Dynamic Analysis and Optimal Selection of Parameters of a Finite Element Modeled Lathe Spindle Under Random Cutting Forces

Abstract

The dynamic analysis of an elastically supported lathe spindle-workpiece system subjected to random cutting forces is presented. The stochastic partial differential equation characterizing the behavior of the system was formulated from the Euler-Bernouli equation. Based on free vibration analysis and experimental verification of the natural frequencies, the hinged boundary condition was considered for the running center. A finite element technique in conjunction with the modal analysis method were used to calculate the mean square displacement of the workpiece. The experimentally calculated power spectral density of the cutting forces was used as the input excitation to the mathematical model. The effect of bearing stiffness and damping, and bearing spacing on the mean square displacement were studied. A direct search optimization technique was carried out to select optimal bearing stiffness and the bearing spacing. Results are presented in the form of plots and tables.

On the Natural Frequencies of High-Speed Spindles with Angular Contact Bearings

“…2) is expressed by (3) where m b = ; mass of the ball, d, = diameter of the ball centre trajectory, R = shaft rotational speed and R , = orbital speed of the ball at location j . The gyroscopic moment at each ball location is represented by (4) where J , = polar moment of inertia, R, = the ball angular velocity and p = the angle between the rotational plane and the plane normal to the axis of rotation.…”

“…Wardle et al (3) presented an interesting result on static bearing stiffness change with respect to preload settings for three different angular contact bearing configurations. Sharan et al (4) used a finite element model to study the dynamic behaviour of a lathe spindle by including the effects of bearing stiffness variation. More recently, Aini et al (5) presented a dynamic model of a spindle which includes the bearing preload effects.…”

<title>Experimental analysis and modeling of the dynamic performance of machine tool spindle-bearing systems</title>

“…Such an arrangement leads to the independence of the motions described in the traditional dynamic models [9][10][11][12][13][14][15][16][17]. However, in reality, at least two principal types of indeterminacy resulting from the properties of support bearings are involved in the spindle dynamics.…”

“…These are geometric non-uniformities and errors, and the non-linearity of the elastic spring and damping characteristics of the bearing supports [15][16][17]. The fundamentals of the spindle dynamics [4,[9][10][11][12][13][14][15][16][17] were developed to predict the mean value and variance of the vibrations of the spindle nose under cutting force harmonic excitation, and to estimate their influence on the final surface profile. subsystems such the spindle unit.…”

“…These investigations [1][2][3][4][5][6][7][8] have formed the basis of the mathematical modelling, dynamic analysis and parameter identification of the machine tool mechanical structure, by considering it as a complex, integrated mechanical system. Modal analysis [23, 30-32], finite element methods [10,[32][33][34][35], impedance methods [36] and other advanced methods of dynamic analysis [20, 22, 24-26, 29, 37-40], based on amplitude-frequency response and Fourier analysis, have been utilized. These tools allow the calculation of the mass, stiffness and damping characteristics for the advanced design of machine tools with desired dynamic parameters.…”