INTRODUCTION - Injury Biomechanics and Control: Optimal Protection from Impact - page 8

optimal control problem for a two-component system. The two-component
system involves only the base and the object to be protected. In addition,
the auxiliary problem involves only one control function, the control force
acting on the object, whereas the primary problem has two control func-
tions, the control force acting on the object and the control force acting
on the container. This makes the auxiliary control problem easier to solve.
Knowing the optimal control and the minimum of the performance index
for the auxiliary problem, one can determine the optimal controls and the
minimum value of the performance index for the primary problem, which is
a three-component system. The solutions of these two problems are related
by simple analytical equations. To solve the auxiliary problem, one, as a
rule, needs to use numerical methods. However, if the object to be protected
is modeled by one or two point masses, the auxiliary problem can also be
solved analytically in some cases.
The concept of pre-acting control for active shock isolators was consid-
ered by Balandin, Bolotnik, and Pilkey (2005). With pre-acting control, the
isolation system begins to respond to an impact before the impact has been
applied to the base. The limiting performance of an isolator with pre-acting
control was investigated for a single-degree-of-freedom system subject to
an instantaneous impact. The isolation performance index was defined as
the maximum of the absolute value of the displacement of the object to be
isolated relative to the base, provided that the magnitude of the control force
transmitted to the object does not exceed a prescribed value. It was shown
that there is a substantial advantage in the use of pre-acting isolators over
isolators without pre-action. Particular attention was given to a pre-acting
isolator based on a passive elastic element (a spring) that is separating the
object to be protected from the base. An example illustrated the calculation
of the design parameters of such an isolator.

1.2.2 Best and Worst Disturbance Analyses

A technique to study the sensitivity of impact responses to prescribed crash
test conditions was presented by Crandall et al. (1996). Motor vehicle
impacts were used to illustrate the principles of this sensitivity analysis.
Impact conditions were regulated by specifying either a corridor for the
acceleration time history or other test parameters such as velocity change,
crush distance, and pulse duration. By combining a time-domain constrained
optimization method and a multibody dynamics simulator, the upper and
lower bounds of occupant responses subject to the regulated corridors were
obtained. It was found that these prescribed corridors may be either wide
enough to allow extreme variations in occupant responses or so narrow
that they are physically unrealizable in the laboratory test environment. A
new corridor based on specifications for the test parameters of acceleration,