^{1,2}

^{1,2}

^{1,2}

^{1,2}

^{3}

^{1,2}

^{1}

^{2}

^{3}

When designing mechanical assemblies, assembly tolerance design is an important issue which must be seriously considered by designers. Assembly tolerances reflect functional requirements of assembling, which can be used to control assembling qualities and production costs. This paper proposes a new method for designing assembly tolerance networks of mechanical assemblies. The method establishes the assembly structure tree model of an assembly based on its product structure tree model. On this basis, assembly information model and assembly relation model are set up based on polychromatic sets (PS) theory. According to the two models, the systems of location relation equations and interference relation equations are established. Then, using methods of topologically related surfaces (TTRS) theory and variational geometric constraints (VGC) theory, three VGC reasoning matrices are constructed. According to corresponding relations between VGCs and assembly tolerance types, the reasoning matrices of tolerance types are also established by using contour matrices of PS. Finally, an exemplary product is used to construct its assembly tolerance networks and meanwhile to verify the feasibility and effectiveness of the proposed method.

Assembly tolerance design is one of the research hotspots in the field of computer-aided tolerancing (CAT). Assembly constraints are essentially constraints between assembly feature surfaces of parts. Assembly tolerance is a vector constraint. Its orientation, type, and value are mainly determined by the assembly functional requirements of mechanical assemblies. Assembly tolerances are crucial for many activities in the product’s life cycle, which not only affects assembly qualities but also determines manufacturing costs. Automatic generation of assembly tolerance networks can greatly reduce design complexity and improve design quality. Meanwhile, optimization design of assembly tolerances can remarkably reduce manufacturing cost of mechanical assemblies. The design of assembly tolerance networks is a complex multiscale problem. It involves associations between the multiple scales, such as the assembly functional requirement, part positioning, datum reference frame, assembly sequence, assembly feature, and tolerance specification. The essences of many important issues are multiscale problems in medicine, physics, computers, chemistry, materials science, robotics, and other disciplines. Multiscale modeling and operations are widely used in these research fields. Kou et al. [

In recent years, research on assembly tolerance design has made plentiful and substantial achievements. Based on TTRS theory, Clement et al. [

Establishing assembly tolerance networks in CAD systems is a complex design problem, in which there are still many unresolved issues. Some commercial CAD systems have realized the automatic generation of dimensional tolerances. However, automatic design of assembly tolerance networks cannot completely be achieved. Current researches on tolerance networks do not meet the integration requirements of CAD/CAM systems. The aim of this paper is to propose a new mathematical method for establishing assembly tolerance networks based on PS theory [

The paper is organized as follows. Section

For mechanical assemblies, their assembly processes are just like constructing objects with building blocks. Firstly, parts are assembled into components according to constraint relations among them, and then these components and other parts are further assembled into high-level components. The above process is repeatedly carried out until the assembling of product is completed. For the majority of assembles, their structures could be seen as follows: an assembly is composed of subassemblies and parts. Likewise, a subassembly is also composed of its subassemblies and parts directly under it. Therefore, an assembly could be disassembled into basic structural units, namely, parts. Different subassemblies and parts can be concurrently assembled. Obviously, the assembly process is a typical nonlinear process. From the above analysis, we can see that some mechanical assemblies have hierarchical structures; therefore, their structures can be expressed by product structure trees (PST). For example, the spindle box of a NC milling machine shown in Figure

Spindle box of a NC milling machine.

Cutaway view of spindle box

Product structure tree model

In this paper, a hierarchical structure tree model, namely, assembly structure tree (AST), is established to express assembly structure and assembly sequence of product. PST possesses a tree structure, which consists of all the parts of a product and reflects its functional relations and assembly relations. Based on the characteristics of PST and basic requirements of product assembly, we establish the following rules to reconstruct PST to obtain AST of product.

Rule 1: connection relations between parts can be divided into two types: contact and fitting. Contact includes physical contact and virtual contact. Fitting includes plane fitting, column fitting, conical fitting, spherical surface fitting, prism fitting, screw thread fitting, welding fitting, riveting fitting, and bonding fitting. Among them, connections between parts, formed by welding, riveting, and bonding, are not allowed to be dismantled; otherwise, it will result in the failure of connection form. Assembly tolerance mainly reflects assembly position precision between parts. Parts, which are connected together by bonding, riveting, or welding, should be considered as an independent part to be studied. This paper mainly considers plane fitting, column fitting, conical surface fitting, spherical surface fitting, prism fitting, and screw thread fitting.

Rule 2: in AST, subassemblies or parts, which belong to a same parent node, cannot interfere in each other’s assembling. If the interferences occur when reconstructing product structure tree, the interference components should further be broken into smaller ones at the same level, or all the subassemblies and parts in the same parent node are redivided and recombined until the assembling interferences disappear.

Rule 3: assembling positions of parts can be determined by other parts or components’ constraints in the same subassembly, and they can be assembled without interferences. Meantime, subassemblies, as a whole, can also be positioned and assembled through the contacting and fitting between them and other subassemblies or parts. They belong to the same parent node.

