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High frequency vibrations and waves in laminated orthotropic plates
Journal of Sound and Vibration (1973) 30(1), 33-44
HIGH FREQUENCY VIBRATIONS AND WAVES IN LAMINATED ORTHOTROPIC PLATES R. B. NELSONAND S. B. DONG
School of Enghteering and Applied Science, University of California, Los Angeles, California 90024, U.S.A. (Received 3 AIarch 1973) The extended Ritz technique is used to study high frequency vibrations and waves in infinite homogeneous and laminated orthotropic plates. This technique utilizes exact expressions for the wave form along the extent of the plate, but models the plate's behavior through the thickness with a large number of generalized coordinates. A procedure is shown for computing the group velocities from the modal data. A number of examples are presented to' illustrate the modeling capability of this solution technique in the high frequency range. Frequencies, modal displacements and group velocities are obtained in the range 10-30 times above the lowest non-zero cut-off frequency and for a wide range of wave numbers. In addition, modal stress patterns are obtained and compared with analytical results. The method is shown to give accurate information over a large region of the frequency spectrum. Thus, it is a versatile and efficient, as well as computationally very efficient, means for generating data for investigating the transient response of composite plates including problems of impact.
1. INTRODUCTION Wave propagation in layered media is a subject of considerable importance to a number of areas ranging from geophysics to pavement design, and more recently to laminated composite structures. Conceptually, th e theory and formulation of the problem of wave propagation in a given layered system is straightforward. The analytical procedure involves the solution to the field equations for each layer and the imposition of stress and displacement continuity at interfaces and prescribed traction or displacement conditions at tile free surface(s) to yield a frequency equation. The difficulty lies not in the formalism of this approach, but in the cumbersome algebra necessary to generate and solve the frequency equation. Therefore, most analyses have been restricted to two- or three-layered systems and no analytical treatment of a completely arbitrary layered system is available. Recently, a numerical approach based on the extended Ritz technique was presented for the problem of plane waves in a general layered plate [I]. In this method, the infinite plate is discretized through the thickness into a number of subregions called laminas. Each lamina can have distinct mechanical and inertial properties as well as thickness. Explicit analytical expressions for the appropriate wave form along the extent of the plate are adopted at the outset and a quadratic interpolation function is taken through the lamina thickness. An algebraic eigenvalue problem results from which the modal information is extracted by use of a very efficient direct-iterative eigensolution technique [2]. This method was shown to give a very accurate representation of the physical behavior of isotropic plates for the lowest branches of the frequency spectrum for real wave numbers. Numerical results for a limited number of orthotropic and laminated plates were also presented in reference [1] to indicate the versatility of the procedure. 33
34
R. B. NELSON AND S. B. DONG
In this paper, a modification of the solution method for the algebraic eigensystem model of the layered plate in reference [1] is made to permit exploration of the frequency spectra in the high frequency ranges. This extension involves a shifting procedure [3] which is applied to the governing equations before using the direct-iterative eigensolution technique. Because higher frequencies and modal patterns are sought, the modeling capability used in the present approach must be examined. A detailed study is presented showing the accuracy and range of effectiveness for various diseretized models. In addition, it is shown how group velocities can be computed subsequent to the solution of the eigensystem. Group velocities are of importance in two respects. They represent the rates at which energies are being propagated, and since group velocities are also slopes of the branches in the frequency spectra, this information is highly useful in determining the branch-layout. Finally, in order to answer questions concerning the accuracy of the method for other than isotropic plates, some additional comparisons between analytical and numerical results for orthotropic and laminated plates are given.
2. EXTENDED RITZ FORMULATION For completeness, the derivation of the equations for planar waves in laminated plates is presented here, but in a different manner than that in reference [I], to emphasize the traveling form of the waves. Consider an infinite plate composed Of an arbitrary number of bonded elastic, homogeneous layers. Each layer can have distinct thickness, density and orthotropic material properties with its elastic axes coincident with the coordinate axes. Let xj be rectangular Cartesian coordinate measures and let t denote time. For plane harmonic waves propagating in the xl direction of this plate, the displacements uk (k = 1,2) are of the form
u,(x,, x~, t) = W,(x~) e~"o'-~x, ), (1)
Ua(Xl, x2, t) = iU2(x2) e u~t-ex'),
where o~is the circular frequency and ~ is the wave number, which is related to the wavelength 2 by =
~/~..
(2)
The focus of the analysis is the determination of the frequencies o and the associated modal distributions Uk(x2) over the entire thickness of the plate. In the extended Ritz method, the plate is initially divided into N subregions termed laminas. Note that a single layer of the plate can be modeled by a number of laminas. For the kth lamina, the functions Uk(x2) are approximated by interpolations of the form W.(x2) = Wkb{1-- 311+ 2q z} + Wk,.{411 -- 4112}+ Wkf{2q: -- q}
(k = 1,2),
(3)
where the coefficients U~b, U~.,, U , f are generalized coordinates representing displacement amplitudes at the lamina's back, middle and frontal surfaces (see Figure 1) and t1 is the normalized thickness measure defined by I1 = (x2 - x2sb)/hs,
(4)
with hj as the thickness and xzso the distance to the bottom of thejth lamina. Hamilton's principle in the form [3]
i
tO
' 6L dt = 0
(5)
~VAVES IN LAbIINATED ORTHOTROPIO PLATF.S
9
35
Typical lamina
....
i). /
(
,
Figure I. Ritz representation oflaminated orthotropic plate. provides the basis for generating the governing equations of the problem, where L is the Lagrangian. The contribution to L from thejth lamina is
L, = ~ f f r
,;~ + ,;,,;~J- k , , , , , . , , , * ,9
+ + ~,=r ,1 : 2,2 + t/I,I-112,2) * + c22(u2.2 it2.2) + c66(u,.2 + u2.1)(tq.2 + u*, a)]} dxl dx2,
(6)
where p is the mass density and c u are the elastic moduli. Commas and dots are the usual notations for spatial and time differentiations and the asterisk denotes the complex conjugate. Substituting equations (1) and (3) into equation (6) yields X2JD+hJ
Lj = 89f
dxl f
{p~2[u,= + u ] ] - [c. r v~ +
x2jb
+ 2c12~U1 0"2,2 + c22 U2,2 + %6(Uz.2 + r
(7)
Note that the integral involving x2 in equation (7) after integration contains only the generalized coordinates UAb, Uk,, and Uk: as unknowns: i.e., Lj = 89f d x t [(.o 2 rTmr -- rtkr],
(8)
where r r = [U~b, U2b, Ua,,, U2,,, Ut:, U2:] and m and k are lamina mass and stiffness matrices which are given in Appendix I. The summation of Lj for the N laminas and imposition of interlamina displacement continuity leads to the Lagrangian function L for the plate:
f
L = -~ dxa [co2 U r M U - U "rKU],
(9)
where M and Kare the plate mass and'stiffness matrices and U is a vector o f 4 N + 2 generalized coordinates. For the determination of the modal distribution U, the integral on xt can be factored outside of the variation of L in equation (5) since the xt dependence is explicit. Thus, the variational equation of motion by Hamilton's principle takes the form K U - to2 31U = 0.
