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Problem 3.52PP
A measure of the degree of instability in an unstable aircraft response is the amount of time it 
takes for the amplitude of the time response to double (see Fig), given some nonzero initial 
condition.
(a) For a first-order system, show that the time to double is 
bi2
where p is the pole location in the RHP.
(b) For a second-order system (with two complex poles in the RHP), show that 
lo2
*2 = —;:— ■
fai2
*2 = —;:— ■
Figure Time to double Time Amplitude 2A
Anqditude
______
Ite w
L *2 »
Step-by-step solution
w
step 1 of 4
For the 1* order system.
Let the Transfer Function is 0 (e ) = ——— 
s— p
By taking Inverse L^ lace Transform we can get 
g {t)= ke ’ ‘
Now consider the initial time as /g
And consider at t = tgthe an^litude of the response is as ‘A’ 
Then,
g ( g = t e