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Chatter reliability prediction of turning process system with uncertainties
Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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Chatter reliability prediction of turning process system with uncertainties Yu Liu a,n, Tian-xiang Li b, Kuo Liu c, Yi-min Zhang a a b c
School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819, China School of Mechanical Science & Engineering, Huazhong University of Science and Technology, Wuhan 430074 China State Key Laboratory of Advanced Numerical Control Machine Tool, Shenyang Machine tool (Group) Co., Ltd., Shenyang 110142, China
a r t i c l e i n f o
abstract
Article history: Received 4 November 2014 Received in revised form 12 June 2015 Accepted 30 June 2015
In this paper, reliability analysis of dynamic structural system is introduced into chatter vibration prediction of a turning process system. Chatter reliability is defined to represent the probability of stability (no chatter occurs) for a turning process system. Probability model (reliability model) of chatter vibration is established to predict turning chatter vibration, in which structural parameters m, c, k and spindle speed Ω are considered as random variables. Choosing chatter frequency ωc as an intermediate variable, reliability model is developed from a model impossible to solve to a new model related to chatter frequency, and the new model can be solved. The first-order second-moment, fourth moment method are adopted to solve the turning process system reliability model and obtain the reliability probability of the system. An example is used to demonstrate the feasibility of the proposed method. The reliability probability of turning chatter system was calculated using the FOSM method, fourth moment and compared with that calculated by Monte Carlo simulation method. The results using the three methods were consistent. Reliability lobe diagram (RLD) is proposed to identify the chatter and no chatter regions for chatter prediction instead of stability lobe diagram (SLD). Comparing with the traditional SLD method, chatter reliability and RLD can be used to judge the probability of stability of turning process system. The RLD and the index of chatter reliability have better prospects in workshop application. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Turning process system Chatter reliability Stability lobe diagram First-order second-moment Fourth moment Reliability lobe diagram
1. Introduction Regenerative chatter is a type of self-excited vibration with a time delayed displacement feedback in the turning process system. An important characteristic property is that external periodic forces do not induce chatter vibration. However, the forces, which bring it into being and maintain it, are generated during the vibratory process (dynamic cutting process). The disastrous nature of chatter vibration creates numerous problems such as poor surface finish, excessive noise, breakage of machine tool components, along with reduced tool lifetime and productivity. Extensive research have been carried out to avoid regenerative chatter using prediction methods, real time detection, or simply controlling chatter vibrations with active or passive strategies. However, chatter is still among the most complicated problems for a machinist. n Correspondence to: School of Mechanical engineering and Automation, Northeastern University Box 319, NO. 3-11, Wenhua Road, Heping District, Shenyang, P. R. China,110819. Tel.: þ 86 2483691002; Mobile: þ86 13072498580. E-mail address: [email protected] (Y. Liu).
http://dx.doi.org/10.1016/j.ymssp.2015.06.030 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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Tobias [1] proposed the idea that chatter vibrations stem from the instability of the machining system. They used an orthogonal cutting model to analyze turning stability. One technique used for pre-process chatter prediction and avoidance is the well-known stability lobe diagram (SLD). The stability lobe diagram identifies stable and unstable cutting zones based on parameters such as cutting width in turning and spindle speed. Altintas and co-workers [2] developed an analytical method to predict stability, where the stability lobes were directly determined in the frequency domain. This method is known as zero-order approximation. To improve prediction accuracy, Budak and Altintas suggested a higher-order model to predict the stability of the cutting process. However, Stépán et al. [3–5] employed a semi-discretization scheme determine stability parameters in discrete time domain. Ding et al. [6–8], proposed a full-discretization method to obtain the stability lobes diagram in time domain. Various groups have reported in literature regarding the various aspects of analytical models based on the degrees of freedom (DOF) [9] and flexibility of tool–workpiece system [10]. In this paper, the orthogonal cutting model and the SLD theory proposed by Altintas have been adopted to obtain the SLD because of the factors of the analytical formulations. When we evaluate the stability of the turning process system using any of the above methods, the structural parameters and the spindle speed are known. However, as suggested by Schmitz et al. [11], the measurement result is an approximation or estimation of the value of a specific measurand Thus, the result can be considered complete only when it is accompanied with a quantitative value expressing the measurement uncertainty. Based on this, we can conclude that the influence of the uncertainty on the structural parameters of turning process system to determine the SLD should be studied. Duncan et al. [12] studied the influence of random parameters on the stability lobes diagram in the milling process. For the first time, they used the mean value along with the lower and upper limit values based on the standard deviation to determine three curves representing the lobes. However, a quantitative index to represent the influence of uncertainty was not provided. Graham et al. [13] developed the robust chatter stability model taking into consideration the uncertainty in the natural frequency and the cutting coefficient. Sims et al. [14] applied the fuzzy arithmetic techniques to the chatter stability problem. It is shown that the fuzzy arithmetic can be used to solve process design problems with robustness to the uncertain parameters. For the uncertain factors in a practical milling process, Zhang [15] developed a speed optimization formulation, in which the upper bound of surface location error and lower bound of lobe diagram are adopted as the optimization object and the constraint condition, respectively. Random structural system reliability analysis is a method, which incorporates probability analysis along with probability design into structural analysis based on random factors. In reliability analysis, parameters such as reliability index and reliability probability are used to provide a quantitative index to represent the influence of uncertainty. A fundamental problem in structural reliability theory is the computation of the multi-fold probability integral, and difficulty in computing this probability has led to the development of various approximation methods. First-order second-moment method is considered to be one of the most reliable computational methods [16]. In the last decades, researchers have examined the shortcomings of FOSM, primarily accuracy and the difficulties involved in searching for the design point. In order to improve upon FOSM, fourth moment method was proposed and proved to be simple and no shortcomings with respect to design points [17,18]. Until now, reliability analysis studies including static and dynamic structural systems have made significant progress. The issue related to the reliability of dynamic structural systems mainly includes two aspects. The first is the structural response (displacement, stress, etc.) overrun, which is caused by forced vibration [19–22]. The second is fatigue caused by a resonant and non-resonant structure [23]. However, to the best of our knowledge reliability analysis of a dynamic structural system on the instability of self-excited vibration, e.g., regenerative chatter of a turning process system, has not be reported in literature. The goal of this paper is to introduce the ideas of reliability analysis of a dynamic structural system into structural analysis of a turning process system. A turning process system consists of a holder and a support, cutter, workpiece. The interaction between cutter and workpiece is a dynamic system referring to the machining condition. The variation of cutting forces between the cutter and causes the chatter vibration and is an internal system force. In this paper the dynamical model and SLD is reviewed. A chatter probability model is established to predict turning chatter vibration, in which structural parameters m, c, k, and spindle speed Ω are random variables. Chatter reliability is defined to represent the probability of stability (no chatter occurs) of turning process system. Choosing chatter frequency ωc as an intermediate variable, reliability model is transformed from a model impossible to solve to a new model related to chatter frequency, and the new model can be solved. First-order second-moment method is adopted to solve the turning process system reliability model and obtain the reliability of the system. The reliability lobe diagram is proposed to predict the chatter vibration. Finally, an example is used to demonstrate the effectiveness of the chatter reliability analysis and the RLD method. 2. Dynamic modeling of turning chatter 2.1. Turning machining dynamic model Fig. 1 shows the mechanical model for a regenerative chatter vibration cutting system of facing operation in turning. The following assumptions were made for the dynamic model: (1) the workpiece system is rigid and the tool holder system is the weakest link in the cutting system; (2) vibration system is linear and elastic resilience of the vibration system is proportional to vibration displacement; (3) direction of the dynamic cutting force and steady cutting force are the same, and Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
k
c
y m Fn β
h
feed
x
feed
Y. Liu et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
3
x
b
z
F Ft
Fig. 1. Dynamic model for the turning process system.
damping force is proportional to vibration velocity of the main vibration system; (4) the dynamic change in cutting thickness is only generated by regeneration. Using the dynamic model of the turning process system shown in Fig. 1, the expressions for the dynamic cutting force along the tool vibration direction can be expressed as follows: FðtÞ ¼ K s bhðtÞ
ð1Þ
hðtÞ ¼ hm þ xðt T Þ xðtÞ
ð2Þ
T¼
60
ð3Þ
Ω
where F is the cutting force (N), b is the cutting width (m), Ks is the cutting stiffness coefficient (N/m2) and h is the cutting thickness variation between the consecutive revolutions (m), hm is the mean chip thickness or the specified feed per revolution. The dynamic differential equation of machine tool vibration system is as follows: 60 mx€ ðtÞ þ cx_ ðtÞ þ kxðtÞ ¼ kn b hm þx t xðtÞ ð4Þ
Ω
where m is the equivalent mass of the vibration system (N s2/m), c is the equivalent damping of the vibration system (N s/m) and k is the equivalent stiffness of the vibration system (N/m), kn is the normal cutting stiffness coefficient (kn ¼Kscosβ N/m2). 2.2. Stability analysis review The free vibration equation of the motion of the turning process system can be modeled as x€ ðtÞ þ 2ωn ζ x_ ðtÞ þ ω2n xðtÞ ¼ 0
ω2n ¼
ð5Þ
k m
ð6Þ pffiffiffiffiffiffiffi
ζ ¼ c=ð2 mkÞ
ð7Þ
where ωn is the natural frequency of the cutting vibration system (rad/s), and ζ is the equivalent damping ratio of the cutting vibration system. Combining Eqs. (5)–(7) and using the Laplace transform, we get the frequency response function as shown in Eq. (8): 1 HðjωÞ ¼ 2 k ðjωω2Þ þ 2ωζω jþ 1 n
ð8Þ
n
Eq. (8) is split into real part and imaginary parts: ! 1 1 r2 Re ðHÞ ¼ k ð1 r 2 Þ2 þ ð2ζ rÞ2 I m ðHÞ ¼
1 2ζ r k ð1 r 2 Þ2 þð2ζ rÞ2
ð9Þ
! ð10Þ
where r ¼ ω/ωn. Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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The limiting cutting width blim of the cutting vibration system is shown in Eq. (11) [10]. The relationship between the chatter frequency ωc (rad/s) and spindle speed Ω (rev/s) is illustrated in Eqs. (12) and (13), where N ¼0, 1, 2, … is the integer number of vibration waves (lobe number), and ε (rad) is the phase (in rad) for an additional fraction of a wave. blim ¼
1 2K s Re ðHÞ
ωc 2π ¼ Nþ
Ω
ð11Þ
ε
ð12Þ
2π
ε ¼ 2π 2 tan 1
Re ðHÞ I m ðHÞ
ð13Þ
Fig. 2 shows a typical stability lobe diagram where the Ω versus blim family of curves separates the space into two regions. Any (Ω, b) pair that appears above the collective boundary indicates unstable behavior. Whereas, any pair below the boundary is presumed to be stable.
3. Turning process system chatter reliability modeling The typical stability lobe diagram is obtained using known and fixed values of parameters m, c, k. Fig. 3 shows this example stability lobe diagram obtained using the known structural parameters along with their uncertainty. Monte Carlo simulations were used to demonstrate the distribution of the stability lobe diagram. The parameters m, c, k represent normally distributed random variables with a mean value and standard variance. The mean values and standard deviations of each parameter are as follows: m (10.0610, 0.1) kg, c (1832.3, 30) N s/m and k (7.34 106, 1 105) N/m. Twenty of the samples were shown in Fig. 3.
Cutting width blim / (mm)
2.5 N=1
2 N=3
1.5
N=0
N=2 unstable
1 stable 0.5
0
1000
2000
3000
4000
5000
6000
Spindle speed Ω / (r/min) Fig. 2. Typical stability lobe diagram.
1.5
Cutting width blim /mm
1.4 1.3 1.2 1.1 1 0.9 0.8
1000
1500
2000
2500
3000
Spindle speed Ω / (r/min) Fig. 3. Typical stability lobe diagram from uncertain structural parameters.
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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3.1. Reliability modeling with m, c, k and
5
Ω
Generally, a structure fails if it cannot perform its intended function. It is obvious that the term “failure” can have different meanings. Before attempting a structural reliability analysis, failure must be clearly defined. The concept of a limit state is used to help define failure in the context of structural reliability analysis. A limit state is a boundary between desired and undesired performance of a structure. The limit state of a turning process system is defined as a critical stable state. “Failure” means losing machining stability, and the amplitude of vibration displacement of the cutter would be infinitely large due to the variation in the cutting force. However, “safety” or “reliability” implies stable machining, and the vibration of cutter attenuates due to damping. For the turning process, the machinists need to determine, if chatter would occur or in other words what would is probability of no chatter vibration for a given spindle speed Ω and cutting width b. The chatter reliability of the turning process is defined as the probability of no chatter vibration occurring in a dynamic system with random structural parameters m, c, k and manufacturing parameters Ω and b. The performance function of the turning process system in critical stable state can be expressed as g u ðUÞ ¼ b blim
ð14Þ
where b is the given cutting width (m), U is the random variables vector and U¼(m, c, k, Ω) . The three structural parameters m, c, k and the spindle speed Ω are dependent random variables with a known distribution. Then, the turning process system reliability refers to the probability under given process parameters when the given cutting width is less than the limiting cutting width. The reliability model is defined as Z Rs ¼ Pðg u ðU Þ o 0Þ ¼ f U ðuÞdu ð15Þ T
UR
where f U ðuÞ is the joint probability density of random vector, UR is the safe region of the basic variable space, that is, g u ðU Þ o0 in the UR region. P(.) is the probability function. However, g u ðU Þ is hard to be expressed as a function of U¼(m, c, k, Ω)T because of the existence of N. This is because that for a specified chattering frequency ωc, there is one cutting width blim and there are N spindle speeds Ω. Fortunately, we can express the performance function as the function of the new random vector X ¼(m, c, k, ωc)T. 3.2. Reliability modeling with m, c, k and
ωc
From Eqs. (6), (7), (9), (11), the expression for blim is obtained blim ¼
ðk ω2c mÞ2 þc2 ω2c 2K s ðk ω2c mÞ
ð16Þ
The turning process system reliability refers to the probability under the given process parameters when the given cutting width is less than the limiting cutting width. The new reliability model is defined as g X ðXÞ ¼ b blim Rs ¼ Pðg X ðX Þ o0Þ ¼
Z XR
f X ðxÞdx
ð17Þ
where X consists of m, c, k, ωc. Generally speaking, the structural parameters m, c and k are dependent normally distributed, and the ωc is related to the m, c and k. The distribution of ωc is determinated by the random variables m, c, k and Ω. The reliability model related to the chatter frequency is now analytically solvable. 4. First-order second-moment method of solving the model 4.1. Chatter frequency In the turning process system, chatter frequency can be written as a function of structural parameters m, c, k and spindle speed Ω. 1 R ðHÞ ð18Þ ωc ¼ 2πΩ N þ 1 tan 1 e I m ðHÞ π where
ωc is the chatter frequency (rad/s), N is the stable lobe number, N ¼0, 1, 2,……, and Ω is the spindle speed (rev/s).
4.2. Distribution parameters estimation of chatter frequency We assume that structural parameters m, c, k in the turning process system and the processing parameter Ω obey normal distribution. In order to calculate the reliability of turning process system, we need to obtain the mean value and standard variance. From Eqs. (6), (7), (9), (10) and (18), it can be observed that the chatter frequency is a function of random variables Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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m, c, k, Ω. According to the mean value and variance of a given parameter obtained by experimental testing, the random variables of m, c, k and Ω are generated using the random number generator in Matlab. We can calculate ωc for every sampling group. In order to determine the distribution of the samples, a goodness-of-fit test is needed. The examples show that the chatter frequency distribution obeys the normal distribution. We can estimate the overall mean value and variance according to value of ωc and then estimate the correlation coefficients between chatter frequency ωc and variables m, c, k respectively. 4.3. AFOSM method for determination of turning process system reliability Advanced first-order second-moment (AFOSM) method linearizes the performance function at the Taylor expansion points on the failure surface. The method uses the actual distribution of basic random variables. Reliability evaluation of turning system is as follows. The limit state function for turning system is Z ¼ g X ðXÞ ¼ 0
ð19Þ T
Assume that the basic random variable X ¼(X1, X2, X3, X4) are relevant random variables of the normal distribution. Xi (i¼1, 2, 3, 4) is m, c, k and ωc respectively. ωc is related to m, c and k, and the correlation matrix of the turning system is 0 1 1 0 0 ρX 1 X 4 B 0 1 0 ρX 2 X 4 C B C C ρ¼B ð20Þ B 0 0 1 ρX 3 X 4 C @ A ρX 4 X 1 ρX 4 X 2 ρX 4 X 3 1 where ρXiXj is the correlation coefficient of variable Xi and Xj. The standard deviation σi (i¼1, 2, 3, 4) of each random variable is matrix of turning system is 0 1 σ 21 0 0 ρX 1 X 4 σ 1 σ 4 B C B 0 σ 22 0 ρX 2 X 4 σ 2 σ 4 C B C C¼B C B 0 0 σ 23 ρX 3 X 4 σ 3 σ 4 C @ A
ρX 4 X 1 σ 1 σ 4 ρX 4 X 2 σ 2 σ 4 ρX 4 X 3 σ 3 σ 4
σm, σc, σk and σωc respectively, and the covariance
ð21Þ
σ 24
In Eq. (21), matrix C is a 4 4 symmetric positive definite matrix. The matrix has four real characteristic roots and four linearly uncorrelated and orthogonal characteristic vectors. Assuming that the columns of matrix A consist of regularization characteristic vectors of C. Orthogonal transformation is implemented on vector X, which consists of random variables of the system. X ¼ AY
ð22Þ
μY ¼ AT μX
ð23Þ
σ Y ¼ AT CA
ð24Þ
Limit state function can be expressed of the uncorrelated normal random variable Y as Z ¼ g X ðXÞ ¼ g X ðAYÞ ¼ g Y ðYÞ
ð25Þ
The derivative of the random variable Y is calculated using the checking point method, and is given as ∂g Y ðYÞ ∂g ðXÞ ¼ AT X ∂Y i ∂X i
ð26Þ
Choosing an average point X* ¼(m*c*k*ωc*) for the initial check point. Thus, the initial value for Y* Y n ¼ AT μX
ð27Þ
The partial derivatives of random vector X are as follows: ∂g X ðXÞ ω2 ð c2 ω2 þk 2kmω2 þ m2 ω4 Þ ¼ ∂m 2K s ðk mω2 Þ2
ð28Þ
∂g X ðXÞ cω2 ¼ ∂c K s ðk mω2 Þ
ð29Þ
∂g X ðXÞ 1 c2 ω2 ¼ ∂k 2K s 2K s ðk mω2 Þ2
ð30Þ
2
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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∂g X ðXÞ mω c2 kω ¼ þ ∂ωc K s K s ðk mω2 Þ2
7
ð31Þ
Substituting Eqs. (28)–(31) into the Eq. (26), we obtain the derivative of linearly uncorrelated random variables Y. In the space of random variable Y, equation ZL ¼0 is the limit state tangent plane which passes by the point Y*. By using the properties of linear combination of independent random variables, we obtain the mean value and standard deviation of ZL:
μZ L ¼ gðY n Þ þ
σ ZL
4 X ∂gðY n Þ
∂Y i
i¼1
ðμY i Y ni Þ
ð32Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 u X ∂gðY n Þ2 ¼t σ 2Y i ∂Y i i¼1
ð33Þ
We obtain reliability index of the system:
β¼
μZ L σ ZL
ð34Þ
The sensitivity coefficient of variable Yi is defined as ∂gðY n Þ Yi ∂Y
σ
i cos θY i ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 4 h P ∂gðY n Þ
i¼1
∂Y i
ð35Þ
σYi
Thus, the new Y* is Y n ¼ μY i þ βσ Y i cos θY i
ð36Þ
The new X* is X n ¼ AY n
ð37Þ
Setting the error as ε ¼10 , multiple iterations were carried out until the difference between two consecutive iterations ||X*|| was less than ε. The value of β obtained is substituted into Eq. (37). Reliability probability of the system is as follows: 6
Rs ¼ ΦðβÞ
ð38Þ
where Φð:Þ is the cumulative distribution function of standard normal distribution. 5. Fourth-moment method of solving the model The second order partial derivative matrix is as Eq. (22) 0 1 2 6 cω4 c2 ω4 ω 2c2 kω3 K ðk K ðkcωmω2 Þ3 K s K s ðk mω2 Þ3 mω2 Þ2 K s ðk mω2 Þ3 s s B C B C cω4 ω2 cω2 2ckω B C Þ 2 2 2 2 2 2 2 2 K s ðk mω Þ K s ðk mω Þ K s ðk mω Þ K s ðk mω Þ C ∂ g X ðXÞ B C ¼B c2 ωðmω2 þ kÞ B C 2 c2 ω4 cω2 c2 ω2 ∂X B K s ðk mω2 Þ3 C 2 3 3 K s ðk mω2 Þ K s ðk mω2 Þ K s ðk mω2 Þ B C @ω c2 ωðmω2 þ kÞ c2 kð3mω2 þ kÞ A 2c2 kω3 2ckω m Ks Ks K ðk mω2 Þ3 K ðk mω2 Þ3 K ðk mω2 Þ3 K ðk mω2 Þ2 s
s
s
ð39Þ
s
The first four moment of the random variable vector X is
μX 1 ¼ 0
ð40Þ
μX 2 ¼ σ 2X
ð41Þ
μX 3 ¼ C sX σ 3X
ð42Þ
μX 4 ¼ C kX σ 4X
ð43Þ
The mean, standard variance, three-moment, four moment of the performance function can be calculated as Eqs. (27)– (30). ! n 1X ∂2 g μ X μg ¼ E½gðX Þ ¼ g μX þ σ 2X i ð44Þ 2i¼1 ∂X 2i Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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T !2
