STOCHASTIC MODEL IDENTIFICATION OF A BOLTED JOINT...

6th European Conference on Computational Mechanics (ECCM 6)7th European Conference on Computational Fluid Dynamics (ECFD 7)

1115 June 2018, Glasgow, UK

STOCHASTIC MODEL IDENTIFICATION OF A BOLTEDJOINT FOR THE ROBUST CHARACTERISATION OF THE

VIBRATIONAL BEHAVIOUR OF LIGHT ASSEMBLEDSTRUCTURES

M. Ghienne1,2∗, L. Laurent1 and C. Blanze1

1 Structural Mechanics and Coupled Systems Laboratory (LMSSC),Conservatoire national des arts et metiers,

292 rue Saint-Martin, Case courrier 2D6R10, 75141 Paris cedex 03,[email protected] ; [email protected] ; [email protected]

2 (Curent position) Quartz 7393 Laboratory,Institut superieur de mecanique de Paris - Supmeca,3, rue Fernand Hainaut, 93400 Saint-Ouen, France,

[email protected]

Key words: Parametric uncertainties, bolted joints, random stiffness identification,stochastic model reduction

Abstract. Assembled structures involve non-linear phenomena, as contact or friction,localised at each joint interface. In addition, sources of uncertainties are mainly re-sponsible for deviation between the effective behaviour of the structure and results fromdeterministic simulations. This work aims to provide a pragmatic approach to robustlycharacterise the vibrational behaviour of an assembled structure considering the jointsparameters variability. This approach would be useful for robust design of solutions, suchas solutions for damping vibrations, dedicated to assembled structures and taking intoaccount the variability of the real behaviour of each joint. It is based on a non-intrusiveapproach referred as the Stochastic Model Reduction (SMR) which is used to identify asimple but sufficient stochastic model of bolted joint.

1 Introduction

In this work, we focuse on so-called ”light” assembled structures. These structures areusually made of slender sub-structures assembled, for instance using bolts. The numberof joints of these structures is generally significant which make the characteristics of eachjoint more likely to differ between joints in comparison with a given nominal model. Thiswork aims to robustly characterise the vibrational behaviour of a light assembled structureconsidering the joints parameters variability.

The proposed approach is made of three steps. First, a just sufficient nominal model ofa bolted joint is defined. This model aims to take into account the influence of the jointon the dynamical behaviour of the overall structure. This model is then adapted using a

M. Ghienne, L. Laurent and C. Blanze

Laser

Vibrometer

Figure 1: Presentation of the experimental setup used to identified an adapted nominalmodel of a bolted joint

probabilistic approach to take into account the variability of the real behaviour of eachjoint. The parameters of the joint model are then modelled as Gamma distributed randomvariables. Finally, the parameters of this new stochastic model are identified on the basisof the maximum likelihood principle. A straightforward non-intrusive approach referredas the Stochastic Model Reduction (SMR) approach [1] is used to simulate the vibrationalbehaviour of the structure. The variability of this structure is investigate experimentallyand used as a reference in the identification process.

In this work, we assume that highly elaborate models (taking into account strong non-linearity) are very sensitive to uncertainties which could implies higher errors between theobserved and simulated behaviour of the studied assembled structure. In this context,we seek to define a ”just sufficient” nominal model of a bolted joint allowing to take intoaccount the influence of the joint interfaces on the vibrational behaviour of the globalassembled structure.

The studied structure, inspired by the work of Van Buren et al. [7], is a frame madeof three aluminum beams assembled by four bolted joints as illustrated Figure 1. To pickup a ”just sufficient” model adapted to the studied frame, a preliminary experimentaldifferential analysis has been set up. The vibrational behaviour of two frames, a framewith only one bolted joint and an equivalent monolithic frame, are compared to evaluatethe influence of the bolted joint on the vibrational behaviour of the overall structure. Thevibrational behaviour of each frame is characterised using a laser vibrometer and a sweepsine excitation applied using a shaker as illustrated Figure 1.

2 Definition of the nominal model of a bolted joint

This first section aims to define an adapted nominal model to describe the influence ofthe bolted joints on the overall assembled structure.

