Paper - Natural Frequencies of Thick, Layered Composite Plates

8/3/2019 Paper - Natural Frequencies of Thick, Layered Composite Plates

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e> P. . . .m Composites Engineering, Vol. 4, No. 10, pp. 1011-1021, 1994Copyright 1994 Elsevier Science LtdPrinted in Great Britain. All rights reserved09619'26/94 $7.00+ .000961-9526(94) E00054-9

NATURAL FREQUENCIES OF THICK, LAYEREDCOMPOSITE PLATES

Timothy W. TaylorGeneral Electric Aircraft Engines, Evendale, OH 45215, U.S.A.

andAdnan H. Nayfeh

Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati,Cincinnati, OH 45221, U.S .A.(Received 9 May 1994; final version accepted 27 June 1994)

Abstract-We develop analyses and numerical calculations for the free vibration of simplysupported, thick, layered composite plate strips composed of off-axis lamina. It is assumed thateach layer of the composite plate is of arbitrary thickness and possesses up to monoclinicsymmetry. Formal solutions are obtained for the individual layers which relate the field variablesat the upper and lower layer surfaces. Here, linear transformations are used to refer the anisotropyof each layer to a global coordinate system. The solution of the characteristics for the total systemproceeds by utilizing the matrix transfer method, which relies on satisfying appropriate interfacialconditions across the constituents, and results in a third-order eigenproblem, The order of the freevibration eigenproblem of the layered plate strip is thereby rendered independent of the numberof layers in the composite plate. Numerical illustrations are given in the form of normalizednatural frequency vs nondimensionalized mode number. The effect of the plate's microstructureon its vibration characteristics is investigated by examining changes in natural frequencies forvarious laminate lay-ups and material combinations.

INTRODUCTIONThe free vibration of "thin" plates, where the in-plane dimensions of the plate are muchlarger than the thickness, have been extensively studied and the corresponding theory welldeveloped in the past. These studies typically neglected the variation of displacements andstresses through the thickness of the plate, an assumption well suited for "thin" plates.These studies in classical plate theory (CPT) are well represented in many fine texts onelastic vibration theory, as in Meirovitch (1967), for example.Many analyses, built upon CPT, have been presented for the evaluation of themechanics and dynamics of layered thin plates. In classic papers Reissner (1945) andMindlin (1951)were the first to consider transverse shear deformation in plate theory forbending and free vibration, respectively. Since then, many other approximate methodshave been used to account for the variation of the field variables through the platethickness as in the works of Hearmon (1959), Gazis and Mindlin (1960), Whitney andLeissa (1969) and Leissa and Narita (1989), for example.More recently, others have presented higher-order individual-layer theories; Yanget al. (1966), Bert and Mayberry (1969), Whitney (1969, 1972), Sun and Whitney (1973),Srinivas (1973) and Cho et al. (1990), to account for the variations of field variablesthrough the plate thickness in the free vibration analysis of plates. These studies addressedlaminated plates using various assumptions to account for the interfacial transfer ofstresses from one layer to another within the composite plate. However, these works reliedon the thin-plate approximations as the fundamental theory in their development.Exact formal solutions for thick rectangular plates constructed from orthotropic layerswerepresented byPagano (1969,1970), Srinivas and Rao (1970) and Jones (1970),where allthe layers were assumed to be aligned by their chrystallographic axes. Here, the compositeplate was confined to be composed of, in popular composite terminology, 0 and 90lamina. Their "direct" formulation (the explicit solution of each layer's displacementamplitudes) furthermore required the solution of a set of 6N unknowns (where N is thenumber of lamina in the composite plate) for the free vibration characteristics of the plate.