∂g X ðXÞ ∂g X ðXÞ σ m σ ω ρmω ∂X m ∂X ω ∂g ðXÞ ∂g X ðXÞ ∂g ðXÞ ∂g X ðXÞ þ X σ c σ ω ρcω þ X σ σ ω ρkω ∂X c ∂X ω ∂X k ∂X ω k
σ g ¼ Var½gðX Þ ¼
θg ¼ C 3 ½gðX Þ ¼
n X
∂g X ðXÞ ∂X
!3 ∂g μX ∂X Ti
i¼1
ηg ¼ C 4 ½gðX Þ ¼
n X
!4 ∂g μX
i¼1
∂X Ti
σ 2X þ2
ð45Þ
μX 3
ð46Þ
μX 4
ð47Þ
Thus, we obtain the coefficient of skewness of performance function is
α3g ¼ θg =σ 3g
ð48Þ
The coefficient of kurtosis (the fourth dimensionless central moment) of performance function is
α4g ¼ ηg =σ 4g
ð49Þ
The reliability index based on the second moment method is
βSM ¼ μg =σ g
ð50Þ
We can obtain the reliability index and reliability probability based on the fourth moment method are obtained as
3 α4g 1 β SM þ α3g βSM 1 ð51Þ βFM ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
9α4g 5α23g 9 α4g 1
Rs ¼ Φ βFM
ð52Þ
6. Reliability lobes diagram Within the (Ω, b) plane, using a grid with an appropriate increment, the reliability value at the node whose coordinates are (Ωm, bn) can be calculated. If the data is sufficient, a contour line can be obtained for the given reliability level P nr . The contour plots in the (Ω, b) plane are defined as reliability lobe diagram and has a lobe shape. Fig. 4 shows an example of reliability lobe diagram. Compared with the stability lobe diagram, reliability lobe diagram can be used to estimate the reliability at a selected point. As shown in Fig. 4, the red line represents the stability lobe diagram, and the blue line represents the reliability lobe diagram. Any (Ω, b) pair that is above the blue line indicates unreliable behavior, and the probability of stability (no chatter occurs) is greater than P nr However, any pair that appears below the boundary is presumed to be reliable, and probability of stability (no chatter occurs) is less than P nr . 3
SLD
Cutting width blim(mm)
2.5
2
1.5
RLD
1
R= pr
*
0.5
Reliable region 0 2000
4000
6000
8000
10000
12000
Spindle speed Ω (rpm) Fig. 4. An example of reliability and stability lobe diagram. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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7. A case study 7.1. Data acquisition and analysis of turning experiment The experiment was carried out on a turning system (RUDI, China, Model: CJ0625). A vibration signal acquisition system (B&Ks, Denmark, model:3560-B) and Pulse analysis software (B&Ks, Denmark), modal hammer (PCBs, USA.model:086C01) and an accelerometer (PCBs, USA, model:356A24) were used for data collection in the experiments carried out on the turning process system. The frequency response function test was carried out along the x-direction of the tool. The locations of accelerometer and hammer tapping are shown in Fig. 5. The FRF diagram obtained directly from the test system is shown in Fig. 6. When the acquisition system captures the frequency response function at the measurement points on the lathe, we can identify the parameters according to the frequency response function and determine m, c and k. The results from the experiments are listed in Table 1.
z
x
y Fig. 5. The dynamic structure test on the turning process system.
Real/(m/N)
5
-7
0
-5
0
Imag/(m/N)
x 10
50 x 10
100
150
100
150
200
250
300
350
400
200
250
300
350
400
-6
-0.5
-1
50
Freq. / Hz Fig. 6. FRF curve of the X axis-tool from PULSE LabShops.
Table 1 Identified values of m, c and k of the turning system. Direction
ωn (rad/s)
fn (Hz)
m (kg)
c (N s/m)
k (N/m)
(Tool) X
854.1
136
10.0610
1832.3
7.34e6
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As shown in Table 2, modal parameters m, c and k were identified from the tested FRF. The natural frequency in xdirection of the tool was found to be 136 Hz. Since the first two mode amplitudes in the natural frequency, 71 Hz and 93 Hz are much smaller than the amplitude of the third mode in the natural frequency 136 Hz, the third mode had a major influence on the SLD. As a result, the structural parameters m, c and k were selected based on the 3rd mode in x-direction.
7.2. Distribution of the chatter frequency in turning process In the turning process, it is hard to get the exact value of the dynamic parameters due to factors affecting the equipment, sensors, temperatures, operators etc. Therefore, it is reasonable to consider them as random variables. After 20 tests, mean values and standard deviations of each parameter were determined and found to be: m (11.78, 0.68) kg, c (1464.9, 79.6) N s/m and k (8.22 106, 4.20 105) N/m. When the machine works normally, the spindle speed can be considered as normally distributed, and the standard variance is set to Ω (Ω0, 1.2425) rpm. Ω0 is the mean value of spindle speed. Consider a single DOF system, where the dynamic parameters m, c and k are equal to the mean value of the test result. With the spindle speed increasing, the change of the limiting cutting width blim of turning system and the chatter frequency ωc are shown in Fig. 7. The dashed line represents the limiting cutting width, while the solid line represents chatter frequency with respect to the spindle speed. The parameters m, c, k and Ω were sampled 10,000 times and we can obtain the samples of chatter frequency ωc. The distribution histogram under different speeds is shown in Fig. 8. When the dynamic parameters m, c, k and Ω of turning process system are normally distributed, chatter frequency of the turning system will change in accordance with a normal Table 2 20 samples of test modal parameters. fn (Hz)
m (kg)
c (N s/m)
k (N/m)
Ω (rpm)
132.5 133 133.5 133.5 133 133 133.5 133.5 133.5 133.5 134 132.5 132.5 132.5 132.5 133 133 132 132 133
12.54 11.32 10.92 11.45 11.83 11.61 11.49 10.70 10.70 10.84 11.20 12.59 12.33 12.88 12.21 12.16 11.86 12.53 12.19 12.24
1536.07 1422.94 1371.93 1402.60 1486.57 1422.94 1407.56 1344.09 1344.09 1362.52 1407.30 1541.98 1549.94 1578.46 1495.90 1490.27 1490.27 1574.09 1569.99 1499.61
8.69 106 7.91 106 7.68 106 8.05 106 8.26 106 8.11 106 8.08 106 7.53 106 7.53 106 7.63 106 7.94 106 8.72 106 8.55 106 8.93 106 8.46 106 8.49 106 8.28 106 8.62 106 8.38 106 8.55 106
2998.4 3003.6 3000.9 2999.9 3000.8 2999.8 2999.9 3001.8 3001.7 3001.7 3000.8 2998.6 3000.9 3001.9 3000.6 3001.2 3000.9 2999.6 3000.3 2999.1
2500
2000
2
1500
1.5
1000
1
500
0
Cutting width /mm
Chatter freq. ω c /(rad/s)
chatter freq. cutting depth
0.5
1000
2000
3000
4000
5000
6000
Spindle speed /(r/min) Fig. 7. Stability lobe diagram and the chatter frequency diagram of system with mean values.
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
Ω =2000 rpm fitted normal density
400
200
0 980
400
fitted normal density
200
0 985
990
970
Chatter Freq./(rad/s)
980
Chatter Freq./(rad/s)
Ω = 3500 rpm
Ω = 4000 rpm 400
fitted normal density
200
0 980
990
Freq. of occurrence
400
Freq. of occurrence
11
Ω = 2500 rpm
Freq. of occurrence
Freq. of occurrence
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fitted normal density
200
000
0 1098
Chatter Freq./(rad/s)
1100
1105
Chatter Freq./(rad/s)
Fig. 8. Histogram of chatter frequency with fitted normal density at different speeds.
distribution. Although the conclusion is drawn from the cases that spindle speed is 2000, 2500, 3500 and 4000 rpm, it is same for all spindle speed. In Fig. 7 it can be seen that chatter frequency changes sharply near the intersection of the limiting cutting width, so the chatter frequency is not consecutive mathematically. It is not suitable for calculating the reliability of the points near the intersection directly using the FOSM and fourth moment method. To overcome the shortcoming, the reliability can be calculated from both the adjacent leaves of SLD and the minimum value of the results is chosen as the reliability at this point. At a spindle speed Ω ¼ 3500 rpm the Lilliefors goodness-of-fit test on samples of chatter frequency ωc can be determined. With the condition of the significance level being 5%, the test results show that the chatter frequency of turning system obeys normal distribution or approximately obeys normal distribution. When the spindle speed is set to Ω ¼3500 rpm, the correlation coefficients of four random variables samples of the turning system were calculated. The correlation coefficient between m and ωc is 0.6581, the correlation coefficient between c and ωc is 0.2728, and the correlation coefficient between k and ωc is 0.6762. 7.3. Reliability of the turning process system using the Monte Carlo method According to the mean value, standard variance and the correlation coefficients of the random vector X, Monte Carlo sampling was implemented. A million of samples of X were obtained. Substituting the X value into the Eq. (39), we can calculate the ratio RM ¼
nr N
ð39Þ
in which nr is the times of the gX(X)o0, N is the total number of the sampling and RM is the reliability from Monte Carlo method. The stability lobe diagrams are shown in Fig. 9 for the mean value, mean value plus standard variance and the mean value minus standard variance of the random vector. More than 100 samples were verified, and the values of reliability were calculated. Each point represents the data pairs (Ω, b). The reliability is indicated as a “dot” when the value is greater than 0.9 and a “cross” when the value is less than 0.9. 7.4. Reliability of the turning process system using the FOSM method In order to guarantee stability of the turning process system, the condition of gX(X) o0 should be satisfied. If the cutting width b is given, the reliability of the turning process can be calculated using the FOSM method, and the results are compared with those using Monte Carlo method as shown in Table 2. A custom program was implemented in MATLAB (Mathworks, USA). Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
Y. Liu et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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Cutting width b /(mm)
2
mean+standard variance mean
Rs>0.9 + Rs ≤ 0.9
mean-standard variance 1.5
1
0.5
0
1000
2000
3000
4000
5000
6000
Spindle speed Ω /(r/min) Fig. 9. Reliability of the turning process system.
