Experimental and Analytical Investigation of the Subcritical Instability in Metal Cutting

1999 ◽  
Author(s):  
Tamás Kalmár-Nagy ◽  
Jon R. Pratt ◽  
Matthew A. Davies ◽  
Michael D. Kennedy
Keyword(s):  
High Speed ◽  
Metal Cutting ◽  
Stability Boundary ◽  
Analytical Investigation ◽  
Equation Model ◽  
Center Manifold Reduction ◽  
Single Degree Of Freedom ◽  
Subcritical Hopf Bifurcation ◽  
Delay Differential ◽  
Manifold Reduction

Abstract A single-degree-of-freedom dynamic cutting fixture is used to map out a part of the lobed stability boundary in a simple high-speed machining experiment. The experiment reveals the hysteretic nature of the instability. A 1 DOF mechanical model is derived using parameters identified from the experiment. We then show the existence of a subcritical Hopf bifurcation in this delay-differential equation model which corresponds to the observed experimental instability. The calculation is based on center manifold reduction. Then time domain simulation is used to solve the full nonlinear equation of motion that allows for the tool to leave the workpiece giving excellent agreement with the experiment.

Center Manifold Reduction of Periodic Delay Differential Systems

2007 ◽  
Author(s):  
Eric A. Butcher ◽  
Venkatesh Deshmukh ◽  
Ed Bueler
Keyword(s):  
Differential Equations ◽  
Center Manifold ◽  
Perturbation Expansion ◽  
Flip Bifurcation ◽  
Restoring Force ◽  
Center Manifold Reduction ◽  
Periodic Coefficients ◽  
Delay Differential ◽  
Manifold Reduction ◽  
Time Periodic

A technique for center manifold reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations (ODEs) with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. Center manifold reduction on the map is then carried out. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.

Nonlinear Regenerative Machine Tool Vibrations

1997 ◽  
Author(s):  
Gábor Stépán ◽  
Tamas Kalmar-Nagy
Keyword(s):  
Machine Tool ◽  
Analytical Formula ◽  
Chip Thickness ◽  
Domain Of Attraction ◽  
Equation Model ◽  
Center Manifold Reduction ◽  
Nonlinear Part ◽  
Dimensional Phase Space ◽  
Lengthy Calculation ◽  
Delay Differential

Abstract The existence and the nature of the Hopf bifurcation is presented in the delay-differential equation model of the so-called regenerative machine tool vibration. The relevant nonlinearity is considered at the cutting force dependence on the chip thickness. The delayed terms show a special algebraic structure in the nonlinear part of the equation of motion. This results in a surprisingly simple and useful analytical formula in the end of the lengthy calculation based on center manifold reduction in the corresponding infinite dimensional phase space. The result gives a simple way to estimate the domain of attraction of the stable stationary cutting as well as an estimation of that technological parameter domain, where the cutting is globally stable.

scholarly journals Stability and local bifurcation analyses of two-wheeled trailers considering the nonlinear coupling between lateral and vertical motions

2021 ◽  
Author(s):  
Hanna Zsofia Horvath ◽  
Denes Takacs
Keyword(s):  
Linear Stability ◽  
Stability Boundary ◽  
Small Scale ◽  
Hopf Bifurcations ◽  
Nonlinear Coupling ◽  
Center Manifold Reduction ◽  
Mechanical Models ◽  
Tire Forces ◽  
The Stability ◽  
Manifold Reduction

AbstractThe nonlinear dynamics of two-wheeled trailers is investigated using a spatial 4-DoF mechanical model. The non-smooth characteristics of the tire forces caused by the detachment of the tires from the ground and other geometrical nonlinearities are taken into account. Beyond the linear stability analysis, the nonlinear vibrations are analyzed with special attention to the nonlinear coupling between the vertical and lateral motions of the trailer. The center manifold reduction is performed leading to a normal form up to third degree terms. The nature of the emerging periodic solutions, and, thus, the sense of the Hopf bifurcations are verified semi-analytically and numerically. Simplified models of the trailer are also used in order to point out the practical relevance of the study. It is shown that the linearly independent pitch motion affects the sense of the Hopf bifurcations at the linear stability boundary. Namely, the constructed spatial trailer model provides subcritical bifurcations for higher center of gravity positions, while the commonly used simplified mechanical models explore the less dangerous supercritical bifurcations only. Domains of loss of contact of tires are also detected and shown in the stability charts highlighting the presence of unsafe zones. Experiments are carried out on a small-scale trailer to validate the theoretical results. A good agreement can be observed between the measured and numerically determined critical speeds and vibration amplitudes.

scholarly journals Reduction principle at work

2021 ◽  
Vol 8 (1) ◽  
pp. 46-74
Author(s):  
Christian Pötzsche ◽  
Evamaria Russ
Keyword(s):  
Center Manifold ◽  
Reduction Principle ◽  
Critical Stability ◽  
Center Manifold Reduction ◽  
Infinite Dimensional ◽  
Linearized Stability ◽  
Dichotomy Spectrum ◽  
Nonautonomous Case ◽  
Necessary And Sufficient ◽  
Manifold Reduction

