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Dynamic modeling and nonlinear vibration analysis of spindle system during ball end milling process

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Abstract

In cutting process, the insufficiency in grasp of the tool vibration characteristics of spindle system seriously hinders the improvement of machining quality and efficiency. Thus, this paper develops a novel dynamic model of spindle system in ball end milling process considering the nonlinear contact behavior of bearings. For the sake of coupling with the motion differential equations of spindle shaft, a general analytical expression for the nonlinear contact force of matched angular contact ball bearings is proposed. Then, dynamic cutting force model during ball end milling is established with consideration of the influence of tool vibration on the uncut chip thickness. Furthermore, the effectiveness and feasibility of the proposed model is confirmed by some cutting tests. Finally, the effects of rotation speed, bearing preload, and cutting parameters on the tool end vibration response of spindle system are analyzed in detail. The investigations reveal that the main resonance frequency increases and the corresponding resonance amplitude decreases as bearing preload increases. The larger bearing preload can improve cutting stability and machining quality. It is also concluded that the change regarding axial depth of cut considerably affects the vibration behaviors of tool end. The proposed dynamic model can be applied to predict the vibration of spindle system during ball end milling, especially the tool vibration.

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Funding

The work was supported by National Natural Science Foundation of China (Grant No. 52075087) and the Fundamental Research Funds for the Central Universities (Grant Nos. N2003006 and N2203002).

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Huihui Miao, Chenyu Wang, Changyou Li, Guo Yao, Xiulu Zhang, Zhendong Liu & Mengtao Xu

Authors
  1. Huihui Miao
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  2. Chenyu Wang
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  3. Changyou Li
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  4. Guo Yao
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  5. Xiulu Zhang
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  6. Zhendong Liu
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  7. Mengtao Xu
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Contributions

Huihui Miao: conceptualization, software, methodology, validation, writing—original draft, writing—review and editing. Chenyu Wang: conceptualization, investigation, visualization. Changyou Li: funding acquisition, project administration, resources, supervision, writing—review and editing. Guo Yao: resources, writing—review and editing. Xiulu Zhang: formal analysis, software. Zhendong Liu: validation, investigation, formal analysis. Mengtao Xu: resources, writing—reviewing and editing, supervision, writing—review and editing.

Corresponding authors

Correspondence to Changyou Li or Guo Yao.