Rule 4: in subassemblies, assembling positions of some parts are determined by other parts or components which do not belong to the subassemblies. These parts not only can determine the assembling positions of the subassemblies in the product but also are the assembling datums of the subassemblies. They are referred to as basic parts. In most cases, assembling process of subassembly should begin with basic parts. Under certain conditions, they can be concurrently assembled. Assembly sequence planning of subassemblies can begin with basic parts selected in the way of man-machine interaction.

Rule 5: in AST, an assembly subsequence exists among subassemblies belonging to the same parent node. Each subassembly as a whole is assembled with other subassemblies, and they all can independently realize functional requirements of product. The assembling of subassemblies in the same parent node has precedence relations.

The assembly information model established in this paper includes geometry information and fitting information of parts. According to function, structure, and geometric shape of part, geometry information can be divided into four types: axle sleeve, wheel disc, fork, and shell. Fitting information consists of two parts: fitting type and fitting property. Based on Rule 1 mentioned above, fitting type includes plane fitting, column fitting, conical surface fitting, spherical surface fitting, prism fitting, and screw thread fitting. Fitting property can be classified into clearance fitting, transition fitting, and interference fitting.

Firstly, on the premise of automatically extracting assembly information, assembly information of parts is extracted from the three-dimensional model of product in CAD system. And then a mathematical model based on PS is established to describe the assembly information, in which the parts of assembly are used as the elements of PS and their geometric information and fitting information are used as the contour of PS. Finally, the assembly information can be formally expressed as follows:

Its element

Based on the above analysis, a formal model of assembly information is built up as follows.

In Figure

Assembly information model.

The establishment of an assembly information model could provide basic information for subsequently setting up other models and equations.

If without being constrained by other parts, each part has six degrees of freedom (DOF) in free space, namely, three translational DOFs and three rotational DOFs, which translate and rotate, respectively, along the three mutually perpendicular coordinate axes. Except for the DOFs which are used to realize product functions, if all the other DOFs of a part are limited in assembly space, then its position is also determined. This implies that location requirement of the part is satisfied, which is realized by means of other related parts and components in the same subassembly. These parts and components have contact and location relations with the part and can limit its DOFs. When calculating feasible assembly sequences of a subassembly, we only need to consider assembly relations between parts which constitute the subassembly. Therefore, contact and location relations between parts in the directions of six DOFs are needed to be described in the assembly relation model.

In addition, whether a part can be assembled is also affected by other factors. For example, the space occupied by other parts probably interferes with the assembly path of the part, which makes it impossible to move the part to assembling position, interferences between a part and its assembling tools might occur if operational space is not large enough to use the assembly tools, assembling fixture interferes with assembling of parts because of its clamping method and the space occupies by it, and considering factors of man-machine engineering, there are interferences between man, part, assembly fixture, and machine. To simplify the analysis, this paper mainly considers interferences which exist between parts (or components). Consequently, assembly interference relations between parts are also needed to be described in the assembly relation model.

In an assembly, there are connection relations between parts. If not considering concrete forms of connection structures, location relations between parts can also reflect their connection relations. Thus, this paper no longer discusses connection relations of parts in detail.

By making use of hierarchical structure of the assembly structure tree, assembly information models of subassemblies in different layers could be set up. For subassembly,

Assembly relation model.

Binary array (

Positioning information of a part, which is extracted from an assembly relation model, can be used to establish its location relation model shown as Figure

Location relation model of parts.

In the subassembly, if there is a group of parts which constrains 6 DOFs of a part, then a logic equation composed by this group of parts can be regarded as a location relation equation of this part and its logical value is used to judge whether the position of the part is determined. The formal expression for location relation is shown as follows:

Rule 1: it is assumed that

Rule 2: if

Rule 3: if

Rule 4: if

Rule 5: it is prescribed that the position constraints, which are added to Part

In order to simplify the structure of the location relation model, its stipulated that element

Rule 6: location relation equations of all the parts in a subassembly are combined together to constitute the system of location relation equations of the subassembly.

By means of assembly interference information of a part which is extracted from the assembly relation model, its interference relation model is set up shown in Figure

Interference relation model of parts.

In the interference relation model, the function

Rule 1: when assembling Part

Rule 2: it is presumed that

Rule

Rule 4: if

Rule 5: owing to the direction of interference relation in assembly space, the interference constraints imposed on Part

It is obvious that the following expressions are equivalent

In other words, the interference relations between two parts, which are, respectively, along the positive and negative directions of the same coordinate axis, are equivalent.

Rule 6: interference relation equations of all the parts in a subassembly are combined together to constitute the system of interference relation equations of this subassembly.

TTRS theory proposed by Salomons [

Each SVGC is a constraint between a real feature and its corresponding associated derived feature (ADF). Functional surfaces of parts can be divided into seven types. Therefore, constraints between functional surfaces and corresponding ADFs can also be classified into seven types. By means of the contour matrix of PS, the reasoning matrix of SVGCs is established, as shown in Figure

Reasoning matrix of SVGCs.

