Manufacturing processes are not perfect and are subject to deviations. These deviations occur in all forms and are limited by tolerances. Tolerances are described in standards such as ISO 1101 (Ref. 1) or ASME Y 14.5 (Ref. 2). Tolerance ranges are standardized for various manufacturing processes, such as shaft tolerances in DIN ISO 286 (Ref. 3) or center distances in DIN 3964 (Ref. 4).
In most cases, the deviations follow a normal distribution, as described by Equation 1, where σ is the standard deviation and μ is the nominal value. The standard deviation is a measure of the scatter of the values around the mean.
The positions of the planet pins in the planet carrier are also subject to tolerances. It is known that uneven distribution of the planets around the circumference leads to uneven load distribution. According to ISO 6336 (Ref. 5), uneven load distribution due to deviations can affect the load capacity of the planetary stage. In addition to deliberately implemented asymmetries, it can also be assumed that tolerances lead to a change in the load sharing between the planets.
The tolerances are mostly defined as geometric deviations from an ideal position. The definition for the tolerance is mostly done per part. In the context of catalogue gearboxes, the application and the requirements for torque density can change quite drastically. This means that a tight tolerance is not needed for every application.
If the geometric tolerances are not met, the parts are released using a simulation in a special release process. Both the customer’s requirements and the measured production deviations play a role in this process. The simulations are very reliable but generate a comparatively large amount of work.
This study uses a Monte Carlo simulation to investigate the influence of tolerances on the load capacity. This simulation analyzes the safety of the planetary stage with variable planet pin positions and discusses the results, including the possibility of load-dependent tolerances for catalog gearboxes. The resulting surrogate model should facilitate a determination during the measurement process as to whether the part can be utilized for the order, whether it will be incorporated into another order, or whether it necessitates disposal.
Simulation Model
The simulation model and the theoretical approach used are explained below.
System Simulation
The simulation model is based on a quasi-static gearbox simulation with the FVA-Workbench 9.0.2. In this method, analytical models are used to simulate the behavior of gearbox components under load. Shafts are approximated as Timoshenko beams, rolling bearings as Hertzian contacts, and gear teeth as mechanical plates. The planet carriers are too complex to be meaningfully modeled using analytical equations, so they are modeled as reduced stiffness matrices according to Guyan (Ref. 6).
To obtain the displacements in the gear system, and thus the load distribution on the tooth flank, the stiffness is solved iteratively in a system of linear equations. The procedure is based on the RIKOR method (Ref. 7).
The overall system simulation determines the deformation of all components under load. The face load factor KHΒ and the load sharing factor Kγ for the calculation are derived from the stiffness characteristics of the mechanical system.
Conservative calculations are used for the design of the models presented in the Results and Discussion section. It is assumed that the largest face load factor and the largest load sharing factor in the planetary gear stage occur on the same planet. The load capacity is determined for this planet in accordance with ISO 6336. This study is carried out exclusively on one-sided planetary carriers. Errors caused by the assembly of two parts with tolerances are therefore not considered.
Tolerance Simulation
In the mechanical model, the planetary pins are connected to the side plates via coupling elements, which can be displaced in the radial and tangential directions. This changes the position of the pin and the planet. The influence on the gearing is considered, in particular on the center distance, backlash, and tip clearance. This reduces the amount of effort required for the meshing and the static condensation of the planet carrier. The carrier must only be prepared once in a pre-processing step before the simulation, enabling very fast simulations. In this paper, only one-sided planet carriers are investigated. For two-sided planet carriers, additional static misalignments would occur.
It is assumed that a standard deviation for the position of σ = 6 ?m can be achieved in the production process. Figure 1 shows the resulting Gaussian distribution.
To keep the evaluation of the results manageable, it is assumed in this study that no angular errors occur in the manufacturing process.
Statistical Methodology
To determine the influence of the pin positions, 10,000 positions are simulated in a Monte Carlo simulation. In each simulation run, all planetary pins receive a new deviation, which is determined randomly. These deviations are determined in both the radial and tangential directions according to the assumed distribution from Figure 1, resulting in a superposition of two distribution functions.Summary and Future Work
This study presents a method for evaluating systems with tolerances, using an example of a planet carrier to illustrate and statistically evaluate the influence of manufacturing tolerances on the load capacity of planetary gears.
A normal distribution is assumed for the position tolerances at the planet carrier, and the standard deviation of the position tolerance is assumed to be 6 ?m. A Monte Carlo simulation with 10,000 calculations is used to analyze the influence of the deviation. For this purpose, the number of simulations was deemed sufficient.
The results show that meaningful conclusions can only be made in an aggregated form. To do so, the minimum flank safeties of the planetary stage are plotted over the angle deviations. The results clearly show that optimal utilization is possible for gearboxes with small deviations. However, planet carriers with significantly larger manufacturing deviations can also be used if the backlash allows. These can be mounted and are functional, albeit with lower maximum torque.
With this realization, a geometric position tolerance can be reformulated as follows: Are the customer’s torque requirements such that the part can still be used despite greater deviations? Or can I find another customer with lower power density requirements in the near future? Based on this approach, large deviations should only be considered as rejects if the storage costs are greater than the profit on the component.
This is a proof-of-concept study in which the positional tolerance of the planet pins on the planet carrier is the only parameter that is varied. The simple study shows how the tolerances can be evaluated. Scaling to larger tolerance systems is possible in principle, but there are still some open questions:
In the abscissa of the regression, all influencing parameters must be represented in such a way that a sufficiently good regression is achieved.
Definition of these parameters may make up the majority of the work.
Since all calculations are independent, adding an additional tolerance should not result in additional simulations. This would provide a very time-efficient method of assessing the tolerances.
