THORACIC INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 226

sternum and the other end is attached to the output link of the actuator
of the controller. In this case, the force u acting on the sternum can be
represented as

u =− k(x 1 − X), (6.48)

where x 1 is the absolute displacement of the sternum, X is the coordinate
(measured relative to the inertial reference frame) of the point of attachment
of the spring to the vehicle, and k is the coefficient of stiffness of the spring.
The coordinates x 1 and X are introduced so that the spring is unstrained
for x 1 X and, hence, the force acting on the occupant is zero.

The attachment point can be moved so that the level of protection pro-
vided by an elastic restraint modeled by the spring can coincide with that
of the limiting performance. To calculate the corresponding law of motion
of the attachment point, substitute u

=

x 1 (t) ,where u 0 (t)
is the optimal control that resulted from the solution of Problem 6.1 of
Section 6.3 and x 1 (t) is the corresponding law of motion of the sternum,
into Eq. (6.48) and then solve the resulting equation for X to obtain

= u 0 (t) and x 1 =

u 0 (t)
k

x 1 (t) X(t) = + . (6.49)

This is an analogue of Eq. (6.13) obtained in Section 6.2 for
the single-degree-of-freedom model. Figure 6.4 plots the functions
X(t) − x v (t) , which defines the motion of the attachment point relative to
the vehicle, for the limiting performance behavior shown in Fig. 6.2. The
spring stiffness coefficient is identified as k = 10 5 N / m, which is equal in

order of magnitude to the stiffness coefficient of standard car seat belts.
The solid, dashed, and dotted curves correspond to different durations of

0.5

0.4

0.3

0.2

0.1

0.0

-0.1

0 50 100 150 200

Time (ms)

FIGURE 6.4 Time histories of the motion of the restraint system attachment point for
various crash pulses (solid line for T p = 0 . 08 s, dashed line for T p = 0 . 1 s, and dotted
line for T p = 0 . 12 s).