## 3 Analytical Calculation of Force Frequency Components of the Electro-Mechanical System  
The diagnosed electro-mechanical system is an induction motor. The force acting on the separated area of the electric machine system is the sum of the electromagnetic volumetric and surface forces expressed by the products of magnetic quantities: the magnetic induction vector **B** and the magnetic field strength vector **H**. It is the basis for determining the forces acting on the boundary surfaces of ferromagnetic areas. When taking the stator and rotor core surfaces of an electric machine as the ferromagnetic boundary regions, the magnitude of the magnetic tension force acting in the radial direction per unit area (radial Maxwell tension) is described by the expression:  
\[p_{r}(\alpha,t)\cong\frac{1}{2\mu_{0}}b_{\delta}^{2}(\alpha,t) \tag{3}\]  
where:  
\(b_{\delta}(\alpha,t)\), instantaneous value of flux density in the gap of the electric machine;  
\(\mu_{0}\), magnetic permeability of air;  
\(\alpha\), coordinate describing the point of the gap between the rotor and the stator;  
\(t\), time. Radial forces are responsible mainly for the vibration of the stator core and outer casing of an electrical machine [8]. The flux density in the electric machine gap, with the simplifying assumption that the magnetic conductivity of the ferromagnetic areas \(\mu_{\rm F}>>\mu_{0}\) and accepting that the magnetic voltage between the stator and the rotor is equal to the resultant magnetomotive force of the stator and rotor \(\theta(\alpha,t)\), is inversely proportional to the air gap width distribution \(\delta(\alpha,t)\) between the stator and rotor:  
\[b_{\delta}(\alpha,t)=\frac{\mu_{0}}{\delta(\alpha,t)}\theta(\alpha,t)=\lambda (\alpha,t)\theta(\alpha,t), \tag{4}\]  
where \(\lambda\) is the air gap magnetic conductivity per unit area. The geometry of the air gap, which determines the conductivity of the air gap \(\lambda\)(\(\alpha\),\(t\)), depends on the shape of the stator and rotor slots and the distance between the outer and inner surfaces of the rotor and stator [9]. Thus, the induction of \(b_{g}\)(\(\alpha\),\(t\)) depends on the geometry of the air gap, the distribution of the stator and rotor windings and the method of powering the electrical machine. The electromagnetic torque produced in the electro-mechanical system is identical to the electromagnetic tangential forces between the stator and rotor. Tangential forces whose magnitude per unit area is described by the expression:  
\[p_{t}(\alpha,t)=b(\alpha,t)A(\alpha,t) \tag{5}\]  
arise as a result of the interaction of the resultant magnetic field \(b\)(\(\alpha\),\(t\)) of the stator and the rotor with the current sheet of the rotor \(A\)(\(\alpha\),\(t\)). Here, too, there is a dependency on the geometry of the stator and rotor. Dependencies (3)-(5) clearly indicate the modulation processes occurring. It should be noted that the displacements caused by the radial force introduce additional changes in the conductivity of the air gap, while the fluctuations of the electromagnetic (or mechanical) torque affect the instantaneous change of the rotational speed. The mutual influence of the discussed physical phenomena results in the occurrence of appropriate amplitude and frequency modulation processes [10]. The vibrations generated by the forces of electromagnetic and mechanical origin trigger the reaction of the mechanical supports of the rotating element (rotor), such as commonly used rolling bearings. Stresses between rolling elements and raceways are revealed in the form of mechanical vibrations with fundamental frequencies (identical to the so-called operating frequencies of the bearings) determined on the basis of the geometric dimensions of the bearing and the relative rotational frequency of the inner and outer raceways [11; 12; 13].