# Taguchi Techniques as an Effective Simulation-Based Strategy in the Design of Numerical Simulations to Assess Contact Stress in Gerotor Pumps

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## Abstract

**:**

## 1. Introduction

## 2. Trochoidal Gear Set Background

#### 2.1. Gear Sets under Study

#### 2.2. Nomenclature and Notation

#### 2.3. Geometry Generation

## 3. Verification of Contact Stress Numerical Simulation

#### 3.1. Numerical Model Definition

^{®}19.2R (Canonsburg, PA, USA). This general-purpose software environment was selected because of its wide availability among practitioners and also because most of the definitions in terms of mechanical description are analogous and comparable to those of other FEM simulation packages. Moreover, the multiphysics capability of ANSYS

^{®}would allow future enhancement of the model applicability including fluid or thermal transitive simulations, although these steps are far beyond the aim of the current research. In addition, many other researchers have used this software package to study gear contact stresses before, such as Ram Kumar et al. [45], Karthick et. Al. [46], Benaïcha et al. [47], Rao [48], Lahtivirta and Lehtovaara [49], or Lisle et al. [50]. The most relevant information about the implemented numerical models is described in the following subsections.

#### 3.1.1. Geometry and Material Properties

#### 3.1.2. Loads, Contacts, and Boundary Conditions

#### 3.1.3. Elements and Mesh

#### 3.1.4. Analysis

^{®}Core ™ i7-9700 K CPU @ 3.60 GHz with 16 GB RAM memory.

#### 3.2. Simulation Results: Validation and Discussion

- Maximum volume chamber position corresponding to PZ7e377(25°), PZ6e356/1375(0°), PZ6e356/1575(0°), and PZ9e285(0°). The location of the contact point at maximum stress is Pk2 for all gear sets, according to researchers’ and authors’ results. (Note: PZ7e377(25°) gear set in reference [17] does not provide an exact value but it can be gathered from the graphical figures);
- Minimum volume chamber position corresponding to PZ7e377(0°), PZ6e356/1375(25°), PZ6e356/1575(25°), PZ9e285(25°), and MZ9e885(25°). The location of the contact point at maximum stress are Pk and Pki + 1 for all results, researchers’ and author’s, respectively. The exceptions are PZ7e377(0°) and MZ9e885(25°).

## 4. The Taguchi Approach to a Gerotor Pump

^{®}19 software package (State College, PA, USA) complements the data processing.

#### 4.1. The Volumetric Capacity Target and the Dimensional Constraints of the Gerotor Pump

- The PZ9e285 gear set is the chosen prototypical gerotor because it is well-known by the authors regarding, among others, its fluid dynamic performance [54];
- The material properties remain unchangeable in all the experiments: Young’s module, density, and Poisson’s coefficient of the PZ9e285 gear set (refer to Table 2);
- Two specific angular positions as working functions will be under the study: Tip-to-Tip (T2T) and Valley-to-Tip (V2T), both depicted in Figure 6. The V2T corresponds to the maximum volume chamber and the T2T corresponds to the minimum volume chamber. This labeling presumes to enhance the comprehension of the contact points’ action;

#### 4.2. The Taguchi Method: The Designed Experiments

- The number of control factors. The number of parameters is chosen based on the trochoidal gear profile (Z, e, S), external gear (r
_{c}, w_{c}), and working function (reference position RP, shaft keyway SK); - The number of levels. The levels of each parameter are chosen to be significant in the study, from the benchmark gerotor, the literature, the previously presented FEM validation, and the authors’ know-how. In addition, all level combinations have to accomplish the feasible geometry of a trochoidal gear set practicable to be generated with GeroLAB, which has become a challenging task. The number of levels is selected as the Taguchi approach is going forward, from two-levels to a mixed-level design;
- The noise factors. The mechanical operating conditions of the gear pump are described by three factors: material temperature (θ), torque (T), and material friction coefficient (µ), which can be tuned by surface modification methods [56,57]. The designed experiments can use these operating conditions as the input of FEM conditions as prescribed by an outer array only for noise factors (T, µ and θ);
- Improve the quality of contact stress assessment. The goal of the experiments is to maximize the safety factor (SF) as the response of the signal-to-noise ratio (S/N) set to Higher-is-Better (HB) where:$$\mathrm{HB}\text{}\mathrm{S}/\mathrm{N}=-10\mathrm{log}\left(\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\frac{1}{{y}_{i}^{2}}\right)$$
_{i}= each observed value (data points). In all cases, the target is to maximize the HB S/N ratio. The development of a family of matrices based on orthogonal arrays (OA) and the signal-to-noise (S/N) definition as an indicator of the ratio of the mean to the standard deviation are great contributions of the Taguchi method. As a result, they will be used to evaluate and discuss the results of each Taguchi experiment: - Statistical treatment. In all experiments, the significance (alpha) level was set as 0.05, and means and standard deviation were calculated for all volumetric capacities. The p-value inferior to alpha concludes that there is a statistically significant association between the response characteristic and the term. The p-value between the significance levels of 0.05 and 0.1 can be used for evaluating terms, and it can be considered with practical significance. A higher p-value will conclude that there were no statistically significant differences observed.

