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There are many feasible solutions, and there is not a straightforward method to find the optimal one. One obvious way to achieve that is to use the minimum possible number of sections, maximum chip thickness with minimum pitch. The solution is started with the minimum possible tool length using the maximum allowable chip thickness and minimum pitch without violating the main constraints such as power, gullet area, tooth stress etc. The minimum possible number of sections is used as a start. This means that, if it is geometrically possible, the initial solution contains only one section. However, depending on the workpiece geometry, this may not be possible in which case the solution is started with minimum possible number of sections. The section profiles are selected based on the work geometry as explained in section 2.4. Once the sections are defined, each section is optimized separately. When a constraint violation is encountered, the rise is reduced and the pitch is increased. For example, for waspaloy material, the maximum and the minimum chip thickness are set to 0,065 mm and 0,012 mm, respectively based on the production data [17]. For constraint checks, the calculations such as force and stress are carried out for the first tooth of each section using the equations given in section 2.3. These can be repeated for the rest of the teeth which would take time in simulations. An alternative method is to model the stress based on the variations of the tooth geometry with respect to the first tooth. Simulations have been carried out to determine the following equations for stress predictions 2%%%%%%%2%%%0,0059 1,1811 6,86430,3709 0,00170,0002 0,072 0,0832sbbsfshsttCCCCCCCCCC=−+==+=−−
(7) where C%s is the percentage variation of the tooth stress, C%b is the percentage change in the tooth bottom width, C%h is the percentage change in the tooth height and C%t is the percentage change in the tooth top width. The algorithm starts with possible minimum number of sections, minimum number of teeth per sections, i.e. with maximum tooth rise, and minimum pitch value. Then, these parameters are modified according to the constraints. As a result, the solution which is almost optimal is found. In the following, the optimization algorithm is explained step by step. Step 1: (Cutting speed selection) First, the cutting speed must be selected. A proper cutting speed is selected based on the material and the economical tool life considering tool set up time, batch sizes etc. Step 2: (Max. and min. number of cutting teeth) In broaching, the experience and the analysis suggest that there must be at least one cutting tooth at a time in order to reduce the dynamic affects of tooth impact on the part. This means that: max2cutting lengthpitch = (8) From the geometry of the tool: min.pitch a land =× (9) where a is a constant which greater than 1. By using the maximum and minimum pitch values from above equations, we can determine the maximum and minimum number of teeth in the cutting process, mmax and mmin., respectively. Step 3: (Tooth rise option selection.) Option 2 shown in Figure 4 is the best choice if there is no geometrical limitations. That is because the increase in the bottom width compensates the increase in height. Step 4: (Definition of the geometry) Height and width values of the geometry is defined. Step 5: (Number of simultaneously cutting teeth) The number of simultaneously cutting teeth, mpm, is an important factor which directly affect the total cutting force and power, as well as the pitch of the tool. mpm can be determined based on the maximum available power on the machine for the cutting speed used. The part of the tool that has the maximum cutting area must be found out first which is needed to determine the maximum cutting force per tooth. The maximum possible chip thickness, or rise, is used in force calculations. Then, mpm can be determined as follows: maxmaxinttotalpmppFmF=