FUNDAMENTALS OF IMPACT AND SHOCK ISOLATION - Injury Biomechanics and Control: Optimal Protection from Impact - page 46

variables of the system, displacements, and velocities of the object to be
protected relative to the base. For example, a typical passive isolator of the
single-degree-of-freedom system of Eq. (2.75) is a linear spring - dashpot
isolator with control force

u(y, y) =− c y − ky, (2.76)

where c is the damping factor of the dashpot element and k is the stiffness
coefficient of the spring. Hence, passive isolators can be regarded as a
special class of isolators driven by feedback control.
In shock isolation theory, the control force of an isolator represented
as a function of the state variables and time is sometimes referred as the
characteristic of the isolator.

2.4 DOES ISOLATION OF AN OBJECT FROM THE BASE
ALWAYS LEAD TO A REDUCTION IN THE SHOCK LOAD
TRANSMITTED TO THE OBJECT?

A simple example of a passive isolator for a single-degree-of-freedom
model is presented here to demonstrate an important observation: Isola-
tion of an object from a moving base may increase rather than decrease the
transmitted load if the isolator's parameters are not properly adjusted.
Consider an object attached to a movable base by means of a passive
isolator consisting of a linear spring. The object and the base are allowed to
move translationally along the same straight line. The motion of the object
relative to the base in this system is governed by the differential equation

m y ¨ + ky =− mv(t), (2.77)

where m is the mass of the object, y is the coordinate measuring the position
of the object relative to the base, v(t) is the absolute acceleration of the
base due to a shock disturbance, and k is the spring stiffness coefficient.
Equation (2.77) is a particular case of Eq. (2.40) or (2.75) for u

=− ky .

Suppose that at the initial time instant t = 0, when the system has been in

the equilibrium state,

x( 0 ) = 0 , x( 0 ) ˙ = 0 , (2.78)

the base is subjected to the rectangular shock deceleration pulse

− a

≤ t ≤ τ,
0 f t > τ,

if 0

v(t) = (2.79)

where a is the deceleration magnitude ( a > 0) and τ is the pulse duration.