PSI - Issue 83

Mohamed Rjilatte et al. / Procedia Structural Integrity 83 (2026) 208–217 ∂ ଷ w ୠଷ ∂x ୧ ଷ ቤ ୶ ౟ ୀଵ ൌ ܭ ୘ଶ w ୠଷ ሺ1ሻ ∂²w ୠଷ ∂x ୧ ଶ ቤ ୶ ౟ ୀଵ ൌെ ܭ ఏଶ ∂w ୠଷ ∂x ୧ ฬ ୶ ౟ ୀଵ

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w ୠଵ ሺs ଵ ሻൌw ୠଶ ሺs ଵ ሻ, ∂w ୠଵ ∂x ୧ ฬ w ୠଶ ሺs ଶ ሻൌw ୠଷ ሺs ଶ ሻ, ∂w ୠଶ ∂x ୧ ฬ ୧ ଷ ቤ ୶ ౟ ୀୱ భ ൩െቆsinθ൅∂y ଵ ∂x ୧ ฬ ୧ ଷ ቤ ୶ ౟ ୀୱ మ ൩െቆsinθ൅∂y ଶ ∂x ୧ ฬ ∂²w ୠଵ ∂x ୧ ଶ ቤ ∂²w ୠଶ ∂x ୧ ଶ ቤ ୶ భ ୀୱ భ ൌ ∂²w ୶ భ ୀୱ మ ൌ ∂²w ୠଶ ∂x ୧ ଶ ቤ ୠଷ ∂x ୧ ଶ ቤ ଷ w ୠଶ ∂x ଷ w ୠଷ ∂x

୶ ౟ ୀୱ భ ൌ ∂w ୶ ౟ ୀୱ మ ൌ ∂w ୶ ౟ ୀୱ మ , ୶ ౟ ୀୱ మ ,

ୠଷ ∂x ୧ ฬ ୠଷ ∂x ୧ ฬ

୶ ౟ ୀୱ మ , ୶ ౟ ୀୱ మ ,

ɖ൥ ∂ ଷ w ୠଵ ∂x ɖ൥ ∂ ଷ w ୠଶ ∂x

୧ ଷ ቤ ୶ ౟ ୀୱ భ െ ∂ ୧ ଷ ቤ ୶ ౟ ୀୱ మ െ ∂

୶ ౟ ୀଵ cosθ ଵ ቇeො ଵ െ co μ sθ∂w ୡଵ ∂x ୧ ฬ ୶ ౟ ୀଵ cosθ ଵ ቇeො ଶ െ co μ sθ∂w ୡଶ ∂x ୧ ฬ

୶ ౟ ୀଵ ൌ0, ୶ ౟ ୀଵ ൌ0,

2.1. Non-Linear formulation

According to Hamilton’s principle, the nonlinear free dynamic equations of the suspended beam can be derived as follows: Ɂቆන ሺTെVሻ ଶ஠ன ଴ dtቇൌ0 (9) Where V and T represent the potential energy and the kinetic energy, respectively. Furthermore, the kinetic energy of the beam is defined according to [64] as follows: Tൌ 1 2 ρ ୠ S ୠ න ൬ ∂w ୠ ∂ ሺ t x, tሻ ൰ ଶ ୐ ౘ ଴ dx (10) And V is the potential energy of the entire system, including the strain energy due to bending Vb and the energy associated with the normal (axial) forces, which give rise to the geometric nonlinearity of the system, denoted by Va for each beam: