Second order system rise time question

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The discussion revolves around solving the second-order system represented by the equation y'' + 2y' + 4y = u(t) to find the time it takes for the system to reach 75%, 90%, and 95% of its final value. The poster expresses frustration with the complexity of isolating time from the solution, suggesting that it may involve a difficult transcendental equation. They mention that the book provides approximate answers, with a specific example of around 0.9 seconds for 75%. The poster is seeking a method to solve the problem without relying on software like MATLAB, indicating that manual calculations seem overly complicated. Ultimately, they consider the possibility of using a normalized graph to estimate the values instead.
berdan
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Homework Statement



The question is such :
System is modeled as : y''+2y'+4y=u(t)
find the time at which the system goes up 75,90,95 % of its final value.

Homework Equations


The Attempt at a Solution



I have no idea how to touch that,I tried to find the source of the regular rise time (from 10 to 90%),and didn't got it.How the hell do I even start?It looks impossible to isolate expression for time,from PE solution . Also,from the answers in the book,the answers are approximations (t for 75% for example is ~0.9sec).

Thanks in advance.
 
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Solve the ODE for y(t) conventionally or use Laplace transforms.

EDIT: looks like an ugly transcendental equation to get t(y) though. Get out the old Excel spreadsheet!
 
Last edited:
Yea,that is the problem,I wanted to know how to do it without any help of mathlab and such,but it looks like overly nasty equation to solve manually (it has exponent of time coupled with sin and cosine terms)
anyway,thanks.
 
Maybe they just want you to read it off a normalized graph?
 

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