Ω=2000rpm
Ω=2500rpm
1
1 MC FOSM FM
Ω=2000rpm
0.6 0.4 0.2 0 0.86
0.6 0.4 0.2
0.88
0.9
0.92
0.94
0.96
0.98
0 0.84
1
0.86
cutting width/mm
Ω=3500rpm
0.9
0.92
0.94
0.96
0.98
Ω=4000rpm 1 MC SOFM FOSM SOFM
0.8
MC SOFM FOSM SOFM
0.8
0.6
Reliability
Reliability
0.88
cutting width/mm
1
0.4 0.2
0.6 0.4 0.2
0.92 0 0.9
MC FOSM SOFM
Ω=2500rpm
0.8
Reliability
Reliability
0.8
0.92
0.94 0.96 cutting width/mm
0.94
0.96
cutting width/mm
0.98
0.98
1
1
1.3 0 1.25
1.3
1.35 1.4 cutting width/mm
1.35
1.4
1.45
1.45
1.5
cutting width/mm
Fig. 10. Reliability of turning system at specific speed changes versus cutting width.
The mean value of each random variable of the system was chosen as the initial point. When the difference between the consecutive steps was less than 10 6, the iteration process was stopped. As shown in Fig. 10, “dotted line” represents the reliability calculated by the FOSM method versus the given cutting width, and “star line” represents the reliability calculated by Monte Carlo method versus the given cutting width. Table 2 and Fig. 5 show that the results from FOSM method and Monte Carlo method have consistency. Comparing with the traditional method SLD, FOSM method is not only able to judge the stability of the turning system, but also give the probability of stability.
7.5. Reliability lobe diagram The reliability value is set as 0.99. The reliability lobe diagram is shown in Fig. 11 with stability lobe diagram for comparison. In Fig. 11, the red line represents the reliability lobe diagram and the blue line represents the stability lobe diagram with a 0.99 level. Any (Ω, b) pairs below the blue line are prone to be stable and those above the red line are unstable, while any (Ω, b) pairs below the red line are reliable with a level of 0.99. The (Ω, b) pairs locating between the red and blue line are stable but unreliable, because the reliability is less than the level of 0.99 (Table 3). Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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2.5
Stability lobe diagram
Cutting width (mm)
2
Reliability lobe diagram 1.5
1
0.5
0
0
1000
2000
3000
4000
5000
6000
Spindle speed Ω (r/min) Fig. 11. Reliability lobe diagram. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Table 3 Reliability at a specific speed and the given cutting width. Spindle speed (rpm)
Cutting width (mm)
Monte Carlo
FOSM
SOFM
Relative error (%)
Relative error(%)
2000
0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48
0.999992 0.999943 0.999418 0.995576 0.977333 0.916426 0.780964 0.567168 0.334037 0.154406 0.997451 0.983126 0.925611 0.778642 0.538776 0.282741 0.106688 0.028405 0.005169 0.000632 0.999717 0.997555 0.985736 0.9413 0.825741 0.6238 0.380825 0.178348 0.062671 0.016044 0.999984 0.999798 0.997622 0.981674 0.914846 0.742982 0.474095 0.218227 0.06803 0.013859
0.999995988 0.999935931 0.999312594 0.9949823 0.974683778 0.909912456 0.76788194 0.551692621 0.320838344 0.145990286 0.996969971 0.980566336 0.917487828 0.762116804 0.516025255 0.264312865 0.097262972 0.024942432 0.004389717 0.000527374 0.999236865 0.995066455 0.984485544 0.936937314 0.817342802 0.611793755 0.369118503 0.171266086 0.059276682 0.015032765 0.999990381 0.999801932 0.997561275 0.98170982 0.914439328 0.741953400 0.473101636 0.217065441 0.067483736 0.013705943
0.99999421 0.999924114 0.999290965 0.995253121 0.977062263 0.919143292 0.788760239 0.581183392 0.347424436 0.1612121 0.997717095 0.984792802 0.932298736 0.794351545 0.559654994 0.30103559 0.116445791 0.031128411 0.005606638 0.00066947 0.999719249 0.997681595 0.986548366 0.944598935 0.835536596 0.639812444 0.397291966 0.189786557 0.067196183 0.017201182 0.999988766 0.999786567 0.99749934 0.981736255 0.915444204 0.745091177 0.477344008 0.219831302 0.068282474 0.013739408
0.000003988 0.000007069 0.000105479 0.000596694 0.002718032 0.007158429 0.017036551 0.028050727 0.041138026 0.057645712 0.000482491 0.002610394 0.008853711 0.021683285 0.044088433 0.069720916 0.096902529 0.138822389 0.177524656 0.198390516 0.000480502 0.002500883 0.001270162 0.004656326 0.010275001 0.019624661 0.031714739 0.041350358 0.057262281 0.067268729 0.000006381 0.000003933 0.000060873 0.000036487 0.000444723 0.001386340 0.002099684 0.005351193 0.008094750 0.011167200
0.000221025 0.001888722 0.012712524 0.032441941 0.027709337 0.295633104 0.988416778 2.411526598 3.853337517 4.221829385 0.026670404 0.169254095 0.71733829 1.977656522 3.730690255 6.077218323 8.379685165 8.748955615 7.805719055 5.596980885 0.00022492 0.012688948 0.082344233 0.349241867 1.172371893 2.502677739 4.14480208 6.027064061 6.734285345 6.72734038 0.000476584 0.001143575 0.012296714 0.006341347 0.065345721 0.283076361 0.680642838 0.729787858 0.369748666 0.870428599
2500
3500
4000
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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7.6. Experimental validation Cutting test was carried out on a lathe. For chatter criterion, cutting noise was recorded with the microphone for signal analysis. The surfaces with different cutting width were compared in Fig. 12. The spindle speed was 1600 rpm and the cutting width was 0.8 mm and 1.5 mm respectively. When the cutting width is 0.8 mm, the surface is smooth and cutting noise is low and steady, and the reliability is 0.999955. When the cutting width is 1.5 mm, there are a period wave with the wavelength is about 2 mm left on the surface which can be seen in Fig. 12. The reliability is negligible and approximately zero (3.42 10 233). Fig. 13 shows the noise signal recorded using the microphone and the power spectrum where the cutting width is 0.8 mm and 1.5 mm. It can be observed that when the cutting width is 0.8 mm, the magnitude of power spectrum at 344.7 Hz is 0.00112 Pa2, and for the cutting width is 1.5 mm, the magnitude at 346.1 Hz is 0.00603. For a cutting width of 1.5 mm, the magnitude is 6 times the magnitude at 0.8 mm.
b= 0.8mm stable
b= 1.5mm unstable
Fig. 12. Surface finish with different cutting width.
0.5
1
0
0.5
0
0.5
1
1 0 -1
0
0.5
1
0
1.5
1.5
50
Magnitudes (Pa2)
Magnitudes (Pa)
-0.5
2
x 10
-3
0 x 10
500
1000
1500
2000
-3
344.7Hz, 0.00112Pa2 1 0
0
500
1000
1500
2000
0.01
346.1Hz, 0.00603Pa2 0 -50
0.005
0
0.5
1
time（s）
1.5
0
0
500
1000
1500
2000
Freq.（Hz）
Fig. 13. Noise signal recorded using the microphone and the power spectrum for cutting width 0.8 mm and 1.5 mm.
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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SLD
15
stable point unstable point uncertain region
RLD, Rs=0.1 RLD, Rs=0.99
Fig. 14. Experimental validation. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Authors conducted more tests to validate the method mentioned in the article. The spindle speed is set at 1600 rpm and 2000 rpm, and cutting depth varies from 0.6 mm to 1.5 mm. We can obtain the critical values on the reliability lobes curve with a level of 0.99 are 0.853 mm and 0.905 mm at 1600 rpm and 2000 rpm respectively. The stable conditions which are identified through experimental tests are indicated as a “dot”, and the unstable conditions are indicated as a “cross”. Two reliability lobes with 0.99 and 0.1 levels are shown in Fig. 14. It is shown that, due to the uncertainty parameters, test results and prediction results obtained with the stability lobes do not perfectly match in the uncertain region with pink. However, the results from both of the experiment and prediction are same when the cutting depth located below the reliability lobes chart. The points above the “uncertain region” are clearly unstable, and the points below the “uncertain region” are clearly stable. It is shown that the method in this article can determinate the lower limit of the uncertain zone and we can choose the cutting parameters below the reliability lobes chart to avoid the occurrence of chatter. 8. Conclusions (1) Turning process system dynamics model was established and turning process chatter reliability model with random parameters was demonstrated, and FOSM, fourth moment methods was used for the reliability index and the reliability probability calculation. (2) The distribution of random parameters for the turning process system was determined experimentally, and the lobe diagram and chatter frequency curve for turning process system using mean values of random variables was plotted. Chatter frequency is a random variable with the dynamic parameters m, c, k and Ω and was found to be normal distributed. (3) The reliability probability of turning chatter system was calculated using the FOSM and fourth moment method and compared with that calculated using the Monte Carlo simulation. We conclude that results obtained using FOSM method and Monte Carlo methods were found to be consistent. Comparing with the traditional SLD method, chatter reliability method can be used to judge the probability of stability of turning process system. (4) Reliability lobe diagram was proposed to identify the chatter and no chatter regions for chatter prediction instead of stability lobe diagram (SLD). Comparing with the traditional SLD method, chatter reliability and RLD can be used to judge the probability of stability of turning process system.
Acknowledgments This work is supported by the National Natural Science Foundation of China (51105067, 51135003) and the Fundamental Research Funds for the Central Universities (N120403011). References [1] S.A. Tobias, Machine Tool Vibration, Blackie and Sons Ltd., Glasgow, Scotland, 1965. [2] E. Budak, Y. Altintas, Analytical prediction of stability lobes in milling, CIRP Ann. 44 (1995) 357–362. [3] T. Insperger, G. Stépán, Updated semi-discretization method for periodic delay-differential equations with discrete delay, Int. J. Numer. Methods Eng. 61 (2004) 117–141. [4] T. Insperger, G. Stépán, Semi-discretization method for delayed systems, Int. J. Numer. Methods Eng. 55 (2002) 503–518. [5] O. Elbeyli, J.Q. Sun, On the semi-discretization method for feedback control design of linear systems with time delay, J. Sound Vib. 273 (2004) 429–440. [6] Y. Ding, L. Zhu, X. Zhang, H. Ding, Second-order full-discretization method for milling stability prediction, Int. J. Mach. Tools Manuf. 50 (2010) 926–932. [7] Y. Ding, L. Zhu, X. Zhang, H. Ding, A full-discretization method for prediction of milling stability, Int. J. Mach. Tools Manuf. 50 (2010) 502–509.
