PSI - Issue 75

696 6 Robert Goraj et al. / Procedia Structural Integrity 75 (2025) 691–708 Goraj / StructuralIntegrity Procedia (2025) between ( ) = 373 MPa and ( ) = 93 MPa. 3.2. Modal analysis The roots of the denominator of (15) are referred as the poles s n of the transfer function G 3 . They indicate the system natural frequencies and the modal damping ratio according to: 1,2 ( )= √ 1,2 1 ∗ ,2 2 (17) 1,2 ( ) = − Re{ 1,2 } √ 1,2 1 ∗ ,2 (18) The asterisk in (17), (18) indicates the conjugate complex values of s 1,2 . The natural frequencies f 1 and f 2 are plotted in Fig. 4 as 2D-functions of the modal damping and the parameter I y .

Fig. 4. Map of system natural frequencies

The natural frequency f 1 is indicated as: entire motor . It represents a modal form, in which the mass m 1 and m 2 oscillate nearly in phase. The frequency f 1 ranges from approx. 29 Hz to 31 Hz. The curve-indication: rotating components stands for a modal form, in which the displacement of the mass m 1 dominates the oscillation behavior. Depending on the chosen I y , the natural frequency f 2 lies here in the range from 71 Hz to 129 Hz. The horizontal dashed line at 110 Hz represents the excitation frequency f exc of a three-blade propeller at the motor operation speed of 2200 rpm (3*2200/60 rpm = 110 Hz). It cuts one of the dotted curves for the parameter I y = 5.4 cm 4 . 3.3. Frequency response analysis The frequency response analysis gives the steady state amplitude of the longitudinal stress as a function of the excitation frequency f . This amplitude is referred as a high cycle fatigue stress σ HCF . Setting the steady state bending moment: ( , )= 1 √ 3 ( , ) 3 ∗ ( , ) (19) into (16) follows the steady state amplitude longitudinal stress: HCF ( , )= 1 √ 3 ( , ) 3 ∗ ( , ) 2 (20) The σ HCF is shifted in phase regarding to the excitation by the angle: ( , )=arg 3 ( , ) (21) The results of the frequency response analysis are shown in Fig. 5 in the form of a two-dimensional (second moment of area dependent) Bode plot.