Maximum transmissibility of sky hook damper

Question

I have been studying the sky hook damping system.

On page 14 of paper linked above, I found that the maximum transmissibility of sky hook system can be calculated by:

$$T_{max} = \dfrac{1}{2\xi}$$ where $\xi$ is the damping ratio and $T_{max}$ is the maximum transmissibility.

The condition for maximum transmissibility was $\xi>0$, and $\dfrac{\omega_d}{\omega_n}=1$ in the following transmissibility equation.

$$ T = \sqrt{\dfrac{1+\left(2\xi\dfrac{\omega_d}{\omega_n}\right)^2}{\left(1-\left(\dfrac{\omega_d}{\omega_n}\right)^2\right)^2+\left(2\xi\dfrac{\omega_d}{\omega_n}\right)^2}}$$

But, when I simply substitute $\dfrac{\omega_d}{\omega_n}=1$, I get the following.

$$T_{max} = \dfrac{\sqrt{1+\left(2\xi\right)^2}}{2\xi}$$

What is wrong with this equation? Could anyone explain how to correctly induce the maximum transmissibility of skyhook system?

Answer 1

This isn't exactly my field, but I might have an idea. Sometimes people simplify equations and throw away terms that will be insignificant in value or close to zero. It's possible that a series expansion was performed and only the first term (which would be 1/2ξ) is considered.