2.1 Nominal model of the bolted joint

The preliminary differential analysis allowed us to highlight the influence of the boltedjoint on the behaviour of the assembled structure. As expected, the bolted joint affects

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the dynamical characteristics of the assembly for a decrease of its eigenfrequencies. Onlythe stiffness of the bolted joint is considered in this paper. The experimental setup allowsus to study the influence of the excitation magnitude and the bolt tightening on the jointstiffness. Indeed, the bolt is instrumented with deformation gauges to infer the appliedtightening and the excitation magnitude is driven by the input signal applied to the shaker.The experimental observations highlight a particular dependency of the joint stiffness tothe frame eigenfrequencies and therefore to its eigenshapes. For the considered frame,the joints are mostly loaded in bending. The joint stiffness model should consequentlydepend on the local curvature of the frame eigenshapes.

The studied structure is made of slender parts, a beam finite element model is thenadapted to simulate its vibrational behaviour. Numerous interface models can be foundin the litterature as Zero-thickness, thin layer or node-to-node elements [11, 2]. Forbeam like structures, node-to-node elements are well adapted. The influence of jointson the vibrational behaviour of the assembly can then be modeled using generic stiffnessand damping applied between nodes from each assembled part [10, 7] (Figure 2a), usingelements defined between each node pair in the joint interface bounded by a given radiusof influence [7] (Figure 2b) or using dedicated elements formulated to take into accountthe behaviour of the bolted joint [12].

By definition, the bending deformation energy of a beam depends directly on thecurvature and is proportionnal to EI, where E is the Young modulus and I the momentof inertia. In this work, we choose to model the joint stiffness through an equivalentYoung modulus applied to the frame elements corresponding to the joint as illustratedFigure 2c.

(a) (b) (c)

Figure 2: Nominal model of the bolted joint

2.2 Identification of the nominal joint parameter

To identify the Young modulus of the chosen nominal model, we propose to minimizethe least squares error between the experimental and simulated eigenfrequencies of theframe defined equation (1).

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ǫls =N∑

k=1

(ωexpk − ωEF

k )2 (1)

The Figure 3 illustrates the nominal Frequency Response Function (FRF) of the framesimulated with the finite element model defined Figure 2c and the identified Young mod-ulus. The nominal model of the bolted joint is then adapted to simulate the behaviour ofa frame with a single bolt on its vertical beam. It might be reminded that the previousdeveloppement was applied to a bolted joint linked to a vertical beam of the studiedframe. As a bolt positionned on the horizontal beam would not necessarely be loaded ina similar manner, the identification process and the needed experimental results shouldbe reiterated for this second bolt configuation.

0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300

−100

−50

0

50

Frequency (Hz)

FRF

Expe FRFSimulated FRF

Figure 3: Experimental and simulated FRF of the frame with a single bolt

3 A straightforward approach to simulate the vibrational behavior of a struc-

ture with random parameters: the SMR approach

A nominal model of the bolted joint has been identified, it then remains to pick upan adapted method to robustly characterize the vibrational response of a structure in arandom manner. We are particularly looking for the eigenspace characterization of linearsystems with dynamic properties considered as random variables. We assume that a finiteelement model of the structure is available. Methods based on statistical sampling providea good framework to solve the random dynamic problem, nevertheless they need intensivecomputation to remain accurate [3, 4]. Two particular approaches are mainly used in theliterature to approximate the statistical properties of the response of a random system:the perturbation and the spectral methods [6, 5]. The perturbation method is based onan approximation of the random variable of interest through the truncation of its Taylorexpansion while the Spectral Stochastic Finite Element Method (SSFEM), introduced byGhanem and Spanos [6], is based on a discretization of the random variables of interestamong a finite random space.

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In the context of this work, a straightforward non intrusive approach is used to char-acterize random eigenspace with a reduced number of deterministic finite element com-putations [1]. This approach is referred as the Stochastic Model Reduction approach(SMR).

3.1 Solving the random eigenvalue problem using the SMR approach

This section briefly overviews the SMR approach. More details about the basic assump-tions of this approach, comparisons with other classical methods and practical applicationsare available in a dedicated paper [1] and in the PhD thesis manuscript (in french) [?].

The general eigenvalue problem of undamped or proportionally damped systems canbe expressed for a problem with nd degrees of freedom by

λk(θ)M(θ)Φk(θ) = K(θ)Φk(θ) (2)

λk ∈ R and Φk ∈ Rnd are the kth eigenvalue and the kth associated eigenvector.

The eigenvector Φk is assumed to be mass normalized such that Φ⊤

k MΦk = 1. MatricesM(θ) ∈ R

nd×nd and K(θ) ∈ Rnd×nd represent the mass and stiffness matrices of the struc-

ture. Their randomness is due to the physical parameters of the structure such as massdensity, Young’s modulus or geometric properties. (Ω,F , p) is the abstract probabilityspace associated with the underlying physical experiments. θ ∈ Ω is a basic event fromthe complete probability space Ω. The space of square integrable random variables isdenoted by L2(Ω) and forms a Hilbert space with the norm ‖ · ‖L2(Ω). In this paper, E[·],Var[·] and σ[·] denote respectively the mathematical expectation, variance and standarddeviation.

