Identification of Stability Cutting Parameters Using Laser Doppler Vibrometry

Abstract

High-speed milling operations of thin walls are often limited by the so-called regenerative effect that causes poor surface finishing. To optimize the cutting process in terms of quality surface and productivity, the frequency response function (FRF) of the wall needs to be measured in order to identify the modal parameters of the system which are used to obtain the stability lobes that help identify the optimal system’s parameter values to warrant stable cutting conditions. The aim of this work is to experimentally show the variation on the frequency response function (FRF) values obtained by using a laser Doppler vibrometer (LDV) device and accelerometer sensors during a milling operation processes of an aluminum thin-walled workpiece of 1 mm thick and 30 mm height. It is shown that the FRF values variations has strong influence on the stable cutting bounds. To further assess our findings, we used the collected experimental data obtained by using the LDV during milling machine cutting operation processes of several thin-walled workpieces to identify the cutting parameters values that allow us to obtain good quality and acceptable surface finish.

THE EMHPM to PREDICT stability lobes for thin walls machining

In order to predict stability in machining operations, the EMHPM was used. This method developed by the authors, permits faster computation times and high accuracy in comparison with literature review. The workpiece under study has a single degree-of-freedom (flexibility only in the y direction according to the machine tool standard axis nomenclature), with a dynamic model of the following form [9]:

$$\displaystyle{\ddot{y}(t) + 2\zeta \omega _{n}\dot{y}(t) +\omega _{ n}^{2}y(t) = \frac{K_{s}(t,z)}{m_{m}} (y(t) - y(t-\tau ))}$$

(54.1)

Here, m _m represents the modal mass, ζ is the damping ratio, ω _n is the natural angular frequency and K _s (t, z) is the cutting force over the workpiece in the y-direction, which is given by:

$$\displaystyle{ K_{s}(t,z) =\sum \limits _{ j=1}^{z_{n} } \frac{1} {4k_{\beta }}[K_{tc}\cos 2\phi _{j} + K_{rc}\left (\sin 2\phi _{j} + 2\phi _{j}\right )]_{z_{j,1}(\phi _{j})}^{z_{j,2}(\phi _{j})} }$$

(54.2)

where z _n is the number of teeth, K _tc and K _rc are the tangential and the normal cutting force coefficients, $k_{\beta } = 2\tan \beta /D$, β is the helix angle, D is the tool diameter, and ϕ _j (z) is defined as

$$\displaystyle{ \phi _{j}(z) =\phi +(j - 1)2\pi /z_{n} - k_{\beta }z_{j} }$$

(54.3)

here, ϕ is the immersion angle measured from the y axis and ϕ _j is the j tooth angular position. The values of the tangential and normal cutting coefficients can be obtained experimentally [8].

To obtain the stability lobes of Eq. (54.1), a solution procedure based on the EMHPM is proposed. In this methodology, the Eq. (54.1) must be rewritten as:

$$\displaystyle{ \ddot{y}_{i}(T) + 2\zeta \omega _{n}\dot{y}_{i}(T) +\omega _{n}y_{i}(T) \approx \frac{K_{st}} {m_{m}}\left (y_{i}(T) - y_{i}^{\tau }(T)\right ) }$$

(54.4)

where K _st  = K _s (t, a _p ), and x _i (T) denotes de approximate solution of order m in the i-th sub-interval that satisfies the initial conditions $y_{i}(0) = y_{i-1}$, $\dot{y}_{i}(0) =\dot{ y}_{i-1}$.

Next, the Eq. (54.1) is expressed in state space form in accordance with the following matrix form representation

$$\displaystyle{ \dot{\mathbf{x}}(t) = \mathbf{A}(t)\mathbf{x}(t) + \mathbf{B}(t)(\mathbf{x}(t) -\mathbf{x}(t-\tau )) }$$

(54.5)

where $\mathbf{x} = {[x,\dot{x}]}^{T},\quad \mathbf{A}(t+\tau ) = \mathbf{A}(t)$, $\mathbf{B}(t+\tau ) = \mathbf{B}(t)$ and τ is the time delay. By following the EMHPM procedure, the Eq. (54.5) can be represented by its equivalent form as

$$\displaystyle{ \dot{\mathbf{x}}_{i}(T) -\mathbf{A}_{t}\mathbf{x}_{i}(T) \approx \mathbf{B}_{t}\mathbf{x}_{i}^{\tau }(T) }$$

(54.6)

where x _i (T) denotes the m order solution of Eq. (54.5) in the i-th sub-interval that satisfies the initial conditions $\mathbf{x}_{i}(0) = \mathbf{x}_{i-1}$, A _t and B _t represent the values of the periodic coefficients at the time t. In order to approximate the delay term x _− τ (T) in Eq. (54.6), the period [t ₀ − τ, t ₀] is discretized in N points that could be equally spaced as shown in Fig. 54.7. Thus, each sub-interval has a time span equal to $\Delta t = T/(N - 1)$. Here, it is assumed that the function x _i ^τ(T) in the delay sub-interval $[t_{i-N},t_{i-N+1}]$ has a first-order polynomial representation of the form:

$$\displaystyle{ \mathbf{x}_{i}^{\tau }(T) = \mathbf{x}_{ i-N+1}(T) \approx \mathbf{x}_{i-N} + \frac{N - 1} {\mathrm{r}} (\mathbf{x}_{i-N+1} -\mathbf{x}_{i-N})T }$$

