Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
1/34
Organization (S): EDF/EP/AMV
Handbook of Référence
R4.06 booklet: Under-structuring
Document: R4.06.02
Modal calculation by dynamic under-structuring
traditional and cyclic
Summary:
This report/ratio presents the theoretical bases of the methods of calculation modal per modal synthesis. We begin
by the description of the transformation of RITZ and methods of modal recombination which result from this.
Then, we present the modal synthesis which uses the traditional techniques of under-structuring and of
modal recombination.
The modal computational tools by modal synthesis implemented in Code_Aster, then, passed in
review. We present, first of all, the techniques of traditional dynamic under-structuring of
CRAIG-BAMPTON and of MAC NEAL. Then, we approach the methods of dynamic under-structuring
cyclic. Completely dedicated to the study of the structures with cyclic repetitivity, they benefit the best from
geometrical characteristics of the structure. Methods of CRAIG-BAMPTON and MAC NEAL, developed
within this framework, are exposed.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
2/34
Contents
1 Introduction ............................................................................................................................................ 3
2 modal Synthesis ................................................................................................................................... 4
2.1 Transformation of RITZ .................................................................................................................. 4
2.2 Modal recombination ................................................................................................................... 5
2.3 Modal synthesis ............................................................................................................................. 6
2.3.1 Normal modes ............................................................................................................... 7
2.3.2 The static deformations ........................................................................................................ 7
2.4 Conditions of connection between substructures .................................................................................... 8
3 modal Calculation by traditional dynamic under-structuring .................................................................. 11
3.1 Introduction .................................................................................................................................... 11
3.2 Method of Craig-Bampton .......................................................................................................... 11
3.3 Method of Mac Neal ................................................................................................................... 13
3.3.1 First case .......................................................................................................................... 16
3.3.2 Second case ...................................................................................................................... 16
3.4 Implementation in Code_Aster .............................................................................................. 18
3.4.1 Study of the substructures separately ............................................................................... 18
3.4.2 Assembly and resolution ..................................................................................................... 18
3.4.3 Restitution on physical basis .............................................................................................. 18
4 modal Calculation by cyclic dynamic under-structuring .................................................................... 19
4.1 Introduction .................................................................................................................................... 19
4.2 Cyclic repetitivity ....................................................................................................................... 19
4.2.1 Definition .............................................................................................................................. 19
4.2.2 Propagation of wave .............................................................................................................. 20
4.2.3 Concept of diameters and nodal circles ........................................................................... 21
4.2.4 Boundary conditions .......................................................................................................... 22
4.3 Methods of cyclic under-structuring ...................................................................................... 23
4.3.1 Method of Craig-Bampton ................................................................................................. 23
4.3.2 Method of Mac Neal .......................................................................................................... 28
4.4 Implementation in Code_Aster .............................................................................................. 32
5 Conclusion ........................................................................................................................................... 32
6 Bibliography ........................................................................................................................................ 33
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
3/34
1 Introduction
In front of the complexity of the structures mechanical, often made up of an assembly of several
components, numerical or experimental methods traditional of vibratory mechanics
reveal expensive, sometimes even unusable. In perfect coherence with the modular organization
great projects, the methods of under-structuring seem the most effective means
to make a vibratory study of the whole starting from the dynamic behavior of the components
[bib4].
In this report/ratio, we present, first of all, the theoretical bases of the methods of synthesis
modal. They associate techniques of under-structuring and modal recombination. Each
substructure is represented by a base of projection made up of dynamic clean modes
and of static deformations to the interfaces. The study of the conditions of connection between substructures is
simplified by the consideration of compatible grids.
Then, we present the two techniques of modal calculation per traditional under-structuring,
implemented in Code_Aster [bib6]: methods of Craig-Bampton and Mac Neal. They
distinguish by the use of different bases for the substructures.
Lastly, we present the techniques of modal calculation per cyclic under-structuring. Methods
implemented in Code_Aster [bib5], allowing the calculation of the modes of a structure repetitivity
cyclic starting from the study of the one of its sectors are exposed.
General notations:
:
Maximum pulsation of a system (rad.s-1)
m
M
:
Stamp of mass resulting from modeling finite elements
K
:
Stamp rigidity resulting from modeling finite elements
Q
:
Vector of the degrees of freedom resulting from modeling finite elements
F
:
Vector of the forces external with the system
ext.
F
:
Vector of the bonding strengths applied to a substructure
L
:
Stamp containing the vectors of a base of projection organized in column
:
Vector of the generalized degrees of freedom
B
:
Stamp extraction of the degrees of freedom of interface
L
:
Stamp connection
T
:
Kinetic energy
U
:
Deformation energy
Id
:
Stamp identity
:
Stamp diagonal generalized rigidities
R
:
Stamp residual dynamic flexibility
E ()
Re ()
0
:
Stamp residual static flexibility
Note:
The exhibitor K characterizes the sizes relating to the substructure S K and the sizes
generalized are surmounted by a bar: for example M K is the matrix of mass
generalized of the substructure S K.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
4/34
2 Synthesis
modal
The dynamic under-structuring consists in determining the behavior of a structure from
vibratory characteristics of each one of its components ([bib3] and [bib4]). Methods
implemented in Code_Aster, use simultaneously the traditional techniques of
modal recombination and of dynamic under-structuring.
These methods, although different from that of the finite elements, adopt a step enough
comparable. They reveal three essential stages:
Stage 1: numerical study of each component by the determination of their characteristics
vibratory. Work consists in identifying clean modes and static deformations by
traditional techniques of vibratory mechanics. If one compares each substructure to one
super-element, this stage is similar to elementary calculation.
Stage 2: connection of the substructures. The given vibratory characteristics are used
previously for each component, and one takes account of their liaisonnement. This work constitutes
the stage of under-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
locate physical total structure.
2.1
Transformation of RITZ
The transformation of RITZ is the subject of the reference material [R5.06.01]. We recall here
its principle. For the problem of the numerical determination of the real clean modes of the system
not deadened associated the structure, that we will indicate by clean modes, one is reduced to
resolution of the problem of minimization according to:
Either virtual displacement, one seeks: Min
1 T (
2
K - M)
2
whose solution Q checks:
(K - 2M) Q = 0
éq 2.1-1
The method of RITZ consists in seeking the solution of the equation of minimization on a subspace
space of the solutions. Let us consider the matrix containing the vectors of the base of under
space in question, organized in columns. Restricted with this space of reduced size, the equation
of minimization takes the form:
Min
1 T (
2
)
p
p K
M p
=
-
2
That is to say the required solution:
Q =
éq
2.1-2
it checks:
(K - 2M) = 0
éq 2.1-3
where: is the vector of generalized displacements,
K = TK
M = T
and
M
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
5/34
After having solved the system [éq 2.1-3], obtaining the clean modes in the physical base is done with
assistance of the relation [éq 2.1-2]. The transformation of RITZ thus makes it possible to replace the problem with
eigenvalues initial [éq 2.1-1] by a problem of comparable nature [éq 2.1-3], but of dimension
reduced. The new matrices of rigidity and mass remain symmetrical.
However, this transformation must be used with prudence. Indeed, the new base being
incomplete, an approximation is made on the level of projection. The precision of the final result depends
then choice of the basic vectors and the relative error due to this reduction in the number of unknown factors
must be estimated.
