B Spline Based General Force Model For Broaching Commerce Essay


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Due to variety of profiles which are produced by broaching, the geometry of cutting edge will be varied from simple line (key ways and holes) to very complicated curves (fir trees). Variety of cutting edge geometry imposes complexity to chip load distribution along the cutting edge. Hence, introducing a general force model for broaching which is applicable for all of the broaching cutters is very difficult.

In this paper, a general force model for broaching is presented. This model expresses the cutting edge as a B-Spline parametric curve and calculates the chip load and cutting forces in broaching operation. The presented model has a great flexibility to model broaching cutter geometry and it can be applied for the entire broaching cutter.


Machining processes are widely used to produce industrial parts with different shapes and complicated profiles. Broaching is a machining process which can produce a wide range of complex internal and external profiles. It is also most acknowledged because of its high rate of productivity and surface quality in comparison to other machining process.

It is commonly used for the machining of a range of external and internal profiles such as keyways, guide ways, holes and fir-tree slots on turbine discs. Broaching has several considerable advantages over other machining processes. For example roughing and finishing of a complex profiles on a part can be completed in one stroke of the machine which would require many passes with another process such as milling. It can also produce parts with best surface quality and high geometrical and dimensional tolerances in one stroke.

In broaching, the geometry of machined part is derived directly by the inversion of broaching cutter geometry. For this reason unlike other machining processes, broaching cutters have a wide range of geometries as well as parts. Therefore, chip load has a complicated non-uniform 2D or 3D geometry along the cutting edge depending on the profile complexity of workpiece. Consequently, the cutting force along the edge is not uniform.

Proposing a general force model which can be applied for all of the broaching cutter geometries is not as easy as proposing a force model for other machining processes such as milling, turning and boring. However, achieving stable cutting, desired surface quality and higher rate of production needs a reliable force model to predict the cutting forces before performing machining. Well-established force model helps the process designer to estimate the cutting forces and to select the best parameters to reduce the force and vibration and increase the part quality.

Using B-Spline parametric curves enhances geometric modeling with design flexibility, accuracy and generality. These curves can be used in modeling of cutting as well as simulating cutting edge intersection and chip load calculation.

Although broaching is widely used in industry, there is a few number of literatures about this process. In 1960 Monday presents a detailed description of broaching technology. Recently Ozturk and Budak (2003) performed Finite Element Analysis (FEA) to predict the cutting tool stresses during the broaching process. Also Kokturk and Budak (2004) performed an optimization on the geometry of broaching tools cutting edges. In their study the cutting conditions are changed until they can satisfy the constraint. They used the optimized conditions to improve the broaching process. Yussefian et. al (2008) applied B-Spline parametric curves in modeling of boring process. By taking geometric flexibility of B-spline curves, their model was also capable of modeling any arbitrary boring cutting edge geometry as well as computing the chip load for various cutting conditions. Moetakef-Imani and Hosseini (2009) calculated the normal and tangent vector along the serrated cutting edge using B-Spline curves to simulate the cutting forces in rough end milling.

The objective of this paper is to propose a general force model for the broaching process using B-Spline interpolation of cutting edge. This research proposes a new methodology to simulate the cutting force of broaching process. Similar to other machining processes cutting force in broaching can be expressed by tangential , feed and radial components which are directly related to chip load area and the length of contact between cutting edge and workpiece. Each cutting edge is first modeled by B-Spline parametric curves then the chip load is calculated by integration of area between two successive edges. The proposed force model can calculate the chip load for any arbitrary geometry of cutting edge from the simplest to the most complicated.


Similar to other multi tooth cutter such as milling tools, broaching tool is a cutter in which several cutting edges engage with the workpiece simultaneously (Tlusty 1999). Figure (1) shows a typical geometrical representation of a broaching tool.

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Clearance Angle



Rake AngleC:\Documents and Settings\100386071\Desktop\NAMRI Paper Files\New Picture (1)11.bmp


There is no feed motion in broaching process and increasing the height of the teeth causes the tool to go through the workpiece and remove the material (similar to feed motion in other cutting processes). Figure (2) shows the cut per tooth and direction of cutting speed in broaching tool.

C:\Documents and Settings\100386071\Desktop\NAMRI Paper Files\New Picture5.bmp


The differential force components in broaching can be expressed as below:


In equations (1), is differential component of tangential force, is differential component of feed force and is differential component of radial force. and are chip thickness and length of the cut for infinitesimal element along the cutting edge respectively. and are cutting and edge constants along tangential, feed and radial directions. The radial component of force appears only during oblique broaching when cutting edge has an inclination angle with the direction of cutting speed. The total tangential, feed and radial component of cutting force for each edge can be calculated by integrating of those components along the cutting edge. Equation (2) shows the force integration along the cutting edge from the start to the end of engagement.


