List of Figures

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Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey

School of Engineering and Sciences

Semi-Active Magnetorheological Damper Device for Chatter Mitigation during Milling of Thin-Floor Components

A dissertation presented by Santiago Daniel Puma Araujo

Submitted to the

School of Engineering and Sciences

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy In

Engineering Science

Major in Advanced Manufacturing Monterrey Nuevo León, December 4^th, 2020 Instituto Tecnológico y de Estudios Superiores de Monterrey

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Dedication

To my beloved mother Dolores who has supported me unconditionally throughout my life and my father Emerson who has been an example to follow and inspiration in my life despite no longer being with us.

To my family and friends.

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IV

Acknowledgements

To my advisors Dr. Alex Elias, Dr. Oscar Martínez, and Dr. Daniel Olvera who supported me since the beginning of my education at Tecnológico. Thanks for your helpful advice, hold up during my thesis preparation and guidance through my research.

To my co-advisors Dr. Luis Norberto and Dr. Gorka Urbikain. for the support in my thesis and the publication of scientific paper.

I want to thank you for the support given by Tecnológico de Monterrey on tuition and CONACyT with the support for living.

I am grateful to my family who has unconditionally supported me during the time of my life.

Thanks to Natalia for her support and comprehension during this time because for you I have been able to get this point.

Thanks to my colleagues and friends from the engineering graduate program for their friendship and nice moments during our studies: Jorge Palacios, Xavier Sanchez, Oscar Escalera, Saul Rubio, Rigoberto Guzman, and Eduardo Salas with whom I shared moments of joy and stress.

Finally, all of these wouldn’t be possible without the sponsorship granted by Tecnológico de Monterrey through the Research Group of Nanotechnology for Devices Design, and by the Consejo Nacional de Ciencia y Tecnología de México (Conacyt), Project Numbers 242269, 255837, 296176, National Lab in Additive Manufacturing, 3D Digitizing and Computed Tomography (MADiT) LN299129 and the University of the Basque Country. These institutions believed in my potential as a professional and researcher.

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Semi-Active Magnetorheological Damper Device for Chatter Mitigation during Milling of Thin-Floor Components

By

Santiago Daniel Puma Araujo

Abstract

The productivity during the machining of thin-floor components is limited due to unstable vibrations, which lead to poor surface quality and part rejection at the last stage of the manufacturing process. In this thesis, the design of a novel semi-active magnetorheological damper device to suppress chatter conditions during milling operations of thin-floor components is introduced. To validate the performance of the magnetorheological (MR) damper device, a one-degree-of-freedom experimental set-up was designed to mimic the machining of thin-floor components.

To find the milling operation's stability boundaries, a novel cutting force model in which the bull-nose end mill is discretized in disks is introduce, and then, the Enhance Multistage Homotopy Perturbation Method (EMHPM) was applied.

Numerical simulation results obtained by the EMHPM show that predicted stability lobes of the cantilever beam closely follow experimental data. At the end of the thesis, it is discussed how the usage of the MR damper device modifies the stability boundaries that increase at least three times the productivity of the cutting process.

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VI

List of Figures

Figure 1.1. Total income from the gross domestic product. ... 2 Figure 1.2. Percentage of manufacturing revenue to gross domestic product. ... 2 Figure 1.3. Scientific articles on Chatter by country:(a) all countries and (b) zoom excluding China and the United States. ... 4 Figure 1.4. Chatter publications on machining from 1980 to the present. ... 9 Figure 1.5. Monolithic parts: (a) Replacement for a monolithic part; (b) Satellite part machined from aluminum. ... 11 Figure 1.6. Marks on a thin floor and wall of the monolithic piece. ... 12 Figure 1.7. Comparison of cutting with a square and round insert and stress diagram(a) square (b) round (c) 90 ° sharp-edged (d) radius width. ... 13 Figure 2.1. Scheme of the orthogonal cutting process. Adapted from [53]. ... 21 Figure 2.2. Composition of optical microscope photographs of the bull-nose end mill taken with an Alicona microscope: (a) bottom view (diameter measurement); (b) frontal view (nose radius measurement); (c) lateral view (helix angle measurement).

... 24 Figure 2.3. Averaged cutting Coefficients taken from [19]. ... 25 Figure 2.4. Finite difference force model. ... 26 Figure 2.5. Finite difference force model procedure: (a) Linear regression results for simulated forces; (b) parameters regarding the study interval. ... 26 Figure 2.6. Comparison between and models with experiment data at 3000 rev/min and the cutting conditions: a^p = 0.5 mm, f^z = 0.3 mm/tooth (a) Average cutting coefficients; (b) Finite difference model. ... 27 Figure 2.7. Main cutting parameters in milling operation with a bull-nose end mill:

axial discretization and differential cutting force components. ... 28 Figure 2.8. (a) Scheme for characterization of forces; (b) Photographs in experimentation. ... 31 Figure 2.9. Experimental cutting coefficients (circle marks) vs. fitted model (solid lines) along the toroidal section: (a) tangential; (b) radial; (c) axial directions; (d) κ angle along the toroid section. ... 32 Figure 2.10. Comparison between experiment data and the proposed force model at 3000 rev/min and the cutting conditions: (a) ap = 0.5 mm, fz = 0.2 mm/tooth; (b) ap = 1.5 mm, fz = 0.15 mm/tooth. ... 33 Figure 3.1. Critical exponents for autonomous linear ODEs: (a) Hopf bifurcation, (b) and Saddle-Node bifurcation. Adapted from Insperger [15]. ... 36

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VII

Figure 3.2. Critical characteristic multipliers for periodic systems: (a) secondary Hopf bifurcation, (b) cyclic-fold bifurcation, and (c) double period bifurcation.

