Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 66

Applying the boundary conditions w = 0 and at x = 0 in Eq. (2.54) leads to C₁ + C₃ = 0 and −λ₂ C₁ + λ₂ C₃ = 0. These equations are satisfied if C₁ = C₃ = 0.

Applying the boundary conditions w = 0 and at x = L in Eq. (2.54) yields

C₂ sin(λL) + C₄ sinh(λL) = 0 and −λ₂ C₂ sin(λL) + λ₂ C₄ sinh(λL) = 0. Therefore, nontrivial solutions are obtained when C₄ = 0 and sin(λL) = 0, so λ = nπ/L (for n = 1,2,…). Since λ = (ω² ρS/EI)^1/4, we find that the natural frequencies ω_n are given by

(2.64)