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Dynamic analysis and surface morphology prediction for deep cavity turning of bent-blade cutter

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Abstract

Due to the complex integral structure design of the deep cavity parts, large overhang bent-blade cutter is used to machining such parts. Thus, the tools’ complex geometry structure, large overhang, and low stiffness, resulting in lateral, axial, and torsional vibration in the machining process, making the actual machining trajectory deviate from the theoretical value, seriously affect the processing quality, producing violent vibration. Therefore, this paper established a model to predict the surface topography of bent-blade cutter considering the combined vibration effect of bending and torsion, including the complicated process geometries, varying tool profiles, and multi-direction vibrations. The influencing factors of the turning surface topography are studied. Meanwhile, based on the cutting edge trajectory, the surface topography is simulated and analyzed by considering the static offset and dynamic vibration of the tool. Besides, based on the extended Timoshenko beam model (E-TBM) theory, the control equation of bent-blade cutter considering the combined effect of bending and torsion is derived and solved by finite element method (FEM). Finally, a series of turning experiments are carried out to verify the accuracy and efficiency of the presented model. The influence rule of turning parameters on surface roughness was studied by theoretical and experimental results, and the turning parameters were optimized to improve the machining quality and accuracy of deep cavity parts.

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Abbreviations

A :

Area of cross-section, mm2

B 2 :

Width of cutter board II, mm

C :

Equivalent damping

B 3 :

Length of cutter board III, mm

D :

Diameter of cutter board I, mm

G :

S hear modulus

H 1 :

Height of cutter board I and II, mm

H 2 :

Height of cutter board III, mm

I b :

Inertia moment of cutter bar, m4

I p :

Inertia moment of cutter board, m4

K :

Equivalent stiffness

K N :

Element stiffness matrix

M :

Equivalent mass

M N :

Element mass matrix

L 1 :

Total length of bent-blade cutter, mm

L 2 :

Length of cutter board II, mm

T :

Kinetic energy, J

T :

Transformation matrix

δ :

Engagement angle

U :

Strain energy, J

Ω:

Volume of cross-section, mm3

W :

Work done by external force and torque, J

J :

Polar moment of inertia, m4

a p :

Depth of cut, mm

b :

Width of cut, mm

f :

Feed rate, mm/r

h :

Thickness of cut, mm

n :

Spindle speed, rmp

r :

Radius of the circular insert, mm

w :

Transverse displacement, m

ϕ :

Twist angular displacement, rad

θ :

Rotation angular displacement, rad

M :

Torque, N/m

ψ :

Warping displacement, rad

n :

Spindle speed, rmp

v :

Poisson’s ratio

ρ :

Density, kg/m3

l :

Sampling length, mm

Ω:

Volume of cross-section, mm3

k * :

Ratio of the normal cutting force to the tangential cutting force

ε :

Angular location of the infinitesimal segment at the cutting edge

E-TBM:

Extended Timoshenko Beam Model

FEM:

Finite element method

EBM:

Euler-Bernoulli beam model

TBM:

Timoshenko Beam Model

SVT:

Saint Venant Theory

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Funding

The study was financially supported by the National Natural Science Foundation of China (grant numbers 52275445 and 51922066) and the Key Research and Development Plan of Shandong Province (grant numbers 2021JMRH0301 and 2020CXGC010204).

Author information

Authors and Affiliations

  1. Key Laboratory of High Efficiency and Clean Mechanical Manufacture, Ministry of Education, School of Mechanical Engineering, Shandong University, Jinan, People’s Republic of China

    Xiaojuan Wang & Qinghua Song

  2. National Demonstration Center for Experimental Mechanical Engineering Education, Shandong University, Jinan, People’s Republic of China

    Qinghua Song

Authors
  1. Xiaojuan Wang
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  2. Qinghua Song
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Contributions

X.W. and Q.S. contributed the central idea, analyzed most of the data, and wrote the initial draft of the paper. X.W. contributed to refining the ideas, carrying out additional analyses, and finalizing this paper.

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Correspondence to Qinghua Song.