Adapted from Insperger [15]. ... 38

Figure 3.3. Schematic EMHPM solution using first polynomial to approximate delay subinterval. ... 47

Figure 3.4. Approximation of the delay term. (Adapted from Insperger [65]). ... 53

Figure 3.5. Experimental setup used for stability verification. An accelerometer is attached in the workpiece to detect vibrations, while the optical sensor is used to measure the spindle rotation. ... 58

Figure 3.6. Predicted stability lobes obtained from Equation (3-77) together with experimental data in a down-milling operation. (○ stable cut, □ quasi-periodic chatter, Δ double periodic chatter, and ◊ border cutting parameters). ... 59

Figure 3.7. Continuous wavelet transforms (a,b,c), sampled acceleration signals (d,e,f), Poincaré diagrams (g,h,i), and power spectral density (j,k,l) for unstable cut A (a,d,g,j), stable cut B (b,e,h,k), and quasi-periodic unstable cut C (c,f,i,l). ... 62

Figure 4.1. How magnetorheological fluid works: (a) without magnetic field and (b) with the magnetic field. ... 64

Figure 4.2. Bingham Model of Controllable Fluid Damper. ... 68

Figure 4.3. Bouc-Wen Model of MR Damper. ... 68

Figure 4.4. Modified Bouc-Wen model of MR shock absorber. ... 69

Figure 4.5. (a)Force vs Time, (b) Force vs Velocity and (c) Force vs Time. ... 71

Figure 5.1. (a) Optical image of the experimental setup and magnetorheological (MR) damper installation; (b) Schematic of cross-section A–A, the assembling of capsule–MR fluid is placed above the electromagnet. ... 75

Figure 5.2. MR damper device magnetic flux density distribution. ... 76

Figure 5.3. Measured Frequency Response Function (FRF) of the thin floor with the MR damper. ... 76

Figure 5.4. Stability lobes diagrams for three scenarios of magnetic flux density applied to the MR damper device together with experimental data in a down-milling operation. (○ stable cut, □ quasi-periodic chatter, Δ double periodic chatter, and ◊ border cutting. ... 78

Figure 5.5. Analysis of the collected accelerometer signals (D, E and F) for different MR damper magnetic flux density values. Continuous wavelet transforms (a,b,c). Sampled acceleration signals (d,e,f). ... 79

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List of Tables

Table 1.1. List of most productive authors on topics related to machining chatter. . 9

Table 2.1. Characterization test parameters. ... 31

Table 2.2. Cutting coefficients of the tool. ... 31

Table 3.1. Measured modal parameters of the workpiece. ... 58

Table 4.1. Parameters for the modified Bouc-Wen model. ... 70

Table 4.2. Typical Properties MRF-122EG [122]. ... 73

Table 5.1. Modal parameters of the cantilever beam under the action of a semi-active MR damper device. ... 77

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IX

List of Symbols

𝑎_𝑝 Axial depth of cut [mm]

𝑎𝑒 Radial Depth of cut [mm]

𝑨(𝑡) Direction Dynamic milling force coefficient

𝛽 Helix angles

𝐷 Diameter of the tool [mm]

𝑫 Differentiation matrix

Δ𝑥, Δ𝑦, Δ𝑧 Displacement difference vector in x, y y z

𝚫 Difference between present and previous periods

𝐹𝑡, 𝐹𝑟, 𝐹𝑎 Force in directions tangential, radial and axial (N)

𝑓_𝑧 Feed rate per tooth [mm/rev-teeth]

ℋ Homotopy of a function ℎ Chip thickness (mm)

𝑗 Tooth counter in cut

𝐾_𝑡, 𝐾_𝑟, 𝐾_𝑎 Force coefficients in directions tangential, radial and axial (N) 𝑘_𝑧 Stiffness in direction z (N/m)

𝜅(𝑧) Cutting edge angle.

𝐿, 𝑁 Linear and non-linear operator 𝑚_𝑧 Modal mass in direction z (kg.)

𝑛 Spindle speed (rev/min) 𝑝 Auxiliary parameter 𝚽 Transition matrix 𝜙_𝑠𝑡 Start angle

𝜙_𝑒𝑥 Exit angle

𝜙_𝑗 Instantaneous tooth inmmersion angle 𝜙_𝑝 Cutter pinch angle

𝑷_𝒊 , 𝑸_𝒊 , 𝑹_𝒊 System matrix in state space 𝑅 Nose radius [mm]

𝑇 Cutting period 𝜏 Time delay

𝜔_𝑛 Natural frequency [Hz]

𝜁 Damping

𝑁_𝑧 Tool teeth

𝑑𝐹_𝑡, 𝑑𝐹_𝑟, 𝑑𝐹_𝑎 Tangential, radial and axial differential force components respectively

𝐾_𝑐, 𝐾_𝑒 Coefficient of shear and friction ℎ(𝜙, 𝜅) Dynamic chip thickness

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Chapter 1

1.1. Introduction

The development of technology allows society to create research and access various topics of interest in seconds. Search engines' uses represent a fundamental tool for socializing knowledge, and it is allowed to appreciate the global trends in multiple themes. Today, it is possible to identify a topic's relevance by the number of Websites where it can find information about it.

An example is a search for the word COVID a term that became popular due to the global health contingency, which as of the date of this work, yielded 6,390,000,000 results. Another example is the term World Cup which shows 2,700,000,000 results. In the same way, we can appreciate scientific advances and the evolution of language. The term nanotechnology has about 35,900,000 and milling about 108,000,000.

About the terms used in machining, such as High-speed milling, were found 49,100,000 results. Besides a specific concept in machining regarding its dynamics, the search for High-speed milling dynamics has about 3,140,000. The term thin wall milling shows 7,820,000 sites. Moreover, the search for thin-floor milling returns 6,050,000. Based on the above information, it is possible to appreciate that the number of sites where to find information decreased when the topic sought is more specific to a given field of study and, therefore, less addressed.

The above considerations provide the reader with a reference on how search engines will appreciate terms values are current trends. It is important to note that the importance and quality of the information obtained on each site are variable.

These were allowed to establish the dynamic machining were addressed in about three and a half million sites. Which indicates the interest generated in the scientific community by developing work on this issue. However, it also poses the challenge of continuing progress in this area, as there is still much to improve. Thus, it could

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be related what relationship exists between a country and manufacturing specifically in machining dynamics.

For instance, the manufacture of parts represents a fundamental part for the sale and purchase of various goods worldwide, which have a direct relationship with the Gross Domestic Product (GDP) of each country, as can be seen in the Figure 1.1 and Figure 1.2 source World Bank.

Figure 1.1. Total income from the gross domestic product.

Figure 1.2. Percentage of manufacturing revenue to gross domestic product.