(I0)
3. HIGH FREQUENCY ANALYSIS In reference [I], the solution to equation (10) was obtained by means of an efficient direct-iterative eigensolution technique [2]. In this method, a suitable subset of generalized coordinates is selected with a view of using them to reduce the rank of the eigensystem. This set is improved by a static analysis where the loads are those obtained by the matrix product
36
R . B . NELSON AND S. B. DONG
of the mass with the reduced generaiized coordinates, which can be interpreted as tile inertial distributions. Then, after the eigensystem is reduced, a direct solution is effected in the subspace. This procedure is repeated with the eigenvectors of the current cycle being used as the reduced generalized coordinates for the next cycle. Convergence ofthis iterative procedure is assured because the method is seen to be the well-known Stodola-Vianello method applied simultaneously to a group &vectors. This technique can be modified [4] to obtain eigendata for higher frequencies by letting the frequency to be decomposed into to2 = too2 + It,
(I I)
where coo is an arbitrary reference frequency. Upon substitution ofequation (11) into equation (I0), there results K U - id~IU = 0, (12) where g=K-w~3l. (13) The direct-iterative eigensolution technique can then be applied to equation (12). The computed eigendata contains eigenvalues It whose values are the smallest in absolute value of the eigensystem: i.e., values nearest COo,and their associated eigenvectors. (The actual frequencies are then obtained by use of equation (11).) Although the rate of convergence depends on the initial choice of displacement forms, it is quite rapid when more generalized coordinates are used than are required [4]. In this paper, as in reference [1], 16 reduced coordinates are used in order to obtain accurately the I0 eigensolutions nearest 6Oo. 4. GROUP VELOCITIES It is well-known that group velocities express the rate at which energies are transported [5, 6]. The definition of group velocity vz is vg = dw/d~:
(14)
i.e., it is the slope to a branch of the frequency spectrum. A common method of estimating v, has been by graphical differentiation of a frequency branch with respect to ~ since an explicit relation for the branch is not available or is cumbersome to obtain. Group velocities in the present analysis can be extremely useful in the construction of the layout of the frequency branches if they are known a priork Tolstoy [7, 8] and Biot [9] have given an interpretation of group velocities as ratios of energy density and flux, and this provides a simple relation for computing group velocities directly from eigendata. For the mth eigenvector, ~,~, Rayleigh's quotient is 2 ~ Kff~,. tom = ~ "
(15)
From the elemental matrices in Appendix I it is evident that only K depends on 3, so that differentiation of equation (15) gives v~,, 2to,, ~ M ~ , , '
(16)
K, = 0-~K(0.
(17)
where K~ is defined as
The matrix Kg is assembled from the elemental contributions kg = ak/O~. The coefficients of kg are given in Appendix I.
WAVES IN LAMINATED ORTHOTROPIC PLATES
37
Equation (16) can be simplified if the eigenvectors #,~ are normalized with respect to the mass matrix, # ~ M # = 1. Then equation (16) becomes I vv~ = 2r # ~ K , ffJ=.
(18)
In the present eigensolution technique, the normalization condition is built in So that it does not have to be performed aposteriorL 5. EXAMPLES Since equation (10) is an approximation to the actual frequency equation its modeling accuracy should be examined, particularly in the high frequency region where eigenvectors are typically very complicated and therefore difficult to model. Three different plates were investigated, in regions where analytical results can be obtained for comparison with similar results of the Ritz method. The examples, (a) a homogeneous isotropic plate (v -- 0.3), (b) a homogeneous orthotropic plate c22/c~z --0.01, and (c) a four-layer laminated orthotropic plate, were chosen to permit both analytical and numerical treatments for several different type plates. In the homogeneous plates 50 equal-thickness laminas were used, and 60 equalthickness laminas were used for the laminated plate. The density and plate thickness were taken as unity in all three cases. In the following, ~ is redefined as ~ = H/2.
Isotropie plate As an initial check an isotropic plate (v -- 0.30) was examined numerically and the results compared with the corresponding analytical values. The lowest I 0 branches were investigated in reference [1] and shown to be in excellent agreement for a wide range of wave numbers, 0 < r < 10.0. The highest frequency obtained was 12 = coH/(~V'G/p) = 10-0. Here a shift frequency f2o = 17-5 is employed to examine high frequency behavior (modes 24--33 at cut-off) by use ofequations (I0)-(13). The results for the I0 frequencies 12 nearest 17.5 and associated group velocities v~ = d~/d~ are shown in Figures 2(a) and 2(b). Certain analytical values of frequencies in this region may easily be calculated. They are (a) cut-off frequencies, i.e., frequencies at r = 0, and (b) "grid" frequencies, where symmetric and antisymmetric branches intersect and also cross through intersections of Mindlin's bounding grid system [10]. (These points are circled in Figure 20).) As indicated in Tables l(a) and l(b) the numerical values are in very close agreement over a large range of wave numbers r A second check'is made by considering the Lam6 frequencies for the plate [10]. As shown in Table 2 the results for both frequency and group velocity are in very good agreement, even for ~ -- 20.0. The wave forms predicted by the extended Ritz technique also closely resemble the sinusoidal forms in the analytical solution. As a final check the stresses at cut-off were examined for 12 = 15, 20, 25 and 30. Both numerical and analytical stresses were calculated from displacement patterns which were scaled to give a maximum absolute displacement of unity. The results, shown for ali in Figure 3, give a clear indication of both the excellent qualitative modeling ofthe Ritz method and also the increasing errors in stressest which occur as the distances between nodes decrease to values less than two lamina thicknesses. It should be noted that the stresses in Figure 3 are computed by evaluating the strains at the nodal surfaces of a lamina and then using the constitutive relations. Clearly, special procedures, such as quoting stresses at I/4 and 3/4 points of each lamina after averaging the nodal values, give much more satisfactory results, provided the stresses are continuously varying over the lamina. 1"Numerical and analytical displacementforms remain in close agreement, even at n ~ 30.