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Appendices

Appendix 1

$$\Lambda =\left[\begin{array}{c}{k}_{s}GA\frac{{dv}_{1}}{dx}\left(\frac{\partial w}{\partial x}+\phi \right)+\rho {Av}_{1}\frac{{\partial }^{2}}{{\partial }^{2}t}-{v}_{1}F\\ EI\frac{{dv}_{2}}{dx}\frac{\partial \phi }{\partial x}-{v}_{2}{k}_{s}{AGv}_{2}\left(\frac{\partial w}{\partial x}+\phi \right)+\rho {Av}_{2}\frac{{\partial }^{2}\phi }{{\partial }^{2}t}\\ {k}_{s}{GI}_{{\varvec{p}}}\frac{{dv}_{3}}{dx}\frac{\partial \theta }{\partial x}+\rho {I}_{{\varvec{p}}}{v}_{3}\frac{{\partial }^{2}\theta }{{\partial }^{2}t}-{v}_{3}J\end{array}\right]$$
$$\begin{array}{c}{Q}_{1}^{e}=-{\left[{k}_{s}AG\left(\phi +\frac{dw}{dx}\right)\right]}_{x={x}_{a}}=-V\left({x}_{a}\right)\\ {Q}_{2}^{e}={\left(EI\frac{d\phi }{dx}\right)}_{x={x}_{a}}=-M\left({x}_{a}\right)\\ \begin{array}{c}{Q}_{3}^{e}={\left[{k}_{s}AG\left(\phi +\frac{dw}{dx}\right)\right]}_{x={x}_{b}}=V\left({x}_{b}\right)\\ {Q}_{4}^{e}={\left(EI\frac{d\phi }{dx}\right)}_{x={x}_{b}}=-M\left({x}_{b}\right)\\ \begin{array}{c}{Q}_{5}^{e}={\left({k}_{s}{GI}_{p}\frac{d\phi }{dx}\right)}_{x={x}_{a}}=T\left({x}_{a}\right)\\ {Q}_{6}^{e}={\left({k}_{s}{GI}_{p}\frac{d\phi }{dx}\right)}_{x={x}_{b}}=-T\left({x}_{b}\right)\end{array}\end{array}\end{array}$$

Appendix 2

$$\begin{array}{c}{K}_{ij}^{b11}={\int }_{{x}_{a}}^{{x}_{b}}{k}_{s}GA\frac{d{\varphi }_{i}^{\left(1\right)}}{dx}\frac{d{\varphi }_{j}^{\left(1\right)}}{dx}dx\\ {K}_{ij}^{b12}={\int }_{{x}_{a}}^{{x}_{b}}{k}_{s}GA\frac{d{\varphi }_{i}^{\left(1\right)}}{dx}{\varphi }_{j}^{\left(2\right)}dx={K}_{ij}^{b21}\\ \begin{array}{c}{K}_{ij}^{b22}={\int }_{{x}_{a}}^{{x}_{b}}\left(EL\frac{d{\varphi }_{i}^{\left(2\right)}}{dx}\frac{d{\varphi }_{j}^{\left(2\right)}}{dx}+{k}_{s}GA{\varphi }_{i}^{\left(1\right)}{\varphi }_{j}^{\left(2\right)}\right)dx\\ {K}_{ij}^{b33}={\int }_{{x}_{a}}^{{x}_{b}}{k}_{s}{GA}_{\mathrm{p}}\frac{d{\varphi }_{i}^{\left(3\right)}}{dx}\frac{d{\varphi }_{j}^{\left(3\right)}}{dx}dx\\ \begin{array}{c}{K}_{ij}^{b11}={\int }_{{x}_{a}}^{{x}_{b}}\rho A{\varphi }_{i}^{\left(1\right)}{\varphi }_{j}^{\left(1\right)}dx,{M}_{ij}^{b22}={\int }_{{x}_{a}}^{{x}_{b}}\rho I{\varphi }_{i}^{\left(2\right)}{\varphi }_{j}^{\left(2\right)}dx\\ {K}_{ij}^{b12}={M}_{ij}^{b21}=0,{M}_{ij}^{b33}={\int }_{{x}_{a}}^{{x}_{b}}\rho {I}_{\mathrm{p}}{\varphi }_{i}^{\left(3\right)}{\varphi }_{j}^{\left(3\right)}dx\\ {K}_{i}^{b1}={Q}_{2i-1},{F}_{i}^{b2}={Q}_{2i},{F}_{i}^{b3}={Q}_{4+i}\end{array}\end{array}\end{array}$$
$$\begin{array}{c}{K}_{ij}^{p11}={\int }_{xa}^{xb}{k}_{s}GA\frac{d{\varphi }_{i}^{\left(1\right)}}{dx}{k}_{s}GA\frac{d{\varphi }_{j}^{\left(1\right)}}{dx}dx\\ {K}_{ij}^{p12}={\int }_{xa}^{xb}{k}_{s}GA\frac{d{\varphi }_{i}^{\left(1\right)}}{dx}{\varphi }_{j}^{\left(2\right)}dx={K}_{ij}^{p21}\\ \begin{array}{c}{K}_{ij}^{p22}={\int }_{xa}^{xb}\left(EL\frac{d{\varphi }_{i}^{\left(2\right)}}{dx}\frac{d{\varphi }_{j}^{\left(2\right)}}{dx}+{k}_{s}GAd{\varphi }_{i}^{\left(1\right)}d{\varphi }_{j}^{\left(2\right)}\right)dx={K}_{ij}^{p33}=0\\ {K}_{ij}^{p11}={\int }_{xa}^{xb}\rho A{\varphi }_{i}^{\left(1\right)}{\varphi }_{j}^{\left(1\right)}dx,{M}_{ij}^{p22}={\int }_{xa}^{xb}\rho I{\varphi }_{i}^{\left(2\right)}{\varphi }_{j}^{\left(2\right)}dx\\ \begin{array}{c}{K}_{ij}^{p12}={M}_{ij}^{p21}={M}_{ij}^{p33}=0\\ {K}_{ij}^{p1}={\int }_{xa}^{xb}F{\varphi }_{i}^{\left(1\right)}+{Q}_{2i-1},{F}_{i}^{b2}={Q}_{2i},{F}_{i}^{b3}=0\end{array}\end{array}\end{array}$$