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

Hard Nano-Crystalline Coatings for Cutting Tools

2007 ◽  
Vol 567-568 ◽  
pp. 185-188 ◽  
Author(s):  
Miroslav Piska
Keyword(s):  
High Speed ◽  
Metal Cutting ◽  
Cutting Tool ◽  
Cutting Tools ◽  
Dry Cutting ◽  
Nano Crystalline ◽  
The Right ◽  
Machining Productivity ◽  
Hard Cutting ◽  
Cutting Tool Materials

Modern trends in metal cutting, high speed/feed machining, dry cutting and hard cutting set more demanding characteristics for cutting tool materials. The exposed parts of the cutting edges must be protected against the severe loading conditions and wear. The most significant coatings methods for cutting tools are PVD and CVD/MTCVD today. The choice of the right substrate or the right protective coating in the specific machining operation can have serious impact on machining productivity and economy. In many cases the deposition of the cutting tool with a hard coating increases considerably its cutting performance and tool life. The coating protects the tool against abrasion, adhesion, diffusion, formation of comb cracks and other wear phenomena.

Hybrid Induction Plasma Cutting Process - A New Technology for High Speed Metal Cutting

2014 ◽  
Author(s):  
Jerald E. Jones ◽  
Valerie L. Rhoades ◽  
Mark D. Mann ◽  
Todd Holverson
Keyword(s):  
Induction Heating ◽  
Plasma Torch ◽  
High Speed ◽  
Metal Cutting ◽  
New Technology ◽  
Cutting Process ◽  
Plasma Cutting ◽  
Air Plasma ◽  
Aircraft Carrier ◽  
Initial Development

A new cutting process, a hybrid system, uses induction heating to heat the metal ahead of the plasma cutting torch. The process has demonstrated the ability to plasma cut steel parts at speeds of up to 4X the speed of the plasma torch without the induction heating. Although the total heat input per unit time is greater, because of the increase in speed, the heat which is conducted into the cut pieces is less. This causes less potential metallurgical damage, less potential distortion, and reduced coating damage and reduced emissions during cutting, in comparison to the plasma cutting process without the induction heating. The initial development was primarily for use in cutting nuclear submarine and aircraft carrier hulls, for scrapping after decommissioning. The process has been demonstrated cutting steel plates and can be used in ship production as well. The primary motivation of the SBIR project was to reduce the heating of the cut pieces, in order to reduce the particulate matter (PM) emissions which occur when coated ship hull material is cut. An induction coil is positioned in front of the plasma cutting torch, to bring the material to an elevated temperature of at least 1600° F, before the plasma is applied to the metal surface. Induction heating testing has shown that the 35 kW induction system can maintain the 1600° F surface temperature at travel speeds of above 220 inches per minute on steel as thick as 3 inches. Once the steel is at that temperature an air plasma torch can cut the metal much faster than cutting cold steel.

scholarly journals A Delay Differential Equation Model of HIV Infection, with Therapy and CTL Response

2014 ◽  
Vol 9 ◽  
pp. 53-68 ◽  
Author(s):  
B. El Boukari ◽  
N. Yousfi
Keyword(s):  
Reproduction Number ◽  
Equation Model ◽  
Asymptotically Stable ◽  
Differential Equation Model ◽  
Globally Asymptotically Stable ◽  
Intracellular Delay ◽  
Immunodeficiency Virus ◽  
Delay Differential ◽  
Theoretical Results ◽  
Disease Free

In this work we investigate a new mathematical model that describes the interactions betweenCD4+ T cells, human immunodeficiency virus (HIV), immune response and therapy with two drugs.Also an intracellular delay is incorporated into the model to express the lag between the time thevirus contacts a target cell and the time the cell becomes actively infected. The model dynamicsis completely defined by the basic reproduction number R0. If R0 ≤ 1 the disease-free equilibriumis globally asymptotically stable, and if R0 > 1, two endemic steady states exist, and their localstability depends on value of R0. We show that the intracellular delay affects on value of R0 becausea larger intracellular delay can reduce the value of R0 to below one. Finally, numerical simulationsare presented to illustrate our theoretical results.

Thermal Characters of the Air-Cooled High Speed Motorized Spindle for Wood-Working Machine

2013 ◽  
Vol 579-580 ◽  
pp. 568-572
Author(s):  
Da Guo Ma ◽  
Xin Bo Jiang
Keyword(s):  
High Speed ◽  
Metal Cutting ◽  
Heat Dissipation ◽  
Element Model ◽  
Structure Design ◽  
Air Cooling ◽  
Heat Sources ◽  
Motorized Spindle ◽  
Dissipation Structure ◽  
Motorized Spindles

The structure and composition of the air-cooled high speed motorized spindle for wood-working machine and some features relative to the metal cutting motorized spindle are introduced briefly. Then the main heat sources and heat dissipation mechanism of the air-cooled motorized spindle are thoroughly analyzed, finite element model of the air-cooled motorized spindle is built, the motorized spindles temperature distribution under thermal steady state and the influence of speed are analyzed. The results show that air cooling relative to the water or oil cooling has many advantages and reasonable heat dissipation structure design of air-cooled motorized spindle could meet the requirements of the high-speed motorized spindle for wood-working machine.

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