Sum Of Square Roots Research Articles

Neural networks modeling of honing surface roughness parameters defined by ISO 13565

Abstract For decades, the arithmetic average (AA or R a ) and root sum of squares (RSS or R q ) have been the two major surface roughness measures to define a broad range of surfaces for a mechanical product. A number of drawbacks have been identified in recent years with the above measures. To map more scientifically and closely the surface roughness to the product functions and performances, ISO 13565 has defined a different set of measures, including R k , R pk , R vk , M r1 , and M r2 . This has not only made process planning different and much more difficult, but also made modeling of the relationship between these roughness measures and the machining parameters a multiple-input and multiple-output problem. While some companies are trying the traditional trial-and-error method to implement the ISO 13565 standard, this study applies artificial neural networks to develop an empirical model for the honing process of engine cylinder liners in order to help reduce emissions, improve oil efficiency, and prolong engine life. Threefold cross-validation is applied to develop the models. Hypothesis testing and the prediction error statistics are employed to select the best model. Data from industrial experiments based on fractional factorial design will illustrate th goodness of the modeling approach and the models.

Inverse dynamics of hexapods using the natural orthogonal complement method : Fengfeng Xi, Rosario Sinatra, v21, n2, 2002, pp73–82

For decades, the arithmetic average (AA or Ra) and root sum of squares (RSS or Rq) have been the two major surface roughness measures to define a broad range of surfaces for a mechanical product. A number of drawbacks have been identified in recent years with the above measures. To map more scientifically and closely the surface roughness to the product functions and performances, ISO 13565 has defined a different set of measures, including Rk, Rpk, Rvk, Mr1, and Mr2. This has not only made process planning different and much more difficult, but also made modeling of the relationship between these roughness measures and the machining parameters a multiple-input and multiple-output problem. While some companies are trying the traditional trial-and-error method to implement the ISO 13565 standard, this study applies artificial neural networks to develop an empirical model for the honing process of engine cylinder liners in order to help reduce emissions, improve oil efficiency, and prolong engine life. Threefold cross-validation is applied to develop the models. Hypothesis testing and the prediction error statistics are employed to select the best model. Data from industrial experiments based on fractional factorial design will illustrate th goodness of the modeling approach and the models.

Tolerance Design: A Handbook for Developing Optimal Specifications

Tolerance Design: A Handbook for Developing Optimal Specifications

AsymmetricXXZchain at the antiferromagnetic transition: spectra and partition functions

The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size N as and their amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. A similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as . The partition functions for this case are found to be that of a non-relativistic free fermion system. From the symmetry of the lattice model under rotation, several identities between the partition functions are found. The scaling at zero vertical field is interpreted as a feature arising from viewing the Pokrovsky - Talapov transition with the space and time coordinates interchanged.

Injury criteria of the IMO and the Hybrid III dummy as indicators of injury potential in free-fall lifeboats

During certification of free-fall lifeboats, it is necessary to infer the injury potential of the acceleration forces exerted on the occupants during water entry. Such an evaluation is required by the International Maritime Organization (IMO) and most regulatory authorities. The dynamic response model and the square-root sum of the squares method are two criteria included in the IMO recommendation for testing lifeboats. At least one national authority requests use of the Hybrid III human surrogate when inferring injury potential. The purpose of this paper is to compare the potential for injury as indicated with the IMO criteria and that obtained through use of the Hybrid III manikin. The comparison is based on data obtained during prototype tests with full-scale free-fall lifeboats launched from heights of up to 30 m.

Square-Root Sum Rule for Quark and Lepton Masses in a Minimal Composite Model

It is pointed out that the masses of quarks and leptons in the first generation may satisfy the square-root sum rule of m e 1/2 - m ν e 1/2 = m d 1/2 - m u 1/2 , which suggests the quadratic form of Hamiltonian in “hypercharges” and the additivity of “hypercharges” in a minimal composite model of quarks and leptons.

