Power density, also described as load-carrying capacity, is simply the amount of load that can be safely transferred by a gear or other power transfer component without failure. According to ISO 6336, surface durability, root bending fatigue, and material choice are the three fundamental considerations for power density. This article will briefly review the important parameters in determining allowable power density as determined by the International Standards Organization (ISO) and introduce how isotropic superfinishing benefits power density.
Material Selection
A gear’s material is chosen for a range of properties, including resistance to wear, toughness, static compression strength, shear strength, fatigue strength, and elevated temperature resistance. Ease of machining and ability to harden are also key factors in metal selection. Surface roughness and material choice are not closely related, therefore, material choice will not be discussed further in this article.
Bending Fatigue
Bending fatigue is a critical aspect in determining the repeated load a component can endure before failure. Bending-related failures in gears occur after repeated cycles and at the weakest point in the root or root fillet region. In this region, high stress concentrations exist in conjunction with the highest tensile stresses and/or surface initiation sites due to residual machining marks. Roughness of the root fillet is an important factor in determining root bending fatigue and is duly included in the ISO 6336 root bending equations. However, one key issue when considering roughness of the root region relative to bending fatigue is that the roughness as calculated by Rz is not the only relevant consideration; the relationship of the material and, crucially, the roughness’s shape and size also play a critical part in surface-initiated “notch” failures. Therefore, it is difficult to assign a simplistic correlation to significant changes in root bending strength as a consequence of root roughness alone.
Surface Durability and Isotropic Superfinishing
Isotropic superfinishing can greatly improve power density allowable by increasing the durability of a surface. In terms of gears, durability under ISO 6336 considers pitting as the failure mode. Other surface defects such as scuffing, wear, micropitting, and plastic yielding have been overlooked because either there is insufficient understanding of their impact or the stress levels required are beyond the scope of the standard.
ISO 6336 defines a safety factor for surface durability (SH). For this calculation to work, the permissible contact stress (sHG) has to be larger than the actual contact stress (sH) by an industrially determined safety factor, or margin of error. The overall calculation is shown in Figure 1.
Where:
σHG = Permissible contact stress
σH = Contact stress
SH = Safety factor for surface durability
SH min = Minimum safety factor for surface durability
To calculate the permissible contact stress, it is necessary to factor in tooth flank roughness.
σHP = Permissible contact stress
σH lim = Allowable stress number (contact)
ZNT = Life factor
SH min = Minimum safety factor for surface durability
ZL = Lubricant factor
ZV = Velocity factor
ZR = Roughness factor
ZW = Work hardening factor
ZX = Size factor
σHG = Permissible contact stress
As it can be seen, the relationship is not simply one of roughness (ZR). The permissible contact stress considers other factors impacting the lubricant, such as the pitch line velocity (ZV) and what is described as a lubricant factor (ZL), which looks at the viscosity of mineral oils with or without EP additives.
For the roughness factor, the relative mean peak-to-valley roughness (Rz) is utilized, and as the roughness decreases, the permissible contact stress increases. Isotropic superfinishing lowers the Rz of a surface to below 1 µm, which generates a more durable surface.
A key question is: Why does an isotropic superfinish create a benefit to surface durability? Before considering the larger concepts of lubrication effects, one should simply visualize two surfaces in contact under load in a dry environment.
Figure 4
In Figure 3, as the rough surfaces are compressed together under load, they contact through their highest features or peak asperities. This contact acts to resist the load being applied, meaning the load is being concentrated in localized regions. This localized loading raises the contacting pressure at the micro level — reducing surface durability. Contrast this example with two smooth surfaces (Figure 4), and it becomes clear the load would be evenly distributed over the total area in contact. Additionally, at the micro level, the pressure is relieved while critically still bearing the same overall applied load.
To understand how much of a surface is going to be available to distribute the load, consider the material bearing ratio (Rmr) of the surface. Rmr is a way of ascertaining how much of a surface would be in contact with the mating surface. For machined surfaces, the contacting area will be significantly lower than for an isotropic superfinish. How to measure for a parameter like Rmr was covered in January’s Materials Matter column, “Roughness Measurement of Precision Gear Teeth.”
For a lubricated environment, one must also factor in the impact of lubrication on contacting properties. Lubrication in contacting environments is considered in terms of the lambda ratio (l). There are many equations using various roughness parameters that calculate the lambda ratio. One of the earliest is Tallian’s equation:
Where:
h0 is the minimum film thickness [m]
σA or B is the root mean square roughness of body A and B [m]
λ is the parameter characterizing the ratio of the minimum film thickness to the composite surface roughness
The larger the lambda ratio, the more separation between the mating surfaces and the more complete the elastohydrodynamic lubrication layer. A lambda ratio value of greater than unity implies a full film has formed and there is complete separation. When parts separate, generally, a lower coefficient of friction is obtained, as shown by the Stribeck Curve. This concept was covered in the March Materials Matter column, “Optimizing Performance with Surface Finish and Lubrication.”
So how can the lambda ratio be improved? Assuming that altering the mechanical properties is not an option, then two other options exist: selecting a higher viscosity lubricant enabling a larger film to form or reducing the surface roughness. Increasing the lubricant viscosity is often considered undesirable due to the increase in parasitic losses. Therefore, improving surface roughness is the more desirable option. Isotropic superfinishing is a convenient method to make such an improvement, and the improved surface roughness can be factored into the initial design stage or can be introduced at a later point if an unexpected durability issue occurs.
How significant of an improvement can be derived? Durability in the form of pitting resistance is an area that has been extensively researched in regards to isotropic superfinishing technology. One such study by Paul Niskanen et al. reviewed the impact of isotropic superfinishing on spiral bevel gears using a power circulating pitting (contact) fatigue testing rig. This study showed how an isotropically superfinished gear with an Ra of 2-3 µin. exceeded the baseline failure results of a precision ground gear with an Ra of 9-12 µin. by roughly threefold. Similarly significant results have been obtained consistently across a range of formal testing procedures and scenarios encompassing differing gear designs and applications.
Conclusion
Durability is one of three key aspects in determining the allowable power density of a gear. Isotropic superfinishing improves the durability of a gear through the reduction of roughness, allowing more load to be transmitted over the surface area. This opens two design route options: to apply more load to the surface, given its greater durability, or to reduce the flank width to transmit while maintaining the same load but reducing weight — provided that bending fatigue is not on the design limit. Either option yields an improved power transfer system and increases the intrinsic value of the components.
