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Reduction of model in linear and non-linear dynamics: Methods of RITZ Date:
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:
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: 1/18

Organization (S): EDF R & D/AMA, EDF/CNPE of Tricastin
Handbook of Référence
R5.06 booklet: Dynamics bases Modale of it
Document: R5.06.01

Reduction of model in linear dynamics and not
linear: Method of RITZ

Summary:

This document presents the principle of reduction of model by projection on reduced basis (method of Ritz).
The base most usually used is the modal base.

The problems of truncation due to the use of a reduced base are mentioned. Corrections of truncation
are proposed.

The description and properties of the algorithms of resolution of the system of differential equations of the second
command obtained in transitory analysis are presented in the document [R5.06.04].

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Count

matters

1 Introduction ............................................................................................................................................ 3
2 Methods of reduction of Ritz into linear ............................................................................................ 3
2.1 General description ........................................................................................................................ 3
2.1.1 Formulation continues .............................................................................................................. 3
2.1.2 Approximation finite elements ................................................................................................. 4
2.2 Projection on reduced basis .............................................................................................................. 4
2.3 Projection on modal basis ............................................................................................................. 5
2.4 Modal truncation error ............................................................................................................ 7
2.5 Corrections of modal truncation ............................................................................................... 9
2.5.1 Static correction a posteriori .............................................................................................. 9
2.5.2 Addition of static modes to the base ............................................................................. 10
3 Extension of the methods of reduction of Ritz into non-linear ........................................................... 11
3.1 General problem .......................................................................................................................... 11
3.2 Indication of the error of projection ................................................................................................ 12
4 Use in Code_Aster ............................................................................................................. 13
5 Bibliography ........................................................................................................................................ 14
Appendix 1 ................................................................................................................................................. 15
Appendix 2 ................................................................................................................................................. 18
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1 Introduction

Starting from a description of the geometry and materials of the structures, the method of the elements
stop allows to create a precise and reliable but large-sized model. In the case of one
problem of dynamics, one wishes to calculate the response of a system for various moments
(analyzes transitory) where for various frequencies (analyzes harmonic). Size of the model
finite elements obtained is often irreconcilable with the number of calculations necessary to obtain
all desired results.
For a restricted whole of dynamic stresses, there is generally a subspace of
low dimension allowing to describe the dynamic behavior of the structure under
specific stresses.
The projection of the model on a restricted basis is called method of Ritz or Rayleigh-Ritz.

This document comprises the following points:

·
a presentation of the methods of Ritz, their use into linear,
·
a detail of the possible corrections of truncation,
·
generalization into nonlinear of the methods of Ritz,
·
two simple examples of illustration.

2
Methods of reduction of Ritz into linear

2.1
General description

2.1.1 Formulation
continuous
The method of Ritz consists in projecting checking displacement on a restricted basis of functions
the conditions kinematics of the problem:
N
u~ (M, T) = I T () I (M) éq
2.1.1-1
i=1
Displacement is described by a series of independent forms {(M); I
I
= K
1
}
N multiplied by

amplitudes functions of time {T
(); I
I
= K
1
}
N.
The difficulty consists in defining this family of form {(M); I
I
= K
1
}
N which contrary to
functions of form of the finite element method are nonnull on most of
structure.
The quality of the approximation is related to the fact that displacements obtained have good
approximation in the subspace generated parVect {(M), I
I
= K
1
}
N.

Projection on modal basis
It is known that the clean modes {(M); I =
I

K
1
}
generate the space of the fields
kinematically acceptable. Displacement breaks up according to:

U (M, T) = (T) (M)
I
I

éq
2.1.1-2
i=1
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The option most usually used for the method of Ritz then consists in taking as bases
projection N first modes:
N
u~ (M, T = ~
)
I T () I (M)


éq
2.1.1-3
i=1
Displacement obtained is an approximation of real displacement.
It can be interesting to add to N first modes, other forms (see [§2.6.2]).
2.1.2 Approximation finite elements

In the case of an approximation of displacement by finite elements displacement is already approximate
in the space of the functions of forms:
Nh
H
U (M, T) = IQ T () Nor (M)
éq
2.1.2-1
i=1
One notes U the vector of the degrees of freedom of displacement: U (T) = [q1 (T), q2 (T), Q
K Nh (T)] ;

Method of Ritz in finished dimension
If N < Nh, the method of Ritz applied to the field U (M, T) comes then like a second
approximation:
N
~
U T
() = ()
I T
I
éq
2.1.2-2
i=1
with {I
, i= N}
K
1
the base of N vectors independent and kinematically acceptable.

