modal calculation by dynamic under-structuring clas []

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Titre : Calcul modal par sous-structuration dynamique clas[...] Date : 27/02/2015 Page : 1/55Responsable : CORUS Mathieu Clé : R4.06.02 Révision :

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Modal calculation by classical dynamic under-structuring

Summary:

This report presents the theoretical bases of the methods of calculating by modal synthesis. We begin with thedescription of the transformation of RITZ and the methods of modal recombination which result from this. Then,we present the modal synthesis which uses the classical techniques of under - structuring and modalrecombination.

We present, first of all, the techniques of classical dynamic under-structuring of CRAIG - BAMPTON and MAC- NEAL.

Computational tools by modal synthesis implemented in Code_Aster, then reviewed. One will detail modalcalculation by under-structuring then the harmonic calculation of answer and finally the temporal calculation ofanswer.

One presents then dynamic condensation by dynamic macronutrients in static under-structuring.

The last section of this document presents the principle of the calculation of residue in effort, making it possibleto evaluate the quality of a scale model for a given calculation. One also details the methodology developed forthe indicator of quality for modal calculation for the model generalized.

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Contents1 Introduction........................................................................................................................................... 5

2 Modal synthesis.................................................................................................................................... 6

2.1 Transformation of RITZ..................................................................................................................6

2.2 Modal recombination..................................................................................................................... 7

2.3 Modal synthesis............................................................................................................................. 8

2.3.1 Normal modes...................................................................................................................... 9

2.3.2 Static deformations.............................................................................................................10

2.3.3 Harmonic deformations.......................................................................................................11

2.3.4 Deformations of interface reduced or “modes of coupling”.................................................11

2.4 Conditions of connection between substructures.........................................................................12

2.4.1 Introduction.........................................................................................................................12

2.4.2 Case of the compatible grid of interface.............................................................................12

2.4.3 Case of the incompatible grid of interface..........................................................................14

2.4.4 Conclusion.......................................................................................................................... 16

2.5 Elimination of the linear constraints.............................................................................................16

2.5.1 Case of linearly independent constraints............................................................................17

2.5.2 Case of redundant constraints............................................................................................17

2.5.3 Case of a significant number of constraints........................................................................18

3 Methods of classical dynamic under-structuring.................................................................................20

3.1 Introduction.................................................................................................................................. 20

3.2 Method of Craig-Bampton............................................................................................................20

3.3 Method of Mac Neal..................................................................................................................... 21

3.3.1 First case............................................................................................................................ 24

3.3.2 Second case....................................................................................................................... 24

3.4 Method of the modes of interface (method known as reduced)...................................................26

3.4.1 Introduction.........................................................................................................................26

3.4.2 Definition of the modes of interface....................................................................................26

3.4.3 Calculation of the modes of interfaces...............................................................................26

3.4.4 Creation of the reduced macronutrients.............................................................................27

3.5 Implementation in Code_Aster....................................................................................................28

3.5.1 Study of the substructures separately.................................................................................28

3.5.2 Assembly............................................................................................................................ 29

4 Modal calculation by under classical dynamic structuring..................................................................30

4.1 Introduction.................................................................................................................................. 30

4.2 Dynamic equations checked by the substructures separately.....................................................30

4.3 Dynamic equations checked by the total structure......................................................................31






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4.4.2 Assembly and resolution.....................................................................................................32

4.4.3 Restitution on physical basis..............................................................................................32

5 Harmonic response by under classical dynamic structuring...............................................................33

5.1 Introduction.................................................................................................................................. 33

5.2 Dynamic equations checked by the substructures separately.....................................................33

5.3 Dynamic equations checked by the total structure......................................................................35



5.4.2 Assembly and resolution.....................................................................................................36

5.4.3 Restitution on physical basis..............................................................................................36

5.5 Conclusion................................................................................................................................... 36

6 Transitory response by under classical dynamic structuring...............................................................37

6.1 Introduction.................................................................................................................................. 37

6.2 Transitory calculation by projection on the basis of substructure.................................................37

6.2.1 Dynamic equations checked by the substructures separately............................................37

6.2.2 Dynamic equations checked by the total structure.............................................................38

6.2.3 Double dualisation of the boundary conditions...................................................................38

6.2.4 Treatment of the matrix of damping...................................................................................39

6.2.5 Treatment of the initial conditions.......................................................................................39

6.3 Transitory calculation on a total modal basis calculated by under-structuring.............................40

6.3.1 Calculation of the clean modes of the structure supplements by under - structuring.........40

6.3.2 Dynamic equation checked by the total structure...............................................................41

6.4 Comparative study of the two developed methods......................................................................41



6.5.2 Assembly of the generalized model....................................................................................42

6.5.3 Calculation of the modal base of the complete structure and projection............................43

6.5.4 Resolution and restitution about physical base...................................................................43

6.6 Conclusion................................................................................................................................... 43

7 Dynamic condensation by dynamic macronutrients in static under-structuring..................................44

7.1 Introduction.................................................................................................................................. 44

7.2 Classical problem and principle of the method............................................................................44

7.3 Methods of condensation per complete or reduced base............................................................45

7.3.1 Method of calculating by complete base............................................................................45

7.3.2 Method of calculating by modal reduction..........................................................................47

8 Indications for the back testing of the quality of the scale model.......................................................49

8.1 Case general for a calculation on a scale model.........................................................................49

8.2 Typical case of the calculation of the modes of a generalized model..........................................49

8.2.1 Presentation........................................................................................................................ 49

8.2.2 Calculation of the indicators and associated corrections....................................................50




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8.2.2.1 Definition of the coupled problem...........................................................................50

8.2.2.2 Variation with balance.............................................................................................51

8.2.2.3 Corrections associated with the residual efforts.....................................................52

8.2.2.4 Corrections associated with differentials displacement with interfaces..................52

9 Bibliography........................................................................................................................................ 54

10 Description of the versions of the document....................................................................................55




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1 Introduction

In front of the complexity of the mechanical structures, often made up of an assembly of severalcomponents, the digital or experimental methods classical of vibratory mechanics appear expensive,sometimes even unusable. In perfect coherence with the modular organization of the great projects,the methods of under-structuring seem the most effective means to carry out the vibratory study of thewhole starting from the dynamic behavior of the components [bib4].

In this report, we present, first of all, the theoretical bases of the methods of modal synthesis. Theyassociate techniques of under-structuring and modal recombination. Each substructure is representedby a base of projection made up of dynamic clean modes and static deformations with the interfaces.

Then, we present the two techniques of calculation per classical under-structuring, established inCode_Aster [bib5]: methods of Craig-Bampton, Mac Neal and modes of interfaces. They arecharacterized primarily by the use of different bases for the substructures.

The case specific of the modal analysis of structures to cyclic symmetries by dynamic under-structuring is approached in the document [R4.06.03].

General notations:

m : Maximum pulsation of a system ( rad.s−1 ),

M : Matrix of mass resulting from modeling finite elements,

K : Matrix of rigidity resulting from modeling finite elements,

q , q ,q : Vector of displacement, speed and acceleration resulting from modeling finite elements,

f ext : Vector of the forces external with the system,

f L : Vector of the bonding strengths applied to a substructure,

: Matrix containing the vectors of a base of projection organized in column,

B : Matrix of extraction of the degrees of freedom of interface,

L : Matrix of connection,

T : Kinetic energy,

U : Deformation energy,

Id : Matrix identity,

: Diagonal matrix of the generalized rigidities associated with the normal modes,

R e : Matrix of residual dynamic flexibility,

R e0 : Matrix of residual static flexibility.

M=T M : is the matrix of generalized mass,

C=T C : is the matrix of generalized damping,

K=T K : is the matrix of generalized rigidity,

{f ext}o=T{ f ext}o

: is the vector of the generalized external forces applied

{f L}o=T{ f L}o : is the vector of the generalized bonding strengths applied,

, , : is the vector of generalized displacements, speeds and accelerations.

Note:

The exhibitor k characterize the sizes relating to the substructure S k and the generalized

sizes are overcome by a bar: for example Mk is the matrix of mass generalized of the

substructure S k .




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2 Modal synthesis

The dynamic under-structuring consists in determining the behavior of a structure starting from thevibratory characteristics of each one of its components ([bib3] and [bib4]). Methods implemented inCode_Aster, use simultaneously the classical techniques of modal recombination and dynamic under-structuring.These methods, although different from that of the finite elements, adopt an appreciably comparableapproach but introduce however an additional approximation. They reveal three crucial steps:

Stage 1: digital study of each component by the determination of their vibratory characteristics. Workconsists in identifying clean modes and static deformations by classical techniques of vibratorymechanics. If one compares each substructure to a super - element, this stage is similar to elementarycalculation.

Stage 2: connection of the substructures. One uses the previously given vibratory characteristics foreach component, and one takes account of their liaisonnement. This work constitutes the stage ofunder-structuring itself. It is connected with an assembly.

Stage 3: the resolution and a phase of increase makes it possible to obtain the solution sought in thephysical reference mark of the total structure.

2.1 Transformation of RITZ

The transformation of RITZ is the object of the reference material [R5.06.01]. We point out hisprinciple here. For the problem of the digital determination of the real clean modes of the system notdeadened associated with the structure, which we will indicate by clean modes, one is reduced to theresolution of the problem of minimization according to:

That is to say virtual displacement, one seeks: Min

12T .K−2 M .

of which the solution q check:

K− ² M.q=0 (2.1-1)

The method of RITZ consists in seeking the solution of the equation of preceding minimization onunder - space of the space of the solutions. Let us consider the matrix containing the vectors of thebase of the subspace in question, organized in columns. Restricted with this space of reduced size,the equation of minimization takes the following shape:

Minp=

12

pT .K−2 M. p

That is to say the required solution:

q= (2.1-2)

who checks:

K− ² M=0 (2.1-3)

where is the vector of generalized displacements, K=T K and M=

T M matrices ofgeneralized stiffness and mass are called respectively.

After having solved the system [éq 2.1-3], obtaining the clean modes in the physical base is doneusing the relation [éq 2.1-2]. The transformation of RITZ thus makes it possible to replace the problemwith the eigenvalues initial [éq 2.1-1] by a problem of comparable nature [éq 2.1-3], but of reducedsize. The new matrices of generalized rigidity and mass remain symmetrical.However, this transformation must be used with prudence. Indeed, the new base being incomplete, anapproximation is made on the level of projection: one speaks about modal truncation. The precision of




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the final result depends then on the choice of the basic vectors and the relative error due to thisreduction amongst unknown factors must be estimated.

2.2 Modal recombination

A classical use of the transformation of RITZ, is the dynamic analysis by modal recombination. It isusually used for the calculation of the response of a structure to an excitation low frequency. We willlimit ourselves here to the calculation of the answer to an excitation of a conservative structure. In thiscase, the finite element method enables us to be reduced to the following matric differential equation:

M qK q=f ext (2.2-1)

If one applies the transformation of RITZ, with as bases incomplete projection, the first clean modes ofthe structure, the relation [éq. 2.2-1] becomes:

M K =fext

(2.2-2)

where fext=

T fext

is the vector of the generalized forces.

The clean modes are orthogonal compared to matrices of mass and rigidity. The differential equation[éq 2.2-2] thus revealed diagonal matrices: the system then consists of uncoupled equations. Eachone of them is the equation of an oscillator to a degree of freedom of the type mass-arises whichreveals the mass, generalized rigidity and force relating to the mode j (respectively: m j , k j , f j ),to see Figure 2.2- has.

If one considers the transformation of RITZ [éq 2.2-2], on the level of a degree of freedom, one a:

q i=j ij j

where:

q i is i ème coordinate of the vector q ,

j is j ème coordinate of the vector ,

ij is the component of i ème line and of j ème column of the matrix .

It thus appears that the answer of the structure is expressed like the balanced recombination ofanswers of oscillators to a degree of freedom uncoupled. The transformation of RITZ makes itpossible, in this case, to define an equivalent diagram of the structure, which reveals the oscillatorswith a degree of freedom associated with the identified clean modes. Their stiffness and their massare generalized rigidities ( k j ) and generalized masses ( m j ) corresponding modes.




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Figure 2.2- has: Principle of the dynamic analysis by modal recombination.

Reserved primarily being studied which one wrongly describes as low frequencies1, the modalrecombination consists in using the properties of orthogonality of the clean modes of a structure tosimplify the study of its vibratory answer. In addition to the interest to decrease the order of the digitalproblem to solve, the transformation of RITZ into modal base, in this case, also makes it possible touncouple the differential equations and to release a physical interpretation of the got result. Accordingto the frequency of excitation, one will use a more or less truncated modal base. It is howevernecessary to estimate the truncation error to make sure of the validity of the result.

