A New Curve Tracking Algorithm for Efficient Computation of Stability Boundaries of Cutting Processes

2007 ◽  
Vol 2 (4) ◽  
pp. 360-365 ◽  
Author(s):  
Christoph Henninger ◽  
Peter Eberhard
Keyword(s):  
Stability Boundary ◽  
Computational Cost ◽  
Boundary Curve ◽  
Efficient Computation ◽  
Technological Parameters ◽  
Excited Vibration ◽  
The Stability ◽  
Cutting Processes ◽  
Delay Differential ◽  
Curve Tracking

Dynamic stability of cutting processes such as milling and turning is mainly restricted by the phenomenon of the regenerative effect, causing self-excited vibration, which is well known as machine-tool chatter. With the semidiscretization method for periodic delay-differential equations, there exists an appropriate method for determining the stability boundary curve in the domain of technological parameters. The stability boundary is implicitly defined as a level set of a function on the parameter domain, which makes the evaluation computationally expensive when using complete enumeration. In order to reduce computational cost, we first investigate two types of curve tracking algorithms finding them not appropriate for computing stability charts as they may get stuck at cusp points or near-branch zones. We then present a new curve tracking method, which overcomes these difficulties and makes it possible to compute stability boundary curves very efficiently.

Instability of Heated Panels in Supersonic Air Flow

2008 ◽  
Vol 33-37 ◽  
pp. 1101-1108
Author(s):  
Zhi Chun Yang ◽  
Wei Xia
Keyword(s):  
Stability Analysis ◽  
Stability Boundary ◽  
Boundary Curve ◽  
Static Stability ◽  
Static Deformation ◽  
Limit Cycle Oscillation ◽  
Aerodynamic Load ◽  
Aeroelastic System ◽  
Aerodynamic Pressure ◽  
The Stability

An investigation on the stability of heated panels in supersonic airflow is performed. The nonlinear aeroelastic model for a two-dimensional panel is established using Galerkin method and the thermal effect on the panel stiffness is also considered. The quasi-steady piston theory is employed to calculate the aerodynamic load on the panel. The static and dynamic stabilities for flat panels are studied using Lyapunov indirect method and the stability boundary curve is obtained. The static deformation of a post-buckled panel is then calculated and the local stability of the post-buckling equilibrium is analyzed. The limit cycle oscillation of the post-buckled panel is simulated in time domain. The results show that a two-mode model is suitable for panel static stability analysis and static deformation calculation; but more than four modes are required for dynamic stability analysis. The effects of temperature elevation and dimensionless parameters related to panel length/thickness ratio, material density and Mach number on the stability of heated panel are studied. It is found that panel flutter may occur at relatively low aerodynamic pressure when several stable equilibria exist for the aeroelastic system of heated panel.

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.

scholarly journals Modeling of Abradable Coating Removal in Aircraft Engines Through Delay Differential Equations

2013 ◽  
Vol 135 (10) ◽  
Author(s):  
Nicolas Salvat ◽  
Alain Batailly ◽  
Mathias Legrand
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Computational Cost ◽  
Mathematical Framework ◽  
Machining Processes ◽  
Abradable Coating ◽  
Excited Vibration ◽  
Time Marching ◽  
Coating Removal ◽  
Delay Differential

In modern turbomachinery, abradable materials are implemented on casings to reduce operating tip clearances and mitigate direct unilateral contact occurrences between rotating and stationary components. However, both experimental and numerical investigations revealed that blade/abradable interactions may lead to blade failures. In order to comprehend the underlying mechanism, an accurate modeling of the abradable removal process is required. Time-marching strategies where the abradable removal is modeled through plasticity are available but another angle of attack is proposed in this work. It is assumed that the removal of abradable liners shares similarities with machine tool chatter encountered in manufacturing. Chatter is a self-excited vibration caused by the interaction between the machine and the workpiece through the cutting forces and the corresponding dynamics are efficiently captured by delay differential equations. These equations differ from ordinary differential equations in the sense that previous states of the system are involved in the formulation. This mathematical framework is employed here for the exploration of the blade stability during abradable removal. The proposed tool advantageously features a reduced computational cost and consistency with existing time-marching solution methods. Potentially dangerous interaction regimes are accurately predicted and instability lobes match both the flexural and torsional modal responses. Essentially, the regenerative nature of chatter in machining processes can also be attributed to abradable coating removal in turbomachinery.

