THORACIC INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 228

primarily from the high stiffness of the spring (restraint system), as was the
case in Section 6.2 for a single-degree-of-freedom system. The occupant's
excursion is much less sensitive to a change in the pulse duration. The
increase in this characteristic does not exceed 10% as the pulse duration
increases by 50%. The sensitivity analysis indicates that open-loop control
of the motion of the attachment point in accordance with Eq. (6.49) is
practically inapplicable and a feedback control is needed.

6.4.3 Feedback Control of the Restraint Force

Figure 6.2 represents the limiting performance of the system and shows
that the time histories of the optimal control u 0 (t) and the constrained
injury criteria have a low sensitivity to the changes in the duration of the
crash deceleration pulse. This suggests that to avoid high sensitivity of the
constrained injury criteria (especially the chest compression) to the change
in the pulse duration, a feedback is needed in the control circuit of the
restraint system attachment point. This feedback would sustain the nominal
time history of the control force acting on the occupant and can have the
form

X

=− α [ u − u 0 (t) ] , (6.50)

where u is the current control force, u 0 (t) is the nominal control force time
history to be followed, and α is a feedback gain. Equation (6.50) is an
analogue of Eq. (6.20) that defines the feedback control of the attachment
point of the spring in the single-degree-of-freedom system considered in
Section 6.2. To simulate the behavior of the thorax model with the restraint
system attachment point controlled according to Eq. (6.50), it is necessary to
substitute the expression of Eq. (6.48) for u into Eq. (6.50), to augment the
system of Eq. (6.26) with the resulting relation, and to solve the augmented
system subject to the initial conditions

x 1 ( 0 ) = 0 , x 2 ( 0 ) = 0 , x 3 ( 0 ) = 0 , x v ( 0 ) = 0 ,

(6.51)

x 1 ( 0 ) = v 0 , x 2 ( 0 ) = v 0 , x v ( 0 ) = v 0 ,

( 0 ) =

0 .

The initial conditions of Eq. (6.51) are obtained from those of Eq. (6.27) by
eliminating the condition x 3 ( 0 ) ˙ = v 0 and adding the condition X( 0 ) = 0.

0 in the thorax
injury model (Lobdell et al., 1973), which implies a reduction in the order of
the system of differential equations by 1. The condition X( 0 )

The condition x 3 ( 0 ) ˙ = v 0 has been eliminated because m 3 =

0 indicates
that before the impact the restraint system attachment point is in its home
position and the spring is unstrained.

=