頂置式四缸內(nèi)燃機凸輪配氣機構(gòu)設計及運動仿真

48頁 16000字數(shù)+論文說明書+任務書+4張CAD圖紙【詳情如下】

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內(nèi)容摘要 

     該設計是我在學校的最后一個設計，在設計過程運用到了很多的知識，PROE的使用，凸輪的設計，軸的應力計算，氣門頂桿的設計，以及對整體裝配的理解，尤其是運動仿真那一塊，是我的薄弱環(huán)節(jié)，同時在設計過程中，由于很多的事情耽擱了，造成了后期時間緊，同時工作量巨大的結(jié)果，這是自己的一個不足。對于知識的利用與融會貫通，基礎知識不扎實，在設計過程中體現(xiàn)了出來，不過通過同學，老師的幫助，終于是克服了種種困難，使圖形得以完成。

     工作大致內(nèi)容：任務書，開題報告，翻譯，模型設計，計算說明以及最后的運動仿真。

     運用到的知識非常之多，PROE由于之前不怎么熟悉，所以剛接手的時候非常迷茫，后來通過同學，老師的指導，使自己熟悉了怎么運用，同時對于四沖程機構(gòu)有了進一步的了解。

     通過這次設計，發(fā)現(xiàn)了自己很多的不足，在不足中得以成長，同時也認識到應該合理分配時間，分清什么重要的事先做，做事得有調(diào)理。

     關(guān)于配氣機構(gòu)在生活中的應用，非常廣泛，在現(xiàn)代社會中越來越多的人開車，對于車子的安全性能進一步加強，因此配氣機構(gòu)的合理性以及科學性更是重中之重。     

