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(3) which shows that minimizing the length or maximizing the material removal rate are equivalent. There are many constraints due to the physics of metal cutting, machine and geometrical limitations, which are interrelated. These make the optimization problem complicated which will be explained in the following sections. 2.3 Constraints Ozturk [17] presented the important constraints for the broaching process. There are power, force and tool length constraints for a given machine; tool stress, cutting force, chip load, tooth geometry, maximum number of cutting teeth etc. constraints as summarized in the following. Force model The broaching force is one of the fundamental parameters affecting the process, and needs to be modeled. The broaching forces in two fundamental directions can be given as [21]: ()()11mttciiteiimf fc i i fe iiFKtbKbFKtbKb===+=+∑∑
(4) where m is the number of simultaneously cutting teeth, ti and bi are uncut chip thickness and width of cut for the cutting teeth i, respectively. Kc and Ke represent the cutting and the edge force coefficients, respectively, in the tangential and the normal directions. Broaching processes can be in orthogonal or oblique cutting modes based on the tooth geometry, i.e. whether there is an inclination or helix angle between the normal to the cutting speed direction and the cutting edge. In this study, orthogonal cutting process is assumed for simplicity. In addition to the well known differences in the cutting process mechanics between two cutting modes, such as the chip flow angle, oblique cutting may present some advantages. In orthogonal cutting, each tooth enters and exits the cut instantaneously, which results in sudden increases and decreases in the total cutting force. The oblique angle spreads the entry and exit over a longer cutting distance, hence the sudden changes in cutting forces are eliminated. This can also be modeled and simulated similar to the force simulations for helical end milling processes, provided that the oblique force coefficients are known. The cutting force coefficients for oblique cutting can be determined from [21]. ( )()()()()()222222222cos tan tan sinsin cos tan sinsinsin cos cos tan sincos tan tan sinsin cos tan sinnn n stcn nnn nnn sfcn nnn nnn n srcn nnn niKKiiKβα ηβ τφ φβα η ββα τφ φ βα η ββα ηβ τφ φβα η β−+=+− +−=+− +−−=+− + (5) In equation 5, the force coefficients for tangential, feed and radial forces are calculated by using shear stress ( s τ ), shear angle ( n φ ), rake angle ( n α ), friction coefficient ( n β ), oblique angle ( i ), and chip flow angle (η ). The power of the broaching machine is one of the constraints, which can easily be determined once the cutting force is known. Tooth Stress The maximum stress on a broaching tooth may be one of the constraints as it may cause tooth breakage. In general, broaching teeth have complex geometry and it is not possible to model the stresses analytically. Finite element analysis (FEA) can be used to determine the stresses, however considering great variety of the tooth profiles the procedure needs to be simplified. In order to model the stress on a tooth, Ozturk [17] developed a general equation for the tooth stress based on FEA. He used the tooth form shown in figure 2 due to general shape which can be used to represent variety of broach tool forms. Figure 2: General tooth profile used to find stress. It was demonstrated [17] that even for complex geometries such as fir-tree forms this representation results in good predictions for the stress: 0.374 1.09 0.072 0.088 0.082 0.3561 (1.3 ) tFH B T R l σψ −−−=
(6) There are other constraints related to the geometry and the cutting performance of the process as presented by Ozturk. One of them is related to the chip space in the tool gullet area. The gullet area was also formulated [17] as a function of the tool geometry and will be used in this study as well. Another requirement, but not an absolute constraint, is related to the variation of the cutting force from section to section as well as within a particular section. If this variation is minimized, the sudden jumps in the cutting forces would be eliminated which would reduce tool wear and quality problems. 2.4 Broach tool sections Number of sections and their respective profiles are very important decisions in broach tool design. This fundamental decision affects the cost of tooling, process cycle time, surface quality etc. There are almost infinite possibilities for sectional selections. Therefore, there is a need for a method for this selection. As an example, consider the geometry shown in figure 3. There are two basic methods for distribution of the material volume to be machined among the sections: height pisions or width pisions. They have different implications on the process. First of all, in height pisions the tooth stresses are much lower due to the fact that each section starts with the shortest possible tooth height which increases as much as needed to remove the material for that section. As shown in equation 6, the tooth stress increases with height and decreases with the width. Tooth height is one of the most important factors affecting the tooth stress. In width pisions, on the other hand, the tooth height may become too large causing high tooth stress. For the example shown in figure 3, in width pision method, in some sections the width and total cutting length do not vary while the height of the tooth increases resulting in high stresses. But in height pision method, the cutting length decreases as the teeth become higher which decreases the cutting force, and as a result the stress decreases. Therefore, height pision is more efficient way of piding the sections. Figure 3: Volume pisions for the geometry. 2.5 Tooth rise options In broaching, material removal is facilitated by increasing the size of a tooth with respect to the previous one since there is no feed motion as in other machining operations. This enlargement can be achieved in several ways. Figure 4 shows different rise choices for a sample profile. In option 1, the cutting length is kept constant. In the second option, the cutting length and width can be controlled by selecting proper values of rise on the top and the side. In option 3, the side length is kept constant where the top decreases. The best stress control is in option 2 with relatively small rise on the side so that the effect of increasing height is compensated with increasing bottom width. Figure 4: Tooth rise options. 2.6 Optimization Algorithm The main purpose of the optimization procedure is to obtain the minimum total tool length.