2.3 Modal synthesis

In a general way, the methods of modal synthesis consist in using simultaneously under - dynamicstructuring (cutting in substructures) and the modal recombination on the level of each substructure.Often confused, by abuse language, with the dynamic under-structuring, the modal synthesis is onlyone typical case of this one.

dynamic under-structuring consist in considering the displacement of a substructure in the overallmovement, as its response to the bonding strengths who connect it to the other components and to theexternal forces who are applied to him.

The modal synthesis means that one calculates this movement, on the level of each substructure, bymodal recombination. One thus uses a base of projection which characterizes each under - structure.Indeed, if the total structure is too important to be subjected to a modal calculation, dimensions of thesubstructures make it possible to carry out this work. The modal synthesis forces to study eachcomponent initially separately, in order to determine their base of projection.

In the continuation of this chapter, we present the types of modes and static deformations used in themethods of modal synthesis using the following simple example:

1 Beach of frequency on which the Eigen frequencies of the structure are distinguishable, generally first of astructure.




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Figure 2.3- has: Principle of the dynamic under-structuring.

The vector of the degrees of freedom of the substructure is characterized by an exponent who definesthe number of the substructure, and an index which makes it possible to distinguish the internaldegrees of freedom (index i ), degrees of freedom of border (index j ).

qk={q i

k

qjk}

One is brought, to study the substructure k , to define an impedance in the level of the degrees offreedom of connection. Within the framework of the developments carried out in Code_Aster, it iseither worthless, or infinite.

Basic vectors used in the methods implemented in Code_Aster are:

• normal modes,• constrained modes,• modes of fastener,• modes of interfaces,

that one defines below.

2.3.1 Normal modes

The clean modes or normal modes are advantageously used as bases projection of the substructuresfor several reasons:

• they can be calculated ou/et obtained in experiments,• they offer interesting properties of orthogonality compared to the matrices of mass and

rigidity of the substructure,• they are associated with natural Eigen frequencies of the structure.

They can be of two types according to the condition given to the interfaces of connection, cfFigures 2.3- B and 2.3- C:

• modes specific to blocked interfaces,• modes specific to free interfaces.




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Figure 2.3- B: Modes specific to blocked interfaces.

Figure 2.3- C: Modes specific to free interfaces.

Let us note that in the case of a free substructure, the modes of rigid body (or modes overall) existingare part of the base of transformation.

2.3.2 Static deformations

One defines a mode of interface in each degree of freedom of connection of each substructure.According to the cases, they can be constrained modes or modes fastener.

The constrained modes are static deformations which one joint with the normal modes with interfacesblocked to correct the effects due to their boundary conditions. A constrained mode is defined by thestatic deformation obtained by imposing a displacement unit on a degree of freedom of connection,the other degrees of freedom of connection being blocked.

Figure 2.3- D: Constrained modes.

The modes of fastener are static deformations which one joint with the normal modes with freeinterfaces to decrease the effect of modal truncation. A mode of fastener is defined by the staticdeformation obtained by imposing a unit force on a degree of freedom of connection, the otherdegrees of freedom of connection being free.

Figure 2.3- E: Modes of fastener.

In the case of a substructure having of the modes of rigid body (here, substructure 2), its matrix ofrigidity is not invertible any more and it is not possible to calculate its modes of fastener. It is thennecessary to block certain degrees of freedom to make the structure isostatic, even hyperstatic.




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2.3.3 Harmonic deformations

A third base of projection was introduced within the framework of the developments concerning theharmonic calculation of answer per classical dynamic under-structuring. It is the harmonic base ofCraig-Bampton.

The harmonic base of Craig-Bampton consists of modes specific to blocked interfaces and harmonicconstrained modes [bib8]. The latter are joined to the normal modes with interfaces blocked to correctthe effects due to their boundary conditions. A harmonic constrained mode is defined by the responseof the substructure not deadened to a harmonic displacement, of amplitude unit and frequency given,imposed on a degree of freedom of connection, the other degrees of freedom of connection beingblocked.

Figure 2.3- F: Harmonic constrained modes.

This modal base is more particularly appropriate to the problems of interactions fluid - structure, forwhich the static loadings are not applicable in Code_Aster (not taken into account of the effect ofadded mass). It can be used for any type of calculation (modal, harmonic and transitory).

2.3.4 Deformations of interface reduced or “modes of coupling”

One defines a mode of interface for the whole of the degrees of freedom of connection of eachsubstructure.

The modes of interface are deformations dynamics which one joint with the normal modes withinterfaces blocked or free to correct the effects due to their boundary conditions.

Concretely, the calculation of the constrained modes consists to carry out a condensation of thebehavior of the structure on the degrees of freedom of interface and to thus build an operator ofinterface (complement of Schur) which expresses the behavior of the reduced structure to theinterface. By a spectral analysis, one can then represent the deformations of the interface by linearcombination of the clean modes of this operator of interface.




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Figure 2.3- G: Mode of interface.

2.4 Conditions of connection between substructures

2.4.1 Introduction

Let us consider the problem of two substructures S1 and S 2 bonded in a rigid way not subjected

to external efforts. So that the movement of the structure supplements is continuous, it is necessary toimpose the equality of displacements of the two components on the interface and the law of action-reaction:

∀ M∈S1∩S2 q i

k u1M =q i

k u2M et f L

1M =−f L

2M (2.4.1-1)

where:

u1M represent the field of displacements of the substructure S1 ,

u2M represent the field of displacements of the substructure S 2 ,

f L1M represent the field of the bonding strengths applied to the substructure S1 ,

f L2M represent the field of the bonding strengths applied to the substructure S 2 .

According to the nature of the grids to the interface, the preceding problem can be discretizeddifferently.

2.4.2 Case of the compatible grid of interface

In this part, we limit ourselves to the compatible cases of grids. That means that they check thefollowing properties:

• the restriction of the grid of each substructure S1 and S2 on the interface are based strictly

on the same nodes in their intersection,• the finite elements associated with these meshs of connection are of comparable nature

(linear, quadratic) on both sides of the border.

Consequently, the condition [éq 2.4.1-1] is strictly equivalent to the formulation below:

qS 1∩S 2

1=qS 1

∩S 21 et f LS1∩S2

1=−f LS 1∩S2

2 (2.4.2-1)

where:Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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qS 1∩S 2

k is the vector of the degrees of freedom to the nodes of interface S1∩S 2 under - structure

k ,

f S1∩S 2

k is the vector of the bonding strengths to the nodes of interface S1

∩S2 under - structure k.

Indeed, grids of the two substructures S1 and S2 coinciding on the interface, the functions of form

associated with the finite elements are the same ones with the interface. It is thus enough to imposethe equality on the nodes of the interfaces of connection of each substructure to impose the equalityon all the field of connection.

Let us introduce the matrices of extraction of the degrees of libert-piece of interface BS1∩S 2

k :

qS 1∩S 2

k=BS1

∩S 2

k qk (2.4.2-2)

By using the equation of projection [éq. 2.1-2], the condition of continuity of displacements [éq 2.4.2-2 ] and the formulation applied above to the two substructures, one obtains:

BS1∩S 2

1

1

1=BS 1

∩S 2k

1

1

That is to say:

LS1∩S 2

1

1=LS1

∩S 22

2 avec LS 1

∩S2k

=BS 1∩S2

k

k (2.4.2-3)

where:

LS1∩S 2

1 is the matrix of connection of S1 associated with the interface S1

∩S2 , it expresses the

continuity of displacement between the two substructures starting from the displacementsgeneralized with the interface,

LS1∩S 2

2 is the matrix of connection of S2 associated with the interface S1

∩S2 .

In the case of the problem with the eigenvalues of the total structure, provided with its boundaryconditions, one can write:

K− ² M LT=0

L =0 (2.4.2-4)

The matrices of generalized mass and rigidity, the vector of the generalized degrees of freedom andstamps it connection which appear here, are defined on the total structure. They take a form particularto each method (Craig-Bampton, Mac Neal,…) who will be clarified thereafter. The vector of themultipliers of Lagrange be interpreted by the actions of connection to which the interfaces aresubjected.

It is thus about a classical problem of search for eigenvalues, with which is associated a linearequation of constraint. In Code_Aster, this kind of problem is solved by double dualisation of theboundary conditions [bib6].

Thus, one can show that this system is also solution of the problem of minimization of the followingfunctional calculus, known as integral of action :

f =∫a

bF , ,12⋅dt




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where F , ,12=12

T K 12 M 12

T L−121−2

2 is the Hamiltonian.

The variables of this functional calculus are the generalized coordinates , speeds associated and

multipliers with Lagrange 1

and 2 (of number equal to 2 times the number of equations of

connection). The last term of the functional calculus imposes the equality of the coefficients ofLagrange.

The extremum is reached for the values of the variables which cancel the derivative of f , whatever

a and b realities:

−1L 2=0

LT

1

2 K− ² M =0

1L− 2=0

[−Id L IdLT K LT

Id L −Id ]{ 1

2}−

2[0 0 00 M 00 0 0 ]{

1

2}={000 } (2.4.2-5)

The double dualisation thus leads to a real symmetrical matric problem. It is shown [R3.03.01] that itmakes it possible to make the algorithms of triangulation of matrix unconditionally stable.

2.4.3 Case of the incompatible grid of interface

In this part, we consider the incompatible case of grids. That means that they do not check thefollowing properties a priori:

• the restriction of the grid of each substructure S1 and S2 on the interface are based strictlyon the same nodes in their intersection,

• the finite elements associated with these meshs of connection are of comparable nature(linear, quadratic) on both sides of the border.

The selected method to manage the incompatibility of grid is a method derived from that of the PPCM(“Smaller Common Grid”). This method is used within the framework of several types of problems inCode_Aster : management of grid incompatible, projection of field on grids of different nature… Onedefines consequently an interface “Master” on whom one projects the nodes of the interface “slave”.

The interpolation of the fields to the nodes of the interface slave (elements surrounded of S 2 on the

figure below) is done on the main element of the interface on which each node is project, using thefunctions of forms of this element.




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Figure 2.4- has:

One defines e like interfaces it restricted with an element and ne the number of ddl of the

element.By choosing the interface of the substructure S1 like interface Master, the relation (2.4.1-1) rewritesitself:

u1M −∑j=1

ne

∫e

E j2M M dM q j

2 =0

where M are the functions of forms of the element on which the node is project and E j2 the

restriction of the functions of forms of the element j interface of S2 .

This method makes it possible to couple substructures having of the different grids of nature (linearwith quadratic,…) and of the different densities. It will be retained however that the rule of good useconsists in choosing for Master the grid of the lowest density.

Taking into account the conditions defined in the preceding paragraphs, the expression binding the

degrees of freedom must be written again. By taking the interface of the substructure S 1 like interface

Master, one a:

q j1=∑

l

ne

l ql2

Or:

q1=A q2

where A is a matrix of observation between the ddl of the two interfaces.




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The relations of continuity of displacements are expressed, with the generalized coordinates, by thefollowing matric relations:

L j1

1= L j

2

2

L jk=A B k

k

One can write this relation in matric form:

[L j1− L j

2 ]⋅[1

2]=0

The terms of the matrices of connections are not equal any more to zero or one. They have resultingfrom the interpolations of the displacements projected on the elements.

Lastly, by using the double dualisation, the matric system obtained to solve, in the case of the modalproblem on the total structure is:

⇒[−Id A L IdAT LT K AT LT

Id A L −Id ]{ 1

2}−

2[0 0 00 M 00 0 0 ]{

1

2}={

000}

Where A indicate the matrix identity Id if the substructure is that whose interface is Master and

A , the matrix of observation, if it is about that whose interface is slave.

The double dualisation thus leads to a real symmetrical matric problem even for the case of theincompatible grids.

2.4.4 Conclusion

This method thus makes it possible to treat the connection of interfaces corresponding to basic typesmodal different without cost from management of an always delicate elimination. In addition, it isrelatively simple. The major drawback of this formulation is to lead to final assembled systems of sizemore important than in the case of elimination. Indeed, the coupling of the matric equations was madeby introducing a number of additional degrees of freedom equal to twice the number of equations ofconnection. This increase in dimensions of the matrices can thus be very important. Let us note thatthe degrees of freedom of Lagrange introduced are, in this case, the forces applied to the interfaces toensure the connection between the two substructures.

2.5 Elimination of the linear constraints

The technique of the double dualisation makes it possible to obtain a uniformity in the taking intoaccount of the constraints. This approach is indeed usable as well for the linear constraints asnonlinear, on displacements or the constraints. It thus authorizes, with a great flexibility, the realizationof a large variety of calculations.

