Flexural loss factors of sandwich and laminated composite beams using linear and nonlinear dynamic analysis

Composites: Part B 30 (1999) 245 256 Flexural loss factors of sandwich and laminated composite beams using linear and nonlinear dynamic analysis M. Ganapathi a, *, B.P. Patel a, P. Boisse b, O. Polit c a Institute of Armament Technology, Girinagar, Pune 411 025, India b ESEM, University of Orleans, 40572-Orleans, France c University of Paris X-IUT-Dep. GMP, 1 Chemin Desvallieres, 92410-Ville d Avray, France Received 26 June 1998; accepted 23 September 1998 Abstract The purpose of the article presented here is to analyze the flexural loss factors of beams with sandwich or constrained layer damping arrangements and laminated composite beams using a C 1 continuous, three-noded beam element. The formulation is general in the sense that it includes anisotropy, transverse shear deformation, in-plane and rotary inertia effects, and is applicable for both flexural and torsional studies. The geometric nonlinearity based on von Karman s assumptions is incorporated in the formulation while retaining the linear behavior for the material. The finite element employed here is based on a sandwich beam theory, which satisfies the interface stress and displacement continuity and has zero shear stress on the top and bottom surfaces of the beam. The transverse shear deformation in the form of trigonometric sine function is introduced in the formulation to define the transverse shear strain. The governing equations of motion for the dynamic analysis are obtained using Lagrange s equation of motion. The solution for nonlinear equations is sought by using an algorithmdirect iteration technique suitably modified for eigenvalue problems, based on the QR algorithm. A detailed numerical study is carried out to highlight the influences of amplitude of vibration, shear modulus and thickness of the core of the sandwich beam, aspect ratios, boundary conditions, and lay-up in the case of laminates on the system loss factors. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Flexural; A. Laminates 1. Introduction Sandwich laminated composite structures find an increasing use in aerospace, shipbuilding, construction and other industries. In order to control the resonant amplitudes of vibration and thus in extending service life of such structures under periodic load/impact, the damping in the core and/or constrained layer, and in the composite materials plays an important role. It is common practice to use viscoelastic material as a core material in constrained layer and sandwich layer arrangements for increasing the over all damping characteristics of the structures. For fiber reinforced composites, damping value is higher, in general, compared to that of metallic structures and it depends on fiber and resin type, ply-angle and lay-up. A considerable amount of research work has been done on the vibration and damping of beams with constrained layer/sandwich arrangements over the past few decades. Earlier work on this subject was done by Kerwin [1], Ross * Corresponding author. Fax: 91-020-599509. E-mail address: gana@iat.ernet.in (M. Ganapathi) et al. [2], Ungar and Kerwin [3], and Mead and Markus [4] A good exposition on the available literature dealing with the vibration control with viscoelastic material can be found in the survey articles by Nakra [5,6]. Based on these earlier works, some of the important recent contributions are the work of Moser and Lumassegger [7], Vaswani et al. [8], Hajela and Lin [9], He and Rao [10], Rikards [11] and Bhimaraddi [12]. In all these works, a complex modulus, which consists of real part representing elastic stiffness and an imaginary part representing dissipation, has been widely used to model the behavior of linear viscoelastic materials under harmonic vibration, and they are all based on linear vibration analysis. It may be inferred from most of these investigations that the insight into the variation of damping characteristics with aspect ratio of the beam in conjunction with thickness ratio of skin-to-core and material properties of sandwich or constrained layered case has not been clearly brought out. Such parametric studies are essential to the development of structural design strategies. Further, even though a large amount of work has been carried out on the nonlinear dynamics of continuum media, relatively little has been done in the area of vibration and damping of beams 1359-8368/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S1359-8368(98)00063-8