The SMR approach is presented here considering the case of an assembled structurecomposed of n sub-structures with n different Young’s moduli (Y1, Y2, ..., Yn). This casewell fits the equivalent stiffness model of a bolted joint presented in the previous section.Nevertheless, this formulation could also correspond to other problems, for example, ametallic structure provided with piezoelectric patches or a multi-materials glued structure.The Young’s moduli are assumed to be the only random parameters in the consideredproblem. Thus the stiffness matrixK(θ) is a random matrix andM remains deterministic.The global stiffness matrix can be written according to different stiffness matrices relatingto each sub-structure:

K(θ) =n

∑

i=1

Ki(θ) =n

∑

i=1

γi(θ)Ki (3)

where γi(θ) is the random parameter corresponding to the ith Young’s modulus anddefined as:

γi(θ) =Yi(θ)

Y0i

; Y0i = E[Yi(θ)] and E[γi(θ)] = 1

The first assumption of the SMR approach is: considering a structure with randomparameters, its eigenvalues are random but the corresponding eigenvectors are quite de-terministic. First, it is then assumed that Young’s moduli variations around an expected

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value do not change the eigenvector shapes of the structure. Therefore, the matrix of ran-dom eigenvectors corresponds to the matrix of eigenvectors calculated with the expectedvalue parameters. This assumption added to the mass normalization of eigenvectors Φk

allows us to rewrite the kth random frequency as :

λk(θ) = ω2k(θ) =

n∑

i=1

γi(θ)Φk⊤

KiΦk =n

∑

i=1

λkiγi(θ) (4)

The λki coefficients of this expansion are deterministic. The randomness of the eigenvalueis only due to the input random parameters. The main advantage of this first refinementlevel is to obtain a closed form of the approximate eigenvalue for which the two firstcentral moments of the random eigenvalues are obtained with no computational cost.

When the eigenshapes variability increases, it is no longer possible to approximate therandom eigenvectors as deterministic eigenvectors. In order to accurately estimate thecorresponding eigenvalues it is proposed to take into account the eigenshapes variabilitywith a Taylor expansion about the point γ(θ) = γ0. This improvement is referred hereafteras SMR2.

The kth random eigenvalue λk(γ(θ)) estimated using SMR2 can be written as:

λk(θ) =n

∑

i=1

γi(θ)Φ⊤

k (γ(θ))KiΦk(γ(θ)) (5)

with Φk(γ(θ)) = Φk(γ0) +n

∑

i=1

∂Φk

∂γi

∣

∣

∣

∣

γi=1

(γi(θ)− 1) (6)

For details on the computation of the derivatives of each eigenvector, reader could referto [1].

SMR approach has been widely tested in [1] on different test cases. In particular, theapproach has been applied on an industrial structure modeled with a commercial software:the Ariane 5 payload adapter. This example illustrates the SMR efficiency in comparisonwith Monte-Carlo simulation. The first two statistical moments of the payload adapterfirst ten random eigenfrequencies are accurately estimated with a computational timereduction from 16 h to 18min (i.e. a time reduction of 98%).

4 Identification of a stochastic model of a bolted joint

A nominal model of a bolted joint has been defined and a non-intrusive approach allowsus to characterise the random eigenspace of a structure with a reduced computation time.

It remains to define and identify a stochastic model of joint. This model can be definedby randomizing the parameter of the chosen nominal model of joint. This parameter istherefore defined as a random variable with an unknown distribution. As only limited dataare available from experiments, we choose a straightforward approach which, as a firststep, identify an appropriate probability density function (pdf ) according to the maximumentropy principle, and as a second step, identify the parameters of this distribution usingthe maximum likelihood approach [13, 14].

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The joint stiffness (the equivalent Young modulus) is a positive real-valued variableSupp(KLiaison) =]0,+∞[. According to the maximum entropy principle, this parameteris represented in terms of a Gamma random variable. As the Gamma distribution as onlytwo parameters, the first two statistical moments of the random variable associated tothe joint model are sufficient.

