Help with finding roots for transfer functions

[mpm]

I am in a Systems and Vibrations class but am currently doing differential equations.

A problem I am doing requires me to find the transfer function [X(s)/F(s)] and compute the characteristic roots.

So far I have:

X(s)/F(s) = (6s +4)/(s^2+14s+58)

That is the transfer function but now i have to find the roots.

I realize I only concern myself with the s^2+14s+58 part and set it equal to 0.

I can't factor it because of obvious reasons. I tried the quadratic equation but my calculator says its a non-real result which means its complex.

My problem is that I can't figure out how to do the quadratic when there is a complex root.

I know as^2 + bs + c = a[(s + sigma)^2 + omega^2] = 0.

However, I can't figure out how to find the roots using this equation.

Cany anyone help me with this?

Thanks,

Mike

 

[Fermat]

Do you mean that you just want to find the complex roots of the quadratic,

s² + 14s + 58 = 0 ?

Can't you just use the quadratic formula ?

 

[piercas]

Dear Mike,

Thank you for reaching out for help with finding roots for transfer functions. I understand the importance of understanding and solving these types of problems in your Systems and Vibrations class.

Firstly, to find the roots of a transfer function, you need to solve for the values of s that make the numerator of the transfer function equal to 0. In your case, this would mean setting 6s + 4 equal to 0 and solving for s. This will give you one of the roots.

Next, you will need to solve for the remaining root by setting the denominator of the transfer function, s^2 + 14s + 58, equal to 0 and using the quadratic formula. As you mentioned, the quadratic formula may result in complex roots, which are valid solutions in this case.

To find the complex roots, you can use the equation you mentioned, as^2 + bs + c = a[(s + sigma)^2 + omega^2] = 0. This equation represents the general form of a complex number, where sigma represents the real part and omega represents the imaginary part.

Using this equation, you can solve for the complex roots by setting the real part, (s + sigma)^2, equal to the negative of the coefficient of s^2 and the imaginary part, omega^2, equal to the negative of the constant term. This will give you two equations which you can solve simultaneously to find the values of sigma and omega, and thus your complex roots.

I hope this helps you in finding the roots for your transfer function. If you are still having trouble, I would recommend seeking help from your professor or a tutor who can guide you through the problem step-by-step.

Best of luck with your studies.