Question

Consider the Laplace Transformed functions below, please answer the following and show step by step.

Consider the Laplace Transformed functions below a) What is the value of ,(t) as t->e? b) Does the function f2(t) have a final value? If so, compute the value of (t) as t-e? c) Given the Laplace function F3(s), is f(t) under-, critically- or over- damped? Justify your answer? 5 s(s2 +s+2 2 s2 3s +2

Answer 1

(a) f_1 (t) as t rightarrow infinity using final value theorem. f_1 (infinity) = lim_s rightarrow 0 sF_1 (s) = lim_s rightarrow 0 s5/s(s^2 + s + 2) f_1 (infinity) = lim_s rightarrow 0 5/0 + 0 + 2 = 5/2 = 2.5 F_1 (infinity) = 2.5 (b) f_2(t) = cos omega t it does not have final value. (c) characteristic equation is rightwardsdoublearrow s^2 + 3s + 2 Therefore s^2 + 2 zeta omega_n S + omega_n^2 rightwardsdoublearrow omega_n = squareroot 2 2 squareroot 2 zeta = 3 zeta = 1.06 \r\n zeta > 1 thus it is overdamped.