1011


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1012 T. W. Taylor and A. H. NayfehExact formal solutions for thick, single-layer plates of a more general form ofanisotropy were later presented by Jones (1971). In his treatment the plate was assumedto possess monoclinic or greater material symmetry. This extension of the exact analysesof the vibrational characteristics of plates (again, a "direct" solution) allowed forarbitrary orientation of the reference material ply angle relative to the plate boundaries,that is, an off-axis lamina plate. Furthermore, the plate was assumed to be of infinite

extent in one of the reference orthogonal directions while being bounded in the other two.While the material symmetry class was thereby considerably extended by Jones , theplate geometry was somewhat more restricted (plate strips) than that in previous works.Taylor et al. (1994) subsequently extended the analysis for infinite plate strips ofmonoclinic or greater symmetry to include the transient response due to arbitraryloadings. In addition, they analytically separated the symmetric and antisymmetric modesand presented mathematical verification for the mode crossings mentioned, but notexplored, in Jones' work.In this paper, we extend our exact solution for the free vibration characteristics ofthick plate strips to the case of a thick composite plate strip composed of an arbitrarynumber of "off-axis" lamina. The explicittreatment of the lamina allowsus to investigatethe effect of the plate's microstructure on its vibration characteristics. In this treatmentwemake use of the propagator matrix technique (matrix transfer technique), introducedoriginally by Thomson (1950)and later on by Haskell (1953)and others (Shaw and Bugl,1969;Folds and Loggins, 1977;McCammon and McDaniel, 1985;Jackins and Gaunaurd,1986) for applications in the geophysics, acoustics and electromagnetic fields. Thesestudies considered plates of infinite extent which were typically composed of isotropiclayers rigidly bonded to one another. Subsequently, Nayfeh and Taylor (1988, 1990,1992), Nayfeh and Chimenti (1988a, b, 1991)and others (Schoenberg, 1980; Kundu andMal, 1985; Mal and Kundu, 1987;Dayal and Kinra, 1989; Rokhlin and Wang, 1991) madeuse of the propagator matrix technique in wave propagation studies of multilayeredinfinite plate systems composed of both isotropic and anisotropic layers.A common feature of any analysis of the free vibration characteristics of a system isthe solution of an eigenproblem. The "direct" analysis presented in previous exactsolutions required solving a 6N-order eigenproblem (N being the number of lamina). Arecent assessment of computational models for multilayered anisotropic plates by Noorand Burton (1990) expressed the concern that exact solutions were computationally"expensive", and therefore may not be feasible for practical composite plates. Thiscomputational "expense" is a result of the "direct" method of solution whereby a 6Nsystem of equations must be solved for each mode number and frequency combination.Our reformulation for the solution of the free modes for anisotropic plates avoids thiscomputationally intensive "direct" method.An innovative feature of our approach is that the number of layers in the plate doesnot affect the order of the eigenproblem when solving for the free modes. Indeed,regardless of the number of layers in the composite plate, weneed only solve a third-ordereigenproblern to determine the free mode frequencies and modal shapes. This formulationgreatly simplifies the numerical implementation of the solution. We furthermore find ourformulation to be numerically more stable than the "direct" techniques of solution .Finally, we point out that we do not specifically intend to compare, in detail, ourexact solution with CPT. However, we mention that the solution for "thin" plates isencompassed by this exact solution. Indeed, the lower left corners of our plots of naturalfrequency contain the CPT solution. Instead, we concentrate on reformulation of theexact methods of solution in a more computationally "friendly" manner. Our intent hereis to render the exact solutions more useful to the design community in the analysis ofpractical laminates.

