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Preform Optimization for Bevel Gear of Warm Forging Process
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Procedia CIRP 00 (2017) 000–000 Procedia CIRP 72 (2018) 340–345 www.elsevier.com/locate/procedia
51st CIRP Conference on Manufacturing Systems
Preform Optimization forConference, Bevel Gear of Warm Process 28th CIRP Design May 2018, Nantes, Forging France Hong-Seok Parka*, Febriani Risky Ayub,and Saurabh Kumarcarchitecture of A new methodology to analyze the functional physical University of Ulsan, School of Mechanical Engineering, 93 Daehak-ro, Nam-gu, Ulsan 44610, Korea existing products for an assembly oriented product family identification a,b,c
* Corresponding author. Tel.: +82-(0)52-259-2294; fax: +82-(0)52-259-1680. E-mail address: [email protected]
Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat
École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France
Abstract
* Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]
Preform design is an important aspect in consideration of the improvement of quality and reduction of cost in warm forging process. Bevel gear being a key component of drive mechanisms in automotive industry requires lot of attention to its complex shape design optimization. A multiobjective optimization problem in designing preforms of warm forging for bevel gear by using genetic algorithm is solved and presented in this Abstract paper. The main objective of this study is to design an optimal preform part, which can get a better quality and the reliable preform shape systematically and theoretically. In today’s business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of agile andThe reconfigurable production systems B.V. emerged to cope with various products and product families. To design and optimize production © 2018 Authors. Published by Elsevier Peer-review under responsibility the scientific of the 51st CIRP methods Conference Manufacturing Systems. systems as well as to choose the of optimal productcommittee matches, product analysis areon needed. Indeed, most of the known methods aim to analyze a product or one product family on the physical level. Different product families, however, may differ largely in terms of the number and Keywords: Preform optimization; Forging; Beveland Gear nature of components. This factWarm impedes an Genetic efficientAlgorithm; comparison choice of appropriate product family combinations for the production system. A new methodology is proposed to analyze existing products in view of their functional and physical architecture. The aim is to cluster these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and homogeneity of graph deformation conditions. It has been triedthe to Introduction a1.functional analysis is performed. Moreover, a hybrid functional and physical architecture (HyFPAG) is the output which depicts predict preforms for non-proportional loading condition [1]. similarity between product families by providing design support to both, production system planners and product designers. An illustrative Lu et case al. [2] developed topology optimisation method as an example of agear nail-clipper used to explain the proposed methodology. industrial study on two product families of steering columns of Bevel is a keyiscomponent of drive mechanisms in the An Bin efficient approach preform design optimization for parts thyssenkrupp France carried out to give of the proposedfor approach. automotive Presta industry dueis then to high contact ratioa first andindustrial smooth evaluation ©transmission. 2017 The Authors. by Elsevier B.V. manufactured by with complex shape. Similarly, the artificial neural network The Published bevel gears are mainly Peer-review underwhich responsibility the scientific the 28th Conference wasDesign applied to obtain2018. the optimal perform shape for the bevel metal cutting, produceoflarge quantitycommittee of metal of chips as CIRP
waste and requires a lot of manufacturing time. Warm forging as excellent mechanical properties, less raw material, good tolerance, high productivity and less costly as compared to metal cutting. The design of preforms is most important aspect for achieving 1. Introduction adequate metal distribution in forging process. The motive behind preform design is to get an optimum preform shape Due enables to thesufficient fast development the and domain of which filling of dieincavity get flash communication and an ongoing trend of digitization and allowance. digitalization, enterprises are configuration facing important However, manufacturing the determination of preform is a challenges in today’s market environments: continuing very difficult task and it needs knowledge anda skill that is tendency reduction of product times and achievedtowards by years of experience. So, development the implementation of shortened product lifecycles. addition, therehas is an increasing optimal performs shapes inInwarm forging attracted the demand being Most at the of same time global attentionofofcustomization, many researchers. them didinita for hot competition competitors allproduction over the world. This trend, forging thatwith is widely used for of parts from the simpleisshape to the complex. A simple shape preform which inducing the development from of macro to design micro is brieflyresults discussed by applying energy markets, in diminished lotminimum sizes duepotential to augmenting principle and considering volume consistencyproduction) and maximum product varieties (high-volume to low-volume [1]. To cope with this augmenting variety as well as to be able to identify possible optimization potentials in the existing 2212-8271 © 2018 The Authors. Published by Elsevier B.V. production system, it is important to have a precise knowledge Keywords: Design method; Family identification process Assembly; possesses several benefits such
gear [3]. In recent years, simulation using CAE software has been used to analyze the forging process and die stress distribution. The evaluation of process design that indicated the suitable preform design is able to achieve better material flow [4]. Similarly, Knust et al. [5] describes the development of an of the product algorithm range and characteristics manufactured and/or evolutionary designing CWR (Cross Wedge assembled in this system. In this context, the main challenge in Rolling) preforms and amount of flash simulation by software. modelling and analysis is now not only to cope with single However, the aforementioned publications indicated that the products, limited productofrange or existing product complex ashape interplay objective functions, (e.g.families, product but also to be able to analyze and to compare products define quality) with respect to perform parameters is nottoclearly new product yet, families. can be the observed that classical understood whichIt makes optimization processexisting rather product families are regrouped in design functionofofpreform clients or features. inefficient. Furthermore, initial shape and However, assembly optimization oriented product families are hardly to find. preform parameter of the bevel gear warm forging hasOn notthe been conducted. product family level, products differ mainly in two this paper, preform designed an(ii) initial mainIncharacteristics: (i) theisnumber oftheoretically components as and the shape preform related to forging electrical, process and forged part type of of components (e.g. mechanical, electronical). quality. A methodologies genetic algorithm (GA) optimization is Classical considering mainly singlemethod products developed bevel gear preform designfamilies and it is implemented or solitary,foralready existing product analyze the using Excel (Visual level basic(components for Application). An product structureVBA on a physical level) which causes difficulties regarding an efficient definition and comparison of different product families. Addressing this
Peer-review under responsibility of the scientific committee of the 51st CIRP Conference on Manufacturing Systems.