目錄

概述

1配氣機構(gòu)的功用   ……………………………………………… …………………3

    2配氣機構(gòu)的設計要求   …………………………………………  ……………… 4

    3配氣機構(gòu)計算參數(shù)的確定    ………………………………………… …………… 5

1凸輪軸的設計：

1凸輪軸的設計要求  …………………………………………………………………6

2凸輪軸的選材 …………………  …………………………………………………… 7

3凸輪軸的結(jié)構(gòu) ……………………………… ………………………………………… 8

4凸輪軸的支承軸頸軸承的材料    …………………… 9                       

5凸輪軸的定位方式  ………………………………………………………………… 10

6凸輪軸的最小尺寸定位方式    11

7凸輪軸的熱處理 11

8凸輪軸的磨損形式  12

9凸輪軸的計算13

10凸輪軸強度校核計算14

2氣門組的設計

  21氣門的設計 ………………………………… …………………… ………  18

       211氣門設計的基本要求 20

       212氣門的工作條件分析 22

       213氣門材料的選擇 23

       214氣門頭的設計 24

       215氣門桿的設計 25

22氣門旋轉(zhuǎn)機構(gòu)的設計    ………………… …………………………………………26

23氣門座圈的設計   ………………………………………………………………26

24氣門導管的設計   ………………………………………………………………28

25氣門的主要損壞形式和預防措      ………………………………………………29

3氣門彈簧的設計

31氣門彈簧的設計要求   ……………………………………………………………30

32氣門彈簧的作用  ………………………………………………………………………31

33氣門彈簧的工作條件   ……………………………………………………………31

34氣門彈簧的結(jié)構(gòu)   ………………………………………………………………31

35氣門彈簧的選材  …………………………………………………………31

36氣門彈簧特性曲線與氣門慣性力曲線的配合      ……………………32

37氣門彈簧的有關(guān)計算   ………………………………………………33

         371彈簧的最大彈力 33

         372彈簧最小的彈力 34

         373彈簧的剛度  34

         374彈簧變形 34

         375內(nèi)、外彈簧之間的負荷分配 35

         376內(nèi)外彈簧的剛度  35

         377彈簧的尺寸  36

         378提高氣門彈簧疲勞強度的措施 …… … ……… …………… …37

4凸輪軸配氣機構(gòu)建模設計…………………………………………………………   37

41工作裝置零件建模  ………………………………………………………… ………38

42氣門的建模 …………………  …………………………………………………   … 38

43彈簧壓帽的生成 ……………………………… ……………………………………… 40

44箱體的生成    …………… ……… ……… ……………… 42                        

45裝配部件的裝配生成  ……………………………………………………………… 43

5凸輪軸配氣機構(gòu)仿真設計…………………………………………………………   44

51概述  ………………………………………………………… ………… ……………45

52凸輪配氣結(jié)構(gòu)的機械運動仿真 …………………  ……………………   … 46

6參考文獻 ……………………………………………………………………………………47

7致謝 ………………………………………… …48

頂置式四缸凸輪配氣機構(gòu)的設計

概述

1、配氣機構(gòu)的作用：它是完成換氣過程，根據(jù)發(fā)動機氣缸的工作循環(huán)次序，定時地開啟和關(guān)閉進、排氣門，不斷地用新鮮的氣體來交換氣缸內(nèi)上一循環(huán)的的廢氣。氣門的布置型式方式有頂置式和側(cè)置式，如圖1-1所示：

2、配氣機構(gòu)的要求：

     對于一個正常工作的配氣機構(gòu)具有如下的要求：

①　振動、噪聲較小，且工作可靠和耐磨。

②　進、排氣門的時間充足，泵氣損失小，配氣正時恰當，在排氣過程中能較好的排出廢氣，進氣過程中能吸入較多的新鮮空氣，因而使發(fā)動機具有較高的充量系數(shù)和合適的扭矩特性。

③　結(jié)構(gòu)簡單、緊湊。

④　為了減輕慣性負荷，使配氣機構(gòu)運動零件的質(zhì)量減到最小。    

3、配氣機構(gòu)設計的計算參數(shù)確定：

 從確定氣門座處的通過截面   以及確定喉口流通截面 開始。氣閥處的流通截面積根據(jù)氣體不可壓縮連續(xù)流動的條件確定，也即在額定轉(zhuǎn)速I情況，氣門最大升程時，按氣門座截面處假設的平均速度來確定。

已知：氣缸直徑D=95，

   氣道喉口的最大直徑D，配氣機構(gòu)的結(jié)構(gòu)方案以及燃燒是的形式都已給定的情況下，氣門布置在氣缸上可能性的限制。進氣門 的數(shù)值應大于下列規(guī)定的范圍：

      采用氣門頂置式： ，  則可以得到： ， 根據(jù)柴油機的195B的結(jié)構(gòu)，選擇 =36mm,

排氣門的氣道喉口的直徑，通常取得比進氣門的氣道喉口直徑小10%~~20%，氣閥升程h時，具有圓錐密封面之氣門的流通截面為：

     式中a—氣門頭斜面角（現(xiàn)代發(fā)動機上，a=45度);  氣門的升程，取值一般是氣門頭的25%左右，氣門頭的直徑是40mm，

則：    =10mm

   ∴喉口的直徑經(jīng)校核取值正確。

 1凸輪軸的設計

凸輪軸的布置型式：

1、下置：凸輪軸正時齒輪直接與曲軸正時齒輪嚙合。 

2、中置：推桿短，要加入中間傳動裝置。

3、上置：凸輪軸通過搖臂或直接來驅(qū)動氣門，要用惰輪、皮帶、鏈條，及張緊裝置。結(jié)構(gòu)復雜，用于高速強化的轎車發(fā)動機。Step2系統(tǒng)彈出如圖所示“測量結(jié)果”對話框，在該對話框中進行下列操作。

