FUNDAMENTALS OF IMPACT AND SHOCK ISOLATION - Injury Biomechanics and Control: Optimal Protection from Impact - page 37

The first integral characterizes the time history of the absolute motion of
the object, x(t) , and the second integral is the time history of the absolute
motion of the base, z(t) . The solution of the system of Eq. (2.24) subject
to the initial conditions of Eq. (2.26) for arbitrary functions u(t) and v(t)
is given by

t

ξ) u(ξ)
m

x(t) = Vt + (t − dξ,

0

(2.45)

t

z(t) = Vt + (t − ξ) v(ξ)dξ.

0

Consider now the dynamic disturbance, when the time history of the force
applied to the base is prescribed. In this case, the motion of the system
shown in Fig. 2.7 is governed by the equation

m x ¨ = u, M z ¨ = F(t) − u(t), (2.46)

where m is the mass of the object; M is the mass of the base; x and z are
the absolute displacements of the object and the base, respectively; F is
the disturbance force applied to the base; and u is the force exerted on the
object by the shock isolator.
The first relation of Eq. (2.46) coincides with the first expression of Eq.
(2.24) and describes the motion of the object relative to the fixed reference
frame. The second relation of Eq. (2.46) describes the motion of the base
relative to the fixed reference frame. The base is acted upon by two forces,
the force F due to the external disturbance and the force

− u , which is the
reaction of the force u acting on the object. Subtract the second relation of

Object

Base

F ( t )

u m

M

y

z

x

FIGURE 2.7 Object on a moving base.