2212-8271©©2017 2018The The Authors. Published by Elsevier 2212-8271 Authors. Published by Elsevier B.V. B.V. Peer-review under responsibility of scientific the scientific committee theCIRP 51stDesign CIRP Conference Conference2018. on Manufacturing Systems. Peer-review under responsibility of the committee of the of 28th 10.1016/j.procir.2018.03.083
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evolutionary algorithm is proposed to optimize the preform shape with the parameters as form filling, the preform volume and the shape complexity. The rest of this paper is organised as follows: The basic rules of preform design is presented theoretically. The basic rules consist of the guidelines to determine the preform shape of forging based on material and cross section area. Then, the geometry data information of preform shape is described briefly for optimization. Results obtained with genetic algorithm method studied in this preform design optimization is discussed. This is followed by concluding remarks in the end of the paper.
defect of the corner radii in forging process. To get some conditions in the finishing stage such as minimum friction and forging load and reduce wear along die surfaces, the material is squeezed laterally towards the die cavity through forging direction without additional shear at the die-material interface. In Bevel gear being a common gear for changing rotational axis mostly low-alloy steel is used. This material is used with thermal-refining or induction-hardening treatment for producing gears. By this information, guidelines for the material with the recommended preform dimensions is given in table 1.
2. Design of Preform Shape of Bevel Gear
Table 1. Preform dimension for low-alloy steel [7]
Complex shapes with forging are usually produced in a multistage process. In using multistage metal forging, preform design is selected. The design of the preform shape plays a key role for the forged part quality. Parameters like flash ratio, forming load and energy consumption and hence the economic efficiency of the forging sequence are mainly determined by the preform shape [6]. Finite-Elements-Analysis (FEA) is usually used by engineers for building design preform based on their experience. However, the evaluation of preforms shape using FEA is very costly due to long time involved and computing effort. The initial theoretical design of preform shape for forging, plays a great significant role in achieving preform optimization, either using algorithm or another method. The design of preform bevel gear begins with the geometry of part. Then by considering the functionality of the part, material is selected. The preform shape is designed with consideration of flash dimension, fillet and corner radii, and positioning of parting line. Control of material flow and volume distribution are considered for designing a reliable preform for forging process. The objectives of preform design are [7]: • Assuring the metal flow without any defect and adequately die filling • Minimizing the material wastes in the flash • Minimizing die wear in the finish-forging cavity • Obtaining desired grain flow and suitable mechanical properties. In forging steel parts, there are three general basic rules for designing a reliable preform shape as follows [7]: • The cross-section area of the preform must be equal with the cross-section area of the final part • All the concave radii, including the fillet radii and blend-in radii, on the preform must be greater than the corresponding radii on the final part • The thickness of the preform should be greater than the final part along the forging direction. Comparison of the preform shape and final part approximately is a quarter of the height cross section in forging direction. Hence the metal flow is mostly by upsetting rather than extrusion. Based on the basic design rules of preform, the initial design is started from area of cross section of geometry part. By this statement, the cross-section areas along the main axis of part is determined for designing the initial stock distribution. All the concave radii are determined because it is related to forging
Dimensions of the finish forgings
Preform Dimensions
Fillet Radii (RFP)
RFP 1.2 RFF + 3.18 mm
Blend-in radii (RBP)
RBP RBF + C
Height (H), mm (in.) and depth of adjacent cavity (C)
When HR is less than 10 mm (0.4 in.), then C = 2 mm (0.08 in.) When HR is between 10 mm and 25 mm (1. in.), then C = 3 mm (0.12 in.) When HR is between 25 mm and 50 mm (2 in.), then C = 4 mm (0.16 in.) When HR is greater than 50 mm, then C = 5 mm (0.2 in.)
The fillet radius of preform part (RFP) should be larger than the fillet radius of final part (RFF), especially when the rib height (DF) is larger than the rib width (WF) as shown in Fig. 1. The adjacent cavity depth (C) (Table 2) influences the blend-in radii of preform (RBP) in parting line.
Fig. 1. Comparison of the preform and finished part for a quarter of an “H” cross section [7].
The empirical methods in table 1 is summarized in simple formula, which is adequately general for predicting preform shape of low-alloy steel material. This simple method has been proved practically for predicting preform shape.
Fig. 2. Bevel gear Geometry
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Fig. 2 is the shape of preform by considering the cross section of area of bevel gear and thickness of the final part. Based on an analysis of the perform parameters and the reference from previous studies, three key factors, namely, whole height (H), large diameter (D), and chamfer length (B) were considered as design variables [8]. According to preform dimension for low-alloy steel, fillet radius of preform part (RFP) and blend-in radii of preform (RBP) were defined as design variables for optimized preform shape. The information data of the geometry of preform shape bevel gear is given in table 2. and will be processed for optimization. Table 2. Data geometry of design variable. Design Variables Chamfer Dimension (B) Diameter (D) Height (H) RFF 1 RFF 2 RBF
Initial Design 13.6 84 43.6 2 3 2
3. Design of a Genetic Algorithm for Preform optimization of Bevel Gear Genetic algorithm (GA), as powerful and broadly applicable stochastic search and optimization technique, is perhaps the most widely known types of evolutionary computation methods today [9]. The main idea of these algorithms is to imitate the process of biological evolution with the main mechanisms selection, crossover and mutation. GA is mainly used for multi-objective optimization problems with a high number of variable solutions which leads to high computation times for conventional optimization techniques [10, 11]. The process of GA usually consists of an initialization and an iteration. During the initialization an allele of chromosome is created randomly. A set of genes is described as chromosome. Then each chromosome is defined by fitness function. Afterwards the new chromosome is built based on their fitness value and best chromosome is generated by doing crossover and mutation. In the next step iteration is executed until a termination criterion (fitness value) is met. The preform design of bevel gear can be described as a multi-objective optimization problem, where form-filling must avoid the folds and make sure the preform volume and the complexity of the part must be as small as possible. Therefore, Genetic Algorithm is possibility to solve this problem. All steps of the GA is coded using Excel VBA. The first step to realize this genetic algorithm is to define bevel gear preform part by using N boundary points where N=5. Each boundary point in the shape represents a gene of each chromosome. A gene (Gn) contains the information radius r and the position L along the vertical axis of the part (Fig. 3).
3
Where: 𝐶𝐶 = 𝐺𝐺1 , … 𝐺𝐺5 𝐺𝐺1 = 𝑟𝑟1 , 𝐿𝐿1
Chromosome contains the value of radius for final part and the corresponding height of preform. These are the following process parameters that has to be defined:
I : population size pc : Crossover rate m : Mutation rate c : Termination criterion x, y : Radius rmin and rmax Population size (I) describes how many chromosomes are part of each population. The crossover rate is a factor to reproduction and biological crossover. In this more than one parent is selected and one or more off-springs are produced using the genetic material of the parents, one-point crossover is used in here (Fig. 4). Mutation rate (m) is the factor to define the probability of mutating a gene. Selected genes are changed by an addition or subtraction of the value randomly to the present value (𝑟𝑟1 ). Then new generation chromosome is built up out of the parent genes. The position of each section is randomly chosen (Fig. 5).