R. B. NELSON AND S. B. DONG
38 21
i
i
I
I
i
i
20
19
"S--..---~ ~
~"
18
% 17
~ r)r////
16
15
14 1.4
I
i
I
Fo
x"
i
C
1
I
I
I
I
(o)
I
D-..~ ~
"'~~
o-a
I
\....../~/
0.2
(b) 0
I
I
I
I
I
I
I
I
I
I
2
3
4
5
6
7
8
9
I0
C Figure 2. (a)Ten frequencies in vicinity of ~o ~ 17"5 for isotropic plate, v = 0.30. (b) Group velocities for selected branches in (a). , Symmetric; - - - , antisymmetric. TABLE l(a)
Cut-offfrequencies in range 14 < Calculated Analytical
I2 < 21 for
isotropic plate;
v = 0.30
14"967t 15.008 16.011 16"839t 17.015 18.019 18.710? 19.025 20.002 20"582t 14.967? 15.000 16.000 16.837t 17.000 18.000 18.708? 19-000 20.000 20.579?
t Thickness stretch mode. TABLE 1(b)
Selected gridfrequencies for isotropic plate; v = Values ~ Calculated f2 Analytical .Q Values ~ Calculated ~2 Analytical .(2
0"30
3.578 16.397 16.406 16-395
4.626 15.700 15.705 15"697
5-292 14-969 14.972 14-967
6"325 19'086 19.098 19"079
6.856 18.337 18.345 18.330
7.211 17.555 17.560 17.550
7"416 16.737 16.740 16-733
7.483 15.878 15.879 15.875
8.532 16.397 16.399 16.395
8.798 17.393 17.395 17.390
8-899 20-090 20-097 20.080
8.944 18.336 18.339 18.330
8.978 19.233 19.237 19.225
9-839 18.786 18.790 18.783
WAVES IN LAMINATEDORTHOTROPICPLATES
39
TABLE 2
Lamd fiequencies /2 and group velocities vg; isotropic plate v = 0"30 Values ~
2.0
4.0
6.0
8.0
Computed 2.8284 5.6569 8.4854 11.314 /2 Analytical 2.8284 5.6569 8.4853 11.314 /2 Computed 0.7071 0.7071 0-7071 0.7071 vz'p
10.0
12.0
14.0
16.0
18.0
20.0
14.144
16.975
19.807 22.644
25.486 28.334
14.142
16.971
19.799 22.627
25.456 28.284
0"7072
0.7072
0-7073 0.7073
0.7073
0-7073
"p.Analytical value for vs = l/V'2 = 0'7071. O.5 - i
~
o.4-
..~
I
S:>' <2S
o+-
> /
0-2 -
~
/0.1
-
o-' -I'OxlO I
O
)
(0)
l l'~
I
I
I'OglO 8
(b)
(c)
(d)
Figure 3. Comparison of numerical and analytical stress a,l at cut-off. (a)/2 = 15, Co)/2 = 20, (c)/2 = 25, ( d ) / 2 = 30. F r o m this information it is concluded that the extended Ritz method gives an accurate representation o f the modal response as long as the distance between neighboring nodes is greater than two lamina thicknesses. Thus, based on cut-off behavior alone, 100 elements should be used to determine accurately the lowest 60 modal solutions. O f course, as ~ increases the eigenvectors m a y become quite intricate and the accuracy of each eigenvector is a matter which can be determined only by an inspection o f its actual form and taking into account the modeling capabilities admitted by the present method.
Orthotropic plate A highly orthotropic plate with properties representative o f a unidirectional composite plate is considered next. The elastic moduli are cll = 0.36835 x l0 s lbf/in 2,
c12 -- 0.11392 x 107 lbf/in 2,
c2, = 0.33550 x 106 lbf/in 2,
c66 -~ 0.11538 x 10 s lbf/inZ.t
(19)
In order for numerical and analytical comparisons to be made in this case, the bounding grid was first determined by following Mindlin's procedure [10]. Although the details are rather t 1 lbf/in= = 6"9 kN/m 2.
R.B. N E L S O N A N D S. B. D O N G
40
lengthy, the procedure is straightforward, and the bounding grid system is defined by the curves
2poJ2--(ell+c66)~z+
r
(c22+c66)+ .
(c11+c66)~ + ~-
(c2~+c66)
.-
:,,.V
-4[ciic66e-t-if) (ci2+2ei,e66-ciic,1)e z+ciic66t~ / J] , (20) where n is a positive integer. Both the grid system and the numerically calculated frequency branches are shown in Figure 4, with f2 = totti(lzX/c22[p). As is evident, the branches closely
14
12
O
I
2
3
4
eFigure 4. Frequency spectrum and grid system for highly orthotropie plate. TABLE3
Cut-offfrequencies for 15 < f2 < 30for highly orthotropic plate Calculated Analytical Calculated Analytical
15.002 15.000 23.054 23.000
16.004 16.000 23.448 23.457
17.008 17.000 24.068 24-000
17.586 17.593 25.084 25.000
18.012 18-000 26.103 26.000
19.017 19.000 27.125 27-000
20.024 20-000 28.150 28-000
21.032 21.000 29.178 29.000
22.042 22.000 29.311 29-322
follow the grid system and cross at grid intersections, but also the branches form a pattern much more intricate than for the isotropic plate. In fact, in the absence of the grid system, knowledge of group velocities is necessary to determine which lines do or do not cross. As shown in the insert in Figure 4, for this plate the frequency branches typically deviate from the grid system only in the immediate vicinity of the grid intersections. Analytical and numerical comparisons of the cut-off frequencies are given in Table 3 for the range 15 < f2 < 30. The agreement here is quite similar to that found for the isotropic plate. Finally, the analytical solution for the velocity of Rayleigh type waves in [11] is used to check the value predicted by the extended Ritz procedure, obtained here by calculating the lowest values t2 at ~ = 4.0. The numerical value for the lowest branch, VR---t21~ = 2.8049, is in excellent agreement with the analytical value of 2.8043.
WAVES IN LAMINATEDORTHOTROPIC PLATES
41
This information shows that the Ritz technique successfully models even highly orthotropic homogeneouq olates over a large portion of the frequency spectrum.
Lamhtated orthotropic plate As a final example a four layer orthotropic plate is considered. T h e layers are each 0.25 inch thick and each is of density equal to unity. The material properties for layers I and III (see Figure 5) are cll = 25.168 lbf/in 2, ct, = 0"33557 lbf/in 2, c22 = 1.0711 lbf/in 2, c66 = 0"5 lbf/in 2, (21) and for layers II and IV are cll = 1.0711 lbf/in 2, c22 = 1.0711 lbf/in z,
c12 = 0.27114 Ibf/in 2, C66 = 0"2 lbf/in 2.
(22)
This configuration represents a laminated plate constructed by using two different unidirectional composite layers. As is evident by an examination o f the frequency spectrum shown in Figure 6 no simple grid system exists and the spectrum possesses no perceptible pattern, in contrast to homogeneous plates. In fact, the only information which can be determined analytically without inordinate difficulty is on cut-off frequencies and associated eigenvectors. As detailed in Appendix II, the field equations for each layer are solved for the case of harmonic motion at ~ = 0, and combined in accordance with the 12 interface and 4 free surface conditions. Satisfaction of all 16 conditions results in a set of 16 linear homogeneous equations with transcendental coefficients dependent upon the frequency. Setting the determinant of the coefficients equal to zero gives a frequency equation which was solved numerically. The lowest six non-zero frequencies obtained analytically and the similar Ritz results are shown in Table 4 and the corresponding eigenvectors are shown in Figure 5. Again the results are in excellent agreement. 1,00
o.75
\
o,5o
\
/
\ \\
{ \
\
\
/
,J//J
\
025
f,
\
0
(a)
/
/
050 / / ~
0 2 5 \ ' ~ ~, - -
\\ -I0
0 (dl
t
(c)
(b)
!y
)
Z
I0 -I0
0
(e)
f.1'
1.0 - I . 0
0
i-O
(f)
Figure 5. Displacement distributions for cut-off frequencies in laminated plate. (a) o~= 1"6690, (b) ta = 3-2514, (c) to = 3"2878, (d) to = 5-1907, (e) 09 = 6"5028, (f) to = 7"i 326. - - - , UI ; ,U2.
