scholarly journals Milling Stability Prediction with Multiple Delays via the Extended Adams-Moulton-Based Method

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Jianfeng Tao ◽  
Chengjin Qin ◽  
Chengliang Liu
Keyword(s):  
Transition Matrix ◽  
Rotation Period ◽  
Small Time ◽  
Multiple Delays ◽  
Stability Prediction ◽  
Regenerative Chatter ◽  
Milling Stability ◽  
Milling Operations ◽  
The Stability ◽  
Delay Differential

The occurrence of machining chatter may undermine the workpiece surface quality, accelerate the tool wear, and even result in serious damage to the machine tools. Consequently, it is of great importance to predict and eliminate the presence of such unstable and detrimental vibration. In this paper, we present an extended Adams-Moulton-based method for the stability prediction of milling processes with multiple delays. Taking the nonuniform pitch cutters or the tool runout into account, the regenerative chatter for milling operations can be formulated as delay differential equations with multiple delays. The dynamics model for milling regenerative chatter is rewritten in the state-space form. Dividing the spindle rotation period equally into small time intervals, the delay terms are approximated by Lagrange interpolation polynomials, and the Adams-Moulton method is adopted to construct the Floquet transition matrix. On this basis, the milling stability can be derived from the spectral radius of the transition matrix based on Floquet theory. The calculation efficiency and accuracy of the proposed algorithm are verified through making comparisons with the semidiscretization method (SDM) and the enhanced multistage homotopy perturbation method (EMHPM). The results show that the proposed method has both high computational efficiency and accuracy.

A Coupled Theoretical-Experimental Dynamical Model for Chatter Prediction in Milling Processes

2009 ◽  
Author(s):  
Giuseppe Catania ◽  
Nicolo` Mancinelli
Keyword(s):  
High Speed ◽  
Lumped Parameter Model ◽  
Milling Machine ◽  
Beam Shape ◽  
Milling Operations ◽  
Excited Vibration ◽  
Modal Model ◽  
Cutting Force Components ◽  
The Stability ◽  
Delay Differential

Productivity of high speed milling operations can be seriously limited by chatter occurrence. Several studies on this self-excited vibration can be found in the literature: simple models (1 or 2 dofs) are proposed, i.e. a lumped parameter model of the milling machine being excited by regenerative, time-varying cutting forces. In this study, a model of the milling machine is proposed: the machine frame and the spindle were modeled by an experimentally evaluated modal model, while the tool was modeled by a discrete modal approach, based on the continuous beam shape analytical eigenfunctions. The regenerative cutting force components lead to a set of Delay Differential Equations (DDEs) with periodic coefficients; DDEs were numerically integrated for different machining conditions. The stability lobe charts were evaluated using the semi-discretization method [6–7] that was extended to n dofs models (with n >2). Differences between the stability charts obtained by the low dofs models and the stability charts obtained by the new n dofs model are pointed out. Time histories and spectra related to the vibratory behavior of the system were numerically obtained to verify the effectiveness of the stability charts obtained with the n dofs modal model.

A novel stability prediction method for milling operations using the holistic-interpolation scheme

2019 ◽  
Vol 233 (13) ◽  
pp. 4463-4475 ◽  
Author(s):  
Chengjin Qin ◽  
Jianfeng Tao ◽  
Chengliang Liu
Keyword(s):  
Interpolation Method ◽  
Prediction Method ◽  
Small Time ◽  
Time Interval ◽  
Stability Prediction ◽  
Interpolation Scheme ◽  
Computational Accuracy ◽  
Delay Term ◽  
Milling Operations ◽  
Time Periodic

Currently, accurate and efficient determination of chatter-free cutting conditions is becoming increasingly important. This paper proposes a semi-analytical stability prediction method for milling processes using the holistic-interpolation scheme. The dynamics considering regeneration effect for milling operations is formulated as delay differential equations with time-periodic coefficients. The period of milling dynamic system is divided into two time periods according to the value of the coefficient matrix. On each small time interval for the forced vibration time period, the holistic-interpolation method is utilized by approximating the state term, the delay term, and the time-periodic parameter matrix as a whole unit with the second-order Lagrange interpolating polynomials. Then the Floquet transition matrix can be semi-analytically constructed for milling stability prediction according to Floquet theory. Finally, the benchmark examples of milling models are utilized to validate the effectiveness of the proposed method, which shows that the proposed algorithm achieves both high computational accuracy and efficiency.

scholarly journals Uncharted Stable Peninsula for Multivariable Milling Tools by High-Order Homotopy Perturbation Method

2020 ◽  
Vol 10 (21) ◽  
pp. 7869 ◽  
Author(s):  
Jose de la Luz Sosa ◽  
Daniel Olvera-Trejo ◽  
Gorka Urbikain ◽  
Oscar Martinez-Romero ◽  
Alex Elías-Zúñiga ◽  
...  
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Delay Differential Equation ◽  
Homotopy Perturbation Method ◽  
Multiple Delays ◽  
Third Order ◽  
Method Performance ◽  
Homotopy Perturbation ◽  
The Stability ◽  
Delay Differential