Fig. 4. One-point crossover.
Fig. 5. Mutation [10]
Termination criterion c is for example a maximum number of generations, a defined fitness value or if the change of fitness value is below a limit value. The minimum and maximum values for the radii must be determined by equations (1) and (2), x = 0.5 and y = 2 is chosen as a wide range of possible radii. The next step is explanation of evaluation for each chromosome. 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑥𝑥 × √
min(𝐴𝐴𝑛𝑛 ) 𝜋𝜋
max(𝐴𝐴𝑛𝑛 ) 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑦𝑦 × √ 𝜋𝜋
(1) (2)
4. Design of Fitness Function Fig. 3. Representation of bevel preform
Each chromosome has to be evaluated by fitness value. There are three fitness values for this preform shape, they are
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form-filling, preform-volume and complexity.
𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 and the maximum possible preform volume 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 . The minimum and maximum radii and total length of preform are used to calculate the minimum and maximum preform volume.
4.1 Form-Filling Warm forging is one of the forging process that use formfilling as the main quality parameter for forging parts. Amount of flash needed for Form-filling depends on the complexity of the part. Assuming there is no material flow in axial direction of part. The form-filling can be calculated by this equation: 𝐴𝐴𝑖𝑖,𝑛𝑛 𝐴𝐴𝑖𝑖,𝑛𝑛 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎 ≤ ≤ 1 + 𝑎𝑎 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛 𝐴𝐴𝑖𝑖,𝑛𝑛 = 0 𝑓𝑓𝑓𝑓𝑓𝑓 < 𝑎𝑎 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛 𝐴𝐴𝑖𝑖,𝑛𝑛 = 1 𝑓𝑓𝑓𝑓𝑓𝑓 > 1 + 𝑎𝑎 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛
𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓,𝑛𝑛 = −𝑎𝑎 +
𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓,𝑛𝑛 𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓,𝑛𝑛
(3)
Where : : cross-section area of the individual at cutting plane n 𝐴𝐴𝑖𝑖,𝑛𝑛 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛 : cross-section area of final part at cutting plane n. Amount of flash ( 𝑎𝑎) for every cutting plane can be calculated by this following equation: 𝑎𝑎 = 0.6874 × 𝑒𝑒 5.7217×𝐶𝐶/𝑤𝑤
343
(4)
Where 𝐶𝐶/𝑤𝑤 = 0.34 for basic preform low complexity, figure 6 shows the cross-section area for low complexity. Parameter 𝐶𝐶 represents the distance between the axis of symmetry and the center of gravity which is corresponding to component half. The parameter 𝑤𝑤 represents the part width of the component half.
(6)
2 × 𝐿𝐿 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜋𝜋 × 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜋𝜋 × 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 × 𝐿𝐿
The preforming design must reduce preform volume as much as possible. Therefore, the following relationships between 𝑉𝑉𝑖𝑖 and 𝑉𝑉𝐹𝐹𝐹𝐹 are defined. If 𝑉𝑉𝑖𝑖 is smaller than 𝑉𝑉𝐹𝐹𝐹𝐹 the scrap parts as form-filling cannot be received. The equation of preform volume can be determined by this following equation: 𝑉𝑉𝑖𝑖 − 𝑉𝑉𝐹𝐹𝐹𝐹 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑉𝑉𝐹𝐹𝐹𝐹 𝑉𝑉𝐹𝐹𝐹𝐹 − 𝑉𝑉𝑖𝑖 = 𝑉𝑉𝐹𝐹𝐹𝐹 − 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚
𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣,𝑖𝑖 =
𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣,𝑖𝑖
4.3 Complexity
𝑓𝑓𝑓𝑓𝑓𝑓
𝑓𝑓𝑜𝑜𝑟𝑟
𝑉𝑉𝑖𝑖 ≤1 𝑉𝑉𝐹𝐹𝐹𝐹 𝑉𝑉𝑖𝑖 ≥1 𝑉𝑉𝐹𝐹𝐹𝐹
The manufacturing costs and process constraints are derived by this parameter complexity for each individual 𝐶𝐶𝑖𝑖 . In order to avoid the folds sharp edges in final part, calculation of the length of the spline connecting all boundary points can be considered for the complexity. By this equation the complexity 𝐶𝐶𝑖𝑖 of an individual is calculated by using the Euclidean distance of the boundary points. 𝑁𝑁−1
2
𝐶𝐶𝑖𝑖 = ∑ √(𝑟𝑟𝑖𝑖,𝑛𝑛 − 𝑟𝑟𝑖𝑖,𝑛𝑛+1 ) + ( 𝑛𝑛=1
2 𝐿𝐿 ) 𝑁𝑁 − 1
𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚
𝑁𝑁−1
2 𝐿𝐿 = ∑ √(𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 )2 + ( ) 𝑁𝑁 − 1
𝑁𝑁−1
𝑉𝑉𝑖𝑖 = ∑
𝑛𝑛=1
𝐿𝐿 × 𝜋𝜋 2 ) (𝑟𝑟 2 + 𝑟𝑟𝑛𝑛 × 𝑟𝑟𝑛𝑛+1 + 𝑟𝑟𝑛𝑛+1 (𝑁𝑁 − 1) × 3 𝑛𝑛
(5)
In order to calculate the volume fitness of an individual 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑖𝑖 the individual volume 𝑉𝑉𝑖𝑖 is related to the volume of the final part 𝑉𝑉𝐹𝐹𝐹𝐹 , the minimum possible preform volume
(9)
𝑛𝑛=1
Between the minimum and maximum complexity, the fitness value 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 is derived by this following equation. 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 =
4.2 Preform volume Preform volume is a necessary parameter in warm forging process. The total cost of forging is mostly comprised of material and the process. The material can be reduced by making preform, and this preform volume can also reduce the flashing and forming load. The total volume is calculated by the following equation:
(8)
The minimum complexity can be determined by this following equation: 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐿𝐿
Fig. 6. Parameters for determining preform low complexity [5]
(7)
4.4 Fitness evaluation
𝐶𝐶𝑖𝑖 − 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 − 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚
(10)
The total fitness value for each individual 𝐹𝐹𝑖𝑖 is being calculated by using the fitness values 𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 , 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑒𝑒𝑒𝑒 , and 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 . Within the optimization the parameter form filling needs to be maximized where the parameter preform volume and complexity will be minimized (11). The genetic algorithm aims to maximize this function [5]. 𝐹𝐹𝑖𝑖 = 𝑤𝑤1 × 𝐹𝐹𝑓𝑓𝑓𝑓𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 − 𝑤𝑤2 × 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 − 𝑤𝑤3 × 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 (11) 4.5 Flash dimension
Flash dimension must be determined for the realization of
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optimization of preform shape. As mentioned before in the general basic rules of preform shape related to cross section, this cross section of complexity influences the Flash dimension in the final part geometry. By the guideline dimensions of preform shape, some areas of the part like radii for reducing the flash is determined. Flash dimension also related to flash allowance and forging load, that will influence the quality of the forged part. Flash thickness and flash land width on the forging pressure is reasonably well understood from a qualitative point of view. Flash dimensions can be determined by this following equation [7]: (12) 1 ] 𝑡𝑡 = [0.017 × 𝐷𝐷] + [ (13) √𝐷𝐷 + 5 𝑓𝑓𝑓𝑓 30 = 𝑡𝑡 3 2𝑥𝑥𝐷𝐷2 √𝐷𝐷 [1 + ( )] 𝐻𝐻(2𝑅𝑅ℎ + 𝐷𝐷) where: fw : Flash width (mm) t : Flash thickness (mm) H : Height of the preform (mm) D : Outside diameter of preform forging (mm) Rh : Radial distance of the center of a rib from the axis of symmetry of preform forging.
