Abstract
In this paper, the cutting forces and machine-tool vibrations in ball-end mill process were modeled by using mathematical functions. All derivations of cutting forces are directly based on the tangential, radial, and axial cutting force components. The cutting force and vibration models can be formulated by a function of many parameters such as cutting force coefficients, cutter geometry, cutting conditions, and so on. The proposed cutting force model has been successfully verified by both simulation and experiment with very promising results. Besides, the machine-tool vibrations are also predicted for ball-end mill process.
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Abbreviations
- \( {\text{D}} \) :
-
The diameter of cutter [mm]
- \( {\text{R}}_{0} \) :
-
The diameter of cutter [mm]
- \( {\text{N}}_{\text{f}} \) :
-
The number of flutes on the cutter
- \( \upbeta \) :
-
The helix angle on the cutter [deg]
- \( \upphi_{\text{P}} \) :
-
The cutter pitch angle [deg]
- \( {\text{a }} \) :
-
The full axial depth of cut [mm]
- \( {\text{dz}} \) :
-
The differential axial depth of cut [mm]
- \( \Psi \left( {\upphi_{\text{j}} \left( {\text{z}} \right)} \right) \) :
-
The lag angle at an axial depth of cut z [deg]
- \( \Psi _{0} \) :
-
The lag angle at maximum axial depth of cut z = a [deg]
- \( \upphi_{\text{j}} \) :
-
The instantaneous immersion angle of flute number j, \( \left( {{\text{j }} = 1 \sim {\text{N}}_{\text{f}} } \right) \) [deg]
- \( \upphi_{\text{j}} \left( {\text{z}} \right) \) :
-
The instantaneous immersion angle of flute number j in z cutting depth, \( ({\text{j }}=1\sim {\text{N}}_{\text{f}} ) \) [deg]
- \( {\text{h}}_{\text{j}} \left( {\upphi_{\text{j}} \left( {\text{z}} \right)} \right) \) :
-
The instantaneous chip thickness at immersion angle \( \upphi_{\text{j}} \) [mm]
- \( {\text{db}} \) :
-
The chip width [mm]
- \( {\text{dS}}\left( {\upphi_{\text{j}} \left( {\text{z}} \right)} \right) \) :
-
The edge length of the cutting segment [mm]
- \( r\left( {\upphi_{\text{j}} \left( z \right)} \right) \) :
-
The radius of a circle on xy plane at an arbitrary point (P) on cutting edge [mm]
- \( {\text{f}}_{\text{t}} \) :
-
The feed per tooth [mm/tooth]
- \( \upkappa\left( {\text{z}} \right) \) :
-
The axial immersion angle at z axial depth of cut
- \( {\text{K}}_{\text{tc}} , {\text{K}}_{\text{rc}} , {\text{K}}_{\text{ac}} \) :
-
The shearing force coefficient [N/mm²]
- \( {\text{K}}_{\text{te}} , {\text{K}}_{\text{re}} , {\text{K}}_{\text{ae}} \) :
-
The edge force coefficient [N/mm]
- \( {\text{dF}}_{{{\text{i}},{\text{j}}}} \left( {\upphi,{\text{z}}} \right) \) :
-
The differential cutting force [N]
- \( {\text{F}}_{\text{f}} \left(\upphi \right), {\text{F}}_{\text{n}} \left(\upphi \right), {\text{F}}_{\text{a}} \left(\upphi \right) \) :
-
The cutting force in feed, normal, and axial direction [N]
- \( {\text{t}} \) :
-
The rotation time [Sec]
- \( \uptau \) :
-
The tool passing period [Sec]
- \( {\text{w}}_{\text{t}} \left( {\upphi_{\text{j}} } \right) \) :
-
The dynamic displacement of current flute [mm]
- \( {\text{w}}_{{\left( {{\text{t}} -\uptau} \right)}} (\upphi_{\text{j}} ) \) :
-
The dynamic displacement of previous flute [mm]
- \( {\text{x}}_{\text{t}} , {\text{y}}_{\text{t}} ,{\text{x}}_{{\left( {{\text{t}} -\uptau} \right)}} , {\text{y}}_{{\left( {{\text{t}} -\uptau} \right)}} \) :
-
The machine tool vibrations at time t and \( \left( {{\text{t}} -\uptau} \right) \) [mm]
- \( \left[ {{\text{m}}_{\text{x}} , {\text{m}}_{\text{y}} } \right] \) :
-
The mass matrix of machine-tool dynamic structure [kg]
- \( \left[ {{\text{c}}_{\text{x}} ,{\text{c}}_{\text{y}} } \right] \) :
-
The damping constant matrix of machine-tool dynamic structure
- \( \left[ {\upomega_{\text{x}} ,\upomega_{\text{y}} } \right] \) :
-
The natural frequency matrix of machine tool dynamic structure [Hz]
- \( \left[ {{\text{k}}_{\text{x}} ,{\text{k}}_{\text{y}} } \right] \) :
-
The stiffness matrix of machine-tool dynamic structure [N/m].
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Nguyen, NT., Kao, YC., Dung, H.T., Trung, D.D. (2020). A Prediction Method of Dynamic Cutting Forces and Machine-Tool Vibrations When Milling by Using Ball-End Mill Cutter. In: Sattler, KU., Nguyen, D., Vu, N., Tien Long, B., Puta, H. (eds) Advances in Engineering Research and Application. ICERA 2019. Lecture Notes in Networks and Systems, vol 104. Springer, Cham. https://doi.org/10.1007/978-3-030-37497-6_5
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DOI: https://doi.org/10.1007/978-3-030-37497-6_5
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