42
R . B. N E L S O N A N D S. B . D O N G
5
4
0
I
2
I
!
3
4
CFigure 6. Frequency spectrum for laminated plate, (o,,t = 1"66899rad/s.
TABLE4 Lowest six non-zero cut-off fi-equencies to f o r lamhtated plate
Calculated Analytical
1.66899 1.66895
3.25140 3-25142
3.28782 3.28779
5"19072 5-19071
6-50284 6.50284
7-13260 7.13260
Of interest is the fact that for this plate, which possesses no structural symmetry, the frequency branches do not cross. This fact is highlighted by the very abrupt changes in branches 8-10 at ~ = 0.5, an occurrence which can only be evinced by a group velocity calculation. This example indicates the ease and accuracy with which the present method can be used to study even very complicated laminates. 6. CONCLUSIONS The extended Ritz technique has been shown to very accurately represent the natural vibrations and waves in laminated orthotropic plates for a wide range of frequencies and wave numbers. This conclusion is based on direct comparisons with analytical solutions and applies to displacement and stress distributions as well as frequency calculations. Use of the frequency shift procedure and group velocity computation together with the direct iterative eigensolution technique gives a powerful and efficient method for generating required information on any portion of the frequency spectrum. In view of the computational ease by which large quantities of high quality information can be obtained through application of this method, the problem of response of laminated plates subjected to transient Ioadings including impact can now be investigated by means of the normal mode method. REFERENCES 1. S. B. DONG and R. B. NELSON 1972 Journal o f Applied 3Iechanics 39, 739-745. On natural vibrations and waves in laminated orthotropic plates.
43
WAVES IN LAMINATED ORTIIOIROPIC PLATES
2. S.B. DONO,J. A. XvVOLr,JR. and F. E. PETERSON 1972 International Journalfor NumericalMethods hl Engineering 4, 155-161. On a direct-iterative eigensolution technique. 3. Y. C. FUNG 1965 Fottndations of Solid Mechanics. Englewood Cliffs, N. J.: Prentice-Hall. See p. 318. 4. J. A. WOLF, JR. and S. B. DONG 1972 Transactions of the Society of Automotive Engineers 26192626. Use of a reduced system of generalized coordinates in a direct-iterative eigensolution technique. 5. O. REYNOLDS 1901 Papers on Mathematical and Physical Subjects 1,198. Cambridge University Press. 6. LORD RAYLEIGH 1901 Scientific Papers 1,322. Cambridge University Press. 7. I. TOLSTOY 1956 Journal of the Acoustical Society of America 27, 897-907. Dispersion and simple harmonic point sources in wave ducts. 8. I. TOLSTOY 1956Journalofthe AcousticalSociety of America 28, 1182-1192. Resonant frequencies and high modes in layered wave guides. 9. M . A . BZOT1957 PhysicalReview 105, 1129-I 137. General theorems on the equivalence o f g r o u p "velocity and energy transport. 10. R. D. MINDLIN 1955 An Introduction to the AIathematical Theory of Vibrations of Elastic Plates. Fort Monmouth, New Jersey: U.S. Army Signal Corps Monograph. 1 I. I I. DrRrsmwicz and R. D. M INDLXN 1957 Journal of Applied Physics 28,669-671. Waves on the surface of a crystal.
APPENDIX I T h e elemental stiffness m a t r i x is
_z_ 2 lc66 15clt~ h + 3 h
89
3 h
C22
--~(C12 "~ C66)
//
[k] =
Symmetric
~(Cl 2 _~ C66) ~
..1_,, ~21,
_8c29 xs*66~,,-3T
0
__ 3~Cll ~2 h "~- ~ c66
h 9
I~
+c66)~
~cH,~2h
~c66 -~77
--10(C12 "~
C66)
_.~6c66~2h
~c22
+~h
~(c.~+c~)~ (AI)
_~_~. .C2 h -k 1 6 C22 15"66" "'~ 3 y
--~3(C12+C66)~
1-1JC66~ 2h-~c223]1
~c. ~ h + z c~
89
3 h
- c~6)
~c~6 ~2 h + ~c~, 3 h
44
R.B. NELSONAND S. B. DONG
and the elemental contribution to the group velocity is -~c,,
~h
89
- c,9
"i4-gc66{h [kg] =
Symmetric
~"c . ~h { ( c . + c6+) - ~ c l , ~h -](c12 -Fc66) --~5c66~1t {(c12 + c66) ~c,~ {/z 0 ~ c , , {I~ "I-$6C66~h --~(C12 '+ C66) ~ c n {h
- ~ ( c , 2 + ~:+6) - ~ c 6 6 ~h {(c,~ + c66) ~c+6 ~h
89
- c++)
1"~$C66 ~ll
(A2)
The mass matrix is "-4 0 4 [m] = a'~oph
2 0 16
0 -1 2 0 0 2 16 0 Symmetric 4
0-1 0 2 0 4
(A3)
APPENDIX II To obtain cut-off frequencies for the four layer plate, the displacements for the ith layer take the simple form zq = Ul,(x2) e+'%
u, = U~(x2) e '~
(a4)
Substitution of equations (A4) into the field equations for the ith layer results in ' UI,22 ' "~ O,~2fli U[ = 0, C66 ' U2,22 ~ + o)2pt v~ = 0, C22
(AS)
U[ = Ai sin ct+x2 + At2coscdx2, U2t = Bi sin ff x2 + B2~cos ff x,,
(A6)
with the solution
where a' = co](V~d661p') and fl' = o ~ / ( ~ ' ) . The ith layer is bounded by the ith and ( i + l)th faces x; and x~+1, respectively, with x,~ = 0 and x 5, = I I defining top and bottom faces. On the top and bottom faces the stress free boundary conditions are a12 = O, at2 = O, a;, = o, a;2 = 0, (A7) while on the layer interfaces UI12 = O1~ _t41 , U/2 ,.rl+l v22 t,t" 1 _- - ,,t+l ~1
,
ut~ = ut=+1.
(A8)
The four conditions on the three interfaces, equations (A8), together with the four free surface conditions, equations (A7), are easily expressed in terms of the solution equations (A6) and result in a set of 16 linear homogeneous equations on the coefficients A], A2, ~ BI, B',, i = I ..... 4. The determinant of the coefficients set equal to zero gives the frequency equation for the plate.
