Time-Varying Total Stiffness Matrix of a Rigid Machine Spindle-Angular Contact Ball Bearings Assembly: Theory and Analytical/Experimental Verifications

Abstract

A lagrangian formulation is presented for the total dynamic stiffness and damping matrices of a rigid rotor carrying noncentral rigid disk and supported on angular contact ball bearings (ACBBs). The bearing dynamic stiffness/damping marix is derived in terms of the bearing motions (displacements/rotations) and then the principal of virtual work is used to transfer it from the bearing location to the rotor mass center to obtain the total dynamic stiffness/damping matrix. The bearing analyses take into account the bearing nonlinearities, cage rotation and bearing axial preload. The coefficients of these time-dependent matrices are presented analytically. The equations of motion of a rigid rotor-ACBBs assembly are derived using Lagrange's equation. The proposed analyses on deriving the bearing stiffness matrix are verified against existing bearing analyses of SKF researchers that, in turn, were verified using both SKF softwares/experiments and we obtained typical agreements. The presented total stiffness matrix is applied to a typical grinding machine spindle studied experimentally by other researchers and excellent agreements are obtained between our analytical eigenvalues and the experimental ones. The effect of using the total full stiffness matrix versus using the total diagonal stiffness matrix on the natural frequencies and dynamic response of the rigid rotor-bearings system is studied. It is found that using the diagonal matrix affects natural frequencies values (except the axial frequency) and response amplitudes and pattern and causes important vibration tones to be missig from the response spectrum. Therefore it is recommended to use the full total stiffness matrix and not the diagonal matrix in the design/vibration analysis of these rotating machines. For a machine spindle-ACBBs assembly under mass unbalnce and a horizontal force at the spindle cutting nose when the bearing time-varying stiffness matrix (bearing cage rotation is considered) is used, the peak-to-valley variation in time domain of the stiffness matrix elements becomes significant compared to its counterpart when the bearing standard stiffness matrix (bearing cage rotation is neglected) is used. The vibration spectrum of the time-varying matrix case is marked by tones at bearing outer ring ball passing frequency, rotating unbalnce frequency and combination compared to spectrum of the standard stiffness matrix case which is marked by only the rotating unbalnce frequency. Therfore, it is highly recomended to model bearing stiffness matrix to be a time-dependent.