Knowledge discovery in design instances of mechanical structure symmetry

Transcription

1 Special Issue Article Knowledge discovery in design instances of mechanical structure symmetry Advances in Mechanical Engineering 2015, Vol. 7(11) 1 19 Ó The Author(s) 2015 DOI: / aime.sagepub.com Zhiyong Ma, Taixia Zhang, Fan Liu and Juncheng Yang Abstract Structure symmetry is a common phenomenon with a significant effect on mechanical systems. In this article, the technology of knowledge discovery was adopted to mine association rules from the database of mechanical structure symmetry design instances, and design knowledge of structure symmetry was built to guide the application of structure symmetry in mechanical design. The process of the knowledge discovery in design instances mainly includes the following steps. First, more than 1000 design instances were collected. Three standardized systems, including structure symmetry system, design requirement system, and structure design principle system, were built. Then the design instances were analyzed based on the three standardized systems. Second, a physical database was designed, and the information of design instances was stored in the database. Third, the association rules among structure symmetry, design requirement, and structure design principle were mined using association rule mining algorithm. Finally, the design knowledge of structure symmetry in realizing design requirements was proposed by analyzing mining results and relevant instances. The proposed design knowledge can lead to the application of structure symmetry and help to improve the technical, economic, and social performance of the mechanical products. Keywords Structure symmetry, knowledge discovery, design instances, association rules, design knowledge Date received: 21 July 2015; accepted: 27 September 2015 Academic Editor: Teen-Hang Meen Introduction Symmetry is a common phenomenon in the world, with wide existence in various natural systems and artificial systems. Both natural systems, such as mountains, rivers, birds, and animals, and artificial systems, such as buildings, bridges, airplanes, and motorcars, have symmetry. Due to its important effects, symmetry is researched in biology, 1 physics, 2 philosophy, 3 mathematics, 4 computer science, 5 architecture, 6 and so on. The studies of symmetry in above fields show that symmetry is widely existent and has close relationships with system functions. As reference, the study of symmetry in other fields is very useful for mechanical symmetry research. In mechanical systems, symmetry is also a common phenomenon. From abstract functions to specific structures, from large systems to small standard parts, symmetric properties can be observed. In all kinds of mechanical symmetries, structure symmetry is the most common form, which is important to realize the Zhejiang Provincial Key Lab of Part Rolling Technology, Faculty of Mechanical Engineering & Mechanics, Ningbo University, Ningbo City, China Corresponding author: Zhiyong Ma, Zhejiang Provincial Key Lab of Part Rolling Technology, Faculty of Mechanical Engineering & Mechanics, Ningbo University, 818 Fenghua Road, Ningbo City, Zhejiang Province , China. mazhiyong@nbu.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License ( which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages ( open-access-at-sage).

2 2 Advances in Mechanical Engineering Figure 3. A feedway. Figure 1. A clamp. facilitated. There are plenty of design knowledge in design instances. Using the technology of knowledge discovery, useful design knowledge can be developed from design instances. Therefore, we collected and analyzed more than 1000 mechanical structure design instances. Based on these instances, we established a standardized instance database, mined association rules from the database, analyzed and abstracted the mining results, and proposed the design knowledge of structure symmetry in realizing design requirements (including functions, performances, and restrictions). The proposed design knowledge can guide the application of structure symmetry in mechanical design. Figure 2. A guide mechanism. function, improve the performance, and satisfy the restriction of the mechanical products. For example, in a clamp as shown in Figure 1, there are two sets of fix spots (3) translation symmetrically located on the pole (4). When clamping jaws ((1) and (2)) are fixed on different spots, the clamp can grasp materials with different sizes, which results in the conclusion that the performance range of the clamp is extended by the translation symmetric structure. In a guide mechanism as shown in Figure 2, two sector gears and poles (AC and BC) have a mirror symmetry, which leads point C to facilitate the vertical linear motion. In a feedway as shown in Figure 3, the rotation translation symmetric lamina can carry materials from spot a to b. In mechanical structure design, reasonable application of structure symmetry can help to better realize design requirements and improve the technical, economic, and social performance of the mechanical products. How to scientifically apply structure symmetry in mechanical design? Here is a good solution that by establishing design knowledge and abstracting application rules, the application of structure symmetry can be Related works Mechanical structure symmetry The earliest systemic research of mechanical symmetry was proposed by Barrenscheen, 7 who studied the existent type, functions, and application methods of structure symmetry. From 2006, Feng and colleagues 8,9 began to do further systemic research on mechanical symmetry, built a whole concept system of mechanical symmetry, and researched the existence, expression, and application of function symmetry, structure symmetry, and technology symmetry Besides systemic research, many people carried out the research on some specific symmetry. Zhu et al. 13 proposed a machining method aiming at large symmetric free-form surface using symmetric knife tools and the experiments show that this method is very effective. Prasadj et al. 14 built an estimation method of machining errors using the concept of cyclic symmetry, which can availably reduce the machining errors. Tate et al. 15 researched the detection method of symmetric structure in proactive design for assembly and they pointed out that the symmetry degree of assembly parts is an important factor which can impact the level of assembly complexity. Li et al. 16 established a method for detecting component boundary in computer-aided design (CAD) by combining the symmetry information and function information together.

