A low rank update theory for the vibration analysis of uncertain spot welded structures

R.O. De Alba, B.R. Mace, N.S. Ferguson and C. Lecomte Institute of Sound and Vibration Research University of Southampton, Southampton SO17 1BJ, UK email: rda@isvr.soton.ac.uk

Abstract Spot-welded structures contain inherent variability in location and/or stiffness due to the complexity of the manufacturing process. Therefore, an analysis that includes the uncertainty generated in the joints will provide a range of response predictions, adding more value to the design process compared to deterministic results. Finite element (FE) analysis is frequently used in conjunction with Monte Carlo simulations to predict the variability in the vibration response of assembled structures. However, this is usually computationally expensive, especially for large scale models. In this paper the computational time of such analyses is reduced using an analytical update theory in the component mode synthesis framework. This improvement is achieved not only because the FE model size is reduced, but also because when the response of the assembled structure is calculated by updating the response of the unassembled system. In addition, multipoint constraints are used to spatially locate the connections anywhere in the model. It is found that the proposed approach reduces the computational time when compared to other reduction methods.

1 Introduction

The spot weld is one of the most important structural joints in the automotive industry; a vehicle body typically contains thousands of spot-welds. Each spot-weld can have variations in its geometric and physical properties due to location of the electrode, variations in the electrode contact region, welding duration, electrical current changes, electrode-surface contact characteristics, thermal conditions, etc. These variations lead to variations in the dynamic properties of the joint and therefore in the resulting overall dynamic behaviour of the built-up structure.

Therefore, when the dynamic behaviour of the built-up structure is being analysed including the uncertainty in the spot welds, the subsequent numerical simulations provide a range of response predictions which can add more value in the design process compared to deterministic estimations.

The finite element (FE) method is often used to perform the dynamic analysis of mechanical structures and a deterministic response is obtained as a function of a particular set of input parameters [1]. To include variability in the model, Monte Carlo simulation (MCS) can be used. Nonetheless, as the accuracy of this method depends on the number of repeated analyses used during the simulation [2], the computational effort is high, especially for large scale models.

This paper concerns an efficient method to analyze the uncertainties in the vibrational behaviour of built-up structures with variation in the location of the spot welds. This method applies an update theory within the component mode synthesis (CMS) framework.

CMS is an effective method in which the dynamic behaviour of each component is described in terms of the modes of the component. When the higher frequency modes are truncated a reduction in size is achieved. Another advantage arises in substructuring, where it may be cheaper to solve the eigenvalue problems of a number of the components and of the assembled reduced global system compared to solving the complete global eigenvalue problem [3].

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One of the most accurate and frequently used CMS methods is the Craig-Bampton method [4]. When this method is applied, the interface degrees of freedom (DOFs) are kept and the remaining DOFs are reduced. Then the stiffness of the connections can be added to the model using a set of equations called multipoint constraints (MPCs); doing so has a clear advantage since the connection can be spatially located anywhere in the model.

Combining CMS with MPC joints, the response of the system can be evaluated for many joint locations using the same modal representation of the substructures. However, since the interface DOFs are not reduced, if the number of degrees of freedom involved in the connection is large, then the CMS size reduction is less efficient. To further improve the efficiency of this analysis, a low-rank update theory is applied; here the receptance matrix is first calculated for the unassembled system. Then the response of the assembled system is calculated by updating this response with the effect of the connection. Here the efficiency is increased in two ways: (i) most of the CMS dynamic stiffness matrix is inverted only once during the MCS, (ii) the transformation from CMS coordinates to physical coordinates is also calculated only once during MCS.

Following this introduction, in section 2 is a description of how the Craig-Bampton method can be applied to structures with uncertainties in the joints. In section 3 a brief explanation on how the joint information can be added to the CMS model using MPCs is given. In section 4 the analytical update theory is described and the methodology to apply this theory in the CMS framework is explained. In section 5 numerical examples are presented and the results are discussed. Finally the conclusions are given in section 6.

2 CMS applied to structures with uncertain point connections

In the Craig-Bampton method implemented here [4], the component normal modes are calculated with the interface between the components held fixed. These modes are further augmented by static constraint modes to improve convergence, yield the exact static solution and assure the compatibility between components, facilitating coupling of structures.