（1） 選取圖形類型：單擊“圖形類型”區(qū)域中的 ，從彈出的下拉列表中選擇“測量與時間”。

（2） 創(chuàng)建一個測量：單擊創(chuàng)建測量圖標 ，系統(tǒng)彈出如圖5214所示的“測量定義”對話框，在該對話框中進行下列操作。

a 鍵入測量名字：在該對話框中的名字文本框中鍵入測量名字“舉升臂連接軸速度與時間關(guān)系”。

b 選擇測量類型：單擊“類型”區(qū)域中的 ，從彈出的下拉列表中選擇“速度”。

c 選取測量點：選取模型中的凸輪軸。

d 選取評估方法：單擊“評估方法”區(qū)域中的 ，從彈出的下拉列表中選擇“每一時間步距”。

e 單擊“確定”按鈕系統(tǒng)立即將新建測量添加到如圖8213

f 所示的“測量結(jié)果”對話框中。仿真效果如圖8214所示：

6參 考 文 獻

專著：

【1】付白學馬彪藩旭峰現(xiàn)代汽車電子技術(shù)，20083

【2】史紹熙柴油機設計手冊北京：中國農(nóng)業(yè)機械出版社，1984

【3】UTOCAD 2004簡明教程，科學出版社。2004

【4】建新內(nèi)燃機理論與設計北京：人民交通出版社，2009

【5】 李澄，吳天生，聞百橋機械制圖北京：高等教育出版社，2003

【6】孝達金屬工藝學 北京：高等教育出版社，1997

【7】華大年，華志宏，呂靜平連桿機構(gòu)設計上海：上?？萍技夹g(shù)出版社，1995

【8】吳宗澤機械零件設計手冊北京：機械工業(yè)出版社，2003

【9】http://wwwqutoedcation/com/carcare/intro/htm      

7致  謝

本論文的能夠順利的完成，灌注了齊導師誨人不倦的關(guān)懷、指導和教誨，他嚴謹?shù)目茖W態(tài)度，嚴謹?shù)闹螌W精神。從課題的選擇到項目的最終完成，齊從謙老師給我細心的指導和不懈的支持。

在設計過程中，這個也是我在校期間最后的一個設計，感謝所有幫助過我，并且和我一起努力，克服一個個難題，相信在今后的道路中，我們也能一直像現(xiàn)在這樣面對困難。

同時也由于自身原因，在設計過程中，經(jīng)常由于工作而耽誤，導致了無法如期完成設計，但是在導師的幫助下，我也盡自己最大的努力將這個設計完成，這其中傾注了導師太多太多的精力。最后，由衷地向所有在校園曾經(jīng)關(guān)心和幫助過我老師和同學表示最誠摯的謝意！ 

