HEAD INJURY CONTROL - Injury Biomechanics and Control: Optimal Protection from Impact - page 254

7.4.2 Power-Law Deceleration

This law has the form

⎧
⎨

1 1 /( 1 − β)

A
t β

− β

− if 0 ≤ t ≤ t ∗ = ,

u = A (7.62)

⎩

0 if t > t ∗ ,

where A and β are parameters, A > 0and β< 1 [for β ≥ 1 the right-hand

side of Eq. (7.62) has a nonintegrable singularity at t 0]. When subjected

to the control of Eq. (7.62), the object decelerates from the initial velocity
v 0

=

= 1 to a complete stop at the time instant t ∗ in the position

β) ( 2 − β)/( 1 − β)
A 1 /( 1 − β) ( 2

( 1 −

J 1 = x(t ∗ ) = . (7.63)

− β)

The parameters A and β should be determined to minimize the quantity of
J 1 of Eq. (7.63) under the constraint J 2

1.
Substituting the control of Eq. (7.62) into the right-hand side of Eq. (7.9)
and calculating the maximum of the expression in the curly brackets with
respect to t 1 and t 2 (0

≤

≤ − ≤

t 2 t 1 )give

⎧
⎨

2 . 5 1 1 /( 1 − β)

A − β

1 − 2 . 5 β if ≤ ,

1 − β A

=

J 2 (7.64)

1 . 5 /( 1 − β)

if > 1

1 /( 1 − β)

⎩

−

A β

1 − β A

for

2

β ≤

5 .

(7.65)

for β > 5 .
To find the minimum of the function J 1 of Eq. (7.63) with respect to
A and β under the constraints A

It can be shown that J 2 =∞

2

≥ ≤

5 ,and J 2 < 1, where J 2 is
defined by Eq. (7.64), first calculate the minimum with respect to A for
fixed β .Since J 1 decreases whereas J 2 increases as A increases, the desired
minimum is attained when J 2

0, β

= 1. In accordance with Eq. (7.64),

⎨

β) ( 2 . 5 β − 1 )/ 2 . 5 ( 1 − if < 1 ,

A = (7.66)

⎩

A = 1 − β if ≥ 1 ,

⎧
⎨
⎩

− β

1

β ( 1 − 2 . 5 β)/ [2 . 5 ( 1 − β) ]

if < 1 ,

2 −

J 1 = (7.67)

1 − β

if ≥ 1 .

2 − β