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Appendix

Appendix

The stiffness matrix of the spindle system is as follows

$$\varvec{K}_{x}=\left[\begin{array}{cccccccccccccc}{k}_{11}& {k}_{12}& {k}_{13}& {k}_{14}& & & & & & & & & & \\ {k}_{12}& {k}_{22}& {k}_{23}& {k}_{24}& & & & & & & & & & \\ {k}_{13}& {k}_{23}& {k}_{33}& {k}_{34}& {k}_{35}& {k}_{36}& & & & & & & & \\ {k}_{14}& {k}_{24}& {k}_{34}& {k}_{44}& {k}_{45}& {k}_{46}& & & & & 0& & & \\ & & {k}_{35}& {k}_{45}& {k}_{55}& {k}_{56}& {k}_{57}& {k}_{58}& & & & & & \\ & & {k}_{36}& {k}_{46}& {k}_{56}& {k}_{66}& {k}_{67}& {k}_{68}& & & & & & \\ & & & & {k}_{57}& {k}_{67}& {k}_{77}& {k}_{78}& {k}_{79}& {k}_{\mathrm{7,10}}& & & & \\ & & & & {k}_{58}& {k}_{68}& {k}_{78}& {k}_{88}& {k}_{89}& {k}_{\mathrm{8,10}}& & & & \\ & & & & & & {k}_{79}& {k}_{89}& {k}_{99}& {k}_{\mathrm{9,10}}& {k}_{\mathrm{9,11}}& {k}_{\mathrm{9,12}}& & \\ & & & & & & {k}_{\mathrm{7,10}}& {k}_{\mathrm{8,10}}& {k}_{\mathrm{9,10}}& {k}_{\mathrm{10,10}}& {k}_{\mathrm{10,11}}& {k}_{\mathrm{10,12}}& & \\ & & & 0& & & & & {k}_{\mathrm{9,11}}& {k}_{\mathrm{10,11}}& {k}_{\mathrm{11,11}}& {k}_{\mathrm{11,12}}& {k}_{\mathrm{11,13}}& {k}_{\mathrm{11,14}}\\ & & & & & & & & {k}_{\mathrm{9,12}}& {k}_{\mathrm{10,12}}& {k}_{\mathrm{11,12}}& {k}_{12,12}& {k}_{\mathrm{12,13}}& {k}_{\mathrm{12,14}}\\ & & & & & & & & & & {k}_{\mathrm{11,13}}& {k}_{\mathrm{12,13}}& {k}_{\mathrm{13,13}}& {k}_{\mathrm{13,14}}\\ & & & & & & & & & & {k}_{\mathrm{11,14}}& {k}_{\mathrm{12,14}}& {k}_{\mathrm{13,14}}& {k}_{\mathrm{14,14}}\end{array}\right]$$
(42)
$$\varvec{K}_{y}=\left[\begin{array}{cccccccccccccc}{k}_{11}& -{k}_{12}& {k}_{13}& -{k}_{14}& & & & & & & & & & \\ -{k}_{12}& {k}_{22}& -{k}_{23}& {k}_{24}& & & & & & & & & & \\ {k}_{13}& -{k}_{23}& {k}_{33}& -{k}_{34}& {k}_{35}& -{k}_{36}& & & & & & & & \\ -{k}_{14}& {k}_{24}& -{k}_{34}& {k}_{44}& -{k}_{45}& {k}_{46}& & & & & 0& & & \\ & & {k}_{35}& -{k}_{45}& {k}_{55}& -{k}_{56}& {k}_{57}& -{k}_{58}& & & & & & \\ & & -{k}_{36}& {k}_{46}& -{k}_{56}& {k}_{66}& -{k}_{67}& {k}_{68}& & & & & & \\ & & & & {k}_{57}& -{k}_{67}& {k}_{77}& -{k}_{78}& {k}_{79}& -{k}_{\mathrm{7,10}}& & & & \\ & & & & -{k}_{58}& {k}_{68}& -{k}_{78}& {k}_{88}& -{k}_{89}& {k}_{\mathrm{8,10}}& & & & \\ & & & & & & {k}_{79}& -{k}_{89}& {k}_{99}& -{k}_{\mathrm{9,10}}& {k}_{\mathrm{9,11}}& -{k}_{\mathrm{9,12}}& & \\ & & & & & & -{k}_{\mathrm{7,10}}& {k}_{\mathrm{8,10}}& -{k}_{\mathrm{9,10}}& {k}_{\mathrm{10,10}}& -{k}_{\mathrm{10,11}}& {k}_{\mathrm{10,12}}& & \\ & & & 0& & & & & {k}_{\mathrm{9,11}}& -{k}_{\mathrm{10,11}}& {k}_{\mathrm{11,11}}& -{k}_{\mathrm{11,12}}& {k}_{\mathrm{11,13}}& -{k}_{\mathrm{11,14}}\\ & & & & & & & & -{k}_{\mathrm{9,12}}& {k}_{\mathrm{10,12}}& -{k}_{\mathrm{11,12}}& {k}_{\mathrm{12,12}}& -{k}_{\mathrm{12,13}}& {k}_{\mathrm{12,14}}\\ & & & & & & & & & & {k}_{\mathrm{11,13}}& -{k}_{\mathrm{12,13}}& {k}_{\mathrm{13,13}}& -{k}_{\mathrm{13,14}}\\ & & & & & & & & & & -{k}_{\mathrm{11,14}}& {k}_{\mathrm{12,14}}& -{k}_{\mathrm{13,14}}& {k}_{\mathrm{14,14}}\end{array}\right]$$
(43)
$$\varvec{K}_{z}=\left[\begin{array}{ccccccc}{u}_{11}& {u}_{12}& & & & & \\ {u}_{12}& {u}_{22}& {u}_{23}& & & 0& \\ & {u}_{23}& {u}_{33}& {u}_{34}& & & \\ & & {u}_{34}& {u}_{44}& {u}_{45}& & \\ & & & {u}_{45}& {u}_{55}& {u}_{56}& \\ & 0& & & {u}_{56}& {u}_{66}& {u}_{67}\\ & & & & & {u}_{67}& {u}_{77}\end{array}\right]$$
(44)

The element of the stiffness matrix is calculated according to the method of flexibility influence coefficient in material mechanics, as follows

