Eric A. Butcher

Stability of Up- and Down-Milling Using Chebyshev Collocation Method

The dynamic stability of the milling process is investigated through a single degree-of-freedom model by determining the regions where chatter (unstable) vibrations occur in the two-parameter space of spindle speed and depth of cut. Dynamic systems like milling are modeled by delay-differential equations (DDEs) with time-periodic coefficients. A new approximation technique for studying the stability properties of such systems is presented. The approach is based on the properties of Chebyshev polynomials and a collocation representation of the solution at their extremum points, the Chebyshev collocation points. The stability properties are determined by the eigenvalues of the approximate monodromy matrix which maps function values at the collocation points from one interval to the next. We check the results for convergence by varying the number of Chebyshev collocation points and by simulation of the transient response via the DDE23 MATLAB routine. The milling model used here was derived by Insperger et al. [14]. Here, the specific cutting force profiles, stability charts, and chatter frequency diagrams are produced for up-milling and down-milling cases for one and four cutting teeth and 25 to 100 % immersion levels. The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations are found which agree with the previous results found by other techniques. An in-depth investigation in the vicinity of the critical immersion ratio for down-milling (where the average cutting force changes sign) and its implication for stability is presented.Copyright

Chebyshev Expansion of Linear and Piecewise Linear Dynamic Systems With Time Delay and Periodic Coefficients Under Control Excitations

In this paper, a new efficient method is proposed to obtain the transient response of linear or piecewise linear dynamic systems with time delay and periodic coefficients under arbitrary control excitations via Chebyshev polynomial expansion. Since the time domain can be divided into intervals with length equal to the delay period, at each such interval the fundamental solution matrix for the corresponding periodic ordinary differential equation (without delay) is constructed in terms of shifted Chebyshev polynomials by using a previous technique that reduces the problem to a set of linear algebraic equations. By employing a convolution integral formula, the solution for each interval can be directly obtained in terms of the fundamental solution matrix. In addition, by combining the properties of the periodic system and Floquet theory, the computational processes are simplified and become very efficient. An alternate version, which does not employ Floquet theory, is also presented. Several examples of time-periodic delay systems, when the excitation period is equal to or larger than the delay period and for linear and piecewise linear systems, are studied. The numerical results obtained via this method are compared with those obtained from Matlab DDE23 software (Shampine, L. F., and Thompson, S., 2001, ‘‘Solving DDEs in MATLAB,’’ Appl. Numer. Math.,37(4), pp. 441‐458.) An error bound analysis is also included. It is found that this method efficiently provides accurate results that find general application in areas such as machine tool vibrations and parametric control of robotic systems. @DOI: 10.1115/1.1570449#

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.

Normal Forms and the Structure of Resonance Sets in Nonlinear Time-Periodic Systems

The structure of time-dependent resonances arising in themethod of time-dependent normal forms (TDNF) for one andtwo-degrees-of-freedom nonlinear systems with time-periodic coefficientsis investigated. For this purpose, the Liapunov–Floquet (L–F)transformation is employed to transform the periodic variationalequations into an equivalent form in which the linear system matrix istime-invariant. Both quadratic and cubic nonlinearities are investigatedand the associated normal forms are presented. Also, higher-orderresonances for the single-degree-of-freedom case are discussed. It isdemonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2,...) solutions. The discussion is limited to the Hamiltonian case (which encompasses allpossible resonances for one-degree-of-freedom). Furthermore, it is alsoshown how a recent symbolic algorithm for computing stability andbifurcation boundaries for time-periodic systems may also be employed tocompute the time-dependent resonance sets of zero measure in theparameter space. Unlike classical asymptotic techniques, this method isfree from any small parameter restriction on the time-periodic term inthe computation of the resonance sets. Two illustrative examples (oneand two-degrees-of-freedom) are included.

Order reduction of nonlinear time periodic systems using invariant manifolds

The basic problem of order reduction of linear and nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via Time Periodic Center Manifold Theory. A ‘reducibility condition’ is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show applications to real problems. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘combination’ resonances are discussed.Copyright

Influence of Torsional Motion on the Axial Vibrations of a Drilling Tool

The nonlinear dynamics of a tool commonly employed in deep hole drilling is analyzed. The tool is modeled as a two-degree of freedom system that vibrates in the axial and torsional directions as a result of the cutting process. The mechanical model of cutting forces is a nonlinear function of cutting tool displacement including state variables with time delay. The equations of new surface formation are constructed as a specific set. These equations naturally include the regeneration effect of oscillations while cutting, and it is possible to analyze continuous and intermittent cutting as stationary and nonstationary processes, respectively. The influence of the axial and torsional dynamics of the tool on chip formation is considered. The Poincare maps of state variables for various sets of operating conditions are presented. The obtained results allow the prediction of conditions for stable continuous cutting and unstable regions. The time domain simulation allows determination of the chip shape most suitable for certain workpiece material and tool geometry. It is also shown that disregarding tool torsional vibrations may significantly change the chip formation process. DOI: 10.1115/1.2389212

