An analytical formulation of ZOA-based machining stability for complex tool geometries: application to 5-axis ball-end milling

Abstract

The paper describes a computationally convenient analytical formulation of the stability of the cutting process with respect to self-excited vibrations in the case of five-axis milling based on the commonly used zero-order approximation. In the case of five-axis milling with general milling cutters, it is difficult to calculate stable machining process conditions for two main reasons. The first reason is the difficulty of calculating the mean value of the cutting force Jacobian with respect to the regenerative displacement (closely related to a milling directional matrix) for a generally inclined tool, and the second reason is the nonlinearity of this Jacobian with respect to the process parameters, which means that the problem cannot be reduced to a linear eigenvalue problem as is usual for linear cases (e.g. cylindrical milling with respect to the axial depth of cut). In the first part, this paper presents a modification of the calculation of the machining stability limits for a nonlinearly dependent cutting force Jacobian. A new formulation of this Jacobian for a general tool based on the surface integral over the tool and workpiece engagement region is presented which leads directly to the mean value of the Jacobian of the cutting force (direction matrix) without the need to calculate it as a function of time and then calculate the mean value over one revolution. The advantage is that if we can analytically describe the engagement area, we also obtain an analytical relation for the cutting force Jacobian. This is presented with the practical example of a generally inclined ball-end mill. This analytical formulation of the force Jacobian allows the calculation of its derivatives with respect to the technological parameters (depth of cut, step over, tilt and lead angles), which is useful both for the calculation of stability diagrams and for solving optimization problems related to machining stability.

Appendix

1.1 A.1 Derivation of the formula for the calculation of the cutting force Jacobian

Below, a derivation of the transition from the standard method of calculating the mean value of the Jacobian to the calculation via surface integration over the engagement surface (Eq. 8) is outlined. Here, the engagement is assumed to be the same for all blades within a revolution and we can therefore replace the sum by N-times.

$$\begin{aligned}{}[\overline{\nabla _{\mathbf \Delta } \mathbf{F}}]&=\frac{1}{2\pi }\int \limits _0^{2\pi }\sum _{j=1}^N \int \limits _{s_{min}}^{s_{max}} g(s,j,\varphi )\, [T]\left. \frac{\partial {{ {f}}}}{\partial {h}}\right| _{\mathbf \Delta =\mathbf {0}} \mathbf {\hat{n}}^T \frac{d {w}}{d {s}}{\text {d}}{s} d\varphi \\&=\frac{N}{2\pi }\underbrace{\int \limits _0^{2\pi }\int \limits _{s_{min}}^{s_{max}}g(s,0,\varphi )}_{\int _{S_{eng}}}\frac{ [T]\left. \frac{\partial {{ {f}}}}{\partial {h}}\right| _{\mathbf \Delta =\mathbf {0}} \mathbf {\hat{n}}^T}{\rho (\mathbf{x})} \underbrace{\rho (\mathbf{x})\frac{d {w}}{d {s}}{\text {d}}{s} d\varphi }_{d S}. \end{aligned}$$

where $s_{min}, s_{max}$ denote lower and upper limit of cutting edge parametrization. The merit of the formulation is that while the previous procedure works with a parametrization based on tool rotation and a curve parameter for the cutting edge, the reformulation into a surface integral no longer depends on the surface parametrization and it is therefore possible to choose the parametrization in which the calculation is the simplest.

The derivation of the formulation is based on a textbook formula for a surface element in the curvilinear coordinate system, which is

$$\begin{aligned} {\text {d}}{S}&=\left\| \frac{d\mathbf {x}}{{\text {d}}{s}}\times \frac{d\mathbf {x}}{d\varphi } \right\| {\text {d}}{s} d\varphi . \end{aligned}$$

The cross product can be conveniently written as

$$\begin{aligned} \frac{d\mathbf {x}}{d s}\times \frac{d\mathbf {x}}{d\varphi }&=\frac{d\mathbf {x}}{d s}\times (\mathbf {\hat{e}}_\Omega \times \mathbf {x})\\&= \left( \frac{d\mathbf {x}}{d s}\cdot \mathbf {\hat{e}}_b\right) \left\| \mathbf {x}-(\mathbf {x}\cdot \mathbf {\hat{e}}_\Omega ) \mathbf {\hat{e}}_\Omega \right\| \mathbf {\hat{n}}, \end{aligned}$$

which leads to formulation used in the integral above

$$\begin{aligned} {\text {d}}{S}&= \underbrace{\left\| \mathbf {x}-(\mathbf {x}\cdot \mathbf {\hat{e}}_\Omega ) \mathbf {\hat{e}}_\Omega \right\| }_{\rho (\mathbf {x})} \underbrace{\left| \frac{d {\varvec{\gamma }_j}}{d {s}}\cdot \mathbf {\hat{b}}\right| {\text {d}}{s}}_{{\text {d}}\!{w}} d\varphi , \end{aligned}$$

where $\mathbf {\hat{e}}_\omega$ is the tool axis direction unit vector.

