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Comment on “A novel approach for the prediction of the milling stability based on the Simpson method” by Z. Zhang, H. Li, G. Meng, C. Liu, [Int. J. Mach. Tools Manuf. 99 (2015) 43–47]
International Journal of Machine Tools & Manufacture 103 (2016) 53–56
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Letter to the Editor Comment on “A novel approach for the prediction of the milling stability based on the Simpson method” by Z. Zhang, H. Li, G. Meng, C. Liu, [Int. J. Mach. Tools Manuf. 99 (2015) 43–47] Zhang et al. [1] has recently developed a Simpson 1/3 numerical integration method (S1/3NIM) for prediction of milling stability. Their effort is commendable as the resulting map is concise, fast and accurate compared with the well-accepted pre-existing First order Semi-discretization and Second order Full-discretization methods. However it is noticed that Zhang et al. [1] did not compare with the Simpson 1/3 method proposed by Ding et al. [2] four years earlier. It is the opinion in this letter that this gap deserves some comparative comment in terms of computational accuracy, convergence and computational time (CT). The milling model considered by the two methods has the first-order form
ẋ (t )=Ax (t )+B (t ) x (t )−B (t ) x (t−τ )
(1)
The two methods involve discretizing the delay τ of the system into k equal discrete time intervals [ti, ti +1] where i=0,1,2,………(k − 1), τ ti=i k =i∆t =i (ti +1−ti ) and then solving the arising discrete problem via different integration approaches. Ding et al. [2] solved the problem by double-step integration of Eq. (1) spanning [ti − 1, ti ] and [ti, ti +1] for all i=1,2,………(k − 1) to get
x i + 1 = e2A ∆ t x i − 1 +
∫t
ti + 1
e A ( ti + 1− t ) {B (t ) x (t ) − B (t ) x (t − τ )} dt
i −1
(2a)
For i=0 the integration is one-step on discrete time interval [t0, t1] to have the form
∫t
x1 = e A ∆ t x 0 +
t1
e A ( t1− t ) {B (t ) x (t ) − B (t ) x (t − τ )} dt
0
(2b)
The corresponding definite integrals by Zhang et al. [1] are respectively
∫t
xi + 1 = xi − 1 +
ti + 1
{B (t ) x (t ) − B (t ) x (t − τ )} dt
i −1
x1 = x 0 +
∫t
t1
{B (t ) x (t ) − B (t ) x (t − τ )} dt
0
(3a)
(3b)
It is seen that the two methods differ in that Ding et al. [2] is
http://dx.doi.org/10.1016/j.ijmachtools.2016.01.005 0890-6955/& 2016 Elsevier Ltd. All rights reserved.
based on the Convolution integral while Zhang et al. [1] can be said to be based on the Euler approximate method for solution of ordinary differential equations. Thus the S1/3NIM by Ding et al. [2] was derived from exact solution while that of Zhang et al. [1] was derived from an approximate solution. This means that the S1/3NIM of Ding et al. [2] is expected to be more accurate. Also the S1/3NIM of Ding et al. [2] benefits from the resolution and smooth behavior of convolution [3] thus expected to be more numerically stable than S1/3NIM by Zhang et al. [1]. Finally the S1/3NIM of Zhang et al. [1] is expected to save more CT because exponential matrices do not appear in its monodromy matrix. These expectations are confirmed by numerical simulation presented in Figs. 1 and 2. The input parameters for the numerical simulation which were also utilized by both Zhang et al. [1] and Ding et al. [2] are summarized in Table 1. Fig. 1 shows the stability diagrams computed on a 200 by 100 gridded plane of spindle speed and depth of cut (w) for a fullyimmersed system at discretization level of k = 40 and compared with a reference stability diagram computed with S1/3NIM of Ding et al. [2] at discretization level of k = 80 (the red lines). It is seen in Fig. 1 that the S1/3NIM of Zhang et al. [1] reflected less accuracy than the S1/3NIM of Ding et al. [2] being more out of coincidence with the reference diagram especially in the low speed domain. CTs are indicated in the caption of Fig. 1 to show that the Zhang et al. [1] is only slightly faster than the Ding et al. [2]. The rate of convergence plots are given as r (k )= μsr (k )−μrsr (kR ) versus k where μsr (k ) is spectral radius at k and μrsr (kR ) is the reference spectral radius of high accuracy at reference high value of k designated kR . kR=1000 is used in the S1/3NIM of Ding et al. [2] to calculate reference spectral radii μrsr (kR ) at different parameter combinations (Table 2) and used in rate of convergence plots given in Fig. 2. It is seen that S1/3NIM of Ding et al. [2] reflected faster convergence. Also it is seen as expected that S1/3NIM of Zhang et al. [1] lacks the numerical smoothness of convolution. These finding would also explain why milling stability prediction in [4] using the classical fourth-order Runge–Kutta method was of poor accuracy (especially at coarser discretization) relative to the zeroorder and the first-order semi-discretization methods while the generalized Runge–Kutta method based on convolution integral resulted in superior accuracy in all levels of discretization than the semi-discretizations methods.
