How to calculate the \$\mathcal{H}_2\$ norm of a second order transfer function?
From a state-space representation, the \$H_2\$ norm can be computed as \$\sqrt{\text{Trace}\left(b q b^T\right)}\$ or \$\sqrt{\text{Trace}\left(c p c^T\right)}\$, where \$b\$ and \$c\$ are the input and output matrices, and \$q\$ and \$p\$ are the observability and controllability gramians.
I have done the calculations below using Mathematica and the result is \$\frac{1}{2}\sqrt{\frac{\omega _n}{\zeta }}\$.