Efficient Computation of Stability Charts for Linear Time Delay Systems

2005 ◽  
Author(s):  
Dimitri Breda ◽  
Stefano Maset ◽  
Rossana Vermiglio
Keyword(s):  
Time Delay ◽  
Linear Time ◽  
Computational Cost ◽  
Characteristic Root ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Efficient Computation ◽  
Square Grid ◽  
Stability Chart ◽  
The Stability

A new efficient algorithm for the computation of the stability chart of linear time delay systems is proposed and tested on several examples. The stability chart is obtained by investigating the 2d-parameter space by a first coarse square grid which is then adaptively refined by triangulation to match the desired tolerance. This leads to a considerable reduction in computational cost with respect to investigate a uniform fine square grid. Stability of each point is determined by approximating the rightmost characteristic root real part via a numerical scheme recently developed by the authors and based on pseudospectral differencing methods. A Matlab code is included in appendix.

Numerical Research on Machining Stability Subject to Delayed PID Control

2018 ◽  
Author(s):  
Mingjie Li ◽  
Xiaojian Zhang ◽  
Yakun Xie
Keyword(s):  
Pid Control ◽  
Production Efficiency ◽  
Stability Boundary ◽  
Stability Domain ◽  
Integral Parameter ◽  
Controlled System ◽  
Critical Boundary ◽  
Numerical Research ◽  
The Stability ◽  
Delay Differential

Machining stability analysis is important for chatter avoiding and production efficiency improvement. This paper conducts the numerical research on machining stability subject to delayed PID active control which is used to avoid chatter in machine milling. This control strategy is introduced into a two-degree-of-freedom milling system for illustration, the resulting hybrid system with both regenerative and feedback delays is represented as a delay-differential equation with time-periodic coefficients. From the comparison of stability region, it is found that the delayed PID control with proper parameters can lift the stability boundary largely compared with the case without control. To evaluate the stabilizability of the controlled system in cutting, the sensitivity of the stability boundary with respect to the PID parameters is analyzed. The numerical simulation of critical axial depth to PID parameters indicate that the milling stability critical boundary varies drastically with the derivative parameter. It also demonstrates that the stability critical boundary is strongly influenced by the proportional parameter, but it is less effected by the integral parameter. Hence, the stability domain can be expanded drastically with appropriate PID parameters based on the analysis above.

Efficient Stability Chart Computation for General Delayed Linear Time Periodic Systems

2013 ◽  
Author(s):  
Daniel Bachrathy ◽  
Gabor Stepan
Keyword(s):  
Linear Time ◽  
Order Approximation ◽  
Stability Boundary ◽  
Periodic Coefficient ◽  
Periodic Systems ◽  
Stability Chart ◽  
Matlab Package ◽  
The Stability ◽  
Cutting Processes ◽  
Time Periodic

The determination of the stability of systems with time delay is of high importance in many industrial and research applications, like cutting processes, wheel shimmy, traffic jams and even in neural systems, human balancing. A user friendly numerical method was implemented to analyse the general form of delayed linear time periodic systems with time periodic coefficients. The goal is to create a freeware Matlab package which is able to determine automatically the so-called stability chart, which illustrates the parameter range for which the given linear system is stable. The user has to define the governing equation by the time periodic coefficient matrices, the corresponding time delays, the orders of time derivatives of the general coordinate vector, as well as the range of the parameters and the resolution of the stability chart. The method is optimized for 2 parameters, which is a typical case in engineering applications, but 1 and 3 parameter stability charts are also supported and tested. The stability is analysed in frequency domain based on the Nrth order approximation of Hill’s infinite determinant. The parameter points where the number of unstable Floquet multipliers changes are computed by the Multi Dimensional Bisection Method. From these parameter points, another algorithm selects the stability boundary lines. The algorithm is tested by means of numerous examples.

Computation, Continuation and Bifurcation Analysis of Periodic Solutions of Delay Differential Equations

1997 ◽  
Vol 07 (11) ◽  
pp. 2547-2560 ◽  
Author(s):  
Tatyana Luzyanina ◽  
Koen Engelborghs ◽  
Kurt Lust ◽  
Dirk Roose
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Good Accuracy ◽  
Monodromy Matrix ◽  
Efficient Computation ◽  
Floquet Multipliers ◽  
Discrete Delays ◽  
The Stability ◽  
Delay Differential