The countries with the highest per capita income are obtained thanks to manufacturing processes and that invest in this field's development. As proof of this assertion, we have obtained the data from these countries that have the highest GDP growth and which is the percentage that comes from manufacturing and this

19800 1985 1990 1995 2000 2005 2010 2015 2019

0.5 1 1.5 2 2.5x 10¹³

Year

$

United States China Japan Germany India United Kingdom France Italy Brazil Australia Canada South Korea Spain Mexico

19800 1985 1990 1995 2000 2005 2010 2015 2019

0.5 1 1.5 2

x 10¹³

Year

$

1980 1985 1990 1995 2020 2005 2010 2014 2018

50 60 70 80 90 100

Year

% of manufacturing to GDP

Germany Australia Brazil Canada China South Korea Spain United States France India Italy Japan Mexico United Kingdom

1980 1985 1990 1995 2020 2005 2010 2014 2018

50 60 70 80 90 100

Year

% of manufacturing to GDP

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3

represents at least 40% of their income, as verified in the Figure 1.2b source World Banck. In conclusion, the country's wealth is influenced by the direct relationship between the percentage of GDP devoted to the manufacture of goods and the scientific development of manufacturing processes.

In the manufacturing industry, the machining process by a cutting tool and relative movements between the workpieces removes material to give the desired shape. Machining originated mainly oriented to the manufacture of metallic components such as steel alloys, aluminum, cast iron, among others. However, this process is even used to manufacture non-metallic materials such as composite materials (polymers reinforced with carbon fibers ) [1] with great application and boom in the aeronautical industry, and at the other end bio-compatible materials such as titanium alloys and even bones [2] for medical applications. Consequently, The subject of machining is highly researched in various areas, from what we can see in the Figure 1.3 the number of publications made since 1980, source web of science, we can relate that the countries that have the most significant number of scientific publications in the field of specific manufacturing in the chatter coincide with the countries with the highest GDP see Figure 1.1. Also, the countries with the highest percentage of revenue destined for their GDP in the area of manufacturing, as seen in Figure 1.2.

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a)

b)

Figure 1.3. Scientific articles on Chatter by country:(a) all countries and (b) zoom excluding China and the United States.

Investment in machine tools remains constant in stages despite in stages of the recession. Although, it continues to grow year after year, with China, Germany, Japan, and the United States being the leading countries in the field. From a business perspective, the focus is naturally to produce the parts accurately within the time limits and maximum productivity conditions. However, certain factors can influence the ability to do so. Some of the most important that contribute to the efficiency of the process, according to are [3]:

• Assembly and disassembly of the workpiece in the machine tool.

• Clamping systems, including clamping of the part in the machine tool.

19800 1985 1990 1995 2000 2005 2010 2014 2019 500

1000 1500 2000

Year

Scientific papers

Germany Australia Canada South Korea Spain United States France India Italy Japan Mexico United Kingdom Turkey Iran China

19800 1985 1990 1995 2000 2005 2010 2014 2019 50

100 150 200 250

Year

Scientific papers

Germany Australia Canada South Korea Spain United States France India Italy Japan Mexico United Kingdom Turkey Iran China

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• Machining parameters such as spindle speed, depth of cut, and feed rate.

• Strategies and trajectories.

• Selection of tools and tool holder.

• Wear of tools.

• Change of tools.

• Refrigerant management.

• Chip evacuation.

• Tool and workpiece vibration, including self-excited vibrations and errors due to shear forces.

• Measurement of parts (on the machine or in a separate process)

• Accuracy of the machine tool, including geometric errors in its construction,

• thermally induced errors by sources associated with the cutting process and errors in the cutting path caused by structural dynamics and control.

All of the above addressed justifies the relevance and motivation to model and analyze the static and dynamic deformations that arise in the interaction of the tool- machine and the workpiece during the evolution of the cut [4]. For that reason, the challenge that arises for The solution of Differential Equations with Delay (DDE for its acronym in English of Delay Differential Equations) is which model the dynamics of the process. Delayed Differential Equations have been studied for at least 200 years [5], since many other phenomena occur in different fields of science and engineering in which not only the value at time t but also the response in a past or previous state, in other words, you have a dependency on past behavior determined by a delay value. Unfortunately, solving a DDE is a complicated mathematical task that has been limited in the past, but in the past decade, with accelerated computational advancement, interest in DDEs has rekindled and new areas of opportunity have emerged such as lasers subject to optical feedback, delay control

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for container cranes, or real-time synthesis of musical instruments [6]. One of the classic examples for delay systems is the prey-predator model proposed by Lotka and Volterra [7], where the growth rate of a particular species not only depends on the amount of food present but also on the previous amount ( in the gestation period).

The variables that model these processes have variations. They are observed with variations over time. These oscillations can be classified as stable or unstable.

Unstable oscillations (unbounded movements that evolve over time) are associated with a system described by DDEs. The motivation for studying these oscillations depends on the application, while for some, the unstable oscillations are seen as the desired behavior, such as the design of high-frequency optical oscillators, for others it is a limitation to the performance of a process such as the machining process [6].

The autonomous delay systems have an important development [8], the behavior of systems with periodic delay is not predictable, even for simple linear cases.

However, the stability analysis of these systems is a crucial problem. According to the well-known theory of Ordinary Differential Equations (ODE), the stability problem is determined by the characteristic equation's roots: if the real part of all the roots is negative or is on the left side of the complex plane, the system will be asymptotically stable. Nevertheless, for the analysis of DDEs, the approach is different.

One of the most critical applications of DDEs are in mechanical engineering. It is precisely the analysis of the dynamics of the chip removal process since the delay effect is of utmost importance; the so-called regenerative chatter effect is caused by the repercussion of the previous cut's vibration over the current one. This phenomenon has become the commonly accepted explanation for vibration problems in machining [9].

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Since its inception, machining has had the chatter as one of the main problems that hinder the increase in productivity and precision. In machining, chatter is perceived as excessive vibration between the tool and the workpiece, which results in a low-quality surface finish and accelerated tool wear. It even has a negative impact on the machine's life, which affects the safety and reliability of the process.

For this reason, in recent decades, with the development of high-speed machining, a line of research has focused on offline prediction of process stability. This technique is based on the use of design maps for the optimal selection of cutting parameters, commonly called stability lobes, which draw the boundary that separates stable oscillations from unstable ones in the face of different combinations of technological parameters of the process. Knowing these diagrams, industries can easily choose the values of the parameters or cutting conditions under which the process can be performed without the appearance of chatter vibrations, which cause unwanted effects that increase the waste of materials, consumables, and energy.

Therefore, the development of an adequate methodology capable of determining the delay equation's stability properties is used to model the process remains a challenge.

