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# Axial backlash of planetary roller screw mechanisms: geometric modelling and Sobol's analysis of design parameters

Juliette Lepagneul, Loïc Tadrist, Emmanuel Mermoz, Jean-Marc Linares

<figure><img src="image_1.png" /></figure>

## To cite this version:

Juliette Lepagneul, Loïc Tadrist, Emmanuel Mermoz, Jean-Marc Linares. Axial backlash of planetary roller screw mechanisms: geometric modelling and Sobol's analysis of design parameters. CIRP Annals - Manufacturing Technology, 2023, 72 (1), pp.113-116. 10.1016/j.cirp.2023.03.005. hal-04113545

HAL Id: hal-04113545

https://hal.science/hal-04113545v1

Submitted on 21 Feb 2024

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# Axial backlash of planetary roller screw mechanisms: geometric modelling and Sobol's analysis of design parameters

Juliette Lepagneul<sup>a,b</sup>, Loïc Tadrist<sup>a</sup>, Emmanuel Mermoz<sup>a,b</sup> (2), Jean-Marc Linares<sup>a</sup> (1)

Planetary roller screw mechanisms (PRSMs) are promising for the future all-electric aircraft flight controls because they allow transmission of high loads while being compact. However, impacts after axial backlash transit are critical when loads are alternate. In this paper, the influence of PRSM design parameters on the axial backlash is analyzed. First, axial backlash is geometrically modelled. PRSM conformal contacts require multi-precision computation to avoid numerical cancellation. Second, Sobol's analysis is implemented on four industrial PRSMs. Results show that three out of six parameters are influential independently from the geometry considered. Controlling these parameters is thus mandatory for aeronautical applications.

Design optimization, Geometric modelling, Planetary roller-screw mechanism

## 1. Introduction

Planetary roller screw mechanisms (PRSMs) convert rotation into translation under high loads. They are compounded of a screw, a nut, multiple rollers and two carriers. Most PRSMs are standard or inverted (figure 1a) depending whether rollers move with the nut or the screw. PRSM compactness makes them the cornerstone for the future all-electric aircraft flight controls where high alternate loads are expected. This chronic loading may damage the PRSM and reduce its fatigue lifespan [1], specifically at the thread contact points which are the bottleneck of the load transmission path. Alternate loads have not yet been investigated much for PRSMs. However, alternate loads are known to produce undesired brinelling in ball screws (plastic deformations and fatigue-induced flaking) reducing lifespan [2]. Fatigue of a roller screw has been investigated analyzing surface [3] or bulk stresses [1] under static loads. Axial backlash is critical when high alternate loads are applied on the mechanism. At the load reversal, the free part of the mechanism is accelerated during backlash transit until it impacts on the other part of the mechanism. The impact velocity reads $v = \sqrt{2FJ/m}$ where F is the load, J the axial backlash and m the mass of the moving part. For low backlash, the impact velocity remains in the subsonic domain, and the impact can be modelled with a quasi-static theory. Considering a Hertz model for thread contacts, the impact dynamics reads Eq. (1) where $x$ is the indentation depth, $\ddot{x}$ is the acceleration and $k$ is the Hertz coefficient gathering contact geometry and material properties [1].

$$m \ddot{x} = F - kx^{3/2} \quad (1)$$

The first integration of this equation gives the velocity $\dot{x}$, Eq. (2):

$$\frac{1}{2} m (\dot{x}^2 - v^2) = Fx - \frac{2k}{5} x^{\frac{5}{2}} \quad (2)$$

The maximal contact force $F_{max}$ occurs when $\dot{x} = 0$, which gives Eq. (3) when $J \ll (5F/2k)^{2/3}$.

$$F_{max} = \frac{5}{2} F \left( 1 + J \left( \frac{2k}{5F} \right)^{\frac{2}{3}} \right) \quad (3)$$

Thus, the maximal force supported by the contact points depends on the axial backlash $J$ relatively to the quasi-static indentation $(5F/2k)^{2/3}$.

