Keywords

39.1 Introduction

Major technological advances have been achieved in the field of high-speed machining (HSM). Better spindle designs allow for speeds of 30k and higher in micro-milling. Increasing the spindle speed and depth of cut during machining increases the rate of material removal. Nevertheless, high speed machining at larger depths of cut is hindered by the chatter phenomenon, which results in machining process to be instable leading to unsatisfactory finish of the surface and potentially destroying the life of the cutting tool and work pieces. Driven by the current needs of highly automated production, the recognition of higher-productive, higher-quality, low-cost milling necessitates the concern of exact cutting force and chatter models. These days, the primary instruments used in manufacturing are stability lobe diagrams, which visually distinguish stable cutting circumstances from unstable or chatter ones. Using the tool point frequency response function (FRF), tool geometry, cutting factors, and cutting force coefficients, Altintas and Budak [1] demonstrated the models necessary to forecast the stability lobes. Several studies on the milling process parameters related to the chatter studies have been the subject of a number of previous research, including those by Altintas and Weck [2], Bravo et al. [3], Gagnola et al. [4] and Tanga et al. [5] inverse analysis of the self-excited chatter vibration has been recently reported, and it has been used by Suzuki et al. [6] to build a novel method for calculating the transfer function. The accuracy of the SLD was refined by Raphael et al. [7], who also investigated how the spindle’s electronic position and velocity affected SLD accuracy. Lin and Tu [8], Jiang and Zhang [9], Penga et al. [10], Cao et al. [11], Gao and Meng [12] and a few more publications deal with the design of end mill spindles and identify the design elements that largely affect the stability lobe diagrams of these tools. Effects such as bearing preload on the tip of the tool frequency responses were studied by the Ozturk et al. [13]. Preload and spindle speed were shown to have a correlation in the examined instance. A dynamic model of high-speed motorised spindles was investigated by Liu and Chen [14]. To conduct a design sensitivity analysis, a design flowchart is drawn up. In the past, academics have attempted to optimise the milling process by taking a number of variables into account. To improve machining performance, Altintas [15] used simulated annealing and a genetic algorithm, two artificial computing approaches. Experimental investigations verify the combined use of the two methods. In order to foretell the cutting forces in an end milling operation, Jalili Saffar and Razfar [16] created a 3D modelling of the cutting forces are optimised using a genetic algorithm approach and then verified experimentally. Mathematics modelling the interaction between machining dynamics and material behaviour was developed by Palanisamy et al. [17]. The cutting process settings and tool vibration levels are optimised to save machining time using a robust GA approach. Using particle swarm optimization, Hsieh and Chu [18] created a generative tool path in a five axis milling machine using PSO algorithm. Using proactive and well-informed PSOs improves search efficiency. The harmony search (HS) technique was proposed by Zarei et al. [19] to determine face-milling parameters. The optimal depth of cut, feed rate, and cutting speed are achieved by a series of passes using a variety of machining settings in order to minimise the total cost of manufacturing.

Whereas optimum machining studies related to the cutting process is considered as a function of the geometric parameters of the integrated spindle tool unit are only few presented by the previous authors. In the current study, we aim to acquire the best cutting process conditions of the end milling process by taking into account a few critical parameters of the spindle-tool unit and the corresponding flexible body dynamics. End-milling studies with the currently available parametric conditions confirm the accuracy of the machining process.

39.2 Modeling of Cutting Force Simulations

The workpiece is supposed to be stiff, whereas the tool is assumed to be flexible. Cutting process stability evaluation requires consideration of both tool geometry and machining requirements. The parameter like angle between each tooth, teeth numbers, helix angle of the tool, diameter of the tool and run-out are all part of the tool geometry. Spindle speed, axial depth of cut, feed per tooth, and entry/exit angles (a function of radial immersion) are required machining requirements. As b and h(ϕj) increase in the axial direction, the tangential and radial components of the linear cutting force operating on tooth j increase accordingly.

$$ F_{t,j} (\phi ) = K_{t} bh(\phi_{j} ) $$
(39.1)
$$ F_{r,j} (\phi ) = K_{r} F_{t,j} (\phi ) = K_{r} K_{t} bh(\phi_{j} ) $$
(39.2)

where tangential shearing (Kt) and normal shearing (Kr) both contribute to the overall flute coefficient. Both the tangential and normal force components are projected onto the fixed (x, y) coordinate frame in the following ways:

$$ F_{x.j} = - F_{t.j} \cos \phi_{j} - F_{r.j} \sin \phi_{j} $$
(39.3)
$$ F_{y.j} = + F_{t.j} \sin \phi_{j} - F_{r.j} \cos \phi_{j} $$
(39.4)

