Code_Aster ®
Version
6.0
Titrate:
Solution of a differential equation of second command (NIGAM)
Date:
30/01/03
Author (S):
D. SELIGMANN, O. BOITEAU Key
:
R5.05.01-B Page
: 1/6
Organization (S): EDF/TESE, SINETICS
Handbook of Référence
R5.05 booklet: Transitory dynamics
Document: R5.05.01
Solution of a differential equation
of the second command by the method of NIGAM
Summary:
We present in this document, a method of resolution of the linear differential equation of the second
command obtained during the calculation of a spectrum of oscillator.
Handbook of Référence
R5.05 booklet: Transitory dynamics
HR-19/02/027/A
Code_Aster ®
Version
6.0
Titrate:
Solution of a differential equation of second command (NIGAM)
Date:
30/01/03
Author (S):
D. SELIGMANN, O. BOITEAU Key
:
R5.05.01-B Page
: 2/6
1 Introduction
During the calculation of a spectrum of oscillator, one is brought to solve a differential equation of
second command whose solution is an integral of DUHAMEL.
If this integral can be calculated exactly using the transform of LAPLACE for some
simple analytical functions (Dirac function, Sinus, Cosinus, Heavyside,…) [bib1] it must be integrated
numerically in the general case.
This document presents an effective method to solve this problem.
This method is implemented in Code_Aster, in the operator CALC_FONCTION, key word
factor SPEC_OSCI.
2
Analytical solution of the equation
During the calculation of the spectrum of oscillator of a accélérogramme [R4.05.03], one is brought to solve
the linear differential equation of the second command
q&+ Q + 2
2
&
Q =
- (T)
where
Q (T)
is relative displacement
(T) is the acceleration of the movement imposed on the base
is the pulsation of the oscillator
is the reduced damping of the oscillator
With initial conditions on Q and &q.
The solution of this equation is written in the form:
Q (T) = +t H (T -).() D + Q (0) G (T) + q& (0) H (T) éq
2-1
0
where Q ()
0 and &q ()
0 are displacement and speed at the initial moment.
· Expression of (
H T) and G (T) according to the value of reduced damping.
- If
< 1 (damping under criticality)
- T
(
E
H T) =
2
sin
2
(T1)
1 -
éq
2-2
2
G (T) = E T
cos 1
sin 1 2
(T -) + 2 (T -)
1 -
Handbook of Référence
R5.05 booklet: Transitory dynamics
HR-19/02/027/A
Code_Aster ®
Version
6.0
Titrate:
Solution of a differential equation of second command (NIGAM)
Date:
30/01/03
Author (S):
D. SELIGMANN, O. BOITEAU Key
:
R5.05.01-B Page
: 3/6
- If
= 1 (damping criticizes)
(
H T) = you-wt
G (T) = (1 - wt) ewt
- If
> 1 (supercritical damping)
- T
(
E
H T) =
2
.
1
2
(HS T -)
- 1
G (T) = E T
C 2
1
2 1
(
H T
-) +
S
2
(H T -)
- 1
3 Method
numerical
The numerical method established in Code Aster was proposed by NIGAM and JENNINGS [bib2]
in the case of damping under criticality which corresponds to our initial seismic problem
[R4.05.03].
By introducing the formulation [éq 2-2] in [éq 2-1] one is thus led to solve the equation
differential:
Q (&t) + Q (T) + 2
2
&
Q (T) =
- (T)
with null initial conditions, whose solution is written:
1 T
Q (T)
- (T -)
=
E
[
sin D
(T -)]() D
D
0
with
2
D = 1 -
By supposing that (T) varies linearly inside each interval (T), one can then write
:
() = (T - T
) +
[(T) - (T - T)] for [
0, T
]
T
(T)
(t-t)
T
T
Handbook of Référence
R5.05 booklet: Transitory dynamics
HR-19/02/027/A
Code_Aster ®
Version
6.0
Titrate:
Solution of a differential equation of second command (NIGAM)
Date:
30/01/03
Author (S):
D. SELIGMANN, O. BOITEAU Key
:
R5.05.01-B Page
: 4/6
from where the equation to be solved: (expressed in the new variable)
&q (&) +
2 &q () + 2q () = has + B for [
,
0
T
]
where
= (T has - T
)
B = [(T) - (T - T
)] /T
:
initial
conditions
with
Q (0) = Q (T - T
)
q& (0) = q& (T - T
)
The solution of this equation is the superposition of a particular solution and solutions of
homogeneous problem.
has
2 B
B
· a particular solution: qp (T) = -
+
-
2 3 2
Q T = E [C1.cos + C
H
D
2.sin
D
]
· solutions of the homogeneous problem:
()
()
()
has
-
B B.
