# Global Sensitivity Analysis of Chosen Harmonic Drive Parameters Affecting Its Lost Motion

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{N}to the FS and measuring its angular deformation φ

_{1}, φ

_{1}′ (φ

_{2}, φ

_{2}′, respectively) in both directions, with the WG locked and the CS fixed, obtaining the hysteresis curve shown in Figure 1.

- φ—angular deformation by applying the +4% of T
_{N}to the FS; - φ′—angular deformation by applying the −4% of T
_{N}to the FS.

## 2. Materials and Methods

_{N}to the FS measuring its angular deformation in both directions with the WG locked and the CS fixed to obtain the torsional windup of the gear.

_{1}(revolutions per minute n

_{1}), the FS with the teeth number z

_{FS}(−) acting as the output shaft with the angular velocity ω

_{2}(revolutions per minute n

_{2}) and the CS with the teeth number z

_{CS}(−) fixed according the equation:

- the parametric model of the HD finite element analyses was built and the analyses of the prescribed motion state were performed to determine the lost motion values for the generated set of chosen variables;
- the correlation of the input parameters (flexible ball bearing inner and outer ring diameter, the radial bearing clearance and the offset of the FS and CS nominal teeth spline) and the output parameter (lost motion) was observed and the correlation coefficients were calculated;
- the global sensitivity analyses (GSA) of the chosen variable sets were performed based on the obtained dependence of the input and output variables calculating the GSA indices according the generated quasi-random variable sets.

## 3. Parametric Model Building

- d
_{max}—maximum permissible value of the inner bearing diameter d; - d
_{min}—minimum permissible value of the inner bearing diameter d.

## 4. FEA Model Definition

^{−6}). This relative accuracy value allows virtual models to be created that are accurate to ten thousandths of millimeters depending on the maximum dimension of the particular part. This value of the relative accuracy is sufficient enough, because the real production accuracy of the harmonic drive most precise components, e.g., the gearing of the flexspline and circular spline, is manufactured in the range of hundredths of millimeters. The Ansys mesher works with even higher relative accuracy, which guarantees the required level of accuracy also by meshed model.

## 5. FEA Analyses Variable Set Generation

- the bearing inner ring inner diameter d (mm);
- the bearing outer ring outer diameter D (mm);
- the bearing inner radial clearance G
_{r}(mm).

- 4.
- the offset of the nominal shape of the flexspline tooth r
_{FS}(mm) influencing the dimension over the pin; - 5.
- the offset of the nominal shape of the circular spline tooth r
_{CS}(mm) influencing the dimension between the pin.

## 6. Input and Output Variables Correlation Analysis

_{r}, r

_{FS}, r

_{CS}) and the output parameter (Φ) was observed in the presented study by three methods:

- Linear regression;
- Pearson’s linear correlation (detects a linear relationship between two variables);
- Spearman’s rank correlation (detects a monotonic relationship between two variables).

#### 6.1. Linear Regression Model

_{i}denotes the i-th observation of the dependent variable, while X

_{ij}denotes the i-th observation of the j-th independent variable, with j = 1, 2, …, p. The values j denote the estimated parameters, while i is the i-th independently identically distributed normal error [24].

#### 6.2. Pearson’s Linear Correlation

_{xy}and defined as:

- n—sample size;
- x
_{i}, y_{i}—individual sample points indexed with i.

#### 6.3. Spearman’s Linear Correlation

_{s}using the formula:

- n—sample size (number of observations).

_{i}between two ranks (rg) of each observation rg(X

_{i}) and rg(Y

_{i}) is defined as:

## 7. Global Sensitivity Analysis

- Input and output dependence polynoma definition;
- Quasi-random variables data generation;
- First-order and total-effect Sobol indices calculation.

#### 7.1. Input and Output Dependence Polynoma Definition

- A, B, C, D—polynoma coefficents;
- x—variables (d, D, G
_{r}, r_{FS}, r_{CS}); - n—number of variables;
- i = 1 … n—first variable index;
- j = i + 1 … n—second variable index.

