Identification of a stochastic computational dynamical model using experimental modal data

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Identification of a stochastic computational dynamical model using experimental modal data

Anas Batou, Christian Soize, S. Audebert

To cite this version:

Anas Batou, Christian Soize, S. Audebert. Identification of a stochastic computational dynamical model using experimental modal data. International Conference on Uncertainty in Structural Dy- namics, USD2014, Sep 2014, Leuven, Belgium. pp.1-10. �hal-01066535�

Identification of a stochastic computational dynamical model using experimental modal data - Application on an indus- trial built-up structure

A. Batou¹, C. Soize¹, S. Audebert²

1Universit´e Paris-Est, Mod´elisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 Bd Descartes, 77454 Marne-la-Vall´ee, France

e-mail: [email protected]

2Electricit´e de France R&D, Acoustics and Mechanical Analyses Department, 1 avenue du G´en´eral de Gaulle, 92141, Clamart Cedex, France

In the proceedings of the international conference USD 2014, Katholieke Universiteit Leuven, Belgium, September 15-17, 2014

Abstract

This research is devoted to the identification of a stochastic computational model using experimental eigen- frequencies and mode shapes measured on several specimens. The objective here is to construct a one-to-one correspondence between the results provided by the stochastic computational model and the experimental data that takes into account the random modes crossing and veering phenomena that may occurs from one realization to another one. Then such a correspondence allows the computed modal quantities to be com- pared with the experimental data in order to identify the parameters of the stochastic computational model.

The methodology is applied to a booster pump of thermal units.

1 Introduction

We consider the random context for which the available experimental data are related to a family of several experimental configurations of a given dynamical structure. The observed variability between the experi- mental configurations of this family is induced (1) by the uncontrolled differences that can appear during the manufacturing process (manufacturing tolerances) and during the life cycle of the structure (natural damage, incidents, etc) and (2) by some slight differences which are controlled and are related, for instance, to the boundary conditions, the embedded equipments, etc. These two types of variability induce differences for the data measured on two configurations of the given dynamical structure.

In such a random context, we have to construct a stochastic computational model (SCM) for which two sources of uncertainties have to be taken into account (see for instance [11]): (1) the uncertainties relative to some model parameters of the nominal computational model (NCM) and (2) the modeling errors. The stochastic computational model which is constructed with these two sources of uncertainties (and with addi- tional input and output noises if measurement errors are significant) must have the capability of representing the variability of all the measured configurations.

In this paper, the uncertainties are taken into account using a probabilistic approach and then the SCM is constructed including both the model-parameter uncertainties and the model uncertainties in a separate way (using the generalized probabilistic approach of uncertainties proposed in [10, 11]). Usually, a SCM is

controlled by a set of hyperparameters such as mean values, coefficients of variation, and so on. These hyperparameters have to be identified using experimental data and realizations of the SCM. Several types of observation can be used in order to perform such an identification. The objective of this paper consists in identifying the hyperparameters of a SCM using natural frequencies and mass-normalized mode shapes measured for a family of structures. The methodology proposed introduces a random transformation of the computational observations (computational eigenfrequencies and computational mode shapes) in order to match them to the experimental observation of each measured structure. This methodology automatically takes into account the mode crossings and the mode veerings which can appear between two experimental configurations or between two computational realizations of the SCM. In Section 2, the construction of the SCM is summarized. Section 3 is devoted to the identification of the hyperparameters of the SCM. Finally, in Section 4, an application devoted to an industrial pump of a thermal unit is presented.

2 Construction of the stochastic computational model

The NCM is constructed using the FE method and the boundary conditions of the structure are such that there are no rigid body modes. In this section, a parameterized SCM is constructed using of the generalized prob- abilistic approach of uncertainties proposed in [10, 11], for which both the model-parameter uncertainties and the model uncertainties are taken into account and are separately identified.