<footnote><sup>a</sup> Aix-Marseille Univ, CNRS, ISM, Marseille, France</footnote>

<footnote><sup>b</sup> Airbus Helicopters, Aéroport de Marseille Provence, Marignane, 13700 France</footnote>
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The design of PRSM for aeronautic industry should thus insure backlash tolerance as it may increase the maximal load supported by thread contacts. To specify design parameter tolerance from admissible axial backlash uncertainty, numerous technics have been developed in industry. These technics were summarized in the keynote paper [4]. Among the most promising technics, statistical tolerancing based on Monte Carlo simulations, allows to derive the probability to obtain a desired backlash from a numerical modelling of a PRSM. This numerical method is both easy to implement and efficient for non-linear models of over-constrained mechanisms. Previous computations of axial backlash on an ideal PRSM geometry showed that backlash depends on the contact point positions [5,6]. Contact points are defined by thread tangency and numerically computed from a non-linear set of equations involving geometrical design parameters of PRSMs.

Sobol's analysis of variance [7] allows to bring to the fore the relative importance of each parameter of a numerical model from random simulations. It allows to find the most influential parameters on which the most restrictive tolerance interval should be set.

Based on this, the present study proposes an analytical modelling of axial backlash of PRSM and determines the most influential parameters. In the context of alternate load conditions, this will allow to set restrictive tolerance intervals on those parameters to avoid undesired overloads due to backlash transit.

## 2. Geometrical modelling of PRSM axial backlash

### 2.1 Contact positions

Clearances exist between screw-roller and between roller-nut contacts. These local clearances create a backlash along the screw axis. Considering nominal geometries and under the rigid body hypothesis for rollers, screw and nut, the axial backlash depends on the contact positions that are defined by surface tangency between threads [1,5].

<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a) 3D view of an inverted planetary roller-screw mechanism (PRSM). (b) Geometrical parameters of a convex thread.</figcaption></figure>

Screw and nut are described by a straight thread surface ($r_c = \infty$ where $r_c$ is the curvature radius of the thread profile). In the following, capital T refers to top surface, in orange in figures 2 and 3, (and B for bottom, in green), subscript S stands for screw (N for nut and R for roller). The position of the contact on the top surface of the screw and the nut is given by Eq. (4):

$$T_{S,N} = - \left( \frac{p}{4} + (r_1 - r_{S,N}) \frac{\tan(\alpha)}{\cos(\beta)} \right) + \frac{\theta_1 l_{S,N}}{2\pi} \quad (4)$$

<table><caption>**Table 1:** Analytical computation of screw-roller axial backlash $J_{SR}$ considering a top-screw and a bottom-roller contact.</caption><thead><tr><th>PRSM type</th><th>Standard Fig 2 (a)</th><th>Inverted Fig 2 (b)</th><th>Standard -</th><th>Inverted Fig 2 (c)</th></tr></thead><tbody><tr><th>θ<sub>2</sub></th><td>&gt; 0</td><td>&gt; 0</td><td>&lt; 0</td><td>&lt; 0</td></tr><tr><th>J<sub>SR</sub></th><td>2B<sub>S</sub> - 2T<sub>R</sub> - 2l<sub>R</sub></td><td>2B<sub>S</sub> - 2T<sub>R</sub> - 2l<sub>R</sub> - 2p</td><td>impossible</td><td>2B<sub>S</sub> - 2T<sub>R</sub> + 2l<sub>R</sub></td></tr></tbody></table>

where $r_{S,N}$ is the nominal radius of the screw or nut, $\alpha$ the normal pressure angle, $p$ the pitch, $l_{S,N}$ the lead (oriented helix pitch) of the screw or nut, $|l_{S,N}| = np$ with $n$ the number of threads and $\beta = \arctan(l_S/2\pi r_S)$ the helix angle, see figure 1b. Symmetrically, $B_{S,N} = -T_{S,N}$. For the roller, which threads are convex, the contact position on the bottom surface reads Eq. (5):

$$B_R = \left( \frac{p}{4} + r_c \frac{\cos(\alpha)}{\cos(\beta)} - \frac{\sqrt{r_c^2 - r_2^2}}{\cos(\beta)} \right) + \frac{\theta_2 l_R}{2\pi} \quad (5)$$

Symmetrically, $B_R = -T_R$.
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## 2.2 Backlash on the screw-roller side

