The Constrained Least-Square Optimization of Spherical Four-Bar Path Generators

1992 ◽  
Vol 114 (3) ◽  
pp. 394-405 ◽  
Author(s):  
J. Angeles ◽  
Z. Liu
Keyword(s):  
Numerical Procedure ◽  
Orthogonal Decomposition ◽  
Least Square ◽  
System Of Nonlinear Equations ◽  
Overdetermined System ◽  
Optimization Scheme ◽  
Numerical Examples ◽  
Speed Up ◽  
General Basis ◽  
Four Bar Linkage

In this paper, the optimization of the spherical RRRR four-bar linkage for the problem of path generation is addressed. The problem is formulated as a two-loop minimization of the error between the path-generating point in the coupler curve and the prescribed position, while decoupling the linkage parameters from the configuration variables, namely, the input angles. The synthesis problem consists of evaluating a set of input angles {ψk}1m defining m linkage configurations and the linkage parameters independently. This leads to a constrained overdetermined system of nonlinear equations. The orthogonal decomposition algorithm, introduced elsewhere, is employed to solve the problem. Continuation and damping techniques are used in the numerical procedure to ensure convergence and speed up its rate. The optimization scheme is developed on a general basis and can handle the problem for any number of given path points. Three numerical examples are included.

THE CONSTRAINED LEAST-SQUARE OPTIMIZATION OF SPHERICAL FOUR-BAR LINKAGES FOR RIGID-BODY GUIDANCE

1992 ◽  
Vol 16 (1) ◽  
pp. 47-60 ◽  
Author(s):  
Zheng Liu ◽  
J. Angeles
Keyword(s):  
Rigid Body ◽  
Numerical Procedure ◽  
Optimization Procedure ◽  
Least Square ◽  
Optimization Scheme ◽  
Loop Optimization ◽  
Linkage Error ◽  
General Basis ◽  
Set Up ◽  
Guidance Problem

The synthesis of spherical four-bar linkages for rigid-body guidance consists of the computation of the relevant dimensions of this type of linkage so that it can guide its coupler link to attain a set of prescribed orientations, Generalized coupler curves are used to describe the specified orientations and to evaluate the linkage error, defined as the sum of the errors between the corresponding computed and prescribed orientations. A two-loop optimization procedure is set up to minimize this error, results being obtained resorting to the orthogonal decomposition algorithm, an iterative numerical scheme introduced elsewhere. Continuation and damping techniques are used in the numerical procedure to enhance the convergence likelihood and its rate. The optimization scheme is developed on a general basis and can handle the spherical rigid-body guidance problem for any number of prescribed orientations. A numerical example is included in the paper.

scholarly journals Numerical Evaluation of magnetic absolute Measurements with arbitrary distributed DI-Fluxgate Theodolite Positions

2017 ◽  
Author(s):  
Heinz-Peter Brunke ◽  
Jürgen Matzka
Keyword(s):  
Magnetic Field ◽  
Least Square ◽  
System Of Nonlinear Equations ◽  
Overdetermined System ◽  
Data Set ◽  
Individual Measurement ◽  
Fluxgate Sensor ◽  
The One ◽  
Absolute Measurements ◽  
Magnetic Meridian

Abstract. At geomagnetic observatories so called absolute measurements are used to determine the calibration parameters of the main three-axis-magnetometer and their long term drift. This allows to get the vector of the geomagnetic field in an absolute geographic reference frame over long periods of time in order to study the secular variation of the Earth's magnetic field. Absolute measurements of the magnetic declination D and inclination I are done by means of a nonmagnetic theodolite with a fluxgate sensor mounted on its telescope parallel to the optical axis. A fluxgate measures the magnetic field component along its sensor axis. The reading S of this magnetometer vanishes if it points towards a direction perpendicular to the field. For absolute measurements standard measuring scheme using six to eight such theodolite positions are established routines in magnetic observatories. These standard DI schemes allow for a simple numeric evaluation and cancel out the influence of instrument parameters like sensor offset and misalignment angles between fluxgate sensor and telescope. We present a numerical method that allows to evaluate measurements of an arbitrary number (minimum 5 as there are 5 independent parameters) and of arbitrary theodolite positions and exploit it to this end. We implement an instrument model to calculate the fluxgate reading S in dependence of field, instrument parameters and telescope direction (herein after referred to as theodolite position). Inserting actual measured values gives one nonlinear equation for each theodolite position. Eventually this is resulting in an overdetermined system of nonlinear equations. This system is solved in the sense of a least square solution using the Gauss-Newton-method generalized to an overdetermined system. The accuracy of the resulting D, I and base values is given in terms of estimated variances. The accuracy of the resulting D and I values depends on both, the choice of used theodolite directions and on the accuracy of the measurements. The quality of each individual measurement can be assessed by means of calculated residuals. A general approach has significant advantages. The method allows to seamlessly incorporate additional measurements for higher accuracy. Individual erroneous readings are identified and discarded without invalidating the entire data set. We show how a-priory information can be incorporated and how that allows to even evaluate a very reduced data set. We expect the general method to ease requirements for both manual and automated DI-flux measurements. It can reveal certain properties of the DI-theodolite which are not captured by the conventional method. Based on the new method, a new measuring schema is presented. It avoids the need to calculate the magnetic meridian prior to the inclination measurements. Adjustment is always done with the same fine adjustment wheel, the one for the horizontal circle. Leveling of the telescope is not necessary and thus leveling errors are avoided. All these makes the measurements faster and less prone to errors. The option of using measurements off the normal DI positions makes measurements in the vicinity of the magnetic equator possible for theodolites without zenith ocular.