#### 4.3. First Taguchi Experiment: The Screening Case

_{8}(2

^{7}) OA inner array with a resolution number of 1-low (A and BxC are in the same column) without noise factors is selected and presented in Table 4. The resolution number is a measure of the amount of confounding in a column [33]. As the assignment of factors is to all columns, unavoidably, many interactions are confounded (mixed) with the main effects. This is the major compromise of using fractional factorial experiments: by reducing the number of tests, some information must be surrendered.

_{8}(2

^{7}) is: 8 FEM simulations in this screening case (inner array with 8 gear sets trials, without noise factors, corresponding to each row in Table 4), 7 factor parameters (Z, e, RP, S, r

_{c}, SK, and w

_{c}corresponding to each column in Table 4) and 2 levels (low and high values corresponding to the columns of the chosen Taguchi OA family in Table 4). In addition, the table head of Table 4 indicates the volumetric capacity target and the dimensional constraints to remain constant in the gerotor pump, together with the FEM conditions used in the simulation.

_{c}are the strength factors. Conversely, the parameter w

_{c}, with a behavior almost horizontal between the two levels, is the first factor to be discarded, together with the parameter SK.

_{c}, with working functions, such as SK and RP, where the former shows less dominance than the latter, and it will be discarded. Owing to the higher effect of the level ‘Yes’ in the shaft keyway, which is closer to real working function, all the gear sets in the incoming Taguchi experiments have a shaft keyway.

#### 4.4. Second Taguchi Experiment: The Two-Levels-Four-Geometric-Basic-Parameters Case and One Noise Factor

_{8}(2

^{4}) OA inner array with a resolution number of 2 (A and BxCxD, or AxB and CxD are in the same column), with one noise factor at the outer array L1 OA. Here, 16 FEM simulations were carried out in this second Taguchi (8 gear sets trials, 2 working functions noise factors). This second experiment encompasses less factor parameters (Z, e, S, and r

_{c}) corresponding to each column in Table 5. The outer array accommodates the noise factor at two levels, T2T and V2T, corresponding to columns N1 and N2 in Table 5.

_{c}in the gear set contact stresses as well as no significant interaction effect. One might add the well-known opposite effect on the volumetric characteristics of an even Z value, as it can be seen in the numerals of the flow irregularity in Figure 9. Upon debating this controversial effect, it is decided to go with a full factorial experiment by using the main geometric basic parameters: the number of external teeth, Z, the eccentricity, e, and the arc radius of the external gear tooth, S.

#### 4.5. Third Taguchi Experiment: The Two-Levels-Three-Geometric-Basic-Parameters Case and Three Noise Factors

_{8}(2

^{3}) OA inner array with a resolution number of 4-high (full factorial, all items are in separate columns), with 3 noise factors. Here, 32 FEM simulations were carried out (8 gear sets trials, 4 FEM conditions noise factors). This third experiment encompasses three factor parameters (Z, e, and S) corresponding to each column. Owing to the low effect of r

_{c}, it was removed from the inner array, avoiding interactions confounded with the main effects and gaining resolution. The outer array accommodates the 3 noise factors at two levels, corresponding to columns N1, N2, N3, and N4 (refer to Table 6).

#### 4.6. Fourth Taguchi Experiment: The Mixed-Level-Design-Case with Three Noise Factors

_{18}(2

^{1}–3

^{3}) mixed-level OA inner array with a resolution number of 1-low (no specific interaction columns available, except between AxB by using a specific layout). Here, 72 FEM simulations were carried out (18 gear sets trials, 4 FEM conditions noise factors). The outer array accommodates the three operating conditions (environmental-related, q, and function-related, T and μ) at two levels, corresponding to each column N1, N2, N3, and N4. In this mixed-level design, the parameters of the cutting radius and the wall width of the external gear are kept constant for the 18 gear sets, and factor interactions are not studied based on the results of the previous experiments.