By adding the sum across all the teeth (flutes), the total cutting forces may be calculated using the following closed-form expressions:

$$ F_{x} (\phi ) = \sum\limits_{j = 1}^{{N_{t} }} {F_{x,j} } ;\quad F_{y} (\phi ) = \sum\limits_{j = 1}^{{N_{t} }} {F_{y,j} } $$
(39.5)

39.2.1 Effects of Process Damping

Energy is dissipated in a process when a cutting tool’s relief angle interacts with preexisting vibrations on the surface of the machined workpiece. Turning and milling are two processes where process damping has received a lot of attention from researchers. It happens when the cutter’s flank touches with the imprinted vibrations on the cutting process. It may slow down the machining process and affect the dynamic cutting forces.

At the lower speeds of the spindle, the stability of the end milling are spaced closely and the chatter-free depths of cut are shown to decrease significantly. Fortunately, the chip width that may be used at low spindle speeds can be increased because to the process damping effect. The damping force for the milling force (Fd) is denoted by a 90°-phase shift with respect to the displacement, is of the negative sign to velocity component. Based on this information, the process damping force is described as the viscous damping force

$$ F_{d} = - C\frac{b}{V}\dot{y} $$
(39.6)

Here, we define the y-direction process damping force and it is considered as the function of the velocity of the cutter (\(\dot{y}\)), cutting speed (v), width of the chip (b) and damping coefficient (C). The time-dependent cutting force direction is a major challenge in establishing an analytical solution for milling stability. The average force direction is considered together with the average tooth angle of the cut (favg). A time-independent or self-sufficient system may be built using this method. Initially, µx and µy are used to project the force in X and Y directions then, finally, the results are projected on to the normal of the surface. The milling process’s stability lobe boundaries may be calculated using the following equations:

$$ b_{lim } = \frac{ - 1}{{2K_{s} {\text{Re}} (H_{or} )N_{t}^{*} }} $$
(39.7)
$$ \frac{{f_{c} }}{{\Omega N_{t} }} = j + \frac{\upvarepsilon }{{2\uppi }} $$
(39.8)
$$ N_{t}^{*} = \frac{{\phi_{e} - \phi_{s} }}{{360/N_{t} }} $$
(39.9)
$$ \varepsilon = 2\pi - 2\tan^{ - 1} \left( {\frac{{{\text{Re}} [H_{or} ]}}{{{\text{Im}} [H_{or} ]}}} \right) $$
(39.10)
$$ H_{or} = \mu_{x} H_{x} + \mu_{y} H_{y} $$
(39.11)

where, count of the teeth is represented as Nt, Hor is considered as oriented frequency response at the tool tip, j = 0, 1, 2, … is the lobe number, force angle (β) in degrees. The process damping force may be easily used for both vertical and horizontal milling operations. In Fig. 39.1, n represents the surface normal direction, which is used to establish the geometry for the up-milling operation.

Fig. 39.1
A diagram of up-milling geometry. A circle. Feed functions from right to left. n is the surface's normal direction and passes through the center. The angle between F and the direction of n is beta. The angle between the Y axis and the direction of n is Phi average. Here, Phi s equals to 0.

Up-milling geometry based on the average tooth angle

Full size image

The n-axis damping force of a process is projected onto the x-axis as

$$ F_{x} = F_{d} \cos (90 - \phi_{avg} ) = - \left( {C\frac{b}{V}\cos \left( {90 - \phi_{avg} } \right)} \right)\dot{n} $$
(39.12)

The damping force of a process is projected onto the x-axis as

$$ F_{x} = - \left( {C\frac{b}{V}\cos^{2} \left( {90 - \phi_{avg} } \right)} \right)\dot{x} $$
(39.13)

Therefore, the revised damping for x- and y-axis stability calculations is:

$$ c_{new,x} = c_{x} + \left( {C\frac{b}{V}\cos^{2} \left( {90 - \phi_{avg} } \right)} \right) $$
(39.14)
$$ c_{new,y} = c_{y} + \left( {C\frac{b}{V}\cos^{2} \left( {180 - \phi_{avg} } \right)} \right) $$
(39.15)

The damping values in the x and y directions are produced for down milling using the same method as outlined for up milling.

$$ c_{new,x} = c_{x} + \left( {C\frac{b}{V}\cos^{2} \left( {\phi_{avg} - 90} \right)} \right) $$
(39.16)
$$ c_{new,y} = c_{y} + \left( {C\frac{b}{V}\cos^{2} \left( {180 - \phi_{avg} } \right)} \right) $$
(39.17)