In this work, a new method for solving a delay differential equation (DDE) with multiple delays is presented by using second- and third-order polynomials to approximate the delayed terms using the enhanced homotopy perturbation method (EMHPM). To study the proposed method performance in terms of convergency and computational cost in comparison with the first-order EMHPM, semi-discretization and full-discretization methods, a delay differential equation that model the cutting milling operation process was used. To further assess the accuracy of the proposed method, a milling process with a multivariable cutter is examined in order to find the stability boundaries. Then, theoretical predictions are computed from the corresponding DDE finding uncharted stable zones at high axial depths of cut. Time-domain simulations based on continuous wavelet transform (CWT) scalograms, power spectral density (PSD) charts and Poincaré maps (PM) were employed to validate the stability lobes found by using the third-order EMHPM for the multivariable tool.

Countermeasure Against Regenerative Chatter in End Milling Operations With Vibration Absorbers

2010 ◽  
Author(s):  
Y. Nakano ◽  
H. Takahara
Keyword(s):  
End Milling ◽  
Spindle Speed ◽  
The Self ◽  
Depth Of Cut ◽  
Vibration Absorbers ◽  
Regenerative Chatter ◽  
Stability Lobe ◽  
Milling Operations ◽  
Wide Range ◽  
The Stability

Chatter can result in the poor machined surface, tool wear and reduced product quality. Chatter is classified into the forced vibration and the self-excited vibration in perspective of the generation mechanism. It often happens that the self-excited chatter becomes problem practically because this causes heavy vibration. Regenerative chatter due to regenerative effect is one of the self-excited chatter and generated in the most cutting operations. Therefore, it is very important to quench or avoid regenerative chatter (hereafter, simply called chatter). It is well known that chatter can be avoided by selecting the optimal cutting conditions which are determined by using the stability lobe of chatter. The stability lobe of chatter represents the boundary between stable and unstable cuts as a function of spindle speed and depth of cut. However, it is difficult to predict the stability lobe of chatter perfectly because the prediction accuracy of it depends on the tool geometry, the vibration characteristics of the tool system and the machine tool and the material behavior of the workpiece. In contrast, it is made clear that the stability lobe of chatter has been elevated in the wide range of spindle speed by the vibration absorber in the turning operations. However, it should be noted that none of the previous work has actually applied the vibration absorbers to the rotating tool system in the machining center and examined the effect of the vibration absorbers on chatter in the end milling operations to the best of authors’ knowledge. In this paper, the effect of the vibration absorbers on regenerative chatter generated in the end milling operations is qualitatively evaluated by the stability analysis and the cutting test. It is made clear the relationship between the suppression effect of the vibration absorbers and the tuning parameters of them. It is shown that the greater improvement in the critical axial depth of cut is observed in the wide range of spindle speed by the properly tuned vibration absorbers.

Stability of Up-milling and Down-milling Operations with Variable Spindle Speed

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1151-1168 ◽  
Author(s):  
Xinhua Long ◽  
B. Balachandran
Keyword(s):  
Spindle Speed ◽  
Depth Of Cut ◽  
Control Parameters ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Milling Operations ◽  
The Stability ◽  
Delay Differential ◽  
Milling Dynamics

In this article, a stability treatment is presented for up-milling and down-milling processes with a variable spindle speed (VSS). This speed variation is introduced by superimposing a sinusoidal modulation on a nominal spindle speed. The VSS milling dynamics is described by a set of delay differential equations with time varying periodic coefficients and a time delay. A semi-discretization scheme is used to discretize the system over one period, and the infinite-dimensional transition matrix is reduced to a finite-dimensional matrix over this period. The eigenvalues of this finite-dimensional matrix provide information on VSS milling stability with respect to control parameters, such as the axial depth of cut and the nominal spindle speed. The stability charts obtained for VSS milling operations are compared with those obtained for constant spindle speed milling operations, and the benefits of VSS milling operations are discussed.