One poses = [
,

,

,
1
2
,
3
N]
U =
K
. From where the matric writing:

éq
2.1.2-3

2.2
Projection on reduced basis

Let us consider the following differential connection obtained by a method finite elements:
Nh
DRIVEN & + U
C & + KU =
U

F
R éq
2.2-1

The solution sought in the form [éq 2.1.2-3]. By considering the same form for displacement
virtual, it comes:
T
T
T
T
N
M&+
C & + K =


F
R
éq
2.2-2

where: is the vector of generalized displacements, K = T K and M = T M are called
respectively matrices of generalized stiffness and mass.

The system [éq 2.2-2] is generally a coupled differential connection, the generalized matrices which
compose are in the general case full even if at the beginning the matrices M and K were
hollow. One thus loses the structure particular to the profit of a size of problem much more reduced
N * N.

In the general case, the system [éq 2.2-2] provides only one approximate solution of the system [éq 2.2-1].
The error which one makes is called truncation error.
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One has no information on the value of this error. It can be very large if the subspace of
projection is badly selected. It is known only that this error decreases when the size of the base of
projection increases.

If one has information a priori on the form of the solution, one can choose in an effective way
base projection in order to minimize this error. For example, if it is known that the solution is not
constituted that movements of solid body, one can restrict to 6 the dimension of space.

Thereafter, one chooses the base of the clean modes as bases projection.

2.3
Projection on modal basis

Clean modes
The modes are defined like the couples (
{H H
I, I) i= Nh}
K
1
solutions of the equation:
(
2
K - M) = 0 éq
2.3-1

Note:

It is advisable to check that the modes calculated by approximation finite elements are
sufficient representative: (H
H
I, I) (I
, I). One can consider that the approximation
finite elements

is correct when the modal deformations present a length D `wave
higher than the size of the meshs of the grid (the concept wavelength is one
generalization of the concept definite on the equation of the waves, one can define it as two
time the length between two nodes of the modal deformation).
Thereafter one omits, the index H corresponding to the approximation finite elements.

Quotient of Raleigh: energy interpretation
The own pulsations and forms can be defined as the solutions of the problem of
minimization according to:
I
[,
1 Nh]:

RNh - Vect {, J,
0 I - 1
J
}
I minimizes in under space
[
]
functional calculus:
X T KX
T
K
R (X) =
one poses: 2
I
I
=
= ()
éq
2.3-2
X tMX
I
R
I
T
I Semi

Method of reduction
A method of reduction very largely employed for the linear problems is the method of
modal recombination. It consists in choosing as bases projection N first modes
clean of the structure {I, i= N}
K
1
.
N
~
U T
() = ()
I T
I






éq 2.3-3
i=1

Always let us consider the following differential connection:
Nh
DRIVEN & + U
C & + KU =
U

F
R éq
2.3-4
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Clean modes {I, i= Nh}
K
1
have the property to be M and K orthogonal, i.e. one has them
following relations:
T
I M J = I
m ij
T
I K J = ki ij


is the symbol of KRONECKER
I
m is called modal mass or generalized mass of mode I
ki is called modal rigidity or generalized rigidity of mode I

The matrices projected of M and K on the basis of clean mode are thus diagonal; it is one
advantages which justified the use of the modal base as bases projection. The system
[éq 2.3-4] projected on the basis of clean mode first of the system is written:

\ 0 0
\ 0 0




0 m 0
T
T
I
&
+ C & + 0 K 0
I
= ext.
F


éq 2.3-5





0
0
\
0 0 \

The projection of the matrix C does not have any reason in any general information to be also diagonal. If it
system is strongly deadened (presence of damping devices on the structure), this matrix will not be
diagonal.

Note:

As opposed to what do many software, Code_Aster allows in this case
to integrate the system of modal equations coupled without diagonalisation of the matrix
of generalized damping. The method of integration is in this case an implicit method
NEWMARK or clarifies EULER.

On the other hand, if only damping entering concerned is a structural damping (internal dissipation
material for a homogeneous structure) it is then licit to make the assumption of a damping
proportional, still called assumption of BASILE, in this case C expresses itself like combination
linear of M and K (damping of RAYLEIGH), and its projection on the clean modes is
diagonal (cf Doc. [R5.05.04] on the modeling of damping).

