Bayesian inference to damping identiﬁcation of ﬁber-reinforced composites
from experimental modal data
S. Chandra, K. Sepahvand, C. A. Geweth, F. Saati and S. Marburg
Chair of Vibroacoustics of Vehicles and Machines, Technical University of Munich, 85748 Garching, Germany.

Abstract
For stochastic vibration analysis of the composite plate in-situ stochastic data of the elastic parameters anddamping loss factors are essential. Deterministic values of the elastic and damping parameters of the composite plates can be identify using various optimization technique. In recent decades various stochastic inverse
identiﬁcation procedures are shown to evaluate the in-situ stochastic parameters of the elastic constants of thecomposite plate. However, identiﬁcation of the damping loss factor of the composite plate has received lessattention. A stochastic Bayesian inverse identiﬁcation technique is used here to evaluate the in-situ uncertaintyof the damping loss factors of the glass-epoxy composite plate. Experimentally evaluated ﬁrst 4 modal damping
ratio of the glass-epoxy composite plate are incorporated in Bayesian inverse inference formulation throughlikelihood function. Novelty of the study lies in the development Bayesian inverse formulation consideringmodal damping response to identify the stochastic damping loss factors of the glass-epoxy composite plate.
Keywords: Bayesian inverse inference, Loss factors, Composite plate.

1

INTRODUCTION

Inverse identiﬁcation of the material parameters of a composite structure is popularly done by minimization of
the error function between experimentally evaluated response and model response by using some optimization
technique. This error function minimization technique can identify the material parameters of the system as adesign variable. The identiﬁcation of the material parameters using this approach can estimate the parameters
deterministically without considering uncertainty involve in the modeling and material properties [1, 2, 3]. The
randomness in the composite plate is arise due to variability of several factors such as ﬁber orientations, laminate sequences and various manufacturing uncertainty. The randomness of the material parameters should beconsidered while estimating the stochastic dynamic response of the composite plate. Generally an assumptionof the variation of material properties are made prior to the stochastic dynamic analysis instead of consideringin-situ variability of the material parameters. Therefore, a inverse framework for stochastic estimation of thein-situ uncertainty of the material properties of the composite plate is essential. In such cases stochastic representation of the experimentally evaluated modal response in terms of a probability distribution function (PDF)
are necessary.
In recent years, stochastic inverse identiﬁcations of the material parameters such as elastic parameters andPoisson’s ratio are done by generalized polynomial chaos (gPC) expansion method by Sephavand and Marburg [4, 5]. A collocation based gPC expansion method [8, 9, 6] is used for stochastic inverse identiﬁcationof the parameters of the composite plate. The stochastic inverse identiﬁcation of the parameters involve estimation of the unknown deterministic coefﬁcients of the gPC expansion by optimization algorithm [7]. However,
Bayesian inverse identiﬁcation technique is also quite popular among the researchers [10, 11, 12] to identify
the elastic properties of a dynamic system since 1979 [13]. Statistical properties such as mean and standard
deviation (SD) of the identiﬁed parameters can be derived from the posterior distributions. The basic advantageof the Bayesian inverses technique is posterior PDF can be develop from a limited numbers of experimental
observations. The evaluation of integral is most challenging task in multi-parameter Bayesian inverse inference.
However, Markov Chain Monte Carlo (MCMC) [14, 15] became an efﬁcient alternative to determine the posterior density without evaluating the integral. Various sampling based approaches such as, Metropolis-Hasting

