Reputedly magnesium has more internal damping than aluminium, so you might expect an advantage there.
However, E/rho and sigma/rho are quite similar for both materials, so you tend to have to use the same weight of magnesium as aluminium, and end up with similar modal properties.
There is an additional complexity at higher frequencies. Panels have a coincident frequency where the speed of bending waves in the panel matches the speed of sound in air. This gives perfect coupling of vibration to sound. I suspect that magnesium will have a diferent coincident frequency to aluminium, and depending on the gear teeth numbers this may make your radiated gear whine problems different.
I have always argued against using magnesium for gearboxes in RWD cars because they do it to save weight, so the engine/gearbox assembly gets less stiff, so the bending frequency drops, and that bending frequency is the cause of all sorts of NVH issues.
Cheers
GregLocock provides good information on this subject. I would like to elaborate on the point that he makes with respect to specific modulus (E/rho) and specific strength (sigma/rho). These values are useful when the loading is tensile in nature, e.g., when load, stiffness, and length are specified, and the section area is free.
Most complex components like automotive transmission housings have equally complex loading, with a great deal of bending involved, both in the form of global bending of the entire housing, and local bending/buckling of a particular section. When global bending is involved, the beam mode becomes important, i.e. external or self-weight loading, stiffness and length specified, section area is free. For this case, the Performance Index (term used by M. F. Ashby in his book Materials Selection in Mechanical Design, becomes E1/2 / rho or sigma1/2 / rho. This favors Mg over Al due to its much lower density.
When plate bending or buckling is important, i.e., loaded externally or by self weight in bending, stiffness, length, and width specified, thickness free, the Performance Index becomes E1/3 / rho or sigma1/3 / rho, which favors Mg even more than the previous case.
Another factor to keep in mind are the minimum practical wall thickness that can be achieved during casting. In pressure die casting, significantly thinner walls can be achieved with Mg than in Al. This becomes important when the loads are small enough that the Al wall thickness cannot be made any thinner due to manufacturing/casting constraints, and a similar wall thickness of Mg will perform adequately, and hence weigh less.
The weight of a complete gearbox is around 50-80 kg. The weight of the casing alone is 10-20 kg. Therefore a formula based on the self-weight of the casing is going to be very misleading.
The casings in question would be for trucks, classes 6 to 8. A typical class 8 transmission would be around 300kg and the casing, in aluminium, about 40kg. Cast iron cases are about twice that. Although CI is still used extensively in the USA it is unusual in European on-highway trucks, these days, due to a greater sensitivity to weight here (fuel costs etc.).
The load case that I mentioned from Ashby previously is "loaded externally OR by self weight". Either way, the important concept is that the boundary conditions allow for modification of one parameter:
section area (for beam-type bending)
thickness (for plate bending or buckling)
The idea of the Performance Index does not mean that you would only consider the 10-20 kg of the casing and ignore the mass of the rest of the gearbox/transmission. What it does is allow for designing around the most efficient combination of material, shape, etc.
You are right. The curve of system mass vs stiffness will be offset by the dead load, but the shape of the curve will be the same.
By the way, passenger car gearbox casings for rear wheel drives are stiffness limited (except for some details near the fasteners) so magnesium's greater stress/rho is not especially helpful. Current best practice is to cast the bellhousing, trans, and extension housing, in one piece, to get rid of the surprisingly high compliance of the bolted joints. I have seen a paper suggesting that a structure that consists of thin walls and ribs is noisier than a thicker constant section of the same weight. I suspect this is a matter of details. Cheers
