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Research Papers

# Algebraic Algorithm for the Kinematic Analysis of Slider-Crank/Rocker Mechanisms

[+] Author and Article Information
Giorgio Figliolini

DiMSAT, University of Cassino, Via G. Di Biasio 43, 03043 Cassino (Fr), Italyfigliolini@unicas.it

Marco Conte, Pierluigi Rea

DiMSAT, University of Cassino, Via G. Di Biasio 43, 03043 Cassino (Fr), Italy

J. Mechanisms Robotics 4(1), 011003 (Feb 03, 2012) (12 pages) doi:10.1115/1.4005527 History: Received June 24, 2010; Accepted November 03, 2011; Published February 03, 2012; Online February 03, 2012

## Abstract

This paper deals with the formulation and validation of a comprehensive algebraic algorithm for the kinematic analysis of slider-crank/rocker mechanisms, which is based on the use of geometric loci, as the fixed and moving centrodes, along with their evolutes, the cubic of stationary curvature and the inflection circle. In particular, both centrodes are formulated in implicit and explicit algebraic forms by using the complex algebra. Moreover, the algebraic curves representing the moving centrodes are recognized and proven to be Jeřábek’s curves for the first time. Then, the cubic of stationary curvature along with the inflection circle are expressed in algebraic form by using the instantaneous geometric invariants. Finally, the proposed algorithm has been implemented in a MATLAB code and significant numerical and graphical results are shown, along with the particular cases in which these geometric loci degenerate in lines and circles or give cycloidal positions.

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## Figures

Figure 1

Diagram to define the types A, B and C of the slider-crank/rocker mechanism (No closure condition of the mechanism is shown with the gray area)

Figure 2

Singular configurations of the mechanism related to the boundary lines of Eq. 2 and shown in the diagram of Fig. 1: (a) between types A and B; (b) between types B and C; (c) between B and the area of no closure

Figure 3

Kinematic sketch for the position and centrodes analyses: fixed F (O , X , Y ) and moving f(Ω, x , y ) frames

Figure 4

Analysis in the Gauss plane to determine the fixed centrode of offset slider-crank/rocker mechanisms

Figure 5

Analysis in the Gauss plane to determine the moving centrode of offset slider-crank/rocker mechanisms

Figure 6

Fixed and moving centrodes for an offset slider-crank mechanism with Φ  = 2 and Ψ  = 0.5: r = 200 mm, l =400 mm and e = 100 mm

Figure 7

Fixed and moving centrodes for an offset slider-rocker mechanism with Φ  = 0.91 and Ψ  = 0.45: r = 330 mm, l = 300 mm and e =150 mm

Figure 8

Fixed and moving centrodes for an offset slider-rocker mechanism with Φ = 0.54 and Ψ = 0.19: r = 650 mm, l = 350 mm and e = 125 mm

Figure 9

Graphical construction of a Jeřábek’s curve, as shown in page 237 of Ref. [21]

Figure 10

Graphical construction of the moving centrodes as Jeřábek’s curves for Ψ = 0 with Φ ≥ 1 and Φ ≤ 1

Figure 11

Kinematic sketch to determine the cubic of stationary curvature and the inflection circle

Figure 12

Instantaneous geometric invariants b2 , b3 and a3 for a centered slider-crank mechanism with r = 100 mm, l = 300 mm and e = 0, versus the driving crank angle δ

Figure 13

Instantaneous geometric invariants b2 , b3 and a3 for an offset slider-crank mechanism with r = 100 mm, l = 300 mm and e = 100 mm, versus the driving crank angle δ

Figure 14

Fixed and moving centrodes along with the cubic of stationary of curvature and the inflection circle (Φ = 2 and Ψ = 0) for r = 200 mm, l = 400 mm and e = 0: (a) δ = 0 deg; (b) δ = 30 deg; (c) δ = 90 deg; (d) δ = 180 deg

Figure 15

Fixed and moving centrodes along with the cubic of stationary of curvature and the inflection circle (Φ = 0.5 and Ψ = 0) for r = 400 mm, l = 200 mm, e=0 and δ = 30 deg

Figure 16

Fixed and moving centrodes along with the cubic of stationary of curvature and the inflection circle (Φ = 1 and Ψ = 0) for r = l = 400 mm, e = 0 and δ = 30 deg

Figure 17

Fixed and moving centrodes along with the cubic of stationary of curvature and the inflection circle (Φ = 0.67 and Ψ = 0.33) for r = 450 mm, l = 300 mm, e = 150 mm and δ = 35 deg

Figure 18

Fixed and moving centrodes along with the cubic of stationary of curvature and the inflection circle (Φ = 2 and Ψ = 0.5) for r = 200 mm, l = 400 mm, e = 100 mm and δ = 90 deg

Figure 19

Paths of significant coupler points (Φ = 1.55 and Ψ = 0.5) for r = 200 mm, l = 310 mm, e = 100 mm and δ = 30 deg

Figure 20

Fixed and moving centrodes along with their evolutes and osculating circles, the cubic of stationary of curvature and the inflection circle (Φ = 2 and Ψ = 0) for r = 200 mm, l = 400 mm and e = 0: (a) δ = 30 deg; (b) δ = 0 deg

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