Root Locus: Plotting for k=1.33 & Finding Equivalent Damping Coefficient

[JohnielWhite]

I was told to plot a root locus(by hand) of the open loop transfer function:

H(s)G(s)= (s+2)/(s^2 + 2s + 3)

and for a value of k=1.33 determine the location of the root, then find the equivalent damping coefficient.

After several attempts i was able to draw the correct root locus but I am not sure how to approach the other part of the question. Could someone with knowledge of the root locus offer me some assistance? Any comment would be greatly appreciated.

 

[JohnielWhite]

it should be in the numerator...

H(s)G(s)= k(s+2)/(s^2 + 2s + 3)

 

[reddvoid]

form characteristic equation 1+G(s)H(s)=0
and substitute K=1.33
solve the equation u ll get value of roots at K=1.33

 

[reddvoid]

1+G(s)H(s)=0
s²+2s+3+k(s+2) = 0
s²+3.33S+5.66=0
solve this you'll get roots at k=1.33

 

[JohnielWhite]

ok thanks... how would I go about finding the equivalent damping coefficient?
I am thinking to compare it to the standard second order equation and equate like terms and solve for zeta. Is that the right approach?

 

[reddvoid]

ok thanks... how would I go about finding the equivalent damping coefficient?
I am thinking to compare it to the standard second order equation and equate like terms and solve for zeta. Is that the right approach?

that's exact approach
i think you'll get 0.699 zeta

 

[JohnielWhite]

thanks a lot... you really helped me to clear up some misconceptions i had... this will help in my preparation for my finals...

 

[reddvoid]

you're welcome .
and all the beat for your finals. . .

 

[JohnielWhite]

thanks all...

 

[JohnielWhite]

hey reddvoid, I tried solving for zeta however i got 0.58. What I did was to equate the constants for the characteristic equation from the standard second order equation to solve for omega, then i equate the coefficients of the "s" term and substitute omega and solve for zeta. Dnt know why i am not getting 0.699

 

[reddvoid]

S² + 2zW_nS + W_n² = s²+3.33S+5.66
W_n²=5.66
W_n = sqrt(5.66) = 2.38
2zW_n = 3.33
z=3.33 / (2*2.38)
z = 0.699

this is how i got 0.699
correct me if i am wrong . . .

 

[JohnielWhite]

ok I see where i made the error in my calculations. Thanks a lot for making those clarifications.