246 M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 Fig. 1. (a) Sandwich beam co-ordinate system. (b) Description of sandwich beam finite element. with constrained layer/sandwich arrangements. The available work introducing geometrical nonlinearity for beams with viscoelastic core, namely, Kovac et al. [13], and Hyer et al. [14], has been dealt with forced response analysis. These investigations are conducted employing Galerkin s procedure and the method of harmonic balance. However, no knowledge is readily available, about the influence of amplitude of vibrations of the beam on the system damping factors of constrained layered/sandwich structures. Research on the damping analysis of laminated fiber reinforced composites is not so extensive as that of constrained layer arrangement or sandwich case. Gibson and Plunkett [15], and Gibson [16] reviewed experimental and analytical efforts to characterize the damping properties of fiber reinforced materials. Most of the available work is devoted to laminated composite plates, for instance the work of Lin et al. [17], Alam and Asnani [18], Malhotra et al. [19], and Koo and Lee [20], and they are all based on linear dynamic analysis. However, the investigation related to fiber reinforced composite beam seems to be scarce in the literature. In the light of these observations, an attempt is made here to study the problem through linear and nonlinear dynamic analysis. In the nonlinear analysis, the amplitudes of the vibration of the beam are assumed to be moderately large enough to cause geometric nonlinearity but they are within the limit to consider the linear behavior for the material. Further, this study is also meant for bringing out the combined effects of various parameters such as aspect and thickness ratios, material properties, number of plies in the case of laminate, and boundary conditions on the damping behavior of the beam with constrained layer/sandwich arrangement, and reinforced composite laminates. The mathematical model is based on shear flexible theory and is solved using finite element method. Here, complex eigenvalue problem based on complex moduli is formulated using a new beam element developed recently by Ganapathi et al. [21,22]. Geometric nonlinearity arising from moderately large deformation of the beam has been included based on von Karman s theory but the linear behavior for the material is assumed. The formulation includes inertia associated with bending rotations and inplane motion. The beam theory, used for developing this element, satisfies interface transverse shear stress and displacement continuity in the thickness direction, and has a vanishing shear stress on the top and bottom surfaces of the beam. A higher order deformation theory in the form of a trigonometric sine function is incorporated in the beam theory. The nonlinear governing equations obtained here are solved using a direct iteration technique, where the linear mode shape is taken as the starting vector. The nonlinear equation can be degenerated to a linear case by neglecting the nonlinear stiffness matrices in the formulation. Necessary convergence criteria are specified for the displacement vector and also for the frequency/damping value for the fundamental mode. It can be noted here that this is a maiden attempt employing finite element procedure to predict the system loss factors (damping characteristics) by using nonlinear dynamic analysis. The nonlinear frequency values and in turn, system loss factors are obtained for various values of amplitudes, while considering different geometric and material parameters. 2. Formulation A laminated composite beam is considered with the coordinates x along the length, y along the width and z along the thickness directions as shown in Fig. 1(a). The displacements in kth layer u k, v k and w k at a point (x,y,z) from the median surface are expressed as functions of mid-plane displacements u, v, w, independent shear bending rotations u x and u y of the normal in xz and xy planes. They are also the functions of torsional rotation u and independent parameter g for torsional rotation gradient in the length direction as u k x; y; z; t ˆu x; t yv ;x x; t f 2 y v ;x x; t u y x; t Š zw ;x x; t ; f 3 z g k z Š w ;x x; t u x x; t Š f k y; z g x; t ; v k x; y; z; t ˆv x; t zu x; t w k x; y; z; t ˆw x; t yu x; t ; where t is the time, and the subscript comma denotes the partial derivative with respect to spatial coordinate succeeding it. The functions f 2 (y), f 3 (z) and g k (z) are defined as: f 2 y ˆb=p sin py=b ; 2a f 3 z ˆh=p sin pz=h h=p b 55 cos pz=h ; g k z ˆa k z b k ; 1 2b 2c b and h are width and total thickness of the beam. In Eqs. (2a) (2c), coefficients b k are determined such that the contribution to the displacement component u k, because of bending in the xz plane, is continuous at the interface of adjacent layers and is zero at the mid-point of

