Optimization of different parameters related to milling tools to maximize the allowable cutting depth for chatter-free machining

Abstract

Determination of optimal parameters of cutting tool is one of the most significant factors in any operation planning of metal elements, especially in micro-milling process. This article presents an optimization procedure, based on genetic algorithms, to optimize some parameters related to micro-milling tool including number of teeth, shank diameter, fluted section diameter, shank length, taper length, and length of fluted section. The aim of this optimization is maximizing the minimum value of cutting depth on the border of stability lobe diagrams, which is called allowable cutting depth, for chatter-free machining. Cutting tool is modeled as a three-dimensional spinning cantilever Timoshenko beam based on strain gradient elasticity theory. Structural nonlinearity, gyroscopic moment, rotary inertia, and velocity-dependent process damping are also considered in the cutting tool model. The values of natural frequency, damping ratio, and material length scale of the micro-milling tool are calculated using a system identification based on genetic algorithm to match the analytical response with recorded experimental vibration signal. Using beam model, the allowable cutting depth is increased in the optimization process for a specific range of spindle speed to avoid the chatter phenomenon. Analytical study of micro-milling process stability is carried out to determine the cost function of the genetic algorithm. A plot of the greatest fitness in each generation is sketched. In addition, stability lobe diagrams before and after optimization process are presented to show the efficiency of the optimized micro-milling tool. In the presented examples, the results of genetic algorithm may lead to design or find a micro-milling tool that its acceptable cutting depth increases up to 1.9313 times.

Introduction

The most frequently used technique in manufacturing is milling that is the machining process of materials. ¹ Micro-milling is a significant operation of mechanical micro-machining, with realistic application in mold and die, automotive, aerospace, military, biomedical, and micro-electronics packaging industries. Commercially available micro-milling tools have the tool diameter ranging $[\begin{array}{cc}25\hfill & 1000\hfill \end{array}]\mathrm{\mu }\mathrm{m}$ . ² In addition, regenerative chatter is a main topic of immense engineering significance due to its incidence in macro/micro-machining results in poor surface finish and limits the efficiency and productivity. In the past few years, many efforts have been made to forecast chatter vibration in machining process. In the majority of studies, two-dimensional (2D) and single/multi degrees of freedom (DOF) model with many simplifying suppositions have been applied.^3–12 A number of investigations have focused on the continuous structure for macro-machining operation.^13–16 For micro-machining operation, most of the researchers modeled the tool using classical continuum theory,^17–24 and some few others developed the tool model by applying non-classical continuum theory.^25,26

Optimization of different parameters related to milling tools and determination of optimization objective or optimization criterion are very important in the design phase of milling tools. Tolouei-Rad and Bidhendi ²⁷ determined optimal machining factors for milling processes by applying an optimization system. Choudhury and Rao ²⁸ developed the optimized values of velocity and feed rate for improving cutting operation. Sönmez et al. ²⁹ studied an optimization strategy to maximizing production rate of multi-pass milling processes. Shunmugam et al. ³⁰ used a genetic algorithm (GA) to reduce production cost of a multi-pass face-milling, where the number of passes, depth of cut in each pass, spindle speed, and feed rate were the decision variables. Baek et al. ³¹ optimized the feed rate in a face-milling process by applying a surface roughness model. Suresh et al. ³² predicted and optimized surface roughness by developing a mathematical model and GAs, respectively. Tandon et al. ³³ optimized a numerical control (NC) end milling by applying a new evolutionary computation procedure. Juan et al. ³⁴ modeled rough machining of high-speed milling processes using a polynomial network and minimized the production cost of that. Kim and Ramulu ³⁵ applied an optimization approach on the drilling operation to increase hole property and decrease drilling cost. Wang et al. ³⁶ optimized feed rate and spindle speed to reduce the time of production in plain milling. A number of researchers^37–39 have scrutinized the machining operation by applying the Taguchi method to optimize machining parameters such as cutting velocity, feed rate, and cutting depth taking surface roughness as an objective function. Kılıckap et al. ⁴⁰ developed the relationship between cutting parameters such as cutting speeds, feed rate, and depth of cut, and optimization techniques as response surface methodology and artificial neural network for cutting forces, surface roughness, and tool wear in the milling of Ti-6242S alloy.

Regenerative chatter that is a kind of unstable self-generative vibrations is an undesired phenomenon that can occur during micro/macro-milling operations. Incidence of chatter may lead to poor surface finish and limits the efficiency and productivity of machining processes. It comes from coupling between cutting force and vibration of tool/work-piece system. Several stability models have been considered in previous publications, where mostly the stability limit in terms of axial depth of cut is emphasized for chatter-free machining. ⁴¹ Here, the minimum value of cutting depth in stability lobe diagram (SLD) with avoidance of regenerative chatter phenomenon in a certain spindle speed span is called allowable cutting depth. Consequently, maximizing the allowable depth of cut to chatter-free machining is one of the principal factors in micro-milling process, which significantly affects the cost, time, and efficiency of production processes. The determination of optimal parameters of micro-milling tool containing number of teeth, shank diameter, diameter of fluted section, shank length, taper length, and length of fluted section is a very significant problem to maximize the allowable cutting depth. This optimization may lead to design an optimum geometry of a micro-milling tool, which has maximum allowable cutting depth without damage to tool. In practice, all commercially available micro-milling tools on the market have a specific geometry. Using an optimization method, operators can order a micro-tool design with optimal geometry for a specific machining, or at least select a micro-tool that its geometry is close to the optimal values.

One aim of the current work is the system identification based on time domain technique. The values of natural frequency, damping ratio, and material length scale are calculated from matching the analytical response with recorded vibration signal using an optimization algorithm. To the best of the authors’ knowledge, this method of size scale calculation has not yet been presented in any effort. Therefore, one of the novelties of this study is the calculation of material length scale using system identification. The main purpose of this study is the optimization of a micro-milling tool geometry. The allowable cutting depth is selected as a criterion function. Analytical study of a micro-milling process stability is necessary to determine the objective function of this optimization algorithm. For this purpose, a nonlinear three-dimensional (3D) spinning cantilever Timoshenko beam based on strain gradient elasticity theory is used for modeling the cutting tool by considering structural nonlinearity, gyroscopic moment, rotary inertia, and velocity-dependent process damping. Accordingly, another major novelty is the analytical study of micro-milling process stability to determine the objective function of the tool design optimization. In the literature, the GA is one of the pioneer heuristic algorithms that is the most experimented optimization algorithm due to its simplicity and global perspective. ⁴² Therefore, the GA is used as optimization algorithm in this study.