Consequently: Q () = E
[C1.co (S D) +C2.si (N D)]- +2 -
2
3 2
and by deriving Q (compared to) one a:
B
&q () () E (C cos
1
+ C sin
-
= -
D
2
D
) + E
(- C sin
1
+ C cos
D
D
2D
D
) - 2
The coefficients C1 and C2 are then determined by the initial conditions at the beginning of the interval
(i.e. for = 0).
has
2b
C = Q
1
(T - T
) +
-
2
3
1
has
2
2 - 1
C =
Q
2
& (T - T
) +
Q (T - T
) +
-
B
2
D
and by deferring C1 and C2 in the expression of Q and &q one obtains the matric equality for = T:
Q (T)
-
-
= (
Q T
T
T
T
With, T
) (
)
+ (
B, T
) (
)
Q
& (T)
Q
& (T - T
)
(T)
Handbook of Référence
R5.05 booklet: Transitory dynamics
HR-19/02/027/A
Code_Aster ®
Version
6.0
Titrate:
Solution of a differential equation of second command (NIGAM)
Date:
30/01/03
Author (S):
D. SELIGMANN, O. BOITEAU Key
:
R5.05.01-B Page
: 5/6
4
Coefficients of matrices A and B of the system to be solved
Stamp a:
= E T has
sin
cos
11
(T
D
) + (T
D
)
2
1
-
E T
a12 =
(
sin
T
D)
D
has = -
E T
21
(
sin T
D)
2
1 -
= E T cos has
sin
22
(T
D
) -
(T
D
)
2
1
-
Stamp b:
D
2
sin
- T
2
- 1
(T) 2 1
2
B = E
D
.
cos
11
+
+
+
-
2
D
3
2
(T)
T
T
3
T
D
2
sin
- T 2
- 1
(T) 2
1
2
B = E
D
.
.cos
12
+
-
+
2
D
3
(T)
T
T
2
3
T
2
2
-
- 1
B = E
T
. C (
bone
D
D T
) -
sin
T
2
()
21
+
2
T
1 -
-
2
1
+
1
.
3
2
(if
D
(Nd T) Co (sd T)
T
+
+
2 T
2
- T
2
-
1
b22 = - E
. C
2
(
bone D T
) -
(
sin D T
)
T
1 - 2
-
2
1
.
(if
(T)
3
D
(Nd T)
-
T
+
cos
D
2
T
2
with D
= 1 -
Handbook of Référence
R5.05 booklet: Transitory dynamics
HR-19/02/027/A
Code_Aster ®
Version
6.0
Titrate:
Solution of a differential equation of second command (NIGAM)
Date:
30/01/03
Author (S):
D. SELIGMANN, O. BOITEAU Key
:
R5.05.01-B Page
: 6/6
5
Calculation of Q ()
Knowing Q () and &q (), it is consequently possible to give the analytical expression of acceleration
& (&
Q).
&q () = - () E [C1 (
cos D) + C S
2
(
in D)]
1
+ E (- C if
1 D
(Nd) + C C
2D
(
bone D) - 2
2
&q (&) = + () E [C1 (
cos D) + C S
2
(
in D)]
+ () E (- C if
1 D
(Nd) + C C
2D
(
bone D)
- () E (- C if
1 D
(Nd) + C C
2D
(
bone D)
+ E [- C 2 Co
2
1 D
(sd) - C S
2D
(
in D)]
2
&q (&) = [() - 2 -
D] E
. [C1c (
bone D) + C2 (
sin D)]
however
2
2
2
D = (1 -), from where
q& (&)
2
= E [C1 (
cos D) + C2 if (
N D)]
6 Bibliography
[1]
R.J. GIBERT: Vibrations of the structures, Collection of Direction of Etudes and
Search of Electricité de France, n°69, Eyrolles 1988.
[2]
N.C. NIGAM & PC JENNINGS: Calculation off Response will spectra from motion
earthquake Bull. off the Seismological society off America, Vol.59 n°2 p 909 - 922 April
1969.
[3]
D. SELIGMANN, L. VIVAN: Seismic response by spectral method [R4.05.03].
Handbook of Référence
R5.05 booklet: Transitory dynamics
HR-19/02/027/A
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