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(M-H) algorithm and Gibbs sampler [16] algorithm have been developed for improvement of the MCMC algorithm. Memory less M-H algorithm can efﬁciently derive the multi-parameter posterior PDF. More recently,
Nagel [17] discussed Bayesian inverse problem with a direction to overcome the limitations of sampling based
technique for determining the posterior probability density functions of the system parameters. Various advanced techniques and improvements of Bayesian inference technique are proposed in stochastic identiﬁcationof the elastic constant [18, 19]. Gogu et al. [19] identiﬁed the elastic constants of the unidirectional laminate by tensile strength test. Of late, Rappela and Beex [20] identiﬁed material randomness of the discrete
strut using Bayesian inverse updation and corresponding inﬂuences of the geometric randomness of the ﬁber.
Rosić et al. [21] have proposed a linear Bayesian estimation of the unknown parameters in combination with
the Karhunen-Loève and Polynomial Chaos expansion without using any sampling based technique such as,
MCMC. This method can efﬁciently update non-Gaussian uncertainties. However, stochastic identiﬁcation of
the elastic parameters of multilayer bi-directional composite plate using Bayesian inverse inference techniquefrom modal experimental data has not explored yet. Moreover, researchers have paid much attention to identify
the elastic constants of the composite plate, while identiﬁcation of the damping loss factors of the composite
plate is indispensable requirement for the damped vibration analysis of the composite plate.
Yesilyurt and Habibe [22] have deterministically identiﬁed modal damping ratio of a composite beam by using
short time Fourier transformation (STFT). Cherif et al. [23] have determined damping loss factors of a two dimensional orthotropic plate using inverse Wave method and compared with the classical methods. Li et al. [24]have identiﬁed loss factors of a carbon-epoxy composite plate using modal test data using complex modulus
approach. However, stochastic identiﬁcation of the damping loss factor has not been reported earlier. Here,loss factors of the multilayer bi-directional composite plate is identiﬁed using Bayesian inverse technique fromexperimentally evaluated modal damping ratio.

2

MATHEMATICAL FORMULATIONS

2.1 Finite element formulation of laminated composite plate
Classical thin plate theory is used for analysis of a laminated composite plate using ﬁrst order shear deformation
theory (FSDT). The stress σ  and strain ε  relationship for a thin unidirectional lamina is presented by thegeneralized Hook’s law with reference to the principal material axes (1, 2, 3) asσ  = Cε.

(1)Here, C is principal stress-strain relationship matrix [25] along the principal axes of the kth lamina for the multilayer bi-directional laminated composite plate. The elements of the C matrix for the kth orthotropic lamina are
C11 = E11 /(1 − ν12 ν21 ), C12 = ν21 E11 /(1 − ν12 ν21 ), C22 = E22 /(1 − ν12 ν21 ), C33 = G12 , C44 = G23 and C55 = G13 .
Here Eii , Gi j and νi j are the set of elastic constants such as, Young’s moduli, shear moduli and Poisson’s
ratio of the lamina, respectively. The stress σ and strain ε relationship of the lamina with reference to global
laminate axes (x, y, z) is written as
σ = Qε ,(2)

Q = T −1C T .

(3)

in which,
Here T is transformation matrix [25] to relate the principal lamina axes and laminate axes. The element
stiffness matrix is of the laminated is presented as
Ke =



Ae

B T D B dAe .

(4)

In which, B [25] is the strain-displacement matrix and written asε̄ε = B δ .

6278

(5)

where ε̄ε = [εx εy εxy κx κy κxy εxz εyz ]T is the strain and curvature vector and δ = [u0 j v0 j w0 j θx j θy j ]Tis the nodal displacement vector of the composite plate. Eight-nodded element is used for ﬁnite element (FE)σ and strain ε̄ε of the laminate are related byanalysis of te composite plate.The mid-plane stress resultant σ̄
stiffness matrix D as [25]σ = D ε̄ε .
σ̄
σ = [Nx Ny Nxy Mx My Mxy Qx
In which σ̄
⎡
A11 A12 A16⎢ A12 A22 A26⎢⎢ A16 A26 A66⎢⎢ B11 B12 B16⎢⎢ B12 B22 B26⎢⎢ B16 B26 B66
⎢
⎣ 0
0
0
0
0
0

(6)