FORMULATION OF THE PROBLEMFollowing our earlier analysis of a plate strip composed of a single anisotropic layerwe consider an infinite plate strip, shown in Fig. I with the global coordinate system,


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Natural frequencies of composite plates 1013

/2Fig. I. Layered plate strip with global coordinate system.

consisting of an arbitrary number N of anisotropic layers, each layer with thickness d(k)and width L. Furthermore, we assume each layer exhibits no less than monoclinicsymmetry and to be stacked normal to the X3 axis of a global orthogonal Cartesian systemXi = (Xl ' X2 ' x3) . Hence the plane of each layer is parallel to the XCX2 plane, which is alsochosen to coincidewith the bottom surface of the layered plate. To maintain generality weshall allow each layer to be arbitrarily oriented in the xcx2 plane.To describe the relative lamina ply orientation of the layers, we assign an index k foreach layer, k = 1, 2, .. . ,N, as depicted in Fig. 2a, and a reference ply angle ~ measuredrelative to the global Xl axis as shown in Fig. 2b. Here the reference ply angle does notimply measurement from axes of symmetry but is merely a convenience to express themonoclinic material constants in terms of a set of reference material constants.We carry forth our analysis assuming that the constitutive relations matrix for amonoclinic material is fully populated for each layer, as explicitly shown in our previouspaper (Taylor and Nayfeh, 1992). This assumption implies that we conduct our analysiswith the global coordinate system axes formed by normals to the plate boundaries ratherthan by material symmetry axes. We do, however, define a local coordinate X ~ k ) for eachlayer k to conveniently specify arbitrary locations within the layers.We present relevant results from our previous works required for the formalsolutions of each layer of the composite plate. These solutions satisfy in-plane simplysupported boundary conditions from which we obtain appropriate expressions for the

k+1

2- - - - - - -_ . x,

(0)

< ;


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1014 T. W. Taylor and A. H. Nayfehdisplacements and stresses within each layer as

(1a)

(lb)

(1c)

(2a)

(2b)

(2c)where

(3)U ~ k ) a r e the wave amplitudes, OJ is the circular frequency, i = (_1)112, and A(k) is thewavenumber.Hereafter, we suppress the modal subscript n to avoid notational complexity, butwith the understanding that Uq (q = 1, ... , 6), P, A and OJ stand for Uqn, Pn, An and OJn,respectively. Similarly, we suppress the superscript k designating quantities associatedwith layer k, unless needed to avoid ambiguity. In the above relations A satisfies thecharacteristic equation

(4)

(5a)

(5b)

which relates A to OJ for each layer k and whose coefficients AI, A 2 and A 3 are listed inthe Appendix of our single-layer analysis (Taylor et al., 1994). Similarly, for each layerthe displacement ratios and fVq of eqns (1) and (2) arev: = KII(Aq)K23(Aq) - K 13(Aq)KdAq)q K13(Aq)KdAq) - KdAq)K23(Aq)

W _ K II(Aq)K23(Aq) - K dAq)K13(Aq)q - KdAq)K33(Aq) - K23(Aq)KI3(Aq)

where the Kij and the Dijare listed in eqns (11)and (17)of our single-layeranalysis (Tayloret al., 1994).Matrix formulation

We now recast our formal solutions, eqns (1) and (2), in matrix form for subsequentuse in our implementation of the matrix transfer technique. The matrix notation for thedisplacements and stresses is a more formally elegant notation essential for formulationof the problem consistent with the matrix transfer technique. This matrix formulationrelates the displacements U(k) = [u t ) , U ~ k ) , U ~ k ) J T and stresses a(k) = [aW, aW. aWJT ofone face to those of the other of layer k. The formal solutions of eqns (1) and (2) can bewritten. in matrix form, for each layer as

00

cJ)(k)(XI , X ~ k ) , t) = [ C ( X I ) ] [ X ( k ) ] [ E ( k ) ( X ~ k ) J U ( k ) eiwt ,n= I (6)


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Natural frequencies of composite plateswhere cD(k) is the response vector of layer k:

cD(k) _ [U(k)]- a(k) ,[C(XI) ] and [E(X3)(k)] are diagonal matrices with entries

Cqq = cos(mrxtfL), q = 1,2,5,6, Cqq = sin(mrxIIL), q = 3,4,E(k)(X(k = e ) . Q x ~ k )qq 3 ,

where summation is not implied by the repeated index q, and

lOIS

(7)

