Numerical method for nonlinear complex eigenvalues

Mechanics of Advanced Materials and Structures

ISSN: 1537-6494 (Print) 1537-6532 (Online) Journal homepage: http://www.tandfonline.com/loi/umcm20

Numerical method for nonlinear complex eigenvalues problems depending on two parameters: Application to three-layered viscoelastic composite structures Komlan Akoussan, Hakim Boudaoud, El Mostafa Daya, Yao Koutsawa & Erasmo Carrera To cite this article: Komlan Akoussan, Hakim Boudaoud, El Mostafa Daya, Yao Koutsawa & Erasmo Carrera (2017): Numerical method for nonlinear complex eigenvalues problems depending on two parameters: Application to three-layered viscoelastic composite structures, Mechanics of Advanced Materials and Structures, DOI: 10.1080/15376494.2017.1286418 To link to this article: http://dx.doi.org/10.1080/15376494.2017.1286418

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Date: 17 March 2017, At: 13:28

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES http://dx.doi.org/./..

ORIGINAL ARTICLE

Numerical method for nonlinear complex eigenvalues problems depending on two parameters: Application to three-layered viscoelastic composite structures Komlan Akoussana , Hakim Boudaoudb , El Mostafa Dayaa,c , Yao Koutsawad , and Erasmo Carrerae a Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux (LEM), UMR CNRS , Université de Lorraine, Metz, France; b Equipe de Recherche sur les Processus Innovatifs (ERPI), Université de Lorraine, Nancy, France; c Laboratory of Excellence on Design of Alloy Metals for Low-Mass Structures (DAMAS), Université de Lorraine, Metz, France; d Luxembourg Institute of Science and Technology, Esch-sur-Alzette, Luxembourg; e Department of Aeronautics and Aerospace Engineering, Politecnico di Torino, Torino, Italy

ABSTRACT

ARTICLE HISTORY

The design of high-damping composite viscoelastic sandwich structures (CVSS) requires continuous study of damping properties according to some parameters. The common design parameters that directly affect the damping properties of CVSS are the fibers’orientation and the thicknesses of the composite layers. However, until now there has been no method for continuously studying the influences of any design parameter on the damping properties of CVSS. Also, to study the influence of a design parameter on the damping properties of CVSS, computations are made for various discrete values of this parameter. These calculations are qualified incrementally, and are very expensive in computation time and do not allow to follow continuously the effects of the parameter on the damping properties of CVSS. In order to tackle these problems, we propose in this article a numerical method for studying the damping properties variation of CVSS according to a chosen modeling parameter, through the resolution of a generic, residual, complex nonlinear eigenvalues problems having frequency dependence and a modeling parameter that describes a study interval. This method is based on the asymptotic numerical method, automatic differentiation, homotopy technique and continuation. An application is performed to study the variation of the damping properties of a threelayered symmetric viscoelastic sandwich plates according to the fibers orientation of their orthotropic faces layers.

Received  January  Accepted  January 

1. Introduction Viscoelastic sandwich structures are recognized to be effective in reducing noise and vibration. Also, they are used in many industrial fields such as aeronautic, automotive and aerospace. To use a suitable viscoelastic sandwich structure with a required damping, it is important to model the behavior in vibration of this sandwich. The very first works on vibration modeling of viscoelastic sandwich structures were analytical and were realized by Kerwin [1], Ross [2] and DiTaranto [3] on viscoelastic sandwich made with isotropic materials. From these pioneer works, several works followed. Among them, the finite element method is the most used method for modeling damping properties of viscoelastic sandwich structures [4–11]. Recently, the use of composite materials has overtaken the use of isotropic materials in the design of viscoelastic sandwich structures. Also, many work have being done for modeling the damping properties of composite viscoelastic sandwich structures (CVSS) [12– 24]. Generally, the calculation of the damping properties of viscoelastic sandwich structures returns to the resolution of a nonlinear and frequency-dependent eigenvalues problem of the form: K (ω) − ω2 M {U } = 0

(1)

KEYWORDS

Nonlinear eigenvalue problem; asymptotic numerical method; automatic differentiation; homotopy; continuation; viscoelastic structures

where [K(ω)] and [M] are respectively frequency-dependent stiffness and mass matrices of the structure. There are already efficient methods for solving various eigenvalues problems. When the structure is made only with elastic materials, the eigenvalues problem is linear (Eq. 2) and can be solved by the classical methods such as subspace iteration of Leung [25], Lanczos method [26], QR method of Bathe [27] or Krylov– Arnoldi method [28]: (2) K − ω2 M {U } = 0 For sandwich structures containing viscoelastic layers in which Young’s modulus is frequency dependent, the eigenvalues problem is strongly nonlinear and cannot be solved by the classical methods previously mentioned. Three decades before, there was no direct method to solve that kind of eigenvalues problem. So, to determine the damping properties of these structures, the response to harmonic forcing for many frequencies is calculated to yield resonance peaks from which the loss factors and the eigenfrequencies are estimated [29]. This method can be applied to any structural problem, but the computational cost is very high. Abdoun et al. [30] have proposed a new method to solve accurately the response to harmonic forcing using the asymptotic numerical method. This new method is less expensive and applicable to any structure. Approximate

CONTACT Komlan Akoussan [email protected] Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux (LEM), UMR CNRS , Université de Lorraine, Metz, France. Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/umcm. ©  Taylor & Francis Group, LLC

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techniques such as the modal strain energy method [29] are also used, but only to estimate the loss factor of the structure. Then, direct calculation methods are proposed. Among them, there is the nonlinear Arnoldi method [31] that consists of projecting in the Krylov subspaces, the large size nonlinear eigenvalues problems to bring them into reduced size linear problems that are solved iteratively. An asymptotic technique approach was also proposed by Ma and He [5] but the procedure was limited to a small order of truncation and did not use a continuation algorithm, so this method cannot be used for high damping or for a strongly varying Young’s modulus. To tackle this limitation, Chen et al. [32] proposed an algorithm by coupling a first-order perturbation, an iterative algorithm and a reduced basis technique to solve the nonlinear complex eigenvalues problems. In this way, the problem is solved exactly, even for large-scale structures. However, the computation time is rather high. To reduce computation time and have more accurate solution, Daya et al. [33] proposed a numerical method that associates homotopy and asymptotic numerical technique. They also introduced a continuation algorithm. Duigou et al. [7] in their side developed two numerical iterative algorithms to solve nonlinear eigenvalue problems. Their methods associate homotopy, asymptotic numerical technique and Padé approximants. The first one is a sort of high-order Newton method and the second one uses a more or less arbitrary matrix. The advantages of the last two methods [7, 33] are that they can be applied to large-scale structures, large damping, and strongly nonlinear viscoelastic modulus. Recently, Koutsawa et al. [34] developed a toolbox for generic automatic differentiation in Matlab. This toolbox was used by Bilasse et al. [35] to automate the numerical method proposed in [33]. This reduces much computation time and the difficulties of manual calculations of high-order derivatives of complex functions. Boumediene et al. [36] developed a reduction method based on high-order Newton algorithm and reduction techniques to determine the modal characteristics of viscoelastic sandwich structures. However, the design of viscoelastic sandwich structures with high-damping power requires a continuous study of the variation of the damping properties of the structure according to some modeling parameters such as the thicknesses of the layers or the fibers orientation when composite layers are used. These parameters describe a study interval I and introduce a supplementary variable in the eigenvalues problems to solve which is written according to the considered modeling parameter p: K(ω, p) − ω2 M {U } = 0, p ∈ I K(ω) − ω2 M(p) {U } = 0, p ∈ I K(ω, p) − ω2 M(p) {U } = 0, p ∈ I (3) Among the methods to solve eigenvalues problems that we just quoted, none can be directly used to solve an eigenvalues problem having frequency dependence and containing a modeling parameter that varies in a study interval. Indeed, for the former methods, one has to fix the modeling parameter value. So, to study the variation of the damping properties according to a modeling parameter, several calculations must be made for various values of the parameter. These calculations, called incremental, are very expensive in computation time and do not allow to follow continuously the effects of the modeling parameter on