Methodology

In the current research, the mathematical model is derived by applying mode-summation method (MSM) and then method of multiple scales (MMS) on the formulation of a nonlinear 3D spinning cantilever Timoshenko beam based on strain gradient elasticity theory. The micro-milling operation mathematical model is derived to find the SLD, which is based on cutting depth versus spindle speed. The minimum value of the cutting depth on the border of corresponding SLD has been used as a criterion function. The features of tool including number of teeth N, shank diameter $d_s$ , diameter of fluted section $d_f$ , shank length $L_s$ , taper length $L_t$ , and length of fluted section $L_f$ have been used as decision variables. In addition, the optimization was fulfilled with the help of GAs.

Mathematical formulation

A 3D rotating fixed-free non-uniform Timoshenko beam, as presented in Figure 1, is employed to derive the equations of motion of the micro-milling tool.

Figure 1.

Schematic of a micro-milling tool contains shank, taper, and fluted parts.

The tool turns around its longitudinal axis at a constant spindle speed $\mathrm{\Omega }$ with lateral vibrations $w_y(x,t)$ , $w_z(x,t)$ and rotating displacements ${\varphi }_z(x,t)$ , ${\varphi }_y(x,t)$ with respect to the spindle frame. The position vectors ${\mathit{r}}_0$ and ${\mathit{r}}_1$ represent the position of a reference point before and after deformation relative to the spindle frame, respectively, and are given as follows ²⁶

$$\begin{array}{c}{\mathit{r}}_0=x\mathit{i}+{\eta }_0\mathit{j}+{\xi }_0\mathit{k}\hfill \\ {\mathit{r}}_1=(x-\eta S_{{\varphi }_z}+\xi S_{{\varphi }_y}{C}_{{\varphi }_z})\mathit{i}\hfill \\ +(w_y+\eta C_{{\varphi }_z}+\xi S_{{\varphi }_y}{S}_{{\varphi }_z})\mathit{j}+(w_z+\xi C_{{\varphi }_y})\mathit{k}\hfill \end{array}$$

(1)

Here, the characters $S_{\wp }$ and $C_{\wp }$ that recognized as $\sin (\wp )$ and $\cos (\wp )$ , respectively, are employed to make simpler the writing. Furthermore, ℘ is used instead of any functions. In addition, the symbols ${\eta }_0-{\xi }_0$ and $\eta -\xi$ are reference point offsets from the tool elastic axis before and after deformation, respectively.

For the non-axisymmetric twisted geometry of the tip section of the micro-milling tool, the moments of area relative to $\eta$ - and $\xi$ -axes may be not similar, that is $I_{\eta }(x)\ne I_{\xi }(x)$ , and the principal axes of the second moments of area may not match with the $\eta -\xi$ axes of the fluted section, that is $I_{\eta \xi }(x)\ne 0$ .

For the tool with kinetic energy $T^e$ , strain energy $U^e$ , and virtual work done by the non-conservative forces $W_{\mathrm{nc}}$ , the governing equations of motion can be achieved using extended Hamilton’s principle in the time interval $[\begin{array}{cc}{t}_1\hfill & t_2\hfill \end{array}]$ as follows

$$\underset{t_1}{\overset{t_2}{\int }}\delta (U^e-T^e-W_{\mathrm{nc}})\mathrm{dt}=0$$

(2)

The strain energy of the tool is as follows^26,43,44

$$\begin{array}{c}{U}^e=\frac{1}{2}\underset{0}{\overset{L}{\int }}\int \underset{A}{\int }({\sigma }_{\mathrm{ij}}{\epsilon }_{\mathrm{ij}}+p_i{\gamma }_i+{\tau }_{\mathrm{ijk}}^{\left(1\right)}{\eta }_{\mathrm{ijk}}^{\left(1\right)}+m_{\mathrm{ij}}^s{\chi }_{\mathrm{ij}}^s)d\eta d\xi \mathrm{dx}\hfill \\ {\gamma }_i={\epsilon }_{\mathrm{mm},i}:\mathrm{the}\mathrm{dilatation}\mathrm{gradient}\mathrm{vector}\hfill \\ {\chi }_{\mathrm{ij}}^s=\frac{1}{2}({\theta }_{i,j}+{\theta }_{j,i}):\mathrm{the}\mathrm{deviator}\mathrm{stretch}\mathrm{gradient}\mathrm{tensor}\hfill \\ {\eta }_{\mathrm{ijk}}^{\left(1\right)}=\frac{1}{3}({\epsilon }_{\mathrm{jk},i}+{\epsilon }_{\mathrm{ki},j}+{\epsilon }_{\mathrm{ij},k})\hfill \\ -\frac{1}{15}[{\delta }_{\mathrm{ij}}({\epsilon }_{\mathrm{mm},k}+2{\epsilon }_{\mathrm{mk},m})+{\delta }_{\mathrm{jk}}({\epsilon }_{\mathrm{mm},i}+2{\epsilon }_{\mathrm{mi},m})\hfill \\ +{\delta }_{\mathrm{ki}}({\epsilon }_{\mathrm{mm},j}+2{\epsilon }_{\mathrm{mj},m})]\hfill \\ :\mathrm{the}\mathrm{symmetric}\mathrm{part}\mathrm{of}\mathrm{the}\mathrm{rotation}\mathrm{gradient}\mathrm{tensor}\hfill \\ {\sigma }_{\mathrm{ij}}=2\mu {\epsilon }_{\mathrm{ij}}+\lambda {\epsilon }_{\mathrm{kk}}{\delta }_{\mathrm{ij}}:\mathrm{classical}\mathrm{stress}\mathrm{element}\hfill \\ p_i=2\mu l_0^2{\gamma }_i,{\tau }_{\mathrm{ijk}}^{\left(1\right)}=2\mu l_1^2{\eta }_{\mathrm{ijk}}^{\left(1\right)},\hfill & m_{\mathrm{ij}}^s=2\mu l_2^2{\chi }_{\mathrm{ij}}^s:\mathrm{the}\mathrm{higher}-\mathrm{order}\mathrm{stresses}\hfill \\ \mathbf{\theta }\approx 0.5\mathrm{curl}\mathit{u}=0.5(\nabla \times \mathit{u}):\mathrm{diminutive}\mathrm{rotation}\mathrm{vector}\hfill \\ \mathit{u}={\mathit{r}}_1-{\mathit{r}}_0:\mathrm{displacement}\mathrm{vector}\hfill \\ \mathbf{\lambda }=\mathbf{\upsilon }\mathbf{E}/(1+\mathbf{\upsilon })(1-2\mathbf{\upsilon }),\mathbf{\mu }=\mathbf{G}=\mathbf{E}/2(1+\mathbf{\upsilon })\hfill \end{array}$$

(3)

where E, G, $\upsilon$ , A, L, ${\epsilon }_{\mathrm{ij}}$ , $\{l_0,l_1,l_2\}$ , and $\{\lambda ,\mu \}$ represent Young’s modulus, shear modulus, Poisson’s ratio, cross-section area, overhang length, elements of strain tensor, material length scale parameter, and Lame’s constants of the tool, respectively. Experiments on micro- and nano-mechanical systems have explained that their nominal stress differs from the macro-mechanical system made from same material. This different behavior has regularly been seen as a dependence of the material properties with the size of the system, especially diameter. The parameters of material length scale have been used in the strain energy to show the mentioned fact.

