26 Y. Ekici et al. . − ρ2A2+M2δ x2−LM2 ¨w2 +x2 ∂ ¨w1 (L1) ∂x1 − ¨ u1 (L1) −E2I2 ∂ 4 w1 ∂x1 4 + ∂ ∂x2 E2A2 ∂u2 ∂x2 + 1 2 ∂w2 ∂x2 2 ∂w2 ∂x2 =0. (4.9) The PDEs given above can be reduced to two equations by neglecting the acceleration due to the axial motion of the beams ( . ¨ur =0), integrating Eqs. (4.6) and (4.8) along their respective beam lengths, and substituting them into Eqs. (4.7) and (4.9). Then, the reduced PDEs are obtained as follows: . − ρ1A1 +M1δ x1 −LM1 ¨w1 −E1I1 ∂ 4 w1 ∂x1 4 + E1A1 2L1 L1 0 ∂w1 ∂x1 2 dx1 ∂ 2 w1 ∂x1 2 = 0, (4.10) . − ρ2A2 +M2δ x2 −LM2 ¨w2 +x2 ∂ ¨w1(L1) ∂x1 − ¨ w1 (L1) ∂w2 ∂x2 + ρ2A2 x2 − L2 2 +M2 H x2 −LM2 − L2−LM2 L2 ¨ w1 (L1) ∂ 2 w2 ∂x2 2 −E2I2 ∂ 4 w2 ∂x2 4 + E2A2 2L2 L2 0 ∂w2 ∂x2 2 dx2 ∂ 2 w2 ∂x2 2 =0. (4.11) Furthermore, by applying the same substitution to the by-products of the IBP and neglecting the acceleration due to the axial motion of the beams, boundary conditions (BCs) are obtained as follows: .w1(0) =0, (4.12) . ∂w1 ∂x1 x1=0 =0, (4.13) . E1A1 2L1 L1 0 ∂w1 ∂x1 2 dx1 ∂w1 ∂x1 x1=L1 −E1I1 ∂ 3 w1 ∂x1 3 x1=L1 + L2 0 ρ2A2 +M2δ x2 −LM2 dx2 ¨w1 (L1) =0, (4.14) . L2 0 ρ2A2 +M2δ x2 −LM2 ¨w2 +x2 ∂w1 ∂x1 x1=L1 x2dx2 +E1I1 ∂ 2 w1 ∂x1 2 x1=L1 =0, (4.15) .w2(0) =0, (4.16) . ∂w2 ∂x2 x2=0 =0, (4.17) . ρ2A2L2 2 + M2LM2 L2 ¨ w1 (L1) −E2I2 ∂ 3 w2 ∂x2 3 x2=L2 + E2A2 2L1 L2 0 ∂w2 ∂x2 2 dx2 ∂w2 ∂x2 x2=L2 =0, (4.18) . E2I2 ∂ 2 w2 ∂x2 2 x2=L2 =0. (4.19) 4.2.2 Discretization of the Nonlinear PDEs by Using Galerkin’s Method Nonlinear PDEs can be discretized and converted into nonlinear ordinary differential equations (ODEs) by using Galerkein’s method. The following solution for the response is assumed,
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