Minimizing time of contour-parallel pocket end-milling by prescribing limit axial and radial depth pairs

Abstract

The purpose of this study is to minimize pocketing time by exploiting the most beneficial aspects of tool dynamics and the most beneficial chronology of tool passes in contour-parallel toolpath. This is achieved by always prescribing limiting axial and radial depth pairs on a contour-parallel toolpath as a means of minimizing pocketing time within milling machine load specifications. The optimization approach is identified to be of two types: boundary to centre (b→c) execution and centre to boundary (c→b) execution. Expression for pocketing time is developed for each type taking into account the revelation that the lengths of the bivariately optimized cyclic passes between the first and last cycles constitute an arithmetic series. The presented expressions are demonstrated to be useful in systematic determination of choice of progression (either (b→c) execution or (c→b) execution) of contour-parallel pocketing operation and choice of limiting axial and radial depth pairs that guarantee minimum machining time. For example, the studied experimentally characterized case gave that (c→b) execution is more time saving than (b→c) execution by about 3.5565%–15.7690%. This experimentally characterized case also confirms the expectation that continuous tool–workpiece contact of contour-parallel toolpath always ensures a shorter pocketing time than both one-way up- and down-milling toolpaths. It is further seen that it is faster pocketing at high radial depth range of 60%–100% of tool diameter which fortunately is the range of less load restrictions. The benefits of contour-parallel toolpath relative to one-way up- and down-milling toolpaths got more strongly confirmed when the pocket got larger (for example, the time saving of (c→b) execution relative to one-way up-milling attained 51.6733% for a larger pocket, while time saving of 41.3385% was recorded for smaller pocket) and when the slope of limiting radial depth versus axial depth got less.

Introduction

The most violent self-excited vibration in machining operation is called regenerative chatter or secondary chatter. Other less frequently occurring thus less studied self-excited vibrations in machining operations are classified as primary chatter which are further divided into mode coupling effects, ¹ frictional effects and thermo-mechanical effects. ² From the inception of machining processing of materials, it has been in the mind of engineers and machinist to control vibrations which limit productivity, incur loss and result in poorly performing parts. This could be why at the beginning of the 20th century Taylor described it as the most obscure and delicate of all problems facing the machinist. In the absence of theoretical knowledge, chatter control began with trial and error method based on experience and manuals. Systematic control of regenerative chatter began with discovery of its delayed nature and subsequent modelling of orthogonal turning process with autonomous delay differential equation (DDE). This happened during the later part of the sixth decade and early part of the seventh decade of the twentieth century as seen in the work of Tlusty et al. ¹ Analysis led to computation of stability lobes that demarcate the cutting process parameter plane into stable and unstable sub-domains allowing systematic choices of non-chattered productive parameter combinations. Regenerative chatter is basically triggered by unstable cutting force variation that stem from unfavourable phase difference between two adjacent cuts. The basic milling process was subsequently modelled with linear periodic single discrete DDE by Sridhar et al. ³ The non-autonomous nature of milling model required much more challenging stability analysis than what obtained in turning process and a number of methods were developed to surmount the problem. Some of the methods are as follows: frequency methods which include the zeroth-order approximation (ZOA) method (so-called because it utilizes only the first term in the Fourier series expansion of periodic coefficients) ⁴ and the multi-frequency solution (MFS) method (so-called because it utilizes higher order harmonics), ⁵ the semi-discretization methods (they basically entail regular partitioning of the delayed term while leaving the undelayed terms unpartitioned and approximating the periodic coefficient matrices as piecewise constant functions leading to construction of time-domain finite-dimensional monodromy matrix from which stability conditions are extracted via eigenvalue analysis), ⁶ the methods of temporal finite element analysis (they are based on integral minimization of weighted residuals and accordingly ⁷ h-version and P-version exist for general periodic single discrete delayed system), the Chebyshev collocation method (it utilizes the collocation expansion at the Chebyshev collocation points) ⁸ and the full-discretization methods (like the semi-discretization methods, the delayed state in these methods is regularly discretized in addition to discretization of the current state).^9–11 A simple experimental equivalent of the above-mentioned analytical methods was proposed by Quintana et al. ¹² Their method involves tool–workpiece arrangement that guarantees rising axial depth in feed direction such that a point on milling stability diagram initiates chatter.

During the development of the above listed methods, chatter-suppressing passive and active effects were being introduced in the basic milling model. The effects led to more advanced but somewhat more stable models. The passive dampers do not require external supply of energy to suppress vibration. They include geometric alteration of the tool to interfere with regenerative effects, improved design of machine tool structure against vibration or incorporation of extra devices with good material damping properties for energy dissipation. Some of the known chatter-suppressing geometric alterations of the tool include the following: milling tools with non-uniform pitch thus having multiple discrete delays, ¹³ variable helix tools ¹³ and serration of milling tools. ¹⁴ Some alterations in tool geometry such as tool wear, edge radii and flank angles/profiles suppress chatter by process damping at low speeds. ¹⁵ A chatter-suppressing passive device has been suggested by Semercigil and Chen. ¹⁶ A suggestion by Kim et al. ¹⁷ involves introduction of frictional damping material into a cylindrical hole drilled in the centre of end-milling cutter to dissipate chatter energy. The active dampers on the other hand require external supply of energy to limit vibration within a range. Generally active vibration controllers are servomechanisms of electromechanical, electro-fluidic, electromagnetic or piezoelectric types that are composed of the major elements: sensor, signal processor and actuator. Active control of chatter with novel electromagnetic proof-mass actuator was presented by Huyanan and Sims. ¹⁸ Chatter in milling has been suppressed by Brecher et al. ¹⁹ by bringing into a milling machine a piezoelectric-actuated workpiece holder. Other active techniques like spindle speed variation (SSV) work by disrupting regenerative effects. Elaborate work which includes the study by Bediaga et al. ²⁰ has been carried out focussing on SSV as a means of chatter control. SSV is more effective at low speed but Seguy et al. ²¹ have shown that it more effectively suppresses high-speed flip bifurcation than Hopf bifurcation at the first lobe.

The pursuit of more complicated mathematical models for the mentioned stabilizing designs does not mean that the dynamics of the basic model has been fully exploited. The stabilizing designs are unconventional and unpopular thus requiring costly and probably unworthy awareness creation for widespread adoption. The gains are sometimes not considerable enough to galvanize a paradigm shift in the manufacturing industry. Such stabilizing designs do not come without additional technological demand or even dynamic problems like inertia problems in SSV. Worse still, the computed benefits could fall in a technological range that is not preferred for a machine tool in question. Several efforts have been made to make the most of the dynamics of the basic milling model in planning for productive and smooth milling project. Tekeli and Budak ²² optimized material removal rate (MRR) of pocketing process with limiting pairs of axial depth of cut and radial depth of cut. Heo et al. ²³ considered stability limit of the basic milling model in addition to load capacity of the spindle as a constraint in their formulation and evolutionary analysis of time of high-speed pocketing. Ozoegwu and Ezugwu ²⁴ further explored and generalized the optimization strategy of Tekeli and Budak leading to identification and comparison of two alternative pocket execution routines: vertical chronology (VC) and horizontal chronology (HC). They saw that the former is superior in saving time. Existence of a coordinate for maximum limiting MRR ( $\mathrm{MR}{R}_{\lim }$ ) was identified and demonstrated to give minimum pocketing time for large pockets. The studies by Tekeli and Budak ²² and Ozoegwu and Ezugwu ²⁴ considered one-way toolpath but Ozoegwu et al. ²⁵ have extended the optimization method to zigzag toolpath that maintains continuous contact between the tool and workpiece. The zigzag toolpath avoided the delay of return idle passes inherent in one-way toolpath thus hastening the pocketing process. A method of simultaneously improving chatter-free conditions and reducing machining power of slotting using combined-mode end-milling has been proposed. ²⁶ Cao et al. ²⁷ maximized MRR of high-speed milling considering process stability and surface quality of workpiece. Optimal control theory has been applied in tapping the benefits of milling regenerative dynamics for optimizing productivity. Monnin et al. ²⁸ presented two different optimal control strategies that consider milling dynamics while optimizing productivity. Bort et al. ²⁹ conducted – on the optimal control theory – a multi-objective optimization of productivity and surface quality and recorded for a case study reduction of the cycle time by 46% and the surface roughness (Ra) by 22%. This study aims at extending the optimization method of the works^22,24,25 using contour-parallel toolpath which also maintains continuous contact between the tool and workpiece like the zigzag toolpath. The contour-parallel toolpath maintains a single mode (up-milling or down-milling), while the zigzag toolpath changes mode from one pass to the next. For this reason, the contour-parallel toolpath is more attractive because of simpler and more practical implementation routine. This justifies the aim in this work of studying limiting contour-parallel toolpath as a way of minimizing pocketing time.