The first attempt to describe the chatter was made by Arnold (1946), Hann (1953) and Doi & Kato (1956). However, the first work defining the regenerative effect in the mathematical model detailed by Tlusty & Polacek (1957) and defining the stability lobes was carried out by Tobias & Fishwick in (1958).

Since tool forces, for example, in turning, depend on chip thickness, and chip thickness depends on the difference between the current position and a relatively previous revolution between the tool and the part, the dynamic model contains terms time delay. This formulation appeared in the work of Tobias [9] and Koeningber & Tlusty [10], who pioneered stability analysis for regenerative chatter

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in the 1960’s, among the first works explaining chatter with control effects and giving guidelines design of Merrit (1965) and one of the full milling models that can work with interrupted cutting was made by Schridar & Long (1968). Also, note that Sisson and Kegg (1969) studied the effect of low revolutions on the self-chatter.

After, Gabor Stépán is one of the pioneers in developing mathematics concerning chatter [11] and the inclusion of non-linearities. Likewise, Nayfeh in [12] introduced the use of perturbation methods to analyze the non-linear stability of regenerative chatter, with experimental validation in the work of Pratt [13] applied to the drilling process, among others, that of Minis & Yanushevski [13] with the resolution of the multifrequency method under the Hill's being method of the predecessors of Altintas & Budak (1995) [14]. The variables and applications for vibration suppression are extensive and diverse. An extensive and detailed review of various factors involved in the analysis was done by Davies (2000) where is it possible knowledge of the double period chatter. More recently, Insperger in [15] publishes the condensate of his research work on the method, explaining the development of the Semi-Discretization method that is widely cited and used for the stability analysis of dynamic systems with delays excited parametrically. A methodology that has been the basis for developing the stability analysis of modeling of machining processes such as turning and milling for constant and variable spindle speed.

The statistics of the publications in English from 1980 to the present obtained from related to the term chatter in the title, abstract, or keywords in the web of science platform, but limited to the field of machining. It can be noted that year after year the publications on this subject continue to increase Figure 1.4. Undoubtedly, this is due to the rise of new technologies that have promoted the increase in the machines' speed and the software's computing capacity. The trends of

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environmental legislation in several countries, precision and ultra-precision machining due to the marked generalized trend of miniaturization and, above all, the desire to establish a scientific basis for the last decade's machining processes has driven scientific and the consolidation of different research groups.

Figure 1.4. Chatter publications on machining from 1980 to the present.

One of the research groups with the most significant consolidation is the one led by Professor Altintas. Collaboration between different researchers has also been a catalyst that promotes research and supports new technologies. According to the web of science platform, the 20 authors who have reported the highest number of contributions to the state of the art are listed in Table 1.1.

Table 1.1. List of most productive authors on topics related to machining chatter.

Altintas, Y. (54) Ding, H. (26) Dombovari, Z. (22) Liu, Q. (20) Wan, M. (19) Budak, E. (35) Sims, N.D. (24) Campatelli, G. (21) Munoa, J. (20) Zhang, X. (19) Insperger,T. (30) Ismail, F. (23) Scippa, A. (21) Tlusty, J. (20) Shamoto, E. (18)

Stepan, G. (30) Song, Q (23) Grossi, N. (20) Ai, X. (19) Smith, S. (18)

These authors, in others, mainly concentrate their publications in the International Journal of Advanced Manufacturing Technology. Part of a list is:

19800 1985 1990 1995 2000 2005 2010 2015 2019 200

400 600 800

Year of publication

Number of publications

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• International Journal of Advanced Manufacturing Technology (313)

• International Journal of Machine Tools and Manufacture (244)

• Journal of Manufacturing Science and Engineering Transactions of the ASME (208)

• Advanced Materials Research (131)

The data described above makes clear the perspective and context from which this work was developed. Indeed, Motivations, problem statement and context, hypothesis, objectives, solution overview and dissertation organizations are described in the following sections.

1.2. Motivation

The manufacturing of parts such as impellers, ribs, blisks, and turbine blades, among other components for the aerospace industry, requires machining thin-wall and thin-floor features [16]. Titanium, aluminum alloys, and superalloys such as Inconel are used for these applications because of their excellent corrosion resistance, lightweight, and mechanical properties. Most of these components are machined as monolithic parts to improve their performance and their weight- resistant ratio. For example, In the development of monolithic parts, many components that are assembly changed to be manufactured by one part made by high-speed machining as can be seen in the Figure 1.5a. Also, these pieces have an application in the aerospace area where they have minimum thicknesses that must be machined as we can see in the Figure 1.5b where it was machined a piece of aluminum satellite from a block of 2,276 Kg to 32 Kilograms, with wall thicknesses of 0.38 mm and 0.5 mm floor [17].However, these components lightweight design

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requires thin walls and thin floors to achieve an excellent buy-to-fly ratio [18].

Additionally, the rise in monolithic parts use poses the challenge for manufacturers of the aluminum component to optimize machining processes to reduce production costs by defining a process free of problems, which meets cutting conditions that guarantee maximum productivity. However, the low stiffness at specific part locations makes machining prone to unstable regenerative vibrations (chatter), which is detrimental during finishing machining operations, causing adverse effects on surface quality and tool life.In addition, the chip load and cutting forces are kept in conservative values, resulting in decreasing productivity to avoid vibration.

Therefore, the machining of monolithic parts represents a tremendous problem.

a) b)

Figure 1.5. Monolithic parts: (a) Replacement for a monolithic part; (b) Satellite part machined from aluminum.

1.3. Problem statement and context

In machining monolithic parts in aluminum to be efficient and effective, it is necessary to use a cutting speed that is as high as possible, with rotational speeds of up to 40,000 rpm, and a high feed rate, with advances per edge of the order of 0.2 mm in roughing and 0.05 mm in finishing. The very thin thicknesses that are demanded with the interaction of the workpiece-tool make the presence of static,

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dynamic and residual stress problems that appear. Additionally, the error due to static bending of the part or tool translates into final geometries different from those intended, usually appearing an excess of material or walls and floors of variable thickness. On the other hand, if the vibration forced by the striking of the edges is excessive, marks may appear on the piece and dimensional errors as can be seen in the Figure 1.6 during the finishing of thin walls and floors in monolithic pieces since it is thicknesses of 1 mm or less [17].

Figure 1.6. Marks on a thin floor and wall of the monolithic piece.

