FUNDAMENTALS OF IMPACT AND SHOCK ISOLATION - Injury Biomechanics and Control: Optimal Protection from Impact - page 41

From this expression it follows that the center of mass of the system comes
to a complete stop only if Eq. (2.58) holds. In this case, the center of mass
has a constant acceleration − f 0 /(M + m) during the time interval 0 ≤ t<

τ and comes to a complete stop at the instant t = τ . The deceleration path

of the center of mass, P c ,isgivenby

f 0 τ 2
2 (M

P c = X(τ) =

m) .

(2.63)

+

With reference to the last expression, Eq. (2.57) can be represented as

f 0 μ
u 0 M =

M R

P c .

1 + (2.64)

M + m

This relation measures the effectiveness of the isolation in terms of the
ratio of the rattlespace to the deceleration path of the center of mass of the
system. If this ratio is small (close to zero), the effectiveness is low but it
increases as R/P c increases and tends to infinity as R/P c tends to infinity,
which corresponds to complete isolation of the object from the base.
Equation (2.64) is an analogue of Eq. (2.39) for the case of a dynamic
disturbance. The deceleration path of the base, P b , for a kinematic distur-
bance and the deceleration path of the center of mass, P c , for a dynamic
disturbance characterize the deceleration path of the “system as a whole”
due to an impact load. Therefore, based on Eqs. (2.39) and (2.64), it can be
concluded that for both types of shock disturbance the isolation is ineffective
if the deceleration path of the system substantially exceeds the rattlespace.

2.2.2 Shock Isolation of a Fixed Base

In the previous section, the basic concept of isolating an object that is
located on a movable base from a shock load that is applied to the base was
discussed. In this section, shock isolation of a fixed foundation from a piece
of equipment that undergoes shock loads in normal operating conditions will
be considered. This is sometimes referred to as a shock absorption problem.
A simplified single-degree-of-freedom model of equipment attached to a
foundation is shown in Fig. 2.8. In this model, the equipment's inertia
is represented by a lumped mass m , the impact load is characterized by
the force F appliedtothemass m , and the reaction of the foundation is
represented by the force u that also acts on the mass. According to Newton's
third law, the force transmitted to the foundation is u . If mass m moves

along a vertical line, then the motion is governed by the equation

−

m x ¨ = F + u + mg, (2.65)