Model and stability analysis of milling process

A 2-degree-of-freedom (2-DOF) milling model is used because of tool compliance in both feed and feed-normal directions. The model for analysis with the full-discretization method is put in the form

$$\stackrel{·}{\mathit{y}}\left(t\right)=\mathbf{A}\mathit{y}\left(t\right)+\mathbf{B}\left(t\right)\mathit{y}\left(t\right)-\mathbf{B}\left(t\right)\mathit{y}(t-\tau )$$

(1)

where

$$\mathbf{A}=\left[\begin{array}{cccc}0\hfill & 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-{\omega }_{\mathrm{nx}}^2}& -2{\zeta }_x{\omega }_{\mathrm{nx}}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -{\omega }_{\mathrm{ny}}^2\hfill & -2{\zeta }_y{\omega }_{\mathrm{ny}}\hfill \end{array}\right]$$

(2)

$$\mathbf{B}\left(t\right)=\left[\begin{array}{cccc}0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{w{h}_{\mathrm{xx}}\left(t\right)}{m_x}}& 0\hfill & -\frac{w{h}_{\mathrm{xy}}\left(t\right)}{m_x}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\frac{w{h}_{\mathrm{yx}}\left(t\right)}{m_y}}& 0\hfill & -\frac{w{h}_{\mathrm{yy}}\left(t\right)}{m_y}\hfill & 0\hfill \end{array}\right]$$

(3)

The modal parameters with subscript (x) refer to the feed direction, while those with subscript (y) refer to the feed-normal direction. The w is the axial depth of cut. The radial immersion which is the non-dimensionalized radial depth is contained in the model via the screen functions $g_j(t)$ given for up-milling and down-milling in terms of angles ${\theta }_j(t)$ and $\rho$ (radial immersion), respectively, as

$$\begin{array}{c}\multicolumn{1}{c}{g_j\left(t\right)=\frac{1}{2}\{1+\mathrm{sgn}[\sin ({\theta }_j\left(t\right)-\arctan \left\{\frac{-1}{2\rho }\sin \left[\arccos (1-2\rho )\right]\right\})+\sin \left(\arctan \left\{\frac{-1}{2\rho }\sin \left[\arccos (1-2\rho )\right]\right\}\right)]\}}\end{array}$$

(4a)

$$g_j\left(t\right)=\frac{1}{2}\{1+\mathrm{sgn}[\sin ({\theta }_j\left(t\right)-\arctan \left\{\frac{1}{2\rho }\sin \left[\arccos (2\rho -1)\right]\right\})-\sin (\arccos (2\rho -1)-\arctan \left\{\frac{1}{2\rho }\sin \left[\arccos (1-2\rho )\right]\right\})]\}$$

(4b)

The angular position of jth tooth ${\theta }_j(t)$ of N-toothed miller is given by

$${\theta }_j\left(t\right)=\left(\frac{\pi \Omega }{30}\right)t+(j-1)\frac{2\pi }{N}$$

(5)

This means that the spindle speed $\Omega$ (r/min) is brought into the model through the discrete delay $\tau$ and also through the screen functions. The specific force variations seen in matrix $\mathbf{B}(t)$ are given by

$$\begin{array}{c}\multicolumn{1}{c}{h_{\mathrm{xx}}\left(t\right)=C_t\gamma {\left(v\tau \right)}^{\gamma -1}}\\ \multicolumn{1}{c}{\sum _{j=1}^N{g}_j\left(t\right){\sin }^{\gamma }{\theta }_j\left(t\right)[\chi \sin {\theta }_j\left(t\right)+\cos {\theta }_j\left(t\right)]}\end{array}$$

(6a)

$$\begin{array}{c}\multicolumn{1}{c}{h_{\mathrm{xy}}\left(t\right)=C_t\gamma {\left(v\tau \right)}^{\gamma -1}}\\ \multicolumn{1}{c}{\sum _{j=1}^N{g}_j\left(t\right){\sin }^{\gamma -1}{\theta }_j\left(t\right)\cos {\theta }_j\left(t\right)[\chi \sin {\theta }_j\left(t\right)+\cos {\theta }_j\left(t\right)]}\end{array}$$

(6b)

$$h_{\mathrm{yx}}\left(t\right)=C_t\gamma {\left(v\tau \right)}^{\gamma -1}\sum _{j=1}^N{g}_j\left(t\right){\sin }^{\gamma }{\theta }_j\left(t\right)[\chi \cos {\theta }_j\left(t\right)-\sin {\theta }_j\left(t\right)]$$

(6c)

$$\begin{array}{l}{h}_{yy}\left(t\right)=C_t\gamma {\left(v\tau \right)}^{\gamma -1}\\ {\displaystyle \sum _{j=1}^N{g}_j}\left(t\right){\sin }^{\gamma -1}{\theta }_j\left(t\right)\cos {\theta }_j\left(t\right)\left[\chi \cos {\theta }_j\left(t\right)-\sin {\theta }_j\left(t\right)\right]\end{array}$$

(6d)

All the coordinates involved in the derivation of the milling model presented above rotates 90° with the feed when the tool on a contour-parallel toolpath moves from one edge to a perpendicular edge. The direction of modal parameters remains fixed so that orthogonal change in feed direction is taken care of by interchange of ${\omega }_{\mathrm{nx}}$ with ${\omega }_{\mathrm{ny}}$ and ${\zeta }_x$ with ${\zeta }_y$ .

The fourth-order full-discretization method ¹¹ is adopted here for stability analysis because of its accuracy. The monodromy matrix reads $\mathbf{\psi }={\mathit{M}}_{k-1}{\mathit{M}}_{k-2}\dots {\mathit{M}}_0$ where

$${\mathit{M}}_i=\left[\begin{array}{cccccccc}{\mathbf{M}}_{11}^i\hfill & {\mathbf{M}}_{12}^i\hfill & {\mathbf{M}}_{13}^i\hfill & {\mathbf{M}}_{14}^i\hfill & \cdots \hfill & 0\hfill & {\mathbf{M}}_{1\mathit{k}}^i\hfill & {\mathbf{M}}_{1,\mathit{k}+1}^i\hfill \\ \multicolumn{1}{c}{\mathbf{I}}& 0\hfill & 0\hfill & 0\hfill & \cdots \hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \mathbf{I}\hfill & 0\hfill & 0\hfill & \cdots \hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & \mathbf{I}\hfill & 0\hfill \end{array}\right]$$

(7)

$${\mathit{M}}_{11}^i={\mathit{P}}_i({\mathit{F}}_0+{\mathit{G}}_{44}{\mathit{B}}_i+{\mathit{G}}_{49}{\mathit{B}}_{i+1})$$

(8a)

$${\mathit{M}}_{12}^i={\mathit{P}}_i({\mathit{G}}_{43}{\mathit{B}}_i+{\mathit{G}}_{48}{\mathit{B}}_{i+1})$$

(8b)

$${\mathit{M}}_{13}^i={\mathit{P}}_i({\mathit{G}}_{42}{\mathit{B}}_i+{\mathit{G}}_{47}{\mathit{B}}_{i+1})$$

(8c)

$${\mathbf{M}}_{14}^i={\mathit{P}}_i({\mathit{G}}_{41}{\mathit{B}}_i+{\mathit{G}}_{46}{\mathit{B}}_{i+1})$$

(8d)

$${\mathit{M}}_{1k}^i=-{\mathit{P}}_i({\mathit{G}}_{12}{\mathit{B}}_i+{\mathit{G}}_{13}{\mathit{B}}_{i+1})$$

(8e)

$${\mathit{M}}_{1,k+1}^i=-{\mathit{P}}_i({\mathit{G}}_{11}{\mathit{B}}_i+{\mathit{G}}_{12}{\mathit{B}}_{i+1})$$

(8f)

$${\mathbf{P}}_i={[\mathbf{I}-{\mathbf{G}}_{45}{\mathbf{B}}_i-{\mathbf{G}}_{410}{\mathbf{B}}_{i+1}]}^{-1}$$

(9)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{G}}_{41}=\frac{1}{24{\left(\mathrm{\Delta }t\right)}^5}}\\ \multicolumn{1}{c}{\{-2{\left(\mathrm{\Delta }t\right)}^4{\mathit{F}}_2+{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3+3{\left(\mathrm{\Delta }t\right)}^2{\mathit{F}}_4-\mathrm{\Delta }t{\mathit{F}}_5-{\mathit{F}}_6\}}\end{array}$$

(10a)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{G}}_{42}=\frac{1}{6{\left(\mathrm{\Delta }t\right)}^5}}\\ \multicolumn{1}{c}{\{3{\left(\mathrm{\Delta }t\right)}^4{\mathit{F}}_2-2{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3-4{\left(\mathrm{\Delta }t\right)}^2{\mathit{F}}_4+2\mathrm{\Delta }t{\mathit{F}}_5+{\mathit{F}}_6\}}\end{array}$$

(10b)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{G}}_{43}=\frac{1}{4{\left(\mathrm{\Delta }t\right)}^5}}\\ \multicolumn{1}{c}{\{-6{\left(\mathrm{\Delta }t\right)}^4{\mathit{F}}_2+7{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3+3{\left(\mathrm{\Delta }t\right)}^2{\mathit{F}}_4-3\mathrm{\Delta }t{\mathit{F}}_5-{\mathit{F}}_6\}}\end{array}$$