Nondifferentiable fractional programming with Hanson-Mond classes of functions

A class of nonlinear fractional programming problems is considered in which the numerator of the objective function involves a finite sum of square roots of quadratic forms. Inequality and equality constraints are considered. Necessary and sufficient optimality conditions and weak, strong, and converse duality theorems are established in the framework of the Hanson-Mond classes of functions.

Determination of the SI Volt at the PTB

Work on an electrostatic measuring system for the determination of the SI volt and the conversion factor KV at the PTB is reported. The system consists of a so-called voltage balance and a generator for the dc measuring voltage. There is a need for the development of such a device with an uncertainty below 1 part in 106 because measurements with existing voltage balances, and results obtained by other methods show discrepancies of several parts in 106. In order to reduce the uncertainty and to simplify the apparatus a special measuring method was developed. The voltage/force transducer of the PTB voltage balance is formed from two coaxial cylindrical electrodes and the generated force is substituted for the force of gravity on a weight of 2 g. The electrode voltage is composed of a constant part of 10186 V, derived in a 1000-fold stepup from 10 standard cells, and a much smaller variable part used for balancing the scale beam by means of a control loop. Taking the root sum of squares the total relative uncertainty of the SI volt and of KV is expected to be less than 4 parts in 107.

Dielectric properties and a model of biphase water suspension at 9.4 GHz

Measurements of the complex permittivity of clay-sand slurry containing fine-milled mineral particles suspended in water carried out at 9.4 GHz show a marked dependence on both temperature and moisture content. The character of these relations is similar to that obtained for pure water. Permittivity and specific gravity of a slurry in the range interesting for technical activity are little affected by its chemical composition. On the basis of measurements carried out for five kinds of technological slurries a simple model for quick analysis of their properties is presented. The model consists of two separate layers: water and dry mineral inclusions, the thickness of which is proportional to the volume content in the suspension. Resultant permittivity of the biphase water suspension is equal to the sum of square roots from permittivities of both constituents, multiplied by their volume filling factors. Experimental results agree with those calculated from such simple formula within accuracy of measurement (dielectric constant within 2%, loss factor within 5%).

High-accuracy vector-difference and vector-sum circuits

By combining the exact nonlinear capabilities of translinear circuits with the high-gain, low-input-offset, load-driving and level-shifting properties of op-amps, analogue computing structures of great power and precision can be realised. This point is illustrated by circuits which compute the root sum of squares and root difference of squares.

A duality theorem for a special class of programming problems in complex space

Duality relations for the programming problem of a special class where the objective function is a sum of positive-semidefinite quadratic forms, and a sum of square roots of positive-semidefinite quadratic forms, over a convex polyhedral cone in complex space are considered. The duality relations between the primal problem and its dual are established.

A Current Comparator System to Establish the Unit of Electrical Energy at 60 Hz

A compensated current comparator system to establish the United States legal unit of electrical energy at the National Bureau of Standards at energy levels of approximately 30 and 60 kJ is described. Analysis of the system uncertainties and experimental data indicates that the registrations of three standard type watthour meters were determined with total estimated uncertainties of about 30 ppm at unity power factor (PF) and 40 ppm at 0.5 PF. Of these uncertainties, 18 ppm represents the three standard deviation bound for the effects of random errors, and the remainder the root sum of squares of bounds to possible calibration and systematic effects. These results indicate that it should now be possible to disseminate the energy unit with uncertainties less than the presently quoted 500 ppm.

A Duality Theorem for Nonlinear Programming

One reasonable formulation of stochastic linear programming problem leads to a deterministic nonlinear programming problem where the non-linearity occurs in the objective function as the sum of square roots of positive semi-definite quadratic forms. It may, however, be difficult to solve this problem directly because of the nondifferentiability of the terms in the objective function. The present paper establishes a dual to the nonlinear programming problem of which a solution may be easily obtained. A solution of the dual then helps to obtain a solution of the original problem.