We present a new numerical method for the efficient computation of periodic solutions of nonlinear systems of Delay Differential Equations (DDEs) with several discrete delays. This method exploits the typical spectral properties of the monodromy matrix of a DDE and allows effective computation of the dominant Floquet multipliers to determine the stability of a periodic solution. We show that the method is particularly suited to trace a branch of periodic solutions using continuation and can be used to locate bifurcation points with good accuracy.

scholarly journals Modeling of Abradable Coating Removal in Aircraft Engines Through Delay Differential Equations

2013 ◽  
Author(s):  
Nicolas Salvat ◽  
Alain Batailly ◽  
Mathias Legrand
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Computational Cost ◽  
Mathematical Framework ◽  
Machining Processes ◽  
Abradable Coating ◽  
Excited Vibration ◽  
Time Marching ◽  
Coating Removal ◽  
Delay Differential

In modern turbomachinery, abradable materials are implemented on casings to reduce operating tip clearances and mitigate direct unilateral contact occurrences between rotating and stationary components. However, both experimental and numerical investigations revealed that blade/abradable interactions may lead to blade failures. In order to comprehend the underlying mechanism, an accurate modeling of the abradable removal process is required. Time-marching strategies where the abradable removal is modeled through plasticity are available but another angle of attack is proposed in this work. It is assumed that the removal of abradable liners shares similarities with machine tool chatter encountered in manufacturing. Chatter is a self-excited vibration caused by the interaction between the machine and the workpiece through the cutting forces, and the corresponding dynamics are efficiently captured by delay differential equations. They differ from ordinary differential equations in that previous states of the system are involved. This mathematical framework is here employed for the exploration of the blade stability during abradable removal. The proposed tool advantageously features a reduced computational cost and consistency with existing time-marching solution methods. Potentially dangerous interaction regimes are accurately predicted and instability lobes match both flexural and torsional modal responses. Essentially, the regenerative nature of chatter in machining processes can also be attributed to abradable coating removal in turbomachinery.

scholarly journals Comparison of the Dynamics of Low Immersion Milling and Cutting With Varying Spindle Speed

2001 ◽  
Author(s):  
Tamás Insperger ◽  
Gábor Stépán ◽  
Sri N. Namachchivaya
Keyword(s):  
Time Delay ◽  
Autonomous Systems ◽  
Cutting Speed ◽  
Spindle Speed ◽  
Single Edge ◽  
Stability Properties ◽  
The Stability ◽  
Cutting Processes ◽  
Delay Differential ◽  
Edge Tool

Abstract The stability properties of cutting processes are strongly limited by the so-called regenerative effect. This effect is originated in the presence of a time-delay in the dynamical system of the machine tool. This delay is inversely proportional to the cutting speed. Consequently, conventional cutting with a single-edge tool is modeled by an autonomous delay-differential equation (DDE). In case of milling, the varying number of cutting edges results in a kind of parametric excitation, and the corresponding mathematical model is a non-autonomous DDE. In case of low-immersion milling, this affects the stability boundaries in a substantial way. Cutting with varying spindle speed results non-autonomous DDEs where the time delay itself depends on the time periodically. A new semi-discretization method is proposed to handle the stability of these non-autonomous systems. The stability properties and corresponding bifurcations are compared in the above different cases of machining.

Analytical Model for Self-Excited Vibration of an Overflow Flexible Plate Weir

1999 ◽  
Vol 121 (3) ◽  
pp. 296-303 ◽  
Author(s):  
S. Kaneko ◽  
H. Nagakura ◽  
R. Nakano
Keyword(s):  
Analytical Model ◽  
Stability Boundary ◽  
Pressure Rise ◽  
Flexible Plate ◽  
Instability Condition ◽  
Simply Supported ◽  
Plate Vibrations ◽  
Excited Vibration ◽  
The Stability ◽  
Tank Sloshing

An analytical model for the self-excited vibration of an overflow flexible weir as observed in the French demonstration fast breeder reactor, Super Phenix-1, is proposed. The instability condition was derived for the case in which the plate vibrates at the frequency of the downstream tank sloshing. In this analysis, the flexible plate weir is modeled as a simply supported-free-simply supported-clamped rectangular plate. Eigenfunction expansions were applied for analyzing both plate vibrations and the downstream tank sloshing. The effect of an overflow liquid is formulated based on the assumption that the momentum change due to the collision of an overflow liquid partly transmitted to the pressure rise on the free surface. As a result, the characteristic equation of the system yielding the theoretical stability boundary was obtained. The stability boundary thus derived agreed well with experimental results.

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