(10c)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{G}}_{44}=\frac{1}{6{\left(\mathrm{\Delta }t\right)}^5}}\\ \multicolumn{1}{c}{\{6{\left(\mathrm{\Delta }t\right)}^5{\mathit{F}}_1-{\left(\mathrm{\Delta }t\right)}^4{\mathit{F}}_2-10{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3+4\mathrm{\Delta }t{\mathit{F}}_5+{\mathit{F}}_6\}}\end{array}$$

(10d)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{G}}_{45}=\frac{1}{24{\left(\mathrm{\Delta }t\right)}^5}}\\ \multicolumn{1}{c}{\{6{\left(\mathrm{\Delta }t\right)}^4{\mathit{F}}_2+5{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3-5{\left(\mathrm{\Delta }t\right)}^2{\mathit{F}}_4-5\mathrm{\Delta }t{\mathit{F}}_5-{\mathit{F}}_6\}}\end{array}$$

(10e)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{G}}_{46}=\frac{1}{24{\left(\mathrm{\Delta }t\right)}^5}}\\ \multicolumn{1}{c}{\{-2{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3-{\left(\mathrm{\Delta }t\right)}^2{\mathit{F}}_4+2\mathrm{\Delta }t{\mathit{F}}_5+{\mathit{F}}_6\}}\end{array}$$

(10f)

$${\mathit{G}}_{47}=\frac{1}{6{\left(\mathrm{\Delta }t\right)}^5}\{3{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3+{\left(\mathrm{\Delta }t\right)}^2{\mathit{F}}_4-3\mathrm{\Delta }t{\mathit{F}}_5-{\mathit{F}}_6\}$$

(10g)

$${\mathit{G}}_{48}=\frac{1}{4{\left(\mathrm{\Delta }t\right)}^5}\{-6{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3+{\left(\mathrm{\Delta }t\right)}^2{\mathit{F}}_4+4\mathrm{\Delta }t{\mathit{F}}_5+{\mathit{F}}_6\}$$

(10h)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{G}}_{49}=\frac{1}{6{\left(\mathrm{\Delta }t\right)}^5}}\\ \multicolumn{1}{c}{\{6{\left(\mathrm{\Delta }t\right)}^4{\mathit{F}}_2+5{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3-5{(\mathrm{\Delta }t)}^2{\mathit{F}}_4-5\mathrm{\Delta }t{\mathit{F}}_5+{\mathit{F}}_6\}}\end{array}$$

(10i)

$$\begin{array}{c}\multicolumn{1}{c}{{\mathit{G}}_{410}=\frac{1}{24{\left(\mathrm{\Delta }t\right)}^5}}\\ \multicolumn{1}{c}{\{6{\left(\mathrm{\Delta }t\right)}^3{\mathit{F}}_3+11{\left(\mathrm{\Delta }t\right)}^2{\mathit{F}}_4+6\mathrm{\Delta }t{\mathit{F}}_5+{\mathit{F}}_6\}}\end{array}$$

(10j)

$${\mathit{G}}_{11}=\frac{1}{{\left(\mathrm{\Delta }t\right)}^2}\{{\mathit{F}}_1{\left(\mathrm{\Delta }t\right)}^2-2{\mathit{F}}_2\mathrm{\Delta }t+{\mathit{F}}_3\}$$

(10k)

$${\mathit{G}}_{12}=\frac{1}{{\left(\mathrm{\Delta }t\right)}^2}\{{\mathit{F}}_2\mathrm{\Delta }t-{\mathit{F}}_3\}$$

(10l)

$${\mathit{G}}_{13}=\frac{1}{{\left(\mathrm{\Delta }t\right)}^2}{\mathit{F}}_3$$

(10m)

$${\mathit{F}}_0=e^{\mathit{A}\mathrm{\Delta }t}$$

(11a)

$${\mathit{F}}_1=\underset{0}{\overset{\mathrm{\Delta }t}{\int }}{e}^{\mathit{A}(\mathrm{\Delta }t-t)}\mathrm{dt}=({\mathit{F}}_0-\mathit{I}){\mathit{A}}^{-1}$$

(11b)

$${\mathit{F}}_2=\underset{0}{\overset{\mathrm{\Delta }t}{\int }}{e}^{\mathit{A}(\mathrm{\Delta }t-t)}\mathrm{tdt}=({\mathit{F}}_1-\mathrm{\Delta }t\mathit{I}){\mathit{A}}^{-1}$$

(11c)

$${\mathit{F}}_3=\underset{0}{\overset{\mathrm{\Delta }t}{\int }}{e}^{\mathit{A}(\mathrm{\Delta }t-t)}{t}^2\mathrm{dt}=(2{\mathit{F}}_2-{\left(\mathrm{\Delta }t\right)}^2\mathit{I}){\mathit{A}}^{-1}$$

(11d)

$${\mathit{F}}_4=\underset{0}{\overset{\mathrm{\Delta }t}{\int }}{e}^{\mathit{A}(\mathrm{\Delta }t-t)}{t}^3\mathrm{dt}=(3{\mathit{F}}_3-{\left(\mathrm{\Delta }t\right)}^3\mathit{I}){\mathit{A}}^{-1}$$

(11e)

$${\mathit{F}}_5=\underset{0}{\overset{\mathrm{\Delta }t}{\int }}{e}^{\mathit{A}(\mathrm{\Delta }t-t)}{t}^4\mathrm{dt}=(4{\mathit{F}}_4-{\left(\mathrm{\Delta }t\right)}^4\mathit{I}){\mathit{A}}^{-1}$$

(11f)

$${\mathit{F}}_6=\underset{0}{\overset{\mathrm{\Delta }t}{\int }}{e}^{\mathit{A}(\mathrm{\Delta }t-t)}{t}^5\mathrm{dt}=(5{\mathit{F}}_5-{\left(\mathrm{\Delta }t\right)}^5\mathit{I}){\mathit{A}}^{-1}$$

(11g)

By tracking the parameter combinations that ensure existence of the spectral radii of the monodromy matrix on the circumference of the unit circle, the stability limit that demarcates the stable and unstable spaces is determined. The process of tracking these critical parameter combinations on the plane of axial and radial depths is given as detailed flowchart in Ozoegwu et al. ²⁵

Analysis of time of contour-parallel pocketing with limiting depths pairs

The combination of prescription parameters such as feed speed, axial depth of cut, radial depth of cut and geometric restrictions of the pocket, the tool and the toolpath influences pocketing time by influencing the distribution of MRR of active tool passes towards creation of the pocket. ²⁴ Rectangular pockets of dimensions length $l_l$ , width $l_w$ and depth $l_d$ are considered to be produced with contour-parallel toolpath. The contour-parallel toolpath is a combination of fully immersed initiating passes and subsequent passes of either up-milling mode or down-milling mode. The toolpath is similar to the one-way toolpath considered in Tekeli and Budak ²² and Ozoegwu and Ezugwu ²⁴ being that a single mode is maintained after the initiating passes. But the one-way toolpath causes delay because of return idle motions required at the completion of each active pass to start the next active pass during pocketing operation. This is because of the requirement to maintain a constant milling mode (either up- or down-milling) for all the passes in which $\rho <1$ . Minimal pocketing time can be guaranteed by contour-parallel toolpath (that eliminates these idle motions by maintaining continuous contact between tool and workpiece) when all the passes are executed most productively with limiting pairs of axial and radial depths. These limiting pairs of axial and radial depths are computed at the most productive spindle speed of practical range of the machine. VC of passes will be implemented in execution of contour-parallel pocketing because it was identified in Ozoegwu and Ezugwu ²⁴ to be much more time saving than HC of passes. The superiority stems from the fact that VC allows preponderance of bivariate optimization in terms of axial and radial depths while HC allows univariate optimization in most passes in terms of either of the depths. The formulation for pocketing time of contour-parallel toolpath depends on whether $l_w<l_l$ , $l_w=l_l$ or $l_w>l_l$ . A contour-parallel VC is executed in vertical series of cyclic passes, while the contour-parallel VCs are concentric from the pocket boundary to the pocket centre or concentric from the pocket centre to the pocket boundary.