To use CMS with sub-structures assembled with an MPC connection, first the system is divided into components. For example, an assembly of two overlapped plates can be divided into two components: (1) the upper plate and (2) the lower plate.

The second step is to constrain the interface DOFs, ccu , in each component. Here, the interface DOFs are

all the DOFs within the area in which the location of each of the n point connections varies. The group of elements in which each connection might lie is called a patch, then every DOFs in the n patches are constrained. For example, Figure 1 shows the interface DOFs for a system with three connections with each patch comprising an array of 4x4 elements, i.e. the location of each connection might lie anywhere within sixteen elements. The normal modes for each component i are calculated, but only some of these

modes are kept in the reduced normal mode matrix ikΦ ; here a reduction in size is achieved.

Subsequently, the complete set of static constraint modes icΨ is evaluated and assembled in the

component mode matrix

i i ik c B Φ Ψ (1)

The component modal mass iμ and stiffness iκ matrices are given by

i iT i iμ B M B (2)

i iT i iκ B K B (3)

where iM and iK are the ith component mass and stiffness matrices. The next step is component synthesis. To be able to later assemble the component modal matrices for components 1 and 2 using

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elastic and non rigid connections, the transformation matrix S that relates all the component modal coordinates becomes

(1) (1)

(1) (2)

(2) (1)

(2) (2)

0 0 0

0 0 0

0 0 0

0 0 0

k k

c k

k c

c c

Iq q

Iq qq Sp

Iq q

Iq q

(4)

where ( )ikq , ( )i

cq are the kept modal coordinates and the constraint coordinates respectively and p are the

linearly independent component modal coordinates. The global mass and stiffness matrices in the global coordinates RM and RK respectively are given by,

(1)

(2)

0

0T

R

μM S S

μ (5)

and

(1)

(2)

0

0T

R

κK S S

κ (6)

resulting in

(1) (1)

(2) (2)

(1) (1)

(2) (2)

0 0

0 0

0 0

0 0

kk kc

kk kcR T

kc ccT

kc cc

I m

I mM

m m

m m

(1)

(2)

(1)

(2)

0 0 0

0 0 0

0 0 0

0 0 0

kk

kkR

cc

cc

Λ

ΛK

k

k

(7)

where (1)kkΛ and (2)

kkΛ are diagonal matrices of eigenvalues of component 1 and component 2 respectively,

I is the identity matrix of appropriate size and

(1)

(2)

0

0cc

cc

cc

mM

m and

(1)

(2)

0

0cc

cc

cc

kK

k (8)

are the mass and stiffness matrices for the interface DOFs ccu .

Figure 1: Part of FE mesh of a plate with three point connections allowed to lie within the

highlighted areas. Possible joint location, constrained interface DOFs : w , x , y .

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3 MPC JOINT MODELS

Commonly in FE models, point connections such as welds, rivets and bolts are represented by two-noded elements (e.g. beams or springs with lumped masses). The parameters of these simple elements represent the mass and stiffness characteristics of the real joint, and therefore their influence on the rest of the structure. This simple connection can be connected to the substructures in two different ways: (1) by directly connecting the joint nodes to nodes in the substructures (node-to-node connection) and (2) using interpolation elements or MPCs to connect the joint nodes to the substructures. Examples of the last type of connections are the area contact model 2 (ACM2) [5] and the CWELD element [6] in MSC/NASTRAN [7].

The node-to-node connection requires coincident meshes: if the location of the joint changes, then the mesh of both surfaces needs to be modified. This is a clear disadvantage, especially if the analysis needs to be repeated in a MCS for many different positions. In contrast, when interpolation elements or MPCs are used, the existing meshes of the connected substructures can be used for any joint location.

The ability of the MPC connection to be located anywhere in an element was previously validated [8]. In this earlier report, the example of two coupled plates was considered. The plates were modelled using two different formulations: thin rectangular elements [1] and heterosis plate elements [9]. The results showed that the MPC connection is not accurate when thin plate elements are used, due to the non-conforming formulation. In contrast, when the heterosis element was used the results showed that the MPC connection is as accurate as the node-to-node connection. Only discretization errors due to the FE modelling are present, having the same frequency limitation as any FE model [8].

3.1 MPC elastic connection

The MPC elastic connection in this study consists of spring elements connected to the substructures using

MPCs. The model is then a function of the position of the connection points ,x y as shown in Figure 2.