5
Fig. 7. Form-filling value 𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖,𝑛𝑛
The value of the volume fitness value 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑖𝑖 depends on the calculation of individual preform volume 𝑉𝑉𝑖𝑖 . The total volume of each individual 𝑉𝑉𝑖𝑖 is calculated by summing up the volume between the boundary points, which can be illustrated as a truncated cone. The result of volume fitness value 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑖𝑖 is shown in Fig. 8.
The complexity of the cross-section area of a forging part is important parameter for designing preform optimization. Flash dimension is used for cross section area that is related to the material distribution. Material is necessary for achieving formfilling, it is needed more if the cross-section shape is more complex. 5. Results and Discussions The Preform shape plays a great significant role to improve quality, dimensional accuracy of final product and reduction of cost in warm forging process. In the following investigations, a bevel gear is used as optimized part, the part has been divided into N = 5 cutting planes. According to general basic rules of preform design and preform dimensions in forging process, the initial design of preform shape for bevel gear was designed theoretically. Fillet radii (RFF) and blend-in radii (RBF) are added on this preform shape based on the guidelines of preform dimensions. An evolutionary algorithm is introduced to optimize preform shape by considering the main parameter of forging parts and forging process. The following parameters for the algorithm have been used during the optimization: I : 50 pc : 25% m : 1% c : 25 w1 : 2 w2 : 1 w3 : 1 The fitness functions for this genetic algorithm that are form-filling fitness, have some result that is illustrated by graphic. The form-filling fitness 𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖,𝑛𝑛 has a limited value between 0 and 1 as shown in Fig. 7. Because form-filling fitness has no negative value or > 1.
Fig. 8. Preform volume value Fvolume i
Fig. 9. shows the result of the complexity fitness value
𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 i in dependency of the calculated individual
complexity 𝐶𝐶𝑖𝑖 .
Fig. 9. Complexity fitness value 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖
Fillet radii of preform part (RFP) and blend-in radii of preform part (RBP) are added on this optimized preform shape. The values and the shape of optimal preform of bevel gear can be seen in Table 3 and Fig. 10. respectively.
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[2]
Table 3. Data geometry of design variable. Design Variables Chamfer Dimension (B) Diameter (D) Height (H) RFP 1 RFP 2 RBP t
fw
Initial Design 15.5 82.61 40.84 5.58 6.78 7 1.5 7.2
[3]
[4]
[5]
345
Lu B, Ou H, Cui Z. S Shape Optimisation of Preform Design for Precision Close-Die Forging. Struct Multidisc Optim 44:785–796. DOI (2011) 10.1007/s00158-011-0668-1 D.H. Kim, B.M. Kim, Preform Design of the Bevel Gear for the Warm Forging using Artificial Neural Network, Journal of the Korean Society for Precision Engineering, Vol. 20 (2003), pp.36-43 Yu C-H, Sheu J-J Preform and Die Designs for Hot Forging Process of Linear Slide Block. Applied Mechanics and Materials 419: 395-400. Trans Tech Publications, Switzerland. Doi: (2013) 10.4028/www.scientific.net/AMM.419.395 Knust J, Stonis M, Behrens B-A Preform Optimization for Hot Forging Processes Using an Adaptive Flash Dimension Based on The CrossSection Shape Complexity. Prod. Eng. Res. Dev. 10:587–598. DOI (2016) 10.1007/s11740-016-0702-7
[6] Behrens B-A, Nickel R, Mu¨ller S (2009) Flashless precision forging of a two-cylinder crankshaft. Prod Eng Res Dev 3:381–389. doi:10.1007/s11740-009-0185-x [7]
Fig. 10. Optimal preform shape of bevel gear
6. Conclusion In summary, an initial design and optimization of the preform shape of bevel gear have been carried out in this paper. An initial design preform is determined theoretically based on cross section of the final part. The optimization of preform shape was derived by some equations and with several iterations using Excel VBA by considering parameters of forging part and forging process. The derived equation of formfilling, preform volume, and complexity have the output for final preform geometry, radius r and the position L along the vertical axis of the part. These outputs will form the preform shape of bevel gear that based on the cross section of finished part. The results of optimization preform shape are showed with the data geometry of design variables. Moreover, the result of initial design shows the preform that corresponds to general basic rules based on the cross-section area. The guidelines of Preform dimension for low-alloy steel can be considered as process variables to avoid forging defects. These process variables can affect the forging process, such as minimize friction, reduce wear along die surface, minimize forging load, and determine metal flow pattern. Acknowledgements This work was supported by the ICT R&D program of MSIP/IITP. [B0101-17-1081, Development of ICT based software platform and service technologies for medical 3D printing applications] References [1]
Parsa M. H, Asadpour H, Bozorg H. Forging Preform Design for Simple Shapes. Esaform International Conference Proceeding. (2008)
Dieter GE, Kuhn HA, Semiatin SL Handbook of Workability and Process Design. ASM International (2003) [8] D.H. Kim, B.M. Kim, Preform Design of the Bevel Gear for the Warm Forging using Artificial Neural Network, Journal of the Korean Society for Precision Engineering, Vol. 20 (2003), pp.36-43 [9] Gen M, Cheng R Genetic algorithms and engineering optimization. Ashikaga Institute of Technology. Ashikaga, Japan. (2000) [10] Goldberg D Genetic algorithms in search, optimization and machine learning. Addison-Wesley, London. (1989) [11] Haupt RL, Haupt SE Practical genetic algorithms. Wiley, Hoboken. (2004) [12] Brandao T. Genetic algorithms in PHP code example of (evolutionary programming). (2015). At < http://www.abrandao.com/2015/01/simplephp-genetic-algorithm/>
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51st CIRP Conference on Manufacturing Systems
Preform Optimization forConference, Bevel Gear of Warm Process 28th CIRP Design May 2018, Nantes, Forging France Hong-Seok Parka*, Febriani Risky Ayub,and Saurabh Kumarcarchitecture of A new methodology to analyze the functional physical University of Ulsan, School of Mechanical Engineering, 93 Daehak-ro, Nam-gu, Ulsan 44610, Korea existing products for an assembly oriented product family identification a,b,c
* Corresponding author. Tel.: +82-(0)52-259-2294; fax: +82-(0)52-259-1680. E-mail address: [email protected]
Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat
École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France
Abstract
* Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]
