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Research Papers

Elastodynamic Modeling and Analysis for an Exechon Parallel Kinematic Machine

[+] Author and Article Information
Jun Zhang

School of Mechanical Engineering,
Anhui University of Technology,
Ma'anshan 243032, China;
State Key Laboratory
for Manufacturing Systems Engineering,
Xi'an Jiaotong University,
Xi'an 710049, China
e-mail: zhang_jun@tju.edu.cn

Yan Q. Zhao

School of Mechanical Engineering,
Anhui University of Technology,
Ma'anshan 243032, China
e-mail: zhaoyanqin_91@163.com

Yan Jin

School of Mechanical
and Aerospace Engineering,
Queen's University Belfast,
Belfast BT9 5HN, UK
e-mail: y.jin@qub.ac.uk

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received April 23, 2015; final manuscript received June 9, 2015; published online October 1, 2015. Assoc. Editor: Tony Schmitz.

J. Manuf. Sci. Eng 138(3), 031011 (Oct 01, 2015) Paper No: MANU-15-1195; doi: 10.1115/1.4030938 History: Received April 23, 2015; Revised June 09, 2015

As a newly invented parallel kinematic machine (PKM), Exechon has attracted intensive attention from both academic and industrial fields due to its conceptual high performance. Nevertheless, the dynamic behaviors of Exechon PKM have not been thoroughly investigated because of its structural and kinematic complexities. To identify the dynamic characteristics of Exechon PKM, an elastodynamic model is proposed with the substructure synthesis technique in this paper. The Exechon PKM is divided into a moving platform subsystem, a fixed base subsystem and three limb subsystems according to its structural features. Differential equations of motion for the limb subsystem are derived through finite element (FE) formulations by modeling the complex limb structure as a spatial beam with corresponding geometric cross sections. Meanwhile, revolute, universal, and spherical joints are simplified into virtual lumped springs associated with equivalent stiffnesses and mass at their geometric centers. Differential equations of motion for the moving platform are derived with Newton's second law after treating the platform as a rigid body due to its comparatively high rigidity. After introducing the deformation compatibility conditions between the platform and the limbs, governing differential equations of motion for Exechon PKM are derived. The solution to characteristic equations leads to natural frequencies and corresponding modal shapes of the PKM at any typical configuration. In order to predict the dynamic behaviors in a quick manner, an algorithm is proposed to numerically compute the distributions of natural frequencies throughout the workspace. Simulation results reveal that the lower natural frequencies are strongly position-dependent and distributed axial-symmetrically due to the structure symmetry of the limbs. At the last stage, a parametric analysis is carried out to identify the effects of structural, dimensional, and stiffness parameters on the system's dynamic characteristics with the purpose of providing useful information for optimal design and performance improvement of the Exechon PKM. The elastodynamic modeling methodology and dynamic analysis procedure can be well extended to other overconstrained PKMs with minor modifications.

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References

Figures

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Fig. 1

Structure of an Exechon machine

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Fig. 2

Schematic diagram of an Exechon PKM module

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Fig. 3

Assembly of a UPR or SPR limb

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Fig. 4

Simplified force diagram of an individual limb subsystem

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Fig. 5

Free body diagram of the moving platform subsystem

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Fig. 6

Displacement relationship between the moving platform and the limb body

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Fig. 7

FE-based modal analysis of Exechon PKM at the extreme position. (a) CAD model with stiffener and (b) the first-order natural frequency.

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Fig. 8

Procedure for frequency mapping of the Exechon PKM

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Fig. 9

Distributions of lower natural frequencies over work plane of pz = 1300 mm. (a) The first order, (b) the second order, (c) the third order, (d) the fourth order, (e) the fifth order, and (f) the sixth order.

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Fig. 10

Variations of natural frequencies with respect to w1 and h1. (a) The first order, (b) the second order, (c) the third order, (d) the fourth order, (e) the fifth order, and (f) the sixth order.

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Fig. 11

Variations of lower natural frequencies with respect to rp and rb. (a) The first order, (b) the second order, (c) the third order, (d) the fourth order, (e) the fifth order, and (f) the sixth order.

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Fig. 12

Variations of lower natural frequencies with respect to stiffness of the revolute joint. (a) λr1x, (b) λr1y,(c) λr1z, (d) λr1v, (e) λr1w, (f) λr2x, (g) λr2y, (h) λr2z, (i) λr2u, (j) λr2v, and (k) λr2w.

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Fig. 13

Variations of lower natural frequencies with respect to stiffness of the universal joint. (a) λux, (b) λuy,(c) λuz, and (d) λuw.

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Fig. 14

Variations of lower natural frequencies with respect to stiffness of the spherical joint. (a) λsx, (b) λsy, and (c) λsz.

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