(54.7)

Fig. 54.7

First order polynomial representation of the time delay subinterval

To simplify the notation, the term x _i  ≡ x _i (T _i )is defined. Next, substituting the Eq. (54.7) into the Eq. (54.6), results

$$\displaystyle{ \dot{\mathbf{x}}_{i}(T) = \mathbf{A}_{t}\mathbf{x}_{i}(T) + \mathbf{B}_{t}\mathbf{x}_{i-N} -\frac{N - 1} {\mathrm{r}} \mathbf{B}_{t}\mathbf{x}_{i-N}T + \frac{N - 1} {\mathrm{r}} \mathbf{B}_{t}\mathbf{x}_{i-N+1}T }$$

(54.8)

where

$$\displaystyle{ \mathbf{A}_{t} = \left [\begin{array}{l@{\ \ \ }l} 0 \ \ \ &1 \\ -\omega _{n}^{2} + \frac{K_{st}} {m_{m}}\ \ \ &-2\zeta \omega _{n}\\ \ \ \ \end{array} \right ],\mathbf{B}_{t} = \left [\begin{array}{l@{\ \ \ }l} 0 \ \ \ &0 \\ -\frac{K_{st}} {m_{m}}\ \ \ &0\\ \ \ \ \end{array} \right ] }$$

(54.9)

By following our EMHPM procedure, it might be assumed that the homotopy representation of Eq. (54.8) is of the form

$$\displaystyle{ H(\mathbf{X}_{i},p) = L(\mathbf{X}_{i})-L(\mathbf{x}_{i0})+pL(\mathbf{x}_{i0})-p(\mathbf{AX}_{i} + \mathbf{Bx}_{i-N} -\frac{N - 1} {\mathrm{r}} \mathbf{Bx}_{i-N}T + \frac{N - 1} {\mathrm{r}} \mathbf{Bx}_{i-N+1}T) }$$

(54.10)

Later, substituting the m order expansion of X _i

$$\displaystyle{ \mathbf{X}_{i}(T) = \mathbf{X}_{i0}(T) + p\mathbf{X}_{i1}(T) +\ldots {p}^{m}\mathbf{X}_{ im}(T) }$$

(54.11)

in Eq. (54.10) and it is assumed that the initial solution is given by $\mathbf{x}_{i0} = \mathbf{x}_{i-1}$. Then, the proposed EMHPM is applied to obtain a set of first order delay differential equations which after solving, we get:

$$\displaystyle\begin{array}{rcl} \mathbf{X}_{i0}& =& \mathbf{x}_{i-1} \\ \mathbf{X}_{i1}& =& \mathbf{A}_{t}\mathbf{x}_{i-1}T + \mathbf{B}_{t}\mathbf{x}_{i-N}T -\frac{1} {2} \frac{N - 1} {\mathrm{r}} \mathbf{B}_{t}\mathbf{x}_{i-N}{T}^{2} + \frac{1} {2} \frac{N - 1} {\mathrm{r}} \mathbf{B}_{t}\mathbf{x}_{i-N+1}{T}^{2} \\ \mathbf{X}_{i2}& =& \frac{1} {2}\mathbf{A}_{t}^{2}\mathbf{x}_{ i-1}{T}^{2} + \frac{1} {2}\mathbf{A}_{t}\mathbf{B}_{t}\mathbf{x}_{i-N}{T}^{2} -\frac{1} {6} \frac{N - 1} {\mathrm{r}} \mathbf{A}_{t}\mathbf{B}_{t}\mathbf{x}_{i-N}{T}^{3} + \frac{1} {6} \frac{N - 1} {\mathrm{r}} \mathbf{A}_{t}\mathbf{B}_{t}\mathbf{x}_{i-N+1}{T}^{3} \\ & \vdots & \\ \mathbf{X}_{ik}& =& \frac{1} {k!}\mathbf{A}_{t}^{k}\mathbf{x}_{ i-1}{T}^{k} + \frac{1} {k!}\mathbf{A}_{t}^{k-1}\mathbf{B}_{ t}\mathbf{x}_{i-N}{T}^{k} - \frac{1} {(k + 1)!} \frac{N - 1} {\mathrm{r}} \mathbf{A}_{t}^{k-1}\mathbf{B}_{ t}\mathbf{x}_{i-N}{T}^{k+1} + \frac{1} {(k + 1)!} \frac{N - 1} {\mathrm{r}} \mathbf{A}_{t}^{k-1}\mathbf{B}_{ t}\mathbf{x}_{i-N+1}{T}^{k+1}{}\end{array}$$