39.2.2 Tool Run-Out and Cutting Force Modelling

When a mechanical system is in motion, tool run-out occurs when the tool is not perfectly aligned with the axis of rotation. The effect of the tool run-out is a serious problem in the machining process that may arise from a number of different sources, including misalignment between the tool holder and spindle, incorrect radial tooth location on the cutter, inaccurate tool dimensions, thermal tool deflection, improper tool and holder balance, and so on. As a result, the tool is subjected to unequal stresses across its cutting teeth at all times. Chip thickness varies as the tool rotates. The cutting forces were determined by the cutting geometry that was inserted into the workpiece, and the cutting forces were related to the uncut chip thickness using empirical formulae. The Eccentricity called as the run-out distance and the run cut angle (λ) characterize radial run-out. A circular approximation of the cutting tool’s path allows us to use the following formula to get the chip thickness, which accounts for tool run-out.

$$ h_{j} \left( {\upphi _{j} } \right) = f_{t} \sin (\upphi _{j} ) + h_{ro}^{j} $$
(39.18)

Here we describe the chip thickness variation caused by run-out in cutting edge j as

$$ h_{ro}^{j} = \rho_{j} - \rho_{j - 1} $$
(39.19)

The cutting edge with run-out is the difference between the calculated and actual radii of each flute, which causes a shift in the cutting load. Run-out causes a difference between the nominal radius and the real radius, which is the radius along which the flute actually travels.

The effective radius variation along the cutting edge is not a significant factor when working with the tiny values of depth of cut and helix angle examined in this study, therefore the tool run-out is appropriately described by the difference between the actual and nominal trajectories for each flute. Cutting forces in the X and Y directions may be derived from the resolution of the tangential and radial forces, as the particular cutting force is a function of these three forces and the chip thickness along the cutting edge.

$$ \left\{ {\begin{array}{*{20}c} {F_{x,j} \left( {\upphi _{j} } \right)} \\ {F_{y,j} \left( {\upphi _{j} } \right)} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} { - \cos \left( {\upphi _{j} } \right)} & { - \sin \left( {\upphi _{j} } \right)} \\ {\sin \left( {\upphi _{j} } \right)} & { - \cos \left( {\upphi _{j} } \right)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {F_{t,j} } \\ {F_{r,j} } \\ \end{array} } \right\} $$
(39.20)

39.2.3 Cutting Force Modelling Considering the Influence of Pitch

In most cases, it is believed that the cutter’s teeth are evenly spaced (i.e., have a consistent tooth pitch) throughout its rim. When milling tough materials, it’s common practice to use a tool with a variable pitch end mill. Mills with a tunable pitch angle may also have a tunable helix angle. Figure 39.2 depicts how the feed per tooth of a variable pitch tool changes as a function of the tooth angle θi (degrees).

Fig. 39.2
Two diagrams. A circle that is divided into 6 parts. The angle between each part with the center is Theta. A diagram of a rectangle. It is divided into 5 parts. The length is 2.

Edge inclination on the end mill cutter’s rim

Full size image

The feed per tooth ratio is evaluated as given below

$$ f_{t,variable} = \left[ {\frac{{f_{t,mean} \times \theta_{i} \times N_{t} }}{{360^{ \circ } }}} \right] $$
(39.21)

where ft,mean is the mean feed per tooth and Nt is the number of teeth.

39.3 Results and Discussions

Using the information in Table 39.1, a Matlab programme depicts the stability lobe diagram of the frequency response function of the tool tip.

Table 39.1 System’s modal parameters and cutting force coefficeints
Full size table

Coefficients of the system’s modal and cutting modes are listed in Table 39.1.

To examine the impact of process damping with these updates to the classical model, a MATLAB programme is constructed. The radial immersion ratio of 50% is used in the numerical analysis. With the data obtained from finite element analysis approach, the frequency responses are arrived and it is further utilized to plot the stability areas for the up-milling operation are displayed. Stability charts for varying the process damping constant from zero (no process damping) to the highest value (C = 6.5 × 105 N s/m) are shown in Fig. 39.3. The average steady depth of cut is found to rise from 0.04 to 0.3 mm when the process is dampened.

Fig. 39.3
Three graphs of b versus Omega. a. It presents reverse bell curves when process dampening is absent. b. It presents that the curves fluctuate in the presence of process dampening. c. It presents that the curves fluctuate very much when the process dampening is increased.

Analytical stability lobe diagrams. a In the absence of process dampening, b in the presence of process dampening with C = 6.5 × 104 N s/m, c in the presence of process dampening with C = 6.5 × 105 N s/m

Full size image

In order to simulate the cutting force, modal parameters, the tool shape, and machining requirements must be provided. Each mode of the tool in the x and y axes has its own set of modal characteristics, which include its stiffness, damping ratio, and natural frequency. In Fig. 39.4, we see the results of a time-domain simulation of a milling model with two degrees of freedom, both with and without the influence of process damping. When the process damping effect is included into the model, it is shown that the x and y levels of tool displacements decrease. In addition, it offers the constant tool displacements that are crucial for a consistent cut.