Nevertheless, within the framework of the modal analysis in general, and the techniques of under-structuring in particular, the presence of the double multipliers of Lagrange can be a disadvantage, atthe same time related to the increase in the size of the problem with the number of constraint, but sorelated to the digital conditioning of the system thus obtained. The increase in the size of the problemis particularly clear in the case of the approaches by under-structuring, where, for the final problem,the majority of the equations to be solved relates to the constraints. To circumvent this difficulty, oneimplemented a technique of elimination of the constraints kinematics on the basis of a decompositionQR of the matrices of connection.




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2.5.1 Case of linearly independent constraints

The linear constraints associated with a problem of dynamics can be written in the form of the linearsystem:

[L ]⋅q=0 (2.5.1-1)

q indicating the whole of the degrees of freedom of the complete model (i.e. including all under

structures, in our case). The idea of elimination consists in building a base of the core of L , and tosolve the problem project in this base. The construction of this base is done simply starting fromdecomposition QR of LT .

For a matrix L of size n×N , where n is the number of constraints, and N the number of degreesof freedom, this decomposition is written:

[ LT ]N×n=[ Q ] N×N⋅[ R ] N×n=[ [Q1]N× n [Q2]N×N−n ]⋅[ [R1]n×n

[0]N−n ×n] (2.5.1-2)

Q2 is then directly the searched base, as it is shown in the continuation.

2.5.2 Case of redundant constraints

This decomposition also makes it possible to analyze what occurs in the presence of redundantconstraints (constrained nodes several times, substructure “out of star”, etc). When the constraints allare not independent, one can write L in the form of a linear combination of n In linearly

independent constraints L I , that is to say:

[L ]n×N=[A ]n×n I⋅[ LI ]nI×N

Decomposition of L I is written directly:

[L IT ] N×nI

=[[Q I ]N×n I[QK ]N×N−nI ]⋅[ [R I ]n I×nI

[0]N−n I ×ni] (2.5.2-1)

and decomposition of L becomes:

[ LT ]=[ Q I QK ]⋅[R I⋅AT

0 ] (2.5.2-2)

QK a base of Ker L . Indeed, for any element q core of L , there exists a vector y such as

q=[QK ]⋅y . Under these conditions, one a:




5b0a507feb90

[L]⋅q = [ A⋅R IT

0 ]⋅[ QIT

QKT ]⋅q

= [ A⋅R IT

0 ]⋅[ Q IT

QKT ]⋅[QK ]⋅y

= [A⋅R IT]⋅[QI

T]⋅[QK ]⋅y

= 0

since ImL and Ker L are orthogonal. When constraints of L are linearly independent, one

has directly QK=Q2 . In the contrary case, it is necessary to build QK from Q2 and of the columns

n−n I last columns of Q1 .

To identify the row n I matrix L , the higher triangular matrix is used R 1 , resulting from the relation(2.5.1-2). According to the relation (2.5.2-2), it must be indeed put in the form:

[ R 1]n×n=[ [R I ]nI×n I⋅[A ]n×n I

T

[0]n−n I ×n ]

The row nI is the number of nonworthless diagonal terms of Q1 . Consequently, vectors of Q1

belonging to the core of L are of which those corresponding under the worthless diagonal terms of

R . In practice, the search for n I is dependent on the value fixed like threshold for the zero digitalone.

2.5.3 Case of a significant number of constraints

The principles exposed in the preceding paragraphs can be implemented digital such as they are andpreserve correct performances if the matrices associated with the constraints are limited sizes, that isto say to the maximum about the thousand of constraints, utilizing about ten a thousand of degrees offreedom. In the contrary case, one of the methods consists in building it under space in an iterativeway.

The total approach consists in modifying the lines of the matrix successively Q , for all the constraintsof L . The matrix Q of departure is the matrix identity. To illustrate the approach, let us consider a

particular stage by choosing a set of constraints exits of L . One notes Qk the matrix taking of

account the elimination of k first constraints. The construction of Qk1 is done in the following way:

• To extract the line Lk1 of L ,

• To determine the number of nonworthless elements of Lk1 . These nonworthless elements areassociated with the degrees of freedom implied in the linear relation of the type of (2.5.1-1).• If only one element of Lk1 is nonnull, one cancels the diagonal term of Qk 1 associated

with this degree of freedom.• If several nonworthless elements are detected, one seeks, from Lk preceding lines, which

are the degrees of freedom already implied in a linear relation. One builds the submatrix ofQk1 allowing to eliminate the new constraints by taking account of the presence of the

preceding constraints.

For this last point, one notes q i degrees of freedom implied in Lk preceding linear relations, and q l

other degrees of freedom. The linear subset of relation associated with Lk1 is thus written




5b0a507feb90

[L i L l]{qi

ql}=0 , and one seeks the block of Qk 1 who checks [Li Ll ][Qi i Qi l

Ql i Ql l ]=0 . The block

Qii was built beforehand, and does not have to be modified not to impact them Lk preceding

relations. One thus builds Qi l , Ql i and Ql l such as:

• Qi l=0 ,

• Ql i=− Ll +

Li Qi i, where Ll

+ is theopposite one of Ql , allowing to solve

L iQi iL lQl i=0 ,

• Ql l is built by decomposition Q R of L l=Ql l R l l . After decomposition, columns of Ql l

associated with nonworthless diagonal terms of R l ŀ are put at zeros.

Several choices exist for the successive construction of Lk , and consist is to be extracted a fixed

number of lines from L for each step, that is to say to make a research of the blocks of constraints

disjoined in L .




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3 Methods of classical dynamic under-structuring

3.1 Introduction

After having separately studied the various stages of the under-structuring, and the techniques whichthey bring into play, it appears interesting to present the principal methods of under - dynamicstructuring: method of Craig-Bampton and that of Mac Neal and method of the reduced modes ofinterfaces.

Craig-Bampton uses, as bases projection of the substructures, modes constrained and normal modeswith fixed interfaces [bib1].

In addition, Mac Neal uses, as bases projection of the substructures, modes of fastener and normalmodes with free interfaces [bib2].

One will approach a method derived from these methods named methods of the reduced modes ofinterface which uses as bases projection of the substructures, modes of interface and normal modeswith blocked interfaces.

3.2 Method of Craig-Bampton

The following presentation utilizes only two substructures S1 and S2 , but it is generalizable with anunspecified number of components. After having studied separately each under - structure, their basesof projection (normal modes with fixed interfaces and constrained modes) are known. For each one ofthem (identified by the exhibitor k ), one establishes a partition of the degrees of freedom,

distinguishing the vector from the internal degrees of freedom qik

and the vector of the degrees of

freedom of connection qjk

:

qk={qi

k

q jk}

That is to say:

k the matrix of the clean vectors of the substructure S k ,

k the matrix of the constrained modes of the substructure S k .

The base of projection of S k is characterized by the matrix:

k=[

k

k]

The transformation of RITZ (equation [éq 2.1-2]), allows us to write:

qk={q ik

q jk }=[k k ]{ i

k

jk}=k k

(3.2-1)

ik

is the vector of the generalized degrees of freedom associated with the clean modes of S k ,

jk is the vector of the generalized degrees of freedom associated with the constrained modes of

S k .

However the normal modes are given with fixed interfaces, and each constrained mode is obtained byimposing a unit displacement on a degree of freedom of connection, the others being blocked.




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The generalized coordinates relating to the static deformations are then the values of the degrees offreedom of connection:

q jk= j

k

We interest in the contribution of the substructure S k from an energy point of view. The energieskinetic and of deformation are:

T k=

12

qkT

.Mk . qk=

12

k

k T.Mk .k

k=

12

k . Mk . k

U k=12

qkT

. Kk . qk=12

k k T

. Kk .k k=12k . K k . k

These expressions reveal projections of the matrices of mass and rigidity on the basis of substructure.These matrices, known as generalized, check a certain number of properties:

• because of orthogonality of the normal modes compared to the matrices of rigidity and mass,the left higher block of these matrices is diagonal. Thereafter, we will consider that thesemodes are normalized compared to the matrix of mass,

• one can also show that the constrained modes are orthogonal with the normal modescompared to the matrix of rigidity [bib4].

The matrices of generalized rigidity and mass thus have the following form:

K k=[

k 00

kT

K k

k ] Mk=[ Id

kT

Mk

k

k T

Mk

k

kT

Mk

k ] (3.2-2)

where k is the matrix of the generalized rigidities associated with the clean modes of S k .

The choice of the base of projection to blocked interfaces thus leads to a coupling of the normalmodes and static deformations by the matrix of mass.

This method is interesting if one considers a digital study of the substructures. Indeed, the normalmodes with fixed interfaces and the constrained modes lend themselves well to calculation. On theother hand, their experimental determination is delicate, because it is difficult to carry out anexperimental embedding of good quality.

One shows moreover in [feeding-bottle 1] that this method, is of order 2 in /m where m is the

greatest identified own pulsation.

3.3 Method of Mac Neal

It would be possible to present this method in a way similar to that of Craig-Bampton, the singledifference residing in the use of the normal modes at free interfaces and the modes of fastener.However, it appears interesting to adopt a slightly different approach, which makes it possible to leadto a criterion of truncation, and to reveal the modes of fastener like the static contribution of the notidentified clean modes.

As previously, one considers the problem of a structure with 2 components. The method isgeneralizable with an unspecified number of substructures. In the continuation we will identify any sizeassociated with the substructure S k by the exhibitor k .




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We limit ourselves, here, with the modal calculation of the total structure, therefore the external forcesare worthless, in order to reduce the theoretical developments. Also, the displacement of S k checkthe dynamic equilibrium equation following:

Kk− ² Mk

q=f Lk

(3.3-1)

We will consider that the base of projection of S k consists of all its modes specific to free interfaces.Thus, the dimension of the problem project is equal corresponding to problem resulting from modelingfinite elements. We suppose, moreover, that a certain number of these modes was given, the othersbeing unknown:

qk=[1

k2

k]{1

k

2k}=

k

k

where:

1k is the matrix of the identified modal vectors of the substructure S k ,

2k of the not identified modal vectors substructure is the matrix S k ,

1k

is the vector of the generalized degrees of freedom associated with the clean modes identified of

S k ,

2k

is the vector of the generalized degrees of freedom associated with the not identified clean modes

S k .

The equation [éq 3.3-1] becomes, with the previously definite generalized coordinates:

kT

.K k .k−

2

kT

.Mk .k

k=

kT

f Lk

that is to say:

Kk− ² Mk k= kT

f Lk (3.3-2)

The clean modes are orthogonal compared to the matrices of mass and of rigidity and we choose tonormalize them compared to the matrix of mass. One thus has:

K k=[ 1

k 0

0 2k ] Mk

=[Id 00 Id] (3.3-3)

Now let us consider all two substructure. Each one of them checks the equations [éq 3.3-2] and [éq3.3-3]. The whole of these dynamic equations constitutes the following system:

[1

1 0 0 0

0 12 0 0

0 0 21 0

0 0 0 22]−

2[Id 0 0 00 Id 0 00 0 Id 00 0 0 Id

] {1

1

12

21

22}=[

11

12

21

22]

T

{f L1

1

f L1

2

f L2

1

f L2

2 } (3.3-4)

By gathering the identified modes and the not identified modes:




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{1

2}={

11

12

21

22}

coordonnées des modes identifiés

coordonnées des modes non identifiés

(3.3-5)

The equation translating the transformation of RITZ becomes:

q={q1

q2}=[1 2 ] {1

2} (3.3-6)

With these notations, the system of dynamic equations [éq 3.3-4] becomes:

[ 1 0

0 2]−

2[ Id 00 Id ]{1

2}=[1

T 0

0 2T ] {f L1

f L2} (3.3-7)

This system of equations translates the dynamic behavior of the substructures separately. It does notrepresent the overall movements of the structure. For that, it is necessary to associate to him theconditions of connection between the two components.

The equations between substructures which ensure their liaisonnement derive from the equations [éq2.4.1-1] and [éq 2.4.2-4], as of the organization of the base which we chose [éq 3.3-5]:

L⋅=[ L1 L2 ] {1

2}=0 (3.3-8)

f L1=−f L2

(3.3-9)

The equations [éq 3.3-7] and [éq 3.3-9] allow us to express the generalized coordinates relating to thenot identified modes:

2=−2−2 Id −1

2T f L1

(3.3-10)

From the equations [éq 3.3-8] and [éq 3.3-10], one thus obtains:

L11−L22−2 Id−12T f L1

=0 (3.3-11)

However, according to the formula [éq 2.4.2-4], one knows that one can write the matrices ofconnection, in the form:

L k=Bk k (3.3-12)

One thus has, according to [éq 3.3-11] and [éq 3.3-12]:

−B221B222−2 Id−12T f L1

=0 (3.3-13)




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One sees appearing the matrix of residual dynamic flexibility associated with the not identified modes:

R e=2 2−2 Id−1

2T

(3.3-14)

This term shows that the efforts of connection to the degrees of freedom of interface are nonworthlessif one takes account in modeling only known modes.