2.2 Recombination
modal
A traditional use of the transformation of RITZ, is the dynamic analysis by recombination
modal. It is usually used for the calculation of the response of a structure to an excitation
low frequency. We will limit ourselves here to the calculation of the response to an excitation of a structure
conservative. In this case, the finite element method enables us to be reduced to the equation
following matric differential:
Mq
! + Kq = fext
éq 2.2-1
If one applies the transformation of RITZ, with as incomplete projection, the first bases
clean modes of the structure, the relation [éq 2.2-1] becomes:
M! + K = fext
éq 2.2-2
Where: F
T
= F
ext.
ext. is the vector of the generalized forces.
The clean modes are orthogonal relative with the matrices of mass and rigidity. The equation
differential [éq 2.2-2] thus revealed diagonal matrices: the system is then made up
uncoupled equations. Each one of them is the equation of an oscillator to a degree of freedom of
type mass-arises which reveals the mass, generalized rigidity and force relating to the mode J
(respectively: mj, K J, F J).
If one considers the transformation of RITZ [éq 2.2-2], on the level of a degree of freedom, one a:
IQ = ij
J
J
Where:
IQ is the ième co-ordinate of the vector Q,
J is the coordinated jème vector,
ij is the component of the ième line and the jème column of the matrix.
It thus appears that the response of the structure is expressed like the recombination balanced of
answers of oscillators to a degree of freedom uncoupled. The transformation of RITZ allows, in it
cases, to define a diagram are equivalent of the structure, which reveals the oscillators with a degree of
freedom associated with the identified clean modes. Their stiffness and their mass are generalized rigidities
(K J) and generalized masses (mj) of the corresponding modes.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
6/34
F
F
F
1
2
Fm
m1
m2
mm
K
K
K
(K, M)
1
2
…
m
<=>
Held primarily being studied in low frequencies, the modal recombination consists with
to use the properties of orthogonality of the clean modes of a structure to simplify the study of its
vibratory answer. In addition to the interest to decrease the command of the numerical problem to solve,
transformation of RITZ, in this case, also makes it possible to uncouple the differential equations and
to release a physical interpretation of the result obtained. According to the frequency of excitation, one will use
a more or less truncated modal base. It is however necessary to estimate the truncation error for
to ensure itself of the validity of the result.
2.3 Synthesis
modal
In a general way, the methods of modal synthesis consist in using simultaneously
dynamic under-structuring (cutting in substructures) and modal recombination on the level
of each substructure. Often confused, by abuse language, with the under-structuring
dynamics, the modal synthesis is only one particular case of this one.
The dynamic under-structuring consists in considering the displacement of a substructure in
overall movement, as its response to the bonding strengths which connect it to the others
components.
The modal synthesis means that one calculates this movement, on the level of each substructure, by
modal recombination. One thus uses a base of projection which characterizes each substructure.
Indeed, if the total structure is too important to be subjected to a modal calculation, dimensions
substructures make it possible to carry out this work. The modal synthesis forces to study initially
separately each component, in order to determine their base of projection.
In the continuation of this chapter, we present the types of modes and deformations static used in
methods of modal synthesis using the following simple example:
q1
q2
I
I
q1
2
J
Q J
1
Substructure 1
Substructure 2
2
The vector of the degrees of freedom of the substructure is characterized by an exhibitor who defines it
number of the substructure, and an index which makes it possible to distinguish the degrees of freedom intern (index
I), of the degrees of freedom of border (index J).
qk
qk
I
=
qkj
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
7/34
One is brought, to study the substructure K, to define an impedance in the level of the degrees of
freedom of connection. Within the framework of the developments carried out in Code_Aster, it is is null,
that is to say infinite.
The basic vectors used in the methods implemented in Code_Aster are:
· normal modes,
· constrained modes,
· modes of fastener.
2.3.1 Normal modes
The clean modes or normal modes are advantageously used as bases projection of
substructures for several reasons:
· they can be calculated or/and be measured,
· they offer interesting properties of orthogonality compared to the matrices of mass and
of rigidity of the substructure.
They can be of two types according to the condition given to the interfaces of connection:
· modes specific to blocked interfaces,
· modes specific to free interfaces.
Substructure 1
Substructure 2
Modes specific to blocked interfaces
Substructure 1
Substructure 2
Modes specific to free interfaces
Let us note that in the case of a free substructure, modes of rigid body (or overall modes)
existing belong to 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
case, it can act of constrained modes or modes of fastener.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
8/34
The constrained modes are static deformations which one joint with the normal modes with interfaces
blocked to correct the effects due to their boundary conditions. A constrained mode is defined by
static deformation obtained by imposing a unit displacement on a degree of freedom of connection, them
other degrees of freedom of connection being blocked.
q=1
q=1
Substructure 1
Substructure 2
Constrained modes
The modes of fastener are static deformations which one joint with the normal modes with interfaces
free to decrease the effect of modal truncation. A mode of fastener is defined by the deformation
statics obtained by imposing a unit force on a degree of freedom of connection, other degrees of
freedom of connection being free.
f=1
Substructure 1
Mode of fastener
In the case of a substructure having of the modes of rigid body (here, substructure 2), its
stamp rigidity is not invertible and it is not possible to calculate its modes of fastener. It is
then necessary to block certain degrees of freedom to make the structure isostatic.
2.4
Conditions of connection between substructures
Let us consider the problem of two substructures bonded S1 and S 2 in a rigid way. So that
the movement of the structure supplements is continuous, it is necessary to impose the equality of displacements of
two components with the interface and the law of action-reaction:
Mr. S1 S2
1 (M) = 2 (M
1
) and
(M) = - 2
U
U
F
F (M
L
L
)
éq 2.4-1
Where:
u1 (M) represents the field of displacements of substructure 1,
u2 (M) represents the field of displacements of substructure 2,
F 1L (M) represents the field of the bonding strengths applied to substructure 1,
F 2L (M) represents the field of the bonding strengths applied to substructure 2.
In Code_Aster, we limit ourselves to the compatible cases of grids. That means that they check
following properties:
· the meshs of each substructure S1 and S2 rest strictly on the same ones
nodes in their intersection,
· the finite elements associated these meshs of connection are the same ones on both sides of
border.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
9/34
Consequently, the condition [éq 2.4-1] is strictly equivalent to the formulation below:
q1
2
1
2
1
2 = Q 1
2
and
F
= - F
éq 2.4-2
S S
S S
L
L
S1S 2
S1S2
Where:
qk
is the vector of the degrees of freedom to the nodes of S1 interface
S2
S1 S2
substructure K,
F K
1
2
L
is the vector of the bonding strengths to the nodes of interface S S of
S1 S 2
substructure K.
Indeed, grids of the two substructures S1 and S 2 coinciding, functions of form
associated the finite elements are the same ones with the interface. It is thus enough to impose the equality on the nodes
interfaces of connection of each substructure to impose the equality on all the field of connection.
Let us introduce the matrices of extraction of the degrees of freedom of interface Bk
:
S1 S2
qk1 2 = Bk1 2qk
éq 2.4-3
S S
S S
By using the equation of projection [éq 2.1-2], the condition of continuity of displacements [éq 2.4-2] and
the formulation applied above to the two substructures, one obtains:
B1
1 1
2
2 2
1
2 = B
S S
S1S2
That is to say:
L1
1
2
2
1
2 = L 1
2
with
Lk 1 2 = Bk
K
éq 2.4-4
S S
S S
S S
S1S2
where:
L1
is the matrix of connection of S1 associated with the S1 interface
S2
,
S1S2
L2
is the matrix of connection of S 2 associated the S1 interface
S2
.
