BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 73

x , may be absent from the list of arguments
of the function u(x, x,t,α) . For example, for a linear elastic isolator, the
control force is u =− kx and, hence, is independent of the velocity

One of the variables, x or

x .In

addition, in this case, the set α consists only of one component k .
For the parametric optimization of a shock isolator, it is necessary to
solve Problem 3.1 or Problem 3.2, as appropriate, to find the optimal control
among the parametric family of Eq. (3.72). The performance criteria J 1 and
J 2 in this case become functions of the design variables α . To calculate the
functions J 1 (α) and J 2 (α) for a specific value of α , it is necessary to solve
the differential equation

x ¨ = u(x, x,t,α) ˙ + v(t) (3.73)

subject to the initial conditions

x( 0 ) = 0 , x( 0 ) ˙ = 0 , (3.74)

and then to calculate the desired functions

J 1 (α) = max

t

| x(t ; α) | , (3.75)

J 2 (α) = max

t

| u(x(t ; α), x(t ˙ ; α),t,α) | , (3.76)

where x(t ; α) is the solution of the initial-value problem of Eqs. (3.73) and
(3.74).
Then Problems 3.1 and 3.2 become problems of constrained minimization
of functions of several design variables.

Problem 3.3 Displacement Minimization

List of Variables for Problem 3.3

State variable x Displacement of the object relative to the

base

Control variables u Force produced by the shock isolator

divided by the mass of the object, absolute
acceleration of the object

α Set of parameters of the force characteristic

of the shock isolator u

External disturbance v Shock acceleration pulse, the negative of

the acceleration of the base

(continued)