$$\left\{\begin{array}{l}{k}_{11}={b}_{11}\\ {k}_{12}={b}_{21}\\ {k}_{13}=-{b}_{11}\\ {k}_{14}={b}_{21}\end{array}\right.;\left\{\begin{array}{l}{k}_{22}={l}_{1}{b}_{21}-{b}_{31}\\ {k}_{23}=-{b}_{21}\\ {k}_{24}={b}_{31}\end{array}\right.;\left\{\begin{array}{l}{k}_{33}={b}_{11}+{b}_{12}\\ {k}_{34}=-{b}_{21}+{b}_{22}\\ {k}_{35}=-{b}_{12}\\ {k}_{36}={b}_{22}\end{array}\right.;\left\{\begin{array}{l}{k}_{44}={l}_{1}{b}_{21}-{b}_{31}+{l}_{2}{b}_{22}-{b}_{32}\\ {k}_{45}=-{b}_{22}\\ {k}_{46}={b}_{32}\end{array}\right.$$
(45)
$$\left\{\begin{array}{l}{k}_{55}={b}_{12}+{b}_{13}\\ {k}_{56}=-{b}_{22}+{b}_{23}\\ {k}_{57}=-{b}_{13}\\ {k}_{58}={b}_{23}\end{array}\right.;\left\{\begin{array}{l}{k}_{66}={l}_{2}{b}_{22}-{b}_{32}+{l}_{3}{b}_{23}-{b}_{33}\\ {k}_{67}=-{b}_{23}\\ {k}_{68}={b}_{33}\end{array}\right.;\left\{\begin{array}{l}{k}_{77}={b}_{13}+{b}_{14}\\ {k}_{78}=-{b}_{23}+{b}_{24}\\ {k}_{79}=-{b}_{14}\\ {k}_{\mathrm{7,10}}={b}_{24}\end{array}\right.;\left\{\begin{array}{l}{k}_{88}={l}_{3}{b}_{23}-{b}_{33}+{l}_{4}{b}_{24}-{b}_{34}\\ {k}_{89}=-{b}_{24}\\ {k}_{\mathrm{8,10}}={b}_{34}\end{array}\right.$$
(46)
$$\left\{\begin{array}{l}{k}_{99}={b}_{14}+{b}_{15}\\ {k}_{\mathrm{9,10}}=-{b}_{24}+{b}_{25}\\ {k}_{\mathrm{9,11}}=-{b}_{15}\\ {k}_{\mathrm{9,12}}={b}_{25}\end{array}\right.;\left\{\begin{array}{l}{k}_{\mathrm{10,10}}={l}_{4}{b}_{24}-{b}_{34}+{l}_{5}{b}_{25}-{b}_{35}\\ {k}_{\mathrm{10,11}}=-{b}_{25}\\ {k}_{\mathrm{10,12}}={b}_{35}\end{array}\right.;\left\{\begin{array}{l}{k}_{\mathrm{11,11}}={b}_{15}+{b}_{16}\\ {k}_{\mathrm{11,12}}=-{b}_{25}+{b}_{26}\\ {k}_{\mathrm{11,13}}=-{b}_{16}\\ {k}_{\mathrm{11,14}}={b}_{26}\end{array}\right.;\left\{\begin{array}{l}{k}_{\mathrm{12,12}}={l}_{5}{b}_{25}-{b}_{35}+{l}_{6}{b}_{26}-{b}_{36}\\ {k}_{\mathrm{12,13}}=-{b}_{26}\\ {k}_{\mathrm{12,14}}={b}_{36}\end{array}\right.$$
(47)
$$\left\{\begin{array}{c}{k}_{\mathrm{13,13}}={b}_{16}\\ {k}_{\mathrm{13,14}}=-{b}_{26}\\ {k}_{\mathrm{14,14}}={l}_{6}{b}_{26}-{b}_{36}\end{array}\right.$$
(48)
$$\left\{\begin{array}{c}{u}_{11}={d}_{1}\\ {u}_{12}=-{d}_{1}\\ {u}_{22}={d}_{1}+{d}_{2}\\ {u}_{23}=-{d}_{2}\end{array}\right.;\left\{\begin{array}{c}{u}_{33}={d}_{2}+{d}_{3}\\ {u}_{34}=-{d}_{3}\\ {u}_{44}={d}_{3}+{d}_{4}\\ {u}_{45}=-{d}_{4}\end{array}\right.;\left\{\begin{array}{c}{u}_{55}={d}_{4}+{d}_{5}\\ {u}_{56}=-{d}_{5}\\ {u}_{66}={d}_{5}+{d}_{6}\\ {u}_{67}=-{d}_{6}\\ {u}_{77}={d}_{6}\end{array}\right.$$
(49)

where

$${b}_{1i}=\frac{12E{I}_{i}}{{l}_{i}^{3}};{b}_{2i}=\frac{1}{2}{l}_{i}{b}_{1i};{b}_{3i}=\frac{1}{6}{l}_{i}^{2}{b}_{1i},{d}_{i}=\frac{E{A}_{i}}{{l}_{i}},\left(i=\mathrm{1,2},\mathrm{3,4},\mathrm{5,6}\right)$$
(50)

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Miao, H., Wang, C., Li, C. et al. Dynamic modeling and nonlinear vibration analysis of spindle system during ball end milling process. Int J Adv Manuf Technol 121, 7867–7889 (2022). https://doi.org/10.1007/s00170-022-09805-w

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