BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 113

respective trapezoid. For the curve v + (t) of Eq. (3.207) to lie above the
curve v − (t) = ψ of Eq. (3.203), it is necessary that

τ 1 λ ≥ 1: τ 1 + ( 1 − λ)τ 4 ≤ ≤ τ 1 . (3.208)

1 the lower and upper bounds of the corridor
match one another. By varying the parameters λ and τ 1 within the domain
of Eq. (3.208), it can be assured that the sensitivity ratio R of Eq. (3.184)
does not exceed a prescribed value. The variation can be organized in
various ways. For example, to constrain the parameter τ 1 to lie in the
middle of the interval allowed for this parameter by Eq. (3.208), let

It is apparent that for λ =

τ 1

1

= τ 1 +

2 ( 1

− λ)τ 4 . (3.209)

In this case, the variation is performed with respect to only one parameter,
λ .As λ increases, the corridor becomes wider and, therefore, the best dis-
turbance response measure decreases while the worst disturbance response
measure increases. Accordingly, the sensitivity ratio R is a monotonically
increasing function of λ . The search for the maximum λ that assures that
the quantity R does not exceed the prescribed value R d is reduced to the
solution of the equation R(λ) = R d . This equation can be solved by various
methods, for example, by the interval bisection method. For each trial λ ,
one should solve the best disturbance and worst disturbance problems to
calculate R(λ) .
Figure 3.18 presents the curve R(λ) , calculated for the corridor defined
by Eqs. (3.203), (3.205), (3.207), and (3.209), completed by the velocity
change interval of Eq. (3.201). The curve becomes flatter as λ increases.
Note that this curve begins with λ 1, in

which case the upper and lower bounds of the crash pulse corridor coincide.
This is because of the constraint of Eq. (3.201) on the velocity change. The

= λ ∗ = 1 . 14, rather than with λ =

R 2.0

1.8

1.6

1.4

1.2

1.0

1.0 1.2 1.4 1.6 1.8 2.0

FIGURE 3.18 Worst-to-best ratio R for the peak force transmitted to the object relative
to the variation of the impact pulse versus the similarity factor λ .