PSI - Issue 83

Mohamed Berjal et al. / Procedia Structural Integrity 83 (2026) 295–304

300

2 ௜ ∗ ௜ ∗ ௥ ൅3 ௜ ∗ ௝ ∗ ௞∗ ௜ ∗ ௝௞௥ െ2 ∗ଶ ௜ ∗ ௜ ∗ ௥ ൌ0

(20)

The equation was solved using a single-mode reduced modal approach, initially developed by Benamar and Kadiri in (El Kadiri et al., 2002). This method focuses exclusively on the dominant vibration mode, assuming its predominance in the nonlinear response, while neglecting the contributions of higher-order modes. ൬ ఠ ೙∗ ೗ ఠ ೗ ∗ ൰ ଶ ൌ1൅ ଷ ଶ ௜ ଶ ஻ ೔೔೔೔ ௄ ೔೔ (21) 3. Numerical results and discussion 3.1. Linear vibration analysis This section first presents the results of the linear analysis, followed by the nonlinear study. The computed natural frequencies for different boundary conditions show excellent agreement with the results reported in (Cong et al., 2020), confirming the reliability of the adopted approach. Table 1 presents a comparison between the predicted natural frequencies and reference results for the simply supported configuration ( ் ଵ ൌ ் ଶ ൌλ, ఏଵ ൌ ఏଶ ൌ0 ). An excellent agreement is observed for the first vibration modes, confirming the accuracy of the proposed model. Table 1. Comparison of the first five dimensionless natural frequencies for a simply supported (SS) beam and corresponding relative errors. Mode Present study (SS) Error (%) – theory Error (%) – FEM

Theoretical results (SS) (Cong et al., 2020)

FEM results (SS) (Cong et al., 2020)

1st

0.1362 0.2310 0.4354 0.7849 1.2162

0.1355 0.2307 0.4354 0.7848 1.2162

0.1360 0.2307 0.4349 0.7840 1.2147

0.52 0.13 0.00 0.01 0.00

0.15 0.13 0.11 0.11 0.12

2nd 3rd 4th 5th

Following the good agreement observed in Table 1, the corresponding normalized mode shapes are presented to further characterize the dynamic behavior of the system. These mode shapes confirm the dominant beam–cable interaction and validate the adopted modeling assumptions.

Normalized mode shape

Normalized mode shape

Normalized mode shape

Normalized mode shape