54
Letter to the Editor / International Journal of Machine Tools & Manufacture 103 (2016) 53–56
Fig. 1. The stability diagrams of 1DOF fully-immersed milling (the black curves) computed with k = 40 from maps generated from (a) S1/3NIM of Ding et al. [2] with CT¼ 117 s and (b) S1/3NIM of Zhang et al. [1] CT¼ 110secs. It is seen on comparison with the reference stability diagram generated from the S1/3NIM of Ding et al [2] at discretization level k ¼ 80 (the red lines) that the older S1/3NIM of Ding et al. [2] is more accurate than the very recent S1/3NIM of Zhang et al. [1]. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Letter to the Editor / International Journal of Machine Tools & Manufacture 103 (2016) 53–56
(a)
(c)
0.16 S1/3NIM of Ding et al [2] S1/3NIM of Zhang et al [1]
0.14
S1/3NIM of Ding et al [2] S1/3NIM of Zhang et al [1]
0.5
0.1
0.4
0.08
r(k)
r(k)
0.7 0.6
0.12
0.06
0.3 0.2
0.04
0.1
0.02
0
0 20
40
60
80
100
20
120
40
60
80
120
(d)
0.2 S1/3NIM of Ding et al [2] S1/3NIM of Zhang et al [1]
S1/3NIM of Ding et al [2] S1/3NIM of Zhang et al [1]
0.04
0.15
0.03
0.1
r(k)
r(k)
100
k
k
(b)
55
0.02
0.05
0.01
0
0
20
40
60
80
100
120
20
40
60
80
100
120
k
k
Fig. 2. Sample numerical rate of convergence of spectral radii of S1/3NIM of Ding et al. [2] and S1/3NIM of Zhang et al. [1] for fully-immersed 1 DOF milling at (a) Ω = 5000 rpm and w¼ 0.2 mm, (b) Ω = 5000 rpm and w¼ 1 mm, (c) Ω=8500 rpm and w¼ 2.5 mm and (d) Ω=10000 rpm and w¼ 1.5 mm. It is seen that the older S1/3NIM of Ding et al. [2] exhibit better convergence and Numerical smoothness than the recent S1/3NIM of Zhang et al. [1]. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.).
Table 1 The Input parameters for numerical simulations. Source [5]. Natural frequency of 1DOF tool
ωn
5793rads−1
Mass of 1DOF tool
m
Damping ratio of 1DOF tool Feed exponent in the cutting force law tangential cutting coefficient
ξ γ Ct
0. 03993kg 0.011 1
Normal to tangential cutting coefficient ratio Number of teeth
? N
6×108Nm−1 − γ 0.33333 2
Table 2 Reference spectral radii μrsr (kR ) at different parameter points computed with S1/3NIM of Ding et al. [2].
Ω [rpm]
w [mm]
μrsr (kR )
5000 5000 10000 8500
0.2 1.0 1.5 2.5
0.6349þ0.5184i 0.1234 þ 1.4010i 0.7971þ1.0728i 0.8007þ 0.6526i
56
Letter to the Editor / International Journal of Machine Tools & Manufacture 103 (2016) 53–56
References [1] Z. Zhang, H. Li, G. Meng, C. Liu, A novel approach for the prediction of the milling stability based on the Simpson method, Int. J. Mach. Tools Manuf. 99 (2015) 43–47. [2] Y. Ding, L.M. Zhu, X.J. Zhang, H. Ding, Numerical integration method for prediction of milling stability, J. Manuf. Sci. Eng. 133 (2011) 031005–031009. [3] K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering (Low prize), Cambridge University Press (1999), pp. 356–361. [4] J. Niu, Y. Ding, L. Zhu, H. Ding, Runge–Kutta methods for a semi-analytical prediction of milling stability, Nonlinear Dyn. 76 (2014) 289–304. [5] P.V. Bayly, T.L. Schmitz, G. Stepan, B.P. Mann, D.A. Peters, T. Insperger. Effects of radial immersion and cutting direction on chatter instability in end-milling. In: Proceedings of IMECE’02 2002 ASME International Mechanical Engineering
Conference & Exhibition New Orleans, Louisiana, November 17–22, (2002).