Appendix 3

$${\mathbf{F}}^{t\mathrm{^{\prime}}}={\left\{{{F}_{1}}^{t1}{{F}_{1}}^{t2}{{F}_{1}}^{t3}{{F}_{2}}^{t1}{{F}_{2}}^{t2}{{F}_{2}}^{t3}\right\}}^{T}$$
$${\mathbf{K}}^{t\mathrm{^{\prime}}}=\left[\begin{array}{cc}{K}_{11}^{t11}& \begin{array}{ccc}{K}_{12}^{t11}& 0& \begin{array}{ccc}{K}_{11}^{t12}& {K}_{12}^{t12}& 0\end{array}\end{array}\\ {K}_{21}^{t11}& \begin{array}{ccc}{K}_{22}^{t11}& 0& \begin{array}{ccc}{K}_{21}^{t12}& {K}_{22}^{t12}& 0\end{array}\end{array}\\ \begin{array}{c}0\\ \begin{array}{c}{K}_{11}^{t21}\\ \begin{array}{c}{K}_{21}^{t21}\\ 0\end{array}\end{array}\end{array}& \begin{array}{c}\begin{array}{ccc}0& {K}_{11}^{t33}& \begin{array}{ccc}0& 0& {K}_{12}^{t33}\end{array}\end{array}\\ \begin{array}{c}\begin{array}{ccc}{K}_{12}^{t21}& 0& \begin{array}{ccc}{K}_{11}^{t22}& {K}_{12}^{t22}& 0\end{array}\end{array}\\ \begin{array}{ccc}\begin{array}{c}{K}_{22}^{t21}\\ 0\end{array}& \begin{array}{c}0\\ {K}_{21}^{b33}\end{array}& \begin{array}{ccc}\begin{array}{c}{K}_{21}^{t22}\\ 0\end{array}& \begin{array}{c}{K}_{22}^{t22}\\ 0\end{array}& \begin{array}{c}0\\ {K}_{22}^{t33}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$
$${\mathbf{M}}^{t\mathrm{^{\prime}}}=\left[\begin{array}{ccc}{M}_{11}^{t11}& {M}_{12}^{t11}& \begin{array}{ccc}0& {M}_{11}^{t12}& \begin{array}{cc}{M}_{12}^{t12}& 0\end{array}\end{array}\\ {M}_{21}^{t11}& {M}_{22}^{t11}& \begin{array}{ccc}0& {M}_{21}^{t12}& \begin{array}{cc}{M}_{22}^{t12}& 0\end{array}\end{array}\\ \begin{array}{c}0\\ {M}_{11}^{t21}\\ \begin{array}{c}{M}_{21}^{t21}\\ 0\end{array}\end{array}& \begin{array}{c}0\\ {M}_{12}^{t21}\\ \begin{array}{c}{M}_{22}^{t21}\\ 0\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}{M}_{11}^{t33}\\ 0\\ \begin{array}{c}0\\ {M}_{21}^{b33}\end{array}\end{array}& \begin{array}{c}0\\ {M}_{21}^{t22}\\ \begin{array}{c}{M}_{21}^{t22}\\ 0\end{array}\end{array}& \begin{array}{c}\begin{array}{cc}0& {M}_{12}^{t33}\end{array}\\ \begin{array}{cc}{M}_{12}^{t22}& 0\end{array}\\ \begin{array}{cc}\begin{array}{c}{M}_{22}^{t22}\\ 0\end{array}& \begin{array}{c}0\\ {M}_{22}^{t33}\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$