Execution from the pocket boundary to centre

Suppose the contour-parallel VCs are concentric from the pocket boundary to the pocket centre, then analysis of machining time becomes as follows. The VCs are numbered 1, 2, 3, … from the pocket boundary to the last VC at the centre of the pocket. The length of each cycle of pass on nth VC is

$$S_n^{(b\to c)}=2(l_l+l_w)-4D[1+(2n-2){\rho }_{\lim }]$$

(12)

where ${\rho }_{\lim }$ together with $w_{\lim }^{({\rho }_{\lim })}$ represents the limiting radial and axial machining depths, respectively, and the superscript $(b\to c)$ connotes execution from pocket boundary to centre. The number of passes needed to complete the first fully immersed VC is

$$\mathrm{NO}{P}^{(\rho =1)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{(\rho =1)}}\right)+1=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{(\rho =1)}}\right)$$

(13)

The number of cyclic passes needed to complete each of the bivariately optimized VCs lying between the first and last (central) VCs is given as

$${\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\lim }\right)}}\right)+1=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\lim }\right)}}\right)$$

(14)

The total travel distance for executing the first VC is

$$S_{1,\mathrm{total}}^{(b\to c)}=\mathrm{NO}{P}^{(\rho =1)}{S}_1^{(b\to c)}=\mathrm{NO}{P}^{(\rho =1)}[2(l_l+l_w)-4D]$$

(15)

where $S_n^{(b\to c)}$ is given by equation (12) when n = 1 such that $S_1^{(b\to c)}=2(l_l+l_w)-4D$ . The total travel distance for executing each of the bivariately optimized VCs is

$$S_{n,\mathrm{total}}^{(b\to c)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}{S}_n^{(b\to c)}$$

(16)

The overall distance travelled by the tooth executing all the bivariately optimized VCs lying between the first and last VCs is given by

$$S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(b\to c)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}\sum _{n=2}^{K-1}{S}_n^{(b\to c)}$$

(17)

where K is a value that depends on which of the conditions $l_w<l_l$ , $l_w=l_l$ or $l_w>l_l$ holds. Equation (17) can be seen to constitute an arithmetic series of form

$$S_{Bi.opt,total}^{\left(b\to c\right)}=NO{P}_{Bi.opt}^{\left({\rho }_{lim}\right)}\left\{\left[2\left(l_l+l_w\right)-4D\right]\left(K-2\right)-8D{\rho }_{lim}{\displaystyle \sum _{n=1}^{K-2}n}\right\}$$

(18)

which becomes

$$\begin{array}{c}\multicolumn{1}{c}{S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(b\to c)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}(K-2)\{2(l_l+l_w)}\\ \multicolumn{1}{c}{-4D-4D{\rho }_{\lim }(K-1)\}}\end{array}$$

(19)

When the condition $l_w<l_l$ holds, the situation of the first and last VCs becomes as depicted in Figure 1(a) where green colour indicates the tool together with the first VC and dark dotted colour indicates the tool together with the last VC. The remnant radial immersion for execution of the last (central) VC becomes

$${\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}=0.5\frac{l_w}{D}-1-\mathrm{Floor}(\left(\frac{0.5}{{\rho }_{\lim }}\right)\frac{l_w}{D}-\frac{1}{{\rho }_{\lim }}){\rho }_{\lim }$$

(20)

Figure 1.

Depicting the geometric situation of the first and last VCs of contour-parallel pocketing from boundary to centre when (a) $l_w<l_l$ , (b) $l_w=l_l$ and (c) $l_w>l_l$ . Green colour indicates the tool together with the first VC and dark dotted colour indicates the tool together with the last VC.

The inequality ( $l_w<l_l$ ) in the superscript in equation (20) connotes the prevailing condition that $l_w<l_l$ ; change in condition will also be reflected accordingly in superscripts throughout this work. The length of cycle of pass of the last VC is

$$S_K^{\left(b\to c,l_w<l_l\right)}=2\left(l_l+l_w\right)-4D\left[1+\left(2K-2\right){\rho }_{lim}-2\left({\rho }_{lim}-{\rho }_{remn\left(b\to c\right)}^{\left(l_w<l_l\right)}\right)\right]$$

(21)

where $K=2+\mathrm{Floor}((0.5/{\rho }_{\lim })(l_w/D)-(1/{\rho }_{\lim }))$ . The axial depth of cut corresponding to ${\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}$ is designated as $w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}\right)}$ . The number of cycles of passes needed to complete the last VC becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}\right)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}\right)}}\right)+1}\\ \multicolumn{1}{c}{=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}\right)}}\right)}\end{array}$$

(22)

The total length for executing the last VC is

$$S_{K,\mathrm{total}}^{(b\to c,l_w<l_l)}={\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}\right)}{S}_K^{(b\to c,l_w<l_l)}$$

(23)

The machining time becomes

$$t_m=\frac{1}{v}(S_{1,\mathrm{total}}^{(b\to c)}+S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(b\to c)}+S_{K,\mathrm{total}}^{(b\to c,l_w<l_l)})$$

(24)

When the condition $l_w=l_l$ (as shown in Figure 1(b)) holds, the remnant radial immersion for execution of the last VC becomes

$${\rho }_{\mathrm{remn}(b\to c)}^{(l_w=l_l)}=0.5\frac{l_w}{D}-1-\mathrm{floor}(\left(\frac{0.5}{{\rho }_{\lim }}\right)\frac{l_w}{D}-\frac{1}{{\rho }_{\lim }}){\rho }_{\lim }$$

(25)

The length of cycle of pass of the last VC is

$$S_K^{\left(b\to c,l_w=l_l\right)}=4l_l-4D\left[1+\left(2K-2\right){\rho }_{lim}-2\left({\rho }_{lim}-{\rho }_{remn}^{\left(l_w=l_l\right)}\right)\right]$$

(26)

where $K=2+\mathrm{Floor}((0.5/{\rho }_{\lim })(l_w/D)-(1/{\rho }_{\lim }))$ . The axial depth of cut corresponding to ${\rho }_{\mathrm{remn}(b\to c)}^{(l_w=l_l)}$ is designated as $w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w=l_l)}\right)}$ . The number of cycles of passes needed to complete the last VC is given as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w=l_l)}\right)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w=l_l)}\right)}}\right)+1}\\ \multicolumn{1}{c}{=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w=l_l)}\right)}}\right)}\end{array}$$

(27)

The total length for executing the last VC is

$$S_{K,\mathrm{total}}^{(b\to c,l_w=l_l)}={\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w=l_l)}\right)}{S}_K^{(b\to c,l_w=l_l)}$$

(28)

The machining time becomes

$$t_m=\frac{1}{v}(S_{1,\mathrm{total}}^{(b\to c)}+S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(b\to c)}+S_{K,\mathrm{total}}^{(b\to c,l_w=l_l)})$$

(29)

When the condition $l_w>l_l$ (as shown in Figure 1(c)) holds, the remnant radial immersion for execution of the last VC becomes

$${\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}=\frac{0.5l_l}{D}-1-\mathrm{Floor}\left(\frac{0.5l_l-D}{{\rho }_{\lim }D}\right){\rho }_{\lim }$$

(30)

The length of cycle of pass of the last VC is

$$S_K^{\left(b\to c,l_w>l_l\right)}=2\left(l_l+l_w\right)-4D\left[1+\left(2K-2\right){\rho }_{lim}-2\left({\rho }_{lim}-{\rho }_{remn\left(b\to c\right)}^{\left(l_w>l_l\right)}\right)\right]$$

(31)

where $K=2+\mathrm{Floor}(0.5l_l-D/{\rho }_{\lim }D)$ . The axial depth of cut corresponding to ${\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}$ is designated as $w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}$ . The number of cycles of passes needed to complete the last VC is given as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}}\right)+1}\\ \multicolumn{1}{c}{=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}}\right)}\end{array}$$

(32)

The total length for executing the last VC is

$$S_{K,\mathrm{total}}^{(b\to c,l_w>l_l)}={\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}{S}_K^{(b\to c,l_w>l_l)}$$

(33)

The machining time becomes

$$t_m=\frac{1}{v}(S_{1,\mathrm{total}}^{(b\to c)}+S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(b\to c)}+S_{K,\mathrm{total}}^{(b\to c,l_w>l_l)})$$

(34)

Execution from the pocket centre to boundary

Suppose the contour-parallel VCs are concentric from the pocket centre to the boundary, then analysis of machining time becomes as follows

Suppose $l_w<l_l$ , then the first central fully immersed pass of tool centre will be of length $S_1^{(c\to b,l_w<l_l)}=l_l-l_w-D$ as shown in Figure 2(a). The number of passes needed to complete the first (central) VC is as given in equation (13). The total length for executing the first VC is

$$\begin{array}{c}\multicolumn{1}{c}{S_{1,\mathrm{total}}^{(c\to b,l_w<l_l)}=\mathrm{NO}{P}^{(\rho =1)}{S}_1^{(c\to b,l_w<l_l)}}\\ \multicolumn{1}{c}{=\mathrm{NO}{P}^{(\rho =1)}(l_l-l_w-D)}\end{array}$$

(35)

Figure 2.

Depicting the geometric situation of the first and last VCs of contour-parallel pocketing from centre to boundary when (a) $l_w<l_l$ , (b) $l_w=l_l$ where red is used to indicate position of the tool in the second VC and (c) $l_w>l_l$ where green colour indicates the tool together with the first VC and dark dotted colour indicates the tool together with the last VC.

