HEAD INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 239

to that of impact (defined, for example, as the time of the deceleration of
the head's center of mass to come to a stop due to the deformed coating
or padding), one can neglect this interaction for many types of impacts in
view of the relatively high stiffness of the coating or padding, as compared,
for example, with the bending rigidity of the cervical spine.
The quality of the impact isolation will be characterized for the model
under consideration by the peak magnitude of the displacement of the point
mass,

J 1 =

t ∈ [0 , ∞ ) | x(t) | ,

max (7.8)

and the value of the HIC,

1
t 2 − t 1

t 1 ) . dt 2 . 5

t 2

| u(t) |

J 2 = max

t 1 ,t 2 ,t 2 − t 1 ≤

(t 2 − (7.9)

m

t 1

Problem 7.1 Minimization of the Peak Displacement
for Constrained HIC

List of Variables for Problem 7.1

State variable x Displacement of the head relative

to the inertial reference frame

Control variable u Force of interaction of the head

with the impact surface

Functions and parameters of
the model

m Mass of the head

v 0 Initial velocity of the head

The maximum length of the time
interval t 1 ≤ t ≤ t 2 occuring in the

definition of HIC

Performance index J 1 Maximum magnitude of the

displacement of the head relative
to the inertial reference frame

Performance criterion
subjected to a constraint

J 2 Head injury criterion (HIC)

Constraint H Maximum allowable value for the

criterion J 2

Optimal solution J 1 (u 0 ) Optimal value of the performance

index J 1

u 0 Optimal control u

(continued)