Question

Here are three oscillator differential equations: y''[t] + 0.1 y'[t] + 5.8 y[t] = 0 , y''[t] + 5.8 y[t] = 0 , y''[t] + 1.4 y'[t] + 5.8 y[t] = 0 . One is a differential equation of an undamped oscillator; the other two are differential equations of damped oscillators. Which is which? Of the two damped oscillator differential equations, which one is the more heavily damped?

L. 12) Here are three oscillator differential equations: y"[t]0.1 y [t] + 5.8 y[t] = 0 » y"[t] +5.8 y[t] = 0 , y t]14 y t]5.8 y[t] = 0 One is a differential equation of an undamped oscillator; the other two are differential equations of damped oscillators. Which is which? Of the two damped oscillator differential equations, which one is the more heavily damped?

PLEASE INCLUDE EXPLANATION!!!

Answer 1

Given three equations y"Ct) + 0.1 y'( t) + 5-8 yC4) = 0 y" (t) + 5.87%) 0 y''Ct) + 1.4 4'(4) +5.8 4(4):0. Convert them to laplace transforre. 34 CS +5 (0.1) yes) + 5.8 YCS) =... (5 + 0.1)+5.8) 4(s) =0. 015+5.8=0. Characteristic equation =) s This is in the form 3x2sans-wr=0. 2800 5.0.1 Y = we-5.8. = = 0.02) <A. n w

s"c + S 8 9 1) = 0 3 4 5 +5.87(5)=0 (5 + 5.8) 4 (s)=0 For this y=0 Undamped This is an case. B y"(t) + 1.4 y'rt) + 5.8 g (t) = 0 $ 405) + 1.48 YCS) + 5-8 YCS) = 0 ( 5+ 145 + 5.8) YC) = 0 by comparing with Standard eg. Se spuns it wr=0 e son-in wirse en un=240 | 1,50.029 15:024 [83> 4.) : System is nearly damped.