Stability Analysis of a Variable Spindle Speed Milling Process

2005 ◽  
Author(s):  
X.-H. Long ◽  
B. Balachandran
Keyword(s):  
Milling Process ◽  
Spindle Speed ◽  
Depth Of Cut ◽  
Infinite Dimensional ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Milling Operations ◽  
Milling Operation ◽  
The Stability ◽  
Delay Differential

In this effort, a stability treatment is presented for a milling process with a variable spindle speed (VSS). This variation is caused by superimposing a sinusoidal modulation on a nominal spindle speed. The dynamics of the VSS milling process is described by a set of delay differential equations (DDEs) with time varying periodic coefficients and a time delay. A semi-discretization scheme is used to discretize the system over one period, and the infinite dimensional transition matrix is converted to a finite dimensional matrix over this period. The eigenvalues of this finite dimensional matrix are used to determine the stability of the VSS milling operation with respect to selected control parameters, such as the axis depth of cut and the nominal spindle speed. The benefits of VSS milling operations are discussed by comparing the stability charts obtained for VSS milling operations with those obtained for constant spindle speed (CSS) milling operations.

scholarly journals Stability Analysis for Milling Process with Variable Pitch and Variable Helix Tools by High-Order Full-Discretization Methods

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Yang Zhang ◽  
Kenan Liu ◽  
Wuyun Zhao ◽  
Wei Zhang ◽  
Fei Dai
Keyword(s):  
Milling Process ◽  
Small Time ◽  
High Order ◽  
The State ◽  
Multiple Delays ◽  
Variable Pitch ◽  
Discretization Methods ◽  
Variable Helix ◽  
The Stability ◽  
Space Equation

Chatter is one of the significant limitations in the milling process, which may cause poor surface quality, reduced productivity, and accelerated tool wear. Variable pitch and variable helix tools can be used to suppress regenerative chatter. This study extends the high-order full-discretization methods (FDMs) to predict the stability of milling with variable pitch and variable helix tools. The time-periodic delay-differential equation (DDE) with multiple delays is used to model the milling process using variable pitch and variable helix tools. Then, the DDE with multiple delays is reexpressed by the state-space equation. Meanwhile, the spindle rotational period is divided into many small-time intervals, and the state space equation is integrated on the small-time interval. Then, the high-order interpolation polynomials are used to approximate the state term, and the weights related to the time delay are employed to approximate the time-delay term. The second-order, third-order, and fourth-order extended FDMs (2nd EFDM, 3rd EFDM, and 4th EFDM) are compared with the benchmark in terms of the rate of convergence. It is found that the 2nd EFDM, 3rd EFDM, and 4th EFDM converge faster than the benchmark method. The difference between the curves obtained by different EFDMs and the reference curve is very small. There is no need to extend hypersecond FDMs to analyze the stability of milling with variable pitch and variable helix tools.

scholarly journals An updated Simpson-Based Method for Chatter Stability Prediction in Milling

2021 ◽  
Author(s):  
Zhenghu Yan ◽  
Changfu Zhang ◽  
Jianli Jia ◽  
Baoji Ma ◽  
Xinguang Jiang ◽  
...  
Keyword(s):  
Forced Vibration ◽  
Degrees Of Freedom ◽  
Computation Time ◽  
Milling Process ◽  
Small Time ◽  
The State ◽  
Stability Lobe ◽  
The Stability ◽  
The One ◽  
Delay Differential

Abstract An updated Simpson-based method (USBM) is presented for milling stability analysis. Firstly, the delay differential equation (DDE) is employed to describe the milling process mathematically. Then, the tooth passing period is divided into two subintervals, i.e., the free and forced vibration intervals. Only the forced vibration interval is divided into many equal small-time intervals. Subsequently, the DDE in the state space is solved based on direct integration. By combining the two-step Simpson method and the semi-discretization method, the state transition matrix of the milling system is constructed. The comparison of convergence rate is conducted to validate the accuracy of the proposed method. The results show that the proposed method converges faster than the benchmark methods. The stability lobe diagrams for the one degree of freedom (one-DOF) and two degrees of freedom (two-DOF) milling systems are also obtained by different methods for further evaluation. Meanwhile, the computation time analysis is also carried out. It is revealed that the proposed USBM has advantages in both accuracy and efficiency. Besides, the proposed method can accurately and efficiently predict the stability of milling with both large and low immersion conditions.

Stability in Variable-Pitch Milling Regarding Regenerative Chatter

2006 ◽  
Author(s):  
Nejat Olgac ◽  
Rifat Sipahi
Keyword(s):  
Cutting Tool ◽  
Milling Process ◽  
Depth Of Cut ◽  
Multiple Delays ◽  
Regenerative Chatter ◽  
Variable Pitch ◽  
Characteristic Roots ◽  
Analysis Methodology ◽  
Delay Differential ◽  
Critical Aspects

The regenerative chatter in milling process is studied for two different variable-pitch cutters one with (a) four flutes and the other with (b) six flutes. The cutting dynamics of the process is evaluated from stability perspective. Mathematically, the problem is recognized in a general class of delay differential equations (DDE) with multiple delays, whose stability can be analyzed by a recent stability analysis methodology called the Cluster Treatment of Characteristic Roots, CTCR. This method proves to be very beneficial to surface two critical aspects of the process, which maintain chatter-free cutting: (i) the pitch angle geometry of the cutting tool and (ii) admissible cutting conditions to determine the depth-of-cut and the spindle speeds. Case studies are provided to demonstrate the capabilities.

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