Most PRSMs exhibit right-handed leads for screw $l_S > 0$ and nut $l_N > 0$. The roller has a right-hand thread ($l_R > 0$) for standard PRSMs and a left-hand thread ($l_R < 0$) for inverted PRSMs. In this paper, contacts are identified in polar coordinates by a radius $r_i$ and an angle $\theta_i$ ($i = 1$ for the screw or the nut and $i = 2$ for roller). In figures 2 and 3, the screw is shown in blue, the roller in grey and the nut in green. To suppress the ambiguity when the lead equals the pitch, 2-helix rollers (roller with $n = 2$ threads), $|l_R| = 2p$, are depicted in figures 2 and 3. To illustrate the method to compute axial backlash, a contact between top-screw and bottom-roller surfaces is considered with $\theta_2 > 0$ and a right-hand threaded roller $l_R > 0$ in figure 2a. Screw gap reads $g_S = B_S - T_S = 2B_S$. Contact points on the roller, $B_R$ and $T_R$, computed from Eq. (5), are distant from the contact zone because $|\theta_2| \approx \pi$. To determine roller width (distance between contact points enclosing roller thread tip), these points should be shifted, $w_R = T_R + l_R - (B_R - l_R)$. The backlash $J_{SR}$ is the difference between screw gap and roller width (subscript SR refers to screw-roller contact) Eq. (6):

$$J_{SR} = g_S - w_R = 2B_S - 2T_R - 2l_R \quad (6)$$

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Screw-roller contact, top-screw and bottom-roller (green thread) layout. Blue: screw. Grey: roller. The axial backlash $J_{SR}$ is computed from contact point positions ($B_S, B_R, T_S, T_R$) shifted by integer numbers of pitch $p$ and/or lead $l_R$. The calculation proxies, the screw gap $g_S$ and the roller width $w_R$, are shown. (a) Standard PRSM ($\theta_2 > 0$; $l_R = 2p$); (b) Inverted PRSM ($\theta_2 > 0$; $l_R = -2p$); (c) Inverted PRSM ($\theta_2 < 0$; $l_R = -2p$).</figcaption></figure>

In figure 2b, an inverted PRSM (left-hand threaded roller $l_R = -2p$) is considered with $\theta_2 > 0$. One finds Eq. (7):

$$J_{SR} = 2B_S - 2T_R - 2l_R - 2p. \quad (7)$$

A standard PRSM with $\theta_2 < 0$ is impossible [1,5]. In figure 2c, an inverted PRSM ($l_R = -2p$) is shown with $\theta_2 > 0$. One finds Eq. (7):

The four cases are summarised in Table 1. Actually, all possible configurations of PRSMs are considered in figure 2. Here, a top-screw and bottom-roller contact is presented but identical results may be found considering a bottom-screw and top-roller contact.

$$J_{SR} = 2B_S - 2T_R + 2l_R. \quad (8)$$

## 2.3 Backlash on the nut-roller side

The backlash at the contact between the roller and the nut is also given by the contact point positions, see figure 3. Both for standard and inverted PRSMs, the backlash $J_{NR}$ (subscript NR refers to nut-roller contact) reads Eq. (9):

$$J_{NR} = 2(B_R - B_N) \quad (9)$$
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<figure><img src="image_5.png" /><figcaption>**Figure 3.** Nut-roller contact for (a) top-nut, top-roller; (b) bottom-nut, bottom-roller. Green: nut. Grey: roller. The axial backlash $J_{NR}$ is computed from contact point positions ($B_N$, $B_R$, $T_N$, $T_R$). The calculation proxies, the roller gap $g_R$ and the nut width $w_N$, are shown.</figcaption></figure>

## 2.4 Axial backlash of a PRSM

The total backlash of a PRSM is the sum of the backlash from the screw-roller contact and the one of the nut-roller contact, Eq. (10):

$$J = J_{SR} + J_{NR} \qquad (10)$$

This geometrical modelling of PRSM backlash is fully analytical, being accurate and numerically efficient. It corrects the expression given in [5] and expands it to standard and inverted PRSMs.