HIGH FREQUENCY VIBRATIONS AND WAVES IN LAMINATED ORTHOTROPIC PLATES R. B. NELSONAND S. B. DONG
School of Enghteering and Applied Science, University of California, Los Angeles, California 90024, U.S.A. (Received 3 AIarch 1973) The extended Ritz technique is used to study high frequency vibrations and waves in infinite homogeneous and laminated orthotropic plates. This technique utilizes exact expressions for the wave form along the extent of the plate, but models the plate's behavior through the thickness with a large number of generalized coordinates. A procedure is shown for computing the group velocities from the modal data. A number of examples are presented to' illustrate the modeling capability of this solution technique in the high frequency range. Frequencies, modal displacements and group velocities are obtained in the range 10-30 times above the lowest non-zero cut-off frequency and for a wide range of wave numbers. In addition, modal stress patterns are obtained and compared with analytical results. The method is shown to give accurate information over a large region of the frequency spectrum. Thus, it is a versatile and efficient, as well as computationally very efficient, means for generating data for investigating the transient response of composite plates including problems of impact.
1. INTRODUCTION Wave propagation in layered media is a subject of considerable importance to a number of areas ranging from geophysics to pavement design, and more recently to laminated composite structures. Conceptually, th e theory and formulation of the problem of wave propagation in a given layered system is straightforward. The analytical procedure involves the solution to the field equations for each layer and the imposition of stress and displacement continuity at interfaces and prescribed traction or displacement conditions at tile free surface(s) to yield a frequency equation. The difficulty lies not in the formalism of this approach, but in the cumbersome algebra necessary to generate and solve the frequency equation. Therefore, most analyses have been restricted to two- or three-layered systems and no analytical treatment of a completely arbitrary layered system is available. Recently, a numerical approach based on the extended Ritz technique was presented for the problem of plane waves in a general layered plate [I]. In this method, the infinite plate is discretized through the thickness into a number of subregions called laminas. Each lamina can have distinct mechanical and inertial properties as well as thickness. Explicit analytical expressions for the appropriate wave form along the extent of the plate are adopted at the outset and a quadratic interpolation function is taken through the lamina thickness. An algebraic eigenvalue problem results from which the modal information is extracted by use of a very efficient direct-iterative eigensolution technique [2]. This method was shown to give a very accurate representation of the physical behavior of isotropic plates for the lowest branches of the frequency spectrum for real wave numbers. Numerical results for a limited number of orthotropic and laminated plates were also presented in reference [1] to indicate the versatility of the procedure. 33
34
R. B. NELSON AND S. B. DONG
In this paper, a modification of the solution method for the algebraic eigensystem model of the layered plate in reference [1] is made to permit exploration of the frequency spectra in the high frequency ranges. This extension involves a shifting procedure [3] which is applied to the governing equations before using the direct-iterative eigensolution technique. Because higher frequencies and modal patterns are sought, the modeling capability used in the present approach must be examined. A detailed study is presented showing the accuracy and range of effectiveness for various diseretized models. In addition, it is shown how group velocities can be computed subsequent to the solution of the eigensystem. Group velocities are of importance in two respects. They represent the rates at which energies are being propagated, and since group velocities are also slopes of the branches in the frequency spectra, this information is highly useful in determining the branch-layout. Finally, in order to answer questions concerning the accuracy of the method for other than isotropic plates, some additional comparisons between analytical and numerical results for orthotropic and laminated plates are given.
2. EXTENDED RITZ FORMULATION For completeness, the derivation of the equations for planar waves in laminated plates is presented here, but in a different manner than that in reference [I], to emphasize the traveling form of the waves. Consider an infinite plate composed Of an arbitrary number of bonded elastic, homogeneous layers. Each layer can have distinct thickness, density and orthotropic material properties with its elastic axes coincident with the coordinate axes. Let xj be rectangular Cartesian coordinate measures and let t denote time. For plane harmonic waves propagating in the xl direction of this plate, the displacements uk (k = 1,2) are of the form
u,(x,, x~, t) = W,(x~) e~"o'-~x, ), (1)
Ua(Xl, x2, t) = iU2(x2) e u~t-ex'),
where o~is the circular frequency and ~ is the wave number, which is related to the wavelength 2 by =
~/~..
(2)
The focus of the analysis is the determination of the frequencies o and the associated modal distributions Uk(x2) over the entire thickness of the plate. In the extended Ritz method, the plate is initially divided into N subregions termed laminas. Note that a single layer of the plate can be modeled by a number of laminas. For the kth lamina, the functions Uk(x2) are approximated by interpolations of the form W.(x2) = Wkb{1-- 311+ 2q z} + Wk,.{411 -- 4112}+ Wkf{2q: -- q}
(k = 1,2),
(3)
where the coefficients U~b, U~.,, U , f are generalized coordinates representing displacement amplitudes at the lamina's back, middle and frontal surfaces (see Figure 1) and t1 is the normalized thickness measure defined by I1 = (x2 - x2sb)/hs,
(4)
with hj as the thickness and xzso the distance to the bottom of thejth lamina. Hamilton's principle in the form [3]
i
tO
' 6L dt = 0
(5)
~VAVES IN LAbIINATED ORTHOTROPIO PLATF.S
9
35
Typical lamina
....
i). /
(
,
Figure I. Ritz representation oflaminated orthotropic plate. provides the basis for generating the governing equations of the problem, where L is the Lagrangian. The contribution to L from thejth lamina is
L, = ~ f f r
,;~ + ,;,,;~J- k , , , , , . , , , * ,9
+ + ~,=r ,1 : 2,2 + t/I,I-112,2) * + c22(u2.2 it2.2) + c66(u,.2 + u2.1)(tq.2 + u*, a)]} dxl dx2,
(6)
where p is the mass density and c u are the elastic moduli. Commas and dots are the usual notations for spatial and time differentiations and the asterisk denotes the complex conjugate. Substituting equations (1) and (3) into equation (6) yields X2JD+hJ
Lj = 89f
dxl f
{p~2[u,= + u ] ] - [c. r v~ +
x2jb
+ 2c12~U1 0"2,2 + c22 U2,2 + %6(Uz.2 + r
(7)
Note that the integral involving x2 in equation (7) after integration contains only the generalized coordinates UAb, Uk,, and Uk: as unknowns: i.e., Lj = 89f d x t [(.o 2 rTmr -- rtkr],
(8)
where r r = [U~b, U2b, Ua,,, U2,,, Ut:, U2:] and m and k are lamina mass and stiffness matrices which are given in Appendix I. The summation of Lj for the N laminas and imposition of interlamina displacement continuity leads to the Lagrangian function L for the plate:
f
L = -~ dxa [co2 U r M U - U "rKU],
(9)
where M and Kare the plate mass and'stiffness matrices and U is a vector o f 4 N + 2 generalized coordinates. For the determination of the modal distribution U, the integral on xt can be factored outside of the variation of L in equation (5) since the xt dependence is explicit. Thus, the variational equation of motion by Hamilton's principle takes the form K U - to2 31U = 0.