Consequently, the matric problem [éq 3.3-7] can be reduced to the system are equivalent accordingto:

[ 1−2 Id − 1

T B1T

−B11 B2 Re ]{ 1

f L1}={00 } (3.3-15)

This problem has as unknown factors the generalized coordinates associated with the identified cleanmodes and the bonding strengths applied to the first substructure. The term of residual flexibility,which also appears, is not known. One proposes below a modeling which utilizes the modes offastener of Mac Neal.

There are two cases according to whether one takes into account or not the matrix of residualflexibility.

3.3.1 First case

One neglects the matrix of residual dynamic flexibility associated with the not identified modes:

R e=0

The method of resulting dynamic under-structuring is thus very simple. It has the disadvantage to bebased on a method of modal recombination very sensitive to the effects of truncation.

3.3.2 Second case

A limited development is used: R e=Re 0 O2/m

2 .

Let us adopt the following notations:

that is to say:

n the number of modes of the complete base,

m the number of not identified modes,

the matrix of n normal modes with free interfaces,

the matrix of n modes of fastener (definite for all the degrees of freedom of thesystem, not only for the degrees of freedom of interface enters the 2substructures),

i=i2 the diagonal matrix of n eigenvalues.

The complete matrix of the modes of fastener checks: K=Id ⇔ =K−1 .

The complete modal base of the normal modes with free interfaces constitutes an orthonormal base.The matrix of stiffness, expressed in this base is written:

K=T .K .=

Same manner:




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K−1=

T .K−1.=−1

One from of thus deduced a new form of the complete matrix of the modes of fastener:

=K−1= .−1.T

As the clean modes are orthogonal two to two and stamps it eigenvalues is diagonal, the completematrix of the modes of fastener takes the final shape:

=∑i=1

n

ii−1 i

T

Now let us consider the matrix of residual dynamic flexibility, resulting from the method of Mac Neal:

R e = 22−2 Id−1

2T= ∑

i=m1

n

i i−2−1 i

T

When the number of identified modes is sufficiently important, the dynamic contribution becomesnegligible in front of the static contribution:

2

i2≪1⇒Re ≈R e 0= ∑

i=m1

n

ii−1i

T

where R e 0 is the matrix of residual static flexibility.

To highlight the effect of modal truncation, we can approach this matrix by its development with order1:

R e ≈ ∑i=m1

n

ii−11

2

i2 i

T

This matrix can be calculated according to the matrix of the modes of fastener:

R e 0=∑i=1

n

ii−1i

T−∑

i=1

m

ii−1 i

T⇒ R e 0=−∑

i=1

m

ii−1 i

T

The second term of this formulation is calculable in an exact way, since it utilizes only the modes (withfree interfaces) identified.

Lastly, let us note that in the method of Mac Neal, only the contribution of the modes of fastener to thenodes of interface is necessary.

The resolution of the system [éq 3.3-4] enables us to determine the Eigen frequencies of the totalstructure and the generalized coordinates of the clean modes. The increase with the expression of theclean modes in the physical bases of the substructures is done by the following relation:

q1k=1

k1

kR e

k0B1

k T

f L1

T

The method of Mac Neal thus succeeds, in the case of the calculation of the modes of the totalstructure, with a problem with the eigenvalues of reduced size. The matrices of mass and rigidity exitsof the under-structuring are symmetrical. Two methods are actually proposed, according to whether




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one takes into account or not residual flexibility. The literature on the subject tends to show that theuse of residual flexibility is essential to get reliable results [bib2].

3.4 Method of the modes of interface (method known as reduced)

3.4.1 Introduction

The method of the modes of interface is based on work of various authors, one will find a justificationtheoretical supplements in [bib12]. We will not develop all these aspects in this document because theuse which is made by it in Code_Aster is relatively simplified: it is just a question of decreasing thesize of the system reduced by the use of the spectral properties of the dynamic operator condensedto the interface.

3.4.2 Definition of the modes of interface

After having studied each under - structure separately to build the bases of normal modes, it is aquestion of considering the problem coupled to build the bases of modes of interface. Procedurecurrently present in Code_Aster require to consider the problem coupled by the dynamic under-structuring. One thus uses the method of Craig-Bampton without the normal modes with blockedinterfaces, which is thus equivalent to a method of static under-structuring by condensation of Guyan.The following coupled system is obtained:

K k=

kT

.K k . k Mk=

kT

.Mk . k

indicate here the constrained modes. The calculation of the clean modes of the total structurecondensed with its interfaces by the methods of dynamic under-structuring consists in solving aproblem with the eigenvalues matric according to:

K−2 M RL

T=0

L R=0

It goes from either that during this calculation of the modes of interface, that does not have a directionto calculate as many modes as of degrees of freedom of interface because that would be equivalentreconsidering the system of constrained modes. A truncation error of the base of projection is thusintroduced on this level, but which remains physical and controllable.

The clean modes obtained are defined only on the degrees of freedom of interface, it is thusnecessary to extend these modes on each coupled substructure. One obtains as follows:

Rk=

k R

3.4.3 Calculation of the modes of interfaces

In practice, the calculation of the modes of interfaces with the method presented to the paragraph3.4.2 is not realistic. Indeed, the arithmetic operation of the stress patterns is very expensive in

memory and computing times, more especially as one considers then only one under family of Nvectors, N being very weak in front of the number of degrees of freedom of the interface. Approachadopted for the order MODE_STATIQUE [U4.52.14] consists in building the modes of interfaces using apre-conditioner. If one notes i degrees of freedom of interface, and c degrees of complementaryfreedom, by partitionnant the matrix of stiffness K , stress patterns check

{ i

c}={ Id

−K cc−1K ci

}




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Rather than to directly build the stress pattern on this principle, one assembles built matrices of massM ii and of stiffness K ii defined only in the interface. These matrices are built on the basis of model

of beams of Euler Bernoulli by taking again the connectivity of the nodes of the interface. Each pair ofnodes connected of the interface is then connected by a cylindrical beam whose geometricalproperties are adjusted compared to the short distance between two nodes connected interface. Thediameter of the beam is then fixed at 20% of this minimal distance. The material used for the beam isa steel. One calculation then a family of mode i this model of interface. It is advisable here to

calculate a number of mode M rather appreciably higher than N , where N is the number ofmodes of interface which the user wishes to retain.

To calculate these modes, one carries out a first operation which makes it possible to determine thenumber of under disjoined parts of a given interface. For each disjoined part of interface, onecalculates the 6 modes of rigid body. If one notes d the number of disjoined parts, then the fullnumber M of calculated mode is worth:

• M=3E Nd2 E

6d1 if E

Nd7

• M=36E Nd2

where E . corresponds to the whole part of the quantity considered.

These M modes i are then raised statically on the model of under structure, and then form thebase on which one searches the modes of interface:

={ i

c}={ i

−K cc− 1K ci

i}

One projects then the matrices of mass and stiffness of the macro element on these vectors,

K= T .K . M=

T .M .

and one seeks the clean modes of them:

K−2 M R=0

The modes of interface retained for the construction of the macro elements are finally them N firstvectors defined by:

= R

The calculation of the matrices M ii and K ii , and of the associated modes i is fast andinexpensive in memory, and allows to limit the operation of raising to the only vectors of interest tobuild the modes of interface. Naturally, if as much mode is calculated i that DDL of interface, thenunder space described identical to that is described by the stress patterns

3.4.4 Creation of the reduced macronutrients

For each substructure, one has of the bases of normal modes and the bases of the modes ofinterface. The creation of the macronutrients is then identical to the case of the method of Craig-Bampton. One establishes a partition of the degrees of freedom, distinguishing the vector from the

internal degrees of freedom q ik and the vector of the degrees of freedom of connection q j

k :

qk={q i

k

q jk }




5b0a507feb90

That is to say:

k the matrix of the clean vectors of the substructure Sk ,

Rk the matrix of the modes of interface of the substructure Sk .

The base of projection of Sk is characterized by the matrix:

k=[

kR

k]

The transformation of RITZ (equation [éq 2.1-2]), allows us to write:

qk={q ik

q jk }=[k R

k ] { ik

jk }= k k

(3.4.4-1)

ik is the vector of the generalized degrees of freedom associated with the clean modes of Sk ,

jk is the vector of the generalized degrees of freedom associated with the modes of interface of

Sk .

Let us interest in the contribution of the component Sk from an energy point of view. The energieskinetic and of deformation are:

T k=

12qkT

.Mk . qk=

12

kk

T.Mk .k

k=

12k . Mk . k

U k=

12qkT

.Kk . qk=

12

kk

T.Kk .k

k=

12k . K k . k

These expressions reveal projections of the matrices of mass and rigidity on the basis of substructure.These matrices, known as generalized, check a certain number of properties because of orthogonalityof the normal modes compared to the matrices of rigidity and of mass, the left higher block of thesematrices is diagonal. Moreover, we will consider that these modes are normalized compared to thematrix of mass.

The matrices of generalized rigidity and mass thus have the following form:

K k=[

k

k T

Kk

k

kT

K k

k

k T

Kk

k ] Mk=[ Id

kT

Mk

k

k T

Mk

k

kT

Mk

k ] (3.4.4-2)

where k is the matrix of the generalized rigidities associated with the clean modes of Sk .

3.5 Implementation in Code_Aster

3.5.1 Study of the substructures separately

The base of projection of each substructure is made up of dynamic clean modes and staticdeformations.

The dynamic clean modes of the substructure are calculated with the classical operator ofCode_Aster: CALC_MODES [U4.52.02]. In the case of under - structuring of Craig-Bampton, theinterfaces of connection must be blocked. This is carried out with the operator AFFE_CHAR_MECA[U4.44.01].




5b0a507feb90

The operator DEFI_INTERF_DYNA [U4.64.01] allows to define the interfaces of connection of under -structure. In particular, one specifies the type of the interface, which can be is “CRAIGB“(Craig -Bampton), that is to say”MNEAL“(Mac Neal), that is to say finally”NONE“(not of calculated staticmodes).

The operator DEFI_BASE_MODALE [U4.64.02] allows to calculate the base of complete projection ofunder - structure. Thus, the dynamic modes calculated previously are recopied. In addition, the staticdeformations are calculated according to the type defined in the operator DEFI_INTERF_DYNA[U4.64.01]. If the type is “CRAIGB“, one calculates the constrained modes of the interfaces of under -structure. If the type is”MNEAL“, one calculates the modes of fastener of the interfaces of under -structure. If the type is”NONE“, one does not calculate static deformation, which corresponds to a baseof the type Mac Neal without static correction.

The operator MACR_ELEM_DYNA [U4.65.01] calculates the generalized matrices of rigidity and mass ofunder - structure, as well as the matrices of connection.

3.5.2 Assembly

The model of the complete structure is determined by the operator DEFI_MODELE_GENE [U4.65.02].In particular, each substructure is defined by the macronutrient which corresponds to him (resultingfrom MACR_ELEM_DYNA) and the swing angles which make it possible to direct it. The connectionsbetween under - structures are defined by the data of the names of the two implied substructures andthose of the two interfaces in opposite. In the case of the method of the modes of interface, it isnecessary to specify the option ‘REDUCED’ in order to allow the coupling between the macronutrients.

The classification of the complete generalized problem is carried out by the operator NUME_DDL_GENE[U4.65.03]. The matrices of generalized mass and stiffness of the structure supplements areassembled according to this classification with the operator ASSE_MATR_GENE [U4.65.04].




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4 Modal calculation by under classical dynamic structuring

4.1 Introduction

For the modal problems, the studied system is subjected to no external force. One does not takeaccount of dissipation in the solid. Method of calculating modal by under-structuring, programmed inCode_Aster, which makes it possible to replace the total problem by a simplified problem, proceeds infour times. First of all, clean modes and static deformations are calculated on each substructurecomposing the system. Then, the total problem is project on these fields, and one takes account of thecouplings between the substructures, on the level of their interfaces. One can then classically solvethe reduced problem obtained. Finally, it any more but does not remain to deduce the overall solutionby reconstitution from it.