Preform design is an important aspect in consideration of the improvement of quality and reduction of cost in warm forging process. Bevel gear being a key component of drive mechanisms in automotive industry requires lot of attention to its complex shape design optimization. A multiobjective optimization problem in designing preforms of warm forging for bevel gear by using genetic algorithm is solved and presented in this Abstract paper. The main objective of this study is to design an optimal preform part, which can get a better quality and the reliable preform shape systematically and theoretically. In today’s business environment, the trend towards more product variety and customization is unbroken. Due to this development, the need of agile andThe reconfigurable production systems B.V. emerged to cope with various products and product families. To design and optimize production © 2018 Authors. Published by Elsevier Peer-review under responsibility the scientific of the 51st CIRP methods Conference Manufacturing Systems. systems as well as to choose the of optimal productcommittee matches, product analysis areon needed. Indeed, most of the known methods aim to analyze a product or one product family on the physical level. Different product families, however, may differ largely in terms of the number and Keywords: Preform optimization; Forging; Beveland Gear nature of components. This factWarm impedes an Genetic efficientAlgorithm; comparison choice of appropriate product family combinations for the production system. A new methodology is proposed to analyze existing products in view of their functional and physical architecture. The aim is to cluster these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable assembly systems. Based on Datum Flow Chain, the physical structure of the products is analyzed. Functional subassemblies are identified, and homogeneity of graph deformation conditions. It has been triedthe to Introduction a1.functional analysis is performed. Moreover, a hybrid functional and physical architecture (HyFPAG) is the output which depicts predict preforms for non-proportional loading condition [1]. similarity between product families by providing design support to both, production system planners and product designers. An illustrative Lu et case al. [2] developed topology optimisation method as an example of agear nail-clipper used to explain the proposed methodology. industrial study on two product families of steering columns of Bevel is a keyiscomponent of drive mechanisms in the An Bin efficient approach preform design optimization for parts thyssenkrupp France carried out to give of the proposedfor approach. automotive Presta industry dueis then to high contact ratioa first andindustrial smooth evaluation ©transmission. 2017 The Authors. by Elsevier B.V. manufactured by with complex shape. Similarly, the artificial neural network The Published bevel gears are mainly Peer-review underwhich responsibility the scientific the 28th Conference wasDesign applied to obtain2018. the optimal perform shape for the bevel metal cutting, produceoflarge quantitycommittee of metal of chips as CIRP
waste and requires a lot of manufacturing time. Warm forging as excellent mechanical properties, less raw material, good tolerance, high productivity and less costly as compared to metal cutting. The design of preforms is most important aspect for achieving 1. Introduction adequate metal distribution in forging process. The motive behind preform design is to get an optimum preform shape Due enables to thesufficient fast development the and domain of which filling of dieincavity get flash communication and an ongoing trend of digitization and allowance. digitalization, enterprises are configuration facing important However, manufacturing the determination of preform is a challenges in today’s market environments: continuing very difficult task and it needs knowledge anda skill that is tendency reduction of product times and achievedtowards by years of experience. So, development the implementation of shortened product lifecycles. addition, therehas is an increasing optimal performs shapes inInwarm forging attracted the demand being Most at the of same time global attentionofofcustomization, many researchers. them didinita for hot competition competitors allproduction over the world. This trend, forging thatwith is widely used for of parts from the simpleisshape to the complex. A simple shape preform which inducing the development from of macro to design micro is brieflyresults discussed by applying energy markets, in diminished lotminimum sizes duepotential to augmenting principle and considering volume consistencyproduction) and maximum product varieties (high-volume to low-volume [1]. To cope with this augmenting variety as well as to be able to identify possible optimization potentials in the existing 2212-8271 © 2018 The Authors. Published by Elsevier B.V. production system, it is important to have a precise knowledge Keywords: Design method; Family identification process Assembly; possesses several benefits such
gear [3]. In recent years, simulation using CAE software has been used to analyze the forging process and die stress distribution. The evaluation of process design that indicated the suitable preform design is able to achieve better material flow [4]. Similarly, Knust et al. [5] describes the development of an of the product algorithm range and characteristics manufactured and/or evolutionary designing CWR (Cross Wedge assembled in this system. In this context, the main challenge in Rolling) preforms and amount of flash simulation by software. modelling and analysis is now not only to cope with single However, the aforementioned publications indicated that the products, limited productofrange or existing product complex ashape interplay objective functions, (e.g.families, product but also to be able to analyze and to compare products define quality) with respect to perform parameters is nottoclearly new product yet, families. can be the observed that classical understood whichIt makes optimization processexisting rather product families are regrouped in design functionofofpreform clients or features. inefficient. Furthermore, initial shape and However, assembly optimization oriented product families are hardly to find. preform parameter of the bevel gear warm forging hasOn notthe been conducted. product family level, products differ mainly in two this paper, preform designed an(ii) initial mainIncharacteristics: (i) theisnumber oftheoretically components as and the shape preform related to forging electrical, process and forged part type of of components (e.g. mechanical, electronical). quality. A methodologies genetic algorithm (GA) optimization is Classical considering mainly singlemethod products developed bevel gear preform designfamilies and it is implemented or solitary,foralready existing product analyze the using Excel (Visual level basic(components for Application). An product structureVBA on a physical level) which causes difficulties regarding an efficient definition and comparison of different product families. Addressing this
Peer-review under responsibility of the scientific committee of the 51st CIRP Conference on Manufacturing Systems.