the cross-section. Finally, coefficient b 55 and a k in Eq. (2a) (2c) are computed from the requirement that the transverse shear stress owing to bending in the xz plane is continuous at the interface of the adjacent layers and vanishes at the top and bottom surfaces of the beam. The detailed derivation of these constants b 55, a k and b k can be obtained from the work of Ganapathi et al. [21,22] and Beakou and Touratier [23]. The kinematics shown in Eq. (1), in particular for torsion, allows one to represent the constrained torsion where axial stress is not zero, for instance near the clamped support, and free torsion ie. Saint-Venant torsion when g approaches u x which may be realized far away from the support of a thin beam. The torsional warping function f k used in defining the kinematics in Eq. (1) is the solution derived from threedimensional elasticity equations in conjunction with Saint- Venant assumption of torsion, for a composite beam of rectangular cross section made of different layers. The general expression for f k is taken in the form of a harmonic function and is expressed as f k ˆ X Nˆ1;3 C k N sinh az D k N cosh az sin ay yz; 3 where a is defined as Np/b. The coefficients C k N and D k N in the Eq. (3), while defining the warping function for the rectangular cross section, are determined such that the contribution to the displacement component u k owing to torsion is continuous at the interface of adjacent layers, and the transverse shear stress associated with torsion, is continuous at the interface of the adjacent layers and vanishes at the top and bottom surfaces of the beam. Using von Karman s assumption for the moderately large deformation analysis, the strains in terms of mid-plane deformation for kth layer can be written as 8 9L 8 1 p >< >= >< {1 k } ˆ 0 >: >; >: 0 1 k xx 21 k xz 21 k xy 9b 8 >= >< >; >: 1 k xx 21 k xz 21 k xy 9t 8 9 1 p >= >< >= 0 >; >: >; 0 NL : 4 where superscripts b and t denote the strain contributions arising from bending and torsion, respectively. L and NL refer to the linear components of mid-plane strain and nonlinear part of the in-plane strains, respectively. The mid-plane linear strains part {1 p } L, the nonlinear component of the in-plane strain {1 p } NL, strain terms associated with bending and torsion in Eq. (4) are written as {1 p } L ˆ {u; x }; 5a M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 247 8 >< >: 1 k xx 21 k xz 21 k xy 8 9b >= ˆ >; 9 zw ;xx f 3 z g k z Š w ;xx u x;x Š yv ;xx f 2 y v ;xx u y;x >< >= f 3;z g k ;z w ;x u x ; >: f 2;y v ;x u y >; 8 >< >: 1 k xx 21 k xz 21 k xy 5c 9 8 9 t f >= k g ;x >< >= ˆ f k ;zg yu ;x : 5d >; >: f k ;yg zu >; ;x For a composite laminated beam of thickness h k (k ˆ 1, 2, 3 ), and the ply-angle f k (k ˆ 1, 2, 3 ), the necessary expressions for computing the stiffness coefficients, available in the literature [24], are used. As the formulation deals with the damping model, energy dissipation under harmonic vibration arising from a viscoelastic core is taken into account with complex moduli of an orthotropic material of the form as shown in the following: E * 1 ˆ E R 1 ie I 1; E * 2 ˆ E R 2 ie I 2; E * 3 ˆ E R 3 ie I 3; G * 12 ˆ G R 12 ig I 12; G * 23 ˆ G R 23 ig I 23; G * 13 ˆ G R 13 ig I 13: 6 Here, E * and G * are Young s modulus and shear modulus, respectively. The subscripts 1 denotes longitudinal direction whereas 2 and 3 refer to the transverse directions, with respect to the fibers. The superscripts R and I denote the real and imaginary parts of the complex moduli. The material loss factors h 1, h 2, h 3 under tensioncompression and h 12, h 23, h 13 under shear are defined as h 1 ˆ E I 1=E R 1 ; h 2 ˆ E I 2=E R 2 ; h 3 ˆ E I 3=E R 3 ; h 12 ˆ G I 12=G R 12; h 23 ˆ G I 23=G R 23; h 13 ˆ G I 13=G R 13: 7 The stress strain relation for kth layer is written as 2 3 Q k 11 0 Q k 16 {s k } ˆ 0 Q k 6 44 0 7 4 5 {1k } 8 Q k 16 0 Q k 66 where Q k ij (i, j ˆ 1,4,6) are the reduced stiffness coefficients of kth layer and are complex quantities. The total potential energy functional U of the system is given as ZL Zb=2 X Zh k 1 U d ˆ 1=2 {s k } T {1 k }dx dy dz 0 b=2 k h k {1 p } NL ˆ { 1 2 w2 ;x}; 5b ZL {uvwu x u y u}{f x f y f z m y m z m x } T dx; 0 9

248 M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 where d and L are the vector of the degrees of freedom associated to the displacement field in a finite element discretization and length of the beam, respectively. f x, f y, f z are the distributed forces in the x, y and z directions and m x, m y, m z are the moments about the x, y and z axes. Following the procedure outlined by Rajasekaran and Murray [25], the total potential energy functional U is rewritten as U d ˆ d f g T 1=2 KŠ 1=6 N 1 Š 1=12 N 2 ŠŠ d fg fdg T ffg; 10 where [K] is the linear stiffness matrix, and [N 1 ] and [N 2 ] are nonlinear stiffness matrices, which are of complex form, respectively and {F} is the force vector. The kinetic energy of the beam is written as ZL Zb=2 X Zh k 1 T d ˆ1=2 r k { _u k _v