Threedimensional parametric contact analysis of planetary roller screw mechanism and its application in grouping for selective assembly
Huilin HE, Peitang WEI, Huaiju LIU, Xuesong DU, Rui HU, Genshen LIU, Yajun WU
Threedimensional parametric contact analysis of planetary roller screw mechanism and its application in grouping for selective assembly
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 loadcarrying capacity and transmission accuracy. Therefore, studying the contact characteristics of PRSM forms the fundamental basis for enhancing its transmission performance. In this study, a threedimensional parametric analysis method of contact characteristics is proposed based on the PRSM meshing principle and PyVista (a highlevel 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 threedimensional parametric contact characteristics analysis method, the Fuzzy CMeans 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.
planetary roller screw mechanism (PRSM) / contact position / axial clearance / machining error / grouping for selective assembly
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Abbreviations  
CNC  Computer numerical control 
FCM  Fuzzy CMeans 
OBB  Oriented bounding box 
PRSM  Planetary roller screw mechanism 
VTK  Visualization toolkit 
Variables  
A_{i}B_{i} (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 
${d}_{ij}^{\mathrm{w}}$  Weighted Euclidean distance between cluster center cc_{i} and the jth sample x_{j} 
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 x_{ij} of the jth sample is changed to (x_{ij} + 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 E_{sam} 
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 cell_{i} and v 
${p}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{N}}$  Intersection points between v and the nut 
${p}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{R}\mathrm{N}}$  Intersection points between v and the roller on nut side 
${p}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{R}\mathrm{S}}$  Intersection points between v and the roller on screw side 
${p}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{\mathrm{S}}$  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 
${r}_{\mathrm{R}\mathrm{m}}^{\mathrm{N}}$  Roller contact radius on the nut‒roller side 
${r}_{\mathrm{R}\mathrm{m}}^{\mathrm{S}}$  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 cell_{i} and v 
T  Iteration times 
u_{ij} (i = 1,2,...,K)  Membership of the jth sample to the ith group C_{i} 
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 e_{n} 
${Z}_{\mathrm{N}}^{i\mathrm{U}}$, ${Z}_{\mathrm{N}}^{i\mathrm{B}}$, ${Z}_{\text{R}\text{N}}^{i\text{U}}$, ${Z}_{\text{R}\text{N}}^{i\text{B}}$ (i = 1,2,...,n)  zcoordinate 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 
${Z}_{\mathrm{S}}^{i\mathrm{U}}$, ${Z}_{\mathrm{S}}^{i\mathrm{B}}$, ${Z}_{\text{R}\text{S}}^{i\text{U}}$, ${Z}_{\text{R}\text{S}}^{i\text{B}}$ (i = 1,2,...,n)  zcoordinate 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 
Δβ_{i} (i = S, R, N)  Flank angle error of screw, roller, or nut 
Γ_{Ui}, Γ_{Bi} (i = S, R, N)  Upper and lower profile curves of the screw, roller, or nut, respectively 
${\mathrm{\Gamma}}_{\mathrm{S}\mathrm{R}}^{\mathrm{U}}$, ${\mathrm{\Gamma}}_{\mathrm{S}\mathrm{R}}^{\mathrm{B}}$, ${\mathrm{\Gamma}}_{\mathrm{N}\mathrm{R}}^{\mathrm{U}}$, ${\mathrm{\Gamma}}_{\mathrm{N}\mathrm{R}}^{\mathrm{B}}$  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 e_{n} 
${\delta}_{\mathrm{N}\mathrm{R}}^{i\mathrm{U}}$, ${\delta}_{\mathrm{N}\mathrm{R}}^{i\mathrm{B}}$ (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 
${\delta}_{\mathrm{R}\mathrm{m}}^{\mathrm{N}}$  Axial clearances on the nut‒roller side 
${\delta}_{\mathrm{R}\mathrm{m}}^{\mathrm{S}}$  Axial clearances on the screw‒roller side 
${\delta}_{\mathrm{S}\mathrm{R}}^{i\mathrm{U}}$, ${\delta}_{\mathrm{S}\mathrm{R}}^{i\mathrm{B}}$ (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 
${\phi}_{\mathrm{R}\mathrm{m}}^{\mathrm{N}}$  Roller contact angle on the nut‒roller side 
${\phi}_{\mathrm{R}\mathrm{m}}^{\mathrm{S}}$  Roller contact angle on the screw‒roller side 
φ_{Sm}  Screw contact angle 
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