Code_Aster ®
Version
6.0
Titrate:
Harmonic answer
Date:
24/02/03
Author (S):
F. STIFKENS Key
:
R5.05.03-A Page
: 1/8
Organization (S): EDF-R & D/AMA
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
Document: R5.05.03
Harmonic answer
Summary
This document presents the theoretical bases of the calculation of the steady state of the response of a system
complex mechanics, with linear behavior, subjected to a harmonic dynamic stress. Calculation
relate indifferently directly to the system modelled in finite elements, or represented by a base
modal; in this last case if the modal base is the product of the technique of under-stucturation one
will refer to the document [R4.06.03].
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
HT-66/02/004/A
Code_Aster ®
Version
6.0
Titrate:
Harmonic answer
Date:
24/02/03
Author (S):
F. STIFKENS Key
:
R5.05.03-A Page
: 2/8
Count
matters
1 Introduction ............................................................................................................................................ 3
2 harmonic Equation ............................................................................................................................. 3
2.1 Harmonic equation of the structures ............................................................................................... 3
2.1.1 Direct calculation ............................................................................................................................ 3
2.1.2 Calculation on modal basis .......................................................................................................... 4
2.2 Harmonic equation of the acoustic fluids ................................................................................ 5
2.3 Harmonic equation of the systems fluid-structures ................................................................... 5
2.4 General harmonic equation ....................................................................................................... 6
3 Bibliography .......................................................................................................................................... 8
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
HT-66/02/004/A
Code_Aster ®
Version
6.0
Titrate:
Harmonic answer
Date:
24/02/03
Author (S):
F. STIFKENS Key
:
R5.05.03-A Page
: 3/8
1 Introduction
In the harmonic problems, the studied system is subjected to an excitation varying like
product of an unspecified function of space by a sinusoidal function of time.
To seek the answer consists in calculating the field of the sizes represented by the ddls of
modeling in finite elements of the system. When the system has a linear behavior the answer
field of the sizes observed tends quickly (because of extinction of its component
transient by dissipation interns) towards a steady state: the resulting field varies finally
harmonically like the excitation. It is this steady state of the answer that one proposes
to calculate.
General notations:
T
: time
P
: Not running of the model
:
Pulsation (rad.s-1)
J
: Imaginary pure unit (j2 = -) 1
M
: Stamp of mass resulting from modeling finite elements
K
: Stamp rigidity resulting from modeling finite elements
C
: Stamp damping exit of modeling finite elements
Q
: Vector of the degrees of freedom resulting from modeling finite elements
F ext.: Vector of the forces external with the system
: Stamp vectors of the base of the substructures
: Vector of the generalized degrees of freedom
2 Equation
harmonic
We establish the dynamic equation in the case of a harmonic stress for three kinds of
mechanical systems:
· pure structures (without fluid),
· pure fluids (without structure) with linear “acoustic” behavior,
· analog and digital systems structures and fluids in interaction fluid-structure.
2.1
Harmonic equation of the structures
The vibratory behavior of a pure structure results from the external forces which are applied to him.
The size to be calculated is displacement in any point P of the model.
2.1.1 Calculation
direct
In the case of direct calculation on the model in finite elements we can write:
Driven & + C U & + K U = F (P, T)
ext.
éq
2.1.1-1
where:
M
is the matrix (real) of mass resulting from modeling finite elements of S,
C
is the matrix (real) of damping exit of modeling finite elements of S,
K
is the matrix (real) of rigidity resulting from modeling finite elements of S,
F (P, T)
is the vector (complex) of field of the external forces applied to S,
ext.
U, U & and U & are the vectors (complex) displacement, speed and acceleration, functions of P
and T, resulting from modeling finite elements.
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
HT-66/02/004/A
Code_Aster ®
Version
6.0
Titrate:
Harmonic answer
Date:
24/02/03
Author (S):
F. STIFKENS Key
:
R5.05.03-A Page
: 4/8
In a harmonic problem, one imposes a loading dynamic, spatially unspecified, but
sinusoidal in time. One is interested then in the stabilized answer of the system, without holding account
transitory part.