Decomposition-Based Assembly Synthesis ofSpace Frame Structures Using Joint LibraryThis paper presents a method for identifying the optimal designs of components and joints in the space frame body structures of passenger vehicles considering structural characteristics, manufacturability, and assembleability. Dissimilar to our previous work based on graph decomposition, the problem is posed as a simultaneous determination of the locations and types of joints in a structure and the cross sections of the joined structural frames, selected from a predefined joint library. The joint library is a set of joint designs containing the geometry of the feasible joints at each potential joint location and the cross sections of the joined frames, associated with their structural characteristics as equivalent torsional springs obtained from the finite element analyses of the detailed joint geometry. Structural characteristics of the entire structure are evaluated by finite element analyses of a beam-spring model constructed from the selected joints and joined frames. Manufacturability and assembleability are evaluated as the manufacturing and assembly costs estimated from the geometry of the components and joints, respectively. The optimization problem is solved by a multiobjective genetic algorithm using a direct crossover. A case study on an aluminum space frame of a midsize passenger vehicle is discussed. (DOI: 10.1115/1.1909203)Keywords: design for manufacturing, assembly synthesis, structural design, aluminumspace frame1 IntroductionAlthough often ideal from a structural viewpoint, monolithic designs are not a realistic solution for complex structures, such as automotive bodies, considering the cost-effectiveness of manufacturing processes. As a result, most structural products are designed as assemblies of components with simpler geometries. During the conceptual stage of such products, designers need to determine the set of components and the methods of joining the components, by decomposing the entire product geometry. In the automotive industry, for example, a handful of basic decomposition schemes considering geometry, functionality, and manufacturing issues have been used in this process. However, these decomposition schemes depend mainly on the designers’ experience, which may cause the following problems directly related to the component and joint configurations:? Insufficient structural stiffness. Components and joining methods specified by designers cannot meet the desired stiffness of the final assembly because of the inappropriate allocation of joints, which are less stiff than components.? Insufficient manufacturability. Components and joining methods specified by designers cannot be economically manufactured because of the inappropriate geometry in components and joints. Therefore, a cost-effective and systematic optimization method, which can be used in determining components set by considering overall structural characteristics, manufacturability, and assembleability, will be of significant impact on the industry. As such, this paper presents a method for identifying the optimal designs of components and joints in space frame body structures of passenger vehicles, considering structural characteristics, manufacturability, and assembleability. Dissimilar to our previous work based on graph decomposition (1,2), the problem is posed as a simultaneous determination of the locations and types of joints in a structure and the cross sections of the joined structural frames, selected from a predefined joint library (3). The joint library is a set of joint designs containing the geometry of the feasible joints at each potential joint location and the cross sections of the joined frames, associated with their structural characteristics as equivalent torsional springs obtained from the finite element analyses (FEA) of the detailed joint model made of solid and plate elements. To minimize the computational overhead during optimization, the artificial neural network (ANN) associated with the FEA analyses results (4,5) is built for each configuration type in the joint library by using sampled designs of feasible joints and joined frames (6) and utilized in the overall optimization problem.Structural characteristics of the entire structure are evaluated by the finite element analyses of a model made of beam elements (frames) and torsional spring elements (joints), constructed from the selected joints and joined frames. Manufacturability of components is evaluated based on the estimated manufacturing cost consisting of the costs of extrusion die, bending operation, and casting component for each joint. Assembleability is estimated by the cost of welding in the assembly process. The optimization problem is solved by a multiobjective genetic algorithm (7) using a direct crossover (8,9). A case study on an ASF of a mid-size passenger vehicle is discussed.2 Related Work2.1 DFA/DFM and Assembly Synthesis. Design for assembly (DFA) and design for manufacturing (DFM) refers to a collection of design methods that aim to identify and alleviate manufacturing and assembly problems at the product design stage. Boothroyd and Dewhurst (10), who are regarded as major estab-1Corresponding author. Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received June 13, 2004; final manuscript received November 25, 2004. Assoc. Editor: K.K. Choi. lishers of DFA/DFM concepts, suggest to reducing part count first, followed by part redesign to improve manufacturability and assembleability (11). The analyses of manufacturability and assembleability require a targeting product to be decomposed into elementary manufacturing and assembly features, such as surfaces, dimensions, tolerances, and their correlations (12). Therefore, the conventional DFA/DFM methods assume predetermined components with given geometries and suggests improvements by modifying the given geometries.Decomposition-based assembly synthesis {1,2,9,13,14} adopted in this paper, on the other hand, emphasizes the determination of components prior to the manufacturability and assembleability analyses. The method starts with no prescribed components and generates an optimal component set considering the properties, including structural characteristics of the assembled product, manufacturability, and assembleability.2.2 Automotive Body Structure Modeling. During the early design stage of automotive body-in-white (BIW), simple beam models are widely used. Although beam elements can reasonably model structural members, difficulties often arise in modeling the