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Research Papers

Extended Static Modeling and Analysis of Compliant Compound Parallelogram Mechanisms Considering the Initial Internal Axial Force*

[+] Author and Article Information
Guangbo Hao

School of Engineering-Electrical and
Electronic Engineering,
University College Cork,
Cork, Ireland
e-mail: G.HAO@UCC.IE

Haiyang Li

School of Engineering-Electrical and
Electronic Engineering,
University College Cork,
Cork, Ireland

1Corresponding author.

Manuscript received September 10, 2015; final manuscript received January 15, 2016; published online March 7, 2016. Assoc. Editor: Xianmin Zhang.

J. Mechanisms Robotics 8(4), 041008 (Mar 07, 2016) (11 pages) Paper No: JMR-15-1247; doi: 10.1115/1.4032592 History: Received September 10, 2015; Revised January 15, 2016

Extended nonlinear analytical modeling and analysis of compound parallelogram mechanisms are conducted in this paper to consider the effect of the initial internal axial force. The nonlinear analytical model of a compound basic parallelogram mechanism (CBPM) is first derived incorporating the initial internal axial force. The stiffness equations of compound multibeam parallelogram mechanisms (CMPMs) are then followed. The analytical maximal stress under the primary actuation force only is also derived to determine the maximal primary motion (motion range). The influence of initial internal axial forces on the primary motion/stiffness is further quantitatively analyzed by considering different slenderness ratios, which can be employed to consider active displacement preloading control and/or thermal effects. The criterion that the primary stiffness may be considered “constant” is defined and the initial internal axial force driven by a temperature change is also formulated. A physical preloading system to control the initial internal axial force is presented and testing results of the object CBPM are compared with theoretical ones.

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Figures

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Fig. 1

A CBPM with preloaded displacement: (a) before motion stage moves (no buckling) and (b) after motion stage moves (no buckling)

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Fig. 2

A CMPM with preloaded displacement

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Fig. 3

Effect of the initial axial force on the primary motion of the CBPM: (a) case I and (b) case II

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Fig. 4

Effect of the initial internal axial force on the primary motion stiffness of the CBPM: (a) case I and (b) case II

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Fig. 5

Displacement range for an acceptable 10% deviation of primary stiffness

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Fig. 6

Temperature effect on the initial internal axial force magnitude

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Fig. 12

Other representative mirror-symmetrical compliant joints that can be preloaded: (a) compound rotational mechanism and (b) compound double parallelogram mechanism

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Fig. 10

Testing resutls with comparison to analytical models: (a) overall comparison between the analytical and testing results for seven preloading cases, (b) testing results only for seven prelading cases, (c) compasison of analytical and testing results for pin = 0, (d) compasison of analytical and testing results for pin = 5, (e) compasison of analytical and testing results for pin = −5, (f)compasison of analytical and testing results for pin = 10, (g) compasison of analytical and testing results for pin = −10, (h)compasison of analytical and testing results for pin = 15, and (i)compasison of analytical and testing results for pin = −15

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Fig. 9

Testing rig for the primary motion of the object CBPM

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Fig. 8

Motion relationship between the output stage and the input stage of the reduction mechanism along the X-axis

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Fig. 7

Description of preloading system: (a) a piece of compliant mechanism including a displacment reduction mechanism and the object CBPM and (b) assembled physical preloading system with displacement acutation

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Fig. 11

Maximal stress versus normalized primary motion for E = 2.3 GPa and σs = 60 MPa: (a) analytical stress results and (b) comparison between FEA and analytical results

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