the damping properties since no one knows what happens for the other values of the parameter. Nevertheless, the incremental method is still used until now [15, 17, 20, 22]. In an earlier study of composite viscoelastic sandwich plate [24] where the viscoelastic core is assumed having a complex constant Young’s modulus, we developed a numerical method based on the asymptotic numerical method, automatic differentiation, and continuation to study the effects of the fibers orientation of the structure orthotropic faces on its damping properties. In this paper, we propose a new numerical method for solving generic nonlinear complex eigenvalues residual problems having frequency dependence and containing one modeling variable parameter. The resolution allows to study continuously the effects of the modeling parameter on the damping properties of the structure for each mode. This new method is based on the asymptotic numerical method (ANM), automatic differentiation (AD) [34], homotopy technique, and continuation method [33, 35]. This new method allows us first to get the high derivatives of the eigensolution with respect to the chosen modeling parameter, and second to have the exact value of the parameter for an optimal damped frequency or damping. An application is then proposed considering viscoelastic core having different frequency dependence laws for Young’s modulus in the problem earlier studied in [24]. A comparison with incremental calculation is made to validate the obtained results, and the computation times are compared to prove the efficiency of this new method.

2. Numerical method for nonlinear complex eigenvalues problem with two parameters The residual nonlinear complex eigenvalues problem considered is in the following generic form: p∈I Rg U, ω, p = K ω, p − ω2 M p {U } = 0, (4) In this equation, p is the physic or objective variable modeling parameter that is assumed describing the interval I. K(ω, p) ∈ Cq × Cq is the stiffness matrix of the structure having frequency ω dependence and is also a function of the modeling parameter p. M(p) ∈ Cq × Cq is the mass matrix of the structure and is a function of the modeling parameter p. The residue Rg (U, ω, p) and the modal eigenvector U are vectors of Cq , and the frequency ω ∈ C. The specificity of this generic residual eigenvalues problem is that for any value of p taken in the interval I, there exists a unique eigenpair solution (U, ω) per mode. This generic nonlinear problem is solved by the ANM coupled to AD, the homotopy technique, and the continuation method. Solving this generic nonlinear eigenvalues problems (Eq. 4) allows to solve the two others nonlinear eigenvalues problems of Eq. (3). Indeed to obtained the first two problems given in Eq. (3), it is sufficient to consider respectively the mass matrix M(p) and the stiffness matrix K(ω, p) of the generic residual problem of Eq. (4) as a constant function of the parameter p. 2.1. Diamant approach The so called Diamant approach [37] is a generic approach for solving nonlinear residual problems using ANM and AD. The

MECHANICS OF ADVANCED MATERIALS AND STRUCTURES

rule of the AD in this approach is to calculate automatically Taylor series coefficients used in the ANM formulation. In AD, the program implementing any numerical model or analytic function b = f (a) is viewed as a sequence of elementary operations. The automatic derivation of f is performed by applying the chain rule to the sequence of φ comprising f . High-order AD relies in practice on operator overloading as the vehicle of attaching derivative computations to the elementary operations φ provided by the programming language such as the arithmetic operators and intrinsic functions. Let f (a) = φ ◦ x(a) be an analytical function. From a theoretical point of view, considering the Fàa di Bruno recurrence formula written in terms of Bell polynomials (see [34] for details) of the Taylor coefficient fn of f at order n (Eq. 5) is a short and effective way to present the Diamant interpretation of the ANM: n 1 ∂ mφ (x0 ) Bn,m x(1) , . . . , x(n−m+1) m n! ∂x m=1 (5) It may be split into two parts: fn = f1|x1 =1 xn + fn|xn =0 (6)

fn = (φ ◦ x)n =

On one hand, { f1|x1 =1 } is the tangent linear derivative evaluated with x1 = 1. On the other hand, the second term { fn|xn =0 } is the Taylor coefficient fn evaluated with xn = 0 to cancel the contribution of { f1|x1 =1 }. Let us consider the generic nonlinear problem in the following form: R (U, λ) = 0

(7)

where R(U, λ) and U are vectors of Cq and λ ∈ C is a scalar parameter. A path equation depending on a variable a (e.g., arc-length parameter) is usually added (see [34, 38] for details) to solve this under-determined system of equations. The ANM approximates the solution (U (a), λ(a)) = (U, λ)(a) by Taylor expansions truncated at order N when functions are assumed being analytical: (U, λ) (a + δa) =

N

(δa)n (Un , λn ) (a)

(8)

n=0

where (Un , λn ) (a) =

1 ∂n 1 (n) (n) U , λ (a) = (U, λ) (a) (9) n! n! ∂an

are Taylor coefficients of U and λ at order n evaluated automatically by AD at point a. The Diamant approach allows us to write the generic nonlinear problem (Eq. 7) as follows: Rn = R1|U1 =Id,λ1 =0 Un + R1|U1 =0,λ1 =1 λn (10) + Rn|Un =0,λn =0 = 0 where {R1|U1 =Id,λ1 =0 } is the tangent linear matrix. The notation “U1 = Id ” formally expresses the fact that this tangent linear matrix could be computed evaluating (P times) the first Taylor coefficient of R by choosing U1 in the canonical basis of Rq . Similarly, the tangent linear terms {R1|U1 =0,λ1 =1 } and {Rn|Un =0,λn =0 } are calculated by initializing U1 = 0, λ1 = 1, Un = 0 and λn = 0. In {Rn|Un =0,λn =0 }, the use of null Taylor coefficients for Un and λn enables the cancellation of the tangent linear contributions.