2.1 Construction of the probabilistic model of uncertain model parameters

The NCM exhibits np uncertain model parameters denoted h₁, . . . , hnp. Let be h = (h₁, . . . , hnp). The probabilistic model of uncertain model parameters is constructed by replacing vector h of the uncertain model parameters by the random vectorHwith values inR^np, defined on a probability space(Θ^par,T^par, P^par). The first nrandom eigenvalues0 < Λ^par₁ ≤ . . . ≤ Λ^parn associated with the random mode shapes φ^par₁ , . . . ,φ^par_n are the solutions of the following random generalized eigenvalue problem,

[K(H)]φ^par= Λ^par[M(H)]φ^par. (1)

Let [Φ^par] be the m ×n random matrix whose columns are the first n random mode shapes. We then introduce the n×n random mass and stiffness reduced matrices [M^par] = [Φ^par]^T[M(H)] [Φ^par] and [K^par] = [Φ^par]^T [K(H)] [Φ^par]. The random mode shapes are normalized (almost surely) with respect to the random mass matrix such that

[M^par] = [I_n], (2)

and thus, the random diagonaln×nreal matrix[K^par]is written as

[K^par] = diag(Λ^par₁ , . . . ,Λ^par_n ). (3) By construction, the random matrices[M^par]and[K^par]are positive definite (almost surely) and therefore, their Cholesky decompositions yield,

[M^par] = [L_M]^T[L_M], [K^par] = [L_K]^T[L_K]. (4) Letα_parbe the vector whose components are the hyperparameters of the pdfpH(h)which is then rewritten aspH(h;αpar).

2.2 Construction of the generalized probabilistic model of uncertainties

Let(Θ^mod,T^mod,P^mod)be another probability space. To take into account model uncertainties (induced by modeling errors), the dependent random matrices[M^par]and[K^par]are replaced by the dependent random

matrices [M^tot], [K^tot], defined on a probability space (Θ = Θ^par ×Θ^mod,T = T^par ⊗ T^mod,P = P^par⊗ P^mod), such that for allθ= (θ^par, θ^mod)inΘ = Θ^par×Θ^mod,

[M^tot(θ)] = [LM(θ^par)]^T[GM(θ^mod)][LM(θ^par)],

[K^tot(θ)] = [L_K(θ^par)]^T[G_K(θ^mod)][L_K(θ^par)], (5) in which the probability distributions of the random matrices [GM]and [GK], defined on(Θ^mod,T^mod, P^mod), are explicitly given in [9] in the context of the nonparametric probabilistic approach of uncertainties.

The probability distributions of[G_M]and[G_K]depend on the dispersion parametersδ_Mandδ_Krespectively.

Letαmod be the vector of the dispersion parameters such thatαmod = (δM, δ_K).

The random matrices[M^tot]and[K^tot]are not diagonal. In order to calculate the random eigenfrequencies and the random mode shapes of the SCM with both the model-parameter uncertainties and the model un- certainties, the following small-dimension random generalized eigenvalue problem is introduced. Let 0 <

Λ1 ≤ . . . ≤ Λn be the firstnrandom eigenvalues associated with the random eigenvectorsφ^tot₁ , . . . ,φ^tot_n which are the solutions of the following random reduced generalized eigenvalue problem

[K^tot]φ^tot= Λ[M^tot]φ^tot. (6)

Let[Φ^tot] = [φ^tot₁ , . . . ,φ^tot_n ]. These random modes are normalized with respect to the random mass matrix [M^tot],

[M] = [Φ^tot]^T[M^tot] [Φ^tot] = [I_n], (7) and we have

[K] = [Φ^tot]^T [K^tot] [Φ^tot] = diag(Λ1, . . . ,Λn). (8) Then the firstnrandom eigenvalues of the SCM, with both the model-parameter uncertainties and the model uncertainties, are0<Λ₁≤. . .≤Λ_nand the associated random vectors areφ₁, . . . ,φ_nsuch that them×n random matrix[Φ] = [φ₁, . . . ,φ_n]is written as

[Φ] = [Φ^par] [Φ^tot]. (9)

Finally the SCM is parameterized by the vector-valued hyperparameter α = (α_par,α_mod)which has to be identified using experimental modal data. The admissible space for vectorαis denoted byC.

3 Identification of the SCM using experimental modal data

The objective of this section is to identify the parameter α of the SCM using experimental modal data (eigenfrequencies and mass-normalized mode shapes) and realizations of the modal data calculated using the SCM.

