1 Bearing Faults Simulations Through a Parametric Model of a Gearbox 5 Fig. 1.5 Geometrical relationship between the depth (Cd) and the width ( d) of the defect its resultant forces on the rings. From a mathematical point of view, after having defined the maximum circumferential displacement ("wc) of these points, this assumption leads to define a distribution of these displacements in order to locate all of them. If this type of defect is introduced, this implies that angles j are no longer evenly spaced, as assumed in the model of a “healthy” bearing. Similar considerations have been made for the other types of bearings taking into account also, if necessary, the relationships between axial and radial components, in order to derive appropriate analytical models. 1.4 Experimental Frequency Response Function The relationship between forces applied on the housing and the corresponding virtual accelerations is obtained experimentally. Each test consists on hitting the chassis with a dynamometric hammer and measuring the corresponding accelerations through a three axis accelerometer. As a matter of fact, an impulsive test allows driving all the resonances of the system. This means that, with a single test, it is possible to evaluate the dynamics of the system in a wide range of frequency. As the hammer contains a load cell, a measurement of the applied force can be collected. Signal have been acquired using a Daq board Ni 9234 with a sample frequency of 51.2 kHz. Figure 1.6 shows time histories of the force applied to the chassis and the corresponding measured acceleration. The impulsive test is repeated 10 times to improve the quality of the frequency response function, increasing the signal to noise ratio through the so called linear averaging technique. Knowing the forces applied and the accelerations of the chassis it is possible to estimate the frequency response function of the structure. Figure 1.7 shows, for example, the frequency response function between the force applied on the chassis along the vertical direction, and the corresponding acceleration along the same axis. It is important to highlight that the virtual accelerations resulting from this work, are strictly related to the experimental transfer functions of the transmission. Information collected in the frequency response function are extremely useful. Once the forces transmitted to the chassis, due to a defect on a bearing, are known from the mathematical model, the corresponding accelerations can be easily obtained simply multiplying the forces (in frequency domain) to the frequency response function of the system. For example, being TFX,B1(f) the frequency response function of the chassis, close to bearing B1, along the X axis and FX,B1(f) the forces along the same direction estimated by the model for a defect on bearing B1, it results: RxB1.f / DTFX;B1.f / FX;B1.f / where RxB1.f / is the acceleration measured along x axis due to a defect on bearing B1.
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