OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 165

Note that it is not necessary to single out the criterion J 2 in the formula-
tion of Problem 4.8. This criterion could have been included among criteria
J s (u 2 ) . However, this criterion is chosen to be separated in Problem 4.8 as
was the case for Problem 4.5.
As Problem 4.6 was associated to Problem 4.5 for a single-degree-of-
freedom object, an auxiliary problem will be related to Problem 4.8 for a
multi-degree-of-freedom object.

Problem 4.9 Auxiliary Problem Related to Problem 4.8

List of Variables for Problem 4.9

State variables x 0 Displacement of the base relative to

the inertial reference frame

x 2 Displacement of the housing of the

object relative to the inertial
reference frame

y i Displacement of the i th component

(point mass) of the object relative to
the housing of the object,
i = 1 ,...,n

y n -vector of the displacements of the

components of the object relative to
the housing of the object,
y = [ y 1 , ..., y n ]

Control variable u 2 Force F 2 divided by the mass of the

housing of the object, u 2 = F 2 /m

External disturbance σ Shock acceleration pulse

f i

Functions and
parameters of the model

Force f i divided by the mass of the
i th component of the object,
f i = f i /μ i , i =

1 , ..., n

V Initial velocity of the base, container,

and housing of the object

μ i Mass of the i th component of the

object divided by the mass of the
housing of the object, μ i = μ i /m ,

i = 1 ,...,n

J 1

Performance index Maximum magnitude of the

displacement of the housing of the
object relative to the base

Performance criteria
subjected to a constraint

J 2 Maximum magnitude of the control

u 2

(continued)