Page 114 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 4.4 Pulse Excitation and Rise Time 101

r nr
^0
A n
Ü _______________ f

(a) Constant with rise time (b) Rectangular pulse (c) Half-sine pulse
Figure 4.4-1.

f i t ) =Po\-t
1 . ct)n .
h(t) sm co,J = -j- sin a)„t (4.4-1)
mo)^
and the response becomes
Hr, ^
x (0 = ^ / V „ f sin
/C ^0 ÍJ

h)
( t sin coj
t < t, (4.4-2)
T V t o)J^
For the second ramp function starting at the solution can be written by
inspection of the foregoing equation as
i - / i sin a> „(i-i,)
•^(0 = - T - coj
n^\
By superimposing these two equations, the response for t > becomes
sinoj^t 1
^ (0 = -T - f - + ^ sin "«(Í - t > (4.4-3)
nh
Rectangular pulse. The input pulse here can be considered as the sum of
two step functions, as shown in Fig. 4.4-3.
We already have the response to the step function as
kx
-^ = [1 - cos w„i] t < t, (4.4-4)

■«fi
The peak response here is obviously equal to 2.0 at t = \r.