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Research Papers

Instant-Center Based Force Transmissivity and Singularity Analysis of Planar Linkages

[+] Author and Article Information
P. A. Simionescu1

Department of Mechanical Engineering, Texas A&M University, Corpus Christi, 6300 Ocean Drive Unit 5733, Corpus Christi, TX 78412pa.simionescu@tamucc.edu

I. Talpasanu

Department of Electronics and Mechanical, Wentworth Institute of Technology, 550 Huntington Avenue, Boston, MA 02115talpasanui@wit.edu

R. Di Gregorio

Department of Engineering, University of Ferrara, Via Saragat 1, 44100 Ferrara, Italyrdigregorio@ing.unife.it

In planar linkages, the interconnection between two adjacent links may only be either a revolute pair or a prismatic pair. Since such pairs are single-DOF pairs, the IC of the relative motion of the two interconnected links is at a known position.

1

Corresponding author.

J. Mechanisms Robotics 2(2), 021011 (May 03, 2010) (12 pages) doi:10.1115/1.4001094 History: Received January 30, 2009; Revised December 17, 2009; Published May 03, 2010; Online May 03, 2010

The instant centers of velocity (ICs) of most planar mechanisms can be determined as the intersection of the lines of centers, also known as Aronhold–Kennedy lines, along which the ICs of three distinct links in relative motion are located. It is shown how these intersections can be kept track of in matrix form, very suitable to algorithmic implementation on a computer. Solving for the coordinates of the actual instant centers can be also cast in matrix form. Moreover, the singularity and force transmissivity of the mechanism are reflected in the condition numbers of these matrices and the degree of dispersion of the secondary instant centers i.e., the instant centers that cannot be found by inspection.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

An offset quick-return mechanism with its links and primary instant centers labeled

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Figure 2

The mechanism in Fig. 1 with all ICs connected two-by-two with lines, of which some are not AK lines

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Figure 3

The mechanism in Fig. 1 with all its ICs and all AK lines identified and labeled

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Figure 4

Summary of an AK line based IC location of an offset quick-return mechanism

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Figure 5

Procedure for roll center identification of an automobile equipped with a double A-arm suspension (a) for a bounce-rebound position and (b) for a cornering position

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Figure 6

Procedure for roll center identification of an automobile equipped with a McPherson suspension (a) for a bounce-rebound position and (b) for a cornering position

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Figure 7

A Watt 2 linkage with a sliding joint and the following geometry: xP12=−125, yP12=−30, xP14=85, yP14=−30, xP16=−30, yP16=−115, l2=80, l3=280, l4=190, and l6=200; φ2 is the input-link angle, φ6 is the output-link angle, while γ1 and γ2 are the transmission angles of the RRRR and RTRR four-link loops, respectively

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Figure 8

The equations in Iteration 1 transposed graphically for the Watt 2 linkage in Fig. 7, showing newly determined ICs P13, P15, P24, and P46

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Figure 9

The equations in Iteration 2 transposed graphically for the Watt 2 linkage in Fig. 7, showing newly determined ICs P25, P26, P35, and P36

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Figure 10

Plot of the space centrodes of the mechanism in Fig. 7. The loci of ICs P23, P34, P45, and P56 are not shown since they are a full circle (P23), arches of circle (P34 and P56), or located at infinity (P45). Note that the loci of P13, P35, and P36 appear to have double branches; these are actually continuous curves that close at ±∞.

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Figure 11

Semilogarithmic plot of the sum of the condition numbers of matrices M1 and M2

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Figure 12

Semilogarithmic plot of secondary instant center dispersion, i.e., ICdispersion=max[xP13,yP13,xP15,yP15,xP24,yP24,xP46,yP46,xP25,yP25,xP26,yP26,xP35,yP35,xP36,yP36]

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Figure 13

Semilogarithmic plot of the cumulated errors in solving Eqs. 5,5 i.e., Error=max[M1⋅X1-B1,M2⋅X2-B2]

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Figure 14

Plots of transmission angles γ1 and γ2 (a), kinematic coefficient dφ6∕dφ2 (b), and of the discriminants Δ1 and Δ2 of the quadratic equations occurring in solving the position problem of the RRRR loop and of the RTRR loop of the mechanism (c)

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Figure 15

Plots of the tijk(i,j,k=1⋯6) in Appendix . It is to be observed that the graph of t126 in (d) is the same as the graph of dφ6∕dφ2 in Fig. 1.

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