Interesting discussion. I am learning form it. Natural frequencies can be figured out by modal analysis (eigenvalue calculations), but for that to be realistic one MUST know the elasticity of the belt. The method suggested by AdrianS can generate at least rough elasticity data. I do not know if the belt elasticity is linear.
As Billski mentioned one can not base belt sizing on mean torque. Designing a reduction for a piston engine is far more challenging than designing for a turboprop, because piston engines have intermittent torque. The peak torque can easily be two to three times higher than mean torque. Therefore the design must take into account those peak torques. Additionally, intermittent torque means cyclic loads - which produce fatigue. That must be taken into account too. I do not believe that belts are much fatigue-prone, but I would look into it anyway.
Another point related to belt reduction is the temperature effect. If the subject aircraft is to be flown in summer and winter at altitudes up to, say, 15000ft, then it is something the designer must check, to evaluate the need or not of a belt tensioner (hydraulic or mechanical).
The interesting thing about belt drives is that most do not include a deliberately "soft" element. Maybe the belt is torsionally softer than the other parts, but I would not count on it. Do the work and find out, then apply the info to your analysis. If no element has a lot lower spring rate than the rest then inertia and springiness of the assembly really must be included. This is easy if you have most of it modeled in a CAE system that includes a dynamic solver for assemblies. This is applied to the crank, bottom pulley, belt (tension side only) top pulley, and output shaft.
Belt springiness and weight will probably have to be determined by test and an elastic element modeled and run between top and bottom pulleys on the tension side of the system. The belt section between the pulleys should match on density, weight, and springiness of the real thing. While the springiness of the belt may not be strictly linear, it is usually modeled as linear at the belt tension expected for torque expressed at RPM. If the load vs deflection plot has much curve to it, you will want to do runs at low, medium and high torque, knowing the rpm-torque relationship of your prop (fixed pitch). If you are running a constant speed prop, you will have a range of RPM and torque to cover, but that still only means five runs of the Eigen solver - low-low, low-high, high-low, high-high, and med-med - that will give a pretty good map of resonance frequencies and modes.
Another potentially significant spring is the shafts the pulleys are hung on - When cantilevered, these shafts can be particularly springy, and this scheme is used on small V-twins and the like. When you get into fours and auto engines, the tendency is to support both ends of each pulley, and that will be significantly stiffer than cantilever shafts.
The last element is the prop. Bifilar pendulum to get the inertia, and then model a disc or bar to get the same weight and MMOI to use as the "prop".
In dynamic modeling of this sort, the parts are usually modeled simplistically - no splines, no teeth on the belt or pulleys, little of the real part's detail is included.
Billski