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Chatter reliability prediction of turning process system with uncertainties Yu Liu a,n, Tian-xiang Li b, Kuo Liu c, Yi-min Zhang a a b c
School of Mechanical Engineering & Automation, Northeastern University, Shenyang 110819, China School of Mechanical Science & Engineering, Huazhong University of Science and Technology, Wuhan 430074 China State Key Laboratory of Advanced Numerical Control Machine Tool, Shenyang Machine tool (Group) Co., Ltd., Shenyang 110142, China
a r t i c l e i n f o
abstract
Article history: Received 4 November 2014 Received in revised form 12 June 2015 Accepted 30 June 2015
In this paper, reliability analysis of dynamic structural system is introduced into chatter vibration prediction of a turning process system. Chatter reliability is defined to represent the probability of stability (no chatter occurs) for a turning process system. Probability model (reliability model) of chatter vibration is established to predict turning chatter vibration, in which structural parameters m, c, k and spindle speed Ω are considered as random variables. Choosing chatter frequency ωc as an intermediate variable, reliability model is developed from a model impossible to solve to a new model related to chatter frequency, and the new model can be solved. The first-order second-moment, fourth moment method are adopted to solve the turning process system reliability model and obtain the reliability probability of the system. An example is used to demonstrate the feasibility of the proposed method. The reliability probability of turning chatter system was calculated using the FOSM method, fourth moment and compared with that calculated by Monte Carlo simulation method. The results using the three methods were consistent. Reliability lobe diagram (RLD) is proposed to identify the chatter and no chatter regions for chatter prediction instead of stability lobe diagram (SLD). Comparing with the traditional SLD method, chatter reliability and RLD can be used to judge the probability of stability of turning process system. The RLD and the index of chatter reliability have better prospects in workshop application. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Turning process system Chatter reliability Stability lobe diagram First-order second-moment Fourth moment Reliability lobe diagram
1. Introduction Regenerative chatter is a type of self-excited vibration with a time delayed displacement feedback in the turning process system. An important characteristic property is that external periodic forces do not induce chatter vibration. However, the forces, which bring it into being and maintain it, are generated during the vibratory process (dynamic cutting process). The disastrous nature of chatter vibration creates numerous problems such as poor surface finish, excessive noise, breakage of machine tool components, along with reduced tool lifetime and productivity. Extensive research have been carried out to avoid regenerative chatter using prediction methods, real time detection, or simply controlling chatter vibrations with active or passive strategies. However, chatter is still among the most complicated problems for a machinist. n Correspondence to: School of Mechanical engineering and Automation, Northeastern University Box 319, NO. 3-11, Wenhua Road, Heping District, Shenyang, P. R. China,110819. Tel.: þ 86 2483691002; Mobile: þ86 13072498580. E-mail address: [email protected] (Y. Liu).
http://dx.doi.org/10.1016/j.ymssp.2015.06.030 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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Tobias [1] proposed the idea that chatter vibrations stem from the instability of the machining system. They used an orthogonal cutting model to analyze turning stability. One technique used for pre-process chatter prediction and avoidance is the well-known stability lobe diagram (SLD). The stability lobe diagram identifies stable and unstable cutting zones based on parameters such as cutting width in turning and spindle speed. Altintas and co-workers [2] developed an analytical method to predict stability, where the stability lobes were directly determined in the frequency domain. This method is known as zero-order approximation. To improve prediction accuracy, Budak and Altintas suggested a higher-order model to predict the stability of the cutting process. However, Stépán et al. [3–5] employed a semi-discretization scheme determine stability parameters in discrete time domain. Ding et al. [6–8], proposed a full-discretization method to obtain the stability lobes diagram in time domain. Various groups have reported in literature regarding the various aspects of analytical models based on the degrees of freedom (DOF) [9] and flexibility of tool–workpiece system [10]. In this paper, the orthogonal cutting model and the SLD theory proposed by Altintas have been adopted to obtain the SLD because of the factors of the analytical formulations. When we evaluate the stability of the turning process system using any of the above methods, the structural parameters and the spindle speed are known. However, as suggested by Schmitz et al. [11], the measurement result is an approximation or estimation of the value of a specific measurand Thus, the result can be considered complete only when it is accompanied with a quantitative value expressing the measurement uncertainty. Based on this, we can conclude that the influence of the uncertainty on the structural parameters of turning process system to determine the SLD should be studied. Duncan et al. [12] studied the influence of random parameters on the stability lobes diagram in the milling process. For the first time, they used the mean value along with the lower and upper limit values based on the standard deviation to determine three curves representing the lobes. However, a quantitative index to represent the influence of uncertainty was not provided. Graham et al. [13] developed the robust chatter stability model taking into consideration the uncertainty in the natural frequency and the cutting coefficient. Sims et al. [14] applied the fuzzy arithmetic techniques to the chatter stability problem. It is shown that the fuzzy arithmetic can be used to solve process design problems with robustness to the uncertain parameters. For the uncertain factors in a practical milling process, Zhang [15] developed a speed optimization formulation, in which the upper bound of surface location error and lower bound of lobe diagram are adopted as the optimization object and the constraint condition, respectively. Random structural system reliability analysis is a method, which incorporates probability analysis along with probability design into structural analysis based on random factors. In reliability analysis, parameters such as reliability index and reliability probability are used to provide a quantitative index to represent the influence of uncertainty. A fundamental problem in structural reliability theory is the computation of the multi-fold probability integral, and difficulty in computing this probability has led to the development of various approximation methods. First-order second-moment method is considered to be one of the most reliable computational methods [16]. In the last decades, researchers have examined the shortcomings of FOSM, primarily accuracy and the difficulties involved in searching for the design point. In order to improve upon FOSM, fourth moment method was proposed and proved to be simple and no shortcomings with respect to design points [17,18]. Until now, reliability analysis studies including static and dynamic structural systems have made significant progress. The issue related to the reliability of dynamic structural systems mainly includes two aspects. The first is the structural response (displacement, stress, etc.) overrun, which is caused by forced vibration [19–22]. The second is fatigue caused by a resonant and non-resonant structure [23]. However, to the best of our knowledge reliability analysis of a dynamic structural system on the instability of self-excited vibration, e.g., regenerative chatter of a turning process system, has not be reported in literature. The goal of this paper is to introduce the ideas of reliability analysis of a dynamic structural system into structural analysis of a turning process system. A turning process system consists of a holder and a support, cutter, workpiece. The interaction between cutter and workpiece is a dynamic system referring to the machining condition. The variation of cutting forces between the cutter and causes the chatter vibration and is an internal system force. In this paper the dynamical model and SLD is reviewed. A chatter probability model is established to predict turning chatter vibration, in which structural parameters m, c, k, and spindle speed Ω are random variables. Chatter reliability is defined to represent the probability of stability (no chatter occurs) of turning process system. Choosing chatter frequency ωc as an intermediate variable, reliability model is transformed from a model impossible to solve to a new model related to chatter frequency, and the new model can be solved. First-order second-moment method is adopted to solve the turning process system reliability model and obtain the reliability of the system. The reliability lobe diagram is proposed to predict the chatter vibration. Finally, an example is used to demonstrate the effectiveness of the chatter reliability analysis and the RLD method. 2. Dynamic modeling of turning chatter 2.1. Turning machining dynamic model Fig. 1 shows the mechanical model for a regenerative chatter vibration cutting system of facing operation in turning. The following assumptions were made for the dynamic model: (1) the workpiece system is rigid and the tool holder system is the weakest link in the cutting system; (2) vibration system is linear and elastic resilience of the vibration system is proportional to vibration displacement; (3) direction of the dynamic cutting force and steady cutting force are the same, and Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
k
c
y m Fn β
h
feed
x
feed
Y. Liu et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
3
x
b
z
F Ft
Fig. 1. Dynamic model for the turning process system.
damping force is proportional to vibration velocity of the main vibration system; (4) the dynamic change in cutting thickness is only generated by regeneration. Using the dynamic model of the turning process system shown in Fig. 1, the expressions for the dynamic cutting force along the tool vibration direction can be expressed as follows: FðtÞ ¼ K s bhðtÞ
ð1Þ
hðtÞ ¼ hm þ xðt T Þ xðtÞ
ð2Þ
T¼
60
ð3Þ
Ω
where F is the cutting force (N), b is the cutting width (m), Ks is the cutting stiffness coefficient (N/m2) and h is the cutting thickness variation between the consecutive revolutions (m), hm is the mean chip thickness or the specified feed per revolution. The dynamic differential equation of machine tool vibration system is as follows: 60 mx€ ðtÞ þ cx_ ðtÞ þ kxðtÞ ¼ kn b hm þx t xðtÞ ð4Þ
Ω
where m is the equivalent mass of the vibration system (N s2/m), c is the equivalent damping of the vibration system (N s/m) and k is the equivalent stiffness of the vibration system (N/m), kn is the normal cutting stiffness coefficient (kn ¼Kscosβ N/m2). 2.2. Stability analysis review The free vibration equation of the motion of the turning process system can be modeled as x€ ðtÞ þ 2ωn ζ x_ ðtÞ þ ω2n xðtÞ ¼ 0
ω2n ¼
ð5Þ
k m
ð6Þ pffiffiffiffiffiffiffi
ζ ¼ c=ð2 mkÞ
ð7Þ
where ωn is the natural frequency of the cutting vibration system (rad/s), and ζ is the equivalent damping ratio of the cutting vibration system. Combining Eqs. (5)–(7) and using the Laplace transform, we get the frequency response function as shown in Eq. (8): 1 HðjωÞ ¼ 2 k ðjωω2Þ þ 2ωζω jþ 1 n
ð8Þ
n
Eq. (8) is split into real part and imaginary parts: ! 1 1 r2 Re ðHÞ ¼ k ð1 r 2 Þ2 þ ð2ζ rÞ2 I m ðHÞ ¼
1 2ζ r k ð1 r 2 Þ2 þð2ζ rÞ2
ð9Þ
! ð10Þ
where r ¼ ω/ωn. Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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4
The limiting cutting width blim of the cutting vibration system is shown in Eq. (11) [10]. The relationship between the chatter frequency ωc (rad/s) and spindle speed Ω (rev/s) is illustrated in Eqs. (12) and (13), where N ¼0, 1, 2, … is the integer number of vibration waves (lobe number), and ε (rad) is the phase (in rad) for an additional fraction of a wave. blim ¼
1 2K s Re ðHÞ
ωc 2π ¼ Nþ
Ω
ð11Þ
ε
ð12Þ
2π
ε ¼ 2π 2 tan 1
Re ðHÞ I m ðHÞ
ð13Þ
Fig. 2 shows a typical stability lobe diagram where the Ω versus blim family of curves separates the space into two regions. Any (Ω, b) pair that appears above the collective boundary indicates unstable behavior. Whereas, any pair below the boundary is presumed to be stable.
