Nonrigidly Foldability Analysis of Kresling Cylindrical Origami

2017 ◽  
Vol 9 (4) ◽  
Author(s):  
Cai Jianguo ◽  
Liu Yangqing ◽  
Ma Ruijun ◽  
Feng Jian ◽  
Zhou Ya
Keyword(s):  
Dual Quaternion ◽  
Loop Closure ◽  
Single Vertex ◽  
Folding Angle ◽  
Fundamental Model ◽  
Closure Equation

Rigid origami is seen as a fundamental model in many self-folding machines. A key issue in designing origami is the rigid/nonrigid foldability. The kinematic and foldability of Kresling origami, which is based on an origami pattern of the vertex with six creases, are studied in this paper. The movement of the single-vertex is first discussed. Based on the quaternion method, the loop-closure equation of the vertex with six creases is obtained. Then, the multitransformable behavior of the single vertex is investigated. Furthermore, the rigid foldability of origami patterns with multivertex is investigated with an improved dual quaternion method, which is based on studying the folding angle and the coordinates of all vertices. It can be found that the Kresling cylinder is not rigidly foldable.

Analysis of the single toggle jaw crusher kinematics

2015 ◽  
Vol 13 (2) ◽  
pp. 213-239 ◽  
Author(s):  
Moses Frank Oduori ◽  
Stephen Mwenje Mutuli ◽  
David Masinde Munyasi
Keyword(s):  
Design Methodology ◽  
Loop Closure ◽  
Content Type ◽  
Drive Mechanism ◽  
Jaw Crusher ◽  
Closure Equation

Purpose – This paper aims to obtain equations that can be used to describe the motion of any given point in the swing jaw of a single toggle jaw crusher. Design/methodology/approach – The swing jaw drive mechanism of a single toggle jaw crusher is modelled as a planar crank and rocker mechanism with the swing jaw as the coupler link. Starting with the vector loop closure equation for the mechanism, equations of the position, velocity and acceleration of any given point in the swing jaw are obtained. Findings – Application of the kinematical equations that were obtained is demonstrated using the dimensional data of a practical single toggle jaw crusher. Thus, a description of the kinematics of any given point in the swing jaw of a single toggle jaw crusher is realized. Originality/value – The model of the single toggle jaw crusher mechanism as a planar crank and rocker mechanism is a realistic one. The equations obtained in this paper should be useful in further studies on the mechanics and design of the single toggle jaw crusher.

ON THE COUPLER CURVE OF 3R2C LINKAGES

1998 ◽  
Vol 22 (3) ◽  
pp. 251-267
Author(s):  
H.S. Yan ◽  
W.H. Hsieh
Keyword(s):  
Algebraic Curve ◽  
Three Dimensions ◽  
Critical Properties ◽  
Dimensional Synthesis ◽  
Analytical Geometry ◽  
Loop Closure ◽  
Input And Output ◽  
Closure Equation

The purpose of this paper is to investigate the properties of the coupler curves generated by all 3R2C linkages. First, the 3x3 matrix with dual elements is used to derive the loop closure equation, the displacement equations are derived, and all joints variables are expressed in terms of input and output variables. Then, the parametric equations of the coupler curve are found by the D-H matrix. Finally, homogeneous coordinate is introduced to those displacement equations, and the order and some critical properties of the coupler curve are investigated based on the theories of algebraic curve and analytical geometry of three dimensions. In addition, RCRCR and RRCCR linkages are used as examples for illustration. Moreover, the results on the application of dimensional synthesis are discussed.

