I have a system with two torsion and two gears. the ends of the torsion bars are secured and the gears link the two torsion bars together. I am trying to calculate the natural frequency of the system but have thus far been unable to wrap my brain around the maths to solve the problem. I have managed to reduce it to two individual spring mass systems linked by a lever but am still unable to come up with a suitable way to describe this mathematically.

Look at the topic of natural frequency for rotating discs. You will have to work out equations of motion for your system, then plug them into a matrix form, then find the determinent which will give you whats called a characteristic equation (one which uses eigen values and eigen vectors).