Time-periodic Delay Differential Equations Research Articles

Floquet Theory for Linear Time-Periodic Delay Differential Equations Using Orthonormal History Functions

Abstract In the usual approach to determining the stability of a time-periodic delay differential equation (DDE), the DDE is converted into an approximate system of time-periodic ordinary differential equations (ODEs) using Galerkin approximations. Later, Floquet theory is applied to these ODEs. Alternatively, semidiscretization-like approaches can be used to construct an approximate Floquet transition matrix (FTM) for a DDE. In this paper, we develop a method to obtain the FTM directly. Our approach is analogous to the Floquet theory for ODEs: we consider one polynomial basis function at a time as the history function and stack the coefficients of the corresponding DDE solutions to construct the FTM. The largest magnitude eigenvalue of the FTM determines the stability of the DDE. Since the obtained FTM is an approximation of the actual infinite-dimensional FTM, the criterion developed for stability is approximate. We demonstrate the correctness, efficacy and convergence of our method by studying several candidate DDEs with time-periodic parameters and/or delays, and comparing the results with those obtained from the Galerkin approximations.

In-process impulse response of milling to identify stability properties by signal processing

In this work we present a quantitative measurement-based method to describe the stability behaviour of a periodic dynamical system. In-process response for impulse during milling is analysed to provide operational stability prediction. The proposed method combines chatter detection techniques with the theory of time-periodic delay differential equations applied for general milling operations. Signal processing based on dynamic modal decomposition approach not only presents qualitative properties (like stability/instability) but also quantifies a measure of stability through the determination of the Floquet multipliers. The corresponding monodromy operator of the milling process is approximated from the measured system’s response during machining operation. By changing the technological parameters, the variation of the modulus of the Floquet multipliers can be monitored. The stability limit can precisely be interpolated with the unitary multiplier; furthermore, the stability limit can be extrapolated while the manufacturing parameters remain in the chatter-free region. The presented approach is validated by numerical results and laboratory tests.

Investigation of Gyroscopic Effect on the Stability of High Speed Micromilling via Bifurcation Analysis

Catastrophic tool failure due to the low flexural stiffness of the micro-tool is a major concern for micromanufacturing industries. This issue can be addressed using high rotational speed, but the gyroscopic couple becomes prominent at high rotational speeds for micro-tools affecting the dynamic stability of the process. This study uses the multiple degrees of freedom (MDOF) model of the cutting tool to investigate the gyroscopic effect in machining. Hopf bifurcation theory is used to understand the long-term dynamic behavior of the system. A numerical scheme based on the linear multistep method is used to solve the time-periodic delay differential equations. The stability limits have been predicted as a function of the spindle speed. Higher tool deflections occur at higher spindle speeds. Stability lobe diagram shows the conservative limits at high rotational speeds for the MDOF model. The predicted stability limits show good agreement with the experimental limits, especially at high rotational speeds.

Modeling of variable-pitch/helix milling system considering axially varying dynamics with cutter runout offset and tilt effects

In order to meet the growing industrial demand of high quality and high efficiency machining, long cutters with larger axial depth of cut are extensively implemented in peripheral milling of deep pockets, grooves, and thin-wall parts. However, due to their high flexibilities, chatter easily occurs during the machining process. It has been demonstrated that milling cutters with variable pitch and/or variable helix angles can mitigate the chatter, which are gaining increasingly important applications. Although this kind of cutters have been studied in some detail in previous research, few studies have focused on the interaction dynamics complicated by actual cutter runout in long tool-part contact zone.This paper systematically studies the dynamics and stability of variable-pitch/helix milling system with long end cutters. The cutters are divided into a series of axial disks, and each axial disk includes several cutting elements which are equal to the number of teeth. Variations of pitch and helix angles, cutter run-outs including radial offset and axial tilt, and mode shapes with the cross terms in the orthogonal direction, are assigned to these cutting elements. The system, involving multiple regenerative delays as well as the axially varying dynamics, is formulated as a matrix form of time-periodic delay differential equation with multi-modal dynamic parameters. An updated cross-axis and cross-point modal testing method is then proposed to obtain the dynamic parameters, which can effectively avoid multiple hits when impacting the cutter with hammer. The stability lobes are predicted in state space by developing an improved semi-discretization method, by which the pitch variations and actual runout effects on stability are investigated. It is shown that the stability lobes differ significantly when the combined effects of cross-axis and cross-point FRFs and actual cutter runout are considered. A series of cutting experiments is conducted to demonstrate the proposed model and method. Results show that these lobes well agree with the experimental results.

