Since the Young’s modulus and Poisson’s number are density-related, it is important to use the PM relevant numbers in the design phase of the tooth profiles. A thorough study of this was made by WZL RWTH University in Aachen Germany; this article will discuss the methodology and the PM relevant results from this work. Here are two commonly used equations relating density to Young’s modulus and Poisson’s number:
E0, ρ0 and ν0 are the steel reference values.
For a basic, but still high-end PM process, a gear density of 7.2 or higher can be obtained. So inserting the steel reference values and the target density into either equation will render the material data for a linear elastic analysis.
For the micro geometry gear analysis, there are several parameters that influence the working behaviour. Angular deviations, lead, and involute crowning are used in the above referenced analysis, but these are just examples from the ISO/DIN standard. Designers have sometimes defined their own parameters according to best practices and experience.
First, the contact stress is calculated with different micro geometries for the mating gears. The maximum contact stress is then plotted for each mesh cycle, and a stress map is created. The minimum contact stress value is plotted (1234 MPa) and a tolerance box is inserted around the minimum value.
The tolerance box frames all contact stress values that can occur due to variations in production. This means that if the gears are ground, the box will be smaller compared to the tolerance box of a shaved gear. The maximum contact stress value is much higher than what can be accepted. This means that this is not a robust design point since some of the gears will fail prematurely, even though they are within the specified tolerance, and the minimum achievable stress is in the tolerance box. So another design point has to be chosen from iterations where the highest stresses within the tolerance box are acceptable, even though this will not be where we have the minimum achievable stress.
Figure 1 is the summary for many of these stress calculations for different tolerances. The bars depict the mean values for different micro designs. The max-min spread is also shown. The most interesting conclusion from this slide is that if the micro geometry for a steel gear is copied onto a PM gear, the result will yield higher spread for the PM gear (see the bars to the right). Also, the optimized DIN 7 PM gear has lower contact stress compared to the optimized steel gear. This is due to the lower Young’s modulus. Moreover, it seems the fhβ design parameter requires more attention than fhα. The TE doubles when the micro design is copied from a steel gear as opposed to the optimized PM design.
The point of these exercises was to investigate how much a copied steel gear micro design affects the working behaviour of a PM gear. The result from the stresses shows a 16% increase in stress when the design point of steel is chosen, but the worst allowable manufacturing case gives a 40% increase in contact stress and will be detrimental to the service life of the PM gear. For the TE, the result is even worse if the steel design is simply copied.
So the message here is twofold:
1. PM requires its own micro design
2. Check the robustness of the design with respect to allowable tolerances in production
The gear analyzed in this article is the fourth gear in the transmission of a Smart ForTwo. The gearbox has been rebuilt with PM gears and now has more than 100,000 km with no transmission issues. The fourth gear has also passed endurance and NVH testing in a project with the manufacturer of this transmission, so more than 200,000 km are expected out of this PM transmission.
