Time constant of second order transfer function

Question

How can I obtain the Time Constant of the second order transfer function for example:
Transfer Function

Answer 1

This question can not be addressed in general. It depends on what you want to do with the time constant. If you think of it in terms of bandwidth $\omega_0 = \frac{1}{T}$, there is an approximation for second order systems: $$\frac{1}{T²s²+2T\zeta+1}$$

If you want to design your control sample rate by the usual rules of thumb (ten times faster than system time constant), you should better calculate the zeros of the characteristic polynomial (the denominator of the transfer function) and take the eigenvalue which is more negative. This is the reciprocal to the fastest time constant in the system. To adequately control your system, you should be approximately ten times faster than this value.

As always, the right answer is 'it depends'. For both things you will find counter examples and techniques to deal better. However, these require often advanced techniques and are very tighly tuned to the given example.