k _w k }{ _u k _v k _w k } T dx dy dz; 0 b=2 k h k 11 where the dot over the variable denotes the partial derivative with respect to time and r k is the mass density of the kth layer. Substituting Eqs. (10) and (11) into Lagrange s equation of motion, one obtains the governing equation for the vibration of the beam structure as MŠ{ d } KŠ 1=2 N 1 Š 1=3 N 2 ŠŠ{d} ˆ {F} 12 where [M] is the consistent mass matrix. The eigenvalues for the damped structure can be determined from Eq. (12) by letting {F} equal to zero for the free vibrations MŠ{ d } KŠ 1=2 N 1 Š 1=3 N 2 ŠŠ{d} ˆ {0}: 13 Substituting the characteristics of the time function at the point of reversal of the motion {d} max ˆ l * {d} max 14 in Eq. (13), will lead to the following nonlinear algebraic equation of the form KŠ 1=2 N 1 Š 1=3 N 2 ŠŠ{d} l * MŠ{d} ˆ {0}: 15 The complex eigenvalues of the form l * ˆ l R il I ˆ v * 2 where v * ˆ v R iv I are obtained for the above equation by using direct iteration technique suitably modified for the eigenvalue problems based on the QR algorithm. The resonance frequencies v and the system loss factors h are calculated from the eigenvalues [11], corresponding to different amplitude of vibration level as: v ˆ v R ˆ l R 1=2 ; h ˆ l I =l R 16. 3. Description of the element The element used here is based on Hermite cubic functions for transverse displacements, v and w according to the C 1 continuity requirement, quadratic functions for rotations, u x, u y and u, and linear functions for in-plane displacement, u and roation gradient pertaining to torsion g. Further, the element needs nine nodal degrees of freedom u, v, v, x, w, w, x u x u y u and g at both the ends of the threenoded beam element whereas the center node has three degrees of freedom u x u y, and u shown in Fig. 1(b). This choice of the functions allows us to have the same order of interpolation for both w, x and u x, v, x and u y in the definition of shear strain and permits to avoid transverse shear locking phenomena. Similarly, the u x and g in the torsional strain are interpolated with same degree polynomial which recovers the Saint-Venant torsion (g ˆ u x ). The element behaves very well for both thick and thin situation pertaining to flexure and torsion. It has no spurious mode and is represented by correct rigid body modes. 4. Results and discussion The study presented here has been focussed on highlighting the changes in the flexural damping behavior of the sandwich beam or constrained layer case, and the laminated fiber reinforced composite beams with material and geometrical parameters based on the linear and nonlinear dynamic analysis. In the nonlinear analysis, the amplitudes of the vibration of the beam are assumed to be moderately large enough to cause geometric nonlinearity but they are less enough to retain linear behavior for the material. As the element is derived based on field consistency approach, an exact numerical integration scheme is employed to evaluate all the strain energy terms. Also, there is no need of using shear correction factor here, as the transverse strain is represented by cosine function, which is of higher order in nature. Thus, the present development can be verified numerically by comparing the results based on different models, which are used for studying the thin and thick beams. Such comparisons were made through linear and nonlinear dynamic analyses, wherever possible, for the frequency values of single/multi-layered beams with and without viscoelastic core and an excellent agreement was observed. For the sake of brevity, these results are not presented here. In this section, the system loss factors as obtained from this work will be discussed in detail. Further, based on progressive mesh refinement, 16 elements idealization is found to be adequate to model the sandwich beam for the flexural/bending damping analysis. Here, a symmetric beam of three plies is chosen for the constrained layer arrangement or sandwich case and the middle ply/core ply/damp ply is assumed to be of soft viscoelastic material. The material property [11] and the geometrical parameters considered here are given as follows:

