Answer:
overdamped = (-∞,√20) u (√20,∞)
underdamped = (-√20,√20)
critically damped = [√20,√20] u [-√20,-√20]
Step-by-step explanation:
Hi!
In order to answer this question we must first write the charcteristic equation, which is obtained replacing a new variable, lets say ω by the derivatives of s, and putting it to the power of the same order of the derivative, e.g
s''' -> ω^3
The characteristic equation is:
ω^2 + bω + 5 = 0
We can now solve for ω using the quadratic formula:
If b=0, the solution for ω will be a pure complex number, and that means an undamped oscialtor.
The <em>critically damped</em> solution is given when the determinant of ω is zero, that is
b^2 = 20
|b| = √20
The <em>underdamped </em>is given when we have a non-zero imaginary part for ω
that is
b^2<20
|b| < √20
The <em>overdamped </em>is given when the determinant is positive:
b^2>20
|b| > √20