In the case of thin plate substructures with out-of-plane DOFs w , x and y , the elastic element contains

a translational stiffness, wK and two rotational stiffnesses, xK and yK , as shown in Figure 2. The

stiffness matrix of each point connection i in the local connection DOFs iu is

1

1

1

i

i i i iw i x i y ix

iy

w

diag K K K

F K u (9)

The DOFs iu can be related to the nodal DOFs of the substructures using MPCs. The MPC can be defined

as the set of equations that relate each of the connection DOFs iu to the interface DOFs ccu , i.e.

( , )i i i i ccx y u = G u (10)

where iG is the matrix of coefficients of the MPC equations. In this case iG is populated using the

element shape functions. In doing so, the relationship between iu and ccu is made consistent with the FE

formulation and is a function of ( , )x y .

There are many methods available in the literature to apply MPCs to an FE model, e.g. static condensation [10], augmented Lagrange multipliers, Lagrange elimination etc. In this paper, static condensation is used.

In order to add all n connections to the stiffness matrix RK , a global connection matrix in iu coordinates

is defined as

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1 2 3 ndiag K K K K K (11)

Then a transformation matrix Γ that relates the iu to ccu and imposes the coupling conditions between

plates can be written as

Γ G G (12)

where

1 2 3 ndiagG G G G G (13)

A second transformation matrix Ξ is defined to transform from ccu to p coordinates as

0 0Ξ

0 I (14)

where 0 are zeros matrices of appropriate size. Then the stiffness matrix in p coordinates containing n

connections is

T K Γ K Γ (15)

and the reduced matrix RK containing the connection information K can be calculated as

R R KK K K (16)

It can be observed that when the location of any or all of the point connections changes, only the terms in

the matrix K change. This means that to obtain the reduced mass and stiffness matrices RM and RKK

for any different point connection location, only the matrices iG in equation (13) need to be re-calculated

and equations (15) and (16) re-evaluated, offering a reduction in computation time.

Finally, the steady state time harmonic response in physical coordinates at nodal point r due to harmonic excitation at nodal point e can be evaluated as

1* 2 *1r T e Tre R RU i

KB S K M S B (17)

where *rB and *eB are the thr and the row of B , and is the loss factor assuming constant hysteretic

damping. Alternatively a modal solution can be performed from an eigensolution of RM and RKK , then

the response is calculated using modal summation.

The previous method is an efficient way to calculate the response of a FE system with uncertainties in the location and/or stiffness of the connections. MPCs are combined with CMS; MPC connection is accurate when modelling the change in the location of the joint, while CMS gives a sub-structuring framework and a reduction in the number of the DOFs of the model; combining both, the response of the system can be evaluated for any joint location using the same reduced modal representation of the substructures.

The computational cost of equation (17) is greatly associated with the number of interface DOFs especially if this number is greater than the number of kept fixed interface normal modes. Unfortunately in this method there is no reduction of these DOFs as can be observed in equation (7). In such cases there are two possible options to improve the efficiency of this method: (i) To reduce the number of interface DOFs by introducing characteristic constraint modes [11] as applied in [12]; (ii) To apply a low-rank theory in the frequency domain as explained in the following section.

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wK yKxK

(1)x

(1)y

(2)x

(2)y

(1)y

(1)x

(2)y

(2)x

F

w

Figure 2: MPC elastic connection for bending analysis.

4 A low rank update theory in the frequency domain

In this section a method to improve the speed of the estimation of the response in physical coordinates U of a system with uncertainties in the point connections properties is described. This method uses a low rank update approach in the CMS framework. The low rank update is based on the rank-one update theorem developed by Lecomte in [13].

The response of a nominal system, i.e. the unassembled structure in p coordinates in the frequency domain

is calculated as

1x A F (18)

where A is the dynamic stiffness and is given by

21R Ri A K M (19)

When a disturbance is added to the nominal system, equation (18) can be written as

x KA D F (20)

where x K is the updated response and D is the dynamic stiffness of the disturbance. It is

assumed that D is a low rank matrix and can be expressed as the outer product of given left and right

vectors ld and rd as

Tl r D d d (21)

In the current analysis, the disturbance is given by the connection stiffness matrix K in equation (15). Hence

D K (22)

and

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T Tl d Γ (23)

Tr

d K Γ (24)

