Machining Stability

Abstract

Since the early days of regenerative chatter theory, it has been noticed that the phase difference between the current and the previous passes of machining self-excited vibration is correlated closely to the machining stability. However, an analytical proof of this fact has not been investigated, especially based on a nonlinear machining chatter models. In this chapter, an approach for determining the machining stability is presented in terms of the phase difference. The machining stability is demonstrated by a stability criterion in term of the phase difference sensitivity. By investigating the stability of the approximate solution of a nonlinear delay differential equation as the machining chatter model under small perturbations about an equilibrium state, the stability criterion is established. Through this approach, a theoretical proof of the relationship between the machining stability and the phase difference is given in terms of internal energy of the machining process. The analysis is in agreement with the numerical simulations and experimental data. Once the parameters of the machining system are identified, the stability criterion can be employed to predict the onset of machining chatter. The stability criterion identified for specific machining operations is of critical importance especially when using nanostructured coated cutting tools in turning and milling operations.

Derivation of the Stability Criterion with the Phase Difference Sensitivity

The steady-state characteristic equation can be expressed as

$$ R\left(A,\omega, \phi \right)+jI\left(A,\omega, \phi \right)=0 $$

(2.24)

where A and ω are the steady-state amplitude and frequency of the machining chatter, and ϕ is the phase difference angle. Assume that the small perturbations in the chatter amplitude A, the change rate of the chatter amplitude Δσ, and frequency ω are caused by the small deviation of the phase difference angle Δϕ. The chatter amplitude A and frequency ω with small perturbations are given as

$$ A->A+\Delta A, \mathrm{and} \omega ->\omega +\Delta \omega +j\Delta \sigma $$

(2.25)

It is noticed that the small perturbation in the chatter frequency is associated with the change rate of the chatter amplitude, that is, $\Delta \sigma =-\dot{A}/A$, if the approximate solution of the NLDDE machining chatter model is given by the first-order approach. Hence, we have

$$ R\left(A+\Delta A,\;\omega +\Delta \omega +\Delta \sigma,\;\phi +\Delta \phi \right)+I\left(A+\Delta A,\;\omega +\Delta \omega +j\Delta \sigma,\;\phi +\Delta \phi \right)=0 $$

(2.26)

The Taylor series first-order expansion of (2.26) about the equilibrium state yields

$$ \begin{array}{l}\frac{\partial R}{\partial A}\Delta A+\frac{\partial R}{\partial \omega}\left(\Delta \omega +j\Delta \sigma \right)+\frac{\partial R}{\partial \phi}\Delta \phi \\ {} +j\frac{\partial I}{\partial A}\Delta A+j\frac{\partial I}{\partial \omega}\left(\Delta \omega +j\Delta \sigma \right)+j\frac{\partial I}{\partial \phi}\Delta \phi =0\end{array} $$

(2.27)

Both the real and the imaginary parts will vanish separately, if the above equation is satisfied:

$$ \begin{array}{l}\frac{\partial R}{\partial A}\Delta A+\frac{\partial R}{\partial \omega}\Delta \omega +\frac{\partial R}{\partial \phi}\Delta \phi -\frac{\partial I}{\partial \omega}\Delta \sigma =0\hfill \\ {}\frac{\partial R}{\partial \omega}\Delta \sigma +\frac{\partial I}{\partial A}\Delta A+\frac{\partial I}{\partial \omega}\Delta \omega +\frac{\partial I}{\partial \phi}\Delta \sigma =0\hfill \end{array} $$

(2.28)

A single relationship among Δσ, ΔA, and Δϕ can be obtained by eliminating Δω from the equation set (2.28):

$$ \left[{\left(\frac{\partial I}{\partial \omega}\right)}^2+{\left(\frac{\partial R}{\partial \omega}\right)}^2\right]\Delta \sigma =\left(\frac{\partial R}{\partial A}\frac{\partial I}{\partial \omega }-\frac{\partial I}{\partial A}\frac{\partial R}{\partial \omega}\right)\Delta A+\left(\frac{\partial R}{\partial \phi}\frac{\partial I}{\partial \omega }-\frac{\partial I}{\partial \phi}\frac{\partial R}{\partial \omega}\right)\Delta \phi $$

(2.29)

Notice that the phase difference sensitivity S ^ϕ is defined as

$$ {S}^{\phi }=\frac{\partial \phi }{\partial A}\approx \frac{\Delta \phi }{\Delta A} $$

(2.30)

Hence, (2.29) can be rewritten in the form

$$ \left[{\left(\frac{\partial I}{\partial \omega}\right)}^2+{\left(\frac{\partial R}{\partial \omega}\right)}^2\right]\Delta \sigma =\left[\left(\frac{\partial R}{\partial A}\frac{\partial I}{\partial \omega }-\frac{\partial I}{\partial A}\frac{\partial R}{\partial \omega}\right)+\left(\frac{\partial R}{\partial \phi}\frac{\partial I}{\partial \omega }-\frac{\partial I}{\partial \phi}\frac{\partial R}{\partial \omega}\right){S}^{\phi}\right]\Delta A $$

(2.31)

If the equilibrium state is stable, a positive increment ΔA must result in a negative derivative of the chatter amplitude, $\dot{A}$, thus a positive relative change rate of the chatter amplitude Δσ; and similarly, a negative increment ΔA must lead to a negative relative change rate of the chatter amplitude Δσ. Stated another way, for a stable equilibrium state, the following condition must be satisfied:

$$ \left(\frac{\partial R}{\partial A}\frac{\partial I}{\partial \omega }-\frac{\partial I}{\partial A}\frac{\partial R}{\partial \omega}\right)+\left(\frac{\partial R}{\partial \phi}\frac{\partial I}{\partial \omega }-\frac{\partial I}{\partial \phi}\frac{\partial R}{\partial \omega}\right){S}^{\phi }>0 $$

(2.32)

This is the stability criterion with the phase difference sensitivity.