The number of passes needed to complete each of the bivariately optimized VCs lying between the first (central) and last (boundary) VCs is still given by equation (14). The length of each cycle of pass on nth VC is

$$S_n^{(c\to b,l_w<l_l)}=2(l_l-l_w)-2D[1-4(n-1){\rho }_{\lim }]$$

(36)

where $n=2,3,4,\dots$ . The total length for executing each of the bivariately optimized VCs is

$$S_{n,\mathrm{total}}^{(c\to b,l_w<l_l)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}{S}_n^{(c\to b,l_w<l_l)}$$

(37)

The overall distance travelled by the tooth executing all the bivariately optimized VCs lying between the first and last VCs is given by

$$S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(c\to b,l_w<l_l)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}\sum _{n=2}^{K-1}{S}_n^{(c\to b,l_w<l_l)}$$

(38)

where $K=2+\mathrm{Floor}(0.5(l_w-D)/{\rho }_{\lim }D)=2+\mathrm{Floor}((0.5/{\rho }_{\lim })(l_w/D)-(0.5/{\rho }_{\lim }))$ . This constitutes an arithmetic series that is summed to become

$$S_{Bi.opt,total}^{\left(c\to b,l_w<l_l\right)}=NO{P}_{Bi.opt}^{\left({\rho }_{lim}\right)}\left(K-2\right)\left\{\left[2\left(l_l-l_w\right)-2D\right]+4D{\rho }_{lim}\left(K-1\right)\right\}$$

(39)

The remnant radial immersion for execution of the last (boundary) VC becomes

$${\rho }_{\mathrm{remn},(c\to b)}^{(l_w<l_l)}=0.5\frac{l_w}{D}-0.5-\mathrm{Floor}(\left(\frac{0.5}{{\rho }_{\lim }}\right)\frac{l_w}{D}-\frac{0.5}{{\rho }_{\lim }}){\rho }_{\lim }$$

(40)

The length of a cycle of pass of the last VC is

$$S_K^{\left(c\to b,l_w<l_l\right)}=2\left(l_l-l_w\right)-2D\left[1-4\left(K-1\right){\rho }_{lim}+4\left({\rho }_{lim}-{\rho }_{remn\left(c\to b\right)}^{\left(l_w<l_l\right)}\right)\right]$$

(41)

The axial depth of cut corresponding to ${\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}$ is designated as $w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}\right)}$ . The number of cycles of passes needed to complete the last VC is given as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{NOP}}_{\mathrm{Boundary}}^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}\right)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}\right)}}\right)+1}\\ \multicolumn{1}{c}{=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}\right)}}\right)}\end{array}$$

(42)

The total length for executing the last VC is

$$S_{K,\mathrm{total}}^{(c\to b,l_w<l_l)}={\mathrm{NOP}}_{\mathrm{Boundary}}^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}\right)}{S}_K^{(c\to b,l_w<l_l)}$$

(43)

The machining time becomes

$$t_m=\frac{1}{v}(S_{1,\mathrm{total}}^{(c\to b,l_w<l_l)}+S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(c\to b,l_w<l_l)}+S_{K,\mathrm{total}}^{(c\to b,l_w<l_l)})$$

(44)

Suppose $l_w=l_l$ , drilling of depth $l_d$ is first carried out at the centre of the pocket as shown in Figure 2(b), thus a cycle of pass of the first VC (drilling operation) is of length $S_1^{(c\to b,l_w=l_l)}=0$ . A cycle of pass on the second VC (denoted in red) will be of length $S_2^{(c\to b,l_w=l_l)}=8D{\rho }_2$ where it is seen from Figure 2(b) that ${\rho }_2\le 1/\sqrt{2}$ . Suppose axial depth $w_{\lim }^{({\rho }_2)}$ is read on stability diagram at ${\rho }_2$ , then the number of passes needed to complete the second VC is given by

$${\mathrm{NOP}}_2^{\left({\rho }_2\right)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_2\right)}}\right)+1=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_2\right)}}\right)$$

(45)

The total length for executing the second VC is

$$S_{2,\mathrm{total}}^{(c\to b,l_w=l_l)}={\mathrm{NOP}}_2^{\left({\rho }_2\right)}{S}_2^{(c\to b,l_w=l_l)}=8D{\rho }_2{\mathrm{NOP}}_2^{\left({\rho }_2\right)}$$

(46)

Each of the VCs lying between the second and last VCs is bivariately optimized at $w_{\lim }^{({\rho }_{\lim })}$ and ${\rho }_{\lim }$ . The number of cyclic passes needed to complete each of the VCs is given in equation (14). The length of each cycle of pass on nth VC is

$$S_n^{(c\to b,l_w=l_l)}=8D[(n-2){\rho }_{\lim }+{\rho }_2]$$

(47)

where $n=2,3,4,\dots$ The total length for executing each of the bivariately optimized VCs is

$$S_{n,\mathrm{total}}^{(c\to b,l_w=l_l)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}{S}_n^{(c\to b,l_w=l_l)}$$

(48)

The overall distance travelled by the tooth while executing all the bivariately optimized VCs lying between the second and last VCs is given by

$$S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(c\to b,l_w=l_l)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}\sum _{n=3}^{K-1}{S}_n^{(c\to b,l_w=l_l)}$$

(49)

where $K=3+\mathrm{Floor}((0.5(l_w-D)-D{\rho }_2)/{\rho }_{\lim }D)=3+\mathrm{Floor}((0.5/{\rho }_{\lim })(l_w/D)-(0.5/{\rho }_{\lim })-({\rho }_2/{\rho }_{\lim }))$ . Equation (49) is seen to constitute an arithmetic series with sum

$$S_{Bi.opt,total}^{\left(c\to b{l}_w=l_l\right)}=NO{P}_{Bi.opt}^{\left({\rho }_{lim}\right)}\left(K-3\right)\left\{8D\left[{\rho }_2-2{\rho }_{lim}\right]+4\left(K+2\right)D{\rho }_{lim}\right\}$$

(50)

The remnant radial immersion for execution of the last (boundary) VC becomes

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_{\mathrm{remn},(c\to b)}^{(l_w=l_l)}=0.5\frac{l_w}{D}-0.5-{\rho }_2-\mathrm{Floor}}\\ \multicolumn{1}{c}{(\left(\frac{0.5}{{\rho }_{\lim }}\right)\frac{l_w}{D}-\frac{0.5}{{\rho }_{\lim }}-\frac{{\rho }_2}{{\rho }_{\lim }}){\rho }_{\lim }}\end{array}$$

(51)

The length of a cycle of pass of the last VC is

$$S_K^{(c\to b,l_w=l_l)}=8D[(K-3){\rho }_{\lim }+({\rho }_2+{\rho }_{\mathrm{remn},(c\to b)}^{(l_w=l_l)})]$$

(52)

The axial depth of cut corresponding to ${\rho }_{\mathrm{remn},(c\to b)}^{(l_w=l_l)}$ is designated as $w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w=l_l)}\right)}$ . The number of cycles of passes needed to complete the last VC is given as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{NOP}}_{\mathrm{boundary}}^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w=l_l)}\right)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w=l_l)}\right)}}\right)+1}\\ \multicolumn{1}{c}{=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w=l_l)}\right)}}\right)}\end{array}$$

(53)

The total length for executing the last VC is

$$S_{K,\mathrm{total}}^{(c\to b,l_w=l_l)}={\mathrm{NOP}}_{\mathrm{boundary}}^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w=l_l)}\right)}{S}_K^{(c\to b,l_w=l_l)}$$

(54)

The machining time becomes

$$t_m=\frac{1}{v}(S_{2,\mathrm{total}}^{(c\to b,l_w=l_l)}+S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(c\to b{l}_w=l_l)}+S_{K,\mathrm{total}}^{(c\to b,l_w=l_l)})$$

(55)

Suppose $l_w>l_l$ as seen in Figure 2(c), then the first central fully immersed pass of tool centre will be of length $S_1^{(c\to b,l_w>l_l)}=l_w-l_l-D$ along the width direction. The number of passes needed to complete the first (central) VC is as given in equation (13). The total length for executing the first VC is

$$\begin{array}{c}\multicolumn{1}{c}{S_{1,\mathrm{total}}^{(c\to b,l_w>l_l)}=\mathrm{NO}{P}^{(\rho =1)}{S}_1^{(c\to b,l_w>l_l)}}\\ \multicolumn{1}{c}{=\mathrm{NO}{P}^{(\rho =1)}(l_w-l_l-D)}\end{array}$$

(56)

The number of cyclic passes needed to complete each of the bivariately optimized VCs lying between the first (central) and last (boundary) VCs is still given by equation (14). The length of each cycle of pass on nth VC is

$$S_n^{(c\to b,l_w>l_l)}=2(l_w-l_l)-2D[1-4(n-1){\rho }_{\lim }]$$

(57)

where $n=2,3,4,\dots$ . The total length for executing each of the bivariately optimized VCs is

$$S_{n,\mathrm{total}}^{(c\to b)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}{S}_n^{(c\to b,l_w>l_l)}$$

(58)

The overall distance travelled by the tooth executing all the bivariately optimized VCs lying between the first and last VCs is given by

$$S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(c\to b,l_w>l_l)}={\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}\sum _{n=2}^{K-1}{S}_n^{(c\to b,l_w>l_l)}$$

(59)

where $K=2+\mathrm{Floor}(0.5(l_l-D)/{\rho }_{\lim }D)=2+\mathrm{Floor}((0.5/{\rho }_{\lim })(l_l/D)-(0.5/{\rho }_{\lim }))$ . The summation of the arithmetic series of equation (59) is implemented to give

$$S_{Bi.opt,total}^{\left(c\to b,l_w>l_l\right)}=NO{P}_{Bi.opt}^{\left({\rho }_{lim}\right)}\left(K-2\right)\left\{\left[2\left(l_w-l_l\right)-2D\right]+4D{\rho }_{lim}\left(K-1\right)\right\}$$