## 5. Conclusions

^{−1}per gear thickness and a proper flow irregularity of 3%.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c_{v} | Volumetric capacity/displacement |

c_{v,H} | Volumetric capacity/displacement per gear thickness |

DeRi | External diameter of internal gear (DeRi = 2·ReRi) |

d_{s} | Shaft hole, internal diameter located in the inner gear |

D_{c} | External diameter of external gear |

e | Eccentricity (centre distance) |

G | Radius of circle to complete external gear |

H | Gear thickness |

L_{s} | Shaft keyway length |

O_{1} | Inner/internal gear centre |

O_{2} | Outer/external gear centre |

P_{k} | Contact point |

r_{c} | Cutting radius |

r_{1} | Inner pitch circle |

r_{2} | Outer pitch circle |

R_{2} | Distance O_{2}P_{s} |

S | Arc radius of the external gear tooth |

T | Torque |

w_{c} | Wall width of the external gear |

Z, (Z − 1) | Number external (internal) teeth |

λ | Tooth profile height correction coefficient (λ = r_{2}/R_{2}) |

μ | Friction coefficient |

θ | Temperature |

υ | Equidistant index (υ = S·Z/R_{2}) |

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**Figure 1.**Comparison of energy consumption loss between ICE vehicle and EV (adapted from Kawamata et al. [3]).

**Figure 3.**Authors’ FEM results of maximum contact stress [MPa] in each gear set from researchers’ works at 0° and 25° angular positions. (The commas in this specific figure represents decimal dots. For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.).

**Figure 5.**Results comparison Researchers vs. Authors, relative error [%] in maximum contact stress [MPa] and contact point (Pk) of maximum contact stress in each gear set for corresponding angular position in brackets (degree °).

**Figure 6.**The benchmark gerotor in working function: the Tip-to-Tip (T2T) position (left) and the Valley-to-Tip (V2T) position (right), the shaft keyway (‘yes’ shaft keyway, left) and the shaft hole without keyway (‘no’ shaft keyway, right).

**Figure 7.**Results of FEM maximum contact stress [MPa], GeroLAB flow irregularity [%], and contact point (Pk) of maximum contact stress in each gear set for the first Taguchi experiment (see Table 4).

**Figure 8.**Main effects plot for the first Taguchi experiment (see Table 4).

**Figure 9.**Results of FEM maximum contact stress [MPa], GeroLAB flow irregularity [%], and contact point (Pk) of maximum contact stress in each gear set with the working function noise factor for the second Taguchi experiment (see Table 5).

**Figure 10.**Main effects plot for the second Taguchi experiment (see Table 5).

**Figure 11.**Interaction plot between Z and S for the second Taguchi experiment (see Table 5).

**Figure 12.**Results of FEM maximum contact stress [MPa], GeroLAB flow irregularity [%], and contact point (Pk) of maximum contact stress in each gear set with 4 FEM conditions noise factors for the third Taguchi experiment (see Table 6).

**Figure 13.**Main effects plot for the third Taguchi experiment (see Table 6).

**Figure 14.**Interaction plot between Z and S for the second Taguchi experiment (see Table 5).

**Figure 15.**Main effects plot for the fourth Taguchi experiment (see Table 7).

**Table 1.**Gear sets under study (PZXeYYY, where X is the number of teeth of the outer gear and YYY are the non-decimal eccentricity value).

Gear Set | Researchers’ Work | Analysis–Environment | Observation |
---|---|---|---|

PZ7e377 | Biernacki [23] | FEM–ABAQUS | The keyway is removed |

PZ6e356/1375 PZ6e356/1575 | Ivanović et al. [14] | FEM–CATIA/FEMAP | λ’ = 1/λ = 1.375 λ’ = 1/λ = 1.575 |

PZ9e285 | Gamez-Montero et al. [42] | FEM–GiD/COMET | The shaft hole is included |

MZ9e855 | Photoelasticity model | Scaled 3:1 of PZ9e285 |

**Table 2.**Geometrical and technical parameters of gear sets (please refer to Nomenclature section and Figure 2).

Input Gear Set Variable/ Parameter | PZ7e377 | PZ6e356/1375 PZ6e356/1575 | PZ9e285 | MZ9e885 |
---|---|---|---|---|