Qy ]T and D is given by
B11
B12
B16
D11
D12
D16
0
0

B12
B22
B26
D12
D22
D26
0
0

B16
B26
B66
D16
D26
D66
0
0

0
0
0
0
0
0
A44
A45

z

⎤
0
0 ⎥
⎥
0 ⎥
⎥
0 ⎥
⎥,
0 ⎥
⎥
0 ⎥
⎥
A45 ⎦
A55

(7)

z

where Ai j , Bi j , Di j = ∑Nk=1 z kk−1) (Qi j )k (1, z, z2 )dz, i, j = 1, 2, 6 and Ai j = ∑Nk=1 z kk−1) κ(Qi j )k dz, i, j = 4, 5, κ = 5/6.
((
The elemental mass matrix is written as
Me =



Ae

N T ρ N dAe ,(8)

where ρ is the inertia matrix. The global stiffness matrix K and the global mass matrix M are developed after
assembling the elemental stiffness and mass matrices, K e and M e , respectively. Therefore, Undamped modal
analysis involves the solution of[λi2 M + K ]φi = 0,i = 1, 2, ..., n(9)and extract the modal frequency λi and mode shape φi of the laminated composite plate with n numbers ofdegree of freedom (DOF) in FE model.
2.2 Modal damping formulation of laminated composite plate
During vibration, dissipation of the speciﬁc energy in the form of heat and acoustic radiation is responsible forthe passive damping of the composite plate. Mathematical estimation of the damping of the composite plateis necessarily simple and yet accurately represent the damping of the composite plate. Viscoelastic damping
model is efﬁciently represent the energy dissipation from the vibrating composite plate via complex modulus
approach. It is assumed that stress and strain are harmonically time dependent for linear viscoelastic material
and are presented as σ = σ0 eiωt and ε = ε0 eiωt . Linear viscoelastic constitutive relationship is expressed as
σ (t) = t−∞

C∗ (t − τ)dε(τ),(10)

where complex modulus is expressed as a combination of a real and a imaginary component as C∗ = C + iC .
Real component C and imaginary component C of the elastic moduli are termed as storage moduli and loss
moduli, respectively. Complex elastic constants of the composite material in the principal lamina direction areevaluated as
E11 = E11 (1 + iη11 ), E22 = E22 (1 + iη22 ), G12 = E12 (1 + iη12 ).

(11)Here in η11 , η22 and η12 are the damping loss factors along the longitudinal, transverse and shear directionsof the lamina, respectively. Accordingly, complex elastic constants are inserted in Eq. (2) and corresponding
complex stiffness matrix is presented as
K I.
K ∗ = K R + iK

(12)

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Here K R storage stiffness matrix and K R is loss stiffness matrix. Hence, damped dynamic equation of thecomposite plate is written as
M ẍ + K ∗ x = f .

(13)

∗

Assumed solution of the Eq. (13) is x = {φi∗ }eiλi t , where φi∗ and λi∗ are nth complex eigen mode and complex
eigen frequency. Homogeneous solution of the Eq. (13) is written as[λi∗2 M + K ]φi∗ = 0,i = 1, 2, ..., n.

(14)

The complex eigen frequency is written as∗(1 + iηi )1/2 .
λi∗ = λiR

(15)

Thus modal loss factor ηi of the composite plate is written asηi =Im(λi∗2 ).
Re(λi∗2 )(16)

The ideal forward problem to estimate the modal loss factor ηi of the composite plate stated in the following
form with reference to the Eqs. (14), (16)ηi = G(η11 , η22 , η12 , E11 , E22 , G12 )i = 1, 2, ..., n.