Substituting equation (21) into equation (20) and some manipulation leads to

1 Tl rx x

K KA F d d (25)

where x K occurs on both sides of the equation. To solve this equation, expression (25) can be

premultiplied by Trd and manipulated in order to obtain

11 1T T T

r r l rx

Kd I d A d d A F (26)

Substituting this expression into the right hand side of equation (25) gives

11 1 1 1T T

l r l rx

K A F A d I d A d d A F (27)

Finally, the response in p coordinates can be transformed into physical coordinates as

11 1 1 1* *r T T T e T

re l r l rU

B S A F A d I d A d d A F S B (28)

Substituting equations (23) and (24) into this expression and some manipulation leads to

1* *

11 1 1* *

r T e Tre

r T T T T T e T

U

B SA FS B

B SA Γ I K Γ A Γ K Γ A FS B (29)

where the first term represents the transfer function of the nominal unperturbed system. The second term represents the effect of the perturbation on the transfer function. The first factor represents the transfer function between the excitation DOFs to the u DOFs belonging to the first plate of the nominal system; the second factor represents the full transfer function between the first and second u DOFs of the disturbed system and finally the third factor represents the transfer function between the u DOFs in the second plate to the response DOFs of the nominal system, this is illustrated in Figure 3.

This means that this method can also be applied using measured transfer functions of the unconnected plates.

In order to improve the efficiency of equation (29), when applied in a MCS, it can be re-arranged as

1* *

11 1 1* *

r T e Tre

r T T T T T e T

U

B SA FS B

B SA Γ I K Γ A Γ K Γ A FS B (30)

When equation (30) is used to calculate the response of a system with uncertain point connections, the

efficiency is improved in the following ways: (i) 1 A , , B and S are invariant to changes in the

connections, therefore they are calculated only once during a MCS. Hence, the first term and the first and

third factors in the second term are calculated only once in the MCS. (ii) 1 A is calculated in the

CMS co-ordinates, therefore the size of the matrices are smaller in comparison to the original matrices of the system. (iii) Since the first and third factors in the second term are vectors, the final multiplication is computationally cheap. (iii) The computational effort to invert the second factor of the second term is small since the size of the matrix to be inverted is only 3n for out of plane analyses.

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Figure 3. Graphical representation of the second term in equation (29): first factor;

second factor; third factor.

5 Numerical example

The numerical example is a system of two overlapped plates with free edges with five elastic connections as shown in Figure 4. The plates are modelled using a mesh of 22x22 and 22x20 heterosis elements [9], the offset in the plates was ignored, (i.e. both plates share the same centerline). In order to avoid symmetry in the x and y direction, the first plate is 10% wider and 10% thicker than the second plate. The properties for each plate are given in Table 1.

The stiffnesses values of all five connections are 121 10 N/mwK and 41 10 N/radx yK K .

As a baseline, the connections are located at the midpoint of the area in which the position varies, represented as the shaded elements in Figure 4.

Using this configuration, the transfer mobility from coordinate (0.0836, 0.0364) in plate 1 to coordinate (0.0836, 0.0836) in plate 2 was evaluated using a full modal solution and two different approximations: (1) CMS in which only the first 30 normal modes of each component were kept and 710 constraint modes, here the DOFs are reduced from 10686 to 770; (2) CMS and truncation of the characteristic constraint modes [12], in which only 16 modes were kept from a total of 710 constraint modes; (3) CMS matrices solved using a low rank update theory as in equation (30).

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Figure 4: Two overlapped simply supported plates assembled with five elastic point connections.

Plate (1) 0.23 0.2 0.0022 7860 2.07E+11 0.3

Plate (2) 0.23 0.182 0.002 7860 2.07E+11 0.3

mxL

m

yL

m

h 3kg/m

2N/m

E

Table 1: Properties of the assembled plates.

The proposed approach is slightly more accurate since it does not involve further approximations as can be observed in Figure 5, even though all methods give good agreement with the full modal solution, especially below 1000Hz.

The low-rank update approach leads to nearly a 98% reduction in the computational time when compared to a full modal solution for this example (Figure 6). The computational time for this approach mainly depends on the number of connections since it defines the size of the inverse operation, but it is also determined to a lesser extent by the number of interface DOFs because this number defines the size of the rest of the matrix operations (i.e. multiplications and additions) as can be seen in equation (30). On the other hand, when the characteristic constraint method is used to reduce the size of the problem, the computational time depends almost entirely on the number of interface DOFs, since the most expensive operation performed when this method is used is the eigenvalue solution of the mass and stiffness matrices of the constrained DOFs.