The kinetic energy of the tool can be presented as follows ²⁶

$$\begin{array}{l}{T}^e=\frac{1}{2}\underset{0}{\overset{L}{\int }}\underset{A}{\iint }\rho V.Vd\eta d\xi dx\\ V\triangleq r_{1,t}+\mathrm{\Omega }I\times r_1:\mathrm{velocity}\mathrm{vector}\mathrm{of}\mathrm{the}\mathrm{reference}\mathrm{point}\mathrm{after}\mathrm{deformation}\\ \rho :\mathrm{density}\mathrm{of}\mathrm{the}\mathrm{tool}\end{array}$$

(4)

where ${\wp }_{,t}\stackrel{\mathrm{\Delta }}{=}\partial \wp /\partial t$ . The virtual work expression can be obtained as follows ²⁶

$$\begin{array}{c}\delta W_{\mathrm{nc}}=\hfill \\ \underset{0}{\overset{L}{\int }}(L_{w_y}\delta w_y+L_{w_z}\delta w_z-B{w}_{y,t}\delta w_y-B{w}_{z,t}\delta w_z-R{\varphi }_{y,t}\delta {\varphi }_y-R{\varphi }_{z,t}\delta {\varphi }_z)\mathrm{dx}\hfill \\ L_{w_y}=F_y{\delta }_D(x-L)\hfill \\ L_{w_z}=F_z{\delta }_D(x-L)\hfill \\ \left\{\begin{array}{c}{F}_y\hfill \\ \multicolumn{1}{c}{F_z}\end{array}\right\}=\frac{\mathrm{\Delta }\psi }{2\pi }[K_{t}a\left\{\begin{array}{c}{w}_y-w_{y\tau }\hfill \\ \multicolumn{1}{c}{w_z-w_{z\tau }}\end{array}\right\}\sum _{j=1}^N{\mathit{A}}_j+\frac{G_{r}a}{V_c}\left\{\begin{array}{c}{w}_{y,t}\hfill \\ \multicolumn{1}{c}{w_{z,t}}\end{array}\right\}\sum _{j=1}^N{\mathit{B}}_j]\hfill \\ {\mathit{A}}_j=\left[\begin{array}{cc}-k_r{C}_{{\alpha }_j}^2+C_{{\alpha }_j}{S}_{{\alpha }_j}\hfill & -k_r{C}_{{\alpha }_j}{S}_{{\alpha }_j}+S_{{\alpha }_j}^2\hfill \\ \multicolumn{1}{c}{-k_r{C}_{{\alpha }_j}{S}_{{\alpha }_j}-C_{{\alpha }_j}^2}& -k_r{S}_{{\alpha }_j}^2-C_{{\alpha }_j}{S}_{{\alpha }_j}\hfill \end{array}\right]\hfill \\ {\mathit{B}}_j=\left[\begin{array}{cc}{C}_{{\alpha }_j}^2-g_t{C}_{{\alpha }_j}{S}_{{\alpha }_j}\hfill & C_{{\alpha }_j}{S}_{{\alpha }_j}-g_t{S}_{{\alpha }_j}^2\hfill \\ \multicolumn{1}{c}{C_{{\alpha }_j}{S}_{{\alpha }_j}+g_t{C}_{{\alpha }_j}^2}& S_{{\alpha }_j}^2+g_t{C}_{{\alpha }_j}{S}_{{\alpha }_j}\hfill \end{array}\right]\hfill \\ {\wp }_{\tau }\stackrel{\mathrm{\Delta }}{=}\wp (x,t-\tau )\hfill \\ \tau =\frac{2\pi }{N\mathrm{\Omega }}:\mathrm{the}\mathrm{delay}\mathrm{time}\hfill \\ \mathrm{\Delta }\psi \stackrel{\mathrm{\Delta }}{=}{\psi }_{e{x}_j}-{\psi }_{s{t}_j}\hfill \\ V_c=d_f\mathrm{\Omega }/2:\mathrm{cutting}\mathrm{velocity}\hfill \\ {\alpha }_j=2\pi (j-1)/N:\mathrm{the}\mathrm{angle}\mathrm{of}\mathrm{each}\mathrm{tooth}\mathrm{with}\mathrm{respect}\mathrm{to}\hfill \\ y-\mathrm{axis}\mathrm{in}\mathrm{spindle}\mathrm{frame}\hfill \end{array}$$

(5)

where a, $\{G_r,G_t\}$ , $g_t$ , $\{{\mathfrak{C}}_B,{\mathfrak{C}}_R\}$ , and ${\delta }_D$ are assigned for depth of cut, process damping factors, process damping ratio, bending and rotary damping coefficients, and Dirac delta function, respectively. In addition, the parameters $\{{\psi }_{s{t}_j},{\psi }_{e{x}_j}\}$ , $\{K_r,K_t\}$ , and $k_r$ are assigned for entrance and exit angle of the cutter’s teeth, cutting force factors, and cutting force ratio, respectively. ⁴⁵

The third-order nonlinear delay partial deferential equations (DPDEs) can be derived by substituting $T^e$ , $U^e$ , and $W_{\mathrm{nc}}$ into extended Hamilton’s principle equation and using Taylor series $S_{\wp }=\wp -({\wp }^3/6)$ and $C_{\wp }=1-({\wp }^2/2)$ . The delay ordinary deferential equations (DODEs) can be obtained by applying MSM on the corresponding DPDEs as follows€