In this case, the system [éq 2.3-4] is divided into p linear differential equations of the second command
uncoupled. The response of the system is then the recombination of the response of p simple oscillators
associated the clean modes, from where the expression of “modal superposition” used usually.

Each differential equation is written I
m:

I
m I & + here & + kii = fi





éq 2.3-6

or while dividing by the modal mass:

fi

2
I
& +
2 I I
I & + I
I =






éq 2.3-7
I
m
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with:
I
C
I
C
I amortissem

ent
reduced

modal


=

=

C
2
critical
I
m
. I

This equation can be solved very simply in the frequential field:

)
)
I ()
I =
F







éq 2.3-8
I
m (
2
2
. - +.
2 I I
+ I
)

where $ represent the transform of FOURIER and the frequency of excitation.

Particular numerical methods the such integral of DUHAMEL make it possible to pass this
expression of the frequential field to the temporal field. (see for example Doc. [R5.05.01] on one
method of temporal integration).

2.4
Modal truncation error

In the case of the modal recombination with damping proportional, one can put in
obviousness the truncation error which one makes while projecting on the basis of clean mode first
system. Indeed, if one considers the complete base of N clean modes of the discretized problem,
there is equivalence between the initial problem and the projected problem. Thus the exact solution of the problem
discretized by finite elements is written:
Nh
U =
I
I
I

where the generalized co-ordinates are solution of:

fi

2
I
& +
2 I I
I & + I
I =

I
m

summation extending on all the clean modes from the system (of finished size).

By solving the problem with a reduced number of clean modes, N < Nh. The solution obtained is
the following one:
N
~
U =
I
I
i=1

The error made by truncating the base of representation of the solution is thus:

Nh
E =
~
U - U =
I
I




éq
2.4-1
i=n+1

In the frequential field the expression of the error is:

)
Nh
T
) ~)
I (
F)
1
^E () = U - U =
.

. I

éq 2.4-2
m
2
2
I =n+1
I
I - + 2 J I I

the summation is carried out on all the neglected modes of the system.
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Let us study the response relating static of an oscillator to a purely sinusoidal excitation of
variable frequency (diagram below), with statics coefficients of the static answer
correspondent with a static force. One can distinguish three intervals in the spectrum where the oscillator
has a different behavior. In low frequency (<< 0), the oscillator has a static answer.
Around 0 the oscillator has a dynamic response (amplification of the mode), and high frequency
1
the oscillator answers in an inertial way (
dominating term).
2


Response of an oscillator
100
10

/
statics
1
Amplitude
Static answer
Dynamic response
Inertial answer
, 1
0
1
2
Reduced frequency/0


Let us suppose that the excitation of the system, defined by the vector F (), is with narrow tape, in particular
that it null for frequencies higher than max is given.

In this case, to represent the response of the linear system correctly, it is necessary undoubtedly
to take into account all the modes having a pulsation lower than max, because the latter go
to answer in a dynamic way the excitation.

On the other hand, modes such as >>
I
max nevertheless have a static contribution to the answer
system. These are often these modes that one does not take into account.

By making a development limited in in the vicinity of 0. One obtains the principal part of the error
who is:
)
N
T
) ~)
I. (
F)


^E () = U - U =


. 1 - 2 J
+ 0

I




. I

éq
2.4-3
K




I = p+1
I

I
I
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The error is all the more small as generalized rigidities of the neglected modes are large. In
principle thus, it will be necessary to take all the most flexible modes until residual flexibility
of an additional mode is in negligible relative value compared to the sum of the flexibilities
already taken into account:
N
1 << 1
kn+
K
1
i=1 I

However, it is observed that by neglecting the modes of high frequency one makes an error
systematic on the response of the system (even in low frequency). There are various possibilities
that we will detail now to correct the response in the range [,
0 max] where one chose
modes.

2.5
Corrections of modal truncation

To mitigate the problem of truncation due to the neglected modes, it is necessary to try to estimate their effect in
the field of frequency [,
0 max] which interests us. We saw that the neglected modes
having an own pulsation such as >>
I
max have a contribution known as static to the answer of
system in the field [,
0 max]. The techniques of correction consist in calculating this
static contribution.