structural property of joints. Modeling joints as torsional springs (15) is a classic but popular method because of its simplicity, where equivalent torsional spring rates are identified from experiments or detailed FEA models made of shell elements. Lee and Nikolaidis (16) proposed a two-dimensional (2D) joint model in order to consider joint flexibility, the offset of rotation centers, and coupling effects between the movements of joint branches. Kim et al. (17) discussed the accuracy of FEA-based joint rate evaluations regarding transformation error from shell element model to spring rate and proposed their own model (18).Aiming at joint design, Long (6) presented two tools that link the performance targets for a joint in a BIW to its geometry. The first tool, called translator A, predicts the structural performance of a given joint geometry using an artificial neural network (ANN) and response surface method (RSM). The second tool, called translator B, solves the inverse problem of finding a joint geometry that meets the given performance targets, using the translator A and sequential quadratic programing (SQP). Nishigaki et al. (19) proposed a tool based on first-order analysis (FOA) to design basic layouts of automotive structures, considering models of beam and spring elements. The above works, however, are on the analyses of structural properties of joints, separately or as an integral of an overall structure, and do not addresses the automated synthesis of joint locations and designs within a BIW as addressed in this paper.2.3 Aluminum Space Frame (ASF) Design. During the last two decades aluminum has drawn significant attention from the automotive industry because of the increasing demands on highgas- mileage, lightweight, and environmentally friendly vehicles. Although aluminum has been successfully used in drivetrains and heat exchangers, its usage in the chassis and body is still under development. Since a body-in-white (BIW) accounts for approximately one third of the vehicle weight, much effort has been put on the adaptation of aluminum in BIW (20–23), resulting in a number of commercial mass-produced vehicles with the ASF, such as Acura’s NSX _24_, Audi’s A2 and A8 (25) (see Fig. 1), and BMW’s Z8 (26). Ahmetoglu (27) discussed the design of extruded profiles, bending, friction and formability of aluminum components. Chung et al. (28) studied joint designs in the ASF by comparing FE models with experimental results. Powell and Wiemer (29) and Barnes and Pashby (30,31) summarized the joining technologies currently used in aluminum structure vehicles, including resistance spot welding (RSW), gas metal arc welding (GMAW), self-piercing joint, and laser welding. In the present paper, we are providing a way of finding optimized configurations of components in the ASF considering structural response, manufacturing, and assembly process.Fig. 1 (a) Audi A2 and (b) ASF {20}3 ApproachThe proposed method consists of the following two steps (Fig. 2):1. Geometry of a given structure is transformed into a structural topology graph that represents the liaisons between basic members, the smallest decomposable components of the given structure, identified by the potential joint locations specified by the user. For each potential joint location, a corresponding joint library is built.2. The structural topology graph defined in the first step is decomposed, through an optimization process, into subgraphs representing components by assigning to some of the potential joint locations the joint types and cross sections of the joined frames, selected from the joint library. During this optimization process, the components set represented as the subgraphs is evaluated by considering (i) stiffness of the assembled structure, (ii) manufacturability of components and cast “sleeves” for joints, and (iii) assembleability of the components with the selected joints. The rest of the section describes the details of the above steps with a sample space frame structure in Fig. 3. As illustrated in Fig. 3(b), it is assumed that frames are extruded tubes, bent or welded with cast “sleeves” at joints, following a typical construction method of AFS. Fig. 2 Approaches used in this paper3.1 Overview. Step 1: Construction of structural topology graph with joint libraries. Different from our previous approach (1,2) where the basic members are specified by the designer, the present method requires the designer to specify the potential joint locations. This is to guarantee that the final design contains only the joints feasible for the available frame manufacturing and joining methods.Figure 4 illustrates an example of six potential joint locations shown as gray boxes. At each potential joint location, the designer must also specify feasible joint types to be included in the joint library. The joint library is a set of joint designs containing the geometric configurations (types) of the feasible joints, the crosssectional dimensions of the joined frames, and the welding design at each potential joint location.The joint library Ji of potential joint location i is defined as a tripleJi = (Ti,Si,Wi) (1)where Ti , Si, and Wi are the set of the feasible geometric configuration types, the set of feasible cross sections of the intersecting frames, and the set of feasible welding designs, respectively, at potential joint location i. Since multiple frames intersect at a potential joint location, the elements si of Si is a vector where FSk is the set of valid beam cross section designs for frame k in the structure, FJi is the set of the intersecting frames at potential joint location i, and ︱FJi︱= nfi. For example, the joint library J1 at the potential joint location 1 of Fig. 4 is T1 ,S1 ,W1 with T1=｛t1.0 , t1.1 , t1.2 , t1.3｝.The structural property of a joint is determined by a joint configuration type, a cross-section design of the joined frames, and a weld design. As described in Section 3.2, an ANN is constructed for each joint configuration type, in order to represent the mapping between the joint design variables (cross section and weld designs) and its structural property (torsional spring rates).With given potential joint locations, basic members in the structure can be identified as shown in Fig. 5(a). Then, the structural topology graph G=(V,E) is constructed from basic members such that the basic member mi is represented as node ni in V, and the liaison between two basic members mi and mj is represented as edge e={ni ,nj} in E.Figure 5(b) illustrates the structural topology graph G with seven nodes corresponding to the