(I0)
3. HIGH FREQUENCY ANALYSIS In reference [I], the solution to equation (10) was obtained by means of an efficient direct-iterative eigensolution technique [2]. In this method, a suitable subset of generalized coordinates is selected with a view of using them to reduce the rank of the eigensystem. This set is improved by a static analysis where the loads are those obtained by the matrix product
36
R . B . NELSON AND S. B. DONG
of the mass with the reduced generaiized coordinates, which can be interpreted as tile inertial distributions. Then, after the eigensystem is reduced, a direct solution is effected in the subspace. This procedure is repeated with the eigenvectors of the current cycle being used as the reduced generalized coordinates for the next cycle. Convergence ofthis iterative procedure is assured because the method is seen to be the well-known Stodola-Vianello method applied simultaneously to a group &vectors. This technique can be modified [4] to obtain eigendata for higher frequencies by letting the frequency to be decomposed into to2 = too2 + It,
(I I)
where coo is an arbitrary reference frequency. Upon substitution ofequation (11) into equation (I0), there results K U - id~IU = 0, (12) where g=K-w~3l. (13) The direct-iterative eigensolution technique can then be applied to equation (12). The computed eigendata contains eigenvalues It whose values are the smallest in absolute value of the eigensystem: i.e., values nearest COo,and their associated eigenvectors. (The actual frequencies are then obtained by use of equation (11).) Although the rate of convergence depends on the initial choice of displacement forms, it is quite rapid when more generalized coordinates are used than are required [4]. In this paper, as in reference [1], 16 reduced coordinates are used in order to obtain accurately the I0 eigensolutions nearest 6Oo. 4. GROUP VELOCITIES It is well-known that group velocities express the rate at which energies are transported [5, 6]. The definition of group velocity vz is vg = dw/d~:
(14)
i.e., it is the slope to a branch of the frequency spectrum. A common method of estimating v, has been by graphical differentiation of a frequency branch with respect to ~ since an explicit relation for the branch is not available or is cumbersome to obtain. Group velocities in the present analysis can be extremely useful in the construction of the layout of the frequency branches if they are known a priork Tolstoy [7, 8] and Biot [9] have given an interpretation of group velocities as ratios of energy density and flux, and this provides a simple relation for computing group velocities directly from eigendata. For the mth eigenvector, ~,~, Rayleigh's quotient is 2 ~ Kff~,. tom = ~ "
(15)
From the elemental matrices in Appendix I it is evident that only K depends on 3, so that differentiation of equation (15) gives v~,, 2to,, ~ M ~ , , '
(16)
K, = 0-~K(0.
(17)
where K~ is defined as
The matrix Kg is assembled from the elemental contributions kg = ak/O~. The coefficients of kg are given in Appendix I.
WAVES IN LAMINATED ORTHOTROPIC PLATES
37
Equation (16) can be simplified if the eigenvectors #,~ are normalized with respect to the mass matrix, # ~ M # = 1. Then equation (16) becomes I vv~ = 2r # ~ K , ffJ=.
(18)
In the present eigensolution technique, the normalization condition is built in So that it does not have to be performed aposteriorL 5. EXAMPLES Since equation (10) is an approximation to the actual frequency equation its modeling accuracy should be examined, particularly in the high frequency region where eigenvectors are typically very complicated and therefore difficult to model. Three different plates were investigated, in regions where analytical results can be obtained for comparison with similar results of the Ritz method. The examples, (a) a homogeneous isotropic plate (v -- 0.3), (b) a homogeneous orthotropic plate c22/c~z --0.01, and (c) a four-layer laminated orthotropic plate, were chosen to permit both analytical and numerical treatments for several different type plates. In the homogeneous plates 50 equal-thickness laminas were used, and 60 equalthickness laminas were used for the laminated plate. The density and plate thickness were taken as unity in all three cases. In the following, ~ is redefined as ~ = H/2.
Isotropie plate As an initial check an isotropic plate (v -- 0.30) was examined numerically and the results compared with the corresponding analytical values. The lowest I 0 branches were investigated in reference [1] and shown to be in excellent agreement for a wide range of wave numbers, 0 < r < 10.0. The highest frequency obtained was 12 = coH/(~V'G/p) = 10-0. Here a shift frequency f2o = 17-5 is employed to examine high frequency behavior (modes 24--33 at cut-off) by use ofequations (I0)-(13). The results for the I0 frequencies 12 nearest 17.5 and associated group velocities v~ = d~/d~ are shown in Figures 2(a) and 2(b). Certain analytical values of frequencies in this region may easily be calculated. They are (a) cut-off frequencies, i.e., frequencies at r = 0, and (b) "grid" frequencies, where symmetric and antisymmetric branches intersect and also cross through intersections of Mindlin's bounding grid system [10]. (These points are circled in Figure 20).) As indicated in Tables l(a) and l(b) the numerical values are in very close agreement over a large range of wave numbers r A second check'is made by considering the Lam6 frequencies for the plate [10]. As shown in Table 2 the results for both frequency and group velocity are in very good agreement, even for ~ -- 20.0. The wave forms predicted by the extended Ritz technique also closely resemble the sinusoidal forms in the analytical solution. As a final check the stresses at cut-off were examined for 12 = 15, 20, 25 and 30. Both numerical and analytical stresses were calculated from displacement patterns which were scaled to give a maximum absolute displacement of unity. The results, shown for ali in Figure 3, give a clear indication of both the excellent qualitative modeling ofthe Ritz method and also the increasing errors in stressest which occur as the distances between nodes decrease to values less than two lamina thicknesses. It should be noted that the stresses in Figure 3 are computed by evaluating the strains at the nodal surfaces of a lamina and then using the constitutive relations. Clearly, special procedures, such as quoting stresses at I/4 and 3/4 points of each lamina after averaging the nodal values, give much more satisfactory results, provided the stresses are continuously varying over the lamina. 1"Numerical and analytical displacementforms remain in close agreement, even at n ~ 30.