2.5.1 Static correction a posteriori

The truncation error, by considering only the static response of the neglected modes (transformed
opposite of the principal part of the error) is:

Nh
T
~
I. (
F T)
E (T) = U - U

. I

éq
2.5.1-1
K
i=n+1
I

But a priori the neglected modes as their generalized rigidities are unknown. On the other hand, one
can determine the complete static response of the system to a loading F, the latter is worth:

Nh
T

-
.F
1
I
(T)
U = K. (
F T)

. I
K
i=1
I
The correction which should be made is thus:

Nh
T
I. (
F T)
N
T

-
.F
1
I
(T)


. I K. (
F T) -

. I
K
K
i=n+1
I
i=1
I

The corrected solution of the response of the system is thus worth:

N
T
~
~

-
.F
1
I
(T)
U = U + E U + K. (
F T) -

. I éq
2.5.1-2
K
i=1
I

This correction is called a posteriori, because it does not intervene in the dynamic resolution of
linear system and can only be calculated well later on. If (
F T) breaks up into K produced
functions of time by functions of the co-ordinates of space, this correction requires one
factorization of K and K resolutions.
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This method has the advantage of not increasing the number of vectors taken into account in the base.
This method is applicable in the case of an excitation to narrow tape, or at least having one
known cut-off frequency. The correction is exact in the field low frequency but can
to distort the response of the system in high frequency [§Annexe1].

2.5.2 Addition of static modes to the base

Let us suppose that the loading F (T) is written:

F (T) = J (T) F
. J
J

The second way of correcting the truncation error consists in adding to the base of the clean modes
initial of the static modes J defined as the deformation in each Fj effort given:

1
-
J = K F
. J

éq
2.5.2-1

The new base of projection to be considered is as follows:

) = [,…,
,…,
,] = []
1
2
p
1
2
m

éq
2.5.2-2

The components generalized to use are as follows:

) = [,…,
, µ, µ…,
, µ] = [, µ]
1
2
p
1
2
m

éq
2.5.2-3

The problem projected on the supplemented basis is:

\
0
0

\ 0 0



T



T

0 m
0
I
.M
.
&
T


0 K
0
I
.K
.
.F
.




+



=



éq
2.5.2-4
0
0
\
µ


&
0 0
\






µ T .F
.






T .M
.
T .M
.
T .K
.
T .K
.

It is noted that one lost the diagonal character of the generalized matrices, but the advantage obtained
is that the base supplemented with static modes makes it possible to represent it correctly
low frequency behavior of the initial system.

For example it is simple to show that at null frequency the solution of this system is:
=
0 and µ
= which is the exact solution of the initial static problem.

One presents in appendix 1, the comparison on a discrete system with 3 degrees of freedom between
exact solution, the solution projected on 1 mode, that projected on a mode with a correction
statics and the solution consisted 1 clean mode and 1 static mode.

One realizes that the addition of static modes makes it possible to extend beyond the interval
[, 0 = max
max
(J)] the good dynamic representation of the system. This technique thus seems
very interesting, it has the merit to carry out the correction immediately what will be interesting
for the nonlinear methods where one needs the knowledge of physical displacements with
each step of time.
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3 Extension of the methods of reduction of Ritz in
non-linear

3.1 Problem
General

The non-linear problem of dynamics discretized without damping can be generally put
in the following form:
.
MR. X & + G (X) = F (T)
N
X R éq
3.1-1

G (X) is a non-linear function of X which represents the internal forces of the system like
all the other forces which are dependant on displacement, F the vector of the external forces and
M the matrix of mass of the system.
The matrix of tangent rigidity of the system is by definition:
tg
G

K (=
X)
(X) éq 3.1-2
X


It makes it possible to define a modal base at every moment by:

2
- tg

M + tg
K
.



tg
= 0
I
I (X)







éq 3.1-3
(X)



The modes thus defined depend on X, therefore moment T.
Knowing that the calculation of the modal base is very expensive in calculating times, the idea to want to project
with each step of time the model on a modal basis, then to solve, is without interest by report/ratio
with a direct resolution.

The method most usually used consists in defining a base of projection while adding to
modes calculated on an initial configuration of the forms allowing to project nonthe linearity.
Example: if nonthe linearity comes from a specific shock, one proposes to enrich the base
modal with static modes allowing to project the effort undergone by the structure lasting the shock
[R5.06.04].

The method of Ritz remains always relevant in nonlinear calculations, if the selected base allows
to correctly project displacements and the efforts.