Pole Placement for Delay Differential Equations With Time-Periodic Delays Using Galerkin Approximations

Abstract Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems (TDS). In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

Simpson’s 3/8–based method stability analysis for milling processes

Chatter stability prediction based on linear time-periodic delay differential equations (DDEs) of the milling dynamic process is an important issue in achieving high-performance processing operations. To obtain efficient and precise regenerative chatter prediction, a novel method for milling stability analysis based on Simpson’s 3/8 rule is proposed. First, the dynamics model of milling processes, considering the regenerative effect, is modeled by an n-dimensional delay differential equation. Second, a set of algebraic equations is obtained to determine the transition matrix by converting and discretizing periodic DDE into a system of integral equations at the neighboring sampling grid points. Then, the eigenvalues of the transition matrix are utilized to determine stability based on Floquet theory. Finally, the comparisons of lobe diagrams calculated by different methods and discretization interval number for one and two degrees of freedom milling models are discussed, and the consistency of the predicted and experimental stability boundaries is analyzed. Results show that the proposed method achieves high computation accuracy with high efficiency.

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

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.

A Legendre wavelet–based stability prediction method for high-speed milling processes

Induced by the regeneration mechanism, chatter vibration that has adverse effects on surface finish and productivity could occur in most cutting processes. Therefore, accurately predicting chatter and taking timely suppression measures are desperately needed. This work develops a semi-analytical Legendre wavelet–based stability prediction method in high-speed milling, of which dynamics model is commonly modeled by time-periodic delay differential equations. To begin with, the milling dynamics model is transformed into state equation. Based on analysis of the vibration modes for milling dynamic system, the period for coefficient matrix could be divided into two subintervals. Thereafter, variable transformation is introduced to convert the forced vibration interval into the standard definition interval of Legendre wavelets, which can be discretized non-uniformly into the Chebyshev–Gauss–Lobatto sampling points. On this basis, finite series Legendre wavelets are utilized to accurately approximate the state term, while the derivative of state term can be obtained simultaneously via a new Legendre wavelet operational matrix. By employing the Kronecker product and Floquet theory, the transition matrix is eventually acquired for semi-analytically predicting stability. The superiority and versatility for the presented approach are validated through comparisons with existing algorithms. The results demonstrate that it obtains high accuracy, calculation speed, and applicability.

Comparison of the full-discretization methods for milling stability analysis by using different high-order polynomials to interpolate both state term and delayed term

This paper compares different full-discretization methods for milling stability analysis by using different high-order polynomials to interpolate both state term and delayed term (HFDMs) from the aspects of accuracy and efficiency. The dynamic model of milling process with consideration of regeneration effect is described by time periodic delay-differential equation (DDE) in the state-space. Different high-order interpolation polynomials are used to approximate the state term and delayed term. The state transition matrix is obtained on the basis of direct integration scheme. The rates of convergence of different HFDMs are compared with those of the benchmark methods using different process parameter points, the results indicate that it is difficult to evaluate the accuracy of different HFDMs through the convergence rate analysis of limited process parameter points. Then, mean differences and variances between the referenced and predicted critical depths of cut are employed for accuracy analysis. The 3rd-2nd HFDM is, on the whole, proved to be more accurate than the other methods. The efficiencies of different HFDMs are also verified through time-consuming study for both single degree of freedom (1-DOF) and two degree of freedom (2-DOF) milling system. The 3rd-2nd HFDM is proved to be an efficient method by comparing with the other methods. Besides, the HFDMs are available for predicting the stability lobe diagrams under both large immersion condition and low immersion condition.

