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Research Papers

Reduction of Torque-Induced Bending Vibrations in Ball Screw-Driven Machines via Optimal Design of the Nut

[+] Author and Article Information
Chinedum E. Okwudire

Mechatronics and Sustainability Research Laboratory, Department of Mechanical Engineering,  University of Michigan, 2350 Hayward, Ann Arbor, MI 48109okwudire@umich.edu

J. Mech. Des 134(11), 111008 (Oct 03, 2012) (9 pages) doi:10.1115/1.4007400 History: Received April 27, 2012; Revised July 16, 2012; Published October 02, 2012; Online October 03, 2012

As a result of the push for sustainable machine designs, efforts are constantly being made to reduce the mass/inertia of moving machine components so as to minimize material usage and energy consumption. However, the reduction of structural stiffness that often accompanies such efforts gives rise to unwanted vibrations which must be effectively mitigated to ensure satisfactory performance of the designed machine. The ball screw mechanism (BSM) is commonly used in machines for motion and force transmission. Recent research has shown that, due to the coupling introduced by the nut, a torque applied to the shaft of a ball screw mechanism causes undesirable lateral (bending) vibrations of the screw, which adversely affect the fatigue life and positioning accuracy of ball screw-driven machines. In this paper, an analysis of the stiffness matrix connecting the screw to the nut is used to show that the entry/exit angle of the balls and the lead angle of the screw have the greatest influence on the coupling between the torsional and lateral directions. An objective function is proposed to minimize the static coupling between the applied torque and lateral deformations of the screw. The existence of local minima in the objective function is shown to be dependent on the cyclical characteristics of cross-coupling terms in the screw-nut interface stiffness matrix as a function of the entry/exit angle of the balls. Moreover, the sensitivity of the local minima to other nut/screw parameters is shown to highly depend on the lead angle. Simulations conducted on the finite element (FE) model of a single-axis ball screw-driven machine demonstrate that the optimally selected entry/exit angles result in a significant reduction of the low-frequency torque-induced vibrations of the machine compared to the unoptimized case, particularly when the lead angle is small. The proposed method is therefore suitable for reducing the torque-induced lateral vibrations of ball screws without increasing the diameter (i.e., inertia) of the screw, thus leading to more sustainable designs of ball screw-driven machines.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

(a) Spring model of balls in screw-nut interface and (b) orientation of contact normal

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Figure 2

Relationship between intermediate coordinate system (CS) and global CS

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Figure 3

(a) Balls distributed in screw-nut interface and (b) second stage of transformation based on rigid ball screw assumption

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Figure 4

Solution of kzθx  = 0 as a function of α and Φ

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Figure 5

Simple static beam model of ball screw and nut

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Figure 6

Static deflection shapes of beam

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Figure 7

(a) Solution of kxθx  = 0 as function of σ and (b) relationship among, α, β, and σ when kxθx  = 0

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Figure 8

(a) Numerical proof of monotonicity of D (b) Wx 2 /Wy 2 as a function of dimensionless distance ξ

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Figure 9

Plot of objective function J for (a) case 1: Lead = 20 mm and (b) case 2: Lead = 5 mm. (*) indicates nominal value of parameter

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Figure 10

(a) Single-axis ball screw drive (i.e., test bed) and (b) schematic representation of finite element model of test bed

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Figure 11

Mode shapes of first three coupled modes of ball screw drive model

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Figure 12

FRF between torque applied to motor and lateral vibrations of (a) point 1 and (b) point 2 for case 1: lead = 20 mm

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Figure 13

FRF between torque applied to motor and lateral vibrations of (a) point 1 and (b) point 2 for case 2: lead = 5 mm

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Figure 14

Maximum magnitude of FRF from motor torque to lateral vibrations of screw as a function of table position (X) for case 1: lead = 20 mm

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Figure 15

Maximum magnitude of FRF from motor torque to lateral vibrations of screw as a function of table position (X) for case 2: lead = 5 mm

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