The nonlinear problem projected on an unspecified basis is written:

T
T
T
N

M.&+ G (X)
=
F R




éq 3.1-4

Two possibilities are then possible:

·
nonthe linearities are located and one can evaluate nonthe linearity on the basis of projection
: G (X) = G ().
The problem to be solved is a nonlinear differential connection in smaller size.
Various strategies are possible to solve this problem, depending primarily
technique of integration which one wishes to use.
·
nonthe linearities are total, and it is necessary to pass by again in the space of the physical ddls for
to calculate the internal forces: G (X).
This second method is more expensive it is much less current.
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One presents in appendix 2, the comparison on a system with 3 degrees of freedom with nona linearity
in 3
X enters the exact solution and the solution obtained by the method above with 1 then 2 modes.
One realizes that it is necessary to take more modes counts some than for the linear problem.
On the other hand, on this example 2 modes are enough very well to describe the system.

3.2
Indication of the error of projection

For the nonlinear problems the physical direction of the number of modes to be taken into account is
completely lost, and if the methods of reduction always give a solution it is necessary to know it
degree of confidence which one can grant to them. A way of proceeding, which is a little expensive but
essential is to calculate the residue of the initial system to each step of time. It is defined by:

R = MR. X
. & + G (X) - F (T)

This vector residue is unfortunately not null, it is only its projection on the basis used
who is.

A standard can then be calculated for this residue; more the standard of the residue will be small more one will be able
to grant confidence to the solution.

To use a relative value, one may find it beneficial to calculate the following fraction:

R
R =
(


éq
3.2-1
max F (T), G (X), Mr. X &)

Note:

This indicator is not currently established in Code_Aster.
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4
Use in Code_Aster

In Code_Aster, the methods of Ritz are usable in transient primarily by
operator DYNA_TRAN_MODAL [U4.53.21].

A phase of projection of the matrices of rigidity and mass on a basis of vectors is carried out
by operators PROJ_MATR_BASE [U4.63.12] and PROJ_VECT_BASE [U4.63.13].

The generalized dynamic problem is then solved in operator DYNA_TRAN_MODAL by one
diagram of explicit integration (EULER or DEVOGELEARE) or implicit (NEWMARK).
characteristics and properties of the diagrams of integration are presented in the note [R5.06.04]. For
the structures for which the assumption of BASILE does not apply (damping not
proportional) one will project also the matrix of damping which does not become diagonal.
The integration of the coupled system is done then obligatorily with the implicit scheme (NEWMARK) or
clarify (EULER).

Localized non-linearities are specified directly in operator DYNA_TRAN_MODAL. One
can introduce localized non-linearities of the shock type and friction (see [R5.06.03] Modélisation
shocks and frictions), modal forces function of displacement or speed (see
[R5.06.05] on the modeling of a fluid force of blade).
The static corrections of truncation a posteriori are available in the case of an excitation
single (see [R4.05.01] seismic Réponse).

The addition of static modes can be done by using the operators as a preliminary
MODE_STATIQUE [U4.52.14] and DEFI_BASE_MODALE [U4.64.02]. When the problem comprises
non-linearities only the explicit diagrams can be used.

For total non-linearities [éq 3.1-4], it is possible to use command DYNA_TRAN_EXPLI
[U4.53.03] which calculates with each step of time the internal forces according to the physical ddls, then
projète the problem on a modal basis.

An operation of return to the physical base is then necessary to obtain the sizes
physics such as displacement, speed or acceleration on the structure. This operation is carried out
by operator REST_BASE_PHYS [U4.64.01].

More generally, the approach of Ritz can be used in harmonic calculation by the command
DYNA_LINE_HARM [U4.53.22] and of spectral concentration of power by the command
DYNA_ALEA_MODAL [U4.53.23].

Finally the dynamic under-structuring can be regarded as a method of Ritz specific
[R4.06.02].

