## 3 Data Generation
### Modal Analysis  
In the process of obtaining a high-efficient motor designs, the way the system transfers energy over a frequency band, for specific materials and dynamic events, must be taken into account. The dynamical behaviour of the electric motor, dependent on the geometry, mass, stiffness, damping and boundary conditions, is characterized by its modes [23]. Each mode (a deformation pattern) occurs at a specific frequency, so-called natural frequency, and can be calculated by modal analysis technique. Modal analysis is used to show how the structure can vibrate and identify the structure natural frequency. The identification of the mode-shapes and their frequency is of great importance to an electrical machine designer, because a match between the frequencies of the air-gap exciting forces and the natural frequencies causes resonances, where there is an amplification effect of noise and vibrations. The resonance is an undesirable phenomenon and should be avoided. Knowing in advance the values of the natural frequencies, the designer can el  
Figure 5: Electromagnetic torque and back emf harmonic orders obtained by applying FFT on waveforms for JT = 4.39935 mm, TWS = 6.9565 mm, SO = 1.8815 mm, SOAng = 116.65 deg., TGD = 0.98125 mm. modifying the transmission path and shifting the stator structure eigen-frequencies, or by influencing the excitation source, in this case, the air-gap forces. However, in some cases, because the electric machine operates with variable frequency, it is impossible to eliminate the resonance phenomena caused by variable frequency forces that excite the structure. The modal analysis is based on the general motion equation, for a system with N Degrees-Of-Freedom (DOFs):  
\[[M]\ddot{x}(t)+[B]\dot{x}(t)+[K]x(t)=f(t) \tag{1}\]  
where \([M]\) represents the system mass matrix, \([B]\) and \([K]\) represents the system damping matrix and the stiffness matrix, respectively, of size \(N\times N\). The terms \(x(t)\) and \(f(t)\) are the displacement and force vectors of size \(N\times 1\). Modal analysis solves Equation (1) for free vibrations, i.e., without taking into consideration the damping and the force, resulting in the following expression of the motion equation:  
\[[M]\dot{x}(t)+[K]x(t)=0 \tag{2}\]  
For this work, modal analysis is conducted by FEA, where the 3D structural parametric model of the stator is used. A proper mesh type is chosen for the structural model to fulfill the requirements regarding computational time and model accuracy. The structural mesh is built from a 1 mm mesh size on the stator face, resulting in a 14.568 3D eight-noded hexahedral solid (CHEXA(8) in Simcenter 3D terminology) and in 19.521 nodes. Once the mesh is set, the material for the stator core is chosen. An isotropic material is used for the stator steel, while free-free conditions are imposed for the modal analysis. The 3D structural model is presented in Figure 6. The structural modal analysis is performed for a frequency interval set to cover the audible frequency range, from 20 Hz to 20 kHz. The computational time of the full process per discretized 3D structural mesh, on a workstation having an Intel Core i7-9850H CPU running at 2.6 GHz, with 32 GB of RAM takes approximately 2 min (1 min and 38 s). The first six mode-shapes of the base machine, identified by the automated process, with and without longitudinal deflection, are presented in Table 3. Here, the mode-shapes are identified as a combination of (m,n) where m is the order of the circumferential deflection and n is the order of the longitudinal deflection. Figure 6: Parametric stator mesh having 19.521 nodes and the output nodes (with orange). The process of running massive modal analysis simulations and identifying the modes and natural frequencies without human intervention is allowed by developing a set of scripts in Python and NX Open with the help of Simcenter 3D journaling capability.