S1S2
As we will see it in the next chapters, whatever the method chosen, the problem
with the eigenvalues of the total structure, provided with its boundary conditions, can be written under
form:
(K - 2M) + LT = 0
éq 2.4-5
L = 0
Matrices of generalized mass and rigidity, the vector of the generalized degrees of freedom and
stamp connection which appears here, are defined on the total structure. They take a form
particular with each method (Craig-Bampton, Mac Neal, traditional, cyclic) which will be clarified more
late. The vector of the multipliers of Lagrange makes it possible to translate the law of action-reaction to which
the interfaces are subjected.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
10/34
It is thus about a traditional problem of search for eigenvalues, to which is associated one
linear equation of constraint. In Code_Aster, this type of problem is solved by double
dualisation of the boundary conditions [R3.03.01].
Thus, one can show that this system is also solution of the problem of minimization of
following functional calculus [bib6]:
F
B
=
1 dt
F (!,).
has
2
where F (,
! ,
1
T
1
T
T
1
2
1,)
2 =
2
K + 2! M! + (1 +) L
2
- (
2
1 -)
2
The variables of this functional calculus are the generalized co-ordinates and the multipliers of
Lagrange 1 and 2 (in a number equal to 2 times the number of equations of connection). The last term of
functional calculus imposes the equality of the coefficients of Lagrange.
The extremum is reached for the values of the variables which cancel the derivative of F, whatever
has and B realities:
- 1 + L + 2 = 0
LT (
2
1 +) + (K - M)
2
= 0
1 + L - 2 = 0
- Id L
Id
0 0
0
1
1
0
T
T
2
L
K
L - 0 M
0 =
0
Id L - I
D 2
0 0
0 2
0
The double dualisation thus leads to a real symmetrical matric problem. One shows [R3.03.01]
that it makes it possible to make the algorithms of triangulation of matrix unconditionally stable.
This method thus makes it possible to treat the connection of interfaces corresponding to basic types
modal different without cost from management of an always delicate elimination. In addition, it is
relatively simple. The major disadvantage of this formulation is to lead to systems
assembled final of size more important than in the case of elimination. Indeed, it
coupling of the matric equations was made by introducing a number of degrees of freedom
additional equal to twice the number of equations of connection. This increase in dimensions
matrices can thus be very important. Let us note that the degrees of freedom of Lagrange introduced
are, in this case, the forces applied to the interfaces to ensure the connection between the two pennies
structures.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
11/34
3
Modal calculation by traditional dynamic under-structuring
3.1 Introduction
After having separately studied the various stages of the under-structuring, and the techniques
that they bring into play, it appears interesting to present the two principal methods of
dynamic under-structuring: method of Craig-Bampton and that of Mac Neal.
Craig-Bampton uses, as bases projection of the substructures, modes constrained and
normal modes with fixed interfaces [bib1].
In addition, Mac Neal uses, as bases projection of the substructures, modes of fastener and
normal modes with free interfaces [bib2].
3.2
Method of Craig-Bampton
The following presentation utilizes only two substructures S1 and S 2, but it is
generalizable with an unspecified number of components. After having studied separately each
substructure, their bases of projection (normal modes with fixed interfaces and constrained modes) are
known. For each one of them (identified by the exhibitor K), one establishes a partition of the degrees
of freedom, distinguishing the vector from the degrees of freedom intern qki and the vector of the degrees of freedom
of connection qkj:
qk
qk
I
=
qkj
Are:
K the matrix of the clean vectors of the Sk substructure,
K the matrix of the constrained modes of the Sk substructure.
The base of projection of S K is characterized by the matrix:
K [K K
=
]
The transformation of RITZ (equation [éq 2.1-2]), enables us to write:
qk
K
qk
I
K
K
I
K
K
= K = [] =
éq 3.2-1
Q
K
J
J
ki is the vector of the generalized degrees of freedom associated the clean modes of Sk,
kj is the vector of the generalized degrees of freedom associated the constrained modes of Sk.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
12/34
However the normal modes are given with fixed interfaces, and each constrained mode is obtained
by imposing a unit displacement on a degree of freedom of connection, others being blocked.
generalized co-ordinates relating to the static deformations are then the values of the degrees of
freedom of connection:
qk
K
J = J
Let us interest in the contribution of the component S K from an energy point of view. Energies
kinetics and of deformation are:
T
T K
1 K T
K K
1
1
2!
Q M!Q
2 (K K
! )
K
K K
K T
K K
=
=
M! = 2! M!
T
The U.K.
1 K T
K K
1
1
2 Q
K Q
2 (K K
) K K K
K T
K K
=
=
K = 2 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. Moreover, we will consider that these
modes are normalized compared to the matrix of mass,
· one can also show that the constrained modes are orthogonal with the modes
normal compared to the matrix of rigidity [bib6].
The matrices of generalized rigidity and mass thus have the following form:
K
K T
K
K
0
K
K
Id
M
K =
M =
K T
éq 3.2-2
0
K K K
K T
Mk K
K T
Mk K
where K is the matrix of the generalized rigidities associated the clean modes of S K.
The choice of the base of projection to blocked interfaces thus leads to a coupling of the modes
normal and of the static deformations by the matrix of mass.
In the case of the calculation of the clean modes of the total structure, external forces applied to
system are null. In addition, on the level of each connection, because of law of action-reaction and of
continuity of displacements, the work of the bonding strengths is null. This is explained physically by
the fact that the bonding strengths are internal forces with the total structure.
Thus, on the level of the complete structure, only the energies kinetic and of deformation are not
null:
T
T
T = T1 + T2 = 1 1
1 1
! M! + 1 2
2 2
2
2!
M!
T
T
U = U1 +U 2 = 1 1
1 1
K + 1 2
2 2
2
2
K
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
13/34
The work of the bonding strengths being null, equations of Euler-Lagrange relating to the clean modes
total structure are:
T
K1 0 L1
M1
1
0
0
0
T
0
K 2 L2
2
-
0
M2
2
0 = 0
éq 3.2-3
L1
L2
0
0
0
0
0
Consequently, the problem with the eigenvalues of the complete structure can be expressed by the system
[éq 2.4-5] that we studied in chapter 2.4:
(K - 2M) + LT = 0
L = 0
Thus, the calculation of the clean modes of the total structure by the method of Craig-Bampton consists with
to solve a problem with the eigenvalues matric of reduced size. The matrices which it brings into play
are symmetrical and are calculated starting from the bases of the substructures. This resolution gives us
Eigen frequencies and generalized co-ordinates of the required modes. Theirs are obtained
physical co-ordinates on the grids of the substructures by using the relation [éq 3.2-1]. This
stage is called: stage of restitution on physical basis.
This method is interesting if one considers a numerical study of the substructures. Indeed, them
normal modes with fixed interfaces and the constrained modes lend themselves well to calculation. On the other hand, them
experimental determination is delicate.
It is shown moreover that this method, is command 2 in/m where m is the greatest pulsation
clean identified.
3.3
Method of Mac Neal
It would be possible to present this method in a way similar to that of Craig-Bampton, the single one
difference 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 step, which makes it possible to lead to
a criterion of truncation, and to reveal the modes of fastener like the static contribution of
not identified clean modes.
As previously, one considers the problem of the calculation of the clean modes of a structure with 2
components. The method is generalizable with an unspecified number of substructures. In the continuation
we will identify any size associated with the substructure S K by the exhibitor K.
We limit ourselves, here, with the modal calculation of the total structure, therefore the external forces are null.