GeroLAB | 3:1 (PZ9e285) | |||

Z [-] | 7 | 6 | 9 | |

e [mm] | 3.77 | 3.56 | 2.85 | |

S [mm] | 11.11 | 9.79 (/1375) 14.06 (/1575) | 10.85 | |

DeRi [mm] | 53.24 | 46.28 | 65.45 | |

G [mm] | 30.50 | 26.94 | 35.80 | |

r_{c} [mm] | 0 | 0 | 0 | |

H [mm] | 10.40 | 16.46 | 9.25 | |

Geometrical | 3:1 (PZ9e285) | |||

w_{c} [mm] | 7.0 | 4.1 | 4.2 | |

D_{c} (O_{2}) [mm] | 75 | 62 | 80 | |

d_{s} (O_{1}) [mm] | 25 | 16 | 44 | |

Material properties | POM * | Steel | Sintered metallic powder | Epoxy |

Young’s module [GPa] | 3 | 200 | 115 | 3 |

Density [kg/m^{3}] | 1410 | - | 6800 | 1160 |

Poisson’s coefficient [-] | 0.43 | 0.30 | 0.25 | 0.35 |

Friction coefficient, m [-] | 0.40 | 0.40 | 0.40 | 0.30 |

Torque, T [N·m] | 7.16 | 0.621 (/1375) 0.632 (/1575) | 18.75 | 37.5 |

**Table 3.**D80d40cv1 benchmark gerotor. (Please refer to Nomenclature section and Figure 2).

Parameter | Value | Significance/Target |
---|---|---|

c_{v,H} = c_{v}/H [cc/(rev·mm)] | 1 | Volumetric capacity/Flow rate |

D_{c} [mm] | 80 | Housing/Dimensional constraint |

d_{s} [mm] | 40 | Internal diameter located in the inner gear to accommodate the shaft/Through-shaft application |

H [mm] | 9.25 | Casing/Dimensional constraint |

**Table 4.**The parameter design table of the screening case of the first Taguchi experiment. (Please refer to Nomenclature section and Figure 2).

8 gear sets in common | |||||||

Vol. capacity target: Dimensional constraints: FEM conditions: | ${\overline{c}}_{v,H}$ = 1 cc·rev^{−1}·mm^{−1} (0.093), mean (standard deviation)D _{c} = 80 mm; d_{s} = 40 mm; H = 9.25 mmT = 18.75 N·m; µ = 0.4; θ = 20° | ||||||

L8 OA inner array | |||||||

(control factors) | A | B | C | D | E | F | G |

Factor parameter (column) Trial gear set::Taguchi experiment (row) | Z | e [mm] | Reference position (RP) (Figure 6) | S [mm] | r_{c}[mm] | Shaft keyway (SK) (Figure 6) | w_{c}[mm] |

1::1 | 8 | 2.50 | T2T | 5.86 | 0.0 | No | 3.0 |

2::1 | 8 | 2.50 | T2T | 13.96 | 3.0 | Yes | 6.0 |

3::1 | 8 | 2.97 | V2T | 5.86 | 0.0 | Yes | 6.0 |

4::1 | 8 | 2.97 | V2T | 13.96 | 3.0 | No | 3.0 |

5::1 | 9 | 2.50 | V2T | 5.86 | 3.0 | No | 6.0 |

6::1 | 9 | 2.50 | V2T | 13.96 | 0.0 | Yes | 3.0 |

7::1 | 9 | 2.97 | T2T | 5.86 | 3.0 | Yes | 3.0 |

8::1 | 9 | 2.97 | T2T | 13.96 | 0.0 | No | 6.0 |

**Table 5.**The parameter design table of the second Taguchi experiment. (Please refer to Nomenclature section and Figure 2).

8 gear sets in common | |||||||

Vol. capacity target: Dimensional constraints: FEM conditions: | ${\overline{c}}_{v,H}$ = 0.997 cc·rev^{−1}·mm^{−1} (0.023), mean (standard deviation)D _{c} = 80 mm; d_{s} = 40 mm; H = 9.25 mm; (SK = yes)T = 18.75 N·m; µ = 0.4; θ = 20° | ||||||

L8 OA inner array (control factors) | A | B | D | G | L1 OA outer array (noise factor) | N1 | N2 |

Factor parameter (column) Trial gear set::Taguchi experiment (row) | Z | r_{c}[mm] | S [mm] | e [mm] | Reference position (Figure 6) (column) | T2T | V2T |

1::2 | 8 | 0.0 | 5.86 | 2.50 | |||

2::2 | 8 | 0.0 | 13.96 | 2.97 | |||

3::2 | 8 | 3.0 | 5.86 | 2.97 | |||

4::2 | 8 | 3.0 | 13.96 | 2.50 | |||

5::2 | 9 | 0.0 | 5.86 | 2.97 | |||

6::2 | 9 | 0.0 | 13.96 | 2.50 | |||

7::2 | 9 | 3.0 | 5.86 | 2.50 | |||

8::2 | 9 | 3.0 | 13.96 | 2.97 |

**Table 6.**The parameter design table of the third Taguchi experiment. (Please refer to Nomenclature section and Figure 2).

