa pau!waaap salvld 3!qtuo+ D!dorloyjro h]p~auai3 40 (ZH) J hwanbaJj pmlvu aye
103 umoqs Su!aq 0sIa A~pzy~aurnu paurwaiap sayanbaq IaJnieu aqi ‘L alqeL u! uhtoqs a.w A1IaluayJadxa pau!unaiap sapuanbaq lamleu aqL ‘“aZ6c.O = “0 pue “ar8c.O = H ‘uIN 1L8.0 = “a ale rCIpwau+adxa pau!waiap dad aqi 30 say!p!t$ Iernxaj3 au *uru~0.0~1 Jo qi8ual a%pa ue pue uu.u 8Z.c 30 ssauy3rql e aAeq pue ‘(dad) apstqd paDJo3u!aJ .xaqy 30 iaaqs B 30 aplew alaw .Os 30 aI%ulrMaJs e qym sawqd 3!qmoql asaqL -s!xe ~ysala Isd!3upd aql pue aleid aql 30 a%pa aql uaamlaq ‘,OE pue &T ‘OO= + ‘sa@xe aaJqi ~03 pau!uuaiap SBM aleId D!quroql D!do.xioqlJo ua 30 huanbaq ~empu aqi lC~pzu!d ‘(Z1) uoyenba ‘tqntu~o3 awuqxoldda aq$ uxo.13buanbal3 Iemleu aie.xnDw ue uyqo ue3 auo ~eq~smo~~03I! aIqa$ aqi I.IIOI~*l(lpz)uauqladxa pue blp?~!Jauxnu paupuialap se saguanbaq IeJnieu aqi SMoqs 9 a!qeL *pau!wexa alaM unu 0.0~1 30 q$Sual
asard IsaL
(ZH)/
P “‘N E ‘ON Z “‘N I ‘ON
19P 19P 199 9SP
(11) uwnba
T.E9P T.f.9P
(ZI) =o!wnba
~N,_OTU’~~=“apuv .OE = f/I ‘UIUI8.0 = Ot/ ‘UIUIO.OpI = v :(z~) pun (11) suoyvnba Bu!sn hq h1pxyAatunu pa~vtuysa h~~v~uau+adxa paulu4 Puv -.ialap sawid xqtuoy 3!dorlos! Jo (ZH) J hmanbarJ pmlvu aq~ 9 IHSV.\VH
3lfIV~
‘.L aNV
VLVXVS
‘L
P6Z
ORTHOTROPIC PLATE NATURAL FREQUENCIES
I-0
Frequency I
295
I-I
(k&l
Figure 4. The natural frequency f (kHz) of the generally orthotropic skew plate shown in Figure 2 for variousangles 4; D, = 0.871 Nm, H = O-3810,, D, = 0*392D,, a = b = 140-O mm, ho = 3.28 mm and 19= 120”. -O-,
Numerical results; Cl, experimental results.
5. CONCLUSIONS To estimate the natural frequency of a clamped orthotropic skew plate a numerical approach has been proposed and the accuracy of the approach has been examined numerically and experimentally for typical plates. First, for the estimation of the natural frequency of an isotropic plate an approximate formula with three terms has been derived, the three coefficients being determined by solving the governing differential equation with boundary conditions in at least three particular cases. Numerical discussions for typical plates with various aspect ratios and skew angles have shown that one can accurately estimate the natural frequency by using the approximate formula. Second, for estimating the natural frequency of a specially orthotropic plate (one pair of edges coinciding with a principal elastic axis) an approximate formula with three terms has been directly derived from the approximate formula for the isotropic plate by applying the reduction method. Numerical examination of the accuracy of the approximate formula has shown that one can obtain good results by using it. Next, it has been shown that one can estimate the natural frequency of a generally orthotropic plate from the proposed approximate formula for an isotropic plate, but that this approach can be somewhat inconvenient. By using the reduction method one can derive a more convenient approximate formula for the orthotropic plate in which one uses the natural frequencies of the isotropic plate reduced from the orthotropic one. Finally, typical natural frequencies of an orthotropic plate as well as an isotropic one were determined experimentally in order to examine the differences between the natural frequencies determined numerically from the approximate formulae and experimentally. Results have shown that the natural frequency estimated numerically is in good agreement with that determined numerically.
