Position analysis in polynomial form of planar mechanisms with Assur groups of class 3 including revolute and prismatic joints

This paper describes a method for unified parametric kinematic analysis of those planar mechanisms whose geometry can be defined with a set of independent vectorial loops, i.e. solvable independently; this covers a wide range of planar mechanisms. The method is developed by employing the well-known vectorial illustration, and vectorial-loop equations solved with the aid of complex polar algebra leading to a total of only nine unified/generic one-unknown parametric equations consisting of five equations for position analysis and two equations for velocity and acceleration analysis each. Then, the kinematics of joints and mass centers are manifested as resultants of a few known vectors. This method is needless of relative-velocities, relative-accelerations, instantaneous centers of rotation and Kennedy’s Theorem dominantly used in the literature, especially textbooks. The efficiency of the method is demonstrated by its application to a complex mechanism through only eight unified equations, and simultaneously compared to the solution using the textbook common (Raven’s) method which required the derivation of 67 extra equations to get the same results. This reveals the fact that the method is not only a powerful tool for mechanical designers but a most powerful and efficient method for teaching and learning the kinematics of planar mechanisms.

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Dynamics of Closed-Loop Planar Mechanisms with Coulomb Friction in Prismatic Joints

Advances in Asian Mechanism and Machine Science - Mechanisms and Machine Science ◽

10.1007/978-3-030-91892-7_50 ◽

2021 ◽

pp. 531-540

Author(s):

Minh-Tuan Nguyen-Thai

Keyword(s):

Coulomb Friction ◽

Closed Loop ◽

Planar Mechanisms ◽

Prismatic Joints

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Position analysis of Assur kinematic chains using coupler curves

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1243/09544062jmes249 ◽

2006 ◽

Vol 220 (8) ◽

pp. 1249-1259

Author(s):

W-Y Chung

Keyword(s):

Number Of Solutions ◽

Kinematic Chains ◽

Position Analysis ◽

Prismatic Joints ◽

Intersection Points

The configurations of several Assur kinematic chains (AKCs) are analysed with the aid of coupler curves. An analysed linkage is dismantled into two constituent linkages. The intersection points of two coupler curves generated by the constituent linkages are the solutions of the dismantling point. All possible configurations of the linkage can then be obtained with the dismantling point being found by solving two coupler curve equations. The coupler curve equations of Watt-I and Stephenson-I six-bar are derived with orders and circularities being emphasized. All three AKCs with seven-link and several AKCs with nine-link and even with 11-link are analysed. The maximum number of solutions can also be determined easily on the basis of the orders and circularities of the coupler curves. The chains with prismatic joints included are also considered.

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Kinematic Synthesis of Planar, Shape-Changing Rigid Body Mechanisms for Design Profiles With Significant Differences in Arc Length

Volume 6: 35th Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2011-47657 ◽

2011 ◽

Author(s):

Shamsul A. Shamsudin ◽

Andrew P. Murray ◽

David H. Myszka ◽

James P. Schmiedeler

Keyword(s):

Rigid Body ◽

Shape Change ◽

Rigid Bodies ◽

Arc Length ◽

Shape Variation ◽

Planar Mechanisms ◽

Revolute Joints ◽

Prismatic Joints ◽

Plane Shape ◽

A Chain

This paper presents a kinematic procedure to synthesize planar mechanisms capable of approximating a shape change defined by a general set of curves. These “morphing curves”, referred to as design profiles, differ from each other by a combination of displacement in the plane, shape variation, and notable differences in arc length. Where previous rigid-body shape-change work focused on mechanisms composed of rigid links and revolute joints to approximate curves of roughly equal arc length, this work introduces prismatic joints into the mechanisms in order to produce the different desired arc lengths. A method is presented to inspect and compare the profiles so that the regions are best suited for prismatic joints can be identified. The result of this methodology is the creation of a chain of rigid bodies connected by revolute and prismatic joints that can approximate a set of design profiles.

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Mobility Determination of Mechanisms Based on Rigidity Theory

Volume 4: 36th Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2012-71289 ◽

2012 ◽

Cited By ~ 1

Author(s):

Michael Slavutin ◽

Offer Shai ◽

Andreas Müller

Keyword(s):

The Body ◽

Rigidity Theory ◽

Rigidity Matrix ◽

Planar Mechanisms ◽

Prismatic Joints ◽

Unified View ◽

Joint Rigidity ◽

Pebble Game ◽

First Time

Rigidity theory deals mostly with the topological computation in mechanical systems, i.e. it aims at making generic statements. Mechanism theory is mainly concerned with the geometrical analysis but again also with generic statements. Even more so for mobility analysis where one is interested in both the generic mobility and that of a particular mechanism. In rigidity theory the mathematical foundation is the topology representation using bar-joint and body-bar graphs, and the corresponding rigidity matrix. In this paper novel geometric rules for constructing the body-bar rigidity matrix are derived for general planar mechanisms comprising revolute and prismatic joints. This allows, for the first time, the treatment of general planar mechanisms with the body-bar approach. The rigidity matrix is also derived for spatial mechanisms with spherical joints. The bar-joint rigidity matrix is shown to be a special case of body-bar representation. It is shown that the rigidity matrices allow for mobility calculation as shown in the paper. This paper is aimed at supplying a unified view and as a result to enable the mechanisms community to employ the theorems and methods used in rigidity theory. An algorithm for mobility determination — the pebble game — is discussed. This algorithm always finds the correct generic mobility if the mechanism can be represented by a body-bar graph.

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