Influence of Honing Dynamics on Surface Formation

The dynamics of the rotating tool commonly employed in deep hole honing is considered. The mathematical model of the process including the dynamic model of the tool and the interaction of the workpiece surface and honing sticks is analyzed. The honing tool is modeled as a continuous slender beam with a honing mandrel attached at the intermediate cross section. A single row of stones on the tool rotates and has reciprocational motion in the axial direction. The honing stones are expanded to the machined surface by a special rigid mechanism that provides cutting of workpiece cylindrical surface and which vibrates in the transverse and axial directions. The removal of chip and the tool vibrations cause the variation of expansion pressure and depend on the surface state formed by previous honing stone. The equations of new surface formation are separated as a specific set of the dynamic model. These equations inherently consider the regenerative effect of oscillations during cutting. The expansion pressure, tool stiffness, and technology conditions are considered as varying parameters since their influence on the process are different. The process productivity and precision can be improved by choosing rational conditions evaluated by simulation. The corresponding models and results of numerical simulation are presented. All the results are given in dimensionless form and therefore they are applicable to a wide range of real manufacturing process conditions. The model of new surface formation presented allows the simulation of the machined surface variation in time and to predict workpiece accuracy and possible correction of surface errors.Copyright

Optimal control of parametrically excited linear delay differential systems via chebyshev polynomials

Use of Chebyshev polynomials in solving finite horizon optimal control problems associated with general linear time varying systems with constant delay is well known in the literature. The technique is modified in the present manuscript for the finite horizon control of dynamical systems with time periodic coefficients and constant delay. The governing differential equations of motion are converted into an algebraic recursive relationship in terms of the Chebyshev coefficients of the system matrices, delayed and present state vectors, and the input vector. Three different approaches are considered. First approach computes the Chebyshev coefficients of the control vector by minimizing a quadratic cost function over a finite horizon or a finite sequence oftime intervals. Then two convergence conditions are presented to improve the performance ofthe optimized trajectories in terms of the oscillations of controlled states. The second approach computes the Chebyshev coefficients ofthe control vector by maximizing a quadratic decay rate of L, norm of Chebyshev coefficients of the state subject to a linear matching and quadratic convergence condition. The control vector in each interval is computed by formulating a nonlinear optimization program. The third approach computes the Chebysbev coefficients ofthe control vector by maximizing a linear decay rate of L, nom of Cbebysbev coefficients ofthe state subject to a linear matching and linear convergence condition. The proposed techniques are illustrated by designing regulation controllers for a delayed Mathieu equation over a finite control horizon.

Order Reduction of Parametrically Excited Linear and Nonlinear Structural Systems

Order reduction of parametrically excited linear and nonlinear structural systems represented by a set of second order equations is considered. First, the system is converted into a second order system with time invariant linear system matrices and (for nonlinear systems) periodically modulated nonlinearities via the Lyapunov-Floquet transformation. Then a master-slave separation of degrees of freedom is used and a relation between the slave coordinates and the master coordinates is constructed. Two possible order reduction techniques are suggested. In the first approach a constant Guyan-like linear kernel which accounts for both stiffness and inertia is employed with a possible periodically modulated nonlinear part for nonlinear systems. The second method for nonlinear systems reduces to finding a time-periodic nonlinear invariant manifold relation in the modal coordinates. In the process, closed form expressions for “true internal” and “true combination” resonances are obtained for various nonlinearities which are generalizations of those previously reported for time-invariant systems. No limits are placed on the size of the time-periodic terms thus making this method extremely general even for strongly excited systems. A four degree-of-freedom mass- spring-damper system with periodic stiffness and damping as well as two and five degree-of-freedom inverted pendula with periodic follower forces are used as illustrative examples. The nonlinear-based reduced models are compared with linear-based reduced models in the presence and absence of nonlinear resonances.

SIMULATION OF MACHINED SURFACE FORMATION WHILE HONING

The dynamics of deep hole honing is considered. The mathematical model of the process including the dynamic model of the tool and the interaction of the workpiece surface and honing sticks is analyzed. The honing tool is modeled as a continuous slender beam with a honing mandrel attached at the intermediate cross section. A single row of stones tool rotates and has reciprocational motion in the axial direction. The honing stones are expanded to the machined surface by a special rigid mechanism that provides cutting of workpiece cylindrical surface. The removal of chip and the tool vibrations cause the variation of expansion pressure and depend on the surface state formed by previous honing stone. The equations of new surface formation are separated as a specific set of the dynamic model. These equations inherently consider the regenerative effect of oscillations during machining. The numerical algorithm of machined surface generation has been developed which facilitates the 3D graphical representation and evaluation of the topography of the generated surface. The simulation model accounts for not only the nominal tool motion but also takes into account errors during machining such as tool components deformations and vibrations, tool runout, as well as initial surface distortions produced by previous operation. Based on the surface formation model software for evaluation of typical surface quality and accuracy criteria such as eccentricity, out-of-round, conicity, barrel, axial waviness, misalignment, faceting has been developed. The expansion pressure, tool stiffness, and technology conditions are considered as varying parameters since their influence on the process are different. The process productivity and accuracy can be improved by choosing rational conditions evaluated by simulation. The corresponding models and results of numerical simulation are presented. All the results are given in dimensionless form and therefore they are applicable to a wide range of real manufacturing process conditions. The model of new surface formation presented allows the simulation of the machined surface topography variation in time and to predict workpiece accuracy and possible correction of surface errors.Copyright