New Complex-Number Forms of the Euler-Savary Equation in a Computer-Oriented Treatment of Planar Path-Curvature Theory for Higher-Pair Rolling Contact

1982 ◽  
Vol 104 (1) ◽  
pp. 227-232 ◽  
Author(s):  
G. N. Sandor ◽  
A. G. Erdman ◽  
L. Hunt ◽  
E. Raghavacharyulu
Keyword(s):  
Complex Number ◽  
Rolling Contact ◽  
Radius Of Curvature ◽  
Planar Mechanisms ◽  
Kinematic Synthesis ◽  
Synthesis Of Mechanisms ◽  
Curvature Theory ◽  
Path Curvature ◽  
Planar Linkages ◽  
Contact Mechanism

It is well known from the theory of Kinematic Synthesis of planar mechanisms that the Euler-Savary Equation (ESE) gives the radius of curvature and the center of curvature of the path traced by a point in a planar rolling-contact mechanism. It can also be applied in planar linkages for which equivalent roll-curve mechanisms can be found. Typical example: the curvature of the coupler curve of a four-bar mechanism. Early works in the synthesis of mechanisms concerned themselves with deriving the ESE by means of combined graphical and algebraic techniques, using certain sign conventions. These sign conventions often become sources of error. In this paper new complex-number forms of the Euler-Savary Equation are derived and are presented in a computer-oriented format. The results are useful in the application of path-curvature theory to higher-pair rolling contact mechanisms, such as cams, gears, etc., as well as linkages, once the key parameters of an equivalent rolling-contact mechanism are known. The complex-number technique has the advantage of eliminating the need for the traditional sign conventions and is suitable for digital computation. An example is presented to illustrate this.

New Complex-Number Form of the Cubic of Stationary Curvature in a Computer-Oriented Treatment of Planar Path-Curvature Theory for Higher-Pair Rolling Contact

1982 ◽  
Vol 104 (1) ◽  
pp. 233-238 ◽  
Author(s):  
G. N. Sandor ◽  
A. G. Erdman ◽  
L. Hunt ◽  
E. Raghavacharyulu
Keyword(s):  
Complex Number ◽  
Rate Of Change ◽  
Rolling Contact ◽  
Planar Mechanisms ◽  
Digital Computation ◽  
Complex Arithmetic ◽  
Moving Plane ◽  
Curvature Theory ◽  
Path Curvature ◽  
Number Form

New complex number forms of the Euler-Savary Equation (ESE) for higher-pair rolling contact planar mechanisms were derived in a former paper by the authors. The present work, based on the former, deals with the derivation of the cubic of stationary curvature (CSC) in complex-vector form, suitable for digital computation. The CSC or Burmester’s circlepoint curve and its conjugate, the centerpoint curve for four infinitesimally close positions of the moving plane requires taking into account not only the curvature but also the rate of change of curvature of the rolling centrodes in the immediate vicinity of the position considered. The analytical procedure based on the theory developed in the present paper, when programmed for digital computation using complex arithmetic, takes care of the algebraic signs automatically, without the need for observing traditional sign conventions. The analysis is applicable to both higher-pair and lower-pair planar mechanisms. An example using the complex-number approach illustrates this.

Kinematic Synthesis of Adjustable Mechanisms—Part 1: Function Generation

1973 ◽  
Vol 95 (2) ◽  
pp. 417-422 ◽  
Author(s):  
Joseph F. McGovern ◽  
George N. Sandor
Keyword(s):  
Complex Number ◽  
Relative Orientation ◽  
Complex Numbers ◽  
Input Output ◽  
Kinematic Synthesis ◽  
Digital Computation ◽  
Closed Form Solutions ◽  
Function Generation ◽  
Planar Linkages ◽  
Adjustable Mechanisms

One of the major advantages of linkages over cams is the ease with which they can be adjusted to modify their output. Incorporating an adjustment into the design of a linkage makes it possible to make a selection from two, three, or more different outputs by simply making the adjustment. Adjustable mechanisms make it possible to use the same hardware for different input-output relationships. The adjustment considered in this paper is the changing of a fixed pivot location. This paper presents a method of synthesizing adjustable planar linkages for function generation with finitely separated precision points and higher order synthesis involving prescribed velocities, accelerations, and higher accelerations. The method of synthesis is analytical with closed form solutions and utilizes complex numbers. The method is programmed for automatic digital computation. The linkages considered are a four-bar, a geared five-bar, and a geared six-bar mechanism. Examples include adjustable mechanisms which have been successfully synthesized with the method developed here. Future extensions of the complex number method to include adjustment by changing the length of a link and by changing of the relative orientation of the gears in geared linkages are outlined.

