Least Squares Evaluations for Form and Profile Errors of Ellipse Using Coordinate Data

To improve the measurement and evaluation of form error of an elliptic section, an evaluation method based on least squares fitting is investigated to analyze the form and profile errors of an ellipse using coordinate data. Two error indicators for defining ellipticity are discussed, namely the form error and the profile error, and the difference between both is considered as the main parameter for evaluating machining quality of surface and profile. Because the form error and the profile error rely on different evaluation benchmarks, the major axis and the foci rather than the centre of an ellipse are used as the evaluation benchmarks and can accurately evaluate a tolerance range with the separated form error and profile error of workpiece. Additionally, an evaluation program based on the LS model is developed to extract the form error and the profile error of the elliptic section, which is well suited for separating the two errors by a standard program. Finally, the evaluation method about the form and profile errors of the ellipse is applied to the measurement of skirt line of the piston, and results indicate the effectiveness of the evaluation. This approach provides the new evaluation indicators for the measurement of form and profile errors of ellipse, which is found to have better accuracy and can thus be used to solve the difficult of the measurement and evaluation of the piston in industrial production.

Variation analysis ofprecision mechanical assembly consideringform errors and local deformation

A method to analyze the effect of form errors and local deformation on the assembly accu- racy and its stability in a non-rigid assembly system is proposed. The contact finite element method was used to obtain local deformation of mating surfaces, which was superposed onto form errors to obtain real mating surface data of assemblies. Then mating variation was obtained by establishing vir- tual contact planes. Finally, an experiment of the assembly of two cylindrical components was car- ried out to verify the validity of the proposed method. By comparing the calculation accuracies of 3D assembly with and without taking into account local deformation, the results showed that the effects of local deformation of mating surfaces on calculation accuracy of mating variation was not neglect- able compared with form errors.

RESEARCH ON THE MINIMUM ZONE CYLINDRICITY EVALUATION BASED ON GENETIC ALGORITHMS

A genetic algorithm (GA)-based method is proposed to solve the nonlinearoptimization problem of minimum zone cylindricity evaluation. First, the background of the problemis introduced. Then the mathematical model and the fitness function are derived from themathematical definition of dimensioning and tolerancing principles. Thirdly with the least squaressolution as the initial values, the whole implementation process of the algorithm is realized inwhich some key techniques, for example, variables representing, population initializing and suchbasic operations as selection, crossover and mutation, are discussed in detail. Finally, examplesare quoted to verify the proposed algorithm. The computation results indicate that the GA-basedoptimization method performs well on cylindricity evaluation. The outstanding advantages concludehigh accuracy, high efficiency and capabilities of solving complicated nonlinear and large spaceproblems.

FORM ERROR EVALUATION OF CIRCLES BASED ON A FINELY-DESIGNED GENETIC ALGORITHM

A genetic algorithm(GA)-based new method is designed to evaluate thecircularity error of mechanical parts. The method uses the capability of nonlinear optimization ofGA to search for the optimal solution of circularity error. The finely-designed GA (FDGA)characterized dynamical bisexual recombination and Gaussian mutation. The mathematical model of thenonlinear problem is given. The implementation details in FDGA are described such as the crossoveror recombination mechanism which utilized a bisexual reproduction scheme and the elitist reservationmethod; and the adaptive mutation which used the Gaussian probability distribution to determine thevalues of the offspring produced by mutation mechanism. The examples are provided to verify thedesigned FDGA. The computation results indicate that the FDGA works very well in the field of formerror evaluation such as circularity evaluation.

Modeling Method for Assembly Variation Propagation Taking Account of Form Error

The propagation of variations, such as fixture errors and datum errors resulting from assembly and machining processes, has been extensively studied. However, only a few studies that focus on form error propagation in assembly systems have been implemented. Machining errors, especially form errors, have great impact on assembly accuracy and accuracy stability of precision mechanical systems. With form errors being the research object, a method for calculating mating variation and specifying mating coordinate is proposed to improve the accuracy of the variation propagation model. Taking into account the form error of mating surfaces, the assembly variation propagation of a precision mechanical system is analyzed, and the brief derivation procedure of the variation propagation model is introduced afterwards. The variation propagation model involves a new concept of mating variation specified by the two mating surfaces. An innovative method, the difference surface search based method, is proposed to calculate the mating variation amongst the mating surfaces. The obtained mating variation is then utilized to specify the mating coordinate in the variation propagation model. Moreover, FEM is employed to simulate the contact state of the two mating surfaces to demonstrate effectiveness of the proposed method. Meanwhile, the mating variation and mating coordinate obtained are incorporated into the assembly variation propagation model, which is then verified by a following case study through a comparison between the calculated results and the experimental results. The comparing results indicate that the established model improves the prediction of assembly accuracy. The developed model enables the investigation of various fundamental issues in variation reduction, including variation analysis, process monitoring, accuracy prediction, and accuracy control.

