## 4 Conclusions  
We have presented the numerical, analytical and experimental results of modal and rotor-dynamic analysis of interior permanent magnet (IPM) synchronous motor components. The main findings are important in robust design of the IPM motor and can be summarized as follows:  
1. The presence of the permanent magnets results in higher natural frequencies of the IPM rotor configuration compared to the natural frequencies of the rotor without magnets. This finding can be useful for robust design of the IPM rotor as well as for the design of synchronous reluctance machines' rotors without magnets. This is also important for the IMP machine fault detection since the natural frequencies of the rotor with mechanically damaged magnets (and/or partially demagnetized magnets) may be altered (i.e., lower natural frequencies and/or additional natural frequencies might be detected due to the damage). 2. The numerical modeling approach presented in this study proposes a simplified but accurate modeling of the slot winding and end-winding as a solid object with equivalent mechanical properties of the composite material (i.e., conductor and insulations). The proposed model was successfully validated with experimental measurements. The advantage of such a modeling approach is the reduction of the computational time needed for numerical simulations. 3. When an equal fill factor \(k_{cu}\) is applied for both end winding and slot winding, the numerically calculated and experimentally measured natural frequencies of the stator with windings differed by 4.7%. This finding might be particularly important when coupling mechanical analysis with magnetic and thermal analysis of the electrical machine, since the \(k_{cu}\) of the end windings is usually difficult to determine. 4. The presence of the slot windings and end-winding decreases the natural frequency of the stator assembly. This finding can be also important for analysis of mechanical vibrations imposed by insertion of mechanical supporting structures in order to reduce the mechanical vibrations of the whole stator assembly. 5. The presence of the motor housing results in the higher natural frequencies of the motor assembly. The numerical modal analysis of the stator assembly with housing revealed that certain naturalfrequencies of the end-windings were excited even though the mechanical vibrations of the motor housing/frame (dictated by mechanical vibration of the stator) were not observed/detected, which is important in prevention of premature damage of windings. 6. The rotor dynamic analysis, taking into account the gyroscopic effect, showed that the bearings with higher stiffness contribute to the increase of the critical speeds and their elimination from the operation range of the analyzed IMP rotor. The models show that a strong bifurcation was present due to the presence of the rotor with magnets compared to the rotor-dynamic shaft only. The calculated torsional critical speeds of the IPM rotor are significantly higher compared to the maximum operational speed, confirming the safe design of the IPM rotor analyzed in this study. The proposed rotor-dynamic numerical models of the IPM rotors were successfully validated with the analytical solutions. Further mathematical modeling of the IPM motor will focus on coupled physical phenomena including mechanical vibration effects shown in this study coupled with the magnetic, electrical and thermal effects being important in robust and safe design of the e-machine rotors. Further experimental research work will also include the experimental rotor-dynamic analysis. The results reported here may have important implications in design of mechanically safe and reliable operation of the IPM motors and other rotational electrical machines in order to predict excessive mechanical vibrations and avoid the premature fatigue or e-machine failure. Conceptualization, S.C. and D.M. ; Formal analysis, S.C. and D.M. ; Investigation, S.C. and D.M. ; Methodology, S.C. and D.M.