Novel Methodology for Inflection Circle based synthesis of Straight Line Crank Rocker Mechanism

2021 ◽  
pp. 1-13
Author(s):  
Prashant Shiwalkar ◽  
S. D. Moghe ◽  
J. P. Modak
Keyword(s):  
Compliant Mechanisms ◽  
Relative Length ◽  
Dimensional Synthesis ◽  
Straight Line ◽  
Synthesis Of Mechanisms ◽  
Acceptable Deviation ◽  
Curvature Theory ◽  
Path Curvature ◽  
Artificial Neural Network Ann ◽  
Line Path

Abstract Emerging fields like Compact Compliant Mechanisms have created newer/novel situations for application of straight line mechanisms. Many of these situations in Automation and Robotics are multidisciplinary in nature. Application Engineers from these domains are many times uninitiated in involved procedures of synthesis of mechanisms and related concepts of Path Curvature Theory. This paper proposes a predominantly graphical approach using properties of Inflection Circle to synthesize a crank rocker mechanism for tracing a coupler curve which includes the targeted straight line path. The generated approximate straight line path has acceptable deviation in length, orientation and extent of approximate nature well within the permissible ranges. Generation of multiple choices for the link geometry is unique to this method. To ease the selection, a trained Artificial Neural Network (ANN) is developed to indicate relative length of various options generated. Using studied unique properties of Inflection Circles a methodology for anticipating the orientation of the straight path vis-à-vis the targeted path is also included. Two straight line paths are targeted for two different crank rockers. Compared to the existing practice of selecting the mechanism with some compromise due to inherent granularity of the data in Atlases, proposed methodology helps in indicating the possibility of completing the dimensional synthesis. The case in which the solution is possible, the developed solution is well within the design specifications and is without a compromise.

Singular Solutions in Burmester Theory

1973 ◽  
Vol 95 (2) ◽  
pp. 572-576 ◽  
Author(s):  
R. E. Kaufman
Keyword(s):  
Complex Number ◽  
Singular Solutions ◽  
Design Technique ◽  
Planar Mechanisms ◽  
Point Position ◽  
Proper Interpretation ◽  
General Synthesis ◽  
Number Development

A unified complex number development of four position planar finite position theory is presented. This formulation shows that Burmester circlepoint-centerpoint theory specializes to include slider points, concurrency points, poles, and point position reduction by proper interpretation of the trivial roots of the general synthesis equations. Thus a single design technique can be used for the multiposition synthesis of most pin or slider-jointed planar mechanisms. Four position function, path, or motion generating linkages can all be designed in this manner.

A Method for Type Synthesis of Mechanisms Using Phase Diagrams

1994 ◽  
Author(s):  
Sio-Hou Lei ◽  
Ying-Chien Tsai
Keyword(s):  
Phase Diagram ◽  
Degrees Of Freedom ◽  
Practical Application ◽  
Type Synthesis ◽  
Planar Mechanisms ◽  
Simple Loop ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Spatial Mechanisms ◽  
Synthesis Of Mechanisms

Abstract A method for synthesizing the types of spatial as well as planar mechanisms is expressed in this paper by using the concept of phase diagram in metallurgy. The concept represented as a type synthesis technique is applied to (a) planar mechanisms with n degrees of freedom and simple loop, (b) spatial mechanisms with single degree of freedom and simple loop, to enumerate all the possible mechanisms with physically realizable kinematic pairs. Based on the technique described, a set of new reciprocating mechanisms is generated as a practical application.

Coriolis acceleration analysis of planar mechanisms by complex-number algebra

1982 ◽  
Vol 17 (6) ◽  
pp. 405-414 ◽  
Author(s):  
George N Sandor ◽  
E Raghavacharyulu ◽  
Arthur G Erdman
Keyword(s):  
Complex Number ◽  
Planar Mechanisms ◽  
Coriolis Acceleration ◽  
Acceleration Analysis

A frame-independent comparison metric for discrete motion sequences

2020 ◽  
Vol 234 (9) ◽  
pp. 1764-1774
Author(s):  
Ping Zhao ◽  
Yong Wang ◽  
Lihong Zhu ◽  
Xiangyun Li
Keyword(s):  
Moving Frame ◽  
Kinematic Synthesis ◽  
Reference Motion ◽  
Comparison Results ◽  
Synthesis Of Mechanisms ◽  
Rotational Part ◽  
The Difference ◽  
Four Bar Linkage ◽  
The Given ◽  
Discrete Motion

To evaluate the kinematic performance of designed mechanisms, a statistical-variance-based metric is proposed in this article to measure the “distance” between two discrete motion sequences: the reference motion and the given task motion. It seeks to establish a metric that is independent of the choice of the fixed frame or moving frame. Quaternions are adopted to represent the rotational part of a spatial pose, and the variance of the set of relative displacements is computed to reflect the difference between two sequences. With this variance-based metric formulation, we show that the comparison results of two spatial discrete motions are not affected by the choice of frames. Both theoretical demonstration and computational example are presented to support this conclusion. In addition, since the deviation error between the task motion and the synthesized motion measured with this metric is independent of the location of frames, those corresponding parameters could be excluded from the optimization algorithm formulated with our frame-independent metric in kinematic synthesis of mechanisms, and the complexity of the algorithm are hereby reduced. An application of a four-bar linkage synthesis problem is presented to illustrate the advantage of the proposed metric.