^{2}= 0.98 for HD 42 and R

^{2}= 0.99 for HD 120.

#### 7.2. Quasi-Random Variables Data Generation

- Latin Hyper Cube set;
- Halton set;
- Sobol set.

#### 7.3. First-Order and Total-Effect Sobol’s Indices Calculation

_{i}of X

_{i}on Y according the equation:

_{i}on Y. S

_{i}is a number always between 0 and 1. A high value signals an important variable. Total effects are a direct consequence of Sobol’s variance decomposition approach and estimation procedure. The total effect index accounts for the total contribution to the output variation due to factor X

_{i}, i.e., its first-order effect plus all higher-order effects due to interactions [14].

## 8. Discussion

_{r}has an influence rated by the correlation of about 20.23% to 32.17% by the standard data regression methods for the HD 42, or 10.98% to 23.87% for the HD 120, but this is a minor influence considering the Sobol’s indices values.

_{r}) are intertwined, which influences the lower coefficients by the outer bearing ring diameter D for the HD 42 and similar values to the inner bearing diameter d for the HD 120.

## 9. Conclusions

_{FS}and r

_{CS}. They have the largest size among the observed parameters group and thus their criteria weight is highest too. Their Sobol’s indices values are close to each other. The change in r

_{FS}and r

_{CS}affects the dimension over pin for the FS and between pin for the CS, and also the lost motion in the same way; equal changes in the offsets generate equal change in the output parameter, that is, the HD lost motion.

_{FS}and r

_{CS}values will lead to the required low value of the HD backlash during its operation in the desired motion state.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Ball bearing with flexible rings characteristic dimensions: inner ring inner diameter d; inner ring outer diameter d

_{1}; outer ring inner diameter D

_{1}; outer ring outer diameter D; bearing width B.

**Figure 4.**HD 120 FEA model: (

**a**) general view; (

**b**) front view; (

**c**) detail of ball bearing meshing; (

**d**) detail of teeth meshing.

**Figure 5.**Definition of the offset from the nominal FS and CS teeth shape: (

**a**) r

_{FS}influencing the dimension over pin; (

**b**) r

_{CS}influencing the dimension between pin.

**Figure 6.**HD 42 determination histograms of the lost motion values (blue circles) applying Pearson (red line) and Spearman (black curve) to the input parameters: (

**a**) flexible bearing inner diameter d; (

**b**) flexible bearing outer diameter D; (

**c**) flexible bearing radial clearance G

_{r}; (

**d**) offset of the FS teeth spline r

_{FS}; (

**e**) offset of the CS teeth spline r

_{CS}.

**Figure 7.**HD 120 determination histograms of the lost motion values (blue circles) applying Pearson (red line) and Spearman (black curve) to the input parameters: (

**a**) flexible bearing inner diameter d; (

**b**) flexible bearing outer diameter D; (

**c**) flexible bearing radial clearance G

_{r}; (

**d**) offset of the FS teeth spline r

_{FS}; (

**e**) offset of the CS teeth spline r

_{CS}.

**Figure 8.**HD 42 lost motion FEA results (blue circles) vs. fitted curve results (red asterisks): (

**a**) flexible bearing inner diameter d; (

**b**) flexible bearing outer diameter D; (

**c**) flexible bearing radial clearance G

_{r}; (

**d**) offset of the FS teeth spline r

_{FS}; (

**e**) offset of the CS teeth spline r

_{CS}.

**Figure 9.**HD 120 lost motion FEA results (blue circles) vs. fitted curve results (red asterisks): (

**a**) flexible bearing inner diameter d; (

**b**) flexible bearing outer diameter D; (

**c**) flexible bearing radial clearance G

_{r}; (

**d**) offset of the FS teeth spline r

_{FS}; (

**e**) offset of the CS teeth spline r

_{CS}.