3.1 Experimental modal data as observations

It is assumed thatn_expexperimental configurations of the structure have been tested. For each configuration j,n_jmodes have been experimentally identified using an experimental modal analysis method. For two given configurations, the number of modes, the number and locations of the sensors can be different. For each configuration j, n_j experimental eigenfrequencies ω^exp,j₁ , . . . , ωn^exp,j_j associated with n_j mass-normalized experimental mode shapes ϕ^exp,j₁ , . . . ,ϕ^exp,j_nj have been identified form_j degrees of freedom (DOFs). Let [Φ^exp,j] = [ϕ^exp,j₁ . . .ϕ^exp,j_nj ]be them_j×n_j matrix of then_jexperimental mode shapes of the configuration j. It is assumed thatnj < n < mj < mfor allj in{1, . . . , nexp}. The experimental reduced mass matrix and the experimental reduced stiffness matrix are then written as

[M^exp,j] = [I_nj] , [K^exp,j] = diag(λ^exp,j₁ , . . . , λ^exp,j_nj ), (10) in whichλ^exp,j_i = (ω^exp,j_i )².

3.2 Transformation of the modal data

For alljin{1, . . . , n_exp}, let[P^j]be them_j×mmatrix that performs the projection from themDOFs of the SCM to themj DOFs of the experimental configurationj. Then the projected random modal basis[Φ^j] of the SCM is defined by

[Φ^j] = [P^j] [Φ]. (11)

The experimental modes[Φ^exp,j]cannot directly be compared to the computational modes[Φ^j]because, in general, there is not a one-to-one correspondence between the experimental modes and the computational modes. Indeed, some modes may be missing in the experimental modal basis or in the computational modal basis. Furthermore, due to the experimental variability (variability of the configurations) and the computa- tional randomness (uncertainties), some mode crossing and/or mode veering [7, 8, 6] phenomena may occur.

Therefore, the projected computational reduced-order model (ROM), {[Φ^j],[K],[M]} has to transformed into the ROM,{[Φ^j],[K^j],[M^j]}, such that

[Φ^j] = [Φ^j] [Q^opt,j], (12)

[K^j] = [Q^opt,j]^T[K][Q^opt,j], (13) [M^j] = [Q^opt,j]^T[M][Q^opt,j], (14) in which[Q^opt,j]is a random n×nj real matrix for which each realization[Q] = [Q^opt,j(θ)], forθ inΘ, must belong to the Stiefel manifold,OSt(n, n_j), defined by

OSt(n, n_j) ={[Q]∈R^n×nj,[Q]^T[Q] = [I_nj]}. (15) For allθinΘ, orthogonal matrix[Q^opt,j(θ)]is calculated by minimizing the distance between the computa- tional modal basis[Φ^j(θ)]and the experimental modal basis[Φ^exp,j](see [1]),

[Q^opt,j(θ)] = arg min

[Q]∈OSt(n,nj)[Φ^j(θ)] [Q]−[Φ^exp,j]F . (16) The minimization problem (11) is referred as a Procruste problem [4, 5] for which a solution can be calculated iteratively (see [4]). It should be noted that, in [2], the authors have proposed an alternative solution to take into account mode crossings and mode veerings in the identification of the hyperparameters of a SCM.

3.3 Identification of hyperparameterα

Hyperparameterαof the SCM is identified using the maximum likelihood method and experimental modal data. Then the optimal valuesα^optis solution of the following optimization problem

α^opt = arg max

α∈C nexp

j=1

log(p_Wj(W^exp,j;α)), (17)

wherep_Wj(w;α) is the probability density function (pdf) of the random observation vector W^j which is construction using the random transformed ROM{[Φ^j],[K^j],[M^j]}(see [3]).

Figure 1: Industrial mechanical system.

4 Application

4.1 Industrial mechanical system and experimental modal data

We are interested in the dynamical behavior of a one-stage booster pump used by Electricit´e de France (EDF) company in its thermal units (see Fig. 1). This pump is made up of a diffuser and a volute, with axial suction and vertical delivery, and is mounted on a metallic frame. It has been designed forty years ago by Sulzer Pumps. An experimental modal analysis has been carried out on two specimens of this pump located at two different thermal units. Therefore, there aren_exp = 2experimental configurations (denoted as Pump1and Pump2) which are measured. There are slight differences between Pump1and Pump2concerning the joints between the pumps and their metallic frame. The experimental meshes for the two pumps are not the same.