3

2.2. Differentiation of the generic residual problem by Diamant approach By setting λ = ω2 , the global generic nonlinear eigenvalue residual problem (Eq. 4) becomes: Rg U, λ, p = K λ, p − λM p {U } = 0, p ∈ I (11) For each value of p there exists a unique eigenpair solution (U, λ) per mode for the generic residual problem (Eq. 11); this induces that in an interval Ip ⊂ I and for some precision ε, this eigenpair (U, λ) can be defined as a function of p: (U (p), λ(p)) = (U, λ)(p). Applying perturbation technique for this eigenpair by introducing an increment δ p on the modeling parameter p allows one to develop this eigenpair as Taylor expansions of δ p truncated at order N: N n δ p (Un , λn ) p (U, λ) p + δ p = n=0

1 ∂n 1 (n) (n) U , λ p = (Un , λn ) p = (U, λ) p n n! n! ∂ p (12) Introducing the Taylor expansions of the eigenpair (U, λ) in the generic residual problem (Eq. 4) allows writing the problem also as Taylor expansions as follows: Rg U p , λ p , p + δ p N n = δ p Rgn U p , λ p , p = 0 n=0

1 Rgn U p , λ p , p = Rng U p , λ p , p n! 1 ∂ n Rg U p ,λ p , p (13) = n! ∂ pn Making use of the Diamant approach principle (Eq. 10), this generic nonlinear residual problem is sent into a recurrence form for each value of n (0 ≤ n ≤ N): Rg0 = Rg0|U0 =Id U0 = 0 Rgn = Rg1|U1 =Id,λ1 =0 Un + Rg1|U1 =0,λ1 =1 λn (14) + Rgn|Un =0,λn =0 = 0 To find the eigenpair solution (U, λ), the recurrence residual problems defined by Eq. (14) must be solved at each order n = 0, 1, 2, . . . , N. 2.2.1. Resolution of the residual problem at order 0 The residual problem at order 0 obtained using the Diamant approach differentiation of the global generic nonlinear residual problems (Eq. 4), and given by Eq. (14) is written: Rg0 = Rg0|U0 =Id U0 = K λ0 , p − λ0 M p U0 = 0 (15) Generally, the problems at order 0 obtained after differentiating nonlinear problems by the ANM are linear [30, 33, 35]. But in this case, the residual problem at order 0 is a nonlinear and frequency-dependent eigenvalues problem since the stiffness matrix [K(λ0 , p)] contains the unknown λ0 . Moreover, it contains the unknown p since p ∈ I. So, the residual problem at

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K. AKOUSSAN ET AL.

order 0 can be solved by the homotopy technique [33] for a fixed value of the modeling parameter p, which will be assumed as a starting point (initial value). The aim being to study the effect of the damping properties according to the modeling parameter p, which varies in the interval I, it is appropriate to choose for the initial value of p, p(k=0) = pi = in f (I) eventually p(k=0) = ps = sup(I). In practice, the stiffness and mass matrices contained in the generic nonlinear residual problem (Eq. 4) are large size sparse matrices, and it is not judicious to differentiate them to avoid computational memory problems and also a great computation time. For this, the stiffness and the mass matrices are formulated as follows: [K(λ, p)] =

Q

∗ Egq (λ, p)[Kq ] = [Eg∗ (λ, p)][Kg ]

q=1

[M(p)] =

J

Em j (p)[M j ] = [Em (p)][Mg ]

(16)

j=1 ∗ (λ, p) and Em j (p) are analytical function of the modwhere Egq eling parameter p respectively associated to the constant matrices [Kp ] and [M j ]. This decomposition (Eq. 16) is a prerequisite work done by the user during which he/she determines ∗ (λ, p), Em j (p), the constant matrithe analytical functions Egq ces [Kp ], [M j ] and the integers Q and J. Using the decomposition given in Eq. (16), the differentiation of the stiffness [K] and mass [M] matrices at order n with respect to the modeling parameter p returns to differentiate the analytical func∗ (λ, p) and Em j (p) (Eq. 17). This allows reducing contions Egq siderably the computation time since the size of function to be differentiated is reduced. For simplicity, we have supposed that the analytical functions to be differentiated are grouped in the matrices [Eg∗ (λ, p)] and [Em (p)] and their associated constant matrices in [Kg ] and [Mg ] respectively for stiffness and mass matrices:

[Kn (λ, p)] =

Q

J

Em(n)j (p)[M j ] = [Em(n) (p)][Mg ]

(17)

[Kn ], [Mn ], [Eg∗(n) ] and [Em(n) ] being the Taylor coefficient of [K], [M], [Eg∗ ] and [Em ] at order n, respectively. Furthermore [Eg∗ (λ, p)], which is complex, often has the distinction of being rewritten by making out the term representing the delayed elasticity: (18)

Considering the fixed value chosen for p: p(k=0) = pi = in f (I) (eventually p(k=0) = ps = sup(I)), the residual problem at order 0 (Eq. 15) becomes: Rg0 =

[Eg∗ (0,

p(k=0) )Kg + Eg (λ0 , p(k=0) )Kg

− λ0 Em (p(k=0) )Mg ]U0 = 0

Rh (V, γ , p(k=0) , a) = [Eg∗ (0, p(k=0) )Kg − γ Em (p(k=0) )Mg ]V (20) + a[Eg (γ , pi )Kg ]V By perturbing the homotopy parameter a by an increment δa, the defined new residual problem Rh is written as Rh (V, γ , p(k=0) , a( j) + δa) = S (V, γ ) + (a( j) + δa)T (V, γ ) = 0

(21)

A continuation procedure is set up to drive the analytical S into Rh by determining a monotonic increasing sequence of the homotopy parameter a: {a( j) } j=1,...,J ranging from a(1) = 0 to a(J) = 1 (J to be determined). Each subbranch (a( j) , a( j+1) ) is computed solving the nonlinear problem giving by Eq. (21) through the ANM. Let (V( j) , γ( j) ) be the starting solution at the initial point (a( j) ) of the jth subbranch. Variables V and γ are decomposed into series where Taylor coefficients (Vn , γn ) for n = 1, 2, . . . , N are unknowns, whereas (V0 , γ0 ) = (V( j) , γ( j) ). Introducing Taylor series of the functions S and T in Eq. (21): N

(δa) Sn + (a( j) + δa) n

n=0

N

(δa)n Tn = 0

(22)

n=0

with 1 ∂n 1 (n) S , T (n) (a( j ) ) = (S, T ) (a( j ) ) n! n! ∂an j () 1 ∂n 1 (n) V , γ (n) (a( j ) ) = (Vn , γn ) = (V, γ ) (a( j ) ) n! n! ∂an j () (23) (Sn , Tn ) =

and identifying the same powers of δa leads to N + 1 equations: Sn + a( j ) Tn + Tn−1 = 0, n = 1, 2, . . . , N

j=1

[Eg∗ (λ, p)] = [Eg∗ (0, p)] + [Eg (λ, p)]

Rg0 (U0 , λ0 , p(k=0) ) = Rh (V, γ , p(k=0) , a = 1) = 0 Rh (V, γ , p(k=0) , a) = S(V, γ ) + aT (V, γ ) = 0

S0 + a( j ) T0 = 0

∗(n) Egq (λ, p)[Kq ] = [Eg∗(n) (λ, p)][Kg ]

q=1

[Mn (p)] =

and we introduce the homotopy parameter a as follows:

(24)

The shift in the subscript is a consequence of the homotopy. The first equation to be solved in this homotopy technique resolution is for the case where a( j=1 ) = 0 and corresponds to S0 = [Eg∗ (0, p(k=0) )Kg − γ0 Em (p(k=0) )Mg ]V0 = 0