$$\begin{array}{c}\multicolumn{1}{c}{A_1{T}_{w_{y,\mathrm{tt}}}+A_2{T}_{w_{z,t}}+A_3{T}_{w_y}+A_4{T}_{{\varphi }_z}+A_5{T}_{w_y}^3+A_6{T}_{w_y}{T}_{w_z}^2+A_7{T}_{w_y}{T}_{{\varphi }_z}^2+A_8{T}_{w_y}{T}_{{\varphi }_y}^2}\\ +A_9{T}_{{\varphi }_z}^3+A_{10}{T}_{{\varphi }_z}{T}_{{\varphi }_y}^2+A_{11}{T}_{w_z}{T}_{{\varphi }_z}{T}_{{\varphi }_y}+A_{12}{T}_{w_{y,t}}+A_{13}{T}_{{\varphi }_y}{T}_{{\varphi }_z}^2\hfill \\ +A_{14}{T}_{w_y}{T}_{{\varphi }_z}{T}_{{\varphi }_y}=A_{15}\mathrm{\Delta }{T}_{w_y}+A_{16}\mathrm{\Delta }{T}_{w_z}+A_{17}{T}_{w_{y,t}}+A_{18}{T}_{w_{z,t}}\hfill \\ \multicolumn{1}{c}{B_1{T}_{{\varphi }_{z,\mathrm{tt}}}+B_2{T}_{{\varphi }_z}+B_3{T}_{w_y}+B_4{T}_{{\varphi }_z}^3+B_5{T}_{{\varphi }_z}{T}_{{\varphi }_y}^2+B_6{T}_{{\varphi }_z}{T}_{w_y}^2+B_7{T}_{{\varphi }_z}{T}_{w_z}^2+B_8{T}_{w_y}{T}_{{\varphi }_z}^2}\\ +B_9{T}_{w_y}{T}_{{\varphi }_y}^2+B_{10}{T}_{{\varphi }_{z,\mathrm{tt}}}{T}_{{\varphi }_y}^2+B_{11}{T}_{{\varphi }_{y,t}}{T}_{{\varphi }_y}^2+B_{12}{T}_{w_y}{T}_{w_z}{T}_{{\varphi }_y}\hfill \\ +B_{13}{T}_{w_z}{T}_{{\varphi }_z}{T}_{{\varphi }_y}+B_{14}{T}_{{\varphi }_{z,t}}{T}_{{\varphi }_{y,t}}{T}_{{\varphi }_y}+B_{15}{T}_{{\varphi }_{z,t}}+B_{16}{T}_{{\varphi }_y}{T}_{{\varphi }_{y,t}}^2\hfill \\ +B_{17}{T}_{{\varphi }_z}{T}_{{\varphi }_y}{T}_{{\varphi }_{y,t}}+B_{18}{T}_{{\varphi }_y}{T}_{{\varphi }_z}^2+B_{19}{T}_{{\varphi }_{y,\mathrm{tt}}}{T}_{{\varphi }_y}^2+B_{20}{T}_{{\varphi }_y}^3+B_{21}{T}_{w_z}{T}_{{\varphi }_y}^2\hfill \\ +B_{22}{T}_{{\varphi }_y}{T}_{w_z}^2+B_{23}{T}_{w_y}{T}_{{\varphi }_z}{T}_{{\varphi }_y}+B_{24}{T}_{{\varphi }_y}{T}_{w_y}^2+B_{25}{T}_{{\varphi }_{y,\mathrm{tt}}}+B_{26}{T}_{{\varphi }_y}=0\hfill \\ \multicolumn{1}{c}{{€}_1{T}_{w_{z,\mathrm{tt}}}+{€}_2{T}_{w_{y,t}}+{€}_3{T}_{w_z}+{€}_4{T}_{{\varphi }_y}+{€}_5{T}_{w_z}^3+{€}_6{T}_{w_z}{T}_{w_y}^2+{€}_7{T}_{w_z}{T}_{{\varphi }_y}^2+{€}_8{T}_{w_z}{T}_{{\varphi }_z}^2}\\ +{€}_9{T}_{{\varphi }_y}^3+{€}_{10}{T}_{{\varphi }_y}{T}_{{\varphi }_z}^2+{€}_{11}{T}_{w_y}{T}_{{\varphi }_y}{T}_{{\varphi }_z}+{€}_{12}{T}_{w_{z,t}}+{€}_{13}{T}_{{\varphi }_z}{T}_{{\varphi }_y}^2\hfill \\ +{€}_{14}{T}_{w_z}{T}_{{\varphi }_z}{T}_{{\varphi }_y}={€}_{15}\mathrm{\Delta }{T}_{w_y}+{€}_{16}\mathrm{\Delta }{T}_{w_z}+{€}_{17}{T}_{w_{y,t}}+{€}_{18}{T}_{w_{z,t}}\hfill \\ \multicolumn{1}{c}{{\pounds }_1{T}_{{\varphi }_{y,\mathrm{tt}}}+{\pounds }_2{T}_{{\varphi }_y}+{\pounds }_3{T}_{w_z}+{\pounds }_4{T}_{{\varphi }_y}^3+{\pounds }_5{T}_{{\varphi }_y}{T}_{{\varphi }_z}^2+{\pounds }_6{T}_{{\varphi }_y}{T}_{w_y}^2+{\pounds }_7{T}_{{\varphi }_y}{T}_{w_z}^2+{\pounds }_8{T}_{w_z}{T}_{{\varphi }_y}^2}\\ +{\pounds }_9{T}_{w_z}{T}_{{\varphi }_z}^2+{\pounds }_{10}{T}_{{\varphi }_y}{T}_{{\varphi }_{z,t}}^2+{\pounds }_{11}{T}_{{\varphi }_{z,t}}{T}_{{\varphi }_y}^2+{\pounds }_{12}{T}_{w_z}{T}_{w_y}{T}_{{\varphi }_z}+{\pounds }_{13}{T}_{{\varphi }_y}{T}_{{\varphi }_z}{T}_{w_y}\hfill \\ +{\pounds }_{14}{T}_{{\varphi }_{y,t}}+{\pounds }_{15}{T}_{{\varphi }_z}^3+{\pounds }_{16}{T}_{{\varphi }_y}{T}_{{\varphi }_z}{T}_{{\varphi }_{z,t}}+{\pounds }_{17}{T}_{{\varphi }_{z,\mathrm{tt}}}{T}_{{\varphi }_y}^2+{\pounds }_{18}{T}_{{\varphi }_z}{T}_{{\varphi }_y}^2\hfill \\ +{\pounds }_{19}{T}_{{\varphi }_y}{T}_{{\varphi }_z}{T}_{w_z}+{\pounds }_{20}{T}_{{\varphi }_z}{T}_{w_z}^2+{\pounds }_{21}{T}_{w_y}{T}_{{\varphi }_z}^2+{\pounds }_{22}{T}_{{\varphi }_z}{T}_{w_y}^2+{\pounds }_{23}{T}_{{\varphi }_{z,\mathrm{tt}}}+{\pounds }_{24}{T}_{{\varphi }_z}=0\hfill \end{array}(6)$$

(6)

in which

$$\begin{array}{c}\mathrm{\Delta }{T}_{w_y}=T_{w_y}-T_{w_{y\tau }}\hfill \\ \mathrm{\Delta }{T}_{w_z}=T_{w_z}-T_{w_{z\tau }}\hfill \end{array}$$