A Unified Algorithm for Analysis and Simulation of Planar Four-Bar Motions Defined With R- and P-Joints

2014 ◽  
Author(s):  
Xiangyun Li ◽  
Xin Ge ◽  
Anurag Purwar ◽  
Q. J. Ge
Keyword(s):  
Linkage Analysis ◽  
Least Squares ◽  
Efficient Algorithm ◽  
Image Space ◽  
Loop Closure ◽  
Closure Equation ◽  
Unified Algorithm ◽  
Four Bar Linkage ◽  
Best Fit ◽  
Linkage Synthesis

This paper presents a single, unified and efficient algorithm for animating the motions of the coupler of all four-bar mechanisms formed with revolute (R) and prismatic (P) joints. This is achieved without having to formulate and solve the loop closure equation associated with each type of four-bar linkages separately. In our previous paper on four-bar linkage synthesis, we map the planar displacements from Cartesian to image space using planar quaternion. Given a set of image points that represent planar displacements, the problem of synthesizing a planar four-bar linkage is reduced to finding a pencil of Generalized- or G-manifolds that best fit the image points in the least squares sense. The three planar dyads associated with Generalized G-manifolds are RR, PR and RP which could construct six types of four-bar mechanisms. In this paper, we show that the same unified formulation for linkage synthesis leads to a unified algorithm for linkage analysis and simulation as well. Both the unified synthesis and analysis algorithms have been implemented on Apple’s iOS platform.

Optimal Non-Uniform Parametrization for Fourier Descriptor Based Path Synthesis of Four Bar Mechanisms

2018 ◽  
Author(s):  
Shashank Sharma ◽  
Anurag Purwar ◽  
Q. Jeffrey Ge
Keyword(s):  
Input Data ◽  
Design Space ◽  
Fourier Descriptors ◽  
Fourier Descriptor ◽  
Loop Closure ◽  
The Core ◽  
Closure Equation ◽  
Harmonic Decomposition ◽  
Path Synthesis

Fourier descriptor based path synthesis algorithms rely on harmonic decomposition of four-bar loop closure equation to split the design space into smaller subsets. The core of the methodology depends on calculation and fitting of Fourier descriptors. However, a uniform time parametrization is assumed in existing literature. This paper aims to explore the use of non-uniform time parametrization of input data and calculation of an optimal parametrization. Additionally, design-centric constraints have been proposed to give user enhanced control over coupler speed. As a result, this work improves the existing algorithm tremendously.

Optimum Synthesis of Multiloop Planar Mechanisms for the Generation of Paths and Rigid-Body Positions by the Linear Partition of Design Equations

1977 ◽  
Vol 99 (1) ◽  
pp. 116-123 ◽  
Author(s):  
D. H. Bhatia ◽  
C. Bagci
Keyword(s):  
Rigid Body ◽  
Mechanism Design ◽  
Industrial Applications ◽  
Planar Mechanisms ◽  
Loop Closure ◽  
Optimum Synthesis ◽  
Linear Partition ◽  
Closure Equation

The optimum synthesis of multiloop planar mechanisms for the generation of paths and rigid-body positions employing the linear partition of the design equations is presented. Design equations and their applications for the optimum synthesis of the three distinct types of the Stephenson’s six-bar mechanism for the generation of paths and rigid-body positions are presented. The optimum sets of dimensions of a mechanism are determined by minimizing the error in the loop-closure equation of the path dyad and of the loops of the mechanism. Design equations having nine to twelve unknown dimensions are presented. Industrial applications of the design equations are illustrated with design examples.

Optimum Synthesis of Plane Mechanisms for the Generation of Paths and Rigid-Body Positions via the Linear Superposition Technique

1975 ◽  
Vol 97 (1) ◽  
pp. 340-346 ◽  
Author(s):  
C. Bagci ◽  
In-Ping Jack Lee
Keyword(s):  
Rigid Body ◽  
System Design ◽  
Linear Superposition ◽  
Loop Closure ◽  
Numerical Examples ◽  
Optimum Synthesis ◽  
Closure Equation ◽  
Superposition Technique

A method of optimum synthesis of plane mechanisms for the generation of paths and rigid-body positions is presented. The method is developed for the four-bar plane mechanism with six and eight unknown dimensions. Dimensions of the optimum mechanism are determined by minimizing the error in the loop-closure equations for N design points on the path, along with the loop-closure equation of the linkage, where N is not limited by the number of the unknown dimensions of the system. Design equations are linearized by the method of linear superposition. Solution of design equations requires no iterations, and it leads to a series of optimum mechanisms of different efficiency of approximation. Numerical examples are given.

