The spindle-bearing system of a machine tool is modeled for analytical and experimental studies, and an optimal design formulation is proposed based on the studies as a systematic, rational approach of design for performance of the machine tool. A test model of the spindle-bearing system has been built and used for modal testing and identification of bearing coefficients. The impulse excitation method has been applied for the experiments and the complex modal analysis method for the modal parameter identification.
For the dynamic analysis, the spindle is mathematically represented by a Timoshenko beam including the internal damping of beam material, and each bearing by four bearing coefficients; stiffness and damping coefficients in moment and radial directions. And the dynamic characteristics of the spindle-bearing system is theoretically analysed by introducing the transfer matrix method. The influence of the bearing coefficients, material damping factor and bearing span on the dynamic characteristics of the spindle-bearing system is parametrically examined.
A method is proposed to identify the moment and radial stiffness and damping coefficients of bearings with incomplete mode shapes. A complete relationship between the bearing reactions and the incomplete mode shapes is constructed in the form of a system of linear equations by using the transfer matrix method. Theoretically, the bearing coefficients are straightforwardly identified without iteration from the system of linear equations. But, practically, the inaccuracies of measured data and/or model data can make the identified bearing coefficients meaningless, sometimes obtaining negative values. A constrained minimization problem is suggested to artificially remove this kind of unreasonable results. An application to the spindle-bearing test model has indicated the usefulness of this approach.
A multiobjective optimization problem is next proposed as a mean to systematically and quantitatively improve the cutting capability of machine tools. Two general objective functions are considered. One is the negative real maximum component of the transfer function(dynamic compliance) at the cutting point which may be related to the minimum critical chip width. The others is the polar mass moment of inertia of the spindle. This may be related to the possibility of rapid suppression of chatter vibration due to easy variation of spindle speed. Design variables are geometric dimensions such as the outer diameter, inner diameter and length of spindle elements. Realistic conditions related to the design, manufacturing and assembly of the spindle-bearing system are imposed as constraints. The Pareto optimal solutions representing the trade-off relation of the competing objectives are obtained by the weighting objectives method. The shapes and characteristics of the Pareto optimal solutions obtained are examined and discussed including the influence of optimal design problem parameters on the dynamic behaviours of the spindle-bearing system.