4.1 Identification of the first statistical moment of the random variables as-

sociated to the joint parameters

The aim here is to identify the parameters of the Gamma distribution associated toeach bolted joint of the studied frame. The frame has two bolt configurations (bolt linkedto a vertical beam of the frame or linked to the horizontal one). Then two pairs ofstatistical moments have to be identified, one for each single bolt frame configuration.Only the case of the frame with a single bolt on a vertical beam is presented here.

The parameters of the stochastic model of the joint stiffness are here identified basedon the maximum likelihood principle. Proceeding directly in the space of the randomvariables, the parameters of the Gamma distribution could be identified using the first twostatistical moments of the random variable associated to this stiffness which minimize theerror between the statistical moment of the observed experimental eigenfrequencies andthe computed one obtained with the stochastic model of the frame. Available experimentaldata are eigenfrencies of two configurations of frame with only one bolted joint (bolton the vertical or horizontal beam). These eigenfrenquencis are measured for differenttightening values and excitation magnitudes. This allows to characterise the variabilityof these eigenfrencies.

In order to determine the statistical moment of the numerical frame model, SMRapproach is used in place of classical Monte Carlo strategy which is no reliable in thiscase due to a too large computation cost.

We define the following error for each statistical moment:

ǫE =N∑

i=1

[

E[ωexp]− E[ωSMR]]2

and ǫδ =N∑

i=1

[

δ[ωexp]− δ[ωSMR]]2

(7)

Where ǫE is the error on the expection of the frame eigenfrequencies and ǫδ the erroron the coefficient of variation. First, the expectation of the joint random stiffness isobtained minimizing the error ǫE. Then the coefficient of variation of this stiffness isobtained using this expectation and minimizing ǫδ. Scaling and summation properties ofthe Gamma distribution allows to determine the parameters of the Gamma distribution asa function of the statistical moments of the random variable. Writting Yi(θ) the randomstiffness associated to the ith bolted joint of the frame, we define γi the random variablewith Gamma distribution corresponding to this ith bolt and αi and βi the parameters ofthis Gamma distribution. Then, for Yi(θ) = Yiγi and γi Gamma(αi, βi) = Γ(αi, βi),the parameters αi and βi are defined by the following equations :

αi =

(

E[Yi]

σYi

)2

=

(

1

δ[Yi]

)2

and βi =

(

σYi

E[Yi]

)2

= δ2[Yi] (8)

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This identification procedure has been applied for both configurations of the frame withonly one bolt. Results are presented Table 1 where index Jvert refers to joints associatedto a vertical beam of the frame and index Jhor to joint on the horizontal beam.

Statistical mo-ment of the jointstiffness

E[YJvert ] (Pa) δ[YJvert ] E[YJhor ] (Pa) δ[YJhor ]

24 109 0.09 73.5 109 0.22Gamma distri-bution parame-ters

αγ1 βγ1 αγ2 βγ2

123 0.0081 20.7 0.0484

Table 1: Identification of the parameters of the joints stochastic models (Jvert and Jhorcorrespond to joints respectively on vertical and horizontal beams of the frame)

4.2 The SMR approach applied to a single bolt frame

The parameters of the random stiffness models have been identified using the SMR ap-proach. The SMR results quality is validated a posteriori to ensure that, for the identifiedparameters, the vibrational behaviour of the frame belong to the validity domain of theSMR approach. Figure 4 presents the relative error of the first two statistical momentsfor the first ten random eigenfrequencies of the frame with only one bolt. This relativeerror is computed using as reference results from a 10 000 draws Monte Carlo simulation.The expectation is estimated with an error less than a percent for both refinement levelof the SMR approach. The standard deviation is estimated with a maximal error of 2.1%with SMR1 and less than a percent with SMR2. This can be explain by the relatively lowvariability of the random stiffness, indeed, its coefficient of variation is less than 0.1.

These results shows that SMR1, the first refinement level of the SMR approach, istotally sufficient to characterise the first ten random eigenfrequencies of a frame with asingle bolted joint on a vertical beam. The same conclusions are obtained for the case ofa frame with a single bolted joint on the horizontal beam. SMR1 is then used to simulatethe vibrational behaviour of frame with only one bolt taking into account the variabilityof the joint stiffness.

In the remainder of this paper, we choose to present the stochastic FRF of the framesimulated with the SMR approach as follows: (1) the mean FRF is plotted with full line,it is computed using the expectation of the random eigenvalues and the deterministiceigenvectors of the frame ; (2) the envelope curve arround the mean FRF is obtainedusing 1000 drawn of the independant random stiffnesses.