0.20 m

0.23 m

0.182 m

0.02 m

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The dependence the computational time on the number of interface DOFs of both methods is shown in figure 6. The proposed method scales better with the number of interface DOFs, N, growing for large N as N2.25 as compared to N3.12 for the CMS and characteristic constraint modes method. Therefore, the proposed approach reduces the computational time by factor of approximately N when compared to the characteristic constraint modes method [12]. The statistics of the natural frequencies and mode shapes can be calculated at a very low cost following the characteristic constraint modes method. On the other hand, if the update theory approach is followed, the calculation of the natural frequencies and modeshapes would lead to a different analysis, (i.e. would require a further modal analysis) and a considerable increase in the computational cost.

Being the most time efficient method to calculate the FRFs in a MCS, the method proposed in this paper is then applied for an uncertainty analysis. In order to estimate the variability in the vibration response of this system due to missing or broken welds when the position of the remaining welds is uncertain, a MCS with

500 samples is used to estimate the envelope of the transfer mobility. The ,i ix y coordinates of each of

the i point connections are assumed to be independent Gaussian random variables with their means

,xi yi located at the baseline position and standard deviation of the position given by

0.25xi xS and 0.25yi yS (31)

where xS and yS are the lengths of the x and y sides of each element. In this case, the samples in which

the coordinates of the point connection lay outside the element were discarded (< 0.1% of the sampled locations).

The stiffness of a missing or broken weld is set to zero. In doing so, the inverse of the unperturbed system and the CMS matrices can be retained and do not need to be re-calculated. The 5-95% response envelope is computed for five cases; in each case one of the five spot welds is absent. This envelope is then compared to the baseline condition, i.e. no absent connections and all five spot welds with random position. The results are plotted in Figure 7.

102 103-90

-80

-70

-60

-50

-40

-30

-20

Frequency (Hz)

dB (

re 1

m/N

s)

Figure 5: Transfer mobility of the baseline configuration: full modal solution; - CMS; CMS and characteristic constraint modes; CMS and update

theory.

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102

103

10-2

10-1

100

101

102

103

Co

mpu

tatio

nal

tim

e (

s)

Number of interface DOFs Figure 6: Computational time as function of the number of interface DOFs: CMS and

characteristic constraint modes; CMS and update theory; 7 2.251 7 10t N

9 3.122 8 10t N .

When any spot weld is absent, the translational and rotational stiffness of the joint is reduced. This affects especially the first flexural mode since the strain is concentrated in the overlap area. Then, the bounds of the first natural frequency are extended, especially when the outer connections are broken. This effect can be observed in Figure 7.

For the second natural frequency and first torsional mode, this frequency depends on the effective rotational stiffness of the joint. In this example, this stiffness depends upon the area covered by the spotweld. This area is only reduced when the outer spot welds are absent, this effect can be observed in Figure 7 a) and 7 e), where the bounds of the second natural frequency are lowered. On the other hand, when the inner spot welds are broken, the second resonance is not significantly affected as shown in Figures 7 b) - 7 d).

The third natural frequency is described as a second bending mode. In this mode, the strain energy is higher in the middle of the individual plates and much lower in the overlapped area. Hence, this natural frequency is affected very little when any of the connections is absent.

At higher frequencies, the response is not affected when the inner spot welds are missing. However, when the outer spot welds are missing there is a general change in the response envelopes as can be observed in Figure 7 a) and 7 e). The affected modes are presumably modes with significant torsional contributions which are affected in the same manner as the first torsional mode.

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dB (

re 1

m/N

s)

Frequency (Hz)102

103

-20

-30

-40

-50

-60

-70

-80

-90

dB

(re

1m

/Ns)

Frequency (Hz)102

103

-20

-30

-40

-50

-60

-70

-80

-90

dB

(re

1m

/Ns)

Frequency (Hz)102

103

-20

-30

-40

-50

-60

-70

-80

-90

dB (

re 1

m/N

s)

Frequency (Hz)102

103

-20

-30

-40

-50

-60

-70

-80

-90

dB

(re

1m

/Ns)

Frequency (Hz)102

103

-20

-30

-40

-50

-60

-70

-80

-90

a)

b)

c)

d)

e)

Figure 7. 5%-95% response envelopes for the magnitude of the mobility using MCS with 500 samples: a) first; b) second; c)third; d)fourth; e)fifth spot weld being absent; baseline

absent spot weld.