3 Ma et al. 3 Knowledge discovery in database Knowledge discovery in database is used for mining interesting knowledge from databases. The research of knowledge discovery began in 1980s; Fayyad et al. 17 pointed out that knowledge discovery is a nontrivial process of identifying valid, novel, potentially useful, and ultimately understandable patterns in data. Knowledge discovery includes the following steps: data preparation, data mining, pattern evaluation and presentation, knowledge application, and knowledge maintenance. In all the steps, data mining is a core process. 18 With the continuous development of knowledge discovery, it is applied in more and more domains, such as medical science, biology, and business. Horev- Azaria et al. 19 adopted knowledge discovery to research the performance of cobalt-ferrite nanoparticles in different cellular models. Liekens et al. 20 built a knowledge discovery platform to explore and discover the biomedical information and achieved many progresses. Trappey et al. 21 mined consumer dialogs using text mining technology to acquire customer satisfaction and dissatisfaction. Kusiak 22 adopted different data mining algorithms to mine the rules from manufacturing data to support decision-making processes. Liao et al. 23 obtained the product map with the relationships among customer demands, product characteristics, and transaction records by mining association rules from product database. Object of knowledge discovery Mechanical design mainly includes two processes: conceptual design and technological design. 24 The input of conceptual design is the design requirements, and the output is the structure scheme. Structure is used for realizing design requirements, which can be divided into function, performance, and restriction. In conceptual design, the function, performance, and restriction are uniformly described as function. There are many mapping models from function to structure, such as function structure (FS), 25 function effect structure (FES), 24 and function behavior structure (FBS). 26 Pahl and Beitz established a set of structure design principles, which are used for designing better structure to realize functions. The principles can also be used in the mapping model from function to structure. In this article, the proposed design knowledge is based on two mapping models, such as FS and function structure design principles structure (FPS), as shown in Figure 4: 1. The design knowledge based on FS: it is used for describing what design requirements can be realized by certain structure symmetry. This kind of design knowledge can be described as Figure 4. Mapping models from function to structure. the association rules of structure symmetry ) design requirements. 2. The design knowledge based on FPS: it is used for describing what structure design principles can be realized by certain structure symmetry and what design requirements can be realized by certain structure design principle. This kind of design knowledge can be described as three association rules, including structure symmetry ) structure design principle, structure design principle ) design requirements, and structure symmetry ) structure design principle, design requirements. According to the above design knowledge, we adopted the technology of knowledge discovery to mine association rules from the design instance database and the results were the association rules among symmetry structure, structure design principle, and design requirements. By analyzing, summarizing, and abstracting the mining results, we proposed the design knowledge of structure symmetry in implementing design requirements. The proposed design knowledge can guide the application of structure symmetry and help to improve the technical, economic, and social performance of the mechanical products. The system framework is shown in Figure 5. Knowledge discovery from symmetry instances Process of knowledge discovery Based on the common steps of knowledge discovery, the process of knowledge discovery from design instance database is designed as follows: 1. Instance collection: to collect enough mechanical structure symmetry design instances. To the practicability and universality of the mining results, the collected instances should be typical and comprehensive. 2. Instance standardized analysis: to analyze the information of the instances, such as the

4 4 Advances in Mechanical Engineering Figure 5. System framework. symmetric structure, design requirements, structure design principles, and achievement effect. For the unification of the mining results, standardized systems of structure symmetry, design requirements, and structure design principles should be built to unify the analysis. 3. Database establishment: to design the structure of database and store the analysis result of instances into database. A suitable database structure is necessary for data mining. 4. Association rule mining: according to the mining objects shown in section Object of knowledge discovery, to mine association rules from instance database using association rule mining algorithm. 5. Association rule analysis and evaluation: to analyze the mining results and eliminate the unpractical and un-universal results. 6. Design knowledge establishment: to analyze the mining results and relevant instances and to build design knowledge of structure symmetry in realizing design requirements. Instance collection We collected more than 1000 mechanical structure symmetry design instances from specialized books, patents, academic articles, reference manuals, and design drawings The instances include basic mechanisms, machine equipment, lifting equipment, mining machines, transport equipment, measurement equipment, and agricultural machinery equipment. In all instances, there are 12 kinds of structure symmetries, 157 kinds of design requirements, and 6 kinds of structure design principles. The distribution ratios of structure symmetries in all instances are shown in Figure 6. According to Figure 6, the translation symmetry, rotation symmetry, and mirror symmetry are the most widely used symmetries; the distribution ratio of these three symmetries is almost 75%. Besides, rotation translation symmetry, scaling rotation symmetry, and reversal symmetry are also common. The distribution ratios of design requirements in all instances are shown in Figure 7. Because the kinds of requirements are very large, the requirements as shown in Figure 7 have been abstracted. According to Figure 7, in all instances, structure symmetries are commonly used for realizing the requirements of transferring material, transferring energy, transforming energy, and improving performance. The distribution ratios of structure design principles in all instances are shown in Figure 8. According to Figure 8, the principle of the division of tasks for identical functions is adopted by 88.25% of the instances which indicates that this principle is the most important principle in mechanical structure symmetry design. Instance standardized analysis Standardized analysis is a crucial prepositive step in knowledge discovery. When analyzing instances, we need to describe the information of instance, including symmetry type, symmetric structure, function, performance, restriction, structure design principle, and

