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

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.

Key words： planetary roller screw mechanism (PRSM); contact position; axial clearance; machining error; grouping for selective assembly

Nomenclature

Abbreviations
CNC	Computer numerical control
FCM	Fuzzy C-Means
OBB	Oriented bounding box
PRSM	Planetary roller screw mechanism
VTK	Visualization toolkit
Variables
AiBi (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
BiEi (i = S, R, N)	Contact profile segment in the axial thread profile of screw, roller, or nut
ci (i = S, R, N)	Half thread tooth thickness of screw, roller, or nut
cci	Cluster center of the ith group of samples
Ci (i = 1,2,...,K)	ith group group of samples
c	Vertex coordinates matrix of the meshing model’s OBB
celli (i = 1,2,...,n)	Cell of the meshing model’s OBB
di (i = S, R, N)	Nominal diameter of screw, roller, or nut
d_{ij}^{\mathrm{w}}	Weighted Euclidean distance between cluster center cci and the jth sample xj
Δdi (i = S, R, N)	Nominal diameter error of screw, roller, or nut
en	nth sample
Esam	Sample set
EEij (i = 1,2,...,9; j = 1,2,...,Nsam)	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
Lint	Overlapping region between the meshing model and v
Lm	Projection ranges of the meshing model along the axes
Lv	Projection ranges of v along the axes
ni (i = S, R, N)	Number of starts of screw, roller, or nut
Nj (j = 1,2,...,K)	Total number of samples in the jth group
Nsam	Number of samples in Esam
ni (i = 1,2,3)	Normal vectors of meshing model cell
pi (i = S, R, N)	Pitch of screw, roller, or nut
Δpi (i = S, R, N)	Pitch error of screw, roller, or nut
P	Clustering center
Pi (i = S, R, N)	Point sets for the screw, roller, or nut thread profiles
pi1, pi2, pi3 (i = 1,2,...,n)	Three vertices of meshing model cell
pint	Intersection point between celli 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
rNm	Nut contact radius
rroot_i (i = S, R, N)	Root arc radius of screw, roller, or nut
rRar	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
rSa, rRa, rNf	Major diameters of the screw, roller, and nut, respectively
rSm	Screw contact radius
R1, R2	Random numbers within the range [0, 1]
Serror	Weighting coefficients of errors
Sclearance	Weighting coefficients of axial clearance
si (i = 1,2,...,9)	Sensitivity coefficient of 9 errors
sk (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
uij (i = 1,2,...,K)	Membership of the jth sample to the ith group Ci
U	Membership matrix
v = b1 ‒ b2	Ray vector and its starting point vector and ending point vector
xn1, xn2, …, xn9	Sampling values of 9 errors in en
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)	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_{\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)	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
βi (i = S, R, N)	Flank angle of screw, roller, or nut
Δβi (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
{\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
δg	Overall global axial clearance of PRSM
δij, δj (i = 1,2,...,Nj;　 j = 1,2,...,K)	Clearance values of sample within the group and their mean value
δn	Axial clearance value with 9 errors in en
{\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
λi (i = S, R, N)	Helix angle of screw, roller, or nut
μ	Location parameter of errors distribution
μi (i = 1, 2,..., 10)	Sensitivity factor of each error
σ	Scale parameter of errors distribution
φNm	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
φSm	Screw contact angle

Acknowledgement

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

Conflict of Interest

The authors declare that they have no conflict of interest.

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