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

doi:10.1007/s11465-023-0775-x

2024, Vol. 19

Issue (1) : 2 https://doi.org/10.1007/s11465-023-0775-x

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

Huilin HE¹, Peitang WEI¹(

), Huaiju LIU¹, Xuesong DU¹, Rui HU¹, Genshen LIU¹, Yajun WU²

¹. State Key Laboratory of Mechanical Transmission for Advanced Equipment, Chongqing University, Chongqing 400044, China
². Taiyuan Heavy Machinery Group Co., Ltd., Taiyuan 030024, China

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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.

Keywords planetary roller screw mechanism (PRSM) contact position axial clearance machining error grouping for selective assembly

Corresponding Author(s): Peitang WEI

Just Accepted Date: 01 November 2023 Issue Date: 27 February 2024

Cite this article:

Huilin HE,Peitang WEI,Huaiju LIU, et al. Three-dimensional parametric contact analysis of planetary roller screw mechanism and its application in grouping for selective assembly[J]. Front. Mech. Eng., 2024, 19(1): 2.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-023-0775-x
https://academic.hep.com.cn/fme/EN/Y2024/V19/I1/2

Fig.1 Schematic illustration of the planetary roller screw mechanism structure.

Fig.2 Thread profile of the components: (a) screw, (b) roller, and (c) nut.

Fig.3 Initial three-dimensional meshing model of the planetary roller screw mechanism.

Fig.4 Schematic illustration of the machining errors in threads: (a) pitch error, denoted as Δp_i, (b) nominal diameter error, denoted as Δd_i, and (c) flank angle error, denoted as Δβ_i.

Fig.5 Calculation flow for contact characteristics in this study. PRSM: planetary roller screw mechanism.

Fig.6 Mesh division method for the meshing model.

Tab.1 Geometric parameters of 25 × 4 model PRSM

Tab.2 Tolerance range of machining errors

Tab.3 Chemical composition of 95Cr18 stainless steel

Fig.7 Specimens of the 25 × 4 model planetary roller screw mechanism.

Fig.8 MarSurf UD 130 profilometer.

Fig.9 Thread machining error data of the roller: (a) nominal diameter error, (b) pitch error, and (c) flank angle error.

Fig.10 Actual distribution curve of machining errors in rollers: (a) nominal diameter error, (b) pitch error, and (c) flank angle error.

Tab.4 Distribution parameters of errors

Parameter	Symbol
Screw contact radius/mm	$r S m$
Screw contact angle/(° )	$φ S m$
Roller contact radius on the screw?roller side/mm	$r R m S$
Roller contact angle on the screw?roller side/(° )	$φ R m S$
Axial clearances on the screw?roller side/mm	$δ m S R$
Nut contact radius/mm	$r N m$
Nut contact angle/(° )	$φ N m$
Roller contact radius on the nut?roller side/mm	$r R m N$
Roller contact angle on the nut?roller side/(° )	$φ R m N$
Axial clearances on the nut?roller side/mm	$δ m N R$

Tab.5 Key calculated parameters of contact characteristics

Fig.11 Projection of the screw, nut, and roller contact points on the O?XY plane.

Contact characteristics	Parameter	Calculation results of 20 × 10 PRSM in Liu et al.’s model [43]	Calculation results of 20 × 10 PRSM in our model	Calculation results of 30 × 20 PRSM in Fedosovsky et al.’s model [24]	Calculation results of 30 × 20 PRSM in our model
Contact positions	$r S m$ /mm	9.8173	9.8125	15.0500	15.1000
	$φ S m$ /(° )	3.6605	3.6684	4.5800	4.5000
	$r R m S$ /mm	3.2635	3.2685	5.1400	5.0400
	$φ R m S$ /(° )	11.0730	11.0745	13.5100	13.6000
	$r N m$ /mm	16.250	16.246	25.000	25.000
	$φ N m$ /(° )	0	0	0	0
	$r R m N$ /mm	3.250	3.246	5.000	5.000
	$φ R m N$ /(° )	0	0	0	0
Axial clearances	$δ m S R$ /mm	0.01530	0.01540	0	?0.00704
Axial clearances	$δ m N R$ /mm	0.02	0.02	0.00	0.00

Tab.6 Contact positions and axial clearances of error-free PRSM

Fig.12 Deviation of calculation results between this study and two models.

Fig.13 Meshing position variations induced by the nominal diameter errors in the (a) screw–roller meshing region and (b) nut–roller meshing region, pitch errors in the (c) screw–roller meshing region and (d) nut–roller meshing region, and flank angle errors in the (e) screw–roller meshing region and (f) nut–roller meshing region.

Fig.14 Axial clearance variation caused by the given nominal diameter error ranges in the (a) screw–roller meshing region and (b) nut–roller meshing region; [μ ? 3σ, μ + 3σ] range nominal diameter errors in Tab.4 in the (c) screw–roller meshing region and (d) nut–roller meshing region; given pitch error ranges in the (e) screw–roller meshing region and (f) nut–roller meshing region; [μ ? 3σ, μ + 3σ] range pitch errors in Tab.4 in the (g) screw–roller meshing region and (h) nut–roller meshing region; given flank angle error ranges in the (i) screw–roller meshing region and (j) nut–roller meshing region; and [μ ? 3σ, μ + 3σ] range flank angle errors in Tab.4 in the (k) screw–roller meshing region and (l) nut–roller meshing region.

Tab.7 Sensitivity coefficients of error parameters

Fig.15 Main workflow of Fuzzy C-Means.

Fig.16 Distortion measure for different numbers of clusters.

Fig.17 Selective assembly intervals for different errors in the grouping results.

Fig.18 Axial clearance before and after grouping.

Abbreviations
CNC	Computer numerical control
FCM	Fuzzy C-Means
OBB	Oriented bounding box
PRSM	Planetary roller screw mechanism
VTK	Visualization toolkit
Variables
A_iB_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_iE_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
$d i j w$	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
$p i n t N$	Intersection points between v and the nut
$p i n t R ? N$	Intersection points between v and the roller on nut side
$p i n t R ? S$	Intersection points between v and the roller on screw side
$p i n t S$	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
$r R m N$	Roller contact radius on the nut?roller side
$r R m S$	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
R₁, R₂	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 = b₁ ? b₂	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
$Z N i U$ , $Z N i B$ , $Z R ? N i U$ , $Z R ? N i B$ (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
$Z S i U$ , $Z S i B$ , $Z R ? S i U$ , $Z R ? S i B$ (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
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
$Γ S R U$ , $Γ S R B$ , $Γ N R U$ , $Γ N R B$	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
$δ N R i U$ , $δ N R i B$ 　(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
$δ R m N$	Axial clearances on the nut?roller side
$δ R m S$	Axial clearances on the screw?roller side
$δ S R i U$ , $δ S R i B$ 　(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
$φ R m N$	Roller contact angle on the nut?roller side
$φ R m S$	Roller contact angle on the screw?roller side
φ_Sm	Screw contact angle

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