On the other hand, for these thicknesses, the use of bull-nose mill tools is due to its geometry and its edge [19–20] because the projection of its forces along its profile is not the same as in a front tool as you can see in Figure 1.7ab [21]. However, there are very few publications on the use of this tool, so it is a tool with a complex geometry. Also, due to the load that the monolithic pieces, it is appropriate that they do not have finishes with a sharp edge of 90 degrees since a stress concentrator as can be seen in the Figure 1.7cd [22].

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a) b)

c) d)

Figure 1.7. Comparison of cutting with a square and round insert and stress diagram(a) square (b) round (c) 90 ° sharp-edged (d) radius width.

For machining monolithic the literature offers in-line solutions from fixing perspectives [23], workpiece holders [24], stiffening devices [25], and active materials [26] to reduce the vibration during the processes of machining. In fact, the use of these materials, like piezoelectrics, magnetostrictive, shape memory alloys, magnetorheological (MR), and electrorheological (RT) assist in machining process.

The piezoelectrics have been mounted on the cutting tool and acted as a tuned vibration absorber to suppress chatter. One of these is the application in boring an active system reported by Tanaka et al [22], among the most referenced with this intelligent material, we have that of Tarng et al [23] the application in turning.

The Magnetostrictive materials are used in a controlled magnetic field, it induces giant magnetostrictive movements, these movements are used to reduce and/or alter unwanted vibrational movements. Among the oldest publications, we find that of

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Hiller et al (1980) [27], which consisted in verifying the transmissibility between a magnetostrictive material and the workpiece. Similarly, it is worth highlighting the application to eliminate the chatter appeared with Rojas et al in (1996) [28] with the experimental investigation of the active control of vibrations of machine tools.

In the shape memory alloys there is the Dynamic Response of a pseudoelastic in a beam to a moving load carried out by Jafari et al [29].

The magnetorheological (MR), and electrorheological (RT) materials, has been implemented in actuators and sensor devices in manufacturing applications.These are active materials, also called smart materials, the MR and RT materials whose rheological behavior can be controlled externally through a magnetic and electrical field are mainly used as damper and spring devices. Among theses damping systems, the are the passive, semi-active and active. First, the passive systema is wich do not have an external interference in the damping system, one of the first to appear was Hanh in (1951).Second,the semi-active can change their behavior based on some stimulus, among the first to appear was that of Slavieck in (1965). Finally, the active is wich have sensors and actuators to be able to modify their operation according to the needs. One of the first contributions was that of Cowley & Boyle in (1970) [30], which was to eliminate the chatter in machine tools.

The MR fluids have a vast range of applications in different passive, semi-active, and active automotive and civil engineering areas. For instance, the automotive industry has developed semi-active damper devices for automotive suspension, together with the control of the systems [31]. In civil engineering, MR damper devices are used to change buildings' structural behavior during earthquakes [32].

Also, the use of MR fluids was extended in several engineering areas, such as the

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research of vibrations of pipelines reported by Hui et al. [33]. Concisely, MR technology supports many applications in different industrial contexts.

In machining, it is a common practice to instrument MR and RT materials in the clamping systems so that workpiece properties can be modified by adjusting their damping and stiffness [34]. The MR and RT fluids can work at a wide frequency range; their response occurs in a short time and offers flexibility to be adapted to different machining processes. Recently, in [35], Fleischer et al. studied some industrial applications of this technology; however, the use of these smart materials in machining processes has evolved because of their feasibility of being implemented for scale-up production.

Segalman and Redmond [36] used ER fluids for chatter suppression; they applied a cyclic electrical field to the spindle quill, changing the spindle's natural frequency, which disrupts the modulation of tool vibrations. They found significant reductions in vibration amplitudes through simulation of the milling process. Wang and Fei [37] used an ER fluid-filled sleeve to increase the boring bar's stiffness and damping. They concluded that the applied electric field strength should be adjusted on-line according to the detected vibration signals. Mei et al. [38] used MR fluids to mitigate chatter on a boring bar by adjusting the bar stiffness via a magnetic field intensity.

Similarly, Çeşmesi and Engin [39] developed and implemented an MR damper device in a conventional impact machine. Their mathematical model was also able to predict the damper device's behavior by controlling the magnetic field. According to Som et al. [40], the semi-active MR actuator's attenuation rate in a boring process can be up to 30% in wide frequency ranges of 400 to 600 Hz. Furthermore, MR devices can substantially improve surface finish and reducing tool wear in hard-

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turning operations [41]. The MR damper device's design can be assisted by Finite Element Method simulation because it can identify the magnetic field magnitudes needed when located in a tool-holder [42]. It is interesting to point out that most implementations of MR damper devices in machining operations were mainly oriented to boring or turning operations. This could be explained since the MR damper devices could be easily attached to the lathe machines' tool-holder.

However, its implementation in milling operations is very infrequent, but there have been a few attempts to apply MR technology to increase the workpiece's damping.

For instance, Ma et al. [43] demonstrated stability improvement in the thin-wall milling process.

The most common off-line approach is focused on the calculation of chatter-free cutting parameters based on the study of dynamics stability [44]. Special milling tools such as spherical or bull-nose tools are preferred for thin-floor machining in order to obtain a particular curvature between floors and walls; however, the geometry of the cutting edge has a strong influence over the forces in the tool axis direction, which typically coincides with the direction of the low-dynamic stiffness of thin floors. Since thin-floor dynamics are inherently complex due to the modal parameter variation that depends on the tool-workpiece location, three-dimensional stability diagrams for bull-nose milling tool cutting operations were calculated to identify the location at which the spindle speed of the cutting tool is a chatter-free tuning interval value [19]. Of course, the accuracy of the stability zones' location depends on predicting the magnitude values of cutting forces. In this sense, Opitz and Bernardi did a great investigation on directional coefficients 3FD [45] and Budak et al. [46] developed a method to predict the cutting force of any cutter geometry from orthogonal cutting data. The accuracy of the predicted forces was assessed with collected experimental data. The same method was also applied to ball-end

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milling tools [47]. By fitting pressure and friction coefficients on the rake and flank contact surfaces of ball cutters, Yucesan et al. [48] predicted the acting cutting forces in Cartesian directions with reasonable accuracy.

