# Improving Stability Prediction in Peripheral Milling of Al7075T6

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Milling Stability Prediction with the EMHP Method

_{m}represents the modal mass, ζ is the damping ratio, ω

_{n}is the natural angular frequency, and F

_{y}is the cutting force over the workpiece in the y-direction, which is given by:

_{n}is the number of teeth, K

_{tc}and K

_{rc}are the tangential and the normal cutting force coefficients, and φ

_{j}(t) is defined as:

_{j}(t)) is a screen function, and is equal to 1 if the tooth j is in the cut, and is equal to 0 if j is out of the cut. To obtain the stability lobes of Equation (1), we proposed a solution procedure based on the EMHP algorithm [18], which relies on a sequence of subintervals that provide approximate solutions requiring less CPU time in comparison with other methods found in the literature. In this methodology, we rewrote Equation (1) as:

_{yt}= F

_{y}(t), and y

_{i}(T) denotes the approximate solution of order m in the i-th sub-interval that satisfies the initial conditions ${y}_{i}\left(0\right)={y}_{i-1}$, ${\dot{y}}_{i}\left(0\right)={\dot{y}}_{i-1}$. The Equation (1) can be written in state space, in accordance with the following matrix-form representation:

**A**(t − τ) =

**A**(t),

**B**(t − τ) =

**B**(t), and τ is the time delay. By following the homotopic procedure, the equivalent form of Equation (5) can be written as:

**x**

_{i}(T) denotes the m order solution of Equation (6) in the i-th sub-interval that satisfies the initial conditions ${x}_{i}\left(0\right)={x}_{i-1}$,

**A**

_{t}, and

**B**

_{t}represents the values of the periodic coefficients at the time t. In order to approximate the delay term ${x}_{i}^{\tau}\left(T\right)$ in Equation (6), the period [t

_{0}− t

_{0}] is discretized into N equally-spaced points, but the method does not accept strictly-spaced sub-intervals. Thus, each sub-interval has a time span equal to Δt = τ/(N − 1). Here, we assumed that the function ${x}_{i}^{\tau}\left(T\right)$ in the delay sub-interval [t

_{i−N}, t

_{i−N+}

_{1}] had a first-order polynomial representation, of the form:

**x**

_{i}≡

**x**

_{i}(T

_{i}). Equation (7) was then introduced into Equation (6), to get:

**D**

_{i}is a coefficient matrix,

**w**

_{i−}

_{1}is a vector of the form:

**D**

_{i}[8]. We then determined the transition matrix

**Φ**over the period τ = (N − 1)Δt by coupling each approximate solution through the discrete map

**D**

_{i}, i = 1, 2, …, (N − 1), to get:

#### 2.2. Effect of Runout and Cutting Speed

_{j}represents a weighted factor that is related to the cutting force magnitudes for each flute, j. Here, we propose a different alternative to consider the runout effects, by assuming that relating ρ

_{j}is given as:

_{z}is the feed rate. Notice that δ represents the chip variation in the feed direction (x-axis). To determine the value of $\tilde{r}$ experimentally, once the milling tool is attached to in the machine center, a deflection gauge fixed to the table was set to zero once the first flute pushed the gauge. Then, the apparent radius $\tilde{r}$ was estimated by the average deflection between 180°-spaced flutes. The proportional factor ρ

_{j}for the chip load in the actual j teeth was then calculated as the nominal feed rate plus the ratio $\delta =\tilde{r}$/f

_{z}, representing the percentage of the nominal feed rate.

_{j}= 1, Equation (15) becomes periodic in τ. On the other hand, when the runout influences the machining processes, the periodic behavior arises, related to the spindle period τ

_{T}.

_{T}= 60/n. Thus, the determination of the stability lobes by the EMHP method can be done by computing the transition matrix

**Φ**over the spindle rotation period, i.e., by considering that Equation (15) can be written as:

_{q}, B

_{q}, and α

_{q}are obtained for the tangential and normal directions, as: A

_{t}= 53.3, B

_{t}= 201, α

_{t}= 0.0067, A

_{r}= 84.4, B

_{t}= 20, and α

_{t}= 0.0045.

## 3. Results

#### 3.1. Experimental Validation of Stability Prediction

_{n}= 499 rad/s, ξ = 0.04, and m

_{m}= 10.1 kg. The transfer function obtained through the impact test fit well with the theoretical results. By using these values and computing the eigenvalues of the transition matrix, the stability lobes of the end-milling cutting operation were calculated. Figure 5 shows the stability charts for down-milling with 25% radial immersion, feed rate 0.05 mm/teeth, and a measured apparent radius of 17 µm. The discontinued lines correspond to the typical stability lobes when the cutting speed, runout, and helix angle are neglected, while the solid lines plot the stability boundaries, taking these effects into account.

#### 3.2. Accuracy and Computation Times

_{p}), a more general approach has been studied using an 11 × 11 matrix containing an equally-spaced sweep of spindle speeds (2000 rpm to 7500 rpm) and axial depths of cut (0 to 10 mm). The difference between the matrix’s approximate critical eigenvalues

**µ**and exact ones

**µ**is presented as the function of the number of discretizations, or subintervals. The exact matrix of the eigenvalues

_{0}**µ**was determined by each corresponding method with N = 200. Figure 6 shows the results for 25% radial immersion. Note that this figure does not compare the converged values between methods, as no method converges at the same eigenvalues.

_{0}_{p}) is used, which was calculated over the 11 × 11 matrix used previously. Figure 7 shows the average time for each method as a function of the number of discretization or sub-intervals. Once again, it is clear that the fastest methods for the stability performed were the CH and EMHP methods. It is important to note that the immersion does not change the number of mathematical operations. Figure 7 is valid for any value of radial immersion.

## 4. Discussion

## 5. Conclusions

- We observed a significant variation in the cutting forces depending on cutting speeds, and a model was thus proposed where the cutting-force coefficients varied depending on the cutting speed. Using an exponential cutting speed model, we described how the cutting-force coefficient decreased as a function of the increase in cutting speed.
- It was experimentally demonstrated that inclusion of the effects of the helix angle, runout, and characterization dependent on the cutting speed allowed for much more precise stability boundaries. Furthermore, typical stability lobes with constant cutting coefficients were found to only be valid for a narrow spindle speed range, meaning that their applicability is not valid in real practice.
- The convergence of the EMHP was compared with other efficient methods, such as the semi-discretization and full-discretization methods. When considering numeric convergence and computation times, the EMHP and Chebyshev proved to be the most efficient methods.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**Experimental and fitted curves of cutting shear coefficients in the (

**a**) tangential and (

**b**) radial directions.

**Figure 5.**Experimental data from the down-milling operation, comparison between typical and improved stability analysis.

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**MDPI and ACS Style**

Olvera, D.; Urbikain, G.; Elías-Zuñiga, A.; López de Lacalle, L.N.
Improving Stability Prediction in Peripheral Milling of Al7075T6. *Appl. Sci.* **2018**, *8*, 1316.
https://doi.org/10.3390/app8081316

**AMA Style**

Olvera D, Urbikain G, Elías-Zuñiga A, López de Lacalle LN.
Improving Stability Prediction in Peripheral Milling of Al7075T6. *Applied Sciences*. 2018; 8(8):1316.
https://doi.org/10.3390/app8081316

**Chicago/Turabian Style**

Olvera, Daniel, Gorka Urbikain, Alex Elías-Zuñiga, and Luis Norberto López de Lacalle.
2018. "Improving Stability Prediction in Peripheral Milling of Al7075T6" *Applied Sciences* 8, no. 8: 1316.
https://doi.org/10.3390/app8081316