Appendix 4

$${\mathbf{F}}^{g}={\left\{{{F}_{1}}^{p3}-{{F}_{1}}^{p1}{{F}_{1}}^{p2}{{F}_{2}}^{p3}-{{F}_{2}}^{p1}{{F}_{2}}^{p2}\right\}}^{T}$$
$${\mathbf{K}}^{t\mathrm{^{\prime}}}=\left[\begin{array}{ccc}\begin{array}{c}{K}_{11}^{p33}\\ 0\\ 0\end{array}& \begin{array}{c}0\\ {K}_{11}^{p11}\\ {-K}_{21}^{p11}\end{array}& \begin{array}{c}\begin{array}{ccc}0& {K}_{12}^{p33}& \begin{array}{cc}0& 0\end{array}\end{array}\\ \begin{array}{ccc}{-K}_{12}^{p11}& 0& \begin{array}{cc}{K}_{11}^{p12}& {-K}_{12}^{p12}\end{array}\end{array}\\ \begin{array}{ccc}{K}_{22}^{p11}& 0& \begin{array}{cc}{-K}_{21}^{p12}& {K}_{22}^{p12}\end{array}\end{array}\end{array}\\ {K}_{21}^{p33}& 0& \begin{array}{ccc}0& {-K}_{21}^{p33}& \begin{array}{cc}0& 0\end{array}\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}{K}_{11}^{p21}\\ {-K}_{21}^{p21}\end{array}& \begin{array}{ccc}\begin{array}{c}{-K}_{12}^{p21}\\ {K}_{22}^{p21}\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{cc}\begin{array}{c}{K}_{11}^{p22}\\ {-K}_{21}^{p22}\end{array}& \begin{array}{c}{-K}_{12}^{p22}\\ {K}_{22}^{p22}\end{array}\end{array}\end{array}\end{array}\right]$$
$${\mathbf{M}}^{t\mathrm{^{\prime}}}=\left[\begin{array}{ccc}{M}_{11}^{p33}& 0& \begin{array}{ccc}0& {M}_{12}^{p33}& \begin{array}{cc}0& 0\end{array}\end{array}\\ 0& {M}_{11}^{p11}& \begin{array}{ccc}{-M}_{12}^{p11}& 0& \begin{array}{cc}{M}_{11}^{p12}& -{M}_{12}^{p12}\end{array}\end{array}\\ \begin{array}{c}0\\ {M}_{21}^{p33}\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{c}{-M}_{21}^{p11}\\ 0\\ \begin{array}{c}{M}_{12}^{p21}\\ {-M}_{21}^{p21}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}{M}_{22}^{p11}\\ 0\\ \begin{array}{c}{-M}_{12}^{p21}\\ {M}_{22}^{p21}\end{array}\end{array}& \begin{array}{c}0\\ {M}_{22}^{p33}\\ \begin{array}{c}0\\ 0\end{array}\end{array}& \begin{array}{cc}\begin{array}{c}{-M}_{21}^{p12}\\ 0\\ \begin{array}{c}{M}_{11}^{p22}\\ {-M}_{21}^{p22}\end{array}\end{array}& \begin{array}{c}{M}_{22}^{p12}\\ 0\\ \begin{array}{c}{-M}_{12}^{p22}\\ {M}_{22}^{p22}\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$