**Figure 10.**Quasi-random variables scatter plots: (

**a**) Latin Hyper Cube set; (

**b**) Halton set; (

**c**) Sobol set.

**Figure 11.**HD 42 lost motion values of the fitted function calculated for the Sobol data set: (

**a**) flexible bearing inner diameter d; (

**b**) flexible bearing outer diameter D; (

**c**) flexible bearing radial clearance G

_{r}; (

**d**) offset of the FS teeth spline r

_{FS}; (

**e**) offset of the CS teeth spline r

_{CS}.

**Figure 12.**HD 120 lost motion values of the fitted function calculated for the Sobol data set: (

**a**) flexible bearing inner diameter d; (

**b**) flexible bearing outer diameter D; (

**c**) flexible bearing radial clearance G

_{r}; (

**d**) offset of the FS teeth spline r

_{FS}; (

**e**) offset of the CS teeth spline r

_{CS}.

**Figure 16.**HD 120 global sensitivity analysis calculation results: (

**a**) table data; (

**b**) column graph.

HD type | z_{FS}(-) | z_{CS}(-) | I (-) | n_{1}(min ^{−1}) | P_{1}(W) | 4% (of T_{N})(N m) |
---|---|---|---|---|---|---|

HD 42 | 240 | 242 | −120 | 1500 | 31 | 1 (of 24) |

HD 120 | 240 | 242 | −120 | 1500 | 693 | 21 (of 529) |

Bearing Type | d (mm) | d_{1max}(mm) | D_{1min}(mm) | D (mm) | B (mm) | O (mm) |
---|---|---|---|---|---|---|

113-1145TN | 31.00 | 33.10 | 40.10 | 42.00 | 7.00 | 0.30–0.56 |

114-876TN | 90.00 | 95.20 | 114.80 | 120.00 | 18.00 | 1.20–1.60 |

Limit Dimensions | d (mm) | D (mm) | G_{r} (mm) | r_{FS} (mm) | r_{CS} (mm) | |
---|---|---|---|---|---|---|

HD 42 | nominal | 31.000 | 42.000 | 0.015 | 0.000 | 0.000 |

minimal | 31.000 | 41.984 | 0.005 | 0.000 | 0.000 | |

maximal | 31.016 | 42.000 | 0.020 | 0.020 | 0.020 | |

HD 120 | nominal | 90.000 | 120.000 | 0.030 | 0.025 | 0.025 |

minimal | 90.000 | 119.978 | 0.000 | 0.025 | 0.025 | |

maximal | 90.022 | 120.000 | 0.055 | 0.055 | 0.055 |

**Table 4.**Variable sets generated for the HD 42 by the Space Filling Design method with calculated lost motion values.

Data Set | d (mm) | D (mm) | G_{r} (mm) | r_{FS} (mm) | r_{CS} (mm) | Φ (arcmin) |
---|---|---|---|---|---|---|

1 | 31.0080000000 | 41.9914074074 | 0.0119444444 | 0.0107407407 | 0.0100000000 | 0.3387054552 |

2 | 31.0068148148 | 41.9997037037 | 0.0097222222 | 0.0114814815 | 0.0048148148 | 0.1924051335 |

3 | 31.0020740741 | 41.9860740741 | 0.0169444444 | 0.0100000000 | 0.0040740741 | 0.3345637636 |

4 | 31.0115555556 | 41.9973333333 | 0.0141666667 | 0.0188888889 | 0.0114814815 | 1.0883206873 |

5 | 31.0032592593 | 41.9902222222 | 0.0080555556 | 0.0092592593 | 0.0003703704 | 0.1993212189 |

6 | 31.0074074074 | 41.9937777778 | 0.0175000000 | 0.0159259259 | 0.0011111111 | 0.3183620597 |

7 | 31.0056296296 | 41.9967407407 | 0.0125000000 | 0.0070370370 | 0.0196296296 | 0.5170704345 |