Handbook of Référence
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5 Bibliography

[1]
BATHE - WILSON: Finite Element procedures in Engineering Analysis
[2]
Mr. GERADIN, D. RIXEN: Theory of the vibrations: application to the dynamics of the structures ­
Masson 1993
[3]
J.F.IMBERT: Analyzes of the structures by finite elements - cepadues editions 1979
[4]
BELYTSCHKO - LIU - PARK: Innovative methods for not linear problems
[5]
EWINS D.J. Modal Testing: Theory and practice Reserch Studies Press LTD
[6]
R.E. RICKELL: Non-linear dynamics by mode superposition Computer Methods in Applied
Mechanics and Engineering (1976) flight 7
[7]
P. LUKKUNAPRASIT: Dynamic response off year elastic, viscoplastic system in modal
coordinates. Earthquake Engineering & Structural Dynamics (1980)
[8]
G. JACQUART: Methods of RITZ in non-linear dynamics - Application with systems
with shock and friction localized - Rapport EDF-DER HP-61/91.105
Handbook of Référence
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Appendix 1

Let us consider the discrete system with three masses according to:

m
m
m
K
K
K


The matrices of rigidity and mass are:

m 0 0
K
- K
0




M = 0 m 0 K = - K 2k - K

0 0
m

0 - K 2k

K
That is to say: 2
0 =

m

The clean modes and their pulsation are worth:

1


2 = 0 198
,
2
1
0
,
1

m = 1 8
, 41
,
m
1

= 0 802
,




0 4
, 45

1



2 = 1555
,
2
2
0,
2
m = 2 8
, 63,
m2
= - 0 555
,


-

1 2
, 47

1



2 = 3 247
,
2
3
0
,
3

m = 9 2
, 96
,
m
3

= - 2 247
,




1 802
,

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Let us compare the answers of the system modelled by only one clean mode with or without static correction:
Response of the system
1 mode
Exact solution
1 mode + correc
100
10
1
Amplitude
, 1
, 01
0
1
2
Frequency


It is noted that the static correction makes it possible to correct the low frequency answer, the model with 1
mode plus correction sticks perfectly to the exact solution in low frequency. On the other hand, into high
frequency (beyond the first mode), this correction results in over-estimating the answer enormously.
The use of the static correction will have thus to be used with prudence and within the framework of an excitation with
narrow tape.

Let us look at what the method of addition of a static mode gives.

If one applies a unit force to item 1, the static deformation is worth:

3
1
=
S
2
K
1

The projected matrices of mass and rigidity which one obtains are as follows:


049
,
5



841
,
1

m

2

365
,
0
K 1

)
0

)

M =

and K =
049
,
5
m


3


1




14

2
2


K
0
K


this system has as Eigen frequencies:

) 2
2
) 2
2
1
=

198
,
0

0

and
1
=

667
,
1

0

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The response of the system modelled with a clean mode and a static mode is as follows:


Response of the system
Exact solution
1 mode + 1 static mode
100
10
1
Amplitude
, 1
, 01
0
1
2
Frequency


One realizes that one corrects very well low frequency, (effect of correction static), one models
dynamics of the system well beyond the first mode taken into account. On the other hand the effect of the second mode is
badly represented (shift on the frequency).
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Appendix 2

Let us consider the discrete system with three masses according to:

m
F (X)
m
m
K
K
K


The matrices of stiffness and mass are:

m 0 0
K
- K
0




M = 0 m 0 K = - K 2k - K

0 0
m

0 - K 2k

Let us make this system non-linear by adding a term of force interns between 1
X and x2 cubic:

F = K (. 1
X - X) 3
2

Let us seek to evaluate the response of this system to a forced excitation of frequency close to the first
Eigen frequency of the linear system (one chose =

18
,
0

O), with an important amplitude F = K
m
.
3.

In this configuration, the response of the system cannot be evaluated by the response of the linear system (it
cubic term is well too important), it is necessary to implement a non-linear calculation with pseudo-forces
as one showed in [§3.2].

One can see in the report/ratio [bib8] the curves of the transitory results of this method, while taking in
count one or 2 clean modes of the initial linear system.

With only one clean mode, one realizes that one makes a relatively important error (reaching sometimes
50%), on the other hand it are satisfactory to note that the extrema vibrations are rather well envisaged. One
could have hoped that while exciting in on this side first Eigen frequency it would have is enough to only one clean mode
to model the response of the system, it is seen here that it is not the case. As one often notes it for
non-linear systems, the system also answers with the surharmoniques ones of the frequency
of excitation.

On the other hand, by taking 2 clean modes to model the response of this structure to 3 ddl, one obtains one
very satisfactory result (a few % of error on the amplitude), with the eye one has evil to distinguish the difference. This
show that by choosing a sufficiently rich base of projection one can thanks to a method of
pseudo-forces to model a dynamic system very well complexes with non-linearities.

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