BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 52

reduce the transmitted force to an acceptable value. However, in general,
the peak magnitude of the relative displacement does not provide the pre-
cise estimate for the total necessary space since this criterion measures the
relative displacement of the object only in one direction, corresponding
to a larger deviation. It may happen that in response to the shock pulse
the object may oscillate about its home position and, hence, will experi-
ence displacements in both directions. This is the case, for example, for an
object attached to the base by means of a spring or a spring-and-dashpot
passive isolator. Then the total displacement will exceed the quantity J 1 .
The rattlespace measures the total displacement space. Criteria J 1 and

J 1

J 1

are related by the inequality 2 J 1 and, hence, the upper bound for the

rattlespace is 2 J 1 . This estimate, however, may turn out to be rather con-
servative, especially if the forward excursion of the object relative to the
base is substantially less than the backward excursion, as is the case for
an object attached to the base by means of a spring-and-dashpot passive
isolator; then the criterion J 1 should be preferred.
The peak magnitude of the transmitted force ( J 2 ) is a dynamic criterion
that indirectly characterizes the peak stress developed in the object. The
object can be seriously damaged or even destroyed if the peak stress exceeds
a threshold value.
For the criterion J 2 , the notation u(t) indicates that this criterion takes
into account the time history of the control force but does not imply that
the control force is necessarily a function of time alone. In general, it may
depend on the phase variables of the system, the coordinate x and the veloc-
ity

≤

x,t) . For example,
for the spring-and-dashpot isolator, the control force is characterized by
the function u

x , and (for active systems) on time, that is, u ˙ = u(x, ˙

kx .Then u(t) in Eq. (3.7) should be understood
to be the composite function u(x(t),

=− c x ˙ −

x(t),t) ,where x(t) is the solution of
Eq. (3.3) subject to the initial conditions of Eq. (3.4), for u

˙

= u(x, x,t) . ˙

3.1.3 Statements of Optimal Shock Isolation Problems

Optimization problems for the criteria J 1 and J 2 will be formulated. One
of these criteria will be minimized by choosing an optimal control force u ,
provided that the other criterion is constrained by a prescribed quantity. If
necessary, the criterion J 1 can be changed to J 1 .
If the shock acceleration (or deceleration) pulse is known or can be
reliably predicted in advance, typical optimal shock isolation problems are
formulated as follows.