Least-Square Optimization of Planar and Spherical Four-Bar Function Generator Under Mobility Constraints

1992 ◽  
Vol 114 (4) ◽  
pp. 569-573 ◽  
Author(s):  
Zheng Liu ◽  
J. Angeles
Keyword(s):  
Orthogonal Decomposition ◽  
Least Square ◽  
Optimization Scheme ◽  
Function Generator ◽  
Design Error ◽  
Slack Variables ◽  
Constrained Minimization Problem ◽  
Function Generators ◽  
Original Objective ◽  
Output Equation

An optimization scheme for four-bar function generators under mobility constraints, which can be applied to both planar and spherical four bar linkages, is presented in this paper. The design error, defined as the residual in the input-output equation, is minimized over the vector of linkage parameters. The mobility constraints, given as a set of inequalities, are converted into equalities by introducing slack variables. The problem is thus formulated as an equality-constrained minimization problem, which is then solved using the orthogonal-decomposition algorithm, an iterative numerical method introduced elsewhere. To reduce the dimensional unbalance, which often occurs in solving a synthesis problem, a penalty function is combined with the original objective function, whose minimization leads to dimensionally balanced linkages. A numerical example is included.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

scholarly journals Numerical method of identification of an unknown source term in a heat equation

2002 ◽  
Vol 8 (2) ◽  
pp. 161-168 ◽  
Author(s):  
Afet Golayoğlu Fatullayev
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Method ◽  
Minimization Problem ◽  
Unknown Function ◽  
Source Term ◽  
Numerical Procedure ◽  
Numerical Examples ◽  
Unknown Source

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

scholarly journals Minimization of Deviations of Gear Real Tooth Surfaces Determined by Coordinate Measurements

1993 ◽  
Vol 115 (4) ◽  
pp. 995-1001 ◽  
Author(s):  
F. L. Litvin ◽  
C. Kuan ◽  
J. C. Wang ◽  
R. F. Handschuh ◽  
J. Masseth ◽  
...  
Keyword(s):  
Linear Equations ◽  
Least Square Method ◽  
Least Square ◽  
Tooth Surface ◽  
Hypoid Gear ◽  
Overdetermined System ◽  
Entire Surface ◽  
First Order ◽  
Tooth Surfaces ◽  
Coordinate Measurements

The deviations of a gear’s real tooth surface from the theoretical surface are determined by coordinate measurements at the grid of the surface. A method has been developed to transform the deviations from Cartesian coordinates to those along the normal at the measurement locations. Equations are derived that relate the first order deviations with the adjustment to the manufacturing machine tool settings. The deviations of the entire surface are minimized. The minimization is achieved by application of the least-square method for an overdetermined system of linear equations. The proposed method is illustrated with a numerical example for hypoid gear and pinion.