New predictor-corrector methods based on piecewise polynomial interpolation for milling stability prediction

Chatter frequently occurs during cutting operations, which seriously restricts the machining productivity and workpiece accuracy. Consequently, accurate and efficient stability prediction is of great significance to determine stable machining parameters. A cubic Hermite–Newton approximation method which can determine the chatter stability boundaries more efficiently is presented in this paper. The milling dynamic system can be expressed as time-periodic delay differential equations (DDEs) with consideration of the regeneration effect. A typical benchmark example is provided to assess the convergence feature and stability lobes of the cubic Hermite–Newton approximation method and several existing methods. The results indicate that the cubic Hermite–Newton approximation method can achieve satisfactory results. For the sake of developing the cubic Hermite–Newton approximation method with higher convergence rate and computational efficiency, the tooth-passing period is further separated into two distinct phases according to whether the value of coefficient matrix equals to zero. Meanwhile, the linear interpolation polynomial is used to predict milling stability, and then piecewise polynomial interpolation was utilized in two adjacent time intervals to correct this prediction. By adopting the two benchmark examples, the effectiveness of the two new methods can be analyzed using existing methods. The results demonstrate that the two new methods have superior accuracy and efficiency.

Feasibility analysis of active chatter control for stationary and revolving bar boring operations based on magnitude of control forces using fractional order PDλ controller

Due to the factors of limited rigidity and small inherent damping of boring bars, very small depths of cutting can be applied for chatter free machining. Stability lobe diagrams can truly represent that limit. Boring operations are categorized based on the relative motion between the boring bar and workpiece. Modelling of each category is presented in detail using a 3-DOF model. The dynamics of such systems can be successfully represented using ordinary delay differential equations and time periodic delay differential equations. The open loop stability of each type of boring operation is numerically investigated. It has been observed that the stable depth of cut is much more for revolving bar (RB) as compared to stationary bar (SB) boring process. Active chatter control (ACC) techniques can enhance the material removal rates and surface finish of the workpiece. The feasibility of active chatter control for each category is investigated in detail. It is further observed that RB boring process is not a suitable candidate for ACC because much higher harmonic control forces are required for minor improvement in critical depth of cut.

Efficient active chatter mitigation for boring operation by electromagnetic actuator using optimal fractional order PDλ controller

Abstract Due to the factor of limited rigidity of boring bars, small depths of cutting are normally applied for chatter free machining. Stability lobe diagrams can truly represent that limit. However, due to the presence of manufacturing inaccuracies like ovality and eccentricity, these limits are considerably changed. Hence, considering these during modelling the cutting process, stability analysis and controller design is a novel idea, hence implemented in the present work. Dynamic modelling of boring process is presented in detail using a 3-DOF model. Dynamics of such systems is represented using delay differential equations with time periodic coefficients. The system stability is enhanced with active control techniques. The closed loop system considering fractional order PDλ in the loop is a nonlinear time periodic delay differential equation system. Any systematic controller synthesis process for such systems is rarely available in literature. It is observed that a fractional order PDλ controller designed by using combination of pseudo spectral and response optimization techniques is highly efficient in terms of the requirement of low amplitude of the peak force and simplicity of implementation. Transient vibrations can also be quenched in a limited period of time by using this controller. Different control techniques available in literature (H∞ Loop shaping and PD control) are tested and compared to the proposed controller for enhancing the material removal rates and surface finish of the workpiece. By using the active chatter control, the chatter can be efficiently reduced and the material removal rate can be increased. The critical depth of cut is increased from 0.2 mm (open loop) to 0.6 mm (closed loop) with a limited actuator size in the case study.

Improved precise integration method for chatter stability prediction of two-DOF milling system

The motivation of this paper is to update the precise integration method (PIM) by a second-order Taylor formula and make detailed contrasts with the existing PIM, the semi-discretization method (SDM) to exhibit the necessity of developing this improved PIM (IPIM). The dynamics of two-degree of freedom (DOF) milling process with consideration of regeneration effect is first governed by a time periodic delay differential equation (DDE). With time period being evenly divided into a limited set of intervals, the integral non-homogeneous element is approximated by the second-order Taylor formula in every small time segment. After decomposing the exponential factor into a real term with 2N order algorithm, the transition matrix representing the specific machining system state is established in one whole tooth passing period to search for the chatter-free borderline. To investigate the characteristics of the proposed method in convergence rate, prediction accuracy, and computational efficiency, the benchmark example used in the literatures is introduced to develop a battery of comparisons with PIM and SDM. Finally, the experimental verification is also conducted in a CNC machine tool to further confirm the operability of the proposed IPIM, and the results indicate the method is of availability.