# 3. Influence of design parameters using Sobol's analysis

## 3.1 PRSMs of reference

<table><caption>**Table 2:** Geometrical parameters of PRSMs of reference. Cases 1 and 3: Standard PRSMs; Cases 2 and 4: Inverted PRSMs. (p: pitch, n: number of threads, α: normal pressure angle, $r_C$: curvature radius of thread profile, r: nominal radius).</caption><thead><tr><th>Parameters<br>Units</th><th></th><th>p<br>mm</th><th>n<br>-</th><th>α<br>°</th><th>r<sub>c</sub><br>mm</th><th>r<br>mm</th></tr></thead><tbody><tr><td rowspan="3">Case 1<br>[5]</td><td>Screw</td><td rowspan="3">2</td><td>5</td><td rowspan="3">45</td><td>∞</td><td>15 (r<sub>s</sub>)</td></tr><tr><td>Roller</td><td>1</td><td>7.07</td><td>5 (r<sub>R</sub>)</td></tr><tr><td>Nut</td><td>5</td><td>∞</td><td>25 (r<sub>N</sub>)</td></tr><tr><td rowspan="3">Case 2<br>[5]</td><td>Screw</td><td rowspan="3">5</td><td>2</td><td rowspan="3">45</td><td>∞</td><td>15</td></tr><tr><td>Roller</td><td>1</td><td>10.6064</td><td>7.5</td></tr><tr><td>Nut</td><td>2</td><td>∞</td><td>30.1265</td></tr><tr><td rowspan="3">Case 3<br>[5]</td><td>Screw</td><td rowspan="3">2</td><td>5</td><td rowspan="3">68.21</td><td>∞</td><td>9.75</td></tr><tr><td>Roller</td><td>1</td><td>3.5</td><td>3.25</td></tr><tr><td>Nut</td><td>5</td><td>∞</td><td>16.25</td></tr><tr><td rowspan="3">Case 4<br>[8]</td><td>Screw</td><td rowspan="3">1.66</td><td>3</td><td rowspan="3">45</td><td>∞</td><td>10.5</td></tr><tr><td>Roller</td><td>3</td><td>7.07</td><td>3.5</td></tr><tr><td>Nut</td><td>3</td><td>∞</td><td>17.5</td></tr></tbody></table>

The present theory is applied to a set of 4 PRSMs of reference, that have been previously considered in the literature [5,8]. The set is compounded of 2 standard PRSMs (Cases 1 and 3, $l_R > 0$) and 2 inverted PRSMs (Cases 2 and 4, $l_R < 0$). The
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geometrical properties of each PRSM are gathered in Table 2. Corresponding backslashes are given in Table 3, matching literature results.

## 3.2 Numerical precautions for the implementation of backlash

Numerical evaluation of steep non-linear problems such as the contact of almost planar surfaces may create numerical errors that can lead to reverse conclusions [7]. For each PRSM considered, the validity of numerical computations was checked by varying the number of digits considered with the Matlab® symbolic toolbox. The almost exact numerical value was obtained with 256 digits. This value was used to estimate the numerical precision. For cases 1, 2 and 3, the problem solution converges after 10 digits of precision, see figure 4a. Those cases can be efficiently computed with classical double precision floating points. However, for case 4, the backlash computation does not converge before 27 digits of precision. In the following, to get accurate results case 4, was computed with 40 digits of precision. This geometrical modelling may be implemented in a pseudo Monte-Carlo scheme to perform the Sobol's analysis of variance.

## 3.3 Sobol's analysis of variance

Sobol's analysis of variance is based on the analysis of a random (Monte-Carlo) or quasi-random exploration of the design space. The numerical model $Y = f(X_1, X_2, \dots, X_k)$ is analysed considering that all parameters change randomly except $X_i$, that is fixed, to find the mean value $E(Y|X_i)$. This operation is repeated for various $X_i$ to compute the variance $V$ of the mean value $E(Y|X_i)$. This defines the Sobol's first order index, Eq. (10):