Also, the displacement of S K in the movement of the total structure, checks the equilibrium equation
following dynamics:
(K K 2Mk) qk
F K
-
= L
éq 3.3-1
where F kL is the vector of the bonding strengths applied to S K.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
14/34
We will consider that the base of projection of S K consists of all its modes suitable for
free interfaces. Thus, the dimension of the projected problem is equal to the dimension of the problem resulting from
modeling finite elements. We suppose, moreover, that a certain number of these modes was
determined, others being unknown:
K
K
1
qk
[
K
1 2]
K K
=
=
K
2
Where:
k1
is the matrix of the identified modal vectors of the substructure S K,
k2
of the not identified modal vectors substructure S K is the matrix,
K
is the vector of the generalized degrees of freedom associated the clean modes identified of
1
S K,
K
is the vector of the generalized degrees of freedom associated the not identified clean modes
2
of S K.
The equation [éq 3.3-1] becomes, with the previously definite generalized co-ordinates:
(kT K K - 2kT K K) K = kT K
K
M
F L
that is to say:
(K K 2Mk) K
K T
F K
-
=
L
éq 3.3-2
The clean modes are orthogonal compared to the matrices of mass and rigidity and us
let us choose to normalize them compared to the matrix of mass. One thus has:
K
0
0
1
Id
K K
M K
=
=
K
éq 3.3-3
0
0
Id
2
Now let us consider all two substructure. Each one of them checks them
equations [éq 3.3-2] and [éq 3.3-3]. The whole of these dynamic equations constitutes the system
according to:
T
1
1
1
1
Id 0
0
0
F
1
0
0
0
1
1
1
L
0
2
2
2
2
0
Id
0
0
F
1
0
0
2
1 1
1
L
-
=
éq 3.3-4
0
0
1
1
1
1
0
0
Id
0
2
0
2
2
F
2
L
0
0
0
2
2
2
2
0
0
0
Id
2
2
2
F
2
L
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
15/34
By gathering the identified modes and the not identified modes:
1
1
1
2
co-ordinates of the identified modes
1
=
éq 3.3-5
1
2
co-ordinates of the not identified modes
2
2
2
The equation translating the transformation of RITZ becomes:
q1
1
Q = = [1
2
]
Q
2
éq 3.3-6
2
With these notations, the system of dynamic equations [éq 3.3-4] becomes:
F
1
0
Id 0
L
2
1
T
T
1
=
éq 3.3-7
0
-
0
Id
[1 2]
F
2
2
2
L
This system of equations translates the dynamic behavior of the substructures separately. It
do not represent the movement of the total structure. For that, it is necessary to associate the conditions to him of
connection between the two components.
The equations between substructures which ensure their liaisonnement derive as of the equations [éq 2.4-2]
and [éq 2.4-4], as of the organization of the base which we chose [éq 3.3-5]:
1
L = [L L
1
2]
= 0
éq 3.3-8
2
F = - F
L
L
éq 3.3-9
1
2
The equations [éq 3.3-7] and [éq 3.3-9] enable us to express the generalized co-ordinates
relating to the not identified modes:
2
1
T
2 = - 2 -
-
(
)
Id
2 F 1L
éq 3.3-10
From the equations [éq 3.3-8] and [éq 3.3-10], one thus obtains:
L - L
2
- Id 1
-
(
) Tf
1 1
2
2
2 L = 0
éq 3.3-11
1
However, according to the formula [éq 2.4-4], one knows that one can write the matrices of connection, in the form:
L = B
K
K
K
éq 3.3-12
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
16/34
One thus has, according to [éq 3.3-11] and [éq 3.3-12]:
- +
(- 2
-
B
B
Id 1
) Tf
1 1 1
2 2
2
2 L = 0
éq 3.3-13
1
One sees appearing the matrix of residual dynamic flexibility associated the not identified modes:
R () = (
2
1
2 2 - Id -
T
E
) 2
éq 3.3-14
The bonding strength is nonnull with the degrees of freedom of connection of the substructures. We have
thus:
F = BTf
L
1 L
1
1
Consequently, the matric problem [éq 3.3-7] can be reduced to the system are equivalent according to:
2
- Id - TBT
0
1
1
1
1
=
éq 3.3-15
- B
B R E
() F L
0
11
2
1
This problem has as unknown factors the generalized co-ordinates associated the identified clean modes
and bonding strengths applied to the first substructure.
There are 2 cases according to whether one takes into account or not the matrix of residual flexibility.
3.3.1 First
case
One neglects the matrix of residual dynamic flexibility associated the not identified modes:
Re () = 0.
The method of resulting dynamic under-structuring is thus very simple. It has the disadvantage of
to be based on a method of modal recombination very sensitive to the effects of truncation.
3.3.2 Second
case
A limited development is used: R () = R
2
2
E
E ()
0 + O (/m):
Let us adopt the following notations:
Are:
N
the number of modes of the complete base,
m
the number of not identified modes,
the matrix of N normal modes to free interfaces,
the matrix of N modes of fastener (definite for all the degrees of freedom of
system),
2
I = I
the diagonal matrix of N eigenvalues.
The complete matrix of the modes of fastener checks: K = Id = K - 1
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
17/34
The complete modal base of the normal modes with free interfaces constitutes an orthonormal base.
stamp stiffness, expressed in this base is written:
K = TK =
Same manner:
K - 1
T K-1
-
=
=
1
One thus deduces a new form from it of the complete matrix of the modes of fastener:
= - =
-
K 1
1T
As the clean modes are orthogonal two to two and stamps it eigenvalues is
diagonal, the complete matrix of the modes of fastener takes the final form:
N
=
-
1 T
II I
i=1
Now let us consider the matrix of residual dynamic flexibility, resulting from the method of
Mac Neal:
N
R () = (
2
1
2
1
2 2 - Id -
T
-
T
E
) 2 = I
(I -) I
i=m+1
To highlight the effect of modal truncation, we can approximate this matrix by sound
development with command 1:
N
2
1
R
-
T
1
E () I
I
+
2 I
i=m+1
I
When the number of identified modes is sufficiently important, the dynamic contribution becomes
negligible in front of the static contribution:
2
N
-
T
<<
Re
() Re () = II
1
1
0
2
I
I
i=m 1
+
where R E ()
0 are the matrix of residual static flexibility.
This matrix can be calculated according to the matrix of the modes of fastener:
N
m
m
R
1
-
T
1
-
T
()
0 = -
R
1
-
T
E
I I
I
I I
I
E ()
0 = - I
I I
I 1
=
I 1
=
I 1
=
The second term of this formulation is calculable in an exact way, since it utilizes only them
modes (with free interfaces) identified.
Lastly, let us note that in the method of Mac Neal, only the contribution of the modes of fastener to the nodes
of interface is necessary.
The resolution of the system [éq 3.3-4] enables us to determine the Eigen frequencies of the structure
total and generalized co-ordinates of the clean modes. Increase with the expression of the modes
clean in the physical bases of the substructures is done by the following relation:
qk
K K
1
R K (0) Bk Tf K
=
1 +
E
1
L1
The method of Mac Neal thus succeeds, in the case of the calculation of the modes of the total structure, with one
problem with the eigenvalues of reduced size. Matrices of mass and rigidity exits of under
structuring are symmetrical. Two methods are actually proposed, according to whether one takes in
count or not residual flexibility. The literature on the subject tends to show that the use of
residual flexibility is essential to obtain reliable results [bib2].