(25)

Equation (25) is a linear real eigenvalues problem and can be solved by one of the classical method already mentioned [25–28]. The resolution of Eq. (25) allows to get the first eigenpair solution (V0 , γ0 ) of the homotopy technique resolution. For others order, the Diamant approach decomposition of Eq. (10) is used and gathered with respect to Vn , so the following generic linear systems is deduced: [{S1|V1 =Id,γ1 =0 } + a( j) {T1|V1 =Id,γ1 =0 }] Vn = −{Sn|Vn =0 } − a( j) {Tn|Vn =0 } − Tn−1 , ∀ n ≥ 1

(19)

(26)

which is solved by the homotopy technique. To do so, we transform and split the residual problem (Eq. 19) in two subresidues

Variables {Sn|Vn =0 } and {Tn|Vn =0 } are Taylor coefficients of S and T at order n evaluated by initializing Vn to 0. The tangent linear


matrix A is expressed as follows:

where a(J) is the last value of a satisfying the condition of Eq. (34). Otherwise, a new branch of series is started by continuation from the last series:

[A] = [{S1|V1 =Id,γ1 =0 } + a( j) {T1|V1 =Id,γ1 =0 }] = [Eg∗ (0, p(k=0) )Kg − γ0 Em (p(k=0) )Mg + a( j) Eg (γ0 , p(k=0) )Kg ]

5

(27)

(V0 , γ0 ) =

N

(δa)n (Vn , γn )

(36)

n=0

The generic Eq. (26) is not well defined because at each order n, it contains two unknowns (Vn , γn ). So, the orthogonality condition of the modes (Eq. 28) is added to Eq. (26):

until the condition (Eq. 34) is satisfied.

2.2.2. Resolution of the residual problem at order n and continuation The first eigenpair solution (U0 , λ0 ) of Eq. (14) being deterA Lagrange multiplier χ is then introduced to transform mined, it remains to calculate the other eigenpairs (Un , λn ) for Eqs. (26) and (28) to the following system: each series. For reminding, the equation to solve is written as

follows: − Sn|Vn =0 − a( j ) Tn|Vn =0 − Tn−1 A V0 Vn , = t Rg1|U1 =Id,λ1 =0 Un = − Rg1|U1 =0,λ1 =1 λn V0 0 χ 0 − Rgn|Un =0,λn =0 , n ≥ 1 (37) n ≥ 1 (29) t

V0 (V − V0 ) = 0

(28)

However, the linear tangent matrix A is singular, so the system (Eq. 29) cannot be solved without the solvability condition (Eq. 30), which has to be satisfied at each order n:

t V0 Sn|Vn =0 + a( j ) Tn|Vn =0 + Tn−1 = 0, ∀ n ≥ 1 (30) Remind Sn|Vn =0 = Sn|Vn =0,γn =0 + S1|V1 =0,γ1 =1 γn Tn|Vn =0 = Tn|Vn =0,γn =0 + T1|V1 =0,γ1 =1 γn

(31)

Introducing Eq. (31) in the solvability condition (Eq. 30), a generic form for calculating γn is deduced: γn = −

t

V0 [{Sn|Vn =0,γn =0 } + a( j) {Tn|Vn =0,γn =0 } + Tn−1 ] , t V [{S 0 1|V1 =0,γ1 =1 } + a( j) {T1|V1 =0,γ1 =1 }] ∀n≥1 (32)

By putting the calculated value of γn in the system (Eq. 29) and solving it, the value of the eigenmode Vn is obtained. At the end of the series, a path is estimated from the chosen homotopy technique calculation precision εh by the following expression [24, 30, 33, 35]: 1 V1 N−1 δa = εh Vn

(33)

Then, the stopping calculation condition of homotopy technique implementation is verified: a( j+1 ) = a( j ) + δa a( j+1 ) ≥ 1

(34)

If this condition is satisfied, the homotopy technique calculation is ended and the eigenpair solution (U0 , λ0 ) of the global generic nonlinear residual problem (Eq. 15) at order 0 is obtained from the homotopy technique solution as follows: (U0 , λ0 ) =

N n 1 − a(J) (Vn , γn ) n=0

(35)

Equation (37) is also not well defined because it possesses two unknowns: (Un , λn ), so the orthogonality condition (Eq. 38) of the modes is once again added to it. t

U0 (U − U0 ) = 0

(38)

By setting Kt = {Rg1|U1 =Id,λ1 =0 }, the linear tangent matrix, the following generic system is obtained by combining Eqs. (37) and (38) and using Lagrange multiplier χ :

Un Kt U0 t U0 0 χ − Rg1|U1 =0,λ1 =1 λn − Rgn|Un =0,λn =0 , n≥1 = 0 (39) The expression for linear tangent matrix [Kt ] is [Kt ] = Eg∗ 0, p(k) Kg + Eg λ0 , p(k) Kg − λ0 Em p(k) Mg (40) singular and the system (Eq. 39) requires the following solvability condition before being solvable: t (41) U0 Rg1|U1 =0,λ1 =1 λn + Rgn|Un =0,λn =0 = 0 From this new solvability condition, a generic form of λn is deduced by t U0 Rgn|Un =0,λn =0 λn = − t (42) U0 Rg1|U1 =0,λ1 =1 After λn is calculated, it is introduced in the system (Eq. 39), which is therefore solved to get the eigenmode Un . Once all terms of the series are calculated, the precision value ε, which can be different from the precision of the homotopy calculation precision εh , is used to estimate the suitable radius of the series by the following formula: 1 U1 N−1 δp = ε (43) Un In practice, a convergence study is made to choose the values of ε and εh . The continuation of the calculation or the stopping

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K. AKOUSSAN ET AL.

Table . Algorithm of the new numerical method for solving nonlinear eigenvalues problems.

depends on checking the following condition: p(k+1) = p(k) + δ p p(k+1) ≥ ps = sup (I)

(44)

alternatively: p(k+1) = p(k) − δ p p(k+1) ≤ pi = in f (I)

(45)

according to whether one uses p(k=0) = pi = inf (I) or p(k=0) = ps = sup(I) as initial value of p. Until the appropriate condition being satisfied, a new branch of series is built starting from the last one: (U0 , λ0 ) =

N

(δ p) (Un , λn ) n

(46)

n=0

when started with p(k=0) = pi = in f (I) alternatively: (U0 , λ0 ) =

N

(−δ p)n (Un , λn )

(47)

n=0

when started with p(k=0) = ps = sup(I). The AD of the each new series is made for p = p(k+1) . The damping properties of the structure, i.e., damped frequency and loss factor η, are obtained from the eigensolution λ as follows: ω2 = 2 (1 + iη) = λR + iλI = λ √ λI η= R

= λR , λ

(48)