<table><caption>**Table 3:** Axial backlash computed from Eq. (6-10), in black, are compared to data found in [5] in blue for cases 1, 2 and 3. Case 4: No data in literature.</caption><thead><tr><th></th><th>J<sub>SR</sub> (μm)</th><th>J<sub>NR</sub> (μm)</th><th>J (μm)</th></tr></thead><tbody><tr><td>Case 1</td><td>-105.61 / -107.1</td><td>1.5131 / 0</td><td>-104.10</td></tr><tr><td>Case 2</td><td>0.28687 / 0</td><td>0.47953 / 0</td><td>0.7664</td></tr><tr><td>Case 3</td><td>-65.604 / -65.6</td><td>0.00185 / 0</td><td>-65.602</td></tr><tr><td>Case 4</td><td>0</td><td>-64.263</td><td>-64.263</td></tr></tbody></table>

$$S_i = \frac{V_{X_i} (E_{X_{i-1}} (Y|X_i))}{V(Y)} \quad (10)$$

To compute the total effect of a parameter $X_i$, one first considers the variance of the model $Y$ varying $X_i$, $V_{X_i}(Y|X_{i-1})$, all other parameters of the model being fixed. This operation is then repeated, considering various sets of parameters $X_{i-1}: (X_1, \dots, X_{i-1}, X_{i+1}, \dots, X_k)$. The mean of variances gives the Sobol's total effect index, Eq. (11):

$$S_{Ti} = \frac{E_{X_{i-1}}(V_{X_i}(Y|X_{i-1}))}{V(Y)} \quad (11)$$
Sobol's indices have two main properties, $\sum_i S_i < 1$ and if $S_i = S_{Ti}$ then parameters are independent. Note also that if $S_i = 0$ and $S_{Ti} = 0$ then the parameter has no effect on the model output.

<figure><img src="image_6.png" /><figcaption>**Figure 4.** (a) Smooth convergence of backlash computation after 10 digits of precision (Case 1). (b) Simplified view of $(X_i, X_j)$ design space. Dark grey: working PRSM in design space. Blue, green and orange circles: Dimensions of spots chosen to analyse the effect of $(X_i, X_j)$ around the nominal parameter values. Blue and green spots are too large to perform statistical analysis.</figcaption></figure>

## 3.4 Numerical implementation of Sobol's indices

First, a minimal set of 6 geometrical parameters ($r_S, r_R, r_N, p, r_C, \alpha$) for PRSM design is considered. This set is complete: any other parameter may be derived from it. Second, the problem is made dimensionless by considering relative variations of each of these parameters following the scheme Eq. (12):

$$X_i = X_i^0(1 + \Delta(2\epsilon_i - 1)) \quad (12)$$

Where $X_i^0$ is the nominal value of the parameter $X_i$, $\Delta$ is the relative

amplitude variation and $\epsilon_i \in [0,1]$ is the random parameter for the Monte-Carlo simulation. In this statistical tolerancing study, the amplitude of variations is chosen as $\Delta = 10^{-4}$. Large variation amplitude may lead in unrealistic PRSM geometry impairing Sobol's analysis conclusions, see figure 4b.

The efficient numerical method proposed in [9] is implemented. Two matrices of random numbers, **A** and **B**, are built gathering the random parameters $\epsilon_i$ for the 6 parameters and N evaluations. **A** and **B** are of size $N \times 6$. For each parameter, the matrix $A_B^i$ is built by replacing the i-th column of matrix **A** by the i-th column of matrix **B**, conversely for $B_A^i$. For all 14 matrices, the backlash of the PRSM is computed with the numerical model J described in Section 2. This allows to compute the first order and total effect Sobol's indices, Eq. (13) and (14):

$$S_i = \frac{\sum_{k=1}^{N} J(\mathbf{B})_k (J(\mathbf{A}_B^i)_k - J(\mathbf{A})_k)}{\sum_{k=1}^{N} (J(\mathbf{A})_k - \frac{1}{N} \sum_{k=1}^{N} J(\mathbf{A})_k)^2} \quad (13)$$

$$S_{Ti} = \frac{\sum_{k=1}^{N} (J(\mathbf{A})_k - J(\mathbf{A}_B^i)_k)^2}{2 \sum_{k=1}^{N} (J(\mathbf{A})_k - \frac{1}{N} \sum_{k=1}^{N} J(\mathbf{A})_k)^2} \quad (14)$$

These two numerical implementations are fast converging and the most used in the literature [9]. In the sake of generality, we crosschecked this by verifying their efficiency compared to other numerical implementations proposed in [9]. Again, for numerical efficiency, literature reports that quasi-random Sobol or Halton sequence may be considered. Sobol sequences were implemented. Finally, to speed up computations, the code has been parallelized. With a 6-core laptop (Intel® Core™ I7-10850H CPU @2.70GHz) and double precision, the computation of $10^6$ trials was done in 5 hours with Matlab®. For cases 1, 2 and 3, where double precision computation was efficient, $50 \times 10^6$ numerical tests were done. The 50 evaluations of $S_i$ and $S_{Ti}$ were used to determine standard deviations $\sigma_{S_i}$ and $\sigma_{S_{Ti}}$ for uncertainty bounds. For case 4, with 40 digits precision, $50 \times 10^4$ numerical tests were done. The computation time was around 4h for $10^4$ trials.
# 4. Results: Backlash design parameters analysis

## 4.1 Variance analysis of backlash design parameters

The results of the Sobol's analysis of variance are presented in Table 4. For all considered PRSMs, standard as well as inverted, the influence of $S_i$ and $S_{Ti}$ are similar. The most influential design parameters are the nut nominal radius $r_N$, with the larger $S_i$ and $S_{Ti}$, then the screw nominal radius $r_S$ and the curvature radius of the thread profile $r_C$. The three other parameters, namely the pitch $p$, the roller nominal radius $r_R$ and the normal pressure angle $\alpha$, have negligible influence on the axial backlash.

<table><caption>**Table 4:** Sobol's indices of roller-screw design parameters considering axial backlash for the 4 PRSMs of reference.</caption><thead><tr><th></th><th></th><th>r<sub>S</sub></th><th>r<sub>R</sub></th><th>r<sub>N</sub></th><th>p</th><th>r<sub>C</sub></th><th>&alpha;</th></tr></thead><tbody><tr><td rowspan="4">Case 1</td><td>S<sub>i</sub></td><td>0.330</td><td>&lt; 10<sup>-7</sup></td><td>0.439</td><td>2.89 10<sup>-6</sup></td><td>0.090</td><td>&lt; 10<sup>-13</sup></td></tr><tr><td>&sigma;<sub>S<sub>i</sub></sub></td><td>0.159</td><td>&lt; 10<sup>-7</sup></td><td>0.481</td><td>2.07 10<sup>-6</sup></td><td>0.079</td><td>&lt; 10<sup>-13</sup></td></tr><tr><td>S<sub>Ti</sub></td><td>0.330</td><td>&lt; 10<sup>-7</sup></td><td>0.440</td><td>2.88 10<sup>-6</sup></td><td>0.090</td><td>0</td></tr><tr><td>&sigma;<sub>S<sub>Ti</sub></sub></td><td>0.159</td><td>&lt; 10<sup>-7</sup></td><td>0.481</td><td>2.07 10<sup>-6</sup></td><td>0.079</td><td>0</td></tr><tr><td rowspan="4">Case 2</td><td>S<sub>i</sub></td><td>0.208</td><td>&lt; 10<sup>-9</sup></td><td>0.560</td><td>4.67 10<sup>-5</sup></td><td>0.155</td><td>&lt; 10<sup>-9</sup></td></tr><tr><td>&sigma;<sub>S<sub>i</sub></sub></td><td>0.113</td><td>&lt; 10<sup>-9</sup></td><td>0.470</td><td>3.36 10<sup>-5</sup></td><td>0.116</td><td>&lt; 10<sup>-9</sup></td></tr><tr><td>S<sub>Ti</sub></td><td>0.208</td><td>&lt; 10<sup>-12</sup></td><td>0.560</td><td>4.75 10<sup>-5</sup></td><td>0.155</td><td>&lt; 10<sup>-11</sup></td></tr><tr><td>&sigma;<sub>S<sub>Ti</sub></sub></td><td>0.113</td><td>&lt; 10<sup>-14</sup></td><td>0.470</td><td>3.21 10<sup>-5</sup></td><td>0.116</td><td>&lt; 10<sup>-11</sup></td></tr><tr><td rowspan="4">Case 3</td><td>S<sub>i</sub></td><td>0.311</td><td>&lt; 10<sup>-7</sup></td><td>0.475</td><td>6.48 10<sup>-7</sup></td><td>0.095</td><td>&lt; 10<sup>-13</sup></td></tr><tr><td>&sigma;<sub>S<sub>i</sub></sub></td><td>0.173</td><td>&lt; 10<sup>-8</sup></td><td>0.507</td><td>&lt; 10<sup>-7</sup></td><td>0.082</td><td>&lt; 10<sup>-11</sup></td></tr><tr><td>S<sub>Ti</sub></td><td>0.310</td><td>&lt; 10<sup>-7</sup></td><td>0.476</td><td>&lt; 10<sup>-7</sup></td><td>0.095</td><td>0</td></tr><tr><td>&sigma;<sub>S<sub>Ti</sub></sub></td><td>0.173</td><td>&lt; 10<sup>-8</sup></td><td>0.507</td><td>&lt; 10<sup>-7</sup></td><td>0.082</td><td>0</td></tr><tr><td rowspan="4">Case 4</td><td>S<sub>i</sub></td><td>0.307</td><td>&lt; 10<sup>-7</sup></td><td>0.372</td><td>1.28 10<sup>-5</sup></td><td>0.105</td><td>&lt; 10<sup>-30</sup></td></tr><tr><td>&sigma;<sub>S<sub>i</sub></sub></td><td>0.165</td><td>&lt; 10<sup>-7</sup></td><td>0.452</td><td>1.67 10<sup>-6</sup></td><td>0.080</td><td>&lt; 10<sup>-29</sup></td></tr><tr><td>S<sub>Ti</sub></td><td>0.299</td><td>&lt; 10<sup>-7</sup></td><td>0.374</td><td>2.72 10<sup>-6</sup></td><td>0.102</td><td>&lt; 10<sup>-57</sup></td></tr><tr><td>&sigma;<sub>S<sub>Ti</sub></sub></td><td>0.165</td><td>&lt; 10<sup>-7</sup></td><td>0.452</td><td>1.62 10<sup>-6</sup></td><td>0.080</td><td>&lt; 10<sup>-57</sup></td></tr></tbody></table>