Predictive modeling of chatter stability considering force-induced deformation effect in milling thin-walled parts

Abstract Force-induced deformation is an inevitable phenomenon in thin-wall milling operations, which makes the actual radial depth of cut deviate from its nominal value and changes tool-part engagement angle boundaries. This paper presents an accurate modeling of dynamic milling system with the force-induced deformation effect. Both the thin-walled part and the cutter are discretized into differential elements, such that the influences of the force-induced deformation and multi-point contact structure dynamics can be simultaneously considered at the contact zones. A detail flexible iteration strategy is first presented to calculate the force-induced deformation and the resulting tool-part engagement angle boundaries. After that, time-varying multi-modal dynamic parameters with actual material removal effect are obtained by simultaneously modifying structural static and dynamic stiffness at each tool feed position. The system involving regenerative dynamic displacements is formulated as a matrix of time-periodic delay differential equation. Chatter stability of the system is predicted by an extended second order semi-discretization method. Milling experiments are conducted to validate the proposed approach, and two types of parts with different wall thickness are designed. The results show that chatter can be well predicted by using the proposed approach.

Third-order updated full-discretization method for milling stability prediction

Based on third-order Newton interpolation polynomial and direct integration scheme (DIS), this paper proposes a method to generate stability lobe diagram in milling process. The dynamic model of milling process with consideration of regeneration effect is described by time periodic delay-differential equation (DDE). Then, the DDE is rewritten as state space equation by a transformation. After equally discretizing the time delay into a series of small time intervals, the state space equation of milling system is integrated on the small time interval. Both the state term and delayed term are interpolated by third-order Newton interpolation polynomial, and the periodic-coefficient matrix is interpolated by first-order Newton interpolation polynomial. The state transition matrix which reflects the discrete mapping relation of dynamic responses for current tooth pass period and immediate previous tooth pass period is obtained directly. The accuracy of the proposed method is evaluated by comparing with benchmark methods in terms of the rate of convergence. The efficiency of the proposed method is verified through the comparison of computational time with existing methods. The proposed method is proved to be an accurate and efficient method by the comparison results. The distinction between up-milling and down-milling operations is also analyzed by comparing the stability lobe diagrams for these two operations. Besides, according to the analysis of rate of convergence, the number of substitutions, which are used to convert the variables located out of the required range into the required range, may affect the results of stability lobe diagrams. Moreover, the stability lobe diagram cannot be generated by using fourth-order updated full-discretization method.

Orthogonal polynomial approximation method for stability prediction in milling

Based on orthogonal polynomial approximation scheme, this paper presents several stability prediction methods using different kinds of orthogonal polynomials. The milling dynamics with consideration of the regenerative effect is described by time periodic delay-differential equations (DDEs). Firstly, this work employs the classical Legendre and Chebyshev polynomials to approximate the state term, delayed term, and periodic-coefficient matrix. With the help of direct integration scheme (DIS), the state transition matrixes which indicate the mapping relations of the dynamic response between the current tooth pass and the previous tooth pass are obtained. The stability lobe diagrams for single degree of freedom (DOF) and two DOF milling models are generated by using the Legendre and Chebyshev polynomial approximation-based methods. The rate of convergence of the Legendre and Chebyshev polynomial-based methods is compared with that of the benchmark first-order semi-discretization method (1stSDM). The comparison results indicate that the rate of convergence and the numerical stability of the Legendre and Chebyshev polynomial-based methods are both need to be improved. In order to develop new methods with high rate of convergence and numerical stability base on DIS, the monic orthogonal polynomial sequences are constructed by using Gram-Schmidt orthogonalization to approximate the state term, delayed term, and periodic-coefficient matrix. The rate of convergence and the computational efficiency of the monic orthogonal polynomial-based methods are evaluated by comparing with those of the benchmark 1stSDM. The results turn out that the monic orthogonal polynomial-based methods are advantageous in terms of the rate of convergence and numerical stability. The stability lobe diagrams for single DOF and two DOF milling models obtained by the monic orthogonal polynomial-based methods are compared with those obtained by the 1stSDM. Finally, the monic orthogonal polynomial-based methods are proved to be the effective and efficient methods to predict the milling stability.

