Normal modes of oscillation for a double pendulum are motions of the pendulum in which the coordinates $\phi_1$ and $\phi_2$ vary harmonically in time with the same frequency and phase, but not necessarily with the same amplitude.
Once we have linearised equations of motion, it is said that one can substitute in harmonic solutions to identify the two normal mode frequencies. What does this mean in the example below
$\tau_1 = (I + md^2) \ddot{\phi}_1 + I \ddot{\phi}_2, $
$\tau_2 = I \ddot{\phi}_2.$
What are the harmonic solutions and how do they lead to two normal frequencies if we assume $\tau_1$ and $\tau_2$ to be the same, for example?
The form of these equations is the usual one in mechanics of
$ \boldsymbol{\tau} = \textbf{A} \ddot{\boldsymbol{\phi}} + \textbf{b} .$
This can be rewritten as
$\ddot{\boldsymbol{\phi}} = \textbf{f} ( \boldsymbol{\phi}, \dot{\boldsymbol{\phi}}, \boldsymbol{\tau} )$