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
18/34
3.4
Implementation in Code_Aster
3.4.1 Study of the substructures separately
The base of projection of each substructure is made up of dynamic clean modes and of
static deformations.
The dynamic clean modes of the substructure are calculated with the traditional operators of
Code_Aster: MODE_ITER_SIMULT [U4.52.02] and MODE_ITER_INV [U4.52.01]. In the case of
under-structuring of Craig-Bampton, the interfaces of connection must be blocked. This is carried out
with operator AFFE_CHAR_MECA [U4.25.01].
Operator DEFI_INTERF_DYNA [U4.55.03] allows to define the interfaces of connection of
substructure. In particular, one specifies the type of the interface, which can be is “CRAIGB”
(Craig-Bampton), either “MNEAL” (Mac Neal), or finally “AUCUN”.
Operator DEFI_BASE_MODALE [U4.55.04] allows to calculate the base of complete projection of
substructure. Thus, the dynamic modes calculated previously are recopied. In addition, them
static deformations are calculated according to the type defined in operator DEFI_INTERF_DYNA
[U4.55.03]. If the type is “CRAIGB”, one calculates the constrained modes of the interfaces of
substructure. If the type is “MNEAL”, one calculates the modes of fastener of the interfaces of
substructure. If the type is “AUCUN”, one does not calculate static deformation, which corresponds to
a base of the type Mac Neal without static correction.
Operator MACR_ELEM_DYNA [U4.55.05] calculates the generalized matrices of rigidity and mass of
the substructure, as well as the matrices of connection.
3.4.2 Assembly and resolution
The model of the complete structure is determined by operator DEFI_MODELE_GENE [U4.55.06]. In
private individual, each substructure is defined by the macronutrient which corresponds to him (resulting from
MACR_ELEM_DYNA) and the swing angles which make it possible to direct it. The connections enters
substructures are defined by the data of the names of the two implied substructures and those
of the two interfaces in opposite.
The classification of the complete generalized problem is carried out by operator NUME_DDL_GENE
[U4.55.07]. The matrices of generalized mass and stiffness of the structure supplements are
assemblies according to this classification with operator ASSE_MATR_GENE [U4.55.08].
The calculation of the clean modes of the complete structure is carried out by the operators
MODE_ITER_SIMULT [U4.52.02] or MODE_ITER_INV [U4.52.01].
3.4.3 Restitution on physical basis
The restitution of the results on the initial grids of the substructures is carried out by the operator
REST_BASE_PHYS [U4.64.01].
To decrease the duration of the graphic processing during visualizations, it is possible to create one
coarse grid by operator DEFI_SQUELETTE [U4.75.01]. This grid, ignored during calculation,
is used as support with visualizations.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
19/34
4
Modal calculation by cyclic dynamic under-structuring
4.1 Introduction
In this chapter, we make the synthesis of the methods of cyclic dynamic under-structuring.
We give a definition of the cyclic repetitivity (or cyclic symmetry) and we present them
principal incidences of this property on the dynamic behavior of the structure (circles and
nodal diameters, double modes). Then, we expose, in a rather detailed way, both
methods of cyclic dynamic under-structuring, implemented in Code_Aster.
improvements were made to the traditional methods, by the taking into account of the presence of
nodes of the axis.
These methods suppose that the grid of the basic sector is such as its traces on the interfaces
right-hand side and left are coinciding (compatible grids).
Notations specific to the cyclic under-structuring:
NR
=
a number of sectors
=
angle formed by the basic sector
=
dephasing AND element
OZ
=
cyclic axis of symmetry
=
rotation of angle and axis OZ
Re (Z)
=
real part of complex Z
Im (Z)
=
imaginary part of complex Z
=
stamp passage of the nodes of straight line to the nodes of left
=
stamp change of sector for the nodes of the axis
has
Note:
The index
D
is relative
with the degrees of freedom of straight line
“
G
“
“
with the degrees of freedom of left
“
has
“
“
with the degrees of freedom of the axis
“
1
“
“
with the identified clean modes
“
2
“
“
with the unknown clean modes
4.2 Repetitivity
cyclic
4.2.1 Definition
It is said that a structure is with cyclic repetitivity of axis OZ, if there is an angle 0 < < such as
structure is geometrically and mechanically invariant by rotation around OZ of this angle.
If is the smallest angle checking this property, then any angular portion of angle of
structure is called “basic sector” (or “irreducible sector”).
The total structure is then made up of NR sectors:
2
NR =
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
20/34
4.2.2 Propagation
of wave
One notes the rotation of axis OZ and angle defined in R3.
Let us consider a basic sector of a structure with repetitivity of axis OZ, and two similar points of
two contiguous sectors G and D:
G
OZ
D
One with the relation between the points G and D:
G = (D)
It is noticed that the structure is left invariant by any rotation m (with m whole).
One can note that all rotations leaving the invariant structure (geometrically and
mechanically) are in a finished number:
m m {,
0,
1…, NR -}
1
Let us consider a scalar variable of state of the mechanical system studied U, and Z the associated complex:
U = Re (Z) = Re U
(+ jV)
It is possible to show, by the theory of the finished groups, the following relation for the points D and G
[bib5]:
m
NR
0 1
Z G = E jm
{,…,}
()
Z (D)
2
such as
éq 4.2.2-1
Note:
· the quantities are expressed in the cylindrical reference mark (R, Z),
· for an axisymmetric structure (particular case of cyclic repetitivity), m is called
index of FOURIER,
· in the case of a wave planes not deadened, E jm is complex dephasing between two
contiguous sectors; the equation means that this dephasing can take only one number
finished known values,
· it is possible to limit the number of the values of m to the values ranging between 0 and
NR/2; indeed, it is shown that the wave associated with dephasing NR - m is identical to
that associated dephasing m, but progresses in opposite direction [bib5].
If NR is even: m = 0 and m = NR/2 correspond to real modes:
m = 0
D
U ((D)) = U (D)
m = NR/2
D
U ((D)) = U
- (D)
All the other values of m correspond to modes appearing per orthogonal pairs with
a given frequency (one speaks then about degenerated modes):
U = Re (Z) and V = Im (Z)
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
21/34
If NR is odd: m = 0 corresponds to a real mode not degenerated:
m = 0
D
U ((D)) = U (D)
All the other values of m correspond to degenerated modes appearing per pairs
orthogonal:
U = Re (Z) and V = Im (Z)
4.2.3 Concept of diameters and nodal circles
The cyclic property of repetitivity, translated by the equation [éq 4.2.2-1] makes it possible to know a priori
pace of the clean modes of the structure, which strongly approaches what one can observe
for axisymmetric structures. If one considers a clean mode of a structure with symmetry
cyclic, all the sectors have the same deformation but with an amplitude function of their position
angular, which one can translate by a dephasing between substructures. This mode can be classified
starting from the number of nodal diameters and circles which characterize it. A nodal diameter (which is not
confused with a diameter that if the structure is axisymmetric) is a line of points of
null movement passing by the axis of repetitivity; a nodal circle (which with the circular form only for
the axisymmetric structures) is a line of points of null movement, it even with repetitivity
cyclic. It is noted that it is the deformation of the mode of the substructure on which the mode is pressed
structure supplements which determines the number of circle (S) nodal (with). On the other hand, the number of
diameter (S) nodal (with) is defined by dephasing between two consecutive sectors.