Altintas [49] described the chip regeneration mechanism in three directions with a bull nose. The stability lobes were predicted in the frequency domain and verified experimentally. Other methods to predict stability are based on time-domain discretization. For instance, Olvera et al. [50] proposed the Enhanced Multistage Homotopy Perturbation Method to compute stability lobes in one degree of freedom mimicking thin wall machining. Later, Urbikain et al. [51] proposed stability colormaps based on time-domain simulations of peripheral milling in thin-walled parts with barrel cutters. Regarding tool path planning in thin floors, Smith and Dvorak [52] proposed a tool path strategy that provided extra stiffness by leftover material in the corners; however, this strategy is limited for large areas consisting of mostly thin floors.

The implementation of machine operations of thin-floor components needs to consider two aspects. Firstly, high accuracy is required to determine the modal parameters and the associated acting cutting forces for effective and reliable stability prediction. Secondly, it is vital to acknowledge the shortcomings of active or semi- active damping techniques, including excessive setup times and customized designs that are only suitable for specific machine processes. MR fluids still are not mature in cutting operations and require more investigation if one wants to enhance its industrial application.

Therefore, this thesis aims to investigate how a semi-active MR damping device can influence the dynamics of cutting operations in thin-floor components. First, a

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stability prediction model is developed, and its accuracy is assessed by considering the milling operations of a thin-floor component with bull-end mills. Second, a semi- active damping device based on MR fluid is designed for improving chatter-free cutting conditions to increase productivity. In this sense, the proposed damper device is studied experimentally to improve the stability lobes for thin-floor machining operations. In conclusion, this work aims to solve the problem of establishing a solution proposal to mitigate vibration problems in machining components with thin floors. A stability prediction model is proposed to apply in the milling process of thin floor to obtain optimal cutting conditions.

1.4. Hypothesis

The use of a semi-active magnetorheological damper reduces self-excited vibration and increases the axial depth of cut during thin-floor milling processes with a bull-mose end mill by varying its magnetic field to improve cutting parameters without vibration, obtaining the new stability lobes for different operating conditions.

1.5. Objectives

The objectives set out in this thesis are:

• Develop a force model for a bull-nose mill using axial discretization.

• Experimental validation of the force model for a bull-nose mill.

• Apply and experimentally verify the stability model for thin-floor using the improved homotopy method by sub-intervals.

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• Design, build and inspect the performance of a semi-active magnetorheological damper for thin-floor.

1.6. Solution overview

The use of MR fluids is not yet fully developed and should be further investigated to improve its generalizability in industrial practice. To deepen both aspects, this work proposes a double complementary approach. First, an accurate stability prediction model is developed and applied to the milling thin-floor with a bull-nose end mill. Second, the actual application of a semi-active damping device based on an MR damper is proposed to improve cutting parameters without vibration, thus increasing productivity. The study closes both approaches by characterizing the damping device and obtaining the new stability lobes for different operating conditions.

1.7. Dissertation Organization

The dissertation organization of this doctoral thesis developed here has focused on the machining of aluminum components from thin-floor with the application of a semi-active MR damper with a bull-nose end mill. Furthermore, the developed methods are also applied to the engine (casings, blades, turbochargers).Therefore, this document is structured as follows.

• Chapter 2 presents the model of cutting forces for a Bull-Nose End Mills at variable cutting coefficients.

• Chapter 3 describes the method used to calculate the stability analysis in thin floor machining by solution (MPHM) and experimental validation.

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• Chapter 4 illustrates how it works the fluid Magnetherological.

• Chapter 5 reports the development and characterization of an MR buffer to increase stability lobes and productivity.

• Finally, Chapter 6 shows Conclusions and Future Work

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Chapter 2

2.1. Mechanistic models of the machining process

The interest in creating a model that adequately explains the phenomena and contributes to predicting the three fundamental aspects of the material removal process, such as the shape of the chip, the existing forces, and the cutting temperature, was a constant during century XX. However, there is still no accepted model to describe the totality of this phenomenon.

2.2. Orthogonal cutting mechanics

Chapter I established, the regenerative chatter phenomenon was approached through the modeling of delayed differential equations. Is necessary to start with the orthogonal cutting model, to explain the material removal mechanism, in which the material is removed by the edge of the tool that is perpendicular to the direction of advance between the tool and the workpiece shown in Figure 2.1 . Most general cutting operations are performed in three dimensions and the modeling is geometrically complex.

Figure 2.1. Scheme of the orthogonal cutting process. Adapted from [53].

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This model, known as the Merchant Model, considers that the material shear occurs in a very narrow area called the shear plane. The force on the process material𝐹_𝑅 is broken down into 𝐹_𝑡 shear or tangential force, which has the same direction as the cutting speed and the feed force 𝐹𝑓 that is in the direction of the depth of cut 𝑎. The force 𝐹_𝑅 can be decomposed in the direction of the rake face 𝐹_𝑦 (friction force) and its normal 𝐹_𝑦𝑛or in the plane of shear (shear force) and its normal 𝐹_𝑛𝑠.

Although this model is not completely reliable, it allows making qualitative judgments and calculations, it is also possible to make an approximate calculation of the thermal fields [53]. However, research has been oriented towards numerical modeling through semi-empirical models, called mechanistic, also, was used the simulation of the process using finite elements. Technological development and the evolution towards more sophisticated models contributed to the simulation becoming a tool for managing the non-linearities involved.

2.3. Empirical or mechanistic methods

Orthogonal cutting presents limitations in practical cases of tools with corner radius, with chamfer, or geometries with chip breaker. Therefore, it is convenient to carry out a series of experiments to identify the parameters that relate the cutting forces of a specific geometry and the work material, to have a court model more attached to real conditions. It is common to define the cutting forces mechanistically as a function of the cutting parameters (feed per tooth (𝑓𝑧) and axial depth (𝑎𝑝)) and coefficients of shear and friction cuts for each of the tangential and feed directions.

𝐹_𝑡 = 𝐾_𝑡𝑐𝑓_𝑧𝑎_𝑝+ 𝐾_𝑡𝑒𝑎_𝑝 𝐹_𝑓 = 𝐾_𝑓𝑐𝑓_𝑧𝑎_𝑝+ 𝐾_𝑓𝑒𝑎_𝑝̦

( 2-1)

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The constants 𝐾_𝑡𝑐 and 𝐾_𝑓𝑐 as well as the coefficients that do not contribute to shear 𝐾_𝑡𝑒, 𝐾_𝑓𝑒 are directly calibrated with experiments for a specific tool-material combination. It is clear that if the cutting edges wear out, the shear coefficients will also be affected.