Wheel Setting Error Modeling and Compensation for Arc Envelope Grinding of Large-Aperture Aspherical Optics

Precision grinding is a key process for realizing the use of large-aperture aspherical optical elements in laser nuclear fusion devices,large-aperture astronomical telescopes,and high-resolution space cameras.In this study,the arc envelope grinding process of large-aperture aspherical optics is investigated using a CM1500 precision grinding machine with a maximum machinable diameter ofΦ1500 mm.The form error of the aspherical workpiece induced by wheel setting errors is analytically modeled for both parallel and cross grinding.Results show that the form error is more sensitive to the wheel setting error along the feed direction than that along the lateral direction.It is a bilinear function of the feed-direction wheel setting error and the distance to the optical axis.Based on the error function above,a method to determine the wheel setting error is proposed.Subsequently,grinding tests are performed with the wheels aligned accurately.Using a newly proposed partial error compensation method with an appropriate compensation factor,a form error of 3.4μm peak-to-valley(PV)for aΦ400 mm elliptical K9 glass surface is achieved.

Robustness and precision evaluation of the form error of micro-structured surfaces using real coded genetic algorithm

To obtain the form error of micro-structured surfaces robustly and accurately, a form er- ror evaluation method was developed based on the real coded genetic algorithm （RCGA）. The meth- od employed the average squared distance as the matching criterion. The point to surface distance was achieved by use of iterative method and the modeling of RCGA for the surface matching was also presented in detail. Parameter selection for RCGA including the crossover rate and population size was discussed. Evaluation results of series simulated surfaces without form error show that this method can achieve the accuracy of root mean square deviation （ Sq ） less than 1 nm and surface pro- file error （ St ） less than 4 nm. Evaluation of the surfaces with different simulated errors illustrates that the proposed method can also robustly obtain the form error with nano-meter precision. The e- valuation of actual measured surfaces further indicates that the proposed method is capable of pre- cisely evaluating micro-structured surfaces.

Geometric Approximation Technique for Minimum Zone Sphericity Error

The mathematical modeling for evaluation of the sphericity error is proposed with minimum radial separation center. To obtain the minimum sphericity error from the form data, a geometric approximation technique was devised. The technique regarded the least square sphere center as the initial center of the concentric spheres containing all measurement points, and then the center was moved gradually to reduce the radial separation till the minimum radial separation center was got where the constructed concentric spheres conformed to the minimum zone condition. The method was modeled firstly, then the geometric approximation process was analyzed, and finally,the software for data processing was programmed. As evaluation example, five steel balls were measured and the measurement data were processed with the developed program. The average iteration times of the approximation technique is 4.2, and on average the obtained sphericity error is 0. 529μm smaller than the least square solution,with accuracy increased by 7. 696%.

Design and Manufacture of the Shaving Cutter with Unequal Depth Gashes

Gear shaving is a gear finishing operation of high efficiency and high precision. After shaved by shaving cutter of true involute profile, there are the "mid-concave" phenomena around the pitch points of the work gear tooth flanks inevitably. This problem severely affects the shaving accuracy and gear transmission quality, which hasn’t been resolved thoroughly for a long time. Aiming at the problem, based on shaving mechanism and the analysis on gear tooth profile mid-concave, a new-style shaving cutter with unequal depth gashes is designed and manufactured. As compared with common shaving cutter, its depth of gashes is zero on the pitch points of tooth profiles and gradually gets deeper to max. from pitch points to the tops of teeth or the roots. Because of no depth on the pitch points, there are no cutting edges, that is, no cutting action, so the work gear isn’t cutted around its pitch points and is only pressed during shaving operation. Therefore, the gear tooth errors are decreased greatly. And the experimentations have proved the shaved gear has better surface finish that achieves the expectant effect. In addition, this paper analyses the forming of gash on the basis of slotting principle, and proposes a design method of gash: gash bottom is formed by two involutes which intersect on the pitch point and are concentric with the base circle of involute profile of cutter. Furthermore, the equations of the two involutes are deduced and the solution is introduced. This paper analyses the forming of gash on the basis of slotting principle, and proposes a gash-designing method. And the experiment has proved that the shaved gear has better surface finish that achieves the anticipated effect.

ANALYSIS OF TOOTH FORM ERROR OF EQUAL BASE CIRCLE BEVEL GEAR

The basic principle of equal base circle bevel gear (EBCBG) is illustrated simply Thetooth surface equation of EBCBG manufactured by end milling cutter with involute profile is de-rived. The tooth form error is analyzed on the basis of spherical involute