BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 105

3.6.2 Best and Worst Disturbance Analyses for a System with a
Linear Spring-and-Dashpot Isolator

Description of the Model The solution of Problems 3.7 and 3.8 will be
given for a system with a linear spring-and-dashpot isolator characterized
by the function

u(x, x) ˙ =− c x ˙ − kx, (3.185)

where c and k are the damping and stiffness coefficients, respectively. This
system is shown in Fig. 3.13 and is governed by the equation

ω n x x ¨ + 2 ζω n ˙ x + = v(t), (3.186)

where ω n is the natural frequency of the system and ζ is the damping ratio,

√ k,

c
2 √ k .

ω n = ζ = (3.187)

The peak magnitude of the force transmitted to the object is chosen to
characterize the response of the system of Eq. (3.186) to the disturbance
v(t) ,thatis,

2 ζω n x(t) + ω n x(t) | . J(v) = max

t

| u(x(t), x(t)) |= max

t

| (3.188)

The above relations represent a simplified model of the sled testing equip-
ment in which the sled (base) and the dummy (object) are regarded as rigid

Object

v ( t )

(dummy)

Base
(sled)

k

c

x

FIGURE 3.13 System with a linear spring-and-dashpot shock isolator.