Therefore, to take into account the effect of chip thickness on friction and shear angles, the specific shear pressure (𝐾_𝑡,𝐾_𝑓) is occasionally expressed as a non- linear function of chip thickness

𝐾_𝑡 = 𝐾_𝑇ℎ^−𝑝 𝐾_𝑓 = 𝐾_𝐹ℎ^−𝑞

( 2-2)

where 𝑝 and 𝑞 are constants that must be determined by experiments at different linear advances. It’s important to noted that metals exhibit different cutting coefficients at different speeds, so it is necessary to consider this phenomenon in some applications.

2.4. Mechanistic Model for Bull-Nose End Mills

The cutting forces in a flat end milling are dominant in the x− and y−

directions. However, for ball and bull-nose end milling, the cutting forces have significant components in all three Cartesian directions [49–54–55]. For the chatter analysis of thin floors using bull-nose end milling, the accuracy of the cutting force models is critical to predict reliable stability lobes. The semi-empirical models assume that the cutting forces and the undeformed chip section are linked through the cutting coefficients. The shear coefficient represents the amount of force required for a tool to remove an uncut chip, while the edge coefficient is related to the friction forces.

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A bull-nose end milling tool with a tool diameter D, helix angle β, and tooth Nz, is defined by corner radius R (also called the ball radius). The bull-nose end mill used was purchased from 3G tool local supplier, and the main geometry characteristics described earlier were measured with an Alicona microscope model InfiniteFocusG5 (Alicona Imaging GmbH, Pforzheim, Germany) together with a 2.5X objective. As illustrated in Figure 1, optical microscope photographs confirm a tool diameter of 16 mm and a nose radius of 2.5 mm.

Figure 2.2. Composition of optical microscope photographs of the bull-nose end mill taken with an Alicona microscope: (a) bottom view (diameter measurement); (b) frontal view (nose radius

measurement); (c) lateral view (helix angle measurement).

2.5. Cutting Force Model

A force model was developed to obtain an instantaneous cutting force magnitude for a given depth of cut. It is based on a mechanistic approach that assumes a relationship between forces and the uncut chip thickness through the cutting coefficients. Cutting models have been developed for ball-end mills that present similarities with a bull-nose end mill's geometry. For instance, Yucesan et al.

[48] proposed a model based on an analytic representation of ball-shaped helical flute geometry. Budak et al. reported the prediction of forces from the orthogonal cutting data [46], and for better detail, an oblique cut-off model presented by

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Altintas and Lee [56] was evaluated in sections. However, developing a generalized and accurate model for cutting coefficients for a bull-nose end mill has not been set.

Campa et al. [19] developed a model using averaged cutting coefficients in the toroid part, but they recognized that the tool's flank presented a different behavior.

2.6. Averaged cutting Coefficients

They assume an average value of the shear coefficients and the position angle for the edge of the Bull-Nose End Mills tool reported by Campa et al.[19–55]. It is the fastest and most straightforward, but it can also be imprecise. The approximation seems valid for depths of cut around the nose radius since it can be considered that the resulting edge geometry would be a small angle but would lose precision for depths minors. This method iterates with the depth of cut in 𝑎_𝑝. For each value of 𝑎_𝑝, we obtain some mean cutting coefficients and an average position angle of the cutting edge and the solutions close to the starting ap are taken as valid. These methods use a simple force model. Given its nature, the results obtained by this alternative are approximate and will be valid within certain margins.

Figure 2.3. Averaged cutting Coefficients taken from [19].

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2.7. Finite difference force model

Due to the geometry of the bull-nose end mills Figure 2.2, the decision has been made to develop a model in which the increase in the height variation can be had, thereby approaching that of the tool. The following Figure 2.4 shows the model developed.

Figure 2.4. Finite difference force model.

The model was developed by divide the depth ap into intervals. In each of these intervals, the result of the linear regression of the simulated forces was taken.

The shear force coefficients' values will be the slope, and the friction values will be the interception shown in Figure 2.5.

a) b)

Figure 2.5. Finite difference force model procedure: (a) Linear regression results for simulated forces;

(b) parameters regarding the study interval.

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2.8. Comparison between the average force model and the finite force model

A comparison was made between the average force model and the finite force model. It can be seen in figure Figure 2.6 for a depth of 𝑎_𝑝 = 0.5 mm and 𝑓_𝑧= 0.3 mm/tooth. The is an error in the two models because in the case of the average forces model as seen in section 2.6 it is considered with an axial depth 𝑎_𝑝 an approximation.

On the other hand, in the case of the finite force model section 2.7, there is a greater error since it is being taken by intervals and the projection of the forces at different depths is not being taken. For these reasons, a new force model has been carried out for a bull-nose mill.

a) b)

Figure 2.6. Comparison between and models with experiment data at 3000 rev/min and the cutting conditions:

a^p = 0.5 mm, f^z = 0.3 mm/tooth (a) Average cutting coefficients; (b) Finite difference model.

2.9. New cutting force model discretized in disks

The cutting force model is established by introducing two coefficients for the tangential, radial, and axial directions: one is associated with cutting (shearing) 𝐾_𝑐, which is directly related to the dynamic chip thickness ℎ(𝜙, 𝜅) and the second is associated with the friction (rubbing) coefficient 𝐾_𝑒. As shown in Figure 2.7, the

1.01 1.015 1.02 1.025 1.03 -200

-100 0 100 200 300

Time (s)

Force (N)

1.01 1.015 1.02 1.025 1.03 -200

-100 0 100 200 300

Time (s)

Force (N)

Fx Model Fy Model Fz Model Fx Exp Fy Exp.

Fz Exp.

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differential forces components 𝑑𝐹_𝑡, 𝑑𝐹_𝑟, and 𝑑𝐹_𝑎 are orientated for each differential cutting edge according to the cutting edge angle 𝜅(𝑧). The cutting edge of the tool is discretized in disks with a thickness of 𝑏 = 1 mm along the z-direction. With this discretized approach, it is possible to understand in detail the behavior of the cutting coefficients along the cutting edge in the toroid part in the Figure 2.7 illustrates the cutting parameters of thin-floor machining: axial depth of cut ap along the z- direction, radial immersion ae of cut in the y-direction, and spindle speed 𝑛.

