Complete Solution to the Eight-Point Path Generation of Slider-Crank Four-Bar Linkages

2010 ◽  
Vol 132 (8) ◽  
Author(s):  
Hafez Tari ◽  
Hai-Jun Su
Keyword(s):  
Complete Solution ◽  
Solution Process ◽  
Polynomial Equations ◽  
Homotopy Methods ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Augmented System ◽  
Polynomial Curve ◽  
Four Bar Linkage ◽  
Classical Homotopy

We study the synthesis of a slider-crank four-bar linkage whose coupler point traces a set of predefined task points. We report that there are at most 558 slider-crank four-bars in cognate pairs passing through any eight specified task points. The problem is formulated for up to eight precision points in polynomial equations. Classical elimination methods are used to reduce the formulation to a system of seven sixth-degree polynomials. A constrained homotopy technique is employed to eliminate degenerate solutions, mapping them to solutions at infinity of the augmented system, which avoids tedious post-processing. To obtain solutions to the augmented system, we propose a process based on the classical homotopy and secant homotopy methods. Two numerical examples are provided to verify the formulation and solution process. In the second example, we obtain six slider-crank linkages without a branch or an order defect, a result partially attributed to choosing design points on a fourth-degree polynomial curve.

Use of Resultant in the Dimensional Synthesis of Linkage Components: Motion Generation With Prescribed Timing

1992 ◽  
Author(s):  
Chuen-Sen Lin ◽  
Bao-Ping Jia
Keyword(s):  
Angular Displacement ◽  
Solution Process ◽  
Original System ◽  
Polynomial Equations ◽  
Dimensional Synthesis ◽  
Motion Generation ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Closed Form Solutions ◽  
Resultant Theory

Abstract Resultant theory is applied to derive closed-form solutions for the dimensional synthesis of linkage components for a finite number of precision positions for motion generation with prescribed timing. The solutions are in forms of polynomial equations of the exponential of a single unknown angular displacement. The degree of the derived polynomial depends on the number of links in the linkage component and the number of precision positions to be synthesized for, or the number of compatibility equations. The resultant theory is discussed in detail, and the procedure for the derivation of resultant polynomials is demonstrated. This paper shows that, for the case of two compatibility equations, the solution is a six-degree polynomial. For the case of three compatibility equations, the solution is a fifty-fourth degree polynomial. The Bernshtein formula is applied to check the exact number of solutions of the original system of polynomial equations and to verify the validity of the derived resultant polynomials. An algorithm is also proposed for screening out extra solutions which may be generated through the solution process.

Solving a Planar Four-Bar Design Problem Using Continuation

1989 ◽  
Author(s):  
A. P. Morgan ◽  
C. W. Wampler
Keyword(s):  
Polynomial System ◽  
Continuation Method ◽  
Computer Time ◽  
Design Parameters ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Design Elements ◽  
Real Solutions ◽  
Time Limitations

Abstract The problem of synthesizing a planar 4-bar with given pivots such that the coupler curve passes through live precision points is considered. It is shown that the design parameters must satisfy a system of 4 fourth-degree polynomial equations in 4 unknowns which has at most 36 nonzero real solutions. This polynomial system is solved using a continuation method, which thereby generates the collection of all designs that meet the precision-point specification. A computer program that implements this continuation method has been tested on a number of problems. It is reliable and fast enough for the purposes of design. The approach to kinematic design represented by this work is completely general, subject only to computer-time limitations that may arise for problems with many design elements.

Solving a Planar Four-Bar Design Problem Using Continuation

1990 ◽  
Vol 112 (4) ◽  
pp. 544-550 ◽  
Author(s):  
A. P. Morgan ◽  
C. W. Wampler
Keyword(s):  
Polynomial System ◽  
Continuation Method ◽  
Computer Time ◽  
Design Parameters ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Design Elements ◽  
Real Solutions ◽  
Time Limitations

The problem of synthesizing a planar 4-bar with given pivots such that the coupler curve passes through five precision points is considered. It is shown that the design parameters must satisfy a system of 4 fourth-degree polynomial equations in 4 unknowns which has at most 36 nonzero real solutions. This polynomial system is solved using a continuation method, which thereby generates the collection of all designs that meet the precision-point specification. A computer program that implements this continuation method has been tested on a number of problems. It is reliable and fast enough for the purposes of design. The approach to kinematic design represented by this work is completely general, subject only to computer-time limitations that may arise for problems with many design elements.

