FUNDAMENTALS OF IMPACT AND SHOCK ISOLATION - Injury Biomechanics and Control: Optimal Protection from Impact - page 30

It is apparent that the smaller the ratio τ/t , the smaller the relative error.
Therefore, this approximation is acceptable if one considers the motion
of the particle on a large time interval, considerably exceeding the pulse
duration.

Example 2.6 Instantaneous Shock as the Limiting Case of a
Rectangular Pulse for a Particle on a Spring

Consider a linear oscillator, that is, a particle of mass m attached to a
fixed base by means of a spring of stiffness k (Fig. 2.6). The motion of
this system is governed by the equation

m x ¨ + kx = F, (1)

m

k

x

FIGURE 2.6 Linear oscillator.

where x is the coordinate of the particle measured from the equilib-
rium position at which the spring is unstrained and F is the external
force applied to the particle. Unlike the free particle of Example 2.5,
the oscillator has an intrinsic characteristic time, the natural vibration
period

k
m ,

2 π
ω ,

T n = ω = (2)

where ω is the natural frequency.
The system is assumed to be in equilibrium at the initial time instant

t = 0, and hence Eq. (1) is subjected to the initial conditions

x( 0 ) = 0 , x( 0 ) ˙ = 0 . (3)