In the above equation, represents a differential element of chip area which is removed by the cutting edge and is length of engagement between cutting edge and workpiece. Since the chip load varies along the broaching edge, it must be segmented into elements for which local thickness can be assumed constant. The geometry of chip along the broaching tool cutting edge is complicated however. Since there is no relative motion between successive edges the chip area or chip load remains constant. The common approach for simulation of cutting forces is to divide the cutting edge to infinitesimal elements and calculate the area for each element separately. If total chip area can be calculated, cutting forces are obtained without need to dividing the edge to elements. But due to variety and complexity of cutting edge profiles in broaching, it is difficult to express the edge by an explicit function. Hence, calculation of above integration is not easy. Representing the broaching cutting edge by B-Spline curves is a powerful way to express the geometry with parametric relations which makes integration and derivation along the edge easy.

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A series of data point can be obtained by collecting points’ coordinates along the cutting edge using inspection method such as digitizing or CMM. The desired B-Spline of degree defined by control points passes through all those cutting edge data points and expresses the cutting edge by a parametric curve. This parametric representation of edge can be easily applied to perform derivation and integration along the edge to find chip load area and engagement length. The interpolated B-spline cutting edge of degree p can be expressed as below:


Where is interpolating B-Spline curve of degree , is control points which control the geometry of curve and is B-spline Basis functions which can be computed by:


In equation (4), is a B-spline knot which belongs to the knot vector of . The equation (3) has n+1 unknown control points. For this reason, it is necessary to have a parameter like to relate those control points to the data points. Since parameter corresponds to data point , substituting into the equation (3) yields the following:


There are B-spline basis functions and parameters in equation (5). Substituting into , these values can be organized into a matrix as shown as below:


Data points and control points can be expressed in similar way:





In equation (9) matrix is input data points which represents the points along the cutting edge and matrix can be obtained by evaluating B-spline basis functions at the given parameters. and both are known. The only unknown parameter in equation (9) is matrix . Equation (9) is a system of linear equations with unknown , solving for yields the control points and the desired B-spline interpolation curve becomes available.

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Once the cutting edge is presented by B-Spline, chip area and cutting length for each cutting edge can be calculated directly from B-Spline equations as follows:


Where indicate start of the cut, end of the cut, current cutting edge and previous cutting edge respectively.


In order to verify the proposed model with experiments a simple broach cutter is selected and its cutting edges are modeled using B-Spline curves. The geometry of cutter was selected similar to the one previously presented by Kokturk (2004) to compare the results. Figure (5) demonstrates B-Spline interpolation and geometry of the first cutting edges. Geometric features of cutting edge can be found in table (1).


Number of Teeth


Height of the first tooth

4 mm

Upper length of the tooth

4 mm

Base length of the tooth

4 mm

raise on the upper surface

0.06 mm

Rake angle

12 deg


2.9176 mm


5 mm

Oblique angle


Similar to other interpolation methods, B-Spline interpolation is sensitive to the number of data points. Increasing the number of data points yields a better accuracy but it makes the algorithm very slow. Decreasing the number of data points, accelerate the algorithm but it has a negative effect on accuracy. It has been shown that smooth parts of the curve are not very sensitive to the number of data points. The interpolation inaccuracy occurs in the sharp corners where the direction of curve changes suddenly. As a result, it would be better to use more data points at the sharp corners and less data point at the other parts to obtain the accuracy and time efficiency of the algorithm simultaneously.


Figure (6) show the B-Spline presentation of two successive cutting edges.


It can be seen from the above figures that B-Spline curve follows the data point at the sharp corner with an acceptable accuracy. Figure (7) depicts a zoom view of corner.

Control Points

Chip Load

(Area Between two successive edges)



In order to check the validity of the model, the output of proposed model has been compared to the result of the experimental work performed by Kukturk (2004). Figure (8) shows the final geometry of workpiece. Cutting conditions and force coefficients which are used in simulation can be found in table (2).

4 mm

5.2 mm mm

21 mm

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(Kokturk 2004)

Workpiece Material


Length of cut

21 mm

Width of cut

4 mm


0.06 mm

Cutting Velocity

55.88 (mm/s)

5281 (N/mm^2)

2889 (N/mm^2)

Since the employed cutting edge doesn’t have oblique angle, the cutting force has no radial component. The edge coefficients of cutting force are always very small in comparison to cutting coefficients so it has been assumed that , and are negligible.




It can be seen from figures (9), (10) and (11) that the results of newly proposed model are in good agreement with the previously published results of Kokturk (2004).


Although broaching operation can be applied to produce variety of complex internal and external profiles, the main drawback of broaching is low flexibility after tool design. In other machining process such as turning or milling, cutting parameters such as feed, cutting speed, axial and radial depth of cut can be changed during the process. In contrast once a broaching tool is designed for the specific purpose, the only parameter that can be under control is cutting speed. Therefore, accurate simulation of cutting forces acting on broaching cutting edge is very important to enhance the design methodology and improve the cutting performance.

In this paper, a general force model is developed for simulation of cutting forces in broaching using B-spline representation of cutting edge. In comparison to previous model presented by Kokturk (2004), which applies edge subdivision to infinitesimal elements for simple geometry, the new model can interpolate broaching tool cutting edge without any limitations as well as simulation of cutting forces for any desired input geometry. The newly proposed model can be applied for any geometry of cutting edge to obtain chip load. Once the chip load is identified, cutting forces can be predicted with acceptable accuracy.

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