Small Displacement Torsor Research Articles

A Tolerance Analysis Approach Based on Robot Kinematics

This paper presents a tolerance modeling approach based on robot kinematics theory within CAD/CAM system. Based on the differential kinematics theories, each feature of characteristics including size tolerance and geometrical tolerance was established as a corresponding small displacement torsor. In order to express the relation between the small displacement of all functional features and the functional requirement, the final expression was represented to establish the projection of analysis of the tolerance chain. A mathematical expression was obtained by the mean instead of single points, and the constraints can be represented by interval using the standard deviation, not limit deviations. After having identified the unit components and the functional requirement from the tolerance chain, we obtained the percentage contribution of each unit feature to the functional requirement. A percentage contribution can help designer to decide which tolerance is tighten or loosen. The application of the tolerance analysis approach in a simple three-dimensional sample was also discussed in this paper.

Geometrical Deviation Model of product throughout its life cycle

Nowadays, the requirements of customers concerning a product are more and more tight and high. The satisfaction of these, such as quality, reliability, robustness and cost, plays an important role in the context of global and concurrent economy. However, a product, during its life cycle, passes through manufacturing and assembly stages where geometrical deviations are generated and accumulated. These deviations that obviously have an influence on the performances of the product are not considered. Thus, a model based on the small displacement torsor that allows to model the geometrical deviations of the product is proposed in this paper. (Received 18 March 2010; Revised 20 August 2010; Accepted 1 October 2010)

A small displacement torsor model for tolerance analysis in a workpiece-fixture assembly

Workpiece geometric error, locator geometric error, and clamping error are factors that influence workpiece setup in workpiece fixturing. These errors accumulate and propagate during fixturing. They may be the reason for a machined feature being out of tolerance after machining. This paper presents a methodology for modelling and analysing the combined effect of these errors on a machined feature. Deviation of a machined feature due to the combined errors is expressed in terms of the small displacement torsor parameters. Given a tolerance on the machined feature, constraints are specified for that feature to establish a relationship between the tolerance zone of the feature and the torsor parameters. These constraints provide boundaries within which the machined feature must lie. This is used for tolerance analysis of the machined feature. A case study example was used to illustrate the approach. An experimental system was also set up to verify the analytical model. The results show that this approach offers an effective means for fixturing tolerance analysis.

Three-dimensional modelling of manufacturing tolerancing using the ascendant approach

This paper proposes a three-dimensional approach towards manufacturing tolerancing that uses a strategy to strictly consider the definition of a datum system imposed by ISO standards. This approach, called three-dimensional manufacturing tolerancing (TMT), is based on the small displacement torsor, which describes the possible deviations between machined surfaces and nominal surfaces of the part model. Every requirement of the definition drawing is treated separately with its own calculation model. The iterative ascending analysis allows one to establish the influence of the most recent phase and determine the influential parameters from previous phases. The nominal part model is defined on the datum systems of the requirement, and deviations are expressed with respect to both the machining process and the chosen machine adjustment method. The result takes the form of a formula that yields the variation in characteristics specified in the requirement based on quantifiable parameters identified on the machine.

A Generic Deviation-Based Approach for Synthesis of Tolerances

In this paper, a generic model for the synthesis of tolerances for manufactured parts is presented. The model uses a method of transforming traditional tolerance specifications (as defined in ASME Y14.5M) to a generalized coordinate system (hereinafter referred to as deviation space). Small displacement torsors (SDTs) have been used for representing the deviations. The tolerance synthesis method is formulated as a constrained nonlinear optimization process. Three different types of constraints have been considered for the optimization process: 1) representation of assemblability; 2) mapping of tolerance specification to deviations; and 3) functional requirements. A new deviation-based cost of manufacturing model has been proposed. A working module of the scheme has been implemented and the process has been elaborated with two examples. The possibility of extension of the model and scope for further generalization have been discussed. Note to Practitioners-This paper presents an optimization method for finding tolerance values for different features of an assembly of manufactured parts. Determination of different types of tolerances and their exact values for critical features of any part has been an ad-hoc process; it has been mostly experience-based till recent time. In this paper, efforts have been made to establish a strong mathematically oriented method for synthesis of tolerances. The procedure attempts to minimize cost of manufacturing while the functionality and assemblability are satisfied. It is a generalized method and could be applied for designing rigid manufacture parts. For practical usage, this method should be integrated with three-dimensional computer-aided design packages as a tolerance synthesis module for integrated tolerance design.

Manufacturing errors modelling: Two three-dimensional approaches

In the context of manufacturing tolerancing, this paper presents two approaches of three-dimensional machining simulation. The first one uses a computer aided design (CAD) model to extract geometrical data needed by the small displacement torsor model of manufacturing errors. The second one uses a CAD/CAM (computer aided manufacturing) system to virtually manufacture the part by integrating the manufacturing errors.

Geometrical tolerancing in process planning: A tridimensional approach

A method to perform tridimensional analysis and tridimensional synthesis of machining tolerances is presented. The model relies on the small displacement torsor concept and on the computer aided tolerancing approach. The link between process planning and functional tolerances with torsor chains according to workpiece set-ups is exposed. An example shows how the model is used to determine automatically geometrical variations of the workpiece knowing geometrical variations of each part (part-holder, positioning devices, machined surfaces, etc.).