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Ch72-I044963.fm Page 356 Tuesday, August 1, 2006 9:53 PM
Ch72-I044963.fm
356
356 Page 356 Tuesday, August 1, 2006 9:53 PM
Constitutive Model

Mathematical Identification.
The force-displacement relationship is obtained directly from the tests done, and the EDC-
displacement relationship is obtained through the use of the EDC concept using the constant velocity
of the tests and the force obtained from the following mathematical model. The constitutive model is
obtained by mathematical identification of relationships force-displacements gotten from the test.
Power equations, Eqn. (2), have been found in function of the displacement, 5, and electrical current, i.
Figure 3 and Eqn. (3), shown results for an applied current of 3 Amperes.
/ = (2)

Where / is the force required to overcome the resistance to compress the damper. And, 8 is the
displacement given by compression in the damper.

Mathematical Identification, 3 A
Mathematical Identification, 3 A
30
25
N 20

,
e
c 15
r
o
F 10
5
0
0.010 °
.
0 0.005 0 01 0.015 0.020 0.02 0.030
0.005
0.02
0.015
0.030
0.020
Displacement, m 5 5
Displacement, m
Figure 3: Mathematical identification of relationship force-displacement.
/ = 11263S iU257 (3)
Once, all equations have been established, the constants a and b were plotted, as shown in Figure 4, to
obtain general polynomial expressions, Eqn. (4) and Eqn. (5), in function of the current.
a=-0.0079 i 2 + 1.0958 i + 8.1546 (4)
b = 0.011i 2 - 0.0209 i +0.1869 (5)
Finally, a general power equation, Eqn. (6), constituted by two polynomial expressions has been
obtained:
2
/=(-0.0079i +1.0958i+8.1546)£' i.Olli 2 -0.0209i I 0.1869 (6)
EDC is obtained and plotted, as Figure 5a shown, based on the constant velocity used in tests and the
force obtained from equation (6) at 0.005, 0.01, 0.015, 0.02 and 0.025 m displacements. Similar to the
previous analysis a general power equation, Eqn. (7), has been obtained:

•0.01 I;--0.0209;+O.I868
(7)
The connection between the mathematical model and the software can be given by introducing the
equivalent damping coefficient expression in function of the displacement.