Position Tolerance Verification Using Simulated Gaging

1989 ◽  
Author(s):  
F. Etesami
Keyword(s):  
Lagrange Multipliers ◽  
Optimization Problem ◽  
Coordinate Measuring Machines ◽  
Vision Systems ◽  
Dimensional Inspection ◽  
Perfect Form ◽  
Nominal Feature ◽  
Manual Tasks ◽  
Tolerance Verification

Abstract One of the routine manual tasks in dimensional inspection is the assembly verification of circular features by mechanical gaging. With the aid of coordinate measuring machines or vision systems, this task can be performed more efficiently through simulation or soft-gaging. A formulation is presented for interpretation of 2D position tolerance specifications. Simulated gages are constructed from datum features as a set of constraint relationships. The measure of perfect-form position-imperfection is determined as the distance between the measured and the nominal feature positions subject to datum constraint requirements. The derived formulation is applied to an example part with a hole-slot datum-priority-frame. This formulation results in a three-variable optimization problem which is solved by an Augmented Lagrange Multipliers technique. The extension of the formulation to 3D is also discussed, but without reference to a specific representation.

Comparison between a Laser Micrometer and a Touch Trigger Probe for Workpiece Measurement on a CNC Lathe

2012 ◽  
Vol 498 ◽  
pp. 49-54 ◽  
Author(s):  
G. Valiño ◽  
C.M. Suárez ◽  
J.C. Rico ◽  
B.J. Álvarez ◽  
D. Blanco
Keyword(s):  
Coordinate Measuring Machines ◽  
Machine Accuracy ◽  
Dimensional Inspection ◽  
Cnc Lathe ◽  
Touch Trigger Probe ◽  
Repeatability And Reproducibility ◽  
Cylindrical Workpiece ◽  
Machine Control

The current requirements for an efficient dimensional inspection of manufactured parts have lead to development of different in process and on-machine measurement (OMM) techniques. Touch trigger probes (TTP) are the most common technologies utilized, inspired on contact probes used on coordinate measuring machines (CMMs). The on-machine accuracy of TTPs depends upon precision of the tool-machine control as well as upon the procedure for TTP presetting. Taking this into account, a different OMM technique is considered in this work, which consists on a laser micrometer (LM) that is commonly used for in-process measurement of continuous products. The behaviour of TTP and LM is analysed and discussed in terms of repeatability and reproducibility. Results obtained by both techniques are compared each other by measuring a cylindrical workpiece and by checking the results with those obtained on a CMM.

A discussion on electricity prices, or the two sides of the coin

2021 ◽  
Vol 379 (2202) ◽  
pp. 20190428
Author(s):  
Juan Pablo Luna ◽  
Claudia Sagastizábal ◽  
Paulo J. S. Silva
Keyword(s):  
Power System ◽  
Lagrange Multipliers ◽  
Optimization Problem ◽  
Energy Systems ◽  
Thermal System ◽  
Energy Prices ◽  
Theme Issue ◽  
Price Supports ◽  
System Operator ◽  
Two Sides

We examine how different pricing frameworks deal with non-convex features typical of day-ahead energy prices when the power system is hydro-dominated, like in Brazil. For the system operator, requirements of minimum generation translate into feasibility issues that are fundamental to carry the generated power through the network. When utilities are remunerated at a price depending on Lagrange multipliers computed for a system with fixed commitment, the corresponding values sometimes fail to capture a signal that recovers costs. Keeping in mind recent discussions for the Brazilian power system, we analyse mechanisms that provide a compromise between the needs of the generators and those of the system operator. After characterizing when a price supports a generation plan, we explain in simple terms dual prices and related concepts, such as minimal uplifts and bi-dual problems. We present a new pricing mechanism that guarantees cost recovery to all agents, without over-compensations. Instead of using Lagrange multipliers, the price is defined as the solution to an optimization problem. The behaviour of the new rule is compared to two other proposals in the literature on illustrative examples, including a small, yet representative, hydro-thermal system. This article is part of the theme issue ‘The mathematics of energy systems’.

scholarly journals Retrieving Storm Electric Fields from Aircraft Field Mill Data. Part I: Theory

2006 ◽  
Vol 23 (10) ◽  
pp. 1289-1302 ◽  
Author(s):  
W. J. Koshak
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
High Voltage ◽  
Lagrange Multipliers ◽  
Optimization Problem ◽  
Absolute Calibration ◽  
Retrieval Process ◽  
Physical Constraints ◽  
Side Constraints ◽  
Derivatives Of

Abstract It is shown that the problem of retrieving storm electric fields from an aircraft instrumented with several electric field mill sensors can be expressed in terms of a standard Lagrange multiplier optimization problem. The method naturally removes aircraft charge from the retrieval process without having to use a high-voltage stinger and linearly combined mill data values. It allows a variety of user-supplied physical constraints (the so-called side constraints in the theory of Lagrange multipliers) and also helps improve absolute calibration. Additionally, this paper introduces an alternate way of performing the absolute calibration of an aircraft that has some benefits over conventional analyses. It is accomplished by using the time derivatives of mill and pitch data for a pitch down maneuver performed at high (>1 km) altitude. In Part II of this study, the above methods are tested and then applied to complete a full calibration of a Citation aircraft.