3. Turning process system chatter reliability modeling The typical stability lobe diagram is obtained using known and fixed values of parameters m, c, k. Fig. 3 shows this example stability lobe diagram obtained using the known structural parameters along with their uncertainty. Monte Carlo simulations were used to demonstrate the distribution of the stability lobe diagram. The parameters m, c, k represent normally distributed random variables with a mean value and standard variance. The mean values and standard deviations of each parameter are as follows: m (10.0610, 0.1) kg, c (1832.3, 30) N s/m and k (7.34 106, 1 105) N/m. Twenty of the samples were shown in Fig. 3.
Cutting width blim / (mm)
2.5 N=1
2 N=3
1.5
N=0
N=2 unstable
1 stable 0.5
0
1000
2000
3000
4000
5000
6000
Spindle speed Ω / (r/min) Fig. 2. Typical stability lobe diagram.
1.5
Cutting width blim /mm
1.4 1.3 1.2 1.1 1 0.9 0.8
1000
1500
2000
2500
3000
Spindle speed Ω / (r/min) Fig. 3. Typical stability lobe diagram from uncertain structural parameters.
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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3.1. Reliability modeling with m, c, k and
5
Ω
Generally, a structure fails if it cannot perform its intended function. It is obvious that the term “failure” can have different meanings. Before attempting a structural reliability analysis, failure must be clearly defined. The concept of a limit state is used to help define failure in the context of structural reliability analysis. A limit state is a boundary between desired and undesired performance of a structure. The limit state of a turning process system is defined as a critical stable state. “Failure” means losing machining stability, and the amplitude of vibration displacement of the cutter would be infinitely large due to the variation in the cutting force. However, “safety” or “reliability” implies stable machining, and the vibration of cutter attenuates due to damping. For the turning process, the machinists need to determine, if chatter would occur or in other words what would is probability of no chatter vibration for a given spindle speed Ω and cutting width b. The chatter reliability of the turning process is defined as the probability of no chatter vibration occurring in a dynamic system with random structural parameters m, c, k and manufacturing parameters Ω and b. The performance function of the turning process system in critical stable state can be expressed as g u ðUÞ ¼ b blim
ð14Þ
where b is the given cutting width (m), U is the random variables vector and U¼(m, c, k, Ω) . The three structural parameters m, c, k and the spindle speed Ω are dependent random variables with a known distribution. Then, the turning process system reliability refers to the probability under given process parameters when the given cutting width is less than the limiting cutting width. The reliability model is defined as Z Rs ¼ Pðg u ðU Þ o 0Þ ¼ f U ðuÞdu ð15Þ T
UR
where f U ðuÞ is the joint probability density of random vector, UR is the safe region of the basic variable space, that is, g u ðU Þ o0 in the UR region. P(.) is the probability function. However, g u ðU Þ is hard to be expressed as a function of U¼(m, c, k, Ω)T because of the existence of N. This is because that for a specified chattering frequency ωc, there is one cutting width blim and there are N spindle speeds Ω. Fortunately, we can express the performance function as the function of the new random vector X ¼(m, c, k, ωc)T. 3.2. Reliability modeling with m, c, k and
ωc
From Eqs. (6), (7), (9), (11), the expression for blim is obtained blim ¼
ðk ω2c mÞ2 þc2 ω2c 2K s ðk ω2c mÞ
ð16Þ
The turning process system reliability refers to the probability under the given process parameters when the given cutting width is less than the limiting cutting width. The new reliability model is defined as g X ðXÞ ¼ b blim Rs ¼ Pðg X ðX Þ o0Þ ¼
Z XR
f X ðxÞdx
ð17Þ
where X consists of m, c, k, ωc. Generally speaking, the structural parameters m, c and k are dependent normally distributed, and the ωc is related to the m, c and k. The distribution of ωc is determinated by the random variables m, c, k and Ω. The reliability model related to the chatter frequency is now analytically solvable. 4. First-order second-moment method of solving the model 4.1. Chatter frequency In the turning process system, chatter frequency can be written as a function of structural parameters m, c, k and spindle speed Ω. 1 R ðHÞ ð18Þ ωc ¼ 2πΩ N þ 1 tan 1 e I m ðHÞ π where
ωc is the chatter frequency (rad/s), N is the stable lobe number, N ¼0, 1, 2,……, and Ω is the spindle speed (rev/s).
4.2. Distribution parameters estimation of chatter frequency We assume that structural parameters m, c, k in the turning process system and the processing parameter Ω obey normal distribution. In order to calculate the reliability of turning process system, we need to obtain the mean value and standard variance. From Eqs. (6), (7), (9), (10) and (18), it can be observed that the chatter frequency is a function of random variables Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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6
m, c, k, Ω. According to the mean value and variance of a given parameter obtained by experimental testing, the random variables of m, c, k and Ω are generated using the random number generator in Matlab. We can calculate ωc for every sampling group. In order to determine the distribution of the samples, a goodness-of-fit test is needed. The examples show that the chatter frequency distribution obeys the normal distribution. We can estimate the overall mean value and variance according to value of ωc and then estimate the correlation coefficients between chatter frequency ωc and variables m, c, k respectively. 4.3. AFOSM method for determination of turning process system reliability Advanced first-order second-moment (AFOSM) method linearizes the performance function at the Taylor expansion points on the failure surface. The method uses the actual distribution of basic random variables. Reliability evaluation of turning system is as follows. The limit state function for turning system is Z ¼ g X ðXÞ ¼ 0
ð19Þ T
Assume that the basic random variable X ¼(X1, X2, X3, X4) are relevant random variables of the normal distribution. Xi (i¼1, 2, 3, 4) is m, c, k and ωc respectively. ωc is related to m, c and k, and the correlation matrix of the turning system is 0 1 1 0 0 ρX 1 X 4 B 0 1 0 ρX 2 X 4 C B C C ρ¼B ð20Þ B 0 0 1 ρX 3 X 4 C @ A ρX 4 X 1 ρX 4 X 2 ρX 4 X 3 1 where ρXiXj is the correlation coefficient of variable Xi and Xj. The standard deviation σi (i¼1, 2, 3, 4) of each random variable is matrix of turning system is 0 1 σ 21 0 0 ρX 1 X 4 σ 1 σ 4 B C B 0 σ 22 0 ρX 2 X 4 σ 2 σ 4 C B C C¼B C B 0 0 σ 23 ρX 3 X 4 σ 3 σ 4 C @ A
ρX 4 X 1 σ 1 σ 4 ρX 4 X 2 σ 2 σ 4 ρX 4 X 3 σ 3 σ 4
σm, σc, σk and σωc respectively, and the covariance
ð21Þ
σ 24
In Eq. (21), matrix C is a 4 4 symmetric positive definite matrix. The matrix has four real characteristic roots and four linearly uncorrelated and orthogonal characteristic vectors. Assuming that the columns of matrix A consist of regularization characteristic vectors of C. Orthogonal transformation is implemented on vector X, which consists of random variables of the system. X ¼ AY
ð22Þ
μY ¼ AT μX
ð23Þ
σ Y ¼ AT CA
ð24Þ
Limit state function can be expressed of the uncorrelated normal random variable Y as Z ¼ g X ðXÞ ¼ g X ðAYÞ ¼ g Y ðYÞ
ð25Þ
The derivative of the random variable Y is calculated using the checking point method, and is given as ∂g Y ðYÞ ∂g ðXÞ ¼ AT X ∂Y i ∂X i
ð26Þ
Choosing an average point X* ¼(m*c*k*ωc*) for the initial check point. Thus, the initial value for Y* Y n ¼ AT μX
ð27Þ
The partial derivatives of random vector X are as follows: ∂g X ðXÞ ω2 ð c2 ω2 þk 2kmω2 þ m2 ω4 Þ ¼ ∂m 2K s ðk mω2 Þ2
ð28Þ
∂g X ðXÞ cω2 ¼ ∂c K s ðk mω2 Þ
ð29Þ
∂g X ðXÞ 1 c2 ω2 ¼ ∂k 2K s 2K s ðk mω2 Þ2
ð30Þ
2
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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∂g X ðXÞ mω c2 kω ¼ þ ∂ωc K s K s ðk mω2 Þ2
7
ð31Þ
Substituting Eqs. (28)–(31) into the Eq. (26), we obtain the derivative of linearly uncorrelated random variables Y. In the space of random variable Y, equation ZL ¼0 is the limit state tangent plane which passes by the point Y*. By using the properties of linear combination of independent random variables, we obtain the mean value and standard deviation of ZL:
μZ L ¼ gðY n Þ þ
σ ZL
4 X ∂gðY n Þ
∂Y i
i¼1
ðμY i Y ni Þ
ð32Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 4 u X ∂gðY n Þ2 ¼t σ 2Y i ∂Y i i¼1
ð33Þ
We obtain reliability index of the system:
β¼
μZ L σ ZL
ð34Þ
The sensitivity coefficient of variable Yi is defined as ∂gðY n Þ Yi ∂Y
σ
i cos θY i ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2 4 h P ∂gðY n Þ
i¼1
∂Y i
ð35Þ
σYi
Thus, the new Y* is Y n ¼ μY i þ βσ Y i cos θY i
ð36Þ
The new X* is X n ¼ AY n
ð37Þ
Setting the error as ε ¼10 , multiple iterations were carried out until the difference between two consecutive iterations ||X*|| was less than ε. The value of β obtained is substituted into Eq. (37). Reliability probability of the system is as follows: 6
Rs ¼ ΦðβÞ
ð38Þ
where Φð:Þ is the cumulative distribution function of standard normal distribution. 5. Fourth-moment method of solving the model The second order partial derivative matrix is as Eq. (22) 0 1 2 6 cω4 c2 ω4 ω 2c2 kω3 K ðk K ðkcωmω2 Þ3 K s K s ðk mω2 Þ3 mω2 Þ2 K s ðk mω2 Þ3 s s B C B C cω4 ω2 cω2 2ckω B C Þ 2 2 2 2 2 2 2 2 K s ðk mω Þ K s ðk mω Þ K s ðk mω Þ K s ðk mω Þ C ∂ g X ðXÞ B C ¼B c2 ωðmω2 þ kÞ B C 2 c2 ω4 cω2 c2 ω2 ∂X B K s ðk mω2 Þ3 C 2 3 3 K s ðk mω2 Þ K s ðk mω2 Þ K s ðk mω2 Þ B C @ω c2 ωðmω2 þ kÞ c2 kð3mω2 þ kÞ A 2c2 kω3 2ckω m Ks Ks K ðk mω2 Þ3 K ðk mω2 Þ3 K ðk mω2 Þ3 K ðk mω2 Þ2 s
s
s
ð39Þ
s
The first four moment of the random variable vector X is
μX 1 ¼ 0
ð40Þ
μX 2 ¼ σ 2X
ð41Þ
μX 3 ¼ C sX σ 3X
ð42Þ
μX 4 ¼ C kX σ 4X
ð43Þ
