PSI - Issue 83

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

213

௕ ൌ ௕ ௕ 2 න ቆ ଶ w ஻ ∂ ሺ x, tሻ ଶ ቇ ଶ ୐ ౘ ଴

൅ ௖ ௖ ሺsin ሻ ଶ 2 w ஻ ሺx, tሻන w ஻ ሺx, tሻdx ௅ ೎ ଴

(11)

V ୟ ൌ E ୠ I ୠ 8L ௕ ቈන ൬ ∂w ஻ ∂ ሺ x x, tሻ ൰ ଶ ௕ ଴

ଶ

dx቉

(12) In this analysis, the transverse displacement is expanded as a series of fundamental spatial functions, w i (x) with i ranging from 1 to N, where N is the number of linear modes of the beam, along with harmonic temporal functions,     sin i a t q t   which are generalized coordinates dependent on time. Accordingly, the displacement can be expressed as follows: wሺx, tሻൌa ୧ Wሺxሻ sinሺωtሻ (13) With ௜ constant coefficients, replacing expression (10) in the previous energy formulas, we obtain the following formulas: V ୠ ൌ 1 2 a ୧ a ୨ k ୧୨ sin ଶ ሺωtሻ (14) V ୟ ൌ 1 4 a ୧ a ୨ a ୩ a ୪ b ୧୨୩୪ sin ସ ሺωtሻ (15) Tൌ 1 2 ω ଶ a ୧ a ୨ m ୧୨ cos ଶ ሺωtሻ (16)

m

, ij k

, et ijkl b

ij

are the mass matrix and the linear and nonlinear stiffness tensors, respectively. They are

Where

୧୨ ൌE ୠ I ୠ න ቆ ∂ ଶ W ୧ ∂x ଶ ቇ ୐ భ ௜ ௖ ௝ ൌ ௖ ௖ ሺsin ሻ ଶ ௜ න ௝ dx ௅ ೎ ଴ ଴ ቆ ௝ ቇ න ൬ ௞ ൰ ௅್ ௜௝ ൌ ߩ ௕ ௕ න ௜ ௝ ௅ భ ଴ ଴ ቆ ∂ ଶ W ୨ ∂x ଶ ቇdx

expressed as follows: K

(17)

(18)

௜௝௞௟ ൌ E ୠ S ୠ 4 ௕ න ൬ ௜ ൰ ௅್

(19)

଴ ൬ ௟ ൰

(20)

Using Equation (20), the discretization of the kinetic energy and the potential energy yields: 2 ௜ ௜௥ ൅3 ௜ ௝ ௞ ௜௝௞௥ െ2 ଶ ௜ ௜௥ ൌ0 (21) Before solving the problem for the contribution coefficient i a and the frequency  , Equation (32) is expressed in a dimensionless form by substituting the dimensional parameters with their corresponding dimensionless parameters: