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Three-dimensional parametric contact analysis of planetary roller screw mechanism and its application in grouping for selective assembly

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Abstract

The planetary roller screw mechanism (PRSM) is a novel precision transmission mechanism that realizes the conversion between linear and rotary motions. The contact characteristics of helical surfaces directly determine PRSM’s performance in load-carrying capacity and transmission accuracy. Therefore, studying the contact characteristics of PRSM forms the fundamental basis for enhancing its transmission performance. In this study, a three-dimensional parametric analysis method of contact characteristics is proposed based on the PRSM meshing principle and PyVista (a high-level API to the Visualization Toolkit). The proposed method considers the influence of machining errors among various thread teeth. The effects of key machining errors on contact positions and axial clearance, as well as their sensitivities, are analyzed. With excellent solution accuracy, this method exhibits higher calculation efficiency and stronger robustness than the analytical and numerical meshing models. The influence of nominal diameter and pitch errors of the screw, roller, and nut on the axial clearance follows a linear relationship, whereas flank angle errors have negligible effects on the axial clearance. The corresponding influence coefficients for these three machining errors on the axial clearance are 0.623, 0.341, and 0.036. The variations in contact positions caused by individual errors are axisymmetric. Flank angle errors and roller diameter errors result in linear displacements of the contact points, whereas pitch errors cause the contact points to move along the arc of the roller diameter. Based on the proposed three-dimensional parametric contact characteristics analysis method, the Fuzzy C-Means clustering algorithm considering error sensitivity is utilized to establish a component grouping technique in the selective assembly of critical PRSM components, ensuring the rational and consistent clearances based on the given component’s machining errors. This study provides effective guidance for analyzing contact characteristics and grouping in selective assembly for PRSM components. It also presents the proposed method’s potential applicability to similar calculation problems for contact positions and clearances in other transmission systems.

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Abbreviations

CNC:

Computer numerical control

FCM:

Fuzzy C-Means

OBB:

Oriented bounding box

PRSM:

Planetary roller screw mechanism

VTK:

Visualization toolkit

A iBi (i = S, R, N):

Thread root arc segment in the axial thread profile of screw, roller, or nut

axes :

Three coordinate axis vectors of the meshing model’s OBB

B i E i (i = S, R, N):

Contact profile segment in the axial thread profile of screw, roller, or nut

c i (i = S, R, N):

Half thread tooth thickness of screw, roller, or nut

cc i :

Cluster center of the ith group of samples

C i (i = 1,2,…,K):

ith group group of samples

c :

Vertex coordinates matrix of the meshing model’s OBB

cell i (i = 1,2,…,n):

Cell of the meshing model’s OBB

d i (i = S, R, N):

Nominal diameter of screw, roller, or nut

\(d_{ij}^{\rm{w}}\) :

Weighted Euclidean distance between cluster center cci and the jth sample Xj

Δd i (i = S, R, N):

Nominal diameter error of screw, roller, or nut

e n :

nth sample

E sam :

Sample set

EE ij (i = 1,2,…,9; j = 1,2,…,N sam):

Axial clearance variation resulting from consecutive runs in which only the ith error xij of the jth sample is changed to (Xij + Δ)

K :

Number of clusters

L int :

Overlapping region between the meshing model and v

L m :

Projection ranges of the meshing model along the axes

L v :

Projection ranges of v along the axes

n i (i = S, R, N):

Number of starts of screw, roller, or nut

N j (j = 1,2,…,K):

Total number of samples in the jth group

N sam :

Number of samples in Esam

n i (i = 1,2,3):

Normal vectors of meshing model cell

p i (i = S, R, N):

Pitch of screw, roller, or nut

Δp i (i = S, R, N):

Pitch error of screw, roller, or nut

P :

Clustering center

P i (i = S, R, N):

Point sets for the screw, roller, or nut thread profiles

p i1, p i2, p i3 (i = 1,2,…,n):

Three vertices of meshing model cell

p int :

Intersection point between celli and v

\(p_{{\mathop{\rm int}}}^{\rm{N}}\) :

Intersection points between v and the nut

\(p_{{\mathop{\rm int}}}^{{\rm{R - N}}}\) :

Intersection points between v and the roller on nut side

\(p_{{\mathop{\rm int}}}^{{\rm{R - S}}}\) :

Intersection points between v and the roller on screw side

\(p_{{\mathop{\rm int}}}^{\rm{S}}\) :

Intersection points between v and the screw

r Nm :

Nut contact radius

r root_i (i = S, R, N):

Root arc radius of screw, roller, or nut

r Rar :

Roller arc radius

\(r_{{\rm{Rm}}}^{\rm{N}}\) :

Roller contact radius on the nut–roller side

\(r_{{\rm{Rm}}}^{\rm{S}}\) :

Roller contact radius on the screw–roller side

r Sa, r Ra, r Nf :

Major diameters of the screw, roller, and nut, respectively

r Sm :

Screw contact radius

R 1, R 2 :

Random numbers within the range [0, 1]

S error :

Weighting coefficients of errors

S clearance :

Weighting coefficients of axial clearance

s i (i = 1,2,…,9):

Sensitivity coefficient of 9 errors

s k (k = 1,2,…,10):

Weight coefficient of the kth sample attribute

SSE :

Sum of square errors

t :

Intersection coefficient between celli and v

T :

Iteration times

u ij (i = 1,2,…,K):

Membership of the jth sample to the ith group Ci

U :

Membership matrix

v = b 1 − b 2 :

Ray vector and its starting point vector and ending point vector

x n1, x n2, …, x n9 :

Sampling values of 9 errors in en

\(Z_{\rm{N}}^{i{\rm{U}}},\,Z_{\rm{N}}^{i{\rm{B}}},\,Z_{{\rm{R - N}}}^{i{\rm{U}}},\,Z_{{\rm{R - N}}}^{i{\rm{B}}}\) (i = 1,2,…,n):

z-coordinate of the intersection point between v and the upper and lower contact surfaces of the ith pair of meshing threads on the nut–roller side, corresponding to the minimum axial distance

\(Z_{\rm{S}}^{i{\rm{U}}},\,Z_{\rm{S}}^{i{\rm{B}}},\,Z_{{\rm{R - S}}}^{i{\rm{U}}},\,Z_{{\rm{R - S}}}^{i{\rm{B}}}\) (i = 1,2,…,n):

z-coordinate of the intersection point between v and the upper and lower contact surfaces of the ith pair of meshing threads on the screw–roller side, corresponding to the minimum axial distance

α:

Significance level of hypothesis testing

β i (i = S, R, N):

Flank angle of screw, roller, or nut

Δα i (i = S, R, N):

Flank angle error of screw, roller, or nut

ΓUi, ΓBi (i = S, R, N):

Upper and lower profile curves of the screw, roller, or nut, respectively

\(\Gamma _{{\rm{SR}}}^{\rm{U}},\,\Gamma _{{\rm{SR}}}^{\rm{B}},\,\Gamma _{{\rm{NR}}}^{\rm{U}},\,\Gamma _{{\rm{NR}}}^{\rm{B}}\) :

Contact pairs on the upper and lower sides of the roller threads

δ g :

Overall global axial clearance of PRSM

δ ij, δ j (i = 1,2,…,N j; j = 1,2,…,K):

Clearance values of sample within the group and their mean value

δ n :

Axial clearance value with 9 errors in en

\(\delta _{{\rm{NR}}}^{i{\rm{U}}},\,\,\delta _{{\rm{NR}}}^{i{\rm{B}}}\) (i = 1,2,…,n):

Axial clearances between the upper and lower contact surfaces of the ith pair of meshing threads on the nut–roller side, respectively

\(\delta _{{\rm{Rm}}}^{\rm{N}}\) :

Axial clearances on the nut–roller side

\(\delta _{{\rm{Rm}}}^{\rm{S}}\) :

Axial clearances on the screw–roller side

\(\delta _{{\rm{SR}}}^{i{\rm{U}}},\,\,\delta _{{\rm{SR}}}^{i{\rm{B}}}\) (i = 1,2,…,n):

Axial clearances between the upper and lower contact surfaces of the ith pair of meshing threads on the screw–roller side, respectively

ε :

Iterative threshold

λ i (i = S, R, N):

Helix angle of screw, roller, or nut

μ :

Location parameter of errors distribution

μ i (i = 1, 2,…, 10):

Sensitivity factor of each error

σ :

Scale parameter of errors distribution

φ Nm :

Nut contact angle

\(\varphi _{{\rm{Rm}}}^{\rm{N}}\) :

Roller contact angle on the nut–roller side

\(\varphi _{{\rm{Rm}}}^{\rm{S}}\) :

Roller contact angle on the screw–roller side

φ Sm :

Screw contact angle

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Acknowledgement

The work was supported by the National Key R&D Program of China (Grant No. 2023YFB3406404).

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Authors and Affiliations

  1. State Key Laboratory of Mechanical Transmission for Advanced Equipment, Chongqing University, Chongqing, 400044, China

    Huilin He, Peitang Wei, Huaiju Liu, Xuesong Du, Rui Hu & Genshen Liu

  2. Taiyuan Heavy Machinery Group Co., Ltd., Taiyuan, 030024, China

    Yajun Wu

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  1. Huilin He
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  2. Peitang Wei
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Correspondence to Peitang Wei.

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He, H., Wei, P., Liu, H. et al. Three-dimensional parametric contact analysis of planetary roller screw mechanism and its application in grouping for selective assembly. Front. Mech. Eng. 19, 2 (2024). https://doi.org/10.1007/s11465-023-0775-x

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