(60)

The remnant radial immersion for execution of the last (boundary) VC becomes

$${\rho }_{\mathrm{remn},(c\to b)}^{(l_w>l_l)}=0.5\frac{l_l}{D}-0.5-\mathrm{Floor}(\left(\frac{0.5}{{\rho }_{\lim }}\right)\frac{l_l}{D}-\frac{0.5}{{\rho }_{\lim }}){\rho }_{\lim }$$

(61)

The length of a cycle of pass of the last VC is

$$S_K^{\left(c\to b,l_w>l_l\right)}=2\left(l_w-l_l\right)-2D\left[1-4\left(K-1\right){\rho }_{lim}+4\left({\rho }_{lim}-{\rho }_{remn\left(c\to b\right)}^{\left(l_w>l_l\right)}\right)\right]$$

(62)

The axial depth of cut corresponding to ${\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}$ is designated as $w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}\right)}$ . The number of cycles of passes needed to complete the last VC is given as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{NOP}}_{\mathrm{boundary}}^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}\right)}=\mathrm{Floor}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}\right)}}\right)+1}\\ \multicolumn{1}{c}{=\mathrm{Ceil}\left(\frac{l_d}{w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}\right)}}\right)}\end{array}$$

(63)

The total length for executing the last VC is

$$S_{K,\mathrm{total}}^{(c\to b,l_w>l_l)}={\mathrm{NOP}}_{\mathrm{boundary}}^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}\right)}{S}_K^{(c\to b,l_w>l_l)}$$

(64)

The machining time becomes

$$t_m=\frac{1}{v}(S_{1,\mathrm{total}}^{(c\to b,l_w>l_l)}+S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(c\to b,l_w>l_l)}+S_{K,\mathrm{total}}^{(c\to b,l_w>l_l)})$$

(65)

Numerical computations and discussions

The expressions for machining time presented in the section ‘Analysis of time of contour-parallel pocketing with limiting depths pairs’ are expected to be useful in systematic determination of choice of progression (either boundary to centre $(b\to c)$ execution or centre to boundary $(c\to b)$ execution) of pocketing and limiting axial and radial depth combinations that guarantee minimum machining time. Stability computations using the monodromy matrix presented in the section ‘Model and stability analysis of milling process’ are presented in what follows for the experimentally characterized milling system of Weck et al. cited in Tekeli and Budak. ²² The parameters are given as follows: ${\omega }_{\mathrm{nx}}=2\pi \times 600\mathrm{rad}/\mathrm{s}$ , ${\omega }_{\mathrm{ny}}=2\pi \times 660\mathrm{rad}/\mathrm{s}$ , $k_x=5.6\times {10}^6\mathrm{N}/\mathrm{m}$ , $k_y=5.6\times {10}^6\mathrm{N}/\mathrm{m}$ , ${\zeta }_x=0.035$ , ${\zeta }_y=0.035$ , $m_x=k_x/{\omega }_{\mathrm{nx}}^2\mathrm{kg}$ , $m_y=k_y/{\omega }_{\mathrm{ny}}^2\mathrm{kg}$ , $N=3$ , $C_t=6\times {10}^8\mathrm{N}/{\mathrm{m}}^2$ and $C_n=0.07\times {10}^6\mathrm{N}/{\mathrm{m}}^2$ . The feed speed $v=0.0025\mathrm{m}/\mathrm{s}$ utilized for analysis in the work of Ozoegwu and Ezugwu ²⁴ is adopted here for purpose of direct comparison of machining times. In order to establish a productive spindle speed within the technological limit of the system, stability diagrams of the system given as $w_{\lim }$ versus $\Omega$ are computed for up-milling at radial immersion $\rho =0.67$ . These are presented in Figure 3. Figure 3(a) and (b) is generated using $r=40$ on 200 × 100 gridded plane after 45 min of computation for x-feed up-milling and y-feed up-milling, respectively. The productive spindle speeds within the technological limit of the system can be seen from Figure 3 justifying the choice 12,600 r/min in Tekeli and Budak. ²² The stability diagram giving ${\rho }_{\lim }$ as a function of $w_{\lim }$ at the chosen productive spindle speed of 12,600 r/min is then generated for both x-feed up-milling and y-feed up-milling as given in Figure 3(c) and (d), respectively. The limiting MRR $\mathrm{MR}{R}_{\lim }=D{w}_{\lim }{\rho }_{\lim }v$ is plotted both against $w_{\lim }$ and ${\rho }_{\lim }$ , respectively, as presented in Figure 4(a) and (b) for x-feed up-milling. The arrows indicate points of maximum $\mathrm{MR}{R}_{\lim }$ . The $\mathrm{MR}{R}_{\lim }$ is also plotted both against $w_{\lim }$ and ${\rho }_{\lim }$ , respectively, as presented in Figure 4(c) and (d) for y-feed up-milling. About 68 min of computation using $r=80$ on 50 × 50 gridded plane was required to generate any component of Figures 3(c) and (d) and 4(a)–(d). It was reported in Ozoegwu and Ezugwu ²⁴ that the point of maximum $\mathrm{MR}{R}_{\lim }$ becomes more prone to providing the shortest machining time when the pocket gets larger due to suppression of geometric restriction. The maximum $\mathrm{MR}{R}_{\lim }$ for x-feed direction which occurs at the optimal coordinate ( $w_{\lim }=0.0048\mathrm{m},{\rho }_{\lim }=0.856$ ) can be seen in Figure 4(a) and (b) to be $1.0272\times {10}^{-7}{\mathrm{m}}^3/\mathrm{s}$ , while the value for y-feed direction which occurs at the optimal coordinate ( $w_{\lim }=0.0166\mathrm{m},{\rho }_{\lim }=0.32$ ) can be seen in Figure 4(c) and (d) to be $1.3280\times {10}^{-7}{\mathrm{m}}^3/\mathrm{s}$ . Figures 3(a) and (c) and 4(a) are supported by corresponding graphical results in Tekeli and Budak. ²² The slight differences are due to the fact that stability analyses in this work are carried out with the full-discretization method, while those in Tekeli and Budak ²² were carried out with the ZOA method. The differences between Figure 4(a) and the corresponding graphical results in Tekeli and Budak ²² are additionally due to the fact that actual MRR is considered here but Tekeli and Budak ²² considered non-dimensionalized MRR. Other trends are valid by the virtue of the supported graphs; the fact that $\mathrm{MR}{R}_{\lim }=D{w}_{\lim }{\rho }_{\lim }v$ means that a choice of limiting point on Figure 3(c) defined by $w_{\lim }=0.005\mathrm{m}$ and ${\rho }_{\lim }=0.8$ (for example) also implies that $\mathrm{MR}{R}_{\lim }$ corresponding to $w_{\lim }=0.005\mathrm{m}$ in Figure 4(a) is equal to $\mathrm{MR}{R}_{\lim }$ corresponding to ${\rho }_{\lim }=0.8$ in Figure 4(b). This is an indirect validation of Figure 4(b) as this is obeyed by all points in Figure 3(c).

Figure 3.

(a) The stability limits of 2-DOF up-milling of $\rho =0.67$ on the $\Omega -w$ plane for x-feed direction, (b) the stability limits of 2-DOF up-milling of $\rho =0.67$ on the $\Omega -w$ plane y-feed direction, (c) the stability limits of 2-DOF up-milling at $\Omega =12,600\mathrm{r}/min$ on the $w-\rho$ plane for x-feed direction and (d) the stability limits of 2-DOF up-milling at $\Omega =12,600\mathrm{r}/min$ on the $w-\rho$ plane for y-feed direction.

Figure 4.

(a) The stability limits of 2-DOF up-milling for x-feed direction at 12,600 r/min on $w-\mathrm{MRR}$ plane, (b) the stability limits of 2-DOF up-milling for x-feed direction at 12,600 r/min on $\rho -\mathrm{MRR}$ plane, (c) the stability limits of 2-DOF up-milling for y-feed direction at 12,600 r/min on $w-\mathrm{MRR}$ plane and (d) the stability limits of 2-DOF up-milling for y-feed direction at 12,600 r/min on $\rho -\mathrm{MRR}$ plane. Arrows indicate location of maximum $\mathrm{MR}{R}_{\lim }$ .