5 Ma et al. 5 Figure 6. Distribution ratios of structure symmetries. Figure 7. Distribution ratios of design requirements. achievement effect. There are many descriptive methods to obtain the above information. For the standardized analysis, we need to unify the descriptive method of every kind of information. So when mining from instance database, the same kind of information will be handled uniformly. Since the mining targets are the association rules among structure symmetry, design requirement, and structure design principle, we established the standardized systems of structure symmetry, design requirement, and structure design principle, shown as follows. Standardized system of structure symmetry. A complete classification system has been built by Ma et al. 9 We established the standardized system of structure symmetry according to this classification system, as shown in Table 1. In all symmetries, translation symmetry, rotation symmetry, and mirror symmetry are the three

6 6 Advances in Mechanical Engineering Figure 8. Distribution ratios of structure design principles. Table 1. Standardized system of structure symmetry. First layer Second layer Third layer Mechanical static structure symmetry Basic structure symmetry Combined structure symmetry Scaling structure symmetry TS RoS MS RTS GS ReS RMS RRS STS SRoS SMS SRTS SGS SReS TS: translation symmetry; RoS: rotation symmetry; MS: mirror symmetry; RTS: rotation translation symmetry; GS: glide symmetry; ReS: reversal symmetry; RMS: rotation mirror symmetry; RRS: rotationreversal symmetry; STS: scaling translation symmetry; SRoS: scaling rotation symmetry; SMS: scaling mirror symmetry; SRTS: scaling ro translation symmetry; SGS: scaling glide symmetry; SReS: scalingreversal symmetry. basic symmetries. There are five kinds of combined structure symmetries which are the combinations of the three basic symmetries. The scaling structure symmetries are formed by applying scaling changes on the basic symmetries and combined symmetries. In Table 1, we also stated the abbreviations of all the symmetries in the third layer, which would be used in the following. Standardized system of design requirements. The design requirement of the mechanical product can be divided into function, performance, and restriction. Although the concept and effect of performance and restriction are different from that of function, we adopt the same model to describe function, performance, and restriction to get a uniform description of design requirement. There are many express methods of mechanical function including some popular ones, such as natural language method, 24 input output method, 32 and function character method. 33 In the natural language methods, using a pair of verb noun to describe a function is the simplest method and is easy to be understood. In this article, we used the verb noun pairs to express functions. By referring to the research results of Pahl et al. 24 and Stone and Wood, 34 a standardized system of mechanical function was established based on the analysis of design instances. The standardized system has many layers: in the first layer, there are three verbs ( transfer, transform, and provision ) and three nouns ( material, energy, and signal ). Thus, nine basic functions are formed by these three verbs and three nouns. The standardized system of mechanical function is shown in Table 2. Similar to the function system, we also built a standardized performance system which has three layers. In the first layer, there are three basic performances including improve performance, improve economics, and improve sociality. The standardized system of mechanical performance is shown in Table 3. The standardized system of mechanical restriction is demonstrated in Table 4. In the first layer, the restriction is divided into five types consisting of restriction of material, restriction of energy, restriction of signal, restriction of economics, and restriction of sociality.

7 Ma et al. 7 Table 2. Standardized system of mechanical function (part). First layer Second layer Third layer Fourth layer Transfer material Transport material Deliver material Translational deliver material Rotational deliver material Guide material Translate material Rotate material Separate material Divide material Partition material Extract material Transform material Transform material shape Straightening material Curl material Transform material size Amplify length Provision material Store material Stabilize material Transfer energy Transport energy Transport mechanical energy Transform energy Transform energy capacity Amplify capacity Provision energy Store energy Store resilience Transfer signal Transport signal Input signal Transform signal Transform signal capacity Amplify signal Provision signal Store signal Stabilize signal Table 3. Standardized system of mechanical performance (part). First layer Second layer Third layer Improve performance Increase performance Expand functional scope Expand functional intensity Improve functional precision Restrain side-effect Offset side-effect Reduce side-effect Remove side-effect Improve economics Reduce design cost Reduce manufacture cost Improve sociality Improve friendly man machine interface Improve environment-friendly Table 4. Standardized system of mechanical restriction (part). First layer Restrict material Restrict energy Restrict signal Restrict economics Restrict sociality Second layer Restrict material size Restrict material position Restrict material shape Restrict energy quantity Restrict energy type Restrict signal quantity Restrict signal type Restrict cost Restrict man machine interface Standardized system of structure design principle. In structure design, many people set out preferential principles to design reasonable structures. Kesselring proposed the principles of minimum production costs, minimum space requirements, minimum weight, minimum losses, and optimum handling; Leyer researched the principle of lightweight construction. 24 Pahl and Beitz set out a series of structure design principles, relating to force transmission, task division, stability, and self-help, as shown in Table 5. In Table 5, we also stated the abbreviations of all the principles in the second layer, which would be used in the following. Database establishment Database is used for storing the information in design instances and is the basis of knowledge discovery. A fine database should store the information of instances clearly and completely, should convey the relationships among the information definitely, and should be propitious to information extraction. Based on the concept of relational database, we built a physical database using SQL Server There are seven basic tables, which can store all the information of design instances, such as the basic information of instances, the structure symmetry information, the design requirements realized by structure symmetry, the structure design principles realized by structure symmetry, the standardized