R. B. NELSON AND S. B. DONG
38 21
i
i
I
I
i
i
20
19
"S--..---~ ~
~"
18
% 17
~ r)r////
16
15
14 1.4
I
i
I
Fo
x"
i
C
1
I
I
I
I
(o)
I
D-..~ ~
"'~~
o-a
I
\....../~/
0.2
(b) 0
I
I
I
I
I
I
I
I
I
I
2
3
4
5
6
7
8
9
I0
C Figure 2. (a)Ten frequencies in vicinity of ~o ~ 17"5 for isotropic plate, v = 0.30. (b) Group velocities for selected branches in (a). , Symmetric; - - - , antisymmetric. TABLE l(a)
Cut-offfrequencies in range 14 < Calculated Analytical
I2 < 21 for
isotropic plate;
v = 0.30
14"967t 15.008 16.011 16"839t 17.015 18.019 18.710? 19.025 20.002 20"582t 14.967? 15.000 16.000 16.837t 17.000 18.000 18.708? 19-000 20.000 20.579?
t Thickness stretch mode. TABLE 1(b)
Selected gridfrequencies for isotropic plate; v = Values ~ Calculated f2 Analytical .Q Values ~ Calculated ~2 Analytical .(2
0"30
3.578 16.397 16.406 16-395
4.626 15.700 15.705 15"697
5-292 14-969 14.972 14-967
6"325 19'086 19.098 19"079
6.856 18.337 18.345 18.330
7.211 17.555 17.560 17.550
7"416 16.737 16.740 16-733
7.483 15.878 15.879 15.875
8.532 16.397 16.399 16.395
8.798 17.393 17.395 17.390
8-899 20-090 20-097 20.080
8.944 18.336 18.339 18.330
8.978 19.233 19.237 19.225
9-839 18.786 18.790 18.783
WAVES IN LAMINATEDORTHOTROPICPLATES
39
TABLE 2
Lamd fiequencies /2 and group velocities vg; isotropic plate v = 0"30 Values ~
2.0
4.0
6.0
8.0
Computed 2.8284 5.6569 8.4854 11.314 /2 Analytical 2.8284 5.6569 8.4853 11.314 /2 Computed 0.7071 0.7071 0-7071 0.7071 vz'p
10.0
12.0
14.0
16.0
18.0
20.0
14.144
16.975
19.807 22.644
25.486 28.334
14.142
16.971
19.799 22.627
25.456 28.284
0"7072
0.7072
0-7073 0.7073
0.7073
0-7073
"p.Analytical value for vs = l/V'2 = 0'7071. O.5 - i
~
o.4-
..~
I
S:>' <2S
o+-
> /
0-2 -
~
/0.1
-
o-' -I'OxlO I
O
)
(0)
l l'~
I
I
I'OglO 8
(b)
(c)
(d)
Figure 3. Comparison of numerical and analytical stress a,l at cut-off. (a)/2 = 15, Co)/2 = 20, (c)/2 = 25, ( d ) / 2 = 30. F r o m this information it is concluded that the extended Ritz method gives an accurate representation o f the modal response as long as the distance between neighboring nodes is greater than two lamina thicknesses. Thus, based on cut-off behavior alone, 100 elements should be used to determine accurately the lowest 60 modal solutions. O f course, as ~ increases the eigenvectors m a y become quite intricate and the accuracy of each eigenvector is a matter which can be determined only by an inspection o f its actual form and taking into account the modeling capabilities admitted by the present method.
Orthotropic plate A highly orthotropic plate with properties representative o f a unidirectional composite plate is considered next. The elastic moduli are cll = 0.36835 x l0 s lbf/in 2,
c12 -- 0.11392 x 107 lbf/in 2,
c2, = 0.33550 x 106 lbf/in 2,
c66 -~ 0.11538 x 10 s lbf/inZ.t
(19)
In order for numerical and analytical comparisons to be made in this case, the bounding grid was first determined by following Mindlin's procedure [10]. Although the details are rather t 1 lbf/in= = 6"9 kN/m 2.
R.B. N E L S O N A N D S. B. D O N G
40
lengthy, the procedure is straightforward, and the bounding grid system is defined by the curves
2poJ2--(ell+c66)~z+
r
(c22+c66)+ .
(c11+c66)~ + ~-
(c2~+c66)
.-
:,,.V
-4[ciic66e-t-if) (ci2+2ei,e66-ciic,1)e z+ciic66t~ / J] , (20) where n is a positive integer. Both the grid system and the numerically calculated frequency branches are shown in Figure 4, with f2 = totti(lzX/c22[p). As is evident, the branches closely
14
12
O
I
2
3
4
eFigure 4. Frequency spectrum and grid system for highly orthotropie plate. TABLE3
Cut-offfrequencies for 15 < f2 < 30for highly orthotropic plate Calculated Analytical Calculated Analytical
15.002 15.000 23.054 23.000
16.004 16.000 23.448 23.457
17.008 17.000 24.068 24-000
17.586 17.593 25.084 25.000
18.012 18-000 26.103 26.000
19.017 19.000 27.125 27-000
20.024 20-000 28.150 28-000
21.032 21.000 29.178 29.000
22.042 22.000 29.311 29-322
follow the grid system and cross at grid intersections, but also the branches form a pattern much more intricate than for the isotropic plate. In fact, in the absence of the grid system, knowledge of group velocities is necessary to determine which lines do or do not cross. As shown in the insert in Figure 4, for this plate the frequency branches typically deviate from the grid system only in the immediate vicinity of the grid intersections. Analytical and numerical comparisons of the cut-off frequencies are given in Table 3 for the range 15 < f2 < 30. The agreement here is quite similar to that found for the isotropic plate. Finally, the analytical solution for the velocity of Rayleigh type waves in [11] is used to check the value predicted by the extended Ritz procedure, obtained here by calculating the lowest values t2 at ~ = 4.0. The numerical value for the lowest branch, VR---t21~ = 2.8049, is in excellent agreement with the analytical value of 2.8043.
WAVES IN LAMINATEDORTHOTROPIC PLATES
41
This information shows that the Ritz technique successfully models even highly orthotropic homogeneouq olates over a large portion of the frequency spectrum.
Lamhtated orthotropic plate As a final example a four layer orthotropic plate is considered. T h e layers are each 0.25 inch thick and each is of density equal to unity. The material properties for layers I and III (see Figure 5) are cll = 25.168 lbf/in 2, ct, = 0"33557 lbf/in 2, c22 = 1.0711 lbf/in 2, c66 = 0"5 lbf/in 2, (21) and for layers II and IV are cll = 1.0711 lbf/in 2, c22 = 1.0711 lbf/in z,
c12 = 0.27114 Ibf/in 2, C66 = 0"2 lbf/in 2.
(22)
This configuration represents a laminated plate constructed by using two different unidirectional composite layers. As is evident by an examination o f the frequency spectrum shown in Figure 6 no simple grid system exists and the spectrum possesses no perceptible pattern, in contrast to homogeneous plates. In fact, the only information which can be determined analytically without inordinate difficulty is on cut-off frequencies and associated eigenvectors. As detailed in Appendix II, the field equations for each layer are solved for the case of harmonic motion at ~ = 0, and combined in accordance with the 12 interface and 4 free surface conditions. Satisfaction of all 16 conditions results in a set of 16 linear homogeneous equations with transcendental coefficients dependent upon the frequency. Setting the determinant of the coefficients equal to zero gives a frequency equation which was solved numerically. The lowest six non-zero frequencies obtained analytically and the similar Ritz results are shown in Table 4 and the corresponding eigenvectors are shown in Figure 5. Again the results are in excellent agreement. 1,00
o.75
\
o,5o
\
/
\ \\
{ \
\
\
/
,J//J
\
025
f,
\
0
(a)
/
/
050 / / ~
0 2 5 \ ' ~ ~, - -
\\ -I0
0 (dl
t
(c)
(b)
!y
)
Z
I0 -I0
0
(e)
f.1'
1.0 - I . 0
0
i-O
(f)
Figure 5. Displacement distributions for cut-off frequencies in laminated plate. (a) o~= 1"6690, (b) ta = 3-2514, (c) to = 3"2878, (d) to = 5-1907, (e) 09 = 6"5028, (f) to = 7"i 326. - - - , UI ; ,U2.