(7)

where $(T_{w_y},T_{w_z})$ and $(T_{{\varphi }_z},T_{{\varphi }_y})$ are generalized coordinates for lateral displacements and corresponding rotations, respectively. In addition, the parameters $[\begin{array}{ccc}A\hfill & B\hfill & \begin{array}{cc}€\hfill & \pounds \hfill \end{array}\hfill \end{array}]$ are the Galerkin coefficients. The rotary inertia and gyroscopic moment are considered in these ODEs by terms $A_2{T}_{w_{z,t}}$ , $A_3{T}_{w_y}$ , $B_2{T}_{{\varphi }_z}$ , ${€}_2{T}_{w_{y,t}}$ , ${€}_3{T}_{w_z}$ , and ${\pounds }_2{T}_{{\varphi }_y}$ . The terms $A_{15}\mathrm{\Delta }{T}_{w_y}$ , $A_{16}\mathrm{\Delta }{T}_{w_z}$ , ${€}_{15}\mathrm{\Delta }{T}_{w_y}$ , and ${€}_{16}\mathrm{\Delta }{T}_{w_z}$ are used for representing cutting forces. The process damping effects are presented in these ODEs by terms $A_{17}{T}_{w_{y,t}}$ , $A_{18}{T}_{w_{z,t}}$ , ${€}_{17}{T}_{w_{y,t}}$ , and ${€}_{18}{T}_{w_{z,t}}$ . The coefficients $A_{13-14}$ , $B_{16-26}$ , ${€}_{13-14}$ , and ${\pounds }_{15-24}$ are appeared due to existence of area second moment $I_{\eta \xi }(x)$ . By three-dimensional scanning of the fluted section and calculation of the area second moments along that in a computer-aided design (CAD) software and using curve fitting method, the formulation of all area second moments can be derived based on x variable, like Figure 2.

Figure 2.

Curve fitting process of area second moments.

For plotting SLD, the analytical solution of proposed model can be derived by applying MMS on the governing DODEs. ²⁵

Optimization procedure

Optimization procedure is one of the significant aims of manufacturing systems. The GAs are robust and adaptive schemes, which are effectively applied for solving optimization problems. The GA is a heuristic search technique used in artificial intelligence and computing. This method is based on the theory of natural selection and evolutionary biology. These are sagacious utilization of random search provided with historical data to lead the search into the area of improved efficiency in solution region. To ensure optimization results, the optimization process is executed 20 times and the best result is considered. In addition, by reducing or increasing the constraint range, it has reached the same answers again, indicating that the answers are optimal.

In order to maximize the allowable cutting depth to chatter-free micro-milling process, it is necessary to determine the objective and constraint functions as follows

$$\begin{array}{c}\mathrm{Objective}\mathrm{function}:\mathrm{Maximize}{a}_{\min }(N,d_s,d_f,L_s,L_t,L_f)\\ \mathrm{Constraint}\mathrm{functions}:\{\begin{array}{c}{N}_{\min }\le N\le N_{\max }\hfill \\ \multicolumn{1}{c}{{d_s}_{\min }\le d_s\le {d_s}_{\max }}\\ \multicolumn{1}{c}{d_f={d_f}_n}\\ \multicolumn{1}{c}{{L_s}_{\min }\le L_s\le {L_s}_{\max }}\\ \multicolumn{1}{c}{{L_t}_{\min }\le L_t\le {L_t}_{\max }}\\ \multicolumn{1}{c}{{L_f}_{\min }\le L_f\le {L_f}_{\max }}\\ \multicolumn{1}{c}{a_{\min }\le L_f}\\ \multicolumn{1}{c}{a_{\min }\le a_{\mathrm{Fatigue}\mathrm{failure}}}\end{array}\end{array}$$

(8)

where $a_{\mathrm{Fatigue}\mathrm{failure}}$ is a value of depth of cut that can be obtained from a criterion of fatigue failure under variable loading. Since $L_s$ in the formulas is overhang from the tool holder, the operator should consider a tool with a shank length greater than $L_s$ . Generally, the criterion function in equation (8) is intended to maximize the allowable cutting depth to chatter-free machining using the GA. The flow diagram of presented optimization algorithm can be shown in Figure 3.

Figure 3.

The flow diagram of presented GA to find the best tool geometry with maximum allowable cutting depth.

As presented in equation (5), average regenerative cutting forces are considered in the dynamic model using zero-order solution. For the case of forced vibration, dynamic chip load and its corresponding force with respect to fixed reference framework XYZ are described as follows ⁴⁶

$$\begin{array}{c}\mathrm{Dynamic}\mathrm{chip}\mathrm{load}:h=s_f{S}_{\psi }g\left(\psi \right)\hfill \\ \mathrm{Force}\mathrm{in}\mathrm{feed}\mathrm{direction}:F_Y=a{f}_Y=a{K}_t{s}_f(-S_{\psi }{C}_{\psi }-k_r{S}_{\psi }^2)\hfill \\ \mathrm{Force}\mathrm{in}\mathrm{normal}\mathrm{direction}:F_Z=a{f}_Z=a{K}_t{s}_f(S_{\psi }^2-k_r{S}_{\psi }{C}_{\psi })\hfill \\ \mathrm{Torsion}\mathrm{torque}\mathrm{aboat}\mathrm{longitudinal}\mathrm{axis}:T=a\mathcal{T}=a{K}_t{s}_f\frac{d_f}{2}{S}_{\psi }\hfill \end{array}$$

(9)

where $s_f$ is the feed per tooth and $g(\psi )$ is a step function that is nonzero just in the interval $[\begin{array}{cc}{\psi }_{\mathrm{st}}\hfill & {\psi }_{\mathrm{ex}}\hfill \end{array}]$ . Here, zero- and first-order solutions of mentioned force in equation (9) are considered to investigate the fatigue failure resulting from variable loading,^8,46 as follows

$$\begin{array}{c}{F}_Y=a(F_{Y0}+F_{Y1R}{C}_{\mathrm{\Omega }t}-F_{Y1I}{S}_{\mathrm{\Omega }t})=a{\hat{F}}_Y\hfill \\ F_Z=a(F_{Z0}+F_{Z1R}{C}_{\mathrm{\Omega }t}-F_{Z1I}{S}_{\mathrm{\Omega }t})=a{\hat{F}}_Z\hfill \\ T=a(T_0+T_{1R}{C}_{\mathrm{\Omega }t}-T_{1I}{S}_{\mathrm{\Omega }t})=a\hat{T}\hfill \\ F_{Y0}=\frac{N}{2\pi }\underset{{\psi }_{\mathrm{st}}}{\overset{{\psi }_{\mathrm{ex}}}{\int }}{f}_{Y}d\psi ,\hfill \\ F_{Y1}=\frac{N}{2\pi }\underset{{\psi }_{\mathrm{st}}}{\overset{{\psi }_{\mathrm{ex}}}{\int }}{f}_Y{e}^{-i\psi }d\psi =F_{Y1R}+i{F}_{Y1I}\hfill \\ F_{Z0}=\frac{N}{2\pi }\underset{{\psi }_{\mathrm{st}}}{\overset{{\psi }_{\mathrm{ex}}}{\int }}{f}_{Z}d\psi ,\hfill \\ F_{Z1}=\frac{N}{2\pi }\underset{{\psi }_{\mathrm{st}}}{\overset{{\psi }_{\mathrm{ex}}}{\int }}{f}_Z{e}^{-i\psi }d\psi =F_{Z1R}+i{F}_{Z1I}\hfill \\ T_0=\frac{N}{2\pi }\underset{{\psi }_{\mathrm{st}}}{\overset{{\psi }_{\mathrm{ex}}}{\int }}\mathcal{T}d\psi ,\hfill \\ T_1=\frac{N}{2\pi }\underset{{\psi }_{\mathrm{st}}}{\overset{{\psi }_{\mathrm{ex}}}{\int }}\mathcal{T}{e}^{-i\psi }d\psi =T_{1R}+i{T}_{1I}\hfill \end{array}$$