(17)Considering the fact that nominal values of the elastic parameters are known and distributions of the ηi areevaluated experimentally, then stochastic damping loss factors η11 , η22 and η12 along the principal directionsof the lamina can be identiﬁed by inverse stochastic procedure. Corresponding forward model in close form is
written asm),
d = G(m(18)

where set of simulated modal loss factors for ideal case is deﬁne by d = {ηi }T and m = {η11 η22 η12 }T is thevector of identiﬁable parameters of the inverse problem.
2.3 Bayesian inverse model
Considering the forward formulation stated in Eq. (18), classical inverse problem involves identiﬁcation of model
parameter m from the actual observation of data set d̃d . Thus, model lead to relationm) + ε ,d̃d = G(m(19)

where ε is modal additive error account for deviation between simulated value d and observed values d̃d . The
components of ε are i.i.d. random variables. The Bayesian inverse problem concerned with estimating theparameters m given a set of observation data d̃d . The posterior probability density of the model parameters mtakes the formm|d̃d ) =
p(m

m)p(m
m)
p(d̃d |m.
m)dmmp(d̃d |m)p(m(20)m) indicate the prior probability of the parameters m . Likelihood p(d̃d |mm) is deﬁne as a function of
Herein p(mm) = p(d̃d |mm). A typical assumption for likelihood function for observed data d˜j are normally
m , such as L (mm) and can write as
distributed with a SD σ and expected value G(mm) =p(d˜j |m

6280

m))2
(d˜j − G(m1√ exp −
2
2σ
σ 2π

(21)

and likelihood is stated for n numbers of observational data asnm) = ∏ p(d˜j |m
m)
L (m

(22)



m))2 n(d˜j − G(m
1
m) =
√
.
L (m
n exp −
2σ 2
σ 2π

(23)

j=1

The integral at the numerator of Eq. (20) is a normalizing constant and described by c. Inverse identiﬁcation ofparameters m from md set of modal data can be written using the theory of total probability as [26]mdm|d̃i )
m|d̃d ) = ∏ p(m
p(m

(24)

i=1

md

m|d̃d ) ∝ p(mm)md ∏ Li (m
m)
p(m

(25)

i=1

1

 md



,
(26)
n exp ∑
(2π)md/2 ∏md
i=1 σiwhere Ai is deﬁne as ∑nj=1 (d˜i j − G(mi ))2 . The strategy for numerical solution of the stochastic inverse
problem to identify the parameters is proposed using Markov chain Monte Carlo (MCMC) method. The idea
behind the MCMC is to construct posterior distribution without evaluating the normalizing constant c. The
Metropolis-Hastings (M-H) algorithm construct a stationary Markov chain whose stationary distribution equal
to the target distribution. M-H algorithm involve generating new sample point y of Markov chain from a
proposal distribution q(·|x(k) ) conditional to the current state x(k) and then accepts and rejects the new sample
y with certain probability of acceptance. Generation of sufﬁcient numbers of sample of a Markov chain with astationary distribution is equivalent to the target posterior distribution pX (x). The stationary distribution of theMarkov chain is obtained by ignoring the initial burn-in period of the proposal distribution.
Here in m denotes vector of model parameters and d is set of simulated data for ideal case. The forward model
operator G predicts the model output data set d in terms of eigen frequency as a function of model parameters
m. In the present paper, model parameters are Eii and Gi j , and the forward model yield the data output in theform of modal frequency λi .
m|d̃d ) ∝ p(m
m)
p(m

md

3

(−0.5ATi σi−2 Ai )i=1

NUMERICAL EXAMPLE

A spectral modal Bayesian inverse inferences technique is presented herein to identify the stochastic damping
loss factors of the glass-epoxy composite plate. An experimental free vibration analysis is conducted on 12layers glass-epoxy composite plate in free-free boundary conditions. 100 numbers of identical composite plateof dimension 250×125×2 mm3 are used for the free vibration analysis. The modal frequency and modal damping ratio of the composite plate is determine by impulse hammer technique. To identify stochastic damping loss
factors, elastic parameters of the glass-epoxy composite plate are considered as deterministic and nominal valuesof the elastic parameters are identiﬁed in [27] using inverse gPC identiﬁcation. The nominal vales of the elastic
parameters and mass density are given in Table 1. These values were identiﬁed from the ﬁrst 4 experimentally
evaluated eigen frequency. In this paper the loss factors of the composite plate are evaluated from experimentally evaluated ﬁrst 4 modal damping ratios using Bayesian inference technique. An innovative MCMC is
used to construct the posterior distribution of the loss factors. The innovation lies on the construction of thelikelihood function from the 4 modal data and simultaneously identiﬁed 3 loss factors by inverse operation.
Experimentally evaluated modal sampling data is used to update the damping loss factors from uninformative
prior. Random walk of the M-H algorithm, is used to generate posterior samples of {η11 η22 η12 }T using 50