An experimental modal analysis has been carried for each pump. For Pump1, n₁ = 6modes have been identified. The three six mode shapes for Pump1and Pump2are plotted in Figs. 2 and 3 respectively.

Figure 2: Pump1: First three experimental mode shapes (Thick black line).

Figure 3: Pump2: First three experimental mode shapes (Thick black line).

4.2 Construction of the stochastic computational model 4.2.1 Construction of the nominal computational model

The finite element mesh of the NCM is plotted in Fig. 4. The nominal finite element model is made up of 3D

Figure 4: Finite element mesh of the NCM.

solid elements, Kirchhoff plate elements and spring elements. The assembled model has488,220DOFs. The uncertain model parameters of the NCM are the Young modulusysof the steel, the Young’s modulusycof the cast iron, the thicknessest₁,t₂andt₃of the plates1,2and3of the metallic frame and the stiffnessesk₁, k₂,k₃ andk₄of the discrete springs normal to the metallic frame. Let h = (ys, y_c, t₁, t₂, t₃, k₁, k₂, k₃, k₄) be the vector of the uncertain model parameters. For the updated NCM of Pump1and Pump2, the updated values ofhare denoted byh¹andh². For each updated NCM,n= 20modes are calculated . Forh=h¹, the first3projected mode shapes are plotted in Fig. 5. Forh = h², the first 3projected mode shapes are

Figure 5: First3projected mode shapes for Pump1.

plotted in Fig. 6.

Figure 6: First3projected mode shapes for Pump2.

4.2.2 Construction of the stochastic computational model

The vector h is modeled by a random vector H = (Ys, Y_c, T₁, T₂, T₃, K₁, K₂, K₃, K₄). The Maximum Entropy principle has been used for constructing the probability distribution ofH. Taking into account the

available information, it can be deduced that (1) all the components ofHare independent random variables;

(2) positive-valued Young moduli Ys and Yc are Gamma random variables parameterized by their mean values m_Ys andm_Yc, and by their coefficients of variation (standard deviation divided by the mean value) δ_Ys and δ_Yc; (3) positive-valued random thicknesses T₁, T₂ and T₃ are uniform positive-valued random variables parameterized by their mean valuesmT1,mT2 andmT3, and by their coefficients of variationδT1, δ_T2 andδ_T3; (4) positive-valued stiffnessesK₁,K₂,K₃andK₄are Gamma random variables parameterized by their mean valuesm_K1, m_K2,m_K3 andm_K4, and by their coefficients of variationδ_K1, δ_K2, δ_K3 and δ_K4. Thenαpar = (mYs, δ_Ys, m_Yc, δ_Ys, m_T1, δ_T1,m_T2,δ_T2,m_T3, δ_T3, m_K1, δ_K1,m_K2, δ_K2,m_K3, δ_K3, m_K4,δ_K4)has18components to be identified andα= (α_par,α_mod)has20components to be identified.

4.2.3 Identification of the optimal hyperparameterα^opt.

The vector α^opt is given by the optimization problem defined by Eq. (17) which is solved using a genetic algorithm. For each value ofα, the probability density functions p_W1(w;α)andp_W2(w;α)are estimated using800realizations of the observation vectorsW¹andW² calculated with the SCM. The components of α^opt_par are given in Table 1. The two components of α^opt_mod are δ^opt_M = 0.52 andδ^opt_K = 0.43. These optimal dispersions of the probability distributions for model uncertainties are relatively high due to the experimental variability and due to modeling errors introduced in the NCM. These dispersions could be partly decreased by constructing a more accurate NCM.

Mean value Coefficient of variation Young’s modulus steelY_s m_Ys = 1.33×10¹¹Pa δ_Ys = 0.2 Young’s modulus cast ironY_c m_Yc = 7.57×10¹⁰Pa δ_Yc = 0.17

ThicknessT₁ m_T1 = 0.013m δ_T1 = 0.12

ThicknessT₂ m_T2 = 0.011m δ_T2 = 0.57

ThicknessT₃ mT3 = 0.01m δT3 = 0.09

StiffnessK₁ mK1 = 8.01×10⁸ N/m δK1 = 0.19 StiffnessK₂ mK2 = 5.74×10⁹ N/m δK2 = 0.13 StiffnessK₃ mK3 = 6.30×10⁷ N/m δK3 = 0.34 StiffnessK₄ m_K4 = 2.68×10⁸ N/m δ_K4 = 0.42

Table 1: Components ofα^opt_par.