The mean, standard variance, three-moment, four moment of the performance function can be calculated as Eqs. (27)– (30). ! n 1X ∂2 g μ X μg ¼ E½gðX Þ ¼ g μX þ σ 2X i ð44Þ 2i¼1 ∂X 2i Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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8
T !2
∂g X ðXÞ ∂g X ðXÞ σ m σ ω ρmω ∂X m ∂X ω ∂g ðXÞ ∂g X ðXÞ ∂g ðXÞ ∂g X ðXÞ þ X σ c σ ω ρcω þ X σ σ ω ρkω ∂X c ∂X ω ∂X k ∂X ω k
σ g ¼ Var½gðX Þ ¼
θg ¼ C 3 ½gðX Þ ¼
n X
∂g X ðXÞ ∂X
!3 ∂g μX ∂X Ti
i¼1
ηg ¼ C 4 ½gðX Þ ¼
n X
!4 ∂g μX
i¼1
∂X Ti
σ 2X þ2
ð45Þ
μX 3
ð46Þ
μX 4
ð47Þ
Thus, we obtain the coefficient of skewness of performance function is
α3g ¼ θg =σ 3g
ð48Þ
The coefficient of kurtosis (the fourth dimensionless central moment) of performance function is
α4g ¼ ηg =σ 4g
ð49Þ
The reliability index based on the second moment method is
βSM ¼ μg =σ g
ð50Þ
We can obtain the reliability index and reliability probability based on the fourth moment method are obtained as
3 α4g 1 β SM þ α3g βSM 1 ð51Þ βFM ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
9α4g 5α23g 9 α4g 1
Rs ¼ Φ βFM
ð52Þ
6. Reliability lobes diagram Within the (Ω, b) plane, using a grid with an appropriate increment, the reliability value at the node whose coordinates are (Ωm, bn) can be calculated. If the data is sufficient, a contour line can be obtained for the given reliability level P nr . The contour plots in the (Ω, b) plane are defined as reliability lobe diagram and has a lobe shape. Fig. 4 shows an example of reliability lobe diagram. Compared with the stability lobe diagram, reliability lobe diagram can be used to estimate the reliability at a selected point. As shown in Fig. 4, the red line represents the stability lobe diagram, and the blue line represents the reliability lobe diagram. Any (Ω, b) pair that is above the blue line indicates unreliable behavior, and the probability of stability (no chatter occurs) is greater than P nr However, any pair that appears below the boundary is presumed to be reliable, and probability of stability (no chatter occurs) is less than P nr . 3
SLD
Cutting width blim(mm)
2.5
2
1.5
RLD
1
R= pr
*
0.5
Reliable region 0 2000
4000
6000
8000
10000
12000
Spindle speed Ω (rpm) Fig. 4. An example of reliability and stability lobe diagram. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
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9
7. A case study 7.1. Data acquisition and analysis of turning experiment The experiment was carried out on a turning system (RUDI, China, Model: CJ0625). A vibration signal acquisition system (B&Ks, Denmark, model:3560-B) and Pulse analysis software (B&Ks, Denmark), modal hammer (PCBs, USA.model:086C01) and an accelerometer (PCBs, USA, model:356A24) were used for data collection in the experiments carried out on the turning process system. The frequency response function test was carried out along the x-direction of the tool. The locations of accelerometer and hammer tapping are shown in Fig. 5. The FRF diagram obtained directly from the test system is shown in Fig. 6. When the acquisition system captures the frequency response function at the measurement points on the lathe, we can identify the parameters according to the frequency response function and determine m, c and k. The results from the experiments are listed in Table 1.
z
x
y Fig. 5. The dynamic structure test on the turning process system.
Real/(m/N)
5
-7
0
-5
0
Imag/(m/N)
x 10
50 x 10
100
150
100
150
200
250
300
350
400
200
250
300
350
400
-6
-0.5
-1
50
Freq. / Hz Fig. 6. FRF curve of the X axis-tool from PULSE LabShops.
Table 1 Identified values of m, c and k of the turning system. Direction
ωn (rad/s)
fn (Hz)
m (kg)
c (N s/m)
k (N/m)
(Tool) X
854.1
136
10.0610
1832.3
7.34e6
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10
As shown in Table 2, modal parameters m, c and k were identified from the tested FRF. The natural frequency in xdirection of the tool was found to be 136 Hz. Since the first two mode amplitudes in the natural frequency, 71 Hz and 93 Hz are much smaller than the amplitude of the third mode in the natural frequency 136 Hz, the third mode had a major influence on the SLD. As a result, the structural parameters m, c and k were selected based on the 3rd mode in x-direction.
7.2. Distribution of the chatter frequency in turning process In the turning process, it is hard to get the exact value of the dynamic parameters due to factors affecting the equipment, sensors, temperatures, operators etc. Therefore, it is reasonable to consider them as random variables. After 20 tests, mean values and standard deviations of each parameter were determined and found to be: m (11.78, 0.68) kg, c (1464.9, 79.6) N s/m and k (8.22 106, 4.20 105) N/m. When the machine works normally, the spindle speed can be considered as normally distributed, and the standard variance is set to Ω (Ω0, 1.2425) rpm. Ω0 is the mean value of spindle speed. Consider a single DOF system, where the dynamic parameters m, c and k are equal to the mean value of the test result. With the spindle speed increasing, the change of the limiting cutting width blim of turning system and the chatter frequency ωc are shown in Fig. 7. The dashed line represents the limiting cutting width, while the solid line represents chatter frequency with respect to the spindle speed. The parameters m, c, k and Ω were sampled 10,000 times and we can obtain the samples of chatter frequency ωc. The distribution histogram under different speeds is shown in Fig. 8. When the dynamic parameters m, c, k and Ω of turning process system are normally distributed, chatter frequency of the turning system will change in accordance with a normal Table 2 20 samples of test modal parameters. fn (Hz)
m (kg)
c (N s/m)
k (N/m)
Ω (rpm)
132.5 133 133.5 133.5 133 133 133.5 133.5 133.5 133.5 134 132.5 132.5 132.5 132.5 133 133 132 132 133
12.54 11.32 10.92 11.45 11.83 11.61 11.49 10.70 10.70 10.84 11.20 12.59 12.33 12.88 12.21 12.16 11.86 12.53 12.19 12.24
1536.07 1422.94 1371.93 1402.60 1486.57 1422.94 1407.56 1344.09 1344.09 1362.52 1407.30 1541.98 1549.94 1578.46 1495.90 1490.27 1490.27 1574.09 1569.99 1499.61
8.69 106 7.91 106 7.68 106 8.05 106 8.26 106 8.11 106 8.08 106 7.53 106 7.53 106 7.63 106 7.94 106 8.72 106 8.55 106 8.93 106 8.46 106 8.49 106 8.28 106 8.62 106 8.38 106 8.55 106
2998.4 3003.6 3000.9 2999.9 3000.8 2999.8 2999.9 3001.8 3001.7 3001.7 3000.8 2998.6 3000.9 3001.9 3000.6 3001.2 3000.9 2999.6 3000.3 2999.1
2500
2000
2
1500
1.5
1000
1
500
0
Cutting width /mm
Chatter freq. ω c /(rad/s)
chatter freq. cutting depth
0.5
1000
2000
3000
4000
5000
6000
Spindle speed /(r/min) Fig. 7. Stability lobe diagram and the chatter frequency diagram of system with mean values.
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
Ω =2000 rpm fitted normal density
400
200
0 980
400
fitted normal density
200
0 985
990
970
Chatter Freq./(rad/s)
980
Chatter Freq./(rad/s)
Ω = 3500 rpm
Ω = 4000 rpm 400
fitted normal density
200
0 980
990
Freq. of occurrence
400
Freq. of occurrence
11
Ω = 2500 rpm
Freq. of occurrence
Freq. of occurrence
Y. Liu et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
fitted normal density
200
000
0 1098
Chatter Freq./(rad/s)
1100
1105
Chatter Freq./(rad/s)
Fig. 8. Histogram of chatter frequency with fitted normal density at different speeds.
distribution. Although the conclusion is drawn from the cases that spindle speed is 2000, 2500, 3500 and 4000 rpm, it is same for all spindle speed. In Fig. 7 it can be seen that chatter frequency changes sharply near the intersection of the limiting cutting width, so the chatter frequency is not consecutive mathematically. It is not suitable for calculating the reliability of the points near the intersection directly using the FOSM and fourth moment method. To overcome the shortcoming, the reliability can be calculated from both the adjacent leaves of SLD and the minimum value of the results is chosen as the reliability at this point. At a spindle speed Ω ¼ 3500 rpm the Lilliefors goodness-of-fit test on samples of chatter frequency ωc can be determined. With the condition of the significance level being 5%, the test results show that the chatter frequency of turning system obeys normal distribution or approximately obeys normal distribution. When the spindle speed is set to Ω ¼3500 rpm, the correlation coefficients of four random variables samples of the turning system were calculated. The correlation coefficient between m and ωc is 0.6581, the correlation coefficient between c and ωc is 0.2728, and the correlation coefficient between k and ωc is 0.6762. 7.3. Reliability of the turning process system using the Monte Carlo method According to the mean value, standard variance and the correlation coefficients of the random vector X, Monte Carlo sampling was implemented. A million of samples of X were obtained. Substituting the X value into the Eq. (39), we can calculate the ratio RM ¼
nr N
ð39Þ
in which nr is the times of the gX(X)o0, N is the total number of the sampling and RM is the reliability from Monte Carlo method. The stability lobe diagrams are shown in Fig. 9 for the mean value, mean value plus standard variance and the mean value minus standard variance of the random vector. More than 100 samples were verified, and the values of reliability were calculated. Each point represents the data pairs (Ω, b). The reliability is indicated as a “dot” when the value is greater than 0.9 and a “cross” when the value is less than 0.9. 7.4. Reliability of the turning process system using the FOSM method In order to guarantee stability of the turning process system, the condition of gX(X) o0 should be satisfied. If the cutting width b is given, the reliability of the turning process can be calculated using the FOSM method, and the results are compared with those using Monte Carlo method as shown in Table 2. A custom program was implemented in MATLAB (Mathworks, USA). Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
Y. Liu et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
12
Cutting width b /(mm)
2
mean+standard variance mean
Rs>0.9 + Rs ≤ 0.9
mean-standard variance 1.5
1
0.5
0
1000
2000
3000
4000
5000
6000
Spindle speed Ω /(r/min) Fig. 9. Reliability of the turning process system.
