Design of an optimal damper to minimize the vibration of machine tool structures subject to random excitation
The vibration of machine tools during machining adversely affects machining accuracy and tool life, and therefore must be minimized. The cutting forces for stable turning are generally known to be random, and hence excite all the resonance modes. Of all these modes, those that generate relative motions between a cutting tool and a workpiece are of concern.This paper presents a new approach for designing an optimal damper to minimize the relative vibration between the cutting tool and workpiece during stable machining. An approximate normal mode method is employed to calculate the response of a machine tool system with nonproportional damping subject to random excitation. The major advantage of this method is that it reduces the amount of computation greatly for higher-order systems when responses have to be calculated repeatedly in the process of optimization. An optimal design procedure is presented based on a representative lumped parameter model that can be constructed by using existing experimental or analytical techniques. The two-step optimization procedure based on the modified pattern search and univariate search effectively leads the numerical solution to the global minimun irrespectively of initial values even under the existence of many local minima.
Application of time delay resonator to machine tools
Undesirable vibration can negatively affect the performance of machineries. In case of a machine tool, it negatively affects machining accuracy and tool life. Performance may be improved at the design stage by increasing machine stiffness. However, this is difficult to accomplish with existing machines. This paper implements a delay resonator (DR) on model of a typical machine tool to minimize relative vibration between the cutting tool and workpiece during machining process. The DR is a tunable vibration absorber which converts a conventional passive absorber into a marginally stable resonator. This structure absorbs all the vibratory energy at its point of attachment. A strategy called “Stability charts” is used not only to resolve the stability question but also to find out gain and time delay of the absorption. Results show that relative motion between cutting tool and workpiece is reduced significantly.
Dynamic response and optimal design of a lathe spindle under experimentally measured random cutting force excitations
This thesis presents the dynamic response and optimal design of a lathe spindle under experimentally measured random cutting force excitations. The optimal design is based on minimizing the maximum mean square displacement response of the workpiece under the action of random cutting forces. The stochastic partial differential equation of motion characterizing the behavior of a lathe spindle-workpiece system is formulated based on the Euler-Bernoulli equation. A finite element method using beam elements is used for free vibration analysis to compute the undamped mode shapes and the natural frequencies of the spindle-workpiece system. The workpiece support at the running center has been modelled as hinged or fixed support and the theoretical results are compared with the laboratory experiments to classify the nature of the support condition. Based on the results, the end condition at the running center is classified as hinged. The effect of varying the spindle-bearing stiffness on the natural frequencies and mode shapes are also presented. The forced vibration of the spindle-workpiece system is studied by first investigating the nonstationary random response of a workpiece subjected to a constantly varying cutting tool contact in a turning operation. The results indicate that the workpiece response at the cutting tool is not significantly influenced by the tool feed rate for normal turning operations. A modal analysis in conjunction with the finite element technique is then used to calculate the mean square displacement of the workpiece. The experimentally calculated power spectral density of the cutting forces is used as the input excitation to the mathematical model. A parametric study of the effect of bearing stiffness and damping, bearing spacing, sectional rigidity, and external damper on the mean square displacement is presented. An optimal design of a lathe spindle using a direct search optimization technique with bearing stiffness and spacing, and spindle cross-sectional diameter as design variables, has been carried out. The effect of chuck diameter and workpiece slenderness ratio on the dynamic response of an optimized spindle is also studied. Finally, the influence of a third bearing in the spindle on an optimized system and the design of an external damper are also discussed