8 8 Advances in Mechanical Engineering Figure 9. Physical database design. Table 5. Standardized system of structure design principle. First layer Principles of force transmission Principle of the division of tasks Principle of self-help Principle of stability and planned instability Second layer FFPUS PDSFP PMD PBF PDTDF PDTIF PSR PSB PSP PS PPI FFPUS: flowlines of force and the principle of uniform strength; PDSFP: principle of direct and short force transmission path; PMD: principle of matched deformations; PBF: principle of balanced forces; PDTDF: principle of the division of tasks for distinct functions; PDTIF: principle of the division of tasks for identical functions; PSR: principle of selfreinforcing; PSB: principle of self-balancing; PSP: principle of selfprotecting; PS: principle of stability; PPI: principle of planned instability. system of structure symmetry, the standardized system of design requirements, and the standardized system of structure design principle. The physical database design is shown in Figure 9. Association rule mining Mining association rules is an important problem in data mining and many people have focused on it for years. There are many association rule mining algorithms and the most important algorithms are Apriori 35 and FP-growth. 36 In this article, we used the CPM algorithm to mine association rules from database of mechanical structure symmetry design instances. The CPM algorithm is proposed by Ma et al., 37 which is designed for mining association rules from databases of engineering design instances. The mining results are association rules among structure symmetry, design requirement, and structure design principle. Mining results Association rules between structure symmetry and design requirement To get the information of what design requirements can be realized by certain structure symmetry and its frequent degree, the association rules of structure symmetry ) design requirement were mined. The main mining results are shown in Table 6. Association rules between structure symmetry and structure design principle To get the information of what structure design principles can be realized by certain structure symmetry and its frequent degree, the association rules of structure symmetry ) structure design principle were mined. The main mining results are shown in Table 7 (the numbers in the table indicate the support degrees and confidence degrees).

9 Ma et al. 9 Table 6. Association rules of structure symmetry ) design requirement. Symmetry Requirement Support Confidence Symmetry Requirement Support Confidence TS Expand functional range RoS Expand functional strength TS Expand functional RoS Realize function strength continuously TS Realize function RoS Stop turn bilaterally continuously TS Scatter load uniformly RoS Scatter load uniformly TS Transmit force RoS Reduce vibration TS Reduce vibration RoS Transmit force MS Expand functional ReS Expand functional strength strength MS Offset side-effect ReS Transmit rotational motion MS Fix material ReS Change rotational speed MS Scatter load uniformly STS Transmit translational material SRTS Transmit rotational STS Change rotational speed motion SRTS Expand functional STS Expand functional range strength SRTS Realize function STS Scatter load uniformly continuously RTS Output vibration SRoS Change rotational speed RTS Reduce vibration SRoS Transmit rotational motion GS Realize function SRoS Transmit rotational continuously motion GS Transmit rotational SRoS Expand force motion GS Expand functional SRoS Reduce vibration strength GS Reduce vibration SReS Output vibration TS: translation symmetry; RoS: rotation symmetry; MS: mirror symmetry; ReS: reversal symmetry; STS: scaling translation symmetry; SRTS: scaling ro translation symmetry; RTS: rotation translation symmetry; SRoS: scaling rotation symmetry; GS: glide symmetry; SReS: scaling-reversal symmetry. Table 7. Association rules of structure symmetry ) structure design principle. Symmetry Principle TS RoS MS SRoS SRTS PDTIF 23.82%, 97.37% 30.90%, 97.30% 25.75%, 93.75% 2.79%, 76.47% 0.43%, 66.67% PBF 0.21%, 0.68% 0.86%, 3.13% PMD 0.43%, 1.35% PDTDF 0.43%, 1.75% 0.21%, 0.78% PS 0.21%, 0.88% 0.21%, 0.68% 0.64%, 2.34% 0.86%, 23.53% 0.21%, 33.33% TS: translation symmetry; RoS: rotation symmetry; MS: mirror symmetry; SRoS: scaling rotation symmetry; SRTS: scaling ro translation symmetry; PDTIF: principle of the division of tasks for identical functions; PBF: principle of balanced forces; PMD: principle of matched deformations; PDTDF: principle of the division of tasks for distinct functions; PS: principle of stability. Association rules between structure design principle and design requirement To get the information of what design requirements can be realized by certain structure design principle and its frequent degree, the association rules of structure design principle ) design requirement were mined. The main mining results are shown in Table 8. Association rules between structure symmetry and structure design principle, design requirement To get the information of what design requirements can be realized by certain structure design principle and this principle can be realized by certain structure symmetry, the association rules of structure design principle ) structure design principle, design