REFERENCES 1. S. DURVASULA 1969 American Institute of Aeronautics and Astronautics Journal 7, 11641167. Natural frequencies and modes of clamped skew plates. 2. P. S. NAIR and S. DURVASULA 1973 Journal of Sound and Vibration 26, 1-19. Vibration of skew plates. 3. T. MIZUSAWA, T. KAJITA and M. NARUOKA 1979 Journal of Sound and Vibration 62, 301-308. Vibration of skew plates by using B-spline functions. 4. J. R. KUTTLER and V. G. SIGILLITO 1980 Journal of Sound and Vibration 68, 597-607.
Upper and lower bounds for frequencies of clamped rhombical plates.
7‘. SAKATA
296
AND
T.
HAYASHI
5. M. M. BANERJEE 1979 Journal of Sound and Vibration 63, 377-383. On the vibration of skew plates of variable thickness. 6. K. NAGAYA 1980 BulletinofJapanSocietyofMechanicalEngineers 46, 1017-1023. Vibration of a plate with straightline boundaries. 7. A. W. LEISSA 1969 NASA W-160. Vibration of plates. 8. A. W. LEISSA 1977 The Shock and Vibration Digest 9(10), 13-24. Recent research in plate vibrations: classical theory. 9. L. BAUER and E. L. REISS1973 Journal of the Acoustical Society of America 54, 1373-1375. Free vibration of rhombic plates and membranes. 10. R. S. SRINIVASANand K. MUNASWAMY 1975 Journal of Sound and Vibration 39,207-216. Frequency analysisof skew orthotropic point supported plates. 11. M. A. DOKAINISHand K. KUMER 1973 American Institute of Aeronautics and Astronautics Journal 11, 1618-1621. Vibrations of orthotropic parallelogramic plates with variable thickness. 12. P. A. A. LAURA, L. E. LUISONI and G. SANCHEZ SARMIENTO 1980 Journal of Sound and Vibration 70, 77-84. A method for the determination of the fundamental frequency of orthotropic plates of polygonal boundary shapes. 13. T. SAKATA 1976 Journal of Sound and Vibration 48, 405-412. A reduction method for problems of vibration of orthotropic plates. 14. T. SAKATA (to appear) International Journal of Mechanical Sciences. Approximate formulae for natural frequencies of simply supported skew plates. 15. T. SAKATA 1979 The Shock and Vibration Digest 11(6), 19-22. Reduction methods for problems of vibration of orthotropic plates. 16. T. SAKATA 1977 Journal of the Franklin Institute 303, 415-424. Generalized procedure of the reduction method for vibrating problems of orthotropic plates. 17. T. SAKATA and Y. SAKATA 1977 Journal of the Acoustical Society of America 61,982-985. Approximate formulas for natural frequencies of rectangular plates with linearly varying thickness.
APPENDIX By using the reduction method one can derive approximate formulae for estimating the natural frequencies of both orthotropic and isotropic plates. The reduction method has been explained in detail in references [13] and [16], and summarized in reference [15]. The method has been previously applied to a simply supported orthotropic skew plate [14]. In this appendix the reduction method and how it is applied to a skew plate are briefly explained. Consider two kinds of clamped plates. The first is an orthotropic plate with flexural rigidities Ox, H and D,, with boundary shape given by T(x, y) = 0.
(AI)
The typical lengths of the plate in the x and y directions second kind of plate is also orthotropic and has flexural boundary shape is given by
are e and d, respectively. rigidities D,, H* and 0;.
T*(X, y*) = 0.