Using this representation, the experimental and simulated variability of the frame withonly one bolt on its vertical beam are presented Figure 5. It can be noticed that the simu-lated stochastique behaviour of the frame well fit the variability observed experimentally.A shift of the mean of some eigenfrequencies is however observed for highest frequencies.

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1 2 3 4 5 6 7 8 9 100.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

Rela

tive

erro

rSMR1 expectationSMR2 expectationPert2 expectation

1 2 3 4 5 6 7 8 9 10Eigenfrequencies

0.000

0.005

0.010

0.015

0.020

0.025

Rela

tive

erro

r

SMR1 standard deviationSMR2 standard deviationPert2 standard deviation

Figure 4: Relative error of the estimation of the statistical moments of the first ten randomeigenfrequencies of the single bolt frame

This drawback is due to the chosen finite element model of the frame as this misfit is alsoobserved for a monolithic frame. Equivalent results has been obtained for the case of aframe with only one bolt on its horizontal beam. As the conclusions drawn from theseresults are similar, this second case would not be developped here.

0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300

−100

−50

0

Frequency (Hz)

FRF

Experimental FRFFRF SMR1

Figure 5: Comparison of the experimental and simulated FRF of a single bolt frame

4.3 Application to multi-bolt frame

A stochastic model of each configuration of bolted joint belonging to the studied framehas been identified and validated. These models allow us to simulate the behaviour offrames with more than one bolt. Lets first study the capacity of the identified stochastic

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(a) (b)

Figure 6: Multi-bolt frame configurations

model of joint to robustly characterise the vibrational behaviour of a frame with twobolted joints as define Figure 6a. The variability of the FRF of this frame has beencharacterised experimentaly considering three tightening levels for each M8-type jointof the frame and two excitation magnitudes. The bolt tightening are qualified as verylow preload (a hundred Newtons), medium (≃ 1800N) and high preload (≃ 6000N).The highest preload corresponds to a limit value ensuring no plastification at the jointinterface. Figure 7 presents the experimental variability of the frame and the simulatedstochastic FRF obtained using the SMR approach and the identified stochastic modelsof joint. Both refinement levels of the SMR approach have been computed here. Thedifference between the simulated stochastic FRF obtained with SMR1 and SMR2 is quitelow which suggests that SMR1 is again sufficient for this case. Once more the stochasticsimulation of the frame allows to forecast the variability of this frame configuration. Ashift between experimental and simulated nominal eigenfrequencies at high frequency isstill observed and can be attributed again to the limitations of the finite element modelof the frame.

0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300

−100

−50

Frequency (Hz)

FRF

Expe FRFFRF SMR1FRF SMR2

Figure 7: Comparison of the experimental and simulated FRF of a frame with two bolts

Finally, the stochastic vibrational behaviour of a fully bolted frame as presented Fig-ure 6b is studied. The variability of the vibrational behaviour of this frame has beencharacterised experimentaly. The bolt tightening and the excitation magnitude are cho-

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0 100 200 300 400 500 600 700 800 900 1,000 1,100 1,200 1,300

−100

−50

Frequency (Hz)

FRF

Expe FRFFRF SMR1FRF SMR2

Figure 8: Comparison of the experimental and simulated FRF of a fully assembled frame

sen again as parameters of the study. Two tightening levels corresponding to low andhigh preload as previously defined and two excitation magnitudes are considered. Thestochastic behaviour of the frame is simulated using the SMR approach and the identifiedstochastic models of joint. Experimental and simulated results are presented Figure 8.

As observed in the case of the frame with two bolts, the simulation results show thataveraged eigenfrequencies are slightly over-everstimated. Statics behaviour of the frameremains indeterminate and initial stress state of the joints could differ from the initialstate of the joint used for the identification procedure.

5 Conclusion

In this work, we propose a straightforward approach to identify a stochastic model of abolted joint. This approach first identify a ”just sufficient” nominal model of a bolted jointbased on experimental observations. This model is then extended to take into accountthe variability of the joint behavior by defining the parameters of this nominal model asrandom variables. Finally, the parameters of the new stochastic model are identified onthe basis of the maximum likelihood principle using the SMR approach.

The identified joint model is then used to characterise the vibrational behaviour of amulti-bolted structure. Although the just sufficient nominal model (on which the stochas-tic model is based) seems to be sensitive to the bolt tightening procedure, the proposedapproach allows to tackle the variability of the real structure using a pragmatic andstraightforward procedure.

This approach has been implemented considering the case of an assembled structurebut it can be easily extended to other structures, as exemple to multi-material structuresor glued structures.

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