The field of the external forces is written:
F (,) = F
J T
P T
{
(P)}E
ext.
ext.
The field of displacements is written:
U (P, T)
U
{(P)}E J T
=
The fields speed and acceleration are written:
&u (P, T) = J {U (P)}E J T
&
U (P, T) = - 2 {U (P)}E J T
Finally the structure S checks the following equation:
(
2
K + jC - M) {}
U = {F (P)}
ext.
éq
2.1.1-2
Particular case: if damping is of hysteretic type “total” the equation [éq 2.1.1-1]
becomes:
Driven & + K 1
(+ J) U = F (P, T)
ext.
éq
2.1.1-3
where is a total loss ratio (cf [R5.05.04]).
Then the equation [éq 2.1.1-2] is replaced by:
(
2
K - M) {}
U = {
(P)}
C
fext
éq
2.1.1-4
where:
M
is the matrix (real) of mass resulting from modeling finite elements of S,
K = K + jK is a complex matrix of rigidity.
C
2.1.2 Calculation on modal basis
The calculation of the harmonic response by the method of modal synthesis consists has to seek it
field of unknown displacement, resulting from modeling finite elements, on an adapted space, of
reduced dimension (transformation of Ritz).
One will refer to the documents [R4.06.02] and [R4.06.03].
If one rather uses this method the equation [éq 2.1.1-2] is projected on the basis of modal S and one
leads to the following harmonic equation:
(
2
K + J C - M) {} = {F}
ext.
éq
2.1.2-1
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
HT-66/02/004/A
Code_Aster ®
Version
6.0
Titrate:
Harmonic answer
Date:
24/02/03
Author (S):
F. STIFKENS Key
:
R5.05.03-A Page
: 5/8
where:
M = TM
is the matrix (real) of generalized mass of S,
C = TC
is the matrix (real) of generalized damping of S,
K = TK
is the matrix (real) of generalized rigidity of S,
{F}
T
= {F}
is the vector (complex) generalized harmonic external forces
ext.
ext.
applied to S,
is the matrix (real) modal vectors of the base of Ritz of S,
{(P)}
is the vector (complex) generalized harmonic displacements.
Once {(P)} determined by éq 2.1.2-1 one makes a restitution on physical basis (cf [R4.06.02]).
2.2
Harmonic equation of the acoustic fluids
The document [R4.02.01] described modeling by finite elements of a fluid system (without transport)
having a linear acoustic behavior.
The fluid system F undergoes a harmonic stress acoustic speed on part of its
border. The harmonic answer is described by the following equation [éq 2.2-1], where size with
to calculate is the acoustic pressure in any point P of the model.
(
2
K + jC - M) {p (P)} = - J {v (P)}
N
éq
2.2-1
where:
M
is the matrix (complex) of “mass” acoustic exit of
modeling finite elements of F,
C
is the matrix (complex) of “damping” acoustic exit of
modeling finite elements of F, and in the species of the edge F where one
Z
apply an acoustic impedance,
K
is the matrix (complex) of “rigidity” acoustic exit of
modeling finite elements of F,
v (,) = v
J T
P T
{(P) E
}
where {v (P)} is the vector (complex) of field speeds
N
N
N
normal acoustics applied to the border F of F where one
v
apply acoustic speeds,
p (P, T)
p
{(P)}E J T
=
where {p (P)} is the vector (complex) acoustic pressures resulting from
modeling finite elements of F.
2.3
Harmonic equation of the systems fluid-structures
The document [R4.02.02] described modeling by finite elements of a system F + S made up of one
fluid part (without transport) F in interaction with a part structure S (interaction out of F S).
Fluid and structure have a linear behavior.
