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Simpson’s 3/8–based method stability analysis for milling processes

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Abstract

Chatter stability prediction based on linear time-periodic delay differential equations (DDEs) of the milling dynamic process is an important issue in achieving high-performance processing operations. To obtain efficient and precise regenerative chatter prediction, a novel method for milling stability analysis based on Simpson’s 3/8 rule is proposed. First, the dynamics model of milling processes, considering the regenerative effect, is modeled by an n-dimensional delay differential equation. Second, a set of algebraic equations is obtained to determine the transition matrix by converting and discretizing periodic DDE into a system of integral equations at the neighboring sampling grid points. Then, the eigenvalues of the transition matrix are utilized to determine stability based on Floquet theory. Finally, the comparisons of lobe diagrams calculated by different methods and discretization interval number for one and two degrees of freedom milling models are discussed, and the consistency of the predicted and experimental stability boundaries is analyzed. Results show that the proposed method achieves high computation accuracy with high efficiency.

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Abbreviations

a/D:

ratio of radial cut depth

f n :

natural frequency

g(ϕ j(t)):

window function

h :

interval width

h(t):

cutting force coefficient

K c(t):

milling force coefficient matrix

k t :

tangential milling force coefficient

k n :

normal cutting force coefficient

l :

time interval number

M, C, and K :

modal mass, damping, and stiffness matrix

m t :

modal mass

N :

number of tool teeth

P l(x):

Lagrange polynomial

q(t):

modal displacement vector

T :

one tooth passing period

w :

axial depth of cut

x(t):

state items

x(t − T):

delayed state term

ζ :

relative damping

μ :

approximate eigenvalue

μ 0 :

exact eigenvalue

τ :

time interval

ϕ j(t):

angular position of the jth tooth

ϕ st :

start angle

ϕ ex :

exit angle

Φ :

state transition matrix

ω :

angular natural frequency

Ω:

spindle speed

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Authors and Affiliations

  1. College of Mechanical & Vehicle Engineering, Bengbu University, Bengbu, 233030, China

    Chunjing Liu, Shengfeng Li & Guohua Ding

  2. College of Mechanical & Electronic Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China

    Dunbing Tang

Authors
  1. Chunjing Liu
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  2. Dunbing Tang
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  3. Shengfeng Li
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  4. Guohua Ding
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Contributions

Chunjing Liu: methodology, validation, resources, writing—original draft, corresponding author

Dunbing Tang: conceptualization, writing—review and editing,

Shengfeng Li: writing—review and editing

Guohua Ding: investigation, writing—review and editing, corresponding

Corresponding authors

Correspondence to Chunjing Liu or Guohua Ding.

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Liu, C., Tang, D., Li, S. et al. Simpson’s 3/8–based method stability analysis for milling processes. Int J Adv Manuf Technol 114, 671–682 (2021). https://doi.org/10.1007/s00170-021-06732-0

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  • DOI: https://doi.org/10.1007/s00170-021-06732-0

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