8 | 31.0151111111 | 41.9920000000 | 0.0152777778 | 0.0144444444 | 0.0188888889 | 1.4873940404 |

9 | 31.0044444444 | 41.9866666667 | 0.0102777778 | 0.0196296296 | 0.0085185185 | 0.4067272251 |

10 | 31.0038518519 | 41.9979259259 | 0.0191666667 | 0.0129629630 | 0.0122222222 | 0.5806961197 |

11 | 31.0050370370 | 41.9931851852 | 0.0052777778 | 0.0025925926 | 0.0107407407 | 0.1798381857 |

12 | 31.0062222222 | 41.9908148148 | 0.0197222222 | 0.0033333333 | 0.0151851852 | 0.3502783652 |

13 | 31.0002962963 | 41.9949629630 | 0.0136111111 | 0.0048148148 | 0.0070370370 | 0.1918537718 |

14 | 31.0109629630 | 41.9848888889 | 0.0180555556 | 0.0122222222 | 0.0129629630 | 0.7353834863 |

15 | 31.0091851852 | 41.9878518519 | 0.0075000000 | 0.0137037037 | 0.0181481481 | 0.7737438484 |

16 | 31.0103703704 | 41.9890370370 | 0.0158333333 | 0.0018518519 | 0.0018518519 | 0.2691275760 |

17 | 31.0145185185 | 41.9872592593 | 0.0130555556 | 0.0151851852 | 0.0033333333 | 0.3720114037 |

18 | 31.0157037037 | 41.9955555556 | 0.0186111111 | 0.0077777778 | 0.0092592593 | 0.5075938888 |

19 | 31.0014814815 | 41.9854814815 | 0.0108333333 | 0.0055555556 | 0.0144444444 | 0.2953603605 |

20 | 31.0008888889 | 41.9961481481 | 0.0086111111 | 0.0166666667 | 0.0137037037 | 0.4271771907 |

21 | 31.0139259259 | 41.9943703704 | 0.0091666667 | 0.0040740741 | 0.0025925926 | 0.1805950189 |

22 | 31.0127407407 | 41.9896296296 | 0.0113888889 | 0.0011111111 | 0.0166666667 | 0.3117640981 |

23 | 31.0121481481 | 41.9925925926 | 0.0058333333 | 0.0181481481 | 0.0062962963 | 0.4104898389 |

24 | 31.0097777778 | 41.9842962963 | 0.0069444444 | 0.0062962963 | 0.0055555556 | 0.3359137403 |

25 | 31.0133333333 | 41.9991111111 | 0.0063888889 | 0.0085185185 | 0.0159259259 | 0.3315779483 |

26 | 31.0026666667 | 41.9884444444 | 0.0163888889 | 0.0174074074 | 0.0174074074 | 1.1083372325 |

27 | 31.0085925926 | 41.9985185185 | 0.0147222222 | 0.0003703704 | 0.0077777778 | 0.1838389297 |

**Table 5.**Variable sets generated for the HD 120 by the Space Filling Design method with calculated lost motion values.

Data Set | d (mm) | D (mm) | G_{r} (mm) | r_{FS} (mm) | r_{CS} (mm) | Φ (arcmin) |
---|---|---|---|---|---|---|

1 | 90.011000000 | 119.988185185 | 0.016666667 | 0.041111111 | 0.040000000 | 1.244209085 |

2 | 90.009370370 | 119.999592593 | 0.011333333 | 0.042222222 | 0.032222222 | 0.671159038 |

3 | 90.002851852 | 119.980851852 | 0.028666667 | 0.040000000 | 0.031111111 | 0.999485343 |

4 | 90.015888889 | 119.996333333 | 0.022000000 | 0.053333333 | 0.042222222 | 1.866310632 |

5 | 90.004481481 | 119.986555556 | 0.007333333 | 0.038888889 | 0.025555556 | 0.708068406 |