Numerical analysis of machining time requires numerical choice of ${\rho }_{\lim }$ and $w_{\lim }^{({\rho }_{\lim })}$ where $w_{\lim }^{({\rho }_{\lim })}$ is the smaller of the values read from x-feed stability diagram and y-feed stability diagram at the fixed numerical value of ${\rho }_{\lim }$ . The dimensions of the pocket are $l_w/D=15.33$ , $l_l=0.03\mathrm{m}$ and $l_d=0.03\mathrm{m}$ , where $D=0.01\mathrm{m}$ is the tool diameter and $v=0.0025\mathrm{m}/\mathrm{s}$ is the feed speed. A typical numerical choice becomes ${\rho }_{\lim }=0.2$ and $w_{\lim }^{({\rho }_{\lim })}=0.0105$ . It is seen that $l_w>l_l$ , thus equation (34) becomes applicable for contour-parallel pocketing time of boundary to centre $(b\to c)$ execution. It is seen from Figure 3(c) and (d) that $w_{\lim }^{(\rho =1)}=0.0035\mathrm{m}$ . The quantity $\mathrm{NO}{P}^{(\rho =1)}$ is computed as 9 using equation (13) followed by computation of $S_{1,\mathrm{total}}^{(b\to c)}$ as 2.9394 m using equation (15). The number ${\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{({\rho }_{\lim })}$ is computed as 3 using equation (14). The number of the last VC $K=2+\mathrm{Floor}(0.5l_l-D/{\rho }_{\lim }D)$ becomes 4. This allows computation of $S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(b\to c)}$ as 1.8156 using equation (19). Equation (30) is used to compute ${\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}$ as 0.1. The axial depth of cut $w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}$ is the smaller of the values read from x-feed stability diagram and y-feed stability diagram at the fixed numerical value ${\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}$  = 0.1. It is seen that $w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}=0.021\mathrm{m}$ . The distance $S_K^{(b\to c,l_w>l_l)}$ is computed as 0.2866 m using equation (31). The number ${\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}$ is computed as 2 using equation (32). The distance $S_{K,\mathrm{total}}^{(b\to c,l_w>l_l)}$ is computed as 0.5732 m using equation (33). The machining time is then computed from equation (34) as $t_m^{(b\to c)}=2131.3\mathrm{s}$ . The superscript $(b\to c)$ is added to indicate ‘boundary to centre execution’. If the pocket is created instead with one-way toolpaths, the machining times for VC are $t_m^{(1-\mathrm{way},\mathrm{dm})}=2368\mathrm{s}$ and $t_m^{(1-\mathrm{way},\mathrm{um})}=3504\mathrm{s}$ for down-milling and up-milling, respectively. ²⁴ The superscript $(1-\mathrm{way},\mathrm{dm})$ is used to indicate ‘one-way down-milling’, while the superscript $(1-\mathrm{way},\mathrm{um})$ is used to indicate ‘one-way up-milling’. Contour-parallel up-milling toolpaths have reduced the pocketing time of one-way down-milling toolpaths by 9.9958% while at same time minimizing the possibility of shock load characteristic of down-milling. Contour-parallel toolpaths has also reduced the pocketing time of one-way up-milling toolpaths by 39.1752%. The information that contour-parallel up-milling toolpaths with $t_m^{(b\to c)}=2131.3\mathrm{s}$ has reduced the pocketing time of one-way down-milling toolpaths with $t_m^{(1-\mathrm{way},\mathrm{dm})}=2368\mathrm{s}$ by 9.9958% is indicated as 3504/9.9958 in row 13 and column 1 of Table 1. Pocketing time of other limiting parameter combinations ( ${\rho }_{\lim },w_{\lim }^{({\rho }_{\lim })}$ ) are computed following similar procedure and results summarized in Table 1. It is seen that t_m falls with rise in ${\rho }_{\lim }$ until geometric restriction causes minimum value of 1749 s to be attained when ${\rho }_{\lim }\ge 0.6$ . It is seen from the last two rows of Table 1 that contour-parallel toolpath always reduces the pocketing time of both one-way up-milling and down-milling toolpaths except for the deviation at ${\rho }_{\lim }=0.4$ . This deviation (one-way down-milling performing better than contour-parallel) stems from the fact that the slope of the curve ${\rho }_{\lim }$ ( $w_{\lim }^{({\rho }_{\lim })}$ ) is very steep in the neighbourhood of ${\rho }_{\lim }=0.4$ for up-milling, while the slope in same neighbourhood is relatively mild for down-milling. The steeper the negative slope of the curve ${\rho }_{\lim }$ ( $w_{\lim }^{({\rho }_{\lim })}$ ), the more maximization of MRR is disfavoured.

Table 1.

Pocketing time optimization employing contour-parallel VC with $b\to c$ execution.

	$l_w/D=15.33$ , $D=1\mathrm{cm},l_l=3\mathrm{cm}$ , $l_d=3\mathrm{cm}$ , $v=0.0025\mathrm{m}/\mathrm{s}$ , $\Omega =12,600\mathrm{r}/\min$ , $w_{\lim }^{(\rho =1)}=0.0035\mathrm{m}$ , $\mathrm{NO}{P}^{(\rho =1)}=\mathrm{Ceil}(0.03/0.0035)=9$ , $S_{1,\mathrm{total}}^{(b\to c)}=2.9394$
${\rho }_{\lim }$	0.2	0.4	0.6	0.8	0.856	1
$w_{\lim }^{\left({\rho }_{\lim }\right)}$	$0.0105$	0.0065	0.006	0.005	0.0048	0.0035
${\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}$	3	5	5	6	7	9
K	4	3	2	2	2	2
$S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(b\to c)}$	1.8156	1.4730	0	0	0	0
${\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}$	0.1	0.1	0.5	0.5	0.5	0.5
$w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}$	$0.021$	0.021	0.0065	0.0065	0.0065	0.0065
$S_K^{(b\to c,l_w>l_l)}$	0.2866	0.2866	0.2866	0.2866	0.2866	0.2866
${\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w>l_l)}\right)}$	2	2	5	5	5	5
$S_{K,\mathrm{total}}^{(b\to c,l_w>l_l)}$	0.5732 m	0.5732	1.4330	1.4330	1.4330	1.4330
$t_m^{(b\to c)}$	2131.3	1994.2	1749	1749	1749	1749
$t_m^{\left(1-way,um\right)}/\%\mathrm{\Delta }t$	3504/39.1752	2944/32.2622	1984/11.8448	1792/2.3996	1952/10.3996	2160/19.0278
$t_m^{\left(1-way,dm\right)}/\%\mathrm{\Delta }t$	2368/9.9958	1776/−12.2860	1984/11.8448	1792/2.3996	1936/9.6591	2112/17.1875

The studied case, $l_w>l_l$ , can alternatively be executed from centre to boundary $(c\to b)$ making equation (65) applicable for pocketing time analysis. Pocketing times computed for $c\to b$ execution are summarized in Table 2. It is seen that the $c\to b$ execution saves more time than corresponding $b\to c$ execution for the studied system. This is caused by very shallow limiting axial depth of the fully immersed first VC of $b\to c$ execution. The percentage saving in pocketing time of $c\to b$ execution relative to $b\to c$ execution is given by $\%\mathrm{\Delta }t=100(t_m^{(b\to c)}-t_m^{(c\to b)})/t_m^{(b\to c)}$ and computed for the considered limiting pairs as 3.5565%, 6.0175%, 10.2173%, 15.4145%, 15.7690% and 13.5735%, respectively. It is also seen that given the steps of radial immersion considered, the $c\to b$ execution exhibited a clear minimum pocketing time occurring at ( ${\rho }_{\lim }=0.8,w_{\lim }^{({\rho }_{\lim })}=0.005\mathrm{m}$ ). It is seen from the last two rows of Table 2 that just as the case of $b\to c$ execution, contour-parallel toolpath always reduces the pocketing time of both one-way up-milling and down-milling toolpaths except for the deviation at ${\rho }_{\lim }=0.4$ . Same reason for this deviation as given earlier still holds.

Table 2.

Pocketing time optimization employing contour-parallel VC with $c\to b$ execution.

	$l_w/D=15.33$ , $D=1\mathrm{cm},l_l=3\mathrm{cm}$ , $l_d=3\mathrm{cm}$ , $v=0.0025\mathrm{m}/\mathrm{s}$ , $\Omega =12,600\mathrm{r}/min$ , $w_{\lim }^{(\rho =1)}=0.0035\mathrm{m}$ , $\mathrm{NO}{P}^{(\rho =1)}=\mathrm{Ceil}(0.03/0.0035)=9$ , $S_{1,\mathrm{total}}^{(c\to b)}=1.0197$
${\rho }_{\lim }$	0.2	0.4	0.6	0.8	0.856	1
$w_{\lim }^{\left({\rho }_{\lim }\right)}$	$0.0105$	0.0065	0.006	0.005	0.0048	0.0035
${\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}$	3	5	5	6	7	9
K	6	4	3	3	3	2
$S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(c\to b)}$	3.1992	2.7460	1.3730	1.7436	2.0656	0
${\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}$	0.2	0.2	0.4	0.2	0.1440	1
$w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}\right)}$	$0.0105$	$0.0105$	0.0065	$0.0105$	0.0155	0.0035
$S_K^{(c\to b,l_w>l_l)}$	0.3066	0.3066	0.3066	0.3066	0.3066	0.3066
${\mathrm{NOP}}_{\mathrm{boundary}}^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w>l_l)}\right)}$	3	3	5	3	2	9
$S_{K,\mathrm{total}}^{(c\to b,l_w>l_l)}$	0.9198	0.9198	1.5330	0.9198	0.6132	2.7594
$t_m^{(b\to c)}$	2055.5	1874.2	1570.3	1473.2	1479.4	1511.6
$t_m^{\left(1-way,um\right)}/\%\mathrm{\Delta }t$	3504/41.3385	2944/36.3383	1984/20.8518	1792/17.7902	1952/24.2111	2160/30.0185
$t_m^{\left(1-way,dm\right)}/\%\mathrm{\Delta }t$	2368/13.1968	1776/−5.5293	1984/20.8518	1792/17.7902	1936/23.5847	2112/28.4280

