Chatter suppression techniques in metal cutting

Moving-load dynamic problems: A tutorial (with a brief overview)

The dynamic stability of the milling process is investigated through a single degree of freedom mechanical model. Two alternative analytical methods are introduced, both based on finite dimensional discrete map representations of the governing time periodic delay-differential equation. Stability charts and chatter frequencies are determined for partial immersion up- and down-milling, and for full immersion milling operations. A special duality property of stability regions for up- and down-milling is shown and explained.

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Stability of up-milling and down-milling, part 1: alternative analytical methods

In this paper, research on transverse vibrations of axially moving strings and their control is thoroughly reviewed. In the last few decades, there have been extensive studies on analysis and control of transverse vibrations of axially moving strings because of the wide applications of many engineering devices that axially moving strings represent. In the investigations adopting linear models of moving strings, the paper summarizes recent studies on modal analysis, complicatedly constrained strings, coupled vibrations, and parametric vibration, as well as some early results. In the investigations adopting nonlinear models of moving strings, the paper presents the governing equations with large amplitude, and reviews progress on discretized or direct approximate analytical analyses and numerical approaches based on the Galerkin method or the finite difference method. Furthermore, investigations are reported on modeling of damping mechanisms as viscoelastic materials, coupled vibration of power transmission systems, and bifurcation and chaos. The state of the art of active control of moving strings is surveyed on controllability and observability, the Laplace transform domain analysis and the energy analysis, nonlinear vibration control and adaptive vibration control. Finally, future research directions are suggested such as nonlinear vibration of moving strings under complex constraints and couplings, energetics of nonlinear and time-varying strings, bifurcation and chaos in transverse motion of moving strings, control of hybrid systems containing moving strings, robust and adaptive controls of nonlinear moving strings, and experimental investigations. In this review article there are 242 references cited. @DOI: 10.1115/1.1849169#

Analysis and Control of Transverse Vibrations of Axially Moving Strings

Structural optimization with frequency constraints - A review

Dynamic response of a guided circular saw

Nomenclature C = matrix of admixture coefficients for mode shape changes / = identity matrix K = symmetric stiffness matrix . £_ = modification vector M = symmetric mass matrix u = modification vector in modal coordinates x — displacement vector z = displacement vector in modal coordinates a. = magnitude of local stiffness modification A = change between baseline and modified structures X = natural frequency of modified structure $ = modal matrix with columns k k = kih mode shape vector i = Ath mode shape vector of modified structure ft = frequency matrix with diagonals co oo = natural frequency of baseline structure

Natural Modes of Modified Structures

A procedure for obtaining closed-form homogeneous solutions for the problem of vibration of discrete viscoelastic system is developed for the case where the relaxation kernel characterizing the constitutive relation of the material is expressible as a sum of exponentials. The developed procedure involves the formulation of an eigenvalue problem and avoids difficulties encountered with the application of the Laplace transform approach to multi-degree-of-freedom viscoelastic systems. Analytical results computed by using the developed method are demonstrated on an example of viscoelastic beam.

Closed-Form Solutions and the Eigenvalue Problem for Vibration of Discrete Viscoelastic Systems

The stability characteristics of a milling system are investigated in this paper. A model is developed that includes the gyroscopic effects of the rotating spindle and on the basis of this model new insight into the physics of chatter instability is obtained. In the analysis the basic Fourier series method for solving a harmonically time-varying system without time lag is extended to obtain a finite-order characteristic equation for the milling system in terms of the system matrices. The complex eigenvalues of the resulting characteristic equation can be efficiently predicted by using Muller's optimization algorithm with deflation, and the stability lobes can be predicted by using a nonlinear optimization procedure at different rotating speeds based on the Nyquist stability criterion. In this study it was found that including the gyroscopic effects of the rotating spindle increases the real parts of the eigenvalues of the system. This reduces the critical axial depth of cut. It is also found that the milling forces primarily excite the backward-wave modes but stabilize the forward-wave modes of the rotating spindle.

Stanley G. Hutton

Papers

Dynamic response of a guided circular saw

Natural Modes of Modified Structures

Closed-Form Solutions and the Eigenvalue Problem for Vibration of Discrete Viscoelastic Systems

Chatter Instability in Milling Systems with Flexible Rotating Spindles—A New Theoretical Approach

Cutting-induced vibration in circular saws