An Intelligent Planning Environment for Automated Dimensional Inspection Using Coordinate Measuring Machines

1992 ◽  
Vol 114 (2) ◽  
pp. 222-230 ◽  
Author(s):  
Chia-Hsiang Menq ◽  
Hong-Tzong Yau ◽  
Ching-Li Wong
Keyword(s):  
Decision Making ◽  
Basic Structure ◽  
Inspection System ◽  
Coordinate Measuring Machines ◽  
Localization Algorithm ◽  
Sculptured Surfaces ◽  
Inspection Planning ◽  
Dimensional Inspection ◽  
Knowledge Based ◽  
Intelligent Planning

This paper presents a basic structure of an intelligent planning environment for automated dimensional inspection using coordinate measuring machines (CMMs). Three levels of automation technology, ranging from the facility automation to information and decision automation, are discussed. At the facility level, the operations of the CMMs are examined for the dimensional inspection of various manufactured objects. In this research the dimensional inspection of objects having complex and sculptured surfaces is emphasized. At the information level, a CAD-directed inspection system is implemented. The system is composed of three key elements: CAD/CMM inspection planning module, CAD model based localization algorithm, and comparative analysis module. In addition, the concept of inspection attributes is introduced. Inspection attributes are some pieces of information that are stored in the CAD database along with the design model for inspection application. Typical inspection attributes include functional tolerances and manufacturing capability. At the decision making level, an inspection planner is proposed in conjunction with the CAD directed inspection to provide links between the CAD/CMM system and inspection goal. The planner is a knowledge based system which utilizes artificial intelligence technologies to automate the decision making in inspection planning.

A Constrained Variational Method for Two-Dimensional Shape Optimization

1989 ◽  
Author(s):  
R. L. West ◽  
E. Sandgren
Keyword(s):  
Variational Method ◽  
Shape Optimization ◽  
Lagrange Multipliers ◽  
Optimization Problem ◽  
Virtual Work ◽  
Boundary Integral ◽  
Matrix Equations ◽  
Two Dimensional ◽  
Lagrange Equations ◽  
Shape Optimization Problem

Abstract A constrained variational method is presented for the formulation and solution of a class of two-dimensional continuous shape optimization problems with equality constraints. Conceptually, the method casts the shape optimization problem as an analogous application of the principle of virtual work. It is postulated that the optimal shape is that equilibrium shape distinguished by the stationary value of the systems “effective” virtual work. The resulting formulation leads to a direct variational statement of the shape optimization problem, yielding the optimality criteria consisting of the Euler-Lagrange equations, constraints and boundary conditions. The Euler-Lagrange equations are linearized about the current shape and transformed into a set of Poisson’s equations. A direct boundary integral formulation is developed for the solution of Poisson’s equation that results in a continuous expression for the shape in terms of the Lagrange multipliers. The numerical solution procedure involves discretizing the shape into boundary and domain elements and using the direct boundary element method and the linearized constraint set to form a set of matrix equations. The solution to the set of matrix equations yields new estimates of the shape and the Lagrange multipliers. Convergence of the method is achieved when successive iterations of the shape and Lagrange multiplier estimates fail to improve by some prescribed limit. The classical problem of finding the curve with minimum perimeter and a prescribed enclosed area is used to illustrate the method.

scholarly journals Dualities in Nonholonomic Optimization

2016 ◽  
Vol 54 (2) ◽  
pp. 149-166
Author(s):  
Constantin Udrişte ◽  
Mădălina Constantinescu ◽  
Ionel Ţevy ◽  
Oltin Dogaru
Keyword(s):  
Critical Points ◽  
Lagrange Multipliers ◽  
Optimization Problem ◽  
Function Theory ◽  
Nonholonomic Constraints ◽  
Dual Function ◽  
Canonical Coordinates ◽  
Constrained Optimization Problem ◽  
Riemannian Geometries ◽  
Nonholonomic Manifolds

Abstract This article deals with optimizing problems whose restrictions are nonholonomic. The central issue relates to dual nonholonomic programs (what they mean and how they are solved?) when the nonholonomic constraints are given by Pfaff equations. We emphasize that nonholonomic critical points are not the classical ones and that the nonholonomic Lagrange multipliers are not the classical (holonomic) Lagrange multipliers. Topological significance of Lagrange multipliers and dual function theory introduced by EDO and EDP are key results. Also new Riemannian geometries attached to a given nonholonomic constrained optimization problem are introduced. The original results are surprising and include: (i) aspects derived from the Vranceanu theory of nonholonomic manifolds, and from the geometric distributions theory, (ii) optimal problems in Darboux canonical coordinates.

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