FUNDAMENTALS OF IMPACT AND SHOCK ISOLATION - Injury Biomechanics and Control: Optimal Protection from Impact - page 47

To calculate the time history of the motion of the object relative to the
base, substitute v(t) of Eq. (2.79) into Eq. (2.77) and solve the resulting
differential equation subject to the initial conditions of Eq. (2.78) to obtain

⎧
⎨

2 a
ω 2

sin 2 ωt
2

for 0 ≤ t ≤ τ,

y(t) = (2.80)

sin ω t 2 a

ω 2

sin ωτ
2

τ
2

⎩

−

for t > τ,

= √ k/m is the natural frequency characterizing the vibration of
the object relative to the base.
The force transmitted to the object by the spring is

where ω

⎧
⎨
⎩

2 ma sin 2 ωt
2

− for 0 ≤ t ≤ τ,

u(t) =− ky(t) =

sin ω t − 2 ma sin ωτ

2

τ
2

− for t > τ,

(2.81)

and the maximum absolute value of this force is expressed as

⎨

2 ma sin ωτ
2

for ωτ ≤ π,

| u | = max

t

| ky(t) |= (2.82) max

⎩

for ωτ > π. 2 ma

The solid curve in Fig. 2.9 plots the quantity

max versus ωτ . When the
object is rigidly connected to the base, the absolute acceleration of the object
coincides with that of the base and, hence, the peak magnitude of the force
transmitted to the object is ma (the dashed line in Fig. 2.9). It is apparent
from this figure that

| u |

max <ma for ωτ < π/ 3 and, hence, in this case,
using a linear spring to isolate the object from the base reduces the peak
load transmitted to the object. The smaller is ωτ , the more pronounced is the
reduction. For ωτ = π/ 3, the isolated object expresses the same peak load
as an object that is rigidly attached to the base. For ωτ > π/ 3, the isolation

| u |

| u | max

2 ma

ma

ω

0 3

FIGURE 2.9 Maximum load transmitted to the object by a spring isolator.