Fig. 39.4
2 sets of 2 graphs. a. A graph of Y versus t and a graph of Y versus t. These present the variations from the cutter with the increase in t. These curves fluctuate very much. b. A graph of Y versus t and a graph of Y versus t. These present S-shaped waveforms.

Vibrations from the cutter at a depth of 1 mm and spindle speed of 3000 rpm. a In the absence of process dampening, b presence of process dampening with C = 6.5 × 105 N s/m

Full size image

A tool with a 12 mm diameter is used to demonstrate the impact of run-out. Run-out is typically measured along the feed direction. For the case of a run-out offset of 10 µm, simulations are performed in the time domain both with and without run-out effects, as shown in Fig. 39.5. The cutting process is modelled in two dimensions, and all spindle-tool system characteristics are taken into account.

Fig. 39.5
2 sets of 2 graphs. a. A graph of Y versus t and a graph of Y versus t. These present the variations of run out correction level with the increase in t. These curves fluctuate very much. b. A graph of Y versus t and a graph of Y versus t. These present 2 fluctuating line curves for the variations of X and Y with t.

Run-out correction levels for tool displacements. a In the absence of run-out, b in the presence of run-out

Full size image

The amounts of tool displacement are shown to rise as run-out increases throughout the cutting process. The cutting pressures experienced by the virtual tool for run-out values of 10 and 25 µm are shown in Fig. 39.6. As can be seen in Fig. 39.6, the cutting forces for each flute vary by just a little amount (10–12 µm). Cutting forces may be detected with significant amplitude in the variations in load of the chip at each tooth is shown in Fig. 39.6b, leading to unacceptable vibration levels of the tool as cutting progresses.

Fig. 39.6
Two graphs of F versus t. a. It presents variations of force with a run out of 10 micrometers and without run out. These present triangle waveforms. b. It presents variations of force with a run out of 25 micrometers and without run out. These present triangle waveforms.

Force variations as a result of tooth wear. a With a run-out of 10 µm, b with a run-out of 25 µm

Full size image

Using a tool with variable pitch spacing [0 95 180 275] degrees and with equal pitch [0 90 180 270] degrees for two different axial depths of cut, Fig. 39.7 displays the observed resulting forces with a 25% immersion of the radial cut in up-milling of aluminum alloy. It is seen that the forces fluctuate non-linearly due to the varied pitch as the axial depth of the cut raised from 1 to 2.5 mm, the cutting forces also have been raised significantly.

Fig. 39.7
Two graphs of F versus t. a. It presents variations of force with equal pitch and variable pitch. These present triangle waveforms. b. It presents variations of force with equal pitch and variable pitch. These present triangle waveforms. The waveforms for both cases almost overlap with each other.

Variable-pitch time-domain simulation at 25% radial immersion. a Cutting depth = 1 mm, b cutting depth = 2.5 mm

Full size image

At an axial depth of cut as 1 mm example with as equivalent and varied pitch of the tool, the cutting tool displacement values are achieved as illustrated in Fig. 39.8. The cutting tool’s variable pitch clearly raises the displacement levels of the tool dramatically.

Fig. 39.8
2 sets of 2 graphs. a. A graph of Y versus t and a graph of Y versus t. These present the variations of vibration levels of the tool. These curves fluctuate very much. b. A graph of Y versus t and a graph of Y versus t. These present 2 fluctuating line curves for the variations of vibration levels of tools.

Vibration levels of tools at 1 mm depth of cut. a With the same pitch, b with different pitch

Full size image

39.4 Conclusions

As a result, evaluating the cutting process’s stability demands an ideal selection of cutting tool settings with regard to those factors. Errors in machining may arise from a number of different factors, including tool run-out, tool deflection, workpiece displacements, and variable pitch end mills. There are situations when the procedure fails despite having a well-designed spindle and tool. The aforementioned calculations allow one to determine the maximum allowable tool run-out in terms of spindle dimensions. For the proposed spindle-tool configuration, tools with variable pitches are also permitted for use during machining to achieve the greater stable depth of cuts. In an integrated spindle-tool spindle system, the damping force at the interface regions such as collet and tool, chuck and the collet are having a major impact on the system’s stability, process damping has an effect analogous to that of the internal damping. In order to guarantee the steadiness of the cutting process, the findings shown that the proposed approach may be utilized for accurate assessment of tool run-out, process damping, and variable pitch effects.