10 10 Advances in Mechanical Engineering Table 8. Association rules of structure design principle ) design requirement. Principle Requirement Support Confidence Principle Requirement Support Confidence PDTIF Expand functional PDTIF Reduce vibration strength PDTIF Expand functional PBF Reduce second load range PDTIF Realize function PBF Offset side-effect continuously PDTIF Stop turn bilaterally PBF Stop turn bilaterally PDTIF Transmit translational PDTDF Divide material material PDTIF Fix material PDTDF Realize function continuously PDTIF Reduce functional PDTDF Locate material time PDTIF Divide material PS Locate material PDTIF Realize function PS Reduce side-effect synchronously PDTIF Scatter load uniformly PS Bear material PDTIF Improve functional PS Fix material stability PDTIF Transmit force PMD Absorb deformation PDTIF: principle of the division of tasks for identical functions; PBF: principle of balanced forces; PDTDF: principle of the division of tasks for distinct functions; PS: principle of stability; PMD: principle of matched deformations. requirement were mined. The main mining results are shown in Table 9. Design knowledge We mined a lot of association rules from instance database, but not all association rules were interesting. Useful design knowledge must be based on the analysis and abstraction of association rules which should be practical and universal. However, the specific and practical design knowledge always lacks universality; in contrast, the abstract and universal design knowledge lacks practicability. The mined association rules come from the bottom of the information of instances and lack universality. As a result, the association rules at a higher level need to be abstracted. For the practicability of association rules, we needed to analyze the related instances to judge their practicability. The association rules with both practicability and universality can become the design knowledge. Association rule abstraction The mined association rules lack universality, since they come from the bottom of the knowledge of instances. The association rules need to be abstracted to higher layers based on the standardized systems in section Instance standardized analysis. The symmetries and structure design principles in association rules are in the suitable layer, so we only need to abstract the design requirements in association rules. For association rules shown in Tables 6, 8 and 9, the abstracted results are, respectively, demonstrated in Tables Design knowledge statement According to the abstracted association rules, structure symmetries are mainly used for realizing the requirements of transfer material, transfer energy, transform energy, and improve performance, and also principally used for realizing the principle of the division of tasks for identical functions. The principle of the division of tasks for identical functions is most widely applied in mechanical structure symmetry design and can realize the major requirements. By analyzing relevant instances, the association rules that are not practical and universal can be eliminated, and the design knowledge is proposed based on the remaining association rules. The proposed design knowledge can guide the application of structure symmetry in realizing the design requirements and can help to improve the technical, economic, and social performance of the mechanical products. Design knowledge of structure symmetry in realizing the functions with space symmetry. According to the theory of conceptual design, a function can be decomposed into multiple sub-functions. If the sub-functions are all the same or change regularly, when the positions of subfunctions work in space symmetrically, the function has

11 Ma et al. 11 Table 9. Association rules of structure symmetry ) principle, requirement. Symmetry Principle Requirement Support Confidence Symmetry Principle Requirement Support Confidence TS PDTIF Expand functional range MS PDTIF Expand functional range TS PDTIF Expand functional strength MS PDTIF Fix material TS PDTIF Transmit translational material MS PDTIF Expand functional strength TS PDTIF Scatter load uniformly MS PDTIF Scatter load uniformly TS PDTIF Transmit force MS PDTIF Reduce second load TS PDTIF Reduce vibration MS PBF Reduce second load RoS PDTIF Expand functional strength GS PDTIF Realize function continuously RoS PDTIF Realize function continuously GS PDTIF Expand functional strength RoS PDTIF Expand functional range GS PDTIF Transmit translational material RoS PDTIF Stop turn uniaxially SRTS PDTIF Change rotational speed RoS PDTIF Scatter load uniformly SRTS PDTIF Expand functional strength RoS PDTIF Absorb deformation SRTS PDTIF Realize function continuously RoS PDTIF Reduce vibration STS PDTIF Change rotational speed RoS PMD Reduce vibration STS PDTIF Expand functional range ReS PDTIF Expand functional strength STS PDTIF Transmit translational material ReS PDTIF Stop turn bilaterally STS PDTIF Scatter load uniformly ReS PDTIF Change rotational speed SMS PDTIF Expand functional range TS: translation symmetry; PDTIF: principle of the division of tasks for identical functions; MS: mirror symmetry; PBF: principle of balanced forces; RoS: rotation symmetry; GS: glide symmetry; SRTS: scaling ro translation symmetry; STS: scaling translation symmetry; PMD: principle of matched deformations; ReS: reversal symmetry; SMS: scaling mirror symmetry. Table 10. Abstracted association rules of structure symmetry ) design requirement. Requirement Symmetry Transfer material Transform material Provision material Transfer energy Transform energy Provision energy Improve performance Improve sociality Others MS 9.35%, 36.36% 0.14%, 0.55% 0.07%, 0.28% 4.18%, 16.25% 5.52%, 21.49% 0.35%, 1.38% 5.74%, 22.31% 0.07%, 0.28% 0.28%, 1.11% RoS 7.79%, 26.44% 0.21%, 0.72% 4.18%, 14.18% 6.16%, 20.91% 10.84%, 36.78% 0.28%, 0.96% TS 3.54%, 21.93% 0.07%, 0.44% 1.77%, 10.96% 3.12%, 19.30% 0.07%, 0.44% 7.37%, 45.61% 0.21%, 1.32% RTS 1.84%, 16.56% 1.20%, 10.83% 5.03%, 45.22% 2.12%, 19.11% 0.92%, 8.28% ReS 0.64%, 13.24% 1.20%, 25.00% 1.84%, 38.24% 1.13%, 23.53% SRoS 1.84%, 28.57% 0.14%, 2.20% 1.70%, 26.37% 1.42%, 21.98% 0.14%, 2.20% 1.20%, 18.68% STS 0.92%, 41.94% 0.07%, 3.23% 0.42%, 19.35% 0.35%, 16.13% 0.35%, 16.13% 0.07%, 3.23% SRTS 0.50%, 30.43% 0.35%, 21.74% 0.35%, 21.74% 0.07%, 4.35% 0.35%, 21.74% GS 0.21%, 14.29% 0.07%, 4.76% 0.35%, 23.81% 0.14%, 9.52% 0.71%, 47.62% MS: mirror symmetry; RoS: rotation symmetry; TS: translation symmetry; RTS: rotation translation symmetry; ReS: reversal symmetry; SRoS: scaling rotation symmetry; STS: scaling translation symmetry; SRTS: scaling ro translation symmetry; GS: glide symmetry.