(10)

In addition, the maximum bending moment occurs at the fluted part origin is

$$M=a\hat{M}=a{L}_f\sqrt{{\hat{F}}_Y^2+{\hat{F}}_Z^2}$$

(11)

Consequently, normal stress $\sigma$ and shear stress $\tau$ on the surface of fluted section at the origin of this section can be written as follows

$$\begin{array}{c}\sigma =a\hat{\sigma }=a\frac{32\hat{M}}{\pi d_f^3}\hfill \\ \tau =a\hat{\tau }=a\frac{16\hat{T}}{\pi d_f^3}\hfill \end{array}$$

(12)

If the largest stresses are $({\sigma }_{\max },{\tau }_{\max })$ and the smallest stresses are $({\sigma }_{\min },{\tau }_{\min })$ in a cycle of micro-milling tool, then the midrange steady component of stresses $({\sigma }_m,{\tau }_m)$ and amplitude of the alternating component of stresses $({\sigma }_a,{\tau }_a)$ can be constructed as follows

$$\begin{array}{c}{\sigma }_m=a{\hat{\sigma }}_m=a\frac{{\hat{\sigma }}_{\max }+{\hat{\sigma }}_{\min }}{2},{\sigma }_a=a{\hat{\sigma }}_a=a\left|\frac{{\hat{\sigma }}_{\max }-{\hat{\sigma }}_{\min }}{2}\right|\hfill \\ {\tau }_m=a{\hat{\tau }}_m=a\frac{{\hat{\tau }}_{\max }+{\hat{\tau }}_{\min }}{2},{\tau }_a=a{\hat{\tau }}_a=a\left|\frac{{\hat{\tau }}_{\max }-{\hat{\tau }}_{\min }}{2}\right|\hfill \end{array}$$

(13)

The von Mises stress is often used to calculate yielding of materials under complex loading ⁴⁷ as follows

$$\begin{array}{c}{\sigma \prime }_m=a{\hat{\sigma }\prime }_m=a\sqrt{{\hat{\sigma }}_m^2+3{\hat{\tau }}_m^2}\hfill \\ {\sigma \prime }_a=a{\hat{\sigma }\prime }_a=a\sqrt{{\hat{\sigma }}_a^2+3{\hat{\tau }}_a^2}\hfill \end{array}$$

(14)

Here, the Goodman failure criterion is used to find $a_{\mathrm{Fatigue}\mathrm{failure}}$ for designing a micro-milling tool ⁴⁷ as follows

$$\frac{{\sigma \prime }_a}{s_e}+\frac{{\sigma \prime }_m}{s_{\mathrm{ut}}}=\frac{1}{n_d}\to a_{\mathrm{Fatigue}\mathrm{failure}}=\frac{1}{n_d(\frac{{\hat{\sigma }\prime }_a}{s_e}+\frac{{\hat{\sigma }\prime }_m}{s_{\mathrm{ut}}})}$$

(15)

where $s_{\mathrm{ut}}$ is the ultimate tensile strength, $s_e$ is the endurance strength, and $n_d$ is the design factor or factor of safety.

System identification

In this section, some parameters such as natural frequency, damping ratio, and the values of material length scale of a micro-milling tool are calculated by applying the time domain technique to the system identification test. The specifications of the cutting tool used for system identification test are presented in Table 1.

Table 1.

Specifications of cutting tool used for system identification test.

Parameter	Value	Parameter	Value (mm)
Tool material	Carbide tungsten	$d_f$	1
Tool manufacturer	IMPULSE company	$L_s$	23
N	2	$L_t$	15.5
$d_s$	$6(\mathrm{mm})$	$L_f$	1.5

The time domain technique requires an input to be applied to the cutting tool. In the current experiment, a hammer is used to generate an impact force at the free end of the cutting tool, which in fact stimulates the first bending mode of the structure vibration. The output vibrations are recorded at the shank by an accelerometer sensor (see Figure 4(a)).

Figure 4.

Experimental setup (a) for system identification test and (b) for chatter test of micro-milling process.

The specifications of experiment equipments are tabulated in Table 2.

Table 2.

The specifications of experiment equipments.

Item	Description
Signal analyzer	VibroRack 1000 made by ABP corporation
Accelerometer	AP203-100 piezoelectric accelerometer made by Global Test
Hammer	AU-02 made by Global Test

Figure 5 shows a sample of recorded experimental vibration signal of the tool-sensor system.

Figure 5.

The vibration time response of the tool-sensor system recorded by accelerometer sensor.

In order to identify the system parameters, the nonlinear governing PDE for bending motion $w_y$ or $w_z$ is considered. It is because the experiment results are available only in bending mode. For the reason that it is impossible to ignore the mass of the sensor in comparison with the cutting tool mass, the effect of the mass at the end of the shank part is taken into account in this PDE. By applying MSM, the PDE can be converted to an ODE ²⁶ as follows

$$(\mathcal{M}+A_1)T_{w_{y,\mathrm{tt}}}+A_3{T}_{w_y}+A_5{T}_{w_y}^3+A_{12}{T}_{w_{y,t}}=0$$

(16)

where $\mathcal{M}=m{\int }_0^{L}Y(x{)}^2{\delta }_D(x-L_s)\mathrm{dx}$ and $T_{w_y}$ are the effect of sensor mass and generalized coordinate for lateral displacements, respectively. In addition, the parameter m is assigned for mass of the sensor. The analytical solution of equation (16) can be derived by applying MMS on it as follows

$$\begin{array}{c}{w}_y(x,t)=\hfill \\ [2a\cos (\omega t+\beta )-\frac{2A_5}{A_3-9{\omega }^2(\mathcal{M}+A_1)}{a}^3\cos (3\omega t+3\beta )]Y\left(x\right)\hfill \\ a=a_0\exp (-\frac{A_{12}}{2(\mathcal{M}+A_1)}t)\hfill \\ \beta ={\beta }_0-\frac{3A_5}{2A_{12}\omega }{a}_0^2\exp (-\frac{A_{12}}{\mathcal{M}+A_1}t)\hfill \\ \omega =\sqrt{\frac{A_3}{\mathcal{M}+A_1}}\hfill \end{array}$$