4.2 Dynamic equations checked by the substructures separately

We will consider a structure S composed of N S noted substructures Sk . We suppose that eachsubstructure is modelled in finite elements. We saw that in a calculation by dynamic under-structuring,the vibratory behavior of the substructures results from the forces external which are applied to them,and from the bonding strengths which on them the other substructures exert. Thus, on the level of thesubstructure Sk , and in the case of modal calculation we can write:

Mk qkK k qk

=f Lk (4.2-1)

The field of the bonding strengths is written in complex notation:

f Lkt = {f L

k }e j t

The field of displacements is written:

qkt ={qk } e j t

The fields speed and acceleration are written:

qkt = j {qk } e j t

qkt =−2 {qk }e j t

Finally the substructure Sk check the following equation:

K k−

2Mk{qk

}={f Lk} (4.2-2)

The method of modal synthesis consists in searching the field of unknown displacement, resultingfrom modeling finite elements, on an adapted space, of reduced size (transformation of Ritz). We sawthat for each substructure, this space is composed of dynamic clean modes and of staticdeformations:

qk=[ ik j

k ] { ik

jk }=k k

(4.2-3)




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k are the modal vectors associated with the dynamic clean modes of Sk ,

k are the modal vectors associated with the static deformations of Sk ,

ik is the vector of the generalized coordinates associated with the clean modes of Sk ,

jk is the vector of the generalized coordinates associated with the static deformations of Sk ,

k is the vector of the coordinates generalized of Sk .

The equation [éq 4.2-2] is projected on the basis of Sk by taking account of [éq 4.2-3]. This enablesus to write:

Kk−

2 Mk {

k}={f L

k} (4.2-4)

By supposing that the dynamic clean modes and the static deformations are organized as the formula[éq shows it 4.2-3] and by considering that the clean vectors associated with the dynamic modes arenormalized compared to the unit modal mass, the matrices of generalized mass and rigidity take thefollowing shape:

Mk=[ Id

kT

Mk

k

kT

Mk

k

kT

Mk

k ] Kk=[

k

kT

K k

k

kT

K k

k

kT

K k

k ]

where:

Id is the matrix Identity,

k is the diagonal matrix of the squares of the own pulsations of the base.

One shows, in case of method of Craig-Bampton, that the normal modes and the constrained modesare orthogonal with respect to the matrix of rigidity whose diagonal terms are, consequently, worthless.However, this property is not used in the algorithm programmed in Code_Aster.

4.3 Dynamic equations checked by the total structure

The dynamic equations that checks the total structure are:

[K1

. ..

K k

.. .

KN s

]−2[

M1

.. .

Mk

.. .

MN s

] {

1

. ..

k

. ..

N s

}={f L

1

...

f Lk

...

f LN s

}

Which, it is necessary to add the equations of connection (according to [éq 2.4.1-1]):

∀ k , l LS k∩Sl

k { k }=LS k∩S l

l { l }

This system is solved by double dualisation of the boundary conditions [R3.03.01]. Its final formulationthus utilizes the vector of the multipliers of Lagrange and can be written in the condensed form:

K−2 M { }LT=0L { }=0

(4.3-1)




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The problem defined by the equation [éq 4.3-1] is symmetrical. In addition, its dimension isdetermined by the number of modes taken into account (dynamic modes and static deformations).One is thus brought to solve a classical modal problem, of reduced size, with which an equation withlinear constraint is associated. Its resolution thus does not pose a problem.



The dynamic clean modes are calculated with the operator CALC_MODES [U4.52.02]. The conditionswith the interfaces of connection are applied with the operator AFFE_CHAR_MECA [U4.44.01]. Theoperator DEFI_INTERF_DYNA [U4.64.01] allows to define the interfaces of connection of thesubstructure. The operator DEFI_BASE_MODALE [U4.64.02] allows to calculate the base of completeprojection of the substructure.

The operator MACR_ELEM_DYNA [U4.65.01] calculates the generalized matrices of rigidity, mass andpossibly of damping of the substructure, as well as the matrices of connection.

4.4.2 Assembly and resolution

The model of the complete structure is defined by the operator DEFI_MODELE_GENE [U4.65.02]. Itsclassification is carried out by the operator NUME_DDL_GENE [U4.65.03]. The matrices of mass, rigidityand possibly of damping generalized of the structure supplements are assembled according to thisclassification with the operator ASSE_MATR_GENE [U4.65.04].

The calculation of the clean modes of the complete structure is carried out by the operatorCALC_MODES [U4.52.02].

4.4.3 Restitution on physical basis

The restitution of the results on physical basis is identical to the case of modal calculation. It utilizesthe operator REST_GENE_PHYS [U4.63.31].

To decrease the duration of the graphic treatments during visualizations, it is possible to create acoarse grid by the operator DEFI_SQUELETTE [U4.24.01]. This grid, ignored during calculation, isused as support with visualizations.




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5 Harmonic response by under classical dynamic structuring

5.1 Introduction

For the harmonic problems, the studied system is subjected to a force spatially unspecified, butsinusoidal in time. The form of the loading, the frequency of excitation and the properties modal, playeach one a crucial role. It is also necessary to take account of dissipation in the solid, which one cantranslate by the introduction of a matrix of damping. Harmonic method of calculating by under-structuring, programmed in Code_Aster, which makes it possible to replace the total problem by asimplified problem, proceeds in four times.

First of all, clean modes and static deformations are calculated on each substructure composing thesystem. Then, the total problem is project on these fields, and one takes account of the couplingsbetween the substructures, on the level of their interfaces. One can then classically solve the reducedproblem obtained. Finally, it any more but does not remain to deduce the overall solution byreconstitution from it.

5.2 Dynamic equations checked by the substructures separately

We will consider a structure S composed of N S noted substructures Sk . We suppose that eachsubstructure is modelled in finite elements. We saw that in a calculation by dynamic under-structuring,the vibratory behavior of the substructures results from the forces external which are applied to him,and from the bonding strengths which on them the other substructures exert. Thus, on the level of thesubstructure Sk , we can write:

Mk qkCk qk

Kk qk=f ext

kf L

k (5.2-1)

In a harmonic problem, one imposes a loading dynamic, spatially unspecified, but sinusoidal ofpulsation in time. One is interested then in the stabilized answer of the system, without takingaccount of the transitory part.

The field of the external forces is written:

f extkt = {f ext

k }e j t

The field of the bonding strengths is written:

f Lk t ={ f L

k } e j t

The field of displacements is written:

qkt ={qk } e j t

The fields speed and acceleration are written:

qk t = j {qk }e j t

qkt =−2 {qk }e j t

Finally the substructure Sk check the following equation of the movement:

K k jCk

−2Mk

{qk }={f extk } {f L

k } (5.2-2)




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The method of modal synthesis consists in searching the field of unknown displacement, resultingfrom modeling finite elements, on an adapted space, of reduced size (transformation of Ritz). We sawthat for each substructure, this space is composed of dynamic clean modes and of staticdeformations:

qk=[k

k ] { i

k

jk}=

k

k (5.2-3)

k are the modal vectors associated with the dynamic clean modes of Sk ,

k are the modal vectors associated with the static deformations of Sk ,

ik is the vector of the generalized coordinates associated with the clean modes of Sk ,

jk is the vector of the generalized coordinates associated with the static deformations of Sk ,

k is the vector of the coordinates generalized of Sk .

The equation [éq 5.2-2] is projected on the basis of Sk by taking account of [éq 5.2-3]. This enablesus to write:

K k j Ck−2 Mk { k }= {f extk }{f L

k } (5.2-4)

By supposing that the dynamic clean modes and the static deformations are organized as the formula[éq shows it 5.2-3] and by considering that the clean vectors associated with the dynamic modes arenormalized compared to the unit modal mass, the matrices of generalized mass and rigidity take thefollowing shape:

Mk=[ Id

kT

Mk

k

kT

Mk

k

kT

Mk

k ] Kk=[

k

kT

K k

k

kT

K k

k

kT

K k

k ]

One shows, in case of method of Craig-Bampton, that the normal modes and the constrained modesare orthogonal with respect to the matrix of rigidity whose diagonal terms are, consequently, worthless.However, this property is not used in the algorithm programmed in Code_Aster.

We consider, like type of dissipation, only viscous damping (it is the only one which is supported bythe tools for under-structuring in Code_Aster). Two methods are usable to take into account thisdamping:

• the damping of Rayleigh applied at the elementary level which consists in supposing that thematrix of elementary damping Ce associated with each finite element of the model is a

linear combination of the matrices of elementary mass and rigidity K e and Me :

Ce=eKeeMe

The matrix of damping is then assembled Ck then projected on the basis [éq 5.2-3]:

Ck=[

kT

Ck

k

kT

Ck

k

kT

Ck

k

kT

Ck

k ]

• the damping proportional applied to the dynamic clean modes of each under - structure. Theresulting matrix is thus an incomplete diagonal (one cannot associate damping proportional tothe static deformations):




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Ck=[

k 00 0]

5.3 Dynamic equations checked by the total structure

The dynamic equations that checks the total structure are:

[M1

...Mk

...M NS

]{

1

...k

...

N S

}[C1

...Ck

...CN S

] {

1

...

k

...N S

}[K 1

...Kk

...KN S

] {

1

...

k

...

N S

}={

f ext1

...f extk

...f extN S}{

f L1

...f Lk

...f LN S}

maybe in harmonic:

[K1

.. .

Kk

...

KN s

] j[C1

. ..

Ck

.. .

CN s

]−2[

M1

. ..

Mk

.. .

MN s

]{

1

. ..

k

. ..

N s

}={

f ext1

...

f extk

...

f extN s}{

f L1

. ..

f Lk

. ..

f LN s}

Which it is necessary to add the equations of connection (according to [éq 2.4.2-2]):

∀ k , l LS k∩Sl

k { k }=LS k∩S l

l { l }

This system is solved by double dualisation of the boundary conditions [R3.03.01]. Its final formulationthus utilizes the vector of the multipliers of Lagrange and can be written in the condensed form:

K joC−o

2 M { }oLT={f ext}o

L {}o=0 (5.3-1)

The problem defined by the equation [éq 5.3-1] is symmetrical. In addition, its dimension isdetermined by the number of modes taken into account (dynamic modes and static deformations).One is thus brought to solve a classical harmonic problem, of reduced size, with which an equationwith linear constraint is associated. Its resolution thus does not pose a problem.




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Parameters e and e damping of Rayleigh are introduced, if necessary, by the operatorDEFI_MATERIAU [U4.43.01].

The treatments of the substructures are identical to the case of modal calculation.

The operator MACR_ELEM_DYNA [U4.65.01] calculates the generalized matrices of rigidity, mass andpossibly of damping of the substructure, as well as the matrices of connection. The damping ofRayleigh is taken into account by supplementing the operand MATR_AMOR. Damping proportional isintroduced by the operand AMOR_REDUIT.

The harmonic loading is defined, on the level of the substructure, by the operators AFFE_CHAR_MECA[U4.44.01] (application of the force on the grid), CALC_VECT_ELEM [U4.61.02] (calculation of theassociated elementary vectors) and ASSE_VECTEUR [U4.61.23] (assembly of the vector of loading onthe grid of the substructure).

5.4.2 Assembly and resolution

As in the case of modal calculation, the model of the complete structure is defined by the operatorDEFI_MODELE_GENE [U4.65.02]. Its classification is carried out by the operator NUME_DDL_GENE[U4.65.03]. The matrices of mass, rigidity and possibly of damping generalized of the structuresupplements are assembled according to this classification with the operator ASSE_MATR_GENE[U4.65.04].

The loadings are projected on the basis of substructure to which they are applied, then assembledstarting from classification resulting from NUME_DDL_GENE [U4.65.03] by the operatorASSE_VECT_GENE [U4.65.05].

The calculation of the harmonic answer of the complete structure is carried out by the operatorDYNA_LINE_HARM [U4.53.11].

5.4.3 Restitution on physical basis

The restitution of the results on physical basis is identical to the case of modal calculation. It utilizesthe operator REST_GENE_PHYS [U4.63.31] and possibly the operator DEFI_SQUELETTE [U4.24.01](creation of a grid “skeleton”).

5.5 Conclusion

Method of calculating of answer harmonic per modal synthesis available in Code_Aster rest on that ofmodal under-structuring, also programmed. It consists in expressing the whole of the equations in aspace of reduced size, made up by modes of the various substructures, by a method of Rayleigh-Ritz.The definition of these fields is that used for the modal under-structuring and understands normalmodes as well as other statics or harmonics. The procedure employed results in a projection of thematrices and the second member on restricted space.