Ω=2000rpm
Ω=2500rpm
1
1 MC FOSM FM
Ω=2000rpm
0.6 0.4 0.2 0 0.86
0.6 0.4 0.2
0.88
0.9
0.92
0.94
0.96
0.98
0 0.84
1
0.86
cutting width/mm
Ω=3500rpm
0.9
0.92
0.94
0.96
0.98
Ω=4000rpm 1 MC SOFM FOSM SOFM
0.8
MC SOFM FOSM SOFM
0.8
0.6
Reliability
Reliability
0.88
cutting width/mm
1
0.4 0.2
0.6 0.4 0.2
0.92 0 0.9
MC FOSM SOFM
Ω=2500rpm
0.8
Reliability
Reliability
0.8
0.92
0.94 0.96 cutting width/mm
0.94
0.96
cutting width/mm
0.98
0.98
1
1
1.3 0 1.25
1.3
1.35 1.4 cutting width/mm
1.35
1.4
1.45
1.45
1.5
cutting width/mm
Fig. 10. Reliability of turning system at specific speed changes versus cutting width.
The mean value of each random variable of the system was chosen as the initial point. When the difference between the consecutive steps was less than 10 6, the iteration process was stopped. As shown in Fig. 10, “dotted line” represents the reliability calculated by the FOSM method versus the given cutting width, and “star line” represents the reliability calculated by Monte Carlo method versus the given cutting width. Table 2 and Fig. 5 show that the results from FOSM method and Monte Carlo method have consistency. Comparing with the traditional method SLD, FOSM method is not only able to judge the stability of the turning system, but also give the probability of stability.
7.5. Reliability lobe diagram The reliability value is set as 0.99. The reliability lobe diagram is shown in Fig. 11 with stability lobe diagram for comparison. In Fig. 11, the red line represents the reliability lobe diagram and the blue line represents the stability lobe diagram with a 0.99 level. Any (Ω, b) pairs below the blue line are prone to be stable and those above the red line are unstable, while any (Ω, b) pairs below the red line are reliable with a level of 0.99. The (Ω, b) pairs locating between the red and blue line are stable but unreliable, because the reliability is less than the level of 0.99 (Table 3). Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
Y. Liu et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]
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2.5
Stability lobe diagram
Cutting width (mm)
2
Reliability lobe diagram 1.5
1
0.5
0
0
1000
2000
3000
4000
5000
6000
Spindle speed Ω (r/min) Fig. 11. Reliability lobe diagram. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Table 3 Reliability at a specific speed and the given cutting width. Spindle speed (rpm)
Cutting width (mm)
Monte Carlo
FOSM
SOFM
Relative error (%)
Relative error(%)
2000
0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48
0.999992 0.999943 0.999418 0.995576 0.977333 0.916426 0.780964 0.567168 0.334037 0.154406 0.997451 0.983126 0.925611 0.778642 0.538776 0.282741 0.106688 0.028405 0.005169 0.000632 0.999717 0.997555 0.985736 0.9413 0.825741 0.6238 0.380825 0.178348 0.062671 0.016044 0.999984 0.999798 0.997622 0.981674 0.914846 0.742982 0.474095 0.218227 0.06803 0.013859
0.999995988 0.999935931 0.999312594 0.9949823 0.974683778 0.909912456 0.76788194 0.551692621 0.320838344 0.145990286 0.996969971 0.980566336 0.917487828 0.762116804 0.516025255 0.264312865 0.097262972 0.024942432 0.004389717 0.000527374 0.999236865 0.995066455 0.984485544 0.936937314 0.817342802 0.611793755 0.369118503 0.171266086 0.059276682 0.015032765 0.999990381 0.999801932 0.997561275 0.98170982 0.914439328 0.741953400 0.473101636 0.217065441 0.067483736 0.013705943
0.99999421 0.999924114 0.999290965 0.995253121 0.977062263 0.919143292 0.788760239 0.581183392 0.347424436 0.1612121 0.997717095 0.984792802 0.932298736 0.794351545 0.559654994 0.30103559 0.116445791 0.031128411 0.005606638 0.00066947 0.999719249 0.997681595 0.986548366 0.944598935 0.835536596 0.639812444 0.397291966 0.189786557 0.067196183 0.017201182 0.999988766 0.999786567 0.99749934 0.981736255 0.915444204 0.745091177 0.477344008 0.219831302 0.068282474 0.013739408
0.000003988 0.000007069 0.000105479 0.000596694 0.002718032 0.007158429 0.017036551 0.028050727 0.041138026 0.057645712 0.000482491 0.002610394 0.008853711 0.021683285 0.044088433 0.069720916 0.096902529 0.138822389 0.177524656 0.198390516 0.000480502 0.002500883 0.001270162 0.004656326 0.010275001 0.019624661 0.031714739 0.041350358 0.057262281 0.067268729 0.000006381 0.000003933 0.000060873 0.000036487 0.000444723 0.001386340 0.002099684 0.005351193 0.008094750 0.011167200
0.000221025 0.001888722 0.012712524 0.032441941 0.027709337 0.295633104 0.988416778 2.411526598 3.853337517 4.221829385 0.026670404 0.169254095 0.71733829 1.977656522 3.730690255 6.077218323 8.379685165 8.748955615 7.805719055 5.596980885 0.00022492 0.012688948 0.082344233 0.349241867 1.172371893 2.502677739 4.14480208 6.027064061 6.734285345 6.72734038 0.000476584 0.001143575 0.012296714 0.006341347 0.065345721 0.283076361 0.680642838 0.729787858 0.369748666 0.870428599
2500
3500
4000
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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7.6. Experimental validation Cutting test was carried out on a lathe. For chatter criterion, cutting noise was recorded with the microphone for signal analysis. The surfaces with different cutting width were compared in Fig. 12. The spindle speed was 1600 rpm and the cutting width was 0.8 mm and 1.5 mm respectively. When the cutting width is 0.8 mm, the surface is smooth and cutting noise is low and steady, and the reliability is 0.999955. When the cutting width is 1.5 mm, there are a period wave with the wavelength is about 2 mm left on the surface which can be seen in Fig. 12. The reliability is negligible and approximately zero (3.42 10 233). Fig. 13 shows the noise signal recorded using the microphone and the power spectrum where the cutting width is 0.8 mm and 1.5 mm. It can be observed that when the cutting width is 0.8 mm, the magnitude of power spectrum at 344.7 Hz is 0.00112 Pa2, and for the cutting width is 1.5 mm, the magnitude at 346.1 Hz is 0.00603. For a cutting width of 1.5 mm, the magnitude is 6 times the magnitude at 0.8 mm.
b= 0.8mm stable
b= 1.5mm unstable
Fig. 12. Surface finish with different cutting width.
0.5
1
0
0.5
0
0.5
1
1 0 -1
0
0.5
1
0
1.5
1.5
50
Magnitudes (Pa2)
Magnitudes (Pa)
-0.5
2
x 10
-3
0 x 10
500
1000
1500
2000
-3
344.7Hz, 0.00112Pa2 1 0
0
500
1000
1500
2000
0.01
346.1Hz, 0.00603Pa2 0 -50
0.005
0
0.5
1
time（s）
1.5
0
0
500
1000
1500
2000
Freq.（Hz）
Fig. 13. Noise signal recorded using the microphone and the power spectrum for cutting width 0.8 mm and 1.5 mm.
Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i
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SLD
15
stable point unstable point uncertain region
RLD, Rs=0.1 RLD, Rs=0.99
Fig. 14. Experimental validation. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Authors conducted more tests to validate the method mentioned in the article. The spindle speed is set at 1600 rpm and 2000 rpm, and cutting depth varies from 0.6 mm to 1.5 mm. We can obtain the critical values on the reliability lobes curve with a level of 0.99 are 0.853 mm and 0.905 mm at 1600 rpm and 2000 rpm respectively. The stable conditions which are identified through experimental tests are indicated as a “dot”, and the unstable conditions are indicated as a “cross”. Two reliability lobes with 0.99 and 0.1 levels are shown in Fig. 14. It is shown that, due to the uncertainty parameters, test results and prediction results obtained with the stability lobes do not perfectly match in the uncertain region with pink. However, the results from both of the experiment and prediction are same when the cutting depth located below the reliability lobes chart. The points above the “uncertain region” are clearly unstable, and the points below the “uncertain region” are clearly stable. It is shown that the method in this article can determinate the lower limit of the uncertain zone and we can choose the cutting parameters below the reliability lobes chart to avoid the occurrence of chatter. 8. Conclusions (1) Turning process system dynamics model was established and turning process chatter reliability model with random parameters was demonstrated, and FOSM, fourth moment methods was used for the reliability index and the reliability probability calculation. (2) The distribution of random parameters for the turning process system was determined experimentally, and the lobe diagram and chatter frequency curve for turning process system using mean values of random variables was plotted. Chatter frequency is a random variable with the dynamic parameters m, c, k and Ω and was found to be normal distributed. (3) The reliability probability of turning chatter system was calculated using the FOSM and fourth moment method and compared with that calculated using the Monte Carlo simulation. We conclude that results obtained using FOSM method and Monte Carlo methods were found to be consistent. Comparing with the traditional SLD method, chatter reliability method can be used to judge the probability of stability of turning process system. (4) Reliability lobe diagram was proposed to identify the chatter and no chatter regions for chatter prediction instead of stability lobe diagram (SLD). Comparing with the traditional SLD method, chatter reliability and RLD can be used to judge the probability of stability of turning process system.
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Please cite this article as: Y. Liu, et al., Chatter reliability prediction of turning process system with uncertainties, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.06.030i