The system 2d2y/dt2+4dy/dt+8y=8x, has damping ratio______

Q. The system 2d²y/dt²+4dy/dt+8y=8x, has damping ratio______

Answer is 0.5

The given System is a second order system and is given by

2d²y/dt²+4dy/dt+8y=8x

After taking the Laplace Transform we get

2s²Y(s)+4sY(s)+8Y(s)=8X(s)

(2s²+4s+8) Y(s)=8X(s)

Y(s)/X(s) =8/(2s²+4s+8)

so, the transfer function of the system is

T(s)=Y(s)/X(s) =4/(s²+2s+4)

comparing with a general second Transfer function

T(s)= ω_n² / (s²+2ζω_n s+ω_n²

ω_n² =4 so ω_n=2

2ζω_n=2, so ζω_n=1 and ζ=1/ω_n =1/2=0.5

So the ans is 0.5.