2212-8271©©2017 2018The The Authors. Published by Elsevier 2212-8271 Authors. Published by Elsevier B.V. B.V. Peer-review under responsibility of scientific the scientific committee theCIRP 51stDesign CIRP Conference Conference2018. on Manufacturing Systems. Peer-review under responsibility of the committee of the of 28th 10.1016/j.procir.2018.03.083
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evolutionary algorithm is proposed to optimize the preform shape with the parameters as form filling, the preform volume and the shape complexity. The rest of this paper is organised as follows: The basic rules of preform design is presented theoretically. The basic rules consist of the guidelines to determine the preform shape of forging based on material and cross section area. Then, the geometry data information of preform shape is described briefly for optimization. Results obtained with genetic algorithm method studied in this preform design optimization is discussed. This is followed by concluding remarks in the end of the paper.
defect of the corner radii in forging process. To get some conditions in the finishing stage such as minimum friction and forging load and reduce wear along die surfaces, the material is squeezed laterally towards the die cavity through forging direction without additional shear at the die-material interface. In Bevel gear being a common gear for changing rotational axis mostly low-alloy steel is used. This material is used with thermal-refining or induction-hardening treatment for producing gears. By this information, guidelines for the material with the recommended preform dimensions is given in table 1.
2. Design of Preform Shape of Bevel Gear
Table 1. Preform dimension for low-alloy steel [7]
Complex shapes with forging are usually produced in a multistage process. In using multistage metal forging, preform design is selected. The design of the preform shape plays a key role for the forged part quality. Parameters like flash ratio, forming load and energy consumption and hence the economic efficiency of the forging sequence are mainly determined by the preform shape [6]. Finite-Elements-Analysis (FEA) is usually used by engineers for building design preform based on their experience. However, the evaluation of preforms shape using FEA is very costly due to long time involved and computing effort. The initial theoretical design of preform shape for forging, plays a great significant role in achieving preform optimization, either using algorithm or another method. The design of preform bevel gear begins with the geometry of part. Then by considering the functionality of the part, material is selected. The preform shape is designed with consideration of flash dimension, fillet and corner radii, and positioning of parting line. Control of material flow and volume distribution are considered for designing a reliable preform for forging process. The objectives of preform design are [7]: • Assuring the metal flow without any defect and adequately die filling • Minimizing the material wastes in the flash • Minimizing die wear in the finish-forging cavity • Obtaining desired grain flow and suitable mechanical properties. In forging steel parts, there are three general basic rules for designing a reliable preform shape as follows [7]: • The cross-section area of the preform must be equal with the cross-section area of the final part • All the concave radii, including the fillet radii and blend-in radii, on the preform must be greater than the corresponding radii on the final part • The thickness of the preform should be greater than the final part along the forging direction. Comparison of the preform shape and final part approximately is a quarter of the height cross section in forging direction. Hence the metal flow is mostly by upsetting rather than extrusion. Based on the basic design rules of preform, the initial design is started from area of cross section of geometry part. By this statement, the cross-section areas along the main axis of part is determined for designing the initial stock distribution. All the concave radii are determined because it is related to forging
Dimensions of the finish forgings
Preform Dimensions
Fillet Radii (RFP)
RFP 1.2 RFF + 3.18 mm
Blend-in radii (RBP)
RBP RBF + C
Height (H), mm (in.) and depth of adjacent cavity (C)
When HR is less than 10 mm (0.4 in.), then C = 2 mm (0.08 in.) When HR is between 10 mm and 25 mm (1. in.), then C = 3 mm (0.12 in.) When HR is between 25 mm and 50 mm (2 in.), then C = 4 mm (0.16 in.) When HR is greater than 50 mm, then C = 5 mm (0.2 in.)
The fillet radius of preform part (RFP) should be larger than the fillet radius of final part (RFF), especially when the rib height (DF) is larger than the rib width (WF) as shown in Fig. 1. The adjacent cavity depth (C) (Table 2) influences the blend-in radii of preform (RBP) in parting line.
Fig. 1. Comparison of the preform and finished part for a quarter of an “H” cross section [7].
The empirical methods in table 1 is summarized in simple formula, which is adequately general for predicting preform shape of low-alloy steel material. This simple method has been proved practically for predicting preform shape.
Fig. 2. Bevel gear Geometry
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Fig. 2 is the shape of preform by considering the cross section of area of bevel gear and thickness of the final part. Based on an analysis of the perform parameters and the reference from previous studies, three key factors, namely, whole height (H), large diameter (D), and chamfer length (B) were considered as design variables [8]. According to preform dimension for low-alloy steel, fillet radius of preform part (RFP) and blend-in radii of preform (RBP) were defined as design variables for optimized preform shape. The information data of the geometry of preform shape bevel gear is given in table 2. and will be processed for optimization. Table 2. Data geometry of design variable. Design Variables Chamfer Dimension (B) Diameter (D) Height (H) RFF 1 RFF 2 RBF
Initial Design 13.6 84 43.6 2 3 2
3. Design of a Genetic Algorithm for Preform optimization of Bevel Gear Genetic algorithm (GA), as powerful and broadly applicable stochastic search and optimization technique, is perhaps the most widely known types of evolutionary computation methods today [9]. The main idea of these algorithms is to imitate the process of biological evolution with the main mechanisms selection, crossover and mutation. GA is mainly used for multi-objective optimization problems with a high number of variable solutions which leads to high computation times for conventional optimization techniques [10, 11]. The process of GA usually consists of an initialization and an iteration. During the initialization an allele of chromosome is created randomly. A set of genes is described as chromosome. Then each chromosome is defined by fitness function. Afterwards the new chromosome is built based on their fitness value and best chromosome is generated by doing crossover and mutation. In the next step iteration is executed until a termination criterion (fitness value) is met. The preform design of bevel gear can be described as a multi-objective optimization problem, where form-filling must avoid the folds and make sure the preform volume and the complexity of the part must be as small as possible. Therefore, Genetic Algorithm is possibility to solve this problem. All steps of the GA is coded using Excel VBA. The first step to realize this genetic algorithm is to define bevel gear preform part by using N boundary points where N=5. Each boundary point in the shape represents a gene of each chromosome. A gene (Gn) contains the information radius r and the position L along the vertical axis of the part (Fig. 3).
3
Where: 𝐶𝐶 = 𝐺𝐺1 , … 𝐺𝐺5 𝐺𝐺1 = 𝑟𝑟1 , 𝐿𝐿1
Chromosome contains the value of radius for final part and the corresponding height of preform. These are the following process parameters that has to be defined:
I : population size pc : Crossover rate m : Mutation rate c : Termination criterion x, y : Radius rmin and rmax Population size (I) describes how many chromosomes are part of each population. The crossover rate is a factor to reproduction and biological crossover. In this more than one parent is selected and one or more off-springs are produced using the genetic material of the parents, one-point crossover is used in here (Fig. 4). Mutation rate (m) is the factor to define the probability of mutating a gene. Selected genes are changed by an addition or subtraction of the value randomly to the present value (𝑟𝑟1 ). Then new generation chromosome is built up out of the parent genes. The position of each section is randomly chosen (Fig. 5).