Figure 2.7. Main cutting parameters in milling operation with a bull-nose end mill: axial discretization and differential cutting force components.

The mechanism of the bull-nose end mill is modeled using a mechanistic linear model. Here, the differential cutting forces on the local tool system follow the analytical approach in [4]:

𝑑𝐹𝑡(𝜙𝑗) = 𝐾𝑡𝑐ℎ(𝜙𝑗, 𝜅)𝑑𝑧 + 𝐾𝑡𝑒𝑑𝑧 𝑑𝐹_𝑟(𝜙_𝑗) = 𝐾_𝑟𝑐ℎ(𝜙_𝑗, 𝜅)𝑑𝑧 + 𝐾_𝑟𝑒𝑑𝑧 𝑑𝐹𝑎(𝜙𝑗) = 𝐾𝑎𝑐ℎ(𝜙𝑗, 𝜅)𝑑𝑧 + 𝐾𝑎𝑒𝑑𝑧

(2-3)

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where 𝑑𝐹_{𝑡,𝑟,𝑎} are the differential force components, 𝜙_𝑗(𝑡) is the angle of each tooth on the cutter, which change over time, and ℎ(𝜙_𝑗, 𝜅) is the instantaneous chip thickness as described by Compean et al. in [57]:

ℎ(𝜙_𝑗, 𝜅) = 𝑓_𝑧sin 𝜙_𝑗sin 𝜅 (2-4)

Here, 𝑓_𝑧 is the feed rate per tooth, and κ is the cutting-edge angle for a given axial slice. Through a rotation matrix, it is possible to transform the differential cutting forces to Cartesian coordinates

{

𝑑𝐹_𝑥(𝜙_𝑗, 𝑧) 𝑑𝐹_𝑦(𝜙_𝑗, 𝑧) 𝑑𝐹_𝑧(𝜙_𝑗, 𝑧)

} = [

𝑐𝑜𝑠 𝜙_𝑗 𝑠𝑖𝑛 𝜙_𝑗𝑠𝑖𝑛 𝜅 − 𝑠𝑖𝑛 𝜙_𝑗𝑐𝑜𝑠 𝜅 𝑠𝑖𝑛 𝜙_𝑗 − 𝑐𝑜𝑠 𝜙_𝑗𝑠𝑖𝑛 𝜅 𝑐𝑜𝑠 𝜙_𝑗𝑐𝑜𝑠 𝜅

0 − 𝑐𝑜𝑠 𝜅 − 𝑠𝑖𝑛 𝜅

] {

𝑑𝐹_𝑡(𝜙_𝑗, 𝑧) 𝑑𝐹_𝑟(𝜙_𝑗, 𝑧) 𝑑𝐹_𝑎(𝜙_𝑗, 𝑧)

} (2-5)

2.10. Characterization Procedure

The characterization procedure assumes the linear relationship between the averaged cutting forces 𝐹̃ and the feed rate f^z. This relationship is established as follows:

𝐹̃ = 𝑓_𝑧. 𝐹̃ + 𝐹_𝑐 ̃ _𝑒 (2-6)

Here, 𝐹̃ and 𝐹𝑐 ̃ are the cutting shear and edge components, respectively. 𝑒

Developing Equation (2-6) in terms of differential cutting forces, the following expression is obtained:

𝐹(𝜙) = ∫ 𝑑𝐹(𝜙, 𝑧)

𝑧₂ 𝑧1

= ∫ 𝑑𝐹_𝑐(𝜙, 𝑧)

𝑧₂ 𝑧1

. 𝑓_𝑧+ ∫ 𝑑𝐹_𝑒(𝜙, 𝑧)

𝑧₂ 𝑧1

(2-7)

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To determine the average cutting force per tooth, Equation (2-7) is integrated within the limits of the entrance (𝜙_𝑠) and exit (𝜙_𝑒) angles, and it is divided by the pitch angle 2𝜋 𝑁⁄ _𝑧:

𝐹̃ = 𝑁_𝑧

2𝜋∫ 𝐹𝑑𝜙

𝜙𝑒 𝜙𝑠

. (2-8)

Therefore, the Cartesian components of the average forces for the tool are:

𝐹̃_𝑥𝑐 =𝑁𝑡

2𝜋∫ ∫ (𝐾_𝑡𝑐sin 𝜙 cos 𝜙 sin 𝜅 + 𝐾_𝑟𝑐sin²𝜙 sin²𝜅 − 𝐾_𝑎𝑐sin²𝜙 sin 𝜅 cos 𝜅)

𝑧₂

𝑧₁

𝑑𝑧𝑑𝜙

𝜙_𝑒

𝜙_𝑠

𝐹̃_𝑦𝑐=𝑁_𝑡

2𝜋∫ ∫ (𝐾_𝑡𝑐sin²𝜙 sin 𝜅 − 𝐾_𝑟𝑐sin 𝜙 cos 𝜙 sin²𝜅 + 𝐾_𝑎𝑐sin 𝜙 cos 𝜙 sin 𝜅 cos 𝜅)

𝑧2

𝑧1

𝑑𝑧𝑑𝜙

𝜙𝑒

𝜙𝑠

𝐹̃_𝑧𝑐= 𝑁𝑡

2𝜋∫ ∫ (−𝐾_𝑟𝑐sin 𝜙 sin 𝜅 cos 𝜅 − 𝐾_𝑎𝑐sin²𝜙 sin 𝜅)

𝑧₂

𝑧₁

𝑑𝑧𝑑𝜙

𝜙_𝑒

𝜙_𝑠

(2-9)

The final step consists of computing shear and edge coefficients by solving the system of Equation (2-9)(2-7).

2.11. Experimental Procedure

A total of 120 cuttings were performed for 100% radial immersion in aluminum 7075T6 during dry machining. The forces were recorded using a dynamometer 9257B (Kistler Group, Winterthur, Switzerland) for the set of cutting conditions listed in Table 2.1. The spindle speed is set at 3000 rpm based on the dynamometer’s natural frequency to avoid milling forces' amplification. The force signals were acquired using a VibSoft-20 acquisition card (Polytec GmbH, Waldbronn, Germany) at a sample rate of 48 kHz and processed in a custom-made MATLAB app to remove drift and noise phenomena shown in Figure 2.8.

List of Figures

Dedication

Acknowledgements

List of Figures

List of Tables

List of Symbols

Contents

Chapter 1

Chapter 2