Deformation
Phase enters
Deformation
Family
sector
sector
overall
NR sectors
0 circle
Inflection 1
in phase
0 diameter
NR/2 sectors
0 circle
Inflection 1
in phase
1 diameter
NR sectors
1 circle
Inflection 2
in phase
0 diameter
NR/sectors
1 circle
Inflection 2
in phase
1 diameter
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
22/34
4.2.4 Boundary conditions
Let us consider a structure with cyclic repetitivity, and two basic sectors successive of this one:
G
k+1
D
Rk+1
Interface
G
K
D
Rk
The connections between sectors being regarded as perfect, there are the conditions between the sectors:
qk = qk+1
G
D
continuity of displacements
éq 4.2.4-1
F K = - F k+1
L
L
principle of action - reaction
G
D
The exhibitor indicates the number of the sector considered. The preceding conditions of connection are
expressed in the total reference mark.
By the formula [éq 4.2.2-1] relating to the propagation of wave in the structure and while posing: =
m,
one a:
{qk+}
1
J
1 = E
{qk}
K +
K
{F k+}
1
J
1 = E
{F K}
L
K +
L K
The index K means that the quantity is expressed in the reference mark related to the sector K: Rk.
The equations of connection [éq 4.2.4-1], written in the reference mark related to the sector K thus utilize
stamp passage of the sector K to the sector K + 1. This matrix is not other than the matrix of
rotation of the degrees of freedom of straight line towards those of left, is the matrix of rotation of axis OZ and
of angle, noted: .
We thus obtain the following system:
{qk}
J
= E {
qk}
G K
D K
éq 4.2.4-2
{F K}
J
= - E {
F K}
L
K
L
K
G
D
The boundary conditions [éq 4.2.4-2] make it possible to calculate the clean modes of the whole of
structure starting from only one basic sector.
This formalization can be generalized with the case of the nodes of the axis. One obtains then:
{qk}
J
= E {qk}
K has
has
K has
{F K}
J
= - E {F K}
L
K
has
L
K
has
has
It is checked that if is nonnull, the displacement of the nodes of the axis is null (in fact, one notes then
presence of one or several nodal diameters).
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
23/34
4.3
Methods of cyclic under-structuring
4.3.1 Method of Craig-Bampton
One considers the problem with the eigenvalues of the total structure expressed on the basic sector.
This last is thus subjected to the bonding strengths which are applied to him by the contiguous sectors. By
elsewhere, the basic sector checks the equations of connection [éq 4.2.4-2]. We thus have:
(K - 2M) Q = F L
Q
J
G =
E
Q
D
éq 4.3.1-1
F
J
L = -
E
F
L
G
D
We suppose that the base is made up of the dynamic clean modes of the basic sector
embedded with its interfaces, noted, and of the constrained modes relating to the degrees of freedom of interfaces
right-hand side and left, noted D and G.
Taking into account the fact that the only contribution to displacements of a degree of freedom of interface
comes from the constrained mode corresponding, the transformation of RITZ can be written:
Q
I
I
Q = Q
D = [D G] Q
D =
Q
Q
G
G
Consequently, by using the transformation of RITZ, the system of equations [éq 4.3.1-1] becomes:
0
I
T
2
(K - M) Q
D = [D G] fL
D
Q
F
G
Lg
Q
J
G =
E
Q
D
éq 4.3.1-2
F
J
L = -
E
F
L
G
D
The surmounted matrices of a bar are projections of the matrices finite elements on the basis
modal of the basic sector (generalized matrices).
One can show that the constrained modes are orthogonal with the normal modes with respect to
stamp rigidity [bib5]. Thus, the corresponding products are null.
Let us adopt the following notations:
m: index relating to the clean modes of the sector,
D: index relating to the constrained modes of the right interface,
G: index relating to the constrained modes of the left interface.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
24/34
One can thus write these matrices in the form:
K
0
0
M
M
M
mm
mm
Mandelevium
Mg
K = 0
K
K
dd
dg
M = M
M
M
DM
dd
dg
0
K
K
M
M
M
Gd
gg
gm
Gd
gg
Taking into account their definition, the constrained modes check:
di
di
gi
gi
D = dd = Id
G = Gd = 0
0
Id
dg
gg
The second member of the matric equation [éq 4.3.1-2] becomes:
I
0
0
0
0
di Id 0 F
L = F
L
D
D
gi
0
Id F
L
F
L
G
G
By taking account of these notations, let us develop the matric equation checked by the basic sector:
K
2
mm I
- (M + M Q + M Q
mm I
Mandelevium D
Mg G) = 0
K Q + K Q
2
dd D
dg G - (M + M Q + M Q
DM I
dd D
dg G) = F Ld
K Q + K Q
2
Gd D
gg G - (M + M Q + M Q
gm I
Gd D
gg G) = F Lg
Q
J
= E Q
G
D
F
J
= - E F
L
L
G
D
Let us introduce the two last equations of this system into the three first:
(K - 2M
2
J
mm
mm) I
- (M + E M
Mandelevium
Mg) qd = 0
(K
J
+ E K
2
J
dd
dg) qd - (Mdm I
+ (M + E M
dd
dg) qd) = fLd
(K
J
+ E K
2
J
J
Gd
gg) qd - (Mgm I
+ (M + E M
Gd
gg) qd) = - E
F
Ld
The association of the two last equations makes it possible to eliminate the terms from the bonding strengths. One
leads then to a problem with the eigenvalues final which one can put in the form:
(~
2 ~
() -
())~
K
M
Q = 0
éq 4.3.1-3
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
25/34
With:
~ I
Q =
Q
D
~
K
mm
0
K =
I
- I T
T
0
K
dd + E K dg + E
K Gd + K
gg
I
~
M
M
mm
Mandelevium + E M
M =
Mg
M
I T
I
I T
T
DM + -
E
M
M
gm
dd + E Mdg + -
E
Mgd + M
gg
The matrices of mass and rigidity of the final problem are square. Eigenvalues
solutions are thus real. In addition, the problem is of reduced size.
The resolution of the problem to the complex eigenvalues [éq 4.3.1-3] makes it possible to determine them
complex generalized co-ordinates of the clean modes of the total structure. Values
complexes of displacements of the basic sector in the total mode are given, from
generalized co-ordinates, by the following formula:
Q = [
J
~
D + E D] Q
éq 4.3.1-4
To determine the actual values of displacements, it is necessary to distinguish three cases according to values' from
dephasing AND element:
Case n° 1: = 0:
The displacements Q given by the formula [éq 4.3.1-4] are then with actual values. All sectors
deformed even and vibrate in phase. There is then only one real clean mode:
Q = Re (Q) = Q
éq 4.3.1-5
Case n°2: 0 < < (NR + 1)/2:
The displacements provided by the formula [éq 4.3.1-4] are with complex values. With each one of these
complex modes correspond two orthogonal degenerated real modes:
Q = Re (Q)
Q = Im (Q
1
2
)
éq 4.3.1-6
Case n°3: = NR/2 (=> NR is even):
The displacements provided by [éq 4.3.1-4] are then with complex values. There are NR/2 diameters
nodal, two contiguous sectors vibrate then in opposition of phase. Each complex mode is with
the origin of only one real mode:
Q = Re (Q) = - Im (Q)
éq 4.3.1-7
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
26/34
Taking into account of the nodes of the axis:
One supposes in this paragraph that the degrees of freedom carried by the nodes of the axis, with same
titrate that the nodes of interfaces straight line and left, were blocked for the calculation of the modes
dynamic of the basic sector and were the subject of calculations of constrained modes.