Fig. 4. One-point crossover.
Fig. 5. Mutation [10]
Termination criterion c is for example a maximum number of generations, a defined fitness value or if the change of fitness value is below a limit value. The minimum and maximum values for the radii must be determined by equations (1) and (2), x = 0.5 and y = 2 is chosen as a wide range of possible radii. The next step is explanation of evaluation for each chromosome. 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑥𝑥 × √
min(𝐴𝐴𝑛𝑛 ) 𝜋𝜋
max(𝐴𝐴𝑛𝑛 ) 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑦𝑦 × √ 𝜋𝜋
(1) (2)
4. Design of Fitness Function Fig. 3. Representation of bevel preform
Each chromosome has to be evaluated by fitness value. There are three fitness values for this preform shape, they are
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form-filling, preform-volume and complexity.
𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 and the maximum possible preform volume 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 . The minimum and maximum radii and total length of preform are used to calculate the minimum and maximum preform volume.
4.1 Form-Filling Warm forging is one of the forging process that use formfilling as the main quality parameter for forging parts. Amount of flash needed for Form-filling depends on the complexity of the part. Assuming there is no material flow in axial direction of part. The form-filling can be calculated by this equation: 𝐴𝐴𝑖𝑖,𝑛𝑛 𝐴𝐴𝑖𝑖,𝑛𝑛 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎 ≤ ≤ 1 + 𝑎𝑎 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛 𝐴𝐴𝑖𝑖,𝑛𝑛 = 0 𝑓𝑓𝑓𝑓𝑓𝑓 < 𝑎𝑎 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛 𝐴𝐴𝑖𝑖,𝑛𝑛 = 1 𝑓𝑓𝑓𝑓𝑓𝑓 > 1 + 𝑎𝑎 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛
𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓,𝑛𝑛 = −𝑎𝑎 +
𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓,𝑛𝑛 𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓,𝑛𝑛
(3)
Where : : cross-section area of the individual at cutting plane n 𝐴𝐴𝑖𝑖,𝑛𝑛 𝐴𝐴𝐹𝐹𝐹𝐹,𝑛𝑛 : cross-section area of final part at cutting plane n. Amount of flash ( 𝑎𝑎) for every cutting plane can be calculated by this following equation: 𝑎𝑎 = 0.6874 × 𝑒𝑒 5.7217×𝐶𝐶/𝑤𝑤
343
(4)
Where 𝐶𝐶/𝑤𝑤 = 0.34 for basic preform low complexity, figure 6 shows the cross-section area for low complexity. Parameter 𝐶𝐶 represents the distance between the axis of symmetry and the center of gravity which is corresponding to component half. The parameter 𝑤𝑤 represents the part width of the component half.
(6)
2 × 𝐿𝐿 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜋𝜋 × 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 2 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 = 𝜋𝜋 × 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 × 𝐿𝐿
The preforming design must reduce preform volume as much as possible. Therefore, the following relationships between 𝑉𝑉𝑖𝑖 and 𝑉𝑉𝐹𝐹𝐹𝐹 are defined. If 𝑉𝑉𝑖𝑖 is smaller than 𝑉𝑉𝐹𝐹𝐹𝐹 the scrap parts as form-filling cannot be received. The equation of preform volume can be determined by this following equation: 𝑉𝑉𝑖𝑖 − 𝑉𝑉𝐹𝐹𝐹𝐹 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑉𝑉𝐹𝐹𝐹𝐹 𝑉𝑉𝐹𝐹𝐹𝐹 − 𝑉𝑉𝑖𝑖 = 𝑉𝑉𝐹𝐹𝐹𝐹 − 𝑉𝑉𝑚𝑚𝑚𝑚𝑚𝑚
𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣,𝑖𝑖 =
𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣,𝑖𝑖
4.3 Complexity
𝑓𝑓𝑓𝑓𝑓𝑓
𝑓𝑓𝑜𝑜𝑟𝑟
𝑉𝑉𝑖𝑖 ≤1 𝑉𝑉𝐹𝐹𝐹𝐹 𝑉𝑉𝑖𝑖 ≥1 𝑉𝑉𝐹𝐹𝐹𝐹
The manufacturing costs and process constraints are derived by this parameter complexity for each individual 𝐶𝐶𝑖𝑖 . In order to avoid the folds sharp edges in final part, calculation of the length of the spline connecting all boundary points can be considered for the complexity. By this equation the complexity 𝐶𝐶𝑖𝑖 of an individual is calculated by using the Euclidean distance of the boundary points. 𝑁𝑁−1
2
𝐶𝐶𝑖𝑖 = ∑ √(𝑟𝑟𝑖𝑖,𝑛𝑛 − 𝑟𝑟𝑖𝑖,𝑛𝑛+1 ) + ( 𝑛𝑛=1
2 𝐿𝐿 ) 𝑁𝑁 − 1
𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚
𝑁𝑁−1
2 𝐿𝐿 = ∑ √(𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 )2 + ( ) 𝑁𝑁 − 1
𝑁𝑁−1
𝑉𝑉𝑖𝑖 = ∑
𝑛𝑛=1
𝐿𝐿 × 𝜋𝜋 2 ) (𝑟𝑟 2 + 𝑟𝑟𝑛𝑛 × 𝑟𝑟𝑛𝑛+1 + 𝑟𝑟𝑛𝑛+1 (𝑁𝑁 − 1) × 3 𝑛𝑛
(5)
In order to calculate the volume fitness of an individual 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑖𝑖 the individual volume 𝑉𝑉𝑖𝑖 is related to the volume of the final part 𝑉𝑉𝐹𝐹𝐹𝐹 , the minimum possible preform volume
(9)
𝑛𝑛=1
Between the minimum and maximum complexity, the fitness value 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 is derived by this following equation. 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 =
4.2 Preform volume Preform volume is a necessary parameter in warm forging process. The total cost of forging is mostly comprised of material and the process. The material can be reduced by making preform, and this preform volume can also reduce the flashing and forming load. The total volume is calculated by the following equation:
(8)
The minimum complexity can be determined by this following equation: 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐿𝐿
Fig. 6. Parameters for determining preform low complexity [5]
(7)
4.4 Fitness evaluation
𝐶𝐶𝑖𝑖 − 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚 − 𝐶𝐶𝑚𝑚𝑚𝑚𝑚𝑚
(10)
The total fitness value for each individual 𝐹𝐹𝑖𝑖 is being calculated by using the fitness values 𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 , 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑚𝑚𝑒𝑒𝑒𝑒 , and 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 . Within the optimization the parameter form filling needs to be maximized where the parameter preform volume and complexity will be minimized (11). The genetic algorithm aims to maximize this function [5]. 𝐹𝐹𝑖𝑖 = 𝑤𝑤1 × 𝐹𝐹𝑓𝑓𝑓𝑓𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 − 𝑤𝑤2 × 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 − 𝑤𝑤3 × 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 (11) 4.5 Flash dimension
Flash dimension must be determined for the realization of
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optimization of preform shape. As mentioned before in the general basic rules of preform shape related to cross section, this cross section of complexity influences the Flash dimension in the final part geometry. By the guideline dimensions of preform shape, some areas of the part like radii for reducing the flash is determined. Flash dimension also related to flash allowance and forging load, that will influence the quality of the forged part. Flash thickness and flash land width on the forging pressure is reasonably well understood from a qualitative point of view. Flash dimensions can be determined by this following equation [7]: (12) 1 ] 𝑡𝑡 = [0.017 × 𝐷𝐷] + [ (13) √𝐷𝐷 + 5 𝑓𝑓𝑓𝑓 30 = 𝑡𝑡 3 2𝑥𝑥𝐷𝐷2 √𝐷𝐷 [1 + ( )] 𝐻𝐻(2𝑅𝑅ℎ + 𝐷𝐷) where: fw : Flash width (mm) t : Flash thickness (mm) H : Height of the preform (mm) D : Outside diameter of preform forging (mm) Rh : Radial distance of the center of a rib from the axis of symmetry of preform forging.