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6 Transitory response by under classical dynamic structuring

6.1 Introduction

The object of this reference material is to present the theoretical bases of the two methods ofcalculating of transitory answer per dynamic under-structuring available in Code_Aster. The firstconsists in carrying out a transitory calculation by under-structuring for which the equations of theproblem are projected on the bases associated with each substructure. The difficulty lies in the doubledualisation of the boundary conditions which leads to a matrix of singular mass. To use the explicitdiagrams of integration (which require the inversion of the matrix of mass), it is thus necessary tomodify the treatment of the interfaces in the operator of calculation of the transitory answer. Thesecond method consists in determining the clean modes of the complete structure by under -structuring and projecting on this basis the equations of the transitory problem. The stage of restitutionon the basis of physical final generalized result must thus take account of this double projection.

The operator of transitory calculation of answer which receives the under-structuring is the operatorDYNA_TRAN_MODAL [U4.53.21]. Being based on methods of modal recombination, it was conceived tosolve transitory problems in generalized coordinates and it is very effective for the problems of bigsize of which he makes it possible to reduce the number by degrees of freedom. In addition, itsupports the taking into account of localised non-linearities (with the nodes) which one wishes togeneralize with the case of the under-structuring.

In this report, we present the two methods of calculating transient by under - structuring available inCode_Aster, like their implementation.

6.2 Transitory calculation by projection on the basis of substructure

6.2.1 Dynamic equations checked by the substructures separately

That is to say a structure S composed of N S noted substructures Sk . We suppose that eachsubstructure is modelled in finite elements. The vibratory behavior of the substructures results fromthe forces external which are applied to him and from the bonding strengths which on them the othersubstructures exert. Thus, for Sk , we have:

Mk qkCk qk

Kk qk=f ext

kf L

k (6.2.1-1)

The field of unknown displacement, resulting from modeling finite elements, is required on a spaceadapted, of reduced size (transformation of Ritz) according to the expression:

qk=

k

k (6.2.1-2)

The transformation of Ritz [éq 6.2.1-2], applied to real and virtual displacement led to the transitorydynamic equation of the substructure [éq 6.2.1-1], allows to write:

Mk kCk k K k k=f extk f L

k (6.2.1-3)

The problem defined by the equation [éq 6.2.1-3] is symmetrical. In addition, its dimension isdetermined by the number of modes taken into account (dynamic modes and static deformations).One is thus brought to solve a classical transitory problem but of reduced size.




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6.2.2 Dynamic equations checked by the total structure

The matric writing of the dynamic equation checked by the total structure, is written simply startingfrom the dynamic equations checked by each substructure:

[M1

...Mk

...MNS

]{

1

...k

...

N S

}[C1

...Ck

...CN S

] {

1

...

k

...N S

}[K 1

...Kk

...KN S

] {

1

...

k

...

N S

}={

f ext1

...f extk

...f extN S}{

f L1

...f Lk

...f LN S}

Which, it is necessary to add the equations of connection:

∀ k , l LS k∩Sl

k { k }=LS k∩S l

l { l } f LSk∩S L

k=−f LS k

∩S L

l

This system can be written in the condensed form:

Mk kCk k K k k=f extk f L

k (6.2.2-1)

L =0 (6.2.2-2)

∀ k ,l f LSk∩S L

k=−f LS k

∩S L

l (6.2.2-3)

6.2.3 Double dualisation of the boundary conditions

The condensed problem, given above, arises in the form of a transitory system with which isassociated a linear equation of constraint (in force and displacement). In Code_Aster, this kind ofproblem is classical and it is solved by double dualisation of the boundary conditions [R3.03.01],i.e. by the introduction of auxiliary variables still called multipliers of Lagrange for dualiser theboundary conditions. After introduction of the multipliers of Lagrange, the matric system is put in theform:

[0 0 00 M 00 0 0 ] {

1

2}[ 0 0 0

0 C 00 0 0 ] {

1

2}[

−Id L IdLT K LT

Id L −Id ] { 1

2}={

0f ext0 } (6.2.3-1)

where 1 and 2 are the multipliers of Lagrange.

It is noted, the introduction of the multipliers of Lagrange makes singular the matrix of mass.Consequently, the use of the diagram of integration explicit developed in the operatorDYNA_TRAN_MODAL [U4.53.21] of Code_Aster is impossible because they require the inversion of thematrix of mass.




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To make the matrix nonsingular, it is enough to dualiser with the same multipliers of Lagrange, thecondition on the derivative second of the equations of connection.

Thus the condition of continuity of displacements [éq 6.2.2-2] is modified by the equivalent system:

∀ t , L=0⇔∣L =0L °=0 et L °=0 (6.2.3-2)

where ° and ° are the displacements and speeds generalized at the initial moment.

The matric system which results from this is form [bib10]:

[−Id L IdLT M LT

Id L −Id ] {1

2}[0 0 0

0 C 00 0 0 ] {

1

2}[

−Id L IdLT K LT

Id L −Id ] {1

2}={

0f ext0 } (6.2.3-3)

It is noted that the matrix of mass has the same form as the matrix of stiffness. They are thusinvertible. This system is thus perfectly equivalent to the equation [éq 6.2.3-1] (it checks the conditionsof connection at any moment) and it can be treated, in this form, by the operator DYNA_TRAN_MODAL.

6.2.4 Treatment of the matrix of damping

It is noted that the condition of continuity of displacements, formulated in the equation [éq 6.2.3-2],results in an equation of the second order not deadened. At the time of the resolution by step of timeof a transitory problem, any digital error is likely auto--to discuss, thus decreasing the stability of thealgorithm. To optimize the damping of the digital error, it is enough to dualiser the condition on thederivative first of the equations of connection with the same multipliers of Lagrange multiplied by 2 (soas to make this damping critical):

∀ t , L =0⇔∣L 2 =0L °=0 et L °=0

(6.2.4-1)

The matric system which results from this is form [bib10]:

[−Id L IdLT M L

T

Id L −Id ]{ 1

2}[

−2 Id 2 L 2 Id2 L

T C 2 LT

2 Id 2 L −2 Id ]{ 1

2}[

−Id L IdLT K L

T

Id L −Id ]{ 1

2}={

0f ext0 }

It is thus noted that the treatment of the digital error on the equations of connection results inmodifying the matrix of damping of the transitory problem. This modification is completely comparableto that which is carried out on the matrix of mass.

On the other hand, we did not wish to generalize this treatment the case of the resolution of the notdeadened transitory problems. That would have led us to create a matrix of temporary damping. Onecould have feared to increase the computing times, without real benefit. Moreover, it is completelypossible to the user to define a matrix of damping whose coefficients are worthless. The modificationof this one being automatic, the transitory system not deadened will be actually solved, whileoptimizing the treatment of any digital error intervening on the equations of connection.

6.2.5 Treatment of the initial conditions




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Let us consider a substructure Sk characterized by its base of projection k composed of normal

modes and static deformations. It is supposed that initially the substructure Sk is subjected to a field

of displacement or speed (that does not modify of anything the demonstration) noted q0k . The

transformation of Ritz enables us to write:

q0k=

k 0

k

where 0k is the vector displacements (or speeds) generalized (E) S of Sk to determine.

The vector displacements (or speeds) generalized (E) S initial (ales) is given as follows:

q0k=

k0

k⇒

kT

q0k=

k T

k⋅0

k

⇒ 0k=k T

k −1 k T

q0k

6.3 Transitory calculation on a total modal basis calculated by under-structuring

The second developed method consists in solving the transitory problem on the basis of completemodal structure calculated by under-structuring.

6.3.1 Calculation of the clean modes of the structure supplements by under -structuring

Each substructure Sk is represented by a base of projection, is composed of dynamic clean modes

and static deformations, which we noted k . The base of projection of the structure supplements

which results from it is noted .

The modal base of the complete structure is calculated by under-structuring. Each mode obtained isthus linear combination of the vectors of the bases of projection of the substructures:

p=∑k=1

N S

k k= (6.3.1-1)

where:

p is the matrix of the clean modes of the complete structure,

is the matrix of the generalized modal coordinates of the structure.

The projection of the matrices and the vectors constitutive of the transitory problem, on the basis ofclean mode of the complete structure calculated by modal synthesis makes it possible to determine:

• the matrix of generalized mass: M=T M

• the matrix of generalized rigidity: K=T C

• possibly the matrix of generalized damping: C=T C

• the vector of the generalized external forces: f ext=T f ext

Because of orthogonality of the clean modes of the structure calculated by modal synthesis, comparedto the matrices M and K , the matrices of generalized mass and rigidity obtained Ci - above arediagonal:




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M= T M= TM = pTM p

K=T K= TK = pTK p

(6.3.1-2)

6.3.2 Dynamic equation checked by the total structure

The complete structure is subjected to the external forces which are applied to him. Thus, we canwrite:

MqCqK q=f ext (6.3.2-1)

The field of unknown displacement, resulting from modeling finite elements, is replaced by itsprojection on the basis of clean mode of the structure, according to the formula:

q= p (6.3.2-2)

where is the vector of the generalized coordinates of the structure.

The transformation of Ritz [éq 6.3.2-2], applied to the transitory dynamic equation of the structure [éq6.3.2-1], allows to write:

M C K=f ext (6.3.2-3)

The stage of restitution on physical basis requires to take account of the double projection (on thebasis of modal complete structure, then on the basis of projection of under - structures - cf éq 6.3.2-4).

q= p= (6.3.2-4)

The problem defined by the equation [éq 6.3.2-3] is of completely classical form. One is brought tosolve a symmetrical transitory problem whose dimension is determined by the number of modescalculated by under-structuring and whose matrices of mass and rigidity are diagonal.

Let us note finally that the treatment of the initial conditions is identical to the case of transitorycalculation by projection on the basis of substructure (cf § 6.2).

6.4 Comparative study of the two developed methods

Theoretical bases, associated with the two methodologies put in work in Code_Aster to carry out atransitory calculation of answer by using the techniques of under-structuring, were presented in thepreceding chapters. We specify, here, their essential characteristics.

The first methodology consists in doing a calculation of transitory response by under - structuring. Theequation checked by the complete structure is then projected on the basis of substructure. Theprecision of this method thus is directly determined by the extent of these bases. These last can beenriched without leading to prohibitory computing times because the substructures are, in theory, ofrelatively reduced sizes. At all events, it is difficult to estimate the effect of modal truncation with theonly knowledge of the modes of the bases of the substructures. In addition, the bases of projection ofthe substructures are made up of modes which all are not orthogonal between them (clean modes andstatic deformations). The matrices of generalized mass and rigidity constitutive of the final transitoryproblem are thus not-diagonals. All in all, their bandwidth can be given starting from the number ofstatic deformations of the bases of projection of the substructures. Duration of integration in theoperator of transitory calculation DYNA_TRAN_MODAL will be thus all the more long as there will bedegrees of freedom of interface. In addition, the step of acceptable time of integration maximum bythe diagram of integration explicit is given starting from the maximum frequency of the base ofprojection. In the case of a transitory calculation by under-structuring, this frequency results, in theory,




5b0a507feb90

of the static modes whose diagonal terms are high in the matrix of rigidity generalized and weak in thematrix of generalized mass. Consequently, the step of time of integration cannot be a priori given. Theexperiment shows that it is very weak, with the glance as of Eigen frequencies of the bases of thesubstructures and that the use of the diagram of integration to step of adaptive time ofDYNA_TRAN_MODAL is very advantageous.

The second methodology consists in doing a transitory calculation on the basis of complete modalstructure obtained by under-structuring. It is known that the stage consisting in calculating the cleanmodes structure can be expensive in term of computing time. This is all the more true when nonlinearforces are considered because the base of projection must then be sufficiently wide for representingthe dynamics of the system well. In addition, the modal base on which is calculated the transitoryanswer is generally of size lower than that determined by the clean vectors of the substructures (cleanmodes and static deformations). Double projection thus amounts introducing a cut-off frequency. Onemust thus expect that this method is less precise than the preceding one. However, the calculation ofthe clean modes makes it possible to estimate the effect of modal truncation. In addition, it can makeit possible to validate the models of under-stuctures if one has experimental results. Finally, theessential interest of this method is that the matrices of mass and rigidity used in transitory calculationare diagonal. Digital integration is thus very fast.



If one wishes to introduce a damping of Rayleigh, parameters e and e of this damping are defined,by the operator DEFI_MATERIAU [U4.43.01].The treatments of the substructures are identical to the case of modal and harmonic calculation. Thedynamic clean modes are calculated with the operator CALC_MODES [U4.52.02]. The conditions withthe interfaces of connection are applied with the operator AFFE_CHAR_MECA [U4.44.01].

The operator DEFI_INTERF_DYNA [U4.64.01] allows to define the interfaces of connection of under -structure. The operator DEFI_BASE_MODALE [U4.64.02] allows to calculate the base of completeprojection of the substructure (recopy of the clean modes and calculation of the static deformations).