2.3. Diamant overloading routines and utilization of the new method The new numerical method presented in this paper to solve strongly nonlinear eigenvalues problem with frequency and a modeling parameter dependence is divided into two parts. The first part is the resolution of the problem at order n = 0 by using the homotopy technique coupled to ANM and AD. The second part concerns the resolution of the differentiated problem at each order n ≥ 1 and the continuation. In both cases, the use of AD leads to the computation of the series through the Diamant Matlab toolboox [34]. This toolbox is used to evaluate the Taylor coefficients of any Matlab functions. Differentiation stages are hidden to the user by means of operator overloading techniques. For this process, new type, namely TaylorCoef, is introduced. This object (TaylorCoef) has three different properties: value (n = 0), coef (n = 1, 2, . . . , N) and order (maximum order N). Arithmetic operations and elemental functions are overloaded for TaylorCoef objects by implementing classical recurrence formulas to enable higher-order differentiation when interpreting Matlab codes. Unknowns (U, λ), whose Taylor coefficients must be computed, are defined as (TaylorCoef) objects. In Matlab, the type propagation is implicit what enables a hidden propagation of the higher-order differentiation. So, to use this present numerical method, the user upstream work to define the constant matrices [Kg ] and [Mg ], their associated functions to differentiate [Eg∗ (λ, p)], [Em (p)]. The associated functions, which

I. First part: resolution of the problem at order n = 0 by homotopy technique I. Calculation of (V0 , γ0 ) (Eq. ) I. Construction of the tangent matrix [A] (Eq. ) I. Calculation of the denominator of Eq. () using AD I. Evaluation of T0 I. Series computations γn and Vn at each order n = 1, 2, . . . , N I.. Evaluation of {Sn|V =0 } and {Tn|V =0 } using AD n n I.. Calculation of γn following Eq. () I.. Calculation of Vn following Eq. () I.. Update of Sn et Tn using AD I. Convergence test I.. Estimation of the convergence radius following Eq. () I.. If the condition of Eq. () is not satisfied then go to I. I. If Iinf = Isup : stop calculation, else go to II. II. Second part: resolution of residual problem at each order n and continuation II.. Evaluation of the tangent matrix [Kt ] (Eq. ) II.. Series computations Un and λn at each order n = 1, 2, . . . , N: II.. Evaluation of {Rg1|U =0,λ =1 } and {Rgn|U =0,λ =0 } using AD 1 1 n n II... Calculation of λn following Eq. () II... Calculation of Un following Eq. () II... Update of Rgn using AD II.. Estimation of the convergence radius following Eq. () II.. Stopping calculation test If the condition of Eq. () (alternatively Eq. ) is satisfied then stop calculation, else: initialization of series Un and λn following Eq. () (eventually Eq. ) and go to II..

must be analytical, become (TaylorCoef) objects in the calculation, so their differentiation will be fully occulted in Matlab user-defined functions. The user must also input the study interval I by defining the initial value of the modeling parameter (Iinitial ) and the ending value (Iend ). The choice of the free vibration mode to study (Nmode), the homotopy calculation and continuation precisions (εh , ε) and the degree of truncation of the series N are also input data supplied by the user. We recall that the information and the degree of truncation of the series can be defined from a convergence study. The algorithm of this proposed numerical method is presented in Table 1 and the scheme of using the method is reported in Figure 1.

3. Application and validation In order to validate the new numerical method, we propose to study the damping properties variation of a symmetric

Inputs: [K g], [Mg], E g∗ (λ , p) , [E m(p)], I initial , I end, εh, ε , N, Nmode Resolution of the nonlinear eigenvalues problem by the present method (see Table 1) Outputs: U (p), Ω(p), η (p) Figure . Diagram of how to use the new numerical method.


r the top and bottom elastic layers have the same

y

z

hf

θ

l x

hc

7

L

Figure . Symmetric three-layered composite viscoelastic sandwich plate.

three-layered composite viscoelastic sandwich plate, see Figure 2. The modeling parameter is here the fibers orientation of the orthotropic faces layers of the plate. The viscoelastic core contained in the structure is assumed to be isotropic and its Young’s modulus has a frequency dependence law. The coordinate system (O, x, y, z) with the origin O at one corner is the global coordinates system. One denotes by zi the ith middle plane coordinate with respect to z = 0. The external faces subscripted i = 1, 3 are elastic and orthotropic, while the middle layer subscripted i = 2 is viscoelastic. The thickness of the face layers is h f and hc for the core layer. The plate dimensions are L in the (O, x) direction and l in the (O, y) direction. 3.1. Residual nonlinear problem to solve To study the damping properties variation according to the face fibers orientation θ of the symmetric three-layered composite viscoelastic sandwich plate considered, the nonlinear eigenvalues problem depending on frequency and θ must be first determined. This nonlinear eigenvalues problem was established in the previous work [24]. We recall some details on the finite element formulation that allows to obtain the nonlinear eigenvalues problem. 3.1.1. Kinematics model The damping of the structure is assumed to be induced by a large shear strain in the viscoelastic isotropic layer as a direct consequence of the contrast between the elastic and viscoelastic layers. This large shear deformation must be modeled correctly in order to quantify accurately the damping variation of the sandwich plate. In this way, the kinematic analysis used for the sandwich plate models the top and bottom faces using the Kirchhoff–Love [39] plate theory and the viscoelastic middle layer using the Mindlin–Reissner theory [40, 41] plate theory. As common to the authors [10, 32, 42, 43], the present analysis assumes the following hypotheses:

Young’s/shear modulus and the same mass density. Based on these assumptions, the displacement field relationships associated with the classical Kirchhoff–Love plate theory are presented for the face layers: Ui (x, y, z, t ) = ui (x, y, t ) − (z − zi ) w,x (x, y, t ) Vi (x, y, z, t ) = v i (x, y, t ) − (z − zi ) w,y (x, y, t ) i = 1, 3 Wi (x, y, z, t ) = w(x, y, t )

(49)

where ui and v i are in-plane displacements of the middle plane of the ith layer of the face and w the common transverse dish +h placement. The middle plane of faces are given by z1 = f 2 c h +h

and z3 = − f 2 c . The strain field of the ith layer of the face is then written as follows: {ξi } = {i } + (z − zi ) {κ}

i = 1, 3

(50)

The tensors {i } and {κ} represent the generalized linear membrane strain and the bending strain, respectively. Their components are {i } = t ui,x v i,y ui,y + v i,x , i = 1, 3 {κ} = t −w,xx − w,yy − 2w,xy (51) The displacement field relationships of the first-order shear deformation theory related to the viscoelastic layer i = 2 are U2 (x, y, z, t ) = u(x, y, t ) + zβx (x, y, t ) V2 (x, y, z, t ) = v (x, y, t ) + zβy (x, y, t ), W2 (x, y, z, t ) = w(x, y, t )

(52)

where u and v are in-plane displacements, and βx and βy are the rotations of the middle plane of the central viscoelastic layer. The corresponding strain field is given by {ξ2 } = {2 } + z {κ2 }

(53)

The tensors 2 , and κ2 represent respectively the generalized linear membrane strain and bending strain. The generalized linear transverse shear strain is noted to be ζ2 . The components of these three tensors are {2 } = t u,x v ,y u,y + v ,x , {ζ2 } = t w,x + βx w,y + βy {κ2 } = t βx,x βy,y βx,y + βy,x (54)

r all points on a normal to the plate undergo the same trans- 3.1.2. Constitutive law verse deflection;

r no slipping occurs at the inte’rfaces between the three layers of the plate;

r all points on a normal to the undisturbed neutral plane of the top and bottom layers have the same rotation;.