8 gear sets in common | ||||||||

Vol. capacity target: Dimensional constraints: FEM conditions: | ${\overline{c}}_{v,H}$ = 0.997 cc·rev^{−1}·mm^{−1} (0.023), mean (standard deviation)D _{c} = 80 mm; d_{s} = 40 mm; H = 9.25 mm; (SK = yes)Noise factors | |||||||

L8 OA inner array (control factors) | A | B | D | L4 OA outer array (noise factors) | N1 | N2 | N3 | N4 |

Factor parameter (column) Trial gear set::Taguchi experiment (row) | Z | e [mm] | S [mm] | T [N·m] μ [-] θ [°C] (column) | 15.0 0.01 20 | 22.5 0.50 20 | 22.5 0.01 40 | 15.0 0.50 40 |

1::3 | 8 | 2.50 | 5.86 | |||||

2::3 | 8 | 2.50 | 13.96 | |||||

3::3 | 8 | 2.97 | 5.86 | |||||

4::3 | 8 | 2.97 | 13.96 | |||||

5::3 | 9 | 2.50 | 5.86 | |||||

6::3 | 9 | 2.50 | 13.96 | |||||

7::3 | 9 | 2.97 | 5.86 | |||||

8::3 | 9 | 2.97 | 13.96 |

**Table 7.**The parameter design table of the fourth Taguchi experiment. (Please refer to Nomenclature section and Figure 2).

18 gear sets in common | |||||||||

Vol. capacity target: Dimensional constraints: FEM conditions: | ${\overline{c}}_{v,H}$ = 0.996 cc·rev^{−1}·mm^{−1} (0.041), mean (standard deviation)D _{c} = 80 mm; d_{s} = 40 mm; H = 9.25 mm; (SK = yes)Noise factors | ||||||||

L18 OA inner array (control factors) | A | B | C | D | L2 OA outer array (noise facts) | N1 | N2 | N3 | N4 |

Factor parameter (column) Trial gear set::Taguchi experiment (row) | Reference position (RP) (Figure 6) | Z | S [mm] | e [mm] | T [N·m] μ [-] θ [°C] (column) | 15.0 0.0 ** 20 | 22.5 0.4 20 | 22.5 0.0 ** 40 | 15.0 0.4 40 |

1::4 | V2T | 7 | 5.8 | 2.61 | |||||

2::4 | V2T | 7 | 8.0 | 2.77 | |||||

3::4 | V2T | 7 | 10.4 | 2.93 | |||||

4::4 | V2T | 9 | 5.8 | 2.61 | |||||

5::4 | V2T | 9 | 8.0 | 2.77 | |||||

6::4 | V2T | 9 | 10.4 | 2.93 | |||||

7::4 | V2T | 11 | 5.8 | 2.77 | |||||

8::4 | V2T | 11 | 8.0 | 2.93 | |||||

9::4 | V2T | 11 | 10.4 | 2.61 | |||||

10::4 | T2T | 7 | 5.8 | 2.93 | |||||

11::4 | T2T | 7 | 8.0 | 2.61 | |||||

12::4 | T2T | 7 | 10.4 | 2.77 | |||||

13::4 | T2T | 9 | 5.8 | 2.77 | |||||

14::4 | T2T | 9 | 8.0 | 2.93 | |||||

15::4 | T2T | 9 | 10.4 | 2.61 | |||||

16::4 | T2T | 11 | 5.8 | 2.93 | |||||

17::4 | T2T | 11 | 8.0 | 2.61 | |||||

18::4 | T2T | 11 | 10.4 | 2.77 |

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**MDPI and ACS Style**

Gamez-Montero, P.J.; Bernat-Maso, E.
Taguchi Techniques as an Effective Simulation-Based Strategy in the Design of Numerical Simulations to Assess Contact Stress in Gerotor Pumps. *Energies* **2022**, *15*, 7138.
https://doi.org/10.3390/en15197138

**AMA Style**

Gamez-Montero PJ, Bernat-Maso E.
Taguchi Techniques as an Effective Simulation-Based Strategy in the Design of Numerical Simulations to Assess Contact Stress in Gerotor Pumps. *Energies*. 2022; 15(19):7138.
https://doi.org/10.3390/en15197138

**Chicago/Turabian Style**

Gamez-Montero, Pedro Javier, and Ernest Bernat-Maso.
2022. "Taguchi Techniques as an Effective Simulation-Based Strategy in the Design of Numerical Simulations to Assess Contact Stress in Gerotor Pumps" *Energies* 15, no. 19: 7138.
https://doi.org/10.3390/en15197138