BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 80

continuity of the function x(t) and the fact that x(t) ≡ 0 is a solution of

Eq. (3.98). Therefore, if the velocity 0 at some instant of time, it will

remain zero subsequently. For V > 0, the velocity will be positive until the
object comes to a complete stop, and the displacement x will monotonically
increase during the entire motion.
Proceed from Eq. (3.98) to the equation for the phase trajectory defined
by the function

x(t) =

x(x) . Apply the chain rule to differentiate this function to ˙

obtain x ¨ =˙ x(d x/dx) . Substitution of this expression into Eq. (3.98) gives ˙

x d x x r ˙

dx +

c ˙ = 0 , x( 0 ) ˙ = V. (3.99)

The solution of the initial-value problem of Eq. (3.99) leads to the relations

⎧
⎨
⎩

V 2 − r − x 2 − r

if r = 2 ,

c( 2 − r)

x = (3.100)

1

c ln V

if r = 2 .

x ˙

The maximum value of x is attained at the instant when the velocity,

x , vanishes. From Eq. (3.100) it follows that if r ˙ ≥ 2, then x →∞ as

x 0. Hence, if the damping exponent is greater than or equal to 2, the

displacement of the object approaches infinity. For r< 2, the value of the
criterion J 1 is finite and is given by

˙ →

V 2 − r
c( 2

J 1 (c) =

r) .

(3.101)

−

Since the velocity of the object monotonically decreases for this case, the
maximum value of the damping force occurs at the initial time instant and,
hence,

cV r . J 2 (c) = (3.102)

The minimum of the function J 1 (c) under the constraint J 2 (c) ≤ U occurs

for

U
V r

= =

c c 0 (3.103)

and is given by

V 2
U( 2

J 1 (c 0 ) =

r) .

(3.104)

−