A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed.

A spline-based method for stability analysis of milling processes

Based on the idea of collocation methods, a spline-based integration method for the stability prediction of systems governed by time-periodic delay differential equations (DDEs) is presented. A B-spline curve is adopted to approximate the solution of the DDEs using the direct integration technique. The stability of the dynamic system is then predicted using the Floquet theory based on the established state transition matrix. The proposed method can apply to time-periodic DDEs where the delay and the period are incommensurate, and the proposed method is extended to study the stability of milling processes as well. For stability analysis of variable pitch cutters, the proposed method use one or multiple B-spline curves to approximate the solution of the DDEs according to the relationship between the cutter angles and the radial immersion, which can describe exactly the free vibration of the cutter. Compared with the numerical integration method and the Chebyshev collocation method, the simulation results demonstrate that the proposed technique is more efficient and accurate for stability prediction at low radial immersion cases in milling processes with variable pitch cutters.

High-order full-discretization method using Lagrange interpolation for stability analysis of turning processes with stiffness variation

In turning processes, chatter is an unstable vibration which adversely affects surface finish and machine tool components. Stiffness variation (SV) is an effective strategy for chatter suppression by periodically modulating the stiffness around a nominal value. The dynamics of SV turning is governed by a time periodic delay differential equation (DDE) where the time-period/time-delay ratio (TPTDR) can be arbitrary. Recently, first-, second- and higher-order full-discretization methods (FDMs) have been reported as a popular class of methods for milling stability prediction. However, these FDMs can only deal with time periodic DDE where the TPTDR equals one. In this paper, two high-order FDMs using Lagrange interpolation (HLFDMs) are proposed for stability analysis of SV turning. On each discrete time interval, the time delay term is interpolated by the second-degree Lagrange polynomial, and the time periodic term is linearly interpolated. The state term is approximated using linear interpolation and second-degree Lagrange polynomial interpolation, achieving the first- and second-order HLFDM, respectively. Finally, the transition matrix over a single period is deduced for stability analysis via the Floquet theory. Benchmark examples of damped delay Mathieu equations are used to verify the proposed algorithm, which demonstrates that HLFDMs are highly efficient and accurate. In addition, the second-order HLFDM is used to investigate the effects of SV amplitude and frequency parameters. These results provide theoretical insights for the selection of SV parameters.

Stability prediction of thin-walled workpiece made of Al7075 in milling based on shifted Chebyshev polynomials

With the rapid development of aerospace technology, Al7075 has been widely used for structural components. High-speed milling is one of the most effective ways to improve machining efficiency of Al7075. During the milling process, regenerative chatter which restricts the milling quality and productivity often occurs. With the aim of avoiding regenerative chatter, stability lobe diagram (SLD) is widely used to obtain chatter-free parameters. This work presents a stability prediction method by using shifted Chebyshev polynomials. The milling dynamics with consideration of the regenerative effect is described by time periodic delay-differential equations (DDEs). The transition matrix of the milling system is constructed with the help of Chebyshev–Gauss–Lobatto (CGL) points. In order to demonstrate the accuracy of the proposed method, the rate of convergence of the proposed method is compared with that of the classical benchmark methods. On the other hand, in the process of thin-walled workpiece milling, the dynamic behavior of the workpiece depends on the tool position. To study the influence of the tool position dependent dynamics on the chatter stability of the thin-walled workpiece, a three-dimensional SLD is obtained. The verification experiments are conducted to verify the reliability of the proposed method. The results show that the experimental results are consistent with the predicted results.