The fluid system F undergoes a harmonic stress normal acoustic speed on a part
of its border. The harmonic answer is described by the following equation [éq 2.3-1], where sizes
to calculate are:
· acoustic pressure in any point P of the fluid F,
· displacement in any point P of the structure S,
· as an auxiliary potential of displacement in any point P of the fluid F,
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
HT-66/02/004/A
Code_Aster ®
Version
6.0
Titrate:
Harmonic answer
Date:
24/02/03
Author (S):
F. STIFKENS Key
:
R5.05.03-A Page
: 6/8
U
(P)
(
2
3
K - M - J I) p
(P) = + J {v (P)}
N
éq
2.3-1
(P)
where:
M
is the matrix (real) of “mass” fluid-structure resulting from
modeling finite elements of the fields F and S
I
is the matrix (real) of “impedance” fluid exit of modeling
finite elements of the edge F of the field F where one is applied
Z
impedance
K
is the matrix (real) of “rigidity” fluid-structure resulting from
modeling finite elements of the fields F and S
v (,) = v
J T
P T
{(P) E
}
where v (P) are the vector (real) field acoustic speeds
N
N
N
normals applied to the border F of F
v
U (P, T)
U
{(P)}E J T
=
is the vector (complex) field of displacement in the structure
S
p (P, T)
p
{(P)}E J T
=
is the vector (complex) acoustic field of pressure in
fluid F
P T
(,) {(P)}E J T
=
is the vector (complex) field of potential of displacement in
fluid F
2.4
General harmonic equation
With an aim of taking into account all the harmonic cases of equations the operator
DYNA_LINE_HARM of Code_Aster solves the following general harmonic equation (cf [U4.53.11]):
3
2
K
J
I
N
(- J I - M + jC + K) {}
Q =
I
180
hi (F) E
gi (P)
éq
2.4-1
i=1
where:
I
Stamp fluid “impedance” possible exit of modeling finite elements,
M
Stamp of “mass” resulting from modeling finite elements,
C
Stamp “damping” exit of modeling finite elements,
K
Stamp “rigidity” resulting from modeling finite elements,
{Q (P}) Vecteur of the degrees of freedom resulting from modeling finite elements,
{G (P) Vecteur field with the nodes corresponding to one or more loads of force or
I
} acoustic or potential speed or imposed movement,
H (F)
I
Real or complex function of the frequency F,
=
2 F Pulsation
N
Power of the pulsation when the loading is a function of the pulsation,
I
Phase in degrees of each component of the excitation compared to a reference of
I
phase.
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
HT-66/02/004/A
Code_Aster ®
Version
6.0
Titrate:
Harmonic answer
Date:
24/02/03
Author (S):
F. STIFKENS Key
:
R5.05.03-A Page
: 7/8
As example if one takes the case of a system of fluid modelled in accoustics, without ddls
imposed, simply solicited on part of its border by a field normal speed
v (,) = v
J T
P T
{(P) E
}
, the terms of the equation [éq 2.4-1] become:
N
N
I
non-existent,
M
Stamp of mass resulting from acoustic modeling finite elements,
C
Possibly matrix of damping resulting from modeling finite elements
accoustics if impedance on border,
K
Stamp rigidity resulting from acoustic modeling finite elements,
{Q (P}) = {p (P}), vector of the pressures to the nodes,
{G (P) = {v (P)
N
}
I
}
, vector field normal speed to the faces (finite elements)
H (F)
= - 1. (constant),
I
=
2 F Pulsation,
N
= 1,
I
= 0.
I
Note:
In addition to the solution of the harmonic equation [éq 2.4-1], Code_Aster makes it possible to calculate them
derived from this solution compared to the loading {G (P) or to parameters of
I
}
mass, stiffness or damping (M, K, C). The equations whose these derivative are
solutions and the relative theoretical developments are in [R4.03.04].
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
HT-66/02/004/A
Code_Aster ®
Version
6.0
Titrate:
Harmonic answer
Date:
24/02/03
Author (S):
F. STIFKENS Key
:
R5.05.03-A Page
: 8/8
3 Bibliography
[1]
R. DAUTRAY, J-L. LIONS, “mathematical Analyze and numerical calculation for sciences and
techniques ", Tome 2, Masson, 1985.
[2]
G. DHATT, G. TOUZOT, “Une presentation of the finite element method”, Maloine S.A.,
Paris, 1984.
Handbook of Référence
R5.05 booklet: Transitory or harmonic dynamics
HT-66/02/004/A
Outline document