M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 249 Fig. 2. (a) Variation of loss factor (h L ) with aspect ratio (L/h) of clamped sandwich beam with core shear modulus 2.5 10 6 N/m 2 (W: first mode; x: second mode; D: third mode). (b) Variation of loss factor (h L ) with aspect ratio (L/h) of clamped sandwich beam with core shear modulus 25 10 6 N/m 2 (W: first mode; x: second mode; D: third mode). (c) Variation of loss factor (h L ) with aspect ratio (L/h) of clamped sandwich beam with core shear modulus 250 10 6 N/m 2 (W: first mode; x: second mode; D: third mode). (d) Variation of loss factor (h L ) with aspect ratio (L/h) of the clamped sandwich beam with core shear modulus 2500 10 6 N/m 2 (W: first mode; x: second mode; D: third mode). Material properties: For the face or skin: E f ˆ 45.54 GPa, G f ˆ 17.12 GPa, v f ˆ 0.33, r f ˆ 2040 kg/m 3, h 1 ˆ h 2 ˆ h 3 ˆ h 12 ˆ h 23 ˆ h 13 ˆ 0.0. For the core: E c is varied as 7.25, 72.5, 725 and 7250 MPa, v c ˆ 0.45, r c ˆ 1200 kg/m 3, h 1 ˆ h 2 ˆ h 3 ˆ h 12 ˆ h 23 ˆ h 13 ˆ 0.5. Geometrical parameters: The aspect ratio (L/h) is varied from 10 to 200. The ratio of thickness of face-to-core (h f /h c ) is taken as 7, 1 and 1/7. For the laminated fiber reinforced composite beam analysis, the material is assumed as CFRP(HMS/DX-210) and has the following properties

250 M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 Fig. 2. (continued) [17]: E 1 ˆ 172:70 GPa; E 2 ˆ 7:20GPa; E 3 ˆ 7:20 GPa; G 12 ˆ 3:76 GPa; G 23 ˆ 3:76 GPa; G 13 ˆ 3:76 ; n f ˆ 0:30; h 1 ˆ 7:16197 10 4 ; h 2 ˆ h 3 ˆ 6:71634 10 3 ; h 12 ˆ h 23 ˆ h 13 ˆ 1:12204 10 2 ; r ˆ 1566 kg=m 3 : Geometrical parameters: The aspect ratio is varied and all the layers are of equal thickness. The subscript f and c refer to the face or skin and core of the sandwich case. Firstly, based on linear analysis, the variation of the system loss factors (h L ) obtained is shown in Figs. 2 and 3 against aspect ratio (L/h) of the beam, considering different values for the core shear modulus (G-core) and core-toface thickness ratio (h f /h c ). Both the clamped clamped and simply supported boundary conditions are considered. The shear modulus G of the core and thickness ratio (h f /h c ) of the sandwich beam are varied in such a way that one can see the

M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 251 Fig. 3. (a) Variation of loss factor (h L ) with aspect ratio (L/h) of the simply supported sandwich beam with core shear modulus 2.5 10 6 N/m 2 (W: first mode; x: second mode; D: third mode). (b) Variation of loss factor (h L ) with aspect ratio (L/h) of the simply supported sandwich beam with core shear modulus 25 10 6 N/m 2 (W: first mode; x: second mode; D: third mode). (c) Variation of loss factor (h L ) with aspect ratio (L/h) of the simply supported sandwich beam with core shear modulus 250 10 6 N/m 2 (W: first mode; x: second mode; D: third mode). (d) Variation of loss factor (h L ) with aspect ratio (L/h) of the simply supported sandwich beam with core shear modulus 2500 10 6 N/m 2 (W: first mode; x: second mode; D: third mode). behavior of the beams made of constrained layered damping arrangement to sandwich construction. For low values of the core shear modulus ( 25 10 6 N/ m 2 ) considered here and also for the given geometry, it is seen from these figures that the system loss factor increases with increasing in the aspect ratio up to a certain value. Then, a further increase in the aspect ratio decreases the value of the loss factor. Further, one can observe that the rate of increase in the loss factor is more compared to the decreasing rate with respect to the aspect ratio, irrespective of the modes. This also reveals that for the aspect ratio up to a certain value, the value of damping factor decreases with the increase in mode numbers and then increases with the increase in mode numbers. It is further noticed that the rate of increase or decrease in the value of loss factor with respect to the aspect ratio is more for the lowest resonant mode. The effect of increasing the core thickness, in general, enhances the damping values significantly and in

252 M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 Fig. 3. (continued) particular, in the low range of aspect ratio of the beam. With an increase in the shear modulus value from 2.5 MPa to 25 MPa (Figs. 2(a), (b), 3(a) and (b)), it may be concluded that the range of aspect ratio, over which increasing in the damping values occurred, decreases. For a higher aspect ratio, the influence of thickness of core-to-skin on the values of the system loss factors is considerably less compared to those of lower aspect ratio cases. For a high core shear modulus ( 25 10 6 N/m 2 ) case, it can be noted from these figures (Figs. 2(c), (d), 3(c) and (d)) that the value of loss factor for the low aspect ratio case considered here depends on the thickness of the core. However, it decreases drastically in the higher range of aspect ratio. For a high shear modulus and thick core case, the nature of loss factor behavior with respect to aspect ratio is similar to that of other thickness cases but the actual damping value does not approach to the value of low core thickness case with the increase in the aspect ratio. It may be opined, in general, that the core with a low shear modulus and high thickness enhances the loss factors for the given aspect ratio. For a higher aspect ratio, the variation in the loss factor values is less among the resonant frequencies and is more so with the increase in the

