BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 58

v

U

t

U

FIGURE 3.3 Shock pulse with one excursion beyond the upper bound on the control

force.

To minimize the peak relative displacement of the object, the isolator must
act with constant force

x ,
vanishes at some instant t ∗ > τ . At this time instant, the displacement
x of the object relative to the base is a maximum. The control force
is uniquely defined on the interval 0

− U until the relative velocity of the object,

t ∗ . Beyond this interval, any
control strategy can be utilized that satisfies the constraint

≤ t ≤

| u |≤ U and

does not allow the displacement magnitude to exceed | x(t ∗ ) | . For example,

one can use the control u(t) ≡− v(t) that counterbalances the disturbance

on the interval t ≥ t ∗ . In this case, the object will remain at the position

x x(t ∗ ) .

To prove that this is the correct strategy, represent the solution of Eq. (3.3)
subject to the initial conditions of Eq. (3.4) in the form

=

t

x(t) = (t − ξ) [ v(ξ) + u(ξ) ] dξ. (3.17)

0

The constraint J 2 ≤ U of Eq. (3.8) implies that | u(t) |≤ U and, hence,

u(t) ≥− U for all t . Therefore, the inequality

x(t) ≥ ψ(t), (3.18)

where

t

ψ(t) = (t − ξ) [ v(ξ) − U ] dξ, (3.19)

0

is valid. Differentiate the function ψ(t) with respect to time to obtain

t

ψ(t)

= [ v(ξ) − U ] dξ. (3.20)

0