C.G. Ozoegwu Department of Mechanical Engineering, Nnamdi Azikiwe University, PMB 5025, Awka, Nigeria E-mail address: [email protected] Received 14 January 2016; accepted 19 January 2016 Available online 20 January 2016
Contents lists available at ScienceDirect
International Journal of Machine Tools & Manufacture journal homepage: www.elsevier.com/locate/ijmactool
Letter to the Editor Comment on “A novel approach for the prediction of the milling stability based on the Simpson method” by Z. Zhang, H. Li, G. Meng, C. Liu, [Int. J. Mach. Tools Manuf. 99 (2015) 43–47] Zhang et al. [1] has recently developed a Simpson 1/3 numerical integration method (S1/3NIM) for prediction of milling stability. Their effort is commendable as the resulting map is concise, fast and accurate compared with the well-accepted pre-existing First order Semi-discretization and Second order Full-discretization methods. However it is noticed that Zhang et al. [1] did not compare with the Simpson 1/3 method proposed by Ding et al. [2] four years earlier. It is the opinion in this letter that this gap deserves some comparative comment in terms of computational accuracy, convergence and computational time (CT). The milling model considered by the two methods has the first-order form
ẋ (t )=Ax (t )+B (t ) x (t )−B (t ) x (t−τ )
(1)
The two methods involve discretizing the delay τ of the system into k equal discrete time intervals [ti, ti +1] where i=0,1,2,………(k − 1), τ ti=i k =i∆t =i (ti +1−ti ) and then solving the arising discrete problem via different integration approaches. Ding et al. [2] solved the problem by double-step integration of Eq. (1) spanning [ti − 1, ti ] and [ti, ti +1] for all i=1,2,………(k − 1) to get
x i + 1 = e2A ∆ t x i − 1 +
∫t
ti + 1
e A ( ti + 1− t ) {B (t ) x (t ) − B (t ) x (t − τ )} dt
i −1
(2a)
For i=0 the integration is one-step on discrete time interval [t0, t1] to have the form
∫t
x1 = e A ∆ t x 0 +
t1
e A ( t1− t ) {B (t ) x (t ) − B (t ) x (t − τ )} dt
0
(2b)
The corresponding definite integrals by Zhang et al. [1] are respectively
∫t
xi + 1 = xi − 1 +
ti + 1
{B (t ) x (t ) − B (t ) x (t − τ )} dt
i −1
x1 = x 0 +
∫t
t1
{B (t ) x (t ) − B (t ) x (t − τ )} dt
0
(3a)
(3b)
It is seen that the two methods differ in that Ding et al. [2] is
http://dx.doi.org/10.1016/j.ijmachtools.2016.01.005 0890-6955/& 2016 Elsevier Ltd. All rights reserved.
based on the Convolution integral while Zhang et al. [1] can be said to be based on the Euler approximate method for solution of ordinary differential equations. Thus the S1/3NIM by Ding et al. [2] was derived from exact solution while that of Zhang et al. [1] was derived from an approximate solution. This means that the S1/3NIM of Ding et al. [2] is expected to be more accurate. Also the S1/3NIM of Ding et al. [2] benefits from the resolution and smooth behavior of convolution [3] thus expected to be more numerically stable than S1/3NIM by Zhang et al. [1]. Finally the S1/3NIM of Zhang et al. [1] is expected to save more CT because exponential matrices do not appear in its monodromy matrix. These expectations are confirmed by numerical simulation presented in Figs. 1 and 2. The input parameters for the numerical simulation which were also utilized by both Zhang et al. [1] and Ding et al. [2] are summarized in Table 1. Fig. 1 shows the stability diagrams computed on a 200 by 100 gridded plane of spindle speed and depth of cut (w) for a fullyimmersed system at discretization level of k = 40 and compared with a reference stability diagram computed with S1/3NIM of Ding et al. [2] at discretization level of k = 80 (the red lines). It is seen in Fig. 1 that the S1/3NIM of Zhang et al. [1] reflected less accuracy than the S1/3NIM of Ding et al. [2] being more out of coincidence with the reference diagram especially in the low speed domain. CTs are indicated in the caption of Fig. 1 to show that the Zhang et al. [1] is only slightly faster than the Ding et al. [2]. The rate of convergence plots are given as r (k )= μsr (k )−μrsr (kR ) versus k where μsr (k ) is spectral radius at k and μrsr (kR ) is the reference spectral radius of high accuracy at reference high value of k designated kR . kR=1000 is used in the S1/3NIM of Ding et al. [2] to calculate reference spectral radii μrsr (kR ) at different parameter combinations (Table 2) and used in rate of convergence plots given in Fig. 2. It is seen that S1/3NIM of Ding et al. [2] reflected faster convergence. Also it is seen as expected that S1/3NIM of Zhang et al. [1] lacks the numerical smoothness of convolution. These finding would also explain why milling stability prediction in [4] using the classical fourth-order Runge–Kutta method was of poor accuracy (especially at coarser discretization) relative to the zeroorder and the first-order semi-discretization methods while the generalized Runge–Kutta method based on convolution integral resulted in superior accuracy in all levels of discretization than the semi-discretizations methods.