5
Fig. 7. Form-filling value 𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖,𝑛𝑛
The value of the volume fitness value 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑖𝑖 depends on the calculation of individual preform volume 𝑉𝑉𝑖𝑖 . The total volume of each individual 𝑉𝑉𝑖𝑖 is calculated by summing up the volume between the boundary points, which can be illustrated as a truncated cone. The result of volume fitness value 𝐹𝐹𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑖𝑖 is shown in Fig. 8.
The complexity of the cross-section area of a forging part is important parameter for designing preform optimization. Flash dimension is used for cross section area that is related to the material distribution. Material is necessary for achieving formfilling, it is needed more if the cross-section shape is more complex. 5. Results and Discussions The Preform shape plays a great significant role to improve quality, dimensional accuracy of final product and reduction of cost in warm forging process. In the following investigations, a bevel gear is used as optimized part, the part has been divided into N = 5 cutting planes. According to general basic rules of preform design and preform dimensions in forging process, the initial design of preform shape for bevel gear was designed theoretically. Fillet radii (RFF) and blend-in radii (RBF) are added on this preform shape based on the guidelines of preform dimensions. An evolutionary algorithm is introduced to optimize preform shape by considering the main parameter of forging parts and forging process. The following parameters for the algorithm have been used during the optimization: I : 50 pc : 25% m : 1% c : 25 w1 : 2 w2 : 1 w3 : 1 The fitness functions for this genetic algorithm that are form-filling fitness, have some result that is illustrated by graphic. The form-filling fitness 𝐹𝐹𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑖𝑖,𝑛𝑛 has a limited value between 0 and 1 as shown in Fig. 7. Because form-filling fitness has no negative value or > 1.
Fig. 8. Preform volume value Fvolume i
Fig. 9. shows the result of the complexity fitness value
𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖 i in dependency of the calculated individual
complexity 𝐶𝐶𝑖𝑖 .
Fig. 9. Complexity fitness value 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑖𝑖
Fillet radii of preform part (RFP) and blend-in radii of preform part (RBP) are added on this optimized preform shape. The values and the shape of optimal preform of bevel gear can be seen in Table 3 and Fig. 10. respectively.
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[2]
Table 3. Data geometry of design variable. Design Variables Chamfer Dimension (B) Diameter (D) Height (H) RFP 1 RFP 2 RBP t
fw
Initial Design 15.5 82.61 40.84 5.58 6.78 7 1.5 7.2
[3]
[4]
[5]
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Lu B, Ou H, Cui Z. S Shape Optimisation of Preform Design for Precision Close-Die Forging. Struct Multidisc Optim 44:785–796. DOI (2011) 10.1007/s00158-011-0668-1 D.H. Kim, B.M. Kim, Preform Design of the Bevel Gear for the Warm Forging using Artificial Neural Network, Journal of the Korean Society for Precision Engineering, Vol. 20 (2003), pp.36-43 Yu C-H, Sheu J-J Preform and Die Designs for Hot Forging Process of Linear Slide Block. Applied Mechanics and Materials 419: 395-400. Trans Tech Publications, Switzerland. Doi: (2013) 10.4028/www.scientific.net/AMM.419.395 Knust J, Stonis M, Behrens B-A Preform Optimization for Hot Forging Processes Using an Adaptive Flash Dimension Based on The CrossSection Shape Complexity. Prod. Eng. Res. Dev. 10:587–598. DOI (2016) 10.1007/s11740-016-0702-7
[6] Behrens B-A, Nickel R, Mu¨ller S (2009) Flashless precision forging of a two-cylinder crankshaft. Prod Eng Res Dev 3:381–389. doi:10.1007/s11740-009-0185-x [7]
Fig. 10. Optimal preform shape of bevel gear
6. Conclusion In summary, an initial design and optimization of the preform shape of bevel gear have been carried out in this paper. An initial design preform is determined theoretically based on cross section of the final part. The optimization of preform shape was derived by some equations and with several iterations using Excel VBA by considering parameters of forging part and forging process. The derived equation of formfilling, preform volume, and complexity have the output for final preform geometry, radius r and the position L along the vertical axis of the part. These outputs will form the preform shape of bevel gear that based on the cross section of finished part. The results of optimization preform shape are showed with the data geometry of design variables. Moreover, the result of initial design shows the preform that corresponds to general basic rules based on the cross-section area. The guidelines of Preform dimension for low-alloy steel can be considered as process variables to avoid forging defects. These process variables can affect the forging process, such as minimize friction, reduce wear along die surface, minimize forging load, and determine metal flow pattern. Acknowledgements This work was supported by the ICT R&D program of MSIP/IITP. [B0101-17-1081, Development of ICT based software platform and service technologies for medical 3D printing applications] References [1]
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