74 J. Koutsoupakis et al. such as ANNs can be trained in the numerical domain and employed in making predictions in the physical domain, so long as the simulated data hold similar characteristics as those contained in the physical measurements. This similarity between the numerical and physical data is achieved by optimizing the models using a small number of healthy state measurements as well as powerful optimization techniques [11], aiming to minimize an objective function between simulated and real system responses. While this framework has previously been tested, it has become obvious that its capacity is limited by the capacity of the numerical models, that is, especially in cases where machinery is concerned, the MBD models must be able to simulate the various mechanisms of the system with as much accuracy as possible. Gear transmission systems in particular comprise complex mechanisms, where phenomena such as gear meshing dominate the system’s response and are usually the key features in a measured signal. Additionally, wear-related faults in gear transmission systems most commonly appear in components such as bearings (inner/outer race defects, etc.), and gears, where cracks or spalls can appear in gear teeth. To accurately model such phenomena, mechanisms such as the Time Varying Mesh Stiffness (TVMS), dynamic error and backlash between gear teeth must be taken into account in the model, aiming to increase the accuracy and fidelity of the results. In this work, a detailed gear meshing force formulation is developed, aiming to achieve high-fidelity simulations where the time varying effects of gear meshing are accurately presented in the numerical model of an experimental gear drivetrain system. The enhanced model, after being properly optimized, is employed in data generation for different health states of the drivetrain, producing the data necessary to train an ensemble of Deep Neural Networks (DNNs) for classifying the physical system’s health state. The rest of this work is structured as follows: First, the theoretical formulation used in this work is described in the Background section. Afterwards the experimental set-up and corresponding MBD models are shown in the Analysis section, where the results of the numerical simulations and damage classification are also presented, aiming to validate the proposed method experimentally. Finally, the findings of this work are discussed in the Conclusions section and further developments are briefly discussed. Background The stiffness of a tooth in a meshed gear pair can be broken down to five components based on the potential energy method as [12]: Kh = πEL 4(1−ν2) (1) 1 Kb =Z φ2 −φ1 3[1+(φ2 −φ)sinφcosφ1 −cosφcosφ1] 2 (φ2 −φ)cosφ 2EL[(φ2 −φ)cosφ+sinφ)] 3 dφ (2) 1 Ks =Z φ2 −φ1 1.2(1+v)(φ2 −φ)cosφcos 2φ1 2EL[(φ2 −φ)cosφ+sinφ)] dφ (3) 1 Ka =Z φ2 −φ1 (φ2 −φ)cosφsin 2φ1 2EL[(φ2 −φ)cosφ+sinφ)] dφ (4) 1 Kf = cos2φ1 EL [L∗ uf Sf 2 +M∗ uf Sf +P∗(1+Q∗tan2φ2)] (5) where Kh, Kb, Ks, Ka andKf denote the Hertzian, bending, shear, axial and fillet foundation stiffness. The terms E, Land ν are the tooth material modulus of elasticity, the width of the gear tooth and Poisson’s ratio respectively and φ1 and φ2 are explained schematically in Figure 1. Last, the rest of the terms of Equation (5) are derived from [13]. Note that the Line of Action (LoA) is estimated as the tangent between the two gear base circles. As such, depending on the number of gear teeth, the potential energy method may lead to an over or underestimation of the meshing stiffness. To take this error into account, the meshing stiffness estimation must be split to two cases, based on the minimum number of teeth in a gear, which can be estimated as: zmin =2ha/sin 2a (1) where hais the addendum of the gear and ais the nominal pressure angle of the gear. The distinction between the two cases can also be achieved by estimating the root and base radii of the gear as: RB = mzcosa 2 (2)
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