54
Letter to the Editor / International Journal of Machine Tools & Manufacture 103 (2016) 53–56
Fig. 1. The stability diagrams of 1DOF fully-immersed milling (the black curves) computed with k = 40 from maps generated from (a) S1/3NIM of Ding et al. [2] with CT¼ 117 s and (b) S1/3NIM of Zhang et al. [1] CT¼ 110secs. It is seen on comparison with the reference stability diagram generated from the S1/3NIM of Ding et al [2] at discretization level k ¼ 80 (the red lines) that the older S1/3NIM of Ding et al. [2] is more accurate than the very recent S1/3NIM of Zhang et al. [1]. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Letter to the Editor / International Journal of Machine Tools & Manufacture 103 (2016) 53–56
(a)
(c)
0.16 S1/3NIM of Ding et al [2] S1/3NIM of Zhang et al [1]
0.14
S1/3NIM of Ding et al [2] S1/3NIM of Zhang et al [1]
0.5
0.1
0.4
0.08
r(k)
r(k)
0.7 0.6
0.12
0.06
0.3 0.2
0.04
0.1
0.02
0
0 20
40
60
80
100
20
120
40
60
80
120
(d)
0.2 S1/3NIM of Ding et al [2] S1/3NIM of Zhang et al [1]
S1/3NIM of Ding et al [2] S1/3NIM of Zhang et al [1]
0.04
0.15
0.03
0.1
r(k)
r(k)
100
k
k
(b)
55
0.02
0.05
0.01
0
0
20
40
60
80
100
120
20
40
60
80
100
120
k
k
Fig. 2. Sample numerical rate of convergence of spectral radii of S1/3NIM of Ding et al. [2] and S1/3NIM of Zhang et al. [1] for fully-immersed 1 DOF milling at (a) Ω = 5000 rpm and w¼ 0.2 mm, (b) Ω = 5000 rpm and w¼ 1 mm, (c) Ω=8500 rpm and w¼ 2.5 mm and (d) Ω=10000 rpm and w¼ 1.5 mm. It is seen that the older S1/3NIM of Ding et al. [2] exhibit better convergence and Numerical smoothness than the recent S1/3NIM of Zhang et al. [1]. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.).
Table 1 The Input parameters for numerical simulations. Source [5]. Natural frequency of 1DOF tool
ωn
5793rads−1
Mass of 1DOF tool
m
Damping ratio of 1DOF tool Feed exponent in the cutting force law tangential cutting coefficient
ξ γ Ct
0. 03993kg 0.011 1
Normal to tangential cutting coefficient ratio Number of teeth
? N
6×108Nm−1 − γ 0.33333 2
Table 2 Reference spectral radii μrsr (kR ) at different parameter points computed with S1/3NIM of Ding et al. [2].
Ω [rpm]
w [mm]
μrsr (kR )
5000 5000 10000 8500
0.2 1.0 1.5 2.5
0.6349þ0.5184i 0.1234 þ 1.4010i 0.7971þ1.0728i 0.8007þ 0.6526i
56
Letter to the Editor / International Journal of Machine Tools & Manufacture 103 (2016) 53–56
References [1] Z. Zhang, H. Li, G. Meng, C. Liu, A novel approach for the prediction of the milling stability based on the Simpson method, Int. J. Mach. Tools Manuf. 99 (2015) 43–47. [2] Y. Ding, L.M. Zhu, X.J. Zhang, H. Ding, Numerical integration method for prediction of milling stability, J. Manuf. Sci. Eng. 133 (2011) 031005–031009. [3] K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering (Low prize), Cambridge University Press (1999), pp. 356–361. [4] J. Niu, Y. Ding, L. Zhu, H. Ding, Runge–Kutta methods for a semi-analytical prediction of milling stability, Nonlinear Dyn. 76 (2014) 289–304. [5] P.V. Bayly, T.L. Schmitz, G. Stepan, B.P. Mann, D.A. Peters, T. Insperger. Effects of radial immersion and cutting direction on chatter instability in end-milling. In: Proceedings of IMECE’02 2002 ASME International Mechanical Engineering
Conference & Exhibition New Orleans, Louisiana, November 17–22, (2002).
C.G. Ozoegwu Department of Mechanical Engineering, Nnamdi Azikiwe University, PMB 5025, Awka, Nigeria E-mail address: [email protected] Received 14 January 2016; accepted 19 January 2016 Available online 20 January 2016