12 12 Advances in Mechanical Engineering Table 11. Abstracted association rules of structure design principle ) design requirement. Requirement Principle Transfer material Transform material Provision material Transfer energy Transform energy Provision energy Improve performance Improve sociality Others PDTIF 23.95%, 25.00% 0.91%, 0.95% 0.11%, 0.12% 9.58%, 10.00% 18.93%, 19.76% 0.46%, 0.48% 41.05%, 42.86% 0.68%, 0.71% 0.11%, 0.12% PDTDF 0.46%, 50.00% 0.46%, 50.00% PBF 0.11%, 10.00% 0.23%, 20.00% 0.80%, 70.00% PS 1.03%, 56.25% 0.11%, 6.25% 0.68%, 37.50% PDTIF: principle of the division of tasks for identical functions; PDTDF: principle of the division of tasks for distinct functions; PBF: principle of balanced forces; PS: principle of stability. Table 12. Abstracted association rules of structure symmetry ) principle, requirement. Symmetry Principle Requirement Support Confidence Symmetry Principle Requirement Support Confidence TS PDTIF Transfer material RoS PDTIF Transfer material TS PDTIF Transfer energy RoS PDTIF Transfer energy TS PDTIF Transform energy RoS PDTIF Transform energy TS PDTIF Improve performance RoS PDTIF Improve performance TS PDTIF Improve sociality RoS PS Transfer material TS PDTDF Transfer material RoS PS Improve performance TS PDTDF Improve performance RoS PBF Transfer energy TS PS Transform energy RoS PBF Improve performance TS PS Improve performance RoS PMD Transfer energy MS PDTIF Transfer material ReS PDTIF Transfer energy MS PDTIF Transfer energy ReS PDTIF Transform energy MS PDTIF Transform energy ReS PDTIF Improve performance MS PDTIF Provision energy GS PDTIF Transfer material MS PDTIF Improve performance GS PDTIF Transfer energy MS PS Transfer material GS PDTIF Transform energy MS PS Improve performance GS PDTIF Improve performance MS PBF Transform energy RTS PDTIF Transfer material MS PBF Improve performance RTS PDTIF Transform energy STS PDTIF Transfer material RTS PDTIF Improve performance STS PDTIF Transfer energy SRoS PDTIF Transfer material STS PDTIF Transform energy SRoS PDTIF Transform energy STS PDTIF Improve performance SRoS PDTIF Improve performance SRTS PDTIF Transfer energy SRoS PS Transfer material SRTS PDTDF Transfer material SRoS PS Improve performance TS: translation symmetry; PDTIF: principle of the division of tasks for identical functions; RoS: rotation symmetry; PS: principle of stability; PDTDF: principle of the division of tasks for distinct functions; PBF: principle of balanced forces; PMD: principle of matched deformations; MS: mirror symmetry; ReS: reversal symmetry; GS: glide symmetry; RTS: rotation translation symmetry; STS: scaling translation symmetry; SRoS: scaling rotation symmetry; SRTS: scaling ro translation symmetry.