The base of projection is thus made up of the dynamic clean modes of the basic sector
embedded with its interfaces, noted, and of the constrained modes relating to the degrees of freedom of interfaces
right-hand side, left and axis, noted D, G and A.
As we saw in chapter 4.2.4, if is nonnull, the displacement of the nodes of the axis is
no one (presence of at least a nodal diameter). The taking into account of the nodes of the axis thus does not have a direction
that if = 0. In this demonstration, we will limit ourselves to this case.
The problem with the eigenvalues of the total structure and equations of connection, expressed on this
base are worth then:
I
0
Q
F
D
Ld
(K - 2M) = D G has
Q
[
] F
G
Lg
éq 4.3.1-8
Q
F
has
HQ = Q
and
Q
D
= Q has
, F
has has
L = - F
and
F
L
L = - F
L has
G
D
has
has
Let us adopt the following notations:
m: index relating to the clean modes of the sector,
D: index relating to the constrained modes of the right interface,
G: index relating to the constrained modes of the left interface,
a: index relating to the constrained modes of the interface centers.
One can thus write the matrices in the form:
K
mm
0
0
0
M
M
M
M
mm
Mandelevium
Mg
my
0 K
K
K
M
M
M
M
K =
dd
dg
da
M DM
dd
dg
da
=
0
K
K
K
M
M
M
M
Gd
gg
ga
gm
Gd
gg
ga
0
K
K
K
M
M
M
M
AD
Ag
aa
amndt
AD
Ag
aa
Taking into account their definition, the constrained modes check:
di
di
gi
gi
have have
Id
dd
Gd
0
0
AD
D =
=
G =
=
=
=
0
has
dg
gg
Id
Ag 0
0
da
ga
0
aa
Id
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
27/34
The second member of the matric equation [éq 4.3.1-8] becomes:
0 0
I
0
0
0
F
F
di
Id
0
0 L L
D
D
=
F
F
gi
0
Id
0
L
L
G G
F F
have
0
0
Id L
L
has has
By taking account of these notations, let us develop the matric equation checked by the basic sector:
K
2
mm I
- (M + M Q + M Q + M Q
mm I
Mandelevium D
Mg G
my A) = 0
K Q + K Q + K Q
2
dd D
dg G
da has - (M + M Q + M Q + M Q
DM I
dd D
dg G
da has) = F Ld
K Q + K Q + K Q
2
Gd D
gg G
ga has - (M + M Q + M Q + M Q
gm I
Gd D
gg G
ga has) = F Lg
K Q + K Q + K Q
2
AD D
Ag G
aa has - (M + M Q + M Q + M Q
amndt I
AD D
Ag G
aa has) = F
HQ = Q
and
F
D
L = - F L
G
D
qa = Q
and
F
has has
L = - F
L has
has
has
Let us replace, in the first four equations of the system HQ and F L by their expressions
G
respective, according to qd and F L and let us rewrite, in another form, the two last
D
equations of the system, relating to the nodes of the axis.
(K - 2M
2
mm
mm) I
- ({M + M
Mandelevium
Mg) Q + M Q
D
my A} = 0
(K +K
2
dd
dg) Q + K Q
D
da has -
({Mdm I + (M +M
dd
dg) qd) + M Q
da has} = fLd
(K +K
2
Gd
gg) Q + K Q
D
ga has -
({Mgm I + (M +M
Gd
gg) qd) + M Q
ga has} = - F
Ld
(K +K
2
AD
Ag) Q + K Q
D
aa has -
({Mam I + (M +M
AD
Ag) qd) + M Q
aa has} = fLa
Q
Q
has
has = (1+ A) 2
(T
1+ A) fL = 0
has
One replaces, in the first four equations of the system qa by his given expression
in before last equation. In addition, the association of the second and the third equation
allows to eliminate the terms from the bonding strengths on the right. Lastly, the fourth equation is multiplied
by (1+ T)
has, which makes it possible to eliminate the terms from the bonding strengths to the axis. One leads then to one
problem with the eigenvalues final which one can put in the form:
~
~
(- 2) ~
K
M Q = 0
éq 4.3.1-9
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
28/34
With:
I
~
Q = qd
Q
/
has 2
K
mm
0
0
~
K = 0
K
T
T
T
dd + K dg + K Gd + K gg
(Kda + Kga) (1+a)
T
T
0
1
1
1
(+ has) (Kad +Kag)
(+ has) Kaa (+ has)
M
M
mm
Mandelevium + M
M
Mg
my (1+ A)
~
M = M
T
T
T
T
DM + M
M
gm
dd + Mdg + M Gd + M gg
(Mda + Mga) (1+a)
T
T
T
1
1
1
1
(+ has) Mam
(+ has) (Mad +Mag)
(+ has) Maa (+ has)
Displacements of the axis, divided by two, make it possible to preserve at the matrices of rigidity and of
mass their square character. One restores modal complex displacements by the formula
following:
Q = [
J
~
D + E
D 2a] Q
4.3.2 Method of Mac Neal
One considers the problem with the eigenvalues of the total structure expressed on the basic sector.
This last is thus subjected to the bonding strengths which are applied to him by the contiguous sectors. By
elsewhere, the basic sector checks the equations of connection [éq 4.2.4-2]. We thus have:
(K - 2M) Q = F L
Q
J
G =
E
Q
D
éq 4.3.2-1
F
J
L = -
E
F
L
G
D
The modal base used to reduce dimensions of the problem to be solved, is a modal base with
free interfaces including/understanding of the dynamic modes and the modes of fastener relating to the degrees of
freedom of the interfaces right and left. Let us suppose that the degrees of freedom of the basic sector are
ordered in the following way:
Q
I
degrésde freedom internal
Q = Q
D degrésde freedom of the right interface
Q
degrésde freedom of the left interface
G
Are data base and Bg, the rectangular matrices of extraction such as:
Q = B Q and Q = B Q
D
D
G
G
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
29/34
The boundary condition on displacements becomes with these notations:
B Q
J
= E B Q B Q = 0 with B
J
= E
B
- B
G
D
dg
dg
D
G
éq 4.3.2-2
For the forces, the boundary condition becomes:
F = BTf + BTf
F
T
- J T T
T
L
D L
L = (B - E
B
G
D) F
= - B F
L
G
L
dg L
G
D
G
G
Let us regard as base, for the transformation of RITZ, the whole of the clean modes
dynamic of the basic sector, by distinguishing the identified modes and the unknown modes:
1
Q = [1 2]
éq 4.3.2-3
2
where index 1 (resp. 2) refers to the known modes (resp. unknown). In the continuation, us
will suppose that the clean modes are normalized with the unit modal mass.
By replacing Q by its expression according to the clean modes, and while multiplying on the left by
transposed of the matrix of the modes, the matric equations [éq 4.3.2-1] and [éq 4.3.2-2] become:
(
2
-
)
T
1
Id 1 = 1 F L
(
2
-
)
T
2
Id 2 = 2 F L
éq 4.3.2-4
Bdg11 + Bdg22 = 0
where is the matrix of generalized rigidities (the generalized masses are unit).