(8)(9)

(k)

VI V2 V3 V4 V5 V6

[X(k) ] = WI W2 W3 W4 W5 W6 (10)D l1 D I 2 Dl3 D I4 D I 5 D l6D21 D22 D23 D24 D25 D26D31 D32 D33 D 34 D35 D36

We relate the displacements and stresses at X ~ k ) = 0 to those at X ~ k ) = d(k) byspecializing (6) to the two locations and eliminating the common amplitude vector U toobtain

cD(k)(d(k = [a(k)]C1>(k)(O(k.Here, [a(k)] is the micro transfer matrix for layer k and

[a(k)] = [X(k ) ] [E(k ) (d (k) ] [X(k ) r l

(11)

(12)Matrix transfer technique

We next relate the plate's lateral boundary responses to one another by exploitationof the matrix transfer technique. This technique requires that the interfacial continuityconditions at the layer interfaces be completely specified. We consider a plate with itslayers rigidly bonded to one another wherein all components of displacement and threecomponents of the stress tensor are continuous through each interface. This rigid bondingassumption is conveniently summarized at the interface between layers k and k + 1 byC1>(k+ 1)(O(k+ I = C1>(k)(d(k. (13)

The matrix transfer technique then yields, through repeated application of the interfacialcontinuity conditions (13)with (11), the response vector at X3 = d in terms of the responsevector at x3 = 0:where [A] is the macro transfer matrix for the plate and

[A] = [a(N)][a(N-I)] . . . [a(l)].Free vibration

(14)

(15)

To proceed with our analysis for free vibration of the plate we now must specify theplate's lateral boundary conditions. The lateral boundary conditions required for freevibration of the plate are that the stress vector is zero at both the plate 's top and bottomsurface. We summarize these zero-stress conditions byand (16)

The characteristic equation for free vibration of the plate with its layers all rigidly bondedis immediately specified by applying the zero-stress conditions of (16) to the surface


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1016response relations of eqn (14)

T. W. Taylor and A. H. Nayfeh

o = [M]u(l)(O), (17)where the entries M jj of [M] are

i , j= I ,2 ,3. (18)Here, the entries of [M ] are functions of n, A. (k ) and CJJ where A. (k ) must be found, foreach layer, from eqn (4). Note that eqn (18) represents an eigenproblem of order three asopposed to the 6N-order eigenproblems formulated in the "direct" methods of solution.This reduction in the order of the eigenproblem is a direct result of using the matrixtransfer technique wherein we no longer need explicitly solve for the displacement ratios

U(k) for each layer k. The N displacement ratio vectors U(k) are "condensed" out ofour formulation by the recursive application of the interfacial continuity conditionsof eqn (13).

RESULTS AND DISCUSSIONFor our first example we choose a "structural" material whose reference Cij aretabulated in the Appendix. Wenote that any material, regardless of its material symmetryclass (with the exception of isotropy), once rotated from its material symmetry axes,presents a fully populated material matrix, numerically indistinguishable from a monoclinic material matrix. We thus lose no generality in our numerical calculations, relativeto our theoretical developments, for any choice of material symmetry class in ourexamples. In addition, as a bonding agent we use an epoxy material whose materialconstants are also given in the Appendix.We investigate the effect of changes in microstructure on the response of the platethrough changes in the natural frequencies of free vibration. We solve eqn (17) by first