M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 253 Table 1 Loss factor ratio (h NL /h L ) with vibration amplitudes (w/h) of clamped sandwich beam for different core properties (geometry and material) G-core ˆ 2.5 10 6 N/m 2 G-core ˆ 25 10 6 N/m 2 G-core ˆ 2500 10 6 N/m 2 h f /h c w/h L/h ˆ 10 30 70 100 200 10 30 70 100 200 10 30 70 100 200 0.1 0.9948 0.9953 0.9973 0.9981 0.9988 0.9954 0.9982 0.9987 0.9990 0.9991 0.9990 0.9999 1.0033 1.0065 1.0242 0.3 0.9571 0.9591 0.9763 0.9825 0.9890 0.9603 0.9818 0.9894 0.9909 0.9923 0.9905 0.9988 1.0259 1.0573 1.2155 0.5 0.8907 0.8940 0.9369 0.9527 0.9701 0.8969 0.9506 0.9714 0.9753 0.9790 0.9739 0.9967 1.0697 1.1561 1.5869 0.7 0.8100 0.8118 0.8834 0.9113 0.9433 0.8168 0.9074 0.9458 0.9530 0.9601 0.9503 0.9938 1.1324 1.2975 2.1179 7 0.9 0.7262 0.7232 0.8209 0.8615 0.9102 0.7303 0.8555 0.9140 0.9253 0.9366 0.9211 0.9902 1.2110 1.4742 2.7809 1.1 0.6451 0.6369 0.7542 0.8071 0.8724 0.6459 0.7993 0.8778 0.8936 0.9097 0.8879 0.9864 1.3017 1.6781 3.5448 1.3 0.5733 0.5544 0.6873 0.7504 0.8319 0.5650 0.7409 0.8391 0.8592 0.8807 0.8524 0.9824 1.4011 1.9012 4.3783 1.5 0.5109 0.4822 0.6231 0.6941 0.7899 0.4941 0.6832 0.7989 0.8241 0.8507 0.8155 0.9789 1.5059 2.1356 5.2522 Linear 0.0164 0.0930 0.1746 0.1606 0.0811 0.0973 0.1635 0.0701 0.0400 0.0115 0.0351 0.0052 0.0010 0.0005 0.0001 0.1 0.9906 0.9906 0.9947 0.9967 0.9985 0.9909 0.9963 0.9989 0.9989 0.9993 0.9993 1.0000 1.0018 1.0027 1.0040 0.3 0.9240 0.9213 0.9556 0.9704 0.9867 0.9246 0.9683 0.9882 0.9910 0.9937 0.9919 1.0006 1.0161 1.0243 1.0358 0.5 0.8211 0.8084 0.8854 0.9216 0.9639 0.8165 0.9166 0.9675 0.9756 0.9827 0.9774 1.0017 1.0439 1.0665 1.0980 0.7 0.7110 0.6836 0.7967 0.8565 0.9317 0.6962 0.8481 0.9382 0.9535 0.9670 0.9567 1.0033 1.0841 1.1275 1.1877 1 0.9 0.6097 0.5640 0.7044 0.7821 0.8921 0.5805 0.7706 0.9018 0.9258 0.9471 0.9309 1.0054 1.1350 1.2046 1.3012 1.1 0.5263 0.4621 0.6151 0.7072 0.8473 0.4815 0.6935 0.8603 0.8937 0.9238 0.9013 1.0080 1.1947 1.2949 1.4341 1.3 0.4593 0.3779 0.5349 0.6338 0.7993 0.3994 0.6187 0.8161 0.8584 0.8981 0.8694 1.0110 1.2611 1.3953 1.5815 1.5 0.4073 0.3100 0.4658 0.5660 0.7501 0.3331 0.5504 0.7701 0.8217 0.8707 0.8359 1.0143 1.3322 1.5028 1.7391 Linear 0.0158 0.0906 0.2308 0.2612 0.2065 0.1002 0.2591 0.1895 0.1282 0.0439 0.1252 0.0242 0.0075 0.0054 0.0039 0.1 0.9376 0.9838 0.9962 0.9979 0.9992 0.9853 0.9977 0.9993 0.9994 0.9996 0.9996 1.0001 1.0002 1.0002 1.0003 0.3 0.6255 0.8692 0.9671 0.9817 0.9930 0.8798 0.9800 0.9938 0.9953 0.9966 0.9966 1.0011 1.0023 1.0024 1.0026 0.5 0.3811 0.6983 0.9130 0.9506 0.9809 0.7194 0.9462 0.9829 0.9872 0.9906 0.9909 1.0031 1.0063 1.0067 1.0071 0.7 0.2536 0.5491 0.8413 0.9070 0.9633 0.5732 0.8992 0.9589 0.9754 0.9819 0.9825 1.0060 1.0121 1.0129 1.0136 1/7 0.9 0.4164 0.7600 0.8543 0.9409 0.4427 0.8429 0.9469 0.9602 0.9707 0.9720 1.0096 1.0195 1.0209 1.0220 1.1 0.6790 0.7959 0.9144 0.3626 0.7809 0.9229 0.9421 0.9573 0.9597 1.0140 1.0283 1.0303 1.0319 1.3 0.6005 0.7348 0.8846 0.7169 0.8958 0.9216 0.9420 0.9460 1.0190 1.0383 1.0410 1.0431 1.5 0.5289 0.6737 0.8523 0.6536 0.8664 0.8990 0.9252 0.9316 1.0244 1.0491 1.0526 1.0552 Linear 0.1737 0.3804 0.4012 0.3661 0.2259 0.3889 0.3735 0.2026 0.1267 0.0398 0.1869 0.0862 0.0717 0.0699 0.0687 Table 2 Loss factor ratio (h NL /h L ) with vibration amplitudes (w/h) of the simply supported sandwich beam for different core properties (geometry and material) G-core ˆ 2.5 10 6 N/m 2 G-core ˆ 25 10 6 N/m 2 G-core ˆ 2500 10 6 N/m 2 h f /h c w/h L/h ˆ 10 30 70 100 200 10 30 70 100 200 10 30 70 100 200 7 0.1 0.9804 0.9883 0.9934 0.9943 0.9950 0.9886 0.9942 0.9951 0.9952 0.9953 0.9967 1.0088 1.0606 1.1142 1.2879 0.3 0.8482 0.9018 0.9439 0.9512 0.9570 0.9056 0.9503 0.9574 0.9583 0.9593 0.9711 1.0744 1.5248 1.9910 3.4952 0.5 0.6706 0.7681 0.8582 0.8755 0.8891 0.7752 0.8734 0.8901 0.8919 0.8947 0.9254 1.1922 2.3575 3.5667 7.4544 0.7 0.5142 0.6261 0.7551 0.7822 0.8035 0.6375 0.7790 0.8051 0.8088 0.8127 0.8679 1.3403 3.4118 5.5470 0.9 0.3962 0.5005 0.6508 0.6840 0.7122 0.5159 0.6799 0.7143 0.7196 0.7245 0.8067 1.4994 4.5446 7.6753 1.1 0.3116 0.4088 0.5549 0.5909 0.6235 0.4171 0.5863 0.6260 0.6308 0.6384 0.7473 1.6568 5.6504 9.7868 1.3 0.2513 0.3202 0.4716 0.5108 0.5425 0.3398 0.5062 0.5453 0.5495 0.5590 0.6932 1.8010 6.6671 1.5 0.2760 0.4012 0.4412 0.4712 0.2799 0.4368 0.4740 0.4769 0.4886 0.6455 1.9294 7.5678 1 Linear 0.0537 0.1753 0.1212 0.0759 0.0232 0.1770 0.0818 0.0192 0.0097 0.0025 0.0095 0.0011 0.0002 0.0001 0.0001 0.1 0.9616 0.9770 0.9909 0.9935 0.9957 0.9783 0.9931 0.9959 0.9963 0.9965 0.9979 1.0060 1.0145 1.0167 1.0184 0.3 0.7424 0.8248 0.9234 0.9451 0.9630 0.8332 0.9424 0.9643 0.9670 0.9693 0.9820 1.0527 1.1268 1.1449 1.1603 0.5 0.5184 0.6262 0.8118 0.8610 0.9038 0.6420 0.8552 0.9067 0.9132 0.9192 0.9530 1.1382 1.3325 