OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 136

The optimal control for Problem 4.4 (for the rigid-body model) is
given by Eq. (3.40) and, accordingly,

V
U .

u 0 (t) ˜ =− U for 0 ≤ t ≤ T = (4)

The corresponding time history of the coordinate x is given by

1

2 Ut 2

x 0 (t) = Vt − for 0 ≤ t ≤ T. (5)

The numerical results will be presented below for

V = 10 m / s ( = 36 km / h ). (6)

This is a typical velocity for vehicle crashes. To calculate the deceleration
time of the object in the rigid-body model, substitute 10 m / sfor V and
400 m / s 2 for U into the expression of (4) for T . This yields

T = 25 ms . (7)

In the case under consideration, the response characteristics satisfy
Assumptions 1 - 3.
The optimal behavior of the system is shown in Figs. 4.2 - 4.6.
Figure 4.2 shows the optimal control force u 0 (t) for Problem 4.2.
The y 0 (t) are

plotted in Figs. 4.3, 4.4, and 4.5, respectively. On the time interval
0

optimal time histories x = x 0 (t) , ξ = ξ 0 (t) ,and y =

≤ ≤

T , the function x 0 (t) is defined by (5) and the functions
ξ 0 (t) and y 0 (t) are defined by Eqs. (4.35) and (4.44), respectively.

t

20

0

-20

-40

0 10 20 30

Time (ms)

FIGURE 4.2 Optimal control u 0 (t) for Problem 4.2.