42
R . B. N E L S O N A N D S. B . D O N G
5
4
0
I
2
I
!
3
4
CFigure 6. Frequency spectrum for laminated plate, (o,,t = 1"66899rad/s.
TABLE4 Lowest six non-zero cut-off fi-equencies to f o r lamhtated plate
Calculated Analytical
1.66899 1.66895
3.25140 3-25142
3.28782 3.28779
5"19072 5-19071
6-50284 6.50284
7-13260 7.13260
Of interest is the fact that for this plate, which possesses no structural symmetry, the frequency branches do not cross. This fact is highlighted by the very abrupt changes in branches 8-10 at ~ = 0.5, an occurrence which can only be evinced by a group velocity calculation. This example indicates the ease and accuracy with which the present method can be used to study even very complicated laminates. 6. CONCLUSIONS The extended Ritz technique has been shown to very accurately represent the natural vibrations and waves in laminated orthotropic plates for a wide range of frequencies and wave numbers. This conclusion is based on direct comparisons with analytical solutions and applies to displacement and stress distributions as well as frequency calculations. Use of the frequency shift procedure and group velocity computation together with the direct iterative eigensolution technique gives a powerful and efficient method for generating required information on any portion of the frequency spectrum. In view of the computational ease by which large quantities of high quality information can be obtained through application of this method, the problem of response of laminated plates subjected to transient Ioadings including impact can now be investigated by means of the normal mode method. REFERENCES 1. S. B. DONG and R. B. NELSON 1972 Journal o f Applied 3Iechanics 39, 739-745. On natural vibrations and waves in laminated orthotropic plates.
43
WAVES IN LAMINATED ORTIIOIROPIC PLATES
2. S.B. DONO,J. A. XvVOLr,JR. and F. E. PETERSON 1972 International Journalfor NumericalMethods hl Engineering 4, 155-161. On a direct-iterative eigensolution technique. 3. Y. C. FUNG 1965 Fottndations of Solid Mechanics. Englewood Cliffs, N. J.: Prentice-Hall. See p. 318. 4. J. A. WOLF, JR. and S. B. DONG 1972 Transactions of the Society of Automotive Engineers 26192626. Use of a reduced system of generalized coordinates in a direct-iterative eigensolution technique. 5. O. REYNOLDS 1901 Papers on Mathematical and Physical Subjects 1,198. Cambridge University Press. 6. LORD RAYLEIGH 1901 Scientific Papers 1,322. Cambridge University Press. 7. I. TOLSTOY 1956 Journal of the Acoustical Society of America 27, 897-907. Dispersion and simple harmonic point sources in wave ducts. 8. I. TOLSTOY 1956Journalofthe AcousticalSociety of America 28, 1182-1192. Resonant frequencies and high modes in layered wave guides. 9. M . A . BZOT1957 PhysicalReview 105, 1129-I 137. General theorems on the equivalence o f g r o u p "velocity and energy transport. 10. R. D. MINDLIN 1955 An Introduction to the AIathematical Theory of Vibrations of Elastic Plates. Fort Monmouth, New Jersey: U.S. Army Signal Corps Monograph. 1 I. I I. DrRrsmwicz and R. D. M INDLXN 1957 Journal of Applied Physics 28,669-671. Waves on the surface of a crystal.
APPENDIX I T h e elemental stiffness m a t r i x is
_z_ 2 lc66 15clt~ h + 3 h
89
3 h
C22
--~(C12 "~ C66)
//
[k] =
Symmetric
~(Cl 2 _~ C66) ~
..1_,, ~21,
_8c29 xs*66~,,-3T
0
__ 3~Cll ~2 h "~- ~ c66
h 9
I~
+c66)~
~cH,~2h
~c66 -~77
--10(C12 "~
C66)
_.~6c66~2h
~c22
+~h
~(c.~+c~)~ (AI)
_~_~. .C2 h -k 1 6 C22 15"66" "'~ 3 y
--~3(C12+C66)~
1-1JC66~ 2h-~c223]1
~c. ~ h + z c~
89
3 h
- c~6)
~c~6 ~2 h + ~c~, 3 h
44
R.B. NELSONAND S. B. DONG
and the elemental contribution to the group velocity is -~c,,
~h
89
- c,9
"i4-gc66{h [kg] =
Symmetric
~"c . ~h { ( c . + c6+) - ~ c l , ~h -](c12 -Fc66) --~5c66~1t {(c12 + c66) ~c,~ {/z 0 ~ c , , {I~ "I-$6C66~h --~(C12 '+ C66) ~ c n {h
- ~ ( c , 2 + ~:+6) - ~ c 6 6 ~h {(c,~ + c66) ~c+6 ~h
89
- c++)
1"~$C66 ~ll
(A2)
The mass matrix is "-4 0 4 [m] = a'~oph
2 0 16
0 -1 2 0 0 2 16 0 Symmetric 4
0-1 0 2 0 4
(A3)
APPENDIX II To obtain cut-off frequencies for the four layer plate, the displacements for the ith layer take the simple form zq = Ul,(x2) e+'%
u, = U~(x2) e '~
(a4)
Substitution of equations (A4) into the field equations for the ith layer results in ' UI,22 ' "~ O,~2fli U[ = 0, C66 ' U2,22 ~ + o)2pt v~ = 0, C22
(AS)
U[ = Ai sin ct+x2 + At2coscdx2, U2t = Bi sin ff x2 + B2~cos ff x,,
(A6)
with the solution
where a' = co](V~d661p') and fl' = o ~ / ( ~ ' ) . The ith layer is bounded by the ith and ( i + l)th faces x; and x~+1, respectively, with x,~ = 0 and x 5, = I I defining top and bottom faces. On the top and bottom faces the stress free boundary conditions are a12 = O, at2 = O, a;, = o, a;2 = 0, (A7) while on the layer interfaces UI12 = O1~ _t41 , U/2 ,.rl+l v22 t,t" 1 _- - ,,t+l ~1
,
ut~ = ut=+1.
(A8)
The four conditions on the three interfaces, equations (A8), together with the four free surface conditions, equations (A7), are easily expressed in terms of the solution equations (A6) and result in a set of 16 linear homogeneous equations on the coefficients A], A2, ~ BI, B',, i = I ..... 4. The determinant of the coefficients set equal to zero gives the frequency equation for the plate.