Abstract
In dry machining of high precision parts shape deviations mainly arise due to thermo-elastic expansion during cutting and machining induced residual stresses. In order to meet higher quality standards when applying the ecologically and economically favourable concept of dry machining, predictive compensation strategies have to be developed which take into account the aforementioned causes for shape deviations. This work presents a newly developed hybrid model integrated in an optimisation algorithm that allows to systematically identify such compensation strategies. Both mechanisms contributing to the shape deviations of machined workpieces are implemented as sub-models applying the Finite Element Method. Aiming at a minimisation of total shape deviations the Simultaneous Analysis and Design approach is utilised. The overall optimisation procedure shows to be time-efficient and is able to find milling strategies leading to a significant reduction of shape deviations. When considered individually, the hybrid model is able to predict each of the two mechanisms, deformation due to machining induced residual stresses and uneven material removal due to thermo-elastic expansion, very well. Experiments based on the predicted milling strategies lead to a substantial reduction of shape deviations and show that additional mechanisms should be implemented for a further process optimisation.
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Notes
- 1.
Also called black-box optimisation.
- 2.
Also known as PDE-constrained optimisation.
- 3.
Also known as Navier-Cauchy-equations or elastostatic equations.
- 4.
The objective value corresponds to the arithmetic mean of the squared deviations [μm²].
Abbreviations
- \( {\mathbf{A}} \) :
-
System matrix
- \( a \) :
-
Width of a premachined and normalised bar
- \( a_{e} \) :
-
Width of cut (\( {\text{mm}} \))
- \( {\mathbf{a}}_{{\mathbf{e}}} \) :
-
Vector of widths of cut
- \( a_{e,i} \) :
-
Width of cut of the \( i \)-th milling path (\( {\text{mm}} \))
- \( a_{i} \) :
-
Coefficient of the \( i \)-th regression model
- \( a_{p} \) :
-
Depth of cut (\( {\text{mm}} \))
- \( b \) :
-
Height of a premachined and normalised bar
- \( {\mathbf{b}} \) :
-
Right-hand side vector
- \( b_{i} \) :
-
Coefficient of the \( i \)-th regression model
- \( c \) :
-
Cost-term factor
- \( c_{i} \) :
-
Coefficient of the \( i \)-th regression model
- \( c_{p} \) :
-
Specific heat capacity (\( {\text{J/(kg}}\,{\text{K)}} \))
- \( d_{i} \) :
-
Coefficient of the \( i \)-th regression model
- \( E \) :
-
Modulus of elasticity (\( {\text{N/m}}^{ 2} \))
- \( e_{i} \) :
-
Coefficient of the \( i \)-th regression model
- \( F \) :
-
Objective function (\( {\text{K}}^{ 2} \) or \( {\upmu}{\text{Microsoft}}^{ 2} \))
- \( {\mathbf{G}} \) :
-
System of equations
- \( f_{i} \) :
-
Coefficient of the \( i \)-th regression model
- \( f_{z} \) :
-
Feed per tooth (mm)
- \( g_{i} \) :
-
Coefficient of the \( i \)-th regression model
- \( h \) :
-
Lever (mm)
- \( h_{i} \) :
-
Coefficient of the \( i \)-th regression model
- \( k \) :
-
Thermal conductivity (\( {\text{W/(m}}\,{\text{K)}} \))
- \( K \) :
-
Plate stiffness (Nmm)
- \( l \) :
-
Length of a premachined and normalised bar
- \( M \) :
-
Number of measured temperatures
- \( M_{x} \) :
-
Bending moment along x-axis
- \( M_{xy} ,\;M_{yx} \) :
-
Torsion moment in xy-plane
- \( M_{y} \) :
-
Bending moment along y-axis
- \( n \) :
-
Rotational speed or number of milling paths
- \( {\mathbf{n}} \) :
-
Normal vector
- \( N \) :
-
Size of \( {\mathbf{T}} \), \( {\mathbf{u}}_{source} \) or \( {\mathbf{u}}_{tot,z} \)
- \( \dot{q}_{wp} \) :
-
Heat flux into the workpiece (W/mm²)
- \( {\mathbf{S}} \) :
-
Vector of model states
- \( T \) :
-
Temperature (\( {\text{K}} \)) or plate thickness (\( {\text{mm}} \))
- \( {\mathbf{T}} \) :
-
Vector of simulated temperatures (\( {\text{K}} \))
- \( {\bar{\mathbf{T}}} \) :
-
Vector of measured temperatures (\( {\text{K}} \))
- \( \bar{T}_{i} \) :
-
\( i \)-th measured temperature (\( {\text{K}} \))
- \( \tilde{T}_{i} \) :
-
\( i \)-th simulated temperature corresponding to \( \bar{T}_{i} \) (\( {\text{K}} \))
- \( u \) :
-
Nodal displacement
- \( u_{source} \) :
-
Source stresses induced displacement vector field (\( {\upmu }{\text{m}}) \)
- \( {\mathbf{u}}_{source} \) :
-
Vector of discrete displacements induced by source stresses (\( {\upmu }{\text{m}}) \)
