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PDE-based spiral machining trajectory planning method without tool feed marks on 2D arrayed multi-island regions

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Abstract

Components with complex arrays of multi-island structures are widely used in aerospace applications, but existing commonly used milling strategies usually have multiple feeds and retreats during machining, and the machined surfaces produce feed marks that do not meet the requirements for high-quality surfaces. In this paper, a machine path generation method for array multi-island machining regions is proposed, which divides the array multi-island region machining area into multiple double-connected regions from outside to inside, establishes a partial differential equation model in each double-connected machining subregion to compute isotherms, generates spiral machining trajectories in the double-connected subregions by identifying suitable isotherms in each double-connected machining subregion, and then connects each sub-region helix trajectory; the overall spiral machining trajectory in the array multi-island region can be obtained to avoid multiple tool feeds and feed marks on the machined surface. Meanwhile, in order to avoid the sudden change of cutting direction at the boundary of two sub-regions, an optimization algorithm for B-sample curve generation at the boundary connection of sub-regions is proposed. Finally, the superiority of the method in improving machining quality is verified by machining experiments.

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Abbreviations

\(Loo{p}_{1},Loo{p}_{2},Loo{p}_{3},\cdots \cdots ,Loo{p}_{n-1},Loo{p}_{n}\) :

The inner and outer rings of the arrayed multi-island region

\({\Omega }_{1},{\Omega }_{2},{\Omega }_{3},\cdots \cdots ,{\Omega }_{n-1},{\Omega }_{n}\) :

Machining sub-regions

\({C}_{1},{C}_{2},{C}_{3},\dots \dots ,{C}_{n-1},{C}_{n}\) :

The sub-region boundaries

\({S}_{\text{max}}\) :

The maximum machining step

\({\varvec{R}}\) :

The radius of virtual circular island boundary

\({\varvec{T}}\) :

The temperature

\({K}_{\text{x}},{K}_{\text{y}}\) :

The thermal conductivity of the medium

\(q\) :

The heat generated per unit area

\({u}_{h}\) :

Collection of triangular grid cells

\(\mathrm{1,2},\dots ,{n}_{1}\) :

The number of internal nodes

\({n}_{1}+1,{n}_{1}+2,\dots ,{n}_{1}+{n}_{2}\) :

The number of external boundary nodes

\({n}_{1}+{n}_{2}+1,{n}_{1}+{n}_{2}+2,\dots ,{n}_{1}+{n}_{2}+{n}_{3}\) :

The number of internal boundary nodes

\({S}_{h}\) :

The finite-dimensional space composed of continuous piecewise linear functions

\({P}_{i},{P}_{j},{P}_{m}\) :

Vertices of triangular cell

\(\nabla u\) :

The gradient vector of \({u}_{h}(x,y)\)

\({\phi }_{i}({x}_{j},{y}_{j})\) :

Basis functions

\({e}_{n}\) :

The nth cell

\({{\varvec{K}}}_{{e}_{n}}\) :

The unit stiffness matrix

\({{\varvec{F}}}_{{e}_{n}}\) :

The unit load vector

\({\varvec{K}}\) :

The total stiffness matrix

\({\varvec{F}}\) :

The total load vector

\({\text{S}}_{j}^{i}\) :

Spiral points

\({N}_{j,k}\) :

The B-spline basis functions

\(D\) :

Distance of the final control points

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Funding

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (No. U20B2033, 51975093), the Natural Science Foundation of Liaoning (No. 2020-YQ-09), and the Changjiang Scholar Program of Chinese Ministry of Education (No. Q2021053, T2017030).

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Authors and Affiliations

  1. Key Laboratory for Precision and Non-Traditional Machining Technology of Ministry of Education, Dalian University of Technology, Dalian, 116024, China

    Songjian Wang, Chengxin Wang, Pengfei Wang, Haibo Liu & Yongqing Wang

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  1. Songjian Wang
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  2. Chengxin Wang
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  3. Pengfei Wang
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  4. Haibo Liu
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  5. Yongqing Wang
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Contributions

Methodology, S.W. and C.W.; Investigation, P.W.; Writing original draft, S.W.; Software, H.L. and Y.W. all authors have read and agreed to the published version of the manuscript.

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Correspondence to Yongqing Wang.

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Wang, S., Wang, C., Wang, P. et al. PDE-based spiral machining trajectory planning method without tool feed marks on 2D arrayed multi-island regions. Int J Adv Manuf Technol 125, 2021–2034 (2023). https://doi.org/10.1007/s00170-022-10702-5

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