r the constitutive materials of the plate layers are homogeneous and orthotropic. The Young/shear modulus of the viscoelastic core is complex frequency dependent, but the Poisson ratio is assumed to be constant;

Using the generalized Hooke’s law, the normal forces Ni and the bending moments Mi per unit length for each layer i are obtained: ⎧ ⎫ ⎨ Nixx ⎬ {Ni } = Niyy = [Cm (θ )] {i } , ⎩ ⎭ Nixy ⎧ ⎫ ⎨ Mixx ⎬ {Mi } = Miyy = C f (θ ) {κ} , (55) ⎩ ⎭ Mixy

8

K. AKOUSSAN ET AL.

for face layers (i = 1, 3) and for the viscoelastic core (i = 2) ⎧ ⎫ ⎨ N2xx ⎬ ∗ {N2 } = N2yy = C2m (ω) {2 } , ⎩ ⎭ N2xy ⎧ ⎫ ⎨ M2xx ⎬ {M2 } = M2yy = C2∗ f (ω) {κ2 } (56) ⎩ ⎭ M2xy

which are applied for this structure. The viscoelastic core being isotropic, the stiffness matrix related to the core is written as [Kc (ω)] = E ∗ (0)[Kv ] + E(ω)[Kv ] with E ∗ (0), the delayed elasticity of the core layer. The discretized bending problem (Eq. 61) is now written as follows: ∗ E (0)Kv + K f (θ ) + E(ω)Kv − ω2 M {U } = 0, θ ∈ I

The transverse shear forces Q2 per unit length due to the core deformation is expressed as follows: ∗ Q2xz {Q2 } = = C2s (57) (ω) {ζ2 } Q2yz

In Eqs. (61) and (62), the stiffness matrix related to the faces layers is in the form: K f (θ ) = K1 f cos4 θ + K2 f sin4 θ + K3 f sin2 θ cos2 θ + K4 f sin θ cos3 θ + K5 f sin3 θ cos θ (63)

where [Cm (θ )] and [C f (θ )] are, respectively, the symmetric fiber orientation dependence membrane and bending stiffness matrices of face layers and are written as follows: Cm (θ ) = A1 cos4 θ + A2 sin4 θ + A3 sin2 θ cos2 θ + A4 sin θ cos3 θ + A5 sin3 θ cos θ h2f C f (θ ) = Cm (θ ) 12

(58)

Expression of the defined matrices Ai , i = 1, . . . , 5 from face layer material elastic constants are given in Appendix B. The ∗ (ω)], the symmetsymmetric membrane stiffness matrix [C2m ∗ ric bending stiffness matrix [C2 f (ω)] and the shear stiff∗ (ω)] are frequency dependent and related ness matrix [C2s to the viscoelastic core. Their expressions are given in Appendix A. 3.1.3. Equation of motion and discretization The free vibration formulation of the plate is obtained by applying the Principe of Virtual Displacements (PVD). The bending problem to be solved is ∂ 2ω t ρ2 h2 + 2ρ f i h f i Mβ {δκ2 } ds δωds = − 2 ∂t s s t {Mw } {δκ} + t {Q2 } {δζ2 } ds (59) − s

where hf hc Mβ = [C2∗ f (ω)] [κ2 ] + hc [Cm (θ )] [κ2 ] + [κ] 2 2 hf hc {Mw } = 2 C f (θ ) [κ] + h f [Cm (θ )] [κ2 ] + [κ] 2 2 (60) By discretizing the bending problem (Eq. 59) with twodimensional four-node finite element and assembling the elementary matrices, the following nonlinear complex eigenvalue problem is obtained [24]: (61) K f (θ ) + Kc (ω) − ω2 M {U } = 0, θ ∈ I where [K f (θ )] is the stiffness matrix related to the faces layers and depends on the fibers orientation angle θ . The stiffness matrix [Kc (ω)] depends on the vibration frequency of the structure and is related to the viscoelastic core. [M] is the mass matrix of the structure. The interval of study I is defined according to the dimension of the structure and the boundary conditions

(62)

3.1.4. Nonlinear complex eigenvalues problem resolution From this decomposition, we determine the integers Q and J equal to 6 and 1, respectively. The constant matrices [Kq ] related to the stiffness in Eq. (16) are then [Kq ] = [K f q ], q = 1, 2, .., 5, [K6 ] = [Kv ], and their associated analytical function to differ∗ (λ, p), q = 1, 2, .., 6 are given in the following entiate are Egq equation: ∗ ∗ λ, p = cos4 p Eg4 λ, p = sin p cos3 p Eg1 ∗ ∗ Eg2 λ, p = sin4 p Eg5 λ, p = sin3 p cos p ∗ ∗ Eg3 λ, p = sin2 p cos2 p Eg6 λ, p = E ∗ (0) + E(λ) Em p = 1 (64) For this study, the face fibers orientation angle has no influence on the mass matrix of the structure, so the matrix is a constant function of this modeling parameter. The constant matrices and their associated functions to differentiate being determined, the resolution of the nonlinear complex eigenvalues problem (Eq. 63) by the new method described above, becomes simply to give the following inputs: [Kg ], [Mg ], [Eg∗ (λ, p)] (given in Eq. 65), N (truncation degree of the series), εh , ε (homotopy technique calculation precision and ANM continuation calculation precision, respectively) and I (study interval) to get the outputs: U (p) (modal vector), (p) (damped frequency) and η(p) (loss factor): Kg = K1 f , K2 f , K3 f , K4 f , K5 f , Kv Mg = [M] ∗ ∗ ∗ ∗ ∗ ∗ (65) Eg∗ λ, p = Eg1 , Eg2 , Eg3 , Eg4 , Eg5 , Eg6 We recall that λ = ω2 and p = θ . The starting point is set to pi = in f (I) and the stopping point to ps = sup(I). 3.2. Numerical results In this section, we present some results obtained using the new numerical resolution method proposed for different types of geometry (plate and beam) of three-layered symmetric viscoelastic sandwich structures. To show the generic form of that resolution method, different types of viscoelastic materials having different types of viscoelastic laws are also considered. In order to show the reduction in CPU time calculation of this new resolution method, discrete calculations are also made with the reference formulations [24]. For the discrete calculations,


Table . Geometrical and mechanical characteristics of the three-layered sandwich beam. Mechanical characteristics Geometrical characteristics (mm) hf = 4 hc = 20 L = 1, 000 l = 25

