2 S. Cinquemani et al. Fig. 1.1 Overview of the steps to generate virtual signals corresponding to malfunctioning Standard operating conditions Kinetostatic model Model of defects Experim. Frequency Response Function Motor torque Forces on bearings Forces on housing Accelerations conditions [7]. The outputs generated, corresponding to different fault conditions associated with the main common failures of the elements that constitute the transmission, will provide a useful data base to properly set the algorithm of condition monitoring. The paper deals with the description of such approach (Fig. 1.1). Virtual signals related to malfunctioning can be obtained using the real working conditions (torque and motor speed) as the inputs of the model thus implementing the dynamics associated with the malfunctioning of different bearings. The output of the model consists on the harmonic forces acting on the bearing housing. Through experimental transfer functions of the gearbox, virtual accelerations can be calculated. The paper is structured as follows. Section 1.2 describes the kinetostatic model of the mechanical transmissions. Section 1.3 introduces the mathematical model of main failures of bearings. For the sake of clarity, the presented only cylindrical roller bearings model is here presented, but the reader will understand that this approach can be easily replicated for spherical, tapered and ball bearings. Section 1.4 introduces how to obtain the frequency response function of the gearbox, while Sect. 1.5 shows how the approach can be usefully applied on a real test case. Finally conclusions are drawn in Sect. 1.6. 1.2 Kinetostatic Model of the Transmission Transmissions installed on rolling mills can be generally divided into two groups: with parallel axis (Fig. 1.2) and with orthogonal axis (Fig. 1.3). For each group a set of parameters has been defined to describe the transmission (i.e. mechanical and geometrical features, type of bearings, gears, etc.). To be able to manage different transmissions, a reduced model with lumped parameters is developed according to the parameterization carried out. Forces on bearings (FX, FY, Fz) can be computed according to [3], as a function of the main features of the transmission and of the operating conditions (i.e. motor torque and angular speed). Then, the displacements of the inner ring of each bearing are calculated with respect to applied forces, as the mathematical models of defects need this input to generate the virtual accelerations. Unfortunately, while the direct relationship between displacement of the shaft and forces transmitted by the bearing exists [2, 8], the inverse relationship can not be solved analytically. For this reason an iterative procedure has been implemented. Even if many bearing types (cylindrical rollers, double row tapered rollers, double row self-aligning, and four-pointcontact bearings) have been analysed and modelled, only cylindrical roller bearings will be presented hereafter in order to give an overview of the adopted approach. Each model is furthermore capable to simulate the more common bearing defects: inner and outer race pits and cages wear resulting in an uneven roller distribution.
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