PSI - Issue 83

Mohamed Rjilatte et al. / Procedia Structural Integrity 83 (2026) 208–217

211

௖௝ ൌ ௖௝ ሺ ሻ ௜ሺఠఠ଴ሻఛ ; ሺ ൌ1,2ሻ ௕௜ ൌ ௕௜ ሺ ሻ ௜ሺఠఠ଴ሻఛ ; ሺ ൌ1,2,3ሻ

(4)

Additionally, the following equality can be established: ௖ ൌ ൌ After substituting equation (13) into the linear vibration equations of the cable and beam, we obtain (Gattulli et Paolone 1997): ௖௝ ଶ ௖௝ ൅ ଶ ௖௝ ௖ ଶ ௝ ൌ 8 ̂ ௝ ሺ ൌ 1, 2ሻ െ ௕௜ ସ ௕௜ ൅ ସ ௕௜ ௜ ସ ൌ 0 ሺ ൌ 1,2,3ሻ (5) In which: eො ୨ ൌw ୡ୨ ሺ1ሻtanθ ୨ ൅න ቆ ∂y ୡ୨ ∂x ୡ୨ ∂w ୡ୨ ∂x ୡ୨ ቇdx ୡ୨ ଵ ଴ (6) ௕௝ ௕௝ ௕ ଶ ଷ ௕௝ ௝ ଷ ൫ ௝ , ൯ െ ௕ሺ௝ାଵሻ ௕ሺ௝ାଵሻ ௕ ଶ ଷ ௕ሺ௝ାଵሻ ௝ ଷ ൫ ௝ , ൯ ൌ ௖௝ ௖௝ ̂ ௖௝ ሺ ሻ ௝ ൅ቈ ܪ ௖௝ ௖௝ ሺ1, ሻ ൅ ௖௝ ௖௝ ̂ ௖௝ ሺ ሻ ௖௝ ௖௝ ሺ1, ሻ቉ܿ ݋ ௝ (7) The boundary conditions for a beam segment supported by linear and rotational elastic supports on each side are detailed in reference (Fakhreddine et al. 2019). w ୡଵ ሺ0ሻൌw ୡଶ ሺ0ሻൌ0,

w ୡଵ ሺ1ሻൌɀ ଵ w ୠଵ ሺs ଵ ሻcosθ, w ୡଶ ሺ1ሻൌɀ ଶ w ୠଶ ሺs ଶ ሻcosθ, ∂ ଷ w ୠଵ ∂x ୧ ଷ ቤ ୶ ౟ ୀ଴ ൌെ ܭ ୘ଵ w ୠଵ ሺ0ሻ ∂²w ୠଵ ∂x ୧ ଶ ቤ ୶ ౟ ୀ଴ ൌ ܭ ఏଵ ∂w ୠଵ ∂x ୧ ฬ ୶ ౟ ୀ଴

(8)