Novel application of mapping method from small displacement torsor to tolerance: Error optimization design of assembly parts

The manufacturing/assembly error of machine parts is a key factor that influences the performance and economy of mechanical systems. To achieve high assembly precision and performance on the basis of low manufacturing accuracy and cost, this study primarily optimizes the assembly error of machine parts. First, the small displacement torsor is used to characterize the small deformation between the mating surfaces of parts. Subsequently, to realize the combination of small displacement torsor and tolerance, the small displacement torsor with manufacturing error and assembly deformation is mapped to the tolerance domain. Second, based on the relationship between small displacement torsor and tolerance, an assembly error optimization model is established on the basis of the conventional tolerance-cost model, considering the emergence of manufacturing error and assembly deformation. Third, aiming at the high-pressure rotor for aeroengines, the error optimization design of assembly is carried out developed with the assembly accuracy requirement as the constraint condition, and the total costs of the manufacturing and assembly processes as the objective. The optimization results indicate that the manufacturing error range of each mating surface after optimization changes, from small to large, under the premise of ensuring the product’s performance, which verifies that the difficulty in processing parts is reduced, and that the efficiency of parts processing is also improved. Meanwhile, the relative manufacturing cost after optimization is reduced by 6.79%, which reflects the economic requirements to a certain extent. The content of this article provides the necessary design basis and reference for the realization of high assembly accuracy of mechanical systems, under low cost requirements from the design perspective.

A CAD model for the tolerancing of mechanical assemblies considering non-rigid joints between parts with defects

During the design stage, the ideal simulation and visualization of the mechanical assemblies behavior require the modeling of parts with dimensional and geometrical defects. However, the deviations caused by parts deformations can generate an important difference between the ideal assembly and the real product. In this regard, this paper proposes a tolerance analysis method of CAD assemblies considering non-rigid joints between parts with defects. The determination of realistic rigid components with dimensional and geometrical defects is based on the worst case tolerancing approach and the Small Displacement Torsor (SDT) parameters. The Finite Element (FE) computation is executed to determine deformations of realistic non-rigid part models under external loads. Sub-algorithms to define non-rigid joints between realistic parts are developed. The tolerance analysis is established using the realistic CAD assembly. A case study is presented to evaluate the proposed model.

A Coordination Space Model for Assemblability Analysis and Optimization during Measurement-Assisted Large-Scale Assembly

The assembly process is sometimes blocked due to excessive dimension deviations during large-scale assembly. It is inefficient to improve the assembly quality by trial assembly, inspection, and accuracy compensation in the case of excessive deviations. Therefore, assemblability prediction by analyzing the measurement data, assembly accuracy requirements, and the pose of parts is an effective way to discover the assembly deviations in advance for measurement-assisted assembly. In this paper, a coordination space model is constructed based on a small displacement torsor and assembly accuracy requirements. An assemblability analysis method is proposed to check whether the assembly can be executed directly. Aiming at the incoordination problem, an assemblability optimization method based on the union coordination space is proposed. Finally, taking the space manipulator assembly as an example, the result shows that the proposed method can improve assemblability with a better assembly quality and less workload compared to the least-squares method.

An iterative method of statistical tolerancing based on the unified Jacobian–Torsor model and Monte Carlo simulation

This paper focuses on exploring an iterative method of statistical tolerance design to guide designers to select tolerances more economically and effectively. After having identified the assembly functional requirement (FR) and the functional elements (FEs) of corresponding tolerance chain, the expression of a unified Jacobian–Torsor model can be derived. Monte Carlo simulation is employed to generate random variables simulating the variations of small displacement torsor associated with the FE pairs with all the generated random values being within the intervals constrained by the corresponding tolerance zones. Then, the real multiplication operations are repeatedly executed to this model, a large number of real torsor component values of FR will be obtained and we can perform statistical analysis for these simulated data to get the statistical limits of the assembly FR in the desired direction. The tolerances of critical FEs may need to be adjusted to satisfy the assembly FR imposed by the designer, and the percentage contribution of each FE to the assembly FR can help determine these critical tolerances that need to be tightened or loosened. Once the calculated FR is in close agreement with the imposed FR, the iterative process can be stopped, and the statistical tolerance redesign is achieved. The effectiveness of the proposed method is illustrated with a case study. Compared with the deterministic tolerancing method, the results show that the proposed method is more economical and that can relax significantly the precision required, manufacturing and inspection costs can then be reduced considerably.

Effect of form errors on oil film characteristics of hydrodynamic journal bearings based on small displacement torsor theory

Purpose This paper aims to investigate how form error of journal affects oil film characteristics, which are composed of several parameters including the maximum film pressure, film moment, frictional coefficient and carrying-load capacity. Design/methodology/approach A new generalized equation based on the small displacement torsor theory is derived, as well as its capability of representing types of form error on the journal, using four specified parameters in a three-dimensional (3D) state. Based on the new generalized equation of form errors, the Reynolds equation is represented and solved numerically using the Swift–Stieber boundary condition. Findings The results show that the form errors of journal have significant influence on all oil film characteristics. However, the film moment remains nearly unchanged as film characteristics, especially eccentricity ratio, become large. All film characteristics investigated vary periodically as the form error. More importantly, it is found that the film pressure distribution transforms to an asymmetric shape along the axial direction of the bearing, no longer a symmetric shape in the case of two-dimensional (2D) form errors. It is necessary to substitute the 3D form error model, which takes the variations of the film characteristics in axial direction into account, for the 2D model in the designing stage of journal bearings. Originality/value First, the effect of the form error of the journal on the performance of hydrodynamic journal bearings is studied in the view of the film characteristics systematically. Secondly, the new generalized equation of form error, derived by SDT theory, is capable of representing any types of form error on the journal, not only representing one type of form error merely.

An Analysis Model to Predict Rotation Accuracy of High-Precision Spindles Considering Part Errors and Deformation

High-precision spindles have significant influence on the machining precision and finishing quality largely due to their motion errors. However, the analysis of rotation accuracy is quite not easy in design stage because of the neglecting of geometric errors and deformations of parts in the traditional dimension chains. Hence, a theoretical analysis model is built in present study to do the prediction. The 3D error accumulation path is recognized by Datum Flow Chain (DFC) and the key tolerances are modeled by Small Displacement Torsor (SDT). Thereafter, the variation propagation is conducted by Homogeneous Transformation Matrices (HTM) and the geometric misalignment in the spindle is calculated. Then, an FEA model is built with Timoshenko beam elements and the deformation is calculated after the geometric misalignment is applied to the model. As spindle rotates, the trajectory of the spindle nose is obtained. Finally, the Monte Carlo (MC) method is used to get the distribution and the range of motion errors. To verify the feasibility and reliability of the analysis model, the radial and axial motion errors of a double supported high-precision spindle are analyzed.