The operator MACR_ELEM_DYNA [U4.65.01] calculates the generalized matrices of stiffness, mass andpossibly of damping of the substructure, as well as the matrices of connection. The damping ofRayleigh is taken into account by supplementing the operand MATR_AMOR. Damping proportional isintroduced by the operand AMOR_REDUIT.

The transitory loading is defined, on the level of the substructure, by the operators AFFE_CHAR_MECA[U4.44.01] (application of the force on the grid), CALC_VECT_ELEM [U4.61.02] (calculation of theassociated elementary vectors) and ASSE_VECTEUR [U4.61.23] (assembly of the vector of loading onthe grid of the substructure).

The operator CREA_CHAMP [U4.72.04] which makes it possible to affect a field on the nodes of amodel makes it possible to describe the field of initial displacement ou/et the initial field speed of thesubstructure.

6.5.2 Assembly of the generalized model

As in the case of modal and harmonic calculation, the model of the complete structure is defined bythe operator DEFI_MODELE_GENE [U4.65.02]. Its classification is carried out by the operatorNUME_DDL_GENE [U4.65.03]. The matrices of mass, stiffness and possibly of damping generalized ofthe structure supplements are assembled according to this classification with the operatorASSE_MATR_GENE [U4.65.04].

The loadings are projected on the basis of substructure to which they are applied, then assembledstarting from classification resulting from NUME_DDL_GENE [U4.65.03] by the operatorASSE_VECT_GENE [U4.65.05].




5b0a507feb90

Initial generalized displacements and initial speeds generalized for each substructure, are calculatedby the operator ASSE_VECT_GENE [U4.65.05]. This operator also carries out the assembly of thesevectors according to classification resulting from NUME_DDL_GENE [U4.65.03].

In the case of a transitory calculation project on the “bases” of the substructures, the assembledmatrices and vectors generalized obtained with resulting from this stage are directly used for transitorycalculation. In the case of a calculation on the basis of complete modal structure calculated by under -structuring, it is necessary to carry out specific operations which are presented to the § 5.3.

6.5.3 Calculation of the modal base of the complete structure and projection

This chapter is specific to transitory calculation on modal basis calculated by under - structuring.

The modal base of the complete structure is calculated with the classical operator of Code_Aster:CALC_MODES [U4.52.02]. One defines a classification of the final problem generalized with theoperator NUME_DDL_GENE [U4.65.03]. The matrices of mass, stiffness and possibly of dampinggeneralized are projected on the basis as of clean modes of the structure with the operatorPROJ_MATR_BASE [U4.63.12]. The generalized vectors corresponding to the external loadings areprojected on the basis of clean mode of the structure with the operator PROJ_VECT_BASE [U4.63.13].

6.5.4 Resolution and restitution about physical base

The calculation of the transitory answer of the complete structure is carried out by the operatorDYNA_TRAN_MODAL [U4.53.21].

The restitution of the results on physical basis utilizes the operator REST_GENE_PHYS [U4.63.31]; it isidentical to the case of modal and harmonic calculation. One can use the operator DEFI_SQUELETTE[U4.24.01] to create a grid “skeleton”. Coarser than the grid of calculation, it makes it possible toreduce the durations of the graphic treatments.

6.6 Conclusion

We presented, in this report, the works completed to introduce, in Code_Aster, the calculation oftransitory answer linear per dynamic under-structuring. The methods which were selected consist, forthe first of them, to project the transitory equations on the “bases” of each substructure, composed ofdynamic clean modes and static deformations and for the second, to calculate the clean modes of thecomplete structure by under-structuring and to project the transitory equations there.

We begin with a talk from the theoretical bases on which rest the first method of under-structuringtransitory to lead to the matric formulation of the final problem. In particular, an original treatment ofthe equation of continuity of displacements to make the matrix of mass invertible and to ensure anoptimal stability of the algorithm of integration, led us to modify the shape of the matrices of mass anddamping of the transitory problem.

For the second method, the essential difficulty consists in restoring the results got in coordinatesgeneralized on the physical basis. Indeed, it is necessary to take account of the double projection: onthe basis of modal structure supplements on the one hand, and on the basis of under - structures onthe other hand.

The developments carried out resulted in modifications of the operators DYNA_TRAN_MODAL[U4.53.21] and REST_GENE_PHYS [U4.63.31]. Their syntax was modified very little, so that their use isidentical during a calculation by under - structuring and of a direct calculation by modal recombination.




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7 Dynamic condensation by dynamic macronutrients in staticunder-structuring

7.1 Introduction

The objective of the dynamic condensation of model per static under-structuring is to reduce the fullnumber of degrees of freedom of a model made up of several under-fields by condensing theresolution of the under-fields of linear behavior on the degrees of freedom of their interfaces withunder-fields of nonlinear behavior.

The idea is then to represent each one of these under-fields by macronutrient of a static type to beable to use it in a mixed model also including finite elements for the parts non-condensed of nonlinearbehavior in order to carry out nonlinear analyses by means of the nonlinear operators of calculationSTAT_NON_LINE [U4.51.03] or DYNA_NON_LINE [U4.53.01].

In the condensation of each under-field, one wants to also integrate the dynamic part of his behaviorobtained by modal analysis and for this reason, one will thus use dynamic macronutrients like thestatic macronutrients necessary in static under-structuring.The principle of this conversion and their application is described in the documents [bib13].

7.2 Classical problem and principle of the method

Figure 7.2-a: Under-fields of the classical problem of condensation.

On the figure 7.2-a above are schematized the possible under-fields which one can meet within theframework of a classical problem of condensation per static under-structuring:

• under-fields of the type E1 : under-fields of structure of constant linear behavior condensedand modelled by dynamic macronutrient,

• under-fields of the type E2 : under-fields of structure of behavior potentially nonlinear non-condensed and modelled by finite elements,

• under-fields of the type E3 : under-field of ground represented by a macronutrient resultingfrom the conversion of a matrix complexes impedance of ground.

The basic principle of condensation by macronutrient consists with partitionner for each under-field ofthe type E1 degrees of freedom j their noted interfaces with the under-fields of the type E2like their other degrees of freedom i said “internal”.

Matrices of rigidity K and of mass M break up thus on these degrees of freedom:




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K=[ K ii K ij

K ji K jj] M=[ M ii M ij

M ji M jj]

i ddl internesj ddl d'interfaces

The static principle of under-structuring returns for the matrix K for example to condense it only on

the degrees of freedom j interfaces by correcting the initial term K jj of a complementary termcalled “complement of Schur” according to the principle of the condensation known as of Guyan[bib14]:

K jj=K jj−K ijT K ii

−1K ij=

T K (7.2-1)

The calculation of the new term K jj cost with a projection of the matrix K on a basis of static

modes of “constrained” type obtained by static answer to the unit displacements imposed in each

degree of freedom j interfaces :

=[ ij

I ]=[−K ii−1 K ij

I ]

Condensation on the degrees of freedom j interfaces matrix of mass M and of the loads of theunder-fields of the type E1 will be done in a similar way by the matric product of the matrix and the

vectors of loading on the basis of static modes .

It is noted whereas the terms T K and

T.M . can be directly obtained for each under-field of

the type E1 by means of the operator of calculation of a dynamic macronutrient MACR_ELEM_DYNA

[U4.65.01] and that the terms of loads projected are obtained T .F by means of the operator

PROJ_VECT_BASE [U4.63.13].

7.3 Methods of condensation per complete or reduced base

According to the way of calculating of the modes of representation of the movements of the interfaces,two methods are currently used:

7.3.1 Method of calculating by complete base

The movement U of an unspecified point of a under-field of the E1 type express yourself starting

from the decomposition on the static modes and on a complement brought by the dynamic part of

the movement expressed by clean modes :

U= .q .q (7.3.1-1)

With the method by complete base, on the interface , clean modes have a worthless

contribution and the static modes have an unit value and thus displacement U of a point of theinterface expresses itself:

U=X=Id .q (7.3.1-2)

In this case, physical degrees of freedom X interface merge with the generalized coordinates

q associated with the static modes .




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Generalized coordinates q associated with the clean modes being worthless on the interface,

their resolution will be uncoupled from that of the generalized coordinates q . In the assembly of the

matrices of the problem condensed, the terms of projection on the clean modes T .K . and

T .M . will be juxtaposed. To be able “to mix” physical degrees of freedom with generalized

coordinates, one uses the artifice transparency for the user to make pass these last for additionalphysical degrees of freedom taken among the internal degrees of freedom to a field of the type E1 .

This alias is active in the model where fields of the type E1 are condensed by macronutrients, but tohave the true displacement of the internal degrees of freedom to these fields, it is obtained inpostprocessing after resolution of the total problem assembled by means of the operatorREST_COND_TRAN [U4.63.33] by applying the initial relation:

U= .q .q (7.3.1-3)

On the level of the assembly of a under-field of the type E1 , the initial storage represented on thefigure 7.3.1-a understands terms at the same time on internal degrees of freedom and the degrees offreedom of interface with a profile of noncomplete filling.

Figure 7.3.1-a: Initial assembly of the under-fields E1 before condensation.

After condensation by the complete method, most internal degrees of freedom disappear, but an anysmall portion is devoted to represent them generalized coordinates associated with the clean modesand the terms of the static macronutrient are added to the degrees of freedom of interface with aprofile of complete filling this time. Resulting storage is represented on the figure 7.3.1-b below.

Figure 7.3.1-b: Assembly of the under-fields E1 with complete method.

The assembly of the terms of the static macronutrient is carried out by simple summation on thedegrees of freedom of interface and in general does not require an additional connection on theinterface when one takes into account all the constrained static modes of the interface. It is not thus inthe case individual of condensation on only one node of interface where there it is necessary to add arelation of connection, solid on the degrees of freedom of interface.




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In the case of under-fields of the type E3 , it postprocessing by REST_COND_TRAN is uselessbecause it displacement does not have a component on dynamic modes and displacement with theinterface is definitively obtained as of the resolution of the total problem.

7.3.2 Method of calculating by modal reduction

With the method by modal reduction, the movement U of an unspecified point of a under-field of thetype E1 always express yourself starting from the decomposition on the static modes and on a

complement brought by the dynamic part of the movement expressed by clean modes :

U= .q .q (7.3.2-1)

But this time, on the interface , clean modes have a nonworthless contribution because the

static modes have a nonunit value and are of number generally much lower than the number of

physical degrees of freedom X interface who, in this case, do not merge with the coordinates

generalized q associated with the static modes . It is thus necessary to distinguish in theassembly and the resolution the terms associated with the degrees of freedom physical and thoseassociated with the generalized coordinates. In the same way that for the coordinates generalized qassociated with the clean modes , one uses the artifice transparency for the user to make pass the

generalized coordinates q associated with the static modes for additional physical degrees of

freedom also taken among internal degrees of freedom to a field of the type E1 .

But this time, the resolution of the physical degrees of freedom X interface will not be

uncoupled from that of the generalized coordinates q . In the assembly of the matrices of the

problem condensed, the terms of projection on the clean modes T .K . and

T .M . will bejuxtaposed. But it will then be necessary to add a connection and thus degrees of freedom ofLagrange between the physical degrees of freedom X interface and generalized coordinates

q . This connection is expressed by the relation:

X= .q (7.3.2-2)

This relation was expressed by the keyword répétable LIAISON_INTERF of the operatorAFFE_CHAR_MECA [U4.44.01], which makes it possible to connect each under-field of the type E1with under field of the type E2 (cf appears 7.2-a) by generating a typical case of LIAISON_DDL withthe coefficients of the preceding linear relation.

Then, to have the true displacement of the internal degrees of freedom to the fields of the type E1 ,one proceeds in the same way in postprocessing after resolution of the total problem assembled bymeans of the operator REST_COND_TRAN [U4.63.33] by applying the initial relation:

U= .q .q (7.3.2-3)

After condensation by the method of modal reduction, most internal degrees of freedom disappear, butan any small portion is devoted to represent them generalized coordinates associated with the cleanmodes, another small portion is devoted to the generalized coordinates associated with the cleanmodes of interface. This time, the terms of the static macronutrient are not added to the degrees offreedom of interface which preserve a profile of noncomplete filling then. But one obtains aquadrangular block then of degrees of freedom of Lagrange enters the physical degrees of freedomX

interface and generalized coordinates q .Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Resulting storage is represented on the figure 7.3.2-a below.

Figure 7.3.2-a: Storage of the under-fields E1 with method of modal reduction.

It will be noticed that there is no choice of calculation imposed for the base of reduction of modes ofinterface . A possible choice will be that used in interaction ground-structure [bib15] by consideringa base of clean modes of the under-field of the type E1 on carpet of springs representative of therigidity of the adjacent fields of type E2 or E3 .