(17)

The system parameters including damping ratio and the values of material length scale, assuming that all of them are equal $l_0=l_1=l_2$ , can be determined in such a way that the time response given in equation (17) should have been matched with that one obtained from experimental test (see Figure 5). Khorshidi ⁴⁸ has shown that the material length scale parameters of a micro/nano-structure are depended on its diameter and material. It is shown in this reference that for an especial material of the structure, the material length scale parameters are approximately a constant coefficient of its diameter. Therefore, this constant coefficient should be determined via the presented system identification. In addition, the natural frequency can be calculated by removing the sensor mass effect of the last relation in equation (17). Therefore, taking the root mean square error (RMSE) as the cost function and applying the GA on five different samples of the impact test, the system parameters can be calculated according to Table 3.

Table 3.

The values of the parameters obtained from the system identification test.

Parameter	Average value	Standard deviation
$\omega$	$6020.152(\mathrm{Hz})$	$1.6788$
${\zeta }_B$	5.503%	0.1064%
$l_0,l_1,l_2$	$0.1501d_f$	$0.0045d_f$

Here, the RMSE is used to measure the differences between oscillation values predicted by presented model (equation (17)) and the oscillation values observed from experimental test illustrated in Figure 5. Indeed, this statistic parameter is a quadratic scoring rule that measures the average magnitude of the error. The RMSE can take on any non-negative value and a magnitude nearer to zero representing that a better fit is accounted by the model. The cost function value in each generation is illustrated in Figure 6 during the searching process to achieve the minimum value of the RMSE. In addition, the values of other statistics parameters, such as the sum of squares due to error (SSE) and the coefficient of determination so-called R-square that is usually used for evaluating the model, are presented in Figure 6 for the optimized model.

Figure 6.

Variation of RMSE as the objective function versus number of generations for system identification.

The analytical response, obtained based on the system parameters in Table 3, is compared with the experimental answer in Figure 7 to show the precision of presented system identification.

Figure 7.

Comparison of analytical and experimental response at the end of the tool shank.

Results and discussion

Validation

To verify the nonlinear continuum beam model of micro-milling, a set of experimental chatter tests are conducted on an in-house micro-computer numerical control (CNC) machine with the previous mentioned micro-milling tool (see Figure 4(b)). Technical specifications of the CNC machine are tabulated in Table 4.

Table 4.

Technical specifications for micro-CNC machine.

Item	Description
Travel limitation	$30\mathrm{cm}(\mathrm{X})\times 30\mathrm{cm}(\mathrm{Y})\times 15\mathrm{cm}(\mathrm{Z})$
Spindle speed	Maximum $24,000\mathrm{r}/\min$
Main spindle power	Delta inverter $2.2\mathrm{kW}$
Linear guides	HIWIN ball screw
Stepper motor	LEADSHINE stepper motor driver
Feed speed	Maximum $6\mathrm{m}/\min$
Control software	Mach 3

The other basic parameters utilized for all experimental tests are tabulated in Table 5.

Table 5.

Realistic values of parameters used for experimental chatter tests by Jin and Altintaş. ⁴⁶

Parameter	Value	Parameter	Value
Work-piece material	AISI 1045 steel	$G_r$	$2.136\times {10}^4(\mathrm{N}/\mathrm{m})$
${\zeta }_R$	–	$g_t$	$0.877$
$K_t$	$4042(\mathrm{MPa})$	${\psi }_{\mathrm{st}}$	$0(\mathrm{deg})$
$k_r$	$0.696$	${\psi }_{\mathrm{ex}}$	$180(\mathrm{deg})$

The calculated SLD is evaluated with the chatter test data in Figure 8. The comparisons between the experimental and calculated results show good agreement.

Figure 8.

Comparison between predicted lobe diagrams and the experimental chatter test data.

Optimization: example 1

For a realistic example of optimization, the micro-milling process mentioned in previous sub-section is selected. It is well known that a considerable generation size improves the efficiency of optimization process and consequently diminishes the variations of premature convergence. For that reason, various numbers of generations are checked to get appropriate convergence to optimized value. To guarantee the correctness of the findings, the limitation values corresponding to constraint functions are considered in a specific range around realistic values of Table 1 (see Table 6).

Table 6.

Considered values of limitations based on the tool used in experiments.

Parameter	Value	Parameter	Value
$\left[\begin{array}{cc}{N}_{\min }\hfill & N_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}2\hfill & 4\hfill \end{array}\right]$	$\left[\begin{array}{cc}{L_s}_{\min }\hfill & {L_s}_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}20\hfill & 26\hfill \end{array}\right]\left(\mathrm{mm}\right)$
$\left[\begin{array}{cc}{d_s}_{\min }\hfill & {d_s}_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}4\hfill & 8\hfill \end{array}\right]\left(\mathrm{mm}\right)$	$\left[\begin{array}{cc}{L_t}_{\min }\hfill & {L_t}_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}13.5\hfill & 17.5\hfill \end{array}\right]\left(\mathrm{mm}\right)$
${d_f}_n$	$1\left(\mathrm{mm}\right)$	$\left[\begin{array}{cc}{L_f}_{\min }\hfill & {L_f}_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}1.3\hfill & 1.7\hfill \end{array}\right]\left(\mathrm{mm}\right)$

Here, the factor of safety $n_d$ and the maximum value of feed rate are considered as $1.5$ and $10\mu \mathrm{m}/\mathrm{tooth}$ , respectively. The value of cost function $a_{\min }$ versus the number of iteration is presented in Figure 9 during the searching process to find the maximum value of minimal cutting depth.

Figure 9.

Variation of objective function with number of generations based on the tool used in experiments.

Figure 9 shows that the best possible value of optimum cutting depth obtained from GA is $a_{\min }=127.054\mu \mathrm{m}$ . As well, the values of variables in the optimum situation are presented in Table 7.

Table 7.

The values of variables in the optimum situation based on the tool used in experiments.

Parameter	Value	Parameter	Value (mm)
N	2	$L_s$	20
$d_s$	$6.8(\mathrm{mm})$	$L_t$	16.1
$d_f$	$1(\mathrm{mm})$	$L_f$	1.3

Figures 10 and 11 show the constructed SLDs of the tool with the unoptimized features in Table 1 and the optimized values in Table 7, respectively.

Figure 10.

Stability lobe diagram based on unoptimized tool with the features presented in Table 1.

Figure 11.

Stability lobe diagram based on optimized tool with the features presented in Table 7.

