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Research Papers: Design of Mechanisms and Robotic Systems

Low Order Static Load Distribution Model for Ball Screw Mechanisms Including Effects of Lateral Deformation and Geometric Errors

[+] Author and Article Information
Bo Lin

Mechatronics and Sustainability Research Lab,
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109;
G.G. Brown Laboratory,
2350 Hayward,
Ann Arbor, MI 48109
e-mail: bolin@umich.edu

Chinedum E. Okwudire

Mem. ASME
Mechatronics and Sustainability Research Lab,
Department of Mechanical Engineering,
University of Michigan,
Ann Arbor, MI 48109;
G.G. Brown Laboratory,
2350 Hayward,
Ann Arbor, MI 48109
e-mail: okwudire@umich.edu

Jason S. Wou

Product Development,
Ford Motor Company,
6200 Mercury Drive,
Dearborn, MI 48126
e-mail: swou@ford.com

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received December 16, 2016; final manuscript received September 19, 2017; published online December 11, 2017. Assoc. Editor: David Myszka.

J. Mech. Des 140(2), 022301 (Dec 11, 2017) (12 pages) Paper No: MD-16-1830; doi: 10.1115/1.4038071 History: Received December 16, 2016; Revised September 19, 2017

Accurate modeling of static load distribution of balls is very useful for proper design and sizing of ball screw mechanisms (BSMs); it is also a starting point in modeling the dynamics, e.g., friction behavior, of BSMs. Often, it is preferable to determine load distribution using low order models, as opposed to computationally unwieldy high order finite element (FE) models. However, existing low order static load distribution models for BSMs are inaccurate because they ignore the lateral (bending) deformations of screw/nut and do not adequately consider geometric errors, both of which significantly influence load distribution. This paper presents a low order static load distribution model for BSMs that incorporates lateral deformation and geometric error effects. The ball and groove surfaces of BSMs, including geometric errors, are described mathematically and used to establish a ball-to-groove contact model based on Hertzian contact theory. Effects of axial, torsional, and lateral deformations are incorporated into the contact model by representing the nut as a rigid body and the screw as beam FEs connected by a newly derived ball stiffness matrix which considers geometric errors. Benchmarked against a high order FE model in case studies, the proposed model is shown to be accurate in predicting static load distribution, while requiring much less computational time. Its ease-of-use and versatility for evaluating effects of sundry geometric errors, e.g., pitch errors and ball diameter variation, on static load distribution are also demonstrated. It is thus suitable for parametric studies and optimal design of BSMs.

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Figures

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Fig. 1

Components of a typical BSM [1]

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Fig. 2

(a) Nominal ball center pathway (helix) and coordinate systems of screw, (b) cross-sectional profile of screw’s groove in z3x3 plane, highlighting screw left (SL) portion, and (c) transformation between three coordinate systems attached to screw

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Fig. 3

Groove profile of the nut

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Fig. 4

A ball in four-point contact with Gothic-arch grooves of BSM

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Fig. 5

Treatment of ball-groove Hertzian contact as: (a) linear spring directly linking the opposing contact points (Refs. [1113] and [21]) and (b) nonlinear spring connecting contact point and ball center with residual force (this paper)

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Fig. 6

Relationship between contact, spring stiffness and residual forces for new ball stiffness matrix formulation

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Fig. 7

Finite element partition of screw and the nodal displacements

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Fig. 8

Flowchart summarizing iterative solution for contact loads under static equilibrium using proposed low order model

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Fig. 9

Setup of BSM used for simulation-based case studies

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Fig. 10

(a) Three-dimensional mesh of the BSM and (b) mesh refinement around the contact region

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Fig. 11

Load distribution for four contact points predicted by the three models under study

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Fig. 12

Axial-force-induced lateral deformation of ball screw shaft centerline as predicted by the three models under study

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Fig. 13

Load distribution with +1 μm ball radius error in 17th ball

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Fig. 14

Load distribution with and without offset

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Fig. 16

The effect of (a) groove profile errors, sinusoidal pitch errors, normally distributed ball radius errors, each acting separately and (b) all three error types combined on load distribution

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