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8 Indications for the back testing of the quality of the scalemodel

One of the principles making it possible to validate the quality of a scale model consists in estimating,a posteriori, the variation with the balance which can exist after calculation.

8.1 Case general for a calculation on a scale model

To characterize this variation with balance, one uses the principle of the residue in effort. This residuein effort F r is calculated

• for the temporal answers:

[ K e T r ]qr t [ D v T r ] qrt [ M T r ] qr t −Bu t =F rt (8.1-1)

• for the harmonic answers:

[K e j KhK v s s Dvs2 M ]T r qr s −B u s=F r s (8.1-2)

where K e is the matrix of stiffness of the structure, Dv the matrix associated with viscous

dissipation, and M the matrix of mass. One notes in addition qr the vector of the degrees of

freedom reduced, and T r the base of associated reduction. The external term of efforts is defined by

the product of a matrix of localization of the efforts B and of a vector u specifying the temporalevolution of the excitation. This writing makes it possible to separate the space segments andtemporal from an often noted effort f (t ) . Concerning the harmonic answer, Kh is the matrix

representing the behavior hysteretic, K v the matrix representing the viscoelastic behavior.

One can also adopt the same approach for the calculation of the complex clean modes, detailed indocumentation on the construction of the scale models in dynamics [U2.06.04]

If the solution obtained is exact, then the residue F r is null, by definition. In the contrary case, onecan build a measurement of the variation to balance starting from the static response of the structureto it loading. One notes Rr the residue in displacement associated with each calculated complexmode. By definition, one has

K e Rr=F r (8.1-3)

A measurement of the variation to balance is given by the elastic potential energy of the structure, thatis to say

E r=12∥Rr

T K e Rr∥2

(8.1-4)

An example of use of this technique is given in documentation [U2.06.04]

8.2 Typical case of the calculation of the modes of a generalized model

The same principle can be put in work, but in a way a little more precise, kind to reveal variationsparticular to various balances.

8.2.1 Presentation




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In the case of a calculation by under structuring with use of modes of interfaces, it is important to beable more precisely to determine the origin of a vague calculation. The sources of inaccuracies are:

• the bad representation of displacements to the interfaces, for each under structure, • the dynamic behavior of the internal parts, for each under structure, • differential displacements (separation) being able to appear with the interfaces between under • structures.

The order CALC_CORR_SSD [U4.52.16] allows, on the basis of calculation of work associated withthese various efforts, to build terms of corrections allowing to separately improve the behaviors ofinterfaces, and the dynamic behaviors internal of each one of under structures.

NB:It will in addition be necessary to take care, in the process of enrichment, with orthogonaliserseparately the vectors with fixed interfaces, and the vectors with free interface, as it is carriedout in the case test SDLS122 [V3.02.122], before the concaténer and to build the modal bases.

8.2.2 Calculation of the indicators and associated corrections

To detail the construction of the various indicators and corrective terms built in the orderCALC_CORR_SSD, let us consider a simple example made up of the assembly of 2 pennies structurealong one only interface. This approach spreads easily, but led to heavy notations.

8.2.2.1 Definition of the coupled problem

For each under structure, one introduces the matrices of mass and stiffness. One notes c degrees offreedom associated with the internal parts (“complementary”), and i degrees of freedom associatedwith the interfaces. One introduces also the base of associated reduction, composed at the same timeof modes with fixed interfaces c , and of modes of interfaces i , statements statically on the rest ofthe structure, so that corresponding displacements on the complementary degrees of freedom are

c=−K cc−1K cii

KS=[K ii K ic

K ci K cc ]S

/ M S=[ M ii Mic

M ci M cc]S

/ T S=[ 0 i

c c]S

(8.2.2-1)

The matrices of reduced mass and stiffness are written

K rS=T ST K S T S / M rS=T S

T M S T S (8.2.2-2)

Because of introduction of modes of interfaces, the relation of compatibility of displacements ofinterfaces between under structures 1 and 2 cannot be written any more q i 2=qi 1 , but becomes, ifunder structure 2 is slave:

T iS 2T T iS 11=T iS 2

T T iS 22 (8.2.2-3)

One defines then a base of the core of the constraints to eliminate the relations from interface, on thebasis of element present in the section 2.5

Qi=Ker [ T iS 2T T iS 1 −T iS 2

T T iS 2 ] ⇒ (8.2.2-4)

The complete model is then assembled, then project on a basis of the core of the constraints:

Kg=QT [K r1 00 K r2

]Q / M g=QT [M r1 00 M r2

]Q (8.2.2-5)




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One notes generalized degrees of freedom associated with the problem thus obtained. One thus

has, if one notes q physical degrees of freedom, q=T rqr=T rQ .

8.2.2.2 Variation with balance If one notes Z0=K g−0 M g dynamic impedance associated with the clean mode of pulsation 0 ,

and 0 the associated reduced clean mode, one has

QT T rT [ Z 0 ]T r Q 0=0 (8.2.2-6)

Under these conditions, the relation making it possible to quantify the variation with the balance of thisparticular mode becomes:

[Z 0cc 1 Z 0ci 1 0 0Z 0ic 1 Z 0ii 1 0 0

0 0 Z0ii 2 Z0ic 2

0 0 Z 0ci 2 Z 0cc 2][c 1 c 1 0 0

0 i 1 0 00 0 i 2 00 0 c 2 c 2

][Id 0 00 Qi 1 00 Qi 2 00 0 Id

]{0c 1

0i

0c 2}={

f 0c 1

f 0i 1

f 0i 2

f 0c 2

} (8.2.2-7)

If the base of projection is of good quality, then the variation with balance must be weak, and, ideally,one must even check

{f 0c 1

f 0i 1

f 0i 2

f 0c 2

}≈{0000} , (8.2.2-8)

the solvor having to ensure

{0c 1 0i 1 0i 2 0c 2 }{f 0c 1

f 0i 1

f 0i 2

f 0c 2

}≈0 / 0=T r Q0 (8.2.2-9)

To ensure oneself to have a model of good quality, one must thus ensure the nullity of each term,independently from/to each other. One must thus check the nullity of work of the residual effortsassociated with the loads in each under structure, that is to say

0c 1T f 0c 1=0 / 0c 2

T f 0c 2=0 (8.2.2-10)

One must also check the nullity of the virtual work associated with the residual efforts with theinterface:

0i 1T f 0i 10i 2

T f 0i 2=0 (8.2.2-11)

Lastly, one must also check the compatibility of displacements, and avoid differential displacements ofinterfaces (separations)

0i 2−C 0i 1 T

f 0i 2=0 (8.2.2-12)




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C Is the operator of projection of displacements of the interface Master on the interface slave. In thecase of compatible interfaces, C is the matrix identity.

In practice, to be more simply exploitable, the various work calculated in CALC_CORR_SSD are dividedby the own pulsation of the mode considered. This handing-over on the scale makes it possible tocompare the levels of errors between the various modes.

8.2.2.3 Corrections associated with the residual efforts

Each one of these term makes it possible to define associated corrections. Thus, for each understructure k , one builds under space dedicated to the enrichment based on the principle of the staticcorrection (cf [U2.06.04]):

T Enrich=[ 0 T i I

T c C T c I] (8.2.2-13)

With,

K cc k T c C= f 0c k (8.2.2-14)

And

[K ii k K ic k

K ci k K cc k]{T i I

T c I}={ f 0i

0 } (8.2.2-15)

If the matrix of stiffness would be singular for under structure k , one regularizes the problem by

building a matrix shiftée in mass K−2 M , where is a low value in front of the own pulsations of

the first flexible modes of the structure.

8.2.2.4 Corrections associated with differentials displacement with interfaces

If the calculated coupled modes do not ensure perfectly the continuity of displacements of interfaces,it is possible to enrich displacements on both sides of the interface. Two cases should however herebe considered, according to whether, for under structure k , the interface considered is Master orslave. If the interface is an interface slave, one is then satisfied to project the main displacement ofthe interface on the interface slave, and to statically note this displacement on the rest of understructure. The base of complete reduction is written then

T Enrich−Slav=[ 0 T i I Ci Mast

T c C T c I T c Slav] / T c Slav=−K cc

−1 K ci C i Mast (8.2.2-16)

If the interface considered is Master, it is then necessary to extrapolate the movements of theinterface slave. The adopted approach consists in building a base of the modes of the interfaceMaster, and searching of it the best selection which makes it possible to rebuild displacements of theinterface slave. This combination of modes of interface is then raised statically on the degrees offreedom complementary to under structure considered. If one notes R the operator of extrapolationthus defines, the complete base becomes, for an interface Master:

T Enrich−Mast=[ 0 T i I Ri Slav

T c C T c I T c Mast] / T c Mast=−K cc

−1 K ci Ri Slav (8.2.2-17)

The construction of R is realized as follows:




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• The modes are calculated i Mast interface of under structure Master • One seeks the best linear combination of these modes which makes it possible to represent

displacements of the interface slave. One seeks of which 0 who realizes

0=ArgMin

∥C i Mast −i Slav∥2

(8.2.2-18)

If the standard of the error is higher than 1e-6, then one calculates a number of modes i Mast

superior, and one reiterates as much as the standard of error is not satisfactory. R is written then formally

R= i Mast C i Mast (8.2.2-19)

These corrections are calculated for each interface of each under structure. One separates then eachfamily from vectors have two families to be orthogonalized:

• vectors whose displacement of interfaces are worthless• vectors whose displacements of interface are nonworthless

NB:It will in addition be necessary to take care, in the process of enrichment, with orthogonaliserseparately the vectors with fixed interfaces, and the vectors with free interface, as it is carriedout in the case test SDLS122 [V3.02.122]




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9 Bibliography

1 R. ROY, J. CRAIG & MR. C. BAMPTON: “Coupling of Substructures for Dynamic Analysis”AIAA Newspaper, (July 1968), vol. 6, N° 7, p. 1313-1319.

2 R.H. MAC NEAL: “With hybrid method of component synthesis mode”, Computers andStructures, (1971), vol. 1, p. 581-601.

3 J.F. IMBERT: “Analysis of the Structures by Finite elements” 1979, Cepadues Edition.

4 P. RICHARD: “Methods of dynamic under-structuring in finite elements”. Report EDF HP-61/90.149.

5 P. RICHARD: “Methods of under-structuring in Code_Aster“. Report EDF HP - 61/92.149.

6 J. PELLET: “Code of Mechanics Aster - Handbook of reference: Finite elements in Aster“Key:R3.03.01 “Dualisation of the boundary conditions”.

7 G. JACQUART: “Code of Mechanics Aster - Handbook of reference: Dynamics in modalbase”. Key: R5.06.01 “Methods of RITZ in linear and non-linear dynamics”.

8 T. KERBER: “Specifications: implementation of the harmonic under-structuring” - ReportD.E.R. HP-61/93.053.

9 T. KERBER: “harmonic Under-structuring in Code_Aster“- Report D.E.R. HP - 61/93.104.

10 C. VARE: “Specifications of the implementation of linear transitory calculation by under -dynamic structuring in Code_Aster“- Report D.E.R. HP - 61/94.135/B

11 C. VARE: “User's documentations and Validation of the operators of transitory calculation byunder-structuring” - Report D.E.R. HP-61/94.208/A.

12 F. BOURQUIN & F. HENNEZEL: “Numerical study of year intrinsic component modesynthesis method” – Mathematical Modelling and Numerical Analysis, 1992.

13 O. NICOLAS: Specification of the development of the translation MACR_ELEM_STAT /MACR_ELEM_DYNA. Report EDF/AMA-05.149.

14 Document Aster [U2.07.02]: Note of use of the static under-structuring.

15 Document Aster [U2.06.07]: Interaction ground-structure in seismic analysis with the interface




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10 Description of the versions of the document

Version Aster

Author (S) Organization(S)

Description of the modifications

4 P.RichardEDF-R&D/AMV

Initial text

5 G.Rousseau,C.VaréEDF-R&D/AMV

8.4 C.BodelEDF-R&D/AMA

10.1 G.DEVESAEDF-R&D/AMA

Addition of the § 7 on dynamic condensation by static under-structuring

10.2 M.CORUSEDF-R&D/AMA

Addition of a section on the elimination of the constraints followingthe development around the generalized models.

10.4 F.VOLDOIREEDF-R&D/AMA

Corrections made to the § 5.3.3, 5.2,6.2.2, 6.2.5, 7.

11.1 M.CORUSEDF-R&D/AMA

Addition of a section on the indicators of quality for the scalemodels following the development around the generalized models.Complements of section 2 on the calculation of the modes ofinterfaces.


modal calculation by dynamic under-structuring clas []

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