6 | 90.010185185 | 119.991444444 | 0.030000000 | 0.048888889 | 0.026666667 | 1.048538552 |

7 | 90.007740741 | 119.995518519 | 0.018000000 | 0.035555556 | 0.054444444 | 1.368646926 |

8 | 90.020777778 | 119.989000000 | 0.024666667 | 0.046666667 | 0.053333333 | 2.513551579 |

9 | 90.006111111 | 119.981666667 | 0.012666667 | 0.054444444 | 0.037777778 | 1.678786696 |

10 | 90.005296296 | 119.997148148 | 0.034000000 | 0.044444444 | 0.043333333 | 1.329323399 |

11 | 90.006925926 | 119.990629630 | 0.000666667 | 0.028888889 | 0.041111111 | 0.644800974 |

12 | 90.008555556 | 119.987370370 | 0.035333333 | 0.030000000 | 0.047777778 | 1.295404011 |

13 | 90.000407407 | 119.993074074 | 0.020666667 | 0.032222222 | 0.035555556 | 0.709814782 |

14 | 90.015074074 | 119.979222222 | 0.031333333 | 0.043333333 | 0.044444444 | 1.965452675 |

15 | 90.012629630 | 119.983296296 | 0.006000000 | 0.045555556 | 0.052222222 | 2.019798297 |

16 | 90.014259259 | 119.984925926 | 0.026000000 | 0.027777778 | 0.027777778 | 0.821133320 |

17 | 90.019962963 | 119.982481481 | 0.019333333 | 0.047777778 | 0.030000000 | 1.417778345 |

18 | 90.021592593 | 119.993888889 | 0.032666667 | 0.036666667 | 0.038888889 | 1.296064918 |

19 | 90.002037037 | 119.980037037 | 0.014000000 | 0.033333333 | 0.046666667 | 1.065814949 |

20 | 90.001222222 | 119.994703704 | 0.008666667 | 0.050000000 | 0.045555556 | 1.354223000 |

21 | 90.019148148 | 119.992259259 | 0.010000000 | 0.031111111 | 0.028888889 | 0.667895757 |

22 | 90.017518519 | 119.985740741 | 0.015333333 | 0.026666667 | 0.050000000 | 1.223274066 |

23 | 90.016703704 | 119.989814815 | 0.002000000 | 0.052222222 | 0.034444444 | 1.416564820 |

24 | 90.013444444 | 119.978407407 | 0.004666667 | 0.034444444 | 0.033333333 | 0.860107056 |

25 | 90.018333333 | 119.998777778 | 0.003333333 | 0.037777778 | 0.048888889 | 1.161955836 |

26 | 90.003666667 | 119.984111111 | 0.027333333 | 0.051111111 | 0.051111111 | 2.272618801 |

27 | 90.011814815 | 119.997962963 | 0.023333333 | 0.025555556 | 0.036666667 | 0.662908446 |

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**MDPI and ACS Style**

Hrcek, S.; Brumercik, F.; Smetanka, L.; Lukac, M.; Patin, B.; Glowacz, A.
Global Sensitivity Analysis of Chosen Harmonic Drive Parameters Affecting Its Lost Motion. *Materials* **2021**, *14*, 5057.
https://doi.org/10.3390/ma14175057

**AMA Style**

Hrcek S, Brumercik F, Smetanka L, Lukac M, Patin B, Glowacz A.
Global Sensitivity Analysis of Chosen Harmonic Drive Parameters Affecting Its Lost Motion. *Materials*. 2021; 14(17):5057.
https://doi.org/10.3390/ma14175057

**Chicago/Turabian Style**

Hrcek, Slavomir, Frantisek Brumercik, Lukas Smetanka, Michal Lukac, Branislav Patin, and Adam Glowacz.
2021. "Global Sensitivity Analysis of Chosen Harmonic Drive Parameters Affecting Its Lost Motion" *Materials* 14, no. 17: 5057.
https://doi.org/10.3390/ma14175057