| The characteristic equations and the eigenfunctions of the beams with intermediate support are derived. Three cases of boundary conditions are considered at the left end of the beams: pinned support, clamped support and elastic rotational spring support. Using the eigenfunctions derived, the equation of motion of the pipe is discreted by Ritz-Galerkin method and the critical flow velocity of pipes are obtained at each location of the intermediate support in three cases. In pinned support case, it is found that the critical flow value for divergence is in good agreement with the exact result at the location of the intermediate support for ξ_b ∈ [0.72, 1], ξ_b ∈ [0.44 , 1 ], ξ_b ∈ [0.31, 1 ], ξ_b ∈ [0.24, 1 ] by application of one-mode, two-mode, three-mode, four-mode Galerkin approximation and the critical flow value for flutter is in good agreement with the exact result at the location of the intermediate support for ξ_b ∈ [0, 0.4] by application of two-mode, three-mode Galerkin approximation. Inclamped support case, it is found that the critical flow value for divergence is in good agreement with the exact result at the location of the intermediate support for ξ_b ∈ [0.51, l] and the critical flow value for flutter is in agreement with the exact result at thelocation of the intermediate support for ξ_b ∈[0, 0.4] by application of two-mode Galerkinapproximation. Because the emphasis is placed on analyzing the primary critical flow velocity and instability mechanism, the two-mode Galerkin approximation is employed. Based on the matric equation obtained by application of the two-mode Galerkin approximation, the effect of the elastic rotational spring constants, mass ratio and the tension of the system on the values of critical flow velocity and stability is discussed. The dynamics of pipes is studied by using the method of numerical simulations. Appling Differential Quadrature Method (DQM) to check the results obtained above.
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