- \( u_{source,i} {\kern 1pt} \) :
-
\( i \)-th entry of \( {\mathbf{u}}_{source} \) (\( {\upmu }{\text{m}}) \)
- \( u_{{source,\hat{i}}} {\kern 1pt} \) :
-
Source stresses induced displacement in z-direction in the unsupported corner (\( {\upmu }{\text{m}}) \)
- \( u_{source,x} \) :
-
Source stresses induced displacement in x-direction (\( {\upmu }{\text{m}}) \)
- \( u_{source,y} \) :
-
Source stresses induced displacement in y-direction (\( {\upmu }{\text{m}}) \)
- \( u_{source,z} \) :
-
Source stresses induced displacement in z-direction (\( {\upmu }{\text{m}}) \)
- \( u_{source}^{\hbox{max} } \) :
-
\( \mathop {{ \hbox{max} }\,}\limits_{i = 1, \ldots ,N} |u_{source,i} | \)
- \( u_{therm} {\kern 1pt} \) :
-
Thermo-elastic displacement (\( {\upmu }{\text{m}}) \)
- \( u_{tot} {\kern 1pt} \) :
-
Total displacement (\( {\upmu }{\text{m}}) \)
- \( {\mathbf{u}}_{tot,z} \) :
-
Vector of discrete total displacements in z-direction (\( {\upmu }{\text{m}}) \)
- \( u_{tot,z,i} {\kern 1pt} \) :
-
\( i \)-th entry of \( {\mathbf{u}}_{tot,z} \) (\( {\upmu }{\text{m}}) \)
- \( u_{tot}^{\hbox{max} } \) :
-
\( \mathop { \hbox{max} }\limits_{i = 1, \ldots ,N} |u_{tot,z,i} | \)
- \( v_{c} \) :
-
Cutting velocity (\( 100\,{\text{m/min}} \) or \( {\text{mm/min}} \))
- \( {\mathbf{v}}_{{\mathbf{c}}} \) :
-
Vector of cutting velocities
- \( v_{c,i} \) :
-
Cutting velocity of the \( i \)-th milling path (\( {\text{m/mm}} \))
- \( v_{f} \) :
-
Feed velocity (\( {\text{m/min}} \) or \( {\text{mm/min}} \))
- \( {\mathbf{v}}_{{\mathbf{f}}} \) :
-
Vector of feed velocities
- \( v_{f,i} \) :
-
Feed velocity of the \( i \)-th milling path (\( {\text{mm/min}} \))
- \( w \) :
-
Displacement in z-direction
- \( {\mathbf{X}} \) :
-
Vector of SAND optimisation variables
- \( {\mathbf{X}}^{*} \) :
-
Vector of final SAND optimisation variables
- \( z_{0} \) :
-
Layer thickness of source stresses
- \( {\mathbf{Z}} \) :
-
Vector of NAND optimisation variables
- \( \alpha \) :
-
Rotation angle of the milling paths (\( {\text{rad}} \)) or coefficient of thermal expenasion (\( 1 / {\text{K}} \))
- \( {\Gamma } \) :
-
Workpiece surface
- \( \lambda \) :
-
Lamé’s first parameter (\( {\text{N/m}}^{ 2} \))
- \( \mu \) :
-
Lamé’s second parameter (\( {\text{N/m}}^{ 2} \))
- \( \nu \) :
-
Poisson’ ratio (\( { - } \))
- \( \rho \) :
-
Density (\( {\text{kg/m}}^{ 3} \))
- \( \sigma_{source.m} \) :
-
Source stress tensor field (\( {\text{N/mm}}^{ 2} \))
- \( {\mathbf{\upsigma }}_{source.m} \) :
-
Vector of discrete source stresses (\( {\text{N/mm}}^{ 2} \))
- \( {\hat{\boldsymbol{\upsigma }}}_{source.m} \) :
-
Fixed source stress tensor (\( {\text{N/mm}}^{ 2} \))
- \( \sigma_{source.m,x} \) :
-
Normal source stress in x-direction (\( {\text{N/mm}}^{ 2} \))
- \( \sigma_{source.m,y} \) :
-
Normal source stress in y-direction (\( {\text{N/mm}}^{ 2} \))
- \( \sigma_{source.wp,i} \) :
-
Workpiece inherent source stress tensor in layer \( i \) (\( {\text{N/mm}}^{ 2} \))
- \( \sigma_{source.wp,i,x} \) :
-
Workpiece inherent source stress in layer \( i \) in x-direction (\( {\text{N/mm}}^{ 2} \))
- \( \sigma_{source.wp,i,y} \) :
-
Workpiece inherent source stress in layer \( i \) in x-direction (\( {\text{N/mm}}^{ 2} \))
- \( \tau_{source.m,xy} \) :
-
Shear source stress in x-y-plane (\( {\text{N/mm}}^{ 2} \))
- \( \tau_{source.wp,i,xy} \) :
-
Workpiece inherent source stress in layer \( i \) in x-y-plane (\( {\text{N/mm}}^{ 2} \))
- \( {\Omega } \) :
-
Workpiece domain
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Acknowledgements
The results were obtained within the DFG priority programme 1480 “Modelling, Simulation and Compensation of Thermal Effects for Complex Machining Processes”. The authors kindly thank the Deutsche Forschungsgemeinschaft (DFG) for the financial support of the projects BR825/65–1, SO1236/1–2, SO1236/1–3, BU1289/6–1, BU1289/6–2 and BU1289/6–3.
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Gulpak, M., Wernsing, H., Sölter, J., Büskens, C. (2018). Compensation Strategies for Thermal Effects in Dry Milling. In: Biermann, D., Hollmann, F. (eds) Thermal Effects in Complex Machining Processes. Lecture Notes in Production Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-57120-1_12
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DOI: https://doi.org/10.1007/978-3-319-57120-1_12
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