Core layer

Face layers

Ec∗ (0) = 2.097026 MPa Ec∗ (ω) = Ec∗ (0)(1 + 0.3i) νc = 0.49 ρc = 970 kg m−3

Exx = 141.2 GPa Eyy = 9.72 GPa Gxy = 5.53 GPa νxy = 0.28 ρf = 1536 kg m−3

the complex nonlinear eigenvalue problem is solved by using the homotopy technique already developed and used by Bilasse et al. [9]. The order of truncation of the Taylor series is set to N = 20 and the precision to εh = 10−6 . The same precision is used for the homotopy technique to solve the problem at order 0; however, for the ANM and continuation, the precision is set to ε = 10−4 . For unidirectional composites, the fibers positioning angle θ varies in [0◦ , 180◦ ], so the study interval I is therefore I = [0◦ , 180◦ ]. However, when the boundary conditions applied to the structure are also symmetric, the study interval of θ becomes [0◦ , 90◦ ] and values of [90◦ , 180◦ ] are obtained by a symmetric with respect to the axis x = 90◦ . For square plates, which are geometrical symmetric, the study interval is then [0◦ , 45◦ ] when applying symmetric boundary conditions. To cover all the interval of θ , symmetries are made with respect to axes x = 45◦ and after to x = 90◦ . The initial value of the modeling parameter is here set to Iinitial = p(k=0) = 0◦ . 3.2.1. Three-layered symmetric sandwich beam We consider first, a 1,000 mm × 25 mm clamped-free sandwich beam with composite face sheets and isotropic viscoelastic core. The face sheets material is graphite-epoxy (T300/5208) as used by Arvin et al. [20]. The core layer is characterized by a viscoelastic constant complex law. Geometric and mechanical properties of the beam are presented in Table 2. To study the evolution of the damping properties according to the varying fibers position angle, Arvin et al. [20] calculated the damping properties of the beam for different angles from 0◦ to 90◦ in step of 10◦ . Using our new method for this beam, the

continuous variation of the damping properties is obtained as a function of the fibers orientation angle (Figure 3). For our study, the beam was meshed using 160 × 4 elements. Comparison of our results with those of Arvin et al. [20] (Figure 3) for the first four modes shows a good agreement. However, for the fourth mode, a large gap is observed for the value θ = 0◦ between the two results. Indeed, it is simply an omission: For the value θ = 0◦ , Arvin et al. [20] took the fifth mode value in place of the fourth mode value. For the fifth mode, we found = 141.13 Hz and η = 0.034, whereas Arvin et al. [20] found = 141.41 Hz and η = 0.034 for the fourth mode. For the fourth mode, the variation curve of the damped frequency reaches a maximum point for θ = 7.32◦ . This observation cannot be made in the case of the discrete calculation of Arvin et al. [20] since the calculation for orientation θ = 7.32◦ was not done. 3.2.2. Three-layered symmetric sandwich plates We consider here a three-layered symmetric sandwich plates with two types of viscoelastic laws for core.

r Rectangular plate with 3M ISD112 at 27◦ C core. We consider a three-layered symmetric viscoelastic sandwich plate (plate 1) which faces sheet are in composite material carbon/epoxy (T700/3234). The viscoelastic core is made with 3M ISD112 at 27◦ C in which Young’s modulus is frequency dependent [44, 45]. Geometrical and mechanical characteristics of the plate 1 are grouped in Table 3. ⎛ ⎞ 3 ω j ⎠ (66) G∗c (ω) = G0 ⎝1 + ω − i

j j=1 where j = [0.746; 3.265; 43.284], j =[468.7; 4742.4; 71532.5], and G0 = 0.5 × 106 Pa. Three boundary conditions are studied for this plate such as clamped (CCCC) (Figure 4) simply supported (SSSS) (Figure 5) and clamped-clamped-freefree (CCFF) (Figure 6)

r Rectangular plate with PVB at 20◦ C core We consider once again the plate 1 but we change the core layer by a Polyvinyl Butyral (PVB) at 20◦ C. Young’s modulus 0.25

100

0.2

Loss factor η

Damped frequency Ω (Hertz)

150 The new method mode 1 Arvin et al., mode 1 The new method mode 2 Arvin et al., mode 2 The new method mode 3 Arvin et al., mode 3 The new method mode 4 Arvin et al., mode 4

0.15

The new method mode 1 Arvin et al., mode 1 The new method mode 2 Arvin et al., mode 2 The new method mode 3 Arvin et al., mode 3 The new method mode 4 Arvin et al., mode 4

0.1

50 0.05

0

0

10

20

30

40

50

Angle θ°

60

70

80

90

9

0

0

10

20

(a) Frequency Figure . Damping properties according to fibers orientation of the three-layered clamped-free sandwich beam.

30

40

50

Angle θ°

(b) Loss factor

60

70

80

90

10

K. AKOUSSAN ET AL.

Table . Geometrical and mechanical characteristics of the plate . Mechanical characteristics Geometrical characteristics (mm) hf = 0.762 hc = 0.254 L = 300 l = 100

Core layer

Face layers

ISD at 27 ◦ C Equation () ρc = 1, 600 kg m−3 νc = 0.5

Exx = 119 GPa, Gyz = 3 GPa Eyy = Ezz = 8.7 GPa Gxy = Gxz = 4 GPa νxy = νxz = 0.32 νyz = 0.3, ρf = 1, 560 kg m−3

of PVB is also frequency dependent and given by the following equation: −β (67) G∗c (ω) = G∞ + (G0 − G∞ ) 1 + (iωτ )1−α where G0 = 479 × 103 Pa, G∞ = 2.35 × 108 Pa, τ = 0.3979, α = 0.46 and β = 0.1946. The density of the PVB is ρ = 999 kg m−3 and its Poisson ratio is ν = 0.4. This rectangular plate with PVB core is called plate 2. For this plate 2, two boundary conditions are considered. First, the plate is clamped and the

results are presented in Figure 7. Second, the plate is assumed to be simply supported. The results obtained in this case are presented in Figure 8. For the clamped rectangular plate 1, which has a core layer made with ISD112 at 27◦ C, the symmetry of the boundary conditions permits to reduce the interval of study to [0◦ , 90◦ ] as explained below. As one can see from Figures 4–8, the damping properties values obtained with the new continuous method are in very good agreement with the reference (discrete calculation) for the two different plates and for the two different viscoelastic laws proposed. In Table 4, we show the computation times with the new solving method and the discrete calculation by homotopy. For symmetric boundary conditions, such as CCCC and SSSS, we only calculate damping values for the fixed angle from 5◦ to 90◦ using a regular 5◦ spacing for the discrete calculation method. In the case of the asymmetric boundary conditions, like the case CCFF, 17 angle values, from 5◦ to 180◦ using a regular 5◦ spacing, are calculated. So, the computation times shown in Table 4 for the discrete calculation concern only the listed values but for the presented method this computation times concern the calculation

750 The new method mode 1 discrete calc. mode 1 The new method mode 2 discrete calc. mode 2 The new method mode 3 discrete calc. mode 3

0.45

650 0.4

600

Loss factor η


700

550 500 450 The new method mode 1 discrete calc. mode 1 The new method mode 2 discrete calc. mode 2 The new method mode 3 discrete calc. mode 3

400 350 300 250

0

10

20

30

40

50

Angle θ°

60

70

80

0.35

0.3

0.25

0.2 90

0

10

20

(a) Frequency

30

40

50

Angle θ°

60

70

80

90

(b) Loss factor

450

0.4

400

0.38

350

0.36

Loss factor η


Figure . Damping properties according to fibers orientation of the three-layered plate  (CCCC).

