Page 116 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 4.5 Shock Response Spectrum 103

To determine the solution for t > we use Eq. (4.4-10) but with t replaced
by {t - ij). However, we choose a different procedure, noting that for t > /j, the
excitation force is zero and we can obtain the solution as a free vibration [see Eq.
(2.6-16)] with t' = {t - i|).
JC( / j )

jr(i) = ^ sin (oj + -^(^i) cos coj (4.4-11)
The initial values x(t^) and i(i,) can be obtained from Eq. (4.4-10), noting that
Pi\ = 7T.
kx(ti) 1 1 p .
sin pt^ — ( A ' |sin w„i| ------sm (oj.

1 -
kx(t^)
------ p cos pt^ - p cos co^t^] = ---------- -— j[ l + cos co^t^]
(¿)
1 1
Substituting these results into Eq. (4.4-11), we obtain
P
xk
-—j [(1 + cos sin coj' + sin cos o)j']
1 -
\ ^ n )
P_
sin + sin co^{t' + ij)]
1 -
\ ^ n 1
1 • 277^ . _
sm ------- h sm 277 t > T (4.4-12)
T T
277 T

4.5 SHOCK RESPONSE SPECTRUM

In the previous section, we solved for the time response of an undamped spring-
mass system to pulse excitation of time duration When the time duration is
small compared to the natural period r of the spring-mass oscillator, the excitation
is called a shock. Such excitation is often encountered by engineering equipment
that must undergo shock-vibration tests for certification of satisfactory design. Of
particular interest is the maximum peak response, which is a measure of the
severity of the shock. In order to categorize all types of shock excitation, the
single-DOF undamped oscillator (spring-mass system) is chosen as a standard.