1.3802 1.4206 0.7 0.3658 0.4622 0.6864 0.7589 0.8271 0.4784 0.7500 0.8322 0.8434 0.8533 0.9155 1.2501 1.6014 1.6886 1.7607 0.9 0.3527 0.5687 0.6546 0.7433 0.3583 0.6437 0.7500 0.7654 0.7791 0.8744 1.3732 1.9013 2.0278 2.1402 1.1 0.2549 0.4680 0.5575 0.6595 0.2740 0.5455 0.6677 0.6854 0.7032 0.8333 1.4983 2.2056 2.3733 2.5254 1.3 0.2231 0.3860 0.4782 0.5810 0.4669 0.5900 0.6091 0.6300 0.7945 1.6202 2.4963 2.7109 2.8934 1.5 0.3205 0.4107 0.5102 0.3946 0.5196 0.5388 0.5623 0.7596 1.7310 2.7627 3.0180 3.2307 Linear 0.0502 0.2164 0.2565 0.2026 0.0830 0.2270 0.2121 0.0704 0.0377 0.0101 0.0407 0.0079 0.0042 0.0038 0.0035 1/7 0.1 0.8902 0.9808 0.9947 0.9965 0.9975 0.9824 0.9962 0.9976 0.9979 0.9982 0.9998 1.0009 1.0011 1.0012 1.0012 0.3 0.4695 0.8472 0.9538 0.9680 0.9779 0.8591 0.9664 0.9789 0.9807 0.9839 0.9986 1.0085 1.0101 1.0103 1.0104 0.5 0.6611 0.8810 0.9153 0.9410 0.6831 0.9116 0.9436 0.9482 0.9568 0.9962 1.0227 1.0267 1.0273 1.0276 0.7 0.4949 0.7897 0.8465 0.8905 0.5211 0.8405 0.8952 0.9034 0.9194 0.9931 1.0417 1.0491 1.0501 1.0506 0.9 0.6932 0.7686 0.8310 0.3964 0.7605 0.8381 0.8500 0.8747 0.9897 1.0634 1.0748 1.0765 1.0772 1.1 0.6009 0.6890 0.7670 0.6795 0.7765 0.7929 0.8257 0.9860 1.0862 1.1018 1.1036 1.1050 1.3 0.5180 0.6123 0.7021 0.6019 0.7138 0.7339 0.7752 0.9826 1.1086 1.1285 1.1306 1.1326 1.5 0.4460 0.5411 0.6391 0.5304 0.6527 0.6749 0.7253 0.9793 1.1305 1.1538 1.1569 1.1587 Linear 0.3414 0.4203 0.2898 0.2036 0.0739 0.4198 0.2169 0.0627 0.0332 0.0093 0.1002 0.0721 0.0690 0.0686 0.0683

254 M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 Table 3 Loss factor ratio (h NL /h L ) with vibration amplitudes (w/h) of laminated beams Clamped clamped Simply supported Layup w/h L/h ˆ 10 30 70 100 200 10 30 70 100 200 0 0.1 0.9969 0.9990 0.9993 0.9993 0.9994 0.9933 0.9963 0.9971 0.9973 0.9974 0.3 0.9726 0.9908 0.9934 0.9938 0.9943 0.9431 0.9677 0.9754 0.9767 0.9777 0.5 0.9276 0.9751 0.9821 0.9833 0.9844 0.8578 0.9171 0.9365 0.9399 0.9426 0.7 0.8680 0.9530 0.9660 0.9683 0.9703 0.7576 0.8541 0.8878 0.8938 0.8985 0.9 0.8003 0.9258 0.9461 0.9496 0.9528 0.6585 0.7876 0.8361 0.8447 0.8516 1.1 0.7303 0.8950 0.9234 0.9284 0.9328 0.5695 0.7239 0.7863 0.7974 0.8063 1.3 0.6620 0.8620 0.8990 0.9054 0.9111 0.4936 0.6664 0.7409 0.7543 0.7650 1.5 0.5980 0.8282 0.8738 0.8816 0.8885 0.4306 0.6162 0.7011 0.7164 0.7287 Linear 0.00740 0.00277 0.00119 0.00097 0.00080 0.00395 0.00124 0.00084 0.00079 0.00075 0 /90 /0 0.1 0.9979 0.9993 0.9995 0.9996 0.9996 0.9954 0.9976 0.9982 0.9984 0.9985 0.3 0.9813 0.9938 0.9958 0.9961 0.9965 0.9606 0.9788 0.9846 0.9856 0.9864 0.5 0.9500 0.9831 0.9884 0.9894 0.9904 0.8985 0.9442 0.9596 0.9623 0.9643 0.7 0.9069 0.9678 0.9778 0.9798 0.9817 0.8206 0.8991 0.9267 0.9314 0.9352 0.9 0.8557 0.9484 0.9644 0.9675 0.9705 0.7379 0.8486 0.8897 0.8968 0.9025 1.1 0.8000 0.9260 0.9487 0.9532 0.9575 0.6580 0.7972 0.8518 0.8613 0.8689 1.3 0.7428 0.9012 0.9313 0.9373 0.9430 0.5853 0.7480 0.8154 0.8272 0.8366 1.5 0.6865 0.8750 0.9128 0.9203 0.9274 0.5215 0.7029 0.7816 0.7956 0.8067 Linear 0.00735 0.00270 0.00118 0.00096 0.00080 0.00384 0.00122 0.00084 0.00079 0.00076 (0 /90 /0 /90 ) s 0.1 0.9983 0.9994 0.9995 0.9996 0.9996 0.9959 0.9977 0.9982 0.9983 0.9984 0.3 0.9849 0.9942 0.9958 0.9961 0.9964 0.9641 0.9802 0.9847 0.9853 0.9859 0.5 0.9593 0.9843 0.9886 0.9894 0.9901 0.9073 0.9480 0.9598 0.9616 0.9629 0.7 0.9236 0.9700 0.9782 0.9797 0.9810 0.8355 0.9062 0.9273 0.9304 0.9329 0.9 0.8804 0.9520 0.9651 0.9675 0.9696 0.7585 0.8597 0.8910 0.8957 0.8993 1.1 0.8326 0.9312 0.9498 0.9532 0.9562 0.6836 0.8126 0.8541 0.8604 0.8652 1.3 0.7825 0.9082 0.9329 0.9374 0.9413 0.6150 0.7679 0.8189 0.8267 0.8326 1.5 0.7321 0.8840 0.9149 0.9205 0.9254 0.5542 0.7270 0.7866 0.7957 0.8027 Linear 0.00655 0.00225 0.00115 0.00100 0.00089 0.00316 0.00118 0.00091 0.00088 0.00086 modulus of the core. The effect of boundary condition is seen little on the value of damping. Coming to the nonlinear analysis, the problems are solved using eigenvalue formulation based on the QR algorithm as employed in the linear analysis. To solve the nonlinear eigenvalue problems, an iterative procedure is used. The iteration starts from a corresponding initial mode shape obtained from linear analysis, with amplitude scaled up by a factor. This gives the initial value denoted by d i. Based on this initial mode shape, the nonlinear stiffness matrices are formed as given in Eq. (15) and an eigenvalue and its corresponding vector are evaluated. This eigenvector is then scaled up again and the iteration continues until the frequency/damping factor and the eigenvector obtained from the subsequent two iterations satisfying the required convergence criteria suggested by Bergan and Clough [26] within the tolerance of 0.01%. Now, detailed numerical experiments are conducted for analyzing the nonlinear damping