One can thus draw a formulation from it from 2:
2
1 T
2 = 2 -
-
(
)
Id
2 fL
éq 4.3.2-5
Consequently, one can eliminate 2 from the system of equations [éq 4.3.2-4]. One obtains the problem then with
eigenvalues according to:
(
2
-
)
T T
1
Id 1 + 1 B F
dg L = 0
G
B
T T
dg
2
1
-
11 - Bdg (
2 - I)
2
D
2 B F
dg L = 0
G
The final system to solve can be written:
~
~
(- 2) ~
K
M Q = 0
éq 4.3.2-6
With:
1
~
Q = F
Lg
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
30/34
The forms of the matrices of rigidity and mass are:
T T
~
1
B
1
dg
~
Id
0
K =
M
T
=
B
dg
0
0
1
- B R () B
dg
E
dg
The matrix [R ()]
E
is the matrix of residual dynamic flexibility of the not identified modes:
R () = (
2
1
2 2 - Id -
T
E
) 2
One approximates residual dynamic flexibility by his static contribution, by taking of account them
modes of fastener. Then, the formula of restitution which makes it possible to calculate the complex values of
displacements starting from the generalized co-ordinates of the solutions modes of [éq 4.3.2-6] is
following:
Q = [
T ~
1
- R (0) B
E
dg] Q
The actual values of displacements are determined, as for the method of Craig-Bampton,
by the formulas [éq 4.3.1-5], [éq 4.3.1-6], [and éq 4.3.1-7].
Taking into account of the nodes of the axis:
We suppose, in this paragraph, that the degrees of freedom carried by the nodes of the axis, with
even title that the nodes of interfaces straight line and left, were the subject of calculations of modes of fastener.
We limit ourselves to = 0 who is the only case modified by the taking into account of the nodes of the axis
(§ 4.2.4 and 4.3.1). We thus have:
(K - 2M) Q = F L
HQ = Q
and F
D
L = - F
L
G
D
Q
éq 4.3.2-7
= Q has
has has (1 - has) qa = 0
F
F
T
L = - F
L F L has =
has
has
has
(1 - has) 2
The organization of the degrees of freedom of the basic sector is similar to that of the preceding chapter:
IQ
degrésde freedom internal
Q
D
degrésde freedom of the right interface
Q =
Q
G degrésde freedom of the left interface
Q
has
degrésde freedom of the interface centers
Are Ba the rectangular matrix of extraction of the degrees of freedom of the axis:
Q = B Q
has
has
The boundary condition on displacements of the axis becomes with these notations:
(1-a) Baq = 0 Baaq = 0 with Baa = (has -) 1Ba
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
31/34
For the forces, the boundary condition becomes:
F
F = BTf
+ BTf + BTf
F = - BT F + BT
T
L
G L
D L
L has
L
dg L
1 has -
G
D
has
G
(A) 2
F
F = - BT F - BT
L
dg L
aa
G
2
The base of the transformation of RITZ is made up of the whole of the free clean modes of
basic sector, by distinguishing the identified modes (index 1) and the unknown modes (index 2) definite
by the equation [éq 4.3.2-3].
The equation [éq 4.3.2-7], written in this base takes the following form:
(
2
-
)
T
1
Id 1 = 1 fL
(
2
-
)
T
2
Id 2 = 2 fL
éq 4.3.2-8
Bdg11 + Bdg22 = 0
Baa11 + Baa22 = 0
The second equation makes it possible to determine 2 (cf [éq 4.3.2-5]), which can thus be eliminated from
system. One then obtains the problem with the eigenvalues according to:
F
(
2
-
)
T T
T T
1
Id 1 + 1 B F
dg L + 1 Baa
= 0
G
2
F
B
T
T
dg 1 - B
R (
dg
E) B
F
dg L - B
R (
dg
E)
1
Baa
= 0
G
2
F
B
T
T
aa 1 - B
R (
aa
E) B
F
dg L - B
R (
aa
E)
1
Baa
= 0
G
2
Thus, by defining the following unknown vector:
1
~
Q = fL
G
F
/
2
The following final system is obtained:
~
~
(- 2) ~
K
M Q = 0
éq 4.3.2-9
with:
T T
T T
1
B
1
dg
B
1
aa
Id 0
0
~
T
T
~
K = B
DG1 - B R () B
dg
E
dg
- B R () B
dg
E
aa
M = 0 0
0
B
T
T
aa
0
0 0
1
- B R () B
aa
E
dg
- B R () B
aa
E
aa
Division by two of the bonding strengths applied to the axis makes it possible to preserve at the matrices of
rigidity and of mass their square character.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
32/34
One restores modal complex displacements by the following formula:
Q = [
T
T
1
- R
B
E (0) dg
- 2R (B
E 0)
~
aa] Q
4.4
Implementation in Code_Aster
The processing of the basic sector is identical to that of the substructures in the under-structuring
traditional. It utilizes the operators: MODE_ITER_SIMULT [U4.52.02] or MODE_ITER_INV
[U4.52.01], DEFI_INTERF_DYNA [U4.55.03] and DEFI_BASE_MODALE [U4.55.04].
The clean modes of the structure with cyclic symmetry are calculated by operator MODE_ITER_CYCL
[U4.52.03] according to the base of projection of the basic sector previously defined and of
a number of sectors of the complete structure.
The restitution of the results on physical basis is identical to the traditional under-structuring. It makes
to intervene the operator REST_BASE_PHYS [U4.64.01] and possibly operator DEFI_SQUELETTE
[U4.75.01].
5 Conclusion
The principles of under structuring make it possible to expose the transformation of RITZ and
modal recombination to lead to the modal synthesis which integrates these two techniques. Rules
of liaisonnement between substructures are clarified.
Two methods were developed in Code_Aster: that of Craig-Bampton and that of Mac
Neal. We present, here, their characteristics, as well in the definition of the initial modal base,
that in its exploitation.
After having exposed the definition of a structure to cyclic symmetry and the properties which result from this,
we presented the methods of cyclic under-structuring implemented in
Code_Aster. They appear very interesting for the calculation of the clean modes of a structure with
cyclic symmetry, such as the rotors of the revolving machines of which they benefit fully from
geometrical and mechanical characteristics.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
33/34
6 Bibliography
[1]
R. ROY, J. CRAIG & Mr. C. BAMPTON: “Coupling off Substructures for Dynamic Analysis”
AIAA Journal, (July 1968), Vol. 6, N° 7, p. 1313-1319.
[2]
R.H. MAC NEAL: “A hybrid method off component mode synthesis” Computers and
Structures, (1971), Vol. 1, p. 581-601.
[3]
J.F. IMBERT: “Analysis of Structures by Eléments Finis” 1979, Cepadues Edition.
[4]
P. RICHARD: “Methods of dynamic under-structuring in finite elements”. Report/ratio EDF
HP-61/90.149.
[5]
P. RICHARD: “Methods of cyclic under-structuring in finite elements”. Report/ratio EDF
HP-61/91.156
[6]
P. RICHARD: “Methods of under-structuring in Code_Aster”. Report/ratio EDF
HP-61/92.149
[7]
J. PELLET: “Code of Mécanique Aster - Manuel of reference: Finite elements in
Key aster ": R3.03.01 “Dualisation of the boundary conditions”
[8]
G. JACQUART: “Code of Mécanique Aster - Manuel of reference: Dynamics in base
modal ". Key: R5.06.01 “Méthodes of RITZ in linear and non-linear dynamics”
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Code_Aster ®
Version
4.0
Titrate:
Modal calculation by traditional and cyclic dynamic under-structuring
Date:
08/12/98
Author (S):
G. ROUSSEAU, C. VARE
Key:
R4.06.02-B
Page:
34/34
Intentionally white left page.
Handbook of Référence
R4.06 booklet: Under-structuring
HP-51/98/016/A
Outline document