1.2 A.2 Material removal simulation SW MillVis

The SW implementation of the material removal simulation MillVis uses voxel and distance field representation of the workpiece. The voxel discretization is to divide the volume into cubic blocks, which are further divided into $3\times 3 \times 3$ cells. Each block is labeled as either outer, surface, or inner, with only surface blocks intersecting the workpiece surface and the distance function values containing only vertices belonging to at least one intersected cell, see Fig. 10a. For all vertices of such a cell, the distance to the workpiece surface is reconstructed using trilinear interpolation — this approach is called a distance field, see Fig. 10b. For the vertices distant from the workpiece surface, the value of the distance function is set to plus or minus infinity for simplicity, since the actual value is not needed.

Fig. 10

a Outer, surface and inner block for voxel representation of the workpiece, b distance of the workpiece boundary from cell vertices $d_k$ for detailed workpiece representation by a trilinear surface

The workpiece represented by voxels and distance fields can be effectively visualized using ray tracing, which provides high fidelity detail of the machined surface.

In the material removal simulation, surface and internal blocks that are close to the tool are identified. Then, for each of their cells, it is checked whether it is intersected by the tool and, if so, the distance function values at the corresponding vertices are updated. In practice, the representation of the workpiece geometry by the distance field has been shown to be sufficiently accurate while maintaining reasonable memory requirements, as lower accuracy is usually required for larger workpieces and vice versa (see Fig. 11).

The tool envelope definition and cutting force formulation is based on the framework by [32, 33]. Over the course of the development of this software tool, the ability to predict cutting forces with sufficient accuracy was repeatedly confirmed by comparison with dynamometer data. Figure 11 shows a comparison of measured and simulated cutting forces during pocket machining (Fig. 12).

Fig. 11

Comparison of real and simulated cutting force (active component) during pocket machining

Fig. 12

Pocket machining for comparison of cutting forces: a virtual machining in material removal simulation software and b real-life machining on the same toolpaths

1.3 A.3 Ball-end milling local coordinate system basis

The LCS basis vectors defined in Eq. 4 in the spherical surface parametrization in the ECS are presented below. However, if possible, it is preferable to calculate the LCS basis formulas based on tool axis vector Eq. (22) and their definition which reduces possibility of an error. The tangential direction is dominantly given by circumferential velocity calculated from tool axis vector and point on the tool envelope (or engagement surface)

$$\begin{aligned} \mathbf {\hat{e}}_\Omega \times \mathbf {x}&=r\begin{pmatrix} \tan \psi _T \cos \theta +\cos \phi \sin \theta \\ -\tan \psi _L \cos \theta - \sin \phi \sin \theta \\ \sin \theta (\tan \psi _T \sin \phi -\tan \psi _L \cos \phi ) \end{pmatrix}. \end{aligned}$$

The LCS basis vectors are shown below

$$\begin{aligned} \mathbf {\hat{t}}&=\frac{\mathbf {\hat{e}}_\Omega \times \mathbf {x}}{\Vert \mathbf {\hat{e}}_\Omega \times \mathbf {x}\Vert },\\ \mathbf {\hat{n}}&=\begin{pmatrix} \sin \phi \sin \theta \\ \cos \phi \sin \theta \\ -\cos \theta \end{pmatrix},\\ \mathbf {\hat{b}}&=\frac{r}{\Vert \mathbf {\hat{e}}_\omega \times \mathbf {x} \Vert }\begin{pmatrix} \cos \phi \sin ^2\theta (\tan \psi _L \cos \phi -\tan \psi _T \sin \phi )+\tan \psi _L \cos ^2\theta +\sin \phi \sin \theta \cos \theta \\ \tan \psi _T \left( \sin ^2\phi \sin ^2\theta +\cos ^2\theta \right) +\cos \phi \sin \theta (\cos \theta -\tan \psi _L \sin \phi \sin \theta )\\ \sin \theta \cos \theta (\tan \psi _T \cos \phi +\tan \psi _L \sin \phi )+\sin ^2 \theta \end{pmatrix}. \end{aligned}$$

1.4 A.4 Script for cutting force Jacobian calculation

The calculation of the Jacobian for milling spherical ends leads to matrices that are too complex to be reasonably presented in text form. The code below in Wolfram Mathematica follows the calculation described in the paper. The division into the functions shown is made with readability in relation to the article in mind, not code efficiency. For faster computation, it is preferable to analytically precalculate some of the functions (e.g. matrix [T]).