12th IFToMM World Congress, Besançon (France), June18-21, 2007 CK-xxx 1 Dynamic analysis of rolling bearing system using Lagrangian model Vs. FEM code H. Rubio * J. C. García Prada † C. Castejón ‡ E. Laniado University Carlos III University Carlos III University Carlos III University Carlos III Madrid, Spain Madrid, Spain Madrid, Spain Madrid, Spain Abstract— The rolling bearings dynamical behaviour analysis is a critical condition to determine the machine vibration response. The rolling bearing, with outer ring fixed, is a multibody mechanical system with rolling elements that transmit motion and load from the inner raceway to the outer raceway. In rolling bearing analytical formulation, the contact between rolling element and raceways is considered as nonlinear springs and their stiffness are obtained by using Hertzian elastic contact deformation theory. The contact model among the rolling element and the raceways will be detailed in the paper due to the great important in the vibration pattern analysis . In the work presented, the simulation of kinematics, dynamic and structural behaviours of the rolling bearings and their vibration response without faults will be presented, a analytical model using Lagrange formulation and a simulation model using Algor ® code (event’s simulation). To evaluate the suitability and compatibility among analytical and simulation models, the results from simulation will be applied to inner ring motion equations of analytical model to obtain a valuable error signal. Keywords: Rolling bearing, analytical model, simulation, multibody and rotor dynamics I. Introduction The rolling bearing dynamic behaviour analysis is a very important issue to know the system vibration response. This response is non-linear, mainly when the rolling element is a rolling element 1 [4, 5]. On condition operating, rolling bearings generate some vibrations and noise related to the movement transmission characteristic of these mechanical elements where elastic properties have a great influence. In this work, in a first step, an analytical model is proposed by using Lagrange formulation [9, 10] besides a 3D rolling bearing simulation model is developed. Also, the vibratory signal, obtained from rolling bearing operation by means of a simulation program (Algor ® code), using the "event simulation" technique [1, 12], is *E-mail: hrubio@ing.uc3m.es † E-mail: jcgprada@ing.uc3m.es ‡ E-mail: castejon@ing.uc3m.es 1 Roller and ball implemented as input data (rolling element position, inner race position, contact displacement) solution in the analytical model. The main goal of this work will consist of characterizing the error signal in the inner ring motion equations. II. Analytical model Figure 1 shows the multibody mechanical system to simulate: the configuration of a motor connected to a rolling bearing, where the outer race is fixed. Fig. 1. Multibody mechanical system In rolling bearings, the rolling element-raceway contact is non-linear, it makes that the faults in the raceways or the rolling elements origin complex vibrations. For this, in the analytical model presented Lagrange formulation is applied to calculate the inner raceway and rolling elements positions (independent generalized coordinates) [10]. The motor-rolling bearing system motion equations will be calculated using: d T T V F dt p p p ⎛ ⎞ ∂ ∂ ∂ − + = ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠  (1) where p is generalized coordinate vector, F is generalized loads (forces), T is total kinetic energy and V is the total potential energy. The coordinates used in the analytical model are referenced to the fixed outer race. Figure 2 shows the equivalent geometric model used: rolling element centre positions and the inner centre position related to the outer centre position.