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1

0.5

0
0

1000

2000

3000

4000

5000

6000

1000

2000

3000

4000

5000

6000

1000

2000

3000

4000

5000

6000

1.6
1.4
1.2
1
0.8
0

4.8
4.6
4.4
0

Figure 1.
MCMC sample for posterior distribution of the loss factors due to proposal SDση = {0.005 0.025 0.1}T .
and 100 experimentals sample data. The choice of proposal distribution is very important to specify the jumptowards the target distribution. Generally normal distribution, centered about the current sample point, is used
as a proposal distribution. However, asymmetric proposal distribution like log-normal distribution [28] is usedhere to avoid the generation of the sample points smaller than 0. The SD of the proposal PDF is proposed asση = {0.005 0.025 0.1}T with log-normal distribution. Same starting points of the identiﬁable parameters are(0) (0) (0)considered here as {xη11 xη22 xη12 }T = {0.2 0.2 0.2}T . However, in Fig. 1 it is observed that each loss factors
are converged towards its target distributions after initial burn-in period for both cases from a uninformative
prior. Sample statistics of the parameters are obtained after discarding initial 400 samples. The stochastic parameters of the identiﬁed loss factors are given in Table 2 discarding the initial burn-in period. Table 2 reveals
that variation of the statistical parameters of the identiﬁed loss factors are remain within a limited range ofvariation for two types of sample data. Therefore, proposed modal Bayesian inverse inferences technique can
efﬁciently identify the stochastic loss factors of the composite plate from the modal experimental data.
Table 1. Mean values of the elastic parameters and density of the investigated plate as identiﬁed in [27]Elastic moduli [GPa]Poisson’s ratio [-]Density [gm/cm3 ]6282

E11 = 68.714
ν12 = 0.24
2.1143

E22 = 27.401
ν21 = 0.24−

G12 = 6.122−−

Table 2. Identiﬁed stochastic loss factors of the glass-epoxy composite plate
Parametersη11 [%]
η22 [%]
η12 [%]

4

Prior
Mean
0.8
3
5

SD
0.32
1.20
2

Posterior(50 samples)
Mean SD
0.47 0.12
1.16 0.15
4.46 0.09

Posterior(100 samples)
Mean
SD
0.42
0.11
1.20
0.13
4.56
0.05

CONCLUSION

An stochastic Bayesian inverse identiﬁcation technique is used to identify the loss factors of the glass-epoxy
composite plate from the experimentally evaluated modal damping ratio. In Bayesian inverse formulation thelikelihood function is developed from the sufﬁcient numbers of modal data and the same likelihood function
is used to identify the three loss factors of the composite plate simultaneously. 50 and 100 numbers of ﬁrst 4
modal damping ratios are used to update the posterior distribution via likelihood functions. To avoid calculationof the integration in the denominator of the Bayesian equation MCMC algorithm has been used to determine
the posterior distribution of the loss factors. Sufﬁcient numbers of posterior sample points are generated using
MCMC approach. The efﬁcient learning of the inverse model is indicated in Fig. 1 by suitable convergenceof the identiﬁable parameters with 50 and 100 numbers of sample. Random walk of MCMC is supported lognormal proposal distribution. Thus efﬁcient stochastic identiﬁcation of the set of loss factors of the glass-epoxy
composite plate are shown in this paper using Bayesian inverse identiﬁcation technique.

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