scholarly journals Simulation Tests of a Cow Milking Machine—Analysis of Design Parameters

Processes ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 1358
Author(s):  
Ewa Golisz ◽  
Adam Kupczyk ◽  
Maria Majkowska ◽  
Jędrzej Trajer
Keyword(s):  
Mathematical Model ◽  
Simplified Model ◽  
Design Parameters ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Local Loss ◽  
Linear Drag ◽  
Milking Machine ◽  
Machine Analysis ◽  
The Impact

The objective of this paper was to create a mathematical model of vacuum drops in a form that enables the testing of the impact of design parameters of a milking cluster on the values of vacuum drops in the claw. Simulation tests of the milking cluster were conducted, with the use of a simplified model of vacuum drops in the form of a fourth-degree polynomial. Sensitivity analysis and a simulation of a model with a simplified structure of vacuum drops in the claw were carried out. As a result, the impact of the milking machine’s design parameters on the milking process could be analysed. The results showed that a change in the local loss and linear drag coefficient in the long milk duct will have a lower impact on vacuum drops if a smaller flux of inlet air, a higher head of the air/liquid mix, and a higher diameter of the long milk tube are used.

BENDING OF ORTHOTROPIC RECTANGULAR CANTILEVER PLATE

2021 ◽  
pp. 2-9
Author(s):  
M.V. Sukhoterin ◽  
◽  
A.M. Maslennikov ◽  
T.P. Knysh ◽  
I.V. Voytko ◽  
...  
Keyword(s):  
Iterative Method ◽  
Boundary Conditions ◽  
Trigonometric Series ◽  
Bending Moment ◽  
Partial Solution ◽  
Numerical Results ◽  
Cantilever Plate ◽  
Degree Polynomial ◽  
Fourth Degree ◽  
Beam Function

Abstract. An iterative method of superposition of correcting functions is proposed. The partial solution of the main differential bending equation is represented by a fourth-degree polynomial (the beam function), which gives a residual only with respect to the bending moment on parallel free faces. This discrepancy and the subsequent ones are mutually compensated by two types of correcting functions-hyperbolic-trigonometric series with indeterminate coefficients. Each function satisfies only a part of the boundary conditions. The solution of the problem is achieved by an infinite superposition of correcting functions. For the process to converge, all residuals must tend to zero. When the specified accuracy is reached, the process stops. Numerical results of the calculation of a square ribbed plate are presented.

scholarly journals Regression analysis between sediment transport rates and stream discharge for the Nestos River, Greece

2013 ◽  
Vol 14 (3) ◽  
pp. 362-370
Keyword(s):  
Sediment Transport ◽  
Suspended Load ◽  
Bed Load ◽  
Regression Curve ◽  
Degree Polynomial ◽  
Stream Discharge ◽  
Nestos River ◽  
Polynomial Curve ◽  
Load Transport ◽  
Bed Load Transport

Systematic measurements of sediment transport rates and water discharge were conducted in the Nestos River (Greece), at a place located between the outlet of Nestos River basin and the river delta. This basin area is about 838 km2 and lies downstream of the Platanovrysi Dam. Separate measurements of bed load transport and suspended load transport were performed at certain cross sections of the Nestos River. In this study, relationships between sediment transport rates and stream discharge for the Nestos River are presented. A nonlinear regression curve (4th degree polynomial curve; r2 equals 0.62) between bed load transport rates and stream discharge, on the basis of 63 measurements, was developed. In addition, a nonlinear regression curve (5th degree polynomial curve; r2 equals 0.95) between suspended load transport rates and stream discharge, on the basis of 65 measurements, was developed. The relatively high r2 values indicate that both bed load transport rates and, especially, suspended load transport rates can be predicted as a function of the stream discharge in the Nestos River. However, the reliability of the regression equations would have been higher if more measured data were available.

scholarly journals On Complex-Valued Triple Controlled Metric Spaces and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Nabil Mlaiki ◽  
Thabet Abdeljawad ◽  
Wasfi Shatanawi ◽  
Hassen Aydi ◽  
Yaé Ulrich Gaba
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Spaces ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Fredholm Type ◽  
Type Integral ◽  
Type Spaces ◽  
Complex Valued

In this manuscript, we introduce the concept of complex-valued triple controlled metric spaces as an extension of rectangular metric type spaces. To validate our hypotheses and to show the usability of the Banach and Kannan fixed point results discussed herein, we present an application on Fredholm-type integral equations and an application on higher degree polynomial equations.

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