For Sobol's analysis, the criterion to validate convergence of the simulations is that the standard deviations are in the order of magnitude of the Sobol's indices. As a consequence, 50 repetitions were computed to derive the standard deviations $\sigma_{S_i}$ and $\sigma_{S_{Ti}}$. The computed standard deviations confirms the convergence of the present simulation.

## 4.2 Discussion and model reduction

The present results show that, for each PRSM considered, $\sum S_i < 1$ as expected by the Sobol's theory when results are correctly converged. Further $S_{Ti} = S_i$ which indicates that the effect of each parameter on the axial backlash is independent. A linear model is expected since only small variations around nominal values were considered ($\Delta < 10^{-4}$).

Sobol's analysis of variance for axial backlash of PRSMs faces several numerical problems. First, it is slow converging, which may be hastened considering pseudo random sequences using Halton or Sobol sequences. Second, large number of simulations have to be considered to reduce uncertainty on Sobol's indices. Third, even considering dimensionless variables, in the case of PRSMs where conformal contact occurs, the problem of numerical cancellation arises. This problem has to be considered and solved using a multi-precision computation library. Numerical cancellation and its propagation in the computation may lead to negative Sobol's indices (wrong convergence) and slightly affect their magnitude.

# 5. Conclusion

In the present paper, the influence of design parameters on the axial backlash for standard and inverted PRSMs was quantified using Sobol's analysis of variance. Numerical precautions have to be taken when analyzing PRSMs geometries
where contacts between conformal surfaces occur. Sometimes, these precautions show that a multi-precision algorithm has
to be implemented.

Among the six design parameters considered, the curvature radius of the thread profile, the nominal radii of the nut and
the screw were the most influential on the axial backlash. Conversely, it was not changed by the normal pressure angle, the
nominal radius of the roller and the pitch variation.

From a practical point of view, to control axial backlash, which is critical for aeronautical applications where high alternate
loads are applied on the mechanism, the three influential parameters have to be controlled. The linear behavior of the model
and the independence between the influential parameters will allow easy dimensioning of tolerance intervals on the
curvature radius of the thread profile, the nominal radii of the nut and the screw.

**6. Acknowledgement**

This work was jointly supported by ANRT (CIFRE, N°2021/0010), Airbus Helicopters and Aix-Marseille University.

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