selecting physical dimensions of plate langth L and plate thickness d. We next march inindex n and search a range of circular frequency CJJ for changes in sign of the complexfunction given by the determinant of [M] . Once a change of sign is found for thedeterminant we employ a powerful root solver for complex functions (an IMSL routine)to refine our rough estimate of the root CJJ. After determining the root from the characteristic equation we next employ an eigenproblem solver (again an IMSL routine) todetermine the un-normalized characteristic boundary displacement vector in eqn (17).Equation (17) has three eigenvectors for any natural frequency, one associated with freevibration and the other two not pertinent to our solution. The eigenvector of interest isfinally determined by selecting that eigenvector associated with the zero eigenvalue of [M].We note here that the plots contained in the subsequent figures are "dimensionless"in the sense that the abscissa is nondimensionalized mode number nd/L and the ordinateis normalized frequency f . d. In this way, once the physical dimensions of a plate areselected one need only refer to the correct nondimensional mode number and therebyobtain the frequency from the appropriate modal curve of I " d. Hence the plots areindependent of plate dimensions and given as solid curves as opposed to discrete datapoints. The natural frequencies of plates of arbitrary thickness with identical constituentvolume fractions and materials are therefore completely specified on the figures.We present in Fig. 3a a plot of the natural frequencies of a plate strip composed offive layers of equal thickness (volume fraction 0.2) structural material. The reference plyangles for the layers are rPk = 30, 45, 90, -45, -30, k = 1, ... , 5, respectively.We call special attention to the presence of six distinct thickness modes in the displayedfrequency range of 0 =:; f d =:; 2.4MHz mm.In Fig. 3b we present a plot of the natural frequencies of a plate composed of ninelayers, five layers each of structural material with volume fraction 0.176, alternating withfour layers of epoxy bonding material each of volume fraction 0.03. Each layer of theepoxy bonding material is thus "sandwiched" between layers of the structural material.This construction of the layered plate is intended to simulate a realistic composite layeredplate where the structural layers are bonded together by a more pliant, thin adhesive layer,


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1.0

1017

0.8.4 0.6nd/L(b)0.20.0 ~ - = - - - ' - - - " " " " - - ' - - - - '0.0

~ 0 . 8

1.0.8

Natural frequencies of composite plates2.4 r - - - . . . . , . . - - . . . . . - ~ ~ ......- , . . . ~ - . . . - - - ,

0.4 0.6nd/L(0)

0.20.0 ~ - ~ - - " " ' - " " " " - - - " " " " - - " " - - '0.0

~ 0 . 8

116 r- t _:2....,

Fig. 3. Natural frequencies for (a) a five-layerplate with each layer's vf = 0.2, and (b) a nine-layerplate, five structural layers each with vf = 0.176 and four epoxy bonding layers each withVf = 0.03. The structural material ply angles are 30, 45, 90, -45 and -30 in the plates ofboth (a) and (b).

the epoxy layer. Again, the reference ply angles for the structural layers of Fig. 3b areidentical with those of the layers in the plate of Fig. 3a. In the plot of Fig. 3b we now callattention to the presence of eight distinct thickness modes of free vibration in the samefrequency range as that of Fig. 3a.Figures 3a and 3b reveal striking differences in the natural frequencies of a plate withthe bonding material included in the analysis, Fig. 3b, as compared to a plate with nobonding material between structural layers, Fig. 3a. The "soft" epoxy bonding agentbetween the structural layers of the plate introduces two additional thickness modes offree vibration within the same frequency range as compared to the plate without epoxy.In addition, the magnitude of the natural frequencies for similar modes in the nine-layerplate steadily decreases with mode number relative to the five-layer plate with no bondingmaterial.

We illustrate the change in natural frequency with the change in mode number inFig. 4 for the plates of Fig. 3. Figure 4a is a comparison of the first two naturalfrequencies and the third and fourth in 4b. The natural mode curves for the plate of figure3a are displayed as solid lines, while those of the plate with the epoxy bonding layers aredisplayed as dashed lines. Note the distinct decrease in natural frequencies with increasingwavenumber for the nine-layer plate relative to those of the five-layer plate. Thedifference in frequency is also more pronounced with higher-order thickness modes. Thisbehavior indicates that neglecting the bonding layers for plates composed of laminatedplies can introduce considerable error in free vibration mode calculations. The explicit

