FUNDAMENTALS OF IMPACT AND SHOCK ISOLATION - Injury Biomechanics and Control: Optimal Protection from Impact - page 31

At this time, the particle is subjected to the rectangular force pulse

⎧
⎨

S
τ

if 0 ≤ t ≤ τ,

F(t) = (4)

⎩

0if t > τ.

The response of the system to this pulse has the form

⎧
⎨
⎩

1 − cos ωt if t ≤ τ,

S
mω 2 τ

2sin ωt −

sin ωτ
2

x(t) = (5)

ωτ
2

if t > τ.

To obtain the response of the oscillator to the instantaneous force pulse
F(t)

= Sδ(t) with impulse S , solve the equation

¨ + =

m x kx 0 (6)

subject to the initial conditions

S
m .

x( 0 ) = 0 , x( 0 ) ˙ = (7)

The solution is given by

S
mω sin ωt.

=

x(t) (8)

If the pulse duration τ is much less than the natural vibration period
T n of the oscillator, the relation of (8) is a reasonable approximation for
that of (5). To prove this, assume τ/T n 1,

in accordance with Eq. (2) relating T n to ω , and expand the right-hand
side of Eq. (5) into a Taylor series with respect to the small parameter
ωτ . Thus, retaining only the leading terms in the expansion, we obtain

1, which implies that ωτ

1

2 (ωt) 2

if t ≤ τ,

S
mω 2 τ

x(t) ≈ (9)

ωτ sin ωt if t > τ.

A comparison of Eqs. (8) and (9) indicates that the response of the oscil-
lator to the finite-duration rectangular pulse of (4) is close to the response
to the instantaneous shock pulse F(t) Sδ(t) on the time interval t > τ

if the duration of the pulse is much less than the natural vibration period
of the oscillator.

=