seven basic members in Fig. 5(a) and ten edges connecting the adjacent nodes.Step 2: Creating optimal components set design using optimization procedure. Different from our previous approach [1,2] where structural topology graph G is decomposed by removing its edges, the present method decomposes G by selecting a joint configuration type in the library at each potential joint location. From the selected joint configuration types, the corresponding edges in G are removed.For example, by selecting joint configuration type t1.2 in T1 for the joint location 1 in Fig. 6, the corresponding edges e1={n1 ,n2} and e2={n1 ,n6} are removed. The motivation behind the new approach over the simple removal of edges (as in our previous work) is to establish one-to-one mapping between the topology of G and joint configuration designs. With the simple edge removal, multiple topologies of G can correspond to a joint configuration type. For example, all possible joint configuration types involving three frames (e.g., location 1 in Fig. 6) is 5, while number of possible graphs with three nodes is 2number of edges=23=8. This is because the case where all three frames are connectedcan be represented by four different topologies in the graph. The simple edge removal, therefore, often yields overly connected components, which can be prevented by using the new approach above.The selection of joint configuration types and the removal of the corresponding edges in G result in subgraphs of G, each of which corresponds to a component. The cross-sectional dimensions of a component are then set as the averages of the ones of the joining frames associated with the selected joint configuration types in the component, which are subsequently used for retrieving the precomputed structural properties of the joints from the joint library. Figure 6, for example, shows three subgraphs (Fig. 7(a)), and the corresponding components (Fig. 7(b)) resulted from the selection of the joint types in The optimal decomposed structures with component and joint designs are obtained using the decomposition procedures described above through an optimization loop for three objectives: (i) stiffness of the assembled structure, (ii) manufacturability of components and cast “sleeves” for joints, and (iii) assembleability of the components with the selected joint types.3.1.1 Structural Stiffness. The structural stiffness of the assembled structure is evaluated as a negative of the magnitude of total displacements at specific locations of the assembled structure under given loading conditions. The displacements are calculated with finite element analyses, where the components and joints are represented by beam elements and torsional spring elements, respectively.For example, a T-joint in Fig. 8(a) is modeled as three beam elements connected by torsional spring elements k0 , k1, and k2, each of which has torsional stiffness (rate) around three local orthogonal axes attached to the joint. Note that the relative translationsof these elements are constrained. The section properties of the beam elements are obtained from the cross-sectional dimensions of the components. The rate of the torsion spring elements are estimated by the finite element analyses of the detailed model of a joint, where frames are modeled with plate elements, a cast “sleeve” is modeled as solid elements, and welds joining the frames and the sleeve are modeled as plate elements, as illustrated in Fig. 9.Figure 10 illustrates the loading and boundary conditions for calculating torsional spring rates of in-plane rotation. To facilitate the load application and the measurement of distortion angles, a rigid beam element is added to the center of the frame, subject to rotation. In Fig. 10, distortion angles 0 , 1, and 2 account for the effects of k1 and k2 , k0, and k2, and k0 and k1, respectively, in Fig. 7. Assuming moment arm length L, which is measured as the distance from the rotational center to the point at which the loading P is applied, the following equations are used to estimate k0 , k1, and k2 for in-plane rotation:The other two components of torsional spring rates are calculated in a similar manner. The values of the torsional spring rates for typical joint types, cross-sectional dimensions of the joined frames, and amount of welds are precomputed to produce a set of training data for an artificial neural network (ANN) that implements the joint library. Similar to the translator A’s in (6), this approach allows the spring rates of a joint to simply be retrieved from the joint library without computational overheads during optimization.3.1.2 Component Manufacturability. The manufacturability of components is evaluated as a negative of the total cost of producing components. As stated earlier, it is assumed that frames are extruded tubes, bent or welded with cast “sleeves” at joints, following a typical construction method of AFS. For example, the design in Fig. 11(a) is composed of three frames (Fig. 11(b)) and four cast sleeves (Fig. 11(c)). The cost of producing components is estimated by the sum of the cost of extrusion die (assumed as proportional to the size and complexity of the frame cross sections) and the cost of bending operations (assumed as proportionalto the number of bending). The cost of producing cast sleeves is estimated by the cost of casting, which is assumed as simply proportional to its volume.3.1.3 Component Assembleability. The assembleability of components is calculated as a negative of the total cost of joining. In this paper, the method of joining is assumed to be the GMAW, which is widely used for the ASF (27). The welds are applied between the frames and the cast sleeves at joints. The cost is assumed to be proportional to the volume of total welds, which can be calculated from the total welding length multiplied by weld thickness.3.2 Mathematical Formulation.3.2.1 Definition of Design Variables. A design is uniquely specified by (i) the joint configuration types at all potential joint locations, (ii) the cross-sectional dimensions of all frames (basic members), and (iii) the welding designs at all joints, which are represented by the following three vectors x, y, and z, respectively:x ∈ T0 ×T1 × … ×Tn?1y ∈ FS0 × FS1 × … ×FSB?1z ∈W0 × W1 × … ×Wn?1 (4)where Ti is the set of feasible joint configuration types at potential joint location i (Eq. (1)), n is the number of the potential joint locations, FSk is the set of valid beam cross-section a designs for frame k in the structure (Eq. (2)), B is the number of frames in the structure, and Wi is the set