(54.12)

The m order approximate solution of Eq. (54.8) is obtained by substituting Eq. (54.12) into Eq. (54.13), this yield:

$$\displaystyle{ \mathbf{x}_{im}(T) \approx \sum \limits _{k=0}^{m}\mathbf{X}_{ ik}(T) }$$

(54.13)

To obtain the stability lobes of Eq. (54.1), the Eq. (54.13) is written in the form

$$\displaystyle{ \mathbf{x}_{i}(T) \approx \mathbf{P}_{i}(T)\mathbf{x}_{i-1} + \mathbf{Q}_{i}(T)\mathbf{x}_{i-N+1} + \mathbf{R}_{i}(T)\mathbf{x}_{i-N} }$$

(54.14)

where

$$\displaystyle\begin{array}{rcl} \mathbf{P}_{i}(T)& =& \sum \limits _{k=0}^{m} \frac{1} {k!}\mathbf{A}{_{t}}^{k}{T}^{k}, \\ \mathbf{Q}_{i}(T)& =& \left \{\begin{array}{ll} \sum \limits _{k=1}^{m} \frac{N-1} {(k+1)!(\tau )}\mathbf{A}_{t}^{k-1}\mathbf{B}_{ t}{T}^{k+1} & m \geq 1 \\ \mathbf{0} &m = 0\\ \end{array} \right.. \\ \mathbf{R}_{i}(T)& =& \left \{\begin{array}{cc} \sum \limits _{k=1}^{m} \frac{1} {k!}\mathbf{A}_{t}^{k-1}\mathbf{B}_{ t}{T}^{k} -\mathbf{Q}_{ i}&m \geq 1 \\ \mathbf{0} &m = 0\\ \end{array} \right..{}\end{array}$$

(54.15)

The approximate solution given by Eq. (54.14) can be written as discrete map by using the following identity:

$$\displaystyle{ \mathbf{w}_{i} = \mathbf{D}_{i}\mathbf{w}_{i-1} }$$

(54.16)

where D _i is a coefficient matrix, and w _i − 1 is a vector of the form:

$$\displaystyle{ \mathbf{w}_{i-1} = {[x_{i-1},\dot{x}_{i-1},x_{i-2},...,x_{i-N}]}^{T} }$$

(54.17)

By using Eqs. (54.14) and (54.15), it might be easily showed that the coefficient matrix D _i is given by [9]:

$$\displaystyle{ \mathbf{D}_{i} = \left [\begin{array}{l@{\ \ \ }l@{\ \ \ }l@{\ \ \ }l@{\ \ \ }l@{\ \ \ }l@{\ \ \ }l@{\ \ \ }l} P_{i,(1,1)}\ \ \ &P_{i,(1,2)}\ \ \ &0\ \ \ &0\ \ \ &\ldots \ \ \ &0\ \ \ &Q_{i,(1,1)}\ \ \ &R_{i,(1,1)} \\ P_{i,(2,1)}\ \ \ &P_{i,(2,2)}\ \ \ &0\ \ \ &0\ \ \ &\ldots \ \ \ &0\ \ \ &Q_{i,(2,1)}\ \ \ &R_{i,(2,1)} \\ 1 \ \ \ &0 \ \ \ &0\ \ \ &0\ \ \ &\ldots \ \ \ &0\ \ \ &0 \ \ \ &0\\ 0 \ \ \ &0 \ \ \ &1\ \ \ &0\ \ \ &\ldots \ \ \ &0\ \ \ &0 \ \ \ &0\\ \vdots \ \ \ &\vdots \ \ \ &\vdots \ \ \ &\ddots \ \ \ &\vdots\ \ \ &\vdots \ \ \ &\vdots \ \ \ &\vdots \\ 0 \ \ \ &0 \ \ \ &0\ \ \ &0\ \ \ &\ddots\ \ \ &0\ \ \ &0 \ \ \ &0\\ 0 \ \ \ &0 \ \ \ &0\ \ \ &0\ \ \ &\ldots \ \ \ &1\ \ \ &0 \ \ \ &0 \\ 0 \ \ \ &0 \ \ \ &0\ \ \ &0\ \ \ &\ldots \ \ \ &0\ \ \ &1 \ \ \ &0\\ \ \ \ \end{array} \right ] }$$

(54.18)

Next the transition matrix Φ is calculated over the period T =(N-1)Δt by coupling each approximate solution through the discrete map ${\textbf{D}}_{i},i = 1,2,..,(N - 1)$, to get:

$$\displaystyle{ \Phi = \mathbf{D}_{N-1}\mathbf{D}_{N-2}\ldots \mathbf{D}_{2}\mathbf{D}_{1} }$$

(54.19)

Then, the stability lobes of Eq. (54.1) are determined by computing the eigenvalues of the transition matrix given by Eq. (54.19).