BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 110

0.10

0.05

0.00

Best response
Worst response

-0.05

-0.10

0 20 40 60 80 100

Time (s)

FIGURE 3.16 Displacement of the object relative to the base.

change is defined as

45 . 1km/h ≤ V ≤ 48 . 3km/h ( 28 mph ≤ V ≤ 30 mph ). (3.201)

Figure 3.14 shows the time histories for the best and worst deceleration
pulses. If there were no constraints [such as in Eq. (3.201)] on the velocity
change, both the best and worst disturbance pulses would have switched
between the lower and upper bounds of the corridor since the solution of
a linear programming problem always lies on the boundary of the domain
constraining the design variables. From the time histories shown in Fig. 3.14
it is apparent that both the best disturbance and worst disturbance pulses
do not lie on the walls of the corridor on some time intervals. Therefore,
the velocity change assumes one of the boundary values prescribed by
Eq. (3.201), specifically, V = 45 . 1 km/h for the best disturbance and V =

48 . 3 km/h for the worst disturbance.
Figure 3.15 depicts the time history of the force applied to the dummy
by the seat belt in the case of the worst (solid curve) and best (dashed
curve) disturbances. It is apparent that in both cases the peak magnitude
of the force is attained before the end of the disturbance. This is because
the duration of the pulse is close to the undamped vibration period of the
system. Figure 3.14 shows that the duration of the deceleration pulse lies
between 0.075 and 0.09 s, while the vibration period for the parameters
of Eq. (3.200) is 0.096 s. The time histories of the displacement of the
dummy for the worst and best disturbances are shown in Fig. 3.16. In
fact, the curves of Fig. 3.16 repeat those of Fig. 3.15 scaled by the stiffness
coefficient ( 10 5 N/m ) . This is due to the fact that the damping does not have
much effect during the time interval under consideration. The damping ratio
ζ of Eq. (3.187) calculated for the system with the parameters of Eq. (3.200)
equals 0 . 066 s − 1 and, accordingly, the characteristic time of the damping
equals 15.2 s, which substantially exceeds the undamped vibration period.