300

250

150

0

10

20

30

40

50

Angle θ°

60

70

80

The new method mode 1 discrete calc. mode 1 The new method mode 2 discrete calc. mode 2 The new method mode 3 discrete calc. mode 3

0.32


200

0.34

0.3

90

(a) Frequency Figure . Damping properties according to fibers orientation of the three-layered plate  (SSSS).

0

10

20

30

40

50

Angle θ°

(b) Loss factor

60

70

80

90


0.4 The new method mode 1 discrete calc. mode 1 The new method mode 2 discrete calc. mode 2 The new method mode 3 discrete calc. mode 3

250

0.35

0.3

Loss factor η


300

200

150

0.25

0.2 The new method mode 1 discrete calc. mode 1 The new method mode 2 discrete calc. mode 2 The new method mode 3 discrete calc. mode 3

0.15 100 0.1

50

0

20

40

60

80

100

Angle θ°

120

140

160

0.05

180

0

20

40

(a) Frequency

60

80

100

Angle θ°

120

140

160

180

(b) Loss factor

1500

0.024

1400

0.022

1300

0.02

1200

Loss factor η


Figure . Damping properties according to fibers orientation of the three-layered plate  (CCFF).

1100 1000 900 The new method mode 1 discrete calc. mode 1 The new method mode 2 discrete calc. mode 2 The new method mode 3 discrete calc. mode 3

800 700 600 500 0

10

20

30

40

50

Angle θ°

60

70

80

0.018 0.016 0.014 0.012


0.01 0.008 0.006 90

0.004

0

10

20

(a) Frequency

30

40

50

Angle θ°

60

70

80

90

(b) Loss factor

Figure . Damping properties according to fibers orientation of the three-layered plate  (CCCC).

−3

800

x 10

10

8 600

Loss factor η


9 700

500

400


300

200

0

10

20

30

40

50

Angle θ°

60

70

80

7 6 5


4 3

90

2

0

(a) Frequency Figure . Damping properties according to fibers orientation of the three-layered plate  (SSSS).

10

20

30

40

50

Angle θ°

(b) Loss factor

60

70

80

90

11

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K. AKOUSSAN ET AL.

Table . Calculation times CPU (seconds). Boundary conditions Plate 

The new method

Continuation number Calculation time Discrete calculation Number of increment Calculation time Plate  The new method Continuation number Calculation time Discrete calculation Number of increment Calculation time

Funding CCCC

SSSS

CCFF

(,,) (,,) (,,)  

 

  (,,) (,,)

   —

 

 

— —





—

in the overall interval of study which is [0◦ , 90◦ ] for CCCC and SSSS and [0◦ , 180◦ ] for CCFF. There is no need to calculate the damping value for the fixed angle 0◦ because we start the present calculation method by this value and at the beginning the two methods give exactly the same value. Due to the fact that the structure is geometrical symmetric, the damping properties of the fixed angle 180◦ are the same with the one fixed angle 0◦ . Also, there is no need to calculate this value again by direct calculation. In Table 4, we notice that the computation times found with the presented method are at least four times lower than the discrete calculation times. The difference between the first three modes computation times for the boundary condition CCCC of plate 1 (423 s) and plate 2 (1,023 s) and for SSSS of plate 1 (476 s) and plate 2 (1,142 s) is due to the viscoelastic core used for the structure. Indeed, plate 1 has a core made with ISD112 at 27◦ C modeled with a Maxwell generalized fractional form law, whereas plate 2 has a core made with PVB modeled with a fractional derivatives law. So, the difference is due to the automatic differentiation computation times of the two different viscoelastic laws. In Table 4, the calculation times is given for the first three modes calculations in each case. The continuation number is written in the form (a, b, c) where a, b, c represent respectively the continuation number for the first three modes. Remind that all calculations were done on a computer ASUS Intel Core i3-2350M CPU @ 2.30 GHz.

4. Conclusion A numerical method for solving nonlinear complex eigenvalues problem depending on two parameters has been proposed. This method is based on the asymptotic numerical method (ANM) coupled with the automatic differentiation (AD), the homotopy technique and the continuation. The application has been performed for instance to study the variation of the natural frequency and the modal loss factor of a three-layered symmetric viscoelastic sandwich structures according to the fibers orientation angle of the unidirectional composite faces. The advantages of the proposed method consist on obtaining the modal solutions as continuous function of the design parameter (e.g., fiber orientation in the present study), while other methods yield to solutions at discrete points. This method can be applied to solve free vibration problem of any viscoelastic structure depending on other design parameters such as thickness and temperature.

This work is supported by Region Lorraine and Labex Damas (Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures, Île du Saulcy, F-57045 Metz cedex 01, France).

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13

Appendix A The behavior law of the isotropic viscoelastic core is defined from the frequency dependence Young’s modulus Ec∗ (ω): +∞ ∗ C2m C2m (t ) e−iωt (ω) = iω 0

⎡

1 hc Ec∗ (ω) ⎢ dt = ⎣ νc 1 − νc2 0

⎤ 0 ⎥ 0 ⎦

νc 1

(A.1)

1−νc 2

0

h2c ∗ C (ω) 12 2m +∞ ∗ C2s (t ) e−iωt C2s (ω) = iω

C2∗ f (ω) =

0

hc Ec∗ (ω) dt = 2 (1 + νc )

$

1 0

(A.2)

% (A.3)

0 1

Appendix B Constant matrices expressing faces fibers orientation in the reference of the plate are defined from the elastic constants of the material: ⎡

c11

⎢ A1 = ⎣ c12 0 ⎡ c22 ⎢ A2 = ⎣ c12 0 A3

⎡

c12 c22 0 c12 c11 0

0

⎤

⎥ 0 ⎦ c66 ⎤ 0 ⎥ 0 ⎦ c66

2 (c12 + 2c66 )

⎢ = ⎣ c11 + c22 − 4c66 0 ⎡ ⎢ A4 = ⎣

⎡ ⎢ A5 = ⎣

(B.1)

(B.2)

⎤

c11 + c22 − 4c66

0

2 (c12 + 2c66 )

0

0

c11 + c22 − 2 (c12 + c66 )

0

0

0

0

c11 − c12 − 2c66

c12 − c22 + 2c66

0

0

0

0

c12 − c22 + 2c66

c11 − c12 − 2c66

c11 − c12 − 2c66

⎥ ⎦

⎤

(B.3)

⎥ c12 − c22 + 2c66 ⎦ 0 ⎤

(B.4)

c12 − c22 + 2c66 ⎥ c11 − c12 − 2c66 ⎦ 0

(B.5) with

⎧ ⎪ ⎪ c11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨c 22 ⎪ ⎪ ⎪ ⎪ ⎪ c12 ⎪ ⎪ ⎪ ⎪ ⎩ c66

h f Exx 1 − νxy νyx h f Eyy = 1 − νxy νyx νyx h f Exx νxy h f Eyy = = 1 − νxy νyx 1 − νxy νyx = h f Gxy =

(B.6)