13 Ma et al. 13 Figure 10. Symmetric structure and space symmetric function. static space symmetry to realize the time symmetric function, as shown in Table 13. The end-to-end translation symmetry, rotation symmetry, and rotation translation symmetry are all continuous static space symmetries. In an excavator as shown in Figure 12, multiple digging buckets are translation symmetrically set end-to-end, which can realize the function of excavate material continuously. Figure 11. A supply equipment. space symmetry. A space symmetric function can be realized by a structure with corresponding static space symmetry (Figure 10). Each component of symmetric structure realizes a sub-function. If the sub-functions vary regularly, the realized structure should have scaling symmetry and the components of the structure should vary regularly too. Taking a supply equipment, for example, as shown in Figure 11, to improve the efficiency, the supply equipment needs to carry multiple materials at one time. So the function of transfer material needs to be decomposed into multiple same sub-functions, and the space positions of these sub-functions are translation symmetrically arranged above the pedestal. Due to the space symmetry of the function, there are multiple translation symmetric bevels (b) and push rods (a) in the supply equipment; each set of bevel and push rod can transfer a material independently. Design knowledge of structure symmetry in realizing the functions with time symmetry. If a function needs to be continuously realized or to be realized with a regular intermittence, the function has time symmetry. We can use a structure with dynamic symmetry or continuous Design knowledge of structure symmetry in transferring material. The function of transfer material is timecontinuous and can be realized by a structure with continuous static space symmetry. There are two methods to realize the function of transfer material. The first one is to design multiple parts with same geometry and set them on the path of transferring material. These parts cannot realize the function separately, and they must work together to achieve the material transferring function. The second method also consists of many designed parts which are fixed on the path of transferring material. However, every part can realize the function separately. Thus, the material can be transferred by these parts one by one. According to the form and shape of the material as well as the transferring distance, there are different structure symmetries for material transferring, as shown in Table 14. In a gear pump demonstrated in Figure 13, the function of transfer liquid is realized by two rotation symmetric gears with the cooperation of rotation symmetric shell. In a feedway shown in Figure 3, the function of transfer granule is realized by a rotation translation symmetric lamina. Design knowledge of structure symmetry in scattering load uniformly. If a load is large enough to exceed the bearable range of the supporter, the supporter will be damaged. By scattering load uniformly, a large load can be converted into many small loads within the bearable range of the supporter, and the uniform loads can make

14 14 Advances in Mechanical Engineering Table 13. Symmetric structure and time symmetric function. Time symmetric function Structure symmetry Sketch map Continuous function Continuous end-to-end translation symmetry Continuous rotation symmetry Continuous rotation translation symmetry Intermittent function Discrete end-to-end translation symmetry Discrete rotation symmetry Discrete rotation translation symmetry Table 14. Symmetric structure and transferring material. Characteristic of function Structure symmetry Sketch map Form of material Solid Powder Continuous symmetry Granule and big material Discrete symmetry Liquid and gas Continuous symmetry Transferring distance Long Translation symmetry Short Translation, rotation, or rotation translation symmetry the supporter to sustain force equably. By designing many parts with same geometry, arranging these parts in symmetric locations, and making them to output or transmit the load together, a large load can be scattered uniformly. According to the different surface shapes of the load works on, there are different structure symmetries used in scattering load uniformly, as shown in Table 15. In a flatcar shown in Figure 14, there are 32 double translation symmetric wheels which can scatter the load uniformly and reduce the damage for the road. Figure 12. An excavator. Design knowledge of structure symmetry in reducing vibration. In a running machine, the vibration is unavoidable. Too much vibration can cause negative influence on the working of the machine and even can

15 Ma et al. 15 damage the parts of the machine. The appropriate application of symmetric elastic parts can effectively dampen the vibration. By designing multiple parts and arranging them symmetrically, the force that causes the vibration can be dispersed, reduced, or counteracted. Figure 13. A gear pump. In Table 16, some common structures that can weaken or dampen vibration are listed. In a viscous frictional absorber as shown in Figure 15, multiple isolate chambers, which are filled with viscous liquid, are rotation symmetric. These symmetric isolate chambers can make the axle and the shell to sustain force equably and reduce the vibration during the axial rotation process. Design knowledge of structure symmetry in improving strength of part. Although the strength of a part mainly depends on its material, the structure of the part can also influence its strength. There are many methods to improve the strength of a part. From the view of structure design, there are mainly two methods to achieve the improvement of strength. The first one is to design a part with symmetric structure, such as rotation symmetry and mirror symmetry. Thus, the strength of the part can be distributed uniformly and the weakest area of strength can be avoided. As a result, the bearing capacity of the part can be improved. The second method is to add the assistant structures symmetrically (such as strengthening rib) on the part which can also improve the strength effectively. Figure 16 shows a strengthening rib design, which consists of three sets of strengthening ribs arranging mirror symmetrically and translation symmetrically. These symmetric ribs can improve the strength of the part significantly. Figure 14. A flatcar. Table 15. Symmetric structure and scattering load uniformly. Characteristic of function Structure symmetry Sketch map Surface shapes of the function works on Line Straight Translation symmetry Curve Scaling translation symmetry Flat surface Translation, rotation, or glide symmetry Curved surface Cylindrical Mirror, rotation, or multiple rotation symmetry Spherical Mirror or rotation symmetry

