Gear noise reduction by numerical optimization of macro-geometrical and micro-geometrical design parameters

Picture 1: Bending stress and contact pressure computed by Helical3D

Picture 2: Peak to peak transmission error definition

A numerical optimization process has been carried out in order to improve the performances of two helicoidal gears (here called pinion and gear) used to move the oil pump of a four cylinder diesel engine for a light commercial vehicle application. In particular the aim of this activity has been to reduce the noise generated by the gears under the normal operating conditions without penalizing their strength and reliability.
To do this modeFRONTIER has been used together with Helical3D which is a commercial software performing three-dimensional finite element analyses of gears taking into account the contact problem. Among the several outputs deriving from a Helical3D analysis the most significant ones have been chosen as objectives of the optimization: the maximum bending stress at the tooth root of each gear, the maximum contact pressure and the peak to peak transmission error (PPTE). The first quantities are related to the gear reliability whereas the last one can be considered as a noise indicator. At the end of the optimization activity the predicted PPTE has been reduced of 70% without affecting the calculated gear reliability.

Input and output variables
The input variables for the optimization has been divided into two categories:

• the macro-geometrical variables are those which affect the gear size and the tooth shape;
  
• the micro-geometrical variables are used to modify the theoretical tooth profile in order to improve the gear functioning.

Picture 3: Feasible designs in comparison with the whole design space. On the left the effect of the only transmission ratio constraint, on the right the effect of both constraints

The quantities chosen as output variables are:

• the maximum pinion and gear bending stress at the tooth root (Pic. 1).
  
• the maximum contact pressure (Pic. 1).
  
• the peak to peak transmission error. It is computed as the amplitude of the gears angular tilting, that is the amplitude of the difference between the actual and the theoretical angular displacement 0 (Pic. 2). Generally speaking, the PPTE depends on the tooth stiffness: the more flexible the tooth is the more the tooth can bend and follow the gear movement. This kind of behaviour smoothes the sharpness of the tilting reducing the PPTE.

Objectives and constraints
The optimization process has been performed with the following objectives:

• to minimize the maximum bending stresses;
  
• to minimize the maximum contact pressure;
  
• to minimize the PPTE.

Moreover some constraints have been set, in particular:

• the maximum bending stresses below the material bending strength;
  
• the maximum contact pressure below the material pressure limit.

Picture 4: Section of the Pareto frontier in the plane PPTEPinion stress

Optimization strategy
A first attempt to run an optimization process has been tried using the macro-geometrical variables. As the Pic. 3 shows, this definition of the input variables causes the feasible design space to be too small compared to the whole design space defined by the complete range of the input variables. This involves some problems for the DOE and optimization algorithms.

So the model has been modified changing some discrete type constrained variables into new ones characterized by continuous ranges. This kind of variable transformation stretches the original feasible space into a new one where no constraint is necessary.

The new model has been used to carry out an optimization in two steps: in the first step only the macro-geometrical parameters has been considered to be variable, whereas the micro-geometrical ones has been kept constant at the current value; the second optimization step has been done considering both the macro- and micro-geometrical parameters as variables.
Each optimization step was done following an iterative procedure, performed until the response surfaces accuracy reached a target value.

Picture 5: PPTE comparison between the current design and the best designs after the first and the second step

Results
The contact pressure response surface shows in the first step smaller errors than in the second step. This is due to the micro-geometrical variables which affect mostly the contact pressure.

The PPTE has shown itself to be a very difficult quantity to be interpolated as the high errors found for its response surface demonstrate. At the end of each step the Pareto frontier has been found (Fig. 4) and the decisional procedure applied.

As result it is possible to compare the PPTE computed for the current design to the best designs after the first and the second step (Fig. 5). It can be noted that the optimization process has lead to a PPTE reduction of about 70%.

Conclusions
modeFRONTIER has been used for a gear optimization in order to reduce the predicted noise as much as possible, preserving the structural reliability of the current design. It has been easily interfaced with Helical3D which is a commercial software for three-dimensional analyses of gears.
The response surface tool has proved to be a very useful one because of the very long calculation time necessary to perform a great number of simulations.

In conclusion it can be stated that the use of a multi-disciplinary optimization software like modeFRONTIER permitted the computed performances of the analysed gears to improve considerably.

Filippo Lachina - Iveco FPT, Industrial & Marine, Product Development
Laurent Bailly Maître - Politecnico di Torino