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6 Conclusions

In this paper multipoint constraints (MPC) in combination with component mode synthesis (CMS) were used to model uncertainties in the joint location in a finite element (FE) model. In order to further improve the solution, the CMS matrices are solved using a low-rank update theory. When these approaches are combined, the size of the system matrices is reduced. The response of the system can be evaluated for any joint location using the same modal representation of the substructures and the response of the system is calculated updating the inverse of the dynamic stiffness matrix of the unperturbed system.

The vibration response was calculated using these approaches for different connection positions and compared to a full modal solution. The predictions obtained gave a good agreement and the computational time was reduced by approximately 98% when compared to the full modal solution. When compared to the characteristic constraint mode method [12], this approach leads to a reduction in the computational time t at a rate of t2/3.

Finally, this approach was used in a Monte Carlo simulation (MCS) to evaluate the variability in the vibration response due to missing or broken connections and/or uncertainty in the location of the spot-welds in a model of two plates with five spot welds. Results show that for the example considered, when any of the inner spot welds is missing and the location and size of the remaining connections is uncertain the vibration responses lie approximately within the bounds of the case in which all the connections are present. On the other hand, when any of the outer connections are absent the variability in the vibration response is greater.

Acknowledgements

Ricardo de Alba thanks CONACYT for sponsoring his PhD programme of study.

References

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[2] G.I. Schueller, B.E. Spencer, A. Sutoh, T. Takada, W.V. Wedig, S.F. Wojtkiewicz, I. Yoshida, B.A. Zeldin, and R. Zhang, A State-of-the-Art Report on Computational Stochastic Mechanics, Probabilistic Engineering Mechanics, vol. 12, 1997, pp. 197-321.

[3] L. Hinke, F. Dohnal, B.R. Mace, T.P. Waters, and N.S. Ferguson, Component mode synthesis as a framework for uncertainty analysis, Journal of Sound and Vibration, vol. 324, 2009, pp. 161-178.

[4] R.R. Craig Jr. and M.C. Bampton, Coupling of substructures for dynamic analyses, AIAA Journal, vol. 6, 1968, pp. 1313-1319.

[5] D. Heiserer, M. Charging, and J. Sielaft, High Performance Process Oriented Weld Spot Approach, 1st MSC Worldwide Automotive User Conference, Munich 1999.

[6] J. Fang, C. Hoff, B. Holman, F. Mueller, and D. Wallerstein, Weld modelling with MSC.Nastran, 2nd MSC Worldwide Automotive User Conference,Dearborn, MI. 2000.

[7] MacNeal-Schwendler Corporation., MSC/NASTRAN user's manual, Los Angeles, Calif., 2004.

[8] R. De Alba, N. Ferguson, and B. Mace, A multipoint constraint model for the vibration of spot welded structures, ISVR Technical Memorandum, No 982, June 2009. University of Southampton.

[9] T. Hughes and M. Cohen, The `heterosis' finite element for plate bending, Computers and Structures, vol. 9, 1978, pp. 445-450.

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[10] M.S. Shephard, Linear multipoint constraints applied via transformation as part of a direct stiffness assembly process, International Journal for Numerical Methods in Engineering, vol. 20, 1984, pp. 2107-2112.

[11] M.P. Castanier, Y.C. Tan, and C. Pierre, Characteristic constraint modes for component mode synthesis, AIAA Journal, vol. 39, 2001, pp. 1182-1187.

[12] R.O. De Alba, B.R. Mace, and N.S. Ferguson, Prediction of response variability in uncertain point connected structures using component mode synthesis and characteristic constraint modes, Proceedings of the 10th International Conference on Recent Advances in Structural Dynamics 12-14 July, Southampton, UK, paper no. 56.

[13] C. Lecomte, J. Forster, B. Mace, and N. Ferguson, Bayesian inference for uncertain dynamic systems, Proceedings of the 10th International Conference on Recent Advances in Structural Dynamics 12-14 July, Southampton, UK, paper no. 162.

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