16 16 Advances in Mechanical Engineering Table 16. Symmetric structure and reducing vibration. Types of parts Structure symmetry Sketch map Elastic parts Translation, mirror symmetry, or their combination Rotation translation symmetry Nonelastic parts Rotation, mirror, or translation symmetry Figure 15. A viscous frictional absorber. Figure 17. An expansion clamp. combined symmetric structure should be applied. In an expansion clamp shown in Figure 17, by pulling pole 1 upward, expansion forces on poles 1 and 5 can be generated through pole 2 to grip object 6. There are three translation symmetric poles (poles 2, 3, and 4) between poles 1 and 5. Under the synchronous effects of poles 2, 3, and 4, the expansion forces can be enhanced evidently, so the strength of the function of grip material is improved. Figure 16. A strengthening rib design. Design knowledge of structure symmetry in improving strength of function. By setting repeated structures to synchronously work on the object, the strength of function can be improved. According to the different surface shapes of the function works on, the positions of multiple structures can be arranged as forms of translation symmetry, rotation symmetry, and glide symmetry, as shown in Table 17. If the working surface is very large, Design knowledge of structure symmetry in expanding working range of function. The working range of function includes space range and time range. To expand the working space range of function, we can design many repeated structures and set them on several discrete or continuous locations, which can expand the working space range by letting the structures working together or sequentially. The type of symmetry of the structures is determined by the shape of working space and

17 Ma et al. 17 Table 17. Symmetric structure improving strength of function. Characteristic of function Structure symmetry Sketch map Surface shapes of the function works on Flat surface Small Translation or mirror symmetry Big Glide symmetry or multiple translation symmetry Curved surface Cylindrical Mirror, rotation, or reverse symmetry, multiple rotation symmetry Spherical Mirror or rotation symmetry Table 18. Symmetric structure expanding working range of function. Working range of function Structure symmetry Sketch map Space range Continuous Continuous symmetry Discrete Discrete symmetry Time range Continuous Continuous symmetry Discrete Discrete symmetry usually is translation symmetry or rotation symmetry, as shown in Table 18. There are multi-layer translation symmetric brackets in a crane as shown in Figure 18. By varying the quantity of bracket layer, the height of the crane can be changed, which can expand the working space range. To expand the working time range of function, we can design many parts with same geometry, locate them translation symmetrically or rotation symmetrically, and let them work sequentially. In a digging bucket in Figure 19, multiple buckets are arranged rotation symmetrically, which come to the working spot one by one during operating process. Their continuous work style can expand the working time range effectively. Although some design knowledge of structure symmetry in realizing design requirements was introduced, other design knowledge still exists which can be summarized from association rules and will be introduced in other articles. Conclusion Symmetry, a commonly observed phenomenon, is widely researched in many fields. In mechanical systems, there are also many existences of symmetry, especially for structure symmetry. Reasonable application of structure symmetry can make design requirements to be better realized. In this article, the technology of knowledge discovery is used to mine association rules from mechanical structure symmetry design instances, and some design knowledge is proposed to guide the application of structure symmetry. The authors collected more than 1000 of structure symmetry design

18 18 Advances in Mechanical Engineering requirement, structure symmetry ) structure design principle, structure design principle ) design requirement, and structure symmetry ) structure design principle, design requirements were mined from instance database. By analyzing mining results and relevant instances, the practical and universal design knowledge of structure symmetry to meet the design requirements was proposed, which can conduct the application of structure symmetry and improve the technical, economic, and social performance of the mechanical products. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Science Foundation of China (No ) and the K.C.Wong Magna Fund in Ningbo University. Figure 18. A crane bracket. Figure 19. A digging bucket. instances; built standardized systems of structure symmetry, design requirement, and structure design principle; and analyzed the instances on the basis of the standardized systems. A physical database was designed and the information of design instances was stored. The association rules including structure symmetry ) design References 1. Marcellini S. When Brachyury meets Smad1: the evolution of bilateral symmetry during gastrulation. Bioessays 2006; 28: Yang CN. Conceptual beginnings of various symmetries in twentieth century physics. Chinese J Phys 1994; 32: Fernandez E. Symmetry: key to nature and natural philosophy. Metascience 2004; 13: Dolgy DV, Kim DS, Kim T, et al. Identities of symmetry for Carlitz q-bernoulli polynomials arising from q-volkenborn integral on Zp under symmetry group S3. Adv Stud Theor Phys 2014; 8: Yubin K, Zheng Y and Astrom K. Partial symmetry in polynomial systems and its applications in computer vision. In: 2014 IEEE conference on computer vision and pattern recognition (CVPR), Columbus, OH, June 2014, pp New York: IEEE. 6. Nikolic V, Radovic L and Markovic B. Symmetry of twins. Symmetry 2015; 7: Barrenscheen J. Die systematische Ausnutzung von Symmetrieeigenschaften beim Konstruieren. Doctoral Thesis, Institut fu r Konstruktionslehre, Maschinen- und Feinwerkelemente, Technischen Universita t, Berlin, Feng PE, Ma ZY and Qiu QY. Research on symmetry ontology: from natural science to engineering science. Prog Nat Sci 2008; 18: Ma ZY, Qiu QY and Feng PE. Concept system and application method of mechanical symmetry. J Zhejiang Univ 2010; 44: Feng PE, Zeng LB and Qiu QY. Research on mechanical function symmetry architecture and its application. J Mech Eng 2012; 48: 1 10.