A remarkable result from these figures is that the optimum cutting depth for the optimized tool is $a_{\min }=127.054\mu \mathrm{m}$ while for the unoptimized one is $a_{\min }=65.7865\mu \mathrm{m}$ . It means that the results of GA may lead to design or find a micro-milling tool that its acceptable cutting depth increases $1.9313$ times.

Optimization: example 2

In this sub-section, another realistic example of optimization for a micro-milling process introduced by Jin and Altintaş ⁴⁶ is presented. The nominal values of cutting parameters presented by Jin and Altintaş ⁴⁶ for full immersion milling are tabulated in Table 8.

Table 8.

Nominal values of cutting parameters illustrated by Jin and Altintaş. ⁴⁶

Parameter	Value	Parameter	Value
Tool material	Carbide tungsten	$K_t$	$4042(\mathrm{MPa})$
Work-piece material	AISI 1045 steel	$k_r$	$0.696$
${\zeta }_B\left(\mathrm{shank}\mathrm{section}\right)$	3.62%	$G_r$	$2.136\times {10}^4(\mathrm{N}/\mathrm{m})$
${\zeta }_B\left(\mathrm{taper}\mathrm{section}\right)$	0.41%	$g_t$	$0.877$
${\zeta }_B\left(\mathrm{fluted}\mathrm{section}\right)$	0.41%	$\left[\begin{array}{cc}{\psi }_{\mathrm{st}}\hfill & {\psi }_{\mathrm{ex}}\hfill \end{array}\right]$	$\left[\begin{array}{cc}0\hfill & 180\hfill \end{array}\right]\left(\mathrm{deg}\right)$
${\zeta }_R$	–	$s_f$	$5\mathrm{\mu }\mathrm{m}/\mathrm{tooth}$

In addition, the details of the tool used by Jin and Altintaş ⁴⁶ including number of teeth and tool geometry are presented in Table 9.

Table 9.

Details of the tool used by Jin and Altintaş. ⁴⁶

Parameter	Value	Parameter	Value (mm)
N	2	$L_s$	16
$d_s$	$4(\mathrm{mm})$	$L_t$	6.3
$d_f$	$0.6(\mathrm{mm})$	$L_f$	1.2

In current GA, various numbers of iterations are examined to get proper convergence to optimized value. To ensure the accuracy of the results, the values of constraint functions are selected in a range around realistic nominal values of Table 9, as illustrated in Table 10.

Table 10.

Selected values of constraint functions based on the tool used by Jin and Altintaş. ⁴⁶

Parameter	Value	Parameter	Value
$\left[\begin{array}{cc}{N}_{\min }\hfill & N_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}2\hfill & 4\hfill \end{array}\right]$	$\left[\begin{array}{cc}{L_s}_{\min }\hfill & {L_s}_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}13\hfill & 19\hfill \end{array}\right]\left(\mathrm{mm}\right)$
$\left[\begin{array}{cc}{d_s}_{\min }\hfill & {d_s}_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}2\hfill & 6\hfill \end{array}\right]\left(\mathrm{mm}\right)$	$\left[\begin{array}{cc}{L_t}_{\min }\hfill & {L_t}_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}4.3\hfill & 8.3\hfill \end{array}\right]\left(\mathrm{mm}\right)$
${d_f}_n$	$0.6\left(\mathrm{mm}\right)$	$\left[\begin{array}{cc}{L_f}_{\min }\hfill & {L_f}_{\max }\hfill \end{array}\right]$	$\left[\begin{array}{cc}0.9\hfill & 1.5\hfill \end{array}\right]\left(\mathrm{mm}\right)$

In addition, the value of $1.5$ is considered for the design factor $n_d$ . In addition, the range [0 15] $\mu \mathrm{m}/\mathrm{tooth}$ is considered for the feed rate of the machining. By considering constraints function presented in Table 10, the optimization program is ran using GA to find the best achievable $a_{\min }$ . Graphical representation of objective function $a_{\min }$ versus the number of generation is illustrated in Figure 12.

Figure 12.

Variation of the objective function with number of generations based on the tool used by Jin and Altintaş. ⁴⁶

As can be seen from Figure 12, the optimum value of cost function obtained from GA is $a_{\min }=29.814\mu \mathrm{m}$ . In addition, the values of variables in the optimum situation are presented in Table 11.

Table 11.

The values of variables in the optimum situation based on the tool used by Jin and Altintaş. ⁴⁶

Parameter	Value	Parameter	Value (mm)
N	2	$L_s$	14.2
$d_s$	$4(\mathrm{mm})$	$L_t$	5.9
$d_f$	$0.6(\mathrm{mm})$	$L_f$	0.9

The constructed SLDs based on data before applying optimization in Table 9 and after that in Table 11 are shown in Figures 13 and 14, respectively.

Figure 13.

Stability lobe diagram based on unoptimized tool with the features presented in Tables 8 and 9.

Figure 14.

Stability lobe diagram based on optimized tool with the features presented in Tables 8 and 11.

An interesting result from these figures is that $a_{\min }$ for the optimized tool is equal to $29.814\mu \mathrm{m}$ while $a_{\min }$ for the unoptimized one is equal to $21.980\mu \mathrm{m}$ . It means that GA may lead to design or select a micro-milling tool that its allowable cutting depth increases $1.3564$ times.

Conclusion

To maximize the allowable cutting depth to chatter-free micro-milling process, an optimization procedure based on GAs has been presented in this article. For this reason, the tool has been modeled as a nonlinear 3D spinning cantilever Timoshenko beam based on strain gradient theory. To identify the system parameters, an optimization method has been presented to calculate the natural frequency, damping ratio, and material length scale of the cutting tool. The comparison between the experimental chatter test and predicted results shows the accuracy of presented model.

The procedure explained in this study has made an important improvement in micro-milling efficiency and designing of micro-milling tool. An optimization method using GA has been applied to micro-milling process with decision variables including number of teeth, shank diameter, diameter of fluted section, shank length, taper length, and length of fluted section. The analytical solution has been carried out by applying MSM and MMS on the DPDEs of the system to obtain the SLDs. The cost function of the optimization procedure has been defined as maximization of the minimal cutting depth on the border of SLD. Results show that the obtained cutting depth for the optimized tool is much more than the cutting depth for unoptimized tool. This optimization method may help the operators to order a specific design of micro-milling tool with optimal geometry for a special machining. In addition, machining operators can select a micro-milling tool that its geometry properties are near the optimized one from the accessible tools in the markets.

In summary, the main superiorities of this study are as follows:

1. The calculation of material length scale using system identification.

2. The optimization of a micro-milling tool geometry to find the maximum allowable cutting depth for chatter-free machining.

3. The analytical study of micro-milling process stability to determine the allowable cutting depth as the objective function of the tool design optimization.