behavior of both simply supported and clamped beams by considering different values for the thickness ratio, aspect ratio and shear modulus for the core material. The results, concerning the first resonant mode, are presented in Tables 1 and 2. It is evident from these tables that, in general, a decrease in the system loss factor ratio (h NL /h L ; h NL loss factor obtained from nonlinear analysis) is seen with the increase in the amplitude of vibration (w/h) of the beam. Further, it is more so when the aspect ratio and shear modulus are less. Also, it can be seen that the rate of decrease in the damping ratio is less, with respect to amplitudes, with the increase in the aspect ratio. It may be further viewed from these tables that the loss factor ratios may increase with the aspect ratio, beyond certain values, and the occurrence of this phenomenon depends on the values of the shear modulus of the core and aspect ratio. When the core thickness is very high, the phenomenon of increasing in the damping value against amplitudes, with respect to the aspect ratio, occurs early compared to the case of a very thin core. Also, for certain aspect ratios, it appears that the change in the damping value may almost be negligible with respect to amplitude of vibration levels. This type of trend in the damping behavior is because of the change in the shear energy owing to shear of the sandwich/constrained layer case and depends not only on the modulus of core and thickness ratio but also, on the level of vibration amplitude. The reduction in the damping ratio is more for the constrained type of arrangement (h f /h c ˆ 7) with respect to amplitudes compared to the case of sandwich type of construction (h f /h c ˆ 1/7). The rate of decrease in the system loss factor is, in general, more for the simply supported case compared to the clamped beams. However, for certain combinations of core shear modulus, aspect and thickness ratios, the results could not be obtained for higher amplitudes of vibration cases owing to convergence problem.

M. Ganapathi et al. / Composites: Part B 30 (1999) 245 256 255 A similar investigation is carried out for the laminated fiber reinforced composite beam and is shown in Table 3 for both clamped clamped and simply supported beams. The damping characteristics are qualitatively the same as those of sandwich beam having viscoelastic layer. It is evident from this table that the loss factor ratio is more with increase in the number of layers and is more so for clamped laminated beams. Further, the reduction in the loss factor ratio for the case of clamped beam, is less with the increase in the aspect ratio in comparison to that of the simply supported case. The damping ratio is more for symmetric laminate compared to those of the single layered orthotropic case. 5. Conclusions The effect of the damping behavior, based on linear and nonlinear vibration analyses, is demonstrated here considering the study of sandwich beam or constrained layer beam arrangements, and laminates with the fiber reinforced composite material. The amplitudes of the vibration of the beam are assumed to be moderately large enough to cause geometric nonlinearity but are within the limit wherein linear behavior for the material is considered. Numerical studies are carried out, to highlight the influences of various parameters such as aspect ratio, thickness ratio of face-tocore, shear modulus of the core, boundary conditions and amplitude of vibration on the damping characteristics of constrained layered/sandwich beam and laminated anisotropic beam. This study will be useful for designers/engineers while designing the optimal constrained layer damping treatment or sandwich beam construction and composite laminate for the flexural response under dynamic situations. Some general observations are made as follows: (i) For the aspect ratio up to a certain value, the value of damping factor decreases with an increase in mode numbers and then it increases with an increase in the mode numbers. (ii) The effect of increasing the core thickness, in general, enhances the damping values significantly, especially, in the low range of aspect ratio. (iii) For a higher aspect ratio and core modulus, the difference in the loss factor values is less among the resonant frequencies. (iv) The system loss factor ratio decreases with an increase in the amplitude of vibration and is more so when the aspect ratio and shear modulus are less. (v) An increase in the loss factor ratios may occur beyond certain values of the aspect ratio and this depends on the values of the shear modulus and thickness of the core. (vi) The rate of decrease in the system loss factor is, in general, more for the simply supported case compared to the clamped beams. 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