1.0.8.4 0.6nd/L(b)

0.2

-5 LAYER- - - - II LAYER

0.0 .. . " " " ' -.................-- ' - - -lo-. . . . . . . . . . . . . . - - - J0.0

~ 0 . 8

1.0.8.4 0.6nd/L(0)0.2

-5 LAYER- - - - II LAYER

0.50.0 l ' o o ~ : : : . . . . . . .........--'- -lo-__-..._---I

0.0

Fig. 4. Comparison of natural frequencies for plates in Figs 2a and 2b for (a) first and secondthickness modes, and (b) third and fourth thickness modes. Solid curves represent the plate withfive structural layers (no epoxy), and the dashed lines represent the nine-layer plate (five structuraland four epoxy).


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1018 T. W. Taylor and A. H. Nayfeh

0.4 p.8ndfL 0.8 1.0

1.0.8.4 p.8nd/L0.20.0 " " ' 1 I l I = : : I : : . . . . . . - . . I - ~ ~ ~ . & : - - ' ~0.0

(b)

1.0.8.20.0 _ i lC l . - - < .........-- '..........- - ' - - - ' -..........--'0.0

EI" I . - - - - ? . . . - -0.8...

(c)

0.4 p.8nd/LFig. 5. Natural frequencies for plates composed of the individual structural layers of the plates ofFigs 3a and 3b. Each plate iscomposed of a singlestructural layer of (a) 30 ply angle, (b) 45 plyangle, and (c) 90 ply angle.

modeling of the plate's microstructure, the layers, thus points out an important feature ofrealistic laminated plates, that is, the considerable "softening" effect of a bonding agentbetween layers. This "softening" effect is typically glossed over or ignored in calculationsfor the dynamic response of layered plates.For comparison with the composite plates of Fig. 3, Fig. 5 presents plots of frequencyfor plates composed of a single layer of the structural material in the plates of Fig. 3.Figure 5a shows the frequency response of a 30 lamina plate, (b) a 45 lamina and (c) a90 lamina. Note the presence, in each case, of six thickness modes in the displayedfrequency range. Finally, Fig. 6 presents comparisons of (a) the first thickness mode and

1.2 r - - - ~ - ~ - , - " " " " - ~ - ' - - - r - ~ . . . . . ,

1.0.8.4 0.6nd/l(b)

0.2

-5 LAYER- - I I LAYER- - - - 30" LMNA

0.0 - - ' - - - - - ' _ ~ ~ - - - L _ ~ - ' -..........___'0.0

0.5

,...1.5EE~ 1 . 0.....

1.0.8.4 0.6nd/L(0)0.2

---5 LAYER- - I I LAYER- -30" lAlollNA---- 45 lAlollNA- - - lIO" lAlollNA

0.0 _ K : ; . , - - ' - - - - - ' _ ~ ~ - - - L _ ~ - ' -.........____'0.0

~ 0 . 4

10.8J::2.....

Fig. 6. Comparisons of natural frequency for the plates in Figs 3a, 3b, Sa, 5b and 5c for (a) thefirst thickness mode, and (b) the second thickness mode.


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Natural frequencies of composite plates 1019

1.0.8.4 0.6nd/L(b)

0.20.0 we:;;..=s:::::::::::......... .........--'-.........--' '--.. . . . . . . . .J0.0

e2.4E! 1.6 ~ - - - ' : : : l " ~ : : : : , _ . - = = ; = - - - "~ 0 . 8 L - - - = = ~ ~ " " ' - -

0.8.4 0.6nd/L(o)

3.2eUE~ 1 . 6....,

~ O . 80.00.0 0.23.2

e2.4E~ 1 . 6....,

~ 0 . 80.0 0.0 l I l l l I ! ! ! ! : : : : : : ; C : : = ~ = . . . - - - ' I - - ' O " - . . . . L - .........J0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0nd/L nd/L(c) (d)Fig. 7. Natural frequencies of plates with (a) a single steel layer and sandwich plates with twoequal-thickness steel layers bonded with a single epoxy layer with epoxy volume fract ion Vf = (b)0.Q2, (c) 0.04 and (d) 0.1.

