Study On Three Dimensional Tolerance Analysis Method Based On Jacobian-Torsor Model And Its Application In Engine Assembly

The engine is composed of many precise parts, such as cylinder block, crankshaft, connecting rod, piston, valve and cylinder head. Its structure is complex and functional requirement is abundant. Accuracies of parts have directly influences on engine’s performance, such as power, torque and oil consumption. Understanding the relation between the accuracy of part and the performance is important to engine design and manufacturing. So far, traditional one or two dimensional(1/2D) tolerance chain model is the main method for tolerance analysis of engine assembly. It is difficult to deal with geometric tolerances of features, such as orientation and form tolerances, and their interactions. Three dimensional(3D) tolerance analysis methods represent variations of features in three dimensional space and transfer these variations to the functional requirements. They can consider these geometric tolerances and their interactions which can not be handled by traditional tolerance analysis methods. Correspondingly, their results are more accurate and their analysis process is more reasonable. 3D tolerance analysis methods can analyze the relation between the accuracy of part and the performance of engine from a tolerance chain perspective.Jacobian-Torsor model is a classical 3D tolerance analysis model. The torsor model is suitable for tolerance representation, while the Jacobian matrix is suitable for tolerance propagation. The Jacobian-Torsor model combines both advantages of them. It has a concise mathematical expression and an accurate result, as well as an abundant engineering semantic. Theoretically, for complex assemblies which contain lots of geometric tolerances, and planar and cylindrical features, and clearance pin-hole fits, such as the engines, the Jacobian-Torsor model is a relatively suitable 3D tolerance analysis model.This paper brings the Jacobian-Torsor model into the tolerance analysis of engine assemblies. The vectors’ variations and constraints of features’ torsors constrained by combinational tolerances are established, and their algorithm of propagation is presented. Meanwhile, for partial parallel structures of tolerance chain in engine assemblies, the representation and calculating method of the torsor model for these structures are proposed. In addition, the algorithm of statistical percent contribution and its subdivision of tolerances are presented to distinguish and quantify each participated tolerance. An accurate and reliable 3D tolerance analysis for engine assemblies is achieved by virtue of these methods and algorithms.The main contents are shown as follows:(1) Modeling of 3D geometric tolerances for classical features based on Jacobian-Torsor modelFirstly, In light of the features and tolerances of parts of engines, the torsor models of planar and cylindrical features are introduced, and the mathematical deduction of conical feature is carried out in details. Meanwhile, the characteristic and deficiency of the torsor model is presented. Then, the variations and constraints of vectors in the torsor model for planar and cylindrical and conical features specified by tolerances, especially by combinational tolerances, are presented. These variations and constraints make sure that tolerance representation based on the torsor model is in line with tolerancing standards. Lastly, these variations and constraints are taken into the Jacobian-Torsor model by virtue of Monte Carlo simulation, which improve the accuracy and reliability of the Jacobian-Torsor model.(2) Construction and solution of partial parallel tolerance chain in Jacobian-Torsor modelFor the characteristics of engine assemblies’ structure, the connections’ structure forming partial connection loop in a serial connection route of the Jacobian-Torsor model is defined as a partial parallel structure of tolerance chain. Several classical partial parallel structures of tolerance chain composed of planar and cylindrical features are summarized. Deep research for these structures is carried out, and a method used to deal with these special structures in the Jacobian-Torsor model is presented. This method avoids modeling and solution of the parallel Jacobian matrix and improves the practicability of the Jacobian-Torsor model.(3) Algorithm and subdivision for tolerances’ percent contribution in Jacobian- Torsor modelFor the limitation of the tolerances’ percent contributions of the Jacobian-Torsor model in a deterministic way, the algorithm of tolerances’ percent contributions for the Jacobian-Torsor model in a statistical way is presented. But it can only quantify contributions of all tolerances specified on features. In other words, for a feature specified on a combinational tolerance, this algorithm can not further distinguish the contributions of each tolerance. Therefore, based on a data fit method, a subdivision method is presented to achieve the target that distinguishes and quantifies the percent contributions of each tolerance inside the combinational tolerance.(4) 3D tolerance analysis in engine assemblies based on Jacobian-Torsor modelBased on the above theoretical research, the Jacobian-Torsor model is applied in 3D tolerance analysis of two mechanisms of engine, i.e., the crank-slider mechanism and the valve mechanism. For the crank-slider mechanism, the modeling process of variations and constraints of vectors in torsor models and the solution of partial parallel tolerance chains are illustrated in details. Comparative results testify the accuracy and reliability of the method we proposed. For the valve mechanism, the emphasis is placed on the illustration of algorithm process before and after grouping assembly of tappet. Results from production reality verify the accuracy of the method again.This paper is in the face of complex engine assemblies containing a lot of geometric tolerances, and presents a 3D tolerance representation and propagation method based on the Jacobian-Torsor model. The solution used to deal with partial parallel tolerance chains in the Jacobian-Torsor model is proposed. Meanwhile, the algorithm of tolerances’ percent contribution and their subdivision for the Jacobian-Torsor model in a statistical way is introduced. The method illustrated in this paper can not only be used to offer new theoretical orientation and technical instruction for tolerance analysis of vehicle engines’ assembly, but also can be applied in tolerance analysis of aerospace engines’ assembly.