According to the definition of SVGCs, it is obvious that there are corresponding relations between SVGCs, form tolerances, and dimensional tolerances. By using contour matrix of PS, the reasoning matrix of tolerance types corresponding to SVGCs is set up as shown in Figure

Reasoning matrix of tolerance types corresponding to SVGCs.

Each CVGC is a constraint between two ADFs, in which the two ADFs belong to the same part. Geometric features of a part can be decomposed into point, line, or plane. The interrelations between point, line, and plane can be divided into 27 kinds according to spatial positions of each other. Therefore, 27 kinds of CVGCs can be generated in accordance with them. The reasoning matrix of CVGCs is built shown in Figure

Reasoning matrix of CVGCs.

As with SVGCs, there are corresponding relations between CVGCs and assembly tolerances. Using CVGCs as contour of PS and assembly tolerance types as elements of PS, the reasoning matrix of assembly tolerance types corresponding to CVGCs is set up, as shown in Figure

Reasoning matrix of tolerance types corresponding to CVGCs.

Constraints, which exist between two real features and, meanwhile, respectively, belong to two different parts, constitute MVGCs. They are actually constraints between contact surfaces of two parts. MVGCs with lower pairs, which are often seen in assembly design, can be divided into seven types. By means of contour matrix of PS, the reasoning matrix of MVGCs is established, shown in Figure

Reasoning matrix of MVGCs.

In an assembly structure tree, each mating relation uniquely corresponds to two SVGCs and one MVGC. Four features between two assembly parts are taken as nodes, and three VGCs are taken as arc curves. The tree model is called mating tree, which can be represented with the following equation:

By using the reasoning methods described above, we can reason out assembly sequence of all the subassemblies in different layers of assembly structure tree and tolerance types between parts or subassemblies and then add them into the assembly structure tree. This kind of structure tree with assembly sequence and assembly tolerance information is defined as assembly tolerance network. In order to simplify the process of constructing the tolerance network, we take a subassembly of the spindle box of an NC milling machine (shown in Figure

Step 1: extract assembly information. On the premise of automatically recognizing features, the basic information of the main shaft component is extracted from its three-dimensional model in CAD system.

Step 2: establish assembly structure tree. According to the rules in Section

Step 3: set up assembly information model. According to the rules in Section

Step 4: establish assembly relation model. On the basis of Section

Step 5: establish the system of location relation equations. Extract location information from the assembly relation models in Figure

Main shaft component.

Assembly structure tree of main shaft component.

Assembly information model of main shaft component.

Assembly related models of main shaft component and shaft component. (a) Assembly relation model of main shaft component. (b) Assembly relation model of shaft component.

The system of location relation equations of the main shaft component is listed as follows:

The system of location relation equations of the shaft component is listed as follows:

Step 6: establish the system of interference relation equations. Like Step 5, extract the location information from the assembly relation models in Figure

The system of interference relation equations of the main shaft component is listed as follows:

The system of interference relation equations of the shaft component is listed as follows:

Step 7: generate assembly sequences. Using the reasoning equations generated above and related reasoning method [

The assembly sequences of the shaft component are shown as follows:

Step 8: determine mating tree and datum reference frame. SVGCs and MVGCs are combined into mating trees. ADFs of all the datums belonging to the same part are combined together to form a datum reference frame (DRF), between which there are no SVGCs. Taking the shaft component as an example, we use its assembly sequence, namely, _{14}, DRF_{15}, DRF_{16}, and DRF_{17}.

Step 9: generate assembly feature chains. Taking the mating trees and datum reference frames as nodes, the subassembly sequence is reconstructed, which starts from

By means of the features of the parts in Figure _{1} and ADF_{2} are two ADFs which belong to the same part. Likewise, RF_{1} and RF_{2} are two RFs which also belong to the same part.

Step 10: reason VGC types between features. Using the reasoning matrices of SVGCs and CVGCs in Sections

Step 11: reason tolerance types corresponding to VGCs. It can be seen that the same VGCs can correspond to different tolerance types in the reasoning matrices of tolerance types in Sections

Step 12: establish assembly tolerance network. In the light of the reasoning rules mentioned above, the assembly tolerance types of the spindle box and its all subassemblies can be reasoned out to form assembly tolerance chains. All the assembly tolerance chains reasoned out in Step 11 and the assembly sequence are combined to construct the assembly tolerance network of the product. Figure

Assembly tolerance network of the spindle box.

Assembly tolerance network design is determined by many factors, and these factors associate with each other. Therefore, the design is a multiscale issue. Many scholars have applied various methods to resolve the multiscale issues in different research fields, such as multiscale time schemes [

The authors are particularly grateful to Editor Carlo Cattani for his continual encouragement, patience and sincere help on earlier drafts of this paper. The authors also would like to thank two anonymous reviewers whose insightful comments and constructive suggestions have substantially improved the paper. The authors gratefully acknowledge the support from the major project of the National Natural Science Foundation of China (NSFC) under Grant number 50935006.