Suppose a lengthier pocket with $l_l=20\mathrm{cm}$ that guarantees the condition, $l_w<l_l$ , is now machined. Equation (24) becomes applicable for contour-parallel pocketing time of $b\to c$ execution. This case can alternatively be executed from $c\to b$ making equation (44) applicable for pocketing time analysis. The computational results and the pocketing times of $b\to c$ execution are summarized in Table 3, while the results of $c\to b$ execution are summarized in Table 4. It is seen that the $c\to b$ execution also saves more time than corresponding $b\to c$ execution of lengthier pocket. The percentage saving of pocketing time of $c\to b$ execution relative to $b\to c$ execution is computed for the considered limiting pairs as 11.2833%, 10.0197%, 13.4727%, 12.7450%, 4.2598% and 2.2403%, respectively. It is seen for the studied system that minimum pocketing time always occurs at ( ${\rho }_{\lim }=0.8,w_{\lim }^{({\rho }_{\lim })}=0.005\mathrm{m}$ ) which is close to the point of x-feed maximum $\mathrm{MR}{R}_{\lim }$ . Minimum time has not occurred at the point ( $w_{\lim }=0.0048\mathrm{m},{\rho }_{\lim }=0.856$ ) of maximum $\mathrm{MR}{R}_{\lim }$ for x-feed direction because of geometric restrictions. ²⁴ The results of the studied system seem to suggest that it is faster pocketing at high radial depth range of 0.6D–D. This is a favourable result because it is more likely to meet the load specifications of the tool system in this range of productive limiting parameter combinations than the less-productive limiting parameter combinations of low radial depth range. The reason derives from the fact that cutting force amplitude is proportional to axial depth, while tool forcing duration is affected by radial depth. It is seen from the last two rows of Tables 3 and 4 that contour-parallel toolpath always reduces the pocketing time of both one-way up-milling and down-milling toolpaths when the pocket gets large. This is because larger pockets mean longer tool–workpiece contact, thus more possibility of suppressing any deviation of geometric/dynamics restrictions. That is also why reduction in pocketing time of both one-way up-milling and down-milling toolpaths due to contour-parallel toolpath is stronger for larger pocket.

Table 3.

Pocketing time optimization of large pocket employing contour-parallel VC with $b\to c$ execution.

	$l_w/D=15.33$ , $D=1\mathrm{cm},l_l=20\mathrm{cm}$ , $l_d=3\mathrm{cm}$ , $v=0.0025\mathrm{m}/\mathrm{s}$ , $\Omega =12,600\mathrm{r}/\min$ , $w_{\lim }^{(\rho =1)}=0.0035\mathrm{m}$ , $\mathrm{NO}{P}^{(\rho =1)}=\mathrm{Ceil}(0.03/0.0035)=9$ , $S_{1,\mathrm{total}}^{(b\to c)}=5.9994\mathrm{m}$
${\rho }_{\lim }$	0.2	0.4	0.6	0.8	0.856	1
$w_{\lim }^{\left({\rho }_{\lim }\right)}$	$0.0105$	0.0065	0.006	0.005	0.0048	0.0035
${\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}$	3	5	5	6	7	9
K	35	18	13	10	9	8
$S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(b\to c)}$	39.0654	31.5680	20.8230	18.1728	19.2413	20.8764
${\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}$	0.0650	0.2650	0.0650	0.2650	0.6730	0.6650
$w_{\lim }^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}\right)}$	0.027	0.008	0.027	0.008	0.0053	0.0055
$S_K^{(b\to c,l_w<l_l)}$	0.1334	0.1334	0.1334	0.1334	0.1334	0.1334
${\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(b\to c)}^{(l_w<l_l)}\right)}$	2	4	2	4	6	6
$S_{K,\mathrm{total}}^{(b\to c,l_w<l_l)}$	0.2668	0.5336	0.2668	0.5336	0.8004	0.8004
$t_m$	18,133	15,240	10,836	9882.3	10,416	11,070
$t_m^{\left(1-way,um\right)}/\%\mathrm{\Delta }t$	33,288/45.5269	27,968/45.5092	18,848/42.5085	17,024/41.9508	18,544/43.8309	20,520/46.0526
$t_m^{\left(1-way,dm\right)}/\%\mathrm{\Delta }t$	22,496/19.3946	16,872/9.6728	18,848/42.5085	17,024/41.9508	18,392/43.3667	20,064/44.8266

Table 4.

Pocketing time optimization of large pocket employing contour-parallel VC with $c\to b$ execution.

	$l_w/D=15.33$ , $D=1\mathrm{cm},l_l=20\mathrm{cm}$ , $l_d=3\mathrm{cm}$ , $v=0.0025\mathrm{m}/\mathrm{s}$ , $\Omega =12,600\mathrm{r}/min$ , $w_{\lim }^{(\rho =1)}=0.0035\mathrm{m}$ , $\mathrm{NO}{P}^{(\rho =1)}=\mathrm{Ceil}(0.03/0.0035)=9$ , $S_{1,\mathrm{total}}^{(c\to b)}=0.3303$
${\rho }_{\lim }$	0.2	0.4	0.6	0.8	0.856	1
$w_{\lim }^{\left({\rho }_{\lim }\right)}$	$0.0105$	0.0065	0.006	0.005	0.0048	0.0035
${\mathrm{NOP}}_{\mathrm{Bi}.\mathrm{opt}}^{\left({\rho }_{\lim }\right)}$	3	5	5	6	7	9
K	37	19	13	10	10	9
$S_{\mathrm{Bi}.\mathrm{opt},\mathrm{total}}^{(c\to b)}$	37.9470	30.7190	19.8770	17.3472	21.3674	24.7842
${\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}$	0.1650	0.3650	0.5650	0.7650	0.3170	0.1650
$w_{\lim }^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}\right)}$	0.014	0.0067	0.0064	0.0051	0.0071	0.014
$S_K^{(c\to b,l_w<l_l)}$	0.6466	0.6466	0.6466	0.6466	0.6466	0.6466
${\mathrm{NOP}}_{\mathrm{central}}^{\left({\rho }_{\mathrm{remn}(c\to b)}^{(l_w<l_l)}\right)}$	3	5	5	6	5	3
$S_{K,\mathrm{total}}^{(c\to b,l_w<l_l)}$	1.9398	3.2330	3.2330	3.8796	3.2330	1.9398
$t_m$	16,087	13,713	9376.1	8622.8	9972.3	10,822
$t_m^{\left(1-way,um\right)}/\%\mathrm{\Delta }t$	33,288/51.6733	27,968/50.9690	18,848/50.2541	17,024/49.3492	18,544/46.2236	20,520/47.2612
$t_m^{\left(1-way,dm\right)}/\%\mathrm{\Delta }t$	22,496/28.4895	16,872/18.7233	18,848/50.2541	17,024/49.3492	18,392/45.7791	20,064/46.0626

Conclusion

Optimization of contour-parallel toolpath with limiting axial and radial depth pairs is investigated in this work as a means of minimizing pocketing time within milling machine load specifications. This optimization approach is identified to be of two types: boundary to centre $(b\to c)$ execution and centre to boundary $(c\to b)$ execution. The presented expressions for machining time are useful in systematic determination of choice of progression (either $b\to c$ execution or $c\to b$ execution) of pocketing operation and choice of limiting axial and radial depth combinations that guarantee minimum machining time. The studied numerical case gave that $c\to b$ execution is more time saving than $b\to c$ execution; for example, the percentage time savings of machining typical small pocket by $c\to b$ execution relative to $b\to c$ execution are 15.7690%, 15.4145% and 13.5735% at depth pairs ( ${\rho }_{\lim },w_{\lim }$ ) of (0.8, 0.005), (0.856, 0.0048) and (1, 0.0035), respectively. The results of the studied system suggest that it is faster pocketing at high radial depth range of 0.6D–D (D is the diameter of the tool) which fortunately also is the range of reduced limiting force and power demand. It is confirmed from the study of the illustrative numerical case that as far as the slope of the curve ${\rho }_{\lim }$ ( $w_{\lim }^{({\rho }_{\lim })}$ ) is not too steep in neighbourhood of chosen limiting depths, both up- and down-milling contour-parallel toolpaths reduce the pocketing time of any of one-way up- and down-milling toolpaths; this conclusion gets more strongly guaranteed the larger pockets get. The known fact that the coordinate of maximum $\mathrm{MR}{R}_{\lim }$ for both down- and up-milling one-way toolpaths will not necessarily provide the minimum time because of geometrical constraints also holds for contour-parallel toolpath.