Appendix 5

$${\mathbf{F}}^{g}+{\mathbf{F}}^{b{\prime}}=\left\{\begin{array}{c}{{F}_{1}}^{p3}+\\ -{{F}_{1}}^{p1}+\\ {{F}_{1}}^{p2}+\\ {{F}_{2}}^{p3}+\\ -{{F}_{2}}^{p1}+\\ {{F}_{2}}^{p2}+\end{array}\begin{array}{c}{{F}_{1}}^{b1}\\ {{F}_{1}}^{b2}\\ {{F}_{1}}^{b3}\\ {{F}_{2}}^{b1}\\ {{F}_{2}}^{b2}\\ {{F}_{2}}^{b3}\end{array}\right\}$$
$${{\varvec{K}}}_{N}^{b\mathrm{^{\prime}}}+{{\varvec{K}}}_{N}^{g}=\left[\begin{array}{ccc}{K}_{11}^{b11}& {K}_{12}^{b11}& \begin{array}{ccc}0& {K}_{11}^{b12}& \begin{array}{ccc}{K}_{12}^{b12}& 0& \begin{array}{ccc}0& 0& 0\end{array}\end{array}\end{array}\\ {K}_{21}^{b11}& {K}_{22}^{b11}& \begin{array}{ccc}0& {K}_{21}^{b12}& \begin{array}{ccc}{K}_{22}^{b12}& 0& \begin{array}{ccc}0& 0& 0\end{array}\end{array}\end{array}\\ \begin{array}{c}0\\ {K}_{11}^{b21}\\ \begin{array}{c}{K}_{21}^{b21}\\ 0\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{c}0\\ {K}_{12}^{b21}\\ \begin{array}{c}{K}_{22}^{b21}\\ 0\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}{K}_{11}^{b33}\\ 0\\ \begin{array}{c}0\\ {K}_{21}^{b33}\\ \begin{array}{c}0\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{c}0\\ {K}_{11}^{b22}+{K}_{11}^{b33}\\ \begin{array}{c}{K}_{21}^{b22}\\ 0\\ \begin{array}{c}{K}_{21}^{p33}\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}0\\ {K}_{12}^{b22}\\ \begin{array}{c}{K}_{22}^{b22}-{K}_{11}^{b11}\\ {-K}_{21}^{b11}\\ \begin{array}{c}0\\ {K}_{11}^{p21}\\ {-K}_{21}^{p21}\end{array}\end{array}\end{array}& \begin{array}{c}{K}_{12}^{b33}\\ 0\\ \begin{array}{c}{-K}_{12}^{b11}\\ {K}_{22}^{b33}+{K}_{22}^{p11}\\ \begin{array}{c}0\\ {-K}_{22}^{p21}\\ {K}_{22}^{p21}\end{array}\end{array}\end{array}& \begin{array}{ccc}\begin{array}{c}0\\ {K}_{12}^{b33}\\ \begin{array}{c}0\\ 0\\ \begin{array}{c}{K}_{22}^{p33}\\ 0\\ 0\end{array}\end{array}\end{array}& \begin{array}{c}0\\ 0\\ \begin{array}{c}{K}_{11}^{b12}\\ {-K}_{21}^{p12}\\ \begin{array}{c}0\\ {K}_{11}^{p22}\\ {-K}_{21}^{b22}\end{array}\end{array}\end{array}& \begin{array}{c}0\\ 0\\ \begin{array}{c}{-K}_{12}^{b12}\\ {K}_{22}^{p12}\\ \begin{array}{c}0\\ {-K}_{12}^{p11}\\ {K}_{21}^{b33}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right]$$

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Wang, X., Song, Q. Dynamic analysis and surface morphology prediction for deep cavity turning of bent-blade cutter. Int J Adv Manuf Technol 129, 4435–4455 (2023). https://doi.org/10.1007/s00170-023-12606-4

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