Finding All Solutions to Unconstrained Nonlinear Optimization for Approximate Synthesis of Planar Linkages Using Continuation Method

1999 ◽  
Vol 121 (3) ◽  
pp. 368-374 ◽  
Author(s):  
A.-X. Liu ◽  
T.-L. Yang
Keyword(s):  
Continuation Method ◽  
Optimization Method ◽  
Global Optimum ◽  
Kinematic Synthesis ◽  
All Solutions ◽  
Approximate Synthesis ◽  
Unconstrained Nonlinear Optimization ◽  
Optimum Solution ◽  
Planar Linkages ◽  
Four Bar Linkage

Generally, approximate kinematic synthesis of planar linkage is studied using optimization method. But this method has two defects: i) the suitable initial guesses are hard to determine and ii) the global optimum solution is difficult to find. In this paper, a new method which can find all solutions to approximate kinematic synthesis of planar linkage is proposed. Firstly, we reduce the approximate synthesis problem to finding all solutions to polynomial equations. Polynomial continuation method is then used to find all solutions. Finally, all possible linkages can be obtained. Approximate syntheses of planar four-bar linkage for function generation, rigid-body guidance and path generation are studied in detail and three examples are given to illustrate the advantages of the proposed method.

Optimum Synthesis of Rigid Mechanisms Using a Dynamic Ant-Search Method With Sensitivity Analysis

2019 ◽  
Author(s):  
Nadim Diab ◽  
Omar Itani ◽  
Ahmad Smaili
Keyword(s):  
Sensitivity Analysis ◽  
Optimization Technique ◽  
Design Parameters ◽  
Neighborhood Search ◽  
Intensity Level ◽  
Path Generation ◽  
Exploration And Exploitation ◽  
Planar Mechanisms ◽  
Optimum Synthesis ◽  
Synthesis Of Mechanisms

Abstract Four-bar linkages are commonly used mechanisms in various mechanical systems and components. Several techniques for optimum synthesis of planar mechanisms have been suggested in literature such as the Genetic, Tabu, Simulated Annealing, Swarm-Based and many other algorithms. This paper covers optimization of four-bar mechanisms with path generation tasks using a Dynamic Ant Search (DAS) algorithm. Unlike the Modified Ant Search (MAS) technique where ants unanimously moved between the exploration and exploitation phases, in the proposed algorithm, each ant is free to travel between the two aforementioned phases independent of other ants and as governed by its own pheromone intensity level. Moreover, sensitivity analysis is conducted on the design parameters to determine their corresponding neighborhood search boundaries and thus improve the search while in the exploitation mode. These implemented changes demonstrated a remarkable impact on the optimum synthesis of mechanisms for path generation tasks. A briefing of the MAS based algorithm is first presented after which the proposed modified optimization technique and its implementation on four-bar mechanisms are furnished. Finally, three case studies are conducted to evaluate the efficiency and robustness of the proposed methodology where the performances of the obtained optimum designs are benchmarked with those previously reported in literature.

A New Approach for Locating the Instantaneous Centers of Zero Velocity for 1-DOF Planar Linkages

2018 ◽  
Author(s):  
Nadim Diab
Keyword(s):  
Degree Of Freedom ◽  
Planar Mechanisms ◽  
New Approach ◽  
Zero Velocity ◽  
Graphical Technique ◽  
Instantaneous Center ◽  
Planar Linkages ◽  
Consistent Approach

This paper presents a new graphical technique to locate the secondary instantaneous centers of zero velocity (ICs) for one-degree-of-freedom (1-DOF) kinematically indeterminate planar mechanisms. The proposed approach is based on transforming the 1-DOF mechanism into a 2-DOF counterpart by converting any ground-pivoted ternary link into two ground-pivoted binary links. Fixing each of these two new binary links, one at a time, results in two different 1-DOF mechanisms where the intersection of the loci of their instantaneous centers will determine the location of the desired instantaneous center for the original 1-DOF mechanism. This single and consistent approach proved to be successful in locating the ICs of various mechanisms reported in the literature that required different techniques to reach the same results obtained herein.

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