Network Redundancy: A Key Design Factor for Cooling Networks

2020 ◽  
Author(s):  
Reza Pejman ◽  
Ahmad Raeisi Najafi
Keyword(s):  
Shape Optimization ◽  
Design Optimization ◽  
Cooling Efficiency ◽  
Temperature Uniformity ◽  
Optimization Scheme ◽  
Design Factor ◽  
Numerical Examples ◽  
Optimization Framework ◽  
Hybrid Topology ◽  
Operating Constraints

Abstract Microvascular composite offers a variety of multi-functionality based on the choice of fluid flowing through the embedded microchannels. The design of the microchannel network in microvascular composites is quite challenging. Indeed, the design is often expected to have high cooling efficiency, satisfy the manufacturing and operating constraints, and also have redundancy to increase the temperature uniformity and alleviate the destructive effects of potential microchannel blockage. In this study, we present a design optimization framework to satisfy these requirements. We use the Hybrid Topology/Shape (HyTopS) optimization scheme to design a redundant blockage-tolerant cooling network. In this method, the optimizer can change the topology of the design during the shape optimization process. Being able to modify the topology of the network enables the optimizer to provide network redundancy to effectively optimize the design for blockage tolerance. We also solve several numerical examples to show the unique features of the proposed method.

scholarly journals Finite Element Least Square Technique for Newtonian Fluid Flow through a Semicircular Cylinder of Recirculating Region via COMSOL Multiphysics

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ilyas Khan ◽  
Abid A. Memon ◽  
M. Asif Memon ◽  
Kaleemullah Bhatti ◽  
Gul M. Shaikh ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Fluid Flow ◽  
Newtonian Fluid ◽  
Rectangular Channel ◽  
Numerical Procedure ◽  
Least Square ◽  
Comsol Multiphysics ◽  
Velocity Magnitude ◽  
Increase Reynolds Number ◽  
Flow Through

This article aims to study Newtonian fluid flow modeling and simulation through a rectangular channel embedded in a semicircular cylinder with the range of Reynolds number from 100 to 1500. The fluid is considered as laminar and Newtonian, and the problem is time independent. A numerical procedure of finite element’s least Square technique is implemented through COMSOL multiphysics 5.4. The problem is validated through asymptotic solution governed through the screen boundary condition. The vortex length of the recirculating region formed at the back of the cylinder and orientation of velocity field and pressure will be discussed by three horizontal and four vertical lines along the recirculating region in terms of Reynolds number. It was found that the two vortices of unequal size have appeared and the lengths of these vortices are increased with the increase Reynolds number. Also, the empirical equations through the linear regression procedure were determined for those vortices. The orientation of the velocity magnitude as well as pressure along the lines passing through the center of upper and lower vortices are the same.

Cashless Payments and its Impact on Inflation

2020 ◽  
Vol 7 (9) ◽  
pp. 524-532
Author(s):  
Pisi Bethania Titalessy
Keyword(s):  
Credit Card ◽  
Least Square ◽  
Research Approach ◽  
Electronic Money ◽  
Ordinary Least Square ◽  
Cash Transaction ◽  
Speed Up ◽  
Using Data ◽  
Relationship Of ◽  
The Relationship

Payment with a non-cash system can simplify transactions and are increasingly used. The advantages of non-cash payments are not only due to convenience, speed up transaction time, and time savings but also the benefits that can reduce the circulation of money in the community. The less the amount of physical money in circulation, it will indirectly affect the inflation rate. However, there are inconsistency of research results regarding the relationship of non-cash transactions and inflation. These issues constitute a research gap on cashless payments and inflation in Indonesia. This study aims to prove the relationship between cashless payments and inflation in Indonesia. Using data from Central Bureau of Statistics Republic of Indonesia and Bank Indonesia over the period 2019-2020Q2, the results confirm that electronic money decrease inflation. The research approach in this study focuses on quantitative analysis using the Ordinary Least Square (OLS) method. The results of this study indicate that partially the relationship between debit card transactions and inflation has no significant effect. Credit card transactions have no significant effect on inflation, while electronic money transactions have a significant effect on inflation in Indonesia. Non-cash transactions intensified by Bank Indonesia through the cash-less society need to be considered more with the public's understanding of the use of non-cash transaction instruments so that the use of non-cash transactions in Indonesia is not only used for cash withdrawals but is used in every transaction.

A Steepest-Ascent Method for Solving Optimum Programming Problems

1962 ◽  
Vol 29 (2) ◽  
pp. 247-257 ◽  
Author(s):  
A. E. Bryson ◽  
W. F. Denham
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Calculus Of Variations ◽  
Numerical Procedure ◽  
Nonlinear Ordinary Differential Equations ◽  
Point Boundary ◽  
Numerical Examples ◽  
Steepest Ascent ◽  
Maximum Range

A systematic and rapid steepest-ascent numerical procedure is described for solving two-point boundary-value problems in the calculus of variations for systems governed by a set of nonlinear ordinary differential equations. Numerical examples are presented for minimum time-to-climb and maximum altitude paths for a supersonic interceptor and maximum-range paths for an orbital glider.

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