4.3 Validation of the results

Forα = α^opt, the marginal pdf of the first three eigenvalues of the matrices [K¹]and[K²]are shown in Figs. 7 and 8 respectively. It can be seen in these figures that the first3experimental eigenvalues for Pump1 and Pump2are predicted by the SCM with a high probability level (very high for Pump1).

For Pump1, the mean value of the MAC matrix between the random modal basis[Φ¹](before transforma- tion) of the SCM and the experimental modal basis[Φ^exp,1]is plotted in Fig. 9, while for Pump2, the mean value of MAC matrix is plotted in Fig. 10. Figures 9 and 10 show that the randomness of the SCM introduces random mode crossings and random mode veerings which modify the correspondence. For Pump1, the mean value of the MAC matrix between the random modal basis[Φ¹](after transformation) of the SCM and the experimental modal basis[Φ^exp,1]is plotted in Fig. 11 while for Pump2, the mean value of MAC matrix is plotted in Fig. 12. In Figs. 11 and 12, it can be seen that the random transformation of the random mode shapes of the SCM allows to achieve a good correspondence between the random computational modes of the SCM and the experimental modes.

0 5 10 15 x 10⁵ 0

1 2 3 4 5x 10⁻⁶

eigenvalue 1

pdf ^{0 0.5 1 1.5} _{x 10}²⁶

0 0.5 1 1.5

2x 10⁻⁶

eigenvalue 2

pdf 0 1 2 3 4

x 10⁶ 0

0.2 0.4 0.6 0.8 1 1.2x 10⁻⁶

eigenvalue 3

pdf

Figure 7: Probability density function for the first three eigenvalues of[K¹]. Vertical lines: corresponding experimental values (eigenvalues of[K^exp,1]).

−2 0 2 4 6 8

x 10⁵ 0

1 2 3 4 5 6x 10⁻⁶

eigenvalue 1

pdf ^{0 2 4 6 8 10} _{x 10}¹²⁵

0 0.5 1 1.5 2 2.5

3x 10⁻⁶

eigenvalue 2

pdf 0 0.5 1 1.5 2

x 10⁶ 0

0.5 1 1.5

2x 10⁻⁶

eigenvalue 3

pdf

Figure 8: Probability density function for the first three eigenvalues of[K²]. Vertical lines: corresponding experimental values (eigenvalues of[K^exp,2]).

SCM

Experiments

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

2 3 4 5

6 0.2

0.4 0.6

Figure 9: For Pump1, mean value of the MAC matrix between the random mode shapes of the SCM and the experimental mode shapes before transformation.

5 Conclusion

A methodology for the construction and the identification of a stochastic computational model using ex- perimental eigenfrequencies and mode shapes has been presented. The model-parameter uncertainties and the modeling errors are taken into account in the framework of a generalized probabilistic approach of un- certainties. A transformation of the computational modal quantities is introduced in order to construct a correspondence between the experimental modal data and the computational modal quantities. This method allows us to take into account mode crossings and mode veerings that may occur. The methodology has been applied to the construction a stochastic computational model representing a family of booster pumps of thermal units.

SCM

Experiments

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1

2 3 4 5 6 7 8 9 10

11 0.1

0.2 0.3 0.4 0.5 0.6

Figure 10: For Pump2, mean value of the MAC matrix between the random mode shapes of the SCM and the experimental mode shapes before transformation.

SCM

Experiments

1 2 3 4 5 6

1

2

3

4

5

6 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 11: For Pump1, mean value of the MAC matrix between the random mode shapes of the SCM and the experimental mode shapes after transformation.

Acknowledgements

This research work has been carried out in the context of the FUI 2012-2015 SICODYN Project (pour des SImulations cr´edibles via la COrr´elation calculs-essais et l’estimation des incertitudes en DYNamique des structures). The support of the FUI (Fonds Unique Interminist´eriel) is gratefully acknowledged.

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