Modelling rigid origami with quaternions and dual quaternions

2010 ◽  
Vol 466 (2119) ◽  
pp. 2155-2174 ◽  
Author(s):  
Weina Wu ◽  
Zhong You
Keyword(s):  
Numerical Methods ◽  
Mathematical Modelling ◽  
Three Dimensional ◽  
Vector Model ◽  
Dual Quaternion ◽  
Linkage Mechanism ◽  
Single Vertex ◽  
Closure Conditions ◽  
Flat Pattern ◽  
Rule Out

This paper examines the mathematical modelling of rigid origami, a type of origami where all the panels are rigid and can only rotate about crease lines. The rotating vector model is proposed, which establishes the loop-closure conditions among a group of characteristic vectors. By building up an explicit relationship between the single-vertex origami and the spherical linkage mechanism, the rotating vector model can conveniently and directly describe arbitrary three-dimensional configurations and can detect some self-intersection. Quaternion and dual quaternion are then employed to represent the origami model, based on which two numerical methods have been developed. Through examples, it has been shown that the first method can effectively track the entire rigid-folding procedure of an initially flat or a non-flat pattern with a single vertex or multiple vertices, and thereby provide judgment for its rigid foldability and flat foldability. Furthermore, its ability to rule out some self-intersecting configurations during folding is illustrated in detail, leading to its ability of checking rigid foldability in a more or less sufficient way. The second method is especially for analysing the multi-vertex origami. It can also effectively track the trajectories of multiple vertices during folding.

Concurrent Type and Dimensional Synthesis of Planar Mechanisms Using Graph-Theoretic Enumeration, Automatic Loop Closure Equation Generation, and Descent-Based Optimization

2011 ◽  
Author(s):  
Carl A. Nelson
Keyword(s):  
A Priori ◽  
Goal Function ◽  
Dimensional Synthesis ◽  
Planar Mechanisms ◽  
Loop Closure ◽  
Graph Theoretic ◽  
Optimal Mechanism ◽  
Spatial Mechanisms ◽  
Closure Equation

An approach to concurrent type and dimensional synthesis of planar mechanisms is presented. Using graph-theoretic enumeration of mechanisms and determination of loops, automatic loop closure equations are generated. Using constrained optimization routines based on descent methods, and given an appropriate goal function, optimal mechanism designs can be determined. These are not limited to traditional problems such as rigid-body guidance and path generation, but can be more flexibly expressed according to designer needs. This method can be effective at finding mechanism solutions when topology has not been determined a priori and may also be extensible to synthesis of spatial mechanisms. This paper presents the data flow and algorithm outline, simulation results, and examples of nonstandard synthesis problems solvable with this method.

On the Coupler Curve of RCPCR Linkages

1998 ◽  
Author(s):  
Hong-Sen Yan ◽  
Wen-Hsiang Hsieh
Keyword(s):  
Algebraic Curve ◽  
Three Dimensions ◽  
Critical Properties ◽  
Dimensional Synthesis ◽  
Analytical Geometry ◽  
Loop Closure ◽  
Input And Output ◽  
Closure Equation

Abstract The purpose of this paper is to investigate the properties of the coupler curves generated by all RCPCR linkages. First, the 3 × 3 matrix with dual elements is used to establish the loop closure equation, the displacement equations are derived, and all joints variables are expressed in terms of input and output variables. Then, the parametric equations of the coupler curve are found by the D-H matrix. Finally, homogeneous coordinate is introduced to those displacement equations, and the order and some critical properties of the coupler curve are investigated based on the theories of algebraic curve and analytical geometry of three dimensions. In addition, RCPCR and RRCPC linkages are used as examples for illustrations. Moreover, the results on the application of dimensional synthesis are discussed.

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