BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 108

Eq. (3.186) with v(t) = δ(t) ,where δ(t) is the Dirac delta function, subject
to the initial conditions x( 0 ) =

0. This function can also be
defined as the solution of Eq. (3.186) with zero right-hand side [ v(t) ≡

0and x(t) =

0]

subject to the initial conditions x( 0 ) = 1.

Substitute the relations of Eq. (3.195) into Eq. (3.188) to obtain

0and x( 0 ) =

τ)v(τ) dτ

t

ω n g(t),
(3.197)
The discrete-time approximation of the response measure of Eq. (3.197)
has the form

J = max

t

K(t − , (t) = 2 ζω n ˙ g(t) +

0

min[ i, N ]

J(v) = max

i ∈ [1: L ]

p ij v j ,

j = 1

(3.198)

jh

p ij = K(ih − τ)dτ, j ≤ i, v = [ v 1 ,...,v N ] ,

(j − 1 )h

where L is the number of discrete intervals on the time axis on which the
response to the impact pulse is to be calculated. To obtain reliable results,
it is necessary to consider the response on the time interval substantially
exceeding that of the crash pulse. Since the crash pulse has a rather short
duration, the response measure can attain its maximum after the disturbance
has ceased to act. For this reason, the number of time instants at which the
response is measured should exceed the number of the discretization points
in the crash pulse interval [0 ,T ] and, hence, L > N .
The criterion of Eq. (3.198) is a function of N variables v = [ v 1 ,...,v N ].

The constraints of Eq. (3.192) are discretized as

N

v − (j h) ≤ v j ≤ v + (j h), j ∈ [1 : N ] , − ≤ h v j ≤ V + .

j = 1

(3.199)
In the discrete-time formulation, the worst disturbance problem is
reduced to the maximization of the function of Eq. (3.198) and the
best disturbance problem to the minimization of this function subject
to the constraints of Eq. (3.199). These constraints are linear relative to
the design variables v 1 ,...,v N , and the function to be maximized or
minimized is a maximum of the absolute values of linear functions of
these variables. In this case, the constrained minimization (maximization)
problem can be reduced to that of linear programming, which substantially
facilitates the solution, since there are rapidly converging reliable linear