The The
(AZ)
The typical lengths of this plate in the x and y* directions are e and d*, respectively. The co-ordinate axes are taken along the principal elastic axes of the respective plate materials. The thickness and mass density of both plates are assumed to be respectively equal, and are denoted by ho and p, respectively. The natural frequency is denoted by w(D,, H, D,) for the first plate, and by o*(D,, H*, 0:) for the second. When the relations H* = (d*/d)*H,
0;
= (d*/d)4D,,
r*(x,
y*) = r(x,
dy*/d*)
(A3, A4)
ORTHOTROPIC
PLATE
NATURAL
297
FREQUENCIES
are satisfied the governing equation and boundary conditions of the first plate reduce to those of the second under the transformation y* = (d*/d)y.
(W
Thus one has w*(Dx,H*,D~)=w(D,,H,D,).
(Ah)
On the other hand, the natural frequency o(D,, H, DY) of the first plate can be considered to be a function of the flexural rigidities D,, I-Z and D, when the remaining material properties, boundary shape and boundary conditions are given. By using Taylor’s theorem and neglecting higher order differentiations, one has o(D*, H, D$
= (l/phoe4)[C;D,
+ c;IY+ GO,],
(A7)
where Ci (j = 1,2,3) are constants. Applying the transformation (AS) to the first plate reduces it to the second under the conditions given by equations (A3) and (A4). Hence the natural frequency w*(D,, H*, 0:) of the second plate is given by w*(D,, H*, 0:)” = (l/phoe4)[C;D,
+ C;H*(d/d*)2+
C;D; (d/d*)4].
(AS)
Since the second plate is made by lengthening or shortening the first plate in the y direction only, the value of e/d is a constant. Equation (AB) therefore can be rewritten as w*(D,, H*, 0;)’ = (l/phoe4)[CTD,
+ CzH*(e/d*)2+
CTD;(e/d*)4],
(449)
where CT (j = 1,2,3) are constants. Equation (A9) shows that the natural frequency w(D,, H, D,) of a set of orthotropic plates made by lengthening or shortening a given orthotropic plate in the y direction only is approximated by w(D,, H, D,)2 = (ll&oe4)[GDX
+ C2H(eld)2+
C9,(eld)41,
(A101
where Cj (j = 1,2,3) are constants. The reduction method can be applied to a clamped skew plate and approximate formulae for the natural frequency desired as follows. First, consider the specially orthotropic skew plate shown in Figure 1. As the typical lengths in the n and y directions, one can choose e = b sin 4,
d = b cos t,k
(All)
From equations (AlO) and (Al 1) the natural frequency o (D,, H, D,) of the plate is given by w(D,, H, D,)2 = (l/phoe”)[B;D,
+BkH tan2 $+BiD,
tan4 $1.
(A12)
Since the value of e is a constant in this case, one can rewrite equation (A12) as w (Ox, H, D,)2 = (l/ph0a4)[BIDX
+ B2H tan2 $ + BsD, tan4 ~1.
(A13)
Second, consider the generally orthotropic skew plate shown in Figure 2. As the typical lengths in the x and y directions one can choose e=bcos(r-0+4),
d=bsin(r-8+r$).
(A14)
By substituting e and d into equation (AlO), the natural frequency o(D,, H, D,) of the plate can be expressed as w(D,, H, D,)’ = (l/phoe4)[GD,
+ GH cot2 (0 -4) + GD,
cot4 (8 -4)].
(A15)
298
T. SAKATA
AND
T. HAYASHI
In this case 8 and C$are not independent variables, but there is a relation between them as follows. Assume 8 = &, 9 = &, a = a0 and b = b. for the given plate. Between the given plate and the plate made by enlarging or shortening the given plate in the y direction only, there are the relationships a cos C$= a0 cos do,
bc0s(~-~+~)=b0c0s(~-~0+~0),
a sin O/a0 sin &, = b sin (7~- 0 + d)/bo
sin (T -
Bo+
~$0).
(‘416)
From equations (A16) one has tan (4 - 0) = tan 4 tan (40- eo)/tan CJ~~.
(A17)
Equations (AH) are valid only for a set of skew plates for which the angles 0 and 4 are related by equation (A17).