(b) the second thickness mode for the five plates of Figs 3 and 5. Interestingly, consideringthe first mode of vibration, the 90 plate is the stiffest while the nine-layer plate, asexpected, is the least stiff. In contrast, comparison of the second mode shows the 45plate to be the stiffest while the 90 lamina is the least stiff. In both thickness modes thefive-layer plate is nearly the stiffest, showing that it may be the best choice (assuming thedesire for a high natural frequency response) when all modes of vibration are considered.Our second example is a sandwich plate with a layer of epoxy bonding materialsandwiched between two equal-thickness layers of steel. We use this example so that ourcalculations may be verified by the interested reader and to point out the fundamentaldifference between plates composed of relatively light vs dense structural layers. In thisexample, the density of the steel layers is 7.9 g cm? as compared to 1.6 g ern- 3 for thestructural material of our first example. The density of the epoxy bonding material is1.25 g em-3 in both examples. Figure 7 presents plots of natural frequency for plates with(a) all steel, and the sandwich plate with the epoxy Vr = (b) 0.02, (c) 0.04 and (d) 0.10.Note that for this laminated plate arrangement no new higher-order thickness modesare introduced with increasing epoxy volume fraction in the displayed frequency range asnoted in the first example. Indeed, the six frequency curves are merely compressedtowards lower frequency with the increase of epoxy volume fraction. The remarkabledifference in the behavior of the steel sandwich plate as compared to our first plateexample lies in the relative densities of the structural materials. In the first example thestructural material is 60070 more dense that the epoxy, whereas the steel is 532070 moredense than the epoxy. Thus, the steel sandwich plate's frequency response is mass controlled as the steel layer's mass is relatively far greater than the bonding layer's mass.Figure 8 presents the first and third frequency curves for the steel and sandwich plate ofFig. 7. Here the reduction of frequency with mode number is more pronounced than inour first example, a result of the relative values in density of the steel and epoxy layers.

SUMMARY AND CONCLUSIONSWe have derived expressions for the normal modes of simply supported thick,layered anisotropic plate strips of monoclinic and higher material symmetry. In addition,we have solved the characteristic equation for free modes of vibration using an innovative


10/11

1020 T. W. Taylor and A. H. Nayfeh

1.0.8

---0.4 0.6

nd/L(b)0.2.0 " " - - .........--"---.........--'--.....0.0

0.6

1.0.8.4 0.6nd/L(0)0.2

-srm- - EPOXY- - - - .()4 EPOXY- - - .10 EPOXY

~ 0 . 4

10.8J::2......

1.2 r--........- - r - - . . . . - - ~ - - . . . . . - -7"I

Fig. 8. Comparisons of natural frequencies of the four plates in Fig. 7 for (a) the first thicknessmode, and (b) the third thickness mode.

formulation, considerably more well-behaved than in previous treatments. Our formulation renders the order of the eigenproblem in free vibration independent of the number oflayers in a laminated composite plate, considerably decreasing computational expense forthe exact method of solution. We have investigated the effects of microstructure andbonding agents on the plate's vibratory response by examining the changes in naturalfrequency of free vibration. This investigation shows behavior of layered composite platesthat no "mixture" theory can possibly predict.Acknowledgements-This work is partially supported by NASA Grant NSG1I74 and by NATO GrantCRG9003l8.

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APPENDIXStructural material constants (Cij in GPa):

Cn = 128Cll = 72C33 = 32C13 = 6C23 = 